ANALYSIS OF STUDENTS’ MATHEMATICAL COMMUNICATION IN DRAWING GRAPHS OF MULTIVARIABLE FUNCTIONS

Received: October 29, 2020 Accepted: November 15, 2020 Published: November 30, 2020 Mathematical communication competencies help students to convey mathematical ideas orally and in writing. The purpose of this study was to analyze students' communication skills in drawing graphs of multivariable functions in advanced calculus courses. This type of research is qualitative with descriptive methods. The subjects in this study were 28 students of Mathematics Education class 4B. This study used a stratified random sampling technique to obtain 3 students consisting of high, medium, and low mathematics competencies. The data collection instruments used were the competency mapping test, evaluation test, and interview guidelines. The results of this study are (1) students with high competence excel at writing and making mathematical models, (2) students with moderate competence have advantages in three stages, but there are few errors in applying the agreement, (3) students with low competence make mistakes at the third stage due to wrong in formulating problems and designing problem-solving strategies.

Based on the results of the researchers' observations about the student's communication skills are still low in solving multivariable function graph problems as shown in Figure 1.  Figure 1 shows that students are still unable to construct their knowledge in drawing graphs of multivariable functions which is proven by students still thinking about cartesian diagrams in two-dimensional space. The student cannot describe the x-axis, y-axis, and zaxis in three-dimensional space. Then the chosen strategy is also not properly used so that the wrong answer is obtained. Thus, it is necessary to communicate to students through indepth interviews to find out what they are thinking and taking in this way so that scaffolding can be done to improve student thinking.
Other research about the weakness of student mathematical communication can be overcome by applying project-based learning, problem-based learning model, and combining it with collaborative learning [12] [13] [14] or using the video-tracker analysis to measure communication skills [15]. Other studies use the Cabri 3D application to overcome the communication limitations of students [16]. This is similar to this research which uses a similar application, namely Geogebra 3D, but the difference lies in the provision of scaffolding in the form of in-depth interviews to explore more students' knowledge about what they think and why they choose this strategy.

METHOD
The purpose of this study was to analyze students' communication skills in drawing graphs of multivariable functions in advanced calculus courses. This research approach is qualitative with a descriptive method. The descriptive method is a research method that describes a scientific condition or condition by describing it [23]. The research was conducted in the 2019/2020 academic year at the University of Mataram. The research subjects chosen were students of Semester 4 Mathematics Education for the 2019/2020 academic year who took the Advanced Calculus course totaling 28 students. Then 3 students were taken based on the calculation of their mathematical ability levels as presented in Table 1.  Information: = score of the mathematical ability of each student Mean = average score of students' mathematical ability SD = standard deviation of students' mathematical ability scores Source: [24] In detail, this research procedure can be seen in Figure 2. Activities undertaken include the selection of research subjects, giving test questions, and online interviews. Indicators of students' mathematical communication skills in drawing graphs of multivariable functions are presented in Table 2. Provide scaffolding with in-depth interviews to students who make misconceptions, technical, and theoretical.
Reflect on problem-solving activities with Geogebra's assistive tools.
Analyze data from interviews and scaffolding

RESULTS AND DISCUSSION
The results of this study indicate that the research subjects in drawing multivariable graphs have various mathematical communication skills. The diversity of mathematical communication skills possessed by students shows the level of student understanding of the Multivariable Function material.
Based on the mapping of mathematical abilities of 4A students using the formula presented in Table 1, there are 12 students with high mathematical ability categories, 11 students with moderate mathematical ability categories, and 5 students with low mathematical ability categories. Furthermore, 3 students were selected from each level to represent each category of student mathematical ability. A discussion of mathematical communication skills from each category of mathematical abilities can be stated as follows.

Mathematical Mommunication Skills Obtained from The Subject of High Mathematical Abilities
The description of students' mathematical communication skills with mathematical communication skills can be shown through 3 indicators, namely writing, making mathematical models, and drawing.   Figure 4 shows that M1 has implemented the strategy chosen to draw the graph. Next, M1 plots the coordinates of the points obtained, namely points (0, 0, 0), (1,1,5), and (1, 0, 2). However, there is a new point that is plotted in the three-dimensional space without any prior explanation at the writing stage and creating a mathematical model, namely point (0, 1, 3). However, if checked again with the substitution to z = 2x + 3y, this point satisfies the equation. At the drawing stage, M1 made a mistake in plotting points (1, 0, 2) and points (1,1,5). Thus, the final answer from M1 is still wrong because it is wrong in plotting points (1, 0, 2) and points (1,1,5) so that the graphical image obtained is also wrong. The stage of making a mathematical model can be seen from M1's answer in Figure  1 and operations correctly. M1 can substitute the selected points to z = 2x + 3y then plot the points and are connected to form a function graph. R : What are the first steps you take when drawing charts? M1 : First draw the x-axis, y-axis, and z-axis lines complete with the diagram components. R : OK good. How do you plot the points? M1 : If point (0, 0, 0) is right in the middle, while point (1, 1, 5) starts by shifting 1 unit from 0 to 1 on the x-axis. Then it is shifted 1 unit to the right parallel to the y-axis then shifted upward 5 units parallel to the zaxis. R : Take a look back at your picture, is it by the steps that you have revealed earlier? M1 : Ohh ... yes ma'am, I'm in a hurry

Mathematical Communication Skills Obtained from The Subject of Moderate Mathematical Abilities
The second subject in this study was a student with moderate mathematical ability, namely M2. Furthermore, the mathematical communication skills possessed will be analyzed at the writing, drawing, and writing stages of mathematical models. M2's answer at the writing stage is presented in Figure 5.  Figure 5 shows that at the writing stage, M2 can already identify the problem well. This is shown by writing and the M2 interview results in writing down what was known and asked in the questions. However, at the stage of writing the mathematical model, M2 did not write down the complete completion procedure. It is shown from M2 not writing the substitution process for several points obtained. M2 also did not check the points obtained, but overall M2 did not make any mistakes in the calculation. M2's answer at the drawing stage is presented in Figure 4. R : What do you know about question no.1? M2 : In the problem, the function z = 2x + 3y has been given then I was asked to describe the graph. R : How do you solve it? M2 : I took several points and then substituted them for the equation z = 2x + 3y. R : What do you gain from taking these points? M2 : I obtained 3 points that fulfill the equation, namely points (0, 0, 0), (1,2,8), and (2, 2, 10).

Indonesian Journal of Science and Mathematics Education
Analysis of Student's Mathematical …. Everything is correct ma'am ... has fulfilled the equation z = 2x + 3y. R : OK ... make it a habit to check your answer again. To make it easier to check these points, also make it a habit to write down the complete answer, including the process when substituting each of these points into the equation.  Figure 6 shows that M2 is quite good at point plotting. The three points plotted are points (0,0,0), (2,0,8), and (2, 2, 10). There is 1 new point that was not found at the writing stage, namely point (2, 0, 8) but that point also satisfies the equation = 2 + 3 . M2 draws a graph using the left-hand rule as shown in Figure 7b). This is also permissible, although it is generally agreed that drawing graphs on the Three-Dimensional Cartesian diagram use the right-hand rule as shown in Figure 7a) to obtain a uniform answer. Overall, M2 can graph the multivariable function = 2 + 3 correctly. R : Did you use the right-hand rule or the left-hand rule to draw a threedimensional Cartesian coordinate system? M2 : I forgot the order, actually, I also know by agreeing to use the right-hand rule but I put the x and y wrong. R : Yes, it is okay. Using the left-hand rule is also okay, but so that our answers are all uniform, we agree to use the right-hand rule. OK… Check again whether you have substituted the three points for z = 2x + 3y? M2 : The three points have fulfilled the point, if substituted, I replace the point (1,2,8) with (2, 0, 8) R : Why did it change? M2 : Because when I drew the point, the lines formed almost coincide, so I changed the point.  Figure 8 shows that M3 cannot formulate the problem well so that the problemsolving strategy chosen is also not right. At the written text stage, M3 is asked to graph the multivariable function z = 2x + 3y, but M3 divides it into 3 stages, namely by finding the equation for the resulting line when x = 0, y = 0, or z = 0 then combining the three the equation that has been obtained and describes the result like the fourth image but the fourth image obtained is also wrong. At the drawing stage, M3 can describe the three equations of the resulting lines but when asked to graph a multivariable function it produces an incorrect answer. While at the stage of making a mathematical model (a mathematical expression), M3 did not make a mistake in operating the algebraic form that was formed and there was no mistake in making point substitution, but because the solution strategy is chosen was not correct, the resulting final answer was also incorrect. Figure 9. Screenshot of = 2 + 3 Graph via 3D Geogebra Application Figure 9 shows the scaffolding assistance to students with the 3D Geogebra application. This application aims to reinforce the answers of M1, M2, and M3 in drawing multivariable graphics because there is a feature of rotating objects so that they can develop students' imagination in drawing multivariable graphs in three-dimensional space. Thus, the results of this study show that students with high mathematical abilities also have an impact on good communication skills by fulfilling all indicators but make few mistakes due to the rush factor. Meanwhile, students with moderate mathematical abilities tend to have fairly good communication skills on the condition that they have to pay attention to several problem requirements. Meanwhile, students with low math abilities tend to have poor communication skills because they are weak in formulating problems and designing problem-solving strategies. In other words, it can be concluded that students with low communication skills tend to be less good at planning problem solving so that they show wrong results. Another study only uses computer aids in the form of Cabri 3D to help students draw 3D graphics but this study not only using Geogebra's computer aids but also providing scaffolding to students who have low mathematical skills until the correct answer is obtained [25].

CONCLUSION
The results of this study can be concluded that students' mathematical communication skills with high mathematical abilities have an advantage in the writing stage and making mathematical models, but there are a few errors at the drawing stage, communication skills Mathematical students with moderate mathematical abilities have excellences in all three stages, but there are a few errors in following the agreement which should use the right-hand rule to the left-hand rule, the mathematical communication skills of students with less mathematical abilities, make mistakes in all three stages due to mistakes in formulating problems and designing problem-solving strategies so that the final wrong answer is obtained. For other researchers who want to carry out similar research, it is better if they pay attention to the mathematical communication skills of each student not only based on their mathematical abilities but also in terms of gender, learning style, motivation and interest in learning, and can test this relationship with further experimental research. to obtain significant results.