The Analysis of the Improvement of Student's Mathematical Communication Skills in Solving Problems of Circumference and Area of Circles under the Implementation of Inquiry-Based Learning

: Mathematical communication skills convey and interpret mathematical ideas through drawings, tables, diagrams, formulas, or demonstrations. The inquiry-based learning model allows students to discover data, facts, and information from various sources to provide experiences. This study aims to analyze students' mathematical communication skills, improve them using the inquiry-based learning model, and then deepen them through a description. This study uses an explanatory sequential method combining quantitative and qualitative research. The research subjects were 23 VIB students in the control class and 23 VIA students in the experimental class. The study results show that students' mathematical communication skills are divided into weak, moderate, and strong skills. The independent sample t-test results showed a significant influence of applying the inquiry-based learning model with a sig. 0.003 (p <0.05). Strong and moderate mathematical communication skills showed an increase of 47.83% and 13.04%, while weak skill levels showed a decrease of 60.87%. The improvement of mathematical communication skills is presented as a description of observation and interview results. The research conclusion reveals that applying the inquiry-based learning model can improve students' mathematical communication skills in solving problems of circumference and area of circles in elementary school students.


INTRODUCTION
At an early education level in the Indonesian education system, elementary school is one of the places that provide essential knowledge and skills.The elementary school level is the beginning for someone to learn everything.The stronger the foundation of critical knowledge and skills obtained at this level, the easier it will be to develop them in the subsequent levels.
Mathematics is one of the mandatory subjects taught in elementary school.The scope of mathematics is not only related to calculations, rules, and proofs.More than that, mathematics is a requirement for creating order in daily activities, developing science and technology, and also serving as a pinnacle of knowledge and a servant for other sciences [ (Manalu & Zanthy: 2020); (Asih, Sunardi & Kurniati: 2015); and (Sunardi: 2016)].The benefits of mathematics in daily life lead to essential skills, one of which is mathematical communication skills.
Mathematical communication skills refer to the ability to convey mathematical ideas by expressing and interpreting those ideas in the form of pictures, tables, formulas, diagrams, or demonstrations (Melati, Sunardi & Trapsilasiwi: 2017).This skill is related to students' mathematics learning outcomes.Good mathematical communication skills are one of the guarantees of achieving students' learning outcomes.Conversely, students' mathematics learning outcomes also reflect their level of mathematical communication skills.Both are essential parts of the literacy and numeracy process.
The mathematical communication indicators in this study adopt the written mathematical communication indicators used by Melati, Sunardi & Trapsilasiwi (2017).These indicators include expressing and writing problem-solving processes, converting problems into mathematical sentences, writing mathematical calculations, and using mathematical symbols.These indicators are suitable for elementary school students as they can measure students' mathematical communication skills in a simple way that is appropriate to their developmental characteristics.Many factors, including the implementation of appropriate learning models, influence the improvement of students' mathematical communication skills.The keyword "interaction" is essential to improving mathematical communication skills.Gulo (in Sanjani: 2019) stated that interaction is one way to stimulate and activate critical thinking skills and solve problems.Interaction during the learning process is expected between students and teachers or among students themselves and with other learning sources.
Samsidar, Coesamin & Bharata: 2019, stated that the learning model that prioritizes communication and student interaction as the main subject is the inquiry-based learning model.[ (Hapsari & Kristin: 2021); and (Rahmawati: 2022)] agree that the inquirybased learning model can improve students' motivation and creativity and encourage the achievement of numeracy literacy.Improving mathematical communication skills as one of the indicators of numeracy literacy can be done by presenting relevant material and dividing it into sub-materials (Ardiyani, Saputro & Subekti: 2021).
As a comparison, the expository learning model emphasizes the mastery of learning material.The material is packaged neatly and systematically so students can digest it well (Hidayati, Fahmi & Farida: 2019).Student achievement is expressed by asking students to convey their understanding according to their characteristics.Roy Killen (in Safriadi: 2017) described four principles of the expository learning model, including orientation to objectives, prioritizing communication, student readiness to learn, and the continuity of the learning process.
Delving deeper into the inquiry-based learning model, [ (Sutarti & Wibawa: 2018); (Mbari, Yufrinalis & Nona: 2018); and (Rohayani: 2018)] agree that in addition to emphasizing the process of searching and discovering, the inquiry-based learning model also prioritizes systematic, critical, and logical investigation activities.Reinforcing this, Chususiyah, Dafik & Prastiti (2020) assert that the inquiry-based learning model can develop and activate student thinking abilities if implemented with good planning at each stage.Directing the learning process using actual practices and discussions as learning activities is essential to learning planning (Priansa, 2017;Fitriyah, Suratno, Prastiti, Dafik & Hobri, 2020).
Abidin (2020) advises achieving maximum results by applying the inquiry-based learning model.The inquiry-based learning model will provide top results, especially regarding learning achievement, by connecting the material to daily life contexts.In response to this advice, researchers maximize it by connecting the material and utilizing supporting teaching aids on familiar and known things in students' lives.During the inquiry-based learning model implementation phase, Banchi and Bell (Jazimah, 2020) stated that there are four levels of inquiry-based learning model implementation.These four levels of implementation are confirmed inquiry, structured inquiry, guided inquiry, and open inquiry or free inquiry.These four levels of implementation are related to the consideration of problems raised, as well as the procedures and solution aids provided in their implementation.
In implementing the inquiry-based learning model in primary schools, using the available or accessible inquiry level is unnecessary.In this study, a modified level of structured inquiry implementation was used.The emergence of problems and procedures has been conditioned, while the solution takes place naturally.However, when there are obstacles in the solution process, the teacher plays a role in triggering the emergence of solution ideas from students to ensure that the process runs smoothly.
The syntax of the inquiry-based learning model in this study adopted the syntax presented by Sanjaya (Sinesis, 2016).This syntax includes six stages: problem orientation, formulating problems, formulating hypotheses, collecting data, testing hypotheses, and formulating conclusions.The following table explains the syntax of the inquiry-based learning model.

METHODS
This research design uses a mixed-methods approach of sequential explanatory type.The sequential explanatory design combines quantitative research followed by qualitative research.According to Creswell (2016:299), "The sequential explanatory mixedmethods approach is a design in mixed-methods that is attractive to individuals with a strong quantitative background or from fields that are relatively new to qualitative approaches."Two phases of research are used to classify the potential in the quantitative phase to be further explored in the qualitative phase.The quantitative method conducted first in the sequential explanatory research serves to determine the problem/potential and prove the improvement of the problem/potential.The problem/potential is deepened and expanded using qualitative methods.Sugiyono (2016) explained that the combination method with a sequential explanatory design is carried out to answer the research problem formulation of quantitative and qualitative research that complement each other.The modeling of this research method can be better understood through the following flowchart diagram.
The initial quantitative research phase uses the pretest-posttest control group design experiment method.It begins with administering a pretest, followed by the learning process.The inquiry-based learning model is used in the experimental class, while the expository learning model is applied in the control class.The final session of the quantitative research phase involves conducting a post-test.The analysis of the pretest, learning process, and post-test serves as the basis for determining the influence of the inquiry-based learning model on mathematical communication skills.This also forms the basis for sampling in the qualitative phase.In addition to observation, guided interviews were conducted with representatives from each skill level.
The research population consists of all sixth-grade students in elementary schools in Banyuwangi Regency.The sample consisted of 46 students, divided into 23 from class 6A and 23 from class 6B at State Elementary School 1 Panderejo.The sampling technique used was simple random sampling.Meanwhile, for the learning process sessions, the sampling was done using the purposive sampling technique by considering the preliminary activities related to a mathematics learning process that had taken place in both classes.In addition, daily assessments and mid-semester math report grades were also analyzed.
It was determining and selecting data sources in the qualitative phase using a disproportionate stratified random sampling technique on the category of mathematical communication skill levels in the experimental class.The research data sources for guided interviews were representatives from the experimental class.Six students were selected, with two students representing each skill level.
The instruments used in the quantitative phase are tests (pretest and post-test) and non-test instruments (observation sheets and supporting teaching materials).The pretest and post-test instruments are designed based on the blueprint and are tested for validity and reliability using Pearson correlation and Cronbach's alpha.The non-test instruments are communicated and validated by experts.
The instruments used in the qualitative phase are non-test instruments in the form of guided interviews.Guided interviews are conducted to obtain in-depth information about the student's learning process and mathematical communication skills.In guided interviews, the data sources are also given confirmation post-test questions.
Before analyzing the data, the researcher checks the validity of the data using normality and homogeneity tests.Then, an independent sample t-test is performed to describe the effect of applying the inquiry-based learning model on students' mathematical communication skills and learning outcomes.The observation results of the learning process are presented in a descriptive analysis.
In the qualitative phase, the study results describe the inquiry-based learning model applied in the experimental class.The results are obtained from observing the learning process in the experimental class.The observation results are confirmed in the guided interview session with the data sources representing the three levels of mathematical communication skills.

RESULTS
Based on the conducted research, the results are divided into two parts.The first part is the effect of the inquiry-based learning model on students' mathematical communication skills.The second research result describes the implementation of mathematics learning using the inquiry-based learning model on the topic of the circumference and area of a circle.
The research results showing the influence of the application of the inquiry-based learning model on mathematical communication skills were obtained from the independent sample t-test on the post-test and pretest scores of both classes.Meanwhile, the description of the learning process using the inquiry-based learning model in the experimental class is shown by the analysis of the learning process.Moreover, it is also supported by the confirmation of guided interviews with data sources representing each level of skill.

A. The Effect of Inquiry-based learning Model on Mathematical Communication Skills and Student Learning Outcomes
The Proof of the Effect of the Inquiry-based learning Model on Mathematical Communication Skills is Presented Through Analysis of Pretest and Post-test Scores.The presentation of the pretest and post-test analysis results describes the changes in mathematical communication skills.Moreover, the pretest and post-test analysis results also serve as the basis for conducting an independent sample t-test, which indicates whether or not there is an influence of the inquiry-based learning model on students' mathematical communication skills.Based on the results of the pretest of the control class shown in Fig. 2, mathematical communication skills are divided into three levels.These three levels are strong, moderate, and weak.Strong mathematical communication skills account for 13.04%.The moderate and weak categories are at 17.39% and 69.57%, respectively.

Fig. 3. Distribution of Mathematical Communication Skills in the Experimental Class Pretest Session.
In more detail, Fig. 2 shows that the indicator of expressing problem-solving processes in order of 6, 4, and 13 students is in the strong, moderate, and weak categories, respectively.For the indicator of converting problems into mathematical sentences, 8 and 5 students are in the strong and moderate categories, while 10 students are still in the weak category.In the indicator of writing mathematical calculations, weak mathematical communication skills still dominate with 15 students, while moderate and strong show 5 and 3, respectively.The indicator of using mathematical symbols shows 3, 6, and 14 for the strong, moderate, and weak categories, respectively.Furthermore, Fig. 3  Fig. 3 shows that for the indicator of expressing problem-solving processes, 3, 6, and 14 students are categorized as strong, moderate, and weak, respectively.Meanwhile, for the indicator of translating problems into mathematical sentences, 5 students are categorized as strong, 5 as moderate, and 13 as weak.As for the indicator of writing mathematical calculations, 19 students have weak mathematical communication skills, while only 2 students each have moderate and strong skills.The indicator of using mathematical symbols shows 2, 3, and 13 for strong, moderate, and weak categories, respectively.The results of the pretest scores of both classes in Table 4 above show a relatively balanced condition.The control class pretest shows an average of 57.13 with a standard deviation of 26.68 and a standard error of 5.56.Meanwhile, the experimental class shows an average of 51.91, a standard deviation of 25.6797, and a standard error of 5.3546.
Based on Table 4, the difference in the mean scores between the control and experimental classes is insignificant.The control class has a slightly higher average score of 57.13, slightly higher than the experimental class score of 51.91.However, the slight difference in mean scores must still be tested with inferential statistical tests.
The inferential statistical test is carried out by conducting an independent sample t-test.The independent sample t-test above is performed after the prerequisite test (normality) has been completed.Meanwhile, the homogeneity prerequisite test is conducted simultaneously with the independent sample t-test.
As shown in Table 5 below, in the homogeneity test section, the results of the independent sample t-test show a Sig.Value of 0.682>0.05(not significant).This indicates and proves that the two data groups are identical/homogeneous.Thus, the two prerequisite tests were fulfilled before conducting the independent sample t-test.Next, an independent sample t-test is conducted on the pretest scores of the 46 students in the control and experimental classes.The results of the independent sample t-test in Table 5 below show a Sig.(2-tailed) value of 0.503>0.05(not significant).Therefore, the hypothesis decides to reject HA or, in other words, accept H0.There is no difference in the average pretest scores between the control and experimental groups.This is the beginning of the improvement of students' mathematical communication skills.Özsoy and Ataman (2009) stated that a non-significant result allows different treatments to be given to groups because they are equal.
After implementing each learning model, further testing was conducted to prove the results.The post-test average scores in both classes were relatively higher than the pretest.The following diagram shows students' mathematical communication skills distribution in the post-test session.

Fig. 4. The distribution of mathematical communication skills in the post-test session control class
The expression indicators of the problem-solving process showed that 8, 6, and 9 students were in the strong, moderate, and weak categories, respectively.For the indicator of converting problems into mathematical sentences, 10 and 6 students were in the strong and moderate categories, while seven were still in the weak category.Regarding the indicator of writing mathematical calculations, many students still showed weak mathematical communication skills, 12 students, while 7 and 4 students showed moderate and strong skills, respectively.The last indicator, using mathematical symbols, showed 6, 5, and 12 students for the strong, moderate, and weak skills categories, respectively.
Next, figure 5 below shows the results of the post-test scores in the experimental class with the implementation of the inquirybased learning model.The strong mathematical communication skills category showed a percentage of 60.87%.The moderate category of mathematical communication skills was in the range of 30.42%.Meanwhile, the weak category of mathematical communication skills still accounted for 8.70%.

Fig. 5. The distribution of mathematical communication skills in the post-test session experimental class
The distribution of mathematical communication skills of the experimental class in the post-test session shown in Figure 5 indicates that for expressing problem-solving processes, 14, 8, and 1 students are categorized as strong, moderate, and weak, respectively.For the indicator of translating problems into mathematical sentences, 15 and 7 students are categorized as strong and moderate, respectively, while one student remains in the weak category.For the indicator of writing mathematical calculations, four students are categorized as weak, while 5 and 14 students are categorized as moderate and strong, respectively.The last indicator, using mathematical symbols, has 13, 8, and 2 students in the strong, moderate, and weak categories, respectively.
The descriptive statistical analysis of the post-test scores of the control and experiment classes is presented in Table 6 below.The results show a noticeable difference between the two classes.mean score of the control class is only 69.84, with a standard deviation of 19.84 and a standard error of 4.14.In contrast, the mean score of the experiment class is 86.09, with a standard deviation of 15.66 and a standard error of 3.27.This suggests that the experimental intervention, which aimed to improve mathematical communication skills, has positively impacted the students' learning outcomes.However, further analysis may be necessary to draw more conclusive results.Based on table 6 above, there is a significant difference in the average values obtained by the control and experimental groups.The experimental group's average score is higher than the control group's.However, from a generalization perspective, this can only be confirmed after conducting a test for the difference in means.Prerequisites for normality and homogeneity tests must be carried out before conducting an independent sample t-test.
Similar to what was done for the pretest scores, a normality test has also been conducted for the post-test scores.The normality test was carried out before conducting an independent sample t-test, and the data was found to be normally distributed.Meanwhile, the homogeneity test was conducted simultaneously with the independent sample t-test.The independent t-test was conducted to test whether there was a significant difference in the mean between the two data groups.The null hypothesis (H0) was that there was no significant difference in the mean between the post-test scores of the control and experimental groups.The alternative hypothesis (HA) was that there was a significant difference in the mean between the post-test scores of the control and experimental groups.
The results of the independent t-test conducted using SPSS v.25 showed a Sig.(2-tailed) [t(46)=0.003<0.05](significant at the 5% level of significance).This result indicates that the null hypothesis was rejected, and the alternative hypothesis was accepted.Thus, there was a significant effect of the inquiry-based learning model applied in the experimental group on the mathematical communication skills of sixth-grade students on the circumference and area of a circle.Several observations can be conveyed based on the results of observations during the experiment class using the inquiry-based learning model.Which stage of the inquiry-based learning model has an influence will be discussed.The following are the results of observations that will be presented in a general description in paragraph-by-paragraph order.
In the problem orientation stage, the influence is mainly seen in the first indicator of mathematical communication skills.This stage has a positive impact in that it can provide a good impact on how students express the problem-solving process.The first indicator of mathematical communication skills, the expression of the problem-solving process, emerges and is honed well during group discussions and individual problem-solving.
The second stage of the inquiry-based learning model is slightly different in its influence.A specific finding emerged at this stage in the form of differences in the influence on students with low and moderate abilities who struggle to translate problems into mathematical sentences.This is because, in addition to using logic, students must use their previous knowledge.However, this stage presents no significant obstacles for students with strong abilities.Instead, this stage becomes an excellent opportunity for students with strong skills to hone their skills.
Formulating hypotheses as the third phase influences improving skills, especially in the indicators of expressing and writing problem-solving processes.Students with strong and moderate skills can combine mathematical symbols, such as the approximation of pi (π) and the relationship between diameter and radius, to formulate hypotheses for the formula for the circumference and area of a circle.Meanwhile, students with weak skills still require adaptation with the help of teachers to combine and connect symbols to form a hypothesis for the formula.
The fourth stage, the data collection stage, is the most famous stage among all students, regardless of their skill level.In this stage, all learning resources can be used to support the hypothesis.The teacher also acts as a learning resource by guiding students to obtain accurate data that supports their hypothesis.This stage mainly influences the indicators of writing mathematical calculations, using/recognizing mathematical symbols, and confirming the proposed hypothesis.In general, the data collection stage can confirm all indicators of students' mathematical communication skills while also answering the hypothesis.
Furthermore, the fifth stage, which is the hypothesis testing stage, has an impact on many indicators of mathematical communication skills.This stage is the right time to see students' understanding, especially their mathematical communication skills.At least four indicators improve in this stage: expressing the problem-solving process, transforming problems into mathematical sentences, writing mathematical calculations, and using mathematical symbols in problem-solving.The discussion section will discuss a more precise explanation of the impact of the hypothesis testing stage.
Finally, the application stage of the inquiry-based learning model is the stage that can depict students' achievements related to all indicators of mathematical communication skills.The learning process that provides guided freedom for students to think actively, confirm from various sources, and give opinions will be seen when students present their conclusions.In this stage, the conclusions are presented verbally and demonstrated through students' problem-solving performance.
After the six stages and their overall impact according to the researcher's observations have been presented, the qualitative research phase will be conducted.Guided interviews will be conducted with selected samples.This will provide an overview of the level of students' mathematical communication skills after going through the learning process by applying the inquiry-based learning model.The following are sample questions to be answered during the guided interview to confirm the students' understanding and absorption during the learning process with the inquiry-based learning model.Fig. 6.Guided interview assistant questions 1.A bicycle wheel has a diameter of 75 cm.The wheel rotates 14 times, what is the distance traveled by the wheel in meters? 2. A swimming pool has a circular shape with a diameter of 40 meters.The pool is surrounded by a path that is 2 meters wide.If the path will be paved and the installation cost is Rp22,000 per meter, how much does it cost to install the paving?
the given problem Determining what is known in the problem Determining what is asked in the difficulty Performing the problem-solving process 2 Transforming the problem into a mathematical sentence Understanding the situation of the problem Understanding the appropriate symbols to use Understanding the order of the problem-solving process Writing the mathematical sentence 3 Writing mathematical calculations Understanding the types of mathematical operations Understanding the meaning of mathematical sentences Performing calculations according to the order of operations Checking the calculation results 4 Using mathematical symbols Understanding the meaning of mathematical symbols Understanding the writing of mathematical symbols Selecting and using mathematical symbols Source: Adaptation from Melati, Sunardi & Trapsilasiwi (2017)

Fig. 2 .
Fig. 2. Distribution of mathematical communication skills in the control class pretest session.
of the Learning Process by Applying the Inquiry-based learning Model The description of the learning process is based on the six stages contained There are six stages or syntaxes in implementing the inquiry-based learning model.These six syntax stages include the problem orientation stage, the problem formulation stage, the hypothesis formulation stage, the data collection stage, the hypothesis testing stage, and the conclusion formulation stage.These six stages influence students' mathematical communication skills.

Table 2 .
The syntax of implementing an inquiry-based learning model

Table 3 .
Pretest-posttest control group design O2 : Pretest result of the control group X1 : Treatment for the experimental group -inquiry-based learning model X2 : Treatment for the control group -expository learning model O3 : Post-test result of the experimental group O4 : Post-test result of the control group

Table 4 .
Results of Pretest in Control and Experimental Classes.

Table 5 .
The results of the different tests on the average pretest value used an independent t-test sample

Table 6 .
Post-test results of control and experimental classes

Table 7 .
The results of the post-test mean difference test using an independent t-test sample