Enhancing on mathematical communication for secondary students in Vietnam

HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1075.2017-0126 Educational Sci., 2017, Vol. 62, Iss. 6, pp. 37-44 This paper is available online at ENHANCING ONMATHEMATICAL COMMUNICATION FOR SECONDARY STUDENTS IN VIETNAM Hoa Anh Tuong1, Nguyen Huu Hau2 1Sai Gon University, 2Hong Duc University Abstract. This paper examines a lesson that may enhance mathematical communication for secondary students in Vietnam. The paper focuses on four parts: the term “mathematical communication”; the role of mathematical communication in classrooms; the basic way of mathematical communication which is suitable for Vietnamese students; some results of the research activities about mathematical communication. A case study was analyzed by using the data from observing classroom and reading the students’ written papers. Keywords: Mathematical communication, Expressing initial idea, Explaining, Argumentation and Proving. 1. Introduction Mathematical communication has been much interested by researchers and countries: In international Symposium 2008 Innovative Teaching Mathematics through Lesson Study III focused on mathematical communication, Emori (2008) [3], Isoda (2008) [5], Lim (2008) [6], Vui (2008) [11] and Tuong (2014) [10] interested that “Mathematical communication itself is necessary to develop mathematical thinking”. Programme for International Student Assessment (PISA, 2003) [9] talked about mathematical communication in some core principles of their test design. In Vietnam, after launching the national standard mathematics curriculum in 2006, the classroom mathematics teachers have learnt more on the generative teaching strategies that encourage mathematical communication to implement more effective lessons focusing on mathematical thinking. The reformed curriculum supports active learning that creates opportunities for students’ mathematical communication at different levels of thinking. Especially it supports students to think about, discuss, extend, write, listen and read. During studying mathematics in classroom, students want to communicate with classmates and teachers to understand problem-solving and share their own solving ways. In our opinion, students will study mathematics well in classroom when they are put in a positive social environment where they have the opportunities to create a mathematical understanding in their own ways. Teachers should teach mathematics through organizing activities for students to create intellectual skills and positive attitude. So, lessons should become a mutual learning process rather than just the passive reception from teachers. Received date: 5/2/2017. Published date: 10/4/2017. Contact: Nguyen Huu Hau, e-mail: nguyenhuuhau@hdu.edu.vn 37 Hoa Anh Tuong, Nguyen Huu Hau 2. Content 2.1. Why do we discuss Mathematical communication? Isoda (2008) [5] mentioned: Mathematical communication itself is necessary competence when using mathematical representation and the special format of explanation such as proof. In classroom, communication in mathematics is useful because students themselves can express mathematical ideas. In Standards by The National Council of Teachers of Mathematics (NCTM, 2000) [8], “Representation”, “Argumentation” and “Proof and Reasoning” are process standards as well as “communication”. Emori (2008) [3] mentioned: The meaning of narrow sense of communication includes speaking, listening, writing and reading. The meaning of broad sense of communication includes narrow sense of communication, reasoning, problem-solving and connecting – in which connecting means representation, constructing concepts or ideas, cognitive experiences and personally isolated thoughts. Gallery Walk et al. (2010) [4] mentioned “Mathematical communication is an essential process for learning mathematics because through communication, students reflect upon, clarify and expand their ideas and understanding of mathematical relationships and mathematical arguments (Ontario Ministry of Education, 2005)”. From above mentioned research, in summary, the communication process helps to build meaning and permanence ideas. Through communication, ideas become the objects of reflection, refinement, discussion, and amendment. Consequently, mathematical communication is a way of sharing ideas and clarifying understanding. Mathematical communication connects mathematical representation, explaining, argumentation and presenting the proof. 2.1.1. The role of mathematical communication in classrooms Emori (2008) [3] stated “Mathematical communication is a key idea because it is important not only for the improvement of learning mathematics but also for the development of necessary skills in the development of sustainable society knowledge. 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”. “Communication process helps students understand mathematics more deeply” (NCTM, 2007) [8]. “Communication has been identified as one of the core competences for students to develop” (Luis Radford, 2004) [7]. According to the mentioned research, it is realized that mathematical communication is necessary to develop mathematical thinking. 2.1.2. Which the basic ways of mathematical communication are suitable for Vietnamese students? Cheng Chun Chor Litwin (2007) [2] mentioned “The main components of mathematical communication are representation and proof”. There are three kinds of mathematical communication: The first kind of communication is the internal understanding of mathematics of individuals. This is a thinking process. Students 38 Enhancing on mathematical communication for secondary students in Vietnam have different solutions by using the representation in diagram, in word or in symbol. They obtain the understanding of the process. The second kind of communication is the explanation of the solution to others. This involves the understanding of other thoughts in understanding questions and solutions. By using a good representation or an acceptable representation, students understand the mathematical content. The third kind of communication is internal extension of the mathematics content of individuals. Students try to extend that mathematics content that is convinced by others and extend the difficulty of the problem or pose new problem (Cheng Chun Chor Litwin, 2007) [2]. Brenner (1994) [1] mentioned: mathematical communication had three distinct aspects. Communication about mathematics entails the need for individuals to describe the processes of problem-solving and their own thoughts about these processes. Communication in mathematics means using the language and symbols of mathematical conventions. Communication with mathematics refers to the uses of mathematics which enables students to deal with meaningful problems. Table 1 summarizes the communication framework for Mathematics. Table 1. Communication framework for Mathematics Communication About Mathematics Communication In Mathematics Communication With Mathematics 1. Reflection on cognitive Processes. Description of Procedures, reasoning. Metacognition—giving reasons for procedural Decisions. 1. Mathematical register. Special vocabulary. Particular definitions of everyday vocabulary. Modified uses of everyday vocabulary. Syntax, phrasing. Discourse. 1. Problem-solving tool. Investigations. Basis for Meaningful action. 2. Communication with others about cognition. Giving point of view. Reconciling differences. 2. Representations. Symbolic. Verbal. Physical manipulatives. Diagrams, graphs. Geometric. 2. Alternative solutions. Interpretation of arguments Using mathematics. Utilization of mathematical problem - solving in conjunction with other forms of analysis. In this research, we choose the basic ways of mathematical communication: Representation, explaining, argumentation, presenting the proof (Tuong, 2014) [10]. 2.1.3. The scale levels of mathematical communication The scale levels of mathematical communication Level 1. Expressing initial idea - Students describe and present methods or algorithms to solve the given problems (the mentioned method can be right or wrong). - Students know how to use mathematical concepts, terminologies, symbols and conventions formally. Level 2. Explaining - Students explain the validity of the method and present reasons why they choose that 39 Hoa Anh Tuong, Nguyen Huu Hau method. - Students use mathematical concepts, terminologies, symbols and conventions to support their logical and efficient ideas. Level 3. Argumentation - Students argue the validity of either the method or the algorithm. Students can use examples or counter-examples to test the validity of them. - Students can argue mathematical concepts, terminologies, symbols and conventions which are suitable. Level 4. Proving - Students use mathematical concepts, mathematical logic to prove the given result. - Students use mathematical language to present the mathematical result. 2.1.4. Our research activities about mathematical communication of Vietnamese students Traditionally, the teacher initiates the exchange by asking a question about a known fact or an idea, a student replies and the teacher evaluates the response as to whether it is correct. This is not suitable in modern classroom. To develop mathematical thinking, teachers must provide students with opportunities to acquire mathematical knowledge and skill through mathematical activities. To organize classroom to promote mathematical communication should have a combination of the following factors: Situations contain the conflict between old and new knowledge. There is active cooperation between the members in classroom and how lesson plan designs. During working in group, students can exchange and express ideas by their written paper and oral communication. Students express ideas by using specific signs such as diagrams, drawings, letters, symbols, icons ... This means they use the mathematical representation. Lesson plans integrate to solve real life situations to help students feel excitement and like to study mathematics. We illustrated a lesson on October 3, 2010 at class 6A3 (51 students) of Saigon Practical High school, district 5, Ho Chi Minh City. The content of problem Vi’s birthday is on Friday August 26th 2011. Question a: After 7 days, it is her mother’s birthday. What day is it? What date is it? Why? Question b: What day is 52 days after Vi’s birthday? What date is it? Why? Question c: Birthday of Vi’s father is on November 20th 2011. What day is it? Why? How to organize classroom Teachers create learning environment for students: To argue with teachers and to discuss, debate with classmates to find out the free answer and solve the posed problem in their own ways. To work in groups and have the teachers’ needful help in time. To look for connection problems, apply old knowledge to solve new problems, and find different ways to solve. To explain the understanding of mathematics as well as their solutions by working in group. To have self-regulation and consolidate mathematical knowledge. To argue needful solution using to solve problem. 40 Enhancing on mathematical communication for secondary students in Vietnam Analysis of the tasks Question a: We want students to realize that: seven days equals a week, after 7 days the day will be repeat. In addition, students remember which months have 30 days or 31days in a year. Question b and c: The problem is more difficult than question a. We want students to use different ways and use the remainder of division by 7 to solve the problem. After students finish the task, teacher has more questions: “Question 1. What day is 520 days after Vi’s birthday? What date is it? Why? Question 2. Birthday of Vi’s father is on November 20th 2014. What day is it? Why?” Analysis of the result Next, we analyze the results of the students from data collected by observing classroom and reading the students’ written papers. * Question a: Students argue: Seven days equals a week .Vi’s birthday is on Friday so her mother’s birthday is also on Friday. In addition, students have their own views on the birthday of Vi’s mother that are summarized as follows: Method 1: Students argued 26 + 7 = 33. The August has 31 days, we have two remaining days, so birthday of Vi’s mother is on September 2nd. Method 2: Students argued: It is 5 days from August 26th to August 31st. After 7 days, it is birthday of Vi’s mother so they plus 2 days in September, then birthday of Vi’s mother is on September 2nd. Method 3: Students use calendar sheet from August 26th to September 2nd. Next days 1 2 3 4 5 6 7 Friday Saturday Sunday Monday Tuesday Wednesday Thursday Friday August 26 August 27 August 28 August 29 August 30 August 31 September 1 September 2 Then, birthday of Vi’s mother is on Friday, September 2nd, 2011. * Question b: Students have their own views that are summarized as follows: Method 1: The August 26th is Vi’s birthday. It is 5 days from August 26th to August 31st. The remaining days are: 52 – 5 = 47 days. The September has 30 days, so the remaining days are: 47 – 30 = 17 days. This is on October 17th. Then, the day which is 52 days after the Vi’s birthday is on October 17th. 52 days = 7 weeks + 3 days. The August 26th, 2011 is Friday. Then, the date 52 days after is Monday. Method 2: 26 + 52 = 78 days 78 – 31 = 47 days (August has 31 days) 47 – 30 = 17 days (September has 30 days) 41 Hoa Anh Tuong, Nguyen Huu Hau 52 divided by 7 is 7 with remainder 3. Then, 52 days after Vi’s birthday is on Monday, October 17th, 2011. * Question c: Students have their own views that are summarized as follows: Method 1: Using the result of question b: Monday is on October 17th, 2011. It is 14 days from October 17th 2011 to October 31st 2011. It means 2 weeks. Then, October 31st 2011 is on Monday. It is 20 days from November 1st to November 20th. 20 days = 2 weeks + 6 days. Then, November 20th is on Sunday. Method 2: It is 86 days from September 26th 2011 to November 20th 2011 because (31 – 26) + 30 +31 + 20 = 86 days. 86 days = 12 weeks + 2 days. Then November 20th 2011 is on Sunday. Method 3: Students use calendar sheet from October 17th to November 21st. Monday Monday Monday Monday Monday Monday October 17 October 24 October 31 November 7 November 14 November 21 Then, November 20th is on Sunday. Analysis of the basic way of mathematical communication Student showed the basic way of mathematical communication as follows: Representation: Students used the calendar sheet to find the date in a week from August 26th to September 2nd; Monday schedule of months from October 17th to November 21st to find a solution. Explaining: Students tried to find a solution. Depending on their abilities, they had different ideas. The easiest way is writing a schedule in a week. If you change the assumptions of the problem, such as question b changes from 52 to 520 and change the question c from November 20th 2011 to December 27th 2014; this way will be not suitable. Students understand which solution is the best way to use when the problem changes more difficultly. It means that students are aware of the rationality of solutions. Table 2. The percentage of students expresses communication Students’ communication % Question a Method 1 80.00 Method 2 6.67 Method 3 13.33 Question b Method 1 33.33 Method 2 26.67 Question c Method 1 13.33 Method 2 13.33 Method 3 13.33 42 Enhancing on mathematical communication for secondary students in Vietnam Argumentation: They knew that 7 days equals a week and which months have 30 days or 31 days. Students know how to use every 7 days as a rule, the day is repeated. From there, students will know how to find the remainder in the division by 7 to find solution. In addition, students remember how many days are in August and September, then to perform suitable operations to find the day in a month. Especially, students recognize these problems may have linked to each other. Presenting the proof: Students themselves understood how to solve the problem by listening to peers who were demonstrating the problem. Analysis of Evaluating the scale levels of mathematical communication competence Expressing initial idea: Students expressed problem-solving by writing the schedule in the week from August 26th to September 2nd, Monday schedule from October 17th to November 21st. They applied the algorithm based on the remainder in the division by 7 to find the solution. They used reasonable mathematical operations: addition, subtraction, division and multiplication. Explaining: Students recognized the validity of each solution. Students realized algorithm in finding the remainder in the dividing by 7 is more reasonable than other solutions. Argumentation: Students expressed logical reasoning and solution for each problem clearly. Proving: Students used not only algorithms to find the remainder in the dividing by 7 but also the language of mathematics and logical reasoning in the presenting the proof. 3. Conclusion During the lesson, teachers not only taught the content but also organized classroom for students to express activities. Then teacher evaluated students’ mathematical understanding. Students used different representations to communicate with peers when they worked in group to form and consolidate new mathematical knowledge. When students were asked to explain the understanding of mathematics as well as the result of working in group, they could have self-regulation and consolidate mathematical knowledge. The situations containing the conflict between old and new knowledge really impacted on students’ cognition. That urged students to perceive the benefits of learning mathematics from mathematical communication. Students themselves drew experiences when they solved this problem depending on the difficulty of the given problem, it showed that we selected suitable problem - solving. Evaluating mathematical communication process of student through: Students could express how to solve problems and refer reasoning for each solution that caused them to think how to solve it. Students selected and used appropriately mathematical representations. Students expressed reasonable inferences in finding results. Students explained the rationale of each solution. Students used mathematical concepts, conventions, mathematical language in presenting the proof. 43 Hoa Anh Tuong, Nguyen Huu Hau REFERENCES [1] Brenner (1994), Language and learning: educating linguistically diverse students. [2] Cheng Chun Chor Litwin (2007), Mathematical Communication through Proof and Representation, Paper presented at APEC-TSUKUBA International Conference III, December 8-15, 2007, Tokyo and Kanazawa, Japan. [3] Emori Hideyo (2008), We Shall Overcome Dysfunctional Beliefs For Introducing Communication Study, Proceedings of APEC – Khon Kaen International Symposium in 25-29 August 2008 at Khon Kaen University "Innovative Teaching Mathematics through Lesson Study III - Focusing on Mathematical Communication", pp.70-91. [4] Gallery Walk, Math Congress and Bansho (2010), Communication in the mathematics classroom, The literacy and Numeracy Secretariat, Ontario, ISSN 19138490. [5] ISODA Masami (2008), Japanese Problem Solving Approach for Developing Mathematical Thinking and Communication: Focus on Argumentation with representation and reasoning, Proceedings of APEC-Khon Kaen International Symposium 2008 Innovative Teaching Mathematics through Leson Study III - Focusing on Mathematical Communication. Khon Kaen Session, T