AN IN I ERNATIONAL STUDY OFGENDER DIFFERENCES IN MATHEMATICS ACHIEVEMENTbyXIN MAB.Sc., Beijing Normal University, 1985A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENT FOR THE DEGREE OFMASTER OF ARTSinTHE FACULTY OF GRADUA I^ E STUDIES(Department of Mathematics and Science Education)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIASeptember 1993© Xin Ma, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature) Department of MatIA mr,d,Sc ienciz_ The University of British ColumbiaVancouver, CanadaDate 6Feftem6ex- IS, leicl DE-6 (2/88)ABSTRACTThis study examined gender-related issues in mathematicsbased on achievement data of Populations A and B from TheSecond International Mathematics Study (SIMS).The purposes of this study, which involved two Canadianeducational systems and two Asian educational systems, were 1)to investigate potential interaction effects between gender andeducational system in two populations and in two mathematicalareas, algebra and geometry; 2) to analyze the variability ofmathematics achievement between male and female students ineach mathematical area within and across educational systems; 3)to compare and contrast situations of gender differences betweenalgebra and geometry within and across educational systems.Factorial design, Hartley's Fmax test, and box plots were themajor statistical approaches used in this study.The results showed that no two-factor interaction effectbetween gender and educational system in each mathematicalarea was of statistical significance in either population.Further, there were no statistically significant genderdifferences in algebra. In geometry, gender differences werestatistically significant in Population B. Male studentsoutperformed females from a perspective of across educationalsystems.Within each educational system, no gender differences ineach mathematical area were found to be statistically significantin either population. Therefore, reported gender differences ingeometry were more likely to be a general rather than a localphenomenon.The results of investigation on the variability of mathematicsachievement illustrated two patterns. One pattern involvedBritish Columbia and Ontario. No significant differences on theachievement variability between males and females were found.In general, the majority of both male and female students in thetwo Canadian provinces performed equally well in algebra andi igeometry. For the two populations, within gender gaps wereserious, especially for Population B.Another different pattern was found in Hong Kong and Japan.In Population A, no significant differences on the achievementvariability between boys and girls were found. Generally, themajority of both boys and girls performed equally well in algebraand geometry, although slightly more boys than girls were foundat the bottom end of the achievement distribution. The withingender gaps were serious for this population, although they werenot as wide as those found in British Columbia and Ontario.In Population B, a statistically significant difference on thevariability of algebra achievement between male and femalestudents was found in Hong Kong. Although male and femalestudents equally dominated the top end of the achievementdistribution, males in the lowest 10% of the male distribution andfemales in the lowest 10% of the female distribution tended toperform unequally in algebra and geometry. Female studentsdominated the bottom end of the achievement distribution onevery subtests for this population. The within gender gaps werenarrow in this population.Finally, findings in this study did not support the opinion of abiological explanation of gender differences in mathematics.Furthermore, findings suggested that each educational systemaffected the academic development of both male and femalestudents in the same ways or directions, although one gendermight be affected more seriously than the other.111TABLE OF CONTENTSAbstract^ iiTable of Contents^ i vList of Tables v iList of Figures^ viiiChapter I. Statement of the Problem^ 1Introduction^ 1Need for the Study 3Research Questions 7Limitations of the Study^ 8Chapter II. Literature Review 10Overview^ 1 0Historical Background^ 10Gender Differences across Grades and Topics^ 1 1Gender Differences across Educational Systems^ 17Gender Differences on Variability of Achievement^ 2 4Potential Causations^ 2 5Summary^ 2 8Chapter III. Procedures^ 2 9Data Source 2 9Design of the Study 3 1The Sample^ 3 3Data Analysis 3 4Chapter IV. Results and Discussions^ 3 6Aptness of the ANOVA Model 3 6The ANOVA Results^ 3 6Interaction Effects and Gender Differences^ 4 6ivVVariability of Achievement Distribution^ 5 0Other Findings^ 6 3Chapter V. Summary 6 5Summary of Findings^ 6 5Policy Implications 6 7Suggestions for Further Research^ 6 9References^ 7 1Appendix I 8 4Appendix II^ 8 8List of TablesTable 2.1^Mathematics Achievement of Population Ain SIMS^ 1 9Table 2.2^Mathematics Achievement of Population Bin SIMS^ 2 0Table 2.3^Mathematics Achievement of Population ALongitudinal Study in SIMS^ 2 2Table 3.1^A Factorial Design (A 4 x 2 ANOVA)for the Study^ 3 2Table 3.2^The Sample Distribution forthe Study^ 3 4Table 4.1^Computational Results of A Two-Factor A=4by B=2 Fixed-Effects ANOVA on the AlgebraSubtest of Population A^ 3 7Table 4.2^ANOVA Table for the Algebra Subtestof Population A^ 3 7Table 4.3^Computational Results of A Two-Factor A=4by B=2 Fixed-Effects ANOVA on the AlgebraSubtest of Population B^ 3 8Table 4.4^ANOVA Table for the Algebra Subtestof Population B^ 3 8Table 4.5^Computational Results of A Two-Factor A=4by B=2 by C=2 Fixed-Effects ANOVA on theGeometry Subtest of Population A^ 3 9viviiTable 4.6^ANOVA Table for the Geometry Subtestof Population A^ 3 9Table 4.7^Computational Results of A Two-Factor A=4by B=2 by C=2 Fixed-Effects ANOVA on theGeometry Subtest of Population B^ 4 0Table 4.8^ANOVA Table for the Geometry Subtestof Population B^ 4 0Table 4.9^Set of Means for A 4 by 2 FactorialDesign^ 4 2Table 4.10^Set of Standard Deviations for A 4 by 2Factorial Design^ 4 3Table 4.11^Results of Hartley's Fmax Test forBritish Columbia^ 51Table 4.12^Results of Hartley's Fmax Test forHong Kong^ 5 3Table 4.13^Results of Hartley's Fmax Test forJapan^ 5 6Table 4.14^Results of Hartley's Fmax Test forOntario^ 5 9List of FiguresFigure 4.1^A Graphic Illustration ofTable 4.9^ 4 4Figure 4.2^A Graphic Illustration ofTable 4.10^ 4 5Figure 4.3^A Graphic Representation ofGender Differences and Cell Meansin Population A^ 4 7Figure 4.4^A Graphic Representation ofGender Differences and Cell Meansin Population B^ 4 8Figure 4.5^A Box Plot Illustration of AchievementDistribution in Algebra and Geometryin British Columbia (Population A)^ 5 2Figure 4.6^A Box Plot Illustration of AchievementDistribution in Algebra and Geometryin British Columbia (Population B)^5 3Figure 4.7^A Box Plot Illustration of AchievementDistribution in Algebra and Geometryin Hong Kong (Population A)^ 5 4Figure 4.8^A Box Plot Illustration of AchievementDistribution in Algebra and Geometryin Hong Kong (Population B)^ 5 6Figure 4.9^A Box Plot Illustration of AchievementDistribution in Algebra and Geometryin Japan (Population A)^ 5 7v iiiFigure 4.10 A Box Plot Illustration of AchievementDistribution in Algebra and Geometryin Japan (Population B)^ 5 8Figure 4.11^A Box Plot Illustration of AchievementDistribution in Algebra and Geometryin Ontario (Population A)^ 6 0Figure 4.12 A Box Plot Illustration of AchievementDistribution in Algebra and Geometryin Ontario (Population B)^ 61Figure I.1^A Cell Plot for the Algebra Subtestfor Population A^ 8 4Figure 1.2^A Cell Plot for the Algebra Subtestfor Population B^ 8 5Figure 1.3^A Cell Plot for the Geometry Subtestfor Population A^ 8 6Figure 1.4^A Cell Plot for the Geometry Subtestfor Population B^ 8 7Figure II. 1^Box Plots for the Algebra Subtestfor Population A^ 8 8Figure 11.2^Box Plots for the Algebra Subtestfor Population B^ 8 9Figure 11.3^Box Plots for the Geometry Subtestfor Population A^ 9 0Figure 11.4^Box Plots for the Geometry Subtestfor Population B^ 91ixChapter I. Statement of the ProblemIntroduction"Gender differences in mathematics learning continue to attractmuch research attention" (Leder, 1992, p. 597) and have become animportant component of both local research and internationalstudies.Many studies demonstrate that, on the average, female studentsdo as well as male students in mathematics achievement inelementary schools, but male students tend to outperform femalestudents in secondary schools (Armstrong, 1981; Burton et al., 1986;Ethington & Wolfle, 1984; Fox, 1980; Fennema, 1980, 1984; Leder,1985; Peterson & Fennema, 1985). This pattern, however, has beenchallenged by findings from more current studies which showed anarrowing trend in the gender gap over time (Becker & Hedges,1988; Braine, 1988; Friedman, 1989; Hyde et al., 1990; Kimball, 1989;Rosenthal, 1988; Willms & Kerr, 1987).A number of theoretical models have been proposed to explaingender differences. Several researchers (Deaux & Major, 1987; Eccleset al., 1985; Fennema & Peterson, 1985) have examined social orcultural interaction with gender. Others, like Reyes and Stanic(1988), focused on the effects of the mathematics teacher andclassroom environment on gender differences in mathematicsachievement. A model posed by Leder (1986) favored the influenceof multiple variables, such as environment, curriculum, andcognition, upon gender differences.An examination of these models showed that there was overlapof some factors, such as the impact of social or cultural environmenton gender differences (Leder, 1992).Although the models were proposed with the aim of beinggeneral or universal in terms of gender differences in mathematics,all were based largely on local research rather than internationalstudies. An attempt is needed to develop and test models from across-national perspective.Several international studies on mathematics achievement havecontributed to our understanding of gender differences. For1example, The First International Mathematics Study (FIMS)conducted by The International Association for the Evaluation ofEducational Achievement (IEA) between 1966 and 1973 found that,in mathematics, boys outperformed girls at the ages of 10 and 14(Comber & Keeves, 1973).The Second International Mathematics Study (SIMS) confirmedthe general superiority of boys over girls, although there were anumber of mathematical areas where girls outperformed boys(Robitaille & Garden, 1989).Findings are not always consistent in terms of when genderdifferences begin and whether gender differences exist in every areaof school mathematics. Little systematic research has examinedgender differences across school grades and mathematical areas.Furthermore, very few researchers have investigated potentialinteractions among some crucial factors which might have importantaffects on gender differences. Statistically, an interaction is said toexist if the effect of one variable on another variable is not the samefor all levels of the second (Glass & Hopkins, 1984). One of the majorgoals in the present study was to search for possible interactioneffects between gender and educational system across populationsand mathematical areas by employing the SIMS data.In SIMS, an educational system referred to a province (such asBritish Columbia, Canada), a special region (such as Hong Kong), or anation (such as Japan). Two different populations were involved inSIMS. Population A was defined as "all students in the grade wherethe majority have attained the age of 13.00 to 13.11 years by themiddle of the school year" (Travers et al., 1989, p. 6). Population Bwas defined as "all students who are in the normally acceptedterminal grade of the secondary education system and who arestudying mathematics as a substantial part of their academicprogram" (Travers et al., 1989, p. 7).Internationally, we would not necessarily expect consistency ingender differences because of discrepancies across differenteducational systems. Therefore, international studies of interactionscan provide researchers with a better understanding of the2formation of gender differences and their relationships to otherfactors such as the educational system.International, large scale assessments employ mathematicsachievement tests that have been shown to be highly reliable andvalid. They have also been conducted with large random samples.In this sense, international, large scale assessments provideresearchers with extensive data to study gender differences. Inparticular, they provide an opportunity to make internationalcomparisons on gender differences. This is where universal patterns,if any, can be found and unique gender differences in variouseducational systems can be studied.However, very little literature has addressed internationalcomparisons on critical issues such as the variability of mathematicsachievement between male and female students or genderdifferences across school grades and mathematical areas. This studyattempted to fill the gap in these areas by searching for potentialpatterns of variability of mathematics achievement (between gendergroups) and gender differences across populations, mathematicalareas, and educational systems.Need for the StudyResearch in the last decade has shown that mathematicsachievement was influenced by many factors, of which gender wasone. Gender differences in mathematics are very complicated innature. Not only do influential factors such as age, mathematicalarea, family background, socioeconomic status (SES), and teacher'sattitude exist; but, more importantly, possible interactions existamong those factors. In the past, however, few studies haveconsidered multiple influential factors simultaneously."In recent years behavioral and educational research has becomeincreasingly concerned with assessing interaction effects" (Glass &Hopkins, 1984, p. 403). In addition to studying the effect of onevariable on the other, researchers now pay more attention towhether this effect remains the same for every level of the secondvariable. The reason for this is that when interactions do existamong certain factors, the research focus should be on that group of3factors as a whole rather than on each individually separated factor.It becomes pointless, to some extent, to discuss an individual factor.For example, when the academic achievement of male andfemale students from two different nations are examined, it is notenough to report which national group of students achieves at ahigher level or whether female students outperform males.Researchers sometimes are more interested in whether one genderdoes better in one nationality, and the other gender does better inthe other nationality. When there is an interaction betweennationality and gender, researchers should conduct furtherexaminations to illustrate the nature of this interaction.Studies of interactions among some crucial factors can suggesttheoretical structures of how the factors are linked together and ofthe role each factor plays. Therefore, by looking for possibleinteractions, researchers are able to provide evidence to develop ortest theories and models in the area of gender differences inmathematics. Researchers are also able to compare and contrastfindings concerning interaction effects with previous findingsignoring interaction effects, so that further research questions can beraised and studied.Statistically, the more factors researchers take into account, thericher the information they can get (Glass & Hopkins, 1984). What ismore important, however, is that by considering multiple factorssimultaneously, researchers have an opportunity to examinepotential interaction effects among the factors selected. A factorialdesign was chosen for this study to explore interaction effectsbetween gender and educational system across populations andmathematical areas.The factors of age (population) and mathematical area are verybasic in terms of gender differences in mathematics achievement,and they have been the research foci of many mathematicseducators. Studies of these factors may provide practical suggestionsand guidance for teachers, and they may also furnish researcherswith insights into gender differences for theoretical development.When gender differences are considered from an internationalperspective, educational system is an important factor. Educational4researchers were encouraged to include different educationalsystems in their studies of mathematics achievement because cross-system comparisons enrich the understanding and theories abouthow students learn mathematics and promote educational progressin each educational system (Stigler & Perry, 1988). Furthermore,research on potential interactions of educational system with gendershould contribute to a better understanding of gender differences inmathematics achievement.They are the major reasons why the factors of gender,population, mathematical area, and educational system were selectedfor examination in this study. Instead of treating them separately,however, this study examined potential relationships among them.Four educational systems were involved in this study: BritishColumbia, Canada; Hong Kong; Japan; and Ontario, Canada. Therewere two main reasons to include them in this study.Recently, the differences in mathematics achievement betweenAmerican students and Asian students have been investigated inseveral international assessment projects such as Attributions ofSuccess in Mathematics, Correlates of Achievement, Dallas Times-Herald Survey, Development of Mathematical Thinking, Japan/IllinoisStudy, and Mathematics Achievement in the Elementary Grades: TheMichigan Studies. Many individual researchers in America (Schaub& Baker, 1991; Sowder & Sowder, 1988; Stigler, 1987; Uttal, 1988;Westbury, 1992) also focused on the same issue by using data fromFIMS, The International Assessment of Educational Progress (IAEP),and SIMS, although they did not have the financial possibilities toconduct cross-national projects. All these practices clearly reflect theimportance of understanding the differences in learning betweenAmerican students and Asian students. However, the differences inmathematics achievement between Canadian students and Asianstudents were hardly addressed in previous studies. The presentstudy attempted to add this important part to the literature.Therefore, British Columbia and Ontario were selected for this studyfrom the three North American participants in SIMS (United Stateswas another participant).5On the other hand, Hong Kong and Japan were selected for thisstudy from the three Asian participants in SIMS (Thailand wasanother participant) because they are in the similar level of economicdevelopment as the two provinces of Canada. Similar economicbackgrounds increase the comparability of students' achievement indifferent educational systems.Another area where further exploration is necessary is theextent to which male and female students differ in mathematicsachievement. A greater degree of variability of male achievementhas been found across many school grades (Hieronymus, Lindquist, &Hoover, 1982; Martin & Hoover, 1987; Sabers, Cushing, & Sabers,1987). Benbow and Stanley (1980) reported 14 times as many malesas females scoring over 660 on the mathematics test of the ScholasticAptitude Test (SAT), showing a male domination of the top positionsin the achievement distribution. Is this a universal phenomenonacross school grades? How does the variability of male and femaleachievement change across mathematical areas? These are thequestions which need further studies. This thesis addressed thesematters from a perspective of cross-system comparisons. It was alsoone of the purposes of this study to compare and contrast situationsof gender differences in two populations (Populations A and B) andin two mathematical areas (algebra and geometry) based on randomsamples taken from SIMS.The issue of gender differences has become one of the mostcontroversial topics in mathematics education (Leder, 1992). Forinstance, Willms and Kerr (1987) reported a decreasing trend in thegender gap at the intermediate level over the past decade. Someresearchers (Fennema & Sherman, 1977, 1978; Lee & Ware, 1986)have attributed the gender gap at the high school level to themathematics courses taken by male and female students.On the other hand, other research shows that not only did thegender gap still exist at the intermediate level, but it tended tobecome much larger at the high school level (Burton et al., 1986;Peterson & Fennema, 1985). Still other researchers suggests that thecourses taken by students were not a critical factor (Armstrong,1980; Crosswhite, Swafford, Mcknight & Cooney, 1985). There is a6need to clarify the controversy. One solution to this controversy is tofully examine the gender gaps across populations, mathematicalareas, and educational systems. This is where the SIMS data isapplicable.Research QuestionsThis study compared mathematics achievement of male andfemale students, searching for interactions between gender andeducational system and looking for patterns, if any, of genderdifferences across populations, mathematical areas, and educationalsystems using the SIMS data. There were two populations in SIMS:Populations A and B. Data of mathematics achievement tests fromthose populations were utilized in this study.As mentioned in the previous section, there were three majorresearch purposes in this study: 1) investigating possible interactioneffects between gender and educational system across populationsand mathematical areas; 2) analyzing the variability of mathematicsachievement between male and female students; and 3) comparingand contrasting situations of gender differences in two areas ofschool mathematics. There were two clusters of research questions.Question Cluster 1 combined the first and the third researchpurposes, and Question Cluster 2 reflected the second researchpurpose. The detailed research questions were:Question Cluster 1: Are there any gender differences in mathematicsachievement across populations, mathematical areas, and educationalsystems?a. Are there any interaction effects between genderand educational system?b. In algebra, are there any gender differences in Population Aand Population B?c. In geometry, are there any gender differences in PopulationA and Population B?d. What is the situation of gender differences in eacheducational system?7Question Cluster 2: Are there any gender differences in thevariability of mathematics achievement?a. In Population A, how does the variability differbetween male and female students in algebra and ingeometry across the four educational systems?b. In Population B, how does the variability differbetween male and female students in algebra and ingeometry across the four educational systems?c. Are there any patterns in this area?Limitations of the StudyThe main purpose of this study was to find out if there were anysignificant interaction effects between gender and educationalsystem, or any significant gender differences, in algebra andgeometry by using achievement data of Populations A and B fromSIMS. A search for possible causations of significant interaction ormain effects, if any, was beyond the scope of this thesis. In otherwords, this thesis aimed to answer the "if" questions. The "why"questions are left for further studies. Although potential causationswere conjectured occasionally in this study, the purpose was tosuggest the directions for further studies rather than to furnishexplanations for existing gender differences in mathematics.Although this study took multiple influential factors intoaccount, it looked at the affects of those factors on genderdifferences only in terms of two mathematical areas, algebra andgeometry. Mathematical competence as a whole was not theresearch focus in this study. Without attaching the word "algebra" or"geometry" to modify the meaning of "mathematics", it is notappropriate to say, for example, boys outperform girls inmathematics. Because of this, findings in this study were limited intheir generalizability for more general situations in mathematics.Finally, certain factors should be kept in mind when interpretingand generalizing the results of the achievement in algebra andgeometry. Under the context of SIMS, a group of items related to,say, the algebra part in the mathematics curriculum was used toreflect algebra as a mathematical branch or area. It is possible for8some educational systems to have different contexts under whichthey form their opinions of what algebra should be. Therefore,generalizations are limited to only those educational systems whichhave the similar context to that of SIMS in terms of algebra as amathematical area.9Chapter II. Literature ReviewOverviewThis chapter is a review of the literature on gender differencesin mathematics achievement. Since interaction effects among factorsconcerned in this thesis were rarely addressed in previous studies,the focus is on features of gender differences across grades,mathematical areas, and educational systems separately. A briefhistorical review is first provided to form the basis for discussion.Historical BackgroundHistorically, there was a long-standing, common belief thatacademic work was not appropriate for females (See Tyack & Hansot,1988.). In the early years of coeducational schooling, some (Brooks,1903; Gay, 1902) questioned if girls were physically and mentallysuitable for secondary schooling. Some people believed that normalschooling could damage a female's physical and mental health (SeeTyack & Hansot, 1988.). When girls began attending schools,arguments about the different needs of boys and girls arose.Mathematics always was a major issue in this debate. For example,Kroll (1985) reported one American writer's view as: "Woman...isorganized both bodily and mentally for dealing with an entirelydifferent set of functions, in which mathematics plays a small part"(p. 8). In England, the Board of Education in 1899 claimed thatit... "was prepared to argue that girls did in fact lack the samecapacity for mathematics as boys" (Hunt, 1987, p. 13). Girls inelementary schools in Australia were thought to be suited to sewingand needlework, and boys to arithmetic studies (Zainu'ddin, 1975).At the secondary level in Australia, only boys from rich familieswere thought to be suited to studying mathematics (Clements, 1979).Historically, there were different explanations for genderdifferences in mathematics. Morrison (1915) and Thomas (1915)attributed gender differences in mathematics to differences in theinnate abilities of men and women. Highly related to this was thebelief that boys were naturally good at mathematics, while girls weresuited best to literary subjects and modern languages (See Clement,1 01989.). Others such as Armstrong (1910) and Dean (1909) thoughtthat gender differences in mathematics were caused by thedifferences in the interests and perceptions in future applications ofmathematics between men and women.This brief, historical background is intended to provide a basisfor understanding gender differences in mathematics incontemporary contexts. Although we can still see the shadow oftheoretical explanations of causations posed a century ago, moderneducators have gone into in-depth observations and analyses ofgender differences.Gender Differences across Grades and Mathematical AreasThe traditional belief of males' superiority over females inmathematics achievement leads contemporary researchers toquestion whether or not males outperform females across all schoolgrades and in all mathematical areas. These two questions areconnected and can usually be answered together. Therefore thisreview of literature looks at gender differences across grades andmathematical areas at the same time. By putting those findingstogether, some common conclusions can be generalized and a pictureabout gender differences in mathematics can be created.Early in 1968, Glennon and Callahan reported a prevalent malesuperiority over females in mathematics achievement fromkindergarten to high school. This absolute outperformance of malesover females was soon questioned by other researchers (e.g.Fennema, 1974). In 1976 and 1978, Fennema again questionedfindings from large-scale studies which showed that genderdifferences were not always consistent in lower elementary schoolsbut were usually obvious in upper elementary and secondary schools(Maccoby & Jacklin, 1974; Hilton & Berglund, 1974). Her point wasthat large-scale comparisons ignored the background of males andfemales, namely, the mathematics courses they had taken in the past.She proposed that if the amount of time males and females spent inlearning mathematics was roughly equal, no distinct genderdifferences would be found (Fennema, 1980). This belief was11supported by other research findings (Fennema & Sherman, 1977,1978; Schonberger, 1978; Wise, 1978).However, some researchers later suggested that the number ofmathematics courses taken in the past by male and female studentswas not a critical factor in gender differences, because they stillfound significant gender differences at the high school level evenwhen they controlled for the mathematics background of students(Armstrong, 1980; Crosswhite, Swafford, Mcknight & Cooney, 1985).Recently, many studies were conducted in different grades andin various mathematical areas to investigate the characteristics andpatterns of gender differences in mathematics. The purpose of thesestudies was to go deeply into the specific factors (different gradesand different mathematical areas) to clarify the kinds ofcontradictions mentioned above. Some important findings aredescribed below in the hope that readers can make sense of thedifferent patterns which are illustrated at the end of this section.In arithmetic, two studies done in North America are worthmentioning. One is a five-year study conducted by Callahan andClements (1984) on counting skills of pupils in grade 1. Only slightgender differences in this area were reported in favor of boys. Theother is a large-sample study where results of assessment tests,which were administered to all pupils in grade 6 in California publicschools from 1976 to 1979, were examined. On the arithmetic tests,girls outnumbered boys in using irrelevant rules, choosing anincorrect operation, and making mistakes in negative transfer. Onthe other hand, boys tended to make more errors of perseveranceand formula interference (Marshall, 1983).In Britain, Shuard (1986) examined data from the Assessment ofPerformance Unit (APU) between 1980 and 1982. Students of 11years old from England and Wales were tested for their mathematicsachievement. She found that girls outperformed boys in only onearea, computation of whole numbers and decimals. On the otherhand, boys performed better in other arithmetic areas such asconcepts of whole numbers, decimals and fractions, and applicationsof numbers.12In elementary geometry, Schultz and Austin (1983) investigatedthe effects of transformation types and directions on pupils'understanding of transformations. They found no significantdifferences between boys and girls in grades 1, 3, and 5. In grade 4,there were no gender-related differences in graph comprehension(Curio, 1987). Shuard (1986), however, reported a superiority ofboys over girls in measurement and geometry in grade 5. As far asusing learning aids is concerned, Noss (1987) conducted a study inwhich pupils learned geometrical concepts through Logoprogramming. He found that Logo benefited girls much more thanboys. He described the difference as a "consistent trend." In grade 6,Marshall (1983) found more misuse of spatial information by girlsthan boys in his study.In secondary algebra, girls performed as well as boys and insome topics they outperformed boys in the achievement of basicalgebra (Swafford, 1978). In terms of problem solving in algebra,gender differences in grade 7 were "either very few or nonexistent"(Szetela & Super, 1987). However, Linn and Pulos (1983)investigated the proportional reasoning of boys and girls in grades 7,9, and 11, and found that boys tended to be more successful thangirls in proportional reasoning. Also, the computational estimationand related mathematical skills were reported to favor boys in grade8 (Rubinstein, 1985).Hanna (1986) used data from The Second InternationalMathematics Study (SIMS) to examine gender differences in fivemathematical areas in Population A, and found no significant genderdifferences in the achievement of algebra. Another analysis(Robitaille, 1989) using the same data source indicated that, on thealgebra subtest, mean scores of girls were significantly higher thanthose of boys in 4 out of 19 participating educational systems. ForPopulation B in SIMS, a superiority of boys over girls was reportedby Garden (1989) on the algebra oriented tests in 4 out of 15participating educational systems.In secondary geometry, Curio (1987) found no gender-relateddifferences in graph comprehension in grade 7. In terms of logicalreasoning and use of problem-solving strategies in geometry, Battista13(1990) did not find any gender differences in spatial visualization,geometry achievement, and geometric problem-solving tasks.An analysis of data from SIMS showed statistically significantgender differences in geometry and measurement in Population A(Hanna, 1986). Robitaille (1989) again examined data of PopulationA from SIMS, and reported boys superior to girls in geometry in 5out of 19 educational systems and in measurement also in 5 out of19 educational systems. At the same time, a parallel study onPopulation B of SIMS by Garden (1989) indicated an outperformanceof girls by boys in geometry in 5 out of 15 participating educationalsystems.For other mathematical areas such as probability and statistics,no significant differences between males and females were found inPopulation A from SIMS (Hanna, 1986). Garden (1989), on the otherhand, reported that in 14 out of 15 educational systems participatingin Population B of SIMS, males did better than females in probabilityand statistics. At the same time, in 6 out of 15 participating systems,males outperformed females in calculus.In the past decade, some large-sample studies on generalmathematics achievement continued to draw much attention fromresearchers. Leder (1982) used the Tests of Reasoning inMathematics (TRIM) to measure mathematics performance ofstudents in grades 7, 10, and 11. She reported higher performanceof males in grades 7 and 10, but no significant gender differences ingrade 11. From a series of studies, Benbow and Stanley (1980, 1983)found that highly gifted boys and girls showed large genderdifferences favoring boys on the SAT-M examinations in grades 7and 8. Smith and Walker (1988) studied data of 1979 New YorkState Regents mathematics examinations and found that femalesoutperformed males in mathematics achievement in grades 9 and 11,but in grade 10, males did better.In terms of larger national or international research, FIMSshowed that in mathematics boys outperformed girls at the age of 10and 14 (Comber & Keeves, 1973). A study named Women inMathematics and The National Assessment of Educational Progress(NAEP) (1977-1978) stated that:14Females enter high school with mathematical skills thesame as or greater than males. Sometime during the highschool years, males catch up with and even surpassfemales in certain areas of achievement. (p. 364)Findings similar to those of NAEP studies were reported from ananalysis of SIMS data (Crosswhite et al., 1985). The third NAEP in1983 released a report which identified a trend that genderdifferences in mathematics could occur as early as grade 4. Thisfinding awaits more evidence and support from other studies.One of the most important aspects of international or national,large-scale assessments is that their more extensive data allowresearchers to discover potential common patterns involved in somecritical issues. Robitaille (1989) found some patterns of genderdifferences in mathematics from analyzing achievement scores ofPopulation A in SIMS:Results show that girls are more likely to achieve betterthan boys in computation-level arithmetic, wholenumbers, estimation and approximation, and in algebra.Boys tend to be better in geometry and measurementsubtests, and in proportional thinking. The most markedgender differences appear to lie in the domains oftransformational geometry, proportional thinking, andstandard units of measurement, all of which favor boys ina majority of systems. (p. 121-122)Garden (1989) analyzed mathematics achievement of PopulationB in SIMS and found that boys tended to achieve higher in numbersystems, calculus, probability and statistics. They were also morelikely to succeed in algebra, geometry (including trigonometry andanalytic geometry), and elementary functions. Girls, on the otherhand, might outperform boys in set relations and operations. Itseems that the findings draw an uneven picture which shows malesuperiority over females at the high school level.15In light of these studies, girls are generally weak in spatialproblem solving and mathematical reasoning. It is usually the casethat males tend to outperform females in measurement,proportionality, geometry, spatial reasoning, analytic geometry, andtrigonometry (Battista, 1990; Fennema, 1980; Fennema & Carpenter,1981; Garden, 1989; Hanna, 1986; Linn & Pulos, 1983; Marshall,1983; Pattison & Grieve, 1984; Robitaille, 1989; Shuard, 1986; Wood,1976), as well as in applications of mathematics (Martin & Hoover,1987; Sabers et al., 1987).On the other hand, girls have been found to perform better incomputational skills (Brandon, Newton, & Hammond, 1987; Fennema,1974; Jarvis, 1964; Johnson, 1987; Meece, Parsons, Kaczala, Goff, &Futterman, 1982; Robitaille, 1992), set operations (Robitaille, 1989;Wood, 1976), and symbolic relationship reasoning (Pattison & Grieve,1984).Since the previous findings did not always agree with oneanother, it was difficult and controversial to try to reach a generalconclusion about gender differences in mathematics. However, onepattern was proposed by quite a few researchers, that is, girls wereat least as able as boys in mathematics in early school years; butboys began to catch up with girls in mathematics during theintermediate school years (around grades 7 to 9) and during highschool years boys tended to outperform girls in many mathematicalareas (Armstrong, 1981; Block, 1976; Burton et al., 1986; Ethington &Wolfle, 1984; Fox, 1980; Fennema, 1974, 1980, 1984; Leder, 1985;Maccoby & Jacklin, 1974; Peterson & Fennema, 1985).This pattern has been challenged by more recent studies.Willms and Kerr (1987) reported an evidently decreasing trend inthe gender gap at the intermediate level over the past decade. Inhigh schools, Friedman (1989) found that, although male students didslightly better than females, gender differences appeared to benarrowing with time. Other researchers also reported either nogender differences in the performance of mathematics or a decreasein gender differences over time from their more current studies(Becker & Hedges, 1988; Braine, 1988; Friedman, 1989; Hyde et al.,1990; Kimball, 1989; Rosenthal, 1988). Therefore, in the opinions of16these researchers, a narrowing trend in the gender gap wasbecoming more evident over time.Gender Differences across Educational SystemsAlthough gender differences in mathematics are repeatedlyreported in local assessments, only a few international studies haveaddressed this matter. In those studies, however, gender differenceswere found not to be consistent across different educational systems.FIMS, involving 12 educational systems, was conducted between1966 and 1973. It showed "clear differences in favor of boys in allpopulations in both verbal and computational scores" (Husen, 1967,p. 241). However, gender differences within educational systemswere found not to be significant. This caused Husen to predict thatgender differences were a system phenomenon and might even tendto favor girls from an international perspective. Comber and Keeves(1973) also analyzed data from FIMS and found that in mathematicsboys outperformed girls at the ages of 10 and 14 only whenconsidering all topics as a whole. The findings were consistent withthose of Husen.SIMS also provided an opportunity to examine genderdifferences across different educational systems. Hanna andKundiger (1986) examined mathematics performance of students inPopulation A in 20 educational systems and described their findingsin this way:Most of the differences did not reach statisticalsignificance at the 1% level.^Moreover, the differencesthat did reach statistical significance were not large,ranging from +5 to -7. Looking at each subtest separatelyit appears that for two of the five topics, Measurement andGeometry, the significant differences occurredconsistently in the boys' favor: in 7 of 20 countries boyshad higher p-value and in 10 countries boys had higher 1)-value in Measurement and Geometry, respectively. (p. 6-7)17Hanna et al. (1988) did another study, also using data fromSIMS, and found significant male superiority over females on at leastone subtest of seven in 14 out of 15 educational systems. In all, theymade 105 comparisons and 62 of them were found statisticallysignificant favoring boys.In the last several years, SIMS data have continued to attract theattention of researchers. Robitaille (1989) examined results ofmathematics achievement of students in Population A in SIMS. Thedifferences between boys' mean scores and girls' mean scores wereexpressed as percentage points. Robitaille (1989) stated the criteriaof the analysis:A difference of three percentage points was chosen as athreshold level of practical significance, and only thosewhich exceeded this level in absolute value have beenincluded. Since samples were large in all systems, such adifference would be statistically significant at a very highlevel; but for some systems, the large design resultingfrom cluster sampling means that a difference of 3percentage points would not be statistically significant. (p.120)18Robitaille's findings are reported in the following table.Table 2.1 Mathematics Achievement of Population A in SIMS(boys' mean - girls' mean)Arithmetic Algebra Geometry Measurement StatisticsBelgium^(Fl) -6 -5 - - -4Belgium^(Fr) -4 -5 - - -5B.C. - - + + +Ontario + + + +4 +Finland -4 -6 0 - -5France +4 + +6 +6 +Hong Kong + + + + +Hungary - -5 + + -Israel + - + +4 +Japan - - + + +Luxembourg +4 - +4 + +Netherlands +4 + +5 +7 +New Zealand + - +4 +6 +Nigeria + + +4 + +Scotland + + + + +Swaziland + + + + -Sweden -4 - - - -Thailand - - - - -U.S.A. - - + + -Some common characteristics could be concluded from the table.Most educational systems showed that males were more likely toscore higher than females in geometry and measurement. However,male students did not have more opportunities than females to besuccessful in arithmetic. In algebra and descriptive statistics, allsignificant gender differences which appeared favored girls. Hanna(1989) had findings similar to these results.19Overall, male students did not show significant superiority overfemales at this stage, female students achieved higher than males insome mathematical areas.For Population B in SIMS, Garden (1989) did a similar analysisand reported the findings in the following table:Table 2.2Mathematics Achievement of Population B in SIMS(boys' mean - girls' mean)Set Number System Algebra Geometry FunctionsBelgium^(F1) + +8 +4 + +6Belgium^(Fr) +5 +5 +8 +6 +8British^Columbia -6 + + + +Ontario +5 + + +England & Wales - + + + +Finland +5 +5 + +8 +6Hong Kong. + +5 + +5 +9Hungary +6 +10 +12 +8 +9Israel - +4 +6 + +4Japan + +5 + +5 +12New Zealand - + + + +Scotland + +5 + + +Sweden - 0 + + +Thailand - + + + +United^States + + + + +As with Population A, the differences between boys' mean scoresand girls' mean scores were expressed as percentage points. Becausethe sample size for Population B was smaller than that for PopulationA, a difference of four percentage points was taken as being ofpractical significance. Compared to the table for Population A, "themost noteworthy difference...is that mathematics has become apredominantly male domain in most systems" (Garden, 1989, p. 145).20Gender differences appeared to be evident in this population. Of the29 comparisons which showed significant gender differences, 28were in favor of boys. Although gender differences in someeducational systems did not reach statistical significance, genderdifferences in other educational systems such as Belgium, Finland,and Hungary were quite large. It seems that the table painted apicture of more opportunities for boys than for girls to succeed inmathematics at the senior high school level.Other findings from international large-scale assessments areavailable from Schmidt and Kifer (1989) who used regressioncoefficients as indexes to illustrate gender differences. ForPopulation A in SIMS, they reported that:...across subtests, France shows the largest differencefavoring boys. Hungary, Sweden, England and Wales, andBelgium (Flemish) have differences which favor girls.Only one of these differences, however, is statisticallysignificant.^Such patterns give rise to questions about theextent to which cultural differences and societalexpectations are related to different levels of achievementfor girls and boys. (p. 223)A totally different statistical method called median polishing(Tukey, 1977; Velleman & Hoaglin, 1981) was used by Ethington(1990) to examine gender differences in Population A based on theSIMS longitudinal data. Median polishing does not test hypothesisdirectly but decomposes data into relevant effects. The meaning isderived from the relationships among those effects.Eight educational systems in the longitudinal study wereexamined. The results of analysis are listed in Table 2.3 where"Same" means no gender differences; "Boys" means males superiorover females; and "girls" vice versa.21Table 2.3 Mathematics Achievement of Population ALongitudinal Study in SIMSU.S. Belgium B.C. Thai France N. Z. Ontario JapanAll items same girls girls girls boys boys same sameFractions same girls girls girls boys boys boys sameRatio same same same girls boys boys same boysAlgebra same girls same girls boys boys boys boysGeometry same same same girls boys boys boys sameMeasurement same girls same boys boys boys same boysEthington (1990), using the above table, wrote:The results of the analyses reported above indicate nosubstantial gender effects in any of the content areas, andthe slight effects shown favored girls more often thenboys. ... For example, previous studies found males toperform better than females on problems dealing withproportionality, yet these results show females in Thailandscoring almost 5 percentage points higher than males onthe ratio/proportion/percent items.^Furthermore, withinno content area were males found to persistentlyoutperform females across countries or vice versa. (p. 79)However,The results reported here should not be interpreted interms of significant differences among the groupsdescribed. The median polish method used in the analyseswas an exploratory method that dose not test a priorihypotheses, but rather decomposes the data producing22patterns of effects or deviations therefrom that are notnecessarily apparent in the summary data. (p. 79)Overall, gender differences, in Ethington's opinions, were weak,not always in favor of boys, and not consistent across differenteducational systems.Besides studies using IEA data, Walberg, Harnisch, and Tsai(1986) examined gender differences in mathematics achievement intwelve educational systems by carefully controlling productivityfactors and reported significant superiority of boys over girls in eightsystems and no significant gender differences in the other four.Among the few international assessments, two distinct pointscan be made. First of all, the findings were not consistent. In someeducational systems, an overall trend, from cross-system studies,was that the gender gap in mathematics achievement tended tobecome large and more evident as students approached the end ofhigh schools. In other systems, however, gender differences weresmall and weak all the time.Also, no clear, common patterns of gender differences haveappeared. Different educational systems showed uniquecharacteristics of their own. For instance, research studies have notprovided strong evidence on how the East and the West differ interms of gender differences in mathematics. Within the samegeographical region, on the other hand, the results of analysis werenot consistent either (See Japan and Thailand in the East; France andBelgium in the West.).A need is obvious to conduct further international comparisonsso as to discern potential patterns, if any, and to reach reasonableconclusions.23Gender Differences on Variability of AchievementBesides the controversial pattern that male students are morelikely to outperform their female counterparts somewhere duringthe course of mathematics learning in high school, there existsanother important aspect of gender differences.Bentzen (1966) studied disabled students who had suffered mildmental handicaps, learning disabilities and other handicaps whichcould cause different kinds of learning difficulties. He found theratio of the number of boys to girls was seven to one, indicating thatthere were many more boys than girls located on the bottom line ofthe achievement distribution.Benbow and Stanley (1980), on the other hand, reported that thenumber of boys who achieved above 660 on the mathematicsachievement test of the Scholastic Aptitude Test (SAT) was fourteentimes that of girls.Therefore, a picture in which male students dominate both thetop end and the bottom end of the achievement distribution appears.Other studies also emphasized the male domination at the top end(Fox, Brody & Tobin, 1980; Kerr, 1989).Another group of researchers, whose focus was not on theborder of the distribution but on the spread or variability of theachievement distribution between male and female students, studiedmathematics achievement of students based on the 1977 nationalstandardization of the Iowa Test of Basic Skills (ITBS), and reported alarger degree of variability or spread of achievement scores for boys(Hieronymus, Lindquist, & Hoover,1982). A similar result was foundin the work of Sabers, Cushing and Sabers (1987).Most evident was the finding from Martin and Hoover (1987)who examined another group of achievement scores in the ITBS.They reported a greater degree of variability of male scores on everysubtest and, most importantly, in every grade from three to eight.Recently, Willms and Jacobsen (1990) also found that, during theintermediate years of schooling, "the extent of this 'fan-spread' wasgreater for males than females" (p. 157).24Although there is not a large body of research on this matter, itseems that the findings obtained in the past are somewhatconsistent. That is, the variability of male achievement scores isusually larger than that of female achievement scores, and malestudents tend to dominate both the top and the bottom end ofmathematics achievement distribution.Potential CausationsThree theoretical models are frequently used to consider andexplain the potential reasons of gender differences in mathematics.The first model, the social interaction model (Deaux & Major,1987; Eccles et al., 1985; Fennema & Peterson, 1985), emphasizessocial or cultural interventions for gender differences. Robitaille andTravers (1992), when writing to introduce international large-scaleassessments, also emphasized that "sociocultural factors...may affectstudents' and teachers' attitudes and behaviors in highly significantways" (p. 701). The individual goals and self-schemata, theexpectations and attitudes of society, and the social context whereinteractions happen are other major concerns of this model.Other influential factors such as cultural milieu, perceptions ofthe value of mathematics, motivation, the desire to succeed,achievement behaviors, task-specific beliefs, past events, anddifferent abilities of students are also listed and considered in thismodel. Mathematics courses which students had taken wereemphasized by many researchers (Alexander & Pallas, 1983; Eccleset al., 1985; Ethington & Wolfle, 1984; Fennema & Sherman, 1977;Hanna, 1988; Jones, 1987; Moss, 1982; Olson & Kansky, 1981, Rosier,1980; Wise, Steel, & MacDonald, 1979). This factor is also taken intoaccount in this social interaction model. The model admits thatfactors are not stable because of the different characteristics ofstudents. It is the combination of several influential factors thattends to limit the students' ability to do mathematics.The second model, Reyes and Stanic's model (1988), focuses onthe affect of a mathematics teacher and a classroom on genderdifferences in mathematics achievement. This model pays attentionto social influences, teacher attitudes, mathematics curricula,25teaching methods, classroom processes, students' attitudes, and theirachievement motivations and behaviors.Many recent studies have shown the importance of taking thesefactors into consideration (Becker, 1981; Dunkin & Doenau, 1982;Eccles & Blumenfeld, 1985; Grieb & Easley, 1984; Hart, 1989; Harris &Rosenthal, 1985; Koehler, 1990; Leder, 1987, 1989; Moore & Smith,1980; Reyes, 1984; Spender, 1982; Stallings, 1979; Walden &Walkerdine, 1985). This model also takes into account the SES whichwas emphasized by other researchers as well (See Willms & Kerr,1987.).The third model by Leder (1986) favors the influence ofmultiple variables, such as environment, curriculum, and cognition,on gender differences. This model describes five variables whichLeder believes have the greatest affect on mathematics achievement.The first variable is environment which includes the society,home, school, and classroom. The importance of this variable wasaddressed by other researchers as well (Lee & Bryk, 1986). Thesecond variable links with personal interactions. Parents, peers,teachers, and socializers are within this variable. This variable wasconsidered to be important by many other researchers as well (Eccles& Jacobs, 1986; Lockheed, 1985; Ryan, Macphee, & Taperell, 1983).The third variable deals with curriculum and includes contents ofmathematics, item types, and methods of instruction and evaluation.Walkerdine (1988) argues that mathematics curriculum was one ofthe factors that created a set of social practices inside schools whichwere related to a set of social practices outside schools thatdisadvantaged women. She believed that was the reason for femalestudents falling behind males in mathematics. Spatial ability, verbalability, and reasoning ability belong to the fourth variable, thecognition variable. This is the way in which some researchersexplained gender differences (See Connor & Serbin, 1985.). The fifthvariable is called the psycho-social variable and includesachievement motivation, confidence, conformity, self-esteem andindependence. Joffe and Foxman (1988) also emphasized the affectof this variable on mathematics achievement.26An examination of the models shows overlap of some factors. Inthe words of Leder (1992), all the models...emphasize the social environment, the influence ofother significant people in that environment, students'reactions to the cultural and more immediate context inwhich learning takes place, the inclusion of learner-related affective, as well as cognitive, variables. (p. 609)Finally, it is worth mentioning that there is a controversialattempt to explain gender differences in mathematics. Stafford(1972) believed that biological variables should be considered whenexplaining gender differences.We, therefore, cannot discount the very importantinteraction effects of the environment (such as home life,choice of school, adequacy of teacher, proper attitude, andfamily support) with genetic endowment which results instudents with varying degrees of quantitative reasoningability. (p. 198-199)In support of this position, Petersen (1980) thought thatdifferent hormone levels might be one of the factors which affectedmathematics abilities. Brain lateralization was also posed as anotherpotential cause (Levi-Agresti & Sperry, 1968; Hubbard, Hanifin, &Fried, 1982; Sherman, 1977).However, to involve biological variables in theoretical models isquite controversial. Some researchers (Bryden, 1979; Buffery &Gray, 1972; Shepherd-Look, 1982) questioned the appropriateness ofa biological explanation of cognitive differences, but all admitted thatmore studies were necessary before final judgment about thereasonability of involving biological variables in theoretical modelscould be made.27SummaryOne of the suggested patterns of gender differences inmathematics is that females are as able, if not better, than males inmathematics in early school years but males begin to catch up withfemales in mathematics during intermediate school years (aroundgrades 7 to 9), and during high school years males tend tooutperform females in many mathematical areas. As mentionedearlier, more recent studies have suggested a narrowing trend ingender differences or gender gaps over time.With respect to large-scale studies, situations of genderdifferences were not always consistent across educational systems.The trend of gender differences suggested by each pattern can onlyapply to some educational systems. Although factors such asacademic area and educational system are frequently involved incross-system comparisons, very few researchers have takeninteraction effects into account. There is a need to add interactioneffects among some crucial factors into any literature concerninggender differences in mathematics. Question Cluster 1 was aimed toexplore this area.Despite the fact that there is not a large body of research on thevariability of mathematics achievement, the findings appearsomewhat similar. The variability of male achievement is usuallylarger than that of female achievement. Meanwhile, male studentstend to dominate both the top and the bottom end, while femalestudents tend to stay in the middle, of the mathematics achievementdistribution. However, this pattern has never been examined from across-system point of view. The present study attempted to fill inthe gap. Question Cluster 2 was expected to investigate some of thebasic concerns in this matter from a cross-system perspective.28Chapter III. ProceduresData SourcesThe goal of this section is to provide some basic informationneeded to understand SIMS.SIMS was conducted in 20 educational systems under thesponsorship of IEA. Its mathematics achievement tests wereadministered to more than 125,000 students in those 20 educationalsystems.Quantitative studies always require a high quality of sampling.SIMS provides such a sampling frame. According to Travers, Gardenand Rosier (1989),Sampling procedures for the second study were designedwith dual purposes: to allow population parameters to beestimated to a reasonable degree of precision, and to enablecross-national comparisons of these populationparameters, given sufficient background information, tobe made. (p. 11)The suggested sampling design was:(i) stratification based on groupings seen by eachNational Center as having some significance foreducation in their system;(ii) random selection of schools with probabilityproportional to size of the target group within eachschool;(iii) random selection of two classes within each school atthe target grade level. (p. 11)Under the assumption that all participating educational systemscarried out the sampling work according to the sampling instructionsby SIMS, large, stratified random samples of high quality could beexpected from that study.29It is also important and critical for quantitative studies todevelop or employ achievement tests with high reliability andvalidity. In SIMS, the International Item Grids (IIG) were developedfor this purpose. Garden and Robitaille (1989) described the work oftest constructions as following:Based on the results of a preliminary survey of themathematics curriculum in several systems, a content-by-cognitive-behavior grid was developed for each populationlevel. The content dimension of the Population A grid wassubdivided into five strands: Arithmetic, Algebra,Geometry, Descriptive Statistics, and Measurement. Thesewere then subdivided into a total of 133 finer categories.Population B content was subdivided into 9 strands: Set,Relations and Functions; Number System; Algebra;Geometry; Elementary Functions and Calculus; Probabilityand Statistics; Finite Mathematics; Computer Science; andLogic. These categories were further subdivided into 131finer categories. (p. 85)The "item pools" for all populations were then developed.The initial item pool for Population A consisted of 480items, and that for Population B consisted of 400 items.Items were collected form National Centers, from itemsreleased by the National Assessment of EducationalProgress in the United States, from items used in the firstIEA mathematics survey. A number of items also weredeveloped especially for use in this study. All of the itemswere presented in multiple-choice format with fivealternative responses. (p. 88)Based on the item pools, items were selected according to severalcriteria to form pilot tests for each population.30Three rounds of pilot testing of items were conducted ineight of the participating systems, and the results of thesetrials were used to select the final pool of items for use inthe study. (p. 89)In addition to the criteria used to form pilot tests, several othercriteria for the selection of items were set up to construct final forms.In light of all these efforts, mathematics achievement tests in SIMSmay be considered relatively high in reliability and validity.Design of the StudyThis study analyzed gender differences across two populations(Populations A and B), two mathematical areas (algebra andgeometry), and four educational systems (British Columbia, Canada;Hong Kong; Japan; and Ontario, Canada). Mathematics achievementdata from Population A and Population B in SIMS were utilized inthis study. A 4 x 2 (educational systems x gender) ANOVA wasdesigned to analyze achievement scores of students on the algebrasubtest for each population. The design is illustrated in Table 3.1.In geometry, a similar factorial design was considered for eachpopulation.For each of the planned 4 x 2 ANOVAs, there were twoindependent variables which were educational system and gender.Mathematics achievement on the algebra or geometry subtest foreach population was the dependent variable. The results of eachANOVA was used to answer the research questions in QuestionCluster 1.31Table 3.1 A Factorial Design (A 4 x 2 ANOVA) for the StudyEducational System(P)Gender^(Q)Male (Q2) Female (Q1)British Columbia (P1)Hong Kong (P2)Japan (P3)Ontario (P4)To answer Question Cluster 2, Hartley's Fmax test was used for asimple and quick examination for homogeneity of variance. The teststatistic Fmax is the ratio of the largest variance estimate and thesmallest variance estimate (Glass & Hopkins, 1984). S large 2Fmax =S small 2The box plots of male and female achievement scores in eachmathematical area were also examined and compared for QuestionCluster 2. The reason for using box plots is that they "have theadvantage over the mean and variance of not being sensitive tooutliers" (Myers & Well, 1991, p. 16).Before results of each ANOVA were interpreted, the design wastested for violations of ANOVA assumptions to make sure that theknowledge claims made in this study were credible.32The SampleThe sample size for each educational system was based upon twoconsiderations. The first, that the effect size, one of the mostimportant factors in deciding on sample size, has to be reasonable inorder to make findings in this study practical enough to be useful toeducators. The second, that the power of the statistical test shouldbe high enough for findings to be credible.According to general standards in educational research, an effectsize of 0.50 allows researchers to pick up differences of "middle size"which are strong enough in practice to be taken seriously. A largereffect size increases a danger of inflating Type II errors(overconservatism) while a smaller effect size tends to pick up slightdifferences which in practice should not be taken seriously.An a level of 0.05 was chosen in this study in order to get abalance between Type I and Type II errors. If a is too small, Type IIerrors (0) become large and power (1-(3) becomes small, which canmake results of the analysis questionable. On the other hand, if a i slarge, the possibility of Type I errors (a) become large. Type I errors(overoptimism) can mislead the direction of follow-up studiesseverely. Special care has to be taken to make a study reflect thephenomena under study.A power of 0.90 results in a Type II error of 0.10. To controlType II error and to avoid a danger of overpower, this power levelwas adopted for this study.To make an ANOVA design more robust to possible violations ofassumptions, it is also important to equate the number of subjects ineach group or cell of the design (Glass & Hopkins, 1984).For each planned ANOVA, when the effect size is 0.50, the poweris 0.90, and a is 0.05, the sample size can be decided by the formula33$ —and Pearson-Hartley power function charts. Thus, the sample size(n) for each cell is 50. For safety, a sample size of 60 subjects percell was finally decided.For each educational system, 60 boys from Population A wereselected. Their corresponding achievement scores in algebra andgeometry were located. Another 60 boys from Population B werechosen as well. They also provided achievement scores in algebraand geometry. In the same manner, 120 girls (60 from Population Aand 60 from Population B) were selected from each system. Theachievement scores in algebra and geometry were selected in thesame way as boys did. The selection was done by first separatingboys and girls in each educational system and then taking a randomsample of 60 subjects from the list of both boys and girls.By random sampling, subjects in each of the four educationalsystems were selected as shown below:Table 3.2The Sample Distribution for the StudyB.C.Male FemaleHong KongMale FemaleJapanMale FemaleOntarioMale FemalePopulation A 60^60 60^60 60^60 60^60Population B 60^60 60^60 60^60 60^60Data AnalysisFor clarity, it is necessary to describe how mathematicsachievement tests from which the data (scores) were obtained wereadministered in SIMS. Five test forms were prepared for students inPopulation A. There was one Core Test, four Rotated Forms for cross-sectional study and four other Rotated Forms for longitudinal study.The Core Test was required for all students and one Rotated Formwas assigned randomly to students in Population A in eacheducational system. For Population B, there were eight Rotated34Forms. Two randomly assigned Rotated Forms were required forstudents in population B in each educational system.Based on the distribution of items, the algebra and geometrysubtest scores for each student were calculated and then wereconverted to percentage points which became the data analyzed inthis study.35Chapter IV. Results and DiscussionsIn this chapter, the matter of ANOVA assumptions is brieflyaddressed, results of data analyses are presented by means of tablesand figures, and interpretations of ANOVA tables are provided.The Aptness of ANOVA ModelsIt is an important step in statistical analysis to check the aptnessof an ANOVA model used in a study in order to make the knowledgeclaims of the study more credible. There are three ANOVAassumptions which have to be met before any ANOVA model isapplied to analysis of data. They are the assumptions of normality,homogeneity of variance, and independence of residuals.The aptness of ANOVA models was checked in this study bymeans of cell plots and box plots (See Appendixes I and II.). Noserious violations of ANOVA assumptions were found. Normality,homogeneity of variance, and independence of residuals were allsatisfactorily fulfilled. Therefore, there were no serious, negativeeffects of violations on the level of significance.Furthermore, because the sample size of each cell is equal in thisstudy, ANOVA becomes robust with respect to violations ofassumptions (Glass & Hopkins, 1984). Hence, the aptness of ANOVAmodels used in this study was basically acceptable and theknowledge claims made in the following sections are credible.The ANOVA ResultsA two-factor analysis of variance (ANOVA) was used to analyzeachievement data on the algebra and geometry subtests for eachpopulation. All statistical null hypotheses were tested at an a levelof 0.05. The ANOVA results on the algebra subtest for Population Aare displayed in Tables 4.1 and 4.2.36Table 4.1 Computational Results of Means of A Two-Factor P=4 by Q=2Fixed-Effects ANOVA on The Algebra Subtest for Population A(Cell variances are given in parentheses; cell size = 60)Male FemaleBritish^Columbia 5 2 5 1(21.5) (21.9)Hong Kong 4 7 4 8(23.2) (21.3)Japan 67 68(23.8) (24.8)Ontario 46 41(17.7) (17.6)Table 4.2ANOVA Table for The Algebra Subtest for Population ASOURCE SS d f MS FBetweengroups (P)39303.79 3 13101.26 28.08*Betweengroups (Q)135.47 1 135.47 0.29P by Q 705.16 3 235.05 0.50Remainder 220184.78 472 466.49*p < 0.05The ANOVA results on the algebra subtest for Population B aredisplayed in Tables 4.3 and 4.4.37Table 4.3 Computational Results of Means of A Two-Factor A=4 by B=2Fixed-Effects ANOVA on The Algebra Subtest for Population B(Cell variances are given in parentheses; cell size = 60)Male FemaleBritish^Columbia 5 1 5 8(29.1) (29.3)Hong Kong 8 6 8 1(14.6) (19.9)Japan 73 67(21.2) (19.5)Ontario 5 6 5 6(23.9) (22.3)Table 4,4ANOVA Table for The Algebra Subtest for Population BSOURCE SS d f MS FBetweengroups (P)66113.79 3 22037.93 41.86*Betweengroups (Q)48.77 1 48.77 0.09P by Q 2986.71 3 995.57 1.89Remainder 248469.98 472 526.42*p < 0.05The ANOVA results (Tables 4.2 and 4.4) show that, on thealgebra subtest, there were no significant two-factor interactioneffects between gender and educational system in either PopulationA or Population B.As far as main effects were concerned, gender differences, factorQ, did not approach statistical significance in either Population A or38Population B. Significant differences were found, however, betweenthe means for factor P, educational system, at both the Population Aand Population B levels.Tables 4.5 and 4.6 summarize the ANOVA results for thegeometry subtest for Population A.Table 4.5 Computational Results of Means of A Two-Factor A=4 by B=2Fixed-Effects ANOVA on The Geometry Subtest for Population A(Cell variances are given in parentheses; cell size = 60)Male FemaleBritish^Columbia 4 5 4 6(21.0) (21.0)Hong Kong 45 4 3(19.2) (18.0)Japan 62 59(21.2) (19.5)Ontario 49 45(19.5) (20.3)Table 4.6ANOVA Table for The Geometry Subtest for Population ASOURCE SS d f MS FBetweengroups (P)21498.65 3 7166.22 17.96*Betweengroups (Q)496.13 1 496.13 1.24P by Q 526.95 3 175.65 0.44Remainder 188329.43 472 399.0039*p < 0.05The ANOVA results for the geometry subtest for Population Bare displayed in Tables 4.7 and 4.8.Table 4.7 Computational Results of Means of A Two-Factor A=4 by B=2Fixed-Effects ANOVA on The Geometry Subtest for Population B(Cell variances are given in parentheses; cell size = 60)Male FemaleBritish^Columbia 3 6 3 4(31.9) (30.7)Hong Kong 7 3 6 7(17.1) (20.0)Japan 56 46(23.8) (23.3)Ontario 4 6 4 5(23.4) (23.8)Table 4.8 ANOVA Table for The Geometry Subtest for Population BSOURCE SS d f MS FBetweengroups (P)78337.54 3 26112.51 42.85*Betweengroups (Q)2669.63 1 2669.63 4.38*P by Q 1376.05 3 458.68 0.75Remainder 287632.37 472 609.39*p < 0.05The ANOVA Tables 4.6 and 4.8 on the geometry subtests showsimilar results, in terms of interaction effects, as were found for the40algebra subtests. There was no statistically significant, two-factorinteraction effect in either Population A or Population B.As far as main effects were concerned, there were statisticallysignificant differences between the means for factor P, educationalsystem, for both Population A and Population B. In addition, factor Q,gender, was also of statistical significance in Population B. Malestudents outperformed female students on the geometry subtest forPopulation B from a perspective of across educational systems. Sincethere were no significant interaction effects between gender andeducational system, an interpretation of gender differences ingeometry could be developed for Population B.The means and standard deviations for each mathematical areaare listed in Tables 4.9 and 4.10. A graphical illustration for eachtable is also displayed (See Figures 4.1 and 4.2.). By putting the foursubtests together, these tables and figures can provide acomprehensive picture of achievement results for the students inand illustrates the variability of score distributions. Explanations aregiven in the next two sections.41Table 4.9Set of Means from Four Factorial Designs(Cell Size = 60)EducationalSystem (P)Gender(Q)Population A Population BAlgebra Geometry Algebra GeometryB. C. (P1)Male 52 45 51 36Female 51 46 58 33H.K. (P2)Male 47 45 86 73Female 48 43 81 67Japan (P3)Male 67 62 73 56Female 68 59 67 46Ontario (P4) Male 46 49 56 46Female 41 45 56 4542Table 4.10Set of Standard Deviations from Four Factorial Designs(Cell Size = 60)EducationalSystem (P)Gender(Q)Population A Population BAlgebra Geometry Algebra GeometryB. C. (P1)Male 21.5 21.0 29.1 31.9Female 21.9 21.0 29.3 30.7H.K. (P2)Male 23.2 19.2 14.6 17.1Female 21.3 18.0 19.9 20.0Japan (P3)Male 23.8 21.2 21.2 23.8Female 24.8 19.4 19.5 23.3Ontario (P4)Male 17.7 19.5 23.9 23.4Female 17.6 20.3 22.3 23.843Population A^MaleFemaleGeometryBC HK JAPAN ONTARIO HK JAPANJA ONTARIO^pi-BCAlgebra^GeometryPopulation BFigure 4.1 A Graphic Illustration of Table 4.944Male13-Year-Olds^Female 0Algebra^GeometryHK JAPAN ONTARIO^BC HK JAPAN ONTARIO4518-Year-OldsFigure 4.2 A Graphic Illustration of Table 4.10Interaction Effects and Gender DifferencesThe ANOVA results (Tables 4.2, 4.4, 4.6, and 4.8) show that whathappened in algebra and in geometry was the same in terms of two-factor interaction effects. There were no statistically significantinteraction effects between gender and educational system across thetwo mathematical areas and across the two populations. Therefore,gender, as a factor which influences the learning of mathematics,may have little interaction with the factor of educational system, atleast in algebra and in geometry.These findings suggest that gender differences in algebra andgeometry are not clearly affected by the educational system whichstudents attend. That is, mathematics teachers in the foureducational systems may face similar situations in terms of genderdifferences in algebra and in geometry.When considering main effects, different patterns in algebra andgeometry were found. In algebra, the achievement differencesamong the four educational systems were found to be statisticallysignificant for both populations. No further post hoc analyses werecarried out because the research interests were in genderdifferences. What is more important is that no statisticallysignificant gender differences in algebra were found in eitherpopulation. Both male and female students performed at about thesame level in algebra.In geometry, statistically significant differences were foundbetween the means of the four educational systems. Since theresearch focus was on gender differences, no further interpretationsabout the main effects of factor P were considered. What is differentin geometry is that gender differences reached statistical significancefor Population B in favor of boys. Males and Females in Population Bdiffered significantly in their ability to deal with geometry.A little explanation is needed here. Gender differences appearedstatistically significant only if all male students were put into onegroup and all female students were put into the other group, ignoringeducational system. But what of gender differences within eacheducational system? This matter was also examined in this study.Figures 4.3 and 4.4 present the achievement of male and female46students in each educational system. These graphs illustrate the 95percent confidence intervals of algebra and geometry means for maleand female students in each educational system. Gender differencesare not statistically significant in a mathematical area if male andfemale confidence intervals overlap in that topic (See Glass &Hopkins, 1984.). In this case, only half an interval bar is graphed inorder to simplify the diagrams.11111111A GA GA GA GBC^HK^Japan^OntarioFigure 4.3 A Graphic Representation of Gender Differencesand Cell Means in Population A4711111111AGAGA GAGBC^HK^Japan^OntarioFigure 4.4 A Graphic Representation of Gender Differencesand Cell Means in Population BIn British Columbia, male Population A students had a highermean in algebra than females while female Population A studentshad a higher mean in geometry than males (See Figure 4.1.). Genderdifferences, however, were not statistically significant because Figure4.3 showed that the 95 percent confidence intervals of achievementmeans for male and female students in British Columbia overlappedin both algebra and geometry. On the other hand, female PopulationB students had a higher mean in algebra than males while malePopulation B students had a higher mean in geometry than females(See Figure 4.1.). Actually, the gender gap in algebra was the largestin Population B. All these gender differences, however, were notstatistically significant (See Figure 4.4.).In Hong Kong, female Population A students had a higher meanin algebra than males, but male Population A students had a higher48mean in geometry than females (See Figure 4.1). However, genderdifferences on either algebra or geometry subtest were notstatistically significant (See Figure 4.3.). On the other hand, malePopulation B students had higher means in both mathematical areasthan females (See Figure 4.3.). But all gender differences did notreach statistical significance (See Figure 4.4.).In Japan, female Population A students had a higher mean inalgebra and a lower mean in geometry than male students (SeeFigure 4.1.). Gender differences were not statistically significant,however (See Figure 4.3.). On the other hand, male Population Bstudents had higher means in both mathematical areas than females(See Figure 4.1.). Gender differences were not statisticallysignificant, although the differences between the male and femalemeans were 6 in algebra and 10 in geometry. The gender gap ingeometry was, in fact, the largest for Population B in the foureducational systems (See Figure 4.4.).In Ontario, male Population A students had a higher mean inalgebra than female students with a difference of 5. Male studentsalso had a higher mean in geometry than females (See Figure 4.1.).Gender differences were not statistically significant, but were thelargest in both mathematical areas for Population A (See Figure 4.3.).In Population B, male students had a slightly higher mean ingeometry than females, but female students were very close to malestudents in algebra (See Figure 4.1.). The two achievement curvesalmost coincided (See Figure 4.4.). Gender differences were notstatistically significant. Actually, those gender gaps were thesmallest in both mathematical areas in Population B.The change of gender differences or gender gaps in Ontario fromthe largest in Population A to the smallest in Population B wasunique in the four participating educational systems. Moreinvestigation is needed to find out if educational policies andpractices in Ontario help narrow gender gaps in secondary schools.49Variability of Achievement DistributionSpecial attention should also be given to the variability ofmathematics achievement. Hartley's Fmax test was employed for asimple and quick test of the variability of mathematics achievementbetween male and female students in each population. All statisticalnull hypotheses were tested at an a level of 0.05.Box plots were also used to analyze the male and femaleachievement distribution. For each population in an educationalsystem, the achievement distribution for algebra is displayed on theleft-hand side of the box plot while the achievement distribution forgeometry is on the right-hand side. Within each mathematical area,the female achievement distribution is presented first, followed bythe male achievement distribution.The "box" describes the middle 50% of the distribution, and themedian falls in the box. The whiskers usually extend to 10 and 90percentiles, beyond which the lowest and highest 10% of thedistribution locate (Glass & Hopkins, 1984).In this study, a range of 90 percentile and 100 percentile wasregarded as the highest position in the achievement distribution foreach gender group. On the other hand, a range of 0 percentile and 10percentile was considered as the lowest position in the achievementdistribution for each gender group.Referring to the test statistic Fmax , relevant box plots, and Table4.10 or Figure 4.2, interpretations and discussions are depictedsystem by system.In British Columbia, the differences of the variability betweenmale and female achievement distribution were tested for eachmathematical area in each population.50Table 4.11 Results of Hartley's Fmax Test for British ColumbiaFmax 0.95Fmax 2, 59Population A Algebra 1.04 1.67Population A Geometry 1.00 1.67Population B Algebra 1.01 1.67Population B Geometry 1.08 1.67*p < 0.05For Population A, the standard deviation (SD) of girls in algebra(21.9) was very close to that of boys (21.5). The SD was the same ingeometry with 21.0 for both boys and girls.No statistically significant differences were found between thevariability of boys' achievement and the variability of girls'achievement on either the algebra subtest or the geometry subtest.Figure 4.5 illustrates the achievement distribution for PopulationA. The box plot shows that the algebra achievement curves werenormally distributed for both boys and girls with one outlierappearing at the top end of the female achievement distribution andtwo outliers appearing at the bottom end of the male achievementdistribution. The box plot also indicates that the top end of theachievement distribution was occupied equally by both boys andgirls. On the other hand, girls dominated the bottom end of theachievement distribution.Meanwhile, the box plot shows normal distribution for both boysand girls without any outliers in geometry. The box plot alsosuggests that both the top end and the bottom end of theachievement distribution were occupied evenly by both boys andgirls.510099KEY• Median- 25%, 75%X High/LowO OutlierE Extreme52+-+-+^+- -1VariableP012Figure 4.5 A Box Plot Illustration of Achievement Distributionin Algebra and Geometry in British Columbia (Population A)For Population B, the standard deviation of female scores inalgebra (29.3) was almost the same as that of male scores (29.1). Ingeometry, male standard deviation (31.9) was slightly larger thanfemale standard deviation (30.7).Again, no statistically significant differences were foundbetween the variability of male achievement and the variability offemale achievement in either algebra or geometry.Figure 4.6 illustrates the British Columbia achievementdistribution in Population B. The box plot shows normal distributionwithout any outliers for both male and female students in algebra,and suggests that both the top end and the bottom end of thedistribution were dominated equally by both male and femalestudents.Students performance in geometry showed that male studentshad the lowest mean and the largest standard deviation of the fourmale groups in Population B. As did female students in BritishColumbia. Both male and female students did not perform well ingeometry because male and female achievement distribution was allnegatively skewed. The box plot indicates that the top end and thebottom end of the achievement distribution were equally occupiedby both male and female students.99KEY53+-+-+^+-+-+1^11 2• Median- 25%, 75%X High/Low0 )utlierE Extreme0VariableP0Figure 4.6 A Box Plot Illustration of Achievement Distributionin Algebra and Geometry in British Columbia (Population B)In Hong Kong, the differences of the variability between maleand female achievement distribution were tested for eachmathematical area in each population.Table 4.12Results of Hartley's Fmax Test for Hong KongFmax 0.95Fmax 2, 59Population A Algebra 1.19 1.67Population A Geometry 1.14 1.67Population B Algebra 1.86* 1.67Population B Geometry 1.36 1.67*p < 0.05For Population A, the standard deviation of boys in algebra(23.2) was slightly larger than that of girls (21.3). In geometry, adifference of 1.2 between SD of boys' scores (19.2) and SD of girls'scores (18.0) could be found.No statistically significant differences were found between thevariability of boys' achievement and the variability of girls'achievement on either the algebra or the geometry subtests.Figure 4.7 illustrates the achievement distribution for PopulationA in Hong Kong. The box plot shows that the algebra achievementcurves were normally distributed with no outliers for both boys andgirls, and indicates that the top end of the achievement distributionwas dominated equally by both boys and girls. On the other hand,more boys than girls were found at the bottom end of thedistribution.For geometry, the achievement curves were normallydistributed without any outliers for both boys and girls. The box plotsuggests that the top end of the achievement distribution wasdominated equally by both boys and girls. On the other hand, moreboys than girls were found at the bottom end of the distribution. Asa matter of fact, similar situations were found on both the algebraand the geometry subtests for this population.99KEY• Median- 25%, 75%X High/LowO OutlierE Extreme1VariableP0Figure 4.7 A Box Plot Illustration of Achievement Distributionin Algebra and Geometry in Hong Kong (Population A)5421For Population B, in contrast to Population A, the standarddeviation of female scores became larger than that of male scores.The differences were 5.3 between females (19.9) and males (14.6) inalgebra and 2.9 between females (20.0) and males (17.1) ingeometry.A statistically significant difference was found between thevariability of male algebra achievement and the variability of femalealgebra achievement. Female algebra achievement was significantlymore variable than male algebra achievement. On the other hand,the variability of male geometry achievement did not differsignificantly from the variability of female geometry achievement.Figure 4.8 illustrates the Hong Kong achievement distribution forPopulation B. In algebra, male and female achievement distributionwere positively skewed with outliers and extreme values. The boxplot indicates that the top end of the achievement distribution wasoccupied evenly by both male and female students. On the otherhand, female students dominated the bottom end of the achievementdistribution.In geometry, female achievement curve was slightly positivelyskewed while male achievement curve was normally distributedwith two outliers and one extreme value appearing at the bottomend. The box plot shows that the top end of the achievementdistribution was occupied equally by both male and female students.On the other hand, female students dominated the bottom end of thedistribution exclusively. Again, similar situations were found in bothmathematical areas.555699KEY' Median- 25%. 75%X High/Low0 OutlierE Extreme XI+-+-+-+ -02EE1Variable-P 2 2 2Q 2 1 2Figure 4.8 A Box Plot Illustration of Achievement Distributionin Algebra and Geometry in Hong Kong (Population B)In Japan, the differences of the variability between male andfemale students were tested for each mathematical area in eachpopulation.Table 4.13 Results of Hartley's Fmax Test for JapanFmax 0.95Fmax 2, 59Population A Algebra 1.09 1.67Population A Geometry 1.19 1.67Population B Algebra 1.18 1.67Population B Geometry 1.04 1.67*p < 0.05For Population A, girls had a larger standard deviation (24.8)than boys (23.8) with a difference of 1 in algebra. In geometry, boyshad a larger standard deviation (21.2) than girls (19.4) with adifference of 1.8.No statistically significant differences were found between thevariability of boys' achievement and the variability of girls'achievement on either the algebra or the geometry subtests.Figure 4.9 illustrates the achievement distribution for PopulationA in Japan. In algebra, the achievement distribution for girls waspositively skewed with no outliers. Meanwhile, the achievementcurve for boys showed a slightly positive skewness without anyoutliers. The box plot suggests that more girls than boys could befound at the top end of the achievement distribution. On the otherhand, the bottom end of the achievement distribution was dominatedequally by both boys and girls.In geometry, the achievement curves were normally distributedfor both boys and girls with one outlier appearing at the bottom endof female achievement distribution while two outliers appearing atthe bottom end of male achievement distribution. The box plotindicates that basically boys dominated both the top and the bottomend of the achievement distribution.5799KEYMedian- 25%, 75%X High/Low0 OutlierE Extremex0^020VariableOX+-+-+3132 3^31 2Figure 4.9 A Box Plot Illustration of Achievement Distributionin Algebra and Geometry in Japan (Population A)For Population B, although male students had a larger standarddeviation (21.2) than females (19.5) with a difference of 1.7 inalgebra, the difference of standard deviations between males (23.8)and females (23.3) in geometry was very small.No statistically significant differences were found between thevariability of male achievement and the variability of femaleachievement in either algebra or geometry.Figure 4.10 illustrates the Japanese achievethent distribution forPopulation B. In algebra, female achievement curve was normallydistributed while male achievement distribution was slightlypositively skewed with four outliers at the bottom end. The box plotindicates that male and female students equally dominated the topend of the achievement distribution. On the other hand, more femalestudents than males appeared at the bottom end of the distribution.In geometry, both male and female achievement curves werenormally distributed with two outliers at the top end of femaledistribution and four outliers at the bottom end of male distribution.The box plot shows that male students dominated the top end of theachievement distribution. On the other hand, female studentsdominated the bottom end of the achievement distribution.58ggKEY• Median- 25%, 75%X High/Low0 OutlierE ExtremeVariable-02X^030003P^3^3^ 30 1 2 2Figure 4.10 A Box Plot Illustration of Achievement Distributionin Algebra and Geometry in Japan (Population B)In Ontario, the differences of the variability between male andfemale students were tested for each mathematical area in eachpopulation.Table 4.14Results of Hartley's Fmax Test for OntarioFmax 0.95Fmax 2, 59Population A Algebra 1.01 1.67Population A Geometry 1.08 1.67Population B Algebra 1.15 1.67Population B Geometry 1.03 1.67*p < 0.05For Population A, the standard deviation of boys (17.7) inalgebra was almost the same as that of girls (17.6). In geometry,boys had a smaller standard deviation (19.5) than girls (20.3).No statistically significant differences were found between thevariability of boys' achievement and the variability of girls'achievement on either the algebra or the geometry subtests.Figure 4.11 illustrates the achievement distribution forPopulation A in Ontario. In algebra, the achievement curve for girlswas basically normally distributed with one extreme value at the topend and one outlier at the bottom end of the distribution.Meanwhile, the achievement distribution for boys was slightlypositively skewed with one extreme value and two outliers at the topend and one outlier at the bottom end of the distribution. The boxplot indicates that more girls than boys appeared at the top end ofthe achievement distribution. On the other hand, the bottom end ofthe distribution was occupied equally by both boys and girls.In geometry, the achievement curves were normally distributedfor both boys and girls with one outlier at the top end of theachievement distribution for girls. The box plot shows that the top59E^E^0O+-+-+1^•^11^•^1X^X0099KEY• Median- 25%, 75%X High/LowO OutlierE Extreme1VariablePa4^4^41 2 1end of the distribution was basically dominated by boys. On theother hand, the bottom end of the distribution was occupied equallyby both boys and girls.60Figure 4.11 A Box Plot Illustration of Achievement Distributionin Algebra and Geometry in Ontario (Population A)For Population B, the standard deviation of male scores (23.9) inalgebra became larger than that of females (22.3) with a differenceof 1.6. The standard deviations of geometry scores were nearly thesame for both males (23.4) and females (23.8).No statistically significant differences were found between thevariability of male achievement and the variability of femaleachievement in either algebra or geometry.Figure 4.12 illustrates the Ontario achievement distribution forPopulation B. In algebra, the female achievement curve wasnormally distributed with one outlier at the bottom position whilethe male achievement curve was basically normally distributed withtwo outliers at the bottom end of the distribution. The box plotindicates that the top end of the achievement distribution wasoccupied equally by both male and female students. On the otherhand, female students exclusively dominated the bottom end of thedistribution.In geometry, female and male achievement curves were allnormally distributed with no outliers. The box plot shows that malestudents basically dominated the top end of the achievementdistribution. On the other hand, the bottom end was occupiedequally by both male and female students.6199KEY' Median- 25%. 75%X High/LowO OutlierE Extreme-+-+^ -+-•- + -02Variable-4^ 40^ 2 2Figure 4.12 A Box Plot Illustration of Achievement Distributionin Algebra and Geometry in Ontario (Population B)From a perspective of across educational systems, whencombining findings on the variability of mathematics achievementbetween male and female students with results of gender differencesin each educational system, the pattern between British Columbiaand Ontario was similar. First of all, no gender differences werefound to be statistically significant on all the algebra and geometrysubtests. Second, Hartley's Fmax test did not show any statisticallysignificant differences between the variability of male achievementand the variability of female achievement on all the algebra andgeometry subtests. Third, in general, the majority of both male andfemale students in the two Canadian provinces performed equallywell in algebra and geometry. On most subtests for both PopulationA and Population B, usually male and female students occupiedequally the top end and/or the bottom end of the achievementdistribution. Finally, if an achievement gap in a mathematical areabetween high achievers and low achievers in one gender group canbe called "within gender gap", the gaps were serious for the twopopulations, especially for Population B.Another different pattern was found in Hong Kong and Japan,the two Asian participants. In Population A, no gender differenceswere found to be statistically significant on the algebra and geometrysubtests. Second, Hartley's Fmax test did not show any statisticallysignificant differences between the variability of boys' achievementand the variability of girls' achievement on the algebra and geometrysubtests for this population. Third, in general, the majority of bothboys and girls performed equally well in the two Asian educationalsystems. But high achievers or low achievers in each gender groupmight be at somewhat different levels of performance. Slightly moreboys than girls were found at the bottom end of the distribution onalmost every subtest for this population, despite the fact that nogender group occupied the top end consistently. Finally, the withingender gaps were serious, although they were not wider than thosefound in British Columbia and Ontario.In Population B, no gender differences were found to bestatistically significant on all the algebra and geometry subtests.Second, Hong Kong showed a statistically significant difference on thevariability of algebra achievement between male and femalestudents. Female algebra achievement was significantly morevariable than male algebra achievement. On the other hand,Hartley's Fmax test did not show any statistically significantdifferences between the variability of males and the variability offemales on all other subtests. Third, although male and femalestudents equally dominated the top end of the achievementdistribution on most subtests, females in the lowest 10% of thefemale distribution tended not to perform as well as males in thelowest 10% of the male distribution in both algebra and geometry.Female students dominated the bottom end of the achievement62distribution on every subtest for this population. Finally, most of thewithin gender gaps in this population were reasonably narrow.Furthermore, they were smaller, sometimes very smaller than thefindings for British Columbia and Ontario.Other FindingsIn this section, other relevant findings in the present study arereported.Figure 4.3 illustrates that the achievement curve of malestudents and the achievement curve of female students were almostthe same shape. In Figure 4.4, the shapes of male and femaleachievement curves were still very similar. Similar achievementcurves suggest that no matter how different each educational systemmight be, no matter how much each educational system might affectstudents' achievement, or regardless of the gender gap, everyeducational system tended to affect the academic development ofboth male and female students in roughly the same way or inroughly the same direction.For instance, in Hong Kong, both male and female studentsachievement went down in each population when moving fromalgebra achievement to geometry achievement. The findings suggestthat every educational system has either positive or negative affectson both male and female students although one gender may beaffected more than the other. In other words, although a gender gapmay be quite large in an educational system, the educational affectstend to be in the same direction for both male and female students.This is one of the unique findings of the study. But obviously moreinvestigation is needed. It is not clear at this stage if this is acommon phenomenon or just a local coincidence.Figure 4.3 also shows that male and female achievement curveswere very close to each other. In other words, they almost becamethe same curve. Male and female achievement curves were stillclose to each other in Figure 4.4. These facts suggest that genderdifferences at those stages tend to be so weak in each educationalsystem that they can hardly be detected by some assessmentprograms. The findings do not seem to support the idea that63biological factors might reasonably result in gender differences(Stafford, 1972). If, up to the age of 13, the gender gaps in algebraand geometry are still fairly small, it is reasonable to suggest thatmale and female students were not born with different competencein mathematics.64Chapter V. SummaryIn this chapter, research questions are answered based on theresults and findings that were listed above. Implications for policyand suggestions for further research are also discussed.Summary of FindingsThe first question cluster asked whether there were genderdifferences across populations, mathematical areas, and educationalsystems.The results of ANOVA showed no statistically significantinteraction effects between gender and educational system acrosspopulations and mathematical areas. The findings suggest that froma cross-system perspective gender does not correlate significantlywith educational system. This is the case at least in algebra andgeometry.In algebra, there were no system-wide significant genderdifferences across populations. These findings not only confirm theconclusion made by Hanna (1986) who found no significant genderdifferences in algebra in Population A of SIMS, but also extend hisconclusion to include Population B of SIMS.An interesting pattern was also found from the algebra subtests.In British Columbia and Ontario, boys had higher means than girls inPopulation A. But in Population B female students had higher meansthan males. On the other hand, this was totally reversed in HongKong and Japan; girls had higher means than boys in Population A,but male students had higher means than females in Population B.In geometry, as far as Population A is concerned, of the foureducational systems in this study, only girls in British Columbia had aslightly higher mean than boys with a difference of only 1 mark interms of means. All other tests showed that boys had higher meansthan girls in all other educational systems. Higher geometry meansfor male students were common in Population B in every educationalsystem. These findings suggest a tendency that male students aremore likely to have a higher mean than their female counterparts inthis mathematical area. More studies are needed to prove ordisprove this tendency.65The present study found no statistically significant genderdifferences in geometry in Population A, which is in contrast toHanna's finding (1986). This contrast can be reasonably explainedby the different sample sizes used by Hanna and the present study.The only gender differences which were found to be statisticallysignificant in this study were in geometry. Male studentsoutperformed females in Population B from a cross-systemperspective. However, gender differences were not statisticallysignificant locally in each educational system.^These results suggestthat in each educational system male and female students may notdiffer significantly in geometry throughout the school years.Therefore, the findings indicate that gender differences ingeometry in Population B are more likely to be a whole rather than alocal phenomenon. That is, significant gender differences can befound for a whole population, but gender differences may not besignificant within each educational system. FIMS reported a similarfinding (Husen, 1967).The second question cluster explored gender differences in thevariability of mathematics achievement.In Population A, similarities were found between BritishColumbia and Ontario. No statistically significant differencesbetween the variability of boys' achievement and the variability ofgirls' achievement were found on any subtests. The majority of boysand girls performed equally well in the two mathematical areas.Finally, the within gender gaps were serious.Another pattern was shared by Hong Kong and Japan. Nodifferences on the variability of the achievement between boys andgirls were found to be statistically significant on any subtests. Themajority of boys and girls performed equally well in algebra andgeometry. But high achievers or low achievers in each gender groupmight be at slightly different levels of performance. Finally, thewithin gender gaps were serious in this population.In Population B, similarities were also found between BritishColumbia and Ontario. No statistically significant differences of thevariability between male achievement and female achievement werefound on any subtests. The majority of male and female students66continued to perform equally well in algebra and geometry. Finally,the within gender gaps at this stage were more serious than thosereported in Population A, and were much larger than those found inHong Kong and Japan.Hong Kong and Japan, on the other hand, shared a differentpattern. A statistically significant difference was found on thevariability of algebra achievement between male and femalestudents. On all other subtests, however, no statistically significantdifferences between the variability of male achievement and thevariability of female achievement were found. Although both maleand female students occupied the top end of the distribution equallyon most subtests, males in the lowest 10% of the male distributionand females in the lowest 10% of the female distribution tended toperform unequally in algebra and geometry. Female studentsdominated the bottom end of the achievement distribution on everysubtest for this population. Finally, most within gender gaps werenarrow in this population.The suggested pattern (See Willms & Jacobsen, 1990.) wheremale students dominate both the top and the bottom end of theachievement spectrum while female students stay in the middle wasnot totally confirmed in this study. It seems that there is not acommon pattern throughout the entire schooling in terms of genderdifferences on the variability of mathematics achievement. Thisstudy suggests that, from a view point of gender differences, thevariability of mathematics achievement between male and femalestudents depends, at least, upon age, mathematical area, andeducational system.Policy ImplicationsThe findings obtained in this study have policy implications.First of all, this study found no statistically significant interactioneffects between gender and educational system across populationsand mathematical areas. Since the factor of gender has littleinteraction with the factor of educational system, mathematicsteachers in the four educational systems may face similar situationsof gender differences, although the educational policies and practices67may be different. This provides an opportunity for educationalsystems to learn from one another without creating significantgender differences by adopting educational policies or practicesemployed in other educational systems.Second, this study did not find any statistically significantgender differences across populations and mathematical areas ineach educational system. This result indicates that locally both maleand female students have the same potential to study algebra andgeometry, the two basic and important areas in school mathematics.Therefore, female students should be made aware that theirpotential to learn mathematics and succeed in mathematics is thesame as for boys. Both male and female students should be givenample opportunities to meet and talk to female mathematicians orfemale scientists in order to overcome the traditional belief thatmales do better in mathematics.Third, this study found statistically significant genderdifferences on the geometry subtest for Population B. High schoolmathematics teachers through all the educational systems should beinformed of this result so as to provide equal opportunities for bothmale and female students to progress in geometry learning.Mathematics teachers should create a positive environment whichencourages female students to study geometry actively. Curriculumand instruction developers should also be made aware of this result,and study the relationships between mathematics curriculum orinstruction and gender differences in high schools.Forth, this study found a problem in Asian educational systemswith respect to the variability of mathematics achievement. Moreboys than girls were found at the bottom end of the achievementdistribution on almost every subtest for Population A. In PopulationB, female students dominated the bottom end of the achievementdistribution on every subtest. It seems that some measures shouldbe taken there to improve the mathematics performance of lowachievers both in the male group of Population A and in the femalegroup of Population B.Finally, a problem which the Canadian educational systems facedin terms of the variability of mathematics achievement was also68found. The within gender gaps in each gender group were serious inboth populations. Although the causation can not be inferred fromthis study, findings suggest that the focus for improvement may beon the low achievers in each gender group.Suggestions for Further ResearchBesides the recommendations for follow-up studies made in theprevious discussions, this section offers more suggestions for furtherresearch.First, as mentioned before, potential interaction effects amongsome critical factors related to gender differences are seldomaddressed in literature. Overall, more investigation on interactioneffects is needed to discern the relationships between gender andother factors. This study examined a two-factor interaction effect,gender by educational system. Further research is encouraged tocarry out factorial designs which involve more than two factors.Multiple factor examinations will provide mathematics educatorswith richer results and implications than the two-factor analysisused in this study.Second, as noted in the limitations of the study, this thesisexamined gender differences only in two mathematical areas, algebraand geometry. Further studies may continue to investigate genderdifferences in other mathematical areas such as arithmetic,measurement, trigonometry, and coordinate geometry from a cross-system perspective, so that they can specify the needs of students inthese mathematical areas. Mathematics teachers will benefit directlyfrom the studies by applying evidence from research to theirpractice. Meanwhile, researchers will also benefit from the studiesby applying evidence from research to their theoretical work.Third, because there is not a large body of research on thevariability of mathematics achievement, further studies may takethis direction to enrich our knowledge in this matter. This studysuggests that it is probably too early to report a general pattern onthe matter of the variability of mathematics achievement. Morestudies are needed to develop and test patterns or theories in thisfield.69Finally, the statistical approach used in this study was limited toANOVA only. Further research may analyze gender differences byemploying multiple statistical methods such as ANOVA, MANOVA,and factor analysis. Each statistical application has its ownadvantages. By incorporating more statistical approaches, extensiveand accurate relationships between gender and other factors may befound.70ReferencesAlexander, K. L., & Pallas, A. M. (1983). Reply to Benbow andStanley. 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Reflections on the history of women'seducation in Australia. Education News, 15, 4-13.83Appendix I. Checking the Aptness of ANOVA Models:Results of Cell PlotsMeans VS Std.•Devs.^for SCORE ACHIEVEMEN124 +• 1E•• 1L^22.5 +S 1 •1D 121 +•EVS^19.5 +18 +••44^52^60^6840^48^56^64CELL MEANS84Figure 1.1 A Cell Plot for the Algebra Subtest for Population A85Means VS Std. Devs. for SCORE^ACHIEVEMEN•••••27 +11 ••••••••+C ••E •. ••L . L :24 + 1 +S ••T ••0 •• 121^+ 1 +DE 1V 1S :18^+ +••••••15^+ +. 1 :55^65^75^8550^60^70^80CELL MEANSFigure 1.2 A Cell Plot for the Algebra Subtest for Population BFigure 1.3 A Cell Plot for the Geometry Subtest for Population A86Figure 1.4 A Cell Plot for the Geometry Subtest for Population B87Appendix II. Checking the Aptness of ANOVA Models:Results of Box Plots88Figure II.1 Box Plots for the Algebra Subtest for Population ABox-Plots For99Variable .. SCORE^ACHIEVEMENT- + -KEY* Median- 25%, 75%X High/Low0 OutlierE Extreme_ + _- + -0VariableP0E- + -E+ -X_+ - ++ -xX+ - +- + --+- X 030X023 3 41 2 22^21 289Figure 11.2 Box Plots for the Algebra Subtest for Population BFigure 11.3 Box Plots for the Geometry Subtest for Population A90Figure 11.4 Box Plots for the Geometry Subtest for Population B91
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An international study of gender differences in mathematics achievement Ma, Xin 1993
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Title | An international study of gender differences in mathematics achievement |
Creator |
Ma, Xin |
Date Issued | 1993 |
Description | This study examined gender-related issues in mathematics based on achievement data of Populations A and B from The Second International Mathematics Study (SIMS). The purposes of this study, which involved two Canadian educational systems and two Asian educational systems, were 1)to investigate potential interaction effects between gender and educational system in two populations and in two mathematical areas, algebra and geometry; 2) to analyze the variability of mathematics achievement between male and female students in each mathematical area within and across educational systems; 3)to compare and contrast situations of gender differences between algebra and geometry within and across educational systems. Factorial design, Hartley's F max test, and box plots were the major statistical approaches used in this study. The results showed that no two-factor interaction effect between gender and educational system in each mathematical area was of statistical significance in either population. Further, there were no statistically significant gender differences in algebra. In geometry, gender differences were statistically significant in Population B. Male students outperformed females from a perspective of across educational systems. Within each educational system, no gender differences in each mathematical area were found to be statistically significant in either population. Therefore, reported gender differences in geometry were more likely to be a general rather than a local phenomenon. The results of investigation on the variability of mathematics achievement illustrated two patterns. One pattern involved British Columbia and Ontario. No significant differences on the achievement variability between males and females were found. In general, the majority of both male and female students in the two Canadian provinces performed equally well in algebra and geometry. For the two populations, within gender gaps were serious, especially for Population B. Another different pattern was found in Hong Kong and Japan. In Population A, no significant differences on the achievement variability between boys and girls were found. Generally, the majority of both boys and girls performed equally well in algebra and geometry, although slightly more boys than girls were found at the bottom end of the achievement distribution. The within gender gaps were serious for this population, although they were not as wide as those found in British Columbia and Ontario. In Population B, a statistically significant difference on the variability of algebra achievement between male and female students was found in Hong Kong. Although male and female students equally dominated the top end of the achievement distribution, males in the lowest 10% of the male distribution and females in the lowest 10% of the female distribution tended to perform unequally in algebra and geometry. Female students dominated the bottom end of the achievement distribution on every subtests for this population. The within gender gaps were narrow in this population. Finally, findings in this study did not support the opinion of a biological explanation of gender differences in mathematics. Furthermore, findings suggested that each educational system affected the academic development of both male and female students in the same ways or directions, although one gender might be affected more seriously than the other. |
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Language | eng |
Date Available | 2008-09-15 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0086253 |
URI | http://hdl.handle.net/2429/1932 |
Degree |
Master of Arts - MA |
Program |
Mathematics Education |
Affiliation |
Education, Faculty of Curriculum and Pedagogy (EDCP), Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1993-11 |
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Scholarly Level | Graduate |
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