A Cognitive Analysis of Students’ Mathematical Communication Ability on Circle Equation

DN Afianti* and BAP Martadiputra
Departemen Pendidikan Matematika, Sekolah Pascasarjana, Universitas Pendidikan Indonesia, Jl. Dr.Setiabudi No. 229, Bandung 40154, Indonesia 

*dewinurafianti@gmail.com
Abstract.This study aims to analyze the difficulties of mathematical communication ability of students in one of secondary school on “circle equation” topic. This research conducted by using quantitative approach with descriptive method. The population in this research was all students of that school and the sample was thirty students that was chosen by purposive sampling technique. Data of mathematical communication were collected through essay test. Furthermore, the data were analyzed with a descriptive way. The results of this study indicate that the percentage of achievement of student mathematical communication indicators as follows 1) Declaring a situation into language, symbols, ideas, or mathematical models is 53%; and 2) Explain Mathematical ideas, situations and relations orally and in writing using real objects, pictures, graphs and algebraic expressions 30%. Based on the percentage of achievement on each indicator, it can be concluded that the level of achievement of students' mathematical communication ability is still low. It can be caused the students were not used to convey or write their mathematical ideas systematically. Therefore students' mathematical communication ability need to be improved.
1.	Introduction
Mathematics learning in schools plays an important role to prepare students to have cognitive abilities such as problem solving, reasoning, mathematical communication, and critical thinking through improving the quality of learning in order to face global developments. Mathematical communication is an essential basic math skill that high school students need to have. Some of the reasons underlying the statement of the importance of having mathematical communication skills are: a) This is in line with the formulation of the objectives of mathematics learning namely: learning to communicate (mathematical communication), learning to reason (mathematical reasoning), learning to solve problems (mathematical problem solving), learning to associate ideas (mathematical connections) and learning to present ideas (mathematical representation) [1]; b) The other objectives of mathematics learning are also as follows: understanding the mathematical concepts, explaining the correlation inter concept and applying the concept or algorithm flexibly, accurately, efficiently and appropriately in problem solving, solving problems, and communicate ideas with symbols, tables, diagrams, or other media to clarify circumstances or problems [2]; c) Mathematical Communication Is The Essence Of Teaching, Learning, And Accessing Mathematics [3]; and d) Even the mathematical communication community is a central force in formulating mathematical strategic concepts [4]. Based on these four reasons, mathematical communication skills are one of the cognitive abilities that must be mastered by students. The aim of developing mathematical communication skills in learning are: a) organize and combine mathematical thinking, encourage learning new concepts by drawing objects, using diagrams, writing and using mathematical symbols; b) communicate mathematical thinking in a logical and clear way so that it is easy to understand; c) analyze and evaluate mathematical thinking and other strategies in solving problems; and d) use the language of mathematics to express ideas correctly [1].
Communication is a process of delivering and receiving the results of individual thoughts through symbols to others [5]. The definition of mathematical communication capability is the ability of a person to write mathematical statements, to write reasons or explanations of every mathematical argument that he uses to solve mathematical problems, using terms, tables, diagrams, notations, or mathematical formulas appropriately, and examine or evaluate the others mathematical thoughts [6]. Armed with these abilities, the students will be easier to convey mathematical ideas, so that the mathematical difficulties faced by the students can be easily understood by themselves and others.
Learning must be able to help students communicate mathematical ideas through five aspects of communication, namely representing, listening, reading, discussing and writing [7]. However, in fact the students' mathematical communication ability is still not as expected. Some studies have found poor students' mathematical communication ability. Most students are less able to propose an idea or a mathematical situation into images, symbols or mathematical models and solve them. In addition, another study also found that the mathematical communication ability of grade XI students is still low, that is the average of only about 17% of the class only students who can complete the mathematical communication tests provided [8]. The result of other study in the preliminary observation also found that in general the mathematical communication ability of grade XI students is still low, as evidenced by: a) The ability of students in conveying the mathematical idea is still worse; and b) the students' ability to use mathematical symbols / notations is not accurate [9]. The results of this study indicate that many students do not have good mathematical communication ability. Therefore, in this study researcher want to analyze the answers of students in answering the questions on the matter of circle equation to find out what kind of mistakes students made in answering mathematical communication test.
There are several indicators of mathematical communication ability such as: a) expressing ideas or opinions, situations and mathematic relations both orally and in writing, and describing them visually in the form of real objects, images, graphs, and algebraic expressions; b) represents real objects, images, and diagrams in the form of mathematic ideas and / or mathematic symbols; c) declare or create mathematical models of daily events or other mathematic problems in the language of mathematical symbols, terms, and structures; d) construct conjectures, arguments, formulate definitions and generalizations; e) reexpress a mathematical description or paragraph in her/his own language [10],[1]. As in this research used some indicators of mathematical communication as follow:
a.	Declaring a situation into language, symbols, ideas, or mathematical models.
b.	Explain Mathematical ideas, situations and relations orally and in writing using real objects, pictures, graphs and algebraic expressions.

2.	Experimental Method
This research conducted by using quantitative approach with descriptive. The population in this research was all students in one of secondary school and the sample was thirty students that was chosen by purposive sampling technique. Data of mathematical communication were collected through essay test for 40 minutes duration. Topic of the essay test was circle equation. Furthermore, the data were analyzed with a descriptive way.

3.	Result and Discussion
The results of this study indicate the percentage of student mathematical communication indicators achievement as following: 1) Declaring a situation into language, symbols, ideas, or mathematical models is 53%; 2) Explain Mathematical ideas, situations and relations orally and in writing using real objects, pictures, graphs and algebraic expressions is 30%. Based on the percentage of achievement on each indicator, it can be concluded that the level of achievement of students' mathematical communication ability is still low with the average percentage of each indicator is 41.5%. Some students’ answers are analyzed as follows:
1)	The second communication question with Stating a situation, ideas, and mathematic correlation into images, graphics, or algebraic expressions, as follows: Create a circle equation from figure 1

Figure 1. Graph of Circle Equations

Figure 2 shows that students have not been able to determine a, b and r, so the calculation of the equations is not accurate. The lack of understanding expressed the situation in a mathematical model. 

Figure 2. Student’s answer for communication question indicator 1

Figure 3 shows some students who already understand how to state the situation in the mathematical model, which is only slightly less complete in the section that concludes the equation obtained

Figure 3. Student’s answer for communication question indicator 1

2)	The second communication question with the indicator state daily experience into the form of a mathematic language/symbol, or mathematic model, as follows: A pendulum swing moves back and forth as shown in figure 4. The pendulum swing path (arc AB) satisfies the circular equation. Write down the P coordinates and the length of the pendulum swing

PP

AP

BP

Figure 4. Pendulum swing

For the second indicator 2, there were a lot of students who did not give the answer, only 10 students from 30 students. Many students have complained about the difficulty of linking drawings to mathematical ideas. But several students answered well can be reviewed in Figure 5 which shows the student can find the P coordinate but less accurate in counting. Furthermore, figure 6 shows students can find the P coordinate but it is less precise to calculate the length of the rope.

Figure 5. Student’s answer for communication question indicator 2

Figure 6. Student’s answer for communication question indicator 2

The mistakes that students make in answering those questions reflect that students have difficulties in communicating mathematically. This can be caused students are not used to convey ideas or mathematic ideas and write them in the form of graphs, tables, diagrams, or mathematic models of a problem systematically. One effort that can be done by teachers in the classroom to improve students' mathematical communication is to familiarize the students to openly expose mathematic ideas on the topics being studied, and to create discussion forums so that students can ask each other and listen to each other's answers in the class . This is supported that give attention carefully to the questions of friends in a group can also help students to construct math knowledge more and to set up more effective answer strategies [11]. Furthermore, after the students are used to convey their ideas, teachers should also train students to write the ideas systematically for the expected mathematical communication indicators can be achieved maximally.

4.	Conclusion
Based on the results of student achievement and analysis of each indicator of mathematical communication, it can be concluded that students' mathematical communication ability is still low with average achievement of each indicator is 41.5%. Thus, it is necessary to increase the ability of students' mathematical communication by developing teacher competence in educating and choosing the appropriate learning approach and model to be applied in class so that students can develop their mathematical communication ability well.

Acknowledgments
The highest appreciation are presented by the researcher to lecturer, Mr. Dr. Bambang Avip Priatna, M.Si, that have guided researcher from beginning to the end until the completion of this article.

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