Overall, Grade 12 mathematics performance was low, with the average standard-grade mathematics score being 25% and the average higher-grade mathematics score being 43%. The mean score in mathematics in Subsystem M schools was close to double that of students in Subsystem P schools at the standard grade, and more than double that at the higher-grade level. The TIMSS scores were exceedingly low by international standards. The international mean has been set at 500 and the standard deviation across countries at 100. On average, including those who did not reach matric, the South African performance in TIMSS was more than two standard deviations below the international mean. Furthermore, there was a fair degree of correlation between the mean TIMSS and matric mathematics scores in the different cells. Notably, the correlation was better in Subsystem M schools and was greater for higher-grade than for standard-grade mathematics. It is also notable that students in Subsystem P schools with low Grade 8 mathematics scores often enrol for mathematics at the higher grade level. The mean TIMSS performance of students from Subsystem P schools who chose to do higher-grade mathematics in matric (TIMSS score 285) was considerably lower even than those Subsystem M students who elected to take standard-grade rather than higher-grade mathematics in matric (TIMSS score 425). TIMSS score and passing Grade 12 examinations Our initial hypothesis was that the TIMSS mathematics scores of students who reach matric, select mathematics as a subject and pass matric and mathematics would be higher than the scores of students who are not successful. We identified three distinct groups in the TIMSS data set, (1) those identified in the matric 2006 data set, (2) those identified in the matric 2007 data set and (3) those not identified in either data set (i.e. those that did not reach matric). The kernel density graphs of the TIMSS mathematics scores for the three groups allow a more detailed and nuanced picture of the mathematics starting point of the students (Figure 1). As expected, the TIMSS modal mathematics score for those identified in the matric data sets was higher than that for those who could not be tracked to matric. The graph for the matric 2006 group displays a wide tail to the right, indicating that, in general, students who reached their matric year with consistent grade progression had higher TIMSS mathematics scores. An unexpected finding was the range of mathematics scores amongst the three groups, and the degree of overlap of the three graphs. It would seem that students starting with similar TIMSS mathematics scores at Grade 8 can have quite different outcomes 4 years later. Disaggregating the kernel density of TIMSS scores for Subsystem P and Subsystem M schools reveals a different pattern for these two sets of schools (Figure 2 and Table 2). 2Mean Trends in International Mathematics and Science Study mathematics scores by school subsystem and identification. http://sajs.co.za/index.php/SAJS/article/downloadSuppFile/620/3422

Students in Subsystem P schools, for both those identified and those not identified in matric year, had low TIMSS scores, with the difference of the mean TIMSS scores being 27 points (approximately one-quarter of a standard deviation). As indicated, scores were normalised to an international mean of 500 and a standard deviation of 100. The South African standard deviation is similar in magnitude. Subsystem M schools had higher TIMSS scores, and the difference between the mean TIMSS scores of those who did and those who did not reach matric was 36 points. In general, within both groups, it would seem that TIMSS Grade 8 mathematics scores did not differentiate clearly between those who did and those who did not continue to the matric year. Although, it should be remembered that most of those in Subsystem M reached Grade 12, even though they may not have been identified in the study’s data. The analysis was extended to examine the patterns of TIMSS score for those who ‘passed matric’ and those who ‘did not pass’, in Subsystem P and Subsystem M schools (Figure 3). There was a high degree of overlap of the TIMSS scores between those who ‘passed matric’ and those who ‘did not pass matric’ in Subsystem P schools. The mean TIMSS scores were extremely low (226) for those who did not pass matric and 261 for those who passed matric. There was thus a small difference of 35 points between the two groups. In the Subsystem M schools, there was a higher degree of differentiation. The mean TIMSS score was 324 for those who did not pass matric and 444 for those who did pass (Table 3). There was thus a sizeable difference of 120 points between the two groups. 3 Mean Trends in International Mathematics and Science Study mathematics score by school subsystem and matric passing. http://sajs.co.za/index.php/SAJS/article/downloadSuppFile/620/3423

To further explore the relationship between TIMSS mathematics scores and those who passed matric, the TIMSS scores of those who passed matric were disaggregated into deciles, and the extent to which students from Subsystem P and Subsystem M schools converted their TIMSS scores to matric passes was examined (Figure 4). As expected, students in the higher deciles (deciles 8 to 10) of TIMSS scores had higher pass rates than those in the lower deciles. Students in the higher performance deciles from both subsystems converted to matric passes at an almost similar rate. The pass rates of students in the same TIMSS decile (deciles 5 to 7) were different for students from Subsystem P and Subsystem M schools, with students from Subsystem M schools converting to matric passes at a higher rate. Thus students starting with the same mathematics capability in Grade 8, measured by TIMSS score, converted to passing matric at a different rate in Subsystem P and Subsystem M schools. A further point of significance is that two out of every ten students who fell into the lowest four TIMSS mathematics deciles did pass matric. TIMSS scores and matric selection and performance We analysed the extent to which TIMSS mathematics scores were associated with the choice of mathematics as a matric subject and the performance in matric mathematics. Firstly, we plotted the kernel density of TIMSS mathematics scores for students identified in the matric data set, according to whether they took mathematics in matric or not (Figure 5); secondly, we plotted a graph of average matric mathematics marks by TIMSS decile positions in order to examine their correlation (Table 4). 4Mean Trends in International Mathematics and Science Study mathematics scores by mathematics selection in Grade 12 and school subsystem. http://sajs.co.za/index.php/SAJS/article/downloadSuppFile/620/3424

The kernel density plots of TIMSS scores of students from Subsystem P and Subsystem M schools who either took or did not take matric mathematics as a subject reflect different patterns of choice. In Subsystem P schools, there was little difference in the prior TIMSS mathematics performance between students who did and students who did not choose mathematics as a matric subject. In contrast, students in Subsystem M schools who took mathematics at matric generally had higher TIMSS mathematics scores in Grade 8 than those who did not continue with mathematics. As noted in Table 1, matric mathematics performance and TIMSS mathematics performance were low. The relationship between the average matric mathematics mark and TIMSS mathematics scores is illustrated by a plot of these two sets of scores by the TIMSS deciles into which the student scores fall (Figure 6). Although low, the average matric mathematics mark increased in higher TIMSS deciles, and there was a strong correlation between Grade 8 TIMSS and Grade 12 matriculation mathematics performance. Thus the TIMSS Grade 8 mathematics mark strongly correlated with the mathematics performance in Grade 12. Key findings We examined the correlation between Grade 8 mathematics performance and the mathematics pathways in high schools and performance in Grade 12 examinations. Grade 8 mathematics scores are a good indicator of analytical capabilities, and one would expect that those with higher mathematics scores would have progressed to Grade 12 and achieved success in the Grade 12 examinations. The expectation would also be that their subject choices in the senior secondary level would have included mathematics at the higher-grade levels, and that those with better TIMSS mathematics performance would have achieved higher matric mathematics scores. The findings of the study indicate, firstly, that educational achievement in South Africa, measured by TIMSS mathematics scores, is extremely low. The participation, performance and progression rates in Subsystem M and Subsystem P are significantly different, with Subsystem M students performing at a higher level than those in Subsystem P. Secondly, we found that students starting with similar TIMSS Grade 8 mathematical scores may have quite different educational outcomes 4 years later. Grade 8 mathematics scores appear not to predict who will or will not reach matric, although this result may at least partly be attributable to our not being able to successfully identify all those who actually reached matric. However, Grade 8 mathematics scores are a good indicator of who can pass matric in Subsystem M schools. For Subsystem P schools, although the higher TIMSS scores can predict who has a higher probability of passing matric examinations, this relationship is not as strong. Students who come to secondary school with high Grade 8 mathematics scores, whether from Subsystems M or P, are able to convert to passing matric. For those in the middle bands of performance, the rate of conversion is different in the two subsystems, with Subsystem M achieving higher rates of conversion than Subsystem P schools. A surprising finding was that, in Subsystem P schools, one in five students (20%) whose TIMSS score was in the lowest four deciles was nevertheless able to convert that low demonstrated capability into passing matric. Thirdly, for Subsystem M schools, TIMSS Grade 8 scores are a good sorter for the choice of matric mathematics as a subject, but, for the majority in Subsystem P schools, the subject choice of mathematics has little to do with earlier mathematics performance in TIMSS. Many students with weak TIMSS scores have high aspirations for participation and performance in mathematics, and, even with low scores, register for higher-grade rather than for standard-grade mathematics. Lastly, there is a high correlation between the mean Grade 8 mathematics score and the matric mathematics scores, with this correlation being higher in Subsystem M schools than in Subsystem P schools. Students with higher Grade 8 mathematics performance scores tend to achieve success in matric mathematics. However, it would seem that for students who have low mathematics scores in Grade 8, schooling cannot provide the necessary inputs to overcome their low mathematics scores achieved in earlier grades and cannot improve their mathematical competencies. Conclusion: Talking back to theory and policy In our unequal, low performing educational system, Grade 8 mathematics performance predicts Grade 12 mathematics performance for all students. Across the two subsystems, Grade 8 performance does not predict equally strongly who will or will not reach matric. The two subsystems also behave differently with respect to mathematics subject selection and passing the matriculation examination. In Subsystem P, selection of mathematics for further study is not influenced by earlier mathematics performance, whilst in Subsystem M students with higher TIMSS scores select mathematics to study further. For students from schools historically serving middle-class households, Grade 8 mathematics performance is strongly correlated to passing matric; however, Grade 8 mathematics performance is poorly correlated with passing matric in students from lower-resourced schools situated in poorer areas and serving poorer students. The strong relationship between Grade 8 and Grade 12 mathematics scores corroborates findings in the literature that earlier mathematics performance and strong foundational knowledge form the base for subsequent learning. Analytical skills in mathematics need to be built up from the early years. Mathematical knowledge is hierarchical in nature, and strong prior knowledge is therefore critical for conceptual development. The acquisition of these capabilities is shaped in the early years by the nature and quality of interactions in the home and community, and by the quality of inputs from the school. In Subsystem P, the progression from Grade 8 to Grade 12 does not fit the expected pattern, that is, that those with high Grade 8 mathematics scores will reach and pass matric and those with lower mathematics scores may not do so. Students starting with similar mathematics scores at the Grade 8 level may have different educational outcomes 4 years later. The reason why students with low TIMSS mathematics scores from poorer schools pass at matric level may be that TIMSS mathematics scores are not an adequate indicator of requirements for passing matric, or that students with weaker mathematics background are nonetheless successful in passing matric because of better performance in other subjects. Educational investments made post Grade 8 may enable students to improve their performance in subjects besides mathematics, and to pass matric despite failing mathematics. The pathways of students post Grade 8 in Subsystem P schools, that is, whether or not they select mathematics as a subject, shows that there is little relationship between demonstrated ability and choice of subjects. Students do not seem to be using information about their prowess in mathematics to make appropriate subject choices, perhaps because they do not receive enough accurate feedback at school about their mathematics performance. The policy implication from these findings is that raising the mathematics scores at Grade 12 requires raising the scores at Grade 8. Extrapolating from this, and linking to the literature on cognitive development, we need to raise the mathematics and numeracy scores in the earlier years of schooling. High levels of attention paid to the early years of learning (reception year and foundation phase) for children from environments of lower household and parental resources would contribute to breaking the cycle of poor academic performance. Without this, both the background and school will continue to let the children down and the reproduction of inequality will continue. Students must know and understand earlier concepts; only when they do understand these early concepts, will they progress. We have shown that by the time students reach the secondary level, it is too late to significantly improve matric mathematics performance. Ideally, the study would have used data of cognitive scores from the early years and tracked the cohort to later years, but as the only available cohort cognitive performance data is for Grades 8 and 12, only the relationship between Grades 8 and 12 could be examined. How cognitive development is shaped, in mathematics and in other subjects, can be assisted by panel studies research, by collecting data from the earlier years of schooling, and by paying greater attention to obtaining cognitive data. These issues should therefore be on the education research agenda for future studies. Acknowledgements Competing interests We declare that we have no financial or personal relationships which may have inappropriately influenced us in writing this article. Authors’ contributions V.R. was the principal author and conceptualised the study with S.v.d.B. S.v.d.B. provided econometric input and contributed to the writing of the manuscript. 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