THE INTERNATIONAL JOURNAL OF HUMANITIES & SOCIAL STUDIES Time Series Modeling of Students’ Performance in Mathematics among the Selected Four Senior High Schools in Bongo District, Ghana

: The study paper aims at examining the pattern in the performance of students in mathematics using analysis of the time series. Analysis of the time series was carried out on a longitudinal data of the performance of students’ in mathematics for four consecutive years, from 2015 to 2018. The research employed the Box-Jenkins methodology. The study revealed that the performance of the students in mathematics was best represented with a quadratic pattern and ARIMA (0,0,0). Further, the results showed that the students ' performance in mathematics is likely to stagnate in the near future as seen in the forecast curve. The study concluded that the average mathematics performance of the students over the four-year period is very low and this suggests a decrease in the standard of education within the district of Bongo.

students, represents a step toward rising get admission to and equality of secondary schooling. The secondary faculty level in Ghana presents its customers with in depth and technical / vocational education. Thus, even though some excessive school students take the comprehensive secondary education, which in Ghana constitutes the easy majority, others receive handiest technical / vocational training. High faculty college students in Ghana study four core subjects and a selection of elective topics (as a minimum three), which is determined with the aid of the form of curriculum college students chose to learn.
The newcomers are encouraged to be essential and analytical in these studying surroundings of the twenty first century. Thus, mathematics provides the sort of foundation for college students to assume objectively and analytically. Therefore it is not shocking that secondary colleges throughout the globe are making arithmetic a mandatory part of the curriculum for this academic level. Ghana's Government acknowledges the role that science, math, and technology play in attaining the nation's development agenda.
This policy recognition has contributed to the acknowledgement of arithmetic as one of the foundations of country wide growth (Ghana Ministry of Education, 2014) .To this end, the Ghana Government has introduced a few interventions and injected incentive schemes aimed at enhancing the participation and fulfillment of college students in technological know-how and arithmetic, mainly on the secondary training level since the 1990s. For e.g., the Science Resource Center Project, the Ghana Investment Fund for Electronic Communication (GIFEC), ICT-related programs, the MASTESS (Mathematics, Science and Technology Scholarship Scheme), and the SEIP (Secondary Education Improvement Project) are just a fewof these initiatives. Some similar and interrelated factors are stated to be contingent on achievement at the secondary schooling level (Chua & Mosha 2015).
According to Chua and Mosha (2015), of the variables that stand out inside the present day literature are those relating to the science and mathematics curriculum and the teacher and coaching variables .It is known that the curriculum guides the coaching, learning, and evaluation activities in order to occur in the schools.
Aspects inclusive of the syllabus and textbooks specifically have an effect on academic achievement (Dembélé & Lefoka, 2007;Pridmore, 2007;UNESCO, 2016;World Bank, 2008). The curriculum's reach, consistency, and significance also have an effect on what stakeholders do with the program, as well as their capability to gain the expected outcome. In addition to concerns with the curricula, the pleasant way to train, assign and hire teachers has been delivered to the fore. The maximum promising path is the fundamental circulate in attracting strong trainer candidates by way of investing in superinstructor schooling programmes (Boyd, Grossman, Lankford, Loeb, & Wyckoff, 2008;Coalition for Teaching Quality, 2016;Darling-Hammond, Bransford, LePage, Hammerness, & Duffy, 2005). Furthermore, it was discovered that college districts hiring uncertified teachers often deployed them to underserved areas, and these instructors usually had underserved backgrounds themselves (Darling-Hammond, 1999).
As Hanushek, Kain, and Rivkin (2004) have pointed out, teacher mobility is likewise a large trouble in underserved regions, main to pockets of inequality in these areas in particular rural faculties in contrast with their city counterparts. Moreover, it was located that college districts using uncertified instructors frequently deployed them to underserved areas, and these instructors commonly had underserved backgrounds themselves (Darling-Hammond, 1999). As Hanushek, Kain, and Rivkin (2004) have pointed out, trainer mobility is likewise a large trouble in underserved regions, leading to wallet of inequality in these areas in particular (Darling-Hammond, 1999;Hanushek et al., 2004;Wilson, Floden, & Ferrini-Mundy, 2001). Simply put, if instructors are qualified and capable in acting their obligations and responsibilities while sure variables are stored constant, there may be a highlikelihood of excessive achievement on the a part of college students. Teachers ' pedagogical practices rely in large part on their expertise of content, pedagogical enjoy, experience of pedagogical material, and abilities that they follow to teaching inside the classroom (Borko, 2004;Wilmot, 2009). Literature also claims that underserved and hard-to-reach secondary colleges result in the disparity in achievement observed in the rural -city divide. As a result, assigning seasoned, trained instructors to low-performing schools and college students is possibly to pay off in improved results and decreased gaps. One of the effects that made the hassle even greater obvious is inside the research of Laczko- Kerr and Berliner (2002), which indicates that the fulfillment of mathematics by means of elementary pupils taught via new, uncertified teachers made the fulfillment tests significantly decrease than those taught by using newly qualified teachers did.
In addition to the effect of instructor education on the overall performance of the students, the elements causing terrible or constantly bad overall performance in mathematics are multifaceted. (Swars, Daane, & Giesen, 2006;Swars, Hart, Smith, Smith, & Tolar, 2007;West African Examinations Council [WAEC], 2014, 2015, 2016. The central feature of these research is that instructors and teachers are critical to modern educational trends, in particular as learning outcomes and student achievements come to the forefront.
Of example, most of the useful instructor-related studies have concentrated on elementary or fundamental school. Although a few studies imply that the teaching factor is somewhat less in high colleges, much more look at and studies need to be carried out before delineating which regions of the secondary education gadget suffer from research deficit in Ghana. There is a need to pressure the education machine a little extra in phrases of opportunities and working situations that might draw exceedingly successful teachers to historically hard-to-staff faculties and underserved Ghana population, to organize access and equity at Ghana's secondary level.
The current poor performance of college students in technological know-how and arithmetic at West African Secondary School Certificate Examination necessitates the want to examine the education system at secondary college level (WASSCE; WAEC, 2014WAEC, , 2015WAEC, , 2016WAEC, , 2017WAEC, ,2018. Education stakeholders and the general public have raised questions about student fulfillment in arithmetic coaching and learning across the world. Because of the poor overall performance in arithmetic, most extensively in core mathematics at the West African Senior School Certificate Vol 8 Issue 3 DOI No.: 10.24940/theijhss/2020/v8/i3/HS2003-074 March, 2020 Examinations (WASSCE), many college students cannot meet the entry requirements as mentioned above each year. Place, for instance, in August 2016 .City Fmonline.Com finished an overview of the findings of WASSCE released through Rev. Nii Nmai Ollenum, Head of the WAEC National Office of Ghana, and made some tragic conclusions. The outcomes of 234,871 candidates have been published out of a total of 274,262 candidates who had sat for the examination that year. Of that total, 77,108 applicants representing 32.83 percent of A1-C6 grade applicants. Aside from that, 65,007 applicants representing 27.68 percentage had received grades While a few stakeholders notion the students ' overall performance in arithmetic changed into strong, others notion it became bad. With the sort of divided opinion on the topic of pupil effects, while there is little to no proof to support such arguments and conclusions, it is crucial to observe the outcomes styles of students in the four decided on senior excessive college in Bongo District. However, without clean proof of overall performance styles, it would, for the maximum part, be impossible for state organizations in price of schooling and coverage makers to implement effective progressive education policies. Thus, this study aimed to provide information on overall performance styles at secondary school degree in mathematics, which might be currently not available inside the literature. Mathematics passes the core necessities for entering better establishments in Ghana. As mathematics is a full-size subject within the educational curriculum, research into the overall performance of mathematics students within the4selectedexcessive schools inside the District of Bongo is appropriate. The consequences of the college are higher or higher even lower than the state's country wide averages, this means that that there may be something particular about the district while it comes to scholar fulfillment in WASSCE mathematics. This work therefore aims to assess students' achievement in mathematics between 2015 and 2018. In addition, inside the wake of the interventions brought in current years into the Ghana secondary education system, the take a look at also sought to explore feasible explanations for the trends discovered by using the WAEC outcomes. Three studies questions had been therefore formulated to direct the analysis, which had been given as follows:

Specific Objectives
 To determine the trend of students' performance in mathematics Ghana District.
 To develop appropriate model that best fit students' performance in mathematics Ghana District.  To make a five year forecast of the students' performance in mathematics Ghana District.

Research Questions
 Is there any trend of students' performance in mathematics Ghana District?  What is the appropriate model that best fit students' performance in mathematics Ghana District?  What is the five year forecast of the students' performance in mathematics Ghana District?

Research Methods
The research made use of the annual report of WASSCE results. The data were collected from the selected four Senior High Schools identified in the Bongo District. The data covered periods ranging from January 2015 through December 2018. The performance of students in mathematics in Ghana District was predicted using autoregressive integrated moving average (ARIMA) models to create a predictive tool for the performance of students in mathematics. The non-seasonal model order defined by ARIMA (p, d, q) represents auto-correlation over a maximum order of p months; differentiating order of d adjacent months and moving averages of q annual cycle order. The Box and Jenkins methodology has been used to define the trends best representing the time series of results of the students. Performance incidence plot was created for the students to detect and correct issuance of non-stationarity. Models of varying orders were fitted after stationarity and compared using standardized Bayesian information criterion (BIC), mean absolute error (MAE), and Stationary-R Square. Autocorrelation (ACF) function measuring the correlation between the time series data and the Partial Autocorrelation (PACF) function showing the correlation between the autoregressive time lags used to determine the parameters of the ARIMA model. The values of correlation fell within the confidence limit set for the ACF and PACF; an indication of reasonable forecasting performance. When evaluating the parameters p and q of the ARIMA model, ACF and PACF graphs of normalized data were used. The method of trial and error was used to determine the final structure of the predictive model. For forecasting reason the model with the least BIC, AIC, HQ and the highest Stationary-R Square was chosen. Review of the data was conducted using minitab, gretel, and R-program.

Results and Analysis
An exploratory data analysis on the success of students in mathematics was also used for the fifteen consecutive year, using primarily Minitab software, R system and Gretel software, and the time series analysis Box-Jenkins techniqueSome calculations were made to obtain first the descriptive statistics regarding the success of the students in mathematics, followed by time series plots and a trend analysis. The minimum performance of students in mathematics was found to be 0 and a maximum of 496 while the average performance was 32.68 with a corresponding standard deviation of 6.14, which suggests that the data is widely distributed around the mean. The 18.79 per cent coefficient of variation also indicates that the data has a very high variance .The output of the students in mathematics distribution also shows positive skewedness of 3.59 suggesting that most of the scores are clustered to the right of the mean and have a positive kurtosis value of 14.87 also suggesting that the data is platykurtic, thus having flattened and that there are more of the students' scores at either extreme of the distribution thus flattened than normal peak. This means the quality of education within the district have dropped

Figure 1: Time Plot of Students' Performance in Mathematics
The plot in Figure 1 shows the output of the students with respect to time variations. From figure 1 it can generally be observed that no decreasing trend in the plot is significantly sharp over the time of analysis. But the success of students .In 2015, 2016, 2017 and 2018, respectively, there was no major downward or upward transition. The series displayed a typically no-increasing non-decreasing trend in the time graph implying that there is no change in mean while the fluctuations indicate constant variance over time suggesting that the series is stationary.

Tests for Stationarity
A stationary cycle has mean and variance that doesn't change over time and there are no patterns in the cycle. The series must be stationary in order to continue with the estimation of an ARIMA model, as this analysis used the Augmented Dickey-Fuller (ADF) test and the Kwiatkowski, Phillips, Schmidt and Shin (KPSS) test for proof of stationarity in the performance of students in quantitative methods.

Augmented Dickey-Fuller Test
We test the hypothesis for the ADF test that;  H0: the sequence is not stationary.
This means that a p-value of less than 0.05 at a meaning point of 95 percent means we deny H0 and assume the series is not stationary, then it is stationary .
TEST TEST STATISTIC P-VALUE KPSS 0.021298, 0.1 ADF -7.1264, 0.01 Table 2: Tests for Stationarity The series differentiated four times in order to achieve stationarity and the results are shown in table 3 after the fourth discrepancy. From the KPSS test values in table 2, with a significance amount of 5 percent, the inference is that the sequence is stationary since the p-value (0.1) is greater than 0.05. The ADF check with an inverse null hypothesis, however, shows that the data is stationary at p-value 0.01. In all, the data is concluded stationary on the basis of time map, correlogram, KPSS, and ADF test proof.  The most suitable model from Table 3 representing the pattern in the output of the students is the one with minimal errors. A closed observation of the errors generated by three models, the quadratic model has the minimum MAPE, MAD and MSD, is therefore considered the best model to explain the trend in the performance of students in mathematics

Figure 4: ACF and PACF Plot of Students' Performance in Mathematics
Further analysis and tests on the Autocorrelation Function (ACF) plots and the Partial Autocorrelation Function (PACF) plots were performed. It can be noted that the data tend to be stationary with an interval of confidence of 95 per cent. The ACF and PACF are not major spikes at any lags as shown in Figures 4. It means that both ACF and PACF reaffirmed that the sequence is stable, or non-decreased. The model with the minimum Akaike Information Criteria (AIC), Bayesian Information Criterion (BIC) and Hannan-Quinn (HQ) is the most suitable for the sequence. Therefore, by testing all the competing models in table 4, the ARIMA model (0, 0, 0) has the lowest values and therefore the best predictive model Table 5 below displays estimates of System ARIMA parameters (0, 0, 0). The constant parameters are important with coefficients and p-values of 5 per cent (32.6806, 1.04e-07) respectively. P-value of less than 0.05 shows the parameters ' importance.

Type
Coefficient Standard error Z value P-value Constant 32.6806 6.14405 5.319 1.04e-07 Table 5: shows Parameter Estimation

Model Diagnosis
The following diagnostics are performed to ensure that the model selected is the best one suited to the data

Residuals Plots
The residual trends over time around the zero mean as shown in figure 5 below demonstrate that the residuals are random and independent of each other, thereby suggesting that the model is suitable. www.theijhss.com

The Normal Q-Q Plot
The Standard Q-Q Plot is yet another residual diagnostic test to determine whether it fits regular distribution. The standard likelihood map (Q -Q map) is used to do this. It is a plot, a plot based on quantile estimates. The normal Q-Q plots are used to equate a sample distribution to a theoretical distribution.

Figure 6: Q-Q Plot of Students' Performance in Mathematics
From figure 6 almost all the points are on the straight line suggesting a good fit for the pattern.

Ljung-Box Q Statistics
A test of the overall adequacy of the model is performed using the statistics of the Ljung-Box Q. This shows that the model is generally satisfactory with a p-value of 0.834 which is much greater than 0.05.

MODEL
Statistics DF Sig. Arima(0,0,0) 5.8 10 0.734 Table 6: Ljung-Box Q Statistics 3.9.1. Forecast Since the model tests to be in good shape, we can now predict the next 3 observations for future values in this case

Summary and Conclusion
Findings revealed that the minimum performance of students in mathematics was found to be 0 and a maximum of 496 while the average performance was 32.68 with a corresponding standard deviation of 6.14, which suggests that the data is widely distributed around the mean. The 18.79 per cent coefficient of variation also indicates that the data has a very high variance. The output of the students in mathematics distribution also shows positive skewedness of 3.59 suggesting that most of the scores are clustered to the right of the mean and have a positive kurtosis value of 14.87 also suggesting that the data is platykurtic, thus having flattened and that there are more of the students' scores at either extreme of the distribution thus flattened than normal peak. This means the quality of education within the district have dropped From the forecast curve the performance of the students in mathematics is likely to stagnate in the near future if the police makers are unable to address the issue at stake.
The model with the minimum Akaike Information Criteria (AIC), Bayesian Information Criterion (BIC) and Hannan-Quinn (HQ) is the most suitable for the sequence. Therefore, by testing all the competing models in table 4, the ARIMA model (0, 0, 0) has the lowest values and therefore the best predictive model The most suitable model from Table 3 representing the pattern in the output of the students is the one with minimal errors. A closed observation of the errors generated by three models, the quadratic model has the minimum MAPE, MAD and MSD, is therefore considered the best model to explain the trend in the performance of students in mathematics The series were not differentiated in order to achieve stationarity and the results are shown in table 3. From the KPSS test values in table 3, with a significance amount of 5 percent, the inference is that the sequence is stationary since the p-value (0.1) is greater than 0.05. The ADF check with an inverse null hypothesis, however, shows that the data is stationary at p-value 0.01. In all, the data is concluded stationary on the basis of time map, correlogram, KPSS, and ADF test proof.
The plot in Figure 1 shows the output of the students with respect to time variations. From figure 1 it can generally be observed that no decreasing trend in the plot is significantly sharp over the time of analysis. But the success of students .In 2015, 2016, 2017 and 2018, respectively, there was no major downward or upward transition. The series displayed a typically no-increasing non-decreasing trend in the time graph implying that there is no change in mean while the fluctuations indicate constant variance over time suggesting that the series is stationary.

Conclusion
The research paper concluded that the students' performance in mathematics can best described by a quadratic trend and ARIMA(0,0,0).From the forecast curve the performance of the students in mathematics is likely to stagnate in the near future if the policies makers are unable to address the issue at stake. The research further concluded that the overall performance of the students' in mathematics over the four years period is very poor and this implies that the quality of education within the district has dropped.