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Students’ mathematics achievement in Africa: a preliminary look at gender differences in sub-saharan… Frempong, George 1994

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STUDENTS' MATHEMATICS ACHIEVEMENT IN AFRICA A PRELIMINARY LOOK AT GENDER DIFFERENCES IN SUB-SAHARAN CULTURES THE CASE OF NIGERIA AND SWAZILAND by GEORGE FREMPONG B.Sc. (Hons), University of Science and Technology, Kumasi, 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in THE DEPARTMENT OF CURRICULUM STUDIES Faculty of Education We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1994 © George Frempong, 1994 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) Department of Curriculum Studies The University of British Columbia Vancouver, Canada Date 13th Oct. 1994 DE-6 (2/88) i i ABSTRACT This study compared gender differences in mathematics achievement in two sub-Saharan Africa cultures (Swaziland and Nigeria) using data from Second International Mathematics Study (SIMS). Confirmatory Factor Analysis (CFA) was also used to determine the appropriateness of SIMS hypothesized mathematics measurement model in assessing students of the two cultures. MANOVA with a follow up discriminant function analysis were used to investigate gender differences in means while differences in skewness and kurtosis of mathematics achievement distribution scores for males and females were compared using 95% confidence interval graphs of skewness and kurtosis. The analysis of the data indicated that gender differences in mathematics achievement in both countries were statistically significant, but substantively trivial (eta squared<0.013). Achievement distribution scores in all mathematical areas (Arithmetic, Algebra, Geometry, Statistics and Measurement) were similar for males and females in the two countries except Geometry and Algebra in Swaziland. Gender differences in skewness and kurtosis were statistically significant (p = 0.05). For Algebra, there were gender differences in kurtosis only (p = 0.05). The CFA showed that the SIMS mathematics model did not fit well to the Swaziland and Nigeria data (GFI<0.803). The study concluded that there was little justification in assessing students in the two countries based on the SIMS model. However, based on the model it appeared that culture seemed to have little influence on gender differences in mathematics achievement in both countries. The two countries could therefore share educational policies without creating a gender i i i gap in mathematics achievement. Further research using locally developed test items and involving more than two factors such as IQ and SES has been suggested. i v TABLE OF CONTENTS Abstract ii List of Tables vii List of Figures viii Acknowledgments ix CHAPTER I: THE PROBLEM 1 Introduction 1 Significance of Study 3 Purpose of the Study 9 Research Questions 10 Limitations 10 CHAPTER II: LITERATURE REVIEW 12 Overview 12 Historical Perspective 12 Gender Differences in Mathematics Enrollment 14 Gender Differences Across Mathematical Areas 17 Gender Differences Across Educational Systems 20 Gender Differences in the Variability of Achievement 22 Potential Causation: Why Gender Differences in Mathematics 23 Biological and Genetic Factor 23 Psychological Factor 25 Cultural and Societal Influences 27 Conclusion 29 Summary 30 CHAPTER IH: METHODOLOGY 31 Settings 31 Swaziland 31 Nigeria 32 Data Source 33 Design of the Study 34 Appropriateness and Significance of Statistical Analysis 35 MANOVA/Discriminant Functional Analysis 35 Effect Size and Educational Significance 37 Skewness and Kurtosis 37 Confirmatory Factor Analysis 39 Population and Generalizability of the Study 40 CHAPTER IV: RESULTS AND DISCUSSION 43 Statistical Assumptions 43 MANOVA Randomized Groups Design 44 Discriminant Functional Analysis 47 Discussion on Nigeria Data 48 Discussion on Swaziland Data 50 Effect Size 52 Varaiability of Achievement Distribution 52 Confirmatory Factor Analysis 59 Nigeria Data 60 Swaziland Data v i CHAPTER V: SUMMARY AND CONCLUSIONS 71 Research Questions and Findings 71 Implications for Practice 74 Suggestions for Further Research 75 REFERENCES 77 APPENDIX 90 Tables for Figures in Chapter IV 90 V l l LIST OF TABLES Table 2.1. Achievement on subtests 19 3.1 Distribution of test items on the five mathematical constructs 40 4.1 Descriptive statistics for the five mathematical areas 45 4.2 MANOVA table 45 4.3 Results of discriminant function analysis on Nigeria data 48 4.4 Correlation matrix for subtests (Nigeria) 49 4.5 Classification results with discriminant function (Nigeria) 50 4.6 Results of discriminant function analysis on Swaziland data 51 4.7 Classification results with discriminant function (Swaziland) 51 4.8 Effect size 52 4.9 Skewness and kurtosis by gender for Swaziland and Nigeria 53 4.10 Distribution of Geometry test scores for Swaziland 55 4.11 LISREL Estimates (MAXIMUM LIKELIHOOD) for Nigeria 61 4.12 Correlation among factors for the Nigeria data 64 4.13 Goodness-of-fit Indices for Nigeria data 65 4.14 LISREL Estimates (MAXIMUM LIKELIHOOD) for Swaziland data 66 4.15 Goodness-of-fit Indices for Swaziland data 70 vii i LIST OF FIGURES FIGURE 4.1 A comparison of variance in various mathematical areas by gender in Swaziland and Nigeria 44 4.2 An illustration of mean scores 47 4.3 The 95% confidence intervals of skewness for Nigeria 54 4.4 The 95% confidence intervals of skewness for Swaziland 55 4.5 Distribution of Geometry test scores in Swaziland 56 4.6 The 95% confidence intervals of kurtosis for Nigeria 57 4.7 The 95% confidence intervals of kurtosis for Swaziland 58 4.8 Distribution of Algebra test scores in Swaziland 59 Nigeria Data 4.9 The correlation of Arithmetic test items with all factors 62 4.10 The correlation of Algebra test items with all factors 63 4.11 The correlation of Geometry test items with all factors 63 4.12 The correlation of Statistics test items with all factors 64 Swaziland Data 4.13 The correlation of Arithmetic test items with all factors 67 4.14 The correlation of Algebra test items with all factors 68 4.15 The correlation of Geometry test items with all factors 69 4.16 The correlation of Statistics test items with all factors 69 4.17 The correlation of Measurement test items with all factor 70 ix Acknowledgements Numerous individuals have contributed in diverse ways to the successful completion of this study and I am gratefully thankful. These include ... my committee members, Jim Sherrill, Ann Anderson and Robert Schutz. Your constructive criticisms, clarity of thought and devotion to the study have left an indelible impression in me. "From the bottom of my heart, thank you all indeed." ... Tony Clark. Your keenness in the study right from its 'infancy to adulthood' and positive feedbacks helped me a lot to overcome my insecurities. I wish therefore to express my sincerest appreciation and gratitude. ... David Robitaille. You gave me permission to use SIMS data and also spent your precious time to make sure that I got the right data for the study. I appreciate your assistance. ... fellow graduate students. You all supported and assisted me in many ways. Renee Fountain, Jennifer Khamasi and Joel Zapata deserve special mention in this regard. You are friends indeed. ... staff in the Curriculum Studies Department especially Diana, Saroj, Bob and Brian. Your support to graduate students is invaluable and should be highly acknowledged. Thank you Guys. ... the Almighty God for the guidance and direction through the program. My parents, brothers, sisters and all African women are also acknowledged in this work. To them I dedicate this thesis. 1 CHAPTER I THE PROBLEM Introduction In the school year 1980-81, the International Association for the Evaluation of Educational Achievement (IEA) conducted the Second International Mathematics Study (SIMS). The objective of the study was to compare and contrast, in an international context, the varieties of curricula, instructional practices and student affective and cognitive outcomes across the schools in twenty countries. The twenty countries were made up of twenty-two educational systems. SIMS defined "educational system" to mean educational jurisdiction, which may refer to educational administration of a province (such as British Columbia, Canada) or a nation (like U.S.A). With the exception of Canada and Belgium with two educational jurisdictions each, all other countries in the study had one jurisdiction. The SIMS study gathered detailed information from each of the educational systems on three interrelated aspects of mathematics teaching: the intended curriculum, the implemented curriculum and the attained curriculum. Achievement tests were designed to measure students' knowledge and skills in five mathematical areas: arithmetic, algebra, geometry, measurement, and statistics. The IEA mathematics tests were a reflection of the "international mathematics curriculum". The tests could therefore be described as tests of mathematics achievement in general, not intended to measure the objectives of particular schools. As Dockrell (1980) suggests, test scores reflect among 2 other things, the effect of school and the general impact of culture. However, "the more general a test, the less it reflects the effect of schooling, the more it reflects general culture variables and the more important are likely to be measures of home backgrounds and other general cultural influence" (Dockrell, 1980, p. 7). The question is, to what extent does the SIMS mathematics achievement data reflect the cultural influences on students' achievement in various countries? The only sub-Saharan countries involved in the SIMS study (Swaziland & Nigeria) are also two cultures in which the author has personal experiences as a mathematics teacher at the secondary level. The author's experiences in these two countries led him to believe that in the secondary schools, girls were better mathematicians than boys in Swaziland while he believed the reverse was true for Nigeria. However, there was nothing in the literature to either confirm or dispute this belief. This study apart from satisfying this personal curiosity will also be an important resource for further research into gender differences in mathematics achievement in Africa. Subjects were sampled from similar population groups in Swaziland and Nigeria. In IEA terms, this population was defined as the grade in which the modal student age lies between 13.0 and 13.11 years by the middle of the school year. The aim of this definition was to enable the appropriate grade to be identified in each of the participating countries. This study explores gender differences in mathematics achievement in various mathematical areas in the two sub-Saharan countries using SIMS data. Studies concerning gender differences in mathematics date back more than two decades. The findings of these studies, mostly carried out in developed countries, indicate that there is little distinction between the genders in elementary school mathematical performance. However, they do 3 indicate significant differences in favor of males by late adolescence (Fennema & Carpenter, 1981; Gwizdala & Steinback, 1990). Other studies indicate a trend of the narrowing of the gender gap over time (Becker & Hedges, 1988; Friedman, 1989; Hyde et a l , 1990; Kimball, 1989; Willms & Kerr, 1987). Explanations for this phenomenon include genetics (Benbow & Stanley, 1980, 1981; Benbow, 1988), student attitudes (Fennema & Sherman, 1976; Joffe & Foxman, 1984), differential coursework (Benbow & Stanley, 1983; Pallas & Alexander, 1983), and cultural factors (Fennema & Sherman, 1978; Saxe, 1991). Linn and Hyde (1989) argue that gender differences are not general but specific to particular cultures. Saxe (1991) also suggests that cultural practices affect the development of cognitive functions. This leads to the question of whether gender differences in mathematical performance will manifest themselves in different cultures. Significance of study There is an increasing attention to research on cultural variation in school childrens' performance in mathematics. For instance, it has been reported that students from different Asian cultures (China, Japan, and Taiwan) perform significantly higher on mathematics and science achievement tests than their U.S. counterparts (Hess, Mei & McDevit, 1987). Most recently Ma (1993) also used SIMS data to compare gender differences in Asian and Canadian cultures. His findings indicated a pattern where there was statistically significant gender differences in variability of Algebra achievement in two Asian countries (Honkong and Japan) but not in the two educational systems (British Columbia and Ontario) in Canada. 4 It would be interesting to compare students' mathematics achievement in African cultures. Stigler and Baranes (1989) suggest that "one of the more significant benefits of cross-cultural research is that, it confronts us with our beliefs, and shows us that they are culturally based, not naturally required" (p.299). The purpose of this study is to explore gender differences in mathematics achievement in two cultures in sub-Saharan Africa. Two international studies by IEA have addressed gender differences in mathematics across cultures. Husen (1967) analyzing the first IEA study, found differences in overall achievement to favor males however, differences within countries were not always statistically significant. This led Husen to suspect that gender differences were a within-culture phenomenon. According to Ethington (1990), curricular and instructional differences which are reflections of cultural values are some possible reasons for the between-country gender differences in mathematics performance. Leder (1986), suggests that: A clear recognition of the values, expectations and beliefs of the wider society within which learning takes place is required for a full appreciation of the currently found gender differences in mathematics participation and performance, (p.6) SIMS provided another opportunity to examine gender differences across 20 countries. Schmidt and Kifer (1989) analyzed SIMS data using regression coefficients as indices to explore gender differences and reported that: ...across subtests, France shows the largest difference favoring boys. Hungary, Sweden, England and Wales, and Belgium (Flemish) have differences which favor girls. Only one of these differences, however, is statistically significant, (p. 223) These findings seem to indicate certain patterns and raises the question: To what extent do cultural differences among these countries relate to different levels of achievement for boys and girls? While quite a few studies have been done, how the genders differ in mathematics achievement in different cultures was not adequately addressed and therefore not well understood. By documenting the gender differences in mathematics achievement in Nigeria and Swaziland, two countries with divergent cultural attitudes towards womens' education, this study may provide a clue as to the nature of these differences. For example while parents in Swaziland would encourage their daughters to go to school in order for them to attract very high bride wealth (Lobola), parents in Nigeria tend to discourage their daughters from going to school because of among other factors, the fear of the daughters getting pregnant. It will be interesting to find out if gender differences in the mathematical areas as well as overall performance will be manifested in the two cultures as suggested by Leder (1986). Because of the cultural diversity across educational systems, one would not expect consistency in gender differences in mathematics achievement across the two sub-Saharan cultures. However, the potential interactions of gender with educational system would contribute to understanding of gender differences in mathematics performance. Such cross-culture comparisons would also enrich understanding of the theories about how culture affects students' mathematics learning and therefore promote educational progress in each culture. 6 Culture also seems to influence the level of participation of males and females in mathematics. Swetz (1989) notes that "the institutional permissiveness by which females are allowed to or prevented from pursuing mathematical studies is a societal decision and intrinsically reflects cultural expectations" (p. 139). Stromquist (1989) also writes: In much of the Third World, women still face substantial educational disadvantages. For both men and women, participation in schooling is highest in primary, lower in secondary, and lowest in tertiary levels. However, the participation of women shows a steeper pyramidal distribution, given that-for a variety of reasons-they face considerable obstacles during their trajectory through the educational system, (p. 144) Factors that account for the diminishing participation of women as they move up the educational ladder are predominantly those related to home and culture (Stromquist, 1989). There is, however, an inconsistency across the two cultures that are the focus for this study with regard to male/female participation in education. For example, it is estimated by UNESCO (1986) that in 1982, only 14% of the 12-17 year-old female age group was enrolled in secondary schools in Nigeria, compared to 42% of males of the same age. In Swaziland, the estimate in 1982 indicated that 42% of females of the same age group were enrolled in secondary schools compared to 43% of males. Thus, there were almost equal numbers of males and females enrolled in secondary schools in Swaziland while in Nigeria, there is very wide disparity in the enrollment of male and female students in secondary school. This suggests that, if the population distributions for male and female participation in secondary schooling are not identical, perhaps the mathematics achievement distributions by gender are not identical either. 7 Ethington's (1990) analysis of the SIMS data seems to confirm this suspicion. He used the statistical method called median polishing to examine SIMS data. Median polishing does not test hypotheses directly but rather decomposes data into relevant effects. According to Ethington (1990), the median polishing model used in his study "is similar to the additive model of analysis of variance but uses medians rather than means to describe common effects, row effects, and column effects" (p. 76). The results of the analyses indicated in the words of Ethington: No substantial gender effects in any of the content areas, and the slight effects shown favored girls more often than boys. These findings differ from those cited previously, wherein consistencies were found in gender differences across problem type. For example, previous studies found males to perform better than females on problems dealing with proportionality, yet these results show females in Thailand scoring almost 5 percentage points higher than males on the ratio/proportion/percent items. Furthermore, within no content area were males found to persistently outperform females across countries or vice versa, (p. 79) Ethington cautioned that his results should not be interpreted in terms of significant differences among the groups since the median polish used in the analysis was an exploratory method that did not intend to test a priori hypotheses. The fact, however, still remains that his findings differed from previous works where the mean scores of the genders were compared instead of the median. It could therefore be possible that there is enough of a difference between the mean and median of some of the population groups to make the distribution of scores in these populations non-normal. For example, according to a UNESCO (1986) statistical report on school enrollment in Thailand for 1980, there was almost universal primary 8 education (99% of primary school age group in school) but only 29% of the secondary age group (30% male and 28% female) were in school. One cannot, therefore, be sure of the normality of the male and female populations in this case which may explain the inconsistency in the findings on SIMS data, especially on Thailand. This means that in comparing mathematics achievement of the genders especially in developing countries, where the percentage of enrollment in secondary schools is very low, there is the possibility of the population distributions being non-normal. It is therefore necessary to look at the whole population distribution by reporting all the statistical indices related to the first four moments about the mean of the distribution. The first moment is the mean, a measure of central tendency; the second moment is the variance, a measure of variability of scores around the mean; the third moment represents skewness, a departure from symmetry; the fourth moment reflects the kurtosis, the deviation (in peakedness) from the normal curve. Mathematics achievement is broken down into five mathematical areas as in the SIMS study. The construct mathematics achievement is thus multi-dimensional. Researchers on gender differences in achievement in mathematical areas have always focused on the question, "Do the genders differ on one or more dependent variables (e.g. algebra, arithmetic etc.)"? This study asks the question, "What combination of mathematical area distinguishes the genders, and which mathematical area(s) contribute most to gender differences in mathematics"? Rephrasing the research question as indicated above is very important because apart from using group means the answer to the question involves considering the intercorrelations among the dependent variables. Since 9 various mathematical areas are interrelated, such a research question seems more appropriate. Finally, in analyzing SIMS data most researchers have assumed that the tests were equally appropriate for all students in the participating educational systems. Since educational systems have diverse cultures and curriculum emphasis, it may be possible for a country like Swaziland or Nigeria to have different contexts under which they form their opinion on what, for example, geometry or mathematics should be. This study uses confirmatory factor analysis (CFA) to determine if there was any justification to assess students in the two cultures involved in this study based on the mathematics structure hypothesized by SIMS. Purpose of the Study The main goal of this study is to explore the gender differences in mathematics achievement in two sub-Saharan cultures. The study compares the mathematics achievement of males and females in two sub-Saharan African countries: Nigeria & Swaziland. The study further determines if mathematics achievement of males and females is a function of educational system. Are gender differences in Nigeria different from Swaziland? The study also looks into the gender differences in various mathematical areas and finds out whether in these two sub-Saharan African countries, gender differences vary across mathematical area. Do males and females perform equally well in algebra, arithmetic, geometry, statistics, as well as measurement? Also gender differences in the variability of achievement in Nigeria and Swaziland are compared using skewness and kurtosis. 10 The last part of the study addresses the question of whether the mathematics structure defined by SIMS was valid in both educational systems. Is there any justification to talk about achievement of separate components of mathematics such as algebra in the two educational systems? Research Questions The study is guided by the following research questions: 1. Are there gender differences in mathematics achievement in sub-Saharan Africa (Nigeria & Swaziland)? 2. Are gender differences in mathematics achievement in Nigeria different from Swaziland? 3. Do gender differences in achievement depend on mathematical area? 4. Are gender differences in the variability of achievement scores the same in Nigeria and Swaziland? 5 Is there any justification in assessing students in Swaziland and Nigeria based on SIMS hypothesized mathematical structure? Limitations When considering results from this study, it is important to acknowledge difficulties with the execution of the study that bear upon the interpretation of results. These difficulties impose demand for caution about conclusions from the study. One prominent difficulty pertains to differences in sampling. In Swaziland, the intended national sample was not achieved, and instead a volunteer sample was used. In Nigeria, the target population was originally 11 intended to include students from all states; logistical and financial constraints caused this to be reduced to the ten southern states, which include 90% of the country's school enrollment; of these acceptable data were received from eight states. Another difficulty was in the definition of the target population. SIMS defined the target population as the grade in which the modal student age lies between 13.0 and 13.11 years by the middle of the school year. In Swaziland and Nigeria, the 13 year-olds were distributed across many grade levels. The target populations were therefore redefined as the grade in which 13 year-old children would have been found if they had started at the legal entry age and progressed through the grades year by year. This led to an inflation of the mean ages of the two countries. The mean ages of the sampled population in Swaziland and Nigeria were 15.7 and 16.7 years respectively. These difficulties as discussed above, complicate the process of drawing valid conclusions and must be considered in any attempt to generalize the findings of this study. Finally, it was beyond the scope of this thesis to use affective variables to explain gender differences in mathematics in the two educational systems. Although culture, as a potential causation was occasionally conjectured, the purpose was to provide a context to the study and also to postulate directions for further research rather than to furnish explanations for gender differences in mathematics. 12 CHAPTER II LITERATURE REVIEW Overview To begin, a brief historical background is given to provide the basis for understanding gender differences in mathematics in a contemporary context. This is followed by literature on gender differences in mathematics which is reviewed under the following headings; gender differences in the enrollment of mathematics courses, gender differences across mathematical areas, gender differences across educational systems, and gender differences in the distribution of mathematics achievement. Gender differences in enrollment is included in the literature review because according to Erickson, Erickson, and Haggerty (1980) enrollment in school mathematics courses could be considered a form of achievement in mathematics, along with scores on tests. Although, this study does not combine these two measures in this way, their mutual importance in the assessment of performance in mathematics is considered essential. While enrollment might be considered a less direct measure, it does indicate among other things, the level of school level prerequisites. Finally, the literature on possible reasons for these gender differences is reviewed. This information is important in providing a context for the study. Historical Perspective There had been a common belief that academic work was not suitable for females (Tyacks & Hansot, 1988), to the extent that many people believed 13 that normal schooling could damage the physical and mental health of females (Brooks, 1903; Gay, 1902; Tyack & Hansot, 1988). According to Ellis and Cordeau, (1988) women in Athens in the fourth and fifth centuries B.C. were regarded as inferior to men. Aristotle defined a female as a misconceived male. Women of the higher classes received some training in household arts and were sometimes given acquaintance with music and letters, but men controlled the world of politics, law and intellectual pursuit. The Pythagoreans were unique in proclaiming liberal ideas concerning women and, under the influence of Pythagoras, some women did devote themselves to the study of philosophy and mathematics. Pythagoras had women mathematics teachers in his school. Hypathia (370-415) of Alexandria was a co-author, with her father, of a treatise on Euclid (Ithurriague, 1931). Although women were discouraged from studying mathematics for centuries, some of them persevered. Sophie Germain (1776-1831) of France was a noted mathematician in spite of the fact that her parents thought the pursuit of such a subject was dangerous to her health. They even denied her heat and light and hid her clothing to prevent her from attending classes. The Scottish mathematician, Mary Fairfax-Sommerville (1780-1872) took advantage of lessons given to her brother by his tutor, but her parents also considered mathematics strange and improper for a girl and disapproved of her studying it. To avoid criticism, she had to hastily hide her notebook under her embroidery whenever company unexpectedly arrived (Lafortune, 1986). Arguments about the different needs of boys and girls arose when girls began studying mathematics in recent times. Kroll (1985) reported one American writer's view as: "Woman... is organized both bodily and mentally for dealing with an entirely different set of functions, in which mathematics 14 plays a small part" (p.8). In England, the Board of Education in 1899 claimed that it"... was prepared to argue that girls did in fact lack the same capacity for mathematics as boys" (Hunt, 1987, p. 13). Girls in elementary schools in Australia were thought to be suited for sewing and needlework, and boys to arithmetic studies (Zainu'ddin, 1975). At the secondary level in Australia only boys from rich families were thought to be suited to studying mathematics (Clements, 1979). There was controversy over the causes of gender differences in mathematics in the past. Morrison (1915) and Thomas (1915) attributed gender differences in mathematics to differences in the innate abilities of men and women. Others such as Armstrong (1910) and Dean, (1909) thought that gender differences in mathematics were caused by the differences in the interest and perceptions in future applications of mathematics for men and women. In the words of Ellis and Cordeau (1988): A little needle work to employ a dainty hand, and a little music to amuse a husband in an idle hour were all that the upper classes desired for their daughters. A knowledge of mathematics would not assist a young lady in her primary endeavor of attracting a suitable husband. (P-7) Thus, it appears that mathematics bears a long tradition of being considered not only inappropriate for girls, but somehow disadvantageous to them. Gender Differences in Mathematics Enrollment Although mathematics is now considered an appropriate subject for both boys and girls, boys tend to take more elective mathematics courses in high school than girls and this proportion is greater for more advanced 15 courses (Kerr, 1989). The British Columbia Mathematics Assessment of 1977 (Robitaille & Sherrill, 1977) reported that 64 percent of Grade 12 students with no mathematics beyond Grade 10 were females, and that only 43 percent of Grade 12 mathematics students were female. Similar results have been reported in the U.S. (Fennema & Sherman, 1977, 1978; Armstrong, 1980; Lee & Ware, 1986; Leder, 1986) and internationally (Husen, 1967). The third NAEP report in mathematics (NAEP, 1983) showed the male-female course differential in the U.S. was larger in favor of males in the advanced courses. Girls showed a consistent improvement in relative enrollment figures from 1978 to 1982 across courses in the U.S. In the U.K., figures for 1984-85 for England and Wales show female enrollment lagging considerably behind in mathematics, chemistry, physics, and computer science for both O-level and A-level (Department of Education and Science, 1985). However, the enrollment gap in secondary mathematics between boys and girls in developed countries seems to be decreasing (Kerr, 1989). Armstrong (1980) found only slight differences in participation. Chipman and Thomas (1985) in a review of participation of mathematics, reported that the difference is no longer as great as earlier research had reported. Literature on the enrollment of females in mathematics in Africa is almost non-existent. However, it is known that fewer girls than boys attend school in developing countries (Lee & Lockheed, 1990). According to Stromquist, (1989), in sub-Saharan Africa, 57.7% of females and 46% of males receive no schooling, 35% of females and 43.6% of males complete schooling, 3.3% of females and 6.2% of males complete secondary education, 0.2% of females and 0.7% of males complete tertiary education. This shows that, in sub-Saharan Africa, participation in schooling is highest in primary, lower in 16 secondary, and lowest in tertiary. However, girls have lower participation at all levels of the education ladder. Stromquist (1989) has argued that factors that account for the diminishing participation of women as they move up the educational ladder are predominantly those related to home and culture. These include family decisions to remove girls from schools particularly in traditional societies that question the effect of education on female roles and behaviors (Anderson & Browman, 1982). A study, based on a survey of all female students in a northern Nigerian university (Beckett & O'Connell, 1976), found that 66% were Christians while 32% were Muslims. The same study found that Muslim girls reported that their parents were "afraid that education would spoil them" and that as many as 90% of the girls reported pressure to leave school and marry young. In Swaziland, there is greater female than male participation in secondary education (Stromquist, 1989). This may be attributed to two main reasons. The first is bride wealth (i.e., the groom pays a price for the bride). Robertson's (1985) study of 44 African countries, including Swaziland, found that, in countries where there is bride wealth, the girls reached parity with boys in educational enrollment. In Swaziland, the bride wealth depends on the level of education of the bride. Parents are therefore encouraged to send their daughters to school in order to demand a very high 'Lobola' (bride wealth) from the groom. Another reason may be because men are either given responsibilities for cattle herding (the main occupation in Swaziland) or leave their country for work in South Africa. 17 Gender Differences Across Mathematical Areas Most researchers treat mathematics as a single factor and therefore miss some important features of gender differences (Willms & Jacobson, 1990). For example, among 17-year-old students, Fennema and Carpenter (1981) found that, the male superiority in achievement increased as the difficulty level of mathematics increased, even when the number of courses taken was held constant. Various research studies also found differences in favor of females in tests that required computational skills, and in favor of males in tests of mathematical concepts and applications (Brandon, Newton, & Hammond, 1987; Johnson, 1987; Martin & Hoover, 1987; Sabers et al., 1987; Shuard, 1986; Ward, 1979). It is, therefore, important to look at gender differences in various mathematical areas. For example, in California public schools, an assessment test administered to all pupils in Grade 6 indicated that, in the arithmetic tests, girls outnumbered boys in using irrelevant rules, choosing an incorrect operation, and making mistakes in negative transfer. On the other hand, boys tended to make more repeated errors and formula interference (Marshall, 1983). In Britain, Shuard (1986) examined data from the Assessment of Performance Unit (APU) between 1980 and 1982 when 11 year old students from England and Wales were tested for their mathematics achievement. She found that girls outperformed boys in computation of whole numbers and decimals. On the other hand, boys performed better in other arithmetic areas such as concepts of whole numbers, decimals and fractions, and applications of numbers. 18 In elementary geometry, Schultz and Austin (1983) investigated the effects of transformation types and directions on pupils' understanding of transformations. They found no significant differences between boys and girls in grades 1, 3, and 5. In grade 4, there were no gender-related differences in graph comprehension (Curio, 1987). Shuard (1986), however, reported a superiority of boys over girls in measurement and geometry in grade 5. In secondary algebra, girls performed equally well as boys and, in the achievement of basic algebra they outperformed boys (Swafford, 1978). In terms of problem solving in algebra, gender differences in grade 7 were "either very few or non existent" (Szetela & Super, 1987). However, Linn and Pulos (1983) investigated the proportional reasoning of boys and girls in grades 7, 9, and 11 and found that boys tended to be more successful than girls in proportional reasoning. Also, the computational estimation and related mathematical skills were reported to favor boys in grade 8 (Rubinstein, (1985). In secondary geometry, Curio (1987) found no gender-related differences in graph comprehension in grade 7. In terms of logical reasoning and use of problem-solving strategies in geometry, Battista (1990) did not find any gender differences in spatial visualization, geometry achievement, and geometric problem-solving tasks. Lewis and Hoover's report (1986) on the standardization of the Iowa Test of Basic Skills (ITBS) showed that boys outperformed girls on the problem-solving subtest in the second grade, but this difference decreased in the fifth grade and disappeared by the eighth grade. Throughout the high school years, differences favoring males are common (e.g., Jones, Burton, & Davenport, 1984; Ramist & Arbeiter, 1986). They are apparent in problem-solving tasks and application (Carpenter, Lindquist, Matthews, & Silver, 1983; Swafford, 1980). 19 Using SIMS data Robitaille (1989) found some patterns of gender differences in mathematics. His analyses showed that girls are more likely to achieve better than boys in computational-level arithmetic, whole numbers, estimation and approximation, and in algebra; while boys tend to be better in geometry and measurement subtests, and in proportional thinking. According to Robitaille, "the most marked gender differences appear to lie in the domains of transformational geometry, proportional thinking, and standard unit of measurement, all which favor boys in a majority of systems" (p. 121-122). Hanna (1990) also used SIMS data to examine gender differences in five mathematical areas; arithmetic, algebra, geometry, statistics, and measurement. She used the table below to summarize her results. Table 2.1 Achievement scores on subtests Mathematical Area Arithmetic Algebra Geometry Statistics Measurement Mean (%) 50 45 42 55 49 Girls S.D. 15 13 15 17 20 Mean(%) 50 45 44 55 51 Boys S.D. 13 12 15 16 18 The above table indicates that in the subtests of arithmetic, algebra, and statistics, girls and boys did not differ in their average marks. However, in the two other subtests, geometry and measurement, the average marks for boys were slightly higher. A further statistical analyses showed that in the case of measurement and geometry, the gender differences were greater than could be expected by chance. The level of significance used was 0.01. Gender Differences Across Educational Systems Only a few international studies have addressed gender differences across educational systems. A study by Walbery, Harnisch, and Tsai (1986) examined gender differences in mathematics achievement in twelve educational systems. They reported a significant superiority of boys over girls in eight systems and no significant gender differences in the other four. Between 1966 and 1973, the IEA conducted a First International Mathematics Study (FIMS) involving 12 educational systems. FIMS data analyses showed clear differences in favor of boys in all populations in both verbal and computational scores (Husen, 1967). However, gender differences within educational systems were found not to be significant. This led Husen to suspect that gender differences were a system phenomenon. Comber and Keeves (1973) also using FIMS data confirmed Husen's results. SIMS provided another opportunity to examine gender differences. This involved 20 different countries. Hanna and Kundiger (1986) analyzed the SIMS data and state: Most of the differences did not reach statistical significance at the 1% level. Moreover, the differences that did reach statistical significance were not large, ranging from +5 to -7. Looking at each subtest separately it appears that for two of the five topics, Measurement and 21 Geometry, the significant differences occurred consistently in the boys' favor: in 7 of 20 countries boys had higher p-value and in 10 countries boys had higher p-value in Measurement and Geometry, respectively, (p. 6-7) Robitaille (1989) also examined SIMS data and concluded that most educational systems showed that males were more likely to score higher than females in geometry and measurement. Robitaille (1989) expressed gender differences in terms of differences in percentage points, "a difference of three percentage points was chosen as a threshold level of practical significance"(p. 120). He further added that in algebra and descriptive statistics, all significant gender differences appeared to favor girls. Hanna (1990) analyzed the SIMS data using MANOVA with a follow up univariate analysis and came out with some interesting findings. There were no significant differences between boys and girls in their achievement in algebra, arithmetic, and statistic. However, on everyone of the subtests, there were statistically significant differences among educational systems. The educational system-by-sex interaction was also significant for all five subtests and thus indicating that gender differences are not consistent among educational systems. Her study also showed that for each of the subtests, at least half of the educational systems showed no statistical superiority of one gender over the other. On each subtest, more educational systems had statistically higher mean scores for boys than for girls. In no educational system did girls exceed boys in either measurement or geometry. 22 Gender Differences in the Variability of Achievement The findings that there is little distinction between the genders in the elementary school mathematical performance but significant differences in favor of males in high school has been alluded to earlier. There exists another important feature of gender related differences in achievement; gender differences in the variability of achievement. Quite a few studies in this area show the male domination at the higher end of the distribution (Fox, Brody & Tobin, 1980; Benbow & Stanley, 1980; Kerr, 1989). Benbow and Stanley (1989) reported that the number of boys who achieved above 660 on the mathematics achievement test of the Scholastic Aptitude Test (SAT) was fourteen times that of girls. On the other hand Bentzen's (1966) study of the disabled students who had suffered mild mental handicaps, learning disabilities and other handicaps which could cause different kinds of learning difficulties showed that the ratio of the number of boys to girls was seven to one. This indicates that there are many more boys than girls located on the bottom line of the achievement distribution for disabled students. It therefore appears that male students dominate both the upper and the lower ends of the achievement distribution. Research on the spread of the achievement distribution between male and female students indicates a larger degree of spread or variability of achievement scores for boys (Hieronymus, Lindquist, & Hoover, 1982; Martin & Hoover, 1987; Feingold, 1992). Examining the achievement scores in the ITBS, Martin and Hoover (1987) reported a greater variability of male scores on every subtest and in every grade from three to eight. Hanna's (1989) findings on SIMS data do not seem to support the above pattern. Her results as summarized in Table 2.1 (see p.19) indicate that 23 variability among girls was slightly greater than or equal to the variability among boys on all subtests. To address this contradiction, research on gender differences, like the present study may have to take into account gender differences in variability based on variance, skewness, and kurtosis together with gender differences in central tendency. It is incomplete to describe the variability of population distribution with only variance especially when the distribution may not be normal . Potential Causation: Why gender differences in mathematics? The search for factors accounting for gender differences in mathematics performance has covered many areas of research from physiology to educational practices (Badger, 1981). Many theories put forward are based broadly either on the idea that such differences are innate or they are socially and culturally learned. Whether the differences are innate or learned or the result of a combination of factors remains inconclusive. In this section, some of the major findings are briefly outlined. Biological and Genetic factors There has been an attempt to explain gender differences in mathematics achievement using biological variables. The biological argument is often used by Benbow and Stanley as their explanation for their findings on gifted junior high school students (e.g., Benbow & Stanley, 1982). Their argument is that socialization cannot explain the differences found in junior high school. According to Benbow and Stanley since 24 differential coursework hasn't taken place and gender-role socialization is just beginning to operate in junior high school, the two factors could not be used to explain the large gender differences found in their study. This explanation has been challenged (e.g., Eccles & Jacobs, 1986). Sherman (1977) has examined several suggested biological causes of mathematical ability and gender differences, including hereditary differences linked to the X-chromosome, gender differences in serum uric acid, hormone differences, and difference in brain lateralization and cerebral organization and concluded that only gender differences in brain lateralization had reasonable support. Peterson (1980) also advanced hormone differences as a potential cause of gender differences in achievement. Carrying out an experimental research in both these area is extremely difficult (Sheperd-Look, 1982; Bryden, 1979). The results to date are inconclusive. One group of theorists (Levi-Agresti & Sperry, 1968) has maintained that male superior spatial ability stems from a lesser degree of specialization (For reviews see McGlone, 1980). Sherman (1967) made the point that there is experiential basis for using spatial ability to explain gender differences. Later investigations however, indicate that spatial visualization does not account for differences in mathematical performance (Armstrong,1981; Fennema & Sherman, 1977; Fennema & Tartre, 1985; Pattison & Grieve, 1984). Most research and theory today does not favor a strictly biological explanation of cognitive differences (Shephered-Look, 1982; Bryden, 1979). "Knowledge of basic biological mechanisms has less and less predictive value about the behaviors of animals the more complicated, flexible, and unpredictable the animals' behaviors are and the more complex and heterogeneous their environment and culture are" (Bleier, 1984, p. 106). While gender-related differences in cognitive functioning may turn out to 25 have some biological component (more research is needed), such influences, according to Jacklin (1983) would not limit any cognitive ability to one gender or another and would produce small differences at most. Psychological and social factors are now favored in the literature as explanations of gender differences in mathematics (Linn & Peterson, 1985; Peterson, 1979; Fox, Tobin, & Brody, 1979). Psychological Factor Psychological factors include such variables as feelings, beliefs, and attitudes. The list of variables is long and is further complicated as the possible interactions are, in practical terms, limitless over time and age. Confidence in learning mathematics and achievement seem to be correlated. In grades 6 through 11, Fennema and Sherman (1978) found that, males were consistently more confident than females at equal achievement levels. For females, this can lead to a cycle of failure, with reduced expectations and achievement following reduced effort. Woleat, Pedro, Becker and Fennema (1980) found that females at all achievement levels more strongly attributed failure to lack of ability, and success to effort or circumstance. For males, the situation is reversed; i.e., they tend to attribute success to ability and failure to lack of effort or circumstance. It therefore appears that females feel less in control of their mathematical learning and are less apt to persist. Although confidence seems to be more highly related to achievement than any other affective variable (Fennema, 1984), there is nothing in the literature about the effect of confidence or how such feelings are developed. There is also a significant positive correlation between the perceived usefulness of mathematics for future goal attainment and mathematics achievement (Fennema & Sherman, 1977, 1978; Meece, Parsons, Kaczala, Goff, & Futterman, 1982). Studies indicate that high school girls generally tend to consider mathematics as less essential to their plans and possibilities than boys (Fox, 1977; Fennema & Sherman, 1977, 1978). Fox (1977) also looked at students' general enjoyment of mathematics and found that gender differences were small, with a low but significant correlation with achievement. Girls also tend to rate mathematics as more difficult than boys (Eccles-Parsons, Adler, Futterman, Goff, Kaczala, Meece, & Midgley, 1983). However, since students may enjoy subjects in which they perform well or vice versa, no definite conclusion can be drawn about the direction of causation for both enjoyment and perceived difficulty of mathematics. Another area where the relationship with achievement is not clearly understood is sex-typing of mathematics. Mathematics, when stereo-typed, is seen as a male achievement domain by both male and female students. Hilton and Berglund (1974) in their study found that males regarded mathematics as more useful and more interesting than females. Fennema and Sherman (1977) also studied sex-role socialization as manifested by students' assessment of the usefulness of mathematics and the degree of agreement that mathematics is male domain. The females in their study did not agree that mathematics is a male domain. However, the females were less convinced than the males that mathematics would be useful to them personally. Females surprisingly do not always characterize participation or competence in mathematics as unfeminine (Eccles-Parsons, 1984). It seems that mathematics perceived as a male domain might not be important as a predictor of later achievement (Meyer & Fennema, 1986). Cultural and Societal Influences The entire social milieu of schooling influences how well one learns or feels about mathematics. The most important influences seem to occur within the classroom where mathematics is taught. Learning environments for boys and girls within classrooms, while appearing to be the same, differ a great deal. Fennema (1984) stated categorically that the "causation of sex-related differences in mathematics rests within the school" (p. 161). Often, instead of altering traditional gender stereotypes, schooling tends to reinforce the sex-role lessons of infancy and early childhood (Frazier & Sadker, 1973). In theory, most teachers believe education should be a liberating and democratic influence (Skolnick, Langbort, & Day, 1982), but in practice mathematics teachers are still seen to reinforce traditional behavior and occupational plans for both boys and girls independent of where student interest or talents may lie (Eccles & Hoffman, 1984), and at times even to actively discourage nontraditional (mathematical) female interests (Fox, Brody, & Tobin, 1980). Mathematics teachers also tend to interact more with boys (Sadker & Sadker, 1979). Boys receive more criticism for behavior and more praise for work and behavior (Fennema, 1984). Boys' behavior captures adult attention and implicitly protects the male intellectual ego (Skolnick, Langbort, & Day, 1982). While boys are usually praised for intellectual works, girls are mostly praised for behaving properly and obeying rules. Teacher praise for compliant behavior may pressure girls to adhere to this role (Skolnick, Langbort, & Day, 1982) and forgo attempts to achieve recognition for their intellectual work. Positive interactions which build self-confidence and mathematical interest are especially favorable to boys among high ability students (Brophy, 1985). Differential treatment may be due to differential expectations but the literature is inconclusive here. One study found no differences in teacher expectations of mathematics success (Parsons, Heller, Meece & Kaczala, 1979) while others have shown greater expectations for boys (Ernest, 1976; Levine, 1976). Higher expectations for males result in teachers helping girls but encouraging boys to persist on their own, calling on boys more often for high-level questions (Fennema, 1984), and different teacher 'wait-times' (the pause for student response) which could affect achievement (Fennema & Sherman, 1978). Reyes (1982) found variation in the behavior of grade 12 teachers. Over all the effect of student-teacher interactions on mathematics achievement remains unclear (Erickson, Erickson, & Haggerty, 1980). Adolescence is the time when the clash between principles and realities, between achievement and social fulfillment, makes its first impact on the lives of young women (Skolnick, Langbort, & Day, 1982). Peers are important "socializers" in a students' life. Girls may experience intense peer pressure against achievement in mathematics in high school (Nash, 1979; Fox, Tobin, & Brody, 1979; Fennema & Sherman, 1977; Ernest, 1976). Gifted girls are often judged less favorably by peers than gifted boys (Solano, 1976). Textbooks, test design, school materials, and the media have all been shown to reinforce traditional sex-role stereotypes (Delgaty, Getty, lies, & Pointon, 1980; Sadker & Sadker, 1979; Tobias, 1978; Kepner & Koehn, 1977; Saflios-Rothschild, 1979). Long term impact of these factors on gender-related differences in mathematics would appear likely but these effects are difficult to measure in the short term (Fox, Tobin, & Brody, 1979). Finally, parents, family, and other adults may influence student achievement and participation in mathematics; by their own attitudes and behaviors towards mathematics, their expectations and goals for their 29 children, and the activities they provide or encourage. Several studies have discovered gender differences in these areas (see Eccles-Parsons, 1984; Eccles et al., 1986) but few have assessed the causal impact of these socialization experiences. Encouragement from parents appears to influence girls' decision to continue in advanced mathematics (Armstrong, 1980; Fennema & Sherman, 1977, 1978; Parsons, Adler, & Kaczala, 1982). Also students' beliefs about their parents' perception of them as mathematics learners can be a predictor of enrollment intentions (Fennema & Sherman, 1977). In general, however, the direction of causality with the parent variable is still open to debate; for example, expectations may arise from previous achievement or vice versa. Conclusion It appears that, in the words of Sharman & Meighan (1980) "the dominant cause of sex differences in mathematics attainment may not be that of potential or capability, but that of sex-stereotyping". These observations are supported by Samuel's (1983) experiment where the mathematics performance of girls in all girls school was compared with that of both boys and girls in an equivalent mixed set taught by the same teacher. By the middle of the second year, the girls in the all girls set were still scoring on tests nearly as high as the boys, but the scores of the girls in the mixed set had dropped behind those of the boys by an average of more than 20%. Boys continually monopolized the teacher's time in many ways in the mixed set while the girls in the all-girls set received much more individual attention. Although such studies can be criticized for the undue influence of the teachers on the results, some other findings seem to support it. For example, the findings of a study by Lee & Lockheed (1990) indicate that "single-sex schools affect Nigerian girls positively in both increasing mathematics achievement and in engendering less stereotypic views of mathematics" (p. 227). Summary The literature on gender differences in mathematics achievement seems to indicate that the gender gap is in favor of males. Boys score slightly higher than girls on average in mathematics performance but are largely over-represented among high scorers. Males mathematics test scores have greater variability and mean. This is based on research done in developed countries. Most of these studies report the mean and standard deviation and therefore assume normality of population distributions. However, population distributions are not always normal. It is therefore important to look at the whole population distributions in order to determine the statistical reasons for gender differences. This study compares the population distributions of males and females in secondary schools in two cultures in sub-Saharan Africa with the view to determine the nature of gender differences in mathematics achievement in Africa. 31 CHAPTER III. METHODOLOGY This chapter outlines the methodology used to collect, analyze, and interpret data for the purpose of answering the research questions. But first information is provided about the problem setting: important characteristics of the educational systems in Swaziland and Nigeria. The appropriateness and significance of the statistical procedures used in the study are also discussed. The last section of the chapter deals with the generalizability of the findings of the thesis. Settings Swaziland The Kingdom of Swaziland is a landlocked country. It lies between the Republic of South Africa and Mozambique. Swaziland covers an area of about 17 368 km 2 and has a population of about 520 000 (29.9/km2). It is among the smallest countries in Africa. Since 1973, Swaziland's educational system has expanded rapidly, so that as of 1983, about 130 000 students, or 111% of the primary school age population, were in school (World Bank, 1987). This means that there is almost universal primary education but the actual school age distribution goes beyond the official school age population. Enrollment in secondary education, at 29 000 was equivalent to 43% of the relevant aged population in 1983. Participation of male and female students at all levels was approximately equal. The formal education system in Swaziland consists of seven years of primary education, three years of lower secondary school, two years of upper secondary education and two to five years of higher education. The IEA SIMS sample involved 25 mathematics teachers in secondary schools and their 856 Form Two students. The population in Swaziland included all students in Form Two, the grade level in which 13-year-old students would be found if they had entered school at age 5 and proceeded through without repeating a class. Form Two was also the grade level for which the IEA mathematics test was judged most appropriate for the curriculum. One Form Two class from each school was selected at random to be tested. Nigeria Nigeria, a federation of 19 states, is one of the largest countries in Africa, with an area of 923 800 km^ and an estimated population of over 90 million (97.4/km2). The education system is commensurably large, with approximately 15 million primary students and 3.5 million secondary students enrolled in 1983. It is estimated that in 1982, 97% of the primary age group and 28% of the secondary age group were enrolled in school. Discrepancies between male and female secondary school enrollment rates are apparent, however, with only 14% of the 12-17-year-old female age group enrolled, compared to 42% of the same age males. Female students represented 43% of primary and 26% of secondary students (UNESCO, 1986). Until 1976, the formal education system consisted of nursery preschool institutions, primary schools, secondary educational institutions of different kinds of duration, and a variety of different higher education institutions. Primary education was of six to seven years of duration, with entry age being five or six. Basic secondary education lasted for five years. The National 33 Policy on Education adopted in 1976 introduce a uniform six-year primary education and three year upper secondary program. As these data were collected in 1980-81 academic year, students in Form Three would have attended school under both old and new plans. The IEA SIMS sample comprised 41 mathematics teachers and their 1073 Form Three students in state-owned Secondary Grammar Schools which prepare students for the West Africa School Certificate Examination. The sample was derived from a three-stage stratified random sample. The primary sampling units were the ten southern states in Nigeria. Within each state, a proportional random sample of schools was selected. At the second stage, a random sample of one class per school was selected, and at the final stage, 30 students were randomly selected in each class. Data Source The data for this thesis were from SIMS. In the school year of 1980-81, the IEA conducted a research study in 20 countries including two sub-Saharan African nations: Nigeria and Swaziland. Its mathematics achievement test were administered to more than 125 000 students in these 20 countries. The mathematics test used as the dependent variable in this study was the 40-item SIMS "core" test, which contained items covering five curriculum areas (arithmetic, algebra, geometry, statistics and measurement). The SIMS mathematics achievement test had to be a compromise to try to match all the national curricula and part of the IEA survey assesses that match. McLean, Wolfe, and Wahlstrom (1986, p.16) note that "How well the SIMS item pool matched a system's intended curriculum was measured by calculating the percentage of items in each topic subset that educators said were either highly appropriate or acceptable to that system." The reliability and validity of the SIMS mathematics achievement test as described by Garden and Robitaille (1989) were very high. Complex criteria were used to pilot and select final items for the test. SIMS also provided a high quality of sampling. According to Travers, Garden and Rosier (1989), sampling procedures for the second study were designed with dual purposes: to allow population parameters to be estimated to a reasonable degree of precision, and to enable cross-national comparisons of these population parameters, given sufficient background information, to be made. (p.11) The sampling design was: (1) stratification based on groupings seen by each National Center as having some significance for education in their system; (2) random selection of schools with probability proportional to size of the target group within each school; (3) random selection of two classes within each school at the target grade level, (p. 11) Design of the Study This study examined gender differences within and across two educational systems (Nigeria and Swaziland). A 2x2 (educational system and gender) multivariate analysis of variance (MANOVA) was designed to analyze achievement scores of students for the five mathematical areas (algebra, arithmetic, geometry, statistics and measurement). Achievement in each mathematical area was considered as a dependent variable and educational system and gender as the independent variables. The outcome of the MANOVA gender main effect answered the research question one while the gender by educational system interaction effect answered question two. The MANOVA was followed by discriminant functional analysis with the standardized discriminant function weights providing an indication of the relative importance of the dependent variables ( achievement in each of algebra, arithmetic, geometry, statistics and measurement). This answered research question three. To answer research question four tests for skewness and kurtosis of achievement distributions of both males and females in each of the educational systems for each mathematical area were conducted (the SPSS software program was used for this purpose). Research question five was answered by performing a confirmatory factor analysis using the DOS-LISREL program by Joreskog and Sorbom (1989). Appropriateness and Significance of Statistical Analysis Cohen (1965) writes, "Statistical analysis is a tool, not a ritualistic religion. It is for use not reverence, and it should be approached in the spirit that it was made for psychologists rather than vice versa" (p. 95). This means that, researchers have available powerful statistical tools but the use of appropriate statistical techniques to analyze and interpret research questions is very important. MANOVA/Discriminant Functional Analysis The multivariate analysis of variance (MANOVA) and a follow up discriminant functional analysis were selected to answer some of the research 36 questions in this study. According to Gnanadesikan and Kettenring (1984), the objective of using the multivariate analysis is to increase the "sensivitity of the analysis through the exploitation of the intercorrelations among the response variables so that indications that may not be noticeable in separate univariate analysis stand out more clearly in the multivariate analysis "(p. 323). "Indication" here meaning the relative importance of variables. The utilization of multivariate statistical procedures also reduces what Schutz & Gessaroli (1993) called "probability pyramiding". Type 1 error is controlled by such method. Schutz & Gessaroli (1993) also notes: If, ... the primary research question is of the form, "What combination of these dependent variables distinguishes these groups, and which variables contribute most to the between-group variance?" then a preliminary MANOVA is appropriate. However the follow up analyses in this case should be a discriminant analysis, with the F-to-remove or the standardized discriminant function weights providing an indication of the relative importance of the dependent variables, (p. 904) In a two-group discrimination problem as in this study, the discriminant functional analysis uses a single linear composite of the predictor variables that can distinguish between the groups. The single linear discriminant function is attained by maximizing the ratio of between-group to within-group variance (F-ratio) and also produce the smallest misclassification error rates possible. Basically, the linear composite results in a new axis along which the groups are maximally separated. Effect Size and Educational Significance Power and sample size are directly related. The greater the sample size, the higher the power. Because of the large sample size used in this study very small effects were expected to be statistically significant at 0.05 level. Tests of statistical significance were therefore inadequate for making inferences about practical significance of research results. Among other ways of getting at practical significance, the effect size measures were chosen because of its independence on sample size. The effect size according to Glass and Hopkins (1984) is simply the difference between two means in standard deviation units. Skewness and Kurtosis One major purpose of this research is to compare the distribution of mathematics achievement test scores of male and female students. This is done by comparing the variability and mean scores of the genders. Researchers have traditionally been less interested in differences between variabilities than in differences between means (Games, Keselman, & Clinch, 1979). Researchers on gender differences implicitly assume population homogeneity of variance, as evidenced by the use of inferential statistics (t and F ratios) that assume homogeneity to test the significance of male and female differences in means (Feingold, 1992). Studies done in developed countries where primary education is generally universal and secondary education participation is quantitatively equal for male and female students indicate a larger degree of variability of achievement scores for boys (Feingold, 1992; Hieronymus, Linquist, & Hoover, 1982; Martin & Hoover, 1987). This suggests that the statistical reasons of the achievement difference cannot be attributed to gender differences in central tendency alone, as gender difference in variability, or gender differences in both central tendency and variability could be potential factors for the differences. This means therefore that gender differences in central tendency and gender differences in variability have to be considered together to form a correct decision about the magnitude of gender differences in mathematics achievement. The central tendency and variance are generally the only statistics presented in most educational research reports involving gender differences. This is because, in most of these studies, the underlying assumption is that the population distribution is normal. However, it is often the case that, there is some degree of asymmetry(skewness) and peakedness(kurtosis) in the population distribution curve (Glass & Stanley, 1970). This means that the statistical indices related to the first four moments are always important in describing any distribution. The first moment is the mean; the second moment is the variance , the third moment is the skewness (departure from symmetry); the fourth moment reflects the kurtosis which represents deviation from the normal curve as a result of concentration of scores at either the center (peakedness) or the extreme ends of the distribution According to Hopkins and Weeks (1990): The mean has limited descriptive value when distributions are highly skewed (e.g. 75% or more of cases may be above or below the "average") It is only when a distribution is normally distributed that the conventionally reported mean and standard deviation provide accurate descriptive information about the distribution, (p. 718) Researchers have argued that non-normality rarely has any serious practical consequences on the accuracy of the probability statements regarding population means when conventional significance tests are used (Hopkins & Weeks, 1990). As a result of the robustness of ANOVA and t-test to non-normality, research on mean differences has ignored the third (skewness) and fourth (kurtosis) moments of the distribution (Hopkins & Week, 1990; Newell & Hancock, 1984). Newell and Hancock (1984) argue that, "reliance on only the mean and standard deviation of a distribution...may lead to erroneous inferences concerning such distribution when skewness and kurtosis are ignored" (p.320). Their conclusion is consistent with other research of the effect of non-normality on Type 1 and Type 2 errors for inferences about means (Box, 1953; Glass et al, 1972; Hopkins & Durand, 1989). Measures of skewness and kurtosis are therefore informative for both descriptive and inferential purposes. Confirmatory Factor Analysis (CFA) The main objective of confirmatory factor analysis is to assess the degree to which the plausibility of factor models is empirically confirmed (Kim & Muller, 1978). This is done by introducing a specific hypothesis about the factor structure. It is then assumed that if some factorial causation is in operation, then there is the likelihood that such a specific hypothesis will be supported by a given covariance structure. CFA therefore provides self-validating information. It also helps to assess the appropriateness of a factor analytic model for given data. In the words of Kim and Mueller, (1978), "If a given factorial hypothesis is supported by the data, we will in general also have greater confidence in the appropriateness of the factor analytic model for the given data." This study uses CFA to find out whether the SIMS hypothesized mathematics structure is supported by the Swaziland and Nigeria data and thus check the appropriateness of using the particular items specified in SIMS to represent the five mathematical constructs. SIMS used 40 test-items to define the five mathematical constructs; Arithmetic, Algebra, Geometry, Statistics and Measurement. Table 3.1 shows the number of items and the particular test-items used to identify each mathematical construct. All test-items were of multiple choice format (5 options to each question). Each correct option to a test-item was awarded one mark and a wrong one, zero. The raw scores were used for all the analysis. Table 3.1 Distribution of test items on the five mathematical constructs. Total Mathematical test Test items constructs items Arithmetic Algebra Geometry Statistics Measurement 11 9 11 4 5 Q3, Q14, Q17, Q18, Q20, Q23, Q26, Q31, Q33, Q34, Q35 Ql , Q4, Q10, Q12, Q16, Q22, Q25, Q39, Q40 Q2, Q5, Q6, Q9, Q13, Q19, Q28, Q29, Q32, Q36, Q38 Q7,Q15,Q21,Q27 Q8,Q11,Q24,Q30,Q37 Population and Generalizability of the Study Because of the problems in sampling as discussed in chapter 1, the findings of this study can only be statistically generalized to the Form Three student population in southern Nigeria and Form Two students population 41 in the 25 voluntary schools that participated in Swaziland. However, the findings of this study may be generalized on a logical basis to the entire Form Three student population in Nigeria and the Form Two student population in Swaziland. In Nigeria, the sample population constitutes about 90% of the entire Form Three students population in Nigeria. Although, the Nigeria population is quite heterogeneous, different ethnic groups usually live together as a community. The structure of the school system is quite uniform throughout the country. One would therefore not expect the remaining 10% of the Form Three students population to be completely different from the 90%. Since gaining independence from Britain in 1968, Swazis have been ruled by the monarchy. "At the head is a hereditary king, titled by his people Ngwenyama (Lion) and a queen mother, Ndlovukazi (lady elephant)" (Kuper, 1986. p. 3). The people of Swaziland hold on to their traditional beliefs and owe allegiance to their king and queen. They are a very homogeneous set of people belonging to one ethnic group that speak the same language. "Close" families live in homes that are clustered together into homesteads surrounded by farmlands. These homesteads form communities and schools are usually built for such communities. Students from such schools are therefore closely related in many respects. Although all the schools in Swaziland are controlled by the Swazi Ministry of Education, most of the schools are administered by the churches. Despite the church's influences, cultural values are still adhered to in schools. According to Kuper (1986), one of the kings of Swaziland (King Sobhuza) has argued that: 42 education should not be secularized but it should also not be confused with Christianity. If education is developed apart from religion, it will undermine the unity of the sacred kingdom. The worst allegation against communism was that it was against religion, (p. 144) The Swazi education is based on promoting the Swazi way of life and the Christian way of life. There is not much difference in the lives of students in school and outside school. The schools, apart from infrastructural differences in a few cases, are quite similar in many respects. The findings in this study may therefore not be significantly different from the other schools that did not take part in the study. CHAPTER IV RESULTS AND DISCUSSION This chapter reports the results of statistical analysis conducted, but first a check of violations of statistical assumptions are briefly addressed. Summary of results of data analyses are presented by means of tables and figures. Interpretations of the tables and figures are also provided. For brevity on figures and tables, some words are abbreviated; Arith for Arithmetic, Alge for Algebra, Geom for Geometry, Stats for Statistics, Measure or Meas for Measurement, M for Male, F for Female and Swazi for Swaziland. Statistical Assumptions As a preliminary check for violations, sample variances for each of the five dependent variables in each of the four independent variable groups were examined. For all dependent variables the difference in variance between male and female was not large. The ratio was about 1:1 (see Fig 4.1). Sample sizes were quite discrepant, with a ratio of almost 2.6:1 for male to female groups in Nigeria. However, with the minimal differences in variance and the use of a two tailed test, the discrepancy in sample sizes does not invalidate the use of MANOVA. 7 T Swaziland(M) • Swaziland(F) D Nigeria(M) Nigeria(F) Arith Algebra Geometry Statistics Measure Fig 4.1 A comparison of variance in various mathematical areas by gender in Swaziland and Nigeria. MANOVA Randomized Groups Design The primary goal of this analysis was to compare, by gender, achievement in Arithmetic, Algebra, Geometry, Statistics and Measurement in two sub-Saharan African countries (Swaziland and Nigeria). This was done by carrying out a 2 (Gender) by 2 (country) factorial MANOVA using SPSS program. The descriptive statistics for the five mathematical areas and the MANOVA results are presented in tables 4.1 and 4.2 respectively. Table 4.1 Descriptive Statistics for the five mathematical areas. (S.D in brackets) Test Swaziland Nigeria Total (items) Male female Total Male Female Total Male Female n=378 n=434 n=997 n=386 Arith (11) Alge (9) Geom (11) Stats (4) Meas (5) 3.46 (1.88) 2.74 (1.66) 3.76 (1.88) 2.24 (1.08) 1.65 (1.10) 3.28 (1.95) 2.59 (1.74) 3.35 (1.86) 2.24 (1.10) 1.50 (1.01) 3.36 2.66 3.54 2.24 1.57 4.77 (2.52) 3.27 (1.71) 3.07 (1.82) 2.23 (1.16) 1.54 (0.99) 4.66 (2.37) 3.16 (1.60) 2.99 (1.89) 2.09 (1.21) 1.43 (0.89) 4.74 3.24 3.05 2.19 1.51 4.41 3.12 3.26 2.23 1.57 3.93 2.86 3.18 2.17 1.47 Table 4.2 MANOVA table. Effects Gender (G) Country (C) G x C df 1 1 1 Multivariate F 2.442 71.144 1.838 P 0.032* <0.001* 0.102 alpha was set at 0.05 level for all multivariate F's. The MANOVA table (Table 4.2) shows that there was no significant (p= 0.102 ) Gender by Country interaction effect and none of the univariates were significant (although a follow-up univariate analysis indicated that Geometry approached statistical significance with F = 3.64 and p = 0.056). This indicates that the difference in achievement of each or combinations of the five mathematical areas between males and females was constant for each country (see Fig 4.2). The MANOVA table also shows that there was significant multivariate F-ratio for the Gender main effect (p = 0.032) suggesting that males and females differ in their achievement in one or more mathematical areas. Table 4.1 indicates that female mean scores were less than that for males in all mathematical areas. The magnitude of the achievement difference in each mathematical area (see Table 4.1) while small in some cases was quite substantial in others. The country main effect was also significant (p<0.001) indicating that there is achievement difference between the two countries. The two countries differ on one or more of the five mathematical constructs. Table 4.1 shows that in Arithmetic and Algebra Nigeria mean scores were higher while Swaziland mean scores were higher in Geometry, Statistics and Measurement. I Swaziland(M) Arith Algebra Geometry Statistics Measure Fig 4.2 An illustration of mean scores. Discriminant Functional Analysis Although there was no significant interaction effect, the significant gender main effect and the strong country main effect suggested that there was a potential difference in terms of mathematical construct(s) contributing to gender differences in each country. Discriminant function analysis was therefore conducted using SPSS statistical software program to determine which of the mathematical area(s) contributed most to gender differences in achievement in each country. Using the stepwise procedure, it was possible to identify the relative importance of selected variables in discriminating between the genders in each country. Table 4.3 Results of Discriminant Function Analysis on Nigeria Data. F-To-Enter in Each Step p Standardized Coefficients 0 1 2 Arithmetic 0.593 Algebra 1.161 0.040 Geometry 0.485 0.129 0.297 Statistics 4.200 0.041 0.661 Measurement 3.925 2.554 0.034 0.629 Discussion on Nigeria Data Table 4.3 shows that at step 0, Statistics with F = 4.200 was the best dependent variable to discriminate between male and female, and thus entered first into the discriminant function equation. Because of the intercorrelations among the variables, the F values for the remaining variables changed significantly depending on the magnitude of the correlation between the other variables and Statistics. Table 4.4 shows that Statistics has a positive correlation with all the variables with the lowest correlation being with Measurement. Measurement therefore remained the most important predictor of group difference in step 1 (p = 0.034). The stepwise discriminant analysis identified two mathematical areas as the most significant contributors to gender differences in Nigeria. These are Statistics (0.661) and Measurement (0.629) as indicated by the magnitude of their standardized coefficients and their F-To-Enter values of the two variables. 49 Table 4.4 Correlation Matrix for subtests (Nigeria). Arithmetic Algebra Geometry Statistics Measure Arithmetic 1.000 Algebra 0.478 1.000 Geometry 0.517 0.496 1.000 Statistics 0.456 0.373 0.416 1.000 Measure 0.408 0.365 0.436 0.318 1.000 Table 4.5 shows the effectiveness of the discriminant function to correctly classify gender into male and female groups. All 386 (100%) females were incorrectly classified as males while all 997 (100%) males were correctly classified as males. Overall 72% of the "grouped" cases were correctly classified. This is just equal to the prior probability for males. It seems that because the N's are so unequal (386 and 997) and the prediction power of the discriminant function is so weak that the best prediction was achieved by classifying everyone as a male (the dominant group). This suggests that there is a very weak relationship between mathematics achievement and gender. Gender differences in mathematics achievement seem to be very small. Table 4.5 Classification Results with Discriminant Function (Nigeria) No. of Cases Predicted Group Membership Actual Group Female Male Female 386 0 386 0% 100% Male 997 0 997 0% 100% Percent of "Grouped" Cases Correctly Classified: 72% Discussion on Swaziland Data. Table 4.6 shows that Geometry first entered the discriminant function with F = 9.40 and p = 0.002 indicating that geometry was the best predictor of gender differences in Swaziland. The mean for males (3.76) was higher than that for females (3.35). Statistics instead of Measurement (Measure) became the most important predictor but could not enter the discriminant function equation because of its low F ( F = 2.124). The discriminant analysis therefore identified Geometry as indicated by its standardized coefficient (0.94) as the only significant contributor to gender differences in Swaziland. 51 Table 4.6 Results of Discriminant Function Analysis on Swaziland Data. Variable Arithmetic Algebra Geometry Statistics Measure F to enter 0 1.752 1.631 9.400 0.003 4.025 1 0.089 0.075 2.124 0.563 P 0.002 Standardized Coefficients 0.938 The classification Table 4.7 shows that 260 out of 434 (59.9%) were correctly classified as female while 199 out of 378 (52.6%) were correctly classified as male. Overall 56.5% of the "grouped" cases were correctly classified. This is statistically significant (Chi-square (1) = 13.7, p<0.001). Table 4.7 Classification Results with Discriminant Function (Swaziland). Actual Group No. of Cases Predicted Group Membership Female Male Female 434 Male 378 Ungrouped Cases 5 Percent of Grouped Cases correctly 260 59.9% 179 47.4% 3 60.0% Classified: 56.53% 174 40.1% 199 52.6% 2 40.0% Table 4.8 Effect size. Geometry Statistics (Swaziland) (Nigeria) Eta squared 0.012 0.003 Effect Size Table 4.8 shows eta squared requested from SPSS MANOVA at 0.05 alpha level. The eta-squared is an overestimate of the actual effect size but a consistent measure. For interpreting eta squared which represents the proportion of total variance attributable to differences among groups, Stevens, (1992) writes "Cohen (1977) characterizes eta squared = 0.01 as small, 0.06 as medium and 0.14 as large effect size" (p. 177). The effect sizes for Geometry and Statistics as indicated by their eta squared are small. Gender differences in these subject areas are therefore small. Geometry and Statistics are the two subject areas that, according to the discriminant functional analysis discussed earlier, contributed most to gender differences in Swaziland and Nigeria respectively. Variability of Achievement Distribution. Gender differences in the distribution of mathematics achievement scores in Nigeria and Swaziland were compared using skewness and kurtosis. Table 4.9 shows skewness and kurtosis by gender in each country. For further interpretation of data on Table 4.9, figures in the form of graphs are used. Graphs depicting the 95% confidence intervals of the skewness and kurtosis of the distribution of the five mathematical areas for male and female students in each educational system are shown. Gender differences are statistically significant (p=0.05) in a mathematical area if there was no overlap of male and female confidence interval (see Glass & Hopkins, 1984). Table 4.9 Skewness and Kurtosis by gender for Swaziland and Nigeria. (Standard errors in parenthesis) Swaziland Nigeria Skewness Kurtosis Skewness Kurtosis F M F M F M F M Arith Algebra Geometry Stats Measure 0.523 (0.117) 0.961 (0.117) 0.900 (0.117) -0.323 (0.117) 0.631 (0.117) 0.418 (0.125) 0.526 (0.125) 0.360 (0.125) -0.216 (0.125) 0.473 (0.125) 0.026 (0.234) 1.382 (0.234) 1.543 (0.234) -0.576 (0.234) 0.542 (0.234) -0.139 (0.250) 0.166 (0.250) -0.360 (0.250) -0.595 (0.250) 0.028 (0.250) 0.216 (0.124) 0.251 (0.124) 0.741 (0.124) 0.014 (0.124) 0.446 (0.124) 0.087 (0.077) 0.291 (0.077) 0.526 (0.077) -0.201 (0.077) 0.393 (0.077) -0.503 (0.248) -0.062 (0.248) 0.404 (0.248) -0.891 (0.248) 0.127 (0.248) -0.742 (0.155) -0.345 (0.155) 0.189 (0.155) -0.777 (0.155) -0.137 (0.155) 1 -r 0.8 --0.6 •-0.4 •• to s > 0.2 •• s M en 0 -0.2 ---0.4 --+ + + M F Arith M F Alge M F Geom M + Stats M F Meas Fig 4.3 The 95% confidence intervals of skewness for Nigeria. Hi-Limit Lo-Limit Skewness Fig 4.3 shows that with the Nigeria sample data, female scores are more skewed than male scores with the exception in Algebra. Skewness are positive for both male and female score distributions in all mathematical areas except Statistics for male. The skewness of the distribution of Algebra scores for males and females are almost equal. The 95% confidence interval indicates that there are no statistically significant gender differences in skewness for all mathematical areas. This means that the asymmetric distribution of achievement scores are similar for males and females in all mathematical areas in Nigeria. Male and female achievement score distributions are equally skewed. 1.2 T 1 • -0.8 --0.6 I 0.4 « 0.2 Hi-Limit Lo-Limit • Skewness + + + M F Arith M F Alge M F Geom M + Stats. -0.2 -0.4 -0.6 Fig 4.4 The 95% confidence intervals of skewness for Swaziland. M F Meas Table 4.10 Distribution of Geometry test-scores for Swaziland. Score 0 1 2 3 4 5 6 7 8 9 10 11 Freq 13 48 90 100 87 47 25 11 6 3 2 2 Female Pet 3.0 11.0 21.0 23.0 20.0 11.0 6.0 3.0 1.0 1.0 0.5 0.5 Cum pet 3.0 14.0 35.0 58.0 78.0 89.0 94.0 97.0 98.0 99.0 99.5 100.0 Freq 6 36 64 74 75 51 40 20 9 3 0 0 Male Pet 2.0 10.0 17.0 20.0 20.0 13.0 11.0 5.0 2.0 1.0 0.0 0.0 Cum pet 2.0 11.0 28.0 48.0 67.0 81.0 92.0 97.0 99.0 100.0 100.0 100.0 Figs 4.4 indicates that the distribution of scores in all mathematical areas in Swaziland is more skewed for females than males. Skewness is positive in all cases except for Statistics. The 95% confidence interval shows that there is statistically significant gender differences in skewness for only geometry. Table 4.10 indicates that 58% of females compared to 48% of males had a score of 3 or less. The mean score for females and males in Geometry were 3.35 and 3.76 respectively. The table also shows that although, on average, females scored less than males, just as many females as males (3%) scored greater than or equal 8. There are more female low achievers than males in geometry but equal female/male high achievers. For all other mathematical areas, male and female distributions are equally skewed 25 T Male Female 4 5 6 7 Test-Score 10 11 Fig 4.5 Distribution of Geometry test scores in Swaziland 1 T 0.5 --O £ -0.5 -1 ---1.5 -L + M F Ari th M + + Alge M F Geom M F Stats M Meas Lo-Limit Hi - l imi t " Kurtosis Fig 4.6 The 95% confidence intervals of kurtosis for Nigeria Fig 4.6 shows that with the Nigeria sample data, the kurtosis of the distribution of female scores was higher than males in Arithmetic, Algebra, Geometry and Measure but lower in Statistics. However, the 95% confidence interval shows that there is no statistically significant gender differences in kurtosis in any of the mathematical areas indicating that the distribution of extreme scores seem to be similar for both males and females. There are equal number of males and females who either perform very poorly or very well in all mathematical areas. 2.5 •• 2 --1.5 --1 --„ 0.5 --(0 2 0 - -k_ * -0.5 •• -1 •• -1.5 --M Ml M Arith Alge + F Geom M Stats M Hi-Limit Lo-Limit Kurtosis Meas Fig 4.7 The 95% confidence intervals of kurtosis for Swaziland Fig 4.7 shows that with the Swaziland sample data, the kurtosis of the distribution of male scores was lower than females in all mathematical areas except Statistics. They are almost equal. Apart from Statistics, female score distributions are all positive while male distributions are all negative except Algebra and Measurement. The 95% confidence interval indicates that there is a statistically significant gender difference in kurtosis for Geometry and Algebra only. The kurtosis of the distribution of Geometry test scores is higher for females (1.543) than males (0.360). Fig 4.5 shows that gender differences in scores are higher at the lower end of the distribution than the higher end. There are more female than male low achievers. The kurtosis for the distribution of Algebra test scores is higher for females (1.328) than the males (0.166). Fig 4.8 shows greater percentage of females than males at the lower extreme ends of the distribution. More females than males achieve quite low in Algebra. For the other mathematical areas, there are no statistically significant gender differences in kurtosis. The distribution of extreme scores seem to be similar for males and females in these mathematical areas. 59 Fig 4.8 Distribution of Algebra test scores in Swaziland Confirmatory Factor Analysis Confirmatory factor analysis (CFA) was performed on the Nigeria and Swaziland data with the DOS - LISREL 7.20 program (Joreskog and Sorbom, 1989). The parameters were estimated using the two-stage least squares followed by maximum likelihood (ML) option. According to Schutz & Long, (1988), "it is acknowledged that maximum likelihood (and least squares) parameter estimates may be biased when performing CFA on Pearson product-moment correlation matrix derived from ordinal scaled variables"(p. 500). The polychoric correlation matrix from PRELIS was therefore used as input for the LISREL analysis. Babakus et al., (1987) note that, the polychoric correlation matrix provides more accurate estimates of the pairwise correlations and factor loadings. Three criteria were used to assess the goodness-of-fit of the hypothesized measurement model (refer to model p. 40) to the empirical structure from the Nigeria and Swaziland data based on the literature on CFA (see Dillon & Goldstein, 1984; Kim & Mueller, 1978; Tabachnick & Fidel, 1983). These were 1) the magnitude of factor loadings, 2) relative correlations of items with factors (covariance matrix), 3) magnitude and significance of goodness-of-fit indices. For a good fit, items were suppose to have high but almost equal loadings on specific factors. Items were also to have high correlations with specified factors but low correlations with other factors. Lastly, the goodness-of fit indices were to indicate a non-significant chi-square statistical test and a goodness-of-fit index (GFI) greater or equal to 0.90. Nigeria Data The initial CFA conducted on the 40-item 5-factor structure indicated that the correlation between the factor Measurement and its hypothesized items were very poor. The Measurement items loaded highly on all other factors, and the estimated correlation between the Measurement factor and some of the other factors exceeded 1.0. It was therefore not possible to obtain an admissible factor solution. The five Measure items were therefore removed from the final analysis. The 35-item 4-factor model yielded an acceptable but unsatisfactory solution. Factor loadings are given in Table 4.11. It can be seen that eight of the items have low loadings (<0.30) on their original factor, suggesting that for this sample they are not good indications of the hypothesized mathematical construct. This is most marked for algebra in which four of the nine items have loadings of less than 0.23. \o CO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O r - H t ^ t N - ^ -£ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O N C O t N C O * p p o o p o o o p o o o o o o o o o o o o o o o o o o o o o o c o L O ' * ^ CD O O O O O O O O O O O O O O O O O O O O O O O O 0) s o o O O O O O O O O O O O O O O O O O O O O 0 0 r J C D v £ ) T - l 1 - i a N 0 0 ' * O C 3 \ O O O O O O O O O O O O O O O O O O O O O O O O r H t N C O V O v O ^ l ^ v O ' ^ m t ^ O O O O p p p p p p p p p p p p o o p p o o p o c o t N c o c o c o c o c N o c o i n t N p p o p o o o o o o o o o o o o o o o o o o o o o o o o o o S-H •8 O O O O O O O O O O O L O V O L T I O O O T - I C N C O T - I O O O O O O O O O O O O O O O O O O O O O O O O O O T - H C ^ O « N 0 \ O O t ^ C N | O O O O O O O O O O O O O O O O O O O O O O O O O O C S l 0 V 0 L T ) T - i ^ C N C 0 C N O O O O O O O O O O O p O O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o X! 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This means that there is at least a 10% overlap in variance between most variables and their factors. • Arith • Alge • Geom • Stats 0 . 8 T Q20 Q31 Q17 Q14 Q18 Q3 Q35 Q26 Q33 Q23 Q34 Test item Fig 4.9 The correlation of Arithmetic test items with all factors Fig 4.9 shows that most of the Arithmetic test-items correlate almost as highly with other factors as they do with the Arithmetic factor. For example Q34 has a correlation of 0.705 with Arithmetic, 0.587 with algebra, 0.645 with Geometry and 0.616 with Statistics. However all the test-items correlate higher with arithmetic than with any other factor. Even where an item, for example, Q31 has very low correlations with the factors it has the highest correlation with Arithmetic. Although the items appear to be more of a measure of Arithmetic than any other factor, they do not do that very well and seem to measure a "general" mathematics construct. Fig 4.10 - 4.12 show similar pattern for the other three mathematics constructs. • Ar i th • Aige H Geom • Stats Q4 Q40 Q1 Q10 Q25 Q12 Q22 Q39 Q16 Test item Fig 4.10 The correlation of Algebra test items with all factors. • Ar i th D A I g e 1660(11 • Stats Q28 Q9 Q32 Q36 Q6 Q19 Q2 Q38 Q29 Q5 Q13 Test item Fig 4.11 The correlation of Geometry test items with all factors. 64 0.6 j en 0 . 5 - -_c ~% 0 . 4 - -o _ 0.3 ~ 0.2+1 U- 0 .1+ Arith D Alge • Geom • Stats Q7 Q21 Q15 Test item Q27 Fig 4.12 The correlation of Statistics test items with all factors. Table4.12 Correlations among factors for the Nigeria data. Arith Alge Geom Stats Ari th Alge Geom Stats 1.000 0.833 0.916 0.874 1.000 0.896 0.896 1.000 0.905 1.000 Table 4.12 shows the correlation among the four factors. The correlations are very high showing strong relationship among the factors. This may explain why items correlate almost equally well on all factors and, therefore, seem to measure a general mathematics construct. Table 4.13 Goodness-of-fit Indices for Nigeria data. Model 4-factor 35 items N 1383 DF 554 Chi-square GFI 5076.91 0.802 RMSR 0.066 P <0.001 Table 4.13 contains the model fit indices resulting from the CFA performed on Nigeria data. The goodness-of-fit index (GFI) of 0.802 is rather low (0.90 or above is desirable), and the root mean square residual (RMSR) of 0.066 indicates that a number of the interitem correlations cannot be reproduced with the hypothesized model. Chi-square=5076.91 was significant at p< 0.001 which means the rejection of the model. It appears that despite the fact that items clearly identified with their specified factors, the hypothesized model is not a good fit for the Nigeria data. Swaziland Data The preliminary analysis on the Swaziland data showed an item Q33 correlating very highly (e.g. 0.984 with Q38) with all items. This means that whatever was being measured by Q33 could be measured by the other items. The final CFA on the data was therefore carried out without Q33. o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o d o ' o ' o d o o o o o o o ' o d d o d o o o o o o ' d o d o o o ' d o o o o o o v o r - r ~ < N c o ooot~-oo- - i (nON OOOt(N(NT|-in O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o -c ^ m ^ O O O O Q O ro oo in O O O O O ^ • n - » o o o o o o O O O O O O O O O O O O O O O O O O O V £ > O N V O O O \ 0 0 0 - O O s O \ c < - > 0 0 0 0 0 0 0 0 0 o o o o o o o o o o o o o o o o o o o o o o ^ o r ^ o o - o i n o - i n i n o o o o o o o o o p p o o p o o p o o o o o o o o o o o o < N < N O - m T f m c N » n o - > n o o o o o o o o o d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d o ' d d d O O O O O O O O O O < N — i 0 0 —i VO 4 7f rn oo o o o o o o o o o o v o < n o o a \ ^ o o o s ^ M 3 o O O O O O O O O O ^ - ^ j - e n c o - T j - c o m m c N o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o vo o^ TT a\ m o- o\ o o\ ffiinm^H^-H _ _ _ _ _ _ TJ- f O i o - ^ - r o c ^ ^ t O - m i n o o o o c n c s o o o o o o o o o o o o o o o o o o o o o o p o o o o o o v o o o o o o o o o o p o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o ^ K o o o n v o H ^ m o CN ^ M in o\ o co a> oo o\ CN vo oo ID H N ^H T*- <—, r-C . " ^ , z l ^ . ^ . ^ ^ C C C H 3 H H H t N N m ^ t s i r ) ' £ ) 5 \ H H N C N i m r o r r ) N H r J ( N o o - H ( S c n r i a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aaaao Table 4.14 shows that most (32 Of 39) of the factor loadings are above 0.3 and therefore according to Tabachnick, B. G and Fidel, L. S. (1983) are "eligible for interpretation" (p. 411). However, some (especially Q2) clearly do not measure the same construct as the other items with which it is grouped. I Arith D Alge I Geom I Stats H Meas 0.6 T Q20 Q31 Q17 Q14 Q18 Q3 Q35 Q26 Q23 Q34 Test item Fig 4.13 The correlation of Arithmetic test items with all factors. Fig 4.13 shows that the Arithmetic test items seem to correlate equally well with all factors. All the items correlate slightly higher with Geometry, Measurement and Statistics than Arithmetic. Figs 4.14-4.17 show a similar pattern. Items loaded higher on other factors instead of their specified factor. This means that items did not measure the mathematical constructs they were intended to measure. It appeared that 68 the items are a measure more of the other factors than their hypothesized factor, or else some more general global mathematics factor. I Arith • Alge I Geom • Stats H Meas Q4 Q40 Q1 Q10 Q25 Q12 Q22 Q39 Q16 Test item Fig 4.14 The correlation of Algebra test-items with all factors. 69 C TD CO O O CO U . Ar i th D Alge H Geom Stat Meas Q28 Q9 Q32 Q36 Q6 Q19 Q2 Test item Q38 Q29 Q5 Q13 Fig 4.15 The correlation of Geometry test items and all factors. • • • • El Arith Alge Geom Stat Meas Fig 4.16 The correlation of Statistics test items with all factors. 70 O) c Ti CO o .^ o o CO u_ 0.8 -r 0.7 --0.6 --0.5 - , 0.4 -0.3 -0.2 -0 . 1 -o 41 Ari th • Alge • Geom Stat Meas Fig 4.17 The correlation of Measurement test items with all factors. Table 4.15 Goodness-of-fit Indices for Swaziland data. Model 5-factor 39items N 812 DF 692 Chi-square GFI 5175.26 0.792 RMSR 0.067 P <0.001 Table 4.15 contains the model fit indices resulting from the CFA performed on Swaziland data. The goodness-of-fit index (GFI) of 0.792 is low and the root mean square residual (RMSR) of 0.067 is high. Chi-square = 5076.26 was significant at P<0.001 which means the rejection of the model. The hypothesized model does not seem to fit the Swaziland data. 71 CHAPTER V SUMMARY AND CONCLUSIONS This chapter presents answers to the research questions based on summary of results and findings from data analysis. Implications for practice and suggestions for further research are also discussed. Research Questions and Findings The purpose of this study was to compare achievement differences by gender in two educational systems in sub-Saharan Africa. The initial questions addressed were whether gender differences in achievement in sub-Saharan Africa depended on educational system and which mathematical area contributed most to gender differences in each educational system. The results of the MANOVA analysis indicated a significant gender main effect suggesting that males and females differed in their achievement in one or more mathematical areas. Males' mean scores were higher than females' in all mathematical areas. However, there was no significant interaction effect between gender and educational system. This means that gender did not have a significant correlation with educational system. Gender differences in mathematics achievement exist in the two educational systems. Discriminant functional analysis was used to find out the important predictors of gender differences in the two countries. In Swaziland, the discriminant functional analysis indicated that the only predictor of gender differences was Geometry. In Nigeria, Statistics and Measurement were the best predictors of gender differences. However, the effect sizes of these 72 differences in the two educational systems as already discussed are so small that they are not worth considering for educational practice. It appears therefore that there are no serious gender differences in mathematics achievement in the two cultures. This is contrary to what seems to pertain to the developed countries as reported in the literature review section. This suggests that there may be little influence of culture on mathematics achievement in the two educational systems. It may be that students treat mathematics and culture as independent entities (more research is needed in this area). Another question addressed in the thesis was whether there were gender differences in the variability of mathematics achievement scores in Nigeria and Swaziland. This was done by comparing the skewness and kurtosis of male and female achievement distribution scores in five mathematical areas. The comparison indicated that in Swaziland, gender differences in variability appeared for only Geometry and Algebra. Significant gender differences in skewness and kurtosis for Geometry was found in Swaziland. There were more females than men at the lower end of the Geometry score distribution. In Algebra, there was significant gender differences in kurtosis again in Swaziland. More females than males scored very low in Algebra. It seems that females and males have different achievement distributions in Geometry and Algebra in Swaziland. There were no significant gender differences in skewness and kurtosis for all mathematical areas in Nigeria. It appears therefore that the distribution of male and female achievement scores are similar in Nigeria. In Nigeria there are no gender differences in the variability of achievement in all mathematical areas. Lastly, a confirmatory factor analysis was carried out on the Nigeria and Swaziland data to determine whether there was any justification to assess students in the two countries based on SIMS hypothesized mathematical structure. The low goodness-of-fit index (0.802) and the significant chi-square (P<0.001) indicate that the SIMS hypothesized model do not fit very well to the Nigeria data. However, since all test items load highest on their hypothesized factor (Geometry excluded), it seems reasonable to suggest that these items measured more of their hypothesized factors rather than a "general" factor. There is therefore some degree of confidence in the appropriateness of the SIMS factor model for the subjects in Nigeria. The Swaziland data fit was about the same as Nigeria except that the Measurement factor was as good as any and that all the items load just as well on a single "General" factor as they do on their specific factors. There is therefore little justification to talk about the assessment of separate components of mathematics such as Algebra and Geometry for Swaziland. These findings are not surprising given the magnitude of the inter-system differences in culture and curriculum emphasis among the countries that participated in SIMS. According to Robitaille and Garden (1988), "the systems ... represented a broad spectrum of nations ranging from highly developed, industrialized countries, to developing countries with largely agrarian economies" and "varied equally dramatically on most of the educational background variables investigated in the study" (p. 233). Despite these differences Robitaille and Garden (1988) conclude that "comparisons are possible, and indeed should be made, when comparable systems are compared on variables of comparable importance in their respective systems, always bearing in mind the need to attend to the complexities and inter-system sources of variability" (p. 233). 74 Comparison of gender differences in the two sub-Saharan African countries in this study is therefore possible. However, in order to make sense of the findings and draw valid conclusions, the context in which SIMS was carried out must be borne in mind. The results of the confirmatory factor analysis on the Nigeria and Swaziland data seem to confirm this caution. Implications for Practice The findings from this study raise many questions of interest to policy makers and policy "implementers" of mathematics education in both Swaziland and Nigeria. For example, can the two education systems adopt similar educational policies without creating a gender gap? This question and others of similar types are addressed in this section. The study found no statistically significant interaction effects between gender and educational systems suggesting that gender differences in the two educational systems are similar in many respects despite the possible differences in the mathematics educational policies and practices in the two systems. The two educational systems can therefore share mathematics educational policies and learn from each other without necessarily creating significant gender differences. The study also found no educationally significant differences using Arithmetic, Algebra, Geometry, Statistics and Measurement as predictors of gender differences in the two educational systems. This means that both males and females have the potential to achieve in these mathematical areas. Teachers, parents and policy makers of mathematics education could therefore use the findings of this study to convince students that both males and females have equal potential to succeed in mathematics. Another significant finding in this study is the fact that in both educational systems, there were high intercorrelations among various mathematical areas. This means that students in both educational systems do not differentiate very much between mathematical areas. Teachers in both educational systems should therefore try and make references to other mathematical areas when for example teaching Algebra. This could ease students understanding of mathematics since they will be able to see the links among different mathematical areas. In Swaziland there are gender differences in skewness and kurtosis in geometry. There are more females than males at the lower end of the distribution. Teachers in Swaziland should therefore pay particular attention to low achievers when teaching geometry. Suggestions for Further Research This section offers suggestions for further research. The analyses of this study were carried out by assuming that each of the 40 mathematics items were equally important. But as indicated by the loadings of the items on their respective factors certain items seem to be more important than others. This study could be replicated by scoring the 40 items using the loadings on their respective factors. Although Mclean, Wolf and Wahlstrom (1986) claim that according to educators, SIMS item pool matched quite highly with various educational systems' intended curriculum. The very low means and the poor fit of the Nigeria and Swaziland data to the hypothesized model suggest the model did not fit the data for these two countries. It may therefore be appropriate to replicate this study using locally developed test items and possibly involving the entire intended populations in both educational systems. Further research could also be done using other populations (e.g. primary and college students). As mentioned earlier in this study, most researchers make their claims about gender differences based on differences or no differences in central tendency and variance completely ignoring skewness and kurtosis, two important characteristics/properties of a distribution. More research incorporating all the four moments of a distribution might assist us to understand how gender differences in mathematics achievement occur. Further research in this area could also help us refine the theory of gender differences in mathematics achievement. Finally, interaction effects among related factors like educational system, type of school and gender could be studied. 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Journal of Early Adolescence, 7(1), 85-105. Wittig, M. A., & Petersen, A. C. (Eds.), (1979). Sex-related differences in cognitive functioning: Developmental issues. New York: Academy Press. Wolleat, P. L., Pedro, J. D., Becker, A. D., & Fennema, E. (1980). Sex differences in high school students' causal attributions of performances in mathematics. Journal for Research in Mathematics Education. World Bank (1987). Education policies for sub-Saharan Africa: Adjustment, revitalization and expansion. Washington, DC. Zainu'ddin, A. (1975). Reflections on the history of women's education in Australia. Education News, 15, 4-13. APPENDIX Tables for Figures in Chapter 4 Table for Fig 4.3 95% confidence interval of skewness for Nigeria data. Arithmetic Algebra Geometry Statistics Measure F M F M F M F M F M Hi-Limit 0.464 0.241 0.499 0.445 0.989 0.680 0.262 -0.047 0.694 0.547 Skewnes 0.216 0.087 0.251 0.291 0.741 0.526 0.014 -0.201 0.446 0.393 Lo-limit -0.032 -0.067 0.003 0.137 0.493 0.372 -0.234 -0.355 0.198 0.239 Table for Fig 4.4 95% confidence interval of skewness for Swaziland. Arithmetic Algebra Geometry Statistics Measure F M F M F M F M F M Hi-Limit 0.757 0.668 1.195 0.776 1.134 0.610 -0.089 0.034 0.865 0.723 Skewness 0.523 0.418 0.961 0.526 0.900 0.360 -0.323 -0.216 0.631 0.473 Lo-Limit 0.289 0.168 0.727 0.276 0.666 0.110 -0.557 -0.466 0.397 0.223 91 Table for Fig 4.6 95% confidence interval of kurtosis for Nigeria. Arithmetic Algebra Geometry Statistics Measure F M F M F M F M F M Hi-Limit -0.253 -0.432 0.188 -0.035 -0.154 0.499 0.654 -0.467 0.377 0.173 Kurtosis -0.503 -0.742 -0.062 -0.345 -0.404 0.189 0.404 -0.777 0.127 -0.137 Lo-Limit -0.753 -1.052 -0.312 -0655 -0.654 -0.121 0.154 -1.087 -0.123 -0.447 Table for Fig 4.7 95% confidence interval of kurtosis for Swaziland. Arithmetic Algebra Geometry Statistics Measure F M F M F M F M F M Hi-Limit 0.494 0.361 1.796 0.666 2.011 0.860 -0.108 -0.095 1.014 0.528 Kurtosis 0.026 -0.139 1.328 0.166 1.543 0.360 -0576 -0.595 0.546 0.028 Lo-Limit -0.442 -0.639 0.860 0.334 1.075 -0.140 -1.044 -1.095 0.078 0.472 Nigeria Data Table for Fig 4.9 Correlation of Arithmetic test items with all factors. Q20 Q31 Q17 Q14 Q18 Q3 Q35 Q26 Q33 Q23 Q34 Arith 0.654 0.252 0.525 0.506 0.413 0.401 0.525 0.414 0.591 0.505 0.705 Alge 0.545 0.210 0.438 0.422 0.344 0.334 0.437 0.345 0.492 0.421 0.587 Geom 0.599 0.231 0.481 0.463 0.378 0.367 0.481 0.379 0.541 0.463 0.645 Stats 0.572 0.221 0.459 0.442 0.361 0.351 0.459 0.362 0.516 0.442 0.616 Table for Fig 4.13 Correlation of Algebra test items with all factors Q4 Q40 Ql Q10 Q25 Q12 Q22 Q39 Q16 Arith 0.438 0.184 0.179 0.504 0.168 0.440 0.334 0.310 0.166 Alge 0.526 0.221 0.215 0.605 0.202 0.528 0.401 0.373 0.199 Geom 0.471 0.198 0.193 0.542 0.181 0.473 0.359 0.334 0.178 Stats 0.472 0.198 0.193 0.542 0.181 0.473 0.360 0.334 0.178 Table for figure 4.11 Correlation of Geometry test items with all factors Q28 Q9 Q32 Q36 Q6 Q19 Q2 Q38 Q29 Q5 Q13 Arith 0.344 0.335 0.315 0.503 0.305 0.312 0.291 0.255 0.063 0.203 0.331 93 Alge 0.337 0.328 0.308 0.492 0.299 0.305 0.285 0.250 0.061 0.199 0.324 Geom 0.376 0.366 0.344 0.550 0.333 0.341 0.318 0.279 0.068 0.222 0.361 Stats 0.340 0.331 0.311 0.498 0.302 0.309 0.288 0.252 0.062 0.201 0.327 Table for fig 4.15 Correlation of Statistics test items with all factors. Q7 Q21 Q15 Q27 Arith Alge Geom Stats 0.342 0.350 0.354 0.391 0.413 0.423 0.427 0.472 0.469 0.481 0.486 0.537 Swaziland data 0.379 0.388 0.393 0.434 Table for Fig 4.13 Correlation of Arithmetic test items with all factors. Q20 Q31 Q17 Q14 Q18 Q3 Q35 Q26 Q23 Q34 Arith Alge Geom Stats Meas 0.354 0.339 0.367 0.360 0.365 0.410 0.393 0.425 0.418 0.423 0.554 0.532 0.575 0.565 0.572 0.396 0.380 0.411 0.403 0.408 0.439 0.422 0.456 0.448 0.453 0.496 0.476 0.515 0.506 0.512 0.502 0.481 0.521 0.511 0.517 0.449 0.430 0.466 0.457 0.463 0.314 0.301 0.326 0.320 0.324 0.363 0.348 0.377 0.370 0.374 Table for fig 4.14 Correlation of algebra test items with all factors. Q4 Q40 Ql Q10 Q25 Q12 Q22 Q39 Q16 Arith Alge Geom Stats Meas 0.433 0.451 0.464 0.394 0.442 0.257 0.268 0.276 0.235 0.263 0.443 0.462 0.475 0.404 0.453 0.372 0.388 0.399 0.339 0.380 0.378 0.394 0.406 0.345 0.386 0.375 0.391 0.402 0.342 0.383 0.368 0.384 0.394 0.335 0.376 0.490 0.511 0.525 0.446 0.500 0.399 0.416 0.428 0.364 0.408 Table for Fig 4.15 Correlation of Geometry test items with all factors. Q28 Q9 Q32 Q36 Q6 Q19 Q2 Q38 Q29 Q5 Q13 Arith 0.315 0.487 0.570 0.476 0.276 0.465 0.089 0.574 0.259 0.290 0.321 Alge 0.312 0.483 0.565 0.472 0.273 0.461 0.088 0.569 0.257 0.287 0.318 Geom 0.304 0.470 0.549 0.459 0.266 0.448 0.086 0.553 0.250 0.279 0.309 Stat 0.285 0.441 0.516 0.431 0.250 0.421 0.081 0.519 0.234 0.262 0.291 Meas 0.360 0.557 0.652 0.545 0.315 0.532 0.102 0.656 0.296 0.331 0.367 Table for Fig 4.16 Correlation of Statistics with all factors. Q7 Q21 Q15 Q27 Arith Alge Geom Stat Meas 0.448 0.385 0.413 0.440 0.403 0.492 0.422 0.453 0.483 0.442 0.441 0.379 0.407 0.433 0.396 0.567 0.486 0.522 0.556 0.509 Table for Fig 4.17 Correlation fo Measurement test items with all factors. Q8 Q l l Q24 Q30 Q37 Arith Alge Geom Stat Meas 0.491 0.467 0.565 0.436 0.476 0.296 0.281 0.340 0.263 0.287 0.224 0.213 0.258 0.199 0.217 0.445 0.423 0.512 0.395 0.432 0.612 0.581 0.704 0.543 0.593 

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