Vocational High School Students’ Mathematical Communication: Auditory, Visual, and Kinesthetic Learners

. Mathematical communication skills are essential in activating students' problem-solving abilities and activating excellent cognition. This study aims to analyze the mathematical communication skills conceived by vocational students on financial mathematics material according to auditory, visual, and kinesthetic learning styles. The study used descriptive qualitative with nine samples determined using a purposive sampling technique. Each learning style category comprised three quick, moderate, and slow students comprehending the given materials. The data were collected using a mathematical communication test and a learning style questionnaire. The obtained data were analyzed using Miles and Huberman's model. The results showed that students with an auditory learning style whose ability to comprehend material was quick and slow could complete mathematical communication indicators. In contrast, students with a moderate category could not complete the fourth indicator. Visual learners with quick material comprehension did not formulate situations mathematically, and all the categories did not comply with the fourth indicator. Kinesthetic learners with quick material comprehension could not complete the first and third indicators. The kinesthetic learner with slow comprehension could not fulfill the fourth indicator, while one in the moderate category could complete all mathematical communication indicators.


Introduction
Mathematical communication is one of the abilities currently relevant to the 21 st -century education model.Low communication skills obstruct the effective transmission of information.
Thus, it might lead to misconceptions.Mathematical communication is required in solving problems since it represents the problem and its solvency generation (Sari, Syahputra, & Surya, 2020).Based on the National Council of Teachers of Mathematics (NCTM), mathematical communication refers to the capacity to communicate mathematical concepts orally, in writing, and visually through various media (NCTM, 2000).According to NCTM, there are several indicators of students' mathematical communication skills, encompassing (1) arranging and conveying mathematical thoughts based on information; (2) communicating mathematical reasoning that is understandable to other people; (3) evaluating strategy and results employed as well as their mathematical thinking; and (4) using appropriate mathematical language (NCTM, 2000).According to NCTM (2000), students are considered to have good communication skills if they can express mathematical ideas orally, in writing, and visually through various media, comprehend, interpret, and evaluate mathematical ideas and conclusions drawn orally or visually, and use mathematical notation and terms to present ideas, describe relationships, and model situations.Thus, mathematical communication indicators cover (1) communicating ideas coherently and clearly to others, (2) organizing mathematical thinking through communication, (3) using mathematical language to express mathematical ideas correctly, and (4) analyzing and evaluating mathematical thinking and strategies.
Communication skills are important as they include the eight skills required for the 21 st century: critical thinking, innovation, inquiry, self-direction, perseverance, information utilization, systems thinking, communication, and reflection (OECD, 2018).The need for communication skills is also mentioned in P21 (2019).Regarding the need to face the 21 st century, the 4C skills, namely critical thinking, communication, collaboration, and creativity, are compulsory.
Different learning styles have different abilities in mathematical communication.Teachers need to train mathematical communication skills according to students' learning styles so that abilities can develop optimally.Teacher Kolb (2015) explains that learning style is a person's choice in understanding experiences and transformation processes.Porter and Hernacki (2005) state that students' learning styles consist of visual, auditory, and kinesthetic learning styles, and students might perceive one of them.Teachers can teach according to the student's learning styles so that they can easily obtain lessons and improve their learning outcomes (Gilakjani, 2012;Sheromova, Khuziakhmetov, Kazinets, Sizova, Buslaev, & Borodianskaia, 2020).This study uses three learning styles, namely visual, auditory, and kinesthetic learning styles (Porter & Hernacki, 2005).Researchers undertook this study because of the need for an initial diagnosis of students' learning styles before conducting the teaching and learning process to optimize the learning process (Rogowsky, Calhoun, & Tallal, 2020).Based on Porter and Hernacki (2005), visual learners can learn through visual media such as pictures, posters, videos, and movies.
Auditory learners learn through listening to what others say during conversations, group projects, discussions, and oral lectures.Kinesthetic learners can learn through "hands-on" activities.They enjoy engaging in physical activity and interaction.
To the best of the researchers' knowledge, research on mathematical communication skills based on learning styles had been conducted at junior high school (Asgafi, Anwar, Choirudin, Darmayanti, & Usmiyatun, 2023) and senior high school level (Marzuki et al., 2019).
However, no study has been undertaken in vocational high schools.The vocational high school level has a big relationship with several learning practices, so many people often attribute vocational high school students to kinesthetic learners (Chavarría-Garza, Santos-Guevara, Morones-Ibarra, & Aquines-Gutiérrez, 2022).However, based on the results of a questionnaire given by authors to students at a vocational high school in Sleman, Indonesia, they had a variety of learning styles, such as auditory, visual, and kinesthetic learning.Therefore, research must reveal students' mathematical communication skills based on their different learning styles in vocational high school.The finding can be a reference for vocational high school teachers in determining appropriate learning strategies for their students.
Financial mathematics is one of the important mathematics topics to master because it is closely related to students' economic lives (Moreno, Solórzano, Morales, Villegas, & Sánchez, 2021).Sun, Yuen, Zhang, and Zhang (2020) and Berry, Karlan, and Pradhan (2018) provide examples of how financial education helps avoid issues like having bad credit or defaulting on loans.Expectations do not always match the ideal situation.Students still understand financial mathematics but are limited to using formulas (Ekol & Greenop, 2023).
Financial and mathematical communication are interconnected regarding taking appropriate action in managing finances.Individuals communicate several solutions they have to other people so they can decide appropriately (Van Campenhout, 2015).Communication is also needed regarding financial literacy (Wise, 2013)

Method
This study used a qualitative descriptive method at one of the vocational high schools in Depok, Sleman, Indonesia.The subjects of this study were nine students selected from 33 students from a class majoring in Industrial Chemistry.They were selected by using the purposive sampling technique with the consideration that there were three types of learning styles and three categories in comprehending material for each learning type, which were slow, moderate, and quick.From the nine students, there were three students with an auditory learning style (A1, A2, and A3), three students with a visual learning style (V1, V2, and V3), and three students with a kinesthetic learning style (K1, K2, and K3).Three students from each learning style represented students who could quickly, moderately, and slowly comprehend mathematics material.The selection of students for each learning style was based on the results of the questionnaires, and the selection of students in the categories of quick, moderate, and slow in comprehending math material was based on the results of tests and teacher recommendations.
A learning style questionnaire and a test of students' mathematical communication were the instruments used in this study to gather information on the learning preferences of the participants.Table 1 shows the learning style indicator created by Porter and Hernacki (2005).The learning style questionnaire consisted of twenty statements, with A being the auditory learning style option, B being the visual learning style option, and C being the kinesthetic learning style option.Student learning styles were determined from the greatest number of choices of statements for each learning style.
The mathematical communication task was conducted after the learning style questionnaire was administered.One task about financial mathematics was undertaken.Three graduate students who had done teaching internships at vocational high schools validated the learning style questionnaire and mathematical communication test.Validation results were calculated based on the V-Aiken index.The content validity of the mathematical communication test items was 0.92, and the content validity of the learning style questionnaire was 0.83, which was in the high category according to Aiken (1985).The suggestion from the validator for the learning style questionnaire was to replace inappropriate words, and the suggestion for a mathematical communication test was about writing final savings that do not need to be rounded.Figure 1 shows a math communication task for class 11.
Figure 1.Financial math problems As for data collection techniques, the researchers gave a learning style questionnaire to 33 students to determine which students had auditory, visual, and kinesthetic learning styles.After obtaining the data on student learning styles, researchers chose three auditory students with quick, moderate, and slow mathematics comprehension, three visual students with quick, moderate, and slow mathematical comprehension, and three kinesthetic students with quick, moderate, and slow mathematical comprehension.Then, each subject was given a mathematical communication ability test on the financial mathematics topic they had learned.The test was carried out for 10 minutes.
The Milles and Huberman methodology (Miles, Huberman, & Saldana, 2014), which comprised data reduction, data presentation, verification, and decision-making, served as the foundation for the qualitative descriptive analysis used in this study.The stages encompassed (a) the data reduction stage, in which the responses of the nine research subjects were each verified using the indicators of mathematical communication; (b) data presentation, which was to present the data; and (c) drawing conclusions, which was to conclude from the results of the verification.
Rina saves IDR 5,550,000.00 in a bank with a compound interest rate of 11% per year.How long does Rina's savings take to become IDR 12,790,184.6?

Results and Discussion
The findings of the questionnaire analysis of the learning styles of vocational high school students majoring in Industrial Chemistry obtained groupings of student learning styles.Based on the test results, 17 students had a visual learning style, five had an auditory learning style, and 11 had a kinesthetic learning style.Then, nine subjects were selected based on three types of learning styles, with three subjects for each type of learning divided into quick, moderate, and slow abilities in comprehending mathematics material based on test scores and observations.Table 3 conveys the research subjects used in the present study.For the second indicator, A2 could explain the strategy used but was less precise in calculations.
A2 could not use a calculator to calculate the logarithm in code A. The logarithm calculation was not quite right.For the third indicator, A2 could use the correct notation to represent the information.However, for the fourth indicator, the answer chosen did not make sense because the year for the solution to the task was not round, that was 6.4 years in code B. A2 could analyze and evaluate the results.Here is the interview excerpt between A2 and Researcher (R).

R : How do you solve that problem? A2 : I first write down what was known and what was asked. Then, I looked for the compound interest formula, entered the information in the formula, and calculated it. R
: What did you use to calculate?and what did you find?A2 : I used a calculator ma'am and I found 6.39 years.I rounded it to 6.4 years.
In connection with Figure 3, A3 could write down the information mathematically but was less precise in identifying mathematical aspects of problems located in real-world contexts and identifying significant variables because the initial savings known to be incorrect were IDR 5,500,000.00.The initial savings should be IDR 5,550,000.00.So, for the third indicator, A3 could use the right notation but less precise.For the second indicator, A3 could explain the strategy but was less precise in calculations.A3 could not use a calculator to calculate the Automatically, the selected answer was also not quite right, and the name in the conclusion was false.It must be "Rina," not "Andri," on code D.Here is the interview excerpt between A3 and Researcher (R).Regarding the results conveyed by V2, the errors made were the same as those for A2.To convey the information mathematically, V2 could what information is known and asked exactly.
V2 also could explain the strategy to find the results, and V2 could use the correct notation.
However, for the fourth indicator, V2 did not analyze and evaluate the result in the real world.
Here is the interview excerpt between V2 and Researcher (R).

R
: What is your strategy for solving this problem?
V2 : Write down the available information and know what was asked.Then, I looked for the appropriate formula, calculated it, and got the result.R : What are the results?What tool did you use to calculate?V2 : I found 6.4 years using a calculator.immediately wrote down the strategy without writing any information about the problem.For the second indicator, K1 could explain the right strategy to find the right solution.However, the information used was less precise for initial savings.K1 wrote down the initial savings.IDR 5,500,000.00,which should have been IDR 5,550,000.00as shown in code G. So, K1 used the correct notation but was less precise.However, the result was 8 years, which was appropriate in the real-world context obtained from rounding decimals.Therefore, K1 fulfilled the fourth indicator.Here is the interview excerpt between K1 and Researcher (R).

R
: How do you solve that problem?K1 : Find existing information and look for what was asked, then look for a formula according to the existing information, enter the information into the formula, calculate it, and find a solution.R : What did you get? K1 : I got 8.0867 years, but because the interest was given per year and there were no decimal years, I rounded it up to 8 years.
Regarding K2, it could fulfill all indicators of mathematical communication, which are being able to convey the information mathematically, explain the strategy to find solutions and analyze and evaluate the results obtained in the real world.The results obtained were also correct; that was 8 eight years.Here is the interview excerpt between K2 and Researcher (R).K3 : Honestly, ma'am, I forgot how to calculate logarithms, so I am unsure whether my answer is right or wrong.
Based on the results of the vocational high school students' learning style questionnaire, there was an unexpected result that the most common learning style was a visual learning style with 17 students, followed by a kinesthetic learning style with 11 students, and an auditory learning style with five people.It is similar to Baltaci, Yildiz, and Özcakir (2016) and Gilakjani (2012) in that students mostly used a visual learning style.Based on the study results, two out of three students with visual and auditory learning styles were less familiar with non-routine questions, such as calculating logarithms.In contrast to students with a kinesthetic learning style, one out of three students miscalculated logarithms.The following discussed each subject used in the present study for more details.
Students with an auditory learning style who could accept material quickly and slowly, as represented by A1 and A3, could fulfill all indicators of mathematical communication with the correct answers.That result was in line with Pratiwi, Inganah, and Putri (2020), who showed that the auditorial learning style gave good, long, and detailed answers to address the questions.It was different with students with moderate auditory learning styles receiving material represented by A2.He was less able to analyze and evaluate the results.Reinforced by Baiduri, Khusna, and Solikhah (2021), auditory learning style students did not re-check their answers.
Students with visual learning styles had diverse mathematical communication skills.
Students who could quickly accept material, as represented by V1, did not fulfill the indicators of conveying situations mathematically because they immediately wrote strategies to find solutions.
This was contrary to Rivai, Lestari, Munir, and Anas (2023) that students with a visual learning style could write the information mathematically.This could happen because students with a visual learning style took a long time to understand information by reading.In addition, subjects V1, V2, and V3 did not evaluate answers.It was the same result with Sakinah and Avip (2021) that students with a visual learning style were lacking in interpreting solutions to the real world.
Students with a kinesthetic learning style who could accept material quickly, as represented by S1, did not formulate situations mathematically but could use correct notation even with less precision.So, S1 just fulfilled the second and fourth indicators.That result was similar to Sholihah, Riyadi, and Triyanto's (2020) study that kinesthetic learners could express and evaluate mathematical theory but not for other indicators.Students who comprehended material at a moderate level, as represented by S2, could complete all indicators with appropriate answers.
Students could solve the problem correctly using their strategy (Pratiwi et al., 2020).Meanwhile, students with slow comprehension, as represented by S3, only had errors in calculating logarithms, which caused errors in analyzing and evaluating solutions to the real world.Kurniawati, Suyitno, and Mulyono (2021) stated that kinesthetic learners could not explain the solution to daily problems.

Conclusion
Based on the results and discussion, auditory students who can comprehend material in real life, by changing the year it initially obtained from decimal form to an integer to be eight years.Here is the interview excerpt between A1 and Researcher (R).R: What are your steps in working on this problem?A1 : I first wrote down the information in the question and what was asked.Then, I looked for a strategy that suited what was asked and the information provided.R: What about the answers you found?A1 : I found a year with a result of 8.0867.Because the interest was every year and there were no years in decimal form, I rounded 8.0867 to the nearest year, which was 8.

Figure 2 .
Figure 2. The answer of student A2 Regarding Figure 2, student A2 could write down the situation mathematically precisely.
= (1.11)()log 1.11 (2.31) = t t = 6.39So, it takes about 6.4 years for Rina's savings to become Rp.12,790,184.6 at a compound interest rate of 115 per year logarithm in code C. For the fourth indicator, A3 could analyze and evaluate the solution in the real world, but the answer was incorrect because the logarithm calculation was not quite right.

Figure 3 .
Figure 3.The answer of student A3 R : How do you solve that problem?A3 : Writing down what was known and what was asked, looking for the formula, entering what was known, calculating it, and getting the result.R : What did you use to calculate?and what are the results?A3 : I used a calculator and got the result of 12.274 years.Then I rounded it up because there was no decimal in years, so it got 12 years.

Figure 4 .
Figure 4.The answer of student V1 Regarding Figure 4, V1 did not convey the situation mathematically in the first indicator.V1 immediately wrote down the strategy without writing any information about the problem.For the second indicator, V1 explain the strategy to find the right solution.Although V1 did not convey the situation mathematically, V1 could use the correct notation to find the solution.Thus,

Figure 5 .FV3:
Figure 5.The answer of student V3 The results of V3's work depicted in Figure 5 showed that V3 could convey the information mathematically and correctly for the first indicator.V3 wrote down what was known and asked completely.In addition, V3 could explain the strategy to find solutions and use the right notation.However, V3 could not fulfill the fourth indicator.V3 could not use the logarithm calculator as in code F and did not pass on the answer.Here is the interview excerpt between V3 and Researcher (R).R: What is your strategy for solving this problem?

R:
What are your steps in solving this problem?K2 : I wrote down what was known and asked, then looked for a formula corresponding to what was known and asked, calculated it, and finally found the answer.R : What answer did you find? will take 8 years for Rina's savings to become IDR 12,790,184.6 quickly and slowly can complete mathematical communication indicators.However, a student who has slow comprehension carries some calculation errors.A student who has a moderate comprehension level cannot evaluate answers, so they cannot fulfill the fourth indicator.Students with visual learning styles for quick material comprehension do not formulate situations mathematically, and all the categories do not fulfill the fourth indicator since they cannot evaluate the result and strategy.Meanwhile, students with kinesthetic learning styles for quick material comprehension do not fulfill the first and third indicators.Kinesthetic students with slow comprehension cannot meet the fourth indicator, while the one in the moderate category can complete mathematical communication indicators.Based on the study results, it is expected that teachers can emphasize understanding concepts, using calculators, and evaluating the answers obtained for students with auditory, visual, and kinesthetic learning styles.Students should also be accustomed to writing down the information of the problem.The present study is limited to examining vocational high school students in one major with limited subjects.For further research, it is advisable to examine the mathematical communication of vocational high school students in terms of different cognitive and affective skills in dissimilar fields.

Table 1 .
Indicators of learning styles

Table 3 .
Subjects' qualifications One task was given to find out the mathematical communication skills of vocational high school students in financial mathematics.Students took this test, and researchers analyzed the data using student work suitable for auditory, visual, and kinesthetic learning.Table4displays the summary results of students' mathematical communication skills based on learning styles.

Table 4 .
Summary of subjects' results based on mathematical communication test calculation results.A1 could convey the information of the problems by writing down what was known and asked.A1 could explain the strategy for finding solutions.A1 could use the correct notation to represent the information.Then, A1 could analyze and evaluate the appropriate results