Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.Dạy học đại số ở trường trung học phổ thông theo hướng phát triển năng lực giao tiếp toán học cho học sinh.MINISTRY OF EDUCATION AND TRAINING HA NOI NATIONAL UNIVERSITY OF EDUCATTION LUONG ANH PHUONG TEACHING ALGEBRA IN HIGH SCHOOL TOWARDS DEVELOPING MATHEMATICAL COMMUNICATION COMPETENCE FOR STUDENTS Major.
MINISTRY OF EDUCATION AND TRAINING HA NOI NATIONAL UNIVERSITY OF EDUCATTION LUONG ANH PHUONG TEACHING ALGEBRA IN HIGH SCHOOL TOWARDS DEVELOPING MATHEMATICAL COMMUNICATION COMPETENCE FOR STUDENTS Major: Theory and Methodology of Mathematics Teaching Code: 9140111 SUMMARY OF PHD DISSERTATION IN EDUCATION SCIENCES HA NOI - 2023 The dissertation was finished at: HA NOI NATIONAL UNIVERSITY OF EDUCATTION Scientific supervisors: Dr Le Tuan Anh Assoc Prof Dr Nguyen Thanh Hung Reviewer 1: Assoc Prof Dr Nguyen Huu Hau Hong Duc University Reviewer 2: Assoc Prof Dr Nguyen Tien Trung VietNam Journal of Education Reviewer 3: Assoc Prof Dr Trần Việt Cường Thai Nguyen University of Education The dissertation will be defended in the university committee: Ha Noi national university of educattion Time: ……………… Date: ………… The dissertation can be found at: National Library, Ha Noi or the Library of Ha Noi national university of educattion RESEARCHER’S PUBLICATIONS RELATED TO THE THESIS Luong Anh Phuong, Nguyen Thanh Hung (2019), “Developing mathematical communication capacity for students in teaching algebra at some high schools in the Central Highlands”, Educational and social journals, Special Issue No December 2019, pp 277-281 Luong Anh Phuong, Le Tuan Anh, Nguyen Thanh Hung (2021) “Developing mathematical communication competence for high school student in teaching algebra”, Proceedings of 1st Hanoi forum on pedagogical and education sciences, pp 81-96 Luong Anh Phuong (2021) Some measures to improve the confidence of high school students in mathematical communication, Educational and social journals, Special Issue August 2019, pp 85-90 Luong Anh Phuong, Nguyen Thanh Hung, Le Tuan Anh (2021) “The level of expression of mathematical communication ability of high school students”, Vietnam Journal of Educational Science, No 45 September 2021, pp 26-30 Luong Anh Phuong (2022) Applying the Concepts of “Community” and “Social Interaction” from Vygotsky’s Sociocultural Theory of Cognitive Development in Math Teaching to Develop Learner’s Math Communication Competencies, VietNam Journal of Education, Volume 6, Issue 3, December 2022, pp 209- 215 INTRODUCTION Reasons for choosing the research topic 1.1 Requirements of the field of education in the new era Reimers (2018) states, "in an increasingly volatile, unstable and interdependent global world, it is important to train students not only to understand the world in which they live, but also to improve it" In order to prepare human resources for society to withstand the challenges of the times, education is increasingly affirmed by countries, its importance and attention more than ever 1.2 Requirements of the implementation of the general education program in mathematics In the field of mathematics education, the Mathematics Education Program determines: Mathematics helps students form and develop mathematical competencies including the following core elements: mathematical thinking and reasoning ability; mathematical modeling capacity; mathematical problem-solving prowess; mathematical communication competence; ability to use mathematical learning tools and means Mathematical communication and the problems surrounding mathematical communication have recently been mentioned by some educational scientists and researchers when education has changed in goal orientation Thus, educational administrators and math teachers in Vietnam have begun to identify mathematical communication as one of the important skills that students need in the teaching process In order to help teachers in orienting their teaching methods according to the new approach of the Math Program, it is necessary to conduct in-depth research on the mathematical communication capacity of educators and scientists in the country and in the world international 1.3 The current situation of teaching in high schools towards developing mathematical communication competence for students; The potential and opportunities to develop this competence in teaching Algebra According to the new general education program, mathematics is a compulsory subject and has content built integrating three knowledge circuits: Arithmetic, Algebra and Some Analysis Elements (accounting for 44% of the time); Geometry and Measurement (35% of the duration); Statistics and Probability (7% of the duration) In addition, students' ability to communicate in mathematics is shown by the fact that "students effectively use digits, symbols, charts, graphs, logical links, when presenting, explaining and evaluating mathematical ideas", the best environment for students to practice this activity is the process of studying algebra at high school Many teachers have not paid due attention to the formation and training of mathematical communication skills for students As a result, many students are still limited and lack "confidence" when participating in specific mathematical communication situations Thus, developing mathematical communication capacity is the educational task of each school, each math teacher The other issue is the teacher himself well aware of that? How to develop students' mathematical communication skills? What are the difficulties of students when performing mathematical communication when learning math (difficulties in knowledge, skills, attitudes, language, )? The above issues need to be answered appropriately with specific, in-depth research Stemming from the above reasons, we choose the research topic for the thesis: “Teaching algebra in high schools towards developing mathematical communication competence for students” Research Aims On the basis of theoretical research on mathematical communication capacity as well as the current situation of teaching algebra in high school towards developing mathematical communication competence for students, the thesis proposes a number of measures to teach Algebra in the direction of developing mathematical communication for high school students Research tasks - Overview of competence, mathematical communication competence - Determine the requirements for students' mathematical communication competence after studying algebra content in high school - Show the level of expression of high school students' mathematical communication competence - Research on the current situation of teaching to develop mathematical communication competence in teaching algebra in some high schools in Dak Lak province, the difficulties of students when receiving mathematical language and using mathematical language in the process of mathematical communication Thereby identifying difficulties and limitations in teaching algebra to develop mathematical communication competence for students - Propose some measures to teach Algebra in high school in the direction of developing mathematical communication capacity for students - Pedagogical experiments at two high schools (Buon Don High School, Buon Don District and Hong Duc High School, Buon Ma Thuot City) Objects and research subjects The object of the rerearch is the process of teaching Algebra in high schools The research subjects is the ways to organize teaching Algebra in order to help high school students develop mathematical communication competence Limit the scope of research - Research issues related to mathematical communication capacity and how to organize teaching to develop mathematical communication capacity for students in teaching Algebra in high school - Research limitation: The study is limited to the study of algebraic content in 10th and 11th grade Scientific hypothesis If some teaching measures are implemented in the direction that students are given opportunities, actively confidently carry out mathematical communication activities, it will improve efficiency in the process of approaching, understanding and applying mathematical knowledge and will develop mathematical communication capacity for students Research Methods Use the following research methods in combination: Observation, investigation and interview methods; Theoretical research methods; Experimental method of pedagogy; Product research methods; Mathematical statistical methods Contributions of the thesis 8.1 Theoretical contributions Contributes to the elucidation of the concept of communication; mathematical communication; mathematical communication capacity; The role of mathematical communication in the process of learning mathematics of students Analyze and synthesize a number of theories, perspectives and teaching methods related to teaching and developing mathematical communication capacity Recommend the levels of expression of students with mathematical communication abilities during algebraic learning in high school; The requirements should be met for students' mathematical communication abilities when studying algebra in grades 10,11 8.2 Practical contributions Proposing five pedagogical measures applied in teaching algebra in high school in the direction of developing mathematical communication capacity for students Research results contribute to innovating teaching methods, improving the quality of algebraic learning in high school Research results contribute to improving the quality of learning algebra in high school Arguments in defense Mathematical communication capacity consists of indicators and levels of expression of those indicators based on the division of Cai et al (1996) and reference to the division of behavior levels in each field of Lam Quang Hiep Mathematical communication is an activity that plays an important role in the math learning process of students, students only learn real math when they actually say and write about what they The feasibility and effectiveness of measures for teaching algebra in high school in the direction of developing communication capacity for students is proposed in the thesis 10 Thesis structure In addition to the Introduction, Conclusion, List of published works; References and Appendices, the thesis content include chapters: Chapter Theoretical and practical basis; Chapter Some measures to teach algebra at high schools towards developing mathematical communication competence for students; Chapter Pedagogical experiment Chapter THEORETICAL AND PRACTICAL BASIS 1.1 Literature review 1.1.1 Foreign research works Up to now, there have been many studies on mathematical communication ability in terms of reasoning, its role in the process of learning mathematics of students; its relationship to other mathematical competencies; as well as a few studies on mathematical communication patterns Mathematical communication capacity is divided into indicators in addition to the expressions of those indicators (see NCTM, 1989, 2000, 2003; Greenes and Schulman, 1996; Kennedy & Tip, 1994; Qohar, 2003) Besides describing mathematical communication through indicators, some researchers approach the concept of mathematical communication in a different way The Ontario Ministry of Education (2005) describes, "mathematical communication is the process of expressing verbal, visual, written mathematical ideas, solutions, and understandings, using numbers, symbols, images, graphs, diagrams, and words" In parallel with the in-depth analysis of the content, the authors also assert that mathematical communication is an essential process for learning mathematics (see Polya, 1973; NTCM, 1989; NTCM, 2000; Wichelt, 2009; Lim & Chew, 2007; Roland G Pourdavood & Patrick Wachira, 2016; Laney Samson, 2019) On the basis of general descriptions of mathematical communication capacity, scientists have assessed the mathematical communication ability of students when students learn specific content such as solving algebraic problems, solving problems when learning geometry of specific students The main results found in in some documents show the relationship between mathematical communication and the math learning process, between mathematical communication skills and some other skills, thereby confirming that good mathematical communication skills are necessary for every student if they want to well in math Problem-based learning, contextual learning, and research-based learning are considered learning models that improve students' mathematical communication skills And although, mathematical communication capacity has also been described conceptually or through sets of indicators, educational scientists try to describe in detail the manifestations of students' mathematical communication abilities in specific situations: presenting written proofs; solving algebraic problems or geometric problems using problem-solving methods 1.1.2 In Vietnam The results of research on mathematical communication are mainly in terms of teaching practice on the basis of applying theoretical research achievements Hoa Anh Tuong (2014) approached this research direction on junior high school students Similarly, Vu Thi Binh (2016) offers a teaching method to develop mathematical representation and communication skills for students in grades and based on the teacher's design, construction, regularly organize specific mathematical communication activities or mathematical representations Recently there are Vuong Vinh Phat (2021) and Dang Thi Thuy (2022) Vuong Vinh Phat (2021) proposed a math teaching process called Debating - Summarizing and using this process in teaching calculus to develop mathematical communication competence for high school students 1.1.3 General assessment of the overall research situation - The conceptual connotation of mathematical communication has been clearly indicated and although there are different ways when describing mathematical communication - With the view of mathematical communication in the form of activities, researchers and educators around the world including in Vietnam see mathematical communication as an important and necessary skill for students - The introduction of teaching models to develop mathematical communication capacity has also achieved certain results Some learning models: problem-based learning; contextual learning, learning based on research activities, scientific debate are mentioned as teaching methods to develop mathematical communication capacity In addition, the fact that teachers organize interactive, collaborative and exploratory activities is also an opportunity for students to practice these skills - Teaching analysis and geometry for high school students to develop mathematical communication competence has been mentioned, but teaching algebra has not been studied There has been no qualitative assessment of the reciprocal relationship between mathematical communication capacity and other mathematical competencies except problem-solving competence There has been no work on the indicative levels of mathematical communication capacity as well as the requirements for achieving mathematical communication capacity in algebraic content at all levels of study 1.2 Mathematical language and some problems of mathematical language in the algebra program at high school 1.2.1 Mathematical language 1.2.2 Mathematical language in the algebra program at high school 1.2.3 Some aspects of teaching algebra in high school 1.3 Communication and mathematical communication 1.4 Mathematical communication competence 1.4.1 Competence 1.4.2 Mathematical competence 1.4.3 The concept of mathematical communicative competencies In our opinion, mathematical communication is a special communication activity that includes presenting (speaking, writing or performing) mathematical ideas, opinions, and solutions, including explanation, maintenance, and interpretation, defend one's own point of view and questioning, debating, and argumentating the solution ideas of others through the effective use of mathematical and natural language To describe more clearly about mathematical communication skills, Mathematical communication skill indicators are used The general education program in mathematics in 2018 identified indicators and manifestations of each indicator of students' mathematical communication ability at all levels, including high school Through the study of the concept of mathematical communication skills from domestic and foreign literature, we propose an approach to mathematical communication capacity according to indicators and the expression of indicators according to the following description: Table 1.2 Description of the expression of indicators of mathematical communication competence Indicators of mathematical Expression communication competence Listening comprehension, reading 1.1 Listening comprehension, reading comprehension and recording necessary comprehension and writing summarize basic mathematical information presented in mathematical information, the focus in spoken or mathematical text form or spoken or written texts written by others 1.2 Know how to analyze, select and extract necessary mathematical information from spoken or written texts 1.3 Know how to connect, link, synthesize mathematical information from different documents Present and express (oral or written) 2.1 Present and express their mathematical contents, mathematical contents, ideas and solutions ideas and solutions in interaction with others (with appropriate 2.2 Participate in discussions, give questions, requirements for completeness and debate about mathematical contents, ideas and accuracy) solutions with others 2.3 Explain (reasonably) the presentation, expression, discussion, debate of mathematical contents, ideas and solutions in interaction with others Effectively use mathematical language 3.1 Use correctly and exactly mathematical language (numbers, symbols, charts, graphs, logical in combination with ordinary language to express links, ) in combination with ordinary mathematical content language or physical movements when 3.2 Correct and correct use of mathematical language presenting, explaining and evaluating combined with ordinary language to express thoughts, mathematical ideas in interaction arguments, explanations, debates, defense and (discussion, debate) with others evaluation of mathematical ideas and solutions Show confidence when presenting, 4.1 Be confident when presenting and expressing expressing, asking questions, discussing mathematical contents and debating contents and ideas related to 4.2 Be confident when participating in discussions, mathematics debates, defend personal views in not too complicated situations 1.4.4 The role of mathematical communication and the need to develop mathematical communication competencies for students in the process of teaching mathematics in high school In our opinion: mathematical communication are essential in learning mathematics; Students only learn real math when they actually talk and write about what they In addition, students will be actively involved in mathematics as students are asked to think through their ideas, talk, and listen to other students when coming up with ideas, strategies, and solutions Through mathematical communication, students can express, explain, describe, listen through which the understanding of mathematics deepens 1.4.5 Requirements for students' mathematical communication competence when studying algebra in high school 1.4.6 The Levels of Expression of Mathematical Communication Competence of Students Indicators of mathematical communication capacity are given levels of expression on a gradually larger scale that are of great significance to teachers or educational administrators to use the results of the assessment of students' mathematical communication abilities The combination of expression levels to form the level of each indicator according to the following table is our proposal based on the division of Cai et al (1996) and references the division of behavior levels of each field by Lam Quang Hiep Ordinal number Components of the General Education Program 2018 Listening comprehension, reading comprehension and recording necessary mathematical information presented in the form of mathematical texts or spoken or written by others Table 1.4 Expression level of mathematical communication competence Requireme Levels nt Doing Level Level Level - Listen - Understand to others talk about math (Listen) - Reading -Understand math texts (Reading) -Take notes - Take notes on mathemat ical informati on (Notes) Students the act of listening and reading but they not understand and cannot record any information or record some information, but this information is completely fragmented and has no value Students initially understand a part when reading or listening and can record some necessary information However, many information is missing or inaccurate Students understand and record relatively completely and accurately the necessary information or take full notes but there are some incorrect information Level Students fully understand information when reading, listening and recording fully, accurately, logically the information Level Students not only completely understand and accurately record information logically and accurately, but have linked and integrated new information with old knowledge available in the recording process to help the information content be richer and clearer Notes: - Mainly judging based on the amount of information provided in advance and the quality of that information - The information here is the information contained in the text when the student reads it or in the content spoken by others - Mathematical information can be concepts, theorems, methods of solving mathematics, or simply a mathematical idea presented in writing or expressed by another teacher or student - Training this expression of mathematical communication ability is important because students need to be able to listen, read and distill necessary information and take notes for the learning and self-learning process This evaluation of output information is primarily in terms of reproducing the information provided (input) into information at the output From there, there are appropriate pedagogical activities to practice this skill - Pisa offers a framework for assessing reading comprehension divided into levels Present, express Speak Talk Students Initially, Hs can speak his Hs speaks or Not only can (speak or write) cannot students can ideas and solutions writes down students come mathematical (incapably) say or write relatively ideas and up with radical contents, ideas, speak or down one or a convincingly, solutions in solution ideas, solutions in write down few ideas or valuablely, relatively interaction but they can interaction with ideas, solutions in complete content with others also evaluate others (with solutions the interaction, However, there are in a the ideas and Write Writeable appropriate relevant to but they are some errors in the completely solutions requirements for interactive still confused, presentation such as convincing, themselves in 13 Chapter SOME MEASURES OF TEACHING ALGEGERA IN HIGH SCHOOLS TOWARDS DEVELOPING MATHEMATICAL COMMUNICATION CAPACITY FOR STUDENTS 2.1 Orientations for building measures of teaching algebra in high schools towards developing mathematical communication 2.1.1 Ensuring the conformity of some specific goals in the High School Mathematics Program 2.1.2 Ensuring the specificity of the subject of math 2.1.3 Ensuring the suitability with the existing background of students in terms of knowledge, skills, thinking, and attitudes 2.1.4 Ensuring the comprehensiveness 2.1.5 Ensuring the active view 2.2 Some measures of teaching algebra in high schools in the direction of developing mathematical communication competence for students 2.2.1 Measure Enhance math problems, situations, and tasks that have a lot of potential for mathematical language development 2.2.1.1 Purpose of the measure Help students: understand, grasp the semantics of the vocabulary of mathematical language used in Algebra; Master and fluently and accurately use vocabulary, syntax, symbols and mathematical symbols to express ideas and present mathematical content; Start to have a certain confidence when studying and doing math 2.2.1.2 Scientific basis of the measure Students who want to perform mathematical communication must have knowledge of mathematical language, students who want to have good mathematical communication skills must have skills to effectively use mathematical language to reveal their mathematical ideas and solutions, in addition to rearranging their thinking to ensure ensure the logic, system in mathematical communication 2.2.1.3 Content and how to take measures We propose a way to equip mathematical language and forge skills in using mathematical language for students in the process of teaching algebra in high school for students according to the process of activities as follows: - Activity Introduce and approach vocabulary and mathematical rules in teaching Algebra - Activity Train students to effectively use mathematical language through problems, situations, and tasks with great potential for developing mathematical language - Activity Organize the consolidation of mathematical language knowledge for students: This task is designed with a worksheet with the name "Review table, reinforcing mathematical language" in which students need to make a list of the words, phrases, symbols, mathematical rules they have learned, explained or illustrated This activity students can right after accessing that knowledge about mathematical language or can after completing a class session 14 Table 2.1 Table of review and consolidation of mathematical language Lesson title ……………………………………………… …………… (1) New term Signal, Symbol Principles Description No (word, phrase) (3) (4) (5) (2) In which: (1): Students give themselves a general name for the main math content in that class (2); (3); (4): Students re-list the mathematical language they have just learned in the lesson (5) Students self-interpret concepts, symbols and rules according to their own understanding and give illustrative examples (if any) Teachers can ask students to complete the above worksheet right in class or at home Example 2.2 Teaching forms the term Set, Element, Subset Like the concept of Clause, set are one of the primitive concepts of mathematics However, the concept of a set is more intuitive than the concept of a mathematical model, and at the same time, the concept of a set, an element, and a subset has been learned in middle school, so we choose to approach these concepts for students starting with Activities to review and consolidate existing knowledge - Activity Warm-up - Students are activated their existing knowledge about Set, Subset, and Element Students are divided into groups of students to perform the task individually and discuss in groups, the results are presented on paper Question 1: Aggregation is a concept you are familiar with in middle school Give an example of a set in math or in daily life? Or: State a sentence containing the word "set"? (This sentence is for students who not remember the concept of sets) Question 2: In each of the examples you give, specify the element of each set Question 3: Among the groups' work results, which set is subset of which set? Question 4: When is a set A a subset of a set B (If the examples given by the groups not have the case that a set is a subset of another set, the teacher can actively give some examples for students to perform tasks on that example) Knowledge formation + Set is a fundamental concept of mathematics + Usually in mathematical presentation people use the letters A; B; C… stands for the set Lowercase letter a; b; c… symbol for the element + The element a belongs to the set A, denoted a∈A + Set A is a subset of set B if every element of set A belongs to set B Symbol A⊂B - Activity Practice vocabulary, symbols Purpose: Accept the semantics of the term set, a subset of a set; Proficient in the rules of form used to illustrate sets; Use the mathematical language to express the relationship between the objects of the set Exercise a) State the concept of a subset of a set b) Is the above statement a clause? c) If the above statement is a clause, use the mathematical notation that you know to write the clause in the form of an equivalent clause 15 Knowledge formation: A ⊂ B ⇔ ( ∀ x , x ∈ A ⇒ x ∈ B ) Exercise Given the following sets, list the “elements” of each set a) The set of natural numbers greater than and less than 13 b) The set of vertices of triangle ABC c) The set of results of a one-time coin toss Exercise Fill in the " " to get the correct clauses a) Set A is … of set B if every element of set A belongs to set B b) Every natural number is a … of the set of natural numbers c) The set of vertices of a quadrilateral consists of … elements Exercise Let A be the set of rectangles; B is the set of rhombuses What conclusion you have about the relationship of set A and set B - Activity Consolidation - Please list the mathematical language vocabulary, the rules learned in the above lesson (complete the Review table, consolidate the mathematical language - Table 1.10) 2.2.1.4 Notes when taking measures 2.2.1.5 Conclusion of measure When teaching, teachers need to make the most of opportunities to build and strengthen mathematical language for students At the end of each learning content, at the end of each lesson, at the end of each lesson or at the end of each chapter, it is necessary to consider the systematization of vocabulary, symbols and terms learned in that day as a regular exercise that students need to practice Students are regularly trained to apply mathematical language effectively and accurately step by step to gain knowledge of mathematical language 2.2.2 Measure Practice reading comprehension and listening comprehension skills for students through teaching according to the process of receiving - reflecting information 2.2.2.1 Purpose of the measure Help students have methods, skills to receive and absorb information when approaching a certain math content both while reading or listening 2.2.2.2 Scientific basis of the measure Students cannot form a new mathematical knowledge without accessing and partially understanding the meaning of new information The new information here can be a new mathematical content, a symbol, a new mathematical term or even a new mathematical task, etc The communication process will not be able to continue and be effective if the communication object does not know what he has been reading or listening to Reading comprehension and listening comprehension are the first two components of mathematical communication competence 2.2.2.3 Content and how to take measures To practice reading comprehension and listening comprehension skills for students, we propose the process of teaching Receiving – Reflecting information including the following three activities: - Activity Receive information Record words, terms, mathematical content, analyze graphs, drawings (within your ability) as soon as you read or listen to mathematical text (keywords) - Activity Create the original message Connect mathematical symbols, terms, and information obtained in step into original mathematical messages 16 - Activity Reflect information Formal document creation – new message (written or spoken)/task solving Example 2.4 (Type 2) The teaching phase "drawing graphs of functions with absolute values" is given in the form of a training exercise on reading comprehension skills: The teacher gives students access to the following two mathematical texts: Function: y=2|x|−4= ¿ x−4 , x ≥0 ¿−2 x−4 , x< { With the function graph as the figure beside Comment: From the above math content, we can draw graph of the function y=f (|x|) as follow: - Draw function graph y=f ( x ) - Keep graph y=f ( x ) lies to the right of the Oy axis and discards the part of the graph to the left of the Oy axis - Symmetric the part of the graph to the right of Oy over Oy Conclusion: The union of the two parts of the graph above is function graph y=f (|x|) Requirement: Read the texts above and solve the following exercises: Exercise Based on the information in the first paragraph, explain and restate how the drawing was done Exercise Comment on the “features” of the graph in the figure Exercise Let's draw function graph y x in the other way Compare the two ways of drawing and compare with the comment that the text has made Instructions for teaching the above exercises according to the proposed activities process Teacher activities Student activities Exercise Record the basic math information Function y x is the union of you read in the first paragraph Activity The information you can read on two functions y x x ; the picture (Pay attention to the special y 2 x x (1) points) y x is a VFunction graph shaped The two lines on the figure correspond to the graphs of the two functions above Activity Information that you can develop Because (1) so function graph further y x include function graph y x to the right of Oy and function graph y 2 x to the left of Oy Can you graph those functions? “Linear function”, I can graph it as straight lines Activity It's great that you know how to Draw a line y x to the right of y x graph functions Oy and a line y 2 x to the left of Oy The union of the two halves of the above line is function graph 17 y x Exercise Exercise Activity Activity Comment on the features of the graph Instruct students to continue using existing and new information found in exercise If you know one of the two branches, you think you can get the other one? So for the above case, we need to graph function y 2 x ? The graph consists of two branches that are symmetric about Oy (2) Absolutely yes, I just need to get the symmetry over Oy Absolutely no need, I just graph function y x then I take the part of the graph on the right of Oy that is symmetric about Oy and get the part of the graph on the left of Oy That means I have a V-shaped like in the picture Can you tell me the way you graph To graph function y x , we y x 4? function graph function y x to the right of Oy then get symmetry about Oy The union of the two parts of the graph above is function graph Activity y x I find your way is faster Compare Comments in the text are the general what you have just drawn with the y f ( x ) case function graph comment mentioned in the text 2.2.2.3 Some notes when taking measures 2.2.2.4 Conclusion of measure Collect information; Text analysis and interpretation and Assessment feedback are three aspects to assess students' listening and reading comprehension skills It can be said that teaching reading comprehension skills, listening comprehension skills and teaching problem solving skills always go hand in hand in teaching high school math 2.2.3 Measure Developing speaking skills through mathematical discourse activities 2.2.3.1 Purpose of the measure Help students be able to participate in a conversation about a certain math topic Able to present mathematical solutions and views in a clear and coherent manner; be able to argue, criticize, solve other people's mathematical ideas or defend, explain their own solutions, mathematical ideas 2.2.3.1 Scientific basis of the measure “Diễn ngôn” – Discourse” Follow us, “Mathematical discourse is a mathematical communication activity that takes place in the classroom in which students present their mathematical ideas and solutions, argue to defend that point of view as well as consider (agree or disagree; question; discuss; argue) the views of other students and of the teacher” Mathematical discourse is effective when students articulate their ideas and seriously consider the mathematical views of other students as a way to build mathematical understandings 2.2.3.3 Content and conduct of the measure How is mathematical discourse envisioned? Tasks of teachers and students in the process of mathematical discourse