## Najihan Awang @ Ali1\*, Muhammad Aslam Mohd Safari 2, Syadatul Syaeda Mat Saleh3, Nurul Husna Jamian4
> *1,3,4,5 Faculty of Computer and Mathematical Sciences (FSKM), Universiti Teknologi MARA, Perak Branch, Tapah Campus, 35400 Tapah Road, Perak, Malaysia*
>
> *2Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 53600 Bangi, Selangor*
>
> *Corresponding author: [\*najihanawang@gmail.com](mailto:najihanawang@gmail.com)*
>
> Received Date: \*date Accepted Date: \*date
## ABSTRACT
> *Numerous studies reported, the poor performance of students’ in Mathematics affected their major courses. In the same situation, UiTM resulted the higher failure rate nearly every semester in Mathematics courses especially in Pre-Calculus. A research conducted on 108 full-time students Part 1 Diploma of Science who are taking Pre-Calculus course. This research aims to identify the performance of students in each questions in final examination of Pre-Calculus. From the findings, students facing problems in answering Question 2. This revealed, most students facing difficulties in handling sketching in graph of parabola and circles. Further, to investigate between students’ performance in relation to gender using Principal Component Analysis, PCA and t-test analysis. It was discovered female students outperformed male students in this course.*
>
> ***Keywords:** Pre-Calculus, performance, final examination*
# INTRODUCTION
> Learning Mathematics had been documented as an essential variable in the study of university student’s’ achievement (Naccache and Hleiss, 2016; Gynnild et al, 2005; Li, 2004). At the international level, most programs offered in the university required Mathematics as a compulsory course in order to engage with their major degree programme such as engineering where the students need to use practical applications of mathematical equations and in-depth understanding real-world problem solving (Naccache et al., 2016)
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> Past studies discovered that difficulties faced by the students surround poor understanding of basic concepts, weak computational competence, inability to effectively organize known facts and problem in mastering the mathematical language and symbols (Yusoff and Rahman, 2001).
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> Preliminary course like Pre-Calculus can be adventurous, depending on the level of mastery and competence of the students to transfer their prior mathematical knowledge in secondary school. According to Tang et.al (2009) students who entered the university with poor grade in Mathematics struggled in their Mathematics courses initiated with Pre-Calculus course.
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> The Pre-Calculus course covers the additional algebraic and trigonometric skills needed by the students before they venture into the world of calculus. It consists of four major parts: coordinates, graphs and lines;
>
> functions; systems of equations and inequalities; and trigonometry. Besides, the learner will able to solve exponential and logarithmic equations, linear and non-linear equations and trigonometric equations.
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> Numerous studies revealed the student achievement in Mathematics course has affected their main courses. As Naccache et al. (2016) stated for many years, the poor results recorded in Mathematics achievement indirectly affects the students’ overall academic performance in engineering programmes. In addition, (Borba, 2005; Gynnild et al, 2005; Li, 2004) have also reported some underachievement in Mathematic courses.
>
> Inspired by this condition, this study is embarked to identify the achievement of students in their Final Examination Pre-Calculus as an indicator to measure the capability in mastering their mastering Pre- Calculus course. Therefore, this research aims to identify the performance of students in each questions in final examination of Pre-Calculus and to investigate between students’ performance in relation to gender.
# LITERATURE REVIEW
> In every semester, a faculty face higher rate of failure among the students in Mathematic during their first semester. A research conducted by Ahmad et al. (2017) presented the passing and failure rates in Pre- Calculus course of Diploma in Computer Science (CS110) and Diploma in Mathematical Science (CS143) students from UiTM Segamat, Johor. The data was collected for the most six semesters from year 2012 until 2014. The result demonstrated that the failure rate of 7.69% has tremendously increased to 40.6% in semester 2 of 2013/2014 academic session.
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> In another study carried out by UiTM Sarawak, similar concerns raised by Head of Programs of the Mathematics courses passing rate on their full-time diploma students’ academic performance. The data consisted the result of final examination from semester January-May 2004 to semester January-May 2007. Throughout the seven semesters, Pre-Calculus was recognized as one of the high-failure rates in Mathematic courses with the 31.34% (Tang et al., 2009).
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> The role of gender is one of most interesting variable studied in students’ Mathematics performance. Some researchers revealed that, the higher achievement among male students compared to female students (Gurian, Henley & Trueman, 2001; Li, 2004). Meanwhile in China, Tsui (2007) recorded no gender differences in the overall performance in Mathematics of their 1,078 high-school seniors on the 2002 College Entrance Examination. By contrast, Hamzah et al. (2006) reported female students had outperformed male students in the SPM Mathematics for five consecutive years (2000-2004) in Malaysia. Further, Tang et al. (2009) reported that female students tend to perform better than male students in the ‘high-failure rate’ Mathematics courses in Diploma programme. In light of the different output of researchers, this study investigate the trend of gender differences in Pre-Calculus among the students.
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> This study explores the potential and capability of student in answering final examination of Pre-Calculus course. The findings will identify the topic mostly unscored and needs more attention among the lectures. Besides, this study will encourage resource person to evaluate the effectiveness and significant topics covered.
# METHODOLOGY
> This section elaborates the respondents and instruments, data collection and data analysis procedures.
## Respondents and Instruments
> The respondents comprised of 108 full-time students Part 1 Diploma of Science semester September 2019- January 2020 in UiTM Tapah, Perak. The research instruments consisted the report of final examination analysis for those students who had taken Pre-Calculus in Semester 2019- January 2020 and the students’ particulars as noted in the registration database.
## Data Collection Procedures
> The reports of the final examination analysis for the Pre-Calculus course in Semester 2019- January 2020 were acquired from the Student Information Management System (SIMS) – Academic. Based on the report, information such as students’ gender and marks were obtained for each question in final examination Pre- Calculus.
## Data Analysis Procedure
> The data obtained were analyzed by the Principal component analysis (PCA) biplot and it was perfomed using packages *factoextra* and *ggbiplot* that available in R statistical software.
1. *Principal component analysis (PCA)*
> Principal component analysis (PCA) is a useful technique for exploratory data analysis. PCA, allows us to better visualize the variation present in a dataset with many variables (Bro and Smilde, 2014; Jolliffe and Cadima, 2016; Aït-Sahalia and Xiu, 2019). When the first two principal components (PCs) explain a significant portion of the variance in the data, you can visualize the data by projecting the observations onto the span of the first two PCs. In a PCA, this plot is known as a biplot. A biplot overlays a score plot and a loading plots in a single graph. If the data are well-approximated by the first two principal components, a biplot enables you to visualize high-dimensional data by using a two-dimensional graph (Gabriel, 1971; Jackson, 1991; Jolliffe and Cadima, 2016).
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> A biplot is constructed by using the singular value decomposition (SVD) to obtain a low-rank approximation to a transformed version of the data matrix **Y**, whose *n* rows are the samples (also called the cases, or objects), and whose *k* columns are the variables. Let **Y** be an *n* × *k* matrix holding the data. **Y** can be decompose using a singular value decomposition (SVD) into
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> **Y** = **ULV’**
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> where U is *n* × *k*, and both **L** and **V** are *k* × *k*. The elements of **L**, which is diagonal, are the so called eigenvalues.
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> From the singular value decomposition, the coordinates of the observations are given by
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> **G = UL*c ***(1)
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> and the coordinates for the variables are given by
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> **H’** = **L1–*c* V’** (2)
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> In Equations (1) and (2), the scalar *c* can take any value between zero and one. Regardless of the value of
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> *c*, the equation
## GH’ = ULc L1−c V’ = ULV’ = Y
> always holds. However, as **G** is *n* × *k* and **H** is *k* × *k*, all the coordinates have *k* dimensions. To plot these coordinates in a two-dimensional space, we must select two of them. Usually this is done by choosing those columns of **G** and **H** that correspond to the highest eigenvalues in **L**.
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> Choosing a value for *c* defines the coordinates for different types of biplots. Three values for *c* are most commonly used: *c* = 1 (row-metric preserving biplot), *c* = 0 (column-metric preserving biplot) and *c* = 0.5 (symmetric biplot). In this study we used *c* = 1 to conduct analysis using PCA biplot.
# FINDINGS AND DISCUSSION
## Descriptive statistics analysis
> The descriptive statistics of the marks by questions, as well as the descriptive statistics of the overall marks are presented in Table 1. It can be observed that the median of the marks of all questions is slightly higher than the mean of the marks. This indicates that the distribution of the marks is slightly skewed to the right. The variance of the marks of the Question 4 is found lowest while the variance of the marks of the Question 3 is found highest. This suggest that the marks of the Question 3 have the most spread about the mean.
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> It can be seen that the ranges of minimum and maximum values are quite large. From these values, it is shown that the ranges of marks for the different questions are large, which indicates a large dispersion in the data.
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> Table 1: Descriptive statistics
| Q1 |
12.85 |
13
|
11.40
|
6
|
18
|
| Q2 |
8.70 |
9
|
12.73
|
2
|
15
|
| Q3 |
18.72 |
20
|
26.02
|
6
|
25
|
| Q4 |
15.59 |
16
|
9.07
|
7
|
19
|
| Q5 |
16.79 |
17
|
20.09
|
3
|
23
|
| Overall |
72.66 |
74
|
229.42
|
32
|
98
|
## Performance based on Questions in Final Examination
> According to Table 2, students have difficulties in answering Question 2 as it show the greatest value in level of weak performance. This reveals that students face problems in sketching the graph of parabola and circle. While Question 5 represents that most student understand the topics covered in this question. Basically, the topics are revision from the previous subject Mathematics in SPM. It can be observed, students can moderately answer the rest of question.
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> Table 2: Performance student based on Questions in Final Examination
| Q1 |
14 |
52
|
34
|
| Q2 |
31 |
53
|
17
|
| Q3 |
7 |
36
|
56
|
| Q4 |
4 |
39
|
57
|
| Q5 |
4 |
32
|
64
|
## Analysis based on principal component analysis (PCA)
> Figure 1 resulted the vectors of Question 1 and Question 2 are very close to each other. This indicates that the marks of Question 1 and Question 2 have strong association between them. On the other hand, the vectors of Question 1 and Question 5 have almost 90-degree angle between them, which suggest that the is almost no association between the marks of Question 1 and Question 5.
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> The colored ellipses shown in Figure 1 summarize the spread of PCA points of each group. It can be observed that the female students have much smaller ellipse and also their PCA points scatter much closer to the vectors than the male students. From these results, we can say that the female students have better performance in answering the exam questions as compared to the male students.
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> If we analyze in more detail, the pattern is still similar where the female students have much smaller ellipse and also their PCA points scatter much closer to the vectors than the male students for each question as presented in Figure 2. This shows that the female students also have much better ability to answer the sub- questions as compared to the male students.
> Figure 1: PCA biplot of the marks of the different questions for male and female students.
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>  
1. (b)
> 
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> (c) (d)
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> 
>
> (e)
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> Figure 2: PCA biplot of the marks of the question (a) one, (b) two, (c) three, (d) four and (e) five
## Analysis using T-test to compare the average score by questions between male and female students
> The hypothesis testing is:
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> H0: μm – μf = 0, (the average exam scores do not differ significantly across genders)
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> Ha: μm – μf ≠ 0, (there is a significant difference between male and female average exam score)
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> T-test is utilized to compare the average score by questions between male and female students. Table 2 reveals that the p-values for Question 1, Question 2, Question 3 and Question 4 are greater than 0.05. On the other hand, the p-values for Question 5 and overall scores are less than
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> 0.05. Thus, we can state that there is difference between the average score of male and female students at the 5% level of significance. In sum, there is significant difference between male and female average exam score.
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> Table 2: T-test (Table 2)
| Q1 |
12.03 |
13.25
|
0.0989
|
| Q2 |
8.09 |
9.00
|
0.2345
|
| Q3 |
17.40 |
19.36
|
0.0748
|
| Q4 |
14.86 |
15.95
|
0.1011
|
| Q5 |
14.49 |
17.89
|
0.0005
|
| Overall |
66.86 |
75.44
|
0.0091
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> \*Reject Ho when *p*-value \< 0.05
# CONCLUSION AND RECOMMENDATIONS
> This research found that most students are able to answer final examination in Pre-Calculus. However, they have some difficulties in answering Question 2. This section consists of sketching graphs from topic circles and parabola. As a recommendation, instead of explaining in conventional method, the lectures should use visual tools such as Wolframalpha software and Desmos website application. Further, this study showed that female students outperformed male students in final examination in Pre-Calculus course. As conclusion, gender played role in affecting the performance of student in final examination in Pre-Calculus course.
# REFERENCES
> Ahmad, S.N., Mahadi, S., Yusri, M.Y., Yusop, H., Ali, M.N. & Heng C.H. (2017). Factors Related to
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> Students’ Performance in Calculus. Journal of Applied Environmental and Bilogical, 7(6S), 51-56. Aït-Sahalia, Y., & Xiu, D. (2019). Principal component analysis of high-frequency data. Journal of the
>
> American Statistical Association, 114(525), 287-303.
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> Borba, R. (2005). Math Failure High. The Collegian, October 3, 2005. California State University, Fresno. Retrieved August 15, 2020, 2020 from [http://www.csufresno.edu/Collegian/archieve/2005/10/03/news/math.sht.ml](http://www.csufresno.edu/Collegian/archieve/2005/10/03/news/math.sht.ml).
>
> Bro, R., & Smilde, A. K. (2014). Principal component analysis. Analytical Methods, 6(9), 2812-2831. Gabriel, K. (1971). The biplot graphic display of matrices with application to principal component analysis.
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> Biometrika 58(3): 453–467.
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> Gurian, M., Henley, P., & Trueman, T. (2001). *Boys and Girls Learn Differently\!* United States of America: Wiley.
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> Gynnild, V., Tyssedal, J., & Lorentzen, L. (2005). Approaches to study and the quality of learning. Some empirical evidence from engineering education. *International Journal of Science and Mathematics Education*, 3, 587-607.
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> Hamzah, H., Ariffin, S. R., & Yassin, R. M. (2006). Girls’ Overachievement in the National Examination: Can Type of Items be the Cause?. *Journal of Science and Mathematics Education in Southeast Asia, 29*(2), 1-20.
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> Jackson, J. E. (1991). A User’s Guide to Principal Components. New York: Wiley.
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> Jolliffe, I. T., & Cadima, J. (2016). Principal component analysis: a review and recent developments. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 374(2065), 20150202.
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> Li, Q. (2004). Beliefs and gender differences: A new model for research in mathematics education.
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> *Interchange*, 35(4), 423-445.
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> Naccache H. & Hleiss, R. (2016). Multivariate analysis of students’ performance in mathcourses and specific engineering courses. *Int. J. Educ. Pedagog. Sci*., 10, 3. 927–932.
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> Tang H. E., Voon L.L., Julaihi N. H. (2009). A Case Study of 'High-Failure Rate' Mathematics Courses and its’ Contributing Factors on UiTM Sarawak Diploma Students. Conference on Scientific & Social Resource 8, 1.
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> Tsui, M. (2007). Gender and Mathematics Achievement in China and the United States. *Gender Issues, 24*, 1– 11.
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> Yusof, Y.M & Rahman, R.A (2001*). Mathematics education at Universiti Teknologi Malaysia (UTM): Learning from experience*. Jurnal Teknologi, 34:9-24.