Enhancing Mathematical Communication in the Classroom: A Case Study

Mathemat ical communicat ion competence was mentioned the first time in the Mathematics curriculum of Vietnam, which was published in 2018. It includes different skills such as listening, reading, presenting, expressing mathemat ical ideas, and knowing how to use mathemat ical language to communicate, discuss interaction with other people. The problem is how to organize teaching to promote mathematical communicat ion. In 2004, Radford and Demers based on the theory of Zones of Proximal Development [12] to develop a teaching strategy called the fourth strategy [9]. This teaching strategy has seven steps; it provides groups of the student with different activit ies to discuss, exchange, review their learn ing products. We have adopted this teaching strategy in a teaching situation about the concept of function in junior high school in Vietnam. Twelve students who participated in our study case worked in pairs. Th is teaching process focused on each group of two students reviewing products of another and basing on comments from other g roups to improve their products. Furthermore, our teaching situation puts students in front of the various representations of function and thus enhances the use of mathemat ical language and mathematical communication for students. The results of our study showed that the teaching strategy of Radford and Demers increased students’ mathematical communicat ion. Lessons drawn from the study are that this strategy is really an effective teaching method if teachers assign open problems that stimulate the curiosities of students; in the classroom, they feel free and confident in discussion; if these conditions are met, students who learn in this approach will acquire and reinforce the knowledge as a result of a mathematical communication process.


Introduction
On December 26, 2018, the Ministry of Education and Train ing of Vietnam issued a new general education curriculu m and subject curriculu ms, including Mathematics [6]. W ith this new curricu lu m, the Vietnamese National Assembly for the first time passed a resolution on a curriculu m that could have multiple textbooks. It means that other institutions , not only the Educational publisher, can write textbooks. The most significant difference in the mathemat ics curriculu m compared with the prev ious programs is the appearance of the term "competence" in the first objective. "Forming and developing mathemat ical co mpetencies includes the following core elements: competence of thinking and mathematical reasoning, mathemat ical modelling competence, mathemat ical problem-solving competence, mathematical co mmun ication co mpetence, competence in using mathemat ical tools and means" [7, p.6]. The Math curriculu m describes the levels that need to be reached at the end of Primary, Junior high school and high school for the four competencies. Accordingly, mathematical communicat ion competence includes the following expressions:  "Listening comprehension, reading co mprehension and taking notes of necessary mathematical information presented in written or oral form.  Presenting, expressing (speaking or writ ing) mathematical contents, ideas and solutions in interaction with others (with appropriate requirements for completeness and accuracy).


Using mathematical language effectively (nu mbers, letters, symbols, charts, graphs, logical links, etc.) in combination with natural language or physical movements when presenting, exp lain ing and evaluating mathemat ical ideas in interaction (discussion, debate) with others.  Demonstrating confidence when presenting, expressing, posing questions, discussing and debating content and ideas related to Mathematics." [7, p.13].
One challenge is to help teachers organize teaching so that students reach the levels described in this co mpetency. In this context, we will pay attention to the question: How to develop mathemat ical co mmunication co mpetence for junior high school students in teaching function concepts?

Communicati on Standard of NCTM (2000)
National Council of Teachers of Mathematics (NTCM 2000) has published work on "Princip les and Standards for school mathemat ics" for curriculu m fro m pre-kindergarten through grade 12. Related to communicat ion in teaching math, NTCM (2000) suggested that Instructional programs should help all students [8]:  "organize and consolidate their mathematical thinking through communicative activities;  communicate their mathematical thinking coherently and clearly to peers, teachers, and others;  analyze and evaluating the mathematical thinking and strategies of others;  use the mathemat ical language to express mathematical ideas precisely" [8, p. 60].
According to NCTM (2000), during adolescence, students are often reluctant to do anything that makes them stand out from the group, and many of them often feel hesitant to express their thoughts to others. They feel pressured to be co mpared with their peers. They want to find the right friends for themselves. As such, junior high school students are suitable for working in pairs because they can come up with different ideas, discuss carefully in groups before speaking in front of the class [8].

Teaching Process of Radford and Demers
With the goal of teaching to promote positive activities of students, many researchers have approached constructivist theory [3], J. Stanny [10] believed that learning activit ies should include: activating prior knowledge, creating surprise,; applying and evaluating the new knowledge including a reflective closing assignment. Hartle et al (2012) argued that teaching methods based on adaptation must be: elicit ing prior knowledge, creating cognitive dissonance, applying new knowledge with feedback, reflecting on learning [1]. In 2019, Loc [4], and Loc & Hang [5] came up with a 7-step teaching model in which they approached two key processes: "assimilat ion" and "accommodation"; Experimental teaching has shown its feasibility. In the above teaching process, the authors also pay some attention to the social interaction, but not really a solution for strengthening social interaction. A teaching model of the two authors Radford & Demers (2004), with the approach to Vygostky's social constructivism, emphasized the student's communication and interaction during their process of cognitive development or problem-solving [9]. In their work, Radford & Demers (2004) introduced and commented on teaching strategies that promote mathe matical communicat ion. These strategies are understood as pedagogical decisions to give students an environment to work together and to form concepts through communicat ion. In order to enhance mathematical communicat ion in the classroom, we pay special attention to the fourth strategy of Radford & Demers (2004), wh ich was developed on the basis of the Zone of Pro ximal Develop ment concept, which was introduced by Vygotsky.
Below are the steps of the fourth strategy [9]: 1. "Presentation by the teacher of the mathematical activity to be done. 2. Work in s mall groups to discuss and obtain results that must be carefu lly justified with convincing mathematical arguments. 3. Exchange between groups of results obtained and justifications provided. Study of solutions and arguments provided by other groups. 4. Meeting between the groups who exchanged their solutions to discuss the strengths and weaknesses of the solutions and their arguments. 5. Return to work in small groups to re-write the mathematical solutions and arguments in a more refined way, taking into account the discussion with the other groups 6. Discussion conducted by the teacher about the results obtained" [9, p. 30].
These authors argued that teaching with a fourth strategy will be in line with Vygotsky's suggestion in the sense that students should develop their concepts so they can solve problems autonomously by helping them spread through the zones of proximal development (ZPD) [11,12]. The strategy will bring students to face problems where their solutions go beyond the current conceptual level but become feasible through collaboration with classmates or teachers [2].
Accordingly, each step in strategy helps students experience a ZDP in their conceptual development goals (see Figure 1).

Research Purpose
We used a teaching situation about function concept, which is organized according to the 4 th strategy of Radford & Demers (2004) to pro mote mathematical co mmun ication for students. We considered how students demonstrate their ability to commun icate mathemat ically in a teaching situation.

Research Subject
An experiment took place in 90 minutes and was conducted in 12 students of 9th-grade students of Luong Dinh Cua Junior High School (HCM City) around the end of July 2018. The school was a public school in Ho Chi Minh City. In big cit ies of Vietnam, 9th-grade students often prepare for exams to grade 10 public schools so they will voluntarily register some courses during the summer vacation before the school year. We rando mly selected a class which prepare for 9th grade in the summer of 2018 and invited students to voluntarily participate in the experiment. Twelve (over 20) students in this class agreed to participate.

The Problem Used for Experimental Teaching
This context in Figure 2 is used for teaching the "function concept" in 9 th grade in junior secondary schools of Vietnam. This situation can be considered a real situation. The situation calls for the use of a function to describe the variation, depending on the volume, on the length of the square side to be cut at the four corners of the paper. For 9 th grade curriculu m, in order to find out the answer to the above problem, Vietnamese students construct a formu la for calcu lating the volu me of the bo x in a variable x (V= (30-2x)(30-2x)x), find the answer by giving x some d ifferent values and calculating corresponding V; the result of each case will be written down in the following table 1 in the answer sheet 1 (see Appendix).   In order to choose the size of a square paper, we could consider a generalizat ion. If the square paper is sized S cm, the box's volu me will be V = x(S -2x) 2 cm 3 where 0 < x < S/2. Therefore, V reaches its maximu m value when x equals S/6. With the choice S = 30 cm, maximu m of V = 2000 cm3 when x = 5 cm (see Figure 3) It means that the values of x and V to be found are integers. This choice provides an interesting and easy to imagine answer for junior high school students. However, the value of x = 5 is not easy to detect when only four specific bo xes are studied. Hence,it encourages different answers within and between groups. Therefore, the students and the groups will have opportunities to discuss with each other if the teaching p rocess is designed according to the fourth strategy of Radford & Demers.


Data was collected fro m writ ing and recording products of students and textbooks. They were analyzed as follows:  Analyze the textbooks to choose and build a solving problem. The selected problem should have the potential to develop activity with the material. Therefore, the solving problem would be suitable for the aim o f the curriculu m, especially in line with STEM education.
 Analyze a priori. This analysis is conducted before the experiment and allows us to calculate the pedagogic options to archive our goal.  Analyze a posteriori. This analysis shows what happened in the experiment compared with the estimates in the a priori analysis.

How to Organize Teaching Activities
The students are divided into pairs. Each pair of students in the group will be given six sheets of 30 cm s quare paper and 1 set of tools (scissors, a ruler with tape, tape, and pocket calculator).
Students are organized to work as follows: Step 1: (5 minutes): Teacher introduces the problem.
Step 2 (30 minutes): Each pair will create four bo xes , select the box with the most massive volu me, then answer some questions. Groups will submit their best work to the teacher with the answer sheet 1 (see Appendix).
Step 3 (10 minutes): The pairs will exchange group work in the answer sheet 1 fo r the first time. The group that receives the results of the other groups will check the answers and write comments.
Step 4 (10 minutes): Return the answer sheet 1and comments sheet to the original group. Groups will revise the answers (if needed) after reading co mments fro m other pairs.
Step 5 (20 minutes): The pairs continue to work on the volumetric function by converting to other representations.

Results and Discussion
Analysis of data collection (writing and recording products) allows the recognition of signs which correspond to communication standards of NTCM (2000).

Communicati ng Students' Mathematical Thinking Coherently and Clearly to Peers
Students made boxes and measured the box size to calculate the volume. Co mmunicat ion in pairs helps students reinforce the idea of measuring dimensions to calculate the volu me. Th is helps them feel about the change in volume according to the "dimensions" of the box.

Using the Mathematical Language to Express Mathematical Ideas Precisely
Finding algebraic formulas that show the dependence of volume on length x is not easy for every student. Couple exchanges help them adjust and find the exact mathematical formula.
"S9: What's this formula writing like? S10: Calculating V in x must have x in the formula? S9: Yes. In my opinion, it is (30 -x). (30 -x). x S10: Why is that? Why 30 -x? Not understand. S9: That's it, cut each corner x cm. Isn't that 30 -x? S10: So it must be 30 -x -x? Each edge was cut at the two ends.

Analyzing and Evaluating Others' Mathematical Thinking and Strategies
The learn ing process based on Radford&Demers (2004) strategy 4 gives students the opportunity to review and evaluate their peers' opinions.
"S1: The way to calculate the volume of this group is the same as mine.
S2: Oh, that's right. Then calculate the volume of these boxes. S1: Incorrect. Why? S2: I don't get it. Recalculate them. S1: Why is the same formu la for calculating volu metric results different? S2: Yes. They man ipulate wrongly on the calculator. Just recalculate! S1: You see if the statements below are correct. S2: Question 2.3 is wrong. The variable x is greater than 0 and smaller than 16. S1: Yeah, fix it." [Excerpts from the discussion of pairs 1] Moreover, students also have many opportunities to consider their own opinions. As such, students have the opportunity to develop metacognition.
"S3: they predict with x = 10 cm, there is the biggest volume.
S4: How could that be? That is wrong if x = 10 cm then V = 1000 cm 3 .
S3: So that proves the volume formula is wrong? S4: But it's like our formu la. So is our formu la wrong? Let's consider it again." [Excerpts from the discussion of pairs 2]

Conclusions
In order to obtain the educational objective of developing mathemat ical co mmun ication competence for students, the teacher must know how to design and implement teaching process s o that his students have opportunities to discuss and communicate with each other. The fourth strategy of Radford and Demers (2004) applied in our case study provided students with a learning environment in wh ich mathematical co mmun ication was enhanced. Some lessons we drew fro m the case study to apply the fourth strategy successfully are:  Math problems or learning tasks assigned to students must stimulate learn ing curiosity fo r students; they are interested in solving the problem.  Math problems or learning tasks assigned to students must be open and have many answers that students have to discuss to give opinions and arguments to defend their opinions;  The learn ing at mosphere must be open and democratic so that students feel free and confident when expressing their opinions.  Teachers play an advisory and support role; do not impose their own opinion on the students.
Fro m the results of the study, in the next t ime, we will undoubtedly introduce it to mathematics teachers and pedagogical students in educational universities of Vietnam.