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Ls: , \T'u. isfif‘)! ! .. x x a... v . 3!: ~ 25 . 1153. w 2r: :3 ‘ t m 11 :«turiilznfl La...“- 2521 31...; H v V V . , . ill ll 3 1293 00 ll“ ri— ;. ,.,ri__ A L:t;::~ -r- H‘V ..‘ _‘. ..l..,‘_m This is to certify that the dissertation entitled SPATIAL VISUALIZATION SEX DIFFERENCES, GRADE LEVEL DIFFERENCES AND THE EFFECT OF INSTRUCTION ON THE PERFORMANCE AND ATTITUDES OF MIDDLE SCHOOL BOYS AND GIRLS presented by David Ben-Haim has been accepted towards fulfillment ofthe requirements for Ph.D. degreein Department of Administration and Curriculum Major professor ; é f Date <9 9 8 MSU is an Affirmative Action/Equal Opportunity Ithimtian 042771 Dill Tlllllllllll L 676 6061 bViESI.) RETURNING MATERIALS: Place in book drop to LIBRARIES remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. \> ‘12 QM? 3 o 2002 SPATIAL VISUALIZATION SEX DIFFERENCES, GRADE LEVEL DIFFERENCES AND THE EFFECT OF INSTRUCTION ON THE PERFORMANCE AND ATTITUDES OF MIDDLE SCHOOL BOYS AND GIRLS By David Ben-Haim A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Administration and Curriculum 1982 61/7960} ABSTRACT SPATIAL VISUALIZATION SEX DIFFERENCES, GRADE LEVEL DIFFERENCES AND THE EFFECT OF INSTRUCTION ON THE PERFORMANCE AND ATTITUDES OF MIDDLE SCHOOL BOYS AND GIRLS BY DAVID BEN—HAIM Purpose This study had two related purposes: first, to deter- mine existing differences in spatial visualization abilities and in attitudes toward mathematics of fifth through eighth grade students by sex and grade prior to an instructional intervention; second, to analyze the effects of instruction on the spatial visualization skills and attitudes toward mathematics of a sample of sixth, seventh, and eighth grade students by sex and grade. Also, the study compared atti- tudes toward mathematics and spatial visualization and ex— amined differences in attitudes toward spatial visualization by sex and grade. Methodology There were 1327 fifth through eighth grade students from three sites in and around Lansing, Michigan, who participated in the assessment of differences prior to the instruction. Of these, 430 sixth, seventh, and eighth graders participated in the evaluation of the effects of the instruction and comparison of attitudes; of the 430 stu- dents, 238 took part in the evaluation of the persistence of the effects. The instruments used included a spatial visualization test and two semantic differential scales to measure atti- tudes toward mathematics and spatial visualization. The spatial visualization instruction material included ten sequenced activities which required two to three weeks of instructional time. The statistical analyses included multivariate and univariate analysis of variance and re— peated measures. Major Results Prior to instruction, there were significant (1) grade differences in spatial visualization performance (increasing with age) and in attitudes toward mathematics (decreasing with age). (2) sex differences in spatial Visualization performance (favoring boys), but no sex differences in attitudes toward mathematics. (3) site differences in spatial visualization performance; as the socioeconomic status rose, the performance increased. After the instruction: (1) Sixth, seventh, and eighth grade boys and girls performed significantly higher on the spatial Visualization test; however, no change in attitudes toward mathematics occurred. (2) Boys and girls gained similarly from the instruction, in spite of initial sex differences. (3) Students' attitudes toward mathematics and spatial visualization were similar. (4) Grade differences (decreasing with age) in attitudes toward spatial visualiza- tion were found, but no sex differences. (5) Retention of effects persisted. After a four-week period, boys and girls performed higher on the spatial visualization retention test than on the posttest. ACKNOWLEDGMENTS My sincere appreciation is given to the members of my doctoral committee-—Professors William Fitzgerald, Chairman; Glenda Lappan; Perry Lanier; and John Wagner--for their generous contributions of time and talent in assisting me in the writing of this dissertation. Their counsel and judgment were most influential in guiding this study to completion. My thanks are due to the Iraqi-Jewish Educational Development Fund in Israel which gave me a partial scholor- ship. My thanks are also due to my Israeli friends and colleagues who lent me professional and emotional support. For bringing this manuscript through successive drafts to completion, recognition should be given to my typists, my wife Riva and Paula Moan, and to my editor, Sandra Gross. My special gratitude is expressed to my wife and child- ren Orit and Yochai for their encouragement, assistance, and understanding throughout my graduate career. TABLE OF CONTENTS Page LIST OF TABLES . . . . . . . . . . . . . . . . . . . . vi LIST OF FIGURES . . . . . . . . . . . . . . . . . . . xi Chapter I. INTRODUCTION . . . . . . . . . . . . . . . . . . 1 The Problem . . . . . . . . . . . . . . . 1 Purpose of the Study . . . . . . . . . . . . . 3 Research Questions . . . . . . . . . . . . 5 Research Hypotheses . . . . . . . . . . . . . 6 Assumptions . . . . . . . . . . . . . . . . 10 Scope and Delimitations . . . . . . . . . . . 10 Overview of the Study . . . . . . . . . . . . 12 II. REVIEW OF RELATED LITERATURE . . . . . . . . . . 14 Introduction . . . . . . . . . . . . 14 Defining and Measuring Spatial Visualization . 15 The Importance of Spatial Visualization . . . 23 Sex and Age Differences in Spatial Visualization . . . . . . . . . . . 29 Hypotheses Concerning Sex Differences in Spatial Visualization . . . . . . . . . 38 Specific Training in Spatial Visualization . . 51 Related Attitudinal Studies . . . . . . . . . 57 Summary . . . . . . . . . . . . . . . . . . . 62 III. DESIGN OF THE STUDY. . . . . . . . . . . . . . . 66 Introduction . . . . . . . . . . . . . . . . 66 Population and Sample . . . . . . . . . . . . 66 Instrumentation . . . . . . . . . . . . . . . 72 Instruction Material . . . . . . . . . . . . . 76 Procedure and Data Collection . . . . . . 80 The Hypotheses and Statistical Design of the Study . . . . . . . . . . . . . . . . . . . 82 Summary . . . . . . . . . . . . . . . . . . . 89 iii Chapter IV. AN ALYSIS OF DATA AND RESULTS Introduction . . . . . . . . . Sex and Grade Level Differences in Spatial Visualization Performance and in Attitudes Toward Mathematics . . . . . . . Comparison of Sixth Grade Data Among Sites 1, 2, and 3 . . . . . . . . . . . . The Effects of Instruction . . . . . . . Comparison of Attitudes Toward Mathematics With Attitudes Toward Spatial Visualization . . Sex and Grade Level Differences in Attitudes Toward Spatial Visualization . . . . . . . Retention Analysis . . . . . . . . . . . Correlation Analysis . . . . . . . . . Summary . . . . . . . . . V. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS APPENDIC Appendix A. B. Summary . . . . . . . . . . . . . . Conclusions Disucssion . . . . . . . . . . . . . Implications and Recommendations ES BROCHURE OF MIDDLE GRADES MATHEMATICS PROJECT (MGMP) DEPARTMENT OF MATHEMATICS MICHIGAN STATE UNIVERSITY . . . . . . . . . . . . . MATHEMATICS AND SPATIAL ATTITUDE SCALES-- RELIABILITY AND TEST-RETEST CORRELATION COEFFICIENTS . . . . . . . . . . . . . SPATIAL VISUALIZATION TEST--SAMPLE ITEMS, RELIABILITY COEFFICIENTS, TEST-RETEST EFFECT, AND CORRELATION COEFFICIENTS SPATIAL VISUALIZATION AND MATHEMATICS ATTITUDE PRETEST INSTRUCTIONS . SCHEFFE'S POST HOC COMPARISONS-~SITE l SCHEFFE'S POST HOC COMPARISONS-—SITE 2 iv Page 91 91 91 113 120 126 127 128 137 139 143 143 160 163 171 180 184 190 Appendix Page G. TESTING INTERACTION FOR EACH PLANNED COMPARISON——SITE 3 . . . . . . . . . . . . . 203 H. SCHEFFE'S POST HOC COMPARISONS—-SITE 3 . . . . 204 I. SCHEFFE'S POST HOC COMPARISONS--SIXTH GRADERS BY SITE BY SEX . . . . . . . . . . . . . . . 209 J. PRE- AND POSTTEST STANDARD DEVIATIONS . . . . 213 K. STEP DOWN TESTS——EFFECTS OF THE INSTRUCTION . 214 L. POST- AND RETENTION TEST STANDARD DEVIATIONS . 215 M. STEP DOWN TESTS--RETENTION ANALYSIS . . . . . 216 N. CORRELATION MATRIX BY GRADE LEVEL . . . . . . 217 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . 218 v Table 3.1 3.2 3.3 3.5 3.6 3.7 3.8 4.2 4.3 4.4 LIST OF TABLES Distribution of the Entire Sample by Grade by Sex in Each Site . . . . . Distribution of the Subsample from Site 3 by Grade by Sex Grade 6—-Grade Equivalents and Distribution of Reading and Mathematics SAT Scores for Each Site 0 u o o o o o o o 0 c o o I o . o . . Distribution of Seventh Graders in the Entire Sample on Reading and Math on the Michigan Assessment Test 1981-82 Grade 8~-Grade Equivalents and Distribution of Reading and Mathematics SAT Scores Distribution of the Representative Sample for the Retention Testing by Grade by Sex The 4x2 Crossed Multivariate Design for Site 3 . . . . . . . . . . . The Multivariate Analysis of Repeated Measures Design for the Subsample from Site 3 Means and Standard Deviations of MGMP SVT and MAT Scores for Site 1 by Grade and by Sex A Summary of Multivariate and Univariate Analysis of Variance for the 2x2 Design of Site 1 . . . . . . . . . . . . . . . Means and Standard Deviations of MGMP SVT and MAT Scores for Site 2 by Grade and by Sex . A Summary of Multivariate and Univariate Analysis of Variance for the 2x2 Design of Site 2 . . . . . . . . . . . vi Page 69 7O 7O 71 71 82 83 86 94 97 100 104 Table 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 Means and Standard Deviations of MGMP SVT and MAT Scores for Site 3 by Grade and by Sex A Summary of Multivariate and Univariate Analysis of Variance of the 4x2 Design of Site 3 . . . . . . . . . . . . . . . . Means and Standard Deviations of MGMP SVT and MAT Scores for the Sixth Graders in the Entire Sample by Site and by Sex Summary of Multivariate and Univariate Analysis of Variance for the 3x2 Design of Sixth Graders in the Entire Sample . . . . . . . . Pre— and Post-Test Means of MGMP SVT and MAT Scores for the Subsample from Site 3 by Grade and by Sex A Summary of Multivariate and Univariate Analysis of Repeated Measures for the Subsample from Site 3 . . . . . . . . . . . . Means and Standard Deviations of MAT and SVAT Posttest Scores for the Subsample from Site 3 by Grade and by Sex Analysis of Variance Summary Table for Mean Difference Between the MAT and SVAT Scores Analysis of Variance Summary Table for SVAT Scores . . . . . Post- and Retention Test Means of MGMP SVT and SVAT Scores for the Retention Subsample from Site 3 by Grade and by Sex . . . . . . . A Summary of Multivariate and Univariate Analysis of Repeated Measures for the Retention Subsample from Site 3 . . . . . . . Pearson Correlation Matrix for the Retention Subsample by Sex . Summary of Results for Sex and Grade Level Differences Prior to Instruction by Site Summary of Results of the Immediate Effects of the Instruction and the Retention Analysis . . . . . . . . . . . . . . . . . vii Page 107 110 115 118 122 125 129 130 130 132 136 138 141 142 'F—W Table B01 B.2 O ,_| Reliability Coefficients--Cronbach a for Mathematics Attitude Scale in Each Site by Time by Grade by Sex . . . . . . . . . Pearson Correlation Coefficients Between the Pretest and Posttest Scores on the Mathematics Attitude Scale by Grade by Sex . . . . . . . . . . . . . . . . . . . . . Reliability Coefficients--Cronbach a for the Spatial Visualization Attitude Scale in Site 3 by Time by Grade by Sex . Pearson Correlation Coefficients Between the Posttest and Retention Test Scores on the Spatial Visualization Attitude Scale by Grade by Sex . . . . . . . . . . . . . . . Reliability Coefficients--Cronbach a for the MGMP Spatial Visualization Test in Each Site by Time by Grade by Sex . . . . . . . . . . . Cells Sizes, Mean Scores, and Standard Deviations on the MGMP Spatial Visualization Test for Treatment/Nontreatment Students by Grade . . . . . . . Analysis of Variance Summary Table for Test- Retest Effect--Grade 7 . . . . . . . . . . . Analysis of Variance Summary Table for Test- Retest Effect--Grade 8 Pearson Correlation Coefficients Between Pre—Post and Post-Retention Scores on the MGMP Spatial Visualization Test by Grade by Sex . . . . . . . . . . . . . . . . . . . . Summary of Scheffé's Posteriori Comparisons of Grade Levels for Males' Means and for Females' Means on the MGMP Spatial Visualization Test--Site 1 Summary of Scheffé's Posteriori Comparisons of Grade Levels for Males' and for Females' Means on the Mathematics Attitude Scale-— Site 1 . . . . . . . . . . . . . . . . Summary of Scheffé's Posteriori Comparisons of MGMP Spatial Visualization Test Means of Sex for Grade Six and for Grade Seven--Site 2 viii Pace 186 187 188 189 194 195 196 196 197 200 200 202 Table G.1 H H. .1 2 H.3 H I I. I. .4 .1 2 3 J01 K. 1 Summary of Scheffé's Posteriori Comparisons of Grade Levels for Males' and for Females‘ Means on the Mathematics Attitude Scale-— Site 2 . . . . . . . . . . . . . . . A Summary of Multivariate and Univariate Analysis of Variance for the 4x2 Design of Site 3--A1ternative . . . . . . . . . . . Summary of Scheffé's Posteriori Comparisons of MGMP Spatial Visualization Test Means of Sex for Each of the Grade Levels Five Through Eight-—Site 3 . . . . . . . . . . . . . . . . Summary of Scheffé's Posteriori Comparisons of Mathematics Attitude Scale Means of Sex for Each of the Grade Levels Five Through Eight-— Site 3 . . . . . . . . . . . . . . . . . . Summary of Scheffé's Posteriori Comparisons of Grade Levels for Males' Means and for Females' Means on the MGMP Spatial Visualization Test—-Site 3 Summary of Scheffé's Posteriori Comparisons of Grade Levels for Males' Means and for Females' Means on the Mathematics Attitude Scale--Site 3 . . . . . . . . . . . . . . . . Summary of Scheffé's Posteriori Comparisons of MGMP Spatial Visualization Test Means of Sex for Each Site, 1,2, and 3--Sixth Grade Data Summary of Scheffé's Posteriori Comparisons Between Sites for Male Means and for Female Means on the MGMP Spatial Visualization Test--Sixth Grade Data . . . Summary of Scheffé's Posteriori Comparisons Between Sites for Male Means and for Female Means on Mathematics Attitude Scale-~Sixth Grade Data . . . . . . . . . . . . . . . . Pre- and Posttest Standard Deviations of MGMP SVT and MAT Scores for the Subsample from Site 3 by Grade and by Sex . . . . . . . . . A Summary of Univariate and Step Down Analysis of Repeated Measures for the Subsample from Site 3 . . . . . . . . . . . . . . . . . . . ix 202 203 205 206 207 208 210 211 212 213 214 Table L.1 N.1 Post- and Retention Test Standard Deviations of MGMP SVT and MAT Scores for the Retention Subsample from Site 3 by Grade and by Sex A Summary of Univariate and Step Down Analysis of Repeated Measures for the Retention Subsample from Site 3 . . . . . Pearson Correlation Matrix for the Retention Subsample by Grade . . . . . . . . . . . Page 215 Figure 4. 4. 1 2 4.3 4 4. 4. 4 .4 5 6 .7 4.8 4 .9 4.10 .11 4.12 4. 13 LIST OF FIGURES MGMP Spatial Visualization Profiles of Grade Levels Mathematics Attitude Scale Profiles of Grade Levels MGMP Spatial Visualization Profiles of Sex by Grade Mathematics Attitude Scale Profiles of Sex by Grade MGMP Spatial Visualization Profiles of Grade Levels Mathematics Attitude Scale Profiles of Grade Levels MGMP Spatial Visualization Profiles of Sex by Grade Mathematics Attitude Scale Profiles of Sex by Grade MGMP Spatial Visualization Profiles of Grade Levels Mathematics Attitude Scale Profiles of Grade Levels MGMP Spatial Visualization Profiles of Sex by Grade Mathematics Attitude Scale Profiles of Sex by Grade MGMP Spatial Visualization Sixth Graders--Profiles of Sites 1, 2, 3 by Sex . . xi Page Test (SVT) Means—— by Sex at Site 1 . . 95 (MAT) Means-— by Sex at Site 1 . . 95 Test (SVT) Means-- Levels at Site 1 . . 96 (MAT) Means-— Levels at Site 1 . . 96 Test (SVT) Means-- by Sex at Site 2 . . 101 (MAT) Means-- by Sex at Site 2 . . 101 Test (SVT) Means-- Levels at Site 2 . . 102 (MAT) Means-- Levels at Site 2 . . 102 Test (SVT) Means-- by Sex at Site 3 . . 108 (MAT) Means-- by Sex at Site 3 . . 108 Test (SVT) Means-- Levels at Site 3 . . 109 (MAT) Means-- Levels at Site 3 . . 109 Test (SVT) Means of . . 116 Table 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 5.1 'J'l N Mathematics Attitude Scale (MAT) Means of Sixth Graders--Profi1es of Sites 1,2,3 by sex MGMP Spatial Visualization Test (SVT) Means of Sixth Graders——Profiles of Sex by Site Mathematics Attitude Scale (MAT) Means of Sixth Graders--Profi1es of Sex by Pretest-Posttest Means on the Visualization Test (SVT) by Pretest-Posttest Means on the Visualization Test (SVT) by Pretest-Posttest Means on the Visualization Test (SVT) by Posttest-Retention Test Means Site . . . . . MGMP Spatial Grade MGMP Spatial Sex . . MGMP Spatial Grade and Sex . on the MGMP Spatial Visualization Test (SVT) by Grade Posttest-Retention Test Means on the MGMP Spatial Visualization Test (SVT) by Sex . . . Posttest-Retention Test Means Spatial Visualization Test (SVT) by Grade and Sex . . . . Pre-Post-Retention Test Means on the MGMP on the MGMP Spatial Visualization Test (SVT) by Grade Pre-Post-Retention Test Means on the MGMP Spatial Visualization Test (SVT) by sex xii Pace 0 116 117 117 123 123 124 133 133 159 ,11, CHAPTER 1 INTRODUCTION The Problem Cognitive and affective differences between boys and girls have received much attention in the lay press as well as in professional, educational and psychological litera— ture. A comprehensive review of the literature regarding sex differences by Maccoby and Jacklin (1974) reports that several popularly held beliefs about sex differences are not supported by research; but they conclude that some factors remain as probable distinguishing characteristics between males and females. One of these distinguishing factors which is of a special interest to those involved in the instruction of mathematics, science, and art to children is spatial ability. Maccoby and Jacklin conclude that males are superior in spatial ability in adolescence and adult- hood, but probably not in childhood. Spatial visualization, a particular subset of spatial skills, is mentioned by Fennema (1980) as one of the cogni- tive variables that may help explain sex-related differences in mathematical achievement. The aptitude of spatial vis- ualization emerges as a component of mathematics ability in most factor analytic studies (Schonberger, 1976), and it 1 shows about the same magnitude of correlation with mathematics achievement as verbal ability does (Fennema & Sherman, 1977). Fennema (1975) also indicates that even though the existence of many sex-related differences is currently being challenged, the evidence is still persuasive that in American culture, male superiority in tasks that require spatial visualization is evident beginning during adolescence. Other scholars have pointed out the importance of spa- tial visualization ability for successful study of science and mathematics (Bishop, 1978; Tobias, 1978; Harris, 1981). Because of the importance of this cluster of skills, ques- tions have been raised regarding the interaction of sex and instruction in affecting spatial ability (Eliot & Fralley, 1976; Maccoby & Jacklin, 1974; Harris, 1981). It is a generally accepted view that the cognitive and affective components are so intertwined that it is difficult if not impossible to separate them. There are a number of problems with the work in the affective domain, the most serious being the definitions of the construct ”attitude." Even taking these definitional problems into consideration, the literature suggests several tentative conclusions (Fennema, 1977). 1. There is a positive relationship between atti- tude and mathematics achievement which seems to increase as learners progress in school. 2. Attitudes toward mathematics are fairly stable-- particularly above the sixth grade, although one logitudinal study showed a marked decrease from sixth grade to twelfth grade (Anttonen, 1969). 3. Grades six through eight seem to be critical in the development of attitudes. 4. Extremely positive or negative attitudes appear to be better predictors of achievement than more neutral feelings. 5. There are sex—related differences in attitudes toward mathematics (p. 104). Aiken (1976) states that the correlations between attitude and achievement generally are somewhat higher for girls than for boys and that significant differences in attitudes are frequently found to favor males over females. Even though there is consensus that sex-related differences in attitudes toward mathematics exist, the magnitude and specific dimensions of these differences are unclear. Suydam and Weaver (1975) quote studies with contradictory results and say that in other studies no significant sex- related differences were found. Purpose of the Study The study of sex differences has become value-laden both inside and outside the scientific community. Conse- quently it is important to know where sex differences do exist and why. But, before one can attempt to explain why, it is essential to have accurate and detailed knowledge concerning the nature of existing sex differences and the change these differences undergo at successive ages or grade levels. Agreeing with this point of view, Fennema (1979) indicates that the recent interest in studing sex-related differences in mathematics has resulted in two somewhat different emphases: 4 l. A re-examination of sex—related differences to determine if, and where they still exist; 2. A deeper look to find possible explanations or variables related to any differences that exist. With regard to spatial visualization and attitudes to- ward mathematics, this study attempts to deal with both of the above emphases, focusing on the first one and possibly providing an explanation for the second. Therefore, this investigation has two related purposes. The first is to de- termine existing differences in spatial visualization skills and in attitudes toward mathematics of students in grades five through eight by sex and by grade prior to an instruc- tional intervention. The second is twofold: first, to ana- lyze the effects of instruction in activities involving spa- tial visualization tasks on the skills and attitudes toward mathematics of a sample of sixth, seventh, and eighth grade students by sex and grade level; second, after instruction, to compare attitudes toward mathematics and spatial visuali- zation and to examine differences in attitudes toward spa- tial visualization by sex and grade level*. The spatial visualization test and instruction material used in this study were developed by the Middle Grades Math- ematics Project (MGMP), Department of Mathematics, Michigan State University. The MGMP is a curriculum development pro- ject funded by NSF-DISE (National Science Foundation-Devel- opment in Science Education) to develop units of high qual- ity mathematics instruction for grades five through eight. The staff of the project: Glenda Lappan, Director; William M. Fitzgerald; Elizabeth Phillips; Mary Jean Winter; Pat Yarbrough; and David Ben-Haim. Appendix A includes the bro- chure of the MGMP Project. The test and the instruction material are described in Chapter III. Research Questions Since, there are two different but interrelated pur- poses to this study, two sets of major questions are con- sidered. The first set of questions focuses on documenting the existence of differences in spatial visualization and attitudes toward mathematics of fifth, sixth, seventh, and eighth grade students by sex and grade level before an instructional intervention: 1. What effect, if any, does grade level have on performance of spatial visualization tasks and/or on attitudes toward mathematics? 2. What effect, if any, does sex have on performance of Spatial visualization tasks and/or on attitudes toward mathematics? 3. Do differences between boys and girls in perfor- mance of spatial visualization tasks and/or in at— titudes toward mathematics change with grade level? The second set of research questions examines the effects of instruction in activities involving spatial visualization tasks, as well as the existence of differences in attitudes toward mathematics and spatial visualization by sex and grade level (sixth through eighth), after the instruction. 1. Will instruction in spatial visualization tasks effect the spatial visualization performance and/or attitudes toward mathematics of sixth, seventh, and eighth grade students? -Will these effects be different for boys than for girls? - Will these effects differ by grade level? 2. After the instruction, do differences exist between the attitudes toward mathematics in general and the spatial visualization activities in particular? -Will these differences exist for both sexes? -Will these differences exist for each grade level in the study? 3. After the instruction, do differences exist between the sexes in their attitudes toward spatial visuali— zation activities? -Will these differences exist for each grade level in the study? And finally a question about retention: 4. Do the effects of instruction on spatial visualiza- tion performance and attitudes toward spatial visu- alization tasks persist over time for each grade level and sex? Research Hypotheses The research hypotheses in this study are specifically designed to test differences among a sample of fifth through eighth grade boys and girls in their performance on spatial visualization tasks and their attitudes toward mathematics by sex and grade level prior to the instruction. Further— more, the hypotheses are designed to test the phenomenon involving the effects of selected instruction in spatial visualization tasks on the performance and attitudes of sixth through eighth grade boys and girls and on the differ— ences in spatial visualization performance and attitudes to- ward mathematics by sex and grade level. In addition, some of the hypotheses are designed to test differences in atti- tudes toward mathematics and toward spatial visualization activities by sex and grade level after the instruction. To facilitate the interpretation of the research hypotheses for this study it was necessary to divide them into two separate groups. The first group of hypotheses (H01-H03) are designed to test for any significant differ- ence among the grade levels or between the sexes on measures of spatial visualization abilities and attitudes toward mathematics. The second group of hypotheses (Hoj-H13) are designed both to test for any significant difference of the effects of instruction on the spatial visualization perfor- mance and attitudes toward mathematics of sixth through eighth grade students by sex and by grade, as well as to test for significant difference in attitudes toward spatial visualization, or between the attitudes toward mathematics and toward spatial visualization after the instruction. Group One The following research hypotheses in a multivariate form will be tested to assess the effect of sex and grade 8 level (five through eight) on performance of spatial visual- ization tasks and on attitudes toward mathematics: H01: There will be a difference among the mean scores for each of the four grade levels tested, five, six, seven, and eight, on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. H02: There will be no difference between the mean scores for boys and for girls in grades five through eight on both the Middle Grades Math- ematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. H03: There will be no interaction of grade by sex among the mean scores for fifth through eighth grade students on both the Middle Grades Math— ematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. Group Two The following research hypotheses, some of them in a multivariate form, will be tested both to assess the effects of instruction in spatial visualization tasks on the perfor- mance and attitudes toward mathematics of sixth, seventh, and eighth grade students by grade and sex and to compare attitudes toward mathematics vs. spatial visualization, as well as examine differences in attitudes toward spatial visualization. H04: There will be a difference between the posttest means and pretest means of sixth, seventh, and eighth grade students on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. H05: There will be a difference between the mean gain scores (posttest minus pretest) for each of the three grade levels tested, six, seven, and eight, on both the Middle Grades 1110: Mathematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. There will be no difference between the mean gain scores (posttest minus pretest) for boys and for girls in grades six, seven, and eight on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathema- tics Attitude Scale. There will be no difference between stu- dents' mean scores on the Mathematics Attitude Scale and Students' mean scores on the Spatial Visualization Attitude Scale. There will be no interaction of grade by sex among the mean difference scores-~Math- ematics Attitude Scale score minus Spatial Visualization Attitude Scale score. There will be no difference between the mean scores for boys and for girls in grades six, seven, and eight on the Spatial Visuali- zation Attitude Scale. There will be no difference between the mean scores for each of the three grade levels tested, six, seven, and eight, on the Spatial Visualization Attitude Scale. There will be no difference between the retention test means and the posttest means of sixth, seventh, and eighth grade students on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Spatial Visualization Attitude Scale. There will be no difference between the mean gain scores (retention test minus posttest) for each of the three grade levels tested, six, seven, and eight, on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Spatial Visualization Attitude Scale. There will be no difference between the mean gain scores (retention test minus posttest) for boys and for girls in grades six, seven, and eight on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Spatial Visualization Attitude Scale. 10 The .05 level of significance was the minimum value accepted in testing the research hypotheses, although find- ings below the .05 level are reported. Assumptions For the purposes of this study the following assump- tions were made: 1. It was assumed that a paper-pencil, multiple-choice response instrument was a valid means of assessing student's spatial visualization ability. 2. It was assumed that the sample did not differ significantly from the population with respect to the two variables--attitude toward mathematics and general ability in spatial visualization. 3. It was assumed that the examination materials were kept secure and that students had no opportunity to practice the questions on the tests. It was further assumed that the tests were given at about the same time, that the directions for testing were followed in all groups, and that adequate test conditions were maintained. Scope and Delimitations This investigation was basically concerned with the assessment of sex and grade level differences in spatial visualization ability and attitudes toward mathematics of fifth through eighth grade students prior to instruction. 11 Further, the study attempted to determine the effects of instruction in spatial Visualization tasks on the perfor- mance and attitudes of sixth through eighth grade students by sex and by grade. The extent to which the study can be generalized is limited to the participating schools and students during the time (January - April 1982) when the data were collected. The extent to which the study of existing sex and grade level differences can be generalized is limited to the stu- dents in fifth and sixth grades in a middle school located in the inner city of Lansing (capital of Michigan), to the students in sixth and seventh grades in a middle school located in a suburban area of Lansing and Michigan State University, and to the students in fifth, sixth, seventh, and eighth grades in elementary and middle schools in a rural area in the suburbs of Lansing. The extent to which the study of instruction effects can be generalized is limited to the students in the sixth, seventh, and eighth grades in middle school in a rural area in the suburbs of Lansing. Other limitations which impose constraints upon the generalizability of the findings of this study are 1. All of the experimental instrumentation was administered by the classroom teacher during regularly scheduled class time. 2. The evaluative instruments used in this study had the inherent limitations of paper-and-pencil tests. 12 As pointed out by Glennon (1949), ”the study of the behaviors of each person individually through conversing with him and keeping anecdotal records of his performance on the test items (pp. 394-95)” would be far superior to the paper-and-pencil test. 3. Time was a limitation of the study. Two to three weeks, the length of the instruction, is not suffi- cient to effect any dramatic changes, especially in attitudes. However, mathematics educators are us- ually bound by this or some other similar time limi— tation for a period of instruction or other inter- vention. In contrast, if the length of instruction is extended, the control of confounding variables is more difficult to acheive. Finally, this study did not attempt to evaluate the effect of the instructional model on the teacher behavior, or the superiority of the instruction material and the instrumentation over others in the area of spatial visualization skills. Overview of the Study The study is organized into five chapters. Chapter I gives the background and purpose of the study, research questions and hypotheses to be tested, assumptions, scope, and delimitations of the study. In Chapter II the liter- ature which is significantly related to the study is reviewed. A detailed description of the study is contained ‘4. 13 in Chapter III. Included is a description of the sample, instrumentation, instruction material, and the statistical design of the study. Chapter IV consists of the analysis of the collected data, results and interpretation. Chapter V concludes with a summary, conclusions, implications and recommendations for future research. CHAPTER II REVIEW OF RELATED LITERATURE Introduction One of the main concerns of twentieth century psychol- ogy has been the description and measurement of aspects of intelligence. Within studies of cognitive function, the concept of spatial ability has developed slowly. Even now there is no real agreement on its definition. Much of the work on spatial ability has involved factor analyses. A group of American psychologists in the Aviation Psychology Program used this technique to analyze data on spatial ability. Their experiences were summarized in a symposium held in Washington, D.C. in 1952. Three papers by Fruchter (1954), Zimmerman (1954), and Michael (1954) resulted from the meeting. The consensus was that the Army Air Force tests had repeatedly demonstrated at ”least two, and possibly three or more factors into which the variance of tests formerly appearing on a single factor could be split (Zimmerman, 1954, p. 396).” As a result, two important factors were described: one called ”Spatial Relations and Orientation,‘ and a second factor now called ”Spatial Visualization.” 14 15 Since historically, the similarities and differences between the sexes have been a subject of intense interest, the discovery of sex differences in the factors of spatial ability originated a significant body of research activity. One of the main issues in almost any investigation dealing with spatial visualization has been the sex differences in this ability. The purpose of this investigation was to determine sex differences in spatial visualization skills and in attitudes toward mathematics prior to an instruc- tional intervention as well as to analyze the effects of instruction in activities involving spatial visualization tasks on the skills and attitudes of middle school stu— dents. As a result of the purpose of the investigation, the main focus of this chapter will be on the spatial visualiza- tion factor and the sex differences in spatial visualiza- tion. Review of the related literature, includes (1) The definition and measurement of spatial visualization, (2) the importance of spatial visualization, (3) sex and age differ- ences in spatial visualization, (4) hypotheses concerning sex differences in spatial visualization, (5) specific training in spatial visualization, and (6) related atti- tudinal studies. Defining and Measuring Spatial Visualization Spatial visualization is considered to be one of the two main factors of spatial ability; consequently, the l6 definition and factorial studies related to the spatial ability concept will be dealt with first and then its factors. In the most general terms, spatial ability can be regarded as skill in dealing with the relationship of objects in a spatial context. Evidence of spatial ability does not come from home or classroom observations, but from performance on special tests. Smith (1964) in his comprehensive book on spatial ability, supplied a list of approximately one hundred paper-and-pencil tests. This was by no means an exhaustive report. The list did not include all of the established published tests of Spatial ability, nor did it include performance tasks also used as measurements of this skill. Most of the early studies in spatial abilities dealt with factors deriving from batteries of spatial tests. There are three excellent reviews of the field by Fruchter (1954), Werdelin (1961), and Smith (1964). It seems that only by 1940 was there agreement on the existence of a spatial factor. All three of the above reviews quote Wolfle (1940) who summarized the factorial studies up to 1940. He stated that the space factor was the second most frequently identified, the first being the verbal factor. The space factor appeared prominently in tests requiring the subject to react to spatial relations, to read plans on blueprints, or to tell quickly whether two drawings represent the same or opposite sides of such asymmetrical figures as flags. The same factor seemed to be involved in dealing with both two and three dimensional space (p. 31). 17 There have been other findings over the years which allow comparison of two- and three-dimensional space. Emmett (1949) found that "three-dimensional items have a higher loading in the space factors than two-dimensional (p. 15).” Smith (1964), discussing the same data, thought that Emmett must have identified one of the factors incor- rectly because "a striking feature of these results is the much higher predictive validity of the two-dimensional compared with the three-dimensional section of the test (p. 158).” More recently, Shepard and Metzler (1971) found that the time required to recognize that two perspective drawings portray objects of the same three-dimensional shape was (i) a linearly increasing function of the angular difference in the portrayed orientations of the two objects and (ii) no shorter for differences corresponding simply to a rigid rotation of one of the two-dimensional drawings in its own picture plane than for differences corresponding to a rota- tion of the three—dimensional object in depth (p. 701). Michael, Guilford, Fruchter, and Zimmerman (1957) attempted to synthesize the research findings prior to that year. In the words of Smith (1964): By comparing the results of several factorial investigations, they sought to describe the similarities and differences in the psychological processes underlying those factors which appeared to have been fairly well established (p. 90). They drew on the work of the Air Force group (Fruchter, 1954; Zimmerman, 1954; Michael, 1954), on a Psychometric Monograph by French (1951), and on Thurstone's further research which had found several components of the spatial 18 factor (Thurstone, 1944, 1950). Two major factors (not entirely independent) were described: ”Spatial Relations and Orientation" (SR-0), and ”Visualization.” The SR—O factor seemed to be a composite of French's factors and Thurstone's new factor, S, (closely related to his original space factor) and defined as ”the ability to recognize the identity of an object when it is seen from different angles” (Thurstone, 1950), or as ”the ability to visualize a rigid configuration when it is moved into different position (Thurstone, 1938).” In his manual, French (1954) had suggested the following tests as being marker, or reference, measures of the SR-0 factor: Thurstone's "Flags” (or ”Figures” or ”Cards”) and ”Cubes,” and the ”Spatial Orientation” of the Guilford-Zimmerman Aptitude Survey. Thurstone's "Flags,'l ' and ”Cards” tests require the examinee to indicate Figures,l whether two drawings, usually in different positions, can represent the same side of the object (test description and sample items in Thurstone, 1938, pp. 22-57). In Thurstone's "Cubes” test, the examinee is asked whether two drawings can potentially represent the same cube on each face of which a different design is supposed to exist (description and sample items in Thurstone, 1938, pp. 22-57). The "Spatial Orientation” of the Guilford—Zimmerman Aptitude Survey requires the examinee to ascertain how the position of a boat has changed in a second picture relative 19 to its initial position in the first picture (description and sample items in Guilford and Zimmerman, 1953). In a more recent manual with a kit of 72 factor- referenced cognitive tests for 23 factors published by ETS (Educational Testing Service), the SR—0 factor is identified by S ”Spatial Orientation” and defined as "the ability to perceive spatial patterns or to maintain orientation with respect to objects in space (Ekstrom, French,and Harman, 1976, p. 149).” The authors recommend the ”Card Rotations Test” and the ”Cube Comparisons Test” as measures of Spatial Orientation (description and sample items in Ekstrom et a1, 1976, pp. 150-2). Both tests are revised forms of Thurstone's ”Cards" and Cubes.” The SR-O tests are timed typically so that the subject must respond under speeded conditions. The visualization factor was traditionally thought to be uniform with Thurstone's ”S2" (1950) and Koussy's ”K” (1935), and was later interpreted by Michael (1954) as re- quiring the mental manipulation of visual objects involving a specified sequence of movements, when the objects appear within a more or less complex stimulus pattern. Thurstone (1950) himself gives a clearer definition of his 32: ”It represents the ability to imagine the movement or internal displacement among the parts of a configuration that one is thinking about.” The ETS Manual identifies this factor as VZ ”Visualization” and defines it as ”the ability to manipu- late or transform the image of spatial patterns into other 20 arrangements (Ekstrom et a1, 1976, p. 173).” The tests representing the spatial visualization factor are usually more difficult than those for spatial orientation, and are often power tests. For the V2 factor, French (1954) suggested Thurstone's ' and ”Surface Development” ”Form Board,” ”Punched Holes,’ tests as being reference measures. In the "Form Board” test the examinee, presented with items that consist of several two-dimensional black pieces of various geometrical designs, is asked to draw lines within a large enclosed design to show how the pieces can be placed to fit within it. In the ”Punched Holes" test the examinee, given a series of figures representing a square sheet of paper that has been folded step by step (as communicated by dotted lines in adjacent drawings) into smaller squares, rectangles, or triangles and then punched, is asked to indicate by drawing circles where the holes will be when the sheet is unfolded. In the ”Surface Development” test the examinee attempts to match the edges of a solid figure with the corresponding elements portrayed in a plane figure. (All three tests are described and sample items given in Thurstone, 1938, pp. 22-57). The ETS Manual (1976) still recommends the ”Form Board” and ”Surface Developmentll tests for the ”Visualization” factor and, in addition, the ”Paper Folding” test which is a revision of Thurstone's ”Punched Holes” (description and sample items in Ekstrom et a1, 1976, pp. 174-77). 21 The following tests are also considered to be primarily measures of the spatial visualization factor: The Guilford— Zimmerman ”Spatial Visualization” test, the ”Spatial Rela- tions” portion of the Differential Aptitude Test (DAT), and the more recent Purdue Spatial Visualization Test Battery. In the Guilford-Zimmerman ”Spatial Visualization” test, the examinee, presented with pictures of clocks in various initial positions, is directed by verbal statements concern- ing the nature and sequence of the movements through which the clocks are being put, and is requested to select from several alternative pictures the one that represents the final position of the clock (description and sample items in Guilford—Zimmerman, 1953). The Spatial Relations portion of the DAT requires the examinee to identify the figure of a solid that can be obtained when a given pattern is folded (description and sample items in Bennett, Seashore & Wesman, 1966). The Purdue Spatial Visualization Test requires the subject to imagine the movement of three—dimensional figures according to explicit directions which include a number of axes on which the object is rotated and the differing amount of rotation on each axis (description and sample items in Guay, 1980, pp. 9-10). The authors of the ETS Manual Kit (Ekstrom et a1, 1976) indicate that there has been some difficulty in explaining the difference between the spatial orientation and the visualization factor. They hypothesize that the figure is perceived as a whole in spatial orientation but 22 must be mentally restructured into components for manipula- tion in visualization. In contrast, Cattell (1971) does not accept visualiza— tion as a primary factor. He suggests that it is a second- order factor which includes spatial ability, figural adap- tive flexibility, speed of closure, and flexibility of closure. Royce (1973) suggests both primary and higher order visualization factors. As Carrol (1974) has pointed out, both visualization and spatial orientation require the mental rotation of a spatial configuration in short-term visual memory; visualization requires the additional com- ponent of performing serial operations. Guay (1980) in a paper presented at the annual meeting of the American Educational Research Association (AERA) also emphasized I'that it is the mental manipulation involved, and not the perception or retention, which enable a task to measure spatial ability (p. 5).'' Some subjects may employ an analytic strategy in vis- ualization tests and search for symmetry and planes of reflection as clues to the solution. Shepard (1971) and his colleagues have decribed the processes involved in the men- tal rotation of shapes, while Shepard and Feng (1972) have described the mental processes involved in paper-folding tests. In the 1940s and 19503, especially in England, it was debated as to whether or not any space factor at all could be said to exist in children between the ages of eleven or 23 thirteen. The presence or absence of a space factor in children at these ages hinged on the interpretation of test batteries and the manipulation of the factorial design. The controversy was eventually decided in favor of a spatial factor, but a residue of doubt remained in later reviews which were tentative about the existence of ” spatial ability" as early as age eleven. It seems that the confusion has implied doubt that children can show Spatial ability before age eleven. It is noteworthy that most of the paper-and-pencil spatial visualization tests have been developed for use with adults and high school students (for example, the DAT provides norms for grade eight and higher). The picture, in the eighties is still unclear; Harris (1981) pointed out that our attempts to identify the critical components of various ”Spatial” tests are still part guess work, particularly where we lack factor analyses involving both standard and nonstandard tasks... furthermore, consensus is still lacking on the meaning of the two factors—-orientation and visualization--most commonly identified as the major components of spatial tasks; factor analysts continue to have difficulty in differentiating and interpreting these factors (p. 89). The Importance of Spatial Visualization As mentioned earlier, Spatial visualization is one of two major (not entirely independent) factors of spatial ability. This section will deal with the importance of spatial ability in general and Spatial visualization in particular. It is clear that the importance of spatial ability implies the importance of spatial visualization and 24 vice-versa. Historically, spatial abilities have been of interest ever since Galton (1883) carried out his classical studies of imagery. At various times spatial abilities have been regarded as important in the study of aptitudes. Spatial ability is considered along with verbal and quantitative ability to be one of the main cognitive vari- ables. Wolfle (1940) reported that the space factor was the second most frequently identified after the verbal one. I. MacFarlane Smith (1964), a zealous advocate of spatial abil- ities, described them as ”in some ways more fundamental, more basic and dynamic than verbal abilities (p. 229)." In another section of his book on spatial ability, Smith points out that spatial abilities are necessary for the success— ful study of most practical and technical subjects and of the more advanced branches of mathematics, physics and engineering. High spatial ability is essential in most scientific and technological occupations. For these studies and occupations verbal tests do not measure the appropriate abili- ties at all, and the essay examination may actually have negative validity (p. 298). Early spatial research was developed mainly as a result of practical needs, such as the need to facilitate screening for certain kinds of employment. Many spatial orientation and visualization tests were designed primarily to predict success in occupations such as navigating, piloting, engine— ering, drafting, and art, as well as predicting success in school subjects. Harris (1981) deals with the question of the relevance of high spatial ability for certain disci- plines and professions: 25 According to estimates of trait requirements prepared by the U.S. Employment Service, most technical-scientific occupations require spatial ability in the top 10% or the U.S. population. These fields run the gamut from draftsman, air- plane designer, and architect to chemist, engineer, mathematician, and physicist--including sub- specialties in all fields (p. 117). An indication that spatial visualization is an impor- tant consideration is the concurrent discovery of sex-re- lated differences in favor of males in spatial visualization skills (this topic will be discussed in a later section of this chapter). It has been hypothesized that sex—related differences in space perception might partly account for sex discrepancies in other academic disciplines in which boys excel and which, perhaps significantly, have visual-Spatial components resembling those found in some of the standard visual-spatial tasks. These disciplines include mathema- tics, chemistry, and physics, and to a lesser extent, bio- logy (Harris, 1979a; Kelly, 1978). Three findings that underscore the importance of Spa- tial visualization are that spatial visualization has been shown (1) to relate to achievement in science (Bennett, et a1, 1966, 1973), engineering (Poole & Stanley, 1972) and specifically, chemistry (Baker & Talley, 1972, 1974); (2) to be related to mathematics performance (Fennema & Sherman, 1977, 1978; Sherman, 1979); and (3) to differentiate more among females than males in predicting math performance (Sherman, 1980). 26 In the succeeding paragraphs the relevance of spatial visualization to mathematics will be discussed in more detail. Fennema (1980) argues that the relationship between mathematics and spatial visualization can be logically demonstrated. In mathematical terms spatial visualization requires that objects be (mentally) rotated, reflected and/or translated. These are important ideas in geometry (p. 83). Although the relation between the content of mathema— tics and spatial visualization skills appears logical, re— sults from empirical studies that have explored the rela- tionship are not consistent. In 1967, Very concluded that ”research on spatial ability has failed to produce any significant correlation of [the spatial factor] with any facet of mathematics performance (p. 172)." It seems that Very based his opinion on Murray (1949) whose data were based only on males and Showed a definite contribution of spatial visualization to geometry grades, but showed less of a contribution to Murray's geometry test which was, by his own admission, quite verbal in character. Smith (1964) concluded that although there are several studies which indicate con- sistently that Spatial ability is important in tests which are genuinely mathematical as distinct from those which involve purely mechanical or computational processes... As to the question whether or not the abilities which together con- stitute mathematical ability form a roup factor over and above g [the general factorT, there is some conflict of views (p. 127- -28). Even in geometry, where one would expect to find the strongest relationship with spatial visualization, empirical 27 findings do not indicate clearly that the two are related. Lim reported in 1963, after a thorough review of relevant literature, that the evidence for a relationship between geometric ability and spatial visualization was incon- sistent. Werdelin (1961) was not willing to conclude defi- nitely that spatial—visualization ability and geometric ability were related; however, he felt that ”there [was] strong pedagogical reason to believe in a connection between the ability to visualize and geometrical ability (p. 39).'I Other authors have felt that research indicated a positive relationship. In 1951, Guilford, Green, and Christensen concluded that spatial visualization ability helped in solving mathematics problems. More recently, Moses (1977) concluded her investigation on ”the nature of spatial ability and its relationship to mathematical problem solving by indicating that although spatial ability is a general ability, it is a good predictor of problem-solving perfor- mance. An individual with high spatial ability is often successful at problem solving but will fre- quently not write down visual solution processes as part of his solution even though he may be using them (p. 4640). In a review of literature on ability and creativity in mathematics, Aiken (1973) concluded that spatial-perceptual ability was one of the ”most salient” mathematical factors extracted in various investigations. Schonberger (1976) also indicated that the aptitude of spatial visualization emerges as a component of mathematical ability in most factor analytic studies. Smith (1964) hypothesized that as 28 students move into advanced mathematics courses such as calculus, spatial visualization assumes increasing impor- tance. Fennema (1975), in contrast, hypothesized that spatial visualization is highly important to the learning of mathematics in the primary grades because of increased emphasis on concrete and pictorial representations, all of which have spatial attributes. Elmore and Vasu (1980) investigated the effect of Spatial visualization (with several other variables) on Statistics achievement. They hypothesized that ”since many of the concepts introduced in an inferential statistics course are visual, Spatial ability should be correlated with statistics achievement (p. 457).'' Regression analyses were performed to determine the predictors of success in statis- tics courses. The authors reported that “spatial visualiza- tion ability also made a significant contribution to the prediction of statistical achievement (p. 464).” The relevance of spatial visualization to mathematics is supported by correlational results. The aptitude of spatial visualization shows about the same magnitude of correlation with mathematics achievement as does verbal ability--typica11y r=.45 (Fennema & Sherman, 1977). Bock and Kolakowski (1973) note that spatial ability correlates with school geometry (rho=.57) and with quantitative thinking (rho=.69). In conclusion, the evidence is very clear that spatial visualization is substantially related to many forms of mathematics achievement. 29 Sex and Age Differences in Spatial Visualization A comprehensive review entitled The Psychology of Sex Differences by Maccoby and Jacklin (1974) indicates that many sex differences in cognition have been asserted-—most without strong supporting evidence. Spatial ability appears to be one of the exceptions. The psychological literature on spatial ability has consistently implied that in tests of this ability, males perform better than females. With respect to age, it is claimed that the spatial ability like other intellectual abilities improve with age, from child- hood to adulthood. Additionally there are many allusions to the fact that there are fewer women than men in the occupations such as engineering and drafting that seem to demand spatial abil- ity, and that high school girls seldom take mechanical draw- ing. There are several reviews of the literature on sex differences; the most frequently cited are those of Terman and Tyler (1954), Anastasi (1958), Tyler (1965), Maccoby (1966), Sherman (1971), and the review of Maccoby and Jacklin (1974). These reviews cover a broad range of psych— ological sex differences, of which spatial ability comprises only one part. Each author, however, arrives at a similar conclusion, illustrated by the following quotation from Tyler (1965): To summarize, males are clearly superior on tests of mathematical reasoning, spatial relationship, and science. Females are superior in verbal fluency, rote memory, perceptual speed, and dexterity. Some 30 of these differences develop earlier and appear to be more fundamental than others (p. 247). Another conclusion specifically related to spatial ability and most frequently cited is a quotation from Maccoby and Jacklin (1974): ...that boys excel in visual-spatial ability. Male superiority on visual spatial tasks is fairly consistently found in adolescence and adulthood, but not in childhood. The male advantage on spa— tial tests increases through the high school years up to a level of about .40 of a standard deviation. The sex difference is approximately equal on analy- tic and nonanalytic spatial measures (p. 351-52). Specific reviews on sex differences in Spatial ability (including particularly spatial visualization) and math- ematics are those of Fennema (1974, 1975, 1980) and the recent reviews of Harris, ”Sex-related Differences in Spatial Ability” and ”Sex-related Variations in Spatial Skill” (Harris, 1979a, 1981). These classic papers will be discussed and some of the new studies investigating the sex and grade level differences in spatial visualization will also be presented. Historically, the sex differences in spatial ability were revealed while investigators were dealing with constructing measures of intelligence or spatial ability, and employing factor analysis. Early in this century, Porteus (1918) developed a series of Maze tests (description and sample items in Porteus, 1950, 1965). He compared his test with the Goddard and Stanford revision of the Binet test (Paragraph Memory test, described in French, 1951, 31 p. 76) on a sample of 253 boys and 200 girls, age 7-14. The results showed that on the Porteus Maze tests, as with the Binet test, older children outperformed younger children. But, while boys and girls did about equally well (as ex- pected) on the Binet test, the boys outscored the girls on the Porteus Maze tests. This finding proved not to be coincidental. Porteus (1965) reported the results of fifty years work with his mazes, in which males held the advantage over females in performance on his test, in 99 out of 105 compar- isons. Harris (1979a) based on a review of several studies, points out that ”the sex difference is not peculiar to mazes of Porteus's own design. On other maze tests, it appears just as strongly, and in children as young as 4-6 years of age (p. 134).” The sex difference appears on a great variety of other tasks representing different and not always specifiable mixes of orientation and visualization and sometimes other components. For example, Harris (1979a) mentions such tasks that might require mental rotation (directly related to spatial visualization), imagery, field independence,and sense of direction. An example of a mental rotation task is the spatial subset of the DAT (Bennett et a1, 1959). The DAT Manual does give separate norms for boys and girls. Wesman (1949) in discussing the administration of the DAT to tenth graders pointed out that a raw score of 57 put a girl in the 80th percentile and a boy in the 60th percentile. If 32 no separate norms were given, a boy with a raw score of 57 would be in the 70th percentile and might find himself over his head when looking for a job in engineering. Several studies using the spatial subset test of the DAT as a cri- terion measure report that males excel, at least in the age range from eleven years through college age (Flanagan et a1, 1961, cited in Maccoby & Jacklin, 1974; Harris & Wagner, 1977; Hartlage, 1970; Vanderberg, Stafford, & Brown, 1968). Fennema and Sherman (1977, 1978) conducted a study which specifically investigated the relationship between mathematics achievement and Spatial-visualization Skills. The subjects were females and males (about 2500) in grades six through twelve. They gathered data for three cognitive variables (mathematics achievement, verbal ability, and Spatial visualization); eight affective variables and three other variables (number of mathematics-related courses taken, number of Space related courses taken, and amount of time spent outside of school on mathematics-related activ— ities). The Space relations test of the DAT (Bennett el al, 1973) was the criterion measure for the spatial visuali- zation variable. They concluded that the data from the study did not support the idea that differences in spatial visualization ability are helpful in explaining sex-related differences in mathematics achievement. Few sex—related differences in either mathematics achievement or spatial- visualization skills were found. The two were related (r=.5) similarly for both sexes, and spatial visualization 33 ability appeared to influence both females and males equally to continue studying mathematics. With regard to grade level, Fennema and Sherman found that all three cognitive variables showed significant grade effects, noting that it was expected because of the increased maturity and selectiveness of the subjects (Fennema & Sherman, 1977, 1978). Another large-scale study related to Spatial visualiza— tion which contrasts with the usual finding of sex differ— ences is the Women in Mathematics Study (National Assessment of Educational Progress-—NAEP) reported by Armstrong (1980). The purpose of the study was to identify the most important factors related to the problem of women's participation in mathematics. The vehicle for the Study was a national survey of 1,452 thirteen-year-olds and 1,788 high school seniors. The data collection centered upon achievement and participation in mathematics, student's attitudes toward mathematics and other variables. One of the four tests to assess achievement in mathematics was in spatial visualiza- tion (items from the Revised Minnesota Paper Form Board, Likert & Quasha, 1948). The results of the study reported by Armstrong indicate that ...13 year-old females start their high school mathematics career with at least the same ability as their male contemporaries. In fact, 13 year—old girls are better at computation and spatial visualization than males. Their problem-solving skills are nearly equal. By the twelfth grade, 34 males have overtaken females. Males excel in problem-solving but have abilities in spatial visualization and computation comparable to those of their female counterparts (p. ix). The timing of expression and duration of sex differ- ences in spatial skill is discussed by Harris (1981). In agreement with the conclusion of Maccoby and Jacklin (1974), Harris states that ”the sex difference [in spatial tasks] unquestionably is more common in later childhood, especially after the onset of pubescence (p. 96).” The sex differences on spatial tests are far more reliable in twelve to eighteen-year—olds and in college—age adults than in preadolescent children. An example is the Spatial subset test of DAT mentioned before. But, the absence of sex differences in younger children is not con- sistent. The Porteus Maze tests have already been men- tioned. In another example Roberts (1971) reported on the Block Design subset of the WISC (description of the test in French, 1951, p. 61) which indicated that in a test of over 7000 American six to eleven-year-olds, boys showed a clear advantage over the entire age range. Guay and McDaniel (1977) investigated the relationship between elementary school mathematics achievement (simple mathematics concepts and calculation skills) and high- and low-level spatial abilities among elementary school males and females. The subjects were from grades two through seven, and the instruments employed were four experimenter- developed spatial ability tests. The data Showed 35 significant grade and sex main effects. Although no statistical probes were conducted on the significant grade main effects, inspection of the grade marginal mean scores indicated a trend toward an increase in all four test scores with an increase in grade. With regard to sex differences in spatial abilities, the investigators reported: The data suggested that among elementary school children males had greater high-level spatial ability than females, and males and females had similar low-level spatial ability... the significant sex differences were not found to be a function of any variation in grade level. These observations were consistent with literature reviews indicating sex differences favoring males, but inconsistent with that portion of the reviews suggesting that sex differences become evident only during early adolescence (p. 215). Evidence for the absence of sex differences in spatial visualization prior to adolescent age, was reported by Smith and Schroeder (1979). They devised a spatial visualization test (SVAT—-based on tangrams) which served as the criterion measure. The subjects were fourth grade girls and boys. Smith and Schroeder concluded that ”the results reported here indicated that preadolescent fourth grade girls and boys did not differ significantly in the Spatial visuali- zation ability measured by the SVAT (p. 66).” With regard to changes in the magnitude of the sex difference on standard tasks, Harris (1981), in his review, cited evidence for both increasing and for unchanging magnitude. The first came from Bennett (1969) on the DAT Mechanical Comprehension Test: ...the difference between means of the sexes increases from eighth to twelfth grade from about 36 .8 of a standard deviation to more than 1.2, so that by the time they are high school seniors, only about 10% of girls reach or exceed the mean of high school senior boys (Harris, 1981, p. 93 citing Bennett, 1969). The second, from Wilson et a1 (1975), on the Shepard and Metzler mental rotation test: In a sample of nearly 2500 individuals, the male advantage was large and remarkably consistent over the entire age range from the teens (age 14) to middle age (53 years), both in the period of improvement of performance (ages 14 through the mid-twenties) and decline (late twenties and on) (Harris, 1981, p. 96 citing Wilson et a1, 1975). Nash (1979) suggests that ”the exaggerated, cognitive sex- related differences” will diminish beyond the adult years (p. 290).” Another kind of study is the type designed to compare spatial test performance of men and women in different cultures or geographical locations. Stewart (1974) has provided an extensive review indicating the regular super- iority of males over females in cross-cultural studies of pshcyological differentiation; especially when measured by ”Block Design” (French, 1951, p. 61) and ”Embedded Figures” tests (Witkin, 1962, p. 39). Witkin (1966) found that males from the United States, England, Holland, France, Italy, Israel, Hong Kong, and Siera Leone were more accurate than females in adjusting a luminous rod to the absolute vertical under various conditions of tilt. He also found that males scored higher on an ”Embedded Figures" test. 37 Berry (1966) compared Canadian Eskimos from Baffin Island, Temne tribesmen from Siera Leone, and a reference group from Scotland. He administered four spatial tests to each sample which had an age range from ten years to over forty. Berry found that Temne men outperformed Temne women on most spatial tests, but there were no similar differenes between the Eskimo men and women. The Scottish pattern was Similar to that of the Temne. Berry's data on the Eskimo are reinforced by the findings of MacArthur (1967). He administered Vernon's ”Embedded Figures” test to 167 Eskimo children, age 9-15 years. He found no significant relation between sex and test score at any of the ages. Backman (1972) investigated the relationship of ethnicity, SES (Socioeconomic Status), and sex to patterns of mental abilities of adolescents using almost 3000 twelfth grade subjects who had participated in Project TALENT. The results revealed that ”sex accounted for a much larger proportion (69%) of the total variance than did either ethnicity (13%) or SES (2%). Sex was significantly (P<.001) related to both the shape and the level of the patterns (p. 10)." To conclude this section on age and sex differences in spatial visualization, it is worthwhile to cite Harris' com- ments about the nature of the evidence for sex differences in Spatial ability. Harris (1981) suggests that differences between the sexes are usually smaller than the range of individual differences within a sex; ”the behavioral 38 distinction between the sexes thus seems to be quantitative more than qualitative (p. 90)." In addition, he indicates that questions as to the Size of the ratio of positive to no-difference reports, the minimum number of reports re- quired before an inference of reliable sex differences should be made, and the significance of the numerically ”atypical” reports of superior female performance are still incompletely answered. Harris opposed Jacklin's (1979) suggestion that the differences are ”trivially small” and that the search for explanatory mechanisms ”may be an exercise in futility.” Harris believes ”that the search is far from futile and ultimately will lead to important insights into the nature and origins of human cognition (p. 90).” Some of the theo— retical explanations and hypotheses concerning the sex dif— ferences in spatial visualization (and spatial ability) will be discussed briefly in the following section. Hypotheses Concerning Sex Differences in Spatial Visualization There have been a variety of attempts to account for sex differences in spatial ability in general and spatial visualization in particular. Historically, the attempts to explain these sex differences have ranged broadly, the stress Shifting over time between the classical poles-— biology at one end, the working of culture and socialization at the other. Thus, several different kinds of explanations have been advanced, some emphasizing the contributions of 39 ' others emphasizing the role of biological factors--"nature,' learning and socialization--”nurture.” A brief review of the various theories encountered in the literature, relative to the causes of these sex differ— ences will be attempted below. The major categories of possible biological determinants are genetic, hormonal, and neurological; those of social and cultural determinants are sex-role typing and differential practice. Genetic--X-Linkage Hypothesis The idea of X-linked inheritance of cognitive Skills was proposed by O'Connor (1943). He conducted a number of studies over a period of years and concluded that spatial ”structural visualization,” was ability, which he called ”inherited, not acquired, a unit genetic trait, sex-linked recessive with a fifty percent gene frequency (p. 1).” O'Connor based his declaration upon data obtained from the ”Wiggly Blocks Test” (a three-dimensional jigsaw puzzle test described in French, 1951, p. 110) given to adult men and women. The results indicated that 50 percent of the 562 men could assemble the puzzle in less than 1.71 minutes, while only 25 percent of 180 women tested were able to do so. To put O'Connor's declaration and findings in some per- spective, one must recall that every person has 46 chromo- somes arranged in 23 pairs, including the sex chromosomes. A son receives his single X-chromosome from his mother, while a daughter receives two X-chromosomes, one each from 40 her mother and father. The genes carried on the sex chromo- somes are called sex-linked genes. A gene for a sex—linked recessive trait carried on the X-chromosome, therefore, is only manifest in females if it is present on both X-chromosomes, while it will be manifest in males when it is carried on the single X-chromosome of the male. O'Connor theorized that spatial ability is due to a sex-linked recessive gene on the X-chromosome. He concluded that half of the X-chromosomes which exist among human beings carry the recessive gene. Therefore the probability for a male to Show the trait is 50 percent and for a female 25 percent, which is what his data showed. O'Connor's proposed X-linked inheritance of spatial ability was formulated as a specific hypothesis by Stafford (1961, 1972). The Stafford X—linked hypothesis states that high potential for spatial visualization is carried as a recessive characteristic on the X—chromosome. The X-linkage model predicts both distribution of spatial ability within the population and the direction of correlations of spatial ability within families. Stafford (1961) conducted a cross— generational study comparing the spatial abilities of teen- agers with their parents. He used the ”Identical Blocks” test (described in Stafford, 1962). His results supported his prediction from the model and thus provided support for O'Connor's declaration. Similar results were reported by Hartlage (1970) using the spatial subset of the DAT. Bock and Kolakowski (1973) 41 also obtained similar results when comparing the performance of teenagers and parents (167 families) on the Guilford— Zimmerman Spatial Visualization test (Guilford and Zimmerman, 1953). Studies of twins' abilities by Vanderberg (1969) suggested that spatial abilities may have a consider- able genetic component. Yen (1975) administered four paper-and-pencil spatial tests, measuring two- and three-dimensional Spatial visuali- zation and Spatial orientation, to 2508 white high school students. She followed much the same procedure as Bock and Kolakoswki (1973). Yen's results, unlike those of Bock and Kolakowski, are more complex and do not fully support the X—linked hypothesis. The research on the chromosomal abnor— mality known as Turner's Syndrome (Vanderberg, 1975) did not support the X-linked recessive gene hypothesis and produced a challenge to this hypothesis. The reliability of the early parent-child correlation studies also has been doubted. Boles (1980) in a critical review of the literature on X-linkage of spatial ability reports several recent studies which failed to replicate the earlier results. In contrast with the small sample size used by Stafford (1961, 1965) and Hartlage (1970), the recent studies used large sample sizes. Some of the studies were also factor analytic. Boles (1980) concludes that increased sample Size and factor-analytic logic may be considered improvements in methodology over the original studies, so the subsequent failures to replicate more probably reflect the true state of nature (p. 633). 42 Sex-Limitation Hypothesis and the P3ESIBIE-REIE_5f-SE§"HEPEEHE§____ The following discussion on the sex limitation hypo- thesis is cited from Harris's (1979a, 1979b, 1981) compre- hensive reviews of literature on sex-related differences in spatial ability and the factors influencing spatial abil- ity. Bock (1973) suggested the possibility that spatial deficiency in Turner's Syndrome individuals is associated with the absence of sex hormones. This possibility suggests that expression of the spatial trait in the normal female, and by implication in the normal male as well, would depend on the production of sex hormones above some threshold level. As distinct from the sex-linkage hypothesis, this is called a sex-limitation hypothesis meaning that the sex differences are under the control of the sex hormone (androgen-estrogen) ratio or balance. Harris (1979a) indicates that ”there is, indeed, evidence” for a relationship between the androgenicity (as the index of androgen-estrogen ratio or balance) and the spatial ability in normal females. Among the most consis- tent findings are those of Petersen (1976) in which she infers endocrine status from somatic characteristics. A curvilinear relationship between spatial ability and androgenicity appeared, spatial ability scores being lower in the less androgenized, more ”feminine” girls and higher in the more androgenized, more ”masculinell girls. For boys, 43 Petersen found an unexpected relationship. The boys' spatial scores were inversely related to androgenicity, being higher in the less androgenized, less "masculine” boys and lower in more androgenized, more ”masculine” boys. A similar inverse relationship was reported in boys in an earlier study by Broverman and Klaiber (1969). Harris's (1979b) implication from inferring endocrine status from somatic characteristics is that within the normal range of androgenization there is some middle range, where more males than females fall, within which spatial ability is maximized and beyond which spatial ability is depressed. Hormonal effects also have been studied by analyzing intra-individual changes in performance on cognitive tasks as a function of cyclical changes in hormonal balance. However, the changes in cognitive performance have not been consistently demonstrated. Following the X-linkage model, it was predicted that family correlations for androgenicity likewise ought to follow the patterning predicted by the X-linkage model. Petersen (1976) therefore computed the correlations for somatic androgyny measures for all sibling pairs in a sample of thirteen to eighteen—year—olds. The ordering of correlation values exactly fit the model. Bock (1973), based on studies of physical development, suggested that it is the degree of sexual differentiation within each sex that is influenced by an X-linked gene and that this differentiation in turn influences Spatial ability. Harris 44 (1979a) concluded that ”the evidence thus far at least sug- gests that the X-linked recessive trait is not for spatial ability but for androgenicity, spatial ability itself being an expression of the sex-limiting effect of androgenicity (p. 158).” But more research needs to be done. Neurological Hypothesis For normal, right-handed persons, the two hemispheres of the brain are thought to specialize in somewhat different functions, the left hemisphere specializing in verbal, an- alytical tasks and the right hemisphere in spatial, gestalt tasks. The neurological hypothesis has been raised in light of male superiority in visual-spatial skills, and in view of clinical and experimental evidence pointing to a prominent role for the right hemishpere in the detection and proces- sing of visual-spatial information. The hypothesis is that this functional specialization has proceeded further, or has reached a higher level, in the male than in the female brain. This is a general form of the neurological hypothe- sis. Harris (1981) indicates that taking into account the known left-hemisphere specialization for linguistic skills, the corollary proposal is that in the female brain in contrast to the male, there is further, or higher level of, Specialization of the left hemisphere (p. 101). Thus, while the X-linkage model pertains to visual-spatial ability, the neurOpsychological hypotheses are more ambi- tious in scope and suppose that sex differences in both 45 visual-spatial and language functioning are rooted in cor- tical organization. Several more specific and related forms of neuropsycho— logical hypotheses have been advanced. A comprehensive review of related studies by Harris (1981) considers three as most prominent: (l) the rate of functional maturation of cerebral hemispheres, (2) the extent of lateralization of functions in adulthood, and (3) the rate of physical matur- ation. The first hypothesis is that the fetal sex hormones determine the relative rates of functional maturation which is faster on the left in females (thus favoring earlier language development) and faster on the right in males (thus favoring earlier development of Spatial functions). Harris (1981) concludes from his review that at best the evidence is mixed. The second hypothesis is a straightforward extension of the first hypothesis to adulthood. In light of the per- sistence of sex differences in cognitive skills and sensory sensitivities beyond adolescence and in many instances throughout life, it might be that these sex-differentiated growth trends in childhood culminate, in adulthood, in superior (because further-developed) left-hemisphere specialization in women, and superior right—hemisphere specialization in men. Harris (1981) discusses several studies and concludes that the status of this hypothesis is still very unclear. 46 The third hypothesis, proposed by Weber (1979), builds on the first two but proposes a specific feature of growth whereby the sex differences in extent of lateralization in adulthood are created. His model does not reject the poten- tial sex-differentiating role of fetal hormones mentioned earlier; it proposes, instead, that an important aspect of these early effects will be reflected later in differences between early-and late—maturing adolecents. Harris (1981) cites several conflicting reports on comparisons of early- and late—maturing children. Harris concludes that many more studies ”are needed before these conflicting reports can be resolved (p. 109).” Up to this point hypotheses related to the biological determinants of sex differences in spatial ability were dis- cussed. Now the role of socialization and life experiences will be examined. Differential Practice and Sex-Role Typing The prevailing View of behavioral psychologists, anthropologists, and sociologists has been that innate sex differences are insignificant and, when manifest, are probably reflections of differences in childrearing, learning opportunities, or social expectations. With re- spect to sex differences in spatial ability, the environ- mental hypothesis theorizes that they are a product of sociocultural influences, i.e., that in the course of their upbringing, boys have received more opportunities, encour- 47 agement, and training than girls to acquire visual-Spatial skills. In some traditional cultures, the differences in oppor- tunity may be institutionalized into explicitly different work roles (Harris, 1981). Since hunting and navigating have been a strong male domain, a hypothesis of sex—role typing also has been raised; the children are expected to adopt personality traits ”appropriate” to their sex and culture. Sherman (1977) notes that "the association of female with the hearth, male with the out-of—doors is a persistent and pervasive generalization, extending from primitive societies to our own (p. 186).” The assignment of the major role in hunting and other activities involving the traverse of space to males had profound implications for the development of sex-role socialization. All matter of activ— ities involving aiming, visualizing direction, estimating space, and symbolizing space and spatial relations has been an integral part of male socialization experiences. Sherman (1977) claims that many activities sex-typed as male serve to develop spatial skills. These activities are represented in the toys and games given to children and which they are encouraged to play with directly. Indirectly by their own knowledge of their future adult roles they are also influenced in their choice of activities. Early differences in play activities are later supplemented by a host of other socialization experiences including divergent channeling of the males into particular courses in school, 48 e. ., drafting and industrial arts. In addition, Harris 8 (1981) points to research suggesting that the differences go beyond the toys provided; ”they extend to parents' reactions to their children's manner or style of playing (p. 92).” Boys are allowed to explore objects, to learn about the physical world with less chance of being criticized than are girls. Nash (1979) suggests that the sex typing of the domain of achievement is also a potent factor; boys thus may excel at mathematics because, beginning at age twelve, boys but not girls predict that science and mathematics will be relevant to their work as adults. Another kind of evidence in favor of a socialization explanation pertains to the timing of the sex difference. As mentioned earlier in this chapter, Maccoby and Jacklin (1974) concluded that on the whole the sex differences in spatial ability do not appear until adolescence. One in- ference that socialization theorists have drawn from this is that because society prescribes different experiences for males and femaleS--experiences that differentially contri— bute to Spatial competence--it takes time for these differ— ences to become expressed in actual differences in skill. Understandably, so the reasoning goes, the differences typically first appear in adolescence, since sex-role pre- scriptions are more salient at this time (for examples, see Nash, 1979). It would seem logical to expect that sex differences in spatial abilities would become increasingly enhanced with age as a result of continued sex-typed ‘l 49 experiences. Harris (1981) found evidence supporting this expectation. The socialization view also has been pressed through demonstrations that special training can bring females up to the male level of performance. Such demonstrations, as Maccoby and Jacklin (1974) indicate, would be "strong evi- dence that sex differences in spatial ability are (in large degree) a product of differential training, (p. 129).” Various training experiments have been reported some with success. Those studies which related to specific training in spatial visualization will be discussed in a later sec- tion of this chapter. The evidence for the view that socialization factors are responsible for sex differences in spatial ability looks impressive. In contrast, Harris (1981) raises many doubts and cites studies which do not support the above evidence. For example, apart from reports of specific training showing no improvement on various spatial tests, Harris argues that in any case, the lesson of both successes and failures is unclear. It is not necessarily true that a lack of training is responsible for the females' deficient expression of that skill in the first place. Another main point made by Harris is that even when a choice is given to both sexes with regard to games and toys, boys more often than girls "male” toys, such as blocks, and the choose to play with choice simply does not look as if it is ”compelled” by an outside agent. There is evidence that boys play outdoors 50 more than girls do. Again the differences do not seem to be caused by others. Harris (1981) also describes studies which found that certain relationships between spatial ability and personality factors conventionally associated with male and female sex-typing do not come out as they presumably should according to the sex-role socialization hypothesis. The question of the ”late” appearance of sex differ— ences in spatial tasks is also treated by Harris. Although the sex difference unquestionably is more common in later childhood, there is evidence that sex differences in younger children are by no means absent. Eliot and Fralley (1976) argue that most standard tests of spatial ability are too difficult or too abstract for younger children. It may be that they are based too narrowly upon a psychometric, factor-analytic definition of space. By increasing the range of tests for measuring spatial abilities, sex differ— ences in spatial ability may be found to occur in infancy and early childhood. Harris (1981) concludes his review on the socialization factors related to sex differences in spatial skills by accepting the view that certain experiences are necessary conditions for the development of spatial competence. He also accepts the further proposition that males have more of these experiences than females do. But, he claims there is no clear evidence that 51 (a) these different experiences are sufficient to explain demonstrated sex differences in spatial competence; (b) the different experiences in every case are provided (or imposed) by others; or (c) where opportunities are equal, they are equally useful for the furthering of Spatial ability (p. 96). In conclusion, the view that different life experiences or sex-role typing are sufficient to explain sex differences in spatial ability cannot be accepted. There are also good reasons to question any proposition asserting the suffi- ciency of genetic, hormonal, or neurological factors for explaining those differences. Harris (1979a, 1981) believes that the inescapable conclusion is that both biological and social factors underlie the male's superior visual Spatial ability. Specific Training in Spatial Visualization Studies of training programs to increase spatial abilities in general and spatial visualization in particular are not common in the literature, and those that exist deal mainly with adults. One of several Studies involving train— ing in visualization was conducted by Van Voorhis (1941). His study involved first-year college students in a training program that included estimating linear extent, angles, and areas; three—dimensional tick-tack-toe; and various other visualizing experiences. Compared with the control group, the experimental group scores on a criterion measure (the ”Cards” and ”Figures” section of the Thurstone Test for Primary Mental Abilities) improved significantly. No 52 discussion was included of any reduction of sex differences as a result of training. In another study Brown (1954) compared the spatial vis- ualization ability (as measured by the DAT Space Relations Test) of an experimental group, whose training consisted of one year of plane geometry and specific elements of solid geometry, with a control group at two levels of training: (1) students who completed the regular one-year plane geo- metry course, and (2) comparable students who completed a two-year sequence of plane geometry, advanced algebra, and solid geometry. The gain made by the experimental group was less in every analysis. Several reports on the effects of courses in vocational machine shop, descriptive geometry, and mechanical drawing on spatial relations test scores were considered relevant to this study. Mendicino (1958) conducted a study at the high school level with 150 tenth-grade boys. His study indicated that one school year in vocational machine shop and mechan- ical drawing had no more effect on the space perception test scores of his subjects than did a similar period in a nonvo— cational curriculum. Similarly, Myers (1958) conducted a study to examine the possible effects of mechanical drawing taken during high school on the Spatial test scores of cadets who had taken mechanical drawing with those who had not. Myers found no significant difference and concluded that previous training had very little effect on spatial relations test scores. 53 Sedgwick (1961) conducted a study to determine if instruction in descriptive geometry would improve spatial visualization. Fifty-one matched pairs of engineering, industrial education, and industrial supervision students were divided into an experimental and control group and used to obtain the data. The matching of the students was done on the basis of the preterm score achieved on the DAT Space Relations Test--Form A. The analysis was run between the preterm scores and the postterm scores on Form B of the same test. No statistically significant differences were found. Sedgwick concluded that descriptive geometry does not affect or improve visualization and that visualization is probably an innate capability not modifiable by a specific instruc- tion. Contrary to the findings of Mendicino (1958), Myers (1958), and Sedgwick (1961), Blade and Watson (1955) showed that significant improvement in performance on spatial tests occured in men after one year of engineering studies. No effort was made to determine whether a comparable increase would have been obtained in women. As a result of their study, Blade and Watson suggested the possible need to aid those with undeveloped spatial visualization skills by providing them with maximum opportunity to develop this capacity. Two other investigations (Myers, 1951, 1953) using college students have reported similar gains. An investigation involving programmed instruction as a technique for improving spatial visualization was conducted 54 by Brinkmann (1966). The program entailed a brief course in geometry, but rather than emphasizing formal proofs, Brinkmann stressed a number of exercises requiring pattern folding and object manipulation. Training for eighth graders (14 boys and 13 girls) took place over a three-week period during class time normally devoted to mathematics. When the training program was completed, he readministered the DAT Spatial Relations Test as a posttest. He found that the 27 eighth graders receiving the programmed unit scored significantly higher than the control group. Brinkmann (1966) concluded that in view of the significant gains exhibited by the experimental group, it appears reasonable to assume that the functional skill of individuals in spatial visualization can be improved when appropriate training is provided (p. 184). He pointed out that there were no significant sex differ- ences on the posttest and that girls ”can hold their own when provided an opportunity to learn something about a particular area in which they are often assumed to possess less ability (p. 184).” However Brinkmann's study is criti— cized for not investigating whether the training effects persist over time (Maccoby & Jacklin, 1974; Eliot & Fralley, 1976). Concerning the development of specific training pro- grams, Brinkmann exhibits a reluctance to use programmed instruction for an extended period of time. This stems from his observation that his subjects were reluctant to refer to other training aids that supplemented the programmed 55 booklet. These reactions prompted Brinkmann to suggest caution in developing long and complex programs for relatively immature learners. A research project directed by Carpenter (1965) indicated that in four out of ten experiments in which the experimental group was exposed to program material on visualization, significantly higher scores on the posttest (DAT Space Relations Test) were achieved. He suggests the need to investigate the variable of intensity of instruc— tional input concerning visual thinking. His research effort Showed that the most intense combination (filmstrip and program) yielded slightly less gains on the development of spatial ability than the programs alone. This finding tends to complement Brinkmann's (1966) insistence on brevity and compactness with relation to instructional units on the development of spatial ability. Wolfe (1970) devised a training program Similar to Brinkmann's for students in the seventh, eighth, and ninth grades. Videotaped lessons were interspersed with other student activities over a four-week period. Many of the lessons appear to parallel tasks in the Spatial tests, i.e., cube-counting tasks resembling the test requiring the assembly of plane figures, or cube comparison tasks similar to tests of viSualization. Wolfe found significant gains on problem-related tests, but little gain in scores on a ”transfer” test which presumably measured the same abili- o ties. In another study (Mitchelmore, 1974) school teachers 56 Spent a month designing, constructing, and sketching models of three—dimensional shapes, but afterwards showed no im- provement on various spatial tests. On the elementary level, Smith and Schoreder (1979) investigated whether or not the spatial ability of fourth- grade girls and boys w0uld be affected differentially by instruction. The experimental group took part in a series of 10 half-hour tangram lessons. The criterion measure was a spatial visualization test devised by the authors, and ” The investi— described as similar to the ”Form Board Test. gators concluded that ”fourth grade students, regardless of sex profited strikingly from instruction involving spatial ability, as measured immediately following instruction (p. 66).” They indicated the need for further study to determine whether the effect persists over time and whether both sexes are affected comparably over time. Smith and Litman (1979) in another study employing sixth and seventh grade girls and boys as subjects used Sim- ilar instruction and the same criterion measure as Smith and Schroeder (1979) did in their study. Whereas the Smith and Schroeder (1979) study of fourth graders had shown that both boys and girls improve comparably in Spatial visualization ability as a result of instruction, the Smith and Litman (1979) study ”showed that among slightly older, early ado- lescent children, only the boys significantly improved in this ability as a result of instruction (p. 673).'l The au- thors suggest that their results have important consequences 57 in making decisions regarding the timing of instruction in spatial ability. In light of the above findings it is noteworthy that following their comprehensive review on sex differences Maccoby and Jacklin (1974) concluded that ”it has not been demonstrated that male and female subjects respond differen- tially to training [in spatial tasks] (p. 128)." The au- thors indicate that studies on training in spatial tasks have not investigated how widely the training generalized, or how long the effects of training lasted. In conclusion, the results of these training studies are inconclusive and the field is still open for further research. Related Attitudinal Studies Most educators believe that affective variables are important contributors to the learning of mathematics. The relationship between the two has been the object of many studies. The reviews by Aiken (1970, 1976), Suydam and Weaver (1975), Callahan and Glennon (1975), Fennema (1977), and Kulm (1980) provide an excellent summary of the results of investigations in this area. The causal relationship between attitudes and behavior is far from clear. Although there is evidence that attitudes influence behavior, there is also evidence that behavior influences attitudes (Calder & Ross, 1973). This controversy becomes even more complex with attitudes toward school subjects, in which behavior is related to academic achievement. Either direction of 58 causation can again be supported, with evidence that school behaviors influence achievement and vice versa. Other problems presented by the work done in the area of attitudes toward mathematics are related to the different definitions of the construct ”attitude,” problems with measuring instruments, and what constitutes a global definition of mathematics (Fennema, 1977). Taking into consideration the above problems, only a brief review and summary of results with regard to three areas will be attempted: (1) the relationship between attitudes toward mathematics and achievement in mathematics, (2) sex differences in attitudes toward mathematics, and (3) effects of curricular Structure on attitudes toward math- ematics. Attitude and Achievement The general question asked by current researchers is ”what is the strength of the relationship between attitudes toward mathematics and achievement in mathematics?” Apparently, the common sense feeling that achievement ought to depend heavily on attitudes Stimulates the search for a clear, simple relationship between these variables. Often, the hypothesis is that the relationship is causal, so that attitudes are investigated as predictors of achievement. However, conclusions reached in comprehensive reviews appear to be contradictory. 59 Suydam and Weaver (1975) summarizing elementary school studies state: There is no consistent body of research evidence to support the popular belief that there is a significant positive relationship between pupil attitudes toward mathematics and pupil achievement in mathematics. ...We have little research basis for believing that these two things are causally related (pp. 1-3). Callahan and Glennon (1975) are in agreement with Suydam and Weaver. Also reviewing elementary school studies for the same age, they conclude that the state of the art ”makes it difficult to present compelling research evi- dence...that positive attitudes play an important role in contributing to mathematics achievement (p. 80).” Aiken (1976) argues that ”when attitudes scores are used as predictors of achievement in mathematics, a low but sig- nificant positive correlation is usually found (p. 295)” at the elementary, secondary, college undergraduate and post- graduate levels. Fennema (1977) suggests that part of the contradictory conclusions can be explained by the age of the subjects being considered in the reviews. Two reviews (Suydam and Weaver, 1975; Callahan and Glennon, 1975) were concerned basically with children in grades one through six. Problems of assessing attitude in these grades have not been ad— dressed adequately, and lack of carefully designed measuring instruments may have caused reviewers to seriously question any significant differences reported. Aiken, in his 1976 review, was concerned with a much broader age spectrum. 60 Even while recognizing the serious problems connected with the studies of young children, he was willing to accept the evidence as having some validity because the results coincided with studies having older subjects. Fennema (1977) summarizes the tentative conclusions suggested by the literature: 1. There is a positive relationship between attitude and achievement which seems to increase as learners progress in school. 2. Attitudes toward mathematics are fairly stable--particu1arly after about the sixth grade, although one longitudinal study Showed a marked decrease from 6th grade to 12th grade (Anttonen, 1969). 3. Grades 6-8 seem to be critical in the development of attitudes. 4. Extremely positive or negative attitudes appear to be better predictors of achievement than more neutral feelings (p. 104). Fennema indicates there is a fifth conclusion related to sex differences in attitudes toward mathematics which will be discussed in the succeeding paragraphs. Sex Differences in Attitudes Toward Mathematics On the whole Fennema (1977) concludes that ”there are sex-related differences in attitudes toward mathematics (p. 104).'' But, even though there is consensus that sex-related differences in mathematics attitude exist, the magnitude and specific dimensions of these differences are unclear. Although denoting some studies which failed to find significant sex differences in attitudes and achieve- ment in mathematics, Aiken (1976) indicates that ”differ- ences in both attitudes and achievement in mathematics are 61 frequently found to favor boys over girls at junior-high level and beyond (p. 296).” With regard to sex differences in attitudes, Suydam and Weaver (1975) quote studies with contradictory results and say that in other studies no significant sex-related differences were found. Aiken (1976) states that the correlation between attitude and achievement varies not only with grade level but also with the sex of the student and is generally somewhat higher for girls than for boys. Basic agreement with the conclusion that significant differences in attitudes are frequently found to favor males over females, was reported by Fennema and Sherman in their study (1977, 1978) with learners in grades six through eleven. It has also been reported that mathematics test anxiety is significantly higher for eighth grade girls than for eighth grade boys (Szetela, 1973). Effects of Curricular Structure on Attitudes Toward Matfiematics Over the years a great deal of research in mathematics education has focused on practical questions such as whether an approach to instruction which is ”experimental," student- centered, discovery-oriented, and innovative results in greater achievement and more positive attitudes than an ap- proach which is ”controlled,'l teacher-centered, expository- oriented, and traditional. Taking into consideration that the generalizability of the results of these investigations has been frequently questioned, Aiken (1976) summarizes some 62 tentative findings of these kinds of investigations in mathematics education: 1. Modern mathematics programs do not improve attitudes more than traditional programs. 2. Compared to regular classes, "continuous progress” classes do not have a different effect on attitudes toward mathematics. 3. Discovery methods are not superior to expository methods in their effects on attitudes toward mathematics. 4. Neither follow-up instructions nor flexible scheduling improves attitudes more than traditional instruction. 5. An individual approach to instruction in elementary and junior high mathematics sometimes has a more positive effect on attitude than a traditional approach; other times no difference in the effects of the two types of programs is found. 6. Certain units or topics in mathematics have a more positive or a more negative effect on at- titudes than other units or topics (pp. 300-1). Summary The main concern of this chapter was the review of the literature related to the spatial visualization factor and the sex differences in spatial visualization. ”Spatial Orientation” and ”Spatial Visualization” are considered to be the two major factors of ”Spatial Ability.” However, factor analysts continue to have difficulty in differenti- ating and interpreting these factors. In addition, there has been no final consensus on the number of factors in- volved with spatial ability, and the instruments used for measurement differ in form, content, and power. "Spatial Visualization” is an aptitude that deals with the mental manipulation of rigid figures. This aptitude emerges as a component of mathematics ability in most factor 63 analytic studies (Schonberger, 1976), and it shows about the same magnitude of correlation with mathematics achievement as the verbal component does (Fennema & Sherman, 1977). The review of related literature Showed that clear evidence exists to indicate that spatial visualization is substan- tially related to many forms of mathematics achievement. The review also cited findings that show the relationship of spatial visualization to achievement in science, engineer- ing, and specifically in chemistry. An indication that Spatial visualization is an impor- tant consideration is the discovery of sex-related differ- ences in spatial visualization Skills. The review of sex differences in spatial visualization revealed a widespread assumption that male performance on Spatial visualization is superior to female performance. However, detailed examina- tion of investigations in the field showed a certain amount of confusion about the uniformity and significance of the superiority. Some investigations have noted that sex dif— ferences on Spatial visualization tests appear at early ages; others have indicated that differences do not appear until adolescence or early adulthood. Additionally, there were very few developmental Studies investigating sex dif- ferences in spatial visualization over several ages (espe- cially in the middle school) with a single test. Apparent sex differences in Spatial visualization vary according to ethnicity, social class, and culture. 64 The generalization of male superiority in spatial ability in general and spatial visualization in particular has overlooked the great complexity of the data and has become so pervasive that a body of literature has developed to explain the apparent differences. The theoretical ex- planations and hypotheses concerning the sex differences in spatial visualization (and spatial ability) range broadly; some emphasize the contributions of biological factors-- ”nature,” others emphasize the role of learning and social- ization--”nurture.” Present knowledge about the herita- bility of mental traits and the nature of the effects of hormonal or neurological factors on Specific abilities is far from certain. There are also good reasons to question the sufficiency of life experiences or sex-role typing for explaining the sex differences in spatial visualization. It seems an inescapable conclusion that both biological and social factors underlie the male's superior visual spatial ability. Studies of training programs to increase Spatial visualization are very few. Of the existing Studies, most have been done with adults. No large-scale Study to imple- ment a unit of instruction in spatial visualization for a wide range of grade levels was found. Results from these studies are inconclusive and the field is still open for further research. The final section of the review of related literature was devoted to a brief discussion of attitudes studies which 65 were relevant to this investigation. Although there is agreement that there are sex-related differences in atti- tudes toward mathematics, further research might uncover the magnitude and specific dimensions of these differences. CHAPTER III DESIGN OF THE STUDY Introduction This study had two related purposes: the first was to determine existing differences in spatial visualization abilities and in attitudes toward mathematics of fifth through eighth grade students by sex and grade level prior to the instruction. The second was twofold: first to analyze the effects of instruction on the spatial visualiza- tion skills and attitudes toward mathematics of a sample of sixth, seventh, and eighth grade students by sex and grade; second, after instruction, to compare attitudes toward math- ematics and spatial visualization and to examine differences in attitudes toward spatial visualization by sex and grade level. This chapter includes a description of (l) the popula- tion and sample, (2) the instruments used in the study, (3) the instruction material, (4) procedure and data collection, and (5) the hypotheses and the statistical design of the Study. Population and Sample The subjects of the study attended schools in three sites in and around Lansing, the state capital of Michigan. 66 67 Site 1 is a school in the inner city of Lansing, which has only fifth and sixth grade students and two kindergarten classes. The district demographic data for 1980-81 shows 65 percent White, 23 percent Black, and 10 percent Latino. The school demographic data for the same year shows 56 percent White, 17 percent Black, and 23 percent Latino. Site 2 is a middle school in a suburb of Lansing and Michigan State University. This school serves a community of upper-middle class university and state government professionals and is predominantly White. Site 3 comprises a middle and two elementary schools in a rural area in the suburbs of Lansing. The schools serve a cross-section of middle class families and are predominantly White. For Site 1, with the exception of one fifth—sixth combination class, all of the fifth and sixth grade students were included in the sample. For Site 2, all sixth grade students and 6 of the 11 seventh grade classes were included in the sample (2 of the other seventh grade classes were in an advanced math program, another class took part in an earlier trial of the spatial visualization material, and the other 2 regular classes were not tested. No reasons were given by their math teachers). For Site 3, all the students from the middle school (except small groups of special education students) and all the fifth graders from the two elementary schools (two out of four feeders to the middle school) were included in the sample. 68 The entire sample described above took part in the assessment of existing differences in spatial visualization and attitudes toward mathematics by sex and grade level prior to the instruction. A subsample, which was drawn only from Site 3, participated in the evaluation of the effects of instruction and examination of differences in attitudes toward mathematics and toward spatial visualization by sex and grade level. Descriptive information of the subjects appears in Tables 3.1 - 3.5. Table 3.1 shows the distribution of the entire sample by grade level (five through eight) and by sex in each of the three sites. Table 3.2 shows the distribu- tion of the subsample from Site 3 by grade level (six through eight) and by sex. Table 3.3 provides comparative data on standardized testing of the sixth grade students in the sample using the reading and mathematics Stanford Achievement Tests. The grade equivalents and the distribu- tion of the SAT scores are given for each participating site in the study (The students from Site 1 were tested at the end of the previous school year in May 1980, while the others were tested in September and October 1981). Table 3.4 provides comparative data for the seventh graders in the entire sample based on the 1981—82 test report of the Michigan Educational Assessment Program (MEAP). Table 3.5 shows the grade equivalents and the distribution of the SAT scores for the eighth grade students in the sample from Site 3. 69 TABLE 3.1 DISTRIBUTION OF THE ENTIRE SAMPLE BY GRADE BY SEX IN EACH SITE N N N N _ Classes Boys Girls Total Site 1 8 113 106 219 Grade 5 4 55 49 104 Grade 6 4 58 57 115 Site 2 17 221 206 427 Grade 6 11 150 124 274 Grade 7 6 71 82 153 Site 3 28 338 343 681 Grade 5 4 48 46 94 Grade 6 8 104 104 208 Grade 7 8 90 80 170 Grade 8 8 96 113 209 I I 70 TABLE 3.2 DISTRIBUTION OF THE SUBSAMPLE FROM SITE 3 BY GRADE BY SEX N N N N Classes Boys Girls Total Grade 6 4 54 54 108 Grade 7 7 74 68 142 Grade 8 7 79 101 180 _Total 18 207 223 430 TABLE 3.3 GRADE 6—-GRADE EQUIVALENTS AND DISTRIBUTION OF READING AND MATHEMATICS SAT SCORES FOR EACH SITE Achievement Grade Low Average High Equivalent 1-24Z ile 25-74Z ile 75-99% ile Reading Site 1 6.57 18 55 27 Site 2 7.5 7 55 38 Site 3 7.2 6 65 30 Mathematics Site 1 6.8 17 55 28 Site 2 7.0 12 53 35 Site 3 6.7 11 65 24 71 TABLE 3.4 DISTRIBUTION OF SEVENTH GRADERS IN THE ENTIRE ASSESSMENT TEST 1981-82 SAMPLE ON READING AND MATH ON THE MICHIGAN REading Site 2 Site 3 Math Site 2 Site 3 Equivalent 1-24Z ile 25—747 ile Reading Mathematics 00—24% Mastery of Objectives 1.0 3.1 Grade 9.9 8.9 Catergory of Achievement 25-49Z 5.8 7.1 TABLE 3.5 Low 6 10 50-74% 21.2 26.3 Achievement Average 55 64 75-1001 Mastery of Objectives 90.8 85.1 72.0 63.5 GRADE 8--GRADE EQUIVALENTS AND DISTRIBUTION OF READING AND MATHEMATICS SAT SCORES High 75-99% ile 39 26 72 Instrumentation The instruments used in this study were a semantic differential attitudes scale and a spatial visualization performance test. Attitudes The instrument to measure students attitudes toward mathematics was developed by Shumway et a1 (1981). The same questionnaire was adopted to measure students' attitudes toward spatial visualization activities (the only change is the title). The attitude scale is a six-item semantic dif— ferential with five response options. Appendix B includes the Mathematics Attitude Scale (MAT) and the Spatial Vis- ualization Attitude Scale (SVAT). The scales were scored by assigning scores of 5 through 1 to the student responses. Five signified a positive response; one a negative response. For example: --- Score --— (1) (2) (3) (4) (5) BORING ---;---;---;---;--- EXCITING The scores for the six items were added together and divided by 6 giving an attitude score with a maximum of 5 and min- imum of 1. The internal consistency reliability estimates for the Mathematics Attitude Scale (MAT) reported by Shumway et a1 (1981) ranged from 0.82 to 0.92. In this study, using the same attitude scale with fifth, sixth, seventh, and 73 eighth grade students, the computed Cronbach a reliability estimates ranged from 0.79 to 0.91. Table B.1 in Appendix B includes the reliability coefficients for the Mathematics Attitude Scale in each site by time, by grade, and by sex. The stability of the Mathematics Attitude Scale instrument was supported by the high correlation between the pre- and post-administration of the instrument. In this study, the Pearson correlation coefficients between the math attitudes pretest and posttest (a time period of about two to three weeks) ranged from 0.64 to 0.82. Table B.2 in Appendix B contains the Pearson correlation coefficients between the pretest and posttest scores on the Math Attitude Scale for the entire subsample in site 3 by grade and by sex. All the correlations were significant (P < .001). The Cronbach a reliability coefficients for the Spatial Visualization Attitude Scale (SVAT) in this investigation, ranged from 0.74 to 0.91. Table 3.3 in Appendix B contains the reliability coefficients of the Spatial Visualization Attitudes Scale for the posttesting and retention testing in Site 3 by grade and by sex. The stability of the Spatial Visualization Attitude Scale instrument was supported by the high correlation between the scores at the end of the in- struction and the scores four weeks later (retention). In this study, the Pearson correlation coefficients between the spatial visualization attitude posttest and the spatial vis- ualization attitude retention test four weeks later ranged 74 from 0.51 to 0.80. The 0.51 coefficient occurred only for the girls in grade six and other than that, the coefficients ranged from 0.64 to 0.80. Table B.4 in Appendix B includes the Pearson correlation coefficients between the posttest and retention test scores on the Spatial Visualization Attitude Scale for the entire retention subsample in Site 3 by grade and by sex. All the correlations were significant (P < .001). Spatial Visualization Performance The spatial Visualization performance was measured by the Middle Grades Mathematics Project Spatial Visualization Test (MCMP SVT). This test was developed by the Middle Grades Mathematics Project staff, including this investiga— tor, in the Mathematics Department of Michigan State Univer- sity. During summer and fall 1981, prior to this study, the test was administered to hundreds of fifth, sixth, seventh, and eighth grade students in six Michigan school districts. Test validity and reliability were based upon scholarly analyses of test items and factor analyses by mathemati— cians, mathematics educators, and researchers. The test consists of 32 multiple choice items (5 options for each item) and is preceded by a sample sheet which includes 2 sample items to be discussed in class before taking the test. Appendix C includes the Sample Items preceding the MGMP Spatial Visualization Test and some more sample items similar to those in the MGMP SVT. 75 The MGMP Spatial Visualization Test had two functions. It served as a measure of spatial visualization ability in order to assess existing differences by sex and grade level prior to the intervention and as a pre-post-retention test on spatial Visualization in order to evaluate the effects of instruction in activities involving spatial visualization tasks. The test was not timed, but usually did not exceed 25-30 minutes. The items were scored by assigning a 1 for a correct student response, otherwise 0; no correction was made for guessing. The Cronbach a reliability coefficients calculated for the MGMP SVT ranged from 0.72 to 0.88. Table C.1 in Appendix C includes the reliability coefficients for the MGMP SVT in each site by time, by grade, and by sex. The MGMP SVT test-retest effect was examined, using the Pre-Experimental design 3 from Campbell and Stanley (1963): in which X is the treatment of taking the MGMP SVT the first time; 01, the scores on the MGMP SVT for the same group that had X; 02, the scores on the MGMP SVT for equivalent group of students. The testing effect was assessed for seventh and eighth grade students in Site 3. One seventh grade math teacher with 5 classes and one eighth grade math teacher with 3 76 classes were identified. Two classes from each were random— ly selected. Two weeks before the pretesting (01 and 02), the MGMP SVT was administered to one seventh and one eighth grade class. A One—Way ANOVA test was conducted to compare 01 and 02 for each grade level. No significant difference was found between the mean scores 01 and 02 for each grade level tested, seventh and eighth. Table C.2 in Appendix C includes the sample size, mean scores, and standard deviations; Tables C.3 and C.4 include the summary analyses of the ANOVA test. Another indicator of the quality of the instrument was the significant high correlations between the pretest-post- test scores and especially the significant high correlations between the post-retention test scores (a time period of four weeks). In this study, the Pearson correlation co- efficients between the pretest—posttest scores on the MGMP SVT ranged from 0.56 to 0.81. The correlation coefficients between the post-retention test scores on the MGMP SVT ranged from 0.75 to 0.88. Table C.5 in Appendix C contains the Pearson correlation coefficients between pretest—post- test scores and between post-retention test scores on the MGMP SVT for the entire retention subsample from Site 3 by grade and by sex. All the correlations were significant (P < .001). Instruction Material The spatial visualization instruction material used in this study is a unit developed by the Middle Grades 77 Mathematics Project (MGMP), Department of Mathematics, Michigan State University. The MGMP is a curriculum devel— Opment project funded by NSF-DISE (National Science Founda- tion--Development in Science Education), to develop units of high quality mathematics instruction for grades five through eight. The Spatial Visualization unit includes ten care- fully sequenced activities which requires two to three weeks of instructional time. These activities are an extension of previous work reported in two articles: ”Buildings and Plans”, and ”Spatial Visualization” (Lappan & Winter, 1979, 1982). Prior to this study, the unit was taught success- fully to many regular classes ranging from fifth through eighth grade levels and to different ability groups. The Spatial Visualization unit involves representing three-dimensional objects in two-dimensional drawings and, vice versa, constructing three-dimensional objects with blocks from their two-dimensional representations. The activities deal with the flat views of buildings as well as with the isometric drawings on dot paper (paper with dots arranged on diagonals rather than rows). In most of the activities the students are asked to perform some fairly demanding orientation and visualization tasks. They are asked to mentally rotate a building and draw either flat views of the other sides or isometric drawings from other corners. Cubes are always available to help a student who needs to see the concrete object to be successful. 78 The first four activities deal with the flat projection viewed from the front, right, left, and back sides of the building, including a representational picture of the base for the building showing exactly where blocks touch the ground. In order to acquire a formal understanding of this representational scheme, the students are required to (1) match buildings with their flat views and the base, (2) be able to draw the base and the flat views of an existing building, (3) build a building from a given set of views and the base, and finally (4) be able to construct and represent a building and evaluate another reconstruction. These four steps, in the order given, became the guidelines for the development of the first four activities. The next five activities (5 through 9) comprise the second representational scheme in which the buildings are turned and viewed from the corners so that three faces of the building are seen. The students are introduced to the isometric dot paper and the same four steps of the first scheme are requested. The students learn to relate a build- ing and its isometric drawings, to make isometric drawings, to build a building from isometric drawings, and, as before, to construct and represent a building and evaluate another reconstruction. Inadequacies in this scheme are looked at by the students and compared with the first scheme. Simi- larities and differences are noted. In order to emphasize the orientation and visualization of buildings as well as the drawings, some of the activities require the students 79 to modify buildings by adding and removing cubes, or put together two simple buildings, called puzzle pieces, to match a drawing of a building made from the two pieces. Activity 10 provides the summary of the unit. The students are asked to produce architectural views and isometric corner views, draw single layer buildings with stretched out designs, and as a final step, build the Mystery Building used to launch the unit--the extra chal— lenge being to produce an isometric view of the Mystery Building. The spatial visualization unit utilizes an instruction— al model developed by Shroyer and Fitzgerald (1979). The model consists of three phases: launching, exploring, and summarizing. The teacher plays a central role in this instructional model. First, the teacher provides and moti- vates the challenge and then joins the students in exploring the problem. The teacher asks appropriate questions, en— couraging and redirecting where needed. Finally, through the summary, the teacher helps the students to deepen their understanding of both the mathematical ideas involved in the challenge and the strategies used to solve it. To aid the teacher in using the teaching model de— scribed, a detailed instructional guide is provided. It was developed as a result of many classroom trials of the mater— ials and provides help with both the mathematics content and the classroom management of the activities. Specific sug— gestions for important questions to be asked at appropriate 80 stages of the activities are included. Additional questions which involve generalizations and further challenges for the more able students are provided along with suggestions for helping those students who are having difficulty. Procedure and Data Collection The study was conducted during winter 1982 and data were collected from January 20 to April 20 (including the collection of the retention data). The classroom teachers were provided with all the testing material and written instructions concerning the administration of the test. Appendix D includes the Spatial Visualization and Mathe- matics Attitude Pretest Instructions given to the teachers. Similar instructions were given for the post and retention testing. The tests were administered during the regular school day by the classroom math teachers. The time needed for administration of the instruments did not exceed the regular math hour. The teachers from Site 3 who taught the spatial visualization unit were provided with all the instruction material in addition to the testing material. For all the teachers, this was the first time that they taught the unit on spatial visualization. Prior to the instruction, a two-hour workshop for the teachers was conducted at Site 3 which was the only site at which the instruction was measured. 81 Data Collection The following data were collected: General Information. General information on the subjects and the schools participating in the study, including size, type, and standardized test scores. Pretesting. Pretest scores on the Mathematics Attitude Scale and MGMP Spatial Visualization Test from the whole sample in all three sites. Observations. During the instruction in the spatial visualization unit, which required two to three weeks of instructional time, the investigator made several observations in each participating classroom. The purpose of these observations was to record classroom conditions during instruction and to document any extraordinary events. Posttesting. At the end of the instruction in the spatial visualization unit to the subsample from Site 3, the Mathematics Attitude Scale, the Spatial Visualiza- tional Attitude Scale and the MGMP Spatial Visualization Test were administered. Retention Testing. Four weeks after the end of the instruction and posttesting, the Spatial Visualization Attitude Scale and the MGMP Spatial Visualization Test were given to a representative sample of the partici- pants in Site 3. Table 3.6 shows the distribution of the representative sample by grade and by sex. 82 TABLE 3.6 DISTRIBUTION OF THE REPRESENTATIVE SAMPLE FOR THE RETENTION TESTING BY GRADE BY SEX N N N N Classes Boys Girls Total Grade 6 2 26 26 52 Grade 7 4 41 38 79 Grade 8 4 45 62 107 _Total 10 112 126 238 The Hypotheses and Statistical Design of the Study To analyze the data collected during the study, the investigator selected several statistical procedures for the purpose of clarifying some aspects of the study and to test the hypotheses given in the following paragraphs. The analyses included: means, standard deviations, correla- tions, multivariate and univariate analysis of variance and repeated measures, planned comparison, and Scheffe's Post Hoc comparisons. All the analyses were carried out on the 3600 Computer at the Michigan State University Computer Center using the SPSS (Statistical Package for the Social Sciences) and Finn programs. For the purpose of assessing the differences in spatial visualization skills and attitudes toward mathematics by grade level and sex prior to the instruction, the Multivari- ate Model with a Two-Way fixed effects analysis of variance 83 was selected. Since the populations of Sites 1,2 and 3 (described earlier in this chapter) were different in their characteristics, the data from each site was analyzed separately. The designs for Sites 1,2, and 3 were 2x2, 2x2, and 4x2 crossed, respectively. Each design has two criterion measures, complete and with unequal numbers of subjects in the subclasses. The 4x2 crossed, multivariate design for Site 3 appears in Table 3.7. The designs for Sites 1 and 2 were similar except that for Site 1 there were only grades five and six, while for Site 2 there were only grades six and seven. TABLE 3.7 THE 4x2 CROSSED MULTIVARIATE DESIGN FOR SITE 3 Independent Dependent _Grade Sex MGMP SVT Scoresa MAT Scoresb 5 Boys x1 x2____ Girls x1 x2____ 6 Boys x x2 _ Girls x7 x2 7 Boys x x _ Girls x x 8 Boys x] _____ _§2_____ Girls x x2 aThe scores on the MGMP Spatial Visualization Test. bThe scores on the Mathematics Attitude Scale. 84 The hypotheses tested within this design, given in null form, were H01: There will be no difference among the mean scores for each of the four grade levels tested, five, six, seven, and eight, on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. H02: There will be no difference between the mean scores for boys and for girls in grades five through eight on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. H03: There will be no interaction of grade by sex among the mean scores for fifth through eighth grade students on both the Middle Grades Mathematics Project Spatial Visuali- zation Test and on the Mathematics Attitude Scale. For the significance tests, the assumptions were that the subjects were responding independently of one another and that the error vectors had a two-variate normal dis- tribution with mean zero and general variance-covariance matrix. The general conduct of the study made the assump- tions of independence and normality feasible. There were two other restrictions to consider before proceeding further with the analysis. First, the dependent variables must not be linear functions of one another. They may not, for ex- ample, consist of subset scores and a total score for each subject, nor may they have the same total for every subject. Since in the sample data there was no such relationship between the two dependent variables, this restriction was not a problem. 85 The second restriction concerns the number of parameters that may be uniquely estimated and tested, i.e., the degrees of freedom in the model. For example since the model for Site 3 (Table 3.7) had only eight subclass means, it was assumed that these eight means were distributed independently of one another, so that there were eight degrees of freedom among means. This implied being allowed to estimate or test the significance of only eight means, or eight linear functions of them, although the model had more than eight parameters. Thus, the parameters were restricted, so that estimation of only eight independent quantities was required. To analyze the effects of instruction on the spatial visualization skills differences in spatial visualization and attitudes toward mathematics by sex and grade level, the Multivariate Analysis of Repeated Measures Model was selected. For the subsample from Site 3 a Two-Way six-group design with four measures for each subject was used. The design appears in Table 3.8. The hypotheses tested within this design, given in null form were H04! There will be no difference between the posttest means and the pretest means of sixth, seventh, and eighth grade students on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. 86 There will be no difference between the mean gain scores (posttest minus pretest) for each of the three grade levels tested, six, seven, and eight, on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. There will be no difference between the mean gain scores (posttest minus pretest) for boys and for girls in grades six, seven, and eight on the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathematics Attitude Scale TABLE 3.8 THE MULTIVARIATE ANALYSIS OF REPEATED MEASURES DESIGN Grade 7 Grade 8 FOR THE SUBSAMPLE FROM SITE 3 I! MGMP SVT Scores MAT Scores Pretesta Posttestb PretestC Posttestd Boys x x Y] YZ Girls x x y] yz Boys x x y] yz Girls x x y] y2 Boys x x y] yz Girls X x y] yz aThe pretest scores on the MGMP Spatial Visualization Test. bThe posttest scores on the MGMP Spatial Visuali- zation Test. CThe pretest scores on the Mathematics Attitude Scale. dThe posttest scores on the Mathematics Attitude Scale. 87 Finn and Mattsson (1978) indicate that the multivariate approach to the analysis of repeated measures data is less restrictive than the univariate. In the multivariate analy- sis, the separate measures are considered as multiple cri— terion variables, and they may have unequal variances and a general pattern of covariances or correlation. The var- iances of the measures at different times or under different experimental conditions are estimated from the data and utilized in the analysis. Similarly, the pattern of corre- lations is estimated from the data and incorporated in com- puting the multivariate test statistics. The other basic assumptions and restrictions are as in the multivariate model, and as indicated before, it was assumed that they were met. To compare the attitudes toward mathematics and spatial visualization by sex and by grade after the instruction, a Two—Way ANOVA design was employed. The hypotheses tested within this design were H07: There will be no difference between students' mean scores on the Mathematics Attitude Scale and students' mean scores on the Spatial Visualization Attitude Scale. “08’ There will be no interaction of grade by sex among the mean difference scores--Mathe- matics Attitude Scale score minus Spatial Visualization Attitude Scale score. To examine sex and grade level differences in attitudes toward spatial visualization, a Two-Way ANOVA design was applied. The hypotheses tested within this design were 88 H09: There will be no significant difference between the mean scores for boys and for girls in grades six, seven, and eight on the Spatial Visualization Attitude Scale. H10: There will be no difference between the mean scores for each of the three grade levels tested, six, seven, and eight, on the Spatial Visualization Attitude Scale. To evaluate the retention factor of the effects of instruction in spatial visualization, the Multivariate Analysis of Repeated Measures Model was used. The hypotheses tested within this design were H11: There will be no difference between the retention test means and the posttest means of sixth, seventh, and eighth grade students on both the Middle Grade Mathematics Project Spatial Visualization Test and on the Spatial Visualization Attitude Scale. H12: There will be no difference between the mean gain scores (retention test minus posttest) for each of the three grade levels tested, six, seven, and eight, on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Spatial Visualization Attitude Scale. H13: There will he no difference between the mean gain scores (retention test minus post- test) for boys and for girls in grades six, seven, and eight on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Spatial Visualization Atti- tude Scale. Planned comparisons for the grade factor and Scheffé's Post Hoc comparisons were used to identify the source of significant main effects or interactions. Winer (1971) indicates that the method of planned comparisons ”is primarily for meaningful comparisons planned prior to inspection of the data (p. 201)." With respect to Scheffé's Post Hoc, Winer indicates that 89 the Scheffé method is clearly the most conservative with respect to type 1 error; this method will lead to the smallest number of significant differences. In making tests on differences between all possible pairs of means it will yield too few significant results (p. 201). The .05 level of significance was the minimum value accepted in testing the hypotheses. Summary In this chapter the investigation of differences in spatial visualization skills and attitudes toward mathema- tics by sex and grade level (five through eight) prior to the instruction was described. Also investigated and described in detail were the effects of instruction in activities involving spatial visualization tasks on sixth, seventh, and eighth graders' abilities and differences in spatial visualization and a comparison of their attitudes toward mathematics and spatial visualization. The spatial Visualization instruction material including the sequence of activities and the instructional model was described. The instruments used included two semantic differential scales for measuring attitudes toward mathematics and toward spatial visualization and the MGMP Spatial Visualization Test for measuring the spatial visualization performance of students. Reported reliability coefficients for each instrument by grade and by sex ranged from .72 to .91. The Pearson correlation coefficients between successive admin- istration of the instruments were also reported. All the correlations were significant at level P < .01. 90 A detailed description of the sample and the population was given including background information and the distribu- tion by sites, by grade, and by sex. There were 219 fifth and sixth grade students from Site 1, 427 sixth and seventh grade students from Site 2, and 681 fifth through eighth grade students from Site 3 who took part in the assessment of differences in spatial visualization skills and attitudes toward mathematics by sex and grade level prior to the in- struction. There were 430 sixth, seventh, eighth grade stu- dents from Site 3 who took part in the evaluation of the effects of instruction and comparison of attitudes toward mathematics and spatial visualization; of the 430 students, 238 took part in the evaluation of the persistence of the effects of the instruction (retention). The research designs and analyses for statistical testing were indicated. The analysis of the data determined means, standard deviation, correlations, univariate and multivariate analysis of variance and repeated measures, planned comparisons, and Scheffé's Post Hoc comparisons for significance. Results obtained from the different analyses and their interpretation will he presented in the following chapter. CHAPTER IV ANALYSIS OF DATA AND RESULTS Introduction This chapter presents a summary of the data collected during this investigation, the analysis of data, and results based on this analysis. It consists of seven sections: (1) sex and grade level differences in spatial visualization performance and in attitudes toward mathematics (separate analysis for each site), (2) comparison of sixth grade data among the three sites, (3) the effects of instruction, (4) comparison of attitudes toward mathematics with attitudes toward spatial visualization, (5) sex and grade level differences in attitudes toward spatial visualization, (6) retention analysis, and (7) correlation analysis. A summary or results concludes chapter IV. Sex and Grade Level Differences in Spatial Visualization Performance and in Attitudes Toward Mathematics To assess differences in spatial visualization performance and in attitudes toward mathematics by sex and grade level prior to the instruction, the Multivariate Model with a Two-Way fixed effects analysis of variance was used. As noted earlier, since the populations of Sites 1,2, and 3 91 92 were different in their characteristics, the data from each site was analyzed separately. The first three hypotheses stated in Chapter 3 were adjusted for the grade levels at each site and tested for significance at the 0.05 level. The analysis for each site includes the hypotheses to be tested, means and standard deviations, profiles by grade and by sex for each measure, a summary table of multivariate and univariate analysis of variance, results of post hoc tests, and the results of the significance tests for each of the null hypotheses. Site 1 The design for Site 1 was 2x2 crossed with two criterion measures. The multivariate null hypotheses to be tested within this design were H01: There will be no difference between the mean scores of the two grade levels tested, five and six, on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. H02: There will be no difference between the mean scores for boys and for girls in grades five and six on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. H03: There will be no interaction of grade by sex between the mean scores for fifth and sixth grade students on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. Table 4.1 provides means and standard deviations of the MGMP Spatial Visualization Test (SVT) and the Mathematics Attitude Scale (MAT) scores for Site 1 by grade and by sex. 93 The mean scores from Table 4.1 are used in Figures 4.1 and 4.2 to show profiles for grade levels by sex for the MGMP SVT and MAT measures. In Figure 4.3 and 4.4 the same means are used to illustrate profiles of sex by grade levels for the above two measures. Table 4.2 as suggested by Finn and Mattsson (1978, p. 77-78) includes a summary of Multivariate and Univariate analysis of variance for the 2x2 design of Site 1. The multivariate test indicated that the grade by sex inter- action and sex main effects were not significant. The test gave clear evidence of grade level main effects, shown by the univariate tests to be confined to both variables, the MGMP SVT scores and the MAT scores. Sixth graders performed significantly higher than the fifth graders on the MGMP SVT, while their scores on the MAT were significantly lower than the attitude scores of the fifth graders. Since a significant difference was found between grade levels for the main effects, Scheffé's Post Hoc tests were employed to determine whether males were significantly different in the fifth and sixth grades or whether females were significantly different in the fifth and sixth grades or whether both were significantly different in the fifth and sixth grades. Appendix E includes summary tables of the Scheffé's posteriori comparisons of grade levels for males and for females for each measure, the MGMP SVT and MAT. The Post Hoc tests showed a significant difference between the MGMP SVT mean scores of male students in grades 94 TABLE 4. 1 MEANS AND STANDARD DEVIATIONS OF MGMP SVT AND MAT SCORES FOR SITE 1 BY GRADE AND BY SEX N Grade 5 104 Boys 55 Girls 49 Grade 6 115 Boys 58 Girls 57 Total 219 Boys 113 Girls 106 MGMP SVIa M S.D. 7.34 4.15 7.25 4.68 7.43 4.14 8.85 5.21 9.05 5.15 8.65 5.32 8.13 4.90 8.18 4.99 8.08 4.83 3. 3 3. .099 .950 .249 314 .193 442 .917 1.058 .726 .997 1.101 .859 aMGMP SVT-—Middle Grades Mathematics Project Spatial Visualization Test. bMAT--Mathematics Attitude Scale. (Range 0-32). (Range 1-5). 95 MGMPSVTMEANSCORES 5 6 (SRPJJE Fig. 4.1 MGMP Spatial Visualization Test (SVT) means—— Profiles of grade levels by sex at Site 1 MATMEANSCORES 5 6 GRADE Fig. 4.2 Mathematics Attitude Scale (MAT) means—— Profiles of grade levels by sex at Site 1 96 MGMPSVTMEANSCORES G B Fig. 4.3 MGMP Spatial Visualization Test (SVT) means—— Profiles of sex by grade levels at Site 1 MATMEANSCORES Fig. 4.4 Mathematics Attitude Scale (MAT) means-- Profiles of sex by grade levels at Site 1 97 TABLE 4.2 A SUMMARY OF MULTIVARIATE AND UNIVARIATE ANALYSIS OF VARIANCE FOR THE 2x2 DESIGN OF SITE 1 Source of Multivariatea Univariate‘ Variation D.F. F P < F p < Constant (M) 1 1559.00 .0001 SVTb 610.32 .0001 MATC 2566.03 .0001 Grade (A)d 1 8.75 .0003 eliminating M SVT 5.29 .0225 MAT 12.00 .0007 Sex (B) 1 1.99 .1393 eliminating SVT .04 .8447 M and A MAT 3.95 .0483 Grade x Sex 1 .15 .8645 eliminating SVT .19 .6626 M,A, and B MAT .10 .7559 Between Groups 4 Within Groups 215 SVT: MSe = 23.73 __ MAT: MSe = .94 Total 219 __ aMultivariate D.F. = 2 and 214. bSVT--Spatial Visualization Test. CMAT-~Mathematics Attitude Scale. dA reordering of the main effects, testing Grade (A) eliminating M and B resulted in multivariate F = 8.93 and P < .0002. 98 five and six at the .05 level; however, no significant difference was found between the MGMP SVT scores of female students in grades five and six. Both comparisons, between MAT mean scores of male students in grade five and six and between MAT mean scores of female students in grade five and six, were found to be significantly different at the .05 level. To summarize, based on the statistical analysis of the data from Site 1, the following decisions were made with respect to hypotheses H01-H03 (starting with the interaction): 1. The multivariate null hypothesis (H03) of no interaction of grade by sex was not rejected. 2. The multivariate null hypothesis (H02) of no difference between boys and girls in grades five and six was not rejected. 3. The multivariate null hypothesis (H01) of no difference between grade levels five and six was rejected at the .001 level. The corresponding univariate hypothesis for the SVT mean scores was rejected at the .05 level, and for the MAT mean scores was rejected at the .001 level. Site 2 1. The design for Site 2 was 2x2 crossed with two criterion measures. The multivariate null hypotheses to be tested within this design were 99 H01: There will be no difference between the mean scores of the two grade levels tested, six and seven, on both the Middle Grades Mathe- matics Project Spatial Visualization Tests and on the Mathematics Attitude Scale. H02: There will be no difference between the mean scores for boys and for girls in grades six and seven on both the Middle Grades Mathe- matics Project Spatial Visualization Test and on the Mathematics Attitude Scale. H03: There will be no interaction of grade by sex between the mean scores for sixth and seventh grade students on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathemacits Attitude Scale. Table 4.3 provides means and standard deviations of the MGMP Spatial Visualization Test (SVT) and the Mathematics Attitude Scale (MAT) scores for Site 2 by grade and by sex. The mean scores from Table 4.3 are used in Figures 4.5 and 4.6 to show profiles for grade level by sex for the MGMP SVT and MAT measures. In Figures 4.7 and 4.8 the same means are used to illustrate profiles of sex by grade levels for the above two measures. Table 4.4 includes a summary of Multivariate and Uni- variate analysis of variance for the 2x2 design of Site 2. There was no interaction of grade by sex. The multivariate tests of sex and grade main effects were significant at the .01 level. For the sex main effects, only the univariate test of the MGMP SVT scores was significant at the .01 level. For the grade main effects, only the univariate test of the MAT scores was significant at the .01 level. Sixth and seventh grade males performed on the MGMP SVT sig- nificantly higher than sixth and seventh grade females. 100 TABLE 4.3 MEANS AND STANDARD DEVIATIONS OF MGMP SVT AND MAT SCORES FOR SITE 2 BY GRADE AND BY SEX Grade 6 Boys Girls Grade 7 Boys Girls Total Boys Girls 274 150 124 153 71 82 427 221 206 D. .95 .48 .13 .89 .27 .37 .93 42 22 MGMP SVTa M S. 12.15 5 12.91 6 11.23 5 12.90 5 14.07 6 11.88 5 12.42 5 13.28 6. 11.49 5. MATb 3.297 3.259 3.343 3.027 3.039 3.017 3.200 3.188 3.213 S.D. .900 .939 .851 .871 .879 .869 .898 .924 .871 aMGMP SVT--Middle Grades Mathematics Project Spatial Visualization Test. (Range 0-32). bMAT--Mathematics Attitude Scale. (Range 1-5). 101 MGMPSVTMEANSCORES 6 7 GRADE Fig. 4.5 MGMP Spatial Visualization Test (SVT) means—- Profiles of grade levels by sex at Site 2 MATMEANSCORES 6 7 GRADE Fig. 4.6 Mathematics Attitude Scale (MAT) means-- Profiles of grade levels by sex at Site 2 102 MGMPSVTMEANSCORES G B Fig. 4.7 MGMP Spatial Visualization Test (SVT) means-- Profiles of sex by grade levels at Site 2 MATMEANSCORES G B Fig. 4.8 Mathematics Attitude Scale (MAT) means-- Profiles of sex by grade levels at Site 2 103 Seventh graders scored on the MAT significantly lower than the sixth graders. In order to further investigate the results, Scheffé's Post Hoc procedures were employed to test sex main effects with respect to the MGMP SVT means and for grade main effects with respect to the MAT means. Appendix F includes summary tables of the Scheffé's posteriori comparisons of sex within each grade level for the MGMP SVT means, as well as the comparisons of grade levels for males and for females for the MAT means. The Post Hoc tests showed significant difference between the MGMP SVT means scores of boys and girls in each grade level tested, six and seven. The Post Hoc comparisons of MAT means indicated that the drop in attitudes from grade six to seven was significant for girls but not for boys. To summarize, based on the statistical analysis of the data from Site 2, the following decisions were made with respect to hypotheses H01-H03 (starting with the interaction): 1. The multivariate null hypotheses (H03) of no interaction of grade by sex was not rejected. 2. The multivariate null hypothesis (H02) of no difference between boys and girls in grades six and seven was rejected at the .01 level. The corresponding univariate null hypothesis for the SVT mean scores was rejected at the .01 level while the 104 TABLE 4.4 A SUMMARY OF MULTIVARIATE AND UNIVARIATE ANALYSIS OF VARIANCE FOR THE 2X2 DESIGN OF SITE 2 Source of Multivafiate -Ufiivariate Variation D.F. F P < P p < Constant (M) 1 3170.25 .0001 SVTb 1904.31 .0001 MATC 5516.26 .0001 Grade (A)d 1 6.35 .0020 eliminating M SVT 1.61 .2050 MAT 9.02 .0029 Sex (B) l 6.05 .0026 eliminating SVT 10.16 .0013 M and A MAT .29 .5924 Grade x Sex 1 .22 .8012 eliminating SVT .19 .6636 M,A, and B MAT .34 .5596 Between Groups 4 Within Groups 423 SVT: MSe = 34.55 MAT: MSe = .79 Total 427 aMultivariate D.F. = 2 and 422. bSVT--Spatial Visualization Test. CMAT-~Mathematics Attitude Scale. dA reordering of the main effects, testing Grade (A) eliminating M and B resulted in multivariate F = 7.01 and P < .0011. 105 univariate null hypothesis for the MAT mean scores was not rejected. 3. The multivariate null hypothesis (H01) of no difference between grade levels six and seven was rejected at the .01 level. The corresponding univariate null hypothesis for the MAT mean scores was rejected at the .01 level while the univariate null hypothesis for the SVT mean scores was not rejected. Site 3 The design for Site 3 was 4x2 crossed with two criterion measures. The multivariate null hypotheses to be tested within this design were H01: There will be no difference among the mean scores for each of the four grade levels tested, five, six, seven, and eight, on both the Middle Grades Mathematics Projects Spatial Visualization Test and on the Mathematics Attitude Scale. H02: There will be no difference between the mean scores for boys and for girls in grades five through eight on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. H03: There will be no interaction of grade by sex among the mean scores for fifth through eighth grade students on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathematics Attitude scale. Table 4.5 provides means and standard deviations of the MGMP Spatial Visualization Test (SVT) and the Mathematics Attitude Scale (MAT) scores for Site 3 by grade and by sex. The mean scores from Table 4.5 are used in Figure 4.9 and 106 4.10 to show profiles for grade levels by sex for the MGMP SVT and MAT measures. In Figures 4.11 and 4.12 the same means are used to illustrate profiles of sex by grade levels for the above two measures. Table 4.6 includes a summary of Multivariate and Univariate analysis of variance for the 4x2 design of Site 3. Three planned comparisons were used for the grade main effects. The multivariate tests of grade by sex interac- tion, sex main effects and all three planned comparisons for grade levels were significant at the .05, .001 and .001 level respectively. The presence of significant interaction of grade by sex indicates that generalization about one main effect does not apply across levels of the other design factor. However, the profiles of grade levels by sex with respect to MGMP SVT (Figure 4.9) indicate an ordinal inter- action. Consequently, further exploration of the data was needed. The interactions of L1 x Sex, L2 x Sex, and L3 x Sex (L1, L2, and L3 indicate planned comparisons of grade levels defined in Table 4.6) were tested separately with the appro- priate degrees of freedom for the multivariate (2 and 672) and for the univariate (1 and 673) analysis (the results are included in Appendix G, Table G.1). The multivariate test of L3 x Sex interaction was significant (F=5.49) at the .01 level while the L1 x Sex (F=2.15) and L2 x Sex (F=.56) were not significant. Thus, the grade x sex 107 TABLE 4.5 MEANS AND STANDARD DEVIATIONS OF MGMP SVT AND MAT SCORES FOR SITE 3 BY GRADE AND BY SEX MGMP SVTa MATb N M S.D. M S.D. Grade 5 94 8.81 4.57 3.685 .941 Boys 48 9.58 4.49 3.788 1.016 Girls 46 8.00 4.56 3.577 .852 Grade 6 208 10.17 4.86 3.044 .907 Boys 104 11.31 5.18 3.064 .952 Girls 104 9.03 4.24 3.024 .863 Grade 7 170 11.17 5.12 2.858 .945 Boys 90 11.76 5.47 2.696 .992 Girls 80 10.50 4.63 3.061 .854 Grade 8 209 12.97 5.87 2.802 .873 Boys 96 15.27 6.26 2.836 .959 Girls 113 11.02 4.73 2.774 .796 Total 681 11.10 5.40 3.016 .953 Boys 338 12.33 5.82 3.007 1.031 Girls 343 9.89 4.65 3.025 .870 aMGMP SVT--Midd1e Grades Mathematics Project Spatial Visualization Test. (Range 0—32). bMAT--Mathematics Attitude Scale. (Range 1-5). 108 Boys MGMPSVTMEANSCORES Fig. 4.9 MGMP Spatial Visualization Test (SVT) means-- Profiles of grade levels by sex at Site 3 MATMEANSCORES 5 6 7 8 Fig. 4.10 Mathematics Attitude Scale (MAT) means-— Profiles of grade levels by sex at Site 3 109 Grade 5 - ............ MGMPSVTMEANSCORES Fig. 4.11 MGMP Spatial Visualization Test (SVT) means--Profiles of sex by grade levels at Site 3 MATMEANSCORES Fig. 4.12 Mathematics Attitude Scale (MAT) means-- Profiles of sex by grade levels at Site 3 1.... A SUMMARY OF MULTIVARIATE AND UNIVARIATE ANALYSIS 110 TABLE 4.6 OF VARIANCE OF THE 4x2 DESIGN OF SITE 3 Source of Variation D.F. Constant (M) 1 Grade (A)d 3 eliminating M L1=(5,6)vs.(7,8) (1) L2= 5 vs. 6 (1) L3= 7 vs. 8 (1) Sex (B) 1 eliminating M and A Grade x Sex 3 eliminating M,A, and B Between Groups 8 Within Groups 673 Total 681 aMultivariate D.F. Multivariatea "_F______P—? 4707.91 .0001 44.15 .0001 21.11 .0001 6.87 .0012 23.23 .0001 2.73 .0122 2 and 672. bSVT--Spatial Visualization Test. CMAT—-Mathematics Attitude Scale. Univariate E p < SVTb 3324.55 .0001 MATC 7511.13 .0001 SVT 38.47 .0001 MAT 35.20 .0001 SVT 5.16 .0235 MAT 31.67 .0001 SVT 12.10 .0006 MAT .48 .4879 SVT 44.19 .0001 MAT .16 .6807 SVT 3.00 .0300 MAT 2.79 .0400 SVT: MSe = 25.37 MAT: MSe = .82 dA reordering of the main effects, testing Grade (A) eliminating M and B resulted in multivariate F values of 44.76, 21.26, and 8.22 for L1, L2, and L3 respectively and about the same P values as before. 111 interaction was due to the contrast of grade 7 vs. 8 interaction with sex. Scheffé's Post Hoc tests were employed to determine whether the sex difference between the means of each measure was significant within each grade level as well as to compare across grade levels between males' means and between females' means. Appendix H includes summary tables of the Scheffé's posteriori comparisons of sex within each grade level and also comparisons of grade levels for males and for females on each measure, the MGMP SVT and MAT. The Post Hoc tests showed that in sixth and eighth grade the contrasts between the MGMP SVT means of boys and girls were signifi- cant at the .05 level (Table H.l); at the other grade levels no significant contrasts were found. As expected (from Table 4.6), none of the contrasts between the MAT means of boys and girls within each of the grade levels five through eight was significant (Appendix H, Table H.2). The only one that was close to significance at the .05 level was the sex contrast within grade seven. The Post Hoc tests across grade levels for sex resulted in one significant contrast between the MGMP SVT means of grade eight male and grade seven males (Table H.3). In addition, two significant contrasts were found between the MAT means of grade six males and grade five males; and between grade six females and grade five females (Table H.4). The drop in attitudes toward mathematics from grade five to grade six 112 was significant because both sexes scored significantly lower in sixth grade than in fifth grade. To summarize, based on the statistical analysis of the data from Site 3, the following decisions were made with respect to hypotheses H01-H03 (starting with the interaction): 1. The multivariate null hypothesis (H03) of no interaction of grade by sex among the mean scores for fifth through eighth grade students was rejected at the .05 level. A further analysis revealed that the interaction was significant only for the contrast of grade seven vs. eight. This result should be considered in generalizations about the significance of sex or grade main effects across levels of the other design factor. The multivariate null hypothesis (H02) of no difference between the mean scores for boys and for girls in grades five through eight was rejected at the .001 level. The corresponding univariate hypothesis for the MGMP SVT mean scores was also rejected at the .001 level, while the univariate hypothesis for the MAT mean scores was not rejected. The multivariate null hypothesis (H01) of no difference among mean scores for each of the four grade levels tested, five, six, seven, and eight, was rejected. All three planned comparisons between 113 grade levels: five and six vs. seven and eight, five vs. six, and seven vs. eight, were significant at the .001, .001 and .01 level, respectively. The corresponding univariate hypotheses for the MGMP SVT and for the MAT scores were each significant except for the MAT scores for the seven vs. eight compar- ison. Comparison of Sixth Grade Data Among Sites 1,2, and 3 The sixth grade was the only grade level in common for all three sites participating in this investigation. In each site the entire sixth grade population was tested. Thus, even though comparison of data across sites was not planned a priori, this common grade level allowed the inves- tigator to determine differences in spatial visualization skills and in attitudes toward mathematics by site and by sex for students in grade six prior to an instructional intervention. The design was 3x2 crossed with two criterion measures. The Multivariate Model with a Two-Way fixed effects analysis of variance was selected. Table 4.7 provides means and standard deviations of the MGMP Spatial Visualization Test (SVT) and the Mathematics Attitude Scale (MAT) scores for the sixth graders in the entire sample by site and sex. The mean scores from Table 4.7 are used in Figures 4.13 and 4.14 to show profiles for sites by sex for the MGMP SVT and the MAT measures. In 114 Figures 4.15 and 4.16 the same means are used to illustrate profiles of sex by site for the above two measures. Table 4.8 includes a summary of Multivariate and Univariate analysis of variance for the 3x2 design of the sixth graders in the three Sites 1, 2, and 3. Two planned comparisons were used for the site main effects: Site 1 vs. 3 and Site 2 vs. 3. These planned comparisons were chosen taking into account the relative population characteristics that showed Site 3 to be in the middle between Site 1 and 2. The multivariate test of site x sex interaction was not significant, while the sex main effects and the two planned comparisons of sites were significant at the .001 level. The univariate tests of SVT for sex main effects and for the planned comparisons all were significant at the .001 level. The only univariate test of MAT found to be significant at the .05 level was for the comparison of Site 2 vs. 3. Scheffe's Post Hoc tests were employed to further explore the data. Appendix I includes summary tables of the Scheffe's posteriori comparisons of the MGMP SVT means of sex within each site and also comparisons between sites for males and for females on each measure, the MGMP SVT and MAT. The Post Hoc tests showed significant sex contrasts in Site 2 and 3 (Table 1.1), results which are in agreement with previous separate analyses of Sites 1, 2,and 3 (previous section in this chapter). The MGMP SVT means of males and 115 TABLE 4.7 MEANS AND STANDARD DEVIATIONS OF MGMP SVT AND MAT SCORES FOR THE SIXTH GRADERS IN THE ENTIRE SAMPLE BY SITE AND BY SEX MGMP SVTa MATE N M S.D. M S.D. Site 1 115 8.85 5.21 3.098 .917 Boys 58 9.05 5.15 2.950 1.058 Girls 57 8.65 5.32 3.249 .726 Site 2 274 12.15 5.95 3.301 .898 Boys 150 12 91 6.48 3 259 .939 Girls 124 11.23 5.13 3.353 .848 Site 3 208 10 17 4.86 3.040 .890 Boys 104 11.31 5.18 3.050 .917 Girls 104 9.03 4.24 3.029 ____ .866 Total 597 10.84 5.61 3.171 .906 Boys 312 11.67 6.00 3.132 .960 Girls 285 9.93 5.00 3.214 .841 aMGMP SVT--Middle Grades Mathematics Project Spatial Visualization Test. (Range 0-32). bMAT--Mathematics Attitude Scale. (Range 1-5). 116 ..A —L _A (O -‘ 00 U] MGMPSVTMEANSCORES Q SITE 1 SITE 2 SITE3 Fig. 4.13 MGMP Spatial Visualization Test (SVT) means of sixth graders--Profiles of Sites 1,2, and 3 by sex MATMEANSCORES SITE 1 SITE 2 SITE 3 Fig. 4.14 Mathematics Attitude Scale (MAT) means of sixth graders--Profiles of Sites 1,2, and 3 by sex ll7 MGMP SVT MEAN SCORES to 3 IS a N Fig. 4.15 MGMP Spatial Visualization Test (SVT) means of sixth graders--Profiles of sex by site SITE 1 ——————— SITE 2 — -— —— SITE 3 MKTMEANSCORES G B Fig. 4.16 Mathematics Attitude Scale (MAT) means of sixth graders--Profiles of sex by site 118 TABLE 4.8 SUMMARY OF MULTIVARIATE AND UNIVARIATE ANALYSIS OF VARIANCE FOR THE 3X2 DESIGN OF SIXTH GRADERS IN THE ENTIRE SAMPLE Source of Variation Constant (M) Site (A)d eliminating M L1 = 1 vs. 3 L2 = 2 vs. 3 Sex (B) eliminating M and A Site x Sex Between Groups Within Groups Total D.F. l (1) (1) 6 591 597 aMultivariate D.F. = Multivar iatea Univariate F P < F P < 4328.91 .0001 SVTb 2388.07 .0001 MATC 7415.06 .0001 9.64 .0001 SVT 19.25 .0001 MAT .94 .3340 10.95 .0001 SVT 15.42 .0001 MAT 9.93 .0018 8.47 .0001 SVT 13.43 .0003 MAT 1.60 .2058 .99 .4128 SVT 1.13 .3226 MAT 1.17 .3113 SVT: MSe = 29.28 MAT: MSe = .81 2 and 590. bSVT-~Spatial Visualization Test. CMAT--Mathematics Attitude Scale. dA reordering of the main effects, testing Grade (A) eliminating M and B resulted in multivariate F values of 9.32, and 10.54 for L1, and L2, and about the same P values as before. 119 of females from Site 1 were significantly (at the .05 level) lower than those of the corresponding males and females from Site 2 (Table I.2). The significant difference between Site 1 and 3 on the MGMP SVT was due to significant contrast between the males from these sites. The significant difference between Site 2 and 3 on the MGMP SVT was due to a significant contrast between the females from these sites; however the contrast between males from Site 2 and 3 was nearly significant at the .05 level (Table 1.2). The attitude Post Hoc showed only one significant contrast between females from Site 2 and 3, although the correspond- ing male contrast was nearly significant at the .05 level (Table 1.3). To summarize, based on the statistical analysis of the sixth grade data in the entire sample it was concluded that there were significant site differences on the MGMP SVT. Sixth graders from Site 2 performed on the MGMP SVT significantly better than their counterparts from Site 1 and 3. A similar pattern was found between Sites 3 and l in favor of Site 3. Sex differences on the MGMP SVT were significant within sites and the Scheffe's Post Hoc comparisons Showed that they existed in Site 2 and 3 as expected from the previous analyses in each site separately. With respect to MAT means, significant difference was found only between Sites 2 and 3 in favor of Site 2. 120 The Effects of Instruction The Multivariate Analysis of Repeated Measures was used to analyze the effects of instruction on spatial visualiza- tion skills and on differences in spatial visualization and attitudes toward mathematics by sex and by grade level. Only a subsample from Site 3 participated in this part of the study. The design for the subsample from Site 3 was a Two-Way six—group design with four measures on each subject. The multivariate null hypotheses to be tested within this design were H04: There will be no difference between the posttest means and the pretest means of sixth, seventh, and eighth grade students on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. H05: There will be no difference between the mean gain scores (posttest minus pretest) for each of the three grade levels tested, six, seven, and eight, on both the Middle Grades Mathema- tics Project Spatial Visualization Test and on the Mathematics Attitude Scale. H06: There will be no difference between the mean gain scores (posttest minus pretest) for boys and for girls in grades six, seven, and eight on the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathe- matics Attitude Scale. Table 4.9 provides pre— and post—test means of the MGMP Spatial Visualization Test (SVT) scores and the Mathematics Attitude Scale (MAT) scores for the subsample from Site 3 by grade and by sex. The corresponding standard deviations are given in Appendix J. The MGMP SVT means from Table 4.9 are used in Figures 4.17, 4.18, and 4.19 to Show profiles for 121 pretest-posttest by grade, by sex, and by grade and sex. Since the change and differences in attitudes appear to be very small, no profiles are illustrated with respect to the MAT means. Table 4.10 includes a summary of Multivariate and Univariate Analysis of Repeated Measures for the subsample from Site 3. The multivariance test of the SVT Constant and MAT Constant is of little interest since those terms are collapsing over time (averaging pre- and post—test); however, significant values of the Constant in the rows of sex or grade main effects indicate on the overall that sex differences or grade level differences do exist. The test of SVT Linear and MAT Linear is a test of significance for the time effect (in our case the gain from pretest to posttest) and actually it is a test of an independent variable by time interaction. The results of the Multivariate and Univariate analysis of Repeated Measure (Table 4.10) show no significant inter- action of grade by sex by time, or sex by time. The test did show significant interaction of planned comparisons of grade levels by time. G1 X T was significant (due to SVT Linear) at the .001 level, and G2 x T was significant (also due to SVT Linear) at the .01 level. The interpreta- tion is that from the pretest to posttest grades seven and eight gained on the SVT significantly higher than grade six, while grade seven gained on the SVT significantly higher than grade eight. The test of the grand mean of SVT Linear 122 TABLE 4.9 PRE- AND POST-TEST MEANS OF MGMP SVT AND MAT SCORES FOR THE SUBSAMPLE FROM SITE 3 BY GRADE AND BY SEXa MGMP SVTb MATC Pretest Posttest Pretest Posttest N M M M L Grade 6 108 11.04 16.96 3.094 3.153 Boys 54 12.48 18.83 3.156 3.265 Girls 54 9.59 15.09 3.032 3.040 Grade 7 142 11.22 20.02 2.918 2.947 Boys 74 11.74 21.00 2.761 2.815 Girls 68 10.64 18.96 3.089 3.091 Grade 8 108 13.23 20.56 2.779 2.737 Boys 79 15.98 23.06 2.826 2.753 Girls 101 12.22 18.60 2.742 2.725 Total 430 12.01 19.48 2.904 2.911 Boys 207 13.48 21.22 2.889 2.909 Girls 223 10.65 17.86 2.918 2.913 8The standard deviations are given in Appendix J. bMGMP SVT--Middle Grades Mathematics Project Spatial Visualization Test. (Range 0-32). CMAT-~Mathematics Attitude Scale. (Range 1-5). 123 MGMPSVTMEANSCORES PRE POST Fig. 4.17 Pretest—posttest means on the MGMP Spatial Visualization Test (SVT) by grade MGMPSVTMEANSCORES PRE POST Fig. 4.18 Pretest-posttest means on the MGMP Spatial Visualization Test (SVT) by sex 124 Girls Grade 6 --------- Girls Grade 7 ------ Girls Grades — — — Boys Grade 6 Boys Grade 7 I—-————I 25 Boys Grade 8 A———A MGMP SVT MEAN SCORES PRE POST Fig. 4.19 Pretest-posttest means on the MGMP Spatial Visualization Test (SVT) by grade and sex 125 TABLE 4.10 A SUMMARY OF MULTIVARIATE AND UNIVARIATE ANALYSIS OF REPEATED MEASURES FOR THE SUBSAMPLE FROM SITE 3 _ r—‘r—r-———-~—-—-:::::::::::::;:::;::::_ Source of Multivariatea Univariate Variation D.F. F P < F P < SVT CONS . . Grand Mean 1 2082.52 .0001 SVT LINC 1080.91 .0001 (M) MAT GONsd 5928.85 .0001 MAT LINe .05 .8195 Grade (A)f 2 eliminating M G1=6 vs.(7,8) (1) SVT CONS 16.51 .0001 12.32 .0001 SVT LIN 15.41 .0002 MAT CONS 10.95 .0011 MAT LIN 1.01 .3160 02= 7 vs. 8 (l) SVT CONS 4.84 .0284 5.71 .0002 SVT LIN 7.73 .0057 MAT CONS 3.94 .0479 MAT LIN 1.01 .3150 Sex (B) 1 SVT CONS 41.96 .0001 eliminating 11.70 .0001 SVT LIN 1.11 .2927 M and A MAT CONS .19 .6656 MAT LIN .11 .7415 Grade by Sex 2 SVT CONS 3.22 .0410 eliminating M, 1.84 .0660 SVT LIN .62 .5388 A, and B MAT CONS 3.33 .0368 MAT LIN .60 .5508 Bet. Groups 6 W/in Groups 424 SVT CONS 106.58 Univariate SVT LIN 22.17 MS error MAT CONS 2.45 MAT LIN .39 aMultivariate D.F. = 4 and 421. bSpatial Visualization Test averaging over the time dimension (Pre+Posttest). CSVT Time effect over subjects. dMathematics Attitude Scale averaging over the time dimension (Pre+Posttest). eMAT Time effect over subjects. fA reordering of the main effects, testing grade (A) eliminating M and B resulted in multivariate F values of 12.70 and 6.41 for G1 and 62 planned comparisons, and the same P values as before. 126 is a test of time effect over all subjects. This test was signficant at the .001 level, Showing significant difference between the posttest means and the pretest means of the subjects on the MGMP SVT but not on the MAT. None of the univariate tests of MAT Linear was significant. This was because the MAT scores showed no significant change of atti- tudes toward mathematics from pretest to posttest. A fur- ther analysis of the data which includes step-down tests of the transformed repeated measures is given in Appendix K. To conclude, hypothesis H04 was rejected at the .001 level due to significant difference between the post- test means and the pretest means of sixth, seventh, and eighth grade students on the MGMP SVT; however, no signifi— cant difference was found between the MAT means. Hypothesis H05 also was rejected at the .001 level. There was evidence of significantly different mean gain scores among the grade levels, six, seven, and eight, on the MGMP SVT. Hypothesis H06 was rejected. The conclusion was that there was no significant difference in the mean gain scores for boys and for girls in grades six, seven, and eight on both the MGMP SVT and MAT. Comparison of Attitudes Toward Mathematics With ___—_AfEIEEHE§_T5WEEH_SEEEI§I—Visualization To compare by sex and by grade the attitudes toward mathematics and spatial visualization after the instruction, a Two—Way Anova design was employed. The null hypotheses to be tested within this design were 127 H07: There will be no difference between students' mean scores on the Mathematics Attitude Scale and students' mean scores on the Spatial Visualization Attitude Scale. H08: There will be no interaction of grade by sex among the mean difference scores--Mathematics Attitude Scale score minus Spatial Visualiza- tion Attitude Scale socre. Table 4.11 provides means and standard deviations of MAT and SVAT posttest scores for the subsample from Site 3 by grade and by sex. Table 4.12 includes a summary of analysis of variance for sex by grade level for the results of the test on the mean difference between the MAT and SVAT scores. The ANOVA results indicated that all three F values of the interaction and the two main effects were not significant at the .05 level. To conclude, both null hypotheses H07 and H08 were not rejected. The differences between the mean scores of Spatial Visualization Attitute Scale and Mathematics Attitude Scale by grade or by sex level were not significant. Sex and Grade Level Differences in Attitudes Toward Spatial VisuaIizatlon For the purpose of examining sex and grade level differences in attitudes toward spatial visualization after the instruction, a Two-Way ANOVA design was used. The hypotheses to be tested within this design were H09: There will be no difference between the mean scores for boys and for girls in grades six, seven, and eight on the Spatial Visualization Attitude Scale. 128 H10: There will be no difference between the mean scores for each of the three grade levels tested, six, seven, and eight, on the Spatial Visualization Attitude Scale. Table 4.11 includes the means and standard deviations of the SVAT scores (the right side of the table) by grade and by sex. Table 4.13 includes a summary of analysis of variance for sex by grade level for the results of the test on attitudes toward spatial visualization. The results of the analysis showed no significant grade by sex interaction. The sex main effects was also nonsignificant. Only grade main effects was significant at P<.01 level. To conclude, hypothesis H09 was not rejected while H10 was rejected at the .01 level. No sex differences in attitudes toward spatial visualization were found. However, grade level differences in attitudes toward spatial visualization were significant among grade levels six, seven, and eight. Table 4.11 indicates a drop in attitudes toward spatial visualization from grade six to grade eight. Retention Analysis To evaluate the persistence of the effects of instruc- tion in spatial visualization (retention), the Multivariate Analysis of Repeated Measures Model was used. The multi- variate null hypotheses to be tested within this design, were MEANS AND STANDARD DEVIATIONS OF MAT AND SVAT 129 TABLE 4.11 POSTTEST SCORES FOR THE SUBSAMPLE FROM Grade 6 Boys Girls Grade 7 Boys Girls Grade 8 Boys Girls Total Boys Girls 108 54 54 142 74 68 180 70 101 430 207 223 SITE 3 BY GRADE AND BY SEX 3. MATa M 153 .265 .040 .947 .815 .091 1228.1 S.D. .828 .893 .750 .883 .936 .804 SVATb .225 .131 .328 POST S.D. .840 .855 .808 .945 1.070 .782 2 .911 .909 .913 .925 .787 aMAT--Mathematics Attitude Scale. 3. .171 158 (Range 1-5). bSVAT--Spatial Visualization Attitude Scale. (Range 1-5). 130 TABLE 4.12 ANAYLSIS OF VARIANCE SUMMARY TABLE FOR MEAN DIFFERENCE BETWEEN THE MATa AND SVATb SCORES Source of Variation D.F. MS F Grade 2 .067 .077 Sex 1 .021 .024 Grade x Sex 2 .217 .249 Between Groups 5 Within Groups 424 .869 aMAT--Mathematics Attitude Scale. bSVAT—-Spatia1 Visualization Attitude Scale. TABLE 4.13 ANALYSIS OF VARIANCE SUMMARY TABLE FOR SVATa SCORES Source of Variation D.F. MS F Grade 2 6.555 8.128** Sex 1 .008 .010 Grade X Sex 2 1.754 2.175 Between groups 5 Within groups 424 .806 aSVAT--Spatial Visualization Attitude Scale. **Significant P<.01. .926 .878 .779 .001 .919 .115 131 H11: There will be no difference between the retention test means and the posttest means of sixth, seventh, and eighth grade students on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Spatial Visualization Attitude Scale. H12: There will be no difference between the mean gain scores (retention test minus posttest) for each of the three grade levels tested, six, seven, and eight, on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Spatial Visualization Attitude Scale. H13: There will be no difference between the mean gain scores (retention test minus posttest) for boys and for girls in grades six, seven, and eight on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Spatial Visualization Attitude Scale. Table 4.14 provides the post- and retention test means of the MGMP Spatial Visualization Test (SVT) and Spatial Visualization Attitudes Scale (SVAT) for the retention sub- sample from Site 3 by grade and by sex. The corresponding standard deviations are given in Appendix L. The MGMP SVT means from Table 4.14 are used in Figures 4.20, 4.21, and 4.22 to show profiles from post-retention test by grade, by sex, and by grade and sex. Table 4.15 includes a summary of Multivariate and Univariate Analysis of Repeated Measures for the retention subsample from Site 3. As mentioned earlier (in the section on the effects of the instruction) the SVT Linear and SVAT Linear which indicate the time effect are the multivariate tests of interest. The multivariate test of grade by sex interaction was not significant. The multivariate tests of sex main effects and the planned comparison G1 (grade 6 132 TABLE 4.14 POST- AND RETENTION TEST MEANS OF MGMP SVT AND SVAT SCORES FOR THE RETENTION SUBSAMPLE FROM SITE 3 BY GRADE AND BY SEXa Grade 6 Boys Girls Grade 7 Boys Girls Grade 8 Boys Girls Total Boys Girls 52 26 26 79 41 38 107 45 62 238 112 126 MGMP SVTb Posttest Retention M _ M 17.11 18.65 18.54 19.50 15.70 17.81 20.06 20.71 20.93 21.46 19.13 19.89 21.23 22.36 24.40 25.02 18.94 20.42 19.95 21.00 21.77 22.44 18.33 19.72 aThe standard deviations are given in Appendix L. SVATC Posttest Retention M M 3.356 3.107 3.404 3.044 3.308 3.171 3.215 2.876 3.033 2.781 3.412 2.978 2.933 2.861 2.989 2.970 2.893 2.782 3.119 2.920 3.101 2.918 3.135 2.921 bMGMP SVT—-Middle Grades Mathematics Project Spatial Visualization Test. (Range 0-32). CSVAT-—Spatia1 Visualization Attitude Scale. (Range 1-5). 133 27 Grade 6 —————— 25 Grade 7 —— — — — Grade 8 MGMPSVTMEANSCORES POST RETENHON Fig. 4.20 Post—retention test means on the MGMP Spatial Visualization Test (SVT) by grade 27 25 Boys 23 Girls — — — MGMPSVTMEANSCORES POST RETENTION Fig. 4.21 Post-retention test means on the MGMP Spatial Visualization Test (SVT) by sex 134 Girls Grade 6 ----- — Girls Grade 7 — — _ _ Girls Grade 8 — — Boys Grade 6 —— Boys Grade 7 l——._l 27 Boys Grade 8 A——a MGMP SVT MEAN SCORES POST RETENTION Fig. 4.22 Post-retention test means on the MGMP Spatial Visualization Test (SVT) by grade and sex ,.,.._—7777W 135 vs. grades 7,8) were significant. This was due to the existence of sex and grade level differences on the sum of post- and retention test, but not due to the effects of time since both interactions of sex by time and G1 by time were not significant. However, the multivariate test of the planned comparison 02 (grade seven vs. eight) was signifi- cant and the univariate tests showed the only significant interaction to be Oz by time for the SVAT measure. Table 4.14 indicates a greater drop on the SVAT mean score for grade seven than grade eight. The multivariate test of the grand mean of SVT Linear and SVAT Linear indicated significant time effect at the .001 level. From post to retention test the performance on the SVT was significantly higher, but the attitude toward spatial visualization was significantly lower. A further analysis of the retention data which includes step-down tests of the transformed repeated measures is given in Apendix M. In conclusion, hypothesis H11 was ejected at the .001 level due to a significant time effect on both mea- sures, SVT and SVAT. Hypothesis H12 was rejected only for the planned comparison G2 (grade 7 vs. 8) due to the effects of time on the SVAT results. However, hypothesis H13 was not rejected. The conclusion was that there was no significant sex by time effect on either measure, the SVT or SVAT. 136 TABLE 4.15 A SUMMARY OF MULTIVARIATE AND UNIVARIATE ANALYSIS OF REPEATED MEASURES FOR THE RETENTION SUBSAMPLE FROM SITE 3 Source of Multivariatda Univariate Variation D.F. F P < F P < SVI CONSb 3003.08 .UUOI Grand Mean 1 1279.22 .0001 SVT LINC 21.04 .0001 (M) SVAT CONSd 3700.00 .0001 SVAT LINe 22.36 .0001 Grade (A)f 2 eliminating M G1=6 vs.(7,8) (1) SVT CONS 13.42 .0004 7.31 .0001 SVT LIN 1.24 .2670 SVAT CONS 5.10 .0250 SVAT LIN .38 .5374 Gz= 7 vs. 8 (1) SVT CONS 2.71 .1009 3.36 .0108 SVT LIN .82 .3667 SVAT CONS 1.70 .1940 SVAT LIN 7.69 .0060 Sex (B) l SVT CONS 19.26 .0001 eliminating 6.36 .0001 SVT LIN 2.38 .1245 M and A SVAT CONS .14 .7095 SVAT LIN .39 .5353 Grade by Sex 2 SVT CONS 2.16 .1171 eliminating M, 1.19 .3029 SVT LIN .31 .7347 A, and B SVAT CONS 1.78 .1706 SVAT LIN 1.61 .2029 Bet. Groups 6 W/in Groups 232 SVT CONS 132.87 Univariate SVT LIN 12.58 MS error SVAT CONS 2.35 SVAT LIN .42 aMultivariate D.F. = 4 and 229. bSpatial Visualization Test averaging over the time dimension (Post+Retention). CSVT Time effect over subjects. dSpatial Visualization Attitude Scale averaging over the time dimension (Post+Retention). eSVAT Time effect over subjects. fA reordering of the main effects, testing grade (A) eliminating M and B resulted in multivariate F values of 7.69 and 3.79 for G and G2 planned comparisons, and the same P values as be ore. 137 Correlation Analysis Thus far, the data collected from the subsample from Site 3 have been used to determine the effects of instruc— tion in activities involving spatial visualization tasks on both performance and attitudes toward mathematics as well as to compare attitudes toward mathematics and spatial visuali- zation by grade and by sex. In addition, the review of the literature showed considerable interest in the relationship between achievement and attitude, and in the sex and grade level differences in the magnitude of the correlation be- tween attitude and achievement. In this section, attention has been given to the correlation coefficients between pairs of scores from the pre- post- and retention-tests. Seven measures were considered; those included pre- post- and re- tention Spatial Visualization Test, pre-post Mathematics Attitude Scale test, and post-retention Spatial Visualiza- tion Attitude Scale test. The resulting 7x7 intercorrela- tion matrix for the entire group of subjects (N=228) having pre- post- and retention test scores is given in Table 4.16. In addition, Table 4.16 includes the correlation coeffi- cients for each sex level. Appendix N includes the inter- correlation matrix by grade level, six, seven, and eight. The high correlation coefficients between SVT posttest (2) and SVT pretest (l), SVT retention test (3) and SVT posttest (2), MAT posttest (5) and MAT pretest (4), and between SVAT retention test (7) and SVAT posttest (6) indicate the relatively high test-retest reliability for SVTb Pretest (1) SVT Posttest (2) SVT Retention test (3) MATC Pretest (4) MAT Posttest (5) SVATd Posttest (6) SVAT Retention test (7) 138 TABLE 4.16 PEARSON CORRELATION MATRIX FOR THE RETENTION SUBSAMPLE BY SEXa Total Boys Girls Total Boys Girls Total Boys Girls Total Boys Girls Total Boys Girls Total Boys Girls Total Boys Girls (1) 1.000 1.000 1.000 .718 .735 .673 .691 .752 .605 .159 .225 .092 .159 .157 .186 .136 .193 .072 .253 .349 .129 (2) .846 .855 .822 .196 .234 .185 .149 .185 .160 .216 .331 .121 .343 .419 .284 aTotal retention subsample N = #Girls = 122. (3) .175 .236 .135 .121 .181 .100 .176 .284 .083 .259 .338 .188 (4) .730 .781 .672 .262 .198 .344 .225 .147 .331 228, #Boys = bSVT--Spatial Visualization Test. CMAT—-Mathematics Attitude Scale. dSVAT-~Spatial Visualization Attitude Scale. (6) (5) .428 .313 .516 .294 .176 .437 106, .707 .747 .657 139 each instrument. All those coefficients were significant at the .001 level; taking into consideration the Bonferroni inequality for simultaneous multiple test (Morrison, 1976, pp. 116-20) they are still significant at the .05 level. Comparing the test-retest correlation coefficients for boys and for girls, Table 4.16 shows in every case higher values for boys than girls. The correlation coefficients for the entire subsample between measures of SVT and measures of MAT or SVAT--for example between (7) and (3)-—all were positive and significant. Those coefficients indicate the relation- ship between attitudes and achievement as measured by MAT (or SVAT) and SVT. It is obvious from Table 4.16 that the correlations between SVT and the attitude measures were in general higher for boys than girls. In summary, one must be careful not to imply that any of the foregoing significant correlations suggest cause— effect relationships. However, they do pose questions that could possibly be answered by further research. Summary The statistical analysis of the data collected during this investigation has been presented in this chapter. The major results with regard to sex and grade level differences in spatial visualization performance and in attitudes toward mathematics prior to instruction are summarized in Table 4.17. In addition, the comparison of the sixth grade data among Sites 1,2, and 3 indicated no significant interaction 140 of site by sex, significant sex differences on the SVT in favor of boys, and significant site differences on both the SVT and MAT measures. The Multivariate Analysis of Repeated Measures was used twice. First, to analyze the immediate effects of the instruction in spatial visualization; second to analyze the retention factor. The results of both analyses are sum- marized in Table 4.18. In comparing attitudes toward mathematics with atti- tudes toward spatial visualization by sex and by grade, none of the main effects or the interaction was significant. The analysis to examine differences in attitudes toward spatial visualization by sex and grade level revealed significant differences only for grade main effects but no sex differ- ences or interaction of grade by sex. The presentation of the statistical analysis of the data concluded with a table of Pearson Correlation Coeffi- cients between pairs of measures for the subjects who had pre- post- and retention test scores by sex and by grade level. Sex differences were found with respect to the magnitude of the correlation coefficients; in general, the value of the correlation coefficients was larger for boys than for girls. 141 TABLE 4.17 SUMMARY OF RESULTS FOR SEX AND GRADE LEVEL DIFFERENCES PRIOR TO INSTRUCTION BY SITE Site 2 Site 38 Grades 6,7 Grades 5,6,7,8 Grade x Sex SVT' interaction NS NS (8 vs.7)* NS NS (7 vs.8)* Sex (Boys vs.Gir1s)**(Boys vs. Girls)** Differences MAT . NS NS Grade level SVT <7 ,8)vs. <5 ,6)“ Differences (6 vs.5)* (8 vs.7) (5,6) .(7, ** (5vis.6)§; NS aThree planned comparisons were employed for the grade main effects at Site 3:(5,6) vs. (7,8), 5 vs. 6, and 7 vs. 8. bSVT--Spatial Visualization Test. CMAT--Mathematics Attitude Scale. *Significant at the .05 level (the left variable higher than the right variable). **Significant at the .01 level (the left variable higher than the right variable). 142 TABLE 4.18 SUMMARY OF RESULTS OF THE IMMEDIATE EFFECTS OF THE INSTRUCTION AND THE RETENTION ANALYSIS Immediate Effects of the Instruction Retention Analysis SVT NS SVATC NS SVT NS Sex by Time SVAT NS SVT ((7,8)vs.6):: SVT NS Graded by (7 vs.8) Time MAT NS SVAT NS NS (8 vs.7)** 72* *9: 7': (Post vs.Pre) NS SVT (Retention vs.Post)* SVAT (Post vs.Retention) aSVT--Spatial Visualization Test. bMAT--Mathematics Attitude Scale. CSVAT-~Spatial Visualization Attitude Scale. dTwo planned comparisons were employed for the grade main effects: 6 vs. (7,8), and 7 vs. 8. **Significant at the .01 level (the left variable higher than the right variable). CHAPTER V SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS The preceding chapters of this study were devoted to a discussion on the current significance of the problem, a delineation of its purpose, a review of the relevant liter- ature, a presentation of the methodology, and the analysis of the data. Chapter V, the final chapter of this report, is devoted to (l) a general summary of the study including the findings, (2) major conclusions and discussion, and (3) implications for educational issues and recommendations for practice and further research. Summary Purpose of the Study This study had two related purposes: the first was to determine existing differences in spatial visualization abilities and in attitudes toward mathematics of fifth through eighth grade students by sex and grade level prior to an instructional intervention. The second was twofold: first, to analyze the effects of instruction in activities involving spatial visualization tasks on the skills and attitudes toward mathematics of a sample of sixth, seventh, and eighth grade students by sex and grade level; second, 143 144 after the instruction, to compare attitudes toward mathematics and spatial visualization and to examine differences in attitudes toward spatial visualization by sex and grade level. Research Questions Since, there were two different but interrelated purposes to this study, two sets of major questions were considered. The first set focused on documenting the existence of differences in spatial visualization and attitudes toward mathematics of fifth, sixth, seventh, and eighth grade students by sex and grade level before an instructional intervention. The first set of questions included the following: 1. What effect, if any, does grade level have on performance of spatial visualization tasks and/or on attitudes toward mathematics? 2. What effect, if any, does sex have on performance of spatial visualization tasks and/or on attitudes toward mathematics? 3. Do differences between boys and girls in performance of spatial visualization tasks and/or in attitudes toward mathematics change with grade level? The second set of research questions examined the effects of instruction in activities involving spatial visualization tasks, as well as the existence of differences in attitudes toward mathematics and spatial visualization by 145 sex and grade level (sixth through eighth) after the instruction. The second set of questions included the following: 1. Will instruction in spatial visualization tasks affect the spatial visualization performance and/or attitudes toward mathematics of sixth, seventh, and eighth grade students? -Will these effects be different for boys than for girls? -Will these effects differ by grade level? 2. After the instruction, do differences exist between the attitudes toward mathematics in general and the spatial visualization activities in particular? -Will these differences exist for both sexes? -Will these differences exist for each grade level in the study? 3. After the instruction, do differences exist between the sexes in their attitudes toward spatial visualization activities? —Will these differences exist for each grade level in the study? And finally a question about retention: 4. Do the effects of the instruction on spatial visualization performance and attitudes toward spatial visualization tasks persist over time for each grade level and sex? 146 Related Research The review of the literature related to sex differences in spatial visualization skills revealed a widespread assumption that male performance on spatial Visualization is superior to female performance. However, detailed examin- ation of investigations in the field showed a certain amount of confusion about the uniformity and significance of the superiority. Some investigations have noted that sex differences on spatial visualization tests appear at early ages; others have indicated that differences do not appear until adolescence or early adulthood. Additionally, there were very few developmental studies investigating sex differences in spatial visualization over several ages (especially in the middle school) with a single test. It was noted that the theoretical explanations and hypotheses concerning the sex differences in Spatial visualization (and spatial ability) range broadly; some emphasized the contri- butions of biological factors--"nature”, others emphasized the role of learning and socialization--”nurture”. Studies of training programs to increase spatial visualization are very few. Of the existing studies, most have been done with adults. Previous research findings have indicated that a distinct controversy exists concerning the possibility of training individuals to increase their spatial visualization abilities. No large scale study to implement a unit of instruction in spatial visualization for a wide range of grade levels was found. 147 The review of the literature related to attitude studies suggested that there is a positive relationship between attitude and mathematics achievement, but there is no agreement on the magnitude and the direction of causal— ity. It was also noted that the literature suggests the existence of sex-related differences in attitudes toward mathematics in favor of males over females; however, the magnitude and specific dimensions of these differences are unclear. Methodology The study was conducted during winter 1982. Data were collected from January 20 to April 20 and included general information on the subjects and the schools which partici— pated in the study and pre- post- and retention test scores on spatial visualization tests and attitude scales. The subjects of the study attended schools in three sites in and around Lansing, the state capital of Michigan. The three sites differed in the characteristics of their population. There were 219 fifth and sixth grade students from Site 1, 427 sixth and seventh grade students from Site 2, and 681 fifth through eighth grade students from Site 3 who took part in the assessment of differences in spatial visualiza— tion skills and attitudes toward mathematics by sex and grade level prior to the instruction. There were 430 sixth, seventh, and eighth grade students from Site 3 who took part in the evaluation of the effects of instruction in 148 activities involving spatial visualization tasks and comparison of attitudes toward mathematics and spatial visualization; of the 430 students, 238 took part in the evaluation of the persistence of the effects of the instruction (retention). The instruments used included two semantic differential scales for measuring attitudes toward mathematics and toward spatial visualization (MAT and SVAT), and the Middle Grades Mathematics Project Spatial Visualization Test (MGMP SVT) for measuring the spatial visualization performance of stu- dents. The attitude scale was a six—item semantic differen— tial with five response options. Appendix B includes the Mathematics Attitude Scale (MAT) and the Spatial Visualiza— tion Attitude Scale (SVAT). The MGMP Spatial Visualization Test consisted of 32 multiple choice items and was preceded by a sample sheet which included 2 sample items to be dis- cussed in class before taking the test. Appendix C includes the Sample Items preceding the MGMP SVT and some more sample items similar to those in the MGMP SVT. The spatial visualization instruction material used in the study included ten carefully sequenced activities which required two to three weeks of instructional time. The activities involved representing three—dimensional objects in two-dimensional drawings and vice versa, constructing three-dimensional objects with blocks from their two- dimensional representations. The activities dealt with 149 flat views of buildings as well as with isometric drawings on dot paper. The spatial visualization unit used an instructional model consisting of three phases: launching, exploring, and summarizing. To aid the teacher in using the teaching model, a detailed instructional guide was provided. The hypotheses tested were as follows: H012 There will be no difference among the mean scores for each of the four grade levels tested, five, six, seven, and eight, on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. There will be no difference between the mean scores for boys and for girls in grades five through eight on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. There will be no interaction of grade by sex among the mean scores for fifth through eighth grade students on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. There will be no difference between the posttest means and pretest means of sixth, seventh, and eighth grade students on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. There will be no difference between the mean gain scores (posttest minus pretest) for each of the three grade levels tested, six, seven, and eight, on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. There will be no difference between the mean gain scores (posttest minus pretest) for boys and for girls in grades six, seven, and eight on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Mathematics Attitude Scale. 11072 The standard 150 There will be no difference between stu- dents' mean scores on the Mathematics Attitude Scale and students' mean scores on the Spatial Visualization Attitude Scale. There will be no interaction of grade by sex among the mean difference scores-—Math— ematics Attitude Scale score minus Spatial Visualization Attitude Scale score. There will be no difference between the mean scores for boys and for girls in grades six, seven, and eight on the Spatial Visualization Attitude Scale. There will be no difference between the mean scores for each of the three grade levels tested, six, seven, and eight, on the Spatial Visualization Attitude Scale. There will be no difference between the retention test means and the posttest means of sixth, seventh, and eighth grade students on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Spatial Visualization Attitude Scale. There will be no difference between the mean gain scores (retention test minus posttest) for each of the three grade levels tested, six, seven, and eight, on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Spatial Visualization Attitude Scale. There will be no difference between the mean gain scores (retention test minus posttest) for boys and for girls in grades six, seven, and eight on both the Middle Grades Mathematics Project Spatial Visualization Test and on the Spatial Visualization Attitude Scale. analyses of the data yielded the following: means, deviations, reliability and correlation coeffi— cients, multivariate and univariate analysis of variance and repeated measures, planned comparisons, and Scheffé's Post Hoc comparisons. All the analyses were carried out on the 3600 computer at the Michigan State University Computer Center using the SPSS (Statistical Package for the Social 151 Sciences) and Finn programs. The .05 level of significance was the minimum value accepted in testing the hypotheses although findings below the .05 level were reported. Findings Within the parameters of this study, the following findings are presented. First, with regard to the existence of differences in spatial visualization abilities and in attitudes toward mathematics of fifth through eighth grade students by sex and grade level prior to an instructional invervention, the following was found: 1. For hypothesis H01, the multivariate tests indicated that at each site, 1,2, and 3, the overall difference among grade levels on both the Middle Grades Mathematics Project Spatial Visualization Test (MGMP SVT) and Mathematics Attitude Scale (MAT) scores was highly significant (P<.002). Analyses of the univariate hypotheses associated with these tests yielded probabilities that indicated signif- icant grade level differences in favor of the higher grade levels on the MGMP SVT at Sites 1 and 3 (the means for grades five and six at Site 1 were 7.34 and 8.85; the means for grades five, six, seven, and eight at Site 3 were 8.81, 10.17, 11.17, and 12.97 respectively). Even though, there was no signif- icant difference on the MGMP SVT between the grade levels at Site 2, the seventh graders there also 152 scored higher than the sixth graders (12.90 vs. 12.15). With regard to the MAT scores, the univari- ate analyses indicated significant grade level differences in favor of the lower grade levels at all three sites (the means for grades five and six at Site 1 were 3.55 and 3.10; the means for grades six and seven in Site 2 were 3.30 and 3.03; the means for grades five, six, seven, and eight at Site 3 were 3.68, 3.04, 2.87, and 2.80 respectively). For hypothesis H02, the multivariate tests indicated that at each site, 2 and 3, the overall difference between boys and girls on both the MGMP Spatial Visualization Test (SVT) and the Mathematics Attitude Scale (MAT) scores was highly significant (P<.002). There were, however, no significant sex differences at Site 1 (P<.l393). Analyses of the univariate hypotheses associated with the multivar- iate test at Sites 2 and 3 yielded probabilities that indicated significant sex differences in favor of boys on the MGMP SVT in both sites (the means for boys and girls at Site 2 were 13.28 and 11.49, while at Site 3 they were 12.28 and 9.89; the mean score for boys at Site 1 was slightly higher than for girls, 8.18 vs. 8.08). With regard to the MAT scores, the univariate analysis indicated no sig- nificant difference between boys and girls in either Site. 153 3. For hypothesis H03, the multivariate test indi- cated significant grade by sex interaction on both measures only at Site 3 (P<.05); no significant in— teraction at Site 1 (P<.8645) and Site 2 (P<.8012); however, testing the interaction separately for each planned comparison at Site 3 resulted in only one significant test, (7 vs. 8) by sex. Examination of the means (Table 4.5) and the profiles of the means by grade and sex (Figures 4.9 - 4.12) shows ordinal interaction for the MGMP SVT scores mainly because of a relatively higher gain for boys than for girls between grades seven and eight (11.76 to 15.27 for boys while only 10.50 to 11.02 for girls). 4. For the comparison of sixth grade data among Sites 1,2, and 3, the multivariate test of site main effects indicated that the overall difference among the three sites on both the MGMP Spatial Visualiza- tion Test (SVT) and the Mathematics Attitude Scale (MAT) scores was highly significant (P<.0001). The univariate analyses showed significant difference among the sites on the MGMP SVT in favor of Site 2 over 3 and Site 3 over 1 (the means for Sites 1,2 and 3 were 8.85, 12.15, and 10.17 respectively). With respect to the MAT measure, the univariate analyses showed significant difference only between Sites 2 and 3 but not between Sites 1 and 3 (the means for Sites 1,2, and 3 were 3.10, 3.30, and 3.04 154 respectively). The multivariate test of sex main effects indicated that the overall difference be- tween sixth grade boys and girls on both the MGMP SVT and MAT measures was highly significant (P<.0001); however, the univariate analyses indi- cated significant sex differences in favor of boys only on the MGMP SVT (P<.0003) and not on the MAT (P<.2058). No significant interaction of site by sex was found (P<.4128). The following findings are related to the evaluation of the effects of the instruction in activities involving spatial visualization tasks on a sample of sixth, seventh, and eighth graders from Site 3. The findings for the immediate effects are presented with respect to H04 through H06, and for the retention effects with respect to H11 through H13. The findings regarding the comparison of attitudes after the instruction are presented with respect to H07 through H10. 5. For hypothesis H04, the multivariate analysis of repeated measures indicated that the overall differ- ence between pretest and posttest means was highly significant (P<.0001). However, analyses of the univariate hypotheses associated with this test in- dicated significant overall gain between pretest and posttest scores on the MGMP Spatial Visualization Test (P<.0001) but not on the Mathematics Attitude 155 Scale (P<.8195). The overall means on the MGMP SVT for pretest and posttest were 12.01 and 19.48. For hypothesis H05, the multivariate analysis of repeated measures indicated highly significant grade effects (P<.0002); however, the corresponding uni- variate analyses indicated only significant grade by time interaction on the MGMP Spatial Visualization Test but not on the Mathematics Attitude Scale. Planned comparisons indicated that grades seven and eight had a significant higher gain on the MGMP SVT than grade six (P<.0002); also that grade seven had a significant higher gain on the MGMP SVT than eight (P<.0057). The pretest-posttest means for grade seven were 11.22 and 20.02, while for grade eight they were 13.23 and 20.56 (for completeness, the figures for grade six were 11.04 and 19.96). For hypothesis H06, the multivariate and univar- iate analysis of repeated measures showed no signif- icant interaction of sex by time. Boys and girls gained similarly from pretest to posttest on both the MGMP Spatial Visualization Test and the Mathema- tics Attitude Scale (the means on the MGMP SVT pretest and posttest for boys were 13.48 and 21.22, while for girls they were 10.65 and 17.86; the MAT means were similar for boys and girls, and for pretest and posttest, about 2.9). No significant interaction of grade by sex by time was indicated. 10. 11. 156 For hypotheses H07 and H08, the ANOVA test indicated no significant differences between the mean scores of Spatial Visualization Attitude Scale and Mathema- tics Attitude Scale either by grade or by sex. No significant interaction of grade by sex was found among the mean difference scores-—Mathematics Atti- tude Scale score minus Spatial Visualization Atti- tude Scale score. For hypotheses H09 and H10, the ANOVA test indicated significant grade main effects (P<.001) on the Spatial Visualization Attitude Scale (the means for grade six, seven, and eight were 3.401, 3.225, and 2.974 respectively). No significant sex main effects or interaction of grade by sex was found. For hypothesis H11, the multivariate analysis of repeated measures indicated that the overall difference between post- and retention test means was highly significant (P<.0001). The univariate analyses associated with this test indicated significant time effects on both the MGMP Spatial Visualization Test (SVT) and Spatial Visualization Attitude Scale (SVAT). The overall MGMP SVT means for post- and retention test were 19.95 and 21.00; for the SVAT the means were 3.119 and 2.920. For hypothesis H12, the multivariate analysis of repeated measures for the grade main effects indicated significant differences (P<.01); however, 12. 157 the univariate analyses showed only one significant interaction between the planned comparison of grade 7 vs. 8 by time for the SVAT measure. There was a greater drop on the SVAT mean score for grade seven than grade eight between post- and retention test. For hypothesis H13, the multivariate and univar- iate analyses of repeated measures indicated no significant interaction of sex by time on either measure-~the MGMP Spatial Visualization Test or the Spatial Visualization Attitude Scale. No signif- icant interaction of grade by sex by time was indicated. Figures 5.1 and 5.2 present the profiles on the means for the MGMP SVT pre— post- and retention test scores by grade and by sex. 13. Finally, for the correlation analysis, highly significant correlation coefficients were found between successive administration of the measures. These coefficients indicated high test-retest reliability of each measure, MGMP SVT, MAT, and SVAT. In addition, positive correlations were found between the MGMP SVT measure and the attitude measures, MAT or SVAT, all of which were significant and, in general, higher for boys than girls. 158 NKSNH’SVTIAEABISCCNRES PRE POST RETENHON Fig. 5.1 Pre—post-retention test means on the MGMP Spatial Visualization Test (SVT) by grade 159 MGMPSVTMEANSCORES PRE POST RETENTION Fig. 5.2 Pre-post-retention test means on the MGMP Spatial Visualization Test (SVT) by sex 160 Conclusions The analysis of the data gathered in this study and presented in the preceding chapters warrants a number of conclusions. These conclusions are based on evidence obtained from the findings of the present study and the investigator's interpretation of these results. Within the limitations of the study (specified in Chapter 1), the following conclusions seem justified. First, it was concluded prior to an instructional intervention, that 1. Among middle school grade levels five through eight, there were grade differences in spatial visualization performance (as measured by the MGMP SVT). The performance of boys and girls in grades five through eight improved with age. 2. Among middle school grade levels five through eight, there were grade differences in attitudes toward mathematics (as measured by the MAT). The attitudes of fifth through eight grade students decreased with age. 3. Among middle school students (grades six through eight), there were sex differences in spatial visualization performance (as measured by the MGMP SVT). The mean for the boys was significantly higher than the mean for the girls on the Spatial Visualization Test. 4. 6. 161 The sex differences in spatial visualization skills (as measured by the MGMP SVT) increased signifi- cantly from grade seven to eight. 5. Among middle school students (grades five through eight), there were no sex differences in attitudes toward mathematics (as measured by the MAT). There was site (school area) effect on the spatial visualization performance (as measured by the MGMP SVT). As the socioeconomic status (SES) of the site rose, the performance on the Spatial Visualization Test increased. After the instructional intervention the findings indicated the following conclusions: 7. Sixth, seventh, and eighth grade students, regardless of sex, profited strikingly from instruction in activities involving spatial visualization tasks, as measured by the MGMP SVT immediately following instruction. Seventh grade students, regardless of sex, gained more from the instruction than sixth and eighth grade students. The grade level difference in spatial visualization between grades seven and eight decreased dramatically after the instruction (before the instruction the MGMP SVT means were 11.22 vs. 13.23; after the instruction they were 20.02 vs. 20.56). 9. 10. ll. 12. 13. 14. 162 In spite of the initial sex difference in spatial visualization skills, the results indicated that at the middle school level the spatial visualization ability of girls and boys was not differentially affected by the instruction. Attitudes toward mathematics of sixth, seventh, and eighth grade students was not affected by the instruction in spatial visualization tasks. No change occured in attitudes either by grade or by sex. Immediately after the instruction in spatial visualization tasks, sixth, seventh, and eighth grade students, regardless of sex and grade level, had Similar attitudes toward mathematics and toward spatial visualization. However their attitudes toward spatial visualization were slightly higher than their attitudes toward mathematics (3.164 vs. 2.911). After the instruction in spatial visualization tasks there was a grade level difference in attitudes toward spatial visualization. There was a drop in attitudes from grade six to grade eight. After the instruction in spatial visualization tasks there was no sex difference in attitudes toward spatial visualization. The effects of the instruction in spatial visualiza- tion tasks were persistent over time. In fact, 163 after a four-week period, sixth, seventh, and eighth grade students performed higher on the MGMP SVT retention test than on the posttest (21.00 vs. 19.95). 15. The long—term effects of the instruction in spatial visualization tasks were comparable for both sexes. And finally, it was concluded from the results of the correlation analysis that 16. Among sixth, seventh, and eight grade students there were sex differences in the magnitude of the correlation coefficients between the MGMP SVT measure and the attitude measures. The correlation coefficients all were positive, significant and higher for boys than for girls. Discussion The findings of this study prior to the instructional intervention suggest that among middle school students (grades five through eight) there is a trend toward an increase in spatial visualization skills (as measured by the MGMP SVT) with an increase in grade level. This observation might be expected because of the increased maturity and selectiveness of the subjects; it is consistent with common findings of significant grade effects for the cognitive variables. Several studies (such as Fennema and Sherman 1977, 1978; Guay and McDaniel, 1977) report similar results with respect to significant grade effects upon spatial 164 visualization performance. As to the findings of a decrease in attitudes toward mathematics with an increase in grade level, it is also consistent with reviews of the literature (see, for example, Fennema, 1977). The main result of this investigation prior to the in- structional intervention, perhaps surprising to some, is that there are sex differences in spatial visualization among middle school students. In general, boys seem to out- perform girls with respect to spatial visualization ability. This finding is in agreement with the conclusion of the com- prehensive review by Maccoby and Jacklin (1974) indicating that ”male superiority on visual-spatial tasks is fairly consistently found in adolescence and adulthood (p. 351).” The results of the Women in Mathematics study (NAEP) re— ported by Armstrong (1980) and the Fennema-Sherman study (1977, 1978), however, are apparently in conflict with the results Of this study. Both studies conclude that there are no sex-related differences in spatial visualization among middle school students. The discrepancy between these studies and the present study may depend upon a number of factors, one of which is instrumentation. The MGMP Spatial Visualization Test is relatively new, but demonstrated consistently high reliability and content validity (as described in Chapter 3). The items in this test are of different types and not all one type as is the case with the instruments used in the above two studies (the Revised Minnesota Paper Form Board, and the DAT Spatial 165 Relations test). The tests with one type of item might measure only a specific aspect of spatial visualization ability, while the test in the present study measures dif- ferent aspects of this ability. Perhaps the difference be- tween types of items is the main reason for a conflict in the results. It remains for future studies to evaluate the relationship of this test with the other instruments and provide clues to the construct validity of the tests, i.e., which test is measuring what. Maccoby and Jacklin (1974) concluded also that sex differences in spatial ability become evident only during early adolescence. If the fifth grade students are assumed to be preadolescents, the data from Sites 1 and 3 (no fifth graders at Site 2) seem to support Maccoby and Jacklin's notion on the timing of the appearance of sex differences in spatial visualization. The fifth grade data from Site 1 shows almost no difference between boys and girls in spatial visualization performance (7.25 vs. 7.43). In contrast, the fifth grade data from Site 3 presents a sizable difference between the sexes in spatial visualization performance (9.58 vs. 8.00). Given the discrepancy between the results from the two sites, it seems that there is no clear evidence for the existence of sex differences in spatial visualization as early as grade five. Further research using the MGMP SVT in grades four and five should reveal useful information. The data from Site 3 (Table 4.5), where all four grade levels (five through eight) participated in the study, ‘- 166 indicates that from grade six to seven there was a greater increase in girls' scores on the MGMP SVT than in boys'; in contrast, between grade seven and eight boys' scores on the MGMP SVT accelerated more rapidly than girls'. This phenomenon may be related to the fact that girls mature earlier than boys. Also peer pressure, social role expectation, and sex-typed activities might be other factors to be investigated in connection with the relationship between adolescent development and spatial visualization. Regarding the change in the magnitude of the sex difference on spatial tasks, examination of the related literature again showed conflicting reports (see Harris, 1981); however, the significant increase in sex differences in spatial visualization skills from grade seven to eight Should be considered in decision making as to the timing of instructional intervention or remediation. Furthermore, the increase in sex differences in spatial visualization skills from grade seven to eight was accompanied by changes in attitudes toward mathematics. The attitudes of girls decreased while the attitudes of boys increased from grade seven to eight. The timing of the appearance of this combination of trends in attitudes and spatial visualization ability may support the hypothesis of sex-role stereotyping as a possible partial explanation for the sex differences in spatial visualization. The finding of no sex differences among middle grade students in their attitudes toward mathematics in this study ‘r'pn‘ 167 contradicts the conclusion of Fennema's review (1977) that ”there are sex-related differences in attitudes toward mathematics (p. 104).” This result, however, is in agree- ment with reviews of Suydam and Weaver (1975) who quote studies with contradictory results and say that in other studies no significant sex-related differences are found. The results of the comparison of sixth grade data among sites (school areas) suggest a relationship between the kind of community in which the students live and their spatial visualization performance; however, the sex differences in spatial visualization appeared across sites with no inter- action of site by sex. This issue of relevance of school area to spatial visualization skills should be further investigated to uncover possible sources and determinants of differences in spatial visualization performance. One of the main results of this investigation was that after the instruction intervention middle school students, regardless of sex, gained significantly from the training program in spatial visualization tasks. In the literature, few studies aimed at evaluating training programs to increase spatial abilities in general and Spatial visualiza- tion in particular. Of the existing studies, most have been done with adults. The result from the present study is in conflict with Sedgwick's (1961) conclusion that spatial vis- ualization is probably an innate capability not modifiable by a specific instruction. The conflict in the results between the two studies may depend on several factors such 168 as the age of the subjects (Sedgwick's study used older subjects than did the present study), the instructional material (he used descriptive geometry) or instrumentation (he used the DAT-Space Relations Test). All of the above factors should be manipulated in future research to deter- mine their roles. The significant gain from instruction reported in this study is in agreement with Brinkmann's (1966) conclusion that the functional skills of individuals in spatial visual- ization can be improved when appropriate training is pro- vided. Although Brinkmann used programmed instruction emphasizing geometry, there were several features similar to this study: the subjects in his study were eighth graders; he stressed concrete object manipulation similar to the emphasis of the instruction unit used in the present study; and the period of time and setting of the training were similar. All of the above factors may have contributed to the agreement in the results. A comment might be in order regarding the finding that seventh grade students profited more from the instruction in Spatial visualization tasks than the sixth and eighth graders. This result may be partially explained by the findings of the Scheffé's Post-Hoc analysis of the data prior to the instruction. The sex differences in spatial visualization performance in grades six and eight were significant, while in grade seven no significant difference was found; grade seven, in other words, was more homogeneous 169 than six and eight. This is a possible explanation of the above finding. Another main result regarding the effects of the instruction was that boy and girl students did not respond differentially to the training program. This outcome was in spite of the sex differences in favor of males prior to the instruction, which could support the expectation that males might profit more from training than females. Consequently, the present result does not support the hypothesis that the sex difference is an underlying ”ability.” It should be stressed at this point that the instruction in spatial visualization tasks did not intend a priori to eliminate the sex differences in spatial visualization skills. No remedi- ation for girls was planned or occurred. Obviously the sex differences in spatial visualization skills were still present after the instruction. The persistence over time of the effects of the in- struction on the spatial visualization skills was unex- pected. Regardless of grade or sex, there was a gain. If this finding is replicated, it will be evidence that spatial visualization ability is a trait that, once acquired, can be further developed, possibly by the increased awareness of spatial visualization or by other explanations to be posed and investigated. The findings of no effects of the instruction on the attitudes toward mathematics are in accordance with Aiken's (1976) conclusions on the effects of curricular structure on 170 attitudes toward mathematics. Probably, a period of two to three weeks-—the length of the instruction-—was not suffic- ient to effect any dramatic changes in attitudes. The pat— tern of attitudes toward spatial visualization as demon— strated by the subjects after the instruction is similar to that of their attitudes toward mathematics. Grade level differences were found in attitudes toward spatial visuali- zation in favor of younger students; but no sex differences were found. The correlation analysis revealed positive correlation coefficients between performance on the spatial visualiza- tion test and attitudes toward either mathematics or spatial visualization; sex difference in the magnitude of these coefficients was also found. Both findings are consistent with the conclusions of reviews by Fennema (1977) and Aiken (1976); however, the direction of the sex difference is contradictory to that indicated by Aiken. In the present study the correlation coefficients for boys were higher than for girls. Examination of the raw data showed that among the girls and boys who performed low on the spatial visual— ization test, more girls than boys had a high attitude toward spatial visualization. To conclude, this investigation adds another piece of evidence to support the hypothesis that sex differences in Spatial visualization abilities do exist. These differences favor males and appear as early as adolescence; however, the second part of this study on the effects of instruction to 171 increase spatial visualization skills provides evidence to support the notion that these skills are teachable and can be learned. The specific skills involved in the manifesta- tion of the spatial visualization ability improve with practice. Given the opportunity to develop spatial visuali- zation skills, both sexes have equal potential for acquiring significant gain from the training. When those skills have been attained they last and even continue to develop over time. If these findings are replicated, the hypothesis that socialization and differential training are factors that may account for differences in spatial visualization would be indirectly supported, although such findings would not rule out biological determinants. At present the first issue to be concerned with is the replicability and generalizabiliby Of this study. In the following section, recommendations for additional research will include the suggestion to investigate this issue further. Implications and Recommendations The following implications and recommendations are based on the investigator's interpretations of the analysis of data collected in this study, on the conclusions reached, and on the review of the related literature. General and specific implications will be discussed in terms of their educational impact and action needed. Recommendations will r—p—fJ 172 be made regarding possible extension of this study and future research in the area of spatial visualization. Implications for Educational Issues Bishop (1980), in his review on ”Spatial Abilities and Mathematics Education,' examined the interface between two sets of psychological constructs: one concerns the visual/ spatial field and the other, mathematics education. He aimed to review the different research emphases which have contributed to knowledge about spatial ability and to de- scribe what they can, or cannot, offer the mathematics educator. Grouping the spatial research into four areas (factor analysis, developmental psychology, individual dif- ferences, and teaching experiments) Bishop suggests that the factor analysts can offer tests and tasks, various classifications of abilities, and ideas about order and dependence between different abilities. The developmental psychologists can suggest ideas of what can be expected of children at different ages and of stages children will pass through (under present conditions) in learning to comprehend certain features of their world. The individual-difference researchers can offer many ideas about the reasons why these differences exist and how such differences could relate to the conditions in which learning and development takes place. Experiments in teaching can help to clarify just what these conditions are and how they can be exploited by teachers (p. 266). The present investigation was concerned with the last two areas of research—-individua1 differences and teaching experiments--mentioned above by Bishop. In this study, the finding of sex differences in spatial visualization abili- ties among middle school students in favor of boys has 173 implications for other disciplines. The review of the related literature in Chapter 2 indicated the existence of evidence to support the notion that spatial visualization ability is an important consideration in most technical- scientific occupations and especially in the study of science and mathematics. Consequently, it has been hypo- thesized that sex-related differences in science and mathe— matics achievement might partly be accounted for by sex- related differences in spatial visualization. The results of this study suggest that educators in these disciplines should be aware of the tenability of this hypothesis and consider it while taking actions of remediation. Neverthe- less, special remedial instruction in visual—spatial skills is not normally offered in schools. The additional finding that middle school students, re- gardless of sex and grade, can benefit significantly from instruction in spatial visualization tasks implies the like- lihood of success in remediation of these skills. Even though it is very clear that spatial visualization has a strong relationship to many forms of mathematics achievement (as indicated in Chapter 2), spatial visualization skills seems to be neglected in the mathematics curriculum, especi— ally in the elementary and middle schools. To the best of this investigator's knowledge, the mathematics curriculum does not include units of instruction in spatial visualiza- tion per se, even though it is often believed that Euclidean 174 geometry covers spatial visualization. (Geometry is not spatial visualization!) Mathematics education in the middle school has under- gone vast changes during the last two decades, although not all have been successful. Attempts have been made to re— structure the mathematics curriculum to adapt the mathema- tics taught to the developmental level of the child, to em- phasize problem solving, to teach structures of mathematics based on certain key principles, and to integrate mathema- tics with other disciplines. It seems that the time is right for cultivating interest in the area of spatial vis- ualization. Mathematics educators in charge of pre- and in—service teacher education should advise teachers and curriculum decision makers to include spatial visualization in the mathematics curriculum. In the present study, it was demonstrated that middle school mathematics teachers, as indicated by the outcome of their students' performance, successfully taught a unit of instruction on spatial visualization tasks. The teachers were simply provided with the instructional material, in- cluding well-sequenced activities involving spatial visuali- zation tasks and a teacher guide emphasizing the instruc- tional model described in Chapter 3. A two-hour workshop prior to the instruction was also provided. It is note- worthy that this study used a single test for students ranging in grades from five to eight, which is a distinct 175 advantage in an area where different instruments are often used to define a given ability at different ages. Furthermore, another implication is relevant to the timing of the instruction. If the argument that spatial visualization ability is important for success in mathema- tics is accepted, and if attempts are made to improve middle school students' abilities in this area, then the timing of instruction appears to be crucial. Although all three grade levels, sixth, seventh and eighth, gained significantly from the instruction, the data indicated that seventh grade boys and girls gained significantly more than the others. This suggests that seventh grade is the optimal time for the teaching of spatial visualization tasks. Additional support for this suggestion derives from the finding prior to the instruction that sex differences in spatial visualization in eighth grade was significantly higher than those in seventh grade. Finally, mathematics educators in general and middle school mathematics teachers in particular should be con- cerned with the replication of previous results regarding the drop in attitudes toward mathematics from grade five to grade eight. They should take action to identify reasons for this decline. Then, attempts can be made to reverse this trend. 176 Recommendations For Future Research The implications of the results of this study for educational issues and recommendations for action needed with the educational system were indicated in the previous section. Recommendations regarding the extension of this study and future research in the area of spatial visualization are as follows: 1. Because of the important implications of the results obtained in this study, it is recommended that the study should be replicated with other samples of middle school students to verify the generaliza— bility of the results. Even though the MGMP Spatial Visualization Test has demonstrated consistently high reliability and content validity, further evaluation of its char- acteristics is recommended. The test's construct validity should be compared to that of similar tests of spatial visualization. It is suggested that the MGMP SVT be included in a battery of tests to evalu- ate its relationship with other instruments that measure spatial visualization ability and/or in- struments that measure other congitive and affective cognitive and affective behaviors. Furthermore, it is suggested that a parallel form equivalent to the existing MGMP SVT be produced. An additional form could be used for obtaining a reliability 177 coefficient or in situations where several repeated measures are needed. Using the MGMP SVT, it is recommended that there be a further investigation of the existence of differences in spatial visualization performance by sex and/or by grade level. It is suggested that the test also be administered to fourth and fifth graders as well as to high school students and possibly to samples of Older students. Such testing may provide clues as to the timing Of the appearance of sex differences in spatial visualization and also the changes in the magnitude of these differences over time. It is recommended that the relationship between the effects of the entire school environment and the spatial visualization performance of the students be investigated. It is suggested that a comparison be made between schools within a district and between districts (with different demographic character- istics) in order to uncover possible sources and determinants of differences in spatial visualization performance. This study has suggested one possible treatment that has been effective in improving spatial visualiza- tion performance of the participants. Further research is suggested to compare this treatment with others in order to determine whether there is a more 178 efficient method of instruction for increasing spatial visualization skills of middle school students. An additional factor, the timing of the instruction, should be examined to determine the optimal grade (age) level at which students might benefit most from the treatment. Furthermore, it is suggested that the effectiveness of the instruction, indicated in this study, be evaluated in order to provide for remediation of low achievers of both sexes. Such evaluation might suggest a tool which will eliminate sex-related differences in spatial visualization. It is recommended that a longitudinal study of the 430 students who participated in the instruction be initiated to provide answers to the following ques- tions: (1) To what extent (measured in terms of months or years) do these students retain spatial visualization competencies and attitudes toward spatial visualization? (2) Are there any signifi- cant differences between these students and a com- parable group of students who did not participate in the instruction regarding cognitive and affective behaviors toward mathematics and science? (3) Will the group of students who participated in the instruction be more successful than their counter- parts in learning geometry and/or in problem solv- ing? (4) Will these students perform significantly 179 better than their counterparts on standardized tests such as the Stanford Achievement Test? Each of the above comparisons Should also include sex as source of variation to detect possible differences between the sexes on the issues discussed. It is recommended that small groups of students be used during the instruction in order to study in detail how they accomplish spatial visualization tasks. Thus, the emphasis will be on identifying cognitive processes (reasoning patterns) rather than on performance or product. ' behaviors be It is recommended that teachers studied while they teach the spatial visualization activities described in this study. It is suggested that an evaluation be carried out on (1) the impact of the instructional model on teaching style, (2) the use and quality of the teaching guide, and (3) teachers' reactions regarding the relevance of spatial visualization to other subject areas. APPENDICES APPENDIX A BROCHURE OF MIDDLE GRADES MATHEMATICS PROJECT (MGMP) DEPARTMENT OF MATHEMATICS MICHIGAN STATE UNIVERSTIY 180 MIDDLE GRADES MATHEMATICS PROJECT DEPARTMENT OF MATHEMATICS MICHIGAN STATE UNIVERSITY The MGMP is a curriculum development project funded by NSF-DISE, to develop units of high quality mathematics in— struction for grades 5 through 8. Each unit * is based on a related collection of important mathematical ideas, * provides a carefully sequenced set of activities which lead to an understanding of the mathematical challenges, * helps the teacher foster a problem-solving atmosphere in the classroom, * uses concrete manipulatives where appropriate to help provide the transition from concrete to abstract thinking, * utilizes an instructional model which consists of three phases...launching, exploring, and summarizing, * provides a carefully developed instructional guide for the teacher, * requires two to three weeks of instructional time. The goal of the MGMP materials is to help students develop a deep, lasting understanding of the mathematical concepts and strategies studied. Rather than attempting to break the curriculum into small bits to be learned in isolation from each other, MGMP materials concentrate on a 181 cluster of important ideas and the relationships which exist among these ideas. Where possible the ideas are embedded in concrete models to assist the students in moving from this concrete stage to more abstract reasoning. Many of the activities are built around a specific mathematical challenge. The instructional model used in the units focuses on helping the students solve the mathematical challenge. The instruction is divided into three phases. During the first phase the teacher launches the challenge. The launching consists of introducing new concepts, clari- fying definitions, reviewing old concepts, and issuing the challenge. The second phase of instruction is the class explora- tion. During the exploration the students work individually or in small groups. The students may be gathering data, sharing ideas, looking for patterns, making conjectures, or developing other types of problem—solving strategies. The teacher's role during exploration is to encourage the stu- dents to persevere in seeking a solution to the challenge. The teacher does this by asking appropriate questions, en— couraging and redirecting where needed. For the more able students, the teacher provides extra challenges related to the ideas being studied. When most of the children have gathered sufficient da- ta, the class returns to a whole class mode (often beginning the next day) for the final phase of instruction, summariz- ing. Here the teacher has an opportunity to demonstrate 182 ways to organize data so that patterns and related rules become more obvioius. Discussing the strategies used by the children helps the teacher to guide the students in refining these strategies into efficient, effective problem solving tecniques. The teacher plays a central role in this instructional model. First the teacher provides and motivates the chal- lenge and then joins the students in exploring the problem. The teacher asks appropriate questions, encouraging and re— directing where needed. Finally, through the summary, the teacher helps the students to deepen their understanding of both the mathematical ideas involved in the challenge and the stratgies used to solve it. To aid the teacher in using the teaching model de- scribed, a detailed instructional guide is provided. This guide was developed as a result of many classroom trials of the materials. It provides help with both the mathematics content and the classroom management of the activities. Specific suggestions for important questions to be asked at appropriate stages of the activities are included. Exten- sion questions and challenges for the more able students are provided along with suggestions for helping those students who are having difficulty. The units developed include SPATIAL VISUALIZATION FACTORS AND MULTIPLES PROBABILITY SIMILARITY 183 STAFF Glenda Lappan, Director William M. Fitzgerald Elizabeth Phillips Mary Jean Winter Pat Yarbrough David Ben-Haim CONSULTANTS Janet Shroyer (Development) Aquinas College, Grand Rapids, MI Richard Shumway (Evaluation) Ohio State University APPENDIX B MATHEMATICS AND SPATIAL ATTITUDE SCALES-~RELIABILITY AND TEST-RETEST CORRELATION COEFFICIENTS EXAMPLE: DIRECTIONS: BAD SAD BORING 184 MATHEMATICS ATTITUDE SCALE (MAT) NAME BOY/GIRL FOR EACH PAIR OF WORDS BELOW PLACE AN X ON THE BLANK THAT BEST TELLS HOW YOU FEEL ABOUT-- SNOW FOR EACH PAIR OF WORDS BELOW PLACE AN X ON THE BLANK THAT BEST TELLS HOW YOU FEEL ABOUT -- MATHEMATICS GOOD HAPPY EXCITING HOLD BACK EASY LESS 185 SPATIAL VISUALIZATION ATTITUDE SCALE (SVAT) EXAMPLE: DIRECTIONS: BAD SAD BORING NAME BOY/GIRL FOR EACH PAIR OF WORDS BELOW PLACE AN X ON THE BLANK THAT BEST TELLS HOW YOU FEEL ABOUT-- SNOW HATE PRETTY FOR EACH PAIR OF WORDS BELOW PLACE AN X ON THE BLANK THAT BEST TELLS HOW YOU FEEL ABOUT -- SPATIAL VISUALIZATION ACTIVITIES GOOD HAPPY EXCITING HOLD BACK EASY LESS H TABLE B.1 186 RELIABILITY COEFFICIENTS--CRONBACH a FOR MATHEMATICS Grade 5 Boys Girls Grade 6 Boys Girls Grade 7 Boys Girls Grade 8 Boys Girls ATTITUDE SCALE IN EACH SITE BY TIME Site 1 N a 104 .84 55 .85 49 .83 115 .82 58 .83 57 .79 BY GRADE BY SEX Pretest Site 2 N d 274 .84 150 .86 124 .82 153 .86 71 .83 82 .89 Site 3 N a 94 .87 48 .91 46 .83 208 .88 104 .88 104 .88 170 .85 90 .86 8O .83 209 .86 96 .89 113 .83 Posttest Site 3 N a 108 .84 54 .85 54 .82 142 .86 74 .85 68 .85 180 .88 79 .90 101 .86 187 TABLE B.2 PEARSON CORRELATION COEFFICIENTS BETWEEN THE PRETEST AND POSTTEST SCORES ON THE MATHEMATICS ATTITUDE SCALE BY GRADE BY SEX Correlationa ____ N Coefficient Grade 6 108 .76 Boys 54 .82 Girls 54 .67 Grade 7 142 .75 Boys 74 .81 Girls 68 .64 Grade 8 180 .69 Boys 79 .72 ____ Girls , 101 :22. Total 430 .73 Boys 207 .79 Girls 223 .67 aAll significant (P < .001). 188 TABLE B.3 RELIABILITY COEFFICIENTS--CRONBACH a FOR THE SPATIAL VISUALIZATION ATTITUDE SCALE IN SITE 3 BY TIME BY GRADE BY SEX Posttest Retention test N a N a Grade 6 52 .79 Boys 26 .82 Girls 26 .74 Grade 7 79 .89 Boys 41 .91 Girls 189 TABLE B.4 PEARSON CORRELATION COEFFICIENTS BETWEEN THE POSTTEST AND RETENTION TEST SCORES ON THE SPATIAL VISUALIZATION ATTITUDE SCALE BY GRADE BY SEX Correlationa N Coefficients Grade 6 50 .64 Boys 24 .79 Girls 26 .51 Grade 7 74 .76 Boys 38 .80 Girls 36 .68 Grade 8 104 .69 Boys 44 .67 Girls 60 ~____.70 Total 228 .71 Boys 106 .75 Girls 122 .66 aAll significant (P < .001). APPENDIX C SPATIAL VISUALIZATION TEST-—SAMPLE ITEMS, RELIABILITY COEFFICIENTS, TEST-RETEST EFFECT, AND CORRELATION COEFFICIENTS 190 SAMPLE ITEMS PRECEDING THE MGMP SPATIAL VISUALIZATION TEST Do these sample items and then wait for further instructions. This is an example of the mat plan of a building. The number in each square tells how many cubes are to be placed on that square. Use the information in the mat plan to answer the two sample items. Sample Item 1 This is a corner View of the building . ' ' ‘ ' above. Which corner was it drawn from? - IA B c D 'FRONT-RIGHT BACK—RIGHT BACK—LEFT FRONT-LEFT Sample 1. A B C D @@©® These are the views of the same building, when seen straight on from the sides. Which is the FRONT VIEW? A B C D Sample 2. A B oodd STOP: Wait until you are told to begin. Sample Item 2 191 MGMP SPATIAL VISUALIZATION TEST — SAMPLE ITEMS The following 8 items are similar to those included in the MGMP Spatial Visualization Test. 1. You are given a picture of a building drawn from the FRONT—RIGHT corner. Find the BACK VIEW. 2. You are given the mat plan of a building. Find the LEFT VIEW. I A B C D E 192 3. You are given the BACK VIEW of a building. Find the FRONT VIEW. I A B C D E BACK VIEW 4. You are given the BASE, FRONT VIEW, and RIGHT VIEW of a building. Find the mat plan for the building that uses the greatest number of cubes and also fits the given base and views. FRONT RIGHT VIEW VIEW BASE 5. Find the view from the FRONT—LEFT corner. H il" H A B C _ ' D E all”; ' l/VFRONT 193 6. Find the View from the BACK-LEFT corner. less 7. Which of these buildings can be made from the two pieces given? A _ .13, l l 2 1 I 8. Find another View of the first building. .3. . .C.. .D. ..-E 194 TABLE C.1 RELIATILITY COEFFICIENTS--CRONBACH a FOR THE MGMP SPATIAL VISUALIZATION TEST IN EACH SITE BY TIME BY GRADE BY SEX Posttest Retention Pretest Site 2 N a Grade 5 Boys Girls Grade 6 52 .84 Boys 26 .84 Girls 26 .84 Grade 7 79 .82 Boys 41 .81 Girls 195 TABLE C.2 CELLS SIZES, MEAN SCORES,AND STANDARD DEVIATIONS ON THE MGMP SPATIAL VISUALIZATION TEST FOR TREATMENT/NONTREATMENT STUDENTS BY GRADE N M SD_ Grade 7 Group 1 (Treatment) 20 11.95 5.35 Group 2 (Nontreatment) 21 11.24 3.42 Grade 8 Group 1 (Treatment) 26 14.61 5.43 Group 2 (Nontreatment) 28 14.82 7.35 196 TABLE C.3 ANALYSIS OF VARIANCE SUMMARY TABLE FOR TEST-RETEST EFFECT--GRADE 7 Source of Variation d.f. Mean Square F Ratio P Group 1 5.192 .260 .613 Error Term 39 19.968 .260 TABLE C.4 ANALYSIS OF VARIANCE SUMMARY TABLE FOR TEST-RETEST EFFECT--GRADE 8 Source of Variation d.f. Mean Square F Ratio P Group 1 .572 .014 .908 Error Term 52 42.274 197 TABLE C.5 PEARSON CORRELATION COEFFICIENTS BETWEEN PRE-POST AND POST-RETENTION SCORES ON THE MGMP SPATIAL VISUALIZATION TEST BY GRADE BY SEX Correlation** Correlation** N Pre-Post Post-retention Grade 6 50 .63 .86 Boys 24 .67 .88 Girls 26 .56 .83 Grade 7 74 .62 .77 Boys 38 .60 .77 Girls 36 .64 .75 Grade 8 104 .78 .87 Boys 44 .81 .87 Girls 60 .71 .85 Total 228 .72 .85 Boys 106 .74 .85 Girls 122 .67 .82 **All significant (P<.001). APPENDIX D SPATIAL VISUALIZATION AND MATHEMATICS ATTITUDE PRETEST INSTRUCTIONS 198 SPATIAL VISUALIZATION AND MATHEMATICS ATTITUDE PRETEST INSTRUCTIONS Materials: A) B) C) D) E) Semantic Differential Test about Math. Spatial Visualization Test Booklets. Sample Items sheet for the Spatial Visualization test. Answer wheet. #2 pencils. Instructions: A) Give Semantic Differential Test first. About 5 minutes should be sufficient. Collect this paper before distributing next test. Students should ppipp their name and circle girl or boy. Distribute answer sheets and #2 pencils. Complete only the name and sex sections. Distribute the Sample Items sheet for the Spatial Visualization test and discuss it for 2—3 minutes with the students. Distribute Spatial Visualization test booklets. Read cover sheet instructions. Allow as much time as needed for the 32 questions. Please encourage the students to answer each question, even if they are not sure (no penalty for guessing). The easy questions are spread throughout the test. APPENDIX E SCHEFFE'S POST HOC COMPARISONS--SITE 1 199 SCHEFFE'S POST HOC COMPARISONS--SITE 1 Following the multivariate and univariate analysis of the data from Site 1, Scheffé's Post Hoc contrasts of means were tested for significance at the .05 level. II LA) .84 1.96 S = (’1 F.95;1,215 MSe for the MGMP SVT 23.73 (from Table 4.2) MSe for the MAT .94 (from Table 4.2) -<:_> II difference between two compared means. 1 1 MSe (31+ 6% (n1,n2 the cells size from Table 4.1). Q) >6) ll As indicated by Scheffé (1959, p. 71) the contrast 3 is significantly different from zero if and only if|0| >S0g_ The comparisons of the MGMP Spatial Visualization Test means for males in grade five and six, and for females in grade five and six are presented in Table B.1. The comparisons of Mathematics Attitude Scale means are shown in Table E.2. 200 TABLE E.1 SUMMARY OF SCHEFFE'S POSTERIORI COMPARISONS OF GRADE LEVELS FOR MALES' MEANS AND FOR FEMALES' MEANS ON THE MGMP SPATIAL VISUALIZATION TEST--SITE l ‘ Contrast W pa Grade 6 Males to Grade 5 Males 1.796 1.80* Grade 6 Females to Grade 5 Females 1.860 1.22 a; indicates the difference between appropriate MGMP SVT means from Table 4.1. *Significant at the .05 level. TABLE E.2 SUMMARY OF SCHEFFE'S POSTERIORI COMPARISONS OF GRADE LEVELS FOR MALES' AND FOR FEMALES' MEANS ON THE MATHEMATICS ATTITUDE SCALE--SITE 1 _ Contrast Q @a Grade 5 Males to Grade 6 Males .358 .50* Grade 5 Females to Grade 6 Females .370 .42* 89 indicates difference between appropriate MAT means from Table 4.1. *Significant at the .05 level. APPENDIX F SCHEFFE'S POST HOC COMPARISONS--SITE 2 201 SCHEFFE'S POST HOC COMPARISONS--SITE 2 Following the multivariate and univariate analysis of the data from Site 2, Scheffé's Post Hoc contrasts of means were tested for significance at the .05 level. 3 = \’1 F.95;1,423 = 3-84 MSe for the MGMP SVT 1.96 34.55 (from Table 4.4) MSe for the MAT .79 (from Table 4.4) ) V = difference between two compared means. V: (IMSe (%1+ %3 (n1,n2 the cells size from Table 4.3). The contrast V is significant if [@|>S0g ) O > The comparisons of the MGMP Spatial Visualization Test means for males and females in grade six, and for males and females in grade seven are presented in Table F.l. The comparisons of Mathematics Attitude Scale means for males in grade six and seven, and females in grade six and seven are shown in Table F.2. ' i‘w. 202 TABLE F.1 SUMMARY OF SCHEFFE'S POSTERIORI COMPARISONS OF MGMP SPATIAL VISUALIZATION TEST MEANS OF SEX FOR GRADE SIX AND FOR GRADE SEVEN--SITE 2 SO. . _ Contrast W Va Grade 6 Males to Grade 6 Females 1.398 1.68* Grade 7 Males to Grade 7 Females 1.868 2.19* 30 indicates the difference between appropriate MGMP SVT means from Table 4.3. *Significant at the .05 level. TABLE F.2 SUMMARY OF SCHEFFE'S POSTERIORI COMPARISONS OF GRADE LEVELS FOR MALES' AND FOR FEMALES' MEANS ON THE MATHEMATICS ATTITUDE SCALE-~SITE 2 509 2a Contrast W W Grade 6 Males to Grade 7 Males .251 .22 Grade 6 Females to Grade 7 Females .248 .32* aw indicates difference between appropriate MAT means from Table 4.3. *Significant at the .05 level. APPENDIX C TESTING INTERACTION FOR EACH PLANNED COMPARISON—-SITE 3 203 TABLE G.1 A SUMMARY OF MULTIVARIATE AND UNIVARIATE ANALYSIS OF VARIANCE FOR THE 4x2 DESIGN OF SITE 3--ALTERNATIVE _ Source of MdIflvariatea Univariate Variation D.F. F P < F p < Constant (M) 1 4704.91 .0001 SVTb 3324.55 .0001 MATC 7511.13 .0001 Grade (A)d 3 eliminating M L1=(5,6)vs.(7,8) (1) 44.15 .0001 SVT 38.47 .0001 MAT 35.20 .0001 L2= 5 vs. 6 (1) 21.11 .0001 SVT 5.16 .0235 MAT 31.67 .0001 L3= 7 vs. 8 (1) 6.87 .0012 SVT 12.10 .0006 MAT .48 .4879 Sex (B) l 23.23 .0001 SVT 44.19 .0001 eliminating MAT .16 .6897 M and A Grade x Sex 3 eliminating M,A and B L1 X Sex (1) 2.15 .1170 SVT .95 .3295 MAT 2.70 .1007 L2 X Sex (1) .56 .5717 SVT .43 .5126 MAT .51 .4776 L3 x Sex (1) 5.46 .0044 SVT 7.62 .0060 MAT 5.15 .0236 Between Groups 8 Within Groups 673 SVT: MSe = 25.37 MAT: MSe = .82 Total 681 aMultivariate D.F. = 2 and 672. bSVT - Spatial Visualization Test. CMAT - Mathematics Attitude Scale. dA reordering of the main effects, testing Grade (A) eliminating M and B resulted in multivariate F values of 44.76, 21.26 and 8.22 for L1, L2 and L3 respectively, and about the same P values as before. APPENDIX H SCHEFFE'S POST HOC COMPARISONS--SITE 3 204 SCHEFFE'S POST HOC COMPARISONS--SITE 3 Following the multivariate and univariate analysis of the data from Site 3, Scheffé's Post Hoc contrasts of means were tested for significance at the .05 level. The univariate degrees of freedom of Grade x Sex interaction were used to determine 3. S = lI3 F.95;1,673 MSe for the MGMP SVT 7.80 2.79 25.37 (from Table 4.6) MSe for the MAT .82 (from Table 4.6) W = difference between two compared means. A“ 1 1 OT: MSe (fi + d) (n1,n2 the cells size from Table 4.5). 1 2 The contrast 9 is significant if |w|>Sog Tables H.1 and H.2 present comparisons between the MGMP SVT means, and between the MAT means for males and females in each grade level, five through eight. Tables H.3 and H.4 introduce comparisons between the MGMP SVT means, and between the MAT means for males and for females across grade levels, 5-6, 6-7 and 7—8. 205 TABLE H.l SUMMARY OF SCHEFFE'S POSTERIORI COMPARISONS OF MGMP SPATIAL VISUALIZATION TEST MEANS OF SEX FOR EACH OF THE GRADE LEVELS FIVE THROUGH EIGHT--SITE 3 Contrast V V Grade 5 Males to Grade 5 Males 2.900 1.58 Grade 6 Males to Grade 6 Females 1.949 2.28* Grade 7 Males to Grade 7 Females 2.159 1.26 Grade 8 Males to Grade 8 Females 1.951 4.25* a0 indicates difference between appropriate MGMP SVT means from Table 4.5. *Significant at the .05 level. 206 TABLE H.2 SUMMARY OF SCHEFFE'S POSTERIORI COMPARISONS OF MATHEMATICS ATTITUDE SCALE MEANS OF SEX FOR EACH OF THE GRADE LEVELS FIVE THROUGH EIGHT--SITE 3 Contrast 809 9a Grade 5 Males to Grade 5 Females .520 .21 Grade 6 Males to Grade 6 Females .350 .04 Grade 7 Males to Grade 7 Females .388 ' -.36 Grade 8 Males to Grade 8 Females .351 .07 a; indicates difference between appropriate MAT means from Table 4.5. No v was found significant at the .05 level. 207 TABLE H.3 SUMMARY OF SCHEFFE'S POSTERIORI COMPARISONS OF GRADE LEVELS FOR MALES' MEANS AND FOR FEMALES' MEANS ON THE MGMP SPATIAL VISUALIZATION TEST--SITE 3 Contrast 809 9a Grade 6 Males to Grade 5 Males 2.452 1.73 Grade 6 Females to Grade 5 Females 2.488 1.03 Grade 7 Males to Grade 6 Males 2.023 .45 Grade 7 Females to Grade 6 Females 2.900 1.47 Grade 8 Males to Grade 7 Males 2.062 3.51* Grade 8 Females to Grade 7 Females l 322 52 89 indicates difference between appropriate MGMP SVT means from Table 4.5. *Significant at the .05 level. 208 TABLE H.4 SUMMARY OF SCHEFFE'S POSTERIORI COMPARISONS OF GRADE LEVELS FOR MALES' MEANS AND FOR FEMALES' MEANS ON THE MATHEMATICS ATTITUDE SCALE-—SITE 3 Contrast 9 Grade 6 Males to Grade 5 Males .441 -.73* Grade 6 Females to Grade 5 Females .447 -.56* Grade 7 Males to Grade 6 Males .364 -.36 Grade 7 Females to Grade 6 Females .376 .04 Grade 8 Males to Grade 7 Males .371 .14 Grade 8 Females to Grade 7 Females .369 -.29 39 indicates difference between appropriate MAT means from Table 4.5. *Significant at the .05 level. APPENDIX I SCHEFFE'S POST HOC COMPARISONS--SIXTH GRADERS BY SITE BY SEX 209 SCHEFFE'S POST HOC COMPARISONS--SIXTH GRADERS BY SITE BY SEX Following the multivariate and univariate analysis of the data from the sixth graders in the entire sample, Scheffé's Post Hoc contrasts of means were tested for significance at the .05 level. 81 = \l1 F.95;1,59l ll? F.95;2,591 MSe for the MGMP SVT 1.96 (for sex within each site) 82 2.45 (for each sex between sites) 29.18 (from Table 4.8) MSe for the MAT .81 (from Table 4.8) ) W = difference between two compared means. 1 1 M86 (fil + fig O) 5.6) II (n1,n2 the cells size from Table 4.7) The contrast 9 is significant ile|>SGQ. The comparisons of the MGMP Spatial Visualization Test means for males and females in each site are presented in Table 1.1. The comparisons between sites for males and for females on the MGMP SVT and on the MAT are given in Tables 1.2 and 1.3 respectively. 210 TABLE I.l SUMMARY OF SCHEFFE'S POSTERIORI COMPARISONS OF MGMP SPATIAL VISUALIZATION TEST MEANS OF SEX FOR EACH SITE, 1,2, AND 3--SIXTH GRADE DATA S A A Contrast 109 Va Site 1 Males to Site 1 Females 1.978 .40 Site 2 Males to Site 2 Females 1.287 1.68* Site 3 Males to Site 3 Females 1.471 2.28* 39 indicates difference between appropriate MGMP SVT means*from Table 4.7. Significant at the .05 level. 211 TABLE I.2 SUMMARY OF SCHEFFE'S POSTERIORI COMPARISONS BETWEEN SITES FOR MALE MEANS AND FOR FEMALE MEANS ON THE MGMP SPATIAL VISUALIZATION TEST-~SIXTH GRADE DATA Contrast 8209 ;a Site 2 Males to Site 1 Males 2.050 3.86* Site 3 Males to Site 2 Males 1.682 -l.6 Site 3 Males to Site 1 Males 2.173 2.26* Site 2 Females to Site 1 Females 2.121 2.58* Site 3 Females to Site 2 Females 1.763 2.2* Site 3 Females to Site 1 Females 2.185 .38 aw indicates difference between appropriate MGMP SVT means from Table 4.7. *Significant at the .05 level. 212 TABLE 1.3 SUMMARY OF SCHEFFE'S POSTERIORI COMPARISONS BETWEEN SITES FOR MALE MEANS AND FOR FEMALE MEANS ON MATHEMATICS ATTITUDE SCALE--SIXTH GRADE DATA Contrast SZOW Va Site 2 Males to Site 1 Males .341 .31 Site 3 Males to Site 2 Males .281 -.20 Site 3 Males to Site 1 Males .361 .11 Site 2 Females to Site 1 Females .353 .09 Site 3 Females to Site 2 Females .293 -.32* Site 3 Females to Site 1 Females .363 .23 99 indicates difference between appropriate MAT means from Table 4.7. *Significant at the .05 level. APPENDIX J PRE- AND POSTTEST STANDARD DEVIATIONIS 213 TABLE J.l PRE- AND POSTTEST STANDARD DEVIATIONS OF MGMP SVT AND MAT SCORES FOR THE SUBSAMPLE FROM SITE 3 BY GRADE AND BY SEX MGMP SVTa M_A_Tb Pretest Posttest Pretest Posttest N S.D. S.D. S.D. S.D. Grade 6 104 5.40 6.59 .783 .828 Boys 54 5.89 6.57 .863 .893 Girls 54 4.49 6.11 .697 .750 Grade 7 142 5.11 5.57 .916 .883 Boys 74 5.43 5.44 .994 .936 Girls 68 4.72 5.56 .796 .804 Grade 8 180 5.99 6.53 .838 .814 Boys 79 6.35 5.36 .940 .882 Girls 101 4.85 6.72 .751 .761 Total 430 5.65 6.40 .859 .855 Boys 207 6.17 5.94 .950 .925 Girls 223 4.74 6.41 .766 .787 aMGMP SVT—-Midd1e Grades Mathematics Project Spatial Visualization Test. bMAT--Mathematics Attitude Scale. APPENDIX K STEP DOWN TESTS--EFFECTS OF THE INSTRUCTION 214 TABLE K.l A SUMMARY OF UNIVARIATE AND STEP DOWN ANALYSIS OF REPEATED MEASURES FOR THE SUBSAMPLE FROM SITE 3a Source of Univariate Step Dowfi__ Variation D.F. F P < F P < Grand Mean 1 SVT CONSb 4001.31 .0001 4001.31 .0001 (M) MAT CONSc 5928.85 .0001 375.00 .0001 MAT LINd .05 .8195 .45 .5028 SVT LINe 1080.91 .0001 22.97 .0001 Grade (A) 2 elimination M G1=6 vs.(7,8) (l) SVT CONS 16.51 .0001 16.51 .0001 MAT CONS 10.95 .0011 19.70 .0001 MAT LIN 1.01 .3160 .74 .3909 SVT LIN 15.41 .0002 10.75 .0012 G2: 7 V8.8 (1) SVT CONS 4.84 .0284 4.84 .0284 MAT CONS 3.94 .0479 6.94 .0088 MAT LIN 1.01 .3150 .87 .3503 SVT LIN 7.73 .0057 9.87 .0019 Sex (B) 1 SVT CONS 41.96 .0001 41.96 .0001 eliminating MAT CONS .19 .6656 4.41 .0364 M and A MAT LIN .11 .7415 .26 .6083 SVT LIN 1.11 .2927 .04 .8480 Grade by Sex 2 SVT CONS 3.22 .0410 3.22 .0410 eliminating M, MAT CONS 3.33 .0368 2.30 .1013 A and B MAT LIN .60 .5509 .53 .5875 ___ SVT LIN .62 .5388 1.33 .2661 Bet. Groups 6 W/in Groups 424 SVT CONS: MSe = 106.58 MAT CONS: MSe = 2.45 MAT LIN : MSe = .39 SVT LIN : MSe = 22.17 8The Multivariate tests as in Table 4.10. For a discussion of Step down tests see Finn and Mattsson (1978) or Bock (1975). bSpatial Visualization Test averaging over the time dimension (Pre + Posttest). CMathematics Attitude Scale averaging over the time dimension (Pre + Posttest). dMAT Time effect over subjects. eSVT Time effect over subjects. APPENDIX L POST- AND RETENTION TEST STANDARD DEVIATIONS 215 TABLE L. 1 POST- AND RETENTION TEST STANDARD DEVIATIONS OF Grade 6 Boys Girls Grade 7 Boys Girls Grade 8 Boys Girls Total Boys Girls MGMP SVT AND MAT SCORES FOR THE RETENTION 52 26 26 79 41 38 107 45 62 238 112 126 SUBSAMPLE FROM SITE 3 BY GRADE AND BY SEX MGMP SVTa Posttest Retention S.D. S.D. 6.18 6.09 6.08 6.09 6.07 6.09 6.41 5.55 6.38 5.37 6.40 7.70 6.53 6.50 5.33 5.51 6.40 6.48 6.58 6.25 6.31 6.01 6.42 6.48 SVATb Posttest Retention S.D. S.D. .693 .721 .660 .831 .734 .601 .994 .998 1.127 1.093 .796 .886 .832 .668 .891 .682 .791 .651 .877 .805 .948 .884 .812 .732 aMGMP SVT--Middle Grades Mathematics Project Spatial Visualization Test. bSVAT-~Spatia1 Visualization Attitude Scale. u , APPENDIX M STEP DOWN TESTS-—RETENTION ANALYSIS 216 TABLE M.1 A SUMMARY OF UNIVARIATE AND STEP DOWN ANALYSIS OF REPEATED MEASURES FOR THE RETENTION SUBSAMPLE FROM SITE 3a Source of Univariate SteB Dowfi__ Variation F P < F P < Grand Mean I SVT CONSC , . . (M) SVAT CON M3700 00.0001 141.60 .0001 SVAT LIN 22. 36 .0001 .43 .5136 SVT LINe 21.04 .0001 8.24 .0045 Grade (A) 2 elimination M Gl=6 vs.(7,8) (1) SVT CONS 13.42 .0004 13.42 .0004 SVAT CONS 5.10 .0250 13.37 .0004 SVAT LIN .38 .5374 .02 .8825 SVT LIN 1.24 .2670 1.75 .1868 G2= 7 V8.8 (l) SVT CONS 2.71 .1009 2.71 .1009 SVAT CONS 1.70 .1940 3.94 .0483 SVAT LIN 7,69 .0060 5.61 .0187 SVT LIN .82 .3667 .99 .3214 Sex (B) 1 SVT CONS 19.26 .0001 19.26 .0001 eliminating SVAT CONS .14 .7095 3.78 .0531 M and A SVAT LIN .39 .5353 .01 .9845 SVT LIN 2.38 .1245 2.16 .1434 Grade by Sex 2 SVT CONS 2.16 .1171 2.16 .1171 eliminating M, SVAT CONS 1.78 .1706 .89 .4116 A and B SVAT LIN 1.61 .2029 1.43 .2404 SVT LIN .31 .7347 .30 .7436 Bet. Groups 6 - _ w/in Groups 232 SVT CONS: MSe = 132.87 SVAT CONS: MSe = 2.35 SVAT LIN : MSe = .42 SVT LIN : MSe = 12.58 aThe Mu1tivariate tests as in Table 4.15. For a discussion of Step down tests see Finn and Mattsson (1978) or Bock (1975). bSpatial Visualization Test averaging over the time dimension (Post + Retention test) CSpatial Visualization Attitude Scale averaging over the time dimension (Post + Retention test). dSVAT Time effect over subjects. eSVT Time effect over subjects. APPENDIX N CORRELATION MATRIX BY GRADE LEVEL 217 TABLE N.1 PEARSON CORRELATION MATRIX FOR THE RETENTION SUBSAMPLE BY GRADEa SVTb Pre-test (1) SVT Post-test (2) SVT Retention test (3) MATC Pre—test (4) MAT Post—test (5) SVATd Post—test (6) SVAT Retention test (7) 8N of sixth graders = 50; N of seventh graders 104. Grade 6 7 8 6 7 8 6 7 8 6 7 8 6 7 8 6 7 8 6 7 8 N of eighth graders = (1) 1.000 1.000 1.000 .633 .622 .783 .598 .628 .726 .142 -.001 .361 .077 .170 .313 .225 .223 .169 .323 .304 .291 (2) .855 .767 .873 .077 .216 .317 .092 .250 .309 .211 .318 .269 .459 .408 .371 bSVT--Spatia1 Visualization Test. CMAT--Mathematics Attitude Scale. dSVAT--Spatia1 Visualization Attitude Scale. (3) .029 .258 .285 .157 .261 .270 .104 .292 .246 .286 .362 .275 (4) .736 .722 .722 .210 .165 .329 .213 .196 .251 (6) (5) .260 .399 .459 .176 .345 .270 .639 .760 .685 74; BI BL IOGRAP HY BIBLIOGRAPHY Aiken, L.R. ”Attitudes Toward Mathematics.” Review of Educational Research 40 (1970):551-96. . ”Ability and Creativity in Math.” Review of Educational Research 43 (1973):405-32. . ”Update on Attitudes and Other Affective Vari- ables in Learning Mathematics. ” Review of Educational Research 46 (1976): 293- 311. Anastasi, A. Differential Psycholo y: Individual and Group Differences in Behavior. 3rd ed. New York: Macmillan 00., I958. Anttonen, R.G. ”A Longitudinal Study in Mathematics Atti- tude.” Journal of Educational Research 62 (1969): 467-77. Armstrong, J.M. Women in Mathematics. Achievement and Participation 0 Women in Mat ematics: An Overv1ew. Denver: E ucation Commiss1on o t e States, Marc 1980. Backman, M.E. ”Patterns of Mental Abilities: Ethnic Socio- economic and Sex Differences.” American Educational Research Journal 9 (1972):1-12. Baker, S.R., and Talley, L.H. "The Relationship of Visual- ization Skills to Achievement in Freshman Chemistry.” Journal of Chemical Education 49 (1972): 775-76. Baker, S.R., and Talley, L.H. "Visualization Skills as a Component of Aptitude for Chemistry--A Construct Validation Study." Journal of Research in Science Teaching 11 (1974):95-97. Bennett, G.K. Bennett Mechanical Comprehension Test: Manual, Forms S and T. New York: The Psychological Corp. I969. Bennett, G.K.; Seashore, R.G.; and Wesman, A.G. Differen- tial Aptitude Tests. 3rd ed. New York: The Psychologi- cal Corp. I959. 218 219 Bennett, G. K. ; Seashore, H.G.; and Wesman, A. G. Differ— ential Aptitude Test: Manual. 4th ed. New York: Psychologi cal Corp. 1966. Bennett, G.K.; Seashore, H.G.; and Wesman, A.G. Differ- ential A titude Test: Manual. 5th ed. New York: Psyc o og1ca Corp. . Berry, J.W. ”Temne and Eskimo Perceptual Skills.” Inter- national Journal of Psychology 1 (1966):207-29. Bishop, A.J. "Spatial Abilities and Mathematics Education--A Review.” Educational Studies In Mathematics 11 (1980): 157-69. Bishop, J.E. ”Developing Students' Spatial Ability." Science Teacher 45 (1978): 20-23. Blade, M., and Watson, W. S. ”Increase in Spatial Visualiza- tion Test Scores During Engineering Study. " Psycholo- gical Monographs 69, no. 12 (1955) Bock, R. D. ”Word and Image: Sources of the Verbal and Spatial Factors in Mental Test Scores. ” Psychometrika 38 (1973). 437- 57. . Multivariate Statistical Methods in Behavioral Research. New Yor : McGraw-H1 Boo Bock, R.D. and Kolakowski, D. ”Further Evidence of Sex- Linked Major-Gene Influence on Human Spatial Visual- izating Ability.” The American Journal of Human Genetics 25 (1973):l-14. Boles, D.B. ”X-Linkage of Spatial Ability: A Critical Review.” Child Development 51 (1980):625-35. Brinkmann, E. H. ”Programmed Instruction as a Technique for Improving Spatial Visualization. ” Journal of Applied Psychology 50 (1966): 179- 84. Broverman, D.M., and Klaiber, E.L. ”Negative Relationship Between Abilities.” Psychometrika 34 (1969):5-20. Brown, F.R. ”The Effect of an Experimental Course in Geometry on Ability to Visualize in Three Dimensions.” Ph.D. dissertation, University of Illinois, 1954. Calder, B.J., and Ross, M. Attitudes and Behavior. Morristown, N.J.: General Learning Press, 1973. 220 Callahan, L.G., and Glennon, V.J. Elementary School Mathematics: A Guide to Current Research. Washington, D.C.: Assoc1ation for Superv131on and Curriculum Development, 1975. Campbell, D.T., and Stanley, J.C. Experimental and Quasi- Experimental Desi us for Research. Chicago: Rand McNally & Co., 1983. Carpenter, F.; Brinkmann, E.H.; and Lirones, D.S. Eduga- bility of Students in the Visualization of Objects in Space. The Univers1ty of Michigan, Cooperative Research Project No. 1474, Ann Arbor, 1965. Carrol, J.B. Psychometric Tests as Cognitive Tasks: A New ”Structure of InteIIect.' ETS Research Bulletin 74—16 Pr1nceton, N.J.: Educational Testing Service, 1974. Cattell, R.B. Abilities: Their Structure, Growth, and Action. Boston: Houghton Mifflin Co., 1971. Eliot, J., and Fralley, J.S. ”Sex Differences in Spatial Ability.” Young Children 31 (1976):l87—98. Elmore, P.B., and Vasu, E.S. ”Relationship Between Selected Variables and Statistics Achievement: Building a Theoretical Model.” Journal of Educational Psychology 72 (1980):457-67. Ekstrom, R.B.; French, J.W.; and Harman, H.H. Manual for Kit of Factor-Referenced Co nitive Test. Pr1nceton, N.J.: Educational Testing Serv1ce, I976. Emmett, w.G. ”Evidence of a Space Factor at 11+ and Earlier.” British Journal of Psychology. Statistical Section 2 119495: 3-l6. Fennema, E. ”Mathematics Learning and the Sexes: A Review.” Journal of Research in Mathematics Education 5 (1974): . ”Spatial Ability, Mathematics, and the Sexes.” In Mathematics Learnin : What Research Says about Sex Differences, pp. 33-43. Edited By E. Fennema. Columhus, Ohio: ERIC Center for Science, Mathematics, and Environmental Education, 1975. 221 . ”Influences of Selected Cognitive, Affective and Educational Variables on Sex-Related Differences in Mathematics Learning and Studying. ” In Woman and Mathematics: Research Perspectives for Change, pp. 79—135. Edited By L. H. Fox, E. Fennema, and J. Sherman. Washington, D. C. National Institute of Education, 1977. ”Women and Girls in Mathematics - Equity in Mathematics Education.” Educational Studies in Mathematics 10 (1979):389-401. . ”Sex—Related Differences in Mathematics Achievement: Where and Why.” In Women and the Mathematical Mystique, pp. 76-93. Edited By L.H. Fox, E. Brody, and D. Tohin. Baltimore: Johns Hopkins University Press, 1980. Fennema, E., and Sherman, J. ”Sex-Related Differences in Mathematics Achievement, Spatial Visualization and Affective Factors.” American Educational Research Journal 14 (Winter 19775:51-71. Fennema, E., and Sherman, J. l'Sex-Related Differences in Mathematics Achievement and Related Factors: A Further Study." Journal for Research in Mathematics Education Finn, J. D. , and Mattsson, I. Multivariate Analysis In Educational Research. Chicago: National Educational Resources, 1978 Flanagan, J.C.; Dailey, J.T.; Shaycoft, M.F.; Corham, W.A.; Orr, D.B.; Godberg, 1.; and Neyman, C.A., Jr. Counselor's Technical Manual for Interpreting Test Scores. Was ington D.C.: Preject Ta ent 0 ice, 9 1. French, J.W. The Description of Aptitude and Achievement Tests in Terms 0 Rotate Factors. Psyc ometric Monograph, no. 5. Chicago: Universilty of Chicago Press, 1951. Manual for Kit of Selected Tests for Reference Aptitude and Achievement Factors. Princeton: Educational Testing Service, 1954. Fruchter, B. ”Measurement of Spatial Abilities. History and Background. ” Educational and Psycholo ical Measurement 14 (1954): 387-95. Galton, F. Inquiries into the Human Faculty and its Development. London: MacMi[lan & Co., [883. 222 Glennon, Vincent J. ”A Study in Needed Redirections in the Preparation of Elementary School Teachers of Arithmetic." Mathematics Teacher 42 (1949):389-96. Guay, R.B. ”Spatial Ability Measurement: A Critique and an Alternative.” Paper presented at the 64th annual meeting of the AERA, Boston, Mass., 7-11 April, 1980. Guay, R.B., and McDaniel, E.D. ”The Relationship between Mathematics Achievement and Spatial Abilities among Elementary School Children.” Journal for Research in Mathematics Education 8 (19775:2ll—l5. Guilford, J. P. and Zimmerman, W. F. The Guilford- Zimmerman Aptitude Survey: IV. Spatial Visualization Form B. Beverly Hills, Calif. Sheridan Supply Co., 1953. Guilford, J.P.; Green, R.F.; and Christensen, P.R. A Factor Analytic Study of Reasoning Abilities: II. Adminis— tration o Tests an Ana y51s o Resu ts. Report No. 3. Los Angeles: Univer31ty of Southern California Psychology Laboratory, 1951. Harris, L.J. "Sex-Related Differences in Spatial Ability: A Developmental Psychological View.” In Becomin Female: Perspectives on Development, pp. 133-81. Edited By C. Kopp. New York: Plenum, 1979a. . ”Variances and Anomalies.” Review of Sex-Related Differences in Co nitive Functioning, edited 5y M.A. Witting and A.G. Petersen. Selence 206 (1979b):50-52. ”Sex-Related Variations in Spatial Skill.” In Spatial Representation and Behavior across the Life ypan Theory and Application, pp. 83-125. Edited By L. S. Liben, A. H. Patterson, and N. Newcombe, New York: Academic Press, 1981. Harris, L.J., and Wagner, N. ”Performance by Young Adults on the Spatial Subset of Differential Aptitude Test: Sex Differences.” Data, Department of Psychology, Michigan State University, 1977. (Mimeographed). Hartlage, L.C. ”Sex-Linked Inheritance of Spatial Ability.” Perceptual and Motor Skills 31 (1970):610. Jacklin, C.N. ”Epilogue.” In Sex-Related Differences in Co nitive Functionin , pp. 357-71. Edited By M.A. Witting S A.G. Petersen, New York: Academic Press, 1979. 223 Kelly, A. Girls and Science: An International Study of Sex Differences in Sc 00 Seience Ac ievement. Interna— tional Assoc1ation for the Evaluation of Education Achievement Monograph Studies, No. 9. Stockholm: Almqvist and Wiksell, 1978. Koussy, A.A.H. e1. An Investi ation into the Factors in Tests Involving the Visual Perce tion of S ace. The British Journal of Psychology Monograph Supplements, no. 20. Cambridge, Eng.: The University Press, 1935. Kulm, G. ”Research on Mathematics Attitude.” In Research in Mathematics Education, pp. 356-87. Edited by R.J. Shumway. Reston, Va: National Council of Teachers of Mathematics, 1980. Lappan, G., and Winter, M.J. ''Building and Plans.” Mathematics Teaching 87 (June 1979):16-19. Lappan, G., and Winter, M.J. ”Spatial Visualization.” In Mathematics for the Middle Grades (5-9), 1982 Yearbook of the National CounCil of Teachers of Mathematics, 55. [IS-29. Edited By L. Silvey and J.R. Smart, Reston, Va: The National Council of Teachers of Mathematics, 1982. Likert, R., and Quasha, W.H. Revised Minnesota Paper Form Board Test. New York: The Psychological Corp., I948. Lim, H. Geometry and the Space Factors. Stanford: School Mathematics Study Group, Stanford University, 1963. Maccoby, E.E. ”Sex Differences in Intellectual Functioning, In The Development of Sex Differences, pp. 25-55. Edited By E. MaccoBy, Stanford, Calif.: Stanford University Press, 1966. Maccoby, E.E., and Jacklin, C.N. The Psycholo y of Sex Differences. Stanford, Calif.: Stanford University Press, 1974. MacArthur, R. ”Sex Differences in Field Dependence for the Eskimo.” International Journal of Psychology 2 (1967): 139-40. Mendicino, Lorenzo. ”Mechanical Reasoning and Space Percep- tion: Native Capacity of Experience.” Personality Guidance Journal 36 (1958):335-38. Michae1,-W.B. ”A Suggested Research Approach to the Identification of Psychological Processes Associated with Spatial Visualization Factors.‘l Educational and Psychological Measurement 14 (1954):401-6. 224 Michael, W.B.; Guilford, J.P.; Fruchter, B.; and Zimmerman, W.S. ”The Description of Spatial—Visualization Abilities.” Educational and Psychological Measurement 17 (1957):185—99. Mitchelmore, M.C. ”The Perceptual Development of Jamaican Students, with Special Reference to Visualization and Drawing of Three-Dimensional Geometrical Figures and the Effects of Spatial Training.” Ph.D dissertation, Ohio State University, 1974. Morrison, D.F. Multivariate Statistical Methods. 2nd ed. New York: Mc raw-Hi Boo Moses, B.E. ”The Nature of Spatial Ability and Its Relationship to Mathematical Problem Solving Dissertation Abstracts International A38 (1977): 4640. Murray, J.E. ”An Analysis of Geometric Ability.” Journal of Educational Psychology 40 (1949):113-24. Myers, C.T. The Effect of Training or Practice, or Both, pp Scores on CEEB Spatial Relations Test, For VACI. ETS Researc Bu etin. Princeton, . .: a ucationa Testing Service, March 16, 1951. . ”A Note on a Spatial Relations Pretest and Posttest.” Educational and Psychological Measurement 13 (1953):596—600. The Effects of Training in Mechanical Drawing on — Sp atia e ations est cores as Predictors of En gineering Draw1ng Grades. ETS Research Bulletin 58:4f Pr1nceton, N. J. Educational Testing Service, March 1958 Nash, S.C. Sex Role as a Mediator of Intellectual Function- ing. In Sex-Related Differences in Intellectual Functioning: Developmental Issues, pp. 263-302. Edited By M.A. Witting and A.G. Petersen. New York: Academic Press, 1979. O'Connor, J. Structural Visualization. Boston, Mass.: Human Engineering Eahoratory, I943. Petersen, A.G. ”Physical Androgyny and Cognitive Functioning in Adolescence.” Developmental Psychology 12 (1976):524-33. Poole, C., and Stanley, G. ”A Factorial and Predictive Study of Spatial ABilities.” Australian Journal of Ppychology 24 (1972):317-20. 225 Porteus, S.C. ”The Measurement of Intelligence: 653 Children Examined by the Binet and Porteus Tests.” Journal of Educational Psychology 9 (1918):13-31. . The Porteus Maze Test and Intelli ence. Palo Alto, Calif.: Pac1f1c BooEs, I950. . Porteus Maze Test: Fifty Years' Application. Palo Alto, Calif.: Pacific Books, I965. Roberts, J. Intellectual Develo ment of Children by Demographic and Soc1oeconom1c Factors. Department of Health, Education, and Welfare PuElication, No. HSM 72-1012, series 11, no. 110. Washington, D.C.: U.S. Government Printing Office, 1971. Royce, J.R. ”The Conceptual Framework for a Multi-Facotr Theory of Individuality. In Multivariate Analysis of Psychological Theory, pp. 305-407. Edited By J.R. Lon on an Royce. New York: Academic Press, 1973. Scheffé, H. The Analysis of Variance. New York: John Wiley & Sons, 1959. Schonberger, A.K. ”The Interrelationship of Sex, Visual Spatial Abilities, and Mathematical Problem Solving Ability in Grade Seven.” Ph.D. dissertation, Univer— sity of Wisconsin, 1976. Sedgwick, L.K. ”The Effect on Spatial Perception of Instruction in Descriptive Geometry.” Masters Thesis, Southern Illinois University, 1961. Sherman, J. On the Psycholo y of Women. Springfield, 111.: Charles C. Thomas, l97l. . ”Effects of Biological Factors on Sex-Related Differences in Mathematics Achievement.” In Woman and Mathematics: Research Perspectives for Chan e, pp. 137—206. Edited By L.H. Fox, E. Fennema, and J. Sherman. Washington, D.C.: National Institute of Education, 1977. . ”Predicting Mathematics Performance in High School Girls and Boys. Journal of Educational Psychology 79 (1979):242-49. . ”Mathematics, Spatial Visualization, and Related Factors: Changes in Girls and Boys, Grades 8—11.” Journal of Educational Psychology 72 (1980):476-82. Shepard, R.N., and Feng, C. A Chronometric Study of Mental Paper Folding. Cognitive Psychology 3 (1972):228-43. 226 Shepard, R.N., and Metzler, J. Mental Rotation of Three Dimensional Objects. Science 171 (1971):701-3. Shroyer, J., and Fitzerald, W. ”The Mouse and the Elephant.” Oregon Mathematics Teacher (February 1979):10-13. Shumway, R.J.; White, A.L.; Wheatley, G.H.; Reys, R.B.; Coburn, T.G.; and Schoen, H.L. ”Initial Effect of Calculators in Elementary School Mathematics.” Journal for Research in Mathematics Education 12 (1981): _— IIg-EI. Smith, I.M. Spatial Ability: Its Educational and Social Significance. London: UniverSity of London Press, 1964. Smith, W.S., and Litman, C.I. ”Early Adolescent Girls' and Boys' Learning of a Spatial Visualization Skill.” Science Education 63 (1979):671-76. Smith, W.S., and Schroeder, C.K. ”Instruction of Fourth Grade Girls and Boys on Spatial Visualization.” Science Education 63 (1979):61-66. Stafford, R.B. ”Sex Differences in Spatial Visualization as Evidence of Sex-Linked Inheritance.” Perceptual and Motor Skills 13 (1961):428. . Identical Blocks: Form AA. University Park: Office of Student Affairs. Pennsylvania State University, 1962. . ”New Techniques in Analyzing Parent-Child Test Scores for Evidence of Hereditary Components. In Methods and Goals in Human Behavior Genetics, pp. l7l-86. Edited by S.C. Vanderberg. New York: Academic Press, 1965. . "Hereditary and Environmental Components of Quantitative Reasoning.” Review of Educational Research 42 (1972):183-201. Stewart, V.M. ”Sex and Temperament Revisited: A Cross-Cultural Look at Psychological Differentiation in Males and Females." Paper presented at the Second International Conference of the International Association for Cross-Cultural Psychology, Kingston, Ontario, August, 1974. 227 Suydam, M.N., and Weaver, J.F. Usin Research: A Key to Elementary School Mathematics. Columhus, Ohio: ERIC Center for Soience, Mathematics, and Environmental Education, 1975. Szetela, W. ”The Effects of Test Anxiety and Success- Failure on Mathematics Performance in Grade Eight.” Journal for Research in Mathematics Education 4 ll9735:152—60. Terman, L.M., and Tyler, L.E. ”Psychological Sex Differen- ces.” In Manual of Child Psycholo y. 2nd ed., pp. 1064-1114. Edited By L. Carmichael, New York: John Wiley & Sons, 1954. Thurstone, L.L. Primary Mental Abilities. Psychometric Monograph, no. 1. Chicago: University of Chicago Press, 1938. . A Factorial Study of Perce tion. Chicago: UniverSity of Chicago Press, . . Some Primary Mental Abilities in Visual Thinking. Psychometric Laloratory Report, no. 5. C icago: University of Chicago Press, 1950. Tobias, S. Overcoming Math Anxiety. New York: W.W. Norton, 1978. Tyler, L.E. The Psycholo y of Human Differences. 3rd ed. New York: Appleton-Century-Crofts, 1965. Van Voorhis, W.R. ''Improvement of Spatial Ability by Training.” Ph.D. dissertation, Pennsylvania State University, 1941. Vanderberg, S.G. I'Twin Study of Spatial Ability.” Multivariate Behavioral Research 4 (1969):273-94. . ”Sources of Variance in Performance of Spatial Tests.” In Children's Spatial Development, pp. 57-66. Edited by J. Eliot and N. Salkind. Springfield, 111.: Charles C. Thomas, 1975. Vanderberg, S.G.; Stafford, R.B.; and Brown, A. ”The Louisville Twin Study.” In Progress in Human Behavior Genetics, pp. 153-204. Edited by S.C. Vanderberg. _ Baltimore: Johns Hopkins University Press, 1968. Very. P.S. ”Differential Factor Structures Mathematical Ability.” Genetic Psychology Monographs 75 (1967): 169-207. 228 Waber, D.P. Cognitive Abilities and Sex-Related Variations in the Maturation of Cortical Functions. In Sex- Related Differences in Cognitive Functioning, pp. - . E 1te y M. . Witting an A.C. Petersen, New York: Academic Press, 1979. Werdelin, I. Geometrical Ability and the Space Factors in Boys and Gir s. Lun , Swe en: G eerups, Wesman, A.G. ”Separation of Sex Groups in Test Reporting.” Journal of Educational Psychology 40 (l949):223-29. Wilson, J.R.; DeFries, J.C.; McClearn G.E.; Vanderberg, S.G.; Johnson, R.G.; and Rashed, M.N. ”Cognitive Abilities: Use of Family Data as a Control to Assess Sex and Age Differences in Two Ethnic Groups.” International Journal of A ing and Human Development 6 ll9755:26l-76. Winer, B.J. Statistical Principles in Experimental Desi n. New York: McUraw-Hi Boo Co., Witkin, H.A. Differentiation. Studies of Develo ment. New York: John Wiley & Sons, l962. . Cultural Influences in the Development of Cognitive Style. Proceedings of the 18th International Congress of Psychology, Moscow, 1966. Wolfe, L.R. ”Effects of Spatial Visualization Training on Spatial Ability.‘l Ph.D. dissertation, State University of New York at Albany, 1970. Wolfle, D. Factor Analysis to 1940. Psychometric Monograph, no. 3. Chicago: University of Chicago Press, 1940. Yen, W.M. ”Sex-Linked Major—Gene Influences on Selected Types of Spatial Performance." Behavior Genetics 5 (1975):281-98. Zimmerman, W.S. ”Hypotheses Concerning the Nature of the Spatial Factors.” Educational and Psycholo ical Measurement 14 (195452396—406. 'IICHIGRN STQTE UNIV. LIBRARIES I ll lllllll ll Ill l l 31293006766061