Sebuah Kajian Pustaka: Journal of Mathematics Education p-ISSN 2089-6867 Volume 10, No. 1, February 2021 e–ISSN 2460-9285 https://doi.org/10.22460/infinity.v10i1.p69-80 69 MATHEMATICS COMMUNICATION MISTAKES IN SOLVING HOTS PROBLEMS Lelly Nur Rachmawati, Yus Mochamad Cholily, Zukhrufurrohmah* Universitas Muhammdiyah Malang, Indonesia Article Info ABSTRACT Article history: Received Aug 8, 2020 Revised Jan 10, 2021 Accepted Jan 11, 2021 Student mistakes in communicating mathematical ideas are still widely practiced. Therefore, it is essential to analyze students' mathematical communication errors in solving mathematical problems so that learning planning can be better. This study aims to describe students' mathematics communication errors in solving higher-order thinking skills in linear algebra and matrix subject. The type of research is a qualitative descriptive study. They were 155 students as subject research. The data analysis started by collecting students' answers and then grouped them according to mathematics communication skills criteria. Later identified and analyzed the errors made by students of each mathematics communication criteria. The results showed that mathematical communication errors on the indicators of writing mathematical situations were concept errors and principle errors. The declaring idea's mathematical communication error is a concept error, a principle error, and an operation error. Furthermore, mathematical communication errors on the indicator state that solving-problem using the language itself is a concept error and operator error. Keywords: High Order Thinking Skill, Mathematical Communication, Mathematics Mistakes Copyright Β© 2021 IKIP Siliwangi. All rights reserved. Corresponding Author: Zukhrufurrohmah, Department of Mathematics Education, Faculty of Teacher Training and Education Universitas Muhammdiyah Malang, Jl. Raya Tlogomas 246 Malang, East Java 65144, Indonesia Email: zukhrufurrohmah@umm.ac.id How to Cite: Rachmawati, L. N., Cholily, Y. M., & Zukhrufurrohmah, Z. (2021). Mathematics communication mistakes in solving HOTS problems. Infinity, 10(1), 69-80. 1. INTRODUCTION Learning and communication are integral parts. Kleden (2016) emphasized that in learning activities there was an exchange of information in the form of knowledge and experience between lecturers and students, students and students, also between students and teaching materials, so mathematical communication skills were very much needed. Firdaus and Aini (2019) added that learning activities carried out by students were not limited to memorizing formulas and mastering calculations, but also learning through mathematical communication. Kleden (2016) emphasized that with excellent mathematics communication, students can convey ideas or thoughts appropriately to convince themselves and others. The importance of student mathematical communication is currently a problem that must be considered (Angraini, 2019; Argarini, Yazidah, & Kurniawati, 2020; Astuti & https://doi.org/10.22460/infinity.v10i1.p69-80 Rachmawati, Cholily, & Zukhrufurrohmah, Mathematics communication mistakes … 70 Leonard, 2015; Nuraida & Amam, 2019; Prabawanto, 2019; Rosita, Nopriana, & Silvia, 2019). Therefore, the development of students' mathematics communication skills in problem-solving must master at the lecture level (Fatimah, 2012). It will be useful in social life and good problem-solving results for students (Ruswana, 2019). Excellent communication skills are essential for students because, through symbolic language, students can communicate mathematically accurately and accurately (Novianti, 2017). Student's ability to solve problems can be influenced by students' written mathematics communication skills through their answer sheets (Ardina & Sa’dijah, 2016). Umar (2012) emphasized that communication and problem-solving skills are related and essential to have so that mistakes not occurred. Therefore, it is crucial to develop student mathematics communication. The importance of having mathematics communication skills is not in line with the facts which show that many students make mistakes in communicating their ideas (Pratiwi, 2015). Most of them do not master the concepts but only memorize them (Ruswana, 2019). Students make a lot of mistakes in mathematics communication skills. Therefore mathematics communication conducted through tests will determine the students' ability to master the material (Subekti et al., 2016). Based on this information, mathematics communication skills are related to mathematical errors made by students. According to Ardiawan (2015), mathematics errors are systematic errors that occur when the answers are written differently from the real solutions. Firdaus (2019) said that there were some mistakes when solving Matrices about understanding the question of the story, using formulas and concepts, and doing calculations. Besides, Wijaya and Masriyah (2013) said that there were several types of mistakes made, namely concept errors, principle errors, and operating errors. Concept errors are caused by students when misinterpreting or using terms so they cannot understand a given material/problem (Widodo, 2013; Widodo et al., 2020). Principal error is a mistake made because of the inability of students to connect/link several concepts in the right relationship (Astuty & Wijayanti, 2013). Then, Astuty and Wijayanti (2013) said the operation error was an error made due to incorrect calculation. At the same time, Zukhrufurrohmah and Kusumawardana (2019) research states that the types of mathematical errors can be in the form of skills, concepts, and techniques. But in this study, the mathematics errors that be analyzed were concept errors, principle errors, and operating errors. Agree with Asviangga et al. (2018) that one of the problems that can lead to mathematics communication skills is a matter of high-level thinking. Rahayuningsih and Jayanti (2019) said that improving the quality of learning can obtain by solving Higher- Order Thinking Skills (HOTS). Thus when students do problem-solving activities can use a different approach, it will be possible to do so that a high-level thinking process occurs (Rahayuningsih & Jayanti, 2019). HOTS questions are questions that require you to use higher-order thinking skills (Pratiwi et al., 2017). The aspects of the HOTS problem are analyzing, evaluating, and creating (Rahayuningsih & Jayanti, 2019). Based on some of these aspects in this study HOTS Problem indicators in measuring mathematics communication are students can: 1) analyze the mathematical situation by breaking down the information obtained; 2) evaluate by testing against an idea by the procedure to clarify a statement, and 3) create with writing a summary or theory to conclude. According to the explanation, this researcher is interested in describing mathematics communication errors that students have when completing HOTS Questions in Linear Algebra and Matrix subjects. Volume 10, No 1, February 2021, pp. 69-80 71 2. METHOD This research is qualitative descriptive research aiming to describe the written mathematics communication errors made by students in completing the HOTS problem on matrix material. The study begins by conducting field observations with the data obtained is that students make some mistakes in conveying the idea of solving a given problem. Based on this problem, instruments were then arranged in the form of HOTS-style test questions to lead to errors and mathematics communication of students. The test questions were validated by two mathematics education lecturers who experts in teaching Linear Algebra and Matrix subjects. The test was given to the students through the lms.umm.ac.id platform. Students collect their answers by uploading handwritten photos or sending typed results to lms.umm.ac.id through their respective accounts. HOTS problems that valid were served in Table 1. Table 1. Research instruments Number of Question HOTS Aspect Mathematics communication Indicator on the answer to the problem Mathematical Error that Appears in Problem 1. Find matrix 𝐴, matrix 𝐡, matrix 𝐢, so 𝐴𝐢 = 𝐡𝐢 and 𝐴 β‰  𝐡! Analyze Understanding the mathematical situation clearly, students write that the matrix 𝐴𝐢 = 𝐡𝐢 and matrix 𝐴 are different from matrix 𝐡 Concept errors, principle errors and operating errors Evaluate Writing ideas clearly, students will test by multiplying the two matrices to find out whether 𝐴𝐢 = 𝐡𝐢 and served matrix 𝐴 are different from matrix 𝐡 Concept errors, principle errors and operating errors Create Stating the results of problem- solving using their language, students write the conclusion that matrices A and B are different matrices so that 𝐴𝐢 = 𝐡𝐢 and 𝐴 are different from 𝐡 Concept errors, principle errors and operating errors 2. Suppose there is a matrix 𝐴 with the following components: 𝐴 = [ 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 ] If 𝐴2 = 𝐴𝐴, 𝐴3 = 𝐴𝐴𝐴,π‘Žπ‘›π‘‘ Analyze By understanding the mathematical situation clearly, students write down the information obtained from matrix 𝐴, matrix 𝐴2 = 𝐴𝐴, matrix 𝐴3 = 𝐴𝐴𝐴 and 𝐴𝑛 = π΄π΄β€¦π΄βŸ π‘Žπ‘  π‘šπ‘Žπ‘›π‘¦ π‘Žπ‘  𝑛 Concept errors, principle errors and operating errors Evaluate Writing out ideas clearly, students test by calculating the matrices 𝐴2and 𝐴3 whether they produce a zero-matrix or not, then clarify that the results Concept errors, principle errors and operating errors Rachmawati, Cholily, & Zukhrufurrohmah, Mathematics communication mistakes … 72 Number of Question HOTS Aspect Mathematics communication Indicator on the answer to the problem Mathematical Error that Appears in Problem 𝐴𝑛 = π΄π΄β€¦π΄βŸ π‘Žπ‘  π‘šπ‘Žπ‘›π‘¦ π‘Žπ‘  𝑛 , Is it possible that 𝐴𝑛 produces a zero matrix? Provide detailed explanations of the answers! of the matrix 𝐴2 and 𝐴3 are not zero-matrices, and then test for the student matrix 𝐴𝑛 use each idea to find out whether the matrix 𝐴𝑛 will produce a zero matrix Create Using their language, students write the possibility of the matrix 𝐴𝑛 to produce a zero- matrix Concept errors, principle errors and operating errors The result of 155 students work are grouped according to good mathematics communication skill or less mathematics communication skill. The good mathematics communication skills are given when students answer fulfill three indicators of mathematics communication. Otherwise, students have less mathematics communication skill. Data analysis techniques in this study were carried out through 3 stages: data reduction, data presentation, and concluding. The data reduction stage is carried out by grouping the answers of the two questions based on similar answers into several types of the four classes. This is done to facilitate the analysis based on indicators that have been determined. Then the data obtained are presented in tables and diagrams. And the next step is to conclude the result based on data gathered. 3. RESULTS AND DISCUSSION 3.1. Results Data obtained from the four classes were 155 student answers. Figure 1 shows that 148 students have less mathematics communication in the Question 1 or first question, and seven students who have excellent mathematics communication. Judging from the number of answers shows that mathematics communication possessed by students is still lacking. Only 4.5% of the total amount has excellent mathematics communication. It shows that students cannot communicate their ideas well so that they have difficulty in problem-solving. Figure 1. Number of mathematics communication questions 1 7 148 0 100 200 A m o u n t o f S tu d e n t' s A n sw e r Mathematical Communication Skills Question 1 Good Less Volume 10, No 1, February 2021, pp. 69-80 73 Whereas in Figure 2, the number of answers to second question or Question 2 does not differ from Question 1. The number of students with communication is more or less compared to students with excellent communication. But for the second question, there is no difference with the distant numbers, namely 85 students with less mathematics communication and 70 students with excellent communication. It means that students are more able to communicate their ideas in the second question than the first question. Figure 2. Number of mathematics communication questions 2 Figure 1 and Figure 2 shows that students with good mathematics communication skills are minimal. It can happen because students were making many mistakes when solving problems, such that it did not match the mathematics communication indicator. Table 2 shows that most students made mistakes when stating their problem-solving problems by using their language. These mistakes can be taken because students get used to calculating without looking back on the problem. Another possibility that causes students to have less mathematics communication skills understands the multiple matrices such that students cannot write their ideas clearly. The weakness of understanding the problems is also one of the factors that caused students to make mistakes to determine the solution steps. The percentages of each student's mathematics communication criteria were presented in Table 2. Table 2. Percentage of mathematics communication of students each indicator No. Mathematics communication indicator Question 1 Question 2 Average good less good less good less 1 Understand the mathematical situation clearly 84.52% 15.48% 83.23% 16.77% 83.86% 16.13% 2 Write ideas of problem- solving clearly 76.77% 23.23% 73.55% 26.45% 75.16% 24.84% 3 State the results of problem- solving using your language 6.45% 93.55% 58.06% 41.94% 32.26% 67.75% Table 2 shows that not all students meet the three mathematics communication criteria in solving HOTS problems. The third criterion of mathematics communication, declare the answer to solving HOTS's problem using their language, obtain the most significant percentage of 93.55%. This percentage is the highest compared to the other two 70 85 0 20 40 60 80 100 A m o u n t o f S tu d e n t' s A n sw e r Mathematical Communication Skills Question 2 Good Less Rachmawati, Cholily, & Zukhrufurrohmah, Mathematics communication mistakes … 74 indicators. It happens because students are incomplete in writing conclusions of problem- solving. The highest rate in the second question for the category lacking was also found in the indicator, stating the solving results using their language that is 41.94%. Most students did not write the conclusions of the calculation that they obtain. Although students are good at understanding mathematical situations or writing their ideas, it did not mean that students' mathematics communication was excellent. Each rate of mathematics communication skills showed that not all students fulfilled all criteria of mathematics communication. Students in problem-solving with problems may make mistakes by mathematics communication written owned. The mistakes made by students in completing Problems are presented in Table 3. Table 3. Forms of student mathematical errors Mathematics communication indicator Types of mistakes made by students Concept Principle Operation Understand the mathematical situation clearly In Question 1, it is wrong to determine matrix 𝐴,𝐡,𝐢 In Problem 2, the concept of an identity matrix and matrix multiplication is wrong Not found Write ideas of problem- solving clearly In Question 2, it is wrong to apply the concept of multiplication to the 𝐴𝐴 matrix In Question 1, it is wrong to use ideas, so there is no relationship between the matrix 𝐴 𝐡 𝐢 In Question 2 it is wrong to form the matrix 𝐴𝑛 because it does not find a relationship with the matrix 𝐴2 and 𝐴3 Not found State the results of problem-solving using your language In Question 1, it is wrong to conclude that 𝐴𝐢 = 𝐡𝐢. In Question 2, it is wrong to conclude that 𝐴𝑛 will probably produce a zero-matrix Not found In Question 1, it was wrong to calculate the 2π‘₯2 matrix multiplication. 3.1.1. Mathematical Errors when Understanding Mathematical Situations Clearly On the indicators of understanding the mathematical situation clearly, students made mistakes, namely misconceptions and principles, but did not make operational errors. In the second question, students did not make misconceptions when understanding mathematical situations. Students created a mistake in the first question. The mistake made by students were wrong in determining the A, B, C matrices. These errors resulted in students not getting AC = BC. The answers of students who make misconceptions when understanding the mathematical situation are clearly shown in Figure 3. Figure 3 shows that students only determine matrix C and then conclude that AC is the same as AB, while matrix A and matrix B are not explained as components. Volume 10, No 1, February 2021, pp. 69-80 75 𝐢 = [ 1 2 3 4 ] 𝐴𝐢 = 𝐡𝐢 𝐴 = [ 1 2 3 4 ] = 𝐡[ 1 2 3 4 ] Figure 3. Misconceptions when understanding mathematical situations in the first problem Students make a principles mistake in the form of incorrectly linking the concept of the identity matrix and matrix multiplication. Figure 4 is the answer to students who earned a principles error when they did not understand the mathematical situation correctly. Students make a mistake when multiplying the matrix and stating that the matrix A2 is the identity matrix. 𝐴2 = [ 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 ]= Matrix A still produces an identity matrix (fixed matrix). The answer is the matrices does not produce a zero matrix, because the matrix 𝐴𝑛 still produces an identity matrix because the multiplication result of the matrix is the number of columns. Figure 4. Mistakes in principle when understanding mathematical situations clearly 3.1.2. Mathematical Mistakes when Writing Out Problem-Solving Ideas Clearly Students make misconceptions, principles, and operations when writing problem- solving ideas on the second problem. But in the first question, no students made mathematical mistakes. In the second problem, students created a misconception, namely not applying the concept of multiplication to matrix A. Students wrote the conclusion that An produced a null matrix, but did not test the multiplication against A4 matrix. Figure 5 shows students' mistakes, not doing the calculation concept of matrix A, so that they do not write ideas. Yes, 𝐴𝑛 yields a zero matrix Explanation: Suppose 𝐴4= matrix 0 Because 𝐴3 = [ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ] Figure 5. Misconceptions when writing clear problem-solving ideas Rachmawati, Cholily, & Zukhrufurrohmah, Mathematics communication mistakes … 76 Students made a principles mistake in the second problem by not doing the multiplication concept on the matrix A3. Students only perform calculations on matrix A2 so that they do not find the relationship between the A2 and A3 matrices in finding the An matrix. So, it concludes without checking whether An yields a null matrix (Figure 6). Figure 6. Principle mistakes when writing clear problem-solving ideas 3.1.3. Mathematical Errors when Expressing Problem Solving Results in His Language Students make misconceptions and operational errors when clearly stating the results of solving problems. The mistake made by students when reporting the results of solving the problem was that the multiplication matrix did not recognize ranks so that An could not be a zero matrix. In Figure 7 showed that students wrote that An could not be a null matrix because matrix A is not a zero matrix. Figure 7. Misconceptions when expressing problem solving results in his language Operation errors made by students when stating the results of solving the problem in the first problem are wrong to conclude that AC = BC. Students get the same results, but the calculations that have been done are not correct. In Figure 8, students state that the results of problem-solving do not match the calculations obtained. 𝐴 = [ 2 3 2 4 ]𝐢 = [ 2 βˆ’4 βˆ’2 4 ] β†’ 𝐴𝐢 = [ βˆ’4 8 βˆ’2 4 ] 𝐡 = [ 1 3 0 1 ]𝐢 = [ 2 βˆ’4 βˆ’2 4 ] β†’ 𝐡𝐢 = [ βˆ’4 8 βˆ’2 4 ] It proved that the both multiplication (above), the matrix 𝐴𝐢 = 𝐡𝐢 have the same result. Figure 8. Operating errors when stating problem-solving results in his language Matrix operation does not recognize the power form. The power referred to in matrix operation is the repeated multiplication of a matrix with the matrix itself. The condition of a matrix to be raised is that it must be a square matrix or a square matrix. Thus, the exponent of the square matrix itself is the sum of the powers. if the matrix is nonzero, then 𝐴𝑛 is unlikely to produce a null matrix later. For example number 1, if number 1𝑛 then whatever n will not change from 1 to 0. Volume 10, No 1, February 2021, pp. 69-80 77 3.2. Discussion The data of Figure 1 and Figure 2 show that the student's mathematics communication ability was needed to be increased. This finding was in line with Firdaus and Aini (2019) research, which found that students' communication is still in the low category. This research also shows that most students did not fulfill the three indicators of mathematics communication, which indicated their mathematics communication skills need to be improved. Firdaus and Aini (2019) also found that students' weakness in mathematics communication can be caused by student mathematics communication indicators that have not been achieved. Moreover, Syafina and Pujiastuti (2020) found that only students with good mathematics ability can fulfill all mathematics communication indicators. This research found that students' ability to state the problem-solving results using their language is the most difficult mathematics communication indicator for students to fulfill. Students' answers do not give reasons with their language style, so it does not state the conclusions of the given problem. This phenomenon is also found in Mirna's (2018) research. Besides, some students still have difficulty writing ideas and knowing the mathematical situation clearly. So that in this study, students made mistakes in concepts, principles, and operations. Students' types of errors are in line with Zulfah's (2017) research that students make mistakes in the form of concept, process, and principles errors. Student's mistakes influenced the ability of mathematic communication measurement in each indicator. In other words, mathematics communication ability related each other with students' mistakes (Zulfah, 2017). Afifah et al. (2018), in their research, stated that the types of mistakes made by students were concept errors, operation errors, and principles errors. Firdaus (2019) research shows that in solving the Matrix Problem, the type of error made is concept and operation error. Likewise, with this study that students also carried out misconceptions and operations. More specifics, based on Table 3, most students made conceptual mistakes while solving the problems given. This finding can be considered to point out that students may not be able to communicate their idea well because they did not understand the problem. This possibility inline with Abidin et al. (2017) research concluded that a misconception caused another five kinds of students' mistakes. These conceptual mistakes also affect principles mistakes of student's work (Mirna, 2018). For example, in Figure 4, students' misconception about the definition of identity matrix then applied this wrong idea to analyze the situation. Albeit a little, students also make procedural mistakes when they calculate or simplify the algebra form. Students also found this mistake in Parwati and Suharta (2020) research that procedural mistakes are still made frequently by students when solving algebra problems. 4. CONCLUSION Students' mathematics communication errors in completing HOTS problem on matrix material are concept errors, principles errors, and operating errors. These errors can occur in every student mathematics communication indicator. In the first mathematics communication indicator that understands the mathematical situation clearly, mistakes made are concept errors and operating errors. The second indicator of mathematics communication is writing ideas clearly; the mistakes made are concept errors and principles mistakes. As for the third mathematics communication indicator that states the problem-solving results using their language, the mistakes made are concept errors and operating errors. 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