### The Graphic of Life (Applied Math to Life, Research Book 3)

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This approach could assist them in mathematical representation and belief and improve their problem-solving skills. Thus, this study investigates the difference in mathematical representation and belief and problem-solving skills of students who learned with RME and students who were engaged in conventional learning.

## Applied mathematics

This study also investigated the effect of mathematical representation as a mediator between mathematical belief and problem solving. Fig 1 shows that this study was performed to identify the effectiveness of the RME approach in mathematical belief and representation and problem solving.

In addition, this study identified the role of mathematical representation as a mediator between mathematical belief and problem solving. This study was conducted to answer the following research questions:. The study involved Form 1 secondary school students, who were divided into control and treatment groups. RME and traditional approaches were used by and students, respectively. The treatment group had 95 male and female students. Fifty-six students had low ability, 96 had average ability and 57 students had high ability. The control group had male and female students. Sixty of them had low ability, 96 had average ability and 61 students had high ability.

The mathematics ability of students was based on the results of their mathematics achievement in the past semester. The results were then categorised using Anates software into low, moderate and high [ 36 ]. The demographic profile is shown in Table 1. The study used the quasi-experimental design with non-equivalent pre- and post-test control groups.

The control group was created for comparison with the experimental group [ 37 , 38 ]. The quasi-experimental design refers to an experiment that consisted of units with treatment. This approach was utilised because the study used the existing class [ 39 ], which indicated that the research subjects were not selected randomly [ 40 ]. The quasi-experimental design was used to determine the effectiveness of the RME approach in improving problem solving skills, mathematical representation and belief of students.

The research design is shown in Table 2. Pre- and post-tests were conducted in both groups. The pre-test ensured similarity between groups and statistical control by comparing the mean of mathematical belief, representation, and problem solving with significant value of more than 0. The treatment group was given a task using the RME approach in teaching, whereas the traditional method was used as control group.

Students in both groups were taught during 10 two-hour sessions in their respective classrooms. The post-test was given to both groups after they were taught social arithmetic to determine the effectiveness of the RME approach. The test questions for pre- and post-tests were similar.

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The researcher observed each session for both groups throughout the discussion. Observations were conducted for 5 weeks in 10 sessions for both groups. A post-test was given to the two groups after social arithmetic and ratio were taught. Internal and external validities were determined with reference to Johnso and Christensen [ 40 ].

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Internal validity is a controlled variable set by the researcher that aims to identify the actual effect on the treatment variable. External validity sees how far the findings can be applied to individuals and settings other than the ones in the study. Issues, such as selection of research and lost subjects mortality , emotional maturity, intellectual and physical well-being, testing, research instrument and validity of research objects, can arise from the quasi-experimental design of pre- and post-tests.

These issues refer to factors related to the study and the attitude and emotion of students. The experimental group was taught using the RME approach. Teachers followed three main phases to teach this approach. In the first phase, teachers introduced realistic problems to students and helped them understand the problem setting. Teachers revised previous concepts and connected them with the experience of students. In the second phase, students worked in groups.

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Each student had a book that contained contextual questions and constructed situational problems, shared ideas, analysed patterns, made guesses and expanded problem-solving strategies based on knowledge or formal experience. The third phase of assessment showed the progress of students in problem solving. They discussed their problems and discovered useful strategies. Teachers guided and instructed students throughout the discussion on how to solve problems efficiently and effectively. Students in the control group were taught using a marker and whiteboard. They participated in the exercises given by the teachers.

The exercises are based on reference books provided by the school. Each school uses different reference books. Teachers narrated and jotted down information on the whiteboard. The enhanced educational curriculum unit requires every teaching method to be contextual. Thus, all teachings conducted in low secondary schools are traditionally contextual teaching. Six teachers were involved in the RME approach. They were selected based on the criteria of the RME approach training organised by the Ministry of Education in Indonesia.

The teachers underwent training for one month to ensure the success of the study and consistency with the design plan. The study objectives, RME and traditional approaches, planning and execution process and assessment methods were introduced to the teachers. The same teachers were assigned to treatment and control groups. The study was conducted after they understood the entire concept. The researcher observed throughout the study to determine whether the teachers were using the RME approach.

Observation began from the start until the end of class for every session. The teachers were given feedback about their teaching. The researcher observed the traditional class to ensure that the teachers were not using the RME approach or any other teaching method. A pilot study was conducted with students to determine the validity and reliability of the research instrument.

The validity of the research instrument was verified by four experts; two experts for content and two for language. According to the experts, the instrument language is suitable for measuring mathematical belief, representation and problem solving. The data from the pilot study were analysed using SPSS Findings showed that the reliability of the mathematical belief instrument, problem solving and mathematical representation are 0. The discriminant and difficulty index for the mathematical belief test and the mathematical problem solving test are at good and an average levels, respectively.

The discriminant index should be at good and very good levels. The pilot study results indicated that the developed items are solid and strong for the actual study. Sixty statements in the mathematical belief scale were used. The instruments for mathematical representation consisted of a written test set with four questions on the topic of arithmetic.

The instrument was constructed by the researcher to collect information about a representation problem solved by the students and their success in solving mathematical problems.

This instrument had four problem statements with an open-question format. These mathematical problems required students to apply comprehension, analysis and interpretation in the context of daily life. The full score for each item was 4 and 0 was the lowest score. The Mathematical Problem Solving Beliefs Instrument is used to collect information about the method and the success of how the students solve mathematical problems.

This instrument has five problem statements with an open-question format and requires students to comprehend, analyse and interpret these problems in the context of daily life. The full score for each item is 4 and 0 is the lowest score. The problem solving instrument is measured using marking schemes. The total score of the students is changed to a scale of 0 to The marking scheme for each item is shown in Table 3.

## Department of Mathematics - Textbook List

The marking scheme used for levels of mathematical representation and problem solving is the same as that used by [ 43 ], which was adapted to the arrangement outlined by the government. The analysis for the actual study was performed using SPSS Analysis of covariance ANCOVA was performed to identify the difference in mathematical belief, representation and problem solving between the treatment and the control groups where the pre-test is a covariate. This step was followed in the structural equation modelling SEM test to identify the role of mathematical representation as a significant mediator in the relationship between mathematical belief and problem solving.

These requirements include normality and homogeneity of variance between groups. The normality test showed the skewness and kurtosis values for the mathematical belief gain score for the treatment and the control groups are 0. This result shows that normality requirement was met and data were considered normal if the skewness and kurtosis value ranged from Therefore, one-way UNIANOVA can be performed to identify the differences in the mathematical belief gain score of the treatment and the control groups, as shown in Table 4.

This finding means that the RME approach has better effect on the increase in the mathematical belief of students than the use of the traditional method. Fig 2 shows the pre- and post-test means for a two-group design. The normality test showed the skewness and kurtosis values for mathematical representation pre-test for the treatment 0. These results indicated that the normality requirement was met. This result indicated that the RME approach and the traditional method had the same effect on the increase in the mathematical representation of students.

Fig 3 shows the pre- and post-test means for a two-group design. The normality test showed the skewness and kurtosis values of mathematical problem-solving gain scores for the treatment group Results showed that the normality requirement was met. These results prove that the RME approach was better than the traditional method at improving problem solving skills.

Fig 4 shows the pre- and post-test means for a two-group design.