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Sex differences in mathematics in Scotland Kerr, Peter 1988

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SEX DIFFERENCES IN MATHEMATICS IN SCOTLAND by P E T E R K E R R A THESIS SUBMITTED IN T H E R E Q U I R E M E N T S M A S T E R P A R T I A L F U L F I L M E N T OF FOR T H E D E G R E E O F O F ARTS in T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Mathematics and Science Education We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH C O L U M B I A 21 December 1988 ® Peter Kerr, 1988 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. Department of l/VdXfo^*^^' Jbom-vc^ €^<-<^ U^^ ' The University of British Columbia Vancouver, Canada DE-6 (2/88) ABSTRACT This study examines sex differences in mathematics in Scotland. The study is based on longitudinal data from the 1985 Scottish School Leavers Surve3r of 5726 students in one school district. It compares the distribution of scores for boys and girls on the 1984 Ordinary Grade (O-grade) Arithmetic examination, taken by the majority of Scottish students at the end of compulsory schooling, and matches these results with indicators of male and female ability, socioeconomic status (SES), previous arithmetic achievement in primary school, and destinations upon completion of compulson' schooling. The findings suggest that boys slightly outperformed girls on the O-grade Arithmetic examination. Girls were more likely to present for this examination, but more girls than boys scored at the lower end of the distribution. These differences did not vary substantially for pupils at different levels of ability, SES, or prior achievement in arithmetic. The gender gap in mathematics favoring boys, however, did become significant after the period of compulsory schooling. More girls than boys stayed on at school, but fewer of them elected to take further training in mathematics. Boys took more advanced mathematics courses in the last two years of high school and performed better than girls on those courses. Policy implications of these findings and directions for research are discussed. Teachers and counsellors must become informed about the lack of female persistence in mathematics and take steps to alleviate it. Future research should examine why girls in Scotland do not keep up with boys and the factors that have enabled some girls to overcome this general tendency. ii TABLE OF CONTENTS Abstract ii Table of Contents iii List of Tables iv List of Figures v Chapter I. The Problem 1 A. The Scottish Setting 3 B. The Problem Situation, Strengths, and Perceived Need 8 C. Purpose of the Study 11 Chapter II. Review of Literature 13 A. Sex Differences in Mathematics 14 1. Sex differences in mathematics enrolment 15 2. Sex differences on mathematics achievement test scores. 16 3. Problems in measurement of sex differences in mathematics 19 4. Sex differences in the distribution of mathematics achievement 22 5. Current trends in sex differences in mathematics 22 B. Causation: Why Sex Differences Exist 24 1. Cognitive, ability, or biological variables 25 2. Affective variables 32 3. Educational and other societal variables. 34 4. Socioeconomic status 38 5. Conclusions about causation 44 C. Post-secondary destinations 45 Chapter III. Methodology 52 A. Design 52 B. Achieved Sample 55 C. Dependent Variables 57 D. Independent or Control Variables 59 E. Data Set Construction 61 F. Analyses 62 Chapter IV. Results 64 Chapter V. Summary and Conclusions 80 A. Principal findings 83 B. Policy implications 85 C. Directions for Research 89 Appendix 93 References 94 iii LIST OF TABLES Table 1. Parameter estimates and standard errors for regression of O-grade Arithmetic on sex, SES, and a SES-by-sex interaction 71 Table 2. Parameter estimates and standard errors for the regression of O-grade Arithmetic results on sex, Primary 3 arithmetic (P3ARIT), and a P3ARIT—by—sex interaction 72 Table 3. Parameter estimates and standard errors for the regression of O—grade Arithmetic results on sex, Primary 7 verbal reasoning (P7VRQ), and a P7VRQ-by-sex interaction 73 Table 4. Crosstabulations of O — grade Arithmetic results by post—compulsory destinations (Column percentages by sex) 77 Table 5. Highers Mathematics results in S5 and S6 by sex 78 iv LIST OF FIGURES Figure 1. Cumulative frequency distribution, O—grade Arithmetic scores 65 Figure 2. Cumulative frequency distribution, P3 Arithmetic scores. 68 Figure 3. Cumulative frequency distribution, P7 verbal reasoning scores 69 v CHAPTER I. THE PROBLEM Research on sex differences in mathematics is a complex, fascinating field. Fennema and Petersen (1986), two leading writers in this area, admit that the more they study and analyze the problem of sex differences in mathematics, the more complex the problem becomes. As academic, government, and public awareness of the existence of these differences and their negative consequences for women has grown over the last two decades, so has the quantity and quality of research in this area. More powerful studies, many of national and international scope, with better databases and improved, more specific lines of inquiry have been completed. These improvements have helped researchers define the location and the extent of sex differences in mathematics with greater accuracy. National and international studies conducted during the past decade have indicated that sex differences in mathematics still tend to favor males, but that the size of these differences and the age of subjects involved are smaller and more restricted in scope, respectively, than was previously believed. For instance, boys tend to take more elective mathematics courses than girls in the U.S. (NAEP, 1983), especially at advanced levels, but the difference is not great and has been declining. (Chip'man & Thomas, 1985). On tests of mathematics achievement, sex differences usually favor boys but performance gaps are not universally present at any age or grade level, never very large, and not regularly observable prior to Grade 10 (Fennema, 1984). In the last decade or 1 The Problem / 2 so, research has indicated a reduction in the size of the gap between boys' and girls' mathematics performance in the U.S. (Levine & Ornstein, 1983) and the U.K. (Marriott, 1986; Willms & Kerr, 1987) compared with the results of earlier studies. Although the quantity and quality of research on sex differences in mathematics have both increased significantly over the last 10 or 15 years, there remain important areas of research where gaps of information exist or where valuable extensions to current research could be undertaken. If a nation's educational system is inconsistent in terms of instruction, curriculum, or evaluation across geographical boundaries, results of national research may not be readily generalizable. This problem affects research from the U.S., the main source of information on sex differences in mathematics. Because the onset of puberty is now acknowledged to be the approximate starting point for predictable sex differences in mathematics (Fennema, 1984), studies that examine only younger students are poorly targeted. Few studies have provided important background information on prior student achievement or ability (Ethington & Wolfle, 1984) or socioeconemic status (Stanic & Reyes, 1986). Previous research also has failed to examine adequately the distribution of sex differences across various levels of ability, previous achievement, or famity background. Finally, few studies have been able to tie sex differences in high school mathematics to subsequent student academic and occupational movement. The present study addresses some of the areas in the literature where gaps exist or where extensions of current knowledge could be made. The study The Problem / 3 outlines the distribution of achievement of boys and girls on the national government examination in arithmetic, administered at the end of the fourth year of secondarj' school in Scotland, the final year of compulsory schooling for most Scottish students. Sex differences are estimated across various levels of socioeconomic background, ability, and early arithmetic achievement. The analyses also examine the relation between fourth year arithmetic examination results and the performance in higher level mathematics for those who stayed beyond the minimun leaving age, or occupational directions for those who left school after S4. To avoid confusion, the author uses the term "sex differences" instead of "gender differences" throughout this study, although "gender" is often used in the literature and is appropriate in a context describing the social, cultural, or psychological aspects of male and female behavior. The remainder of this chapter provides information about the problem setting: important characteristics of the Scottish educational system, including some pertinent historical developments. The problem situation is then explained, including the current status of sex differences outlined in the literature. Next the importance of this study and its perceived strengths are outlined and finalty, after the setting, background, and motivation for the study are in place, the formal statement of the problem concludes the chapter. A. THE SCOTTISH SETTING The present study examines sex differences in mathematics achievement in the secondary schools of one of Scotland's educational authorities or school The Problem / 4 districts. Scottish schools are administered by a total of nine mainland and three island Education Authorities. Appendix A describes the area examined by the present study. Several important historical changes have influenced the development of the modern Scottish system of education and bear upon this study. Prior to 1962 the Scottish secondary program was designed around the norm of a five-year course for most students. In the Fifth year, usually at age 17, students could sit either Lowers or Highers examinations (or some combination of the two) in a variety of subjects. The Highers examinations were the main qualification for entrance to higher education in Scotland. About half the fifth year students would then stay on for a sixth year to attempt more Highers or to re-sit old ones (Raffe, 1984a). However, the minimum school-leaving age prior to 1973 was onty 15, so many students did not feel compelled to staj' until the fifth year and left without any official certification. In 1962, in response to official worries about substantial early leaving from the five-year academic high school course, Scotland introduced the Scottish Certificate of Education (SCE) Ordinary Grade (or O-grade) examinations for sixteen-year olds in academic programmes, at the end of their S4 year (Scottish Education Department, 1959). Pupils may now leave school at the end of their fourth, fifth, or sixth years. They may take O-grade examinations in any of these years and Highers examinations in their fifth and sixth years. Students deemed capable may sit O-grades in a selection of 48 different The Problem / 5 subjects. English and mathematics ("arithmetic") are two core subjects taken by virtually all students! and about 80% of all students sit an O-grade examination in these two subjects. Highers examinations still retain their essential characteristic as the main qualification for higher education, undertaken voluntarily by students after completion of their compulsory years of schooling. The second significant change occured in 1965. At that time the Scottish Education Department began to reorganize its secondary education system along comprehensive lines, by abolishing selective transfer to secondary school (Scottish Education Department, 1965). Previously, pupils had been streamed into separate schools according to their abilities and interests directly after primary school. Comprehensive reorganization meant that all schools were now designed to serve the needs of all pupils in a designated catchment area. Although catchment areas differed in socioeconomic makeup, and therefore provided differing populations of students for their local schools, comprehensive reorganization did open all secondary schools to a broader range of students (Willms, 1986). By the late 1970's, after comprehensive reorganization, the Scottish secondary system consisted of 470 secondary schools and virtually all pupils (95%) were being admitted to programmes with comprehensive intakes (McPherson and Willms, 1987). Another important change was the raising of the minimum school leaving age to 16 in 1973. This meant that Scottish students, who enter secondary t"Arithmetic" and "Mathematics" are two separate O-level examinations offered by the Scottish Education Department. Mathematics is a more difficult examination aimed at a minority of very capable students. The Arithmetic O-grade examination, taken by the majority of students, is the main measure of achievement employed in this study. The Problem / 6 school at age 12, would end their compulsory education at or around the end of fourth year. Finally, while enrolment in Scottish education authority secondary schools is declining, from a plateau of some 410,000 in 1978 and 1979 to a projected trough of approximately 290,000 in 1991, youth unemployment has risen sharpty since 1979. This shift in employment patterns may mitigate against the effects of falling enrolment b3' motivating students to stay in school, but it also creates new problems for schools. Students now deciding to stay on past fourth year are often non-academic pupils demanding a level of programming which has not previously been available in the post-compulsory courses of most Scottish schools (Burnhill, 1984). One of the net effects of the changes cited above has been a shift in the role of the O-grade examination. The original intention had been that 30% of the 16-year age group should have a reasonable chance of passing the examination in three or more subjects. By 1976, with more students staying on past fourth year and with most students enrolling original^ in comprehensive programmes, some 80% of the age group were attempting at least one O-grade examination and the O-grade had come to. be regarded as the end-of-compulsory-schooling certificate (Gray, McPherson, & Raffe, 1983). As a result, pass rates declined, and in 1973 the original pass/fail distinction was replaced by 5 bands, A to E, with the A, B, C bands corresponding to the original pass grade. About half of the pupils taking O-grades remain in school for a fifth or sixth year, during which time the majority take SCE "Higher-grade examinations," or "Highers" (Gray et al. 1983) The Problem / 7 Beyond the Sex Discrimination Act of 1975, Section 22, which legally protects women's rights to equal access to education, there is little to be found in government policy or legislation in response to the issue of sex differences in education in general, or in mathematics in particular. For instance, the main goal of comprehensive reorganization was not to reduce sex inequality — class inequality being one more obvious target (McPherson & Willms, 1987). However, a report by Her Majesty's Inspectors of Schools (Scottish Education Department, 1975) stated: the process of secondary reorganization along comprehensive lines, frequently with mixed ability grouping — almost invariably coeducational — at the early stages of secondary education ha[d] encouraged reappraisal of the curricular arrangements (p. 37). It is easier normally for coeducational comprehensive schools to minimize differences in the provision for boys and girls, (p. 39) Studies have shown that these changes in the organization of the Scottish school system were indeed accompanied by a general trend towards increased female educational achievement (Scottish Education Department, 1986) and participation in higher education (Burnhill & McPherson, 1984). Burnhill and McPherson showed that, in the earty 1980's, academically capable female students finished secondary school with occupational ambitions that surpassed their female counterparts in the early 1970's, and now more closely resembled those of male students. Both men and women also were more likely to view higher female ambitions as appropriate. Again, these changes seem to have been part of wider social changes and not the result of any major educational reform designed to The Problem / 8 reduce sex differences. B. THE PROBLEM SITUATION, STRENGTHS, AND PERCEIVED NEED This study covers a ten-year period. The data set provides information on a sample of 5726 students, most of whom were born in 1968, as they moved through primary and secondary school. The sample comprises nearly all (97%) the students in this age category in one Scottish school district. Examination results are described for a test of early arithmetic achievement at the Primary 3 (P3) level, a test of verbal reasoning ability at Primary 7 (P7), achievement in arithmetic at Secondary 4 (S4), and achievement in mathematics by students who continued beyond the age of compulsory schooling to their fifth or sixth years of secondary education. Also incorporated in the analyses is survey questionnaire information supplied by the students on such factors as family background and choices of direction once they had passed the minimal school-leaving age. This study is strengthened by the consistent nature of the Scottish educational system across geographical boundaries, and the comprehensive, longitudinal nature of the data available. For instance, the O-grade data, the main achievement indicator and focus of this study, have some advantages over data describing pupil achievement in the U.S., such as the National Assessment of Educational Progress (NAEP) or the High School and Beyond (HSB) data. O-grade examinations, as described above, are national certificate examinations and success on these examinations is one of the major goals for students of secondary school programmes since the institution of comprehensive reorganization. The Problem / 9 Because of the diversity of goals and curricula in U.S. high schools, the NAEP and HSB achievement tests include some items that are insensitive to instruction (Willms, 1985). The Scottish tests, however, are more likely to reflect school-related achievement uniformly across all schools of the district, and those scores will be comparable to those of students across the entire country. Also, the Scottish system has fewer transition points where students may "leak out" of the study of mathematics. In a typical U.S. high school, mathematics instruction beyond Grade 8 (age 13) progresses in a sequence of courses with options for withdrawal along the way: Algebra I, Geometry, Algebra II, Trigonometry, Pre-Calculus, and Calculus (Lee & Ware, 1986). In Scotland, English and arithmetic are considered core subjects for all students deemed capable until the end of S4. Thus the first major transition point in Scottish school mathematics occurs upon completion of the fourth year examinations and with the subsequent decision whether to continue study in the fifth year. The uniformity of Scottish system provides an advantage in the present study. Fraser and Cormack (1987a) feel that this uniformity is reflected in other ways as well, in comparisons within the United Kingdom. In Scotland, mathematics at the "Highers" level (S5 and S6) can be taken along with several other subjects, but in England and Wales students are restricted to only three "A"-level subjects after O-Grades and this leaves no room for mathematics on the schedules of any but the most specialized students. Thus the lack of intense pressure to specialize at an early age (roughly 14 for English students) may encourage more Scottish girls to continue with mathematics. Fraser and Cormack note that almost 50% of recent mathematics graduates at Edinburgh University The Problem / 10 were female, compared to the national average of 30%. While these phenomena of uniformity and female persistence in mathematics in Scotland might engender a hypothesis of no sex differences in secondary mathematics achievement, there is little empirical evidence to support or disprove this claim. Almost no work has been done on sex differences in Scottish primary and secondary school mathematics or on longitudinal analysis of Scottish boys' and girls' progress in mathematics. In 1983 the Assessment of Achievement Programme (AAP) carried out a survey of P4, P7 and S2 students and concluded that "at all three stages boys and girls performed equallj' well" (Assessment of Achievement Programme, 1983). Although there were no further details given, the Times Educational Supplement (1986) reported that these results "...showed once again that Scotland does not have the problem in this area experienced south of the border." It is generally acknowledged that boys often do better than girls at mathematics in secondary school (see Chapter 2). Could Scotland be an exception to this pattern? The present study provides much needed information on this situation. Another area where previous research is inadequate is the consideration of family background along with mathematics achievement. Grant and Sleeter (1986) published a review of studies incorporating the variables of race, sex and socioeconomic status (SES) in educational research. They concluded from their review of 71 studies that few of the investigators had integrated the three factors in their designs and analyses. They argued that this lack of integration of race, sex, and SES oversimplifies the analysis of student behavior in school The Problem / 11 and could perpetuate biases. Such research provides too narrow a focus, treating individuals as if they were members of just one group, and ignores the joint contribution of each of the factors. Similar^, Stanic and Reyes (1986) concluded that, "there has been no definitive study of SES as it relates to mathematics achievement" (pp. 4-5). The present study incorporates data on student family SES into the analysis of sex differences in mathematics. Abilit3' and its effect on mathematics achievement continues to be a controversial issue (see Chapter 2). Ethington and Wolfle (1984) stated that the analysis of the interaction between sex and other important variables in the process of mathematics achievement is complex, and that questions about male-female differences in mathematics achievement "may have no meaning unless one asks the question in relation to specific values of variables that measure prior ability and educational experience" (p. 375). This studjr provides some measure of prior ability. C. PURPOSE OF THE STUDY This study compares the achievement of boys and girls on the pivotal O-grade examination in arithmetic, taken in the fourth year of secondary school, for an entire Scottish school district in 1984. The first part of the stud3' examines SES, student achievement in arithmetic in P3, and student verbal reasoning ability in P7. The study considers the predictive value of each of these three measures on S4 O-grade Arithmetic achievement. The second part of the stud3' examines the relationship between these same O-grade results and the The Problem / subsequent decision to leave school or stay beyond S4. The achievement of "stayers" on mathematics examinations in S5 and S6, and the various destinations of school leavers are related to previous achievement in O-grade Arithmetic. CHAPTER II. REVIEW OF LITERATURE The research questions considered in this stud3' are: What are the size and distribution of sex differences in achievement on the O-grade Arithmetic examinations taken at the end of S4? How do these sex differences vary across levels of student ability and socioeconomic status? How are subsequent educational and employment destinations related to O-grade arithmetic performance? This chapter reviews important studies in the literature that relate to each of these research questions. To begin, evidence from the literature on the size, age of occurrence, distribution, and changing patterns of sex differences in mathematics is presented. Then the literature on possible causes of these sex differences is reviewed. Although this study is designed to address the research questions above in a descriptive fashion, and is not intended to test any of the various theories about the causes of sex differences in mathematics, it may, with its information on student ability and family background, provide valuable information for later investigations of causation. This review of literature therefore includes a summary of research into the possible causes of sex differences in mathematics with special attention to the areas of ability and family background. The review concludes with an examination of the literature on the relationship between sex differences in mathematics and post-secondary destinations. Throughout this chapter, historical trends in research, when apparent, are indicated. 13 Review of Literature / 14 A. SEX DIFFERENCES IN MATHEMATICS To survej' the literature on the size and age of occurence of sex differences in mathematics, it is important to distinguish between mathematics achievement and mathematics attainment or enrolment. This is often overlooked in the way articles are presented in the literature. Erickson, Erickson, and Haggerty (1980) felt that enrolment in school mathematics courses should 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 enrolment might be considered a less direct measure, it does indicate the level of school qualification achieved and these qualifications directly influence future student options. Also, course taking and achievement on mathematics achievement tests appear to be directly related. Thus enrolment is a valid indicator of performance and receives consideration along with achievement in this review. Enrolment and achievement on tests comprise the first two parts of this five-part section. The third part examines problems and difficulties which may occur in large studies and in meta-analyses of research on sex differences in mathematics. The fourth part outlines research evidence on sex differences in the distribution of mathematics achievement, and the last part summarizes the current discernible directions or trends in sex differences in mathematics indicated in the literature. Review of Literature / 15 1. Sex differences in mathematics enrolment. Boys tend to take more elective mathematics courses in high school than girls and this proportion is greater for more advanced courses. 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 female, 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 not large at any time, but that the male advantage was largest in the advanced courses. Girls showed a consistent improvement in relative enrolment figures from 1978 to 1982 across courses. In the U.K., figures for 1984-85 for England and Wales show female enrolment still lagging considerabty behind in mathematics, chemistry, physics, and computer science for both O-level and A-level entries (Department of Education and Science, 1985). A comparison of England and Wales with Scotland (Fraser & Cormack, 1987b) shows that Scottish girls are enrolling for more mathematics than their English and Welsh counterparts, but are still not equivalent to boys in this respect. In general, the enrolment gap in secondary mathematics between boys and girls in developed countries seems to be decreasing. Armstrong (1980) found only slight differences in participation. Chipman and Thomas (1985), in a recent review, reported that the difference is no longer as great as earlier research had Review of Literature / 16 reported. Lee and Ware (1986) make the valuable point that overall lack of persistence in mathematics, regardless of sex, is considerable (p. 13) and this helps underscore the fact that, although some differences in enrolment may still exist, there is considerable overlap in enrolment patterns between the two sex groups. 2. Sex differences on mathematics achievement test scores. The basic pattern to be noticed in the literature on test results is a narrowing or tightening of the size and age of occurrence of regularly observed sex differences. Fennema (1974) was one of the first to question the accepted view that male superiorhVy in mathematics achievement was prevalent throughout the school years, from kindergarten to high school leaving (e.g., Glennon & Callahan, 1968). Subsequent research supported this narrower definition of the usual age and grade levels where sex differences in mathematics tend to occur. Large-scale studies and reviews in the early 1970's concluded that sex differences in young children's mathematics achievement were inconsistent, but that male superiority became evident by upper elementary or junior high school and continued to grow (Maccoby & Jacklin, 1974; Hilton & Berglund, 1974). Also, males were found to be superior in "higher-level", cognitive mathematics areas requiring understanding and application (usually equated with "problem solving"). Fennema again sharpened perceptions (1977, 1978) when she pointed out that many of these large scale studies, such as the 1972-73 NAEP National Mathematics Assessment or the first mathematics study conducted by the Review of Literature / 17 International Association for the Evaluation of Educational Achievement (IEA) (Husen, 1967), failed to control for history of courses taken. Because males often have studied more mathematics than females, the studies compared males and females with unequal mathematics backgrounds. Between 1974 and 1978, when this discrepancy in attainment was taken into consideration, a number of studies showed differences in achievement to be less than previously indicated. For example, Fennema and Sherman (1977, 1978) found differences were small and sometimes nonexistent when coursework was controlled. Schonberger (1978) observed male superiority in problem solving to be limited to the high ability range of students and only to certain problem types, and Wise (1978) noted differences being eliminated when coursework was held constant. This led some to conclude that differential coursework was the major cause of sex-related achievement differences and Fennema (1980) stated, "this author believes strongly that if the amount of time spent learning mathematics were somehow equated for females and males, educationally significant sex-related differences in mathematics performance would disappear" (p. 82). However, subsequent studies showed differences persisting even when controls were instituted for previous courses taken. Armstrong (1980) analyzed data from two large U.S. national studies, including the second NAEP in 1977-78 (NAEP, 1978, 1979) and concluded that 13-year-old girls enter high school mathematics on at least an equal footing with males in terms of ability and grades but that by the end of high school males consistently outperformed females, especially on higher level cognitive items. The second international study by the IEA, of 24 countries in 1981-82, supported these findings (Crosswhite, Review of Literature / 18 Dossey, Swafford, McKnight & Cooney, 1985; see also NAEP, 1983). In light of these and other studies, Fennema (1984) partially rejected her earlier hypothesis that achievement differences might be eliminated if coursework were equalized. This, then, is the current state of affairs: consistent achievement differences favouring boys are not expected to emerge prior to Grade 10, are typically not very large, and are not found universally even in advanced high school years. Girls often excel in lower cognitive level, computational tasks, and boys on higher-order tasks. The situation in the U.K. fits this general pattern of differences in achievement in mathematics. Shuard (1986) anatyzed the Assessment of Performance Unit (APU) data for all of England and Wales, 1980-82, and found that 11-year-old girls performed better than boys in just one category: "computation: whole numbers and decimals." Boys, in contrast, did better in the categories of concepts with whole numbers, decimals, and fractions, and in applications with numbers, measurement, and geometr3'. Similarly, the Schools' Council Project (Ward, 1979) gives a number of examples of questions which call for understanding of number, where boys do better than girls. Computational questions, where girls had an advantage, were of about the same difficulty but could be done by rule alone (Shuard, 1986). Later work in mathematics, of course, builds on understanding, not on memorization of rules. In an important series of studies with results that differed from the generall}7 small sex differences in mathematics found in the literature, Benbow Review of Literature / 19 and Stanley (1980, 1983) found boys exceeded girls, in a highly-gifted population of Grade 7 and 8 students, on the SAT-M examinations, and that the male advantage was quite large at the higher levels of SAT-M scores. 3. Problems in measurement of sex differences in mathematics. Some of the potential problems that may interfere with the measurement of sex differences in mathematics have been indicated. Selection bias should be avoided: if males are more likely to drop out of school, then high school studies may be comparing a more heterogeneous group of females with a more homogeneous, intellectually-motivated group of males. Also, if males who remain in school are likely to take more advanced mathematics courses than females, another form of selection bias is introduced if course-taking experience is not controlled for in analyses of test results describing third or fourth year performance (Fennema & Sherman, 1977). Not controlling for courses taken was a problem in national studies in the U.S. prior to 1974. Since then studies have usually controlled for course-taking but the problem may still exist in a subtler form: course titles at the secondary level are a questionnable indicator of content studied. Detailed definitions of courses and more surveys and research on what is actually taught, and how, are needed (Raizen, 1986); for example, some studies show boys often receive more hours of formal instruction time than girls (Eccles & Jacobs, 1986). In Scotland, because all students are required to take arithmetic through their compulsory years of schooling, and because the same O-grade examination, Arithmetic, is a point of destination for a majority of students across the countrj', the consistent nature of the educational system Review of Literature / 20 overcomes these problems of course-taking and instruction. Another area of difficultj^ in the literature is student family background or SES. Data have been collected in the vast majority of studies from white, middle-class North Americans, and very little cross-cultural data or even upper or lower-class data are available for comparison (Jacklin, 1979). Cross-cultural data would provide information about whether different types of family socialization affect male and female learning of mathematics differently and thereby help to support or disprove environmental theories of causation of sex differences in mathematics. The present study anatyzes data on student achievement across all socioeconomic levels in one school district. Often large-scale studies (meta-analyses and national or international studies), because of their large sample sizes, may find differences which are statistically significant but of questionable practical or substantive significance t (Willms, 1987). Also, the use of mean scores ma3r mask important differences between males and females in the distributions of their mathematics scores. This study, which employs a nationally standardized metric, and describes the cumulative distributions of male and female scores, is able to consider the substantive importance of observed differences as well their distributional qualities. Meta-analyses may suffer from other problems as well. Maccob3r and tFeminists such as Bleier (1984) would go further to suggest that "by their very methods of reporting trivial differences in the mean scores of large groups or populations of boys and girls as sex differences [so-called 'scientific'] disciplines helped, along with other cultural forces, to create the entit3' (gender differences) that they claim to explain and measure scientifically" (pp. 108-109). Review of Literature / 21 Jacklin (1974) published a review of some 1600 studies on sex differences, mostly from 1966 to 1973. Their review was a significant contribution to the literature and is one of the most cited of such reviews to this day. In their massive work are examples of some of the shortcomings of research on sex differences (Block, 1976). More than one third of the documented comparisons between the sexes were based on samples of 60 or less, and about one fifth involved 40 or less. The conceptual quality of psychological measures used often was ignored, evaluations were based predominantly on studies of younger children (75% on 12-year olds or younger, 40% on preschool) even though sex differences in mathematics are known to begin primarily in adolescence, and categories for grouping studies were inconsistently defined - sometimes conceptually disparate studies were combined. Finally, more flaws in documentation were found in the review than are acceptable in a reference volume (Block, 1976). Age is a potentially confounding factor in a different sense. Puberty, the time when sex differences in mathematics most reliably appear, is also likely to be a critical time for intensification of socialization effects and thus for differences between the sexes. Also, there are differences in the physical maturation rates of males and females (Tanner, 1962). Therefore comparing males and females on tests at the same age may be inappropriate because developmental ages may be different. More information is needed about maturation rates of behavioral characteristics (Wittig & Petersen, 1979). These problems are a limitation of the present study as well. Review of Literature / 22 4. Sex differences in the distribution of mathematics achievement. Two more important features of sex-related differences in achievement should be mentioned. First, with regard to differences in the distribution of test scores, boys appear to be more variable than girls; that is, they score more frequently at both ends of the spectrum. Male domination at the higher end of the distribution is widely acknowledged (Fox, Brody & Tobin, 1980; Benbow & Stanley, 1980), but their position at the low end is often overlooked. For example, in special programmes for pupils with learning disabilities, mild mental handicaps, or other educational disadvantages, boys typically outnumber girls by at least seven to one (Bentzen, 1966). Studies.such as those completed by Benbow and Stanley, which fail to consider the entire distribution of students across all levels of ability (or SES), are obviously limited in their application to more general populations. This study looks at the whole distribution of students. 5. Current trends in sex differences in mathematics. The final feature to note in this section is that the achievement and attainment gaps in mathematics appear to be diminishing. In the U.K. for example, there has been a steady improvement in girls' performance in mathematics at both 'O' (Grade 10) and 'A' (Grade 12) levels (Marriott, 1986; Willms & Kerr, 1987). Levine and Ornstein (1983) noted similar trends in the NAEP reports between 1973 and 1978. Stockard and Wood (1984) found that boys were generally underachieving Review of Literature / 23 (receiving lower grades for equal abilitj') more often than girls in high school, although this difference was much smaller for mathematics than for English or total grades. They labelled female underachievement a myth, asserted that school grades appeared to be an appropriate place for females to exercise their valuing of achievement, and claimed that the focus for intervention and remediation might well shift to the poor transfer of achievement to occupational attainment. Similarly, Walden and Walkerdine (1985) examined the classroom performance of girls in mathematics in the U.K. and found that, contrary to common belief, girls' performance did not decline as they progressed through school. Walden and Walkerdine also found that there was not a large difference in examination performance between boys and girls but that the girls' success often came to be seen as failure in the classroom because of widely-held ideas about what constitutes "proper" learning, coupled with traditional notions of masculinity and femininity. Often pupils' actual attainments were not seen as reliable or simple indicators of success. According to Walden and Walkerdine, teachers want to encourage "real understanding," so they watch for and promote those characteristics which they consider to be related to this — confidence, flexibility, risk-taking, and rule-breaking. Because these behaviours are usually found more often in males, females who have actually done well on examinations are less likely to be stretched and challenged by their teachers and also less likely to be entered for O-level examinations (Walden & Walkerdine, 1985). The changing role of women in society, the feminist movement, and the increased quantity and quality of research available to identify differences, their Review of Literature / 24 causes and possible solutions all help to explain the reduction of the enrolment and achievement differentials between the sexes. Also relevant are new legislation and school programmes, instructional material and curricula adjusted to eliminate sexual bias, and new programmes for intervention and prevention, including in-service to help raise school staff and student awareness. B. CAUSATION: WHY SEX DIFFERENCES EXIST School mathematics involves the acquisition of mathematical knowledge and the attitudes, expectations, beliefs and values of students, peers, parents, and educators toward males and females as learners of mathematics. All these areas are intertwined and virtually inseparable so that it is basicalty impossible at present to study the totality of the causation of sex-related differences in mathematics. One can, however, select important variables and look at their developmental interrelationships and effects (Fennema, 1984). One interpretation is hopeful: since there are few clear cut indicators of cause and effect, there appears to be no major barrier to equit}' in mathematics education. Differentiation in learning appears largely dependent on the social environment and this can be changed. Research on causation is examined here in four categories: cognitive, ability, or biological variables; affective variables; educational and other societal variables; and socioeconomic status, followed by a section of conclusions. Review of Literature / 25 1. Cognitive, ability, or biological variables. The chief distinction in the intellectual prowess of the two sexes is shown by man's attaining to a higher eminence, in whatever he takes up, than can women — whether requiring deep thought, reason, or imagination, or merely the use of the senses and hands. If two lists were made of the most eminent men and women in poetry, painting, history, science, and philosophy, with half-a-dozen names under each subject, the two lists would not bear comparison. We may also infer...that if men are capable of a decided preeminence over women in many subjects, the average of mental power in man must be above that in women... (Darwin, 1871, pp. 873-874) With such sweeping condemnations, men of the last century relegated women to a situation of double jeopardy: they were tried twice for the offence of being female — once at birth when they were assigned to a life of mental ineptness and again at death when the lives thus lived were judged inadequate. Scientists of that era created such disciplines as craniology to support the theory of inherent physical differences between the sexes. Women were found to have "a head almost too small for intellect but just big enough for love" (Meigs, 1847). Later this obviously flawed discipline disappeared as a serious scientific field, replaced by the measurement of mental abilities through IQ testing or by other measures of physiological differences, and the scientific case for inherent or genetic differences between the sexes came under increasing scrutiny and attack. Review of Literature / 26 Today it is generally acknowledged that sex differences in general academic ability or intelligence are non-existent or inconsequential (Stockard, 1980). In fact, although females tend to excel in general verbal ability (Macc'oby & Jacklin, 1974), average differences are actually quite small and by high school males are approximately equal on the verbal skills needed for mathematics learning (Fennema, 1984). The current study provides a measure of "verbal reasoning ability" which will permit a re-examination of these conclusions with a population of Scottish high school students. Sex differences in a subset of spatial ability, spatial visualization, are often associated with sex differences in mathematics (Maccoby & Jacklin, 1974), but here again recent research indicates a modification of an earlier viewpoint. Spatial visualization involves the visual imagery of three-dimensional objects, or two-dimensional representations of three-dimensional objects, and the mental manipulation of these objects. In mathematical terms, it requires objects to be rotated, reflected or translated. One popular test used to measure this ability is the Space Relations sub-test of the Differential Aptitude Test (Connor & Serbin, 1980; Wattanawaha, 1977). Maccoby and Jacklin (1974) and Harris (1978) described consistent and increasing mean differences in spatial visualization in favour of males, beginning at adolescence. However, in the Fennema and Sherman studies (1977, 1978) few sex-related differences in spatial visualization were found and the data did not support the theory that these differences explain sex differences in achievement, although the two were highly correlated (see also Fennema, 1983). Armstrong (1980) found no sex differences in spatial ability for Grade 12; among 13-year-olds, females actually did better on the Review of Literature / 27 spatial tasks. Thus, at the very least, a broad causal relationship between mathematics achievement and spatial ability is not indicated. In fact spatial visualization may be a .by-product of mathematics learning and perhaps, as enrolments equalize, any existing sex differences in spatial visualization will be reduced. Research has not yet indicated whether teaching spatial visualization directly improves mathematical performance (Jacklin, 1979). Although neither general intelligence nor spatial ability appear central to an explanation of sex-related differences in mathematics, this by no means has meant the end of efforts to establish the existence of an inherent biological aspect of these differences. Benbow and Benbow (1984) examined data on about 65 000 highly-gifted Grade 7 and 8 students. These students took the College Board Scholastic Aptitude Test — mathematics section (SAT-M), a test designed for above-average 12th graders. Boys outscored girls on these tests, especially in the higher range of scores; for example, there were 14 times as many boys as girls who scored above 660. t Because course-taking differences were not a factor for this sample (the school years of elective mathematics courses were still in the future for these students) and high school mathematics experience was minimal, a biological component to male superiority in "mathematics reasoning ability" was hypothesized. These studies caused a sensation in the U.S. media at the time: Benbow was quoted as saying that many women "can't bring themselves to accept sexual differences in aptitude. -But the difference in math is a fact. The best way to help girls is accept it and go from there" ("Gender factor", 1980) Newsweek of the same date asked in its story headline, "Do Males IThe mean and standard deviation for the SAT-M examination are approximately 475 and 110, respectively (Waters, 1981). Review of Literature / 28 Have a Math Gene?" ("Do males", 1980). Pallas and Alexander (1983) represent critics of these studies. They point out first that the sample was limited to very able and motivated students and results might not be generalizable; other studies are cited with more regular cohorts that do not show achievement differences at such an early age. Also, it would have been preferable to test the effect of course background directly with students who differed on that variable, otherwise its effect remains uncertain. Furthermore, differential coursework is not the only possible "environmental" explanation of sex-related mathematics differences (see below), and even equivalent course work (everyone had completed Grade 6) does not necessarily indicate equivalent mathematical experience. Eccles and Jacobs (1986) cited studies that showed that boys often receive more hours of formal instruction than do girls in primary mathematics classes and also may have more informal mathematically-related experiences outside the classroom (p. 369). Benbow and Benbow (1984) based their hypothesis for a biological difference in mathematical reasoning ability between the sexes on acceptance of a biological base for sex-related differences in spatial abilities. They assume that spatial ability may positively influence mathematical reasoning ability, or at least that the two mental abilities 'may involve similar cognitive processes, and that sex differences in spatial ability and their biological explanations also will apply to mathematical reasoning. As reported earlier, sex differences in spatial ability is not as clear-cut an issue as was formerly believed. Also the biological aspect of spatial differences which do exist has not been proven (Keating, 1976; Serbin, Review of Literature / 29 1980, 1983) and the proposed link between spatial and mathematical reasoning abilities is still just an hypothesis. Large SAT-M sex differences favouring boys are not unique to populations of the highly-gifted. For example, Lee and Ware (1986) looked at the 1982 HSB data on 8321 high school graduates with at least one academic course in mathematics. They found girls had higher overall GPA's and higher grades in mathematics, but lagged behind boys on the SAT-M by an average of 45 points and they were at a loss to explain the contradictory results. Benbow and Stanley's assumption (1980, 1983) that the SAT-M measures mathematical aptitude belies girls' in-class performances. Jackson (1980) asserts that the SAT-M instead measures acquired intellectual skills and is only described as an aptitude test "because, unlike a traditional achievement test, it is not tied to a particular course of study, and because it is designed to assist in predicting future academic performance" (p. 383). Similarly Schafer and Gray (1981) quote a member of the Educational Testing Service (producers of the SAT tests) to the effect that "the developers of the SAT do not view it as a measure of fixed capacities" but instead, "the test is intended to measure aspects of developed ability." Chipman of the National Institute of Education wrote (1981) that the mathematics SAT samples "performance in a domain of learned knowledge and skill...In a fundamental sense, we do not yet know what mathematical ability is...". Kifer (1986), in looking at the Second International Mathematics Stud}', concurred, questioning the existence of mathematical ability and applauding the Japanese teachers who, unlike their North American counterparts, do not blame poor student performance on lack of ability but on their own failure to motivate. Review of Literature / 30 Sherman (1977) examined several suggested biological causes of mathematical ability sex differences, including hereditary differences linked to the X-chromosome, sex differences in serum uric acid, hormone differences, and difference in brain lateralization and cerebral organization. She concluded that only sex differences in brain lateralization had reasonable support. Petersen (1980) felt hormones might also prove important but research in both these areas is extremely difficult (Shepherd-Look, 1982; Bryden, 1979). For example, different metabolic forms of the same family of hormones — say estrogens or androgens — have different behavioural effects as do different dosages or the timing of the injections. Different species of laboratory animals or even particular strains within a species vary in their hormonal profiles and respond differently to hormonal manipulations. Because of these and other fundamental differences in the biological and behavioural responses among different species of laboratory animals it is difficult to extrapolate findings from rat to guinea pig or monkey and, certainly, to human. Then, of course, laboratory circumstances may not have relevance to natural environments where the influence of learning and environmental factors is unavoidable and inseparable from the basic biological mechanisms and structures. Research on the differential rate of brain lateralization, or hemispheric specialization between the sexes has produced inconclusive results. One group of theorists (Levi-Agresti & Sperry, 1968) has maintained that man's superior spatial performance is the result of a stronger lateralization of brain functions while another group (Buffery & Gray, 1972) has argued that man's superior spatial ability stems from a lesser degree of specialization. (For reviews see Review of Literature / 31 McGlone, 1980). There is also a general difficult}' with trying to connect sex differences in brain lateralization and sex differences in behaviour: the way skills are assumed to be divided between the two halves of the brain is not the same way that skills are divided between women and men. For example, the left half of the brain is thought to be more specialized for verbal skills, at which women are better, and for analytical and logical skills, at which men are widely held to be superior. The right hemisphere appears to be more specialized for spatial-visual abilities, a male-dominated area, and intuitive thinking, a trait usually assigned to women (Hubbard, Hanifin & Fried, 1982). "Knowledge of basic biological mechanisms has less and less predictive value about the behaviours of animals the more complicated, flexible, and unpredictable the animals' behaviours are and the more complex and heterogeneous their environment and culture are"' (Bleier, 1984, p. 106). Most research and theory today does not favour a strictly biological explanation of cognitive differences (Shepherd-Look, 1982; Bryden, 1979). While sex-related differences in cognitive functioning may turn out to have some biological component (more research is needed), such influences would not limit any cognitive ability to one sex or another and would produce small differences at most (Jacklin, 1983). Social and psychological factors are now favoured in the literature as explanations of sex differences in mathematics (Linn & Peterson, 1985; Petersen, 1979; Fox, Tobin, & Brody, 1979). Review of Literature / 32 2. Affective variables. The affective domain involves feelings, beliefs, and attitudes; it is complicated, as the list of variables is long and the possible interactions are, in practical terms, limitless over time and age. Confidence in learning mathematics, for instance, is related to general self-esteem, causal attributions, and mathematics anxiety among other factors. Fennema and Sherman (1978) found that, in grades 6 through 11, 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. Wolleat, 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; that is, they tend to attribute success to ability and failure to lack of effort or circumstance. Thus females feel less in control of their mathematical learning and are less apt to persist. Confidence is more highly related to achievement than any other affective variable (Fennema, 1984). However much is unknown about the true effect of confidence or how such feelings are developed. The perceived usefulness of mathematics for future goal attainment also shows a significant positive correlation with mathematics achievement and attainment (Fennema & Sherman, 1977, 1978; Meece, Parsons, Kaczala, Goff, & Futterman, 1982) and 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 Review of Literature / 33 mathematics. She found sex differences were small, with a low but significant correlation with achievement. Similarlj', as girls tend to enjoy mathematics less, they also rate it as more difficult than boys (Eccles-Parsons, Adler, Futterman, Goff, Kaczala, Meece, & Midgley, 1983). However the direction of causation for both enjoyment and perceived difficulty of mathematics remains uncertain; for example, students may enjoy subjects in which they perform well or vice versa. Another area where the relationship with achievement is not clearly understood is sex-typing of mathematics. When mathematics is stereotyped it is seen as a male achievement domain by both male and female students. Males, however, have this perception to a greater extent than females; females surprisingly do not regularly characterize participation or competence in mathematics as unfeminine (Eccles-Parsons, 1984) and mathematics perceived as a male domain might not be important as a predictor of later achievement (Meyer & Fennema, 1986). A closely-related concept, perceived congruency between engaging in mathematics and one's own sex role can, on the other hand, influence performance negatively for females (Fennema & Sherman, 1977). Thus girls may experience conflict between academic pursuit and popularity. In general one should keep in mind that "sex role" is not a set of specific behaviours but a concept, the perceptions of which differ not only between persons but also within a person depending on situations and personal development. Value attached to mathematics is a function of both the perceived qualities of the task and an individual's needs, goals and self-perceptions. Past experiences, social stereotypes, and information from parents, teachers and peers all influence mathematical choices and performance. Review of Literature / 34 3. Educational and other societal variables. While the entire social milieu influences how well one learns as well as how one feels about mathematics, the most important influences occur within the classrooms where mathematics is taught. Learning environments for boys and girls within classrooms, while appearing to be the same, differ a great deal; the differences today are just more subtle than in the past. Fennema (1984) has stated categorically that the "causation of sex-related differences in mathematics rests within the schools" (p. 161). Classroom experiences can contribute to sex differences in mathematics in two wa3's: the differential treatment of boj'S and girls within class or the differential impact of similar treatment. Often, instead of altering traditional sex 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 practise mathematics teachers are still seen to reinforce traditional behaviour and occupational plans for both boys and girls independent of where student interests 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 behaviour and more praise for work and behaviour (Fennema, 1984). The hidden curricular message for boys reads, "if you would only behave yourself and try, you would succeed". Boys' behavior captures adult attention and implicitly protects the male Review of Literature / 35 intellectual ego (Slolnick, Langbort, & Day, 1982). While boys are usually praised for intellectual works, girls are mostly praised for behaving properly and obejdng rules of form, and criticized primarily for intellectual reasons. They learn they are pleasing but not necessarily capable, and teacher praise for compliant behavior pressures girls to adhere to this role (Skolnick, Langbort, & Day, 1982). Positive interactions which build self-confidence and mathematics interest are especially favourable 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 tremendous variation in the behaviour of grade 12 teachers, and overall the effect of student-teacher interactions on mathematics achievements remains unclear (Erickson, Erickson, & Haggerty, 1980). Fennema and Peterson (1986) found boys and girls were engaged in the same activities during mathematics class and were equals engaged in those activities. However, Brush (1980) noted the competitive climate of many mathematics classes and apparently girls are less likely to thrive in such an environment. Research suggests girls' preparation for continuation in mathematics Review of Literature I 36 could be most favorably affected by a non-competitive learning atmosphere, active encouragement, exposure to positive role models, sincere praise, and explicit advice regarding the value of mathematics studies; the so-called 'girl-friendly classroom' (Eccles, Miller, Reumann, Feldhauffer, Jacobs, Midgley, & Wigfield, 1986). These factors are often neglected, and students who come to high school mathematics classes with well-learned sex-role stereotypes leave the same way. Perhaps equal treatment of males and females is not enough and teachers must play an active role in creating a confident, positive female attitude towards the learning of mathematics. Adolescence is the time when the clash between principles and realities, between achievement and social fulfilment, makes its first major impact on the lives of young women (Skolnick, Langbort, & Day, 1982). Peers are important 'socializers' in a student's 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). In more advanced mathematics courses, female registration may fall short of a 'critical mass' necessary for same-sex support. The adolescent years are fraught with insecurities and pressures. Strong support and guidance to pursue mathematics is required but instead North American students are given their first option to drop mathematics in Grade 9 and do not receive adequate information about the possible consequences of this decision. Counsellors operating under sex-biased assumptions report reacting Review of Literature / 37 differently to a girl who wishes to drop back to a less advanced mathematics class than to a boy who "is more apt to need the math" (Skolnick, Langbort, & Day, 1982). 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). Their long-term impact on sex-related differences in mathematics would appear likely but these effects are difficult to measure in the short run (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 children, and the activities they provide or encourage. Several studies have discovered sex 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' decisions to continue in advanced mathematics (Armstrong, 1980; Fennema & Sherman, 1977, 1978; Parsons, Adler, & Kaczala, 1982). A strong relationship was shown between parents' judgement about their child's ability in mathematics and the child's estimates of her own ability, plans for more mathematics courses, and actual achievement (Parsons et al., 1982). Also, student beliefs about their parents' perceptions of them as mathematics learners can predict enrolment intentions (Fennema & Sherman, 1977). In general, however, the direction of causality with the parent Review of Literature / 38 variable is still open to debate; for example, expectations may arise from previous achievement or vice versa. 4. Socioeconomic status. Although socioeconomic status (SES) is another social variable which could conceivably influence the level of sex-related differences in mathematics, it has been noted above that SES has generally been inadequately considered in the literature to date. The combination of sex, mathematics, and SES in the literature is extremely rare and in general the consideration of sex in studies of SES (and vice versa) has been inadequate. Willms (1983) defined SES as "the relative position of a family or individual on an hierarchial social stratification structure, based on their access to, or control over, wealth, prestige, and power" (p. 40). SES is often operationally defined as comprising the income, education, and occupation of household heads (White, 1982), but there is often considerable variation from this basic premise. The traditional definition is often supplemented or replaced by an assortment of variables including size of family, educational aspirations, ethnicity, mobility, presence of reading materials in the home, and amount of travel, as well as such school level variables as teachers' salary, pupil/teacher ratio, per capita expense, and staff turnover. White (1982) found over 70 different variables used (either alone or in some combination) as indicators of SES in the studies he reviewed. The educational significance of the SES of a student's family is considered in different studies via the possible effect of SES on educational Review of Literature / 39 aspiration (how far you would ideally like to go), expectation (how far you realistically think you can go, given your particular set of circumstances), attainment, and achievement. It is widely believed that SES is strongly, and positively, correlated with academic achievement. In the Equality of Educational Opportunity Survey (Coleman et al., 1966) the authors concluded that SES had a major influence on achievement, and that schools did little to alter this effect. Other studies also have found a strong relationship between SES and academic achievement (Klein, 1971; Levine, Stephenson, & Mares, 1973; Baker, Shutz, & Hinze, 1961; and Thomas, 1962). Oakes (1987) found SES differences in achievement and participation in her analysis of the HSB data and other studies. Poor children do less well than their more affluent peers overall, especiallj' on measures of higher level skills and problem solving (National Center for Educational Statistics, 1985). SES differences in achievement are evident at age 9, clearly in place at age 13, and continue to increase during senior high school (Carpenter, Matthews, Lindquist, & Silver, 1984). In the HSB data, with other school and home factors controlled, students' socioeconomic status (defined by educational levels of parents, father's occupation, family income, and household possessions) accounted for a substantial amount of the differences in students' mathematics achievement (Rock, 1984). This strong relationship between SES and academic achievement is in fact often referred to in the literature as an indisputable fact and is not accompanied by further reference or supporting evidence. Review of Literature / 40 The family characteristic that is the most powerful predictor of school performance is socioeconomic status (SES): the higher the SES of the student's family, the higher his academic achievement. This relationship has been documented in countless studies and seems to hold no matter what measure of status is used (occupation of principal breadwinner, family income, parents' education, or some combination of these). (Boocock, 1972, p. 32) To characterize youth according to the social class position of their parents is to order them on the extent of their participation and degree of success in the American Educational System. This has been so consistentlj' confirmed by research that it can now be regarded as empirical law ... SES predicts grades, achievement, and intelligence scores, retentions at grade level, course failures, truancy, suspensions from school, high school dropouts, plans for college attendance, and total amounts of formal schooling. (Charters, 1963, pp. 739-740) However, numerous other studies have reported moderate-to-weak relations between SES and academic achievement (Lambert, 1970; Knief & Stroud, 1959; Fetters, 1975; Hennesy, 1976; and Wright & Bean, 1974) and the obtained correlations between SES and various measures of academic achievement range from 0.100 to 0.800 without adequate explanation for this wide variation (White, 1982). White (1982) examined almost 200 studies that considered the relationship between SES and academic achievement. Results of his meta-analysis indicated that: ...as it is most frequentlj' used (with the student as the unit of Review of Literature / 41 analysis) and traditionally defined (using one or more indicators of parents' income, educational attainment, or occupational level), SES is positively but only weakly correlated with measures of academic achievement, (p.474) White offers explanations for his observations: The relation between SES and academic achievement is generally weak and appears to be declining in more recent studies because of the increased availability to all people of television, movies, community groups and organizations, and preschool, and also because of increased efforts at equalization of educational opportunities. The achievement of older students is observed to be less affected by SES, perhaps partial^ because of a disproportionately high dropout rate among low achievers toward the end of high school. Studies that use aggregated units of analysis for the study of SES (for example, school or district) instead of individual student units, produce much higher correlations between SES and academic achievement and this may account in part for the frequent but misleading conclusion of a strong relationship between SES and academic achievement (White, 1982). Also, "some measures that are used as indicators of SES, such as 'home atmosphere', are much more strongly correlated with academic achievement than are the traditional indicators of SES" (White, 1982, p. 474). These discrepancies in the units of analysis used in different studies also help explain the wide variation in correlations across studies. Finally, academic achievement variables (grade level, year of study, type of achievement measure) can influence the correlations of achievement with SES. Review of Literature / 42 In a three-year study of grades 6 and 9 students and a one-year study of grade 12 students, Brush (1980) reported that SES was an important influence on mathematics attainment among high school students but not among younger students. Younger students were influenced by their feelings about mathematics and were more open to changing their course planning; their socio-economic background was not a restricting factor in these plans. For high school students, ability and, to a lesser extent, socio-economic status were the main predictors of attainment. The effect of SES was stronger for girls than for boys. Brush suggested that one reason for this difference may be that many low-SES girls choose clerical work as their goal and drop mathematics when it becomes optional, but low-SES boys take additional mathematics because it is required for certain career options (for example, carpentry and drafting) or because these boys lack clearly-defined goals. High-SES high school girls will take more mathematics than their low-SES counterparts because they are planning to attend university. Studies by Sewell, Hauser, and Featherman (1976), Marini and Greenberger (1978), and Rehberg and Rosenthal (1978) suggest that any male advantage in educational attainment is greatest amongst low-SES pupils. Marini and Greenberger (1978) cited a number of studies of students who attended high school between 1955 and 1960 that showed the direct effect of SES on educational aspirations to be greater than the effect of academic ability for both sexes. Marini and Greenberger's study (1978) of some 2500 Grade 11 students in Penns3dvania reversed this conclusion; academic achievement (their measure of ability) had a stronger effect than SES on educational ambition for both sexes, Review of Literature / 43 although the difference between the size of the two effects was greater for girls than boys. In between-sex comparisons, a greater effect on the ambitions of boys than girls was found for SES. Thus boj^ s' educational aspirations and expectations were shown to be more strongly tied to their social class origins than those of girls. Difficulties in comparing the Marini and Greenberger study with previous studies were cited and more research was called for to confirm this apparent shift in the relative importance of SES and ability. The reduction in the sex differentiation of adult roles since the late 50's, the increase in participation of married women in the labour force, the decline in fertility rates (Ferriss, 1971), and efforts to achieve greater equalitj' of educational opportunity were all listed as possible explanations for the increased role of academic achievement relative to SES in the determination of educational goals (Marini & Greenberger, 1978). Rehberg and Rosenthal (1978) reached the conclusion that " in total effect neither ability nor social class predict college entry as strongly for women as men" (p. 244). These researchers also found the power of social class to predict attainment and achievement for both sexes to be modest and diminishing. "Social class is no longer the dominant force in schooling; merit has emerged as the larger determinant of the individual's progress through the school and of his or her ultimate schooling attainment" (pp. 261-262). There are, of course, still social and therefore "class" elements attached to the ways in which merit is distributed; for instance, in the way parents who have reached high levels of occupational attainment socialize their children. Lee and Ware (1986) did not find social class to be a contributing factor to step-by-step analysis of U.S. students' participation in high school mathematics courses. Stockard and Wood (1984) found Review of Literature / 44 the influence of SES to differ little between males and females. Both of these studies found ability to exert a stronger influence over achievement outcomes than social class for both males and females. 5. Conclusions about causation. Some of the factors that may cause sex-related differences in mathematics have been discussed. Obviously, getting to the roots of sex differences in mathematics is a very difficult, complex task. Within the category of causation research and theory most of the current emphasis is on environmental or experiential, as opposed to biological, factors. Why is this? Perhaps proponents of environmentally-based theories of causation — educational psychologists, sociologists, mathematics educators, etc., have an emotional investment in eradicating sex differences in mathematics which impairs their receptivity to apparently unchangeable biological explanations. Many current writers in this area are, after all, women. Perhaps too these erstwhile crusaders lack the scientific background necessary for a proper understanding of the biological theories and are forced to try and ignore these ideas in the hope they will atrophy through neglect. In general, I think not. The evidence cited above indicates an ever-strengthening body of support for the environmental, experiential, and attitudinal approaches to understanding sex differences in mathematics. One can and should now look beyond the question of "Why aren't women more like men?" to ask "Why do people behave as they do?". This approach legitimizes the behavior of both males and females, permitting the examination of sex differences from a choice perspective rather than from a deficit perspective, and greatly increases the Review of Literature / 45 possibilities for successful intervention and prevention of sex differences in mathematics. Many such programs are alreadj' producing positive results — final testimony to the validity of this approach (Grayson, 1987; Jump, Heid, & Harris, 1987). Finally, although the current emphasis is on environmental and experiential explanations of sex differences in mathematics, it should also be made clear that further work on possible biological or genetic correlates of these differences remains a necessary and worthwhile component of the effort to improve female performance. Arch (1987) argued that "what must be avoided is a simple causal dichotomy which says that either sex differentiation of behavior precedes differential socialization and thus is biologically set, or it follows such socialization and thus can be altered at will by environmental change. What is missing is the possibility that there are biological potentialities and limits which influence how socialization practices or environmental factors in general come to have differential effects" (p. 7). The current research, with its indicators of prior student ability and famity background, provides valuable information for this investigation. C. POST-SECONDARY DESTINATIONS Schonberger and Holden (1984) state that sex-related differences in mathematics participation and performance have been of considerable interest to researchers since Sells (1978) identified high school mathematics as the 'critical filter' which confined female students to a few, traditionally female college majors at the University of California at Berkeley in 1972. Sells' work and other Review of Literature / 46 related studies will be examined in this final section of the literature review. The problem of mathematics avoidance and the extent to which it can limit career choices will be placed in a sociological context by considering the literature on the career and academic choices of young people (and especially young women), enrolment patterns in post-secondary education, and the current status of women in the labor force. In 1972, Sells, a Berkeley sociologist, found that of a sample of applicants seeking admission to the University of California, 57% of the males, but only 8% of the females, had taken four years of high school mathematics (Sells, 1978). Sells concluded that without the four years of high school mathematics almost all of these women would be ineligible for the introductory courses in such disciplines as economics, mathematics, chemistry, physics, engineering sciences, statistics, and computer science; and that without these courses they would be unable to major in 50% of the fields offered at the university. These women would be confined in their choices to five main areas: the humanities, music, social work, elementary education, and guidance and counselling. Sells' description of restricted patterns of female enrolment and subsequent career choices have been well documented in both the U.S. and Canada (Ernest, 1976; Armstrong, 1980; Ontario Ministry of Labor, 1983). Schonberger and Holden (1984) cited several recent studies that have supported Sells' hypothesis that successful completion of high school mathematics is critical to women's chances for matching the educational and occupational attainment of men after Review of Literature / 47 graduation. For example, Hacker (1983) has documented the increased emphasis on mathematics, especially calculus, in U.S. baccalaureate programs in engineering over the last century and has hypothesized that this has constituted a barrier for women and minorities in these programs. Elkins and Leutkemeyer (1974) found that the mathematics portion of the Scholastic Aptitude Test (SAT-M) differentiated between persisting and non-persisting engineering students at the University of Maryland, while the verbal portion (SAT-V) did not. High school mathematics is the most critical factor in the completion of a degree in science or mathematics (McNamara & Scherrei, 1982). On the other hand, Arnold (1987) found that personal values and perceived social constraints were more important than education (or ability) in determining the future vocational achievement of top female high school mathematics students. Many girls continue to show a preference for traditional careers, are uninformed about the labor market and available careers, and seem to be generally unprepared for a long stay in the work force (Toronto Board of Education, 1983). Studies have shown that the vast majority of males and females usually aspire to "traditional" occupational roles held by their own sex (Sayer, 1980; Looft, 1971; Vondracek & Kirchner, 1974; Iglitzin, 1972; Tibbetts, 1975). Sex-typing of occupations by young people appears to be declining, especially with regards to female occupations; that is, both males and females now'more readily accept female aspiration to non-traditional jobs (White & Ouellette, 1980; Gregg & Dobson, 1980). However, actual expectations of employment, as opposed to aspirations, tend to adhere to traditional lines (Larter & FitzGerald, 1979). Of 1350 students randomly selected from Houston, Texas Review of Literature / 48 high schools, 2-year colleges, and 4-year colleges, 90% of males aspired to male-dominated occupations and 80%; expected to attain their aspiration. By contrast 64% of female high school seniors aspired to non-traditional occupations with only 40% expecting to attain them. Of all females surveyed, 70% expected to enter a female-dominated occupation (Derryberrj', Davis, & Wright, 1979). Rosen and Aneshensel (1978) surveyed 3200 students in New York State and reported that the number of girls who think that they will have a non-traditional feminine occupation decreases with age; the percentage with professional or executive expectations in Grades 7-8 was twice as large as that in Grades 11-12. A great majority of young women plan to combine careers with marriage (Glaze, 1979; Sayer, 1980). One effect of this is reduced or moderate "career salience". Almquist and Angrist (1970) define a career-salient woman as one who places career above all other priorities. Skolnick, Langbort, and Day (1982) tell us that: Making room for family considerations tends to orient girls toward jobs rather than careers; that is, toward occupations which thej' imagine involve less training and less single-minded devotion to professional development. Stereotypically, math and science occupations are thought to fit the career mold, perhaps precisely because they have traditionally been male domains. Although it sways girls away from math and science fields, the notion that traditionally female occupations are more conducive to family life is questionable. Many times these occupations are merely lower-paying, (p.43) Review of Literature / 49 The Glaze (1979) studies also showed inadequacies in young women's awareness of the current status of women in the labor force. For example, the majority of respondents grossly underestimated the marriage rate of college graduates. The implications are significant considering the above-mentionned marital plans of most of these girls. The majority (80%) also mistakenly believed that divorce was more common among career women than among housewives and were poorly informed about the size of the female workforce, the average number of years females work outside outside the home (underestimating both of these), and about reasonable approaches to making career decisions. Larter and FitzGerald (1979), and Sayer (1980) concurred with this finding of low female occupational awareness and expectation and posited that such attitudes were well-entrenched and would be slow to change in future. Over the last two decades women have comprised an increasing percentage of the total enrolment in programs at universities and colleges. In the U.S. in 1981, women earned about 48% of the B.A.'s, 50% of the Master's degrees, and 31% of the Ph.D's; up 6, 10, and 18 percent respectively from 1970 (O'Neill, 1984). In Canada, the Ontario Ministry of Labour Women's Bureau (1983) reports that between 1970 and 1980 female enrolment in full-time undergraduate programs increased from 36% to 46%, in M.A. programs from 24% to 40%, and in Ph.D. programs from 16% to 32%. However, despite these increases women are still not represented in university programs in the same proportion as they are in the general population and those women enrolled in university still are clustered in the Arts programs and in other traditionally female areas. In Ontario universities women represent only 10% of the total enrolment in Review of Literature / 50 engineering and applied sciences and 31% of enrolment in mathematics and the physical sciences. Enrolment in the health sciences is high (56% female) but concentrated in traditional areas such as nursing instead of dentistry (17%), optometry (34%), or medicine (31%). Women have made some inroads into these and other traditionally male programs such as architecture, commerce, law, and business administration (Gaskell, 1980). In graduate school, women are not as well represented as in the undergraduate years. Part-time enrolment and enrolment in Colleges of Applied Arts and Technology are also up for women (Ontario Ministry of Labour Women's Bureau, 1983). However female representation in such college programs as clerical and retailing (99% and 91% respectively) indicates an unfortunate concentration in areas which are not remunerative and which are not projected to grow in the future (Menzies, 1981). Women in this century have dramatically increased their participation in the labour force. During the period 1947 to 1980, the number of women in the labour force in the U.S. increased 173%, compared to an increase of 43% for males. Statistics Canada (1982) reports that from 1972 to 1982 there was a 64% increase in the number of women in Ontario's labour force. The majority of married women now work outside the home and it is predicted that married women with children can expect to spend at least 25 years in the workforce while single women can count on as many as 45 years. ...by the year 2000, 65-78% of all Canadian women are expected to be in the labour force. The majority of these women will spend from 25 to 48 years working outside the home. (Toronto Board of Education, 1983, pp.35-36) Review of Literature / 51 However, although women's work has changed greatly, its status has not. Townson (1980) suggests that "women have exchanged one work ghetto for another". At the beginning of the century, 34% of all Canadian women in the paid labour force were maids or servants. In 1982, about one third of the 4.9 million workers were in clerical positions and 71% were restricted to four occupations: clerical (33%), service jobs (19%), sales (10%), and medicine and health (9%) (Toronto Board of Education, 1983). Within this broad sociological perspective one need hardly point out the position of mathematically related fields of study and employment; these areas remain in the category of traditionally male-dominated endeavour. Paradoxically mathematics has expanded its sphere of influence in recent times; it is no longer used only in the fields of physics, chemistry, engineering, and astronomy but is also increasingly called for in industry, computer technology, business, government, nursing, home economics, and education. This situation, when considered with the current trends in post-secondary destinations outlined above can only serve to underscore the continuing role of mathematics as a "critical filter" in our modern society. C H A P T E R III. M E T H O D O L O G Y This chapter outlines the methodology used to analyze male and female performance in mathematics in Scotland and to compare these results across levels of family background, ability, primary arithmetic achievement, and destinations beyond the compulsory level of secondary schooling. The chapter describes five areas of the study: the research design, the achieved sample, independent or control variables, dependent or outcome variables, data set construction, and analyses. A . D E S I G N Careful scruthry of education requires a regular sampling of leavers from secondary schools. The 1985 Scottish Young Peoples Survey (SYPS) utilized in the present study provides such a sampling frame. It was conducted by the Centre for Educational Sociology (CES), University of Edinburgh, with funding from the Scottish Education Department (SED). The 1985 survey forms part of a series of biennial surveys conducted by CES. These surveys are made more effective by the availability of the powerful government appartus for constructing as complete a sampling frame as possible: for example, through information available to the central government such as the record of examination presentations. Also, the surveys are strengthened because they are administered by a non-government agency (the CES). The potential conflict between government's need to manage and defend its policies and social research's need for open-minded, critical inquir}' is therebj' avoided (Burnhill, McPherson, Raffe, & 52 Methodology / 53 Tomes, 1987). The 1985 SYPS survej'ed all pupils in Scotland who completed S4 with an original sampling frame of 10%. The target population consisted of two overlapping groups: a school-year group of students who were in S4 in 1983-1984, and a school-leaver group who had left school from any school year in the same session (Lamb, 1986). The 1985 SYPS was a large-scale multipurpose postal survey. Questionnaires were mailed to home addresses in April, 1985, thus reaching respondents some nine or more months after their school-leaving dates. The range of topics covered in the questionnaire included the student's educational histor}- (number of primary and secondarj' schools attended), the examined and non-examined school curriculum (special tutoring received, examinations taken, work experience programs, special college-level "link" courses taken in high school, subjects taken in third and fourth years), affective factors (feelings about school, attitudes of parents, peers, and teachers, destinations of friends), a record of post-secondary destinations; reasons for leaving school, parents' job status, and parents' education. When student achievement and ability data from primary school records became available from the school district used in the present study, many new research possibilities presented themselves, including studies of a longitudinal nature. To have as complete a data set as possible, the CES decided to augment the original 10% sample from this district (the "main survey") with an additional 87% sample (the "regional survey"), bringing the final survey fraction for the district to 97%. The remaining 3% of the population had been previously Methodology / 54 committed to another survey and was therefore unavailable. Finallj', the local authority also provided results for O-grade Arithmetic, O-grade Mathematics, and Highers Mathematics examinations taken by students in the present study in their fourth, fifth, and sixth years of secondary school. The present study examines O-grade Arithmetic results for all students who completed their fourth year of secondary school in one Scottish school district in 1984. Analysis is done to establish whether there are male-female differences in achievement in O-grade Arithmetic, and to measure the size of these differences. Often when the achievement scores of two or more groups are considered, only mean scores are compared. Mean scores, like other statistics such as the standard deviation and the range, are summary measures that may mask important characteristics of the groups. In this study, however, the whole distribution of scores is available which allows a comparison of the cumulative distributions of male and female scores. To begin the analysis of O-grade Arithmetic results in relation to previous student behavior and background, the relationship between results on a test of acquired arithmetic knowledge administered to these students eight years previous to their S4 examinations, in Primary 3 (P3ARIT), and their O-grade Arithmetic scores is examined. Next, the relationship between results on a test of verbal reasoning ability administered in Primary 7 (P7VRQ) and the S4 O-grade Arithmetic scores are analyzed, followed by the relationship between the students' socioeconomic status (SES) and their performance on O-grade arithmetic. Finally, to conclude this section of analysis, the study uses multiple regression to examine Methodology / 55 possible interactions between sex and P3ARIT, between sex and P7VRQ, and between sex and SES. The second set of analyses examines how O-grade Arithmetic achievement is related to subsequent student destinations. To this end the study relates the S4 arithmetic results to the critical choice between leaving secondary school after S4 or staying on for S5 and perhaps S6. School leavers' destinations are then further categorized into areas such as employment, further education, and unemployment, and each of these areas is compared according to O-grade Arithmetic performance. Students who stayed on may have taken mathematics in a number of different courses. They may, for instance, have retaken the O-grade Arithmetic examination in S5 and even once more in S6. They also may have elected to take O-grade Mathematics or Highers Mathematics in S5 or S6. All these categories can be related to the O-grade Arithmetic scores to see, for instance, what fraction of girls go on to mathematics Highers and how well girls do compared with boys, given their levels of S4 O-grade Arithmetic. B. ACHIEVED SAMPLE The achieved sample is, of course, rarely 100% of the target population. The major problem with coverage in the 1985 survej' was due to non-contact: not all sample members' addresses were current or accurate, causing a loss of coverage. This non-contact rate was reduced through contact with the original address, through an intermediary, or through an agency. Excluding errors of processing, the remaining non-coverage was due to response to the questionnaire Methodology / 56 itself, either as a whole or to particular items within the questionnaire. Response to the questionnaire was boosted by readministration and reminders. Final coverage takes all of these elements into account and is calculated as a percentage of valid returns over the original target sample (Burnhill et al., 1987, p. 119). Figures for the achieved sample in this study are as follows: a questionnaire was sent to 5726 individuals from a total S4 population in the school district of 5869. Questionnaires were returned by 4519 people; a response rate of nearly 80%. Of this group who returned questionnaires, 80% were successfully matched with test information from their Primarj' 3 year, 89% were matched with P7 information, and 82% had O-grade examination information available (70% wrote the Arithmetic O-grade examination). Students with both P3 and O-grade information totalled 66%, and those in both P7 and O-grade databases 74%. The data are the best available in the U.K. to date for the purpose of estimating the size and distribution of sex differences in mathematics along lines of ability and social class. For example, what may become a comparable sampling arrangement for England and Wales did not appear until 1985. Prior data on pupils were based on Department of Education and Science (DES) questionnaires sent to head teachers and Department of Employment (DE) surveys of labor-market entrants. Both of these inquiries were conducted without contacting the young people themselves and also with little attention to previous education. The SYPS, on the other hand, provides data that describe school Methodology / 57 leavers' own accounts of their experience, and not what teachers or career officers believe about them. Also the SYPS education data may be analyzed in the light of data on the school leavers' transitions to the labor market or to post-school education, and vice versa (Burnhill et al., 1987). Even in recent years the DES has been obliged to use SSLS datat because there were no comparable data available for England and Wales (see for example DES, 1984). Looking beyond the school district examined in this study, the consistent, uniform nature of the Scottish school sj'stem (for example, the same O-grade examinations are written and prepared for across the country) makes the results generalizable to Scotland as a whole. Furthermore, the comprehensive nature of the data base heightens the descriptive power of the results and makes them of potential interest to researchers, educators, and policy-makers outside the U.K. as well. C. DEPENDENT VARIABLES The present study presents an outcome measure of achievement describing pupils' O-grade Arithmetic examination results. These results are described as they fall within each of six percentage categories: A - 70 to 100%; B - 60 to 69; C - 50 to 59; D - 40 to 49; E - 30 to 39; and No Award - below 30%. (For some analyses these categories will be separated further into a total of 14 bands.) Categories A to C are nominally considered to represent the passing levels at O-grade. As in previous years, the 1984 O-grade Arithmetic examination tThe Scottish School Leavers Survey, the predecessor to the SYPS. Methodology / 58 was available to students along with the O-grade Mathematics examination. While O-grade Mathematics is designed to be undertaken by more advanced students, O-grade Arithmetic should not be considered a routine computational exercise. In fact, a cursor.y examination of the items on the 1984 O-grade Arithmetic test reveals a preponderance of higher cognitive level problem-solving questions; these more difficult items outnumber the straight-forward computational questions by approximately 2 to 1. In an earlier study, Willms (1986) described how he had scaled the outcome variables. This same method of scaling was utilized in the present study: Each outcome measure was scaled on a logit distribution using a technique for re-expressing grades described by Mosteller and Tukey (1977). Any scaling, including a simple numbering of ordered categories (in this case from one to fourteen for O-grade bands) is arbitrary, and different methods of scaling can produce different results (Spencer, 1983; Willms, 1986). The logit technique assumes that each attainment category represents a score on an underlying distribution (similar to a normal distribution but with fatter tails), and divides the distribution into pieces according to the percentage of pupils in each attainment category. The scaled score is then the centre of gravit}^ of each piece. Because the two covariates VRQ and SES are approximately normally distributed, the logit scaling of the outcome variables provides a slightly better fit to the data than simply numbering the categories, (p. 229) Methodology / 59 For the post-S4 analyses, subjects were categorized as having stayed in school beyond S4 or as having left. Questionnaire information was then used again to assign each school leaver to 1 of 3 areas: Full-time employment, On the Youth Training Scheme (YTS)t, and Unemployed. Examination results for students who stayed in school past S4 are presented for O-grade Arithmetic, O-grade Mathematics, and Highers Mathematics. D. INDEPENDENT OR CONTROL VARIABLES The Regional Council (school board) for one Scottish Educational Authority (school district) was able to provide background test information from the elementary years for most of the 1984 school leavers surveyed in this study. Two tests provided by the Educational Authority are of particular interest to this study. The first is the Regional Council Education Committee Screening Survey in Mathematics, which was administered in the third year of elementary school (known as Primary 3, or P3). This is an untimed, two-part group test with 17 items and a maximum score of 50. The test was developed locally by the Regional Council and no information on validity or reliability is available. The second test is the Moray House Verbal Reasoning Test (Godfrej' Thomson Unit, 1971) administered in the final year of elementary school (P7). This timed group tThe Youth Training Scheme (YTS). was instituted in 1981 (replacing the Youth Opportunities Program) to provide twelve months of work experience, training, and education for 16 and 17 year olds who have recently left secondary school. Although the scheme welcomes the unemployed, it is also intended to be an integral part of the whole economy and to this end firms are encouraged to place their young, first-year employees in the program for a year of consistent, well-rounded preparation. Trainees are paid a nominal salary while on the YTS (Raffe, 1984b). Methodology / 60 test (45 minutes) includes 100 items. The coefficient of internal consistency, calculated from the Kuder-Richardson formula 20, was 0.968 after allowing for the effects of age. The test correlated positively with the Moray House English Test (MHE 40) (0.902), and the Moray House Mathematics Test (MHM 4A) (0.888). This test provides a measure of previous student ability (verbal reasoning quotient or VRQ) in the present study. The SES of the pupils is an independent variable derived from the SYPS questionnaire information. This is a standardized SES variable calculated from three indicators of SES: father's occupation, mother's education, and number of siblings. In previous research with SES, Willms (1986) selected these variables after examining several combinations of variables, including mother's occupation and father's education. He found father's occupation, mother's education, and number of siblings to be the best of the SES-related predictors of achievement in O-grade arithmetic used in his study. Mother's occupation and father's education were tested in preliminary analyses in Willms' study but were dropped from the model because they did not contribute substantially to the explained variance of the outcome variable after father's occupation, mother's education, and number of siblings were included. The occupations of pupils' fathers are classified using the Registrar General's "Social Class" index (Office of Population Censuses and Surveys, 1970), which contains 7 employment categories (see below). Father's occupation is scaled for each individual using the Hope-Goldthorpe Scale (Goldthorpe & Hope, 1974). The average Hope-Goldthorpe value is then computed for all pupils within each Methodology / 61 of the Registrar General's emp^ment categories. These average prestige ratings constitute the scaled values for each category of the Registrar General's index. The Office of Population Censuses and Surveys employment categories and scaled values are as follows: 1. Professional 72.85 2. Intermediate 59.82 3. Skilled Non-Manual 45.94 4. Skilled Manual 40.17 5. Partly Skilled 32.13 6. Unskilled 20.17 7. No Job or Unclassified 34.53 Mother's education is classified as education up to 15 years of age, and 1 beyond. Sex was coded 0 for males and a dummy variable: 0 represents represents education at age 16 or 1 for females. E. DATA SET CONSTRUCTION To compile the final data base for this study several steps were necessary. First the P3, P7, and S4 (O-grade) data were matched. Approximately 80% of any two sets of data were matched via information in computer files (i.e., surname, initials, birthdate, and school) but the remaining 20% required visual inspections and independent judgements. Next the common variables of main and regional survey data were merged and matched to the P3-P7-S4 data. Variable and value labels were then set; a descriptive label was attached to each Methodology / 62 variable in the combined data (some 150 variables) and the labels were set for each value of each variable, including different types of missing data (for example, "did not answer" or "not part of survey"). The proportions or frequencies of pupils having each value of each variable, including the proportion with missing data, was determined and validated. Finally, as described above, some variables had to be scaled for analysis purposes. F. ANALYSES For the purpose of analysis, the goals of the present study can be stated as eight specific questions, with accompanying analytical techniques: Question 1: Do the distributions of results in O-grade Arithmetic differ between the sexes? For this analysis, separate cumulative frequency distributions for males and females chart the cumulative percentages, from 0 to 100, at or below each of 14 O-grade "bands": A (90-100), A (85-89), A (80-84), A (75-79), A (70-74), B (65-69), B (60-64), C (55-59), C (50-55), D (45-49), D (40-44), E (35-39), E (30-34), and No Award (0-29). Questions 2 and 3: Do the distributions of results on the earlier achievement tests of primary school arithmetic and of verbal reasoning ability differ between the sexes? Cumulative frequency distributions are again employed for these analyses to chart the cumulative percentages of males and females at or below each Primary 3 arithmetic score and each Primary 7 verbal reasoning score, ranging from 70 to 140. With such a wide spread of scores to be included, cumulative frequenc3? distributions provide the advantage of allowing every data point to be analyzed. Methodology / 63 Question 4: How do sex differences on O-grade Arithmetic vary across levels of socioeconomic status; that is, what is the role of SES in arithmetic achievement, and what is the interaction between sex and SES? Analyses regress O-grade Arithmetic results on sex (dummy variable) and on several indicators of SES (father's occupation, mother's education, and number of siblings) In a second analysis, an interaction term for sex by SES is added to the model. Questions 5 and 6: How well do early student achievement in arithmetic and early student ability in verbal reasoning predict later achievement in O-grade Arithmetic? The analysis regresses O-grade Arithmetic results on scores on the Primary 3 arithmetic test and the Primary 7 verbal reasoning test, respectively, and examines the interactions between sex and these two tests. Question 7: Does the relationship between achievement in O-grade Arithmetic and students' destinations following completion of their compulsory schooling differ by sex? Cross-tabulations compare O-grade Arithmetic results in six letter-grade categories (A to E, and 'No Award') with student destinations if they left school after S4 (3 categories: Full-time employment, YTS, or Unemployed), or if they remained in school (4 categories: Repeated O-Grade Arithmetic, O-Grade Mathematics, Highers Mathematics, or No mathematics. Question 8: How do results in Highers Mathematics compare with previous achievement in O-Grade Arithmetic for males and females? Cross-tabulations are again employed for these analyses. CHAPTER IV. RESULTS This chapter reports findings for each of the questions asked at the end of Chapter 3. Question 1 asked whether the distributions of scores on the O-grade Arithmetic examination, taken in S4, differed between the sexes. Figure 1 displays the cumulative frequency distributions of these O—grade results for males and females. Data were available for 5726 students overall, 2937 boys and 2789 girls. Achievement is presented in 15 categories, including students who did not write the O-grade Arithmetic examination ("Did Not Present") and 14 O-grade "bands": A l (90-100), A2 (85-89), A3 (80-84), A4 (75-79), A5 (70-74), Bl (65-69), B2 (60-64), Cl (55-59), C2 (50-54), Dl (45-49), D2 (40-44), E l (35-39), E2 (30 34), and No Award (0-29). Scores of C or higher are considered a passing grade for these examinations. Figure 1 indicates that sex differences on the O-grade Arithmetic examination were small. Boys did slightly better at the higher end of the distribution; that is, boys gained more on girls within the higher categories of results. Results were approximately equally distributed in the middle grades, and girls scored more often than boys at the lower levels. Overall 72% of the girls in the present study presented for the O-grade Arithmetic examination compared with 67% of the boys. Although this difference is not large, it is statistically significant because the sample was large and comprised nearly the entire population of students for that cohort. Thus girls were more likely to attempt O-grade Arithmetic but less likely to perform well. For example, girls received more failing grades (D, E, or No Award) than boys. 64 JO e o 30- | 20 10 0 Sex o Female a Male & E2 El D2 Dl C2 CI B2 Bl A5 A4 A3 A2 Al S4 Arithmetic O-grade Scores Figure 1 Cumulative Irequency distribution, Arithmetic O-grade scores 70 tn C rt 0) Results / 66 Scores on O—grade Arithmetic were quite evenly distributed across the various scoring bands for the student group as a whole. Nine percent of all presenting students scored from 30 to 39 (letter-grades E l and E2), 9% from 40 to 49 (Dl and D2), 12% from 50 to 59 (Cl and C2), 10% from . 60 to 69 (Bl and B2), 11% from 70 to 79 (A4 and A5), and 12% from 80 to 89 (A2 and A3). Four percent of students scored at each of the two extremes, 0-29 (No Award) and 90-100 (Al). However, more than twice as many females (167) as males (79) placed at the lowest scoring level of "No Award". Thus, although more girls took the examination, 6% of all girls who wrote O-grade Arithmetic scored from 0 to 29, compared with only 3% of boys. Girls were also more frequently represented, usually by a small margin, in all the lower categories from E2 (30-34) up to B2 (60-64). For the five grade levels designated as A, comprising examination scores from 70 to 100, boys were more frequently represented than girls at each level except A3 (80—84). Again, differences were generally small. Overall, 28% of boys scored 70 or higher, compared with 24% of girls, and of the 26% of students who scored an A, 54% were males. Males did show a considerable advantage in representation at the highest level of A l : 62% of the students in this categor}' were male (5% of males versus 3% of females). The separate lines for males and females in Figure 1 show at each point the proportion of students scoring at or below a given level. For example, 33% of the males and 28% of the females in the sample did not write the O-grade Arithmetic examination; 43% of both males and females attained a grade level up to E l (or did not present). Figure 1 shows how the female majorities at the Results / 67 lowest levels of achievement overcame the initial male advantage in cumulative percentage for the "Did Not Present" category, and how the differences between the two distributions from then on were small (lines roughly parallel). Question 2 asked whether the distributions of scores on the arithmetic achievement test administered to students in Primary 3 differed by sex. Figure 2 shows the cumulative frequency distributions of the P3 arithmetic scores, ranging from 70 to 140, by sex. Male and female students had similar distributions on this test of Primary arithmetic. Further analysis confirms this conclusion. The means of the male and female scores were 98.88 and 99.45 respectively, and the standard deviations were 14.25 for males and 13.74 for females. The differences between these values were not statistically significant at the 0.05 level (t=1.38). Question 3 asked how the distributions of scores on the test of verbal reasoning ability administered to students in Primary 7 differed by sex. Figure 3 displays the cumulative frequency distributions of the Primary 7 verbal reasoning scores, from 70 to 140, by sex. Males scored more frequently in the lower range, 70 to 84. Males accounted for 62% of the pupils within this range. In the middle range of scores, 85 to 119, which accounts for 90% of the sample, results were evenly distributed between the sexes. Males were also overrepresented in the higher range: 55% of those scoring between 120 and 140 were male. Primary 7 Verbal Reasoning Scores Figure 3.Cumulative frequency distribution, P7 verbal reasoning scores. Results / 70 The means of male and female scores were 95.96 and 97.52 respectively, and the standard deviations were 11.10 for males and 10.26 for females. The differences between the sexes, therefore, was about 15% of a standard deviation, which was statistically significant at the 0.05 level (t=5.18). However, Figure 3 indicates that the distributions for males and females were nearly identical for the major group of students in the middle range of scores, and that the statistically significant difference in average scores stemmed from the presence of more males at the lower end of the distribution. Question 4 asked whether sex differences on O-grade Arithmetic varied across levels of socioeconomic status. O-grade Arithmetic results were regressed on sex, SES, and the interaction of sex with SES. Table 1 shows the parameter estimates and standard errors for this regression. In all analyses, sex was coded 0 for males and 1 for females. SES was scaled to have a mean of 0 and a standard deviation of 1 for the entire school district. Therefore, the intercept is an estimate of the expected attainment, or O-grade score, for a male with an SES score that is average for the population under consideration (all S4 students in one school district in 1984). The parameter estimates for the sex effect are estimates of the female advantage for O-grade Arithmetic. Similarly, the parameter estimates for SES are estimates of the Arithmetic/SES slope for males, while the parameter estimates for the SES-by-sex interaction are estimates of the differences between the male and female Arithmetic/SES slopes. Parameter estimates are statistically significant (p<0.05) if they are approximate!}' two times their standard errors. Table 1 indicates a male Results / 71 advantage ("Sex" = —0.036) for O-grade Arithmetic, after taking account of SES, but this difference was not statistically significant. The interaction effect of sex b}' SES is the difference between the slopes of the regression lines for males and females. This parameter estimate ("SES by Sex") was also not statistically significant. Overall, only 14% of the variance in O-grade Arithmetic achievement was explained by the variables sex, SES, and sex-by-SES interaction. Parameter estimates and standard errors for regression of O—grade Arithmetic on Table 1 sex, SES, and a SES—by—sex interaction. Parameter Estimate Standard Error Intercept (Expected O-grade Arithmetic score of average male) 0.018 0.017 Sex (Difference between females and males for the average pupil) -0.036 0.024 Socioeconomic Status (SES slope for males) 0.395 0.017 SES by sex (Difference between female and male SES slopes) -0.034 0.024 Results / 72 Question 5 asked how well student achievement on a Primary 3 arithmetic test predicted later achievement on O-grade Arithmetic and whether these predictions differed by sex. O-grade results were regressed on sex, Primary 3 arithmetic (P3ARIT), and the interaction of sex with P3ARIT. Table 2 shows the parameter estimates and standard errors for these regressions. For the sex effect, there were no statistically significant differences between males and females in O-grade Arithmetic, after taking account of P3ARIT ("Sex" = -0.022). Parameter estimates for P3ARIT are estimates of the O-grade Arithmetic/P3ARIT slope for males. The parameter estimates for the P3ARIT-by-sex interaction are estimates of the differences between males and females in their 0-grade/P3 slopes. In this case the male slope was 0.047 and the female slope 0.043 (0.047 — 0.004). Thus the interaction effect, although statistical^ significant, was not large; statistical significance is attributable to the large sample size. Table 2 Parameter estimates and standard errors for the regression of O—grade Arithmetic results on sex, Primary 3 arithmetic (P3ARIT), and a P3ARIT—by—sex interaction. Parameter Estimates Standard Error Constant 0.020 0.016 Sex (Difference between females and males for the average pupil) 0.022 0.023 P3 Arithmetic (P3ARIT slope for males) 0.047 0.001 P3 Arithmetic by Sex (Difference between female and male P3ARIT slopes) 0.004 0.002 Results / 73 Question 6 asked whether sex differences on O-grade Arithmetic varied across levels of student ability as measured by a test of verbal reasoning administered in Primarj' 7. O-grade Arithmetic results were regressed on sex, Primary 7 verbal reasoning ability (P7VRQ), and the interaction of sex with P7VRQ. Table 3 shows the parameter estimates and standard errors for these regressions. For the sex effect, males had an advantage in O-grade Arithmetic after taking account of P7VRQ ("Sex" = -0.118). Parameter estimates for P7VRQ are estimates of the O-grade Arithmetic/P7VRQ slope for males. Parameter estimates for the P7VRQ-by-sex interaction are estimates of the differences between males and females in their 0-grade/P7VRQ slopes. In this case, the slope for males was 0.070 and the slope for females was 0.066 (0.070 - 0.004). As in Question 5, the interaction effect was not large and its statistical significance derived from the large sample size. Table 3 Parameter estimates and standard errors for the regression of O—grade Arithmetic results on sex, Primary 7 verbal reasoning (P7VRQ), and a P7VRQ—by—sex interaction. Parameter Estimate Standard Error Constant 0.080 0.013 Sex (Difference between females and males for the average pupil) -0.118 0.019 P7VRQ (Verbal reasoning slope for males) 0.070 0.001 P7VRQ by Sex (Difference between female and male P7VRQ slopes) -0.004 0.002 Results / 74 Question 7 asked whether the relationship between achievement in O-grade Arithmetic and students' destinations following completion of their compulsory schooling differed by sex. Table 4 presents results of crosstabulations matching O-grade Arithmetic examination results in six categories (A - 70 to 100; B - 60 to 69; C - 50 to 59; D - 40 to 49; E - 30 to 39; and No Award - 0 to 29 or missing)t with post-compulsor}' destinations in seven categories (Unemployed, YTS, Working, School without further mathematics, Repeated and passed O-grade Arithemetic, Passed O-grade Mathematics, and Passed Highers Mathematics), t Table 4 displays the destinations of school leavers nine months after completion of S4 (April, 1985) and the highest level of mathematics achievement in S5 or S6, up to Highers Mathematics, for those students who remained in school for one or two additional years. The figures in Table 4 show the percentages of each sex group for each destination category, for each of the various levels of previous O-grade Arithmetic performance. For example, 1.3% of the males who scored an A in O-grade Arithmetic were unemployed nine months later, compared with 1.1% of females. Information was available for both O-grade Arithmetic and post-compulsory destinations for 5445 of the students in the tThe letter grade of A is acknowledged to include a large range of possible scores, larger than that usually accepted as the highest level of achievement in North America. However, it was felt that a more accurate picture of the Scottish grading system and its relationship to subsequent student movement would be provided if the letter-grade system was presented as it is used in Scotland, instead of breaking the A level into smaller groups. tin Scotland, students may take O-grade examinations at the end of their fourth, fifth, and sixth years and Highers examinations in their fifth and sixth years. These examinations may also be repeated if students are returning to school and wish to improve their marks. Therefore, the post-compulsory categories involving the study of mathematics are totals for fifth and sixth year examinations results achieved by students in each of O-grade Arithmetic, O-grade Mathematics, and Highers Mathematics. Results / 75 present study (95%). This group was quite evenly divided between the sexes: 2805 males and 2639 females. Five percent of the original sample did not respond to the question on destinations in their questionnaires. Overall, students with higher grades in O-grade Arithmetic were more likely to stay in school beyond S4 than low achievers. This tendency was more pronounced among females than males. For the letter-grades considered as passing marks; that is, from A through C, 76% of females stayed on, compared with 60% of the males. However, females with these fair-to-excellent O-grade results did not persist in mathematics education as frequently as their male counterparts. In fact, only about one-half of the female "stayers" (51%) from the A to C categories successfully completed another mathematics course in S5 or S6, compared with 64% of the male "stayers". Among students who achieved an A in O-grade Arithmetic, almost half of both the male and female groups were able to pass Highers Mathematics by the end of secondary school. Another 30% of females stayed in school but did not continue with mathematics at all, compared to 20% of males. Among school leavers with an A at O-grade, males (62%) were more likely to find full-time employment within nine months than females (42%). Results / 76 Table 4 Crosstabulations of O-grade Arithmetic results by post-compulsory destinations (Column percentages by sex). S4 O-grade Arithmetic Results Post-S4 Destinations B D E NA;| Unemployed Males Females Pupils who left school 1.3 1.1 7.4 7.6 10.7 6.8 10.7 7.8 9.1 10.0 23.2 23.4 YTS Males Females 5.7 4.3 21.5 11.9 27.7 23.7 41.7 26.4 37.0 35.0 43.4 40.3 Working Males Females 11.2 3.9 17.8 9.5 23.8 16.2 19.5 10.8 30.5 19.4 23.5 18.7 Pupils who stayed in school No more mathematics Males Females 20.4 30.3 26.1 49.9 28.4 41.8 15.9 32.8 20.8 24.7 9.3 16.7 O-grade Arithmetic Males Females 1.3 7.1 16.7 2.6 10.4 0.1 0.1 O-grade Mathematics Males Females 14.2 14.3 21.0 15.8 8.3 9.5 5.1 5.5 0.5 0.1 0.1 Highers Mathematics Males Females 47.2 46.1 6.2 5.3 1.1 0.7 0.4 0.4 Total Number Males Females 737 615 290 259 328 325 220 271 195 236 1035 933 ; :NA= No Award Results / 77 Students with a B in O-grade Arithmetic who persisted in mathematics usually went no further than O-grade Mathematics, with males holding a slight advantage within this category. Females (71%) stayed in school more often than males (53%), but more male "stayers" pursued the study of mathematics. "C" students displayed a similar pattern. Sixty percent of males left school, compared with 47% of females, but over 75% of the female "stayers" dropped mathematics, a greater proportion than males. For each level of O-grade achievement, males subsequently held a greater percentage of full-time jobs and placements on the YTS. The female tendency to remain in school more often than males maj' partially account for the male advantage in these two non-school categories, but it also should be noted that patterns of unemployment were similar for the two sexes and therefore males who left school appear to have been more successful in finding productive activities. Question 8 asked whether achievement in Highers Mathematics differed between males and females across various levels of O-grade Arithmetic achievement. This question was included to consider possible sex differences within the select group of students who went on to advanced mathematics study in secondary school. Overall, data on both O-grade Arithmetic and Highers Mathematics were available for 1114 students, 636 males (57%) and 478 females (43%). Because the original sample was divided almost evenly (51% to 49% for males and females, respectively) it is apparent that a greater percentage of males sat Highers examinations in S5 or S6. Table 4 showed that 47% of the males and 46% of the females who Results / 78 scored an A in O-grade Arithmetic in S4 were able to pass Highers Mathematics in S5 or S6. In terms of Question 8, this meant that the overwhelming majority (93%) of Highers Mathematics passes were achieved by students with an O-grade Arithmetic score of A in S4. Therefore a cell-by cell matching of O-grade results with Highers results was deemed unnecessary. Instead, the figures in Table 5 simply show the number of males and females who placed in each of the Highers Mathematics letter-grade categories. Table 5 Highers Mathematics results in S5 and S6 by sex. Highers Mathematics Males Females A (90-100) 3 A (85-89) 9 — A (80-84) 5 5 A (75-79) 17 10 A (70-74) 35 17 B (65-69) 60 40 B (60-64) 55 49 C (55-59) 70 68 C (50-54) 89 95 D (45-49) 73 53 D (40-44) 92 69 E (35-39) 77 45 E (30-34) 44 25 • F (00-29) 6 2 Although females did not sit Highers examinations as frequently as males, those who did present were somewhat more likely to achieve a passing grade of A to C. Fifty-nine percent of females passed Highers mathematics compared to 54% of males. Among passers, males had an advantage (20% of passing males Results / 79 versus 11% of passing females) within the A letter-grade category, and a slight advantage within the B category (34% of passing males versus 31% of passing females). The female advantage in overall passes is explained by the female advantage at the lowest passing level, C (57% of female passes versus 46% of male passes). Males dominated both extremes of Highers Mathematics performance; all of the twelve students who scored from 85 to 100 were male and 46% of the males received a failing grade (below 50) compared with 41% of the females. CHAPTER V. SUMMARY AND CONCLUSIONS This study has addressed several questions concerning sex differences in mathematics. The study is based on longitudinal data from about 6000 students in one school district of Scotland. The first chapter outlined the purpose of this study: to examine sex differences in achievement on the O-grade Arithmetic examination and to match these results with indicators of male and female ability, SES, previous arithmetic performance, and destinations upon completion of compulsory schooling. The need to include such factors as SES, ability, prior performance, and subsequent academic and career behaviors in a well-informed description of sex differences, and the frequent failure of previous research to do so, helped explain the rationale for undertaking the present study. In the first chapter also contained a description of the unique Scottish background and setting for the study, highlighting those features which strengthened the project's potential for contributing to the existing body of knowledge in this area. The reorganization of Scottish secondary schools along comprehensive lines, the raising of the minimum school leaving age to 16, and increased enrolment at all levels of secondarj' school have combined to give the O-grade Arithmetic examination a pivotal role in the Scottish education system. This is an examination for a "core subject", taken by a majority of students, 80 Summary and Conclusions / 81 and it helps to mark a vital transition point at the end of compulsory schooling. Although the current study is based upon data from one school district, the size and scope of the data base (almost 100% of one age group documented across a ten-year period) and the consistent nature, of the Scottish education system across geographic boundaries give the results a relevance and strength that goes beyond their immediate frame of reference. In the second chapter a review of the literature on each of the areas of interest in the present study was presented, and several important observations emerged. In general, when results of the most recent studies are compared with earlier analyses, sex differences in mathematics achievement and enrolment appear to be declining. Consistent achievement differences favoring boys are not expected to appear until adolescence, are never very large, and are not always present at any grade level. When differences do appear, girls often do better on straight computation or one-step word problems, but boys do better on multiple-step word problems, applications, spatial relationships, more advanced logical reasoning, statistics and graphs (California State Department of Education, 1979; Education Commission of the States, 1978; National Council of Teachers of Mathematics, 1980). Boys tend to score more frequently than girls at both the high and low extremes of the achievement distribution. A review of the literature on the causation of sex-related differences in mathematics showed an ever-strengthening body of work to support environmental theories of causation. Studies to date have not shown general intelligence, spatial Summary and Conclusions / 82 visualization, innate mathematics ability, or other biological explanations to have a sound basis in scientific fact. On the other hand, such "environmental" factors as confidence and the perceived usefulness of mathematics are highly correlated with mathematics achievement and enrolment. The classroom environment is also very important. The role of socioeconomic status in predicting mathematics performance appears to be important but diminishing. The second chapter also included a review of the literature on post-secondary destinations. The review described the increasing importance of mathematics in many people's lives after secondary school and the improved participation and performance of women in mathematics-related fields of work and study. However, the review also indicated that the majority of women still adhere to the restricting path of traditional occupational roles and aspirations and fail to understand the possible consequences of these decisions. The review of literature includes, whenever possible, examples of studies done in the U .K. , especially Scotland, to match the more prominent body of information from North America. Data from the U .K. usually indicated situations similar in strength and direction to those of North American studies. The review also provided in-depth documentation of the perceived need for, and strengths of, the current study when its design was compared with the existing literature. Summary and Conclusions / 83 A. PRINCIPAL FINDINGS Chapters 3 and 4 described the methodology and results for the present study. The principal findings for the study were as follows: 1. Overall, sex differences on the O-grade Arithmetic examination, taken at the end of compulsory schooling by a majority of students, were small. However, a consistent pattern to the differences was apparent. For each of the eight band levels from CI (55-59) to A l (90-100), a greater percentage of overall male scores than female scores was present. Conversely, girls were more frequently represented in each of the six band levels from No Award (0-29) to C2 (50-54). However, a higher percentage of boys (37%) than girls (28%) in our sample did not present for the O-grade Arithmetic examination. Thus boys showed a slight advantage across the spectrum of passing grades and the cumulative male advantage for all these categories combined indicates a definite sex difference in mathematics favoring males that is consistent with the literature. Because the male advantages were consistently small in each category, there was considerable overlap between the sexes, and many females in Scotland are doing well in high school mathematics up to O-grade Arithmetic. 2. Girls and boys performed equally well in the early years of primary school arithmetic, and had about the same overall ability in verbal reasoning at the end of elemental school. Thus the later male advantage in O-grade Arithmetic Summary and Conclusions / 84 indicates a shift from the status quo in elementary school. Females are not realizing their potential in mathematics as well as males. These findings are again consistent with the literature on sex differences in mathematics: these differences are not expected to appear before adolescence. 3. The relationship between SES and achievement is well established (White, 1982). In an earlier study, Willms and Kerr (1987) observed that, although socioeconomic status was related to students' performance on the 1984 O-grade Arithmetic examination, there was no significant interaction between SES. and sex. The present study also found that sex differences in achievement did not vary for different levels of SES. 4. Females lagged behind males in their rate of persistence in advanced mathematics in secondary school. The O-grade Arithmetic examination marks the end of compulsory schooling in the "core" subject of mathematics for most students. The study of mathematics beyond S4 is predicated upon satisfactory performance on the O-grade examination. Thus, the O-grade examination marks an important transition point for Scottish high school students. The results of the present study indicated sex differences in the way this transition was negotiated. It appears that many females are managing to compete reasonably well with their male counterparts in mathematics up to the end of compulsory schooling in S4. Also, females who passed the O-grade Arithmetic examination chose to staj' in school more often than males. However, amongst students who stayed on, a greater percentage of males than females participated and were successful in optional, higher-level mathematics courses. Female ability and potential in Summary and Conclusions / 85 mathematics are not being translated into training in this subject area as readily as for males. 5. Males performed better than females on the Highers Mathematics examinations in S5 or S6. First, a greater percentage of males than females sat these Highers examinations. Also, males and females with good O-grade Arithmetic scores (almost always "A") did not achieve similar results when they attempted the more advanced Highers course. Males dominated the A and B categories of Highers results, and the male advantage was more pronounced at the highest scoring levels (all of the students scoring 85 to 100 were males). B. POLICY IMPLICATIONS How can schools ensure that girls with the ability to perform as well as males on the O-grade Arithmetic examination realize their potential by achieving good results on this important evaluation? How can they ensure that girls who perform creditably on this examination do not limit their future options by failing to persist in mathematics beyond the O-grade level? Given good results on the O-grade examination, how can female performance on the Highers Mathematics examination be improved relative to males? If girls are to produce better results in mathematics, to persist in upper-level mathematics courses, and to make greater inroads into traditionally male-dominated areas of work and study, public educational policies must incorporate an awareness of the inequities centred around the transition point between compulsory schooling and optional mathematics study documented in this study. In this section some suggestions are given for Summary and Conclusions / 86 policies which address these concerns at the student, classroom, school, and school-district levels. These suggestions are based on the review of literature undertaken as part of the present study. Female mathematics students, including those in elemental school, should be made aware of the need for, and the usefulness of, mathematics in their lives, and of the importance of mathematics as a key to many future career options. Female students should be encouraged to continue their high school mathematics studies for at least four years at a level compatible with their interests and abilities. Female students should learn of the tendency of girls in previous years to sometimes leave their full potential in school mathematics unrealized and should be encouraged to discuss this phenomenon. All students should have ample opportunity for success and personal satisfaction in mathematics. At the class level, teachers should create a positive and supportive environment for female mathematics students. Mathematics can be presented as fun, valuable, historic, even beautiful, and not as difficult as is often imagined. Classes should provide a sound fundamental mathematics program that enables students to grasp new concepts and ideas, solve problems, and think critically and creatively. Working through mathematics problems often involves much more than skills application. Students are asked to' take risks, make mistakes without becoming upset, tolerate ambiguity, persist in independent work, and trust in their abilities to find solutions and to effect changes in their environment (Tobias, 1978). Females often lack confidence in their intellectual abilities and feel Summary and Conclusions / 87 inadequate at problem solving (Deaux, 1976). Group work, cooperative learning, an experimental approach, and a deemphasis on right and wrong (and on neatness for girls) are some classroom possibilities for improving this situation. To dissolve the threatening mystique of mathematics, girls need to practice and to experience success. Manipulatives should be used at all levels of mathematics instruction, especially for females, who may have had less experience with manipulative play and exploration than males. Students should experience problems with more than one solution or more than one possible approach. They should be encouraged to take chances, to guess, and to use estimation. These activities, when completed successfully, can build confidence and flexibility as well as problem—solving skills. Students should have an opportunity to meet women working in all kinds of occupations that involve mathematics, including those in what has been considered the male domain. The relevance of mathematics to life beyond the classroom should always be emphasized, and the conception of mathematics should be. expanded to include technology, science, and business, as well as areas traditionally considered the female domain. At the school level, the progress of female students in mathematics should be carefully monitored. Special classes, sections, or courses for girls should be provided when appropriate to foster achievement, to overcome mathematics anxiety, and to address the special needs of females. Students and staff should be made aware of the conflicts in modern womens' lives: the traditional vision of Summary and Conclusions / 88 the full-time wife and mother versus economic realities which dictate that most women must work; child-rearing duties, still seen primarily as a woman's responsibility, versus demands of career; traditional notions of femininity versus competition with males; and a new ethic of independence and competence which goes beyond what many women suspect is really acceptable or even possible. Teacher education programs should be implemented to increase teacher effectiveness with female students. For example, teacher expectations for female students should be high. Teachers should be given evidence from the realms of work, school, and research to support such raised expectations. Examples of intervention programs that have improved teacher awareness and subsequent female performance could be provided. Next, teachers should not interact with females in a stereotypical way. For example, female students would benefit from an equalization in the number of direct, open-ended, and abstract questions, more detailed instructions, and longer "wait-times" after questioning. High schools should hire more female mathematics teachers and inservice should be provided for all women teachers, especially those in elementary school, so that they might present themselves as comfortable and confident users of mathematics. Counsellors and teachers should be kept abreast of changes in society and of career opportunities for mathematics students. Teachers and department heads should be given leave to visit business and industry for varying lengths of time. "Career education" should be a responsibility shared by all school staff including teachers and should present new perspectives on nontraditional careers as well as help both boys and girls to envision a new shared role within the family Summary and Conclusions / 89 (Skolnick, Longbort, and Day, 1982). Counsellors and teachers should give special attention to guiding female students across the compulsory/optional transition point where many "leak" out of mathematics. At the district level, materials, textbooks, and examinations should be free of sexual bias. School districts should monitor female enrolment and achievement in mathematics. Proven intervention programs should be implemented where necessarj', including elementary schools. Innovations and trends in the labor market should be reflected in the mathematics curriculum. C. DIRECTIONS FOR RESEARCH This final section presents some suggestions for future research. Some of the projects mentioned would continue or extend the directions of the present study, whereas others would consider questions that were off limits to the present study because of its design. For example, the present study was observational in nature and several of the recommendations to follow incorporate more controlled, experimental settings for analysis. In Scotland, research should consider the course-taking patterns of girls who stay in secondary school beyond S4. The present shuty has indicated that females are not persisting in mathematics to the same extent as males. A natural hypothesis would be that those girls who remain in school but do not persist in the study of mathematics despite encouraging results in O-grade Arithmetic would probably not study physics or chemistry either; instead they Summary and Conclusions / 90 would choose more traditional courses for females such as English, history, other languages, etc. This hypothesis requires confirmation. If this scenario exists, intervention programs to expand female perspectives and to adjust course-taking patterns away from traditional lines should be instituted and research should document the effects of these programs from their onset. Reseach should continue to examine sex differences at the transition point between O-grade Arithmetic and more advanced mathematics by investigating why girls do not take more mathematics at the higher levels. Comparative studies of the design and inherent difficult}' of the O-grade Arithmetic, O-grade Mathematics, and Highers Mathematics courses and their examinations should be undertaken to ascertain if these courses require noticeably different student skills. Perhaps many girls lack the necessary skills to successfully complete the more advanced mathematics courses. Questionnaire data should also be collected to decide if, consistent with the literature from North America, Scottish girls lack confidence in their abilities at higher levels of of mathematics, perceive these courses as more difficult than boys do, or don't see optional mathematics courses in S5 and S6 as necessary to their future wants and needs. In this study boys did not present as frequently as girls for O-grade Arithmetic, but those who did performed generally better than girls. At the Highers level, boys presented more often than girls and achieved better results. Questionnaires and within-school studies should also examine how boys and girls make their decisions about persistence in mathematics; perhaps boys are not more skilled but are making better, more-informed decisions with the help of Summary and Conclusions / 91 teachers, parents, counsellors and peers, at both the O-grade Arithmetic and Highers Mathematics levels. In applied settings, attempts to remediate or intervene could be conducted under experimental conditions. Studies could consider questions such as the following: What are the effects of short and long term exposures to role models? How are teacher attitudes about women and mathematics related to their mathematics teaching and what effect would training to heighten their awareness have? What would be an effective way to teach about mathematics-related careers? Would compulsory four-year mathematics programs be effective? Would greater emphasis on mathematics problems in the social sciences motivate females to study more mathematics? What is the best time for intervention (Fennema, 1983)? Better instruments to measure teacher attitudes should be developed (Grayson, 1987). More affirmative research is needed (Stage, 1986). Studies should examine why some women have succeeded in mathematical programs and careers. Deficit models have less to offer intervention programs than do models of what is required for success. For instance, case studies of females who persisted in mathematics and performed well in advanced courses should be undertaken. Similarly, more case studies of mathematically gifted students and their backgrounds (parents, families, teachers) should be conducted. How do the attitudes of gifted boys and girls compare? Do gifted children have different backgrounds of play and parental expectations? What are the characteristics of those teachers, male and female, who positively influence highly able girls Summary and Conclusions / 92 (Fennema, 1983)? More evaluation research is also needed. (Stage, 1986). Enough successful models are now in place that much valuable information could be gathered on their most important components for dissemination to a more general audience. Finally, longitudinal research that considers broader social, economic and political conditions beyond schools and families, at various times and in different locations, is required. Research should be done to relate sex differences in mathematics to the changing state of the economy, to the range of jobs available, to incentives and training programs offered to women, and to political priorities which in turn dictate social support services for women such as child care programs. APPENDIX The School District The district encompasses a land area of some 505 square miles and has a population of roughly 345 000, with a total work force of approximately 135 000. Although the land area includes just 1.7% of the total area of Scotland the population of the district exceeds 6% of the county's total population. Employment can be categorized roughly as follows: primary industries 7.3% ; construction 7.2% ; manufacturing 32.3% ; and the service sector 53%. The majority of the population is concentrated in five main towns and the lifestyle of much of the remaining minority is best characterized as rural (Regional Council Publications, 1987). Secondary education in this district was described as a fully comprehensive system as early as 1967. There are presently 21 secondary schools. Most of these have a policy of forming remedial groups where needed and then dividing all other pupils into classes of mixed ability; usually this is a planned, instead of a random, mix. From Secondary 3 onward there are roughly two broad groupings: those students following courses that lead to certification on the Scottish Certificate of Education examination, and those following non-certificate courses. Recently the distinction between the two has become less clear; that is, less able pupils are allowed to present for the Scottish Certificate of Education in some subjects. 93 REFERENCES Almquist, E. M.. & Angrist, S. S. (1970). 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