**MODELING MATHEMATICS PERFORMANCE BETWEEN RURAL AND URBAN SCHOOL USING A FUZZY LOGIC APPROACH** \*\*Double blind review, please do not include authors information in this version \*\* Received Date: \*date Accepted Date: \*date Published Date: \*date **HIGHLIGHTS** - - - ABSTRACT *This study concerns competitiveness in Sijil Pelajaran Malaysia (SPM) performance between two different schools in Kedah, Malaysia, focusing on Mathematics scores. There are two different schools selected namely SMK Sungai Layar and SMK Bandar Sungai Petani. SMK Sungai Layar is a rural school while SMK Bandar Sungai Petani is an urban school. The objectives are to determine which schools between urban and rural schools perform better in mathematics subjects and classify students' performance on Mathematics subject using Fuzzy Logic. It is found that the performance of urban school was better than the rural school. As for rural school, the performance was moderate. The percentage of Mathematics value for SMK Bandar Sungai Petani is higher than SMK Sungai Layar. The number of students from an urban school who got a good score was double from the number of students from rural schools. The results show that the students from the urban school have excellent flexibility and reliability in Mathematics subject.* *Keywords: Sijil Pelajaran Malaysia, mathematics, urban school, rural school, fuzzy logic.* # INTRODUCTION Students in Malaysia begin their learning at the early age of four at kindergarten, even though it is unnecessary. The government offers free education of six years at the primary level and five years at the secondary level. Primary education starts when children reach the age of seven. Secondary Education in Malaysia is a continuation of primary education. The Sijil Pelajaran Malaysia (SPM), or the Malaysian Certificate of Education, is a national examination taken by all fifth-form secondary school students in Malaysia. SPM is the penultimate examination sat by secondary school students before entry into sixth form or technical education. Nowadays, the mathematics subject's concern has received greater attention due to its importance in our country. According to Tan Sri Muhyiddin Yassin (2014), SPM results showed improvement in all subjects except Mathematics. Mathematics is one of the compulsory subjects to enter universities. To improve Mathematics achievement among students, the Ministry of Education should take a course of action. One of them that should be concerned is about education delivery between rural schools and urban schools. Usually, students who live in urban areas mostly receive a better education than those who live in rural areas. Different regions of school provide various types of atmosphere to their students. An analysis is needed to determine competitiveness between rural schools and urban schools regarding Mathematics subject. This paper investigates which school performs better in Mathematics subject and classifies students' performance based on the mathematics subject using the fuzzy application. ## LITERATURE REVIEW ### **Fuzzy Set Theory** The fuzzy set concept has developed in various ways and many disciplines (Zimmermann, 2010). The set of the real numbers that characterize the class of objects is a fuzzy set. A fuzzy set has a graphical explanation that describes how the transition from one to another takes place. The graphic is called a membership function. A membership function is a curve that describes how every point in the input space is mapped to a membership value ranging between zero and one. In fuzzy sets, the set theoretic operations include intersection, union and complement, which correspond to the logical operators (AND, OR). In fuzzy set theory, the linguistic terms are used to elucidate an expert's knowledge and are also the most significant element in fuzzy (Nguyen, 2013). The advantage of using fuzzy set theory is it can be easily evaluated and dealt with uncertainty. The majority of problems in this real world are dealing with uncertainty. However, the uncertain knowledge can be classified into incomplete important knowledge that uses probability techniques and imprecise information. A fuzzy set theory can give verbal statements numerical clarity without losing their imprecise qualities. The linguistic term by fuzzy set theory is easier to measure the performance of the school. For example, the terms of very good, good, average, low and very low are easier to analyze rather than the overall score of the student's performance. It can be created using the linguistic rating to the school's performance related to the results that students achieved in their examinations. ### **Performance Evaluation using Fuzzy Logic** Lotfi Zadeh formalized the fuzzy logic tool at the University of California in 1965 which is a mathematical tool for dealing imprecision and information granularity (Sakthivel, 2013). What Lotfi Zadeh suggested is extremely a paradigm change that first gained acceptance in the Far East, and its successful method has guaranteed its implementation around the world. Ingoley et al. (2012) directed a study to investigate the significance and complexity of the examination questions. By using Fuzzy Logic, they had taken accuracy and time rate to get the complexity. Total time allocated by the students in universities or institution to complete answering the questions was fixed. How much time exactly needed by a student to answer the questions, was difficult to determine. Therefore they considered the importance and complexity of the question in order to carry the study. The difficulties of questions imply the capability of the students to give the correct answers. A fuzzy rule-based approach has been applied to model academic performance evaluation of the students. Yadav et al. (2013) studied about fuzzy to implement the use of membership values and clarify whether the decision of the result was helpful to be understood or not. Their objectives were to help the students in the educational processes and improve their academic performance in the future. The data had been classified into five possible outcomes that were Unsatisfactory, Satisfactory, Average, Good and Excellent. Data collected starts from semester 1, semester 2 and semester 3. They designed a fuzzy-rule based system by using the Fuzzy Logic ToolboxTM 2.2.7 by MathWorks. Kharola et al. (2015) used Mathlab-Simulink model using fuzzy logic toolbox to evaluate performance of the students. The outcomes proved that the advantage of suggested method rather than using traditional average method. They considered focusing more on students' performance instead of attributes that may affect student academic performance. Fuzzy model that had been presented by them closely imitate the performance of traditional average method. Next, further improvement and modification can be used by using fuzzy controllers. The model also can be used to evaluate teachers' performance, employees, faculties and others. They used Stage-wise fuzzy reasoning methodology to reduce issue regarding rule explosion. A fuzzy Expert System experimented with 20 students' marks that they obtained in semester1 and semester2 examinations. Both inputs had similar Triangular Membership Functions. They suggested using fuzzy logic method for students' academic performance evaluation that used a proper fuzzy inference mechanism and associated rule (Yadav, 2011). Some researchers stated that some students frequently got low grades in their first examination results. Even worst, few of them were dismissed or expelled due to low grades. Ajiboye et al. (2013) stated that they used fuzzy logic to prevent the students from being expelled by forecasting the students' risk status based on certain predictive reasons. Their research had allowed the teachers to focus more on students' weaknesses and helped the management in decision-making by offering scholarships to brilliant students to minimize the risk of failures. While those who obtained low grades, should be motivated and enhanced their studies. # METHODOLOGY The data collected are from two schools located in Kedah were SMK Bandar Sungai Petani (urban school) and SMK Sungai Layar (rural school). The data consists of the midterm and trial examination Mathematics results of Form 5 from the science and civil engineering classes. This data used to find out and determine which schools performed better based on Mathematics subjects. The Fuzzy Logic ToolboxTM by MathLab was used to analyze the results. **Fuzzy Logic System** The actual mapping process from a given set of input variables to output depends on a set of fuzzy rules. The general fuzzy inference system's main process includes fuzzification, fuzzy rule-based, fuzzy inference engine, and defuzzification. Fuzzification is the first step in the fuzzy inferencing process. Fuzzification is the process of converting a crisp quantity into fuzzy quantity. Once the input, output variables and membership functions are identified, the next stage is to design the rule-based decision matrix. Fuzzy rule-based keeps the input variables and rules before getting the outputs. These rules are expressed in the If-Then form. The fuzzy inference engine is responsible for taking the human feelings, thought, and logical inference to obtain a rational and reasonable result from fuzzy rule-based. The last stage in fuzzy inference system is defuzzification. Defuzzification is the process of converting or transforming the fuzzy output into a crisp output. Figure 1 shows the process of the fuzzy system. ![](176-1-514-1-2-20210115_media/media/image1.emf) Figure 1: Basic structure of a fuzzy system (Yadav and Singh, 2012) ### **Fuzzy Logic for Academic Performance Evaluation** In this paper, Fuzzy Logic has been proposed for student academic performance evaluation shown in Figure 2. ![](176-1-514-1-2-20210115_media/media/image2.png) Figure 2: Fuzzy logic for academic performance evaluation The two inputs included consist of a midterm examination and trial examination. The output variable is called the performance of the school, which is determined by fuzzy logic. ### **Evaluation of Academic Performance with Fuzzy Logic** Academic performance evaluation with fuzzy logic consists of three steps: 1. Fuzzification of inputs midterm examination, trial examination and output of performance value. 2. Determine input and output membership functions in rules and inference process. 3. Defuzzification of performance value. ### **Fuzzification of Midterm and Trial Examination and Performance Value** Using input variables and their membership functions of fuzzy sets, fuzzification of examination results was carried out. Two examination results were collected from each school participating in this project and both from input variables of the fuzzy logic. Each input variable has five membership functions that are triangular. The fuzzy sets of input variables are given in Table 1, and Figure 3 shows different types of membership functions for inputs. Table 1 : Fuzzy set of input variable | | | | ----------------------- | --------------- | | **Lingusitic Variable** | **Interval** | | Very Low (VL) | ( 0, 0, 25 ) | | Low (L) | ( 0, 25, 50 ) | | Average (A) | (25, 50, 75 ) | | High (H) | (50, 75, 100) | | Very High (VH) | (75, 100, 100 ) | Figure 3: Membership function of midterm and trial examination Figure 4 : Membership function of performance of the schools There are five membership functions for the output attribute, which is the success value. Table 2 shows the fuzzy set of output variables, and Figure 4 shows the various forms of output membership functions. Table 2 : Fuzzy set of output variable | | | | ----------------------- | ------------------- | | **Lingusitic Variable** | **Interval** | | Very Poor (VP) | ( 0, 0, 0.25 ) | | Poor (P) | ( 0, 0.25, 0.50 ) | | Average (A) | (0.25, 0.50, 0.75 ) | | Good (G) | (0.50, 0.75, 1) | | Very Good (VG) | (0.75, 1, 1 ) | Insert the range for the linguistic variables for both input and output variables into Mathlab-Simulink to determine their membership function. Figure 5 shows the result examination based on five linguistic variables which are very low, low, average, high and very high. Figure 6 shows the school's performance based on five linguistic variables that are very poor, poor, average, good and very good. ![](176-1-514-1-2-20210115_media/media/image3.png) Figure 5: Membership function for result examination ![](176-1-514-1-2-20210115_media/media/image4.png) Figure 6: Membership function for performance value ### **Rules and Inference** The rules determine input and output membership functions that will be used in the inference process. ![](176-1-514-1-2-20210115_media/media/image5.png) Figure 7 : Rule Editor If some rules are active for the same output membership function, only one membership value must be chosen. This method is called "fuzzy decision" or "fuzzy inference". In this paper, the method proposed by Mamdami, is given below: \[\mu_{c}\left( y \right) = \max_{k}{\lbrack min\lbrack\mu_{A}(input\left( i \right),\mu_{B}\left( \text{input}\left( j \right) \right)})\rbrack\rbrack,\ \ \ \ \ \ \ \ \ \ \ k = 1,2,3,4\ldots r\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)\] This expression decides an output membership function value for every active rule. While one rule is active, an AND operation is used connecting inputs. The smaller input value is selected and its membership value is decided as membership value of the output for that rule. This procedure is continued, so that output membership functions are verified for every rule. To conclude, AND (min) operations are used between inputs and OR (max) operations are used between outputs. ### **Academic Performance Evaluation with Fuzzy Logic** Once completing the fuzzy decision process, the fuzzy number acquired have to convert to a crisp value. This process is known as Defuzzification. Many techniques have been developed for Defuzzification. In this paper, a centre of area called Centroid method was used.The crisp value is computed using the formula given: \(z = \frac{\int_{}^{}{u_{c}\left( z \right) \times x \times dz}}{\int_{}^{}{u_{c}(z) \times dz}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\) (2) ## RESULT In this paper, the application of Fuzzy Logic for students' Mathematics performance evaluation had been implemented in MATLAB (version 7.6). Fuzzy Logic ToolBox has been used for this research work. The proposed Fuzzy Logic was analyzed based on 60 students' marks from both schools obtained from their midterm and trial SPM examinations. Based on the data, the student's performance value and performance according to their school are calculated, then clusters them into their classes based on random data. 120 students were clustered into five classes which are Very Good, Good, Average, Poor and Very Poor. Both examination scores were fuzzified by means of the triangular membership functions for each student. Active membership functions, using the Mamdami Fuzzy Decision Techniques, were determined according to the rule table. By measuring the centre (centriod) of the resulting geometric shape, the output (performance value) was measured and then defuzzified. This series was replicated for each student using the Midterm and Trial Mathematics scores. Both inputs had same Triangular Membership Functions. For example, let's take the calculated midterm and trial performance 90 and 79. The result shows performance value is 0.797, close to 1. It shows that the more the performance value is close to 1, the better is the result. Figure 8 shows the membership functions for the example given, while Figure 9 shows the surface viewer for students' Mathematic performance evaluation. ![](176-1-514-1-2-20210115_media/media/image6.png) Figure 8: Performance value for Mathematic scores of 90 and 79 ![](176-1-514-1-2-20210115_media/media/image7.png) Figure 9: Surface viewer of Mathematics performance evaluation Table 3 and Table 4 show the classification of student's mathematics performance for both schools. Table 3: Result overall for SMK Bandar Sungai Petani

Average Percentage Mathematics Performance Value

71.32 %

Very Good

0 student

Good

38 student

Average

21 student

Poor

1 student

Very Poor

0 student
Table 4: Result overall for SMK Sungai Layar

Average Percentage Mathematics Performance Value

62.04 %

Very Good

1 student

Good

19 student

Average

30 student

Poor

10 student

Very Poor

0 student
Fuzzified was applied to generate the output. The average percentage for the performance value was used to compare the result. The average used because the method is easy to understand for plotting and comparing data \[11\]. Table 3 and 4 show that students from urban school performed better than students from rural school. The percentage of mathematics value for SMK Bandar Sungai Petani is higher than that of SMK Sungai Layar by 9.28%. The number of students from urban school that got a Good score is 38, which was double the number of students from rural schools that only had 19 students getting similar scores. Information from Table 3 also shows that 21 of the students from urban school achieved an average score while one of them achieved a low score. However, from Table 4, 30 students achieved an average score while ten students scored low results. This category of students deserves special attention so that they can cope well with their studies. Generally, cases with values above 0.5 defuzzied value have satisfactory academic achievement, while cases with values less than 0.5 need to do extra work and efforts to meet the challenges ahead. ## CONCLUSION This study aims to determine which schools between rural and urban schools perform better in Mathematics subject and classify students' performance based on Mathematics subject by using fuzzy application. During the evaluation of students', the result's from urban school and rural school, the urban school were discovered to have great flexibility and reliability in Mathematics subject. The defuzzied or the performance value obtained from the midterm and trial examination result from each student shows the students' performance according to Mathematics subject. It means that by using fuzzy application, the students can be classified according to their performance. Using these results, students with low achievement should get more attention and extra practices and work. Gaps or variations in outcomes can be seen between urban and rural schools during the fuzzy logic process. Both schools used the same membership features in this article, although their mid-term and trial examination as an input variable and performance value as output variables were used. In conclusion, the performance assessment using fuzzy logic is not only relevant for the subject of mathematics, but can also be extended to the performance assessment of theoretical lessons by students such as history and geography subjects. ## RECOMMENDATION For a recommendation to future researchers, they can combine the methods between fuzzy logic and artificial neural networks, named Neuro-Fuzzy Systems to get better results and evaluate students' performance. These techniques can be applied to other subjects too. **REFERENCES** Ajiboye**,** A. R., Arshah, R. A., & Qin, H. (2013[). Risk status prediction and](http://www.unilorin.edu.ng/publications/ajiboyear/link%20to%204.pdf) [modelling of students' academic achievement: A fuzzy logic approach.](http://www.unilorin.edu.ng/publications/ajiboyear/link%20to%204.pdf) *International Journal of Engineering and Science,3*(11), 7-14. Ingoley, S. N., & Bakal, J. W. (2012). Evaluating students performance using fuzzy logic. In *International Conference, IJCA Proceedings on International Conference on Recent Trends in Information Technology and Computer Science, ICRTITCS (9)* (pp. 15-20). Jawahar, J., & Seema, S. (2009). *Cost accounting*. Retrieved from [A147\&dq=average+is+simple+method+to+compare\&source=bl\&ots=Cd4G](https://books.google.com.my/books?id=1KklpFKeT6EC&pg=PA147&lpg=PA147&dq=average%2Bis%2Bsimple%2Bmethod%2Bto%2Bcompare&source=bl&ots=Cd4GNZ_rHA&sig=7VRot0-A0YqTdYxsVL9zJQmBCd4&hl=en&sa=X&redir_esc=y%23v%3Donepage&q=average%20is%20simple%20method%20to%20compare&f=false) [NZ\_rHA\&sig=7VRot0-](https://books.google.com.my/books?id=1KklpFKeT6EC&pg=PA147&lpg=PA147&dq=average%2Bis%2Bsimple%2Bmethod%2Bto%2Bcompare&source=bl&ots=Cd4GNZ_rHA&sig=7VRot0-A0YqTdYxsVL9zJQmBCd4&hl=en&sa=X&redir_esc=y%23v%3Donepage&q=average%20is%20simple%20method%20to%20compare&f=false) Kharola, A., Kunwar, S., & Choudhury, G. B. (2015). Students performance evaluation: A fuzzy logic reasoning approach. *PM World Journal*, *4*(9), 1-11. Nguyen, C. H., Pedrycz, W., Duong, T. L., & Tran, T. S. (2013). A genetic design of linguistic terms for fuzzy rule based classifiers. *International Journal of Approximate Reasoning, 54*(1), 1-21. Sakthivel, E., Kannan, K. S., & Arumugam, S. (2013). Optimized evaluation of students performance using fuzzy logic. *International Journal of Scientific & Engineering Research, 4*(9), 1128-1133. SPM results best in five years. (2014, Mac). Retrieved December 1, 2016, from Yadav, R. S. & Vijendra, P. S.(2011). Modeling academic performance evaluation using soft computing techniques: A fuzzy logic approach. *International Journal on Computer Science and Engineering (IJCSE),3*(2), 676-686. Yadav, R. S., & Ahmed, P. (2013). Modeling academic performance evaluation using subtractive clustering approach. *International Journal of Computer Science and Technology, 4*, 73-80. Yadav, R. S., & Singh, V. P. (2012). Modeling academic performance evaluation using fuzzy c-means clustering techniques. International Journal of Computer Application, 60(8), 15-23. Zimmermann, H. J. (2010). Fuzzy set theory. Wiley Interdisciplinary Review: Computational Statistics, 2(3), 317-332.