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COMPARISON OF MALES AND FEMALES ON MATH ITEM PERFORMANCE: ANALYSIS OF RESPONSE PATTERNS By SONIA FELICIANO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1986 ACKNOWLEDGMENTS I would like to acknowledge my gratitude to several people who have influenced my formal education and/or made this study possible. My special thanks to Dr. James Algina, Chairman of the doctoral committee who contributed to the development of my love for research and statistics. He has been insuperable as professor and valued friend; his help and guidance in the preparation and completion of this study were invaluable. I extend my thanks to Dr. Linda Crocker for her advice and help during my doctoral studies at University of Florida. Thanks also go to Dr. Michael Nunnery, member of the doctoral committee. To Dr. Wilson Guertin, who was a friend for me and my family, I extend my special thanks. Thanks are also extended to Dr. Amalia Charneco, past Undersecretary of Education of the Puerto Rico Department of Education, for her continuous support. To my sister Nilda Santaellar who typed the thesis, I give my sincere thanks. Special thanks go to my family and to those friends who provided encouragement throughout this critical period of my life. TABLE OF CONTENTS Page ACKNOWLEDGMENTS.... .......................... ii LIST OF TABLES .. ................ ................... v LIST OF FIGURES...................... ............... vii ABSTRACT....... ...................................... viii CHAPTER I INTRODUCTION.................................. 1 Purpose of the Study ............... ........ 6 Significance of the Study................... 6 Organization of the Study................... 8 II REVIEW OF THE LITERATURE ...................... 9 Sexrelated Differences in Incorrect Response Patterns........................ 10 Sexrelated Differences in Problem Solving............... ..... ........... .... 14 Cognitive and Affective Variables that Influence Sex Differences in Mathematics Learning and Achievement................ 23 Differences in Formal Mathematics Education....................... 24 Differences in Spatial Ability.......... 26 Differentiated Effect of Affective Variables................... ....... ... 30 Problem Solving Performance and Related Variables.... ..................... ......... 37 Computational Skills and Problem Solving Performance. .......................... 38 Reading and Problem Solving Perfor mance........................... .... .. 44 Attitudes Toward Problem Solving and Problem Solving Performance........... 50 iii III METHOD..... ... ...... ...... ..... ............... 54 The Sample ................ ................... 54 The Instrument............. ................ 55 Analysis of the Data....................... 57 Analysis of Sex by Option by Year Cross Classifications................ 57 Comparison of Males and Females in Problem Solving Performance........... 66 IV RESULTS........... ..... ....... ...... .. ........ 68 Introduction ............................. 68 Sexrelated Differences in the Selection of Incorrect Responses.......... 68 Sexrelated Differences in Problem Solving Performance.................... 81 Summary.................................... 93 V DISCUSSION..................................... 101 Summary and Interpretation of the Results ................................... 101 Implications of the Findings and Suggestions for Further Research.......... 103 Sexrelated Differences in Incorrect Responses................ 104 Sexrelated Differences in Problem Solving Performance................... 105 REFERENCES.......................................... 107 BIOGRAPHICAL SKETCH................................ 116 LIST OF TABLES Table Page 3.1 Hypothetical Probabilities of Option Choice Conditional on Year and Sex.......... 58 3.2 Hypothetical Probabilities of Option Choice Conditional on Year and Sex and Arranged by Sex......... ...... ........... 58 3.3 Hypothetical Joint Probabilities of Yearr Option, and Sex............................. 64 3.4 Hypothetical Probabilities of Option Choice Conditional on Sex and Year.......... 66 4.1 Summary of Significant Tests of Model 3.2..... 71 4.2 Chisquare Values for the Comparison of 75 Models (3.2) and (3.3) with Actual Sample Sizes and with the Corresponding Number of Subjects Needed for Significant Results 75 4.3 Means, Standard Deviations, and tTest for the Eight Mathematical Variables............ 82 4.4 ANCOVA First Year: Multiplication Covariate 85 4.5 ANCOVA First Year: Division Covariate....... 85 4.6 ANCOVA Second Year: Subtraction Covariate... 86 4.7 Reliability of the Covariates for Each of the Three Years of Test Administra tion........................................ 91 4.8 ANCOVA First Year: Other Covariates......... 92 4.9 Adjusted Means on Problem Solving, by Covariate and Sex.......................... 94 4.10 ANCOVA Second Year: Other Covariates........ 95 Table Page 4.11 Adjusted Means on Problem Solving, by Covariate and Sex........................ 96 4.12 ANCOVA Third Year........................ 97 4.13 Adjusted Means on Problem Solving, by Covariate and Sex....................... 98 LIST OF FIGURES Figure Page 3.1 Number of Questions by Skill Area in a Basic Skills Test in Mathematics6.......... 56 4.1 Sex by Multiplication Interaction............. 87 4.2 Sex by Division Interaction ................ 88 4.3 Sex by Subtraction Interaction................ 89 vii Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy COMPARISON OF MALES AND FEMALES ON MATH ITEM PERFORMANCE: ANALYSIS OF RESPONSE PATTERNS By Sonia Feliciano August, 1986 Chairman: James Algina Major Department: Foundations of Education The first objective of this study was to investigate sex differences in the selection of incorrect responses on a mathematics multiplechoice test, and to determine whether these differences were consistent over three consecutive administrations of the test. A second objective was to compare male and female performance in problem solving after controlling for computational skills. The responses of all 6th grade students from the public schools in Puerto Rico who took the Basic Skills Test in Mathematics6 ("Prueba de Destrezas B&sicas en Matem&ticas6") during three academic years were used in the analyses relevant to the first objective. viii Loglinear models were used in the analysis of incorrect responses. The results of the analyses showed that for 100 of the 111 items of the test, males and females selected different incorrect options, and this pattern of responses was consistently found during the three years of test administration. However, for the vast majority of the 100 items the malefemale differences were relatively small, considering the fact that the number of subjects needed to obtain statistical significance was very large. The responses of approximately 1,000 randomly selected students per academic year were analyzed in the comparison of male and female performance in problem solving. Females outperformed males in problem solving and in six of the seven computational variables. Males showed superiority in equivalence in all the three years, but statistical significance was obtained in only one of the years. Analysis of covariance (ANCOVA) was used in the comparison of male and female performance in problem solving after controlling for computational skills. Seven analyses of covariance tests were conducted, one for each of the covariates. Estimated true scores for observed scores were used in the analyses. The results tend to show that for examinees with similar levels of computational skills, sexrelated differences in problem solving performance do not exist. Females retained their superiority in problem solving when equivalence (in all three years) and subtraction (in one year) were the controlling variables. The question of whether malefemale differences in problem solving depend on computational skills was answered, partially, in the affirmative. CHAPTER I INTRODUCTION Sex differences in mathematics learning and achievement have been the subject of intensive research. Research done before 1974 has shown that male performance on mathematical achievement tests is superior to female performance by the time they reach upper elementary or junior high school (Fennema, 1976, p. 2). The literature strongly suggests that at the elementary level females outperform males in computation and males excel in mathematical reasoning (Glennon & Callahan, 1968; Jarvis, 1964; Maccoby, 1966). Since 1974, research findings have been less consistent. Fennema (1974), after reviewing 36 studies, found that during secondary school or earlier, sexrelated differences in mathematics achievement are not so evident, but that when differences are founds they favor males in high level cognitive tasks (problem solving) and females in low level cognitive tasks (computation). As a result of a further review of the literature, Fennema (1977) concluded that at the elementary level, sexrelated differences do not exist at all cognitive levels, from computation to problem solving. Many variables, cognitive affective, and educational, have been investigated since 1974 in relation to sex differences in mathematics learning and achievement. Fennema and Sherman (1977) investigated the effect of differential formal mathematics education. After controlling for the number of years of exposure to the study of mathematics, they found sex differences in only two of the four schools under study. However, in those schools where boys scored higher than girls, differences were also found in their attitudes toward mathematics. Hilton and Berglund (1974) found significant sex differences after controlling for the number of mathematics courses taken, and attributed them to sex differences in interests. "As the boys' interests in science increase related to the girls', their achievement in mathematics increases relative to that of the girls" (p. 234). Wise, Steel, and McDonald (1979) reanalyzed test data collected in a longitudinal study of 400,000 high school students (Project Talent). They found that when the effect of the number of high school mathematics courses was not controlled, no sex differences emerged for 9th graders, but that gains made by boys during the next three years were more than twice that of the girls. These differences between the sexes disappeared when the number of mathematics courses taken was controlled. Results of the 1978 Women and Mathematics National Survey, Survey I, indicated no significant sex differences for 8th grade students on measures of problem solving or algebra. However, females outperformed males in computation and spatial visualization. For the 12th grade students, statistically significant sex differences favoring males were found in problem solving, but not in algebra, computation, or spatial visualization. For males and females who had enrolled in courses beyond general mathematics and who had taken or were enrolled in courses such as precalculus, calculus, or geometry, differences in problem solving or spatial visualization did not exist. Sex differences favoring males were found on a total score obtained summing across the computation, problem solving, and algebra subtests (Armstrong, 1979). The mathematics data collected in the second survey by the 1978 National Assessment of Educational Progress showed significant sex differences for both 13 and 17yearold students. The 13yearold females outperformed males in the computational subtest and males outscored females by 1 1/2 percentage points in problem solving (statistically significant). No statistically significant differences were found in algebra. No sex differences were found for the 17yearold group either in the computation subtest or in the algebra subtest. Males surpassed females in problem solving. A reanalysis of the data from the 17yearold group confirmed male superiority in problem solving after controlling for mathematics preparation. Males who were enrolled or had completed algebra II outperformed the females in computation and problem solving but not in algebra. Males who studied beyond algebra II outscored females on all three subtests: computation, algebra, and problem solving (Armstrong, 1979). Carpenter, Lindquist, Mathews, and Silver (1984) analyzed the results of the Third National Assessment of Educational Progress (NAEP), and compared them with the First and Second Surveys. Between 1978 and 1982, the differences between the average performance of males and females remained stable at each age level. At ages 9 and 13, the overall performance of males and females was not significantly different. At age 17r males scored higher than females by about 3 percentage points. When course background was held constant, achievement differences still existed at age 17. For each category of course background, male achievement exceeded female achievement. Consistent with previous assessments, sex differences in problem solving in favor of males were found for the 17yearold sample. At ages 9 and 13, no large differences were found between the sexes within any level of course background. Marshall (1981, 1984) investigated sex differences in mathematics performance. She found that males and females excel each other in solving different types of problems. Females were better on items of computation and males were more successful on wordstory problem items (problem solving). She also found that females successful performance in the problem solving items was more dependent on their successful performance in the computation items. Males did not need, as much as females, to succeed in the computation items in order to answer correctly the problem solving items. Although the general findings seem to support sex differences in mathematics learning and achievement, the research done does not consistently support superiority for either sex. Most of the research has been concerned with how the sexes differ on subtests or total test scores in mathematics. Moreover, the great majority of the studies deal with correct responses. Sex differences in incorrect responses at the item level have not been fully researched. Only two studies dealing with sex differences in incorrect responses at the item level were found in the research literature (Marshall, 1981, 1983). Marshall investigated whether boys and girls made similar errors in computation and story problems. She analyzed boys' and girls' answers to six mathematics items and found that the sexes made different errors, possibly reflecting different problem solving strategies. Her original findings were supported when she studied the same problem using a large number of items three years later. Purpose of the Study The purpose of this study was twofold. The first was to investigate sexrelated differences in the selection of incorrect responses in a multiplechoice achievement test in mathematics. For each test item, the research questions were as follows: 1. Is there a difference in the proportion of males and females choosing each incorrect option? 2. Is the same pattern of differences found in data obtained in three different administrations of the test? The second objective was to investigate sexrelated differences in test scores in mathematics problem solving. The following questions were studied: 1. Do males and females differ in problem solving performance? 2. Do sex differences in problem solving persist after controlling for computational skills, and does the differential success of males and females on problem solving items depend on their success on the computation items? Significance of the Study Item response patterns are very useful techniques in the assessment of mathematics learning and achievement. Total test scores can be very misleading in the assessment of student performance and provide no diagnostic information about the nature and seriousness of student errors (Harnisch, 1983). Item response patterns are valuable for the identification of large group differences, including districttodistrict, schooltoschool, and classroomtoclassroom variations on different subsets of items. The response patterns can provide diagnostic information about the type of understanding the student has on various mathematics topics (e.g.r problem solving). Marshall (1981, 1984) has used the item response pattern technique and her findings indicate sex differences in mathematics performance at the item level. Females outperformed males in computation and males outscored females in problem solving. Also the success of girls in the problem solving items was dependent upon their success in the computation items; for boys, success in the problem solving items did not depend as much on their computational performance. Marshall (1981, 1983) has also reported that males and females differ in the selection of incorrect responses, reflecting differences in reasoning abilities. In Puerto Rico, a high percentage of children promoted to the 7th grade in the public schools does not master the basic skills in mathematics. If 6th grade male and female children can be diagnosed as having different problem solving abilities, as Marshall found with California children, teachers may need to provide tailormade mathematics instruction for each sex, in order to ensure equal access to formal education and enhance mathematics achievement. Since there are no investigations reported in sex differences in item response patterns in Puerto Rico, research is needed. Organization of the Study A review of the literature on sex differences in mathematics performance is reported in Chapter II. The research methodology is presented in Chapter III. Research questions, sample, instrument, and data analysis are discussed in that chapter. Chapter IV is an exposition of the results of the study. Chapter V contains a summary and interpretation of the results of the study and the implications of the findings together with suggestions for further research. CHAPTER II REVIEW OF THE LITERATURE Sex differences in mathematics learning and achievement have been a subject of concern for educators and psychologists. Many studies found in the literature support the existence of these differences. Boys show superiority in higher level cognitive tasks (problem solving or mathematical reasoning) in the upper elementary years and in the early high school years (Fennemar 1974; Maccoby & Jacklin, 1974). Almost all the research carried out has dealt with analysis of total correct scores in mathematics aptitude and achievement tests or scores in subtests. The literature related to sex differences in incorrect responses, the main subject of the present study, is surprisingly sparse. For the most part, the studies have investigated the differences between the sexes in mathematics learning and achievement and the underlying variables causing the differences. Cognitive and affective variables have been the matter at issue in the establishment of sex differences. Although research in mathematics problem solving, the secondary subject of this investigation, is extensive, most of the studies consider sex differences incidental to the major study findings. The available literature offers very little research directly related to the problem of sex differences in this area. The review of the literature has been divided in four sections. The first section consists of a detailed summary of the available research on sex differences in incorrect responses. The second section deals with sexrelated differences in problem solving performance. These sections are directly related to the objectives of the study. The third section is more peripheral, and contains a discussion of the more prevalent issues about the influence of cognitive and affective variables on sex differences in mathematics learning and achievement. The fourth is a summary of the research dealing with variables considered as influential to mathematics problem solving performance. Sexrelated Differences in Incorrect Response Patterns Research findings tend to suggest that boys and girls may be approaching problem solving differently (e.g., Fennema and Sherman, 1978; Marshall, 1981, 1983; Meyer, 1978, among others). Marshall (1981) investigated whether 6th grade boys and girls approach mathematical problem solving with different strategies. Her specific interest was whether the sexes made the same errors. She analyzed the responses of 9,000 boys and 9,000 girls to 6 selected items, 2 computation items, and 1 story problem item from each of 2 of the 16 forms of the Survey of Basic Skills test administered during the academic year 197879. The Survey is a 30item achievement test administered every year to all 6th grade children in California through the California Assessment Program. There are 16 forms of the test, to which approximately 9,000 boys and 9,000 girls respond each year. Of the 160 mathematics items contained in the 16 forms of the test, 32 are on measurement and graphing, 28 on number concepts, 28 on whole number arithmetics, 20 on fraction arithmetics, 20 on decimal arithmetics, 20 on geometry, and 12 on probability and statistics. The item analysis performed on the 197879 data showed that boys and girls tended to select different incorrect responses. In the first computation item (Form 1 of the test) both sexes reflected similar mistakes in carrying, but in different columns. In the second computation item, both sexes ignored the decimal points and selected the same incorrect response. However, more girls than boys chose this response. In the first computation item (Form 2 of the test) the incorrect choice of both sexes was option cr but the second most frequently selected option was a for boys and b for girls. In the second computation item of this test form, no sex differences were found in response patterns. Approximately 45% of each sex selected option c. The next popular choice for both sexes was option d, selected by approximately 35% of both boys and girls. On the story problem of Form 1, males and females responded alike. Their most popular incorrect response choice was option a for both males and females. The second most popular incorrect choice was option c for both sexes. Response to the story problem in Form 2 showed sex differences in response choice. Including the correct option, 33% of the girls selected option a, 20% chose option c, and 20% option d. For males approximately 25% selected option a and the same percent selected option d. Marshall concluded that although the analysis of incorrect responses does not explain why boys and girls differ in their responses, the analysis shows that boys and girls approach problems in different ways and these varying strategies can be useful in identifying how the sexes differ in reasoning abilities. Two years later, Marshall (1983) analyzed the responses of approximately 300,000 boys and girls to mathematics items contained in the 16 test forms of the Survey of Basic Skills during the years 1977, 1978, and 1979. She used loglinear models (explained in Chapter III) to investigate sex related differences in the selection of incorrect responses, and the consistency of such differences over three years of administration of the test. Based on her findings that sex differences were found in 80% of the items, Marshall classified the students' errors according to Radatz' (1979) fivecategory error classification. The categories are language (errors in semantics), spatial visualization, mastery, association, and use of irrelevant rules. It was found that girls' errors are more likely to be due to the misuse of spatial information, the use of irrelevant rules, or the choice of an incorrect operation. Girls also make relatively more errors of negative transfer and keyword association. Boys seem more likely than girls to make errors of perseverance and formula interference. Both sexes make languagerelated errors, but the errors are not the same. Available research is not extensive enough to make definite judgments about the sexrelated differences observed in incorrect responses. Clearly more research is needed. Sexrelated Differences in Problem Solving It has already been acknowledged that the subject of problem solving has been extensively researched. However, as early as 1969, Kilpatrick criticized the fact that the study of problem solving has not been systematic; some researchers have studied the characteristics of the problem while others have given their attention to the characteristics of the problem solvers. Moreover, differences in the tests used to measure problem solving performance also constitute an obstacle when trying to compare the results of the studies carried out. In order to avoid this pitfall and provide a basis for comparison, the studies reviewed in this section, dealing with sexrelated differences in problem solving have been divided in two groups. The first comprises those studies that used the RombergWearne Problem Solving Test. The second contains other relevant studies in which problem solving performance has been measured by means of other instruments. The RombergWearne Problem Solving Test merits special mention because it was the first attempt "to develop a test to overcome the inadequacies of total test scores in explaining the reasons why some students are successful problem solvers and others are not" (Whitaker, 1976, pp. 9, 10). The test is composed of 23 items designed to yield 3 scores: a comprehension score, an application score, and a problem solving score. The comprehension question ascertains whether a child understands the information given in the item stem. The application question assesses the child's mastery of a prerequisite concept or skill of the problem solving question. The problem solving question poses a question whose solution is not immediately available, that is, a situation which does not lend itself to the immediate application of a rule or algorithm. The application and problem solving parts of the test may refer to a common unit of information (the item stem) but the questions are independent in that the response to the application question is not used to respond to the problem solving question. Meyer (1978) Whitaker (1976), Pennema and Sherman (1978), and Sherman (1979) have used the RombergWearne test in their studies. Meyer (1978) investigated whether males and females differ in problem solving performance and examined their prerequisite computational skills and mathematical concepts for the problem solving questions. A sample of 179 students from the 4th grade were administered 19 "reference tests" for intellectual abilities and the RombergWearne test. The analysis showed that males and females were not significantly different in the comprehension, application, and problem solving questions of the test. The sexes differed in only 2 of the 19 reference tests, Spatial Relations and Picture Group NameSelection. A factor analysis, however, showed differences in the number and composition of the factors. For females, a general mathematics factor was determined by mathematics computation, comprehension, application, and problem solving. For males, the comprehension and application parts determined one factor; problem solving with two other reference tests (Gestalt and Omelet) determined another factor. Meyer concluded that comprehension of the data and mastery of the prerequisite mathematical concepts did not guarantee successful problem solving either for males or for females. Problem solving scores for both sexes were about one third their scores in comprehension and one fourth their scores in application. She also concluded that the sexes may have approached the problem solving questions differently. The methods used by females for solving problem situations may have paralleled their approach to the application parts. Males may have used established rules and algorithms for the application parts, but may have used more of a Gestalt approach to the problem solving situation. Whitaker (1976) investigated the relationship between the mathematical problem performance of 4th grade children and their attitudes toward problem solving, their teachers' attitude toward mathematical problem solving, and related sex and programtype differences. Although his main objective was to construct an attitude scale to measure attitudes toward problem solving, his study is important because his findings support Meyer's regarding the lack of significant sexrelated differences in problem solving performance. Performance in the problem solving questions, for both males and females, was much lower than performance in the application questions, and much lower than performance in the comprehension questions. In fact, the mean score for each part of the item, for both males and females, was almost identical to the mean scores obtained by males and females in Meyer's study. Whitaker noted that each application item is more difficult than its preceding comprehension item, and that each problem solving item is more difficult than its preceding application item. No significant sexrelated differences were found for any of the three parts of the item (comprehension, application, or problem solving). Fennema and Sherman (1978) investigated sexrelated differences in mathematics achievement and cognitive and affective variables related to such differences. They administered the RombergWearne Problem Solving Test to a representative sample of 1320 students (grades 68) from Madison, Wisconsin, predominantly middleclass, but including great diversity in SES. The sample consisted of students who had taken a similar number of mathematics courses and were in the top 85% of the class in mathematics achievement. They were tested in 1976. Four high school districts were included. In only one of the high school districts were sexrelated differences in application and problem solving founds in favor of males. They concluded that when relevant factors are controlled, sexrelated differences in favor of males do not appear often, and when they dor they are not large. Sherman (1980) investigated the causes of the emerging sexrelated differences in mathematics performance, in favor of males, during adolescence (grades 811). She wanted to know if these differences emerge as a function of sexrelated differences in spatial visualization and sociocultural influences that consider math as a male domain. In grade 8, she used the RombergWearne Test and, in grade 11, a mathematical problem solving test derived from the French Kit of Tests. The analysis showed that for girls, problem solving performance remained stable across the years. Mean problem solving performance for boys, however, was higher in grade 11 than in grade 8. No sexrelated differences were found in grade 8, but boys outperformed girls in grade 11, where the Stafford test was used. Sherman found that for both sexes problem solving performance in grade 8 was the best predictor of problem solving performance in grade 11. Spatial visualization was a stronger predictor for girls than for boys. Mathematics as a male domain was a good predictor for girls only; the less a girl stereotyped mathematics as a male domain in grade 8, the higher her problem solving score in grade 11. Attitudes toward success in mathematics in grade 8 was a more positive predictor of problem solving performance for boys than for girls; the more positive the attitudes toward success in mathematics in grade 8, the higher their performance in problem solving in grade 11. None of these four studies, all of which used the RombergWearne Mathematics Problem Solving Test, show statistically significant sexrelated differences in problem solving performance. In later studies other tests were used to measure this variable (Kaufman, 1984; Marshall, 1981, 1984). Kaufman (1983/1984) investigated if sex differences in problem solving, favoring males, exist in the 5th and 6th grades and if these differences were more pronounced in mathematically gifted students than in students of average mathematical ability. The Iowa Test of Basic Skills and a mathematics problemsolving test were administered to 504 subjects. Males in the average group as well as males in the gifted group outperformed females, but only the gifted group showed statistically significant differences. As a result of her investigations, Marshall (1981) concluded that sexrelated differences in mathematics performance may be the result of comparing the sexes on total test scores. If the test contains more computation items than problem solving items, girls will perform better than boys, but if the test contains more problem solving items than computation ones, boys will outperform girls. With this in mind, Marshall investigated sexrelated differences in computation and problem solving by analyzing the responses of approximately 18,000 students from grade 6 who had been administered the Survey of Basic Skills Test: Grade 6, during the academic year 197879. Two of the 16 test forms of the Survey were used to assess skills such as concepts of whole numbers, fractions, and decimals. These skills were tested both as simple computations and as story problems (problem solving). Two computation items and one story problem item were selected because they were particularly related; both computation items required skills needed in solving the corresponding story problem. It was assumed that correct solution of the computation item correlates with solving the story problem because the story problem requires a similar computation. Marshall found that girls were better in computation and boys were better in problem solving. She also found that boys were much more likely than girls to answer the story problem item correctly after giving incorrect responses to both computation items. Apparently, mastery of the skills required by the computation items is more important for girls than for boys. If girls cannot solve the computation items, they have little chance of solving the related story problem item. For girls, the probability of success in the story problem item after giving successful answers to both computation items is almost 2 1/2 times the probability of success after giving incorrect responses to both computation items. For boys, the probability of success in the story problem item after successful responses to the computation items is about 1 3/4 times the probability of success on the story problem item after incorrect responses to the computation items. Three years later, Marshall (1984) analyzed more in depth these phenomena of sexrelated differences. Her interest was twofold. First, she wanted to know if there were differences in the rate of success for boys and girls in solving computation and story problem items. Second, she examined additional factors that interact with sex to influence mathematics performance, such as reading achievement, socioeconomic status (SES), primary language, and chronological age. Two questions were raised: Do the probabilibities of successful solving of computation and story problem items increase with reading score? Are these probabilities different for the two sexes? Approximately 270,000 students from the 6th grade were administered the Survey of Basic Skills of the California Assessment Program, during the years 1977, 1978, and 1979. Responses were analyzed using loglinear models. Successful solving of computation items was positively associated with successful solving of story problems. Girls were more successful in computation than boys, and boys were more successful than girls in solving story problems. This finding supports reports from the National Assessment of Educational Progress (NAEP) (Armstrong, 1979). To investigate the effects of reading, SES, language and chronological ager only those test forms containing 2 computation items and 2 story problems were considered for analysis; 32 items from 8 test forms were included in the analysis. The results of these analyses showed that at every level of reading score, 6th grade children were more successful in computation than in story problems. Although the differences were not larger at every reading score boys consistently had higher probabilities of success in story problems than did girls, and girls consistently showed higher probabilities of success in computation than boys. Also, as the reading score increased, the difference between the probability of success in story problems and the probability of success in computation grew larger. This difference grew larger for girls than for boys. Although SES was a major factor in solving computation and story problem items successfully, the effect was similar for each sex. Sexrelated differences by primary language or chronological age were not large. This research carried out by Marshall with elementary grade children supports previous research findings that males are better than females in mathematics problem solving (a higher order skill) and females are better than their counterpart males in computation (a lower level skill). Marshall's research also brought out a different aspect of this question: the notion that girls find it more necessary than boys to succeed in the computation items in order to successfully solve the story problem items. Cognitive and Affective Factors That Influence Sex Differences in Mathematics Learning and Achievement The research reviewed in the literature does not provide evidence of any unique variable that could serve as an explanation for the observed sexrelated differences in mathematics learning and achievement. However, some issues have been discussed, among which the most prevalent are that sexrelated differences in mathematics learning and achievement are a result of differences in formal education; that sexrelated differences in mathematics learning and achievement arise from sex differences in spatial visualization; and that sexrelated differences result from a differentiated effect of affective variables on the mathematics performance of males and females. Differences in Formal Mathematics Education (Differential Coursework Hypothesis) The basis for the differential coursework hypothesis is the fact that sexrelated differences in mathematical learning and achievement show up when comparing groups which are not equal in previous mathematics learning. Atter the 8th grader boys tend to select mathematics courses more otten than girls. Therefore, girls show lower achievement scores in mathematics tests because their mathematics experience is not as strong as the boys' (Fennemar 1975; Fennema & Sherman, 1977; Sherman, 1979). Fennema and Sherman's study (1977) lends additional support to the feasibility of viewing sex differences in mathematics learning and achievement as reflecting something other than a difference in mathematics aptitude. After controlling for previous study of mathematics, they found significant sex differences in mathematics achievement in only two of the four schools under study, making the attribution to sex per se less likely. Controlling for the number of space visualizationrelated courses, the sexrelated differences which originally emerged in spatial visualization scores became nonsignificant. In the two schools where sex differences in mathematics achievement were found, differences between the sexes were also found in their attitudes toward mathematics. Researchers like Backman (1972), who analyzed data from Project Talent, and Allen and Chambers (1977) have also hypothesized that sexrelated differences in mathematics achievement may be related to different curricula followed by males and females. Allen and Chambers attributed male superiority in mathematics problem solving to differences in the number of mathematics courses taken in high school. This issue has been seriously questioned by Astin (1974) Fox (1975ar 1975b), and Benbow and Stanley (1980), among others. Astin and Fox have reported large differences in favor of males among gifted students taking the Scholastic Achievement Test. These differences occur as early as grade 7, when there are no sex differences in the number of courses taken. Benbow and Stanley (1980) compared mathematically precocious boys and girls in the 7th grader with similar mathematics background, and found sizeable sexrelated differences favoring boys in mathematical reasoning ability. Five years later, they conducted a followup study which showed that boys maintained their superiority in mathematics ability during high school. While Fox attributed sexrelated differences in mathematical achievement to differential exposure to mathematical games and activities outside school, Benbow and Stanley suggested that sexrelated differences in mathematics performance stem from superior mathematical ability in males, not from differences in mathematics formal education. The differential coursework hypothesis is not totally convincing and, as reported before, it has been challenged by researchers such as Benbow and Stanley (1980). However, Pallas and Alexander (1983) have questioned the generalizaoility of Benbow and Stanley's findings based on the fact that they used highly precocious learners. The differential coursework hypothesis can be accepted only as a partial explanation of differences in mathematics performance found between the sexes. Differences in Spatial Ability The basic premise in this issue is that males and females differ in spatial visualization and this explains differential mathematics learning and achievement. Until recently, sex differences in spatial ability in favor ot males were believed to be a fact and were thought by some to be related to sex differences in mathematical achievement. Research findings in this area have been inconsistent. In 1966, Maccoby stated that "by early school years, boys consistently do better (than girls) on spatial tasks and this difference continues through the high school and college years" (p.26). In 1972, Maccoby and Jacklin said that the differences in spatial ability between the sexes "remain minimal and inconsistent until approximately the ages of 10 or 11, when the superiority of boys becomes consistent in a wide range of populations and tests" (p.41). In 1974, after a comprehensive literature search, Maccoby and Jacklin concluded that sex differences in spatial visualization become more pronounced between upper elementary years and the last year of high school, the years when sexrelated differences in mathematics achievement favoring boys emerge. Guay and McDaniel (1977) supported in part Maccoby and Jacklin's 1974 findings. They found that among elementary school children, males had greater high level spatial ability than females, but that males and females were equal in low level spatial ability. This finding is inconsistent with that portion of Maccoby and Jacklin's review that suggests that sex differences become evident only during early adolescence. Cohen and Wilkie (1979) however, stated that in tests measuring distinct spatial tasks, males perform better than females in early adolescence and throughout their life span. Most studies carried out after 1974 have failed to support these sex differences in spatial abilities (Armstrong, 1979; Connor, Serbin, & Schackman, 1977; Fennema & Sherman, 1978; Sherman, 1979). Fennema and Sherman (1978) and Sherman (1979) have explored sexrelated differences in mathematical achievement and cognitive and affective variables related to these differences. In a study involving students from grades 6, 7, and 8, from four school districts, Fennema and Sherman found that spatial visualization and problem solving were highly correlated for both sexes (.59 and .60). Even in the school district where sex differences were found in problem solving, no significant sexrelated differences were found in spatial visualization. When Sherman (1980) compared groups of males and females in two different grades, 8 and 11, she found no sexrelated differences in problem solving or in spatial visualization in grade 8. In grade 11, however, although the sexes differed in their problem solving performance, no sexrelated differences were found in spatial vizualization. Even though spatial visualization in grade 8 was the second best predictor of problem solving performance in grade 11, sex differences in grade 11 were not a result of spatial visualization since no differences were found in that skill. In spite of the fact that no sex differences were found in spatial abilities, it is evident that males and females may use them in a different way. Meyer (1978), with an elementary grade sample, and Fennema and Tartree (1983) with an intermediate level sampler found that the influence of spatial visualization on solving mathematics problems is subtler and that males and females use their spatial skills differently in solving word story problems (problems that measure problem solving ability or reasoning). Fennema and Tartree (1983) carried out a threeyear longitudinal study which showed that girls and boys with equivalent spatial visualization skills did not solve the same number of items, nor did they use the same processes in solving problems. The results also suggested that a low level of spatial visualization skills was a more debilitating factor for girls than for boys in problem solving performance. Landau (1984) also investigated the relationship between spatial visualization and mathematics achievement. She studied the performance of middle school children in mathematical problems of varying difficulty, and the extent to which a diagramatic representation is likely to facilitate solution. She found that spatial ability was strongly correlated to mathematical problem solving and that the effect of spatial ability was more influential for females. Females made more use of diagrams in the solution of problems, reducing the advantage of males over females in problem solving performance. The issue of sexrelated differences in spatial visualization ability as an explanation for sex differences observed in mathematics achievement is less convincing and the findings more contradictory than in the issue of sex differences in formal education. Besides these cognitive issues, other issues, mostly affective in nature have also been studied in trying to explain the origin of these sex differences in mathematics achievement and learning. The studies dealing with these affective variables are reviewed in the next section. Differentiated Effect of Affective Variables Researchers have attempted to explain the effect of sex differences in internal beliefs, interests, and attitudes (affective variables) on mathematics learning and achievement. A brief statement of each explanation precedes the summary of studies conducted that support the explanation. Confidence as lerners of mathematics. Females, more than males, lack confidence in their ability to learn mathematics and this affects their achievement in mathematics and their election of more advanced mathematics courses. Maccoby and Jacklin (1974) reported that self confidence in terms of grade expectancy and success in particular tasks was found to be consistently lower in women than in men. In 1978, Fennema and Sherman reported that in their study involving students from grades 6 through 12, boys showed a higher level of confidence in mathematics at each grade level. These differences between the sexes occurred in most instances even when no sexrelated differences in mathematics achievement were found. The correlation between confidence in mathematics performance and mathematics achievement in this study was higher than for any other affective variable investigated. Sherman reported a similar finding in 1980; in males, the most important factor related to continuation in theoretical mathematics courses was confidence in learning mathematics. This variable weighed more than any of the cognitive variables: mathematics achievement, spatial visualization, general ability, and verbal skill. In the case of females, among the affective variables, confidence in learning mathematics was found to be second in importance to perceived usefulness of mathematics. Probert (1983) supported these findings with college students. A variable that needs discussion within the context of sex differences in confidence as learners of mathematics is causal attribution. Causal attribution models attempt to classify those factors to which one attributes success or failure. The model proposed by Weiner (1974) categorizes four dimensions of attribution ot success and failure: stable and internal, unstable and internal, stable and external, and unstable and external. For example, if one attributes success to an internal, stable attribute, such as ability, then one is confident of being successful in the future and will continue to strive in that area. If one attributes success to an external factor such as a teacher, or to an unstable one, such as effort, then one will not be as confident or success in the future and will cease to strive. Failure attribution patterns work this way: if failure is attributed to unstable causes, such as effort, failure can be avoided in the future and the tendency will be to persist in the task. However, if failure is attributed to a stable cause, such as ability, the belief that one cannot avoid failure will remain. Studies reported by BarTal and Frieze (1977) suggest that males and females tend to exhibit different attributional patterns of success and failure. Males tend to attribute their success to internal causes and their failures to external or unstable ones. Females show a different pattern; they tend to attribute success to external or unstable causes and failures to internal ones. The pattern of attributions, success attributed externally and failure attributed internally, has become hypothesized to show a strong effect on mathematics achievement in females. Kloosterman's (1985) study supported these findings. According to Kloosterman, attributional variables appear to be more important achievement mediators for females than for males, as measured by mathematics word problems. More research is needed in this area. Mathematics as a male domain. Mathematics is an activity more closely related to the male sex domain than to the female sex domain (Eccles et al., 1983). Thus, the mathematical achievement or boys is higher than that of girls. According to John Ernest (1976) in his study Mathematics and Sex, mathematics is a sexist discipline. He attributed sexrelated differences in mathematical achievement to the creation by society of sexual stereotypes and attitudes, restrictions, and constraints that promote the idea of the superiority of boys in mathematics. Ernest reported that boys, girls, and teachers, all believe that boys are superior in mathematics, at least by the time students reach adolescence. Bem and Bem (1970) agree and argue that an American woman is trained to "know her place" in society because or the pervasive sexrole concept which results in differential expectations and socialization practices. Plank and Plank (1954) were more specific. They discussed two hypotheses related to this view: the differential cultural reinforcement hypothesis and the masculine identification hypothesis. The differential cultural reinforcement hypothesis states that society in general perceives mathematics as a male domain, giving females less encouragement for excelling in it. The masculine identification hypothesis establishes that achievement and interest in mathematics result from identification with the masculine role. A study related to the differential cultural reinforcement hypothesis is that of Dwyer (1974). Dwyer examined the relationship between sex role standards (the extent to which an individual considers certain activities appropriate to males or females) and achievement in reading and arithmetic. Students from grades 2, 4, 6, 8, 10, and 12 participated in this study. She found that sex role standards contributed significant variance to reading and arithmetic achievement test scores and that the effect was stronger for males than for females. This led to her conclusion that sexrelated differences in reading and arithmetic are more a function of the child's perception of these areas as sexappropriate or sexinappropriate than of the child's biological sex, individual preference for masculine and feminine sex roles, or liking or disliking reading or mathematics. In a study which agrees with the masculine identification hypothesis, Milton (1957) found that individuals who had received strong masculine orientation performed better in problem solving than individuals who received less masculine orientation. Elton and Rose (1967) found that women with high mathematical aptitude and average verbal aptitude scored higher on the masculinity scale of the Omnibus Personality Inventory (OPI) than those with average scores on both tasks. It is not until adolescence that sex differences in the perception of mathematics as a male domain are found (Fennemar 1976; Stein, 1971; Stein & Smithless, 1969; Verbeker 1983). In a study with 2nd, 6th and 12th graders, Stein and Smithless (1969) found that students' perceptions of spatial, mechanical, and arithmetic skills as masculine became more defined as these students got older. Fennema (1976) considers that the influence each sex exerts upon the other on all aspects of behavior is stronger during adolescence. Since during these years males stereotype mathematics as a male domain, they send this message to females who, in turn, tend to be influenced in their willingness to study or not to study mathematics. Before that stage, girls consider arithmetic feminine, while boys consider it appropriate for both sexes (Bobber 1971). Usefulness of mathematics. Females perceive mathema tics as less useful to them than males do, and this perception occurs at a very young age. As a results females exert less effort than males to learn or elect to take advanced mathematics courses. Many studies reported before 1976 found that the perception of the usefulness of mathematics for one's future differs for males and females, and is related to course taking plans and behavior (Fox, 1977). If females do not perceive mathematics as useful for their future, they show less interest in the subject than counterpart males. These differences in interest are what Hilton and Berglund (1974) suggest to account for sexrelated differences in mathematics achievement. Although the perception of the usefulness of mathematics is still an important predictor of course taking for girls, there is a growing similarity between males and females regarding the usefulness of mathematics (Armstrong & Pricer 1982; Fennema & Sherman, 1977; Moller, 1982/1983). Armstrong and Price investigated the relative influence of selected factors in sexrelated differences in mathematics participation. Both males and females selected usefulness of mathematics as the most important factor in deciding whether or not to take more mathematics in high school. Moller's study revealed that both males and females based mathematics coursetaking decisions on career usefulness. A Fennema and Sherman (1977) study showed only slight differences between males and females in their feelings about the usefulness of mathematics. In her study of this variable among college students, Probert (1983/1984) did not find any sexrelated differences either. These have been the main affective variables researched in attempting to explain the underlying causes of sexrelated differences observed in mathematics learning and achievement. In spite ot the great diversity of studies dealing with both cognitive and affective variables, there are no clearcut findings to render unequivocal support to a particular variable as accounting for these sexrelated differences. However, everything seems to point to the fact that affective, rather than cognitive variables play a more significant role in the sexrelated differences observed in mathematics performance and learning. In most of the studies dealing with affective variables, findings consistently show that these factors influence mathematics performance in females more than in males. In at least one area confidence as learners of mathematics, Sherman (1980) found that this variable influenced course election more than all the cognitive variables previously discussed. The case for the societal influences on sex roles and expectations to account for the differences in mathematics learning is also supported in one way or another in the studies reported in the literature. Problem Solving Performance and Related Variables Problem solving has been perhaps the most extensively researched area in mathematics education. Published reviews by Kilpatrick (1969) Riedesel (1969), and Suydam and Weaver (19701975) attest to this. Much of the research done has focused on identifying the determinants of problem difficulty and the problem features that influence the solution process. At presents no set of variables has been clearly established as a determinant of problem difficulty. Several researchers have investigated the effect or reading and computation on problem solving performance. Others have studied the effect of student attitudes toward problem solving in problem solving learning and achievement. Typically, correlational methods have been used to investigate these questions. Computational Skills and Problem Solving Performance One of the first researchers to study the effect of computation and reading on problem solving performance was Hansen (1944). He investigated the relationship of arithmetical factors, mental factors, and reading factors to achievement in problem solving. Sixth grade students were administered tests in problem solving and categorized as superior achievers (best problem solvers) and inferior archievers (poorest problem solvers). The two groups were compared in selected factors believed to be related to success in arithmetic problem solving: arithmetical, mental and reading factors. After controlling for mental and chronological age, the superior achievers in problem solving surpassed the inferior achievers in mental and arithmetical factors. The superior group did better in only two of the six items under the reading factors: general language ability and the reading of graphs, charts, and tables. The findings suggest that reading factors are not as important as arithmetic and mental factors in problem solving performance. However, these findings should be taken cautiously, as the content of the Gates tests (used to measure reading) is literary and does not include mathematical material. Chase (1960) studied 15 variables in an effort to find out which ones have significant influence on the ability to solve verbal mathematics problems. Only computation, reading to note details, and fundamental knowledge were primarily related to problem solving. Computation accounted for 20.4% of the 32% variance directly associated with problem solving. Chase concluded that a pupil's ability in the mechanics of computation, comprehension of the principles that underline the number systems, and the extent to which important items of information are noticed when reading, are good predictors of the student's ability in solving verbal problems. Balow (1964) investigated the importance of reading ability and computation ability in problem solving performance. He objected to the approaches used by other researchers who in their analyses dichotomized research subjects as "poor" or "good" students, and who ignored the recognize effect of intelligence on reading and on mathematics achievement. Balow administered the Stanford Achievement Test (subtests of reading, arithmetic, and reasoning) and the California ShortForm test of mental ability to a group of 1,400 children from the 6th grade. All levels of achievement were included in the analysis. Analysis of variance and covariance were used and compared. He confirmed the findings of other researchers to the effect that there is a direct relationship between I.Q. and reading ability and between I.Q. and computational skills. The results of the analysis of variance revealed that increases in computation ability were associated with higher achievement in problem solving. A relationship between reading ability and problem solving was also found, but it was not as strong. Significant differences in problem solving performance associated with computational ability were found when intelligence was controlled. Balow concluded that computation is a much more important factor in problem solving than reading ability, and that when I.Q. is taken into consideration, the degree of the relationship between reading and problem solving ability becomes less pronounced. Intelligence tends to confound the relationship between these two variables. Knifong and Holtan (1976, 1977) attempted to investigate the types of difficulties children have in solving word problems. They administered the word problem section of the Metropolitan Achievement Test to 35 children from the 6th grade. Errors were classified in two categories. Category I included clerical and computational errors. Category II included other types of errors, such as average and area errors, use of wrong operation, no response, and erred responses offering no clues. It the student's work indicated the correct procedure and yet the problem was missed because of a computational or clerical errors it was assumed that the problem was read and understood. An analysis of frequencies showed that clerical errors were responsible for 3% of the problems incorrectly solved, computational errors accounted for 49%, and other errors for 48% of the erred problems. Knifong and Holtan concluded that "improved computational skills could have eliminated nearly half or the word problem errors" (p. 111). These computational errors were made in a context where other skills such as reading, interpretation of the problem, and integration of these skills necessary for the solution of word problems, might interact. However, Knifong and Holtan state that their findings neither confirm nor deny that improvement or reading skills will lead to improvement in problem solving. They conclude that "it is difficult to attribute major importance to reading as a source of failure" (p. ll). In a later analysis, looking for evidence of poor reading abilities affecting children's success in word problems, Knifong and Holtan (1977) interviewed the children whose errors fell under the category or "other errors." Students were asked to read each problem aloud and answer these questions: What kind of situation does the problem describe? What does the problem ask you to find? How would you work the problem? Ninety five percent of the students read the problem correctly; 98% explained the kind of situation the problem described in a correct manner; 92% correctly answered what the problem was asking them to find, and 36% correctly answered the question of how to work the problem. The fact that a large percent of the students whose errors were classified as "other errors" (in which reading skills might have been a factor) correctly stated how to work the problem, is strong evidence of their ability to read and interpret the problems correctly. The errors made by this group of students had a distinct origin, unrelated to reading ability. Zalewski (1974) investigated the relative contribution of verbal intelligence, reading comprehension, vocabulary, interpretation of graphs and tables, mathematical concepts, number sentence selection, and computation to successful mathematical word problem solution, and the relationship of the dependent variable to the eight independent variables. She worked with a group of 4th grade children who were administered the subtests of the Iowa Test of Basic Skills (ITBS) and the Weschler Intelligence Scale for Children (WISC). Multiple regression analysis was performed. A correlation of .769 was found between word problem solving and the eight independent variables. Correlations between word problem solving and the independent variables ranged from .363 (verbal intelligence) to .674 (mathematical concepts). Correlations between the independent variables ranged from .369 (verbal intelligence and computation) to .749 (reading comprehension and vocabulary). Mathematical concepts, computation, and number sentence selection were almost as effective as all eight independent variables in predicting achievement in mathematical word problem solving. Mathematical concepts, computations number sentence selection, and reading comprehension accounted for 58% of the variance, whereas all eight predictors accounted for 59% of the variance. The two best predictors were mathematical concepts and computations which accounted for 54% variance. Other variables accounted for about 40% of the variance. The author recommends that the findings of this study be interpreted cautiously because the correlation between the eight independent variables was high, and, according to Zalewski, in a study of this nature where the interest is primarily in the influence of several variables on one dependent variable, a low correlation between the independent variables is required. (p. 2804) In a more recent investigations Exedisis (1983) studied the contribution of reading ability, vocabulary, mathematical concepts, computation, sex, and race on problemsolving performance. The Iowa Test of Basic Skills was administered to a group of 6th, 7th, and 8th grade anglo and black Chicago male and female adolescents. Problem solving was highly correlated to an understanding of basic mathematical concepts, somewhat correlated to race, and weakly correlated to computational and vocabulary skills, sex, and reading ability. Although the findings of these studies show a relationship between computational skills and problem solving achievement, this relationship is not strong enough to be considered the most determinant factor in problem solving achievement, as some of the researchers have been careful to point out. In spite ot the dismissal of reading as a determinant factor in problem solving achievement by some of these same researchers, more recent studies in this area have led others to hold different views. Reading and Problem Solving Performance Martin (1964) studied the contribution of reading comprehension, computation, abstract verbal reasoning, and arithmetic concepts to arithmetic problem solving performance. Fourth and 8th grade students were administered the Iowa tests of Basic Skills and the LorgeThorndlike intelligence test (verbal). He found that in the 4th grade the correlations between problem solving and abstract verbal reasoning, reading comprehension, arithmetic concepts, and computation were .61, .64, .66, and .60 respectively, and .56, .68, .69, and .63 in the 8th grade. When computation was held constant, the correlation between problem solving and reading was .52 in grade 4 and .54 in grade 8. When reading was held constant the correlation between problem solving and computation was .43 in grade 4 and .42 in grade 8. Creswell (1982) worked with a sample of anglo and black adolescents from Chicago. Each subject was administered the California Achievement test. Multiple regression was used to analyze the data. The analysis showed that reading is more important than computation in predicting student performance in problem solving. Reading accounted for 49.5% of the variance; computation accounted for 14.6% of the variance. Ballew and Cunningham (1982) worked with 6th grade students in an attempt to find what proportion of students have as their main source of difficulty with word problems each of the following factors: a) computation skills, b) interpretation of the problem, c) reading and, d) integrating these skills in the solution of problems. They also wanted to know if a student can be efficiently diagnosed as having one of the four categories as his/her main difficulty with mathematics word problems. Their study is important because it represents an attempt to demonstrate that multiple factors can interact in the correct solution of a mathematics word problem. They constructed three graded tests from a basal mathematics series for grades 3 through 8. For test 1i the problems were set in pure computational form (the effects or reading, interpretation, as well as the necessity for integration were removed in an effort to measure the computational skills required by the word problems). For test 2, the effects of reading and computation were removed by reading the problems to the students and by giving scores based on whether or not the students set them up properly, in an attempt to measure problem interpretation alone. For test 3, the effect of computation was removed. The test yielded two scoresone by grading the students on whether or not they set up the problems properly and another by grading on the basis of the correct answer. The tests were administered to all 244 students from the 6th grade in two different schools. A diagnostic profile was obtained for each of the 217 students for which complete data were available: a computational score, a probleminterpretation score, a reading score, and a readingproblem solving score. They assumed that if the readingproblem interpretation score was lower (one or more levels lower) than the probleminterpretation score, the difficulty was due to reading ability. If the score of the lowest of the three areas (computation, problem interpretation, and readingproblem interpretation score) was the same as the readingproblem solving score, the student's area of greatest immediate need was either computation, problem interpretation, or reading. If the readingproblem solving score was lower than the lowest of the other three scores, the student's area of greatest immediate need was integration. Analysis of the data revealed that for 19% of the students, problem interpretation was their major difficulty; for 26% of the students, integration (total problem solving) was their greatest immediate need; for another 26%, computation was the major weakness; and for 29%, reading was their greatest immediate need. Seventy five percent of the students demonstrated clear strength in computation, 21% in problem interpretation, and 4% in readingproblem interpretation. An analysis across all students (including those without complete data) showed that 26% of the subjects could not work word problems at a level as high as that at which they could computer interpret problems, and read and interpret problems, when those areas were measured separately. This led them to conclude that knowing the skills or the components of solving word problems is not sufficient for success, since the components must be integrated into a whole process (mastery learning of the components cannot assure mastery of the process). Their analysis also led them to conclude that, in the case of 6th graders, inability to read problems is a major obstacle in solving word problems. Only 12% ot the subjects could read and set up problems correctly at a higher level than they could computer while 60% could compute correctly at a higher level than they could read and set up problems; 44% could set up problems better when they heard them read than when they read the problem themselves. Only 13% could set up problems better when they read them than when they heard them read. Muth (1984) investigated the role of reading and computational skills in the solution of word problems. A group of 200 students from the 6th grade were administered a test of basic skills and a mathematics word problem test. The word problem test consisted of 15 sample items supplied by the National Assessment of Educational Progress. The items were adapted to include some extraneous information and complex syntactic structure. Four versions of the test were constructed by combining two versions of problem information (absence vs. presence of extraneous information) with two versions of syntactic structure (simple vs. complex syntax). Task performance was measured by means of the number of problems answered correctly, number of problems set up correctly, and amount of time spent taking the test. Reading ability and computational ability were both positively correlated with number of correct answers and with number of problems correctly set up, and negatively correlated with testtaking time. Presence of extraneous information was negatively correlated with correct answers and correct set ups and positively correlated with testtaking time. Syntactic complexity was not significantly correlated with any of the performance measures. Results of a multiple regression analysis showed that reading accounted for 46% of the variance in total correct answers and computation accounted for 8%. Reading ability and computational ability uniquely accounted for 14% and 8% of the variance in the number of correct answers, respectively. Extraneous information added significantly to the variance explained in the number of correct answers, but syntactic structure did not. Reading ability accounted for 5% of the variance in testtaking time, but computation did not add significantly to the variance explained by reading. Muth concluded that reading and computation both contribute significantly to success in solving arithmetic word problems, but that reading plays a more significant role than does computation. The studies reviewed in this section show a positive relationship between reading and problem solving performance, but in the case of Ballew and Cunningham (1982) this relationship is not viewed singly but rather as one among the interacting factors that produce successful problem solving. The third variable reviewed is the effect of student attitudes toward problem solving on problem solving performance. Many researchers have tried to demonstrate that this variable is a determinant factor in problem solving achievement. Attitudes Toward Problem Solving and Problem Solving Performance Research studies support the existence of positive and rather stable relationships between student attitudes and achievement in mathematics. Aiken (1970) has suggested that an individual's attitude toward one aspect of the discipline (mathematics), such as problem solving, may be entirely different from his/her attitude toward another phase of the discipline, such as computation. Research, however, has been directed to the use of single, global measures of attitudes toward mathematics rather than to the investigation of attitudes toward a particular phase of the discipline. The studies described below are only part of the few investigations which have examined the relationship between student attitudes and performance in the area of problem solving. Carey (1958) constructed a scale to measure attitudes toward problem solving. Her interest was in general problem solving rather than in mathematical problem solving. Her work constitutes the first attempt to construct a measure of attitudes toward problem solving. The scale was used with a group of college students, and she found, among other things, that problem solving performance is positively related to problem solving attitudes and that, in the case of females, positive modification of attitudes toward problem solving brings a significant gain in problem solving performance. Lindgren, Silvar Faraco, and DaRocha (1964) adapted Carey's scale of attitudes toward problem solving and applied it to a group of 4th grade Brazilian children. Students also answered an arithmetic achievement test, a general intelligence test, and a socioeconomic (SE) scale. A low but significant positive correlation was found between arithmetic achievement and attitudes toward problem solving. A near zero correlation was found between attitudes toward problem solving and intelligence. Since problem solving is one aspect of the discipline of mathematics, this correlation between attitudes and arithmetic achievement can lead to a conclusion or a strong correlation between attitudes toward problem solving and problem solving performance. Whitaker (1976) constructed a student attitude scale to measure some aspects of 4th grade student attitudes toward mathematic problem solving. He included statements reflecting children's beliefs about the nature of various types of mathematical problems, the nature of the problem solving process, the desirability or persevering when solving a problem, and the value of generating several ideas for solving a problem. He correlated student attitudes toward problem solving with their scores in a mathematical test which yielded a comprehension score, an application score, and a problem solving score. He found a significant positive relationship between problem solving performance and student attitude scores on the subscale which measured reactions to such things as problem solving techniques or problem situations, or to the frustration or anxiety experienced when confronted with problem solving situations. In another part of this study, Whitaker investigated the relationship between the attitudes of 4th grade teachers toward problem solving and their students' performance in problem solving. A very weak and nonsignificant negative correlation was found between the teacher's attitudes toward problem solving and student performance. The studies reviewed have confirmed the relationship between problem solving performance and attitudes toward problem solving (Carey, 1958; Lindgren et al., 1964; Whitaker, 1976). However, the results reported in the studies that investigated the relationship between problem solving performance and computation and between reading and problem solving fail to be consistent in their conclusions. Hansen (1944), Chase (1960) Balow (1964), Knifong and Holtan (1976, 1977), and Zalewski (1974) concluded that computation is more strongly related to problem solving than is reading. Martin (1964), Creswell (1982), Ballew and Cunningham (1982), and Muth (1984), concluded that reading ability and mathematical problem solving show a stronger relationship than computation and problem solving. Exedisis's (1983) findings led to the conclusion that the effect or reading and computation in problem solving performance is unimportant. CHAPTER III METHOD The first objective of this study was to investigate sex differences in the selection of incorrect responses in a mathematics multiplechoice achievement test, and to determine whether these differences were consistent over three consecutive administrations of the test. The second objective was to investigate whether males and females differ in problem solving performance, if these differences persist after accounting for computational skills, and if the malefemale differences depend on the level of computational skills. This chapter contains descriptions of the sample, the test instrument, and the statistical analysis used in achieving the above mentioned objectives. The Sample To achieve the first objective of the study, all the students who took the Basic Skills Test in Mathematics6 ("Prueba de Destrezas Basicas en Matem&ticas6") in each or the three years were included in the study. To achieve the second objective of the study, approximately 1,000 students were selected randomly for each year. For the first year, 1,002 were selected (492 boys and 510 girls); for the second year, 1,013 students were selected (504 boys and 509 girls); and, for the third years 1,013 students were selected (509 boys and 504 girls). The student population in Puerto Rico includes children from the urban and rural zones and comprises children from low and middle socioeconomic levels. Findings can be generalized only to this population. The Instrument The Basic Skills Test in Mathematics6 is a criterionreferenced test used in the Department ot Education of Puerto Rico as part of the annual assessment program. The test measures academic achievement in operations, mathematical concepts, and story problems. It has a reported split half reliability or .95. The test was designed specifically for Puerto Rico. Its contents and the procedures followed for its development were formulated and reviewed by educators from the mathematics department of the Department of Education of Puerto Rico, in coordination with the Evaluation Center ot the Department of Education and mathematics teachers from the school districts. The emphasis placed on each skill area is depicted in Figure 3.1. Fig. 3.1 Number of Questions by Skill Area in the Basic Skills Test in Mathematics 6 Analysis of the Data Analysis of Sex by Option by Year Cross Classifications Loglinear models were used to analyze the sex by option by year cross classifications for each item. Two topics are addressed in this section, the hypotheses tested using loglinear models and a comparison of the hypotheses tested in this study with those tested by Marshall (1981, 1983). The object of the analysis was to test two hypotheses: 1. The proportion of males and the proportion of females choosing each incorrect option does not vary from year to year. Note that this hypothesis is stated in the null form. 2. Assuming that the first hypothesis is correct, the proportion of males who choose each incorrect option is different from the proportion of females who choose each incorrect option. This hypothesis is stated in the alternate form. In Table 3.1 a hypothetical cross classification of sex, option, and year is presented. Hypothesis 1 is true for this three dimensional contingency table. In Table 3.2 the three dimensional contingency table is rearranged to show the year by option contingency table for each gender. Table 3.1 Hypothetical Probabilities of Option Choice Conditional on Year and Sex Option Year Sex 1 2 3 First M .7 .2 .1 F .5 .3 .2 Second M .7 .2 .1 F .5 .3 .2 Third M .7 .2 .1 F .5 .3 .2 Table 3.2 Hypothetical Probabilities of Option Choice Conditional on Year and Sex and Arranged by Sex Option Sex Year 1 2 3 M First .7 .2 .1 Second .7 .2 .1 Third .7 .2 .1 F First .5 .3 .2 Second .5 .3 .2 Third .5 .3 .2 Inspection of the year by option contingency tables shows that year and option are independent for each gender. Thus, hypothesis 1 is equivalent to the hypothesis that year and option are independent conditional upon gender. Hypothesis 2 is also true for Table 3.1 and Table 3.2. Therefore, when hypothesis 1 is correct, hypothesis 2 is equivalent to the hypothesis that sex and option choice are dependent. In his discussion of the analysis of three dimensional contingency tables, Fienberg (1980) presents the following saturated model for the data: log mijk = Ui+Uj+Uk+Uij+Uik+Ujk+Uijk. (3.1) In this model, mijk is the expected value of the frequency in cell ijk of the three dimensional table. The model states that all three classification factors for a three dimensional contingency table are mutually dependent. In the present research i is the year index, j is the option index, and k is the sex index. Fienberg shows that deleting the terms Uij and Uijk yields a model in which year and option are independent conditional upon sex. This model is log mijk = Ui+Uj+Uk+Uik+Ujk. (3.2) Fienberg also shows that deleting the Ujk term from (3.2) to obtain log mijk= Ui+Uj+Uk+Uik. (3.3) yields a model that specifies that option is independent of sex. Based on Fienberg's presentation, an appropriate analysis for testing the hypotheses is 1. Conduct a likelihood ratio test of the adequacy of model (3.2). If this test is nonsignificant, then the data support the adequacy of the model, and the hypothesis that conditional on gender, options and year are independent. Because model (3.1) is a saturated model, testing the adequacy of model (3.2) is the same as comparing the adequacies of models (3.1) and (3.2). 2. Conduct a likelihood ratio test comparing .the adequacies of models (3.2) and (3.3). If this test is significant, then model (3.2) fits the data better than model (3.3) and the data support the hypothesis that the choice of incorrect option is dependent on sex. To summarized if the first test is nonsignificant and the second test is significant, then the choice of option is dependent on sex, and the pattern of dependency is the same for all three years. A problem arises in interpreting this analysis with the data used in this research. Over the three years, there were responses available from 135,340 students. Even if only 5% of the students answered an item incorrectly, the responses of 7,767 students would be used in analyzing this item. On the other hand, if 90% answered an item incorrectly, the responses of 121,806 students would be used in analyzing the data. As a result of the large sample size the tests described above are likely to be very powerful. In step 1 of the analysis, then, even a very small change from year to year in the proportion of males or females who choose an option is apt to be detected, and the results will indicate that hypothesis 1 is not supported. For step 2, even a very small dependence of option choice on sex is likely to be detected and hypothesis 2 is likely to be supported. In brief, the problem caused by the large sample size is that practically insignificant differences may yield statistical significance. Fortunately, the form of the test statistic used in the likelihood ratio test suggests a reasonable solution to this problem. The test statistic is G2 = 2 ZFo loge (Fo/Fe). (3.4) Here the summation is over all the cells in the contingency table Fo refers to the observed frequency in a cell, and Fe refers to the estimated expected frequency in a cell. Denoting the observed proportion in a cell by Po and the estimated expected proportion in a cell by Per the test statistic can be written G2 = 2 N TPo loge (Po/Pe) (3.5) where N is the total number of subjects. This form of the test statistic suggests the following strategy. For any significant G2, using Po and Pe calculated from the total data set available for an item, calculate the minimum N required for G2 to be significant. If the minimum N is very larger this suggests that the statistically significant result is not practically significant since it can only be detected in very large samples. Of course, the question remains as to what can be considered a minimum large N. Although there is room for argument, it seems reasonable to claim that if an average of 1000 subjects per year is required to show significance, then the result is not likely to be practically significant. On the basis of this reasoning, it was proposed to ignore all significant results that would be nonsignificant if there were less than 3000 subjects available. In addition, all loglinear model tests were conducted using a .01 level of significance. Since this research is based on Marshall (1981, 1983), it is important to compare the method of analysis used in this study to the one used by Marshall. Marshall also used a twostep analysis. In the first step of her analysis she deleted the Uijk term from (3.1) and tested the adequacy of the model, log mij k = Ui+Uj+Uk+Uij+Uik+Ujk. (3.6) Following this, she deleted the Ujk term to obtain log mijk = Ui+Uj+Uk+Uij+Uik. (3.7) and compared these two models using a likelihood ratio test. If the first test was nonsignificant and the second significant, Marshall claimed that option choice was dependent on sex and the pattern of dependency was the same from year to year. This is the same claim that this study sought to establish. However, the approach used here was to present evidence that model (3.2) fits the data while Marshall tried to show that model (3.6) fits the data. The major difference between the two approaches concerns the operationalization of the concept that the 64 genderoption dependency is the same from year to year. In this study, a three dimensional table was considered to exhibit the same year to year pattern of genderoption dependency if, conditioned on gender, the same proportion of students chose each incorrect response over the three year period. This seems to be a straightforward and natural way to operationalize the concept. To illustrate how Marshall operationalized the concept in question, a hypothetical set of probabilities was constructed conforming to the pattern specified by Marshall. This is displayed in Table 3.3. Table 3.3 Hypothetical Joint Probabilities of Year, Option, and Sex Option Year Sex 1 2 3  First M .120 .045 .015 F .060 .060 .030 I I Second M .144 .022 .019 F .072 .030 .039 I~ Third M .132 .040 .016 F .066 .054 .039 Note: Probabilities reported are truncated to three places. One important characteristic of this table involves twobytwo subtables of sex cross classified with two of the options for each year. For example, consider the three twobytwo tables obtained by cross classifying sex and option choices one and two for each year. These tables are indicated by dotted lines on Table 3.3. For this table, the ratio of the odds of a male choosing option one to the odds of a female choosing option one is the same (within truncating error) for each year. For example, this odds ratio for the first year is (.120/.045) / (.060/.060) = 2.67. Within the error caused by reporting truncated probabilities the odds ratio for years two and three is the same as that for year one. The equality of these odds ratios is Marshall's operationalization stage of the year toyear genderoption dependency. To show that the odds ratio can be constant over years, but that the probabilities of option choice conditional on sex and year can change from year to year, for both males and females, the probabilities in Table 3.3 were converted to the probabilities of option choice conditional upon sex and year. These conditional probabilities are reported in Table 3.4. Unlike the probabilities in Table 3.2, those in Table 3.4 change from year to year for both males and females. Table 3.4 Hypothetical Probabilities of Option Choice Conditional on Sex and Year Option Year Sex 1 2 3 First M .666 .250 .083 F .400 .400 .200 Second M .774 .120 .104 F .510 .212 .276 Third M .698 .214 .087 F .431 .352 .215 Note: Probabilities reported are truncated to three places. Which procedure is more appropriate to test the claim that the pattern of male and female option choice remains the same from year to year? It seems more reasonable to test for a pattern like that in Table 3.2 than to test for a pattern like that in Table 3.4, and, consequently, this was the strategy adopted in this study. Comparison of Males and Females in Problem Solving Performance One object of the study was to compare the performance of males and females on problem solving. Two questions were addressed. First, do males and females differ in problem solving performance? Second, do these differences persist when computational skill is controlled for, and do these differences depend on the level of computational skill? Seven analyses of covariance (ANCOVA) tests were conducted, one with each of addition, subtraction, multiplication, division, addition of fractions, decimals in subtraction, and equivalence as covariates. A well known problem that arises in the use of ANCOVA is that an unreliable covariate can cause spurious differences between the sex groups. To solve this problem, Porter (1967/1968) proposed the use of estimated true scores for observed scores. Porter (1967/1968) conducted a simulation that gave empirical support to the adequacy of this strategy. Hunter and Cohen (1974) have provided theoretical support for this strategy. CHAPTER IV RESULTS Introduction The data gathered from 6th grade students from the public schools in Puerto Rico who took the Basic Skills Test in Mathematics6 during three consecutive years were analyzed in this study. The first objective was to investigate whether boys and girls differ in the selection of incorrect responses, and if the pattern of differences was consistent throughout the three years in which the test was administered. The second objective was to investigate whether males and females differ in problem solving performance, if these differences persist after accounting for computational skills, and if the malefemale differences depend on the level of computational skills. The results are discussed in two sections. Study findings in the area of sex differences in the selection of incorrect responses are discussed in the first section. The second section is devoted to an exposition of the findings in the area of sexrelated differences in problem solving. SexRelated Differences in the Selection of Incorrect Responses As indicated in Chapter III, there are two models of interest. The first model indicates that year and option are independent, conditional upon sex, log mijk = Ui+Uj+Uk+Uik+Ujk. (3.2) Substantively, this model implies that the pattern of male and female option choices is consistent over the three years of test administration. The second model indicates that option is independent of sex, log mijk = Ui+Uj+Uk+Uik (3.3) This model implies that the pattern of option choice is the same for males and females. In order to determine if males and females differed in the selection of incorrect responses and if these differences were stable across the three years of test administration, a twostep test was performed. First, a likelihood ratio test of the adequacy of model (3.2) was conducted for each of the 111 items of the test. Under this test a model fits the data when the X2 value obtained is nonsignificant at a specified alpha level. For 50 (45%) of the items, model (3.2) fitted the data adequately. For these items the pattern of male and female option choices was consistent for the three years. The X2 values for the 61 items for which the likelihood ratio tests were significant at the .01 level are reported in Table 4.1. Also reported in this table are the actual sample sizes and the minimum sample sizes necessary for the likelihood ratio tests to be significant. Of the 61 items, 59 had minimum sample sizes greater than 3,000. Thus, although for the three years both males and females samples had inconsistent option choices on these 61 items, on 59 of the items the inconsistency was relatively minor. Consequently, these 59 items were included in step 2 along with the initial 50 items. In step 2, a likelihood ratio test was performed comparing the adequacy of the models in (3.2) and (3.3). The X2 values associated with models (3.2) and (3.3) for each of the 109 items subjected to step 2 are reported in columns b and c of Table 4.2. Also reported in Table 4.2 is the difference between the two X2 values (see column d). This latter figure is the test statistic for comparing the adequacies of models (3.2) and (3.3). Significant X2 statistics are indicated by asterisks. The X2 statistics were significant for 100 of the items indicating a malefemale difference in option choice for these items. In column e of Table 4.2 the actual sample sizes are reported. In column f, the minimum sample sizes necessary for significance are reported. For those 100 items which had significant X2 statistics reported in column d, 94 (94%) had minimum sample sizes greater than 3,000. TABLE 4.1 Summary of Significant Tests of Model 3.2 Item Xe for Actual Sample Min. Sample Number Model 3.2* Size Size for Significance (a) (b) (c) (d) 8 13 23 25 28 29 30 31 34 36 37 42 45 48 49 52 54 56 57 46.15 25.74 29.25 35.90 46.00 61.48 25.19 23.26 58.25 84.36 23.70 23.26 22.41 179.25 27.82 26.01 41.41 21.62 28.51 16,796 15,718 60,377 26,707 84,847 88,396 87,473 32,870 82,257 86,137 37,068 59,140 63,098 97,593 16,766 39,858 74,502 44,107 53,075 3,352 5,624 19,011 6,852 16,988 13,242 31,982 13,015 13,006 9,404 14,405 23,417 25,932 5,014 5,550 14,114 16,570 18,789 17,146 TABLE 4.1 Continued Item Xe for Actual Sample Number Model 3.2* Size (a) (hi ( 50.96 66.43 21.39 55.03 158.36 88.01 56.28 50.51 34.28 48.01 75.29 44.39 14,425.97 24.61 22.75 99.68 79.38 64.56 23.98  Min. Sample Size for Significance (d) 7,560 8,387 15,342 8,485 3,028 7,660 6,943 14,352 13,935 12,128 7,647 14,011 54** 19,969 34,077 6,485 7,693 10,957 18,700 41,829 60,494 35,632 50,699 52,068 73,196 42,429 78,708 51,862 63,223 62,511 67,528 84,368 53,358 84,175 70,183 66,304 76,806 48,690 TABLE 4.1 Continued Item XZ for Actual Sample Min. Sample Number Model 3.2* Size Size for Significance (a) (b) (c) (d) 80 75.01 55,277 6,787 81 95.51 57,710 5,565 82 51.99 71,315 12,633 83 20.12 81,021 37,088 84 38.62 78,959 18,830 85 1240.78 41,180 306** 86 102.85 39,903 3,573 87 66.75 48,194 6,650 88 63.01 38,742 5,663 89 43.26 39,648 8,441 90 85.53 53,557 5,767 91 82.53 62,954 7,025 93 78.39 80,723 9,484 94 70.15 87,241 11,454 95 40.59 83,969 19,053 96 29.31 74,287 23,343 98 29.87 69,052 21,291 101 23.77 83,782 32,462 102 61.06 75,708 11,419 104 40.47 72,517 16,503 TABLE 4.1 Continued Item XZ for Actual Sample Min. Sample Number Model 3.2* Size Size for Significance (a) (b) (c) (d) 105 28.66 60,712 19,510 108 26.56 86,261 29,912 111 30.71 86,190 25,849 * p<.Ol ** Minimum sample size less than 3,000 TABLE 4.2 Chisquare Values for the Comparison of Models (3.2) and (3.3) with Actual Sample Sizes and with the Corresponding Number of Subjects Needed for Significant Results Item Xk for X2 for X1 for Actual Num Model Model Difference Sample ber 3.2 3.3 Size lal 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 (b) 16.11 17.54 20.06 7.54 14.75 11.86 4.60 46.15 10.13 9.54 5.89 12.58 25.74 11.19 16.63 14.51 17.90 (c) 30.94 103.22 47.51 21.87 70.40 23.13 35.73 329.98 109.99 65.29 19.06 198.46 62.37 80.50 17.33 84.29 44.20 (_d) 14.83* 85.68* 27.45* 14.33* 55.65* 11.27* 31.13* 283.83* 99.86* 55.75* 13.17* 185.88* 36.63* 69.31* 0.70 69.78* 26.30* (e) 7,427 11,018 11,497 6,016 7,706 13,046 8,832 16,796 23,075 35,060 24,908 40,423 15,718 30,238 42,889 35,114 43,962 Minimum Sample Size for Signif. (f) 10,061 2,583** 8,733 8,414 2,782** 23,256 5,700 1,189** 4,642 12,634 37,996 4,369 8,621 8,765 1,230,914 10,109 33,582 TABLE 4.2. Continued Item Xe for X for X2 for Actual Minimum Num Model Model Difference Sample Sample ber 3.2 3.3 Size Size for Signif. (a) (b) (c) (d) (e) (f) 18 11.71 15.68 3.97 52,340 264,864 19 9.37 18.77 9.40* 41,363 88,402 20 12.37 13.46 1.09 55,707 1,026,746 21 11.51 19.03 7.52 49,240 131,547 22 9.63 83.74 74.11* 51,724 14,022 23 29.25 153.74 124.49* 60,377 9,744 24 10.35 42.93 32.58* 68,426 42,194 25 35.90 147.40 111.50* 26,707 4,812 26 18.84 47.30 28.46* 56,717 40,037 27 8.78 222.87 214.09* 78,353 7,353 28 46.00 242.78 196.78* 84,847 8,662 29 61.48 170.22 108.74* 88,396 16,331 30 25.19 53.84 28.65* 87,473 61,338 31 23.26 47.42 24.16* 32,870 27,333 32 15.51 41.69 26.18* 48,777 37,430 33 7.57 48.84 41.27* 39,470 19,214 34 58.25 324.67 266.42* 82,257 6,203 35 6.91 225.66 218.75* 88,069 8,088 TABLE 4.2 Continued Item X2 for a for Xk for Actual Minimum Num Model Model Difference Sample Sample ber 3.2 3.3 Size Size for Signif. (a) ( (c) (d) (e) (f) 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 84.36 23.70 13.64 13.35 6.67 12.65 23.26 16.29 3.55 22.41 6.56 7.82 179.25 27.82 14.23 16.37 26.01 6.69 41.41 134.56 76.85 84.24 757.60 701.39 603.42 189.11 21.65 115.84 55.90 16.30 76.15 452.64 59.75 240.27 224.54 96.14 27.49 95.41 50.20* 53.15* 70.60* 744.25* 694.72* 590.77* 165.85* 5.36 112.29* 33.49* 9.74* 68.33* 273.39* 31.93* 226.04* 208.17* 70.13* 20.80* 54.00* mE 86,137 37,068 38,870 61,057 56,438 57,145 59,140 33,712 67,192 13,098 56,727 56,555 97,593 16,766 25,410 24,715 39,858 44,694 74,502 34,472 14,011 11,0618 1,648** 1,630** 1,943** 7,154 126,357 12,021 37,851 117,007 16,628 7,172 10,549 2,258** 2,385** 11,418 43,168 27,718 TABLE 4.2 Continued Item Xz for Xa for Xl for Actual Minimum Num Model Model Difference Sample Sample ber 3.2 3.3 Size Size for Signif. (a) (b) (c) (d) (e) (f) 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 72 73 90.87 70.14 35.78 51.69 73.68 148.39 394.54 300.95 278.34 246.48 94.11 113.88 180.60 225.46 251.66 271.35 208.90 37.80 163.01 140.26* 71.34* 48.52* 7.27 34.20* 22.72* 81.96* 373.15* 245.92* 119.98* 158.47* 37.83* 63.37* 146.32* 177.45* 176.37* 226.96* 191.49* 13.19* 19.53 21.62 28.51 17.49 50.96 66.43 21.39 55.03 158.36 88.01 56.28 50.51 34.28 48.01 75.29 44.39 17.41 24.61 11,770 18,263 146,668 25,873 36,987 14,828 1,918** 4,142 8,719 9,279 22,532 24,953 7,121 7,158 7,121 5,977 8,831 81,271 41,796 44,107 53,075 44,044 41,829 60,494 35,632 50,699 52,068 73,196 42,429 78,708 51,862 63,223 62,511 67,528 84,175 53,358 74 22.75 84,175 12,057 TABLE 4.2 Continued Item X1 for X1 for Xa for Actual Num Model Model Difference Sample ber 3.2 3.3 Size (aL 75 76 77 78 79 80 81 82 83 84 86 87 88 89 90 91 92 93 94 (b) 8.69 99.68 79.38 64.56 23.98 75.01 95.51 51.99 20.12 38.62 102.85 66.75 63.01 43.26 85.53 82.53 16.36 78.39 7015 (c) 237.07 167.72 432.36 127.04 72.59 128.13 106.82 108.12 95.18 69.05 433.47 430.40 126.27 289.14 405.04 85.48 93.12 229.02 137.27 (d) 228.38* 68.04* 352.98* 62.48* 48.61* 53.12* 11.31* 56.13* 75.06* 30.43* 330.62* 363.65* 63.26* 245.88* 319.51* 2.95 76.76* 150.63* 67.12* (e) 67,783 70,183 66,304 76,806 48,690 55,277 57,710 71,315 81,021 78,959 39,903 48,194 38,742 39,648 53,557 62,954 88,231 80,723 87,241 Minimum Sample Size for Signif. (f) 5,963 20,723 3,774 24,696 20,123 20,906 102,511 25,525 21,685 52,129 2,425** 2,662** 12,304 3,240 3,368 428,727 23,092 10,766 26,113 TABLE 4.2 Continued Item X1 for Xa for X for Actual Minimum Num Model Model Difference Sample Sample ber 3.2 3.3 Size Size for Signif. (a) (b) (c) (d) (e) (f) 68.41 212.34 126.96 41.77 9.94 41.48 315.62 437.09 7.97 283.94 48.59 1,682.71 99.20 79.33 18.33 49.11 477.44 27.82* 183.03* 118.36* 11.90* 0.50 30.13* 291.85* 376.03* 1.85 243.47* 19.93* 1,669.34* 86.07* 52.77* 12.60* 31.31* 446.73* 83,969 74,287 50,028 69,052 72,647 60,931 83,782 75,708 50,663 72,517 60,712 71,946 72,482 86,261 76,800 70,980 86,190 60,638 8,154 8,492 116,576 2,918,956 40,627 5,767 4,045 550,173 5,984 61,199 866** 16,918 32,840 112,453 45,544 3,876 * P<.01 ** Minimum sample size less than 3,000 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 40.59 29.31 8.60 29.87 9.44 11.35 23.77 61.06 6.12 40.47 28.66 13.47 13.13 26.56 5.73 17.80 30.71 Sexrelated Differences in Problem Solving Performance In this part, results for the two questions related to the second objective of the study are presented. These questions serve as the framework for the presentation. Each question is stated, followed by the results pertaining to that question. Question 1: Do males and females differ in problem solving performance? The responses of 492 males and 510 females who took the Puerto Rico Basic Skills Test6 during the spring of the first year were analyzed in this study. Also, data from 504 males and 509 females tested in the second year and from 509 males and 504 females tested in the third year were included in the analysis. The mean performance scores and the standard deviations for each of the eight variables are presented in Table 4.3. Results of ttests are also presented in this table. Females outperformed males in problem solving, a finding consistently present in all the three years of test administration. Over the threeyear period the mean differences favored females in all variables except equivalence. The sexrelated differences in problem solving were significant (p<.01) for all three years. TABLE 4.3 Means, Standard Deviations, and tTests for the Eight Mathematical Subtests Year/ Males Females Subtest X SD X SD t First Problem Solving Addition Subtraction Multi pl i ca ti on Division Fracadd Decsub Equivalence Second Problem Solving Addition Subtraction Multiplication Division Fracadd 3.436 2.32 5.412 1.00 4.550 1.66 4.014 1.80 3.136 1.87 2.475 1.75 2.648 1.83 .77 .77 N = 492 3.632 5.470 4.843 4.148 3.242 2.565 2.28 .98 1.52 1.73 1.77 1.74 3.862 5.592 4.852 4.433 3.580 2.743 3.321 .73 2.45 .86 1.56 1.68 1.91 1.81 1.86 .76 2.82* 3.05* 1.22 3.81* 3.72* 2.32** 5.77*  .83 N = 510 4.015 5.603 4.923 4.550 3.632 2.903 2.36 .86 1.44 1.67 1.86 1.91 2.65* 2.29** .86 3.76* 3.42* 2.94* TABLE 4.3 Continued Year/ Males Females Subtest X SD X SD t Decsub 2.863 1.87 3.440 1.80 5.00* Equivalence .76 .76 .65 .75 2.32** N = 504 N = 509 Third Problem Solving 3.927 2.49 4.341 2.44 2.67* Addition 5.536 .84 5.704 .64 3.59* Subtraction 4.836 1.52 4.958 1.46 1.30 Multiplication 4.168 1.74 4.541 1.68 3.47* Division 3.343 1.88 3.795 1.83 3.87* Fracadd 2.819 1.89 3.117 1.85 2.53** Decsub 3.021 1.93 3.448 1.85 3.60* Equivalance .830 .82 .800 .78 .60 N = 509 N = 504 Note: The number of items in the problem solving subtest was 9. In each computation subtest, the number of items was 6. An item was included in the computation subtest only if it measured a computation skill required to solve a problem solving item. * p <.01 ** p <.05 Consistent significant differences were also found for addition, multiplication, division, addition of fractions, and subtraction of decimals. For subtraction the difference was not statistically significant. Question 2: Do sexrelated differences in problem solving persist when computational skills are controlled for, and is the malefemale differences in problem solving dependent on level of computational skills? To address the question of dependence of malefemale problem solving differences in computational skills for each year and computation subtest, the possibility of an interaction was investigated. For the first year, statistically significant interactions were found between sex and multiplication, F (1,998) = 8.59, p<.01; and sex and division, F (1,998) = 4.25, p<.01. A significant interaction was found between sex and subtraction in the second year, F (1,1009) = 6.39, p<.05. No significant interactions were found in the third year. Analysis of covariance summary tables are shown as Tables 4.4, 4.5, and 4.6. Also, the three interactions are depicted in Figures 4.1, 4.2, and 4.3. Each figure indicates that at lower levels of computational skills, males outperformed females in problem solving, with the reverse happening at higher levels of computational skills. TABLE 4.4 ANCOVA Summary Table: Multiplication Covariate First Year Source df SS MS F Multiplication (M) 1 1195.90 1195.90 265.66* Sex (S) 1 31.95 31.95 7.07* M x S 1 38.80 38.80 8.59* Error 998 4509.00 4.51 p <.01 TABLE 4.5 ANCOVA Summary Table: Divison Covariate First Year Source df SS MS F Division (D) 1 1195.90 1195.90 264.66* Sex (S) 1 13.94 13.94 3.22 D x S 1 18.39 18.39 4.25* Error 998 4317.00 4.32 * p<.01 TABLE 4.6 ANCOVA Summary Table: Subtraction Covariate Second Year Source df SS MS F Subtraction (S) 1 93.14 593.14 122.80* Sex (S) 1 18.58 18.58 3.85* SxS 1 30.85 30.85 6.39* Error 1009 4873.40 4.83 *p <. 5 87 9 8 7 6 . 0 I~ 5 o ,. 4 3 2 1 1 2 3 4 5 6 Multiplication Fig. 4.1 Sex by Multiplication Interaction. 88 9 8. 7 6 5 4 3 2 1 1 2 3 4 5 6 Division Fig. 4.2 Sex by Division Interaction 89 9 8. 7. 6 : 5 *r* O U) 1 4 1 2 3 4 5 6 Subtraction Fig. 4.3 Sex by Subtraction Interaction The significant interactions found between sex and multiplications and sex and division (first year), and sex and subtraction (second year), answered, in part, the question of whether malefemale differences in problem solving performance depend on computational ability. However, the evidence is quite weak. Of 21 possible interactions, only three were significant. No variable exhibited a significant interaction for each of the three years. The analysis of covariance was also used to determine if sexrelated differences exist after controlling for computational skills. Analyses were conducted for those variables that did not exhibit significant interactions with sex. As discussed in Chapter III, estimated true scores were used for observed scores to adjust for unreliability of the covariates (the computational subtests). Reliability coefficients calculated for each covariate are shown in Table 4.7. Summaries of the analyses of covariance for the first year are reported in Table 4.8. The results show that females retained their superiority in problem solving performance when equivalence was the controlling variable in the analysis of covariancer the only variable in which males outperformed females (nonsignificant). When the controlling variables were addition, subtraction, addition of fractions, and subtraction with decimals, female 
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