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Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 75-11,398 MITCHELMQRE, Michael Charles, 1940- THE PERCEPTUAL DEVELOPMENT OF JAMAICAN STUDENTS, WITH SPECIAL REFERENCE TO VISUALIZATION AND DRAWING OF THREE-DIMENSIONAL GEOMETRICAL FIGURES AND THE EFFECTS OF SPATIAL TRAINING. The Ohio State University, Ph.D., 1974 Education, psychology

Xerox University Microfilms, Ann Arbor, Michigan 48io6

0 1975

MICHAEL CHARLES MITCHELMQRE

ALL RIGHTS RESERVED

THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. THE PERCEPTUAL DEVELOPMENT OF JAMAICAN STUDENTS, WITH SPECIAL REFERENCE TO VISUALIZATION AND DRAWING OF THREE-DIMENSIONAL GEOMETRICAL FIGURES AND THE EFFECTS OF SPATIAL TRAINING

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By

Michael Charles Mitchelmore, M.A., Cert, Ed.

*****

The Ohio State University 1974

Reading Committee: Approved by

F. Joe Crosswhite

Fred L. Damarin, Jr.

Dean H. Owen Adviser Richard J. Shumway îollege of Education ACKNOWLEDGEMENTS

This study, although executed in one country (Jamaica), was planned in a second (the United States), supported by funds from a third (the United Kingdom), and based on ideas conceived in a fourth (Ghana). It therefore owes its suc­ cessful completion to an international band of colleagues, advisers, encouragers, supporters and cooperators.

I should like first to thank all those school and college principals, teachers and students in Jamaica who so willingly gave me their time and interest. The eager cooper­ ation of the eight schools involved in the pilot-testing and the forty schools involved in the developmental and Grade

Nine surveys turned a lengthy task into a most enjoyable one,

I am particularly grateful to the staff of Mico College for making me feel so much a part of their institution— especial­ ly to the principal. Dr. Errol Miller, whose early interest made a Jamaican study feasible and who gave his continual support to the spatial training experiment; to the head of the mathematics department, Mr. A. Hibbert, who undertook all the administrative details of the experiment; and to all the other mathematics tutors of the college.

ii In the planning of the tv: i surveys, I received what

I now regard as typically Jamaican cooperation from officials

of the Examinations Section of the Ministry of Education,

the Overseas Examinations Centre, and the Department of Town

Planning, who not only granted me access to confidential and

unpublished data but also gave me considerable help in inter­

preting it for my peculiar purposes. Members of the School

of Education in the University of the West Indies at Mona

were also most helpful. I am grateful to Dr. L. H. E. Reid

for accepting me as an Occasional Student and to Dr. A. S.

Phillips for providing me with a letter of introduction

authorizing me Lo conduct research in the schools. My

deepest gratitude goes to Mr. Ian Isaacs for sharing with me not only his life-long knowledge of Jamaica and its

schools and his more recent experiences in educational re­

search, but also (with his wife Pat) the hospitality and

friendship of his home. It was because of my previous

association with Mr. Isaacs as adapter of the Caribbean

edition of the Join Schools Project mathematics series that

I was invited to Jamaica in 1972, and I shall always be thankful to him for encouraging me to return.

For their instruction, assistance and comradeship during my years at OSU, I am deeply grateful to all the

iii members of the Faculty of Science and Mathematics Education;

especially to the chairman, Robert Howe; my adviser, Joe

Crosswhite; Richard Shumway, who was also on my reading

committee; Jon Higgins, who served on my advisory committee;

and Alan Osborne and Harold Trimble. I also thank James

Duncan of Curriculum and Foundations for serving on my advisory committee.

This study would not have been possible without the exceptional flexibility of my adviser. In mathematics education research, studies of the visualization aspect of geometry are rare, of drawing even more so, and cross- cultural studies unheard of; yet "Papa Jre" gave me nothing but encouragement as my planned study grew and took its final shape. I am also very grateful to the two psychologists on my committee for contributing from their specialities; Fred

Damarin in testing and measurement and Dean Owen in per­ ception.

This dissertation was written and typed while I was still in Jamaica, and I am indebted to my committee for making time to see me on my one trip to Columbus, for reviewing draft chapters by mail at short notice, and for completing in my absence many of the formal arrangements for graduation which are normally the candidate's responsibility. My headaches

iv at this stage were greatly alleviated by the known relia­ bility of my typist, Jane Burrows, who deserves my sincere thanks for working so speedily and accurately.

I would like to take this opportunity to thank the psychologists and educators all over the world who have taken time to correspond with me about their work, sent reprints, and commented on my research proposals. I am especially grateful to John Berry, Alan Bishop, John Dawson, Jan

Deregowski, Hall Duncan, John Eliot, I. Macfarlane Smith, and Philip Vernon. Their willingness to help the distant, unknown neophyte was a fine example to follow should I ever attain their stature.

It was an Educational Development Award from the

Ministry of Overseas Development, London, which allowed me to spend the year in Jamaica which was needed to carry out the present study. Without that substantial support, only a shorter and more poorly planned spatial training experi­ ment would have been possible. A further grant, from the

Society for the Psychological Study of Social Issues, covered the cost of test papers and other equipment. Mico

College kindly bore the cost of producing the workcards and other materials used in the spatial training experiment (no small contribution) and the School of Education at the

V University also helped with the duplication of test papers.

For all this financial assistance, I am most grateful.

I must also thank Educational Testing Service and Longman

Group Ltd. for allowing me to reproduce material for which

they hold the copyright, the American Institutes for Research

for sending me background material on their I-D test series,

and the Australian Council for Educational Research for per­ mission to use the Pacific Design Construction Test.

My thinking on the aims of mathematics education in developing countries took shape during my early years of teaching in Ghana. Conversations on geometry with other mathematics teachers, especially Hugh Benzie, John Fletcher,

Neil Harding, Marion Harbourn, and Kenneth Snell, were part of my education. Our conclusions were put into print in the Joint Schools Project mathematics series, and all those associated with that project will long be remembered for their kindness and friendship. I am particularly grateful to Dr. Mary Hartley, who gave the project life and resus­ citated it at several critical moments, and to Brian Raynor, my co-editor for ten years, who has frequently taught me how to turn unworkable ideals into teachable practice.

My wife June and daughters Margaret, Sylvia and

Catherine have accompanied and supported me in all four vi countries. Actually, they seemed rather to enjoy it, and thrived wherever we pitched. For this study, they acted as guinea pigs during test development and collated thousands of test papers. My wife also went through the entire manuscript with a fine toothcomb, as a result of which many muddled thoughts and sentences were clarified.

I am very grateful for all their help.

Vll VITA

July 25, 1940 . . . Born at Dartmouth, England

196 1 ...... B.A., University of Cambridge, England

196 2 ...... Certificate in Education, University of Bristol, England

1962-66 ...... Assistant Mathematics Master, Mfantsipim School, Cape Coast, Ghana

1965...... M. A., University of Cambridge, England

1966-67 ...... Senior Mathematics Master, Tema Secondary School, Tema, Ghana

1967-71 ...... Technical Adviser, Mathematics Educa­ tion Research Unit, University of Ghana, Legon, Ghana

1971-72 ...... Research Associate, ERIC Clearing­ house for Science, Mathematics and Environmental Education, The Ohio State University, Columbus, Ohio

July 1972 ...... Visiting professor, OAS Mathematics Institute, University of the West Indies, Kingston, Jamaica

1972-73 ...... Teaching Associate, Faculty of Science and Mathematics Education, The Ohio State University, Columbus, Ohio

1973-74 ...... Holder, ODM Educational Development Award, Kingston, Jamaica

vxix PUBLICATIONS

Mitchelmore, M. C., & Raynor, B. (Eds.) Joint Schools Project mathematics. London: Longman, 1967-75. 5 vols

Mitchelmore, M. C., & Raynor, B. The Joint Schools Project of the Mathematical Association of Ghana. In A. G. Howson (Ed.), Developing a new curriculum. London: Heinemann, 1970.

Mitchelmore, M. C., Raynor, B., & Isaacs, I. (Eds.) Joint Schools Project mathematics. Caribbean edition. London: Longman, 1970-75. 5 vols.

Mitchelmore, M. C. The Joint Schools Project. Mathematics Teaching, 1971, 54, 30-32.

Mitchelmore, M. C. A saga of neglected opportunities. Mathematics Teaching, 1971, 55, 6-9.

Mitchelmore, M. C. Spatial ability and three-dimensional drawing. Magazine of the Mathematical Association (Kingston, Jamaica), 1973, 5(2), 15-40.

Mitchelmore, M. C. Catherine's discovery. The Arithmetic Teacher, 1974, 21, 90-91.

Mitchelmore, M. C. A matter of definition. American Mathematical Monthly, 1974, 81, 643-647.

Mitchelmore, M. C. Performance in a modern mathematics curriculum. West African Journal of Education, 1973, 17, 295-305.

Mitchelmore, M. C. Development and validation of the Solid Representation Test in a cross-sectional sample of Jamaican students. Paper presented to the Second Inter­ national Conference of the International Association for Cross-cultural Psychology, Kingston, Ontario, August 1974.

IX Mitchelmore, M. C., & Raynor, B. Joint Schools Project mathematics. Metric edition. London: Longman, 1974.

Mitchelmore, M. C., Raynor, B., & Jarman, E. Joint Schools Project mathematics. East African edition. London: Longman, 1975.

FIELDS OF STUDY

Studies in Mathematics. Professor James L. Leitzel

Studies in Mathematics Education. Professors F. Joe Crosswhite and Alan H. Osborne

Studies in Psychology. Professors Fred K. Damarin and Dean L. Owen

Studies in Research Methods. Professors John J. Kennedy and Richard J. Shumway

X CONTENTS

Page

ACKNOWLEDGEMENTS...... ii

VITA...... viii

LIST OF TABLES...... xviii

LIST OF FIGURES ...... xxiii

LIST OF ABBREVIATIONS...... xxvi

CHAPTER

1 INTRODUCTION ...... I

1.1 Geometry education...... 1 1.2 Spatial ability ...... 4 1.21 Some definitions...... 4 1.22 The factorial structure of spatial ability ...... 7 1.23 The relation of spatial abil­ ity to mathematical and technical ability ...... 9 1.3 Spatial ability in developing c o u n t r i e s ...... 13 1.31 Difficulties in learning geometry...... 13 1.32 Cross-cultural psychological studies of perception ...... 14 1.33 Implications and research needs . 15 1.4 The present study ...... 18 1.41 Location...... 18 1.42 Research questions...... 19 1.43 The research program...... 20 1.44 Preview ...... 22

XI CONTENTS (continued)

CHAPTER Page

2 PERCEPTUAL DEVELOPMENT ...... 24

2.1 Depth perception...... 24 2.11 The basic problem ...... 24 2.12 Empiricist theories ...... 25 2.13 Nativist theories ...... 27 2.2 Form perception and representation. . . 29 2.21 Form discrimination and copying . 29 2.22 Obliques...... 31 2.23 Perception versus performance . . 32 2.3 Pictorial depth perception and representation...... 34 2.31 Pictorial depth perception. . . . 34 2.32 Pictorial depth representation. . 35 2.33 Representation of spatial r e l a t i o n s ...... 39 2.4 Development of spatial visualization. . 41 2.41 Sectioning of solids...... 41 2.42 Coordination of perspectives. . . 42 2.43 Surface development ...... 43 2.44 Psychometric tests of visuali­ zation...... 44 2.5 Perceptual development...... 45 2.51 The Piagetian model ...... 45 2.5 2 Perceptual learning ...... 47 2.5 3 A mathematical model of per­ ceptual development...... 50 2.54 Representational learning .... 57 2.6 Cross-cultural studies of perception and representation...... 62 2.61 Psychometric studies...... 62 2.62 Perceptual development...... 63 2.63 Pictorial depth perception. . . . 65 2.64 Pictorial depth representation. . 65 2.7 Influences on perceptual development. . 67 2.71 Cultural pressures...... 68 2.72 The physical environment..... 72 2.73 Physiological factors ...... 75 2.74 Sex differences...... 81

X I 1 CONTENTS (continued)

CHAPTER Page

2 (continued) 2.8 Geometric illusions ...... 85 2.81 Illusion susceptibility ...... 85 2.82 Inappropriate constancy scaling . 87 2.83 Field independence...... 90 2.84 Other factors ...... 91 2.85 Developmental patterns...... 93

3 BACKGROUND TO JAMAICA...... 99

3.1 A brief geography and history ...... 99 3.11 Geography ...... 99 3.12 H i s t o r y ...... 100 3.2 The educational system...... 104 3.21 Overview...... 104 3.22 Elementary education...... 104 3.23 Secondary education ...... Ill 3.24 Post-secondary education...... 118 3.3 Some social factors affecting ed u c a t i o n ...... 120 3.31 Language...... 121 3.32 E t h n i c i t y ...... 122 3.33 Family structure...... 123 3.34 Sex bias...... 125 3.35 Environment...... 128

4 TEST SELECTION AND DEVELOPMENT...... 130

4.1 Psychological testing in developing c o u n t r i e s ...... 130 4.11 Test procedures...... 131 4.12 Item content...... 132 4.13 Score interpretation...... 133 4.14 Implications for this study . . . 134 4.2 Test selection...... 138 4.21 Group tests ...... 138 4.22 Individual tests...... 142 4.23 Personality characteristics . . . 145

xixi CONTENTS (continued)

CHAPTER Page

4 (continued) 4.3 Development of group tests...... 150 4.31 Pilot-testing procedures...... 150 4.32 Three-Dimensional Drawing test. . 153 4.33 Hidden Shapes Test...... 155 4.34 Boxes test...... 168 4.35 Personal Data Questionnaire . . . 171 4.36 The group test battery...... 173 4.4 Development of individual tests .... 174 4.41 Pilot-testing procedures...... 174 4.42 Pacific Design Construction Test. 176 4.43 Hidden Figures T e s t ...... 176 4.44 Horizontal-Vertical Test...... 180 4.45 Solid Representation Test .... 181 4.46 Geometric Illusion Measures . . . 187 4.47 Eye-Hand Dominance Schedule . . . 186 4.48 Personal Data Interview...... 190 4.49 The individual test battery . . . 191

5 THE DEVELOPMENTAL SURVEY ...... 194

5 .1 Background...... 194 5.11 Purpose ...... 194 5.12 Delimitations...... 195 5.2 Method...... 196 5.21 Design...... 196 5.22 Selection of schools...... 197 5.23 Procedures...... 199 5.3 Results of spatial ability tests. . . . 203 5.31 Design Construction Test...... 203 5.32 Hidden Figures T e s t ...... 207 5.33 Horizontal-Vertical Test...... 212 5.34 Solid Representation Test .... 219 5.35 Relations between spatial test scores...... 230 5.4 Geometric illusion results...... 234 5.41 Hypotheses...... 234 5.42 Order effect...... 236 5.43 Developmental analysis...... 237

XIV CONTENTS (continued)

CHAPTER Page

5 (continued) 5.44 Correlations between illusions. . 240 5.45 Relation to spatial ability . . . 241 5.45 Three-dimensional interpreta­ tion of illusion diagrams .... 244 5.5 Influence ofbackground variables . . . 245 5.51 Background data ...... 245 5.52 Relation to spatial ability . . . 250 5.53 Relation to illusion suscepti­ bility...... 254

6 THE GRADE NINE SURVEY...... 256

6.1 Background...... 256 6.11 Purpose...... 256 6.12 Delimitations...... 257 6.2 Method...... 259 6.21 Design...... 259 6.22 Identification of schools .... 260 6.23 Environmental classification. . . 261 6.24 Final definition of survey population...... 264 6.25 Selection of school sample. . . . 264 6.26 Procedures...... 266 6.3 R e s u l t s ...... 269 6.31 Numbers ...... 269 6.32 Test characteristics...... 271 6.33 Method of analysis...... 273 6.34 School type comparison...... 275 6.35 Environmental comparison...... 280 6.36 Influence of background variables 284

7 THE SPATIAL TRAINING EXPERIMENT...... 291

7.1 Review of previous research ...... 291 7.11 Technical programs...... 291 7.12 Mathematical programs ...... 295 7.13 Perceptual training programs. . . 297 7.14 Spatial training programs .... 299

XV CONTENTS (continued)

CHAPTER Page

7 (continued) 7.2 Theory and practice of spatial training...... 308 7.21 Principles of spatial training. . 308 7.22 The JSP spatial training program. 312 7.3 Method...... 317 7.31 The setting ...... 317 7.32 Development of instructional m a t e r i a l s ...... 319 7.33 Pilot-testing...... 323 7.34 Experimental designs...... 328 7.35 Hypotheses...... 333 7.36 Procedures...... 334 7.4 Results...... 342 7.41 Numbers ...... 342 7.42 Test characteristics...... 344 7.43 Treatment effects on 3-dimen­ sional drawing and visualization. 348 7.44 Treatment effects on field independence...... 356 7.45 Treatment effects on achievement. 35 7 7.46 Conclusions...... 363 7.5 Further analyses of teachers' college data...... 366 7.51 Influence of background variables...... 366 7.52 College-school comparisons. . . . 371 7.53 Results of individual testing . . 371 7.54 Validity of spatial ability tests 373

8 SUMMARY AND CONCLUSIONS...... 376

8.1 Aims and procedures ...... 376 8.2 Results and conclusions...... 378 8.21 Reliability of the spatial tests. 379 8.22 The structure of spatial ability. 383 8.23 Sex differences in the develop­ ment of spatial a b i l i t y ...... 385

XVI CONTENTS (continued)

CHAPTER Page

8 (continued) 8.24 Influence of background variables on spatial ability 390 8.25 Training of spatial ability 394 8.26 Geometric illusions . . . 3 9 6 8.3 Implications and recommendations 397 8.31 Technical education 397 8.32 Geometry in schools 403 8.33 Teacher education . 406 8.34 Test development. . 408 8.35 Further research. . 412

BIBLIOGRAPHY. 418

Note Appendixes A-N are included in Volume II. A detailed list of contents is to be found on page xxvii in that volume.

xvxi LIST OF TABLES

TABLE Page

Introduction

1.1 Principal space factors: Notation, definition and common marker tests...... 8

1.2 Factor loadings of selected tests (Werdelin, 1951)...... 11

Perceptual development

2.1 Piagetian stages in the development of spatial perception...... 46

Background to Jamaica

3.1 Principal public examinations ...... 106

3.2 Numbers of elementary schools and total enrollment in 1972-73, by type...... 110

3.3 Numbers of secondary and vocational insti­ tutions and total enrollment in 1972-73, by type...... 117

3.4 Enrollment in post-secondary education in 1972-73, by institution ...... 120

3.5 Ethnic composition (1960) ...... 122

Developmental survey

5.1 Mean age and length of education, by sex and grade...... 203

5.2 Frequencies of correct SRT responses, by solid and c o n d i t i o n...... 224

xviii LIST OF TABLES (continued)

TABLE Page

5.3 Comparison of SRT responses in three conditions, by solid...... 225

5.4 Frequency distribution of SRT scores in Condition 1, by grade and solid ...... 227

5.5 Summary of spatial test r e s u l t s ...... 231

5.5 Correlations between spatial test scores, by sex...... 232

5.7 Partial correlations between spatial test scores controlling for age, by sex...... 233

5.8 Correlations between geometric illusion susceptibilities, by age...... 241

5.9 Correlations between spatial test scores and geometric illusion susceptibilities, by sex . 242

5.10 Summary statistics on background variables. . 245

5.11 Correlations between background variables, by sex...... 249

5.12 Correlations of spatial test scores with background variables, by sex...... 251

5.13 Partial correlations of spatial test scores with background variables controlling for age, by s e x ...... 25 3

5.14 Correlations of geometric illusion suscepti­ bilities with background variables, by sex. . 255

XIX LIST OF TABLES (continued)

TABLE Page

Grade Nine Survey

5.1 Numbers of schools in survey population, by environment and school t y p e ...... 265

6.2 Numbers of students in survey sample, by school...... 270

6.3 Correlations between spatial test scores, by sex...... 273

6.4 Percentages of students scoring over 50% on BOX test, by sex and school type...... 279

6.5 Summary statistics on background variables in city and town schools, by school type. . . 286

6.6 Summary statistics on background variables in refined junior secondary and all-age school data, by environment ...... 287

6.7 Correlations between selected background variables, by s e x ...... 288

6.8 Correlations of background variables with spatial test scores, by s e x ...... 289

Spatial training experiment

7.1 Assignment of first year classes and tutors to treatments...... 330

7.2 Research design for first year experiment and numbers of students available for each cell...... 331

7.3 Research design for second year experiment and numbers of students available for each cell...... 332

XX LIST OF TABLES (continued)

TABLE Page

7.4 Sources of students for individual testing. . 336

7.5 Numbers of first year students with full data, by level, sex, treatment and pretest condition...... 343

7.6 Numbers of second year students with full data, by tutor, sex, treatment and pretest c o n d i t i o n ...... 343

7.7 Reliabilities of spatial test scores, by year...... 345

7.8 Test-retest correlations for spatial tests in experimental classes, by year...... 347

7.9 Correlations between spatial pretests in both years combined, by s e x ...... 347

7.10 Significance levels of main effects in analyses of variance of 3DD and BOX pretest scores, by y e a r ...... 35 2

7.11 Significance levels of main effects in analyses of covariance of 3DD and BOX post­ tests using corresponding pretest as covariate, by year...... 356

7.12 Significance levels of main effects in analyses of variance of achievement test scores, by y e a r ...... 358

7.13 Summary statistics on background variables, by y e a r ...... 368

7.14 Correlations between background variables, by sex...... 369

7.15 Correlations of background variables with spatial pretest scores, by sex...... 370

XXX LIST OF TABLES (continued)

TABLE Page

7.15 Correlations between group and individual spatial tests...... 373

7.17 Correlations of spatial pretest scores with learning potential, mathematics and industrial arts scores, by sex and year. . . 375

Note Supplementary tables are to be found in Volume II (Appendixes A-D).

X X I 1 LIST OF FIGURES

FIGURE Page

1.1 A hypothetical spatial test item...... 6

2.1 How the image of a 3D figure is produced on the retina of the e y e ...... 25

2.2 Stages in children's drawings of a house. . . 38

2.3 "Sameness" in an undifferentated set...... 51

2.4 "Sameness" in a differentiated set...... 52

2.5 Stages in the development of form perception. 53

2.6 Differentiation of mirror images...... 54

2.7 Four well-known geometric illusions ...... 85

2.8 Age trends in illusion susceptibility in three cultures...... 94

3.1 Topography of J a m a i c a ...... 101

3.2 Jamaican parishes and parish capitals .... 101

3.3 The Jamaican educational system, 1973 .... 105

3.4 Distribution of major economic activities in Jamaica...... 129

4.1 Steps involved in the solution of a 3-dimensional trigonometry problem...... 157

4.2 Common errors on Cubes item of 3DD...... 164

4.3 Sample item from I-D Boxes t e s t ...... 169

4.4 Top view of SRT apparatus...... 184

xxxii LIST OF FIGURES (continued)

FIGURE Page

4.5 Subjects'-eye-view of SRT apparatus ...... 184

5.1 Research design for the developmental survey...... 197

5.2 Mean DCT scores by grade and sex...... 205

5.3 DCT item No. 9 and some typical errors. . . . 208

5.4 Mean HFT scores by grade and sex...... 209

5.5 HFT practice item 3 and test item 3, with errors typical of younger students...... 211

5.6 Mean Poles scores by grade and sex...... 215

5.7 Mean Bottles scores by grade and sex...... 217

5.8 Typical SRT responses at each stage of development, by solid ...... 223

5.9 Mean SRT drawing and selection scores by grade and sex...... 230

5.10 Mean GIM scores by grade and sex...... 238

6.1 Research design for the Grade Nine survey . . 260

6.2 Location of schools sampled in Grade Nine survey...... 268

6.3 Mean spatial test scores in city and town schools, by sex and school t y p e ...... 277

6.4 Mean spatial test scores in refined junior secondary and all-age school data, by sex, environment and school t y p e ...... 283

7.1 Mean scores on 3DD posttests by level/tutor, pretest condition, treatment, sex and year. . 350

XXIV LIST OF FIGURES (continued)

FIGURE Page

7.2 Mean scores on BOX posttests by level/tutor, pretest condition, treatment, sex and year. . 351

7.3 Mean scores on 3DD posttests adjusted for covariance with 3DD pretests, by level/tutor, treatment, sex and y e a r ...... 354

7.4 Mean scores on BOX posttests adjusted for covariance with BOX pretests, by level/tutor, treatment, sex and y e a r ...... 355

7.5 Mean scores on HST posttest in first year adjusted for covariance with HST pretest, by level, sex and treatment...... 357

7.6 Mean scores on Solids achievement test, by level/tutor, treatment, sex and year...... 359

7.7 Mean scores on Statistics achievement test, by level/tutor, treatment, sex and year . . . 360

XXV LIST OF ABBREVIATIONS

Numbers in brackets indicate the sections in which defini­ tions may be found.

BOX I-D Boxes test (4.34)

DCT Pacific Design Construction Test (4.42)

GCE General Certificate of Education (3.23)

GIM Geometric Illusion Measures (4.46)

GNAT Grade Nine Achievement Test (3.23)

HFT Hidden Figures Test (4.43)

HST Hidden Shapes Test (4.33)

HVT Horizontal-Vertical Test (4.44)

JSP Joint Schools Project (7.22)

PDI Personal Data Interview (4.48)

PDQ Personal Data Questionnaire (4.35)

SRT Solid Representation Test (4.45)

2D 2-dimensional (1.21)

3D 3-dimensional (1.21)

3DD Three-Dimensional Drawing test (4.32)

11+ Common Entrance Examination (3.23)

13+ Technical Schools Common Entrance Examination (3.23)

XXVI CHAPTER ONE

INTRODUCTION

1.1 Geometry education

Geometry may be defined as the study of the logical

relations between various spatial concepts. Accordingly,

every geometry course taught in school or college calls on

two major facets of the intellect; logical reasoning and

spatial ability. Elementary courses emphasize spatial in­ tuition, but logic still enters in the classification of

figures and grows in importance as more and more properties and the relations between figures are studied. Similarly, advanced courses, although they may be more concerned with the inferential structure of the subject, rely on spatial intuition in their exposition and application. An elementary geometry textbook which does not ask "Why?" occasionally is as misguided in its approach as an advanced textbook which never asks "Where?"

Geometry education at the secondary level has been drawn in two directions as first logic and then spatial intuition has been emphasized. In England, only "Euclid" was taught until the foundation of the Association for the Improvement of Geometry Teaching (now the Mathematical Association) in

1870, after which there was a steady lessening of the emphasis on formal logic and a decrease in the number of theorems required for examinations, reaching zero within the past decade (Combridge, 1972)> This process was accom­ panied by a growing emphasis on the development of spatial intuition as first practical geometry and then transforma­ tion geometry were introduced. The latest phase of the swing was accelerated by the foundation of the Association for

Teaching Aids in Mathematics (now the Association of

Teachers of Mathematics) in 195 2.

In the United States, a trend toward a more practical approach to secondary mathematics reached its peak in about

1935, but the Grade 10 formal geometry course was largely unaffected. The so-called "New Math" revolution of circa

1960 brought an increase in rigor and a corresponding re­ duction in the role of spatial intuition (Osborne and

Crosswhite, 1970). Some geometry was introduced into the elementary school course, but this consisted very largely of nomenclature. A reaction against the abstract nature of their first courses led the School Mathematics Study

Group to produce a more practical secondary school course which had the development of spatial intuition as one of

its specific aims (School Mathematics Study Group, 1972).

How well this latest swing will catch on remains to be

seen, as the latest considerations of the geometry curriculum

are still mostly geared to an axiomatic, but not necessarily

euclidean, geometry course in Grade 10 (Henderson, 1973).

Similar swings between emphases on logic and space in

school geometry have evidently also taken place in Russia

(Chetverukhin, 1971). Current emphases in continental

Europe seem to be on the logical aspects (Servais & Varga,

1970).

However the pendulums may swing, both logic and spatial intuition will continue to be important in the learning and

teaching of geometry. This study is concerned only with the

spatial aspect, which is likely to be more important in the general education of the individual and in the applica­

tions of mathematics to the physical world (Klarnkin, 1971) .

The spatial aspect is also of more immediate concern in developing countries, where there is evidence for a defi­ ciency in spatial ability which could have serious effects on the learning of geometry (see Section 1.3). Therefore, nothing more will be said about logic in geometry. The discussion of spatial ability will range far and wide, but always with the aim of clarifying those problems that students have in learning and applying geometry which can be ascribed to difficulties of visualization rather than of reasoning.

1.2 Spatial ability

1.21 Some definitions

There are two characteristics which differentiate spatial ability from other intellectual constructs; It is concerned with space figures, and it is concerned with their visualization.

A figure is any set of points in space. In this study, figures will either be 3-dimensional (3d ), in which case they will be called objects, configurations, or scenes, or they may be 2-dimensional (2D), when they will be called drawings, pictures, diagrams, or forms. Diagrams are frequently intended to depict objects; such diagrams will be called representations of the objects. The process of producing a representation of an object will be called 3D drawing. A transformation of a figure is a mapping of the figure from one space into the Scime cr a different space.

Thus 3D drawing approximates a transformation of the object from 3D space into the 2D space of the paper on which the diagram is drawn.

Visualization can be described as the formation of an image of a figure and the mental manipulation of the image according to specified transformations. In behavioral terms, visualization consists of matching a figure to its transform, usually by identification, discrimination, or reproduction. This definition includes 3D drawing. The important distinction is that the transformation is specified in advance, and any reasoning employed involves only prop­ erties of that transformation. If the transformation is not specified in advance, the ability involved would better be described as spatial reasoning.

An example may make clearer the distinction between spatial ability and spatial reasoning. In Fig. 1.1, the question may be, "Which of the figures (B)-(E) can be ob­ tained from (A) by turning it around but not over?" Since the transformation is specified, this is a test of spatial ability. A subject may note, for example, that (B) can be obtained from (A) by turning it around, that (C) can be obtained from (B) by turning it over, and that (A) must therefore be turned over to obtain (C); this reasoning in­ volves only the properties of the given transformations.

Suppose instead the question was, "Which one figure is 1 p Gl o r d P

(A) (B) (C) (D) (E)

Fig» 1.1 A hypothetical spatial test item. Spatial ability is tested by asking, "Which of the figures (B)-(E) can be obtained from (A) by turning it around but not over?" Spatial reasoning is tested by asking, "Which one figure is different from the other four?"

different from the other four?" The transformation is not

given, so this is a test of spatial reasoning: the subject

must search his repertoire of transformations until he finds

one which classifies one figure differently from the other

four. Figurai items of the form "A:B as C:?" are also tests

of figurai reasoning. It is seen that spatial reasoning

depends on spatial ability: A subject who could not

visualize rotations would certainly not be able to find the

"odd man out" in Fig. 1.1. On the other hand, a subject who

could visualize rotations would not necessarily be able to

find the odd man out either. The two skills are not

identical. 1.22 The factorial structure of spatial ability

The work of factor analysts such as Thurstone, Burt,

Vernon and Guilford has lent empirical validity to the con­

struct of spatial ability. To Thurstone (1938), spatial

ability was a "primary mental ability," one of the seven

clearly-defined factors to emerge from his analysis of over

sixty tests. Burt (1949) and Vernon (1951) are among the

many British psychologists who have obtained a single

spatial factor k remaining after extraction of a general

intelligence component. American psychologists, using a

different method, have frequently obtained several spatial

factors. Guilford (1967) has most clearly separated spatial

ability from spatial reasoning in his Structure of the Intel­

lect model, which contains 120 factors. The full list of

factor-analytic studies is far too long to review here, even

if it were restricted to those primarily concerned with clarifying the nature of the space factors. Relevant re­

views have been published by French (1951), Guilford (1967),

Michael et al. (1957), Smith (1964), and Werdelin (1961).

Table 1.1 summarizes the space factors which appear to be well-established, the notations used by French et al.

(1953) and Guilford (1967), and some tests which load strong­

ly on each factor (marker tests). It should be noted 8

Table 1.1

Principal space factors: Notation, definition and common marker tests

Notation Definition Market Tests French Guilford EFU Perceptual Speed : The abil­ Identical Pictures ity to hold a configuration Finding A 's in mind so as to make identi­ Number Comparisons fication of its image under a translation, in the ab­ sence of any distracting visual information.

Cf NFx Field Independence; The Hidden Figures ability to hold a configura­ Gottschaldt Figures tion in mind so as to make Copying Patterns identification of its image under a translation, in tlie presence of distracting visual information

CFS-V Spatial Orientation: The Cube Comparisons ability to visualize a total Card Rotations rigid configuration moved Flags to a new position and to appreciate spatial order or arrangement of objects, primarily with respect to one's own body as a frame of reference

Vz CFT Spatial Visualization: The Paper Folding ability to mentally manipu­ Punched Holes late parts within a total Form Board configuration, and to Surface Development recognize the new position, location, or appearance of the obj ect that the space factors are not uncorrelated; in fact, a large number of spatial tests have to be included in the test battery in order to separate the space factors, especially S and Vz. The definitions in Table 1.1 indicate how close the various abilities are, and, as Myers (1957) has pointed out, it is not clear whether the different factors arise from distinct abilities or from differences in subjects or in test format or instructions. Zimmermann (1954a, 1954b) showed that the same item type could be used to measure P,

Cf, S, or Vz simply by increasing its complexity. In a recent study, Borich and Bauman (1972) used Guilford's tests of S and Vz and French's tests of the same factors, and found that the pairs of tests produced by each author were more highly intercorrelated than the pairs of tests purporting to measure each factor.

1.23 The relation of spatial ability to mathematical and technical ability

Several factor-analytic studies have included mathe­ matical tests in the battery. Extraction of a general intel­ ligence factor then allowed the researcher to estimate the independent contributions of numerical, verbal and spatial ability to mathematical achievement. Results differ widely, depending on the type of mathematical tests used as well as 10 the range of marker tests in the battery and the precise method of analysis.

The most extensive studies available are those of

Barakat (1951), Wrigley (1958) and Werdelin (1961). Barakat and Wrigley included tests of Algebra, Geometry, Problem

Arithmetic, and Mechanical Arithmetic in their battery, as well as verbal and non-verbal intelligence tests and various other marker tests. They both found that the mathematical tests loaded heavily on the general intelligence factor, negligibly on the numerical, verbal and spatial factors, and moderately on a fifth factor which seemed to refer to aca­ demic achievement. Werdelin (1961) concentrated on geometry tests; his factor loadings, in three samples of secondary age children, are summarized in Table 1.2. The lower loadings on the general factor are probably due to the fact that Werdelin did not include general intelligence tests in his battery. The results show clearly that mathe­ matical ability is not a unitary trait, but that different tests require different skills. The loadings for the geo­ metrical tests are particularly interesting; as the problems become more complex, the spatial loading decreases and the general reasoning loading increases. Numerical and verbal loadings are largest where expected, but they 11

Table 1.2

Factor loadings of selected tests (Werdelin, 1951)

Factor^ Test Content Sample^ R NV S Sections Plane sections A 13 -13 18 28 of mathematical B 18 -23 08 41 solids C -12 -09 14 44

Geometrical Sketching A 14 07 15 52 Construc­ diagrams from B 18 -09 26 53 tion instructions C 27 -03 28 51

Geometrical Elementary A 29 20 14 32 Problems I metric B 33 21 08 44 geometry C 37 16 23 26

Geometrical Metric geometry A 53 08 06 34 Problems II of triangles B 55 23 04 32 C 49 16 14 28

Geometry Proving theorems A 35 01 11 13 B 10 17 29 34 ' C 46 08 29 23

Equations Solving linear A 32 33 05 08 equations B 33 37 -02 06 C 34 46 -11 20

Arithmetic Word problems A 62 00 09 11 B 56 20 17 16 C 52 28 32 17

^Sample A: 127 boys, B: 232 boys, C : 183 girls.

Factor R: General reasoning, N: numerical, V: verbal, S: spatial. 12 remain generally smaller than the spatial loadings.

Many further studies relating mathematical and spatial abilities are summarized by Smith (1964) and Werdelin (1961),

Smith concludes:

It is clear, however, from the trend of the results in the majority of studies mentioned that the numerical factor has little in common with what is normally understood by "mathematical ability." ... The spatial factor appears to have a greater claim for consideration as an essential basis for aptitude for mathematics (1964, po 126).

Smith (1964) also quotes a long list of validity studies. Validities of spatial tests for predicting per­ formance in mathematics hover around 0.20 to 0.30. For technical drawing and similar studies, the validity tends to be higher, about 0.40 to 0.50. Smith again:

Pupils who perform well in spatial tests (rela­ tive to verbal and English tests) tend to do well in subjects for which the criterion of success depends on the execution of an actual job of work, whether in technical drawing, metalwork, wood­ work, mechanical science, mathematics, or art (1964, p. 177).

More recent studies have continued to bear out Smith's conclusion. For example, Sowder (1973) found that the SMSG

Paper Folding test (Wilson et al., 1968) was a good dis­ criminator for geometry achievement in Grade 10. Other recent studies are those of Baker and Lalley (1972), 13

Caldwell (1970), Hanna (1965), McCallum (1970),

Siemankowski and MacKnight (1971), and Westbrook (1965).

1.3 Spatial ability in developing countries

1.31 Difficulties in learning geometry

That students in developing countries might be prevented from realizing their full mathematical potential by short­ comings in spatial ability first occurred to the author about ten years ago while teaching mathematics in Ghana, especially while working as an assistant examiner for the

West African Examinations Council. For example, 0-level students (aged about 17) who appeared to have a sound grasp of 2D trigonometry were hopeless at solving 3D problems:

Most of them could not even draw a diagram to represent the problem situation. Teachers frequently left 3D geometry until the very end of the five-year course (or even omitted it altogether) because of its difficulty, to themselves as well as to their students. University lecturers in such subjects as geology and engineering were frequently heard grumbling about students' inability to "think in three dimensions."

Similar observations have been made by others. Vernon

(1967b, p. 340) states that such comments are common among 14 expatiate teachers in Africa, but gives no details. Skerap

(1971, pp. 33-34) reports a case of a Ugandan student study­ ing Pythagoras' theorem who could draw the squares on the two sides enclosing the right angle (which were horizontal and vertical) but not the square on the hypotenuse (which was, of course, oblique). In Jamaica, the geometry ques­ tions on the national Grade Nine Achievement Test (see

Section 3.23) are extremely simple and the most unlikely dis- tractors are frequently chosen.^

1.32 Cross-cultural psychological studies of perception

The above informal observations are supported by a growing body of cross-cultural research on the psychology of perception (Dawson, 1971; Lloyd, 1972; Serpell, 1972). The consensus of this research is that Africans and natives of other developing countries generally do less well than

Europeans of comparable age and education, on both paper- and-pencil and performance tests of spatial ability

(Biesheuval, 1949; Cryns, 1962; Vernon, 1969). Signifi-

I am grateful to Mr. Ian Isaacs for bringing the Jamaica observation to my attention and for supplying the item analysis on which it was based. 15

cantly for the present study, the retardation appears

greater in spatial than verbal ability (Smith, 1970). It

has also been demonstrated that they have greater difficul­

ties in drawing or interpreting diagrams of 3D objects

(Deregowski, 1972b; Hudson, 1967). The only exceptions

to these generalizations so far found are Eskimos and North

American Indians (Kleinfeld, 1973).

Cross-cultural differences in perception are also revealed by studies of susceptibility to geometric illusions.

However, the direction of the difference between Western and non-Western cultures varies from illusion to illusion

(Segall et al., 1956).

Cross-cultural studies of perception and possible ex­ planations for the results obtained are reviewed in detail in Sections 2.6-2.8.

1.33 Implications and research needs

The developing countries are all trying to hasten their development by a certain degree of industrialization. These efforts were initially hindered by the shortage of local engineers and skilled and semiskilled workers, and subsequent training programs have generally failed to produce the man­ power needed to replace expatriate technicians and to expand and develop new industries. In the midst of massive unem­ 16 ployment, there is a great shortage of middle-level technical manpower in the developing countries (Williams, 1965).

There are many reasons for this state of affairs, not the least of which is the reluctance of parents and students to regard manual labor as worthy of education and ambition.

In view of the clear relation of spatial ability to success in technical fields that has been found in Europe and North

America (Section 1.23), however, it seems likely that a major factor could be the retardation in spatial ability in developing countries. As Hudson (1962a) has pointed out, poor pictorial depth perception and representation could cause difficulties at all levels from the factory floor

(Winter, 1963) to the high school classroom (Section 1.31) and beyond. The practical issues are: How serious is the presumed handicap in spatial ability in the present gener­ ation of students? and: How successfully can this handicap be overcome by specially-designed remedial treatment?

Previous cross-cultural studies have done little to answer these questions. For good reasons, researchers have concentrated on populations comparatively untouched by westernization, mainly illiterate adults and elementary age children; they have therefore had to use individual tests with small and possibly unrepresentative samples. Relevant 17 published research on secondary school children, who are the engineers, technicians, machinists and mechanics of the futur'., is strictly limited (Irvine, 1959; McFie, 1961).

The recent development of pencil-and-paper tests which can be reliably used with unsophisticated subjects of secondary age

(Schwarz, 1961, 1963; Schwarz & Krug, 1972) suggests that problems of testing larger samples can now be overcome.

Also, because of their high face validity, spatial ability tests should be least susceptible to the difficulties of interpretation which plague cross-cultural studies (Berry,

1969).

The matter of the training of spatial ability is also relatively unexplored. Studies in the United States indicate that although simply taking a course in geometry or mechan­ ical drawing has little effect on spatial ability (Smith,

1964), special training programs can be more successful

(Brinkmann, 1966; Carpenter et al., 1965; Van Voorhis, 1941).

Only two small studies have been found which deal with spatial training in developing countries (Dawson, 1957a;

Serpell & Deregowski, 1972); both tend to support the train- ability hypothesis. The writer and his colleagues have developed a spatial training program which has been incorpo­ rated into a regular mathematics course now in use in 18

secondary schools in West Africa (Mitchelmore & Raynor,

1957-75) and the Caribbean (Mitchelmore et al., 1970-75), but this program has never been subjected to experimental

testing. If it is an effective program, then it could be of immediate usefulness in schools and colleges in developing countries.

There is a paucity of research into the 3D drawing of mathematical objects, so there is a particular need to in­ vestigate how children in developing countries learn to draw.

1.4 The present study

1.41 Location

The country chosen for the present study was Jamaica.

The writer visited the country in July 1972, and gained the impression from local educators (Bennett, unpublished) and his own preliminary work (Mitchelmore, 1973b) that

Jamaicans also suffer retardation in spatial ability. This impression was also gained by Vernon (1965, 1969) who in­ cluded a small sample of 11 yr old Jamaican boys in a wide cross-cultural study.

Jamaica is one of the most advanced of the developing countries, and as such has a most urgent need for middle- 19 level technical manpower. Because of the wide variation in development (the rural people are still very poor), educa­ tional provision (there are five types of secondary school), social conditions (it is estimated that half the children are brought up without a permanent father-figure in the home), and racial type (about 75% are Negro, the remainder being various mixtures of Negro, European, Chinese, and Indian) in a relatively small area, Jamaica is an excellent laboratory for the study of the relative weight of economic, social, and genetic factors in the development of spatial ability.

1.42 Research questions

The considerations outlined above led to the formula­ tion of the following general research questions:

1. How are the various facets of spatial ability related in Jamaican students?

2. How do the various facets of spatial ability develop in Jamaican students?

3. How is spatial ability in Jamaican students related to their physical charac­ teristics, environment, upbringing, and educational experience?

4. What is the effect of a training program involving practical model-making activities on the spatial ability of Jamaican students? 20

Because of the particular interest in the practical appli­ cations of geometry, 3-dimensional visualization and drawing were to be included as facets of spatial ability to be in­ vestigated .

It was also decided to include geometric illusion sus­ ceptibility, which is not considered to be a facet of spatial ability, as a further indicator of perceptual development. The question to be investigated was as follows;

5. How is geometric illusion susceptibility in Jamaican students related to the various facets of their spatial ability and to their physical characteristics and upbringing?

Reluctantly, two further important questions could not be included in the present study:

6. How is spatial ability in Jamaican students related to other aspects of their intelli­ gence and achievement?

7. Are Jamaican students retarded in spatial ability relative to North American students of comparable educational level, and if so, in what way and by how much?

1.43 The research program

In order to study the relation of spatial ability to environment and educational experience (research question

3), a survey was planned of the spatial ability of students in different types of secondary schools in various parts of 21

Jamaica. It was decided to conduct this survey at the

Grade 9 level, the last year in which free education was

available to all Jamaicans in 1973-74. To study developmental patterns and the relation between spatial ability and illusion

susceptibility (research questions 2 and 5), a cross-

sectional survey of Grades 1-9 was planned. To limit the

sources of variation, this survey had to be restricted to high--ability students in the capital city, Kingston. Finally, the opportunity arose of testing the spatial training program

(research question 4) with students at a teachers' college in

Kingston. A three-phase program was therefore executed, as

follows :

1. Developmental survey Four tests of spatial ability and one of illusion susceptibility were administered individually to small samples of high-ability children in Grades 1, 3, 5, 7 and 9 from public schools in Kingston.

2. Grade Nine survey Three tests of spatial ability were administered to classes of Grade 9 students in four types of secondary school in three types of environment in eastern Jamaica.

3. Training experiment The effectiveness of an individualized unit in practical, elementary 3D geometry for improving the spatial ability of prospective primary school teachers was tested. 22

All three phases provided information regarding the structure of spatial ability in Jamaican students (research question 1). Personal data collected from students also provided information on the relation of spatial ability

(research question 3) and illusion susceptibility (research question 5) to age, sex, skin color, hand and eye dominance, ambition, social class, family size, and family composition.

It proved possible to collect a small amount of data relative to research question 6 during the training experi­ ment. Research question 7 was studied by repeating parts of the developmental study and the school survey in a small number of schools in Columbus, Ohio, but the data became available too late for analysis and inclusion in this dis­ sertation.

1.44 Preview

The structure of the remainder of this dissertation is as follows. Chapter 2 examines the experimental litera­ ture on perceptual development, with particular reference to

3-dimensional perception and representation and the various factors which may influence their development. Chapter 3 outlines the geography and history of Jamaica and describes 23 the present social and educational system. The rationale

for the selection of the tests used in this study is given in Chapter 4, which also contains an account of how the tests were adapted and pilot-tested. Methods, procedures and results of the three phases of the research program are des­ cribed in Chapters 5, 6 and 7; Chapter 7 also contains a review of previous research on spatial training programs.

Finally, Chapter 8 examines the results of the three phases for answers to the five research questions and for implica­ tions in psychological theory and educational practice. CHAPTER TWO

PERCEPTUAL DEVELOPMENT

The purpose of this chapter is two-fold; to review relevant research and to outline a consistent theory of per­ ceptual development. Emphasis is placed on the perception, representation and visualization of 3D figures. The first five sections deal with results obtained in Western (mainly

European and North American) samples. Cross-cultural results are examined in the next two sections, and the final section deals with geometric illusions.

2.1 Depth perception

2.11 The basic problem

Fig. 2.1 illustrates how the image of a 3D figure is formed on the retina of the eye of an observer. The normal person perceives the distance and solidity of the object, even with one eye if the object or the eye moves. But the retinal receptors form a 2D surface and cannot respond to the differing distances of various points on the object by (for example) registering sensations at different depths. The

24 25

image representation on retina

eye object

frontal plane

Fig. 2.1 The retinal image of a figure is produced by a pencil of light rays which travel from the object to the eye. A representation of the object is given by the intersection of this pencil with the frontal plane, an imaginary plane perpendicular to the line of sight. A line drawing of the object is a repre­ sentation of its outline. In this example, the objective shape of the top face of the object is a rectangle; its apparent or subjective shape is a parallelogram, as shown in the line drawing.

problem thus arises: By what means is the third dimension perceived?

2,12 Empiricist theories

Many perceptual theories, generally labeled empiricist, have assumed that stimulation of individual receptors in the retina are the primitive sensations out of which perceptions are built, and that coordination of sensations into per­ ceptions must be learnt. In the case of depth perception, it 26

is held that the learning results from tactile exploration

during early childhood; the tactile and visual experiences

become associated so that a later image "awakes the memory

of everything like it experienced in previous visualiza­

tions ... such as the number of steps needed to reach a man who appeared in the field of view to be of a certain size"

(Helmholtz, 1925, p. 293).

Empiricist theories have been summarized by E. J. Gibson

(1969) and only two will be mentioned here. Brunswik (1956) pointed out that any retinal image gives only partial infor­ mation about the stimulus figure so that the perceived object represents only the figure which the observer's previous experience shows is the most probable. Hence the way in which an ambiguous figure is perceived depends on the subject's place and mode of upbringing; certain interpreta­ tions could have greater "ecological cue validity" than others. Piaget has emphasized the function of tactile- manipulative activity in early childhood, arguing that

"spatial concepts are internalized actions" (Piaget & Inhelder,

1967, p. 454) and that a mental image is "no more than a record or trace of the muscular adaptations involved in an action" (ibid., p. 455). 27

2.13 Nativist theories

The nativist position, that depth perception is primitive and does not have to be learnt by experience, has been revived through the work of J. J. Gibson (1950, 1966), who showed that the pencil of light rays which travel from an object to the eye does contain information about depth which can be picked up by a 2D receptor surface such as the retina. Texture is one cue. Imagine the top face of the cuboid in Fig. 2.1 covered with uniformly spaced dots; on the retina, the farthest dots would be closer together than the nearest dots, and the difference between the two spacings would be directly related to the slope of the face. It is thus possible to postulate an innate mechanism in the cerebral cortex which could recover depth information from retinal stimulation and produce 3D perceptions. There is ample evidence from physiological studies of other mammals that there are cortical cells which respond to abstract features such as orientation or binocular disparity (Ewert, 1974;

Pettigrew, 1972).

Gibson's theory has received support from the work of

Bower (1964, 1966) who has shown conclusively that infants 28

49-60 days old perceive more than just the size and shape of the retinal image of an object. For example, they can dis­ tinguish a cube from one three times the size at three times the distance and a rectangle from a trapezium in a different orientation which gives the same retinal image, but they do not distinguish between the same rectangle in different orientations.

The theory also explains the developmental trend in shape constancy in childhood. In Fig. 2.1, the top face of the object is intended to be a rectangle; this is its objective shape. It is represented in the frontal plane by the 2D figure which would produce the same image as the sloping rectangle, namely a parallelogram; this is its subjective shape. Many experiments have shown that young children perceive only the objective shape of a sloping ob­ ject (Meneghini & Leibowitz, 1966; Piaget & Inhelder, 1967) whereas adolescents and adults perceive both the objective and subjective shape (Brault, 1962; Guignot et al., 1963).

Similar results have been obtained for size constancy; it is not until about 5 years of age that a child notices that an object appears to get smaller as it gets farther away (Eng,

1954, p. 157). 29

It thus seems evident that children have to learn to

abstract the 2D projection from a 3D object, and not vice

versa as proposed by the empiricists.

2.2 Form perception and representation

Before proceeding to a consideration of 3D drawing, it is

necessary to digress briefly to outline the development of the

perception and representation of 2D shapes.

2.21 Form discrimination and copying

Children can be taught to discriminate simple shapes

(commonly called forms) such as a circle, triangle and square

before they are one year old (Ling, 1941) but it is much

later before they can copy these shapes (Lovell, 1959; Peel,

1959; Piaget & Inhelder, 1967; Terman & Merrill, 1937).

Average ages are approximately as follows: A circle is

copied correctly at 3 yr, but other closed forms are also

drawn as circles at this age. A square is copied correctly

at 4 yr, but a triangle is not represented differently from

a square until 5 yr. A diamond is copied as a square until

about 7 yr.

To find if motor skills affected performance, Piaget &

Inhelder (1967) used matchstick representations as well as 30

paper-and-pencil copying. They claimed that there was little

difference in results for the two methods, but Peel (1959)

and Lovell (1959) found that although the same sequence

occurred using the matchstick method, the tasks were about

half a year easier. Maccoby & Bee (1965) also cite

evidence that reconstruction using strips of paper is easier

than copying. The medium of reproduction thus influences

the probability of success at copying a form, but it does not

account for the relative difficulty of different forms.

Graham et al. (1960) and Piaget & Inhelder (1967) examined stages in copying regular and irregular forms and showed that topological concepts such as closure were copied correctly earlier than metrical concepts such as length and angle. Similar observations have been made from the study of haptic perception (Laurendeau & Pinard, 1970; Page, 1959;

Piaget & Inhelder, 1959). There is, however, no sudden progression from topological to metrical— the metrical dis­ tinction between straight and curved is almost as primitive as the topological distinction between open and closed

(Lovell, 1959). 31

2.22 Obliques

It was noted above that, although a diamond is the same shape as a square, children learn to draw a diamond about 2 yr later than a square. The difference seems to lie in the obliqueness of the sides of the diamond. Several experi­ menters have demonstrated the unexpected difficulty of this concept; Rudel & Teuber (1963) showed that all the children aged 4 yr in their sample could form the concepts of hori­ zontal, vertical, up and down, but not sloping right or sloping left; even at 8 yr there were still some children who could not form the oblique concepts. Olson (1970) showed that children learn to construct a diagonal on a checker­ board at age 6 yr on average, although they can construct rows and columns and recognize diagonals well before this point. Campbell (1969) found that children copied a parallelo­ gram as a slightly distorted rectangle from 4-6 yr, but from

7-9 yr their copies became much closer to the correct shape.

It thus seems that obliqueness only becomes an operational concept around 6-7 yr.

Obliqueness is linked to the concepts of left and right.

Rudel & Teuber (1963) found that their 4 yr olds could not form the concepts of facing left or facing right, but 32

Huttenlocher (1957) showed that the problem was more a matter

of differentiating mirror images. Piaget (1924) and others

(see Laurendeau & Pinard, 1970) have shown that the concepts

of left and right continue to develop up to about 11 yr,

whereas the concepts of up and down are learnt by about 7 yr

(Harris, 1972).

Children can discriminate between left and right and

between left sloping and right sloping possibly as early as

6 months old (McGurk, 1972) but the problem is to learn

consistent differential responses to the two directions.

These can be taught as early as 4 yr (Jeffrey, 1958, 1966), but there does not appear to be any need to differentiate mirror images until the child starts to read and has, for

example, to differentiate p from q and b from d (Davidson,

1935; Gibson et al., 1962). A similar result has been found

for orientation (Howard & Templeton, 1966).

2.23 Perception versus performance

Several examples above have shown that children can discriminate one shape from another or recognize a correct copy of a form long before they can copy it; why is there this lag? Maccoby & Bee (1965) discuss several possible 33

explanations: (1) The child lacks motor skills. As discussed

above, changing the medium of reproduction does change the

difficulty of a copying task; but lack of motor skills cannot

explain why children about 6 yr old can copy a square but not

a diamond. (2) The child draws the objective shape, for

example perceiving a diamond as a square. This does not ex­

plain why a square is earlier drawn as a circle. (3) The

child draws his own image of the form. But a child will

usually select a more mature copy in preference to his own when asked to choose the best copy. (4) More attributes of the form must be used in drawing than in discrimination or recognition. But Olson (1970) took children who could not construct a diagonal and successfully trained them to dis­ criminate several attributes of the concept and they could still not construct the diagonal.

Olson's view is that different tasks require different perceptual information for their successful performance.

Thus, "to recognize a diagonal ... it is necessary only to detect the feature or features which will partition the set of alternatives" whereas "performatory activity ... requires perceptual information for ... selecting between all possible alternatives at each point of the performatory act" (1970, 34 p. 184). This view implies that discrimination or recogni­ tion requires only partial knowledge of a concept, whereas reproduction requires knowing practically all there is to know about it. An alternative formulation is therefore that successful copying or drawing of a form requires a more or less complete grasp of the principles of its construction; thus a child who can copy a diamond may be expected to know that the sides are sloping and that opposite sides are parallel and equal in length. Indeed, Olson (1970) found that children who could construct a diagonal could articulate its properties, whereas those who could not construct a diagonal could not state why a figure recognized as a non­ diagonal was not a diagonal.

2.3 Pictorial depth perception and representation

2.31 Pictorial depth perception

A correct line drawing of an object is a representation of its outline (see Fig. 2.1). Several properties of the drawing arise from the depth dimension in an object and are thus used by artists as cues to depict depth. The important depth cues are position (since the ground is usually below, farther objects are drawn higher in the diagram), size 35

(farther objects are drawn smaller), overlap (a nearer object

can block part of a farther object from view), and perspective

(parallel lines in space converge in the diagram). Many other depth cues, most of which are more relevant to paintings and photographs, are discussed by Gibson (1950). Ways in which

line drawings convey information are also treated by Kennedy

(1973).

Now although a line drawing produces the same retinal image as the outline of an object, young infants do not con­ fuse the drawing with the object (Bower, 1965), further evi­ dence for the nativist position (Section 2.13). A line draw­ ing shows the subjective shape of an object, and just as children must learn to perceive subjective shape so they must learn to interpret a line drawing as representing a solid object. Outlines of single everyday objects are recognized as early as 19 months even in an environment severely deprived of pictures (Hochberg & Brooks, 1962), and Brumbaugh (1971) found that 3-4 yr old children could successfully match cubes, cuboids, spheres and ellipsoids with line drawings of the solids. Depth in pictures depicting scenes is recognized later, between 3 and 5 yr for perspective and between 5 and 36

6 yr for overlap and size in middle class children (Oh, 1969)

2.32 Pictorial depth representation

As with the copying of forms, producing a line drawing of a 3D object is much more difficult than simply recognizing what a drawing represents. To a large extent, 3D drawing is dependent on 2D copying, since the child must not only know what shape to draw to represent a given object but he must also be able to draw that shape. For example, a 5-5 yr old cannot be expected to represent a rectangular table-top in perspective if he cannot copy a parallelogram correctly. How­ ever, development of 3D drawing ability continues long after copying problems disappear.

Children's method of representing 3D scenes have been extensively studied, both by observing single children (Eng,

1954; Stern, 1910) and by classifying collected drawings

(Barnhart, 1942; Kerschensteiner, 1905; Munro et al., 1942;

Rouma, 1913). Although there are differences in detail, four stages are generally reported (Lowenfeld & Brittain,

1966; Luquet, 1929) ;

1. Objects float in space not properly related to each other or to any base line (up to 7 yr ).

2. Objects are shown in correct topological rela­ tion to each other, but without depth depiction. 37

Metrical relations are often localized, giving the impression of many viewpoints incorporated in one drawing (7-9 yr).

3. Attempts are made to show depth by multiple base lines, overlapping and size differences, from a single viewpoint (9-11 yr).

4. Objects are correctly represented, related to a base plane with the horizon in the background, and sometimes using convergence to show receding parallels (from 11 yr).

Eisner (1967) developed a detailed 14-category system for classifying drawings according to the presence of base/ horizon lines and how objects were related to them and each other.

There has been much less work on methods of showing depth in single objects. Kerr (1936) found an increase in the proportion of her sample who drew a "solid" house (as opposed to a simple front view) from 10% at age 7 to 60% at age 13-14, but only half of these made acceptable perspective drawings

(Fig. 2.2). Petitclerc (1972) presents sketches of a pyramid and a cube said to be typical of various age levels, but gives no supportive data. Chetverkhin (1971) only shows drawings of a brick and a cube said to be typical in Grades 1-

6, and states that in Grade 1 they are invariably represented by a rectangle and a square respectively.

The only researcher to have provided definite data on 38

According to Kerr

Solid Perspective

According to Lewis

E ffl ffl a

II

III IV

Fig» 2.2 Stages in children’s drawings of a house, after Kerr (1936) and Lewis (1963). jy stages of development by age is Lewis (1962, 1963). She gave

27 classes of elementary school children the task of drawing a model house in the form of a cube. The five stages postulated and used in scoring are shown in Fig. 2.2. Stage III was sug­ gested by Arnheim (1954) from an analysis of depth depiction in classical pictures. The distinction between Stages IV and

V is not clear in Lewis's reports. Ratings by three inde­ pendent judges showed high agreement; the modal methods of representation were Stage I for Grades K-6 and Stage IV for

Grades 7 and 8, but were rather indistinct for Grades 4 and 5.

2.33 Representation of spatial relations

The representation of specific spatial relations was studied by Piaget and Inhelder (1967). They isolated several stages in the drawing of a slanting circular disc: Drawings at first showed the objective view for all slants (4-6 yr); in the second stage, the circle was drawn smaller for the steeper slants (5-7 yr); next, ellipses were used to depict all but the edge-on position (7-9 yr); the correct representa­ tion of the last position by a straight line was not recorded until 8-9 yr.

Piaget & Inhelder (1967) also report stages in the representation of receding railway lines and a tree-lined road. From 4-7 yr, they were drawn by parallel lines. 40 with the "trees" drawn perpendicular to the "road." From

7-9 yr, the lines curved in at the far end and the trees were upright but of equal height. From about 9 yr, the lines were drawn converging and the trees smaller at the narrower

end. Further data on the representation of the vertical were obtained by asking children to draw vertical posts planted across the summit of a model mountain. At about

4 yr, the posts were drawn parallel to the side of the moun­ tain; from 5-7 yr, perpendicular to the mountain; and then upright with increasing frequency. Compared to reports cited in Section 2.32, these ages seem to be decidedly optimistic.

The representation of the water level in tilted bottles was also studied by Piaget & Inhelder (1967). In the first stage, about 5-6 yr, the water level was shown parallel to the base of the bottle in all positions. In the next stage, between 6 and 8 yr, children drew the water level at an oblique angle in all bottles. A transitional stage (about

7^ yr) was observed in which the water level was drawn cor­ rectly in the horizontal bottle only; correct solutions for all bottles started at 7-9 yr on average.

Dodwell (1963) found that the stages of development in the above tasks were much less clearly defined than Piaget & 41

Inhelder (1967) claimed. The last problem in particular is obviously far more difficult than it appears; Pebelsky (1964) found that half of a group of university students drew the water level in a tilted glass between 6° and 25° to the hori­ zontal.

2.4 Development of spatial visualization

2.41 Sectioning of solids

Piaget & Inhelder (1967) investigated the development of children's ability to imagine sections of a solid. In the first stage, children appeared to have no concept of section, but drew instead a general view, a development of the surface, or the separate faces (4-6 yr). In the next stage, the section concept was grasped, but the actual shape drawn was wrong (6-8 yr). Finally, from age 7, the children came to draw more and more difficult sections correctly: sections of a cylinder by age 7 and of a cone by age 8.

Subsequent replications have suggested that these ages are also optimistic. Boe (1968) obtained a mean of only

10.9 correct out of 16 sections amongst Grade 8-12 children, but Davis (1973), who added a short pretraining session. 42 obtained means of 10.5 in Grade 6 and 13.5 in Grade 10. Both researchers found that oblique sections more significantly more difficult than the transverse and longitudinal sections and that the cone was the most difficult solid. Palow (1959), who used instead photographs taken from different viewpoints, concurred with Davis that children achieve mastery of sec­ tioning at about age 12.

2.42 Coordination of perspectives

Imagining what a scene would look like from a different viewpoint is another spatial operation investigated by

Piaget & Inhelder (1967). At first, children gave only their own viewpoint, as if it were the only one possible

(6-7 yr). Next, individual objects were turned, but not the whole scene (about 7-8 yr). In the next stage, some rela­ tions were correctly reproduced but others not; for example, the left-right sequence may have been correct but the front- back wrong (7-9 yr). Consistently correct views were obtained from about age 8 yr.

Similar sequences have been found by Lovell (1959) and

Laurendeau & Pinard (1970), although Dodwell (1963) found the boundaries between stages very much blurred. It is clear 43 from the detailed tabulations of Laurendeau & Pinard (1970) that not only do the tasks vary widely in difficulty accord­ ing to the viewpoint but that most 12 yr olds cannot solve all the problems involved.

2.43 Surface development

The final section of Piaget & Inhelder's treatise (1967) to be mentioned is that dealing with surface development.

Here, the child was given a cylinder, a cube, a pyramid and a cone and asked to draw their surfaces "opened out flat."

This 2D shape is called the net of the solid. The following stages were reported; In the first, the drawings of the net were indistinguishable from drawings of the whole solid, which at that age generally show one face only (4-6 yr). In the next stage, the net and the solid were drawn differently but with no apparent rationale (5-8 yr). Next the separate parts of the net were drawn but either wrongly connected or distorted as if to show the opening-out process (7-10 yr).

In the final stages, correct nets were drawn, firstly for the cylinder and cone, then for the cube, and lastly for the pyramid (8-12 yr).

It appears that this aspect of Piaget & Inhelder's work has not been replicated. The only reference found to this 44

task is given by Tuddenhara (1970), who devised a 5-minute

test of surface development and estimated (for example) that

about 22% of the Grade 4 sample had acquired the concept.

2.44 Psychometric tests of visualization

Piaget & Inhelder (1957) give the impression that every child acquires the concepts of form, verticality and horizontality, sectioning, coordination of perspectives and surface development (inter alia) by the age of 12 yr, after which age nothing more needs to be said. Later researchers cited above have shown conclusively that this is an over­ simplification. It may well be that by age 12 most children have a complete grasp of what it means to make a section or to develop a surface, but it is certainly also true that there are wide individual differences in the ability to per­ form these operations successfully, and that this ability continues to develop through adolescence if not beyond.

Psychometric tests of visualization (see Sections 1.22,

4.21 and 4.32) may thus be seen as measuring spatial ability without regard to whether particular concepts are acquired or not. Even though psychometric tests use many of the same types of item as Piaget & Inhelder used (change of viewpoint, sectioning and surface development are the most popular), it 45

should be no surprise that scores on the two kinds of test

do not correlate highly (De Vries, 1973; Hathaway, 1973;

Mycock, 1969).

2.5 Perceptual development

2.51 The Piaqetian model

Piaget's empiricist theory of perception was alluded to

in Section 2.12. To him, perception involves the assimila­

tion of sensory input to existing schemata (plans of action

relating to various classes of perceptual situations) and the

accommodation of the schemata to the input ; it is the organ­

ization of these schemata which defines the stage of develop­ ment. Four stages, called sensorimotor, preoperational,

concrete operational and formal operational, are postulated;

these stages are general to intellectual development, but

specific definitions for perceptual growth are given in

Table 2.1. Piaget & Inhelder (1967) have further character­

ized perceptual development in terms of the spatial trans­

formations which children appear to have mastered at each

stage: initially topological (relations of proximity, separa­ tion, order, enclosure and continuity), then projective

(linearity, perspective), and finally euclidean (size, angle, 46

Table 2.1

Piagetian stages in the development of spatial perception

Stage and approxi­ Definition^ mate ages I : Sensorimotor Immediately following sensori-motor activity, tied directly to perception, (Up to 4-5 yr) we have action recalled in imagination subsequent to being performed physically. This gives rise to thought which re­ produces action with all its concrete­ ness and irreversibility.

II: Preoperational Growing coordination of physical actions is accompanied by an internal (4-5 to 7-8 yr) coordination of their schemata (that is to say, of their schematic outlines), though this still proceeds by trial and error, anticipating potential actions in a piecemeal fashion.

III; Concrete The schemata are coordinated suffi­ operational ciently to be combined, and consequently, for each one to be mentally explored in (7-8 to 11-12 yr) alternate directions. This type of reversible combination represents the initial equilibrium state reached by internalized actions. IV: Formal With the further development of operational operational coordination, it becomes possible to conceive of several systems (From 11-12 yr) simultaneously. Thus the child now dis­ poses of abstract operations, capable of being expressed in prepositional form.

^Quoted verbatim from Piaget & Inhelder (1967, pp. 45 4-455). 47 similarity, coordinates). Many researchers have questioned this latter formulation (Cousins & Abravanel, 1971; Dodwell,

1963; Esty, 1970; Fisher, 1965; Martin, 1973a, 1973b;

Rivoire, 1961), but it has been vigorously defended by others (Laurendeau & Pinard, 1970).

2.52 Perceptual learning

Whatever the stages of intellectual development, it is important to consider what causes a child to progress from one stage to the next. Although Piaget & Inhelder (1967) state, for example, that "imagining the rotation and development of surfaces depends largely on the actual process of unfolding solids, and the motor skills involved in such actions” (p. 276), they present no experimental evidence to support this view, nor do they suggest any mechanism whereby the "internalized actions" might lead to a change in the organization of a child's perceptual schemata.

Evidence for the importance of movement-produced sensory feedback was adduced by Held (1965), who showed that active subjects who controlled their own movements made superior adaptation to novel perceptual situations than did passive subjects who were moved by the experimenter. Similarly, 48

Festinger et al. (1957) showed that adaptation was greater when a subject was learning a new response, and argued that perception was determined by "efferent readiness" to respond to each situation in a certain way.

It may, however, be argued that the role of the subject's self-controlled activity in such circumstances is merely to draw closer attention to the perceptual learning task. It might also make the task easier by making more information available. Thus Denner & Cashdan (1967) showed for a group of preschool children that recognition of a wooden hexagon was improved by handling the model, but there was a similar improvement when the model was enclosed in a transparent sphere; in both cases, the subject was provided with extra visual information about the hexagon as he turned it around.

Activity is not always effective; for example, Kershner (1971) found that tracing a movement with the finger did not im­ prove the ability of 6 yr olds to remember that movement, and

Kieren (1969) suggested that activity could actually inhibit learning in older or more intelligent subjects who might be able to gain more by reflection on a symbolical level.

Gibson (1969) argues that perceptual learning consists of differentiating between figures with similar properties: 49

Perceptual learning (is) an increase in the ability of the organism to get information from its environment ... There are potential vari­ ables of stimuli which are not differentiated within the mass of impinging stimulation ... As they are differentiated, the resulting per­ ceptions become more specific. ... There is a change in what the organism can respond to not acquisiton or substitution of a new res­ ponse (p. 77).

A differentiation is unlikely to be made unless there is some incentive to make it, and activity may be one means of pro­ viding the incentive. Thus getting around within one's physical environment requires greater differentiation than simply observing the surroundings, so greater perceptual growth inevitably results from locomotion. Olson (1970) writes of Held's (1965) results:

It is not response information, if there is any such thing, that the experience has provided. Rather, it is perceptual information that has been picked up about the world; the attempted per- formatory act merely provided the occasion for picking up new cues (p. 186).

There are many other factors which could influence which dif­ ferentiations are made and which are not (Section 2.7).

Differentiation theory explains a result due to Veronova

(Shemyakin, 1959) which is difficult to explain on the basis of any response-oriented theory. She found that children who had been confined to bed for long periods acquired the 50

right-left concept before the up-down concept, the reverse

of the order in normal children (Section 2.22). The differ­

ences in activity between the two groups offer no obvious

explanation for this reversal. The simplest explanation is

that up-down differentiation is of little value in a bedroom, whereas the distinction between the various sides of the bed

and bedroom is salient. This is the opposite of the situa­

tion of the normal child, who does not have a ceiling per­ manently over his head and who can interchange left and right simply by walking around the other side.

2.53 A mathematical model of perceptual development

Differentiation theory in itself does not posit any particular structure or stages of perceptual development.

It can however be used to account for the observed patterns of development described in previous sections, as follows.

Perceptual development can be thought of as a network of stages, there being an ordered sequence of stages relating to each type of perceptual situation. At any particular stage for any particular situation, there is a set of figures

(objects or configurations) which are not differentiated.

Call this set A. Let Sq be the subgroup of transformation s^ 51

Flg» 2.3 "Sameness" in an undifferentiated set. Object P is regarded as the same as Q because there is a transformation s^ in the identification group of A which maps P onto Q.

in (A) such that s^(P) is not differentiated from P for all

P£ A; will be called the identification group of A cor­ responding to this stage of development. Because any two members of A are "the same, " (A) ; see Fig. 2.3.

Progression from one stage to the next can be conceptu­ alized as follows. For some reason, some of the objects in

A are differentiated from the others. A subset of Sq (call it Dj_) is constructed, consisting of those transformations which change at least one object into a "different" object;

D]_ will be called the differentiation set for this stage.

The set A is now partitioned into subsets ^A^^^^such that all objects in any one A are "the same" and any pair of objects from distinct A ' s are "different," For each A^, the 52

Fig* 2,4 ’’Sameness” in a differentiated set, P and Q are regarded as the same because there is a transformation s-j in the identification group of A which maps P onto Q, but Q and R are regarded as different because all transformations which map Q onto R belong to the differentiation set.

the identification group is thus ^(A^) ; for the entire set A, the new identification group is S^, where

S = TJ J (A ) . XeA ^ Also, S = and SinDj= 0 (see Fig. 2,4).

At the next stage of development, each subset A^ may be partitioned into further subsets A^ , giving a new identi­ fication group and a new differentiation set D 2» Some examples of this process follow, in which A = ^simple closed plane figures^ and = ^^topological transformations of the planej.

Example 1 At about age 3, a child copies a circle, an ellipse, a triangle and a square in the same way. These 53

O O

1 2 5

Fig. 2.5 Stages in the development of form perception. At Stage 1, all simple closed figures look alike. At Stags 2, polygons are differentiated from other figures. At Stage 3» triangles and quadrilaterals are differentiated from polygons with larger numbers of sides.

figures are equivalent under S^. By about age 4, circles and ellipses are drawn differently from triangles and squares.

What has happened? The child has noticed that if you smooth away all the vertices of a figure, you obtain a different figure. He has thus formed a new identification group,

= {topological transformations which preserve curvilinearity and recti linearity}.

Two of the subsets of A are

A^ = {"smooth" simple closed figures^ = ^circles, ellipse ...]

and A = {polygons] = ^triangles, squares, . . -j

At the next stage, A^ is further broken down into

A^^ = ^triangles].

A^„ = ^quadrilaterals]. 22

^ 2 2 ~ ^polygons with 5 or more sides}. 54

Fig. 2.6 Differentiation of mirror images» All Es are initially regeurded as the same but are eventually differentiated into forward and backward Es.

with a corresponding reduction in the identification group.

The process is illustrated in Fig. 2.5.

Example 2 As noted in Section 2.22, children do not regard mirror images as different until they start school; even then, small perspective changes are ignored (Gibson et al.,

1952). It could be said that, during the first year of school, children are forming the subgroup of even perspective trans­ formations. This process is illustrated in Fig. 2.6 for the letter E and its italicizations and reflections.

Example 3 At around the same age, rotated images are also differentiated. Break-and-close and line-to-curve transfor­ mations are picked up earlier, and perspective and similarity transformations later. The final result in the mature person is to reduce the identification set to the group of transla­ tions, T: figures are rarely regarded as different if they 55 have the same size, shape and orientation and differ only in position.

Two further characteristics of perceptual growth can be incorporated into the model.

1. Each new identification group TJ (A.) often has a naturally isomorphic subgroup in the full group S^. In the early stages, the isomorphs may not be so obvious, but they become clearer in later stages. In Fig. 2.5, for example, the identification group for the third stage may be expressed as

^transformations which do not change the number of vertices in a triangle or quadrilateral and which preserve curvilinearity and recti linearity j".

By the time all polygons with different numbers of sides are differentiated, the identification group becomes simply

^perspective transformations}.

The important thing to notice is that this group acts on the whole group A, not just on its partition {a ^}.

2. At some stage, the differentiation set D-]_ is itself differentiated. In the set of transformations relating any two objects from different subsets of A, a subset is formed whose elements relate specific pairs of objects in the respective subsets. This set will be called the construction 56 set, C^. For example, in Fig. 2.6, consists of the transformations relating El to 3 , E to and so on; so is the set of reflections about a vertical axis.

Later, the group generated by is formed; this group will be called the operator group, In Fig. 2.6, the operator group is the group of reflections about a vertical axis together with horizontal translations.

The formation of D]_, S^, C]_ and correspond to levels of performance with regard to the particular concept involved:

Discrimination corresponds to the splitting of the set A into exclusive subsets Aj^ by the formation of the differentiation set D-j_. Recognition corresponds to the formation of the new identification group Sj_, whereby each member of any A^ is related to every other member of the same A^. Reproduction corresponds to the formation of the construction set with­ in D]_, enabling the transform of any given object to be con­ structed. Finally, the formation of the operator group corresponds to knowledge of the principles of the construction.

The above model may be regarded as an extension of

Piaget's model (Section 2.51) by the interpolation of many other stages between the primitive topological and the refined euclidean. The final stage when is formed corresponds to 57

Piaget's formal operational stage. The model is obviously incomplete and needs further study to determine the relation between the stages for different situations; for example, whereas the near-far differentiation should precede the straight-curved distinction. Also, we have not attempted to integrate the development of spatial visualization (Section

2.4) into the model. However, it is not necessary to fill in the details to see one important difference from Piaget's model: So many aspects of perception can develop inde­ pendently that a child's abilities at any given age could present a highly confused picture and no two children are likely to show the same picture. At least this prediction is in line with the experimental facts; the existence of discernible developmental patterns testifies to the relatively uniform influence of cultural pressures (see Section 2.71).

A similar model of perceptual development, expressed in terms of Lie groups, has been proposed by Hoffman (1956).

2.54 Representational learning

The ability to make a 2D representation of a 3D object of scene is quite different from the ability to copy a 2D shape:

It is possible for a copy to match a 2D shape so closely no- one could tell the difference, whereas a drawing of a 3D 58 object could never be confused with the original. A 3D draw­ ing can therefore only represent some of the properties of the real object.

What distinguishes a "good" drawing from a "poor" one is the amount of information which the drawing contains which is faithful to the original object, but the "right" amount of information depends on what the drawing is to be used for.

Thus, if the only purpose of a drawing is to differentiate spiders from insects, a blob with eight protrusions might be a quite adequate drawing of a spider. For a portrait artist, the accurate depiction of facial features would be important, but the background might well be neglected. For the present study, the concern is with the depiction of depth in regular figures in such a way that the drawing can be used to predict geometrical properties of the figure. The "good" drawing in such a case is the perspective projection of the object onto a plane, which is how a representation has been defined (Fig. 2.1).

Now the question is, how does a child learn to make such a representation? It cannot be simply by looking closely at figures, for even when children have developed the subjective attitude (Section 2.13) and can successfully interpret depth 59 cues (Section 2.31), they still do not make "good" perspective drawings. It seems rather that children (and artists) gradu­ ally develop different methods or schemata for drawing various classes of objects which become more and more faithful to the original.

Gombrich (1960) considered the phylogeny of representa­ tional schemata, showing by many examples that each method was invented by one artist (or group of artists) and then adopted by others as they accepted its usefulness in depicting certain aspects of reality more successfully than previous methods. He argues forcefully that schemata are learned not by looking at nature but at other people's methods of repre­ senting nature. This view is confirmed by the present writer's observation that children living in the tropics who are subject to a strong European influence start drawing

European-type houses, complete with chimney, which they have only seen in pictures. Arnheim (1954), who locked rather at the ontogeny of representational schemata, showed that, as a child's perceptual differentiation develops, so he uses more and more complex shapes to represent the same things. He makes the point that children often discover schemata spon­ taneously, for example, finding that a circular scribble can 60 represent a ball, then a wheel, then a body. Piaget & In- helder (1967) note that schemata are initially topological, and that projective and euclidean relations develop together.

The schematic nature of children's drawing is nicely illustrated in an experiment by Freeman & Janikoun (1972).

They had children draw a cup with a flower painted on the front and the handle hidden behind. Children ages 5-7 yr drew the handle (because they "knew it was there") and no flower, whereas 9 yr olds drew the flower but not the handle;

8 yr olds were transitional. The former reaction is similar to the objective perceptual attitude, the latter is clearly subjective; the point is that children come to represent more and more aspects of reality, and the governing influence is the subjective view. In this case, there was no change in the schema (the cup was drawn with a circle or ellipse for the top edge and a straight line for the base) since it was adequate to represent a cup with or without a handle or flower. The schema would be expected to change later when it became desirable to show that the top and bottom edges were both curved.

The development of representation ability (Section 2.32) can be fitted into the model of perceptual development 61

(Section 2.53) as follows. Consider any collection of 3D figures (objects or configurations). The set A consists of all drawings purporting to represent those figures. As 2D shapes these drawings can be differentiated early on; the important step is when all the drawings of a particular type of figure are identified. At this stage, partitioning of A might not be complete, since the same drawing can represent different types of figures; a child can recognize a figure from any ambiguous drawing, but does not differentiate "good" and "poor" drawings. In the next stages, "good" and "poor" drawings of each figure are increasingly differentiated. The part of A consisting of ambiguous drawings is rejected and the remainder is more and more finely partitioned; at any stage, however, only one subclass of drawings of each figure is con­ sidered "good", the remainder are "poor." The identification group is at first primarily topological, then projective, and finally becomes the group of projectivities which preserve the horizon line. For each identification group, the construction group is formed after Sj_, enabling the child to reproduce the simplest of the drawings considered "good" at that stage.

Construction of accurate perspective representations of arbitrary figures follows the formation of the operator group 52

0^ after the child comes to understand the principles govern­ ing the relation of an object to its representation.

Refinement of A is limited by the level of form per­ ception, but lags far behind because of the need to select the appropriate 2D shape to represent each 3D figure.

A major hindrance is the persistence of the objective attitude which has to be overcome before foreshortening of faces of objects and a constant viewpoint in extensive scenes can be successfully represented.

2.6 Cross-cultural studies of perception and representation

2.61 Psychometric studies

Many researchers have found that natives of developing countries perform poorly on psychometric tests which are widely used in Western cultures to assess spatial ability, even when the samples are carefully matched for age and education. Relatively poor performance on group paper-and- pencil tests (Ombredane et al., 1958; Schwitzgebel, 1962) is partly explained by unfamiliarity with the test situation, but the difference persists in school and college groups which 63

should be more sophisticated (Jahoda, 1969, 1971; Mitchelmore,

1973b; Smith, 1970; Vernon, 1965, 1967b, 1969). Similar

results have been obtained using performance tests of intelli­

gence such as block designs or the Bender Gestalt, which are

basically perceptual in nature (Berry, 1966, 1971a;

Biesheuval, 1949; Dastoor & Emovon, 1972; Deregowski, 1972c;

Jahoda, 1956; Kellaghan, 1968; MacDonald, 1944-45;

Maistriaux, 1955; McFie, 1954, 1961; Nissen et al., 1935; Ord,

1970; Shapiro, 1960; Vernon, 1965, 1967b, 1969).

Most of the above results have been obtained in African

samples. By contrast, studies of North American Eskimos and

Indians show very little retardation in spatial ability

(Berry, 1966, 1971a; Berry & Annis, 1974; Blum & Chagnon,

1967; Kleinfeld, 1973; McConnell, 1954; Vernon, 1966, 1969;

Western New Mexico University, 1967).

2.62 Perceptual development

Poor scores on block designs or the Bender Gestalt often indicate retardation in the development of form per­ ception amongst non-Western groups, rotation errors being

frequently reported (Deregowski, 1972c; McFie, 1961; Shapiro,

1960). Olson (1970) reports a similar retardation in the

formation of the diagonal concept. 54

There is also evidence (Beveridge, 1935; Thouless, 1933)

that adults in developing countries may retain the objective

attitude in depth perception (Section 2.13), and even highly

educated adults may have difficulties interpreting line draw­

ings (Hudson, 1960).

There is now an extensive cross-cultural literature based

on Piaget's model of intellectual development (Dasen, 1972).

The usual finding is that children in non-Western cultures

pass through somewhat similar stages as those in the West

but at a slower rate and often not reaching the final stage

at all. Most of the work has been on conservation, and only

two researchers have included any of the tasks discussed in

Sections 2.33 and 2.41-2.43. Cowley & Murray (1962) obtained

the usual results for a wide variety of spatial tasks admin­

istered to Zulus and Europeans. Dasen (1974) found

Australian Aborigines retarded in the water-level task rela­

tive to European Australians; more recently, he has found that

Eskimos are not at all retarded on this task whereas West

Africans are more retarded than the Aborigines (personal com­ munication, 1974). Retardation is also suggested by the

results of Beard (1968), Poole (1968) and Vernon (1965, 1966,

1967b, 1969) who gave tests containing short Piaget-type

items to samples of one or two age ranges only. 65

2.63 Pictorial depth perception

Many studies have been made of the interpretation of

drawings and photographs by young children and illiterate

adults in non-Western societies (Dawson, 1967a; Dawson et

al., 1974; Deregowski, 1968a, 1968c, 1972b; Duncan et al.,

1973; Fonseca & Kearl, 1960; Forsdale & Forsdale, 1966;

Holmes, 1963; Hudson, 1960, 1962a, 1962b, 1967; Kilbride &

Robbins, 1968; Mundy-Castle, 1966; Shaw, 1969; Spaulding,

1956; UNESCO, 1963; Vernon, 1965, 1966, 1967b, 1969; Winter,

1963). These studies concur in four results: (1) familiar objects can be identified fairly well from a simple line drawing or photograph; (2) conventional signs such as those used to express movement are not understood very well; (3) depth cues are not understood at all well; (4) the probability

of correct interpretation increases with increasing education

and urban influence.

2.64 Pictorial depth representation

Although many writers have noticed cultural differences in methods of representation (Gombrich, 1960; Stern, 1909;

Thouless, 1933), few have studied 3D drawing in any detail.

Hudson (1962b) found that illiterate South African mine- 66 workers frequently drew "developed" views of a cow, an ele­ phant and a car, showing several viewpoints in one diagram

(Section 2.32). His report that subjects felt the failure to show hidden limbs implied they had been cut off has been con­ firmed elsewhere (McElroy, 1955). Deregowski (1970) reported that Zambian uneducated women tended to prefer developed views, but Duncan et al. (1973) found that young African schoolchildren had few difficulties interpreting standard drawings of single objects. Deregowski (1969) also found that primary school children and illiterate domestic servants tended in their sketches of a wire model consisting of nine edges of a cube (the edges of the front and back faces and one of the edges joining them) to draw the two square faces side by side instead of overlapping.

Only two studies have examined drawings of 3D scenes.

Deregowski (1972a) estimated that rural Zambian children were 5-6 yr behind English children in their drawings. Lester

(1974) made a more careful comparative analysis. She had schoolchildren aged 4-11 yr in Lagos, Nigeria and New York draw a model of a simple domestic scene from three directions and then scored the drawings for their success in representing the overall spatial arrangement of the objects, the articula- 67 tion of contiguous objects, and the uniformity of aspect

(viewpoint) for individual objects. There was a significant cross-cultural difference only for the first category, the superiority of the American children increasing with age.

2.7 Influences on perceptual development

The results reviewed in Section 2.6 are consistent with the view that the average person in a developing country appears retarded in perceptual development because he has not learnt to make the same differentiations as a person of his age in an industrialized nation. Thus illiterate Africans' tendency to rotate copies of block designs, to see shapes objectively, and to ignore depth cues in line drawings all resemble the behavior of pre-school Western children. On the other hand, there is little doubt that Africans develop other perceptual skills which Westerners do not, for example, in estimating the number of cupfuls of rice in a heap (Gay &

Cole, 1967), copying shapes in wire (Serpell, 1972), or tracking and identifying their cattle (Dyson-Hudson & Dyson-

Hudson, 1969). The present section considers a wide range of factors which could influence perceptual development by en­ couraging or inhibiting relevant differentiations. 68

Following Berry (1971, 1974), environmental influences are broadly divided into those due to culture and those due to the physical surroundings or ecology. A third category dealing with physiological factors is also included, but

"theories" which merely point out the greater importance of genetic as opposed to environmental influences within popula­ tions (Jensen, 1972; Osborne & Gregor, 1968) are not dis­ cussed. Sex differences in perception, which appear to arise partly from cultural pressures and partly from physiological factors, are considered in the final subsection.

2.71 Cultural pressures

Economic activity Berry (1974) has pointed out that sub­ sistence societies can be divided into two main types: gather­ ing, hunting and pastoral societies, which tend to be nomadic, and fishing and agricultural societies, which tend to be sedentary. Comparisons between hunting (Eskimo) and agricul­ tural (West African) societies have shown that the former are superior in spatial ability (Berry, 1966, 1971a; Dawson,

1971). One possible reason for this superiority is that survival amongst the Eskimo depends on successful navigation, which in turn requires the perceptual differentiation of 69

apparently uniform snowfields (Kleinfeld, 1973). Another reason is suggested by the observation (Berry, 1974) that

social roles become more differentiated in sedentary societies;

Such spatially demanding tasks as there are in agricultural

societies (such as building houses) tend to be performed by specialists whereas in a nomadic society each person must be able to perform such tasks himself. Comparisons of other types of nomadic and sedentary subsistence societies are needed to fully test these hypotheses.

Upbringing Barry et al. (1957) uncovered a further social comparison which could influence perceptual development. They found that high food accumulation societies (combining agri­ culture with animal husbandry) and low food accumulation societies (relying on hunting, fishing or gathering) had radically different styles of child rearing procedures: The former placed greater emphasis on responsibility and obedi­ ence in their children, whereas the latter tended to put more emphasis on achievement, self-reliance and independence. How these differences could result in differences in spatial ability is suggested by the work of Witkin et al. (1962):

They found that good performance in a number of perceptual 70 tasks involving operations in the presence of a distracting background (field independence; see Section 1.22) was associ­ ated with a more objective and self-sufficient personality, and that both were associated with an upbringing which en­ couraged independence of thought and action. A child who is encouraged to explore and find out things for himself instead of accepting them on the authority of adults may be expected to achieve greater perceptual differentiation, just as Held's active subjects did (Section 2.52). The hypothesis that child rearing practices influence field independence has been confirmed by Dawson (1957a) who compared two tribes in Sierra

Leone. The Eskimos' encouragement of independence in their children has also been suggested as a factor in their greater perceptual development (Berry, 1966, 1971a; Kleinfeld, 1973).

Social norms The social organization of sedentary agricultural societies also tends to inhibit the development of spatial skills in adults. There is often extensive socio-cultural stratification (Berry, 1974) which generally makes for political conservatism and the discouragement of any activity which does not follow precedent (Cole et al., 1971). In such societies, the learning of correct social practices is prized 71 above Western-style intellectual development (Irvine, 1959a,

1969b, 1970). Thus slower reaction times in perceptual tasks could be the result of social inhibition of competitiveness

(Ombredane et al., 1958).

Language Several researchers have noted that African languages contain relatively few words for shapes or spatial relations (Cole et al., 1971; Gay & Cole, 1967; Olson, 1970;

Stewart, 1971) whereas Eskimos have a rich spatial language

(Berry, 1966; Kleinfeld, 1973). The radical view that "we dissert nature along lines laid down by our native language"

(Whorf, 1956) has been used by some to assert that poor per­ formance on Western spatial-perceptual tests is caused by the lack of the relevant vocabulary (Du Toit, 1966; Littlejohn,

1963) or lexical structure (Greenfield & Bruner, 1966; Whorf,

1940). It is more consistent with differentiation theory to suppose that the causality acts in the opposite direction:

A people's language expands to match the perceptual differen­ tiations which are important in their culture (Deregowski,

1968a; Hudson, 1967; Olson, 1970).

Art The relation of art to perception parallels that of language. Berry (1966) noted that there was a wider occurrence 72

of graphics, sculpture and the decorative arts amongst

Eskimos than amongst his African subjects. It seems unlikely

that a cultural emphasis on the visual arts would cause

greater perceptual development; it is more likely that a greater emphasis on art would accompany an improvement in

spatial ability resulting from some third factor.

Education It has been consistently found that people who have experienced Western-style education are perceptually more developed than unschooled subjects (Bhatia, 1955;

Dastoor & Emovon, 1959; Fonseca & Kearl, 1960; Gay & Cole,

1967; Kilbride & Robbins, 1968; Okonji, 1971; Olson, 1970;

Shaw, 1969). Attendance at school clearly causes children to make the differentiations which are required in Western culture but not in the native culture, for example, while learning to read, write, interpret pictures and make drawings.

2.72 The physical environment

The natural environment That a person's interaction with the world outside his residence affects his perceptual development is shown by two studies (Monroe & Monroe, 1971;

Nerlove et al., 1971) in which subjects who travelled farther during their work and spare time tended to score higher on perceptual tests. In cultures where navigation is important. 73

it might be expected that spatial ability will develop most

where the terrain is relatively uniform, as in deserts or

dense forest, but as far as is known this prediction has not

been tested (but see Section 2.82). The hypothesis that

Africans' perceptual development is retarded by their being

bound to their mothers' backs in infancy (Vernon, 1967b) is

also untested.

The man-made environment There are a large number of man- made objects in the visual environment of both the industrial­

ized and developing worlds. The big difference is that in

the industrialized countries most man-made objects are machined, linear and rectangular to a fine degree; in a developing country, more objects are hand-made and conse­ quently cruder, and circular shapes are frequently preferred

(Allport & Pettigrew, 1957). Segall et al. (1966) call this variable "carpenteredness," but the term "rectangularity” is preferred here. It could be expected that greater rectangu­

larity would provide more opportunities for the acquisition

of spatial concepts, but it would be difficult to argue that greater rectangularity causes greater perceptual development; it is more likely that in rectangular environments there are greater pressures to make the differentiations which lead to 74 perceptual development.

It has also been pointed out that "disadvantaged" homes frequently contain few pictures (Biesheuval, 1943; Hudson,

1950) and manipulative toys (Beard, 1968; Vernon, 1967b).

Again, it does not seem likely that these deficiencies cause poor perceptual development, but rather that their presence is associated with efforts on the part of parents and society to encourage relevant perceptual differentiations.

To explain his finding that Indians showed greater shape constancy than English subjects, Thouless (1933) sug­ gested that culturally-accepted artistic conventions determine how the world is perceived. However, it has been clearly shown in previous sections of this chapter that representa­ tional development is consequent to perceptual development, not vice versa. Gombrich's historical survey (1960) sug­ gests that prevailing methods of representation would most strongly influence the development of representational ability in individuals; this could explain why Indian children were less able than Europeans to pick up depth cues in line drawings (Hudson, 1962a).

The influence of the man-made environment on spatial ability has not been tested directly, as there is usually 75

extensive confounding with other factors (but see Section

2.82).

2.73 Physiological factors

Nutrition Cultures displaying extremes of spatial-perceptual development also differ in their nutrition. For example, the average daily intake of the East African Kikuyu is 22 g

fat, 390 g carbohydrate and 100 g protein whereas the corres­ ponding figures for the Canadian Eskimo are 162 g, 59 g and

377 g respectively (Dawson, 1971). Attempts to show that poor nutrition causes poor intellectual development have generally failed, as there have always been accompanying factors such as infection, sensory and cultural deprivation, and poor motivation which could also have caused the observed retardation (Segall, 1974; Vernon, 1969). It is even tenable that some of the behavioral "effects of malnutrition are in fact causes (Guthrie, 1974). The only evidence to show that malnutrition is particularly related to perceptual development rather than to cognitive development generally is discussed in the next paragraph.

Sex hormones One hypothesis to explain sex differences in spatial ability (Section 2.74) is the differential effects of 76

the sex hormones. Dawson (1966) found that a group of male

workers with a history of kwashiorkor (a disease resulting

from protein deficiency) showed several signs of feminization

in physique and cognitive style, and gave as the cause the

inability of the liver to inactivate the normal amount of

female hormones in the male (Stuart-Mason, 1963). Dawson sug­

gested (1967b) that many males in low protein consumption

areas might suffer the same effect without showing noticeable physical symptoms.

Eye and skin pigmentation The fovea of the eye is covered with a layer of yellow macular pigment; the density of pig­ mentation varies from person to person, but the pattern of

light absorption is constant, with a maximum in the blue region at 460 nm (Bone & Sparrock, 1971). Pollack (1963)

showed that denser macular pigmentation was associated with decreased sensitivity to contours, which could adversely

affect perceptual development. Silvar & Pollack (1967) found that a sample of U.S. black children had consistently denser macular pigmentation than a sample of white children, thus suggesting a physiological cause for the generally poor spatial ability of people of African ancestry. Jahoda (1971) appeared to have confirmed the relation by showing that 77

Ugandan students were poorer on a blue spatial test than on the equivalent red test, whereas there was no difference for

Scottish students.

However, Bone & Sparrock (1971) found that West Indian and European students did not differ significantly in macular pigmentation (in fact, black students had slightly lighter pigmentation than whites), which puts the whole theory into doubt. Further discrepancies have been found in studies of illusion susceptibility (Section 2.84). It is possible that the different findings are related to the methods used:

Pollack used direct opthalmoscopic examination of the retina whereas Bone used a flicker technique to compare the optical absorbancies of foveal and extra-foveal regions of the retina.

Calcium deficiency Smith (1971) has also attempted to explain the apparent association between dark skin color and low spatial ability through a physiological mechanism. He noted that vitamin D is synthesized in the lower levels of the skin under the action of ultraviolet light; so light skin would have an evolutionary advantage in northern latitudes by allow­ ing maximum synthesis under low levels of sunlight whereas dark skin in the tropics could prevent the synthesis of excessive amounts of the vitamin (Loomis, 1957). Now vitamin 78

D mediates the absorption of calcium from the intestine into

the blood, and a low calcium level appears to bring a tendency

to vivid visual imagery (Jaensch, 1930). Thus Europeans' low

average level of blood calcium may be a definite advantage

for performance on spatial tasks.

Lateral dominance Several investigators have shown that in most people the right hemisphere of the brain controls visuo-spatial activity, the left hemisphere being responsible

for verbal activity (Harris, 1973; Kimura, 1973). Since

each hemisphere also controls the opposite side of the body, it might be predicted that right-handers (with a dominant

left hemisphere) would score higher on verbal tests and

lower on spatial tests than left-handers. This prediction is not borne out by the evidence, however. In the first place, about 30% of most peoples are mixed-handed, using different hands for different tasks or both hands for the

same task (Annett, 1957). In the second place, left-handers

appear to have higher verbal ability than right-handers

(Annett, 1970; Levy, 1969). Mixed-handers appear to be

lower than both right- and left-handers on both verbal and

spatial ability (Levy, 1969; Miller, 1971). 79

Annett (1964) proposed the following model to account for these results. Handedness and hemisphere dominance are determined genetically by two alleles, say R and r; R repre­ sents right-handedness and is dominant to r, left-handedness.

The frequency ratio of R:r is about 80:20 (Annett, 1967). The

64% of RRs are consistent right-handers and the 4% of rrs are consistent left-handers, but the remaining 32% of Rrs are mixed-handers, susceptible to social pressures to become right-handed (Dawson, 1972b). The heterozygotes may be des­ cribed as "mixed-up" so that they can develop neither verbal nor spatial ability to the same average level as the homozygotes (Levy, 1969).

Although the distribution of eye dominance matches that of hand dominance, the two characteristics appear to be independent (Dawson, 1972b; Kershner, 1970). Eye dominance is unlikely to be related to hemisphere dominance, since both eyes are connected directly to both hemispheres (Kimura,

1973). Yet similar results have been obtained, with mixed­ eyed subjects showing poorer reading and spatial ability than right-eyed or left-eyed subjects (Oltman & Capobianco, 1957);

Kershner (1970) found that left-eyed children were superior to right-eyed children on a spatial task. 80

The interaction of eye and hand dominance also appears to affect perceptual ability. Dawson (1972b), Oltman &

Capobianco (1957) and Rengstorff (1967) report that mixed hand-eye dominance subjects (mostly left-eyed right-handers) had lower spatial and verbal ability than subjects showing intermodal consistency. Kerschner (1970), however, found just the opposite.

Neural deterioration There is evidence that the neurons in the cerebral cortex of kittens which are tuned to any partic­ ular orientation of lines in the visual field may atrophy in the absence of stimulation at that orientation (Blakemore &

Mitchell, 1973). Annis & Frost (1973) claim to have found a similar effect in humans by comparing acuity to different orientation in Kree Indians (whose tepees present oblique edges) and Euro-Canadians (whose environment presumably con­ tains more vertical and horizontal edges). However, their argument is statistically fallacious: They compare a signifi­ cant orientation difference amongst the Europeans with a similar but non-significant difference amongst the Indians, instead of demonstrating a significant culture-orientation interaction (which does not seem likely from the data pre­ sented) . 81

2.74 Sex differences

In the industrialized nations, there is overwhelming

empirical evidence for a sex difference in spatial ability in

favor of males, at least from age 11 onwards; conversely,

females tend to show greater verbal ability (Anastasi, 1958;

Dwyer, 1973; Maccoby, 1967; Maccoby & Jacklin, 1973;

Sherman, 1967; Smith, 1964; Witkin et al., 1962). However,

cross-cultural research has shown that the difference is not universal (Stewart, 1974b). Explanations for sex differences

fall into cultural and physiological categories.

Cultural pressures It has been noted that, in most cultures, girls are more restricted in spatial experience and are socialized for greater conformity than boys (Stewart, 1974b).

(The importance of these factors in the development of spatial ability was discussed in Sections 2.71 and 2.72.)

Amongst subsistence economies, sex-role differentiation is greatest in sedentary agricultural societies, where the woman's place is strictly in the home, and least in nomadic hunting or fishing societies, where women have greater free­ dom (Barry et al., 1957). Reports of sex differences in spatial ability in the first type of society proliferate

(Berry, 1966, 1971a; Dawson et al., 1974; Monroe & Monroe, 82

1971; Nerlove et al., 1971; Olson, 1970), whereas there are equally consistent reports of no sex differences in the second type of society, for example amongst Canadian Eskimos (Berry,

1966; Kleinfeld, 1970; MacArthur, 1967), Canadian Indians

(Berry & Annis, 1974), and Australian Aborigines (Berry,

1971a). The fact that there are fewer left-handed women in conformist than in permissive societies (Dawson, 1972b) argues for the strength of cultural pressures. The influences of sex-typed spatial activities is underlined by a finding reported by Irvine (1969c) that Mashona females, who tradi­ tionally engage in decorative bead work, excelled males on the Morrisby Compound Series (a colored series test of bead stringing) but were inferior to males on other tests of non­ verbal intelligence.

More advanced societies have in general tended to preserve the social norms of the primitive agricultural societies, with the stereotypic male allowed greater inde­ pendence and expected to be active, technical and protective of the poor female who cannot even mend a fuse (Maccoby, 1967;

Sherman, 1967), and it is reasonable to suppose that the greater restriction and conformity can still act to suppress the development of spatial ability in females relative to 83

males. This view is supported by a recent review of studies

in Western settings which shows that a lesser societal em­

phasis on conformity in women is associated with a smaller

sex difference in field independence (Stewart, 1974b);

indeed, Stewart cites a study due to Meizlik (1973) in which

the usual difference was found to be reversed amongst tradi­

tional Orthodox Jews, where the woman takes a more than

usually active responsibility for family affairs.

Physiological factors Cultural factors cannot explain re­

sults such as those obtained by Stafford (1951), who

compared the spatial ability of parents and their children

and found that same-sex correlations were significantly

lower than opposite-sex correlations. His explanation, that

higher spatial ability was associated with a sex-linked

(X-chromosome), recessive gene, is supported by the finding that females lacking one X-chromosome (Turner's syndrome) have normal verbal ability but are strikingly deficient in

spatial ability (McClearn, 1967). However, not all pre­ dictions from Stafford's model have been confirmed (Garron,

1970).

A physiological mechanism has been proposed by

Broverman et al. (1968). The female sex hormones are said 84 to control the adrenergic activity processes in the central nervous system which allow immediate response to sensory

stimuli whilst the male hormones control the cholinergic processes which allow perceptual restructuring; the former would be more appropriate to verbal, the latter to spatial tasks. Some evidence for this theory was cited in Section

2.73; other evidence comes from studies in which the ten­ dency for male rats to be siperior in maze learning and female rats more active on the treadwheel has been reversed by the administration of oestrogen (a female hormone) to the males and testosterone (a male hormone) to the females

(Dawson, 1971, 1972a). It has also been found (Broverman et al., 1972) that testosterone production in human males increases from about 8 years of age and reaches a peak at puberty, which would nicely explain the fact that sex dif­ ferences only become apparent at about that age (Dawson et al., 1974).

It may also be noted that prolonged administration of female hormones tends to raise the level of blood calcium, whilst male hormones have the opposite effect (White et al.,

1954); this is the same contrast which Smith (1971) used to explain differences in spatial ability relating to skin color, 85

2.8 Geometric illusions

2.81 Illusion susceptibility

Figures such as those shown in Fig. 2.7 have long been

the subject of psychological enquiry, but it is still not known what causes the illusion effect. Cross-cultural dif­

ferences have implicated the physical environment, but no theory has been advanced which adequately accounts for all the differences observed. Culture-free theories are equally unsatisfactory. One problem that any theory must solve is that susceptibilities to the various illusions are practi­ cally independent (Jahoda & Stacey, 1970; Taylor, 1972); theories which successfully explain one type of illusion are usually too successful on other types.

The interpretation of an illusion stimulus figure appears to be quite a different process from other perceptual tasks such as form perception, pictorial perception, or spatial visualization. Without the distracting background, error in matching the two lines would decrease monotonically to zero as differentiation for length improved; with the distracting background, not only does the error not decrease monotonically with age— it does not go to zero at all. Perhaps if illusion figures were a common feature of the human environment the 86

Muller-Lyer Illusion Horizontal-Vertical Illusions

(1 ) (2 )

a

\ a / / b \ / \ \ /

Sander Parallelogram Ponzo (perspective) illusion I A

Fig» 2.7 Four well-known geometric illusions. In each case, when the lines a and b are drawn of equal length (as above), most people report that a looks longer than b.

human environment error would decrease to zero; this is sug­ gested by many experiments showing that the illusion effect decreases (but does not vanish) with lengthened exposure and repetition (Piaget, 1969). But illusion figures are not a common feature in the environment of any cultural group.

Illusion susceptibility is thus not concerned with deriving information from sensory stimulation, as are "normal" per­ ceptual processes, but appears to be a by-product of those 87

processes, whether innate or learned. All the theories to

be discussed below start from that premise.

2.82 Inappropriate constancy scaling

An early theory of illusions, recently restated by

Gregory (1953), was that they are caused by interpreting the

diagrams as if they represent 3D figures; the automatic

size scaling which is imposed is inappropriate to a 2D

figure, so an illusion results. The effect is obvious if the

Horizontal-Vertical and Ponzo illusions (Fig. 2.7) are inter­

preted 3-dimensionally. The Sander Parallelogram could

represent a rectangle drawn in perspective, in which case a would be "really" longer than In the Muller-Lyer figure,

^ could be the back edge of a box, room etc. and b a front

edge; since ^ is therefore "further away" than b, it also is

"really" longer. Evidence for this theory comes from the

finding that subjects apparently perceive ^ as farther away

than b when the distance cue provided by Lhe paper surface

is removed (Gregory, 1963).

The theory would predict that artists and others who have been trained to dissociate subjective and objective views of an object would be less susceptible to the illusions, but this has not been found (Carlson, 1966; Jahoda & Stacey, 88

1970). It would also suggest that subjects who were unable

to see depth in pictures would be less susceptible to the

illusions; however, this is opposite to the age trends

(Fig. 2.8), and Dawson et al. (1973) and Wober (1970) found

only insignificant correlations between pictorial depth

perception and illusion susceptibility. There is neverthe­

less some connection between the two; thus Leibowitz et al.

(1969) and Leibowitz & Pick (1972) found that the provision

of additional depth cues by embedding the Ponzo figure in a photograph led to an increased illusion effect in

sophisticated subjects but made no difference amongst Ugandan villagers. There is also a general finding that greater

spatial ability tends to be associated with lower sus­ ceptibility to illusions (Berry, 1968; Dawson, 1973;

Dawson et al., 1973; see also Section 2.83).

The influence of the physical environment Segall et al.

(1966) used the above theory of depth inference and

Brunswik's (1956) theory of ecological cue validity to predict cross-cultural differences in illusion susceptibility.

Specifically, they predicted that: (1) the habit of depth inference applied to the Sander Parallelogram and Muller- 89

Lyer figures would be of higher validity in rectangular than in non-rectangular environments; so susceptibility to these illusions (and, to a lesser extent, the Ponzo illusion) should be greater in environments with more man-made artifacts and rectangular houses, and therefore greatest in

Western cultures; and (2) the habit applied to the

Horizontal-Vertical illusions would have greater validity where there are open vistas; so dwellers in plains and deserts should be more susceptible than city dwellers, who should in turn be more susceptible than dwellers in thick rain forests. These predictions were confirmed in outline in samples from 16 countries, although there were several inexplicable differences between the non-Western samples and no significant differences for the Ponzo illusion.

Results conforming to Segall et al.'s prediction have been reported by Allport & Pettigrew (1957), Berry (1968,

1971b), Gregor & MacPherson (1965) and Stewart (1973,

1974a), but Davis & Carlson (1970), Jahoda (1966), Jahoda &

Stacey (1970) and Mundy-Castle & Nelson (1962) have reported results contrary to their predictions.

It should be noted that the arguments advanced by Segall et al. (1966) are based on the empiricist view that per- 90 ceptions are learned interpretations of retinal sensations.

But it was shown in Sections 2.13 and 2.31 that even though a line in a picture might produce the same retinal image as a line in the environment, the two lines are not per­ ceived as the same; indeed, it is a sophisticated achievement to comprehend that one line can represent the other. Thus, although environmental factors might well influence the perception of lines in the environment (as shown by dif­ ferences in size constancy, for example), their effect is unlikely to transfer to the interpretation of lines in a picture. In any case, most of the subjects in the more primitive samples tested by Segall et al. would probably have been completely unable to relate pictorial lines to environmental depth features. In the Western samples, subjects would be able to regard a diagram as a plane figure or as a representation of a 3D configuration, but they would never confuse the two.

2.83 Field independence

Illusion susceptibility appears to be related to diffi­ culties in overcoming the distracting effect of the supple­ mentary lines; so a negative correlation may be predicted 91 between susceptibility and field independence as measured by an embedded figures test. Significant correlations have been reported by Gardner (1957, 1951) and Gouch & Olton (1972) for the Muller-Lyer illusion, by McGurk (1965) for a com­ posite index of susceptibility, and by Taylor (1972) for a wide range of illusions. However, Jahoda & Stacey (1970) found a significant correlation for only one out of twelve illusions administered, and Dawson et al. (1973) found significant correlations for both Horizontal-Vertical illu­ sions but not for the Muller-Lyer illusion or the Sander

Parallelogram. Correlations found in the larger samples were in any case small, suggesting a definite relation which is mediated or obscured by other factors.

2.84 Other factors

Piaget (1969) has examined a vast amount of informa­ tion relating illusion strength to their spatial dimensions and come up with a "law of relative centration" which fits the relationship with impressive accuracy over many dif­ ferent illusion types. To explain the law, he postulates that the over-estimation of a figurai element results from fixation (centration) on it more than on other elements; 92 that in normal perception, compensations (décentrations) are made from successive fixations; and that illusion figures cause the process of decentration to become unbalanced.

The explanations of the relationships for each type of illusion have a certain post hoc quality, and it is not known whether specific predictions have yet been put to the experimental test by independent investigators.

Several investigators have proposed that geometric illusions have their source in the way optical stimulation is processed in the retina and the brain. These theories have been reviewed by Over (1958) and Taylor (1972); suf­ fice to say that no theory has yet been found to apply to any wide class of illusions.

Further evidence for some physiological influence is provided by the finding of Pollack & Silvar (1967) that density of macular pigmentation was negatively related to susceptibility to the Muller-Lyer illusions under tachis- copic exposure in blue light. Three conflicting studies have examined the relation between skin color and illusion susceptibility: Berry (1971b) found that mean Muller-Lyer susceptibilities in ten samples from five countries were somewhat more closely related to skin color than to degree of 93 rectangularity, but Armstrong et al. (1970; summarized by

Stewart, 1973) found no difference between black and white

students from Evanston, Illinois. In a beautifully con­ ceived study, Jahoda (1971) related differences more specifically to optical effects; he found that Ugandan students were more susceptible to the Muller-Lyer illusion when it was printed in red on a grey background than when it was printed in blue on grey, whereas there was no dif­ ference amongst Scottish students. In view of the contra­ dictory findings relating skin color to macular pigmentation

(Section 2.73), it is impossible to decide which of these three results need explaining away.

2.85 Developmental patterns

Experimental findings relating illusion susceptibility to age throw further light on the causes of the illusion effects, but they can hardly be said to fully illuminate the scene.

Results from three recent studies are illustrated in

Fig. 2.8. Both the Illinois and Zambian samples were ethnically mixed, but whereas in Illinois different races were selected from the same three schools, the Zambian 94

Muller-Lyer Illusion 10

8 Illinois

6 Zambia

Hong Kong

Horizontal-Vertical illusions

Hong Kong ("1 )

Hong Kong ( J_)

k

Sander Parallelogram

5 Zambia

4 Hong Kong

5 Illinois

46 8 10 12 14 16 Age in years

Fig. 2.8 Age trends in illusion susceptibility in three cultures. Results for Illinois were obtained by Armstrong et al. (1970), for Zambia by Stewart (1971), and for Hong Kong by Dawson et al. (1973). The results of Armstrong et al. and Dawson et al. have been smoothed. All data were obtained using the Segall materials (Herskovitz et al., 1969)} the scales on the vertical axes show the mean number of illusion-supported responses. 95

samples were chosen from a wide variety of schools in dif­

ferent parts of the country. All the Hong Kong students

were Chinese. It may be noted that the cross-cultural dif­

ferences shown by these three studies are consistent with

the Segall et al. (1956) predictions (Section 2.82) for the

Muller-Lyer illusion, but not for the Sander Parallelogram.

These and other results may be considered separately

for each of the illusions in Fig. 2.7.

Muller-Lyer illusion Many earlier studies (summarized by

Wohlwill, 1960) showed the same monotonie decrease in sus­ ceptibility from age 5 or 6 yr to adulthood as in Fig. 2.8.

The slight increase in susceptibility at age 16 obtained by

Armstrong et al. (1970) is similar to that reported by

Walters (1942) and Wapner & Werner (1957), but the original graph shows that the increase was most likely due to sampling error. Frederickson & Geurin (1973) have recently shown that illusion susceptibility increases again in old age.

Horizontal-Vertical illusions Walters (1942) and Winch

(1907) found susceptibility to the inverted T to decrease steadily with age, as for the Muller-Lyer illusion. Wursten

(1947) and Fraisse & Vautrey (1956) found for the inverted L a rise in susceptibility to age 10 yr and then a decrease to 96 adulthood. On the other hand, the Dawson et al. (1973) results show a similar pattern for both forms (Fig. 2.8).

Sander Parallelogram Results for this illusion are also somewhat contradictory. Heiss (1930) found that suscepti­ bility decreased with age, but Piaget (1969) reported an increase from 5 to 8 yr, decreasing thereafter. By contrast, the studies illustrated in Fig. 2.8 all show a decrease up to age 10-12 yr, followed by a slight increase thereafter. ponzo illusion The relatively small amount of research on this illusion is at least consistent. Leibowitz & Heisel

(1958) found an increase in susceptibility between ages 4 and 7 with no further changes beyond that stage, while

Wohlwill (1962) found school children equally susceptible at all ages, but more susceptible than college students.

Age effects As no acceptable theory of illusion susceptibil­ ity exists, it seems futile to try to "explain" the develop­ mental trends which are well established. The matter is further complicated by variations in stimuli, exposure times, methods of measurement, etc., from one study to another, which could easily account for the noted inconsistencies.

Only very general comments can therefore be made: 97

1. There are definitely changes in susceptibility with age.

This seems to rule out physiological factors which do not

change with age.

2. There are definite differences in age trends between

different illusions: For the Muller-Lyer, illusion sus­

ceptibility decreases steadily with no clear minimum whereas

for the Horizontal-Vertical illusions and the Sander Paral­

lelogram there is a clear minimum somewhere between ages 7

and 12. This again seems to rule out general factors such

as age-related improvements in analyticity (Wapner & Werner,

1942), scanning (Zaporozhets, 1965), decentration or form perception (Piaget, 1969).

3. Cross-cultural differences in developmental pattern can­ not safely be deduced from the graphs in Fig. 2.8, which conceal much experimental error. However, it does seem clear that whatever differences there are, they are much

smaller than the absolute differences between cultures. This suggests that cross-cultural differences in susceptibility are primarily the result of some constant factor (such as rectangularity) rather than variable factors (such as level of perceptual development).

4. All the graphs in Fig. 2.8 show an initial decrease in 98 susceptibility, at least between ages 3 and 7. This would suggest some correlation with the development of form per­ ception, the subjective attitude in depth perception, or pictorial depth perception, except that there is no evidence of the sort of cross-cultural differences that one would expect in such a case.

5. Any correlation with perceptual development would pre­ dict a sex difference, increasing from age 8 or so. No such effect has been found. Of the studies illustrated in

Fig. 2.8, Armstrong et al. (1970) found no significant sex differences, Dawson et al. (1973) found significant sex differences on all illusions but the direction of the dif­ ferences was inconsistent from year to year; and Stewart

(1971) found males significantly more susceptible than females, consistently across age groups.

It may be that different influences are operative at different ages, that some influences swamp others, or that some influences cancel others out. Attempts to explain developmental trends on this basis (Dawson et al., 1973;

Segall et al., 1955) will continue to appear arbitrary until the causes of variation in illusion susceptibility are better understood. CHAPTER THREE

BACKGROUND TO JAMAICA

In this chapter, the geography and history of Jamaica and its social and educational systems are described in enough detail to provide the necessary background for the following chapters. Several sources have provided so much information for this chapter that it would be tedious to cite them at every juncture: Clarke (1957), Figeroa (1971),

Gordon (1953), Henriques (1968), Hurwitz & Hurwitz (1971),

Kerr (1952), Ministry of Education (1973), and United

Nations Special Fund Project (1971a). The author has fre­ quently modified or augmented this information from his own experience of living in the country and talking informally to Jamaican teachers and other educators during the period of the present research, and from unpublished statistics collected at source.

3.1 A brief geography and history

3.11 Geography

Jamaica lies in the Caribbean Sea about 200 km south of Cuba. It is about 250 km long and 70 km wide, inhabited 99 100 by close to two million people. The central parts of the island are steeply hilly, heavily wooded, and uninhabited in some places. Fig. 3.1 summarizes the topography of the island.

For administrative purposes, the island is divided into

13 parishes and one corporate area (the capital, Kingston).

The parishes and their capital towns are shown in Fig. 3.2.

3.12 History

Jamaica was "discovered" by Christopher Columbus in

1493, and became in the next century a useful staging post for Spanish ships on their way to the American mainland.

A few planters settled to grow tobacco, but the only results of their activity were the elimination of the native Arawak

Indians and the introduction of the first African slaves from what is now Ghana.

The British captured the island in 1655, and were more successful by changing to the latest luxury crop: sugar. In the eighteenth century, great numbers of slaves were im­ ported from all over West Africa to work on the extensive plantations located on the coastal plains all around the island. Absentee landlords reaped great wealth from sugar sales in London while their managers in Jamaica treated the 101

o S>

Hi :■ ----

Above 1 000 m ■!;li 500 - 1 000 m

200 - 500 m 9 . , , ..JP 0 — 200 m km

FlK. 3»1 Topography of Jamaica,

Fc-liV\C Llh)

UcecL r) , P o rb HANCVER ‘v"'’■ 1 Maria- I TRELWMY I ST- JL \ ST. M A R Y Port weSTMoRELAfJjJ tofMO

So-VO-nrux- \ MfrN- r A jPcRTLAND W-Mar sr. EU-.:.BETH ' f Block ■Rive-r TH ofA ftS Kl^^GSTcW — ST. AkJiREW M o l o . r U : coRfrRATE Boj AREA

Fig, 3»2 Jamaican parishes and parish capitals. 102

slaves like cattle. Families and tribes were broken up to guard against organized insurrection, and the practice of any religious customs which might unite the slaves was also frequently forbidden. Resident planters so often took female slaves to bed that a new race of "coloreds" arose, many of whom were set free by their masters. Because of their relation to the white overlords, this class was re­ garded as being above the blacks— but still below the poor and often criminal whites who were imported as indentured servants.

At the time of Emanicipation in 1833, slaves made up some 90% of the population. Their condition was not much improved by the Act, since the planters made high charges for housing and services previously provided free. Rent was often paid in kind from the produce of the small plots of land the slaves had been given at an earlier date. Many of them left the plantations for the hills, preferring to scratch a living farming on their own account. This action, together with the withdrawal of West Indian preferences by

Britain, led to a steady decline in the sugar business and an end to Jamaica's former prominence in the British Empire. 103

In 1830, the freed coloreds were given some of the civil rights previously enjoyed only by whites. In 1865, with blacks clammering for their civil rights, the Assembly gave up the island's independence and became a Crown Colony, ruled from London. The first black man was not elected to the Governor's Council until 1901. Jamaica's economy developed with the introduction of a wider range of crops

(Jamaica came to be the world's largest producer of bananas), but the gap between the rich and poor (still mainly white versus black) remained as wide as ever. Many

Jamaicans emigrated during this period, to be replaced by thousands of East Indian and Chinese immigrants imported to work the plantations.

Jamaica's final march to full independence started in

1938 with the usual sequence of riots, incarceration of the leader, and his later appointment at the head of a new govern­ ment. After a brief spell in the West Indian Federation

(1958-1962), Jamaica became independent under a democratic government in 1962.

The economy of the island was greatly enhanced by the discovery of bauxite in 1952, and a considerable increase in tourism and industry in the past ten years has also 104

contributed to Jamaica's growth. The present per capita

income is about 0600, which is the highest in the Caribbean

and higher than all the new African states.

3.2 The educational system

3.21 Overview

The structure of the Jamaican educational system is

shown diagrammatically in Fig. 3.3. The system is highly

selective and is dominated by the public examinations used

for this purpose from Grade 5 onwards, summarized in Table

3.1. At each stage, the great majority who are unsuccessful,

if they do not drop out, stay on at the same school (or,

after Grade 6, transfer to a junior secondary school) and try

again the next year, always hopeful of reaching one of the more prestigious secondary schools.

3.22 Elementary education

During the first period of Jamaica's history (1655-1833),

there was little public interest in education since three quarters of the white children were sent back to England to be educated and the blacks were not thought to be worthy of

education. Such education as there was, was provided by private foundations. 105

Approx. rade age

22 Teachers' Colleges (6) 20 •H

JSA Sixth (S c CAST forms

•rl Technical H W high High Ü Æ schools schools (6 ) , (40)

Junior secondary schools (64)

O, Cl

All-age Primary Preparatory schools schools schools (553) (219)

Infant Infant Basic schools departments schools (26 ) (26 ) (1754 )

Fig. 3.3 The Jamaican educational system, 1973. The numbers of institutions of each type are shown in brockets. The blocks are intended to shovf how students transfer from one school to another; their width is not proportional to the numbers of students in each institutions. For abbreviations, see Section 3.34. 106

Table 3.1

Principal public examinations

Name of Taken in: examina­ Abbrev­ For admission to: iation School Grade tion Common 11+ All-age 5&6 High schools Entrance Primary Comprehensive high Preparatory schools Technical 13 + All-age 7&8 Technical high Common Junior schools Entrance secondary Grade Nine GNAT All-age 9 High schools Achievement Junior Technical high Test secondary schools Comprehensive high schools Vocational schools

Jamaica JSC High 10 Teachers' School Technical colleges Certificate high Employment Private

General GCE High 11 Sixth forms Certificate 0-level Technical JSA, CAST of Educa­ high Teachers' tion Private colleges UWI preliminary courses Employment

GCE High 13 UWI degree A-level courses Employment 107

The missionary societies, starting about 1754, provided the first elementary education, the main purpose of which was to teach the slaves to read the Bible. Under the terms of the Emancipation Act of 1832, the British govern­ ment made grants to the churches for running schools, but the Jamaican government refused to continue the subsidy when it ran out. The regression to Crown Colony status brought about greater public involvement in education. At first, grants were given to the church elementary schools on the basis of their pupils' performance at inspection time

("payment by results"); in 1871, however, only six schools out of several hundred qualified for the maximum grant.

Tuition fees were abolished in 1865, payment by results in

1898, and government financing of new church schools in

1908.

There has always been a shortage of qualified teachers in the elementary schools, partly because of poor salaries.

Classes were large and buildings cheaply built. The cur­ riculum, which came to encompass Grades 1-9, concentrated on the three Rs, and rote memorization with corporal punishment for failure was a standard teaching method. The influences of their religious foundation, payment by results, poor 108

financial support, and the highly selective examination

system combined to produce an elementary education which

has been described as dysfunctional (Ruscoe, 1963).

Many of these problems still afflict the elementary

schools today. Buildings are old, half the teachers are

"pretrained" (untrained), and the average class size is 50.

An authoritarian attitude can still be observed, and is betrayed by the admonitions to Be Punctual, Talk Quietly and Be Kind to be found in most classrooms. Extensive programs of inservice education and curriculum development have recently been mounted by the Ministry of Education to upgrade the pretrained teachers and improve the curriculum by giving teachers more guidance on what knowledge, skills and attitudes to promote at each grade level; but given the depth of tradition and the size of the problem, it is likely to be some time before these programs can be fully effective.

Since the development of junior secondary schools

(Section 3.23), many elementary schools have closed their senior departments (Grades 7-9). Schools with Grades 1-9 are now called all-age schools, while those with only Grades

1-6 are called primary schools. There has also been a 109

recent increase in the number of schools offering preschool

and kindergarten education, especially the community-owned

basic schools, only half of which receive financial

assistance from the government.

Children can take the Common Entrance (11+) examination

for entry to high school as long as they will be 11 or 12

years old the following September. They thus normally take

the 11+ in Grade 5 and, if unsuccessful, again in Grade 5.

The papers are in language, composition, arithmetic and mental ability (I.Q.), and their content strongly influences what is taught in the elementary schools. In 1974, 30 699 candidates sat the 11+ examination and only 4 777 gained

entry to high schools.

There has been a rapid growth of private elementary schools (preparatory schools) supposedly giving pupils a better education than the public schools. In the 1974 11+ examination, 26% of the free places were awarded to students from preparatory schools, although only 11% of the entries came from these schools. The higher proportion is more likely due to the higher social class and parental interest of the preparatory school pupils than to better teacher qualifications or school equipment. Since their popularity 110

depends to a large extent on their examination results,

it is doubtful whether the curriculum or instruction in

the preparatory schools is any more liberal than in the public schools.

Table 3.2

Numbers of elementary schools and total enrollment in 1972-73, by type

Type of school Ages of Number of Enroll­ pupils schools ment All-age schools (junior departments) 6-12 553 191 652 Primary schools 6-12 219 132 581 Preparatory schools 6-12 171 23 658 Infant schools and infant departments of primary schools 4—6 49 13 080 Basic schools 4-6 1 754 112 150

Official estimates of the numbers in each type of elementary schools are shown in Table 3.2. Actual attendance at public schools is only about 60% of enrollment. Attendance on Fridays is notoriously low, especially in the rural areas where children are needed to collect crops and take them to market; it can be as low as 20% of enrollment in such cases. Ill

3.23 Secondary education

High schools By 1865, there were 10 high schools in

Jamaica, all following a classical curriculum. Most ware single-sex schools and most had been established by private foundations. No regular government aid was given until

1920, and by 1931 there were 30 high schools. In 1973, there were 40 public high schools: 7 boys' schools, 14 girls" schools, and 19 mixed (coeducational) schools; the number of students in high schools was estimated to be 7.5% of the age group.

The high schools are academic and university oriented.

Students take the General Certificate of Education (GCE) examinations set by London or Cambridge university boards, the ordinary (0-) level at the end of Grade 11 and the advanced (A-) level at the end of Grade 13. Only about 25% do well enough on the 0-level examinations to be admitted into the Sixth Form (Grades 12 and 13). However, several high schools have recently become more diversified. Six schools have unrestricted entry to Grade 7, preferring to select after Grade 9. Fifteen have facilities for teaching industrial arts and eight for business education, and more are expected to add technical and commercial departments in the near future. 112

Before the 11+ examination was introduced in 1957, all

high schools charged fees, although there were a few

scholarships available to very bright students (125 scholar-

schips for 8 000 places in 1954). From 1957-73, 2 000 full

and 2 000 half scholarships were awarded annually; in 1974,

all 4 777 scholarships awarded (representing 95% of the

total number of high school places) cover the full cost of tuition, but students still have to buy their own books

and stationery. The number of free places is expected to rise to 5 000 in 1975. Since entry in now on merit and not on ability to pay, the previous upper social class bias of high schools has been much reduced, although it is still as obvious as in other countries which operate a narrow selective system (Husen, 1967, Vol. 2, p. 113).

Since 1970, high schools have also taken students from all-age and junior secondary schools who have done well on the Grade Nine Achievement Test (GNAT). About a fifth of their students enter this way; they are usually regarded as in need of remedial work and put in a separate class to repeat Grade 9.

High schools also face a teacher shortage. It is estimated that 40% of high school teachers have had no 113

professional training, and expatriate contract teachers are employed extensively, especially in the country schools and for mathematics, physics, Spanish, and industrial arts.

Technical high schools Until very recently, education in

Jamaica has been seen as book learning, and attempts to introduce agricultural, vocational or other practical educa­ tion at the secondary level have met with little public support. Independence and subsequent industrialization brought about a gradual change in this attitude. There are currently six technical high schools in Jamaica, all but one of them founded in the 1960s. The curriculum is still rather academic, but students specialize in vocational subjects just as high school students specialize in arts or science subjects. They also take the same GCE 0-level, but in dif­ ferent subjects, although some students only reach Jamaica

School Certificate (JSC) level. There is a de facto sex distinction in the technical high schools: Mechanical engineering, automechanics, electronics, building and agriculture are taken only by boys, and commerce, secretarial studies and home economics only by girls. Despite efforts to break down this division, there have been only a few exceptions to the rule. 114

Students are selected from the junior secondary and

all-age schools on the basis of their performance in the

Technical Schools Common Entrance examination (13+), taken in Grades 7 and 8, and the Grade Nine Achievement Test

(GNAT). Both examinations are similar to the 11+, consisting of papers in English, mathematics and mental ability. In

1972, 614 places were awarded from 9 818 candidates in the

13+ and a further 231 places from the 15 399 GNAT candidates.

Students who enter via the 13+, who must be 13 or 14 years old on entry, take a four-year course to 0-level.

Students who enter via the GNAT start in the second year of this course. Like their colleagues who transfer to high schools, technical high school students thus reach 0-level an average of one year later than high school students.

All-age schools The situation in the senior departments of the all-age schools is little different from that in the junior departments: Classes are large, buildings old and teachers poorly qualified. It is a rare school which has a laboratory or workshop, or even a secure place to keep any classroom equipment or materials they might have. Until recently, there was little or no guidance on curriculum and no official supply of textbooks; and students are often too 115

poor to afford even exercise books or pencils. Current

Ministry of Education programs of inservice training and curriculum development are directed towards alleviating the worst of these problems.

Junior secondary schools To provide a better secondary education for more Jamaican students, the government started establishing junior secondary schools in 1956. Eighteen senior schools were renamed, and a major building program was undertaken with the aid of a World Bank loan. By 1973, there were 64 junior secondary schools all over the country. Each school is open to all students without examination upon reaching the age of 12 yr. To meet the needs of industrial development, the three-year curriculum is intended to be vocationally oriented, but the lure of a white-collar job and the possibility of winning a high school or technical high school place still dominate students' objectives (Miller, 1967a). All the new schools have workshops and laboratories for industrial arts, domestic science, and commercial studies, but often do not have the staff to take full advantage of these facilities.

Since this study was started, the government announced that junior secondary schools would be extended to Grade 10 116

in 1974-75 and Grade 11 in 1975-76. (They are now to be

called secondary schools.) A much greater emphasis on

vocational training is planned for these two years.

Comprehensive high schools Three new high schools combine

academic, technical and commercial programs in one school.

Entry is by the 11+ or GNAT as for the high schools. As many

of the regular high schools are adding technical and com­

mercial departments, the comprehensive high schools may

become indistinguishable from other high schools in the

near future.

Vocational schools A number of vocational institutions pro­

vide for all-age and junior secondary school leavers. There

are two rural vocational schools with courses in home

economics and agriculture, and two technical institutes

which provide pre-apprenticeship training. A network of

trade training centres has recently been established by the

Ministry of Labour.

Private secondary education There is a small number of prestigious high schools not yet absorbed into the public

system which provide good quality education for those who 117

can afford it. There is also a much larger number of poorer private secondary schools and commercial colleges which pro­ vide education for those who have failed all the public examinations. Many of the high schools and teachers' colleges and all the technical high schools also operate evening classes or extension schools. These students com­ monly aim at the GCE 0-level or the Jamaica School Certifi­ cate (JSC), which is at a lower standard than the 0-level but is accepted as a qualification for certain fields of employ­ ment and further education. Over 55 000 students sat the

JSC examination in 1972.

Table 3.3

Numbers of secondary and vocational institutions and total enrollment in 1972-73, by type

Ages Number institution Full--time Part-time High schools 11-19 40 29 899 a Junior secondary schools 12-15 64 51 200 All-age schools (senior departments) 12-15 553 78 912 - Technical high schools 13-18 6 4 350 3 470 Comprehensive high schools 11-17 3 3 615 527 Private schools and colleges 12-19 140 16 668 a Vocational schools 15-17 2 288 - Technical institutes 15-18 2 285 250 Trade training centres 17-20 21 600 800

^No figures available. 118

Summary The numbers of each type of secondary and voca­ tional school and official estimates of their enrollments are given in Table 3.3.

3.34 Post-secondary education

The University of the West Indies (UWI) was founded in

1948 as a college of London University; it is a cooperative venture of the English-speaking West Indian territories and has campuses in Jamaica, Barbados and Trinidad. About 30% of the students attending the Jamaican campus are from over­ seas. Good GCE A-level results in the relevant subjects are required for entry to the degree courses, but the science faculty on the Jamaican campus accepts students with good

0-levels for a one-year preliminary course. Thus most

Jamaican UWI students come from high schools, a few from technical high schools, and the remainder from the other post-secondary institutions.

UWI offers the usual wide range of undergraduate and postgraduate courses. It is also responsible for training secondary school teachers through a one-year education course taken after gaining a bachelor's degree and some teaching experience; however, the first degree alone is regarded as sufficient by most schools. 119

Elementary and junior secondary school teachers are trained in the six teachers' colleges, most of which have been established in the past 40 years. Students come from the entire range of secondary schools and have a wide variety

of JSC or GCE qualifications, but all have to pass the

Teachers' College Entrance examination; in 1972, there were

2 832 candidates of whom 1 098 obtained places. The course consists of two years of academic and professional studies, followed by a one-year internship. The output from the colleges is barely enough to keep up with the attrition from the profession. Expansion of existing colleges is planned, and one new college is being constructed, but it seems that expansion might be limited by the quality of the applicants unless teaching can be made more attractive.

Skilled technicians are trained at the Jamaica School of Agriculture (JSA) and the College of Arts, Science and

Technology (CAST). Both institutions require good passes in the relevant subjects at GCE 0-level, and CAST also has its own entrance examination. Students come from high and technical high schools, and courses last from one to four years. Besides building and engineering subjects, CAST offers courses in pharmacy, institutional management and commerce; 120

there are many more qualified applicants than places, and

the college is likely to expand considerably in the near

future. JSA and CAST also train teachers of agricultural

science and industrial arts for the technical high and

junior secondary schools.

Table 3.4

Enrollment in post-secondary education in 1972-73, by institution

Enrollment Institution Full--time Part--time University of the West Indies^ Undergraduates 1 821 424 Postgraduates 103 183 Teachers' colleges (6) 2 097 - Jamaica College of Agriculture 204 - College of Arts, Science and Technology 762 1 292

^Figures refer only to Jamaicans attending the Jamaica campus.

Enrollment in post-secondary institutions is summarized

in Table 3.4.

3.3 Some social factors affecting education

In Section 3.2, the historical influences of religious

interests and industrial development on education in Jamaica was noted at various points. Five further social factors 121

which are relevant to the present study are discussed

below.

3.31 Language

The majority of Jamaicans speak a dialect (or creole

or patois) which is a compôte of West African and English

vocabulary and grammar (Cassidy, 1971). As Cleary puts it;

At first the Englishman to our shores is certain that we speak a foreign language although he had been made to believe that English was our tongue. It is only after some months that he begins to be confused that after all we do speak English and then he is wrong. For our language is Jamaican, distinct and different from English. True, we have borrowed some English words; but still it is not. It is Jamaican (1971, p. 5).

This quotation succinctly illustrates the sensitivity of

some Jamaicans to the uniqueness of their language; it also

illustrates, although doubtless this was not Cleary's in­

tention, the influence of the dialect on Jamaican standard

English.

Figeroa (1971, p. 191 ff.) has commented at length on

the difficulties which the difference between the home

language and school language produces in elementary schools.

For the present study, the question was whether the experi­ menter (who speaks English English) could effectively com­ municate with his subjects. 122

3.32 Ethnicity

The ethnic structure of Jamaica's population is shown

in Table 3.5 (West Indies Population Census, 1960). The

historical association between light skin color and high

social class (Section 3.12) is still present, and is shown

Table 3.5

Ethnic Composition (1960)

Percentage of Reported ethnic type total population African 78.8 Afro-European 15.1 European 0.8 Chinese & Afro-Chinese 1.0 East Indian & Afro-East Indian 3.4 Others 3.3

by the fact that there is a far greater proportion of

Chinese and Afro-European Jamaicans in high schools than in the general population (Miller, 1967b). Overt color preju­ dice was prominent until fairly recently, but since inde­ pendence it has died out except in a few extremists.

Jamaica's motto of "Out of many, one people" seems to have been realized. However, there is still a covert preference for European features (Miller, 1969). 123

The color factor offered an opportunity and a danger

for the present study. The opportunity was to study the

relation between skin color and perception (Section 2.73)

along a continuum of color within one culture; the danger

was that the experimenter's ethnic type, European, might be

so untypical as to bias the results (Miller, 1972).

3.33 Family structure

Seventy per cent of Jamaican children are born to

unmarried mothers. This bald statistic is enough to prove

that family life in Jamaica (and in several other West

Indian islands) is quite different from that in other parts

of the world (Smith, 1962). Henriques (1967) described four

types of family:

A. Christian marriage - the standard ideal for Western society.

B. Faithful concubinage, or Common Law marriage.

C. Maternal or grandmother family.

D. Keeper family.

Type D is a temporary cohabitation which may develop into

a permanent association (Type B) and perhaps later into marriage (Type A). When a temporary cohabitation breaks up, the woman may seek her mother's help in bringing up the 124 children or may even leave them entirely in her care

(Type C). Henriques estimated that 25% of families were of

Type A, 25% of Type B, the remainder being divided between

Types C and D "in an inexact proportion." He traces the high proportion of loose unions to the days of slavery.

The influence of these conditions on family life is tremendous. Although the Jamaican ideal appears to be the

Type A family of a rather Victorian nature, with a strict and all-powerful father as head of the household, this is only attained in a quarter of the families— and most of these are in the small middle and upper classes. Most women resist marriage because they do not wish to lose the freedom which they enjoy in the looser union. There may be frequent changes of partner, and the many children in a house may have different fathers and be unrelated to the current male resident, if any.

A great number of children therefore grow up without a strong father-figure, or they may be expected to defer to a man in the household even though it is the woman who runs it. Upbringing is at once harsh and soft, with many threats and much spoiling. This results for boys in "a fixation on the mother combined with a belief that the world in general 125

should look after him with maternal care" (Kerr, 1952,

p. 167); there can be little doubt that the absence of a

father-figure also has a profound effect on girls.

The possible effect of father absence on cognitive

development seemed worth investigating in the present study

(Section 4.22).

3■34 Sex bias

The independence of Jamaican women is reflected in the

magnitude of their contribution to the economic life of the

country. There seem to be many more female teachers,

business executives and government officials than in most

countries, to the point where one sometimes gets the impres­

sion that Jamaica is entirely run by its women! However,

just as in the family, women are expected to defer to any male at roughly the same level in the organizational hierarchy,

and the highest levels of government and commerce are still dominated by men.

One reason why women can exercise their independence is that females are well educated compared to males. In the first place, there is a greater number of girls than boys in regular attendance at public schools. For example, only 36% 126

of the candidates for the 1974 Common Entrance examination were boys, and only 41% of the Grade 9 junior secondary and all-age school students surveyed as part of the present study were boys (Section 6.31). To counter this bias, it is government policy to award equal numbers of high and technical high school places to boys and girls. In the second place, there are more females receiving further educa­ tion in Jamaica than in most other countries. In 1972-73, women formed about 83% of students in teachers' colleges, although only about 28% at CAST (5% in building and engineer­ ing, but 95% in institutional management), and about 20% at

JSA. In the UWI engineering and agriculture faculties (in

Trinidad), only 5% of the Jamaican undergraduates are women; but in other faculties, spread over the three campuses, 50% of the Jamaican undergraduates are women, compared to less than 40% amongst other nationalities.

Many reasons have been suggested to explain the greater school attendance of girls. Manley (1963) suggested that young girls may be more docile than boys and therefore more receptive to the dogmatic teaching they receive. The mainly verbal curriculum in primary schools may also be expected to favor girls (Section 2.74), as may the fact that 127

almost all primary school teachers and most principals are

women. In households containing no adult male, lack of

paternal discipline and example is likely to reduce boys'

incentive to attend school. Kerr (1952, p. 81) pointed out

a more practical factor: Boys are often expected to help

with farm work during the day, whereas girls can do their

household chores before and after school. Another factor is

the differential aspirations of boys and girls: Miller

(1967a) found that girls in junior secondary schools regarded

their practical courses such as sewing, cooking and child

care as relevant to their future, whereas boys would have

preferred academic subjects to the gardening, agriculture

and industrial arts courses they were expected to take. It

also appears that most occupations in the country areas are

female-dominated; apart from farming and roadmaking, men's

jobs are away in the towns or cities (Smith, 1960).

Jamaican women's greater independence may be expected to

enhance their cognitive development compared to women in more

male-oriented societies (Section 2.74). However, in inter­ preting the results of the present study, it must be

remembered that school samples cannot be representative of

the general population. 128

3.35 Environment

The usual urban-rural dichotomy cannot be applied blindly to Jamaica. Only the capital city Kingston (popu­ lation 550 000) can be considered urban in the Western sense, although several of the parish capitals (Fig. 3.2) are fast approaching this status.

Rural Jamaica has been divided into two distinct types: the sugar belts with their large plantations, seasonal em­ ployment and migrant labor force, and the hill areas settled by farmers who grow mixed ground crops on farms often only two or three acres small (Fig. 3.4). This dis­ tinction has existed since Emancipation (Section 3.12), but whereas the continuity has led to stable, conservative, poor and closed hill communities (Smith, 1965), the plantation areas have merely continued to show lack of community feel­ ing, fleeting personal relations, and variable economic conditions.

Three other environmental types of more recent origin can be distinguished: the market towns, the mining areas, and the tourist areas. Market towns, scattered all over the island, form the focus of rural activities (at least once a week) and tend to attract a trading and minor administrative 129

Mining Tourism Sugar plantations Small farmholdings

Fir. Pistribution of major economic activities in Jamaica.

community. Their environment is clearly richer than the village, but is not developed to the extent of including any industry. Mining is mostly of bauxite and besides providing local employment also tends to bring more money into the area. Tourism, which started in the 1920s, has expanded enormously in the last decade; activity is mainly restricted to the north coast. Mining and tourist areas are shown in Fig. 3.4. CHAPTER FOUR

TEST SELECTION AND DEVELOPMENT

In Section 1.42, five general research questions were formulated concerning the perceptual development of Jamaican students. Chapter 2 outlined the empirical and theoretical basis of the study, and Chapter 3 described its setting and outlined some special factors likely to affect spatial ability in Jamaicans. The present chapter comes down to the practical details of choosing suitable tests to use in the three phases outlined in Section 1.43. Since no suitable tests had already been used in Jamaica, it was necessary to carry out extensive pilot testing and adaptation; this process is also described in this chapter.

4.1 Psychological testing in developing countries

The simple fact that nations differ in climate, health, education, occupations, expectations and many other cultural and environmental factors leads to difficulties in using in developing countries psychological tests which were developed in the industrial nations. The difficulties especially relevant to the present study are discussed

130 131 below. For longer discussions covering wider issues, see

Berry (1969), Biesheuval (1949), Campbell (1964), Frijda &

Jahoda (1966), Schwarz & Krug (1972), Strodtbeck (1954) and Vernon (1969).

4.11 Test procedures

The rarity of the group testing situation in some cultures is a possible source of difficulties if paper-and- pencil tests are used without adaptation. The variance introduced by unfamiliar test procedures can be so great that the test task becomes almost incidental (Schwarz,

1961; Schwarz & Krug, 1972). Differences in strategy, perseverance and motivation may also be marked (Vernon,

1969) .

Schwarz (1961, 1963) has formulated several principles for overcoming unfamiliarity with the group testing situ­ ation. These principles lead to the use of oral instruc­ tions, visual aids, and supervised practice. The practice not only improves performance to a more stable level (Vernon,

1967a), but also enables the tester to check that all sub­ jects know what to do and where to mark their answers.

Individual tests are in general easier to explain to subjects, and the tester can usually tell whether the subject 132

has understood the instructions correctly, although extra

items may still be needed to counteract gross unfamiliarity

with item format. The subject's performance may, however,

be seriously affected by his perceptions of the tester and

the test situation (Frijda & Jahoda, 1955). Differences

between the subject's and the tester's cultural background

are particularly influential (Campbell, 1954; Wober, 1959).

4.12 Item content

If an item in a test developed in one country refers

to an object which is unfamiliar in a second country, the

object must usually be replaced by one which is familiar.

Such changes can affect the characteristics of the test.

Finding equivalent objects is often very difficult, if not impossible because of the circular arguments which must be invoked (Frijda & Jahoda, 1955).

Several psychologists have attempted to use abstract

"objects," namely figurai material, for item content; they have assumed that the content would be equally unfamiliar to testees in all countries so that the tests could be used cross-culturally (Cattell, 1944; Raven, 1958). This assumption seems preposterous when one considers the vast differences between, say, the typical rural African and the 133 urban European spatial and figurai environments (Bies­ heuval, 1943). Kidd and Rivoire (1965) exposed the fallacy by showing that some of the Cattell and Raven items reflect cultural and class differences whilst others do not.

Even if demonstrably equivalent familiar or unfamiliar objects could be used in test items, there would still remain differences in familiarity with or use of the operations called for. The presence of these remaining differences has been amply demonstrated (Cole et al., 1971).

4.13 Score interpretation

Suppose that it were possible to overcome the obstacles to cross-cultural comparability described in the last two sections; It still could not be concluded that the same scores should be interpreted in the same way in any two cultures. Although "it seems reasonable to assume that the dimensions of skilled human performance are the same everywhere in the world" (Schwarz & Krug, 1972, p. 56), most recent cross-cultural research has thrown doubt on such assumptions (Dawson, 1971).

In particular, the results cited in Sections 1.22 and

1.23 cannot be taken to prove, without further evidence, that "spatial ability" is an identifiable factor in the 134

intelligence of Jamaican students, or that tests of spatial ability predict performance in spatial tasks. For example, Irvine (1955) concluded from the results of testing

Zambian tradesmen that the I-D Mechanical Information test

(Schwarz & Krug, 1972) was, in effect, a test of general intelligence in that sample. As he warns, it cannot be assumed that "a test designed to test quality X, because it looks like a test of quality X, is in fact such a test"

(p. 15). Empirical evidence is required before it can be said what a test measures.

4.14 Implications for this study

Two phases of this study required the use of paper-and- pencil group tests of spatial ability with Jamaican secondary school and college students. In the third phase, younger children were tested individually, and because

Jamaican assistants were not available, all testing had to be carried out by the author. In view of the cautions sounded above, it is necessary to defend these procedures against possible charges of cultural bias. The defense takes the following lines:

1. The tests and test procedures used minimized variance due to extrinsic factors. 135

2. The test items had high face validity and there was adequate evidence of their pre­ dictive validity.

3. There is sufficient evidence for the existence of a spatial factor or factors in Jamaican intelligence.

Extrinsic variance Because the content of all the tests

used in this study was visual, there was no variance

attributable to verbal ability apart from that needed to understand the test task. This variance was reduced in the group testing (Section 4.3) through observance of the

Schwarz principles (Schwarz, 1951, 1963), which were found to be both necessary and effective for communicating the test task and procedures. The choice of Grade 9 was found to further reduce language differences between the writer and the students (Section 3.31). In the individual testing

(Section 4.4), the interview was reduced to simple one-word- answer questions; the tests were easy to explain, and the tester could easily tell from the subject's actions whether the task had been effectively communicated. It is there­ fore claimed that variance due to misunderstanding the test task and unfamiliarity with the test situation were reduced to the level one would expect in more sophisticated popula­ tions . 136

By deciding to use himself as tester, the writer intro­ duced a systematic bias due to the difference in nationality between himself and his subjects. However, the current absence of strong anti-British feelings among Jamaicans suggests that the bias was small. The alternative, of using

Jamaican testers (had it been possible) would have intro­ duced extra variance due to differences in expertise and personality which would have been difficult to control.

Test validitv In the light of the previous observations, the sole remaining sources of variation in scores on the tests used in this study were familiarity with diagrams, with the objects depicted, and with the operations to be per­ formed on these objects. With the exception of two very familiar objects (bottles and telegraph poles), all the objects used were regular 2D and 3D figures (squares, triangles, cubes, cylinders, etc.). Familiarity with these shapes is intrinsic to those aspects of spatial ability which are of particular interest in this study, and therefore a valid component of the observed variance. The same argument applies to familiarity with diagrams and with spatial opera­ tions . 137

Many of the tests used have high face validity. It is

argued in Section 4.32 that the drawing, interpretation and manipulation of diagrams are essential steps in solving problems which involve the application of mathematics to physical space situations. Tests of low face validity, such as the Hidden Shapes Test (Section 4.22, 4.33 & 4.44) and the Pacific Design Construction Test (Sections 4.23 & 4.45), have extensive documentation supporting their relation to spatial ability.

Evidence of predictive validity is slim. The I-D Boxes

(used in this study) and I-D Figures test (similar to

Hidden Shapes) have shown correlations of 0.30-0.47 and 0.30-

0.39 respectively with ratings in various trades in African populations (Schwarz & Krug, 1972). The NIPR General

Aptitude Battery, which uses performance tests of a dis­ tinctly spatial character, has shown good overall validity with illiterate adults in South Africa (Blake, 1971). The paucity of validity data is a result of the recent development of these test batteries and the expense of validity studies.

Factor structure There is evidence of a spatial factor in the intelligence of Africans and those of African descent.

Irvine (1969a) obtained perceptual-spatial factors in three 138

out of four analyses of results obtained with East African

schoolchildren (although there were some anomolous loadings)

and in two out of three other studies analyzed. Vernon (1965)

obtained a spatial factor for Jamaican 11-year old boys.

Grant (1970, 1972) and Kendall (1971, 1972) both obtained

distinct space factors in illiterate South Africans.

Finally, Jensen (1973) showed similar factor structures in

samples of young American children of European and African

ancestry.

As most Jamaicans are of African or European descent

(Section 3.32), these results made it reasonable to assume

provisionally and cautiously that spatial ability is a viable

construct for Jamaican students. This assumption was hedged

by the inclusion of a wide variety of spatial tests.

4.2 Test selection

4.21 Group tests

In selecting group tests of spatial ability, the

obvious first step was to use tests of the factors field

independence (Cf), spatial orientation (S) and spatial

visualization (Vz) as described in Section 1.22, omitting perceptual speed (P) because it has never been linked with mathematical or technical ability (Section 1.23). This was 139

done in a preliminary study of West Indian mathematics

teachers (Mitchelmore, 1973b), using the NLSMA tests Card

Rotations, Punched Holes, Hidden Figures, and Form Board

(Wilson et al., 1968). There were several reasons why this

strategy was not pursued in the present study: Firstly,

the evidence for separate S and Vz factors is slim

(Michael et al., 1950; Werdelin, 1961; Borich & Bauman, 1972)

and they did not separate in Vernon's Jamaican study (1965).

Secondly, bearing in mind the difficulties of extrapolating

from the immediate test items, it was felt that 2D tests of

S and Vz were inappropriate because of their low face validity. Thirdly, since problems involving 3D objects so frequently call for a drawing of the object, and because 3D drawing is under-researched, it was felt desirable to in­ clude a test of 3D drawing : This left less time to devote to the other tests. It was decided that the group test battery should consist of a test of field independence, a

3D test of spatial orientation/visualization, and a 3D drawing test.

As a test of field independence, embedded figures tests

(EFT) have been widely used in cross-cultural research, usually in individual form (Within et al., 1973). Many 140

studies have shown a relation between EFT scores and strict­

ness of upbringing (Section 2.71) while some have found a

relation between EFT scores and 3D pictorial perception

(Dawson, 1967a; Vernon, 1955). Permission was obtained

from the Educational Testing Service to adapt their group test Cf-1 (French et al., 1963), and the NLSMA version used in Grade 5 (Wilson et al., 1968, pp. 103-108) was selected to start with.

The following 3D tests were examined as possible measures of spatial orientation/visualization: Match Box

Corners, Shapes and Models, Paper Folding, and Block

Building, from the NFER Spatial Test 2 (Watts et al., 1951);

Sections I, Views, Cubes, Sections II, and Faces from NFER

Spatial Test 3 (Smith & Lawes, 1959); Cube Rotations, Cube

Comparisons, Paper Folding, and Surface Development, from the Kit of Reference Tests (French et al., 1963); Spatial

Orientation and Spatial Visualization from the Guilford-

Zimmerman Aptitude Survey (Guilford et al., 1956); and Space

Relations from the Differential Aptitude Tests (Bennett et al., 1966). The Guilford-Zimmerman items were considered unfamiliar to Jamaicans, and insufficient information was available on the reliability and validity of the NFER and 141

French Kit tests. Considerable information was available

on the DAT Space Relations test (Bennett et al., 1965;

Ghiselli, 1955), showing it to be a valid predictor of

higher technical performance. The main problem was its difficulty level: It is only recommended for use in Grades

8-12 in the U.S., and in view of the expected retardation was felt likely to suffer from a floor effect in Jamaica.

Permission was obtained from the Psychological Corporation to adapt Form T for Jamaica by using the easier items.

Before copies of Space Relations became available for pilot testing, however, the I-D Boxes test came to light

(Snider, 1972). Like Space Relations, this is a surface development test; but it had been developed specifically for unsophisticated populations using the Schwarz principles

(1961). The test appeared to be of the appropriate diffi­ culty level, having been used between Grades 6 and 10 in

Nigeria, and had acceptable reliability (0.68-0.85) and validity (0.30-0.47). The Boxes test was adopted when the first pilot-testing indicated that the DAT Space Relations test could not be used without extensive revision.

Development of a test of 3D drawing is described in

Section 4.32. 142

4.22 Individual tests

Because factor-analytic studies have used mainly adoles­

cent subjects, no structural theory was available to guide

the selection of tests for the developmental study. Tests

were therefore chosen on an ad hoc basis to sample a wide

range of spatial activities and to investigate interesting

hypotheses suggested by the literature. Several tests

were extensively modified after pilot-testing, and two tests

that proved unsatisfactory were subsequently omitted.

Embedded figures It was decided to include an EFT for the

same reason that it was included in the group test battery

(Section 4.21). In order to make the test easier for younger children, the format was changed so that the complex

figure and the simple figure contained in it were displayed

on the same page. By selecting items from those used in the group test and giving a small sample both versions, the relation between the two forms could be found. Jackson et

al. (1964) found a very high correlation of 0.80 between scores on group and individual versions of this test, but others (Templer, 1972) have found very low correlations.

Block designs Block design tests (Goldstein & Scheerer,

1941; Kohs, 1923; Wechsler, 1949) have also been used exten- 143

sively in cross-cultural studies (see Section 2.51). In

factor analyses (Dawson, 1957b; MacDonald, 1944-45; McFie,

1951; Vernon, 1959), block design tests have loaded con­

sistently on general intelligence factors, with occasional

smaller loadings on perceptual-spatial factors; they also

have the highest subtest-total score correlations in batteries of performance tests of intelligence (Matarazzo,

1972; Ord, 1970). Although Dawson (1957a) considers block design tests to be tests of field independence because of their high correlations with embedded figure tests (0.50-

0.70), it might be more accurate to consider them as measur­ ing general intelligence in practical matters.

Because the Kohs test is lengthy and the WISC test has given rather low scores in young children in Nigeria

(Dastoor & Emovon, 1972), it was decided to use the Ord test entitled Pacific Design Construction Test (Ord, 1970), as recommended by Vernon (1959, pp. 119-120 footnote).

Piaqetian visualization Vernon (1959) used a battery of thirteen short Piagetian tests cross-culturally, and identi­ fied conservation, visualization and numerical factors.

Tests loading on the visualization factor involved the water level in a tilted bottle, the shadow of a manikin, an insect crawling around a bottle, left and right relations, and 144

copying the position of a dot. It was planned to include

these five items as a Piagetian visualization test, but

pilot-testing led to a complete revision (Section 4.44).

Solid representation As noted in Section 2.32, there have

been very few attempts to categorize stages of development

in the representation of regular 3D objects. It was there­

fore planned to develop a new test which would allow a study of children's drawings of the most common mathematical solids, namely the cube, cuboid, cylinder, pyramid and cone. It was found necessary to administer this test individually in order to control for angle of view.

Geometric illusions Cross-cultural research on geometric illusions was discussed in Section 2.8. They were included in the present study in order to investigate more closely the relation of illusion susceptibility to spatial ability, especially field independence and 3D drawing. It was decided to use the revised version (Herskovits et al., 1969) of the test materials which Segall et al. (1966) used in their study. These included the Muller-Lyer, Horizontal-Vertical

(inverted T and L), Sander Parallelogram, and Ponzo illu­ sions (Fig. 2.7, Section 2.81). An attempt was also made to include the Poggendorff illusion which Segall et al. reported 145

was difficult to administer using the first edition of the

test materials.

Other tests tried The Draw-a-Person test (Harris, 1963) was

included initially as a possible measure of intelligence

and sexual bias (Dawson, 1972b). It was found to be a good test to start with, especially with younger students, but seemed to take rather a long time. It was finally decided that the information derived from the test was not worth the time taken to collect it.

An attempt was also made to construct a pictorial depth perception test (Section 2.63). A 15-item test was constructed, using ideas from Duncan et al. (1973), Hudson

(1960) and Vernon (1969). Two items were constructed for each combination of size, overlap, and perspective cues.

However, the test was found uniformly easy by a small pilot sample of students from Grades 3 and 6 so that, in the absence of time for further development, it was reluctantly dropped from the present study.

4.23 Personality characteristics

Many factors which may influence the development of spatial ability were discussed in Section 2.7, and some factors of particular interest in Jamaica were mentioned in 146

Chapter 3. Efforts to include those factors in the present study are described below.

One important factor is a person's age. This was easy to ascertain in Jamaica, unlike many African countries, and formed a basic variable in the developmental study.

The second important factor is the person's sex. As it appears that many intelligent boys do not attend school in

Jamaica (Section 3.34), it is possible that schoolgirls may equal or even excel schoolboys in spatial ability; if so, there would be obvious implications, particularly for technical education, which is presently restricted to boys

(Section 3.23). Also mentioned in Section 3.34 were aspects of female dominance and independence which could affect the development of spatial ability in Jamaican students. Witkin et al. (1962) showed that an authoritarian upbringing was associated with low EFT scores, and this result has been confirmed in cross-cultural studies (Berry, 1966; Dawson,

1967a). It was decided to attempt to obtain ratings of both male and female dominance.

The influence of home conditions has been studied in

Jamaica by Vernon (1965) and in Sierra Leone by Dawson

(1967b). None of Vernon's ten ratings of home background 147

variables had a large correlation with his spatial factor,

the largest being for linguistic background (0.28), male

dominance (0.27), regularity of schooling (0.26) and cul­

tural stimulus (0.25). Dawson found that EFT, Kohs block,

and 3D pictorial perception scores were positively corre­

lated (0.39 to 0.45) with mother dominance, not correlated with father dominance, and negatively correlated (-0.13 to

-0.28) with number of father's wives. Both these studies involved only males.

What effect does the lack of a father or father-substi- tute (Section 3.33) have on spatial ability? Carlsmith

(1964) found that Harvard undergraduates whose fathers had been absent from home during World War II showed higher verbal and lower numerical ability than those whose fathers had not left home; this suggests a general feminization effect, from which an associated decrease in spatial ability would be pre­ dicted. On the other hand, Bieri (1960) found that subjects who scored high on an EFT tended to identify more with the parent of the opposite sex; from this result, it would be predicted that girls' spatial ability would be more seriously affected by father absence, whereas boys' might even be increased. Vernon (1965) found low correlations between 148

father absence and test scores, but he only tested boys.

It was decided to find whether subjects in this study had had

a father or other adult male present during their upbringing.

The well-known relation of social class to intelligence has also been found in Jamaica (Manley, 1963; Vernon, 1965).

By analogy to Dawson's results with number of wives (1967b), it might also be predicted that large family size, which is associated with low social class (Miller, 1967b), might also reduce spatial ability by restricting father-child contacts.

Attempts were therefore made to collect data on parental occupation and education and on family size.

Many studies have demonstrated a difference in spatial ability between urban and rural samples (Berry, 1966;

Holmes, 1963; Kendall, 1972; Shaw, 1969; Vernon, 1965), a difference which is not, of course, restricted to this aspect of intellectual functioning. None of the studies help in determining what causes the difference. There are no doubt many aspects of urban life which contribute, for example, greater rectangularity, more pictures and other visual symbols, better schools, greater cultural stimulation, better housing, and better nutrition (see Sections 2.71-2.73); also, many children and young adults migrate from village to 149

town or city in search of education, employment or excite­ ment. It was therefore necessary to find where students had lived most of their lives. The influence of the environ­ mental factor was studied most directly in the Grade Nine survey.

The quality of a person's early school experience is also likely to influence the development of his spatial and other intellectual abilities. Because of the selective educational system in Jamaica (Section 3.2), it is more likely that the level of one's intellectual ability influ­ ences the quality of his secondary school experience. It was considered impracticable to rate the quality of a student's previous educational experience, but his present experience was partly controlled by including school type in the Grade Nine survey.

The possible relation of skin pigmentation to spatial ability was discussed in Section 2.73. Because of the wide variety of skin colors in Jamaicans (Section 3.32), it was decided to obtain ratings of skin color for as many subjects as possible.

To provide further information on the relation between handedness, eyedness and spatial ability (Section 2.73), tests of hand and eye dominance were devised. 150

It was also decided to investigate the relation of

ambition to ability. Miller (1967a) found that Jamaican

adolescents had ambitions which far exceeded their probable

ability. Of interest in this study was the question of

whether students' ambitions, however unrealistic, are at

least in line with the general direction of their abilities.

On the basis of the above considerations, attempts were made to collect data on the following variables by interview or questionnaire:

Age Father's occupation Sex Mother's occupation Place of uDbrinaino Father'si. s c>«4U.Cucd wxcn Presence of an adult Mother's education male during childhood Skin color Number of people in home Hand dominance Father's strictness Eye dominance Mother's strictness Ambition

Also studied by incorporation into the research designs were type of secondary school attended and location of the school.

4.3 Development of group tests

4.31 Pilot-testing procedures

Construction and preliminary testing of the 3D drawing test were undertaken during May-June 1973 in Columbus, Ohio, and the revised version and the Hidden Figures test were 151

twice pilot-tested and revised during November-December 1973 in Kingston schools which had not been chosen for the Grade

Nine survey. Practice administrations of the I-D Boxes test were carried out in two Kingston schools in December 1973.

All tests were administered by the writer. The pilot-testing schedule is given in detail in Table A.1 (Appendix A).

Before discussing specific tests, it is valuable to consider certain general lessons which were learned from the pilot-testing in Kingston. The first was that the problems Schwarz reported from Nigeria (1961, 1963) were also present in Jamaica. When the initial versions of the

3D drawing test and the Hidden Figures test were administered to Grade 7 students in an all-age school, several looked around in bewilderment after test instructions had been read out; answered Qu 2 as if it were exactly the same as

Qu 1; did not know how or where to write the answers on the separate answer sheet; and did not turn over when they reached the bottom of the first page of items despite the printed instruction to do so. Grade 9 high school students did not seem to have any difficulties, however. This experi­ ence led to a complete revision of item formats and admin­ istration procedures and a second round of pilot-testing. 152

It also suggested that testing procedures currently in use

in Jamaica for the many public examinations used in educa­

tional selection may be grossly invalid; the uncommonly high

intercorrelations between scores on English, mathematics

and mental ability tests (Isaacs, 1974) may indicate that

all tests are measuring primarily the verbal ability involved

in following printed instructions.

Secondly, most schools did not have a separate room which could be used as an examination room, necessitating

the administration of the tests in the students' regular classroom. These classrooms varied considerably, both in the light and space available inside and the noise and dis­

tractions outside. In two of the seven schools, the classes tested were in a large hall with a class on either side, separated only by standing blackboards. In one of them, students sitting two to a desk could not help seeing their neighbor's responses. For the main study, it would clearly be necessary to take steps to ensure that students from the poorer schools were tested in surroundings approximating those in the better ones.

The third observation made was that all schools were exceedingly cooperative and willingly agreed to assist in 15 3 the pilot testing. Students too were most cooperative, perhaps because of the break in school routine, but apparently also because of the novel and appealing nature of the test content. Once the method of administration had been changed as described above, and providing the tester remembered to speak clearly, difficulties of communication due to tester-student differences in accent, skin color, or position power did not become apparent. It was considered that all students could be motivated to do their best and give valid scores on the tests to be used.

4;32 Three-Dimensional Drawing test

Preliminary testing The present drawing test derives from several others which the writer has tried out since July

1972. The first was used with West Indian secondary mathe­ matics teachers in Jamaica (Mitchelmore, 1973b). The items, most of which were taken from an earlier survey of teachers' college students in Jamaica (Bennett, unpublished), involved drawing the hidden edges of prisms, various views of a milk carton, nets of common solids, sections of a cone, and a simple perspective scene. A longer drawing test was administered to mathematics education seniors at the Ohio State University in November 1972. The extra items 154

included drawing a model building, reflections in a mirror,

and the water level in a tilted bottle, together with four

multiple-choice items from various sources. Several items

were also completed by mathematics teachers at Ashland,

Ohio in April 1973 in the course of a talk on visualization

problems at an inservice institute.

Certain tentative conclusions were reached on the basis

of these preliminary enquiries. The most important was that

a person can find it extremely difficult to make a representa­ tion of even the most common object. People may drink out of

a milk carton and look in a mirror every day of their lives and still be unable to draw cartons and reflections.

Subjects would frequently make a first attempt, smile, frown, erase and refine, and then not be satisfied with their final effort: They seemed able to visualize the objects in question, but not in sufficient detail to enable them to make an accurate drawing. Their representational schemata were not developed to the point where they could construct a drawing from geometric properties of the given object.

Of all people, mathematics-teachers should appreciate the properties of central and orthographic projection used in drawing, but the results did not show this to be the 155

case. Although the U.S. students did better than the West

Indian teachers on most items, their level of performance was still considered far from satisfactory (Mitchelmore,

1973b). This could be because geometry as a study of physical space has not played a significant role in mathe­ matics education in either country.

It also became clear that some items required knowledge of specific conventions, for example, whether to use central projection (perspective) or orthographic projection. This caused difficulties in scoring drawings of views of the milk carton, where only the conventional (orthographic) pro­ jection was accepted as correct. This item appeared to place too great a premium on any instruction in technical drawing that a subject may have received.

Scoring of all items was necessarily subjective, but it appeared that a reasonably reliable system could be devised as part of the test development procedure.

First pilot Construction of a 3D drawing test for use with secondary school students started in May 1973. The items on hidden edges, conic sections, and water level were retained.

The item on views of a milk carton was omitted for the reason just given, and that on reflections was omitted because it 155 was much too difficult. The perspective drawing item was scored for slope only, since the representation of height and spacing did not appear to be discriminating.

To provide a more coherent rationale for the test, an analysis was made of the use of visualization and drawing in solving problems involving physical space. By examining a collection of relatively elementary problems in 3D geometry and its applications, it was concluded that only four types of operation are involved (not all in the same problem);

1. From problem to solution If the problem is very simple relative to the solver's ability, the solver can visualize the problem situa­ tion and find the solution without the aid of a diagram.

2. From problem to diagram In most cases, the solver needs to draw a diagram to illustrate the problem situation. The diagram must be a reasonable likeness if it is to be useful. Like type 1, this operation involves reading and understanding the problem statement.

3. From diagram to diagram This is usually an intermediate step. A diagram which is not immediately useful is transformed into a more usable form, for example by adding construction lines or by drawing sections.

4. From diagram to solution The diagram may be part of the problem statement or it may be the result of operations of type 2 or 3. This final step may involve arithmetic, algebra, trigonometry, calculus, etc.

The use of operations of types 2-4 is illustrated in Fig. 4.1 157

Problem

An aircraft is flying at an elevation of 5 000 metres. From a nearby airport, the aircraft has an angle of elevation of 73° on a bearing of 129°. How far south and east of the airport is the aircraft?

Solution

Step 1: Problem to diagram (type 2 operation)

w 73*

S

Step 2: Diagram to diagram (two type 2 operations)

o c

7 3

DC T»v o

Step 3: Diagram to solution (type 4 operation)

X ■- 5 000 cot 73°

Southing = AC = OA sin 39° = 5 000 cot 73° sin 39° metres

Easting = GO = OA cos 39° = 5 000 cot 73° cos 39° metres

Fig. 4.1 Steps involved in the solution of a three-dimensional trigonometry problem. 158

Operations of types 2 and 3 require the production of a

diagram and are therefore proper topics for a drawing test.

Operations of types 1 and 4 are more suitable for testing

by multiple-choice or short-answer free-response items,

such as those described in Section 4.21. Some common opera­

tions of types 2 and 3, including those used in previous tests, are as follows:

Type 2: Problem to diagram Draw a well-known solid from memory Draw a representation of a visible 3D situation Draw a representation of a 3D situation de­ scribed in words

Type 3: Diagram to diagram Complete a perspective drawing Draw the water level in a tilted bottle Draw reflections in a mirror Draw the hidden edges of a solid Draw sections of a solid Draw the dissection of a rectangular solid into cubes Draw the net of a solid Draw various views of a solid Sketch constructions in a solid

As a result of this analysis, several new items were added, some from Werdelin (1961). The test instrument contained the following items:

1&2: Follow verbal instructions to sketch figures in diagrams of a square pyramid and a cube Draw the hidden edges of a prism Dissect a rectangular solid into cubes Draw sections of a cube 159

6: Draw sections of a cone 7; Draw the water level in tilted bottles 8: Draw telegraph poles alongside a road 9: Represent a 2D physical situation described in words 10: Represent a 3D physical situation described in words 11: Make simple drawings of common objects Bonus: Draw a picture of your school

A copy of the test, referred to as Trial Version 1 of the

Three-Dimensional Drawing test, is given in Appendix E.

The test was administered to three Grade 7 and two

Grade 9 classes in parochial schools in Columbus, Ohio in

May-June 1973. No time limits were imposed, but students were asked to record how long they took on each item. The majority of the students finished in one class period, but analysis was confined to items 1-9 which almost all students answered. Multivariate analysis of variance indicated a strong sex effect (P< 0.001) but a weak grade difference

(P< 0.11). Significant grade differences (P< 0.05) were only recorded on items 1&2 and 4. An item analysis (Table

A.3) indicated that the most discriminating items, as measured by the item-test correlations, were 1&2, 3, 4, 7 and 8, and a revised test was prepared from these items.

From the item analysis, it was found necessary to simplify items 3 and 4; items 1&2 were combined and clarified, and 150

the slopes in item 8 were changed to make scoring easier.

The revised test, referred to as Trial Version 2, is in­

cluded in Appendix B. The items will be called Cubes,

Edges, Midpoints, Poles, and Bottles. It was expected that

this test would take not longer than 20 min, separate time

limits to be determined for each item.

Second pilot Trial Version 2 was administered to Jamaican students in Grade 7 of an all-age school and grade 9 of two high schools in early November 1973. These samples were chosen to represent the extremes of the range expected in the Grade Nine survey. Time limits for each item were determined ex tempore, allowing most students to finish each item without irritating those who finished quickly.

The reaction of the poorer students has already been described (Section 4.31). As a result, the test was revised once more. The highly verbal Midpoints was omitted and the items reordered from the concrete and easy to the abstract and difficult (Poles, Bottles, Cubes, Edges); all printed instructions were deleted from the test paper, and visuals and solid models were constructed to allow an entirely audio­ visual administration procedure. This final version of the

Three-Dimensional Drawing test (to be called 3DD) is given 161 in Appendix F; specifications for the visuals and models,

and instructions for administering the test are given in

Appendix G, and the final scoring system (of which more below) in Appendix J.

As a final check, 3DD was administered to a further two classes late in November 1973. On this occasion, only one or two students did not know what to do, and this only on the more difficult Cubes and Edges items. Explanations for these two items were therefore expanded somewhat and the instructions changed to allow easier identification of those who did not understand the task to be performed.

Item statistics for the schools where Trial Version 2 and the final 3DD were tested are reported in Table A.4.

The versions had different maxima, so scores have been con­ verted to percentages for ease of comparison. The consid­ erable improvement in the scores of the all-age school students is noticeable; although part of the improvement was due to a difference of two grades and part may be attributable to inter-school variation, most of the difference can be attributed to the revised administration procedures.

Three of the four items showed the expected differences be­ tween types of school, but Bottles was weak. Table A.5 confirms that Bottles was the least discriminating item, the 162

the others being entirely satisfactory. (Computing

machinery was not readily available to facilitate the cal­

culation of item-test correlations.) Further evidence for

the consistency of the test is provided by an examination of

item content: They all involve the 2D representation of

parallels and perpendiculars. Again, Bottles is weaker : It

only involves parallels (horizontals) and is essentially a

2D problem. It was decided to retain Bottles for its in­

trinsic interest (Piaget & Inhelder, 1967) but to treat it

as a separate scale if it was found to have a low correlation

to total test score in the Grade Nine survey.

Scoring The results reported by Piaget & Inhelder (1967, pp. 171-193 & 375-418) suggest that Poles and Bottles could be scored for stages. Examination of pilot-test papers quickly revealed the impossibiility of doing this: Res­ ponses to Poles were frequently inconsistent between or even within the three segments of the "road" and responses to

Bottles rarely matched the combinations suggested by Piaget as characteristic of each stage. For this reason, it was decided to score each road section and each bottle separately.

Analysis of responses to Poles indicated that the slopes of the poles drawn in each section tended to fall in a series 163 of rather poorly-defined ranges. It was assumed that these ranges indicated successive stages of development, starting at perpendicular to the road and ending at the vertical, the variation indicating drawing error. A preliminary scoring system was constructed on this assumption, the exact definition being left until data from the developmental survey was available for analysis (Section 5.33).

A preliminary scoring system for each part of Bottles was derived from a similar combination of empirical and theoretical considerations.

Error analysis of responses to Cubes (indicated that three operations are involved in the completion of each block :

1. Connecting the edges drawn on adjacent faces

2. Drawing squares on the frontal faces

3. Representing opposite edges of cubes by parallels.

Examples of errors in these three operations are shown in

Fig. 4.2. The third error resembles the most common error on Poles and Bottles, namely drawing a line perpendicular to a line it joins instead of parallel to another line.

The error seems to be at least partly due to an illusion effect akin to the angle illusion (Piaget, 1969), previously reported by Campbell (1969). 164

1. Drawing instead of

^ Y',. \ 2. Drawing instead of \

3. Drawing or instead of

Fig. 4.2 Typical errors on first block of Cubes item of 3DD,

Forty easily recognized manifestations of success on the above three operations were identified, with a bias toward the third type. Sixty scripts randomly selected from the pilot-test papers (20 from each type of school) were scored for the presence or absence of these characteristics, and the ten features with the best item statistics chosen to define the final scoring system.

Edges was the most difficult item to score objectively.

It was difficult enough to judge when a drawing was correct 165

“within drawing error" but even more difficult to be confident

in assigning part scores. A large number of examples were

therefore used to supplement the verbal definitions for each

score.

The final scoring system for 3DD is given in Appendix J.

4.33 Hidden Shapes Test

The NLSMA Hidden Figures Test (Wilson et al., 1968) was

administered to students in two high schools and one all-age

school in Kingston in early November 197 3, using a somewhat

simplified separate answer sheet (see Appendix E). The re­

action of the all-age school students has already been de­

scribed (Section 4.31); there was clearly a great deal of

random guessing, but their mean score of 4.8 was signifi­

cantly greater than chance.

As with the drawing test, the administration procedures

were completely revised along the lines recommended by

Schwarz (1951, 1963). Extra practice examples were con­

structed, visual aids were made to illustrate the test task

and solutions to the practice examples, all items were printed on a single sheet of paper, and answers were to be marked on the test paper by circling one of the five letters below each item. 166

The items were also revised when the high school results showed a clear ceiling effect: Mean scores were

12.2 (girls) and 13.9 (boys) out of 16, with half the boys

scoring the maximum. From an item analysis (Table A.6), admittedly of rather doubtful value, eight items were selected and put together with seven items from Part 1 of the original Cf-1 test (French et al., 1963) and reordered to form a new test. This test will be referred to as the

Hidden Shapes Test (HST) to distinguish it from the individual Hidden Figures Test (HFT) based on the same items.

The trial version of HST is given in Appendix E and the instructions for administering it in Appendix G.

The trial version of HST was given to three classes of

Grade 9 students in a high school, a junior secondary school, and an all-age school in late November 1973. The mean scores were 8.1, 4.8 and 5.5 respectively. It was clear from observation in the classroom that the great majority of the students knew what to do and were not guessing wildly; the scores, though still low, were therefore consid­ ered much more valid measures of students' ability than those obtained using the previous version of this test. However, the test was now too difficult (only 3 out of 108 students 167

scored more than 11 out of 15), giving a poor spread and poor discrimination between schools.

An item analysis of scores in the second pilot testing is given in Table A.7. The patterns of omits and corrected difficulty indexes confirm that most students did not guess at random but omitted items they could not do. The statistics show a sharp increase in difficulty after item 9:

This marks the break between the NLSMA and the Cf-1 items.

This sudden change seemed to have been responsible for the low spread in scores, as most of those students who got almost all of items 1-9 correct got only one or two of items

10-15 correct.

The final version of HST was obtained by revising items as follows: Firstly, items 10, 11 and 12 were made easier, but not as easy as 1-9 (as far as could be judged in advance) by deleting 4, 6 and 4 line-segments respectively. Secondly, to bring item difficulties nearer to 50%, items 4 and 7 were made slightly easier by deleting one or two lines, and items

2, 6 and 8 were made slightly harder by adding one or two lines. The difficult items 13-15 were retained intact as it was expected that changes in items 1-12 would lead to more of the better students attempting these questions, with sub- 168 sequent improvement in item characteristics. The practice examples were not changed, as it was found that most students were getting 3 or 4 correct, as planned. The method of indicating answers was changed from circling to underlining to be compatible with the Boxes test.

For the final version of HST, see Appendix F, The test was printed on blue paper to enhance possible differences due to variation in macular pigmentation, which shows the maximum absorption at 460 nm (Bone & Sparrock, 1971). The score used was the number of correct answers.

4.34 Boxes test

The I-D Boxes test (BOX) was given in an all-age school and a high school in Kingston in late November 1973. The instructions in the examiner's manual (American Institutes for Research, 1964) were followed closely, but the visuals were adapted to be compatible with those used in the revised

3DD and HST administration procedures (Sections 4.32 and 4.33) by drawing enlargements of the practice sheet on two 50 cm x

70 cm cards.

In this test, each subject is given a pair of small wooden cubes which differ only on one face. The test item consists of a diagram of a net of one of the cubes, and the 169

Fig. 4.3 Sample item from I-D Boxes test.

answer is indicated by underlining the appropriate picture

(see Fig. 4.3). Schwarz (1963) claims that this adaptation effectively eliminates the 3D interpretation of 2D drawings

(the relation of the pictures to the cubes is explained and used repeatedly in the introductory procedures) and con­ centrates on the folding operation. Copies of the practice sheet and the test paper are included in Appendix F.

Because of the speeded nature of the test and the small number of alternatives for each item, a guessing correction was applied, as recommended in the scoring manual (Schwarz,

1964a), by subtracting the number of wrong answers from the number of correct answers.

As a result of the practice administrations, it was found necessary to make three changes to the procedure.

The most important was a greater emphasis on the dis­ tinction between the two parts of the test than that indi­ cated in the manual; many students turned over to find out 170

what was on the other side of the page and did not turn over

to Part 2 when they were told to do so. As the test

developer has himself commented (Schwarz, 1951, p. 15), this

test might have been easier to administer, with more reliable

timing, if the two parts had been printed on separate sheets.

In the present form, students who receive their papers first

are able to study the first few items for some 15-30 seconds

longer than those who receive them last; instructions to look

up when they had written their names were not always followed.

The two other changes were to point out that the alternatives

to each item were identical and in the same order, and that

subjects should work down each column instead of across the

rows.

The lengthy explanatory procedures appeared to be

effective, in that all students thought they knew what to do

at the end and started on the test without hesitation. The mean scores obtained were 10.1 in the all-age school (boys

18.2, girls 7.2) and 24.0 in the high school (boys 31.1, girls 21.5), so the test appeared to be discriminating very well. The only disquieting feature was the number of students in the all-age school who appeared to be guessing.

Two criteria of guessing were applied; Defining a guesser 171

as a student whose number of right answers did not exceed the number one standard deviation above the mean predicted by a binomial model of random guessing for that number of responses, it was found that 12 of the 30 students guessed on both parts of the test. As pointed out by Schwarz &

Krug (1972, pp. 146-147), items 1-6 of the test should be answered correctly by anyone who understands what the diagrams represent, since they require no "mental folding."

Defining a guesser as a student who answered less than 5 of these 6 items correctly, it was found that 15 of the 30 students were guessers. It has to be assumed that these students were totally incapable of visualizing the operation of folding a net into the surface of a cube. The only other reasonable explanation is that they were so overcome by the testing situation that all their concentration was put to learning where to mark answers, when to turn over, etc.; this kind of interference will be much reduced when BOX is admin­ istered after simpler tests of a similar nature.

4.35 Personal Data Questionnaire

A Personal Data Questionnaire (PDQ) was administered orally to one of the all-age school samples during the pilot 172

testing. Variables found difficult to measure in individual testing (see Section 4.48) were omitted, but items on age,

sex, family size, presence of an adult male in childhood, and social class were retained. Some items were also in­ cluded to allow exclusion of students with an unusual history of schooling and those who had only recently moved to the area. Items were later added on handedness and ambition.

The final form used by students is given in Appendix F, and instructions for administration in Appendix G. Ques­ tions, explanations, and examples were read out by the tester, and students filled in the answers one by one. This format was adopted because it was found that many students spent some time writing whole sentences instead of giving short answers as intended.

The classification used to derive social class from parental occupation (Miller, 1967b) is given in Table A.8.

Ambitions were classified as spatial/technical (occupations for which spatial ability is a good predictor) or verbal/ personal, as shown in Table A.9. 173

4.36 The group test battery

As a result of the pilot-testing described in the previous sections, the following test battery was ready for use in the Grade Nine survey. The estimated administration

times of each test were as shown.

3DD Three-Dimensional Drawing test 30 min HST Hidden Shapes Test 30 min BOX I-D Boxes test 40 min PDQ Personal Data Questionnaire 15 min

The indicated order was chosen for the following reasons:

3DD is the most concrete, involves the most familiar geomet­ rical operations, -ives fewest problems in how to show the answers, is least speeded, and requires the shortest explanations. HST introduces an unfamiliar, abstract idea and an unusual way of marking answers, but it is still fairly easy to explain using visual aids and the time limits do not necessitate speedy working (although item analyses show that the test has a definite speed factor). BOX has the longest explanation and is the most speeded, but still uses visual aids. By the time PDQ is reached, students should be used to the tester's accent and well prepared to respond to his oral questioning.

It was intended to start testing early in the morning, and to allow a break between HST and BOX to coincide with the 174

short recess which most schools have about 1^ to 2 hours

after school starts. This timing avoided the after-lunch

period when students are reputedly s|.ower, avoided the

period of greatest noise and distraction outside the class­

room, and avoided possible alienation by using up students' break time.

A short introductory statement developed during pilot

testing is shown in Appendix G. To maintain confidentiality, it was decided to give students code numbers and not to use their names for any purpose. To do this, numbers were written on a pack of blank cards and the pack was shuffled

and dealt to students face down at the start of the testing at each school.

The same tests were used in the training experiment at the teachers' college, but not in a single battery. Proce­ dures for this phase of the study are given in Section 7.35.

4.4 Development of individual tests

4.41 Pilot-testing procedures

Preliminary work on the development of a solid repre­ sentation test took place in Columbus, Ohio in May 1973, and 175

most of the individual tests were tried out with the

author's three children during July-August 1973. Pilot-

testing in Jamaica was carried out in Kingston schools during

November-December 1973. To indicate the likely lower and upper limits for each test, most students were selected at random from Grades 3 and 9 in schools which had not been selected for either the developmental survey or the Grade

Nine survey. Details of the pilot-testing schedule are given in Table A.2.

Once again, school principals and students were most cooperative. With the modifications to be described below, it was found quite feasible for the writer to carry out all the individual testing. As with the group tests, expected difficulties due to tester-student differences in accent, color or position power failed to become apparent in most of the tests piloted. Again, this is ascribed to the novel visual character of the tests, the break from routine, and the tester's precautions to speak slowly and to appear friendly and non-threatening.

Two tests, the Draw-a-Person test and a test of pictorial depth perception, were tried out in the pilot-testing but subsequently dropped (Section 4.22). Development of the re­ maining instruments is described in the following sections. 176

4.42 Pacific Design Construction Test

The Pacific Design Construction Test (DOT) was admin­

istered to four Grade 3 and three Grade 6 students for

practice in December 1973. This test (Ord, 1970) differs

from previous block design tests in using flat "blocks",

actually tiles; also, the pattern is copied the same size

as the stimulus pattern, and trays are provided to make their

assembly easier. The test was easy to administer, attractive

to students and gave a wide range of scores. There was no

time to administer the test to high school students.

4.43 Hidden Figures Test

The items in the NLSMA Grade 5 Hidden Figures Test

(Wilson et al., 1968) were arranged in a small booklet with

one item on each page, the hidden figure being drawn to the

left of each complex figure. Preliminary testing with the

author's three children indicated considerable differences in difficulty between the items, so they were reordered before pilot testing in Jamaica. In the first pilot testing in November 1973, with average Grade 5 students, it was found very difficult even to explain the task and only one usable response was collected. For the next pilot testing, with 177

bright Grade 3 students, one extra practice example was

constructed and the number of items was reduced to eight.

This time, the test was much easier to administer, and all

five students completed more than half the items within the

time limit.

The next testing, with Grade 9 high school boys, showed

that all items of the NLSMA test were far too easy at this

level. (The boys had taken the group version of this test

the previous day, but the speed with which they raced through the items did not seem attributable solely to this practice effect.) The items in Part 1 of the original Cf-1 test (French et al., 1953) were therefore used next, with

Grade 9 high school girls; this test appeared much more satisfactory, giving a 4:1 spread of total times in a random sample of four subjects.

The next problem was to marry the best items from each level into a single test which could be used in Grades 3-9.

It was decided to use a subset of items from the correspond­ ing group test, which had reached its final revision by this time (Section 4.33). A test consisting of these 15 items, plus the demonstration example and three of the four practice examples, was therefore administered in December 1973 178

to four Grade 3 and three Grade 6 students (the imminent

Christmas vacation preventing the testing of Grade 9

students).

The final test (see Appendix H) was obtained by-

selecting and ordering the ten best items with a wide range

of difficulties. The pilot-test data on which the

selection was based was too fragmentary to be worth reporting,

but it was expected that half of the youngest age group would

be able to complete at least half of the items, whereas the

oldest students would complete almost all of them. The test was printed on blue paper to enhance possible differences due to variation in macular pigmentation, which shows the maximum absorption at 460 nm (Bone & Sparrock, 1971).

Scoring the test reliably presented an unexpected diffi­ culty. Initially, Witkin's original procedure (Witkin,

1950) was used, but it was observed that several students claimed they had found the hidden shape almost immediately the figures were displayed, only to make and correct several errors while tracing the outline. The same error-correcting behavior was displayed on isolated items by many other students. Scoring only the time until the student said "I see it" thus gave a false assessment of their performance. On the 179

other hand, scoring the time to complete the tracing would

have introduced variance due to motor control and unduly penalized the younger students. The addition of the several practice examples fortuitously allowed a ready solution to this problem: The mean time to trace the easy practice items was taken to represent the tracing time in all cases where the figure was correctly identified before tracing, and the time for each item was found by subtracting the tracing time from the time to completion. The fact that the five hidden shapes contain different numbers of segments

(4, 4, 6, 5 and 7 segments) introduced a small error between items even though the perimeters are almost all the same

(four have perimeters 4 + 3/2 units long, the fifth has a perimeter of 5 + 2/2 units), but this was not expected to lead to any error between students.

Logarithmic scaling is frequently applied to time scores to reduce skewness (Winer, 1971, p. 400). It was decided to predetermine an appropriate logarithmic scale transforma­ tion from the pilot-test data. This transformation, to be applied to each item separately, is given in the record form in Appendix H. Inspection of the pilot-test data suggested a time limit of 80 s plus tracing time for each itemu 180

Finally-revised instructions for administering the HFT are given in Appendix I.

4.44 Horizontal-Vertical Test

Preliminary testing with the author's three children led to a complete revision and renaming of the Piaget visual­ ization test. The left-right discrimination task appeared to depend more on listening skills than on visualization, since the question "Is X to the right or left of Y?" was repeated six times with various values of X and Y. The dot- copying item v;as confusing because it was not clear whether the dot should be copied in the position seen by the tester or as seen by the subject. The other three items did not seem to have any common thread. It was decided to replace the whole set of items by the Poles and Bottles items from

3DD (Section 4.32), which might be expected to measure how well the concept of a reference system has been acquired

(Piaget & Inhelder, 1967). This short test was renamed the

Horizontal-Vertical Test (HVT).

The two items were administered using the same diagrams, instructions and scoring systems as in 3DD, except that the tester used the subject's test paper instead of a separate visual and allowed the subject as much time as he required 181 to finish each item to his satisfaction. In the pilot- testing, this was found to be a very easy test to explain and most subjects completed the drawings quickly and without hesitation. Most of the students (from Grades 3 and 5) drew the poles perpendicular to the "road" instead of

"vertical," but there was a wider range of responses to

Bottles. The test was not tried in Grades 7 and 9, but in view of the results obtained in the group pilot-testing, it was expected that this test would show strong age trends.

4.45 Solid Representation Test

Preliminary work on the development of the Solid

Representation Test (SRT) took place in May 1973 when a group of student teachers in their junior year at the Ohio

State University were asked to have some pupils draw some models of mathematical solids during an afternoon observa­ tion period at an elementary school. The students used tins, boxes and cans as well as wooden and transparent plastic models of several different solids. Some students gave the models to individuals to draw, others showed the models to the whole class at once. Responses were col­ lected from about 50 pupils in Grades 1-6. An attempt was then made to classify the drawings of each solid into a developmental sequence which was consistent with theory and previous empirical results (Section 2.32).

The most primitive drawings appeared to be those which simply showed one face (e.g., a square for a cube), and the most advanced were the usual perspective drawings (no distinction being made between central and orthographic projection).

Three intermediate stages were identified, the first involv­ ing the use of mixed viewpoints, the second and third in­ volving primitive and more advanced depth cues. However, there were many drawings which could not be fitted into this sequence, especially of the cone and cylinder.

Many difficulties in interpreting subjects' responses were traced to the method of administration. For example, it was not possible to decide whether a subject who drew both end faces of a cylinder did so because he "knew they were both there," because he had been shown both faces and felt obliged to represent both in his drawing, because he could see the normally hidden edge through the transparent plastic and did not use the convention of representing it by a broken line, or because he could imagine the hidden edge and represented it with a full line. Drawings of pyramids 183

and cubes were difficult to classify because different

orientations of these solids seemed to encourage different

kinds of errors; the same phenomenon was observed with

upright and horizontal cylinders. "Real-life" objects were

confusing because of distractions such as ridges, lettering,

and rounded edges. Drawings of a sphere were not very

informative because the depiction of depth seemed to depend

on knowledge of an artistic convention rather than any

geometrical property.

It was therefore decided that, in order to obtain reli­

able results, the test would have to be administered

individually, with opaque, abstract solids in a fixed orientation and with the subject viewing them from a fixed

direction. After some experimentation, the apparatus shown

in top view in Fig. 4.4 was devised. In the box, there were

painted wooden models of (from left to right) a blue cuboid

measuring 10 cm x 5 cm x 2h cm, a red cylinder of height

5 cm and diameter 5 cm, a green square pyramid of height

5 cm and base side 5 cm, a yellow cube of side 5 cm, and an

orange cone of height 5 cm and diameter 5 cm. Colors were

used to allow identification without using the names of the

solids. Each model was in a compartment large enough to 184

70 cm

11 cm

Fig. 4.4 Top view of SRT apparatus allow an unobscured view, and each compartment had a lid so that the objects could be displayed one at a time. The numbers 1 to 5 were painted on the outside of each lid in the same color as the model inside; the remainder of the apparatus was painted white inside and out. Initially, the apparatus was simply placed on the table in front of the subject, who sat in line with the pyramid at a distance of about 50 cm from the front of the apparatus. From this position, the models appeared as shown in Fig. 4.5.

!h

Fig. 4.5 Subject's-eye-view of SRT apparatus. Each compartment has its own lid (not shown). 185

For the present study, the following three-condition procedure was adopted. In Condition 1, the subject was shown each solid for a short time (about 1 s) and then drew it from memory. In Condition 2, each solid was displayed for as long as the subject wished while he drew it. In both these conditions, subjects were instructed to draw each block exactly as they saw it and to make their drawings "look solid, like a photograph." In Condition 3, the subject was shown six drawings of each solid alongside the model and asked which he thought was the best drawing of that solid.

This procedure was devised partly to validate the scoring system (it was expected that Condition 2 would be easier than Condition 1 but harder than Condition 3) and partly to find how much a view of an object influences the drawing which is made and how much performance differs in free- response and multiple-choice modes. Tasks were administered in the assumed order of difficulty in order to minimize interference between them.

Scores from Condition 3 gave a measure of pictorial depth perception to replace the test which had to be dropped

(Section 4.22). The alternatives were selected from draw­ ings made in the preliminary phase. For each solid, the 186

most common type was chosen from each stage, with an extra

one from the nearly-correct category. The six drawings were then arranged in a random order, labelled A-F, and copied in black ink in two rows of three on a file card measuring 204 mm x 127 mm. The five cards were covered with

transparent plastic to enable them to be kept clean.

The SRT was pilot-tested in Jamaica with 11 elementary and 8 high school students during November 1973. As a result, certain modifications were made to procedures. All models were rubbed down to a matt finish to eliminate spurious reflections which were sometimes represented as edges. The procedure for displaying the solids in Condition

1 was formalized in order to obtain a more uniform exposure time. To obtain a better control on viewing angle, three chin rests were constructed, for three sizes of child. Each chin rest was attached to one end of an extension arm, the other end of which fitted under the SRT box and included a wedge to tilt it. The wedges compensated for the different heights of the rests so that all subjects viewed the central pyramid from a distance of about 50 cm at an angle of elevation of about 30°. Finally, after one student who did exceptionally well for his age reported watching Sesame 187

Street on television, it was decided to ask students if

they knew the names of the solids and what previous drawing

experience they had had. The final instructions for admin­

istering the test are given in Appendix I.

In view of the small size of the pilot-test sample,

it was decided to defer the definition of an exact scoring

system until all drawings had been collected in the develop­

mental survey (Section 5.34). The final scoring system and

the alternatives used in Condition 3 are given in Appendix J.

4.46 Geometric Illusion Measures

Five of the six illusions used by Segall et al. (1966)

were administered to four Grade 3 and four Grade 9 students

in November 1973. Segall et al. found that the Ponzo

illusion gave little cross-cultural discrimination, so this was initially omitted. An attempt was made to explain the

Poggendorff illusion using a small visual aid, but the

large number of Guttman errors showed that this effort was not successful; the other four illusions were quite satis­

factory. It was therefore decided to reinstate the Ponzo illusion (which had shown an age difference in the Segall et al. study) in place of the Poggendorff. A further trial 188

with four high school subjects indicated that the five

illusions (Muller-Lyer, Sander Parallelogram, Horizontal-

Vertical inverted T and L, and the Ponzo illusion) could be

easily administered in quite a short time, no break being

necessary. Language problems were minimal because each item

required only a judgment of the longer of two lines. The

test will be referred to as the Geometric Illusions

Measures (GIM).

Herskovitz et al. (1969) recommend the recording of

the time taken on each item, but this was not possible with only one tester. Total times were recorded for the set of diagrams of each illusion, but showed little vari­ ation. It was therefore decided not to record times in the survey, but to encourage students who wished to contemplate each picture at length to give a more immediate response.

To investigate a proposed cause of illusion susceptibil­ ity, subjects were also asked what the final diagram (of the Ponzo illusion) looked like to them. Responses were classified as 3D (e.g. road, railway line) or 2D (e.g. trapezium, triangle).

4.47 Eye-hand Dominance Schedule

Annett (1967), who used a questionnaire to establish hand dominance, found that cutting with scissors and 189

unscrewing the lid of a jar were particularly discriminating.

These activities, together with the more obvious writing, were therefore used to form a three-item test of hand dominance. A subject was classified as left- or right- dominant if he used the same hand for all three tasks; otherwise, he was classified as mixed.

Gronwall et al. (1971) experimented with sixteen eye dominance tests, and found a "central cluster" of five tests, each of which involved looking through small holes or lining up near and far objects. Attempts to explain lining-up to younger subjects met with failure, so three varied tasks of the former type were used: Looking through a pin-hole, looking into a bottle, and looking through a magnifying glass. A subject was classified as left- or right-dominant according to the eye used for the majority of the three tasks.

Three items were presented to subjects as tests of how well they could see, and they were not aware that it was the hands and eyes they used which were being noted. The items will be referred to as the Eye-hand Dominance Schedule

(EDS); instructions for administering the schedule are given in Appendix I. 190

4.48 Personal Data Interview

On the first day talking to Jamaican primary school

students, it became clear that the planned personal data

interview would have to be curtailed. It was difficult for

the author to phrase his questions simply enough for the

younger students to understand, and difficult for him to

understand students' responses because of their heavy

accents (Section 3.31). In view of his recent arrival on

the island, the author also felt uncomfortable prying into

students' home background. Further restrictions were neces­

sary because of students' ignorance of their parents' educa­

tion and frequently also of their occupations.

The Personal Data Interview (PDI) was therefore

restricted to relatively straightforward questions which could be answered in a few words. The same questions were used as for the group-administered PDQ (Section 4.35)

except for the omission of the question on handedness.

Miller (1967b) rated skin color by observation on a

5-point scale. A similar procedure was adopted for this

study when attempts to measure pigmentation more objectively

(e.g. using a photometer) were frustrated by variations in skin texture and lighting conditions. The five categories 191 used were White (European color). Light Brown (European with a deep sunburn), Mid Brown (coffee-colored, the color of children of an Afro-European union). Dark Brown (light

African color), and Black (dark African color). No way was found of making these categories any more definite than the verbal descriptions imply.

4.49 The individual test battery

As a result of the pilot-testing described in the pre­ vious sections, the following battery of individual tests was ready for use in the developmental study. The estimated administration times were as shown.

Part 1 (34 min)

EDS Eye-hand Dominance Schedule 4 min DCT Design Construction Test 20 min HFT Hidden Figures Test 10 min

Part 2 (27 min)

HVT Horizontal-Vertical Test 4 min SRT Solid Representation Test 12 min PDI Personal Data Interview 4 min GIM Geometric Illusion Measures 7 min

The battery was administered in two parts to avoid inatten­ tion in the younger students. The total time was such that the four subjects chosen from each school could all be tested in one morning. It was initially intended to administer 192

the drawing tests first, since these involved familiar

operations, caught students' interest, and were untimedo

However, there were signs in some of the pilot-testing

sessions that students tested later in the day were less

attentive and slower in their reactions, and this tendency

was confirmed by local educational researchers. It was

therefore decided that the two timed tests should be admin­

istered as early in the day as possible, preceded by the

short untimed EDS as a warm-up. The two-part administration

of the battery enabled all subjects to take the timed tests

within a 2-hour period, providing adequate control over the

hypothesized fatigue effect.

It was only felt necessary to control for order effects

in regard to GIM, since the drawing tests might induce a

set to interpret diagrams 3-dimensionally which could in­

fluence illusion susceptibility. It was therefore decided

that, in Part 2, half the subjects in the developmental

survey would be given HVT and SRT before GIM and half after

GIM. A copy of the tester's record form is given in

Appendix H. 193

The tests EDS, HFT, SRT and GIM were also given to a random sample of students from the teachers' college population involved in the spatial training experiment

(Section 7.53). CHAPTER FIVE

THE DEVELOPMENTAL SURVEY

5.1 Background

5.11 Purpose

The major purpose of the developmental survey was to investigate the growth of various facets of spatial ability during the school years (research question 2, Section 1.42).

Other purposes were to study the interrelations of these facets (research question 1); the relation of spatial ability to illusion susceptibility (research question 5); and the relation of spatial ability and illusion suscepti­ bility to sex, skin color, hand and eye dominance, ambition, social class, family size, and family composition (research questions 3 and 5).

To achieve these aims, it was decided to test students in Grades 3, 5, 7 and 9, taking equal numbers of boys and girls from each grade. Extra samples from Grade 1 were added later. The individual tests developed for this purpose

(DCT, HFT, HVT, SRT and GIM) are described in Section 4.4, as are the EDS and PDI used for collecting background data.

194 195

5.12 Delimitations

Because of the time taken to administer the tests individually, only small samples could be tested. It was therefore necessary to limit the sources of variation, so the survey was restricted to high-ability students in public schools in the capital city, Kingston. A "high-ability" student was defined as one attending a high school or likely to do so in the future. This choice had several advantages:

All the schools were in a relatively small area; the vari­ ance at each grade level was much reduced (only the top 7.5% of secondary age students attend high schools); and results for these students, the brightest in presumably the richest environment, would provide an upper bound on the development of spatial ability in Jamaican students. The narrow selection process at 11+ (Section 3.2) posed some problems in maintain­ ing sample uniformity through the primary and secondary grades, however. Fortunately, because of the widespread practice of streaming (homogeneous grouping), it was found that the larger and better elementary schools contained classes in each primary grade in which the majority of students usually proceeded to high school. It was decided to select primary students from those classes. As a further 195

delimitation, it was decided to avoid testing students

who had not lived in or near Kingston most of their lives,

or were outside the normal age range for the grade.

5.2 Method

5.21 Design

Because of the selection at 11+, there was no possi­ bility of completely crossing the school factor (S) with grade (G). It was therefore necessary to nest S within G.

In order to completely cross schools with sex (X),

<^ver) using the minimal two schools per grade would have meant using all four Kingston mixed (coeducational) high schools for Grades 7 and 9, leaving no alternates in case of lack of cooperation. Moreover, the mixed schools were known to be less prestigious and academically less successful than the single-sex schools in Kingston (see Section 5.22 and Table B.l). It was therefore necessary to nest S within

X as well as G.

The fourth factor in the design, relevant only to the analysis of GIM results, was the order in which the tests were given.

The full design is illustrated in Fig. 5.1. For the

GIM analysis, there were only two subjects per cell; for 197

Grade (G) 1 3 5 7 9 Sex (X) MF MFMF MF M F School (S) 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 HVT & SRT before GIM HVT & SRT after GIM

Fig. 5.1 Research design for the developmental survey. Schools were nested within GX and n = 2 per cell.

the other analyses, data were collapsed over the order factor, giving four students per cell. The error terms had at least

10 d.f., but it was decided to pool error terms if hypotheses of zero between-school or school-order interaction variance were not rejected at the 0.25 level. A dummy run with imaginary data based on pilot studies suggested that the design in Fig. 5.1 would be more powerful in detecting a significant G effect than a design with S and G crossed, but that it would be less powerful for X and GX. Since the sex effect is also studied in the other phases of this study, this weakness was not regarded as catastrophic.

5.22 Selection of schools

The Kingston high schools in which Grades 7 and 9 were to be tested were selected as follows. An academic index was first calculated for each school by expressing the total 198

number of passes in Mathematics and English Language on the

1972 Cambridge Overseas GCE 0-level examination as a per­

centage of the total enrollment of the school (Table B.l).

The six schools containing boys and the six schools contain­

ing girls which had academic indexes nearest the overall

median were then chosen; in order to include the mixed

schools, the schools containing the two sexes were chosen

alternately, working outwards from the median. For each

sex, four schools were selected at random; two were then

randomly assigned for testing at Grade 7 and two at Grade 9.

The remaining two schools for each sex were treated as

alternates.

An academic index for each Kingston elementary (primary

and all-age) school was obtained by expressing the number of

passes on the 1973 Common Entrance examination as a per­

centage of the number of entries from that school (Table B.2).

The twelve schools with at least 35 passes that had the

highest academic indexes were then selected as those most

likely to contain whole classes of students likely to proceed to high school. Four schools were then randomly

assigned for testing at Grade 1, four at Grade 3, and four

at Grade 5. At each grade level, two schools were randomly 199

chosen for testing boys and two for girls. The Grade 1

schools were initially treated as alternates (see below).

The academic indexes used are subject to several

criticisms. In the first place, they are based on verbal

and numerical tests of unknown reliability which may be only

rough measures of school factors likely to affect the

development of spatial ability. Secondly, the number of passes fluctuates from year-to-year due to variations in

student and teacher quality. Finally, the proportion of primary students entered for the 11+ varies from school to

school, and the total high school enrollment may not be proportional to the number of students eligible to take the

0-level examination in any year. Nevertheless, it was felt that these indexes were the best that could be easily obtained, and that they were at least adequate to identify the extreme schools in each category.

5.23 Procedures

The schools selected for Grades 3, 5, 7 and 9 were visited in a random order between 28 January 1974 and 5 April 1974.

After about six schools had been tested, it became clear that whereas the ceiling of most tests was approached in

Grade 9, Grade 3 results were not at the floor. It was 200

therefore decided to add the four Grade 1 schools, and

the opportunity was taken to shuffle the schools to ensure

a more uniform distribution of their academic indexes. One

of the Grade 1 schools was found to be in a severely

depressed area where the writer did not feel safe? this

school was therefore replaced by another which had a high

academic index, even though this school only had 29 passes.

All principals approached willingly agreed to the testing. In the elementary schools, four students were chosen at random from the boys or girls in the top stream class of the appropriate grade. In two schools. Grade 1 was not streamed so the students were selected from the ten students nominated by their teachers as the brightest in their class. In the high schools, a representative class was first chosen; where streaming was used, this was one of the middle classes, otherwise any convenient class was chosen. Four students of the appropriate sex were then chosen at random from the selected class.

As they arrived for testing, students were assigned alternately to one of the two orders shown in Fig. 5.1; the order to which the first student was assigned was different in the two schools within each grade x sex combination. 201

One Grade 9 school presented four "volunteers" for testing, but it was later found that they were the top boys at mathematics in the school; their test results were therefore rejected and the school replaced by one of the alternates. In another school, one Grade 9 girl obtained scores which were so low as to indicate grogs abnormality; she was therefore replaced by another student chosen at random from the same class.

Testing took slightly longer than expected, but was completed in one day at most schools. In two primary schools, two students were tested on one day and two on the following school day. One of the schools was on double shift, and the top stream class was in the afternoon shift.

All others were tested in the mornings, between 9.00 am and

2.00 pm.

Tests were given in libraries, staff rooms, in various other spare rooms and on verandahs. Conditions varied from the sublime to the ridiculous, but all students became absorbed in the test tasks and at no time did the writer feel that a distraction had affected a student's score significantly. A summary of school data is given in Table

B . 3 . 202

Whenever it was felt possible without straining good tester-school relations, students who had not lived most of their lives in Kingston or who were outside the normal age range for the grade were eliminated before testing started. In all, 28 of the 80 students in the final sample could have been said to have been untypical: 16 were out­ side the normal age range (10 of these were in Grade 3 and were only a few months older than usual), 12 had skipped two or more years of education or had repeated a year, and 4 had spent most of their lives outside Kingston

(Table B.4). These deviations were not considered likely to have any serious effects on the results. The mean age and length of education of the students tested at each grade level are given in Table 5.1; it should be noted that age and length of education did not increase linearly with grade, partly because many students regularly enter Grade 7 in high school the year following Grade 5 in elementary school (Section 3.2). 203

Table 5.1

Mean age and length of education, by sex and grade

Grade Variable Sex 1 3 5 7 9

Age at Boys 7.2 9.7 11.0 12.8 14.3 testing (yr) Girls 7.2 9.5 11.3 12.5 14.5

Years of Boys 1.0 3.1 5.2 6.4 7.9 education^ Girls 1.0 3.1 5.1 6.5 8.1

^Not including basic or infant school.

5.3 Results of spatial ability tests

5.31 Design Construction Test

When scored according to the manual (Ord, 1968), DCT gave poor discrimination amongst the high school students.

It was therefore decided to change the scoring on the harder items to spread out the scores more. The distributions of times on items 9-11 were very similar, as were those for items 12 and 13. The scoring for these items was changed to divide the successful students into three approximately equal groups, as follows:

Items 9-11: Up to 60 s, score 4 61 - 120 s, score 3 121 - 180 s, score 2

Items 12, 13: Up to 120 s, score 5 121 - 180 s, score 4 181 - 240 s, score 3 204

No other scoring changes were made, so the new maximum

score was 39. One change was made in the protocol after it

was found that students often omitted the four white tiles

in item 8; as for item 4, they were asked what tiles should

fill the empty spaces.

The frequency distribution of DCT scores, given in

Table B.5, was faintly bimodal but in fact not significantly

different from rectangular (chi-square = 9.50, d.f. = 7,

P > 0.20). An item analysis (Table B.6) showed that item

difficulties for items 1-8, where the original scoring

system was used, were in virtually the same order (rho =

0.98, P < 0.01) as the item difficulties in a standardi­

zation sample of adult illiterate males in Papua New Guinea

(Ord, 1968, p. 19). The reliability (Cronbach alpha) of

this test was calculated to be 0.93, which compares well with the reliability of 0.84 obtained by Ord (1968, p. 20)

in the above sample and is considered highly satisfactory.

The relation of DCT scores to grade level is given by

sex in Table B.7 and illustrated in Fig. 5.2. From a trend

analysis of these scores (Table B.8), it was found that the

linear trend was highly significant (P < 0.001). The sex difference was also clearly significant(P < 0.01), but there 205

39 T

Mean score Boys. 20

/Girls

Grade 1

FlK. 5.2 Mean DCT scores by grade and sex. Crosses refer to boys and circles to girls. The lines connect data points from the main survey; the unattached data points at Grade 5 are from the check testing.

was also a significant interaction (P < 0.001), apparently due to the fact that the sex difference was much greater at

Grade 5 than at the other grades. Using Tukey's post hoc procedure (Winer, 1971, p. 198), it was found that this was the only grade at which the sex difference was significant

(P < 0.01).

An alternative explanation for the marked sex differ­ ence at Grade 5 was that the schools in which boys were tested in this grade were generally superior to those in which girls were tested. This explanation was strengthened by the observation that the Grade 5 girls' mean score was 206

lower than the Grade 3 girls' mean. To check this possibility, the investigator returned to the four Grade 5 schools in

June 1974 and tested four students of the opposite sex in each school. Students were selected in the same way as before, but only DCT and HVT were administered. The means obtained by this check sample are shown by the unattached points in Fig. 5.2. The difference between boys' and girls' mean scores was smaller, supporting the conjecture as to school quality, but still significant (P ■< 0.01); see Table

B.9 for details. It is thus clear that the sex difference in DCT scores was genuine and not attributable solely to differences in school quality.

It has been claimed that block design tests like DCT, since they involve the successful breaking up of a pattern into its components, are measures of field independence

(Dawson, 1967a; Witkin et al., 1973). However, this investi­ gator's observations suggested that, in this sample, DCT tested mainly knowledge of shapes or "geometric intuition."

The critical concept appeared to be that of the diagonal

(Olson, 1970). Only one student rotated an entire design

(Deregowski, 1972c; McFie, 1961), but many rotated individual tiles, especially in items 1, 4, 5 and 8 where the small triangles were isolated. Younger students apparently had no 207

feeling for orientation or shape, being satisfied with hastily assembled designs which barely resembled the stimulus design. Beyond this stage, the chief problem seemed to be in assembling elements of a design; students often turned a red-and-white diagonal tile over and over searching for a way of placing it next to another diagonal tile to make a larger triangle or a stripe (parallelogram).

Item 9 was something of a pons assinorum in this respect:

Students who had been doing earlier items well always slowed down for this design, and many went to pieces completely; some typical errors are shown in Fig. 5.3. Knowledge of some rather elementary properties of right-angled triangles would have prevented most of the errors depicted and many similar ones. It was concluded that although DCT might not have been a valid test of field independence, it was certainly measur­ ing aspects of spatial ability.

5.32 Hidden Figures Test

The scoring system described in Section 4.42 worked quite well, although a few ad hoc adjustments had to be made for the youngest students who could only do the first practice example. One result of the rather inadequate pilot 208

Fig. 5.3 DCT item No. 9 (first diagram) and some typical errors.

testing soon became obvious: The items were not in order of

difficulty. However, the easier items were in such a posi­

tion that, because of the rule to stop testing only after

three successive failures, most students capable of answer­

ing those items correctly probably did so. In fact,

students appeared to be encouraged by finding easier items

late in the order.

The frequency distribution of HFT scores, given in

Table B.IO, was again faintly bimodal, but, combining scores

of 31 and above, the distribution was still not significantly

different from rectangular (chi-square = 2.6, d.f. = 7, P >

0.50). An item analysis (Table B.ll) confirmed that items

4 and 8 were out of order. The calculated reliability of

the test (Cronbach alpha) was 0.92, which was considered most satisfactory.

The relation of HFT scores to grade level is given by

sex in Table B.7 and illustrated in Fig. 5.4; a trend 209

Men 30 - Mean score Women

20 Boys

Girls

i I Grade 1 College

Fig. 5>4 Mean HFT scores by grade and sex. Grosses refer to males and circles to females. The lines connect data points from the developmental survey; the unattached data points are from the spatial training experiment (Chapter 7).

analysis is reported in Table B.8. The results were very similar to those obtained for DCT: There was a significant linear trend (P < 0.001), sex difference (P < 0.01) and inter­ action (P < 0.001), and post hoc analysis using Tukey's pro­ cedure showed that Grade 5 was the only grade at which the sex difference was significant (P < 0.01).

Fig. 5.4 also shows the mean HFT scores of a sample of teacher's college students; their results are discussed in

Section 7.53.

Embedded figures tests like HFT have long been regarded as measuring field independence (Witkin et al., 1962), al- 210

though variations in item format and test administration might modify the relationship (Witkin et al., 1973). In

the present sample, HFT seemed to be measuring something

rather different in the younger students, namely conserva­

tion of shape. Those who made very poor scores responded very quickly but drew figures which bore only a remote resemblance to the stimulus figure; some typical errors are illustrated in Fig. 5.5. One suspected that the students who made these errors would have been incapable of copying the shapes on to a much less distracting background such as a network of squares. By contrast, the better and older students rarely drew incorrect shapes, preferring to draw nothing if they could not find the hidden figure. Appar­ ently, many of the younger children tested were still at a

Piagetian stage II with regard to form copying (Piaget &

Inhelder, 1967, p. 58). The sex difference shown in Fig. 5.4 could then be interpreted as showing that boys reach Stage

III significantly earlier than girls, and are not necessarily more field independent than girls; in fact, the Grade 7 and

9 results suggest that older boys and girls might not differ significantly in field independence (but see Section

6.3, where a marked sex difference is reported for a similar test administered to groups in Grade 9). 211

Practice item 3 Test item 3 Stimulus Response Stimulus Response PI

Fig. 5.5 HFT practice item 3 and test item 3 with errors typical of younger students. Copyright @ 1962 by Educational Testing Service. All rights reserved. Adapted and reproduced by permission.

(Reuning & VJortley (1973, pp. 48-49) report similar

errors on a similar test which they attempted to administer

to Bushmen in South Africa. Their results could also be interpreted as showing that the Bushmen were retarded in

Piagetian perceptual development, and not necessarily field dependent. )

Even among older students, it was clear that success depended on an intuition for shape, as revealed, for example, by strategies of searching for distinctive corners or edges. It was concluded that, although HFT might also not have been a valid measure of field independence through all ages tested, it was certainly measuring aspects of spatial ability. 212

5.33 Horizontal-Vertical Test

As noted in Section 4.32, it was decided to fix the

scoring system for Poles and Bottles on the basis of the

developmental data. These items also come in the 3DD test

used in the Grade Nine survey and the spatial training

experiment.

Poles The median inclination of the poles on each section

of the road was measured for each subject tested, and these data examined for types of error and developmental patterns.

The frequency distributions of the angles of inclina­ tion for the three sections are given in Table B.14. The bottom section was the easiest, with inclinations falling between -9° and 30° in 54 of the 80 drawings. The top section, although the "road" was sloping at the same angle as the bottom section, was more difficult, with 44 drawings having inclinations between 11° and 70° and a subsidiary mode of 12 drawings between -49° and -30°; these subjects seemed to be tending to draw their poles perpendicular or parallel to the road (which sloped at -45°) instead of vertically

(0°). The middle section was the most difficult, with 47 of the drawings showing inclinations of over 70°.

The mean inclinations showed a tendency to decrease from Grade 1 to Grade 9 in all three sections. The actual 213

inclinations did not fall into well-defined groups, but

reasonably satisfactory ranges were eventually devised to

simplify scoring and analysis. The frequencies of each

of these ranges by grade (Table B.13) were next examined.

There was just enough pattern to justify the assumption

that the parallel and perpendicular directions were the

most primitive, but not enough to distinguish between them;

these two directions were therefore both scored zero. The

other ranges were then assigned successive integer scores as

they approached the vertical, a range of ±8° being allowed

about the vertical for the maximum score. The top and

bottom sections were scored out of 3 and the middle section, with its wider variation, out of 4, making a total of 10.

The detailed scoring instructions are given in Appendix J;

a transparency was prepared to speed up scoring.

The frequency distribution of Poles total scores, given

in Table B.14, showed a clear bimodality. The reliability

(Cronbach alpha) of the Poles score was calculated as 0.71, which is considered barely satisfactory. One reason for

the low reliability could be the use of slope ranges which were unsymmetrical about the vertical, forced by the desire to use the same ranges for each section and yet assign a zero 214

score to inclinations of -45° in the top section and 135° in the middle section. There seemed to be no empirical or theoretical basis for deciding relative weights for scoring left- and right-sloping errors.

The relation of Poles scores to grade level is given by sex in Table B.7 and illustrated in Fig. 5.5; a trend analysis is reported in Table B.8 . The linear trend was again significant (P < 0.001) as was the interaction

(P <. 0.01), but the sex difference was not significant over­ all or at any grade in the post hoc analysis. The absence of a significant sex difference could be due to Pole's lower reliability. It is also noticeable that boys showed no improvement from Grade 1 to 7 and that even Grade 9 students showed a poor knowledge of how to represent the vertical in a drawing.

In the check testing of Grade 5, described in Section

5.31, the difference between boys and girls was slightly larger than in the first testing and was significant at

P < 0.01 (see Fig. 5.6 and Table B.9).

Bottles Almost all Grade 1 students drew the water level perpendicular to the sides of all the sloping bottles, as if the water were solid; these responses were therefore scored 215

6 Mean /* , score

4 Boys

2 o - " - Girls

0 Grade 1 3 5 7 9

Fig. 5.6 Mean Poles scores by grade and sex. Crosses refer to boys and circles to girls. The lines connect data points from the developmental survey; the unattached points at Grade 5 are from the check testing and those at Grade 9 from the Grade Nine survey (Chapter 6).

zero. In bottles 3 (lying flat) and 5 (upright but inverted), there were few drawings intermediate between the "solid" representation and the correct one; these two bottles were assigned a maximum of 2, with a score of 1 given for inter­ mediate slopes. In bottles 2 and 4 (titled at 30° to the vertical, one inverted), there was a continuous range of slopes; many drawings seemed nearly correct but influenced by the tendency to enlarge small angles in drawing (Section

4.32). These two bottles were assigned a maximum of 3, with 2 given for a nearly correct response and 1 for other 216

intermediate slopes. The maximum score for the entire item

was therefore 10. The detailed scoring instructions are to

be found in Appendix J; a transparency was prepared to speed

up scoring.

The frequencies of each range of responses for each

bottle are given by grade in Table B.15. The developmental

pattern was much clearer than for Poles. The frequency

distribution of total Bottles scores, given in Table B.14, was clearly bimodal, as was Poles. The reliability

(Cronbach alpha) of the Bottles score was calculated as 0.91, which was considered very satisfactory.

The relation of Bottles scores to grade level is given by sex in Table B.7 and illustrated in Fig. 5.7; a trend

analysis is reported in Table B.8 . The linear trend was

significant (P <0.001) as was the sex difference

(P <0.01), but the interaction was not significant and the

sex difference was not significant at any of the five grade

levels in the post hoc analysis.

In the check testing, the difference between Grade 5 boys and girls was smaller than in the main survey, but it was still significant at P <0.05 (see Fig. 5.7 and

Table B.9). 217

8 Mean score 6 Boys

4 Girls 2

0 ------Grade 1 3 5 7 9

Fig* 5.7 Mean Bottles scores by grade and sex. Crosses refer to boys and circles to girls. The lines connect data points from the main developmental survey; the unattached points at Grade 5 are from the check testing and those at Grade 9 from the Grade Nine survey (Chapter 6).

HVT total score The correlation between Poles and Bottles was only 0.43. Since also Poles had a low reliability, it was decided that there was no value in finding and analyzing

a total HVT score.

There was, however, one notable similarity between

Poles and Bottles: They both had bimodal distributions with one primitive mode (at scores of 1 and 0 respectively)

and one more advanced mode (scores 5 and 7). This phenomenon would be evidence for a Piagetian type of developmental pattern except that no further stages beyond the first were clearly indicated. The pattern appears to be 218

two-stage, with a distinct break between one stage and the

second. However, whereas the first stage is fixed, con­

sisting of purely localized representation of spatial

relations, the second stage is variable as the subject first

realizes that some external factor is involved and then

gradually develops his knowledge of spatial frames of

reference. A similar two-stage pattern of development is

suggested by the DCT and HFT score distributions, although

the bimodality was not significant.

Relation to 3DD items Since Poles and Bottles were also given as part of 3DD (see Chapter 6 ), it is possible to compare group and individual versions of these items.

Fig. 5.5 and 5.7 show the mean scores obtained by the high

school students in the Grade Nine survey. It is remarkable that the group test scores were higher than the individual test scores in all four cases, although the difference was only significant in one case (boys' Bottles scores); see

Table B.16 for details. This is completely contrary to expectations, since the tester seems better able to explain the test task in the individual situation. There were no obvious differences between the school included in the two samples, and there was no reason to suspect better tester- subject rapport in the group test. 219

5.34 Solid Representation Test

Scoring of SRT was also left until all drawings had been collected. All the drawings of each solid (160 from the present survey plus about 40 others collected in the other phases) were first assigned random numbers to avoid scoring bias. They were then sorted into groups of similar drawings using the classification obtained in the pilot testing as a starting point (Section 4.45).

Drawings of the cube and cuboid were scored first, since they showed the clearest pattern of development in the pilot testing. There was little need to change the pre­ liminary definitions, and the rather fuzzy boundary between nearly correct drawings and drawings correct "within drawing error" was clarified by ordering the borderline cases and imposing an arbitrary cut-off point. There was a number of schematic drawings (11), correctly either showing the solid from a different viewpoint or including the hidden edges; these were assigned to the nearly correct category.

Apart from these, only about ten drawings proved difficult to score, but this was eventually done with fair confidence.

Drawings of the pyramid were sorted next. Again the preliminary classification proved applicable, although the 220

assignment of drawings into stages was somewhat less

obvious. The boundaries between primitive and intermediate use of depth cues, and between their intermediate and correct use, were clarified in the manner described for the cube and cuboid -

The cylinder produced the widest variety of drawings and was the most difficult to score because there seemed to be several dimensions of variation. Drawings which showed both end faces were clearly more primitive than those show­ ing only the top face; drawings with a top face shown by a circle were more primitive than those in which it was shown by an ellipse; and drawings with the bottom edge shown by a straight line were more primitive than those in which it was shown by a curve; but what was the relative importance of these factors? Was a drawing with both faces shown by ellipses more or less primitive than one in which the top face was shown by a circle and the bottom edge by a straight line?

An empirical answer to this problem was obtained by using the other drawings made by each student in the same condition. Drawings showing the ten most common types of error (including the most primitive type of a simple out- 221

line) and the obviously correct drawings were examined first. The total score on the other drawings (cube, cuboid and pyramid) was then found for each drawing, and a mean obtained for each type of error in drawing the cylinder. These means showed the correct order for most of the paired comparisons mentioned above, but also showed that some apparent improvements in drawing were not very significant= The mean scores for the drawings of the other solids were therefore used to group and order the drawings of the cylinders; this gave the general outline of the scoring system, and the remaining drawings were fitted in where they were felt to be appropriate.

Drawings of the cone were sorted in a similar way into a progression which did not, however, give an overwhelming sense of aesthetic consistency. One uncomfortable aspect was the fine dividing line between the most primitive drawing (a triangle) and the most advanced (a triangle with a curved base). This discomfort increased when scores were compared with scores on the other solids: Several students who drew cones correctly made primitive drawings of the others. It was also found that there was a much larger number of regressions (more primitive drawings in an easier 222

condition) for the cone than for the other solids, and a much more confused grade progression of scores. The reason

for these results was eventually discovered: Many of the

"correct" sketches could have been showing the general outline of the solid, which was found in the cylinder analysis to be as primitive as a drawing of a single face. It was there­ fore decided to ignore drawings of the cone in scoring the

SRT.

The classification adopted for the cuboid, cylinder, pyramid, and cube showed the following stages:

0. An outline of the solid or one face viewed orthogonally.

1. Several faces shown but not in the correct rela­ tion to each other, often both visible and invisible faces shown, usually no depth depiction.

2. Only visible faces shown, in correct relation to each other but with poor depth depiction.

3. All faces distorted in an attempt to show depth, but not correctly.

4. Solid drawn correctly by representing parallel edges of the solid with parallel or slightly converging lines.

The similarity to stages previously outlined for the repre­ sentation of 3D scenes (Section 2.32) may be noted. Typical drawings at each stage are illustrated in Fig. 5.8. All the 223

Solid Stage ------Cuboid Cylinder Pyramid Cube

0 O

/

o

Fig. 5.8 Typical SRT responses at each stage of development, by solid. 224

most common drawings are shown in the detailed scoring

system given in Appendix J.

Condition 3 was scored using the scores assigned to

each of the alternatives by the above procedure. It was

found that several of the alternatives were rarely chosen

(or drawn in the other conditions), but there was still a good coverage of the five stages for each solid.

It is interesting to compare the difficulties of the four solids. Table 5.2 shows how many drawings or selections in each condition were judged to be correct within drawing error. It is seen that the pyramid was the hardest to draw and to recognize; the cylinder was the easiest to draw, but was no easier to recognize than the cuboid and cube.

Table 5.2

Frequencies of correct SRT responses, by solid and condition

Condition Cuboid Cylinder Pyramid Cube

1 2 17 0 7 2 11 22 6 7 3 65 64 27 65 225

To check the validity of the scoring system, the scores were examined for consistency with expectations regarding the relative ease of the three conditions, for clear developmental progressions, and for consistency for solids.

Table 5.3 shows that the number of cases in which

Condition 1 (drawing after brief exposure) was easier than

Condition 2 (drawing with indefinite exposure) was never greater than 8 out of 80 and that the number of cases where

Condition 2 was easier than Condition 3 (selection) was never greater than 5 out of 80, both small enough proportions to pose no doubt as to the validity of the scoring system.

Detailed cross-tabulations are given in Tables B.17 and B.18.

Table 5.3

Comparison of SRT responses in three conditions, by solid

Solid Comparisons Cuboid Cylinder Pyramid Cube Condition 1 versus Condition 2 Condition 1 easier 4 5 8 8 Scores equal 49 61 43 60 Condition 2 easier 27 14 29 12

Condition 2 versus Condition 3

Condition 2 easier 3 2 5 2 Scores equal 14 28 32 11 Condition 3 easier 63 50 43 67 226

The developmental pattern for Condition 1 is shown in

Table 5.4, and for Conditions 2 and 3 in Tables B.19 and

B.20. There was a fairly clear progression of the modal

score from 0 in Grade 1 to 4 or 5 in Grade 9.

Finally, the scores for the four solids were added to

give an SRT total score for each condition. The reliabil­

ities (Cronbach alphas) of these scores were 0.91

(Condition 1), 0.93 (Condition 2) and 0.53 (Condition 3).

The reliabilities of the scores for the drawing conditions were regarded as most satisfactory. The low reliability of the score for the selection condition, probably due to the poor choice of alternatives but also to the narrow spread of scores, makes it of limited value in the subsequent

analysis.

As seen in Table 5.3, the scores for each solid under

Conditions 1 and 2 were more often equal than not; and in only 12 cases out of 320 did the scores differ by more than one (Table B.17). However, a repeated measures analysis of variance (Table B.21) showed that scores for drawing the cuboid and pyramid were significantly different under the two conditions (P < 0.001), although scores for the cylinder and cube were not significantly different (P > 0.10) 227

Table 5.4

Frequency distribution of SRT scores in Condition 1, by grade and solid

Grade Score 1 3 5 7 9 Cuboid 0 14 10 9 4 1 1 2 5 2 1 0 2 1 3 2 7 3 2 8 7 4 1 Cylinder

0 10 4 4 1 1 5 3 4 3 2 1 5 3 0 1 3 4 3 7 5 4 2 5 10 Pyramid 0 12 6 5 2 1 3 4 4 5 3 2 1 5 4 6 3 3 1 3 3 10 4 0

Cube 0 12 7 9 1 1 3 1 0 0 2 1 5 1 5 3 3 3 5 7 10 4 1 3 3 228

These results show that being able to view a solid could be

a definite help in drawing it, but not a very great help.

Most subjects appeared to look at a solid, classify it, call up a schema for representing that solid, and then draw that schema. Although subjects frequently glanced from model to drawing in Condition 2, in most cases this caused only minor modifications to the drawing; indeed, several in­ stances were noted where subjects erased part of a drawing and then drew exactly the same lines over. Drawing appeared to be more a matter of reproducing a representation which was known from previous experience to be satisfactory (to the subject), rather than constructing a new representation just for that solid. This view, which agrees with the impression gained while developing the Three-Dimensional

Drawing test (Section 4.32), is consistent with the schematic theory of representational learning summarized in Section 2.54.

Subjects who drew a more advanced drawing when the model was visible were probably in a transitional stage of advancing from one schema to the next.

It was consequently decided to combine scores under

Conditions 1 and 2 into a single SRT drawing score (the correlation between the total scores under the two conditions 229

was 0.95); the reliability of this combined score, was

estimated as 0.96, considered highly satisfactory. By con­

trast, except for the pyramid. Condition 3 was much easier

than Conditions 1 and 2 (see Tables 5.2 and 5.3) and the

correlations of scores under Condition 3 with scores under

Conditions 1 and 2 were much lower (both 0.69). Since it

also had a low reliability, the score under Condition 3 was

retained as a separate SRT selection score; this could be

regarded as a measure of pictorial depth perception. The

frequency distributions of both SRT scores are given in

Table B.22. The distribution of SRT drawing scores was again

clearly bimodal and suggests a two-stage pattern of develop­

ment similar to that hypothesized for Poles and Bottles

(Section 5.33). The distribution of SRT selection scores

was strongly negatively skewed.

The relation of the SRT scores to grade level is given

in Table B.7. Because the two scores have different maxima

(32 for SRT drawing and 16 for SRT selection), they are

illustrated in Fig. 5.9 in terms of a mean score per solid.

A trend analysis is included in Table B.8 . For the drawing

score, the linear trend (P < 0.001), the sex difference

(P < 0.01) and the interaction (P< 0.01) were all significant. 230

4 Boys - o

Selection Drawing 3 Mean Girls + Men score per solid 2 Women Boys

Drawing

0 Grade 1 3 5 7 9 College

Fig. 5.9 Mean SRT drawing and selection scores by grade and sex. Crosses refer to males amd circles to females. The lines connect data points from the developmental survey; the unattached data points are from the spatial training experiment (Chapter 7).

and the sex difference was significant post hoc at Grade 3

(P <• 0.05) as well as at Grade 5 (P C 0.01). For the selection score, only the linear trend was significant.

Fig. 5.9 also shows the mean SRT drawing scores of a sample of teacher's college students (Section 7.53).

5.35 Relations between spatial test scores

The results of the previous sections are summarized in

Table 5.5. To guard against escalating probability levels, the trend analysis was repeated using all six scores in a multivariate analysis (see Table B.8 ). The results were very 231

Table 5.5

Summary of spatial test results

HVTSRT Draw­ Selec­ DCT HFT Poles Bottles ing tion Reliability 0.93 0.92 0.71 0.91 0.96 0.53

Significance of; Linear trend 0.001 0.001 0.001 0.001 0.001 0.001 Sex difference 0.009 0.013 0.113 0.006 0.001 0.132 Interaction 0.001 0.001 0.007 0.135 0.002 0.168

Significant post hoc sex difference: Grade 5 5 none none 3, 5 none Significant 0.01 0.01 - 0 .05,0.01

Significance of sex difference in check testing 0.008 - 0.007 0.034 --

similar to those for the univariate tests, with the linear trend (P ■< 0.001), the sex difference (P <0.01) and the interaction (P <0.001) all significant. The deviation from linearity was not significant overall (P > 0.30) or for any of the tests separately.

The correlations between the six spatial test scores are shown in Table 5.6. As expected from the developmental graphs (Fig. 5.2, 5.4, 5.6, 5.7 and 5.9), all scores except 232

Table 5.6

Correlations between spatial test scores, by sex^

Score 1 2 3 4 5 6 1. DCT 69 39 79 77 71 2. HFT 88 36 64 72 70 3. Poles 61 64 25 55 39 4. Bottles 81 79 55 68 66 5. SRT drawing 91 88 60 88 71 6 . SRT selection 69 62 38 70 69

^Decimal points omitted. Correlations above the diagonal refer to boys, below the diagonal to girls. For each sex, N = 40; P(r 3; 0.26) = 0.05 and P ( r ^ 0.36) = 0.01.

poles were closely related. The correlations for girls were

generally higher than those for boys, but a test due to Box

(1949) showed that the difference between the covariance matrices for the two sexes did not quite reach significance

(chi-square = 29.89, d.f. = 21, P

of separate factors for 2D and 3D tests. Poles appeared

to be separate from the other scores in both sexes and the

SRT selection score appeared to be separate in girls, but

these could be artifacts of the lower reliability of these

two scores.

The correlations between the spatial test scores could have been inflated by the wide age range. To find how the

scores were related apart from their common dependence on 233

maturation, partial correlations were calculated and are shown in Table 5.7. The correlations were considerably reduced by controlling for age, on the average from 0.71 to

0.39 for boys and from 0.85 to 0.70 for girls (leaving out the Poles and SRT selection scores). Even allowing for its

Table 5.7

Partial correlations between spatial test scores controlling for age, by sex^

Score 1 2 3 4 5 5 1. DCT 30 08 53 43 41 2. HFT 75 08 35 38 48 3. Poles 43 47 —0 4 38 15 4. Bottles 50 59 31 37 41 5. SRT drawing 80 75 39 72 43 6 . SRT selection 41 32 08 43 37

^Decimal points omitted. Correlations above the diagonal refer to boys, below the diagonal to girls. For each sex, N = 40; P(r^ 0.26) = 0.05, P(r ^ 0.37) = 0.01.

low reliability, the Poles score was clearly unrelated to anything else except SRT drawing amongst the boys, although there were still moderate correlations with other scores amongst the girls. These results again show that the boys' spatial ability was more differentiated than that of the girls. 234

5.4 Geometric illusion results

The following analysis is based on the number of illusion-supported responses given by each student on each illusion. The number of students who made two or more

Guttman errors (Segall et al., 1966, pp. 116-121) are given by grade in Table B.23. These students were not eliminated because Segall et al. showed that such a procedure made little difference to the results of their analysis.

5.41 Hypotheses

The design of the survey allowed the testing of several hypotheses concerning the causes of illusion susceptibility

(Section 2.8).

If susceptibility were caused by interpreting a 2D diagram 3-dimensionally (Section 2.82), several results would be expected. Firstly, HVT and SRT might induce a set to interpret diagrams 3-dimensionally (Dinnerstein, 1965) which could be strong enough to temporarily increase illusion susceptibility; if so, there would be a significant order effect. This gave the following hypothesis:

Students who do HVT and SRT before GIM show greater illusion susceptibility than those who do HVT and SRT after GIM. 235

Secondly, those who are better at pictorial depth perception

(as shown by higher scores on SRT selection) would be more susceptible:

H2 : The correlations of the illusion sus­ ceptibilities with the SRT selection score are positive.

Thirdly, students to whom the Ponzo diagram looks like a road or railway line would be more susceptible to the illusion than those who see it as a plane figure:

H 3 : Students who report a 3D interpretation of the Ponzo diagram are more susceptible to the Ponzo illusion than those who report a 2D interpretation.

The theory also predicts that those who can better dis­ sociate subjective and objective views of an object are likely to be less susceptible to the illusions. This gave the following hypotheses:

H^: The correlations of the illusion susceptibil­ ities with the HVT and SRT drawing scores are negative.

Piaget (1959) relates susceptibility to the Horizontal-

Vertical illusions to the development of frames of refer­ ence in children. His theory also predicts negative corre­ lations of susceptibilities to those two illusions with

Poles and Bottles scores (included in H^). 236

The proposed effect of field independence (Section 2.83) led to the following hypothesis;

Hg: The correlations of the illusion susceptibili­ ties with the DCT and HFT scores are negative.

The supposed relation of illusion susceptibility to skin color (Section 2.84) gave the following hypothesis:

H^: Darker skin color is associated with lower illusion susceptibility.

No hypotheses were advanced with regard to sex or age. As for environment, since most of the subjects in the survey come from a Westernized urban environment, susceptibilities were not expected to differ greatly from those obtained in

Illinois (Armstrong et al., 1970).

5.42 Order effect

Multivariate and univariate analyses of variance of the GIM scores are reported in Table B.24. Neither the order effect nor its interaction with grade approached significance overall; the interaction of order with sex just reached significance on the Muller-Lyer (P 0.05) and nearly on the Horizontal-Vertical inverted t (P <, 0.06) and

Sander Parallelogram (P ^0.05) illusions, but the pattern of means was not consistent across illusions. The third- 237 order interaction was also non-significant. Hypothesis was therefore rejected, and it was deduced that any set for the 3D interpretation of diagrams induced by HVT and

SRT had no measurable effect on illusion susceptibility.

However, it is possible that the GIM was not sufficiently reliable to detect such an effect (see below).

5.43 Developmental analysis

The relation of the GIM scores to grade level is given by sex in Table B.25 and illustrated in Fig. 5.10.

Also shown in Fig. 5.10 are the means for a sample of teacher's college students (Section 7.53).

In the multivariate analysis of variance reported in

Table B.24, the difference in GIM scores at the five grade levels was not significant, neither was the sex difference or the interaction. It was concluded that either age is not an important factor influencing illusion susceptibility in Jamaican students or the errors of measurement were so large as to obscure the developmental pattern.

It appears that large samples are required to give clear results using this test. Stewart (1971) tested 575 students from Grades 1-12 in various parts of Zambia, but 238

Muller-Lyer illusion 7 Mean Girls score 6

5 Boys Women 4 % Men

7 Horizontal-Vertical illusion 1 (~~1) Mean score 6

5

Horizontal-Vertical illusion 2 (-L)

Mean 4- score

Sander Parallelogram

Mean score

Ponzo illusion 7 Mean score 6

5

Grade 1 3 5 7 9 College

Fig. 5.10 Mean GIM scores by sex and grade. Crosses refer to males and circles to females. The lines connect data points from the developmental survey; the unattached points are from the spatial training experiment (Chapter 7). 239

even when the results are smoothed by combining Grades 1 and 2, 3 and 4, etc., the graphs still zigzag after age 11

(Fig. 2.8, Section 2.8). Armstrong et al. (1970) tested

120 students from Grades 1-12 and Dawson et al. (1973) tested 45 2 students aged 3-17, but their original graphs

(smoothed in Fig. 2.8) showed a similar zigzag pattern to those in Fig. 5.10. The present writer gained the impression that the test was inherently unreliable in its present form due to variations in length of exposure, which seemed to depend on whether the subject knew he was being deceived by the diagrams. Segall et al. (1966, p. 182) noted the con­ siderable effect of variation in exposure time on suscepti­ bility, and rejected it as a source of cross-cultural dif­ ferences; however, it seems likely to be a major source of individual differences. Unfortunately, apart from Guttman reproducibility indexes (typically high, above 0.90 in most of the samples included in Segall et al., 1966), no relia­ bility estimates are available for this test.

By comparing Fig. 5.10 with Fig. 2.8, and making due allowance for experimental error, it is seen that, for the

Muller-Lyer illusion, susceptibility was about the same as that previously found in Zambia and Hong Kong, but lower 240 than that for Illinois; for the Horizontal-Vertical illusions, similar to Hong Kong; and for the Sander

Parallelogram, similar to Zambia and higher than Hong Kong and Illinois. These comparisons suggest that, as regards illusion susceptibility, even high-ability Jamaican students have more in common with students in other developing countries than with U.S. students.

5.44 Correlations between illusions

Correlations between scores for the five illusions are reported in Table 5.8. They were all low, as previously found by Jahoda & Stacey (1970) and Taylor (1972), and lower than those found by Segall et al. (1956). The only correlations to approach significance were those between the Muller-Lyer and Sander Parallelogram illusions, but the anomolous negative correlation amongst the boys suggests that these were purely chance results, and again points to unreliability in this test. 241

Table 5.8

Correlations between geometric illusion susceptibilities, by sex^

Illusion ML HVl HV2 SP Pz Muller-Lyer -09 19 -32 10 Horizontal-Vertical 1 19 18 09 04 Horizontal-Vertical 2 01 3 05 10 Sander Parallelogram 27 -15 17 08 Ponzo 04 13 -02 05

^Decimal points omitted. Correlations: above the diagonal

refer to boys, below the diagonal to girls. For each sex' f N = 40; P(|rl> 0.30) = 0 .05, P(|r|^ 0.39) =: 0 .01.

5.45 Relation to spatial ability

Table 5.9 gives the correlations between the spatial test scores and the GIM scores. Three of the hypotheses formulated in Section 5.41 concerned these correlations:

Hypothesis H2 was not supported for any illusion. In fact, for the Horizontal-Vertical inverted L illusion (both sexes) and the Muller-Lyer (girls only), the correlations were significantly negative. H2 was therefore decisively rej ected.

Only 10 out of the 30 correlations covered by H4 were significantly positive, and only one of these (Maller-Lyer with Poles) amongst both sexes. Since there was no reason 242

Table 5.9

Correlations between spatial test scores and geometric illusion susceptibilities, by sex^

Illusion Spatial test ML HVl HV2 SP Pz

Boys DCT 06 -28 10 06 31 HFT -10 -26 01 13 09 Poles -30 -10 08 18 26 Bottles 04 -31 02 -11 27 SRT drawing -10 -36 -02 08 23 SRT selection -04 -41 -04 10 11

Girls

DCT -32 -24 -02 -15 -28 HFT -34 -17 -05 -25 -26 Poles -30 -22 09 -31 -20 Bottles -29 -22 08 -18 -21 SRT drawing -32 -22 -01 -23 -32 SRT selection -35 -36 -06 -08 -27

^Decimal points omitted. For each sex, N = 40; P(r ^ 0.26) = 0.05, P(r ^ 0. 36) = 0.01

to expect drawing ability to affect illusion susceptibility differentially, H4 was rejected. Even the more restricted hypothesis relating the horizontal-vertical tests and illusions (Piaget, 1969) was not supported by the present data.

Hypothesis was supported by 6 out of 20 correlations, consistently for the Horizontal-Vertical inverted L illusion 243

in boys and for the Muller-Lyer and Ponzo illusions in girls.

As there was no reason to expect the effect of field inde­

pendence to be restricted to particular illusions, H5 was

also rejected.

In view of the high correlations between the spatial

test scores (Section 5.35), it might be better to look for

correlations which are consistent over all six scores

(possibly omitting the less reliable Poles and SRT selection

scores). It is dangerous to make post hoc inferences from such a large set of correlations, but it seems safe to con­ clude that higher spatial ability was associated with lower susceptibility to the Horizontal-Vertical inverted L illu­

sion in boys and to the Muller-Lyer illusion in girls. It is also clear that spatial ability was unrelated to suscepti­ bility to the Horizontal-Vertical inverted T in both sexes.

No explanation can be given for these apparent contradictions; but they do point out once more the difficulties of finding a general theory of illusion susceptibility: It must not only allow for different responses to similar-looking illusions but also different responses to the same illusion in the two sexes. No reference to any sex difference in correlations has been found in the literature, but Stewart 244

(1973, 1974a) recently reported sex differences in opposite

directions for different illusions.

5.46 Three-dimensional interpretation of illusion diagrams

A final test of the theory that illusions are caused by

a 3-dimensional interpretation of 2D diagrams was provided

by asking students what the final Ponzo diagram looked like

to them. A small number of students (8 boys and 6 girls) were unable to answer the question. Amongst the remainder, it was found that boys who reported a 3D interpretation were significantly more susceptible to the illusion than those who reported a 2D interpretation (t = 3.16, d.f. = 30,

P < 0.01), but no such relation held for girls (t = 0.58).

Thus Hg (Section 5.41) was accepted for boys but rejected for girls.

It should be noted that the choice of a 2D or 3D inter­ pretation was independent of spatial ability: The distribu­ tion of responses was fairly uniform across grades (chi- square = 5.99, d.f. = 4 , P > 0.20), about 70% of the students giving a 3D interpretation. 245

5.5 Influence of background variables

5.51 Background data

The mean ages of the present sample by grade are given in Table 5.1 (Section 5.2). Descriptive statistics on the other background variables are given by grade in Table B.26 and summarized in Table 5.10.

Table 5.10

Summary statistics on background variables

Hand dominance Left Mixed Right Frequency 1 21 58

Eye dominance Left Right Frequency 39 41

Number of Mean S.D. siblings 3.98 2.32 Adult male in home Present Absent Frequency 50 30 Social class 1 2 3 4 5 6 Frequency 1 3 30 23 6 3 Ambition Spatial/technical Verbal/personal Frequency 16 60 Skin color Light Mid Dark White Brown Brown Brown Black Frequency 1 8 16 24 31 246

The frequencies for hand dominance were approximately as

expected (Section 2.73) and were fairly uniform across

grades. Only the solitary left-hander used his left hand

for cutting with scissors. Of the 21 mixed-handers, 16 wrote with their right hand but unscrewed the bottle top with their

left, the remainder vice versa. Thus 6 out of the 80 in this sample wrote with their left hands but only one was definitely left-dominant. This result shows how unreliable a single measure of handedness is likely to be as an indi­ cator of hand dominance.

The frequencies for eye dominance varied widely from grade to grade, which suggests that the measure used was rather unreliable. Although most students did not hesitate about which eye to use for each task, 18 out of the 80 were not consistent from task to task (32 were consistently left- and 30 consistently right-dominant). Eye dominance does not seem to be as clearly defined as hand dominance; Dawson

(personal communication, 1974) has found that seven tasks are necessary to define eye dominance reliably.

The mean number of siblings and the number of students reporting a male figure in their upbringing (father, uncle, grandfather or other male guardian) were again fairly 247

uniform from grade to grade, so were probably reliable.

Ratings of social class, however, must be regarded as rather unreliable. In the first place, students were in general poorly informed about their parents' occupations, so a lot of guesswork was needed on the part of the writer.

Secondly, many students lived with mothers or grandmothers who (they said) did not go out to work, presumably being supported by relatives or friends living elsewhere. For these two reasons, no rating of social class was possible for 14 students. The third, most important, reservation concerns the classification system used (Table A.8 ) which seemed to measure social prestige rather than the likely intellectual stimulus in the home. The only valid dis­ tinction seemed to be between professional and clerical occupations requiring at least a minimum level of academic education (categories 1-3) and manual and unskilled occupa­ tions requiring no such education (categories 4-6). Since most students in each grade fell into either category 3 or

4, no modifications were made to the scale.

All but two of the girls tested gave a verbal/personal ambition such as doctor, nurse or teacher. Only four primary boys gave a spatial/technical ambition, whereas more 248

than half the high school boys said they wanted to be

engineers or the like. Because of this unevenness, this variable was excluded from further analysis.

The rating of skin color must also be regarded as of poor reliability. Although Mid Brown was quite distinct from Black, the boundaries of Dark Brown with Mid Brown and

Black were indistinct and probably changed from day to day.

The categories of White and Light Brown were hardly used; proportions in the other three categories were roughly equal in the five grades tested.

Correlations between the background variables are shown in Table 5.11. (The solitary left-hander was omitted when calculating correlations with hand dominance.) The direction of measurement is indicated by giving the meaning of the higher score(s) on each variable. There appear to have been only weak relations between the background vari­ ables in this sample. The low correlations with age reflect the uniformity across grades of all the background variables except social class. The physical variables (hand and eye dominance and skin color) were completely independent, and the interrelations of the sociological variables (number of siblings, adult male in home, and social class) were incon- 249

Table 5.11

Correlations between background variables, by sex^

Higher Variable 1 2 3 4 5 6 7 score 1. Age Older -00 02 07 -04 -24 07 2 . Hand dominance Right -18 27 40 -03 -38 21 3. Eye dominance Right -01 15 20 -17 -32 -07 4. Number of siblings More -24 31 30 -26 -00 25 5. Adult male in home Absent -11 06 05 -08 33 -09 6 . Social class Lower -34 22 26 42 -12 36 7. Skin color Darker 02 10 04 08 -14 17

^Decimal points omitted. Correlations above the diagonal refer to boys; N = 39 or 40. Correlations below the diagonal refer to girls; N = 40. P(|r|;^ 0.30) = 0.05, P( |r| ^ 0.39) = 0.01

sistent between the two sexes: Higher social class was associated with a father present for boys but not for girls and with a greater number of siblings for girls but not for boys. Correlations of the physical with the sociological variables were also weak and inconsistent: Right-handedness and right-eyedness were both associated with a greater number of siblings in both sexes, but with higher social class in boys and lower social class in girls. Father presence was significantly correlated with no physical variable and skin color was significantly correlated only with social class 250

and then only amongst boys. Because of the large number of correlations examined and the small sample size, it is diffi­ cult to decide whether any of these results are other than chance artifacts. The rather low level of the correlations could be a result of restricting the sample to a narrow ability range as well as unreliability in measurement.

Previous researchers in Jamaica (Miller, 1957a) have not examined sex differences in correlations between sociological variables, although they are quite plausible (Sections 3.3 and 3.4).

5.52 Relation to spatial ability

Correlations of the spatial test scores with the back­ ground variables are given in Table 5.12. The strong corre­ lations with age reflect the linear trends shown in the graphs in Section 5.3. Hand and eye dominance, the presence of an adult male in the home, and skin color were not con­ sistently related to spatial ability, contrary to expecta­ tions. Although number of siblings was unrelated to spatial test performance amongst the boys, amongst the girls a larger family was associated with lower test scores, significantly so for DOT, HFT and SRT selection. Lower social class was 251

Table 5.12

Correlations of spatial test scores with background variables, by sex^

HVT SRT Background DCT HFT Draw- Selec- variable^ Poles Bottles ing tion Boys Age 76 72 42 62 75 62 Hand dominance -13 03 -30 -03 -19 03 Eye dominance -10 11 -09 21 05 03 Number of siblings -17 00 -02 -14 -10 -18 Adult male in home 06 -10 -20 -04 -02 04 Social class -20 -35 -11 -25 -12 -41 Skin color -15 07 -20 -04 -08 -20 Girls Age 73 68 49 74 78 66 Hand dominance 09 -05 -12 -07 -13 -11 Eye dominance 16 17 41 28 27 -05 Number of siblings —36 -40 -15 -20 -27 -32 Adult male in home -13 -19 -10 -18 -16 -17 Social class -42 -40 -07 -12 -35 -30 Skin color 00 13 02 -02 -02 -04

^Decimal points omitted. For boys, Ns range from 30 to 40; P(|r|>/ 0.36) = 0.05, P(l r|2>0.46) = 0. 01. For girls / Ns range from 36 to 40; P(|r( > 0.32) =: 0 .01, P (1 rl > 0.38) = 0 .01.

For direction of measurement, see Table 5.11.

associated with lower spatial test scores amongst both sexes,

but significantly so only for SRT selection amongst the boys

and only for DCT, HFT and SRT drawing amongst the girls. 252

A significant relation of family size and social class to

spatial test performance was expected; the small size of the

correlations could have been due to unreliability and

restricted range. It is doubtful whether one should place

any confidence in the obtained sex differences in correlations,

although they have been found elsewhere (Mebane & Johnson,

1970) and are quite plausible in Jamaica (Section 3.34).

In case the correlations of the spatial test scores with

the background variables were unduly inflated by their corre­

lations with age, partial correlations controlling for age were calculated and are given in Table 5.13. Correlations with social class all became smaller and non-significant but correlations with number of siblings amongst the girls were virtually unchanged. Also amongst the girls, partial

correlations of eye dominance with Poles, Bottles and SRT drawing scores were significant, but the much lower correla­

tions with the other spatial test scores suggest that these may have been chance results.

To investigate the effect of mixed hand-eye dominance,

a multivariate two-way analysis of variance of the spatial

test scores was calculated with factors hand and eye dominance, combining data from both sexes and all five grades. 253

Table 5.13

Partial correlations of spatial test scores with background variables controlling for age, by sex'

Background HVT SRT DCTHFT Draw­ Selec­ variable^ Poles Bottles ing tion Boys

Hand dominance -15 02 -32 04 -19 15 Eye dominance -28 05 -15 13 -19 -20 Number of siblings -17 -20 -05 -24 -31 -23 Adult male in home 03 -15 -18 -02 01 -03 Social class -02 -25 00 -12 11 -34 Skin color -26 -06 -23 -07 -17 -29

Girls

Hand dominance -03 02 09 -01 -10 -19 Eye dominance 23 18 46 40 41 -09 Number of siblings -34 -40 -03 -08 19 -31 ’\dult male in home -02 -15 -09 -14 12 -03 Social class -27 -25 13 20 -14 -12 Skin color -06 17 06 —06 -07 -13

^Decimal points omitted. For boys, N = 27; P(1 r|^ 0 .39) = 0.05. For girls, N = 33; P(( r|> 0.35) = 0.05, P(lr 13>0.45) 0.01. bpor direction of measurement, see Table 5.11.

For the purpose of this analysis, the solitary left-hander was placed with the mixed-handers. The results of the analy­ sis are given in Table B.27. The eye dominance effect was significant overall (P < 0.01) and for Poles (P<^ 0.05) and

Bottles (P < 0.02) separately, and approaching significance 254

for SRT drawing (P < 0.05); these results agree with those derived above. Neither the hand dominance effect nor the interaction was significant. An inspection of the mean scores (Table B.28) showed that, on all tests except SRT

selection, right-handed left-eyed students scored substan­ tially lower than students in the other three categories, who were not very different. However, using Sheffe's post hoc procedure, the contrast only reached significance for

Poles (P ^ 0.05). Nevertheless, the results do seem to confirm the findings of Dawson (1972b), Oltman & Capobianco

(1967) and Rengsdorff (1967) rather than those of Kershner

(1970) (Section 2.73).

5.53 Relation to illusion susceptibility

The correlations of the background variables with illusion susceptibilities are reported in Table 5.14. Most of the correlations with age were non-significant, reflecting the absence of clear linear trends (Fig. 5.10). None of the other correlations was significantly different from zero.

Because of the suspected low reliability of GIM and some of the background variables, this result cannot be interpreted as showing that illusion susceptibility was independent of 255

Table 5.14

Correlations of geometric illusion susceptibilities with background variables, by sex^

round Illusion variable ML HVl HV2 SP Pz Boys Age -01 -33 19 11 15 Hand dominance -03 -06 -00 01 04 Eye dominance -15 -29 -22 -22 23 Number of siblings -23 16 17 13 05 Adult male in home 31 17 -04 -13 12 Social class 10 32 15 -14 03 Skin color 10 -05 12 -02 -16

Girls

Age -39 -07 -03 -27 -31 Hand dominance -29 -04 14 -01 -12 Eye dominance -05 -22 11 -14 -04 Number of siblings 12 12 -16 -02 -03 Adult male in home 04 07 -02 10 16 Social class 03 05 14 05 -03 Skin color -06 -03 -17 -07 -19

^Decimal points omitted. For boys, Ns range from 30 to 40; P(|r|> 0.35) = 0.05. For girls, Ns range from 36 to 40; P(|r|> 0.33) = 0.05.

^For directions of measurement, see Table 5.11.

the background factors considered. However, unlike Table 5.9,

Table 5.14 does not even show any patterns of near-signifi­ cant correlations to suggest possible relations for further investigation. CHAPTER SIX

THE GRADE NINE SURVEY

6 .1 Background

6.11 Purpose

The major purpose of the Grade Nine survey v.’as to

investigate the relation of spatial ability to environment

and educational experience (Section 1.42, research question

3). Other purposes were to study further the relation between various facets of spatial ability (research ques­

tion 1) and the relation of spatial ability to sex, handed­ ness, ambition, social class, family size and family composi­ tion (research question 3).

As indicated in Section 4.23, it was not practicable to measure students' previous educational experience. However,

students' current educational experience, broadly defined by the type of school they are attending, is an important

factor in the Jamaican system. The examinations used for

selection into high and technical high schools (Section

3.23) are mainly verbal and numerical, and of only moderate validity in predicting academic achievement (Manley, 1969).

256 257

The correlation of verbal and numerical ability with

spatial ability in Jamaican students is unknown. If it were

high, most students of high spatial ability would be in high

schools and few in technical high schools. If it were low,

there would be substantial numbers not in high schools who

could benefit from technical education. Either way, there would be obvious educational implications.

Students were therefore selected from representative

secondary schools of four different types, in three widely differing environments. The group tests developed for this

survey (3DD, HST, and BOX) are described in Section 4.3, as is the PDQ used for collecting personal data.

6.12 Delimitations

The great majority of secondary school students are in the high, junior secondary and all-age schools (Section

3.23). It was intended initially to include only these three types in the survey, but after it had been pointed out that the technical high schools provide a large proportion of the lower- and (after further training at

CAST) middle-level technical manpower in Jamaica, this type of school was also included. The three comprehen­ sive schools also supply future technicians but these were 258 too few to include. For design reasons (Section 6.21), single-sex high schools had to be excluded.

Of the six types of environment mentioned in Section

3.35 (the cities, the sugar belts, the market towns, and the hill farming, mining and tourist areas), it was decided that only the cities, market towns and hill farming areas would be investigated. The sugar belts held a transient population, and the mining and tourist areas were small in size and although of great economic importance not typical of the country as a whole.

With four types of school and three environments, using the minimum of two schools per cell meant testing about 24 schools. (In fact, there were no village technical high schools, so only 22 schools were tested.) This number and test limitations on time and size made it possible to test only one class in each school. It was decided to test a representative class in Grade 9, the last grade in which free education was available to all Jamaicans in the school year 1973-74.

Because of the temperature and often poor nutrition,

Jamaican schoolchildren tend to become fatigued by the afternoon. It was therefore decided to administer all tests 259 in the mornings. For this reason, the survey was restricted to places within two hours' drive of Kingston. Schools were therefore selected from the Kingston-St. Andrew Corporate

Area and the parishes of St. Andrew, St. Thomas, Portland,

St. Mary, Manchester, Clarendon and St. Catherine (Fig. 3.2).

This region will be referred to as the survey area.

6.2 Method

6.21 Design

All technical high, junior secondary, and all-age schools in Jamaica are mixed (coeducational), but only half the high schools are. To have tested boys and girls in separate schools (as was done for the developmental survey— see Section 5.21) would have doubled the testing time and given a weak design. It was therefore decided to exclude the single-sex high schools and to test boys and girls within the same school.

The design for the school survey is illustrated in

Fig. 6.1. It was intended to obtain at least ten data points in each cell. Practical considerations later pre­ cluded analysis of the results in the completely crossed design, and only selected comparisons were studied (see

Section 6.32). 250

- Technical Junior High All-age high secondary school school school school

Boys Girls Boys Girls Boys Girls Boys Girls

1 City 2

1 Town 2

1 Village 2

FiK. 6.1 Design for Grade Nine survey. Schools were nested within environments and school types; it was intended to obtain at least 10 data points in each cell.

6.22 Identification of schools

Of the 19 mixed high schools, 15 fell within the survey

area. Of the 6 technical high schools, 5 were within the survey area. Of the 64 junior secondary schools, 43 were in the survey area. All these schools were expected to have at least two classes of at least 35 students, so were likely to provide data for at least the intended 10 boys and 10 girls.

By contrast, all-age schools often have very small enrollments, especially in their senior departments. In 261 order to eliminate the smaller schools, the returns of the

1973 Grade Nine Achievement Test (Section 3.2) were con­

sulted. These showed that about twice as many girls as boys

took the examination, so that in order to obtain at least

10 boys and 10 girls, it would be necessary to select only schools which entered 30 or more students. Of the 553 all­ age schools in the country, 3 39 were in the sample area; of these, 5 3 were deemed large enough for inclusion in the survey.

6.23 Environmental classification

A most convenient classification of towns was available from the work of the United Nations Special Fund Project,

"Assistance in Physical Planning" (1971b). From the 1 150 towns and villages outside Kingston, 215 had been selected as likely urban growth points and surveyed in 1970. In­ cluded were all 126 places with populations of 2 000 and over, and probably most of those with populations of over

1 000. Each place had been ranked on a scale from 0 to 5 on the quality of each of the following factors:

1. Size of population and number of shops 2. Size and importance of market 3. Accessibility 4. Actual and potential economic activity: a. Agriculture b. Industry 262

c. Tourism d . Mining e. Fishing f. Forestry g. Diversification 5. Water supply 6 . Electricity and telephone 7. Post-primary educational facilities 8 . Other public facilities 9. Land Authority centre 10. Government land available for development

Since most of these factors appeared to be related to urban- rural differences in spatial ability (Section 4.23), it was decided to use the Project scales for the environmental classification of the schools previously identified.

Category 4, which already has seven times the weight of the other factors and which was doubled again in the Project's measure of economic growth potential, was reduced to three times the weight of the other factors; the ten ratings were then added to give an urban index for each school location. Kingston was arbitrarily given the maximum possi­ ble index (60), and those not listed were given the minimum of those listed (10).

The locations of the schools previously identified were found to fall into groups as follows: 23 had urban indexes of 10 through 12, 10 of 16 through 19, 9 of 20 through 25,

10 of 28 through 34, 10 of 39 through 5 3, and 1 (Kingston) 263 of 60. The group with urban indexes of 39 through 53 were developing small cities, whereas almost all those with urban indexes of 18 or less were small villages in the hill farm­ ing areas. The range of 20 through 34 could have been used to define the market town, but in order to maintain a clearer distinction between towns and villages, five intermediate places with indexes 20 through 22 were excluded. It was also decided to clarify the town-city distinction by ex­ cluding the small cities.

The following environmental categories were therefore adopted for the purpose of this survey:

City..... Kingston-St. Andrew Corporate Area

Town Urban index of 23 through 34

Village... Urban index of 18 or less

Schools with urban indexes not in these ranges were eliminated from the survey population. Exceptions were made for the two rural technical high schools: Both were situ­ ated in the middle of ribbon developments which linked larger centres, in each case a town and a small city; these two schools were admitted as town schools. 264

6.24 Final definition of survey population

In order to further clarify the characteristics of the rural population to be surveyed, six of the village schools were eliminated; two all-age schools within easy reach of Kingston and judged to be subject to a strong urban influence, and one junior secondary school, two all­ age schools and one technical high school in the sugar belts. All the remaining village schools were in the hill farming areas.

Two junior secondary schools attached to high schools and therefore subject to untypical influences were also eliminated.

The numbers of schools in the survey population, when the delimitations described in the last few sections had been applied, were as shown in Table 5.1. The schools in the survey population are given in Table C.l, and the schools in the survey area which were eliminated for various reasons are listed in Table C.2.

6.25 Selection of school sample

For the technical high schools, both schools in the city and town categories were selected, there being no choice. For the high schools, because of the small numbers. 265

Table 6.1

Numbers of schools in survey population, by environment and school type

School type Envir­ Technical Junior onment High All-age high secondary

City 4 2 11 18 Town 3 2 9 3 Village 3 0 7 19

schools were chosen at random within each category. For the junior secondary and all-age schools, because of the

small number of schools which could be surveyed, it was felt that a purposive sample would be more representative than a random sample. An academic index was therefore obtained for each school and two schools chosen in each category nearest the median for that category.

The academic index for junior secondary schools was the percentage of students entered for the 1973 Grade Nine

Achievement Test who obtained free places in the high, technical high, and comprehensive high schools. Since this percentage was very small for the all-age schools, the percentage of students entered for the 1973 GNAT who scored

50% or more was used as the all-age school index. These 266 indexes were subject to the same criticisms as the elemen­

tary and high school academic indexes used in the develop­ mental survey (Section 5.22), but like them were felt to be

at least adequate to identify the extreme schools in each category.

Schools in five of the six junior secondary and all­

age school categories were quickly chosen by the method described. The small number of town all-age schools were found to have academic indexes all below the medians for

Kingston and village all-age schools, so the two schools with highest indexes were chosen to represent this category.

The location of the 22 survey schools thus selected is shown in Fig. 6.2. The selection of students within each school is described in the next section.

6.26 Procedures

The selected schools were visited for testing between

22 April and 13 June 1974. High schools were tested first because of imminent public examinations, others being tested in an order which depended on geographic proximity and personal convenience. Each school was visited one or two weeks before the testing date to explain to the prin­ cipal the purpose of the research, to obtain his or her 267 cooperation, select the class to be tested, and fix the test date and time.

One of the selected Kingston high schools was replaced by an alternate when the principal appeared uncooperative, and one all-age school in Kingston had no Grade 9 so also had to be replaced. All other principals willingly agreed to the testing. In schools where streaming was employed, it was intended to test a middle class; in two cases, this was not made quite clear and above-average classes were tested. In one school, two small classes had to be combined. One all-age school divided its senior depart­ ment by ability instead of age, so many eighth graders were included in the middle group. In schools where streaming was not used, convenient classes were tested; in two cases, the whole of Grade 9 was tested.

In one school, it was necessary to administer the tests in the afternoon session. In all other schools, tests were given in the morning, typically between 9.00 and 11.30. In most cases, this occupied three class periods, which were chosen with a scheduled recess between the second and third periods wherever possible. In some schools, PDQ was given after 3DD and HST; however, instead 268

t->nvvo^

BelUF

Lcnncn

Fig. 6.2 Location of schools sampled in Grade Nine survey.

of providing relaxation this procedure seemed rather to destroy concentration so the practice was not used unless

PDQ could be followed by a recess. Most tests were ad­ ministered in the class’s regular classroom, but several principals kindly moved classes into quieter areas of the school for the testing.

Tests were scored immediately and results reported to the school principal and students. All data were treated confidentially using code numbers known only to the students.

The two village "high schools" had to be dropped from the survey when visits revealed that they both con­ tained only a small number of students who had passed the 269

Common Entrance examination; they were therefore not com­

parable to other high schools in the sample. One of the

schools set their own lower-level examination for entry to

Grade 7. The other school took all-comers in Grades 7-9

and only selected students for academic courses in Grades

10 and 11; this school had in fact been eliminated as a

junior secondary school because of the high school element

(Section 6.24). The test results of these schools are

given in Table C.3; comparison with the results of the

other schools (Tables C.9-C.11) confirms that they were

nearer junior secondary than high school status.

Summary data on the twenty schools thus included in

the survey are given in Table C.4.

6.3 Results

6.31 Numbers

A full vector of three scores (3DD, HST and BOX) was

obtained for 639 students; 277 boys and 362 girls. The numbers of students in each school are shown in Table 6.2.

Background characteristics of the sample are described in

Section 6.36. Table 6.2

Numbers of students in survey sample, by school

High Technical Junior All-age Envir­ schools high secondary schools onment Boys Girls Boys Girls Boys Girls Boys Girls City 1 10 27 18 16 11 15 15 13 2 16 12 16 15 17 18 11 22

Town 1 17 12 21 15 13 18 8 14 2 11 22 12 15 13 24 16 20

Village 1 — — — 14 24 10 27

2 —— 13 20 15 13

to •o ' j 271

6.32 Test characteristics

The 3DD item scoring systems were developed from the developmental survey data for Poles and Bottles (Section

5.33) and from the pilot-testing data for Cubes and Edges

(Section 4.32). The item reliabilities (Cronbach alphas) in the present sample were calculated as 0.69 (Poles),

0.57 (Bottles), 0.77 (Cubes) and 0.80 (Edges). The low reliability of Bottles seemed to be the result of its narrow range, most students scoring between 7 and 10. The inter-item correlations are shown in Table C.5. Although the inter-item correlations were rather low, the item-test correlations appeared high enough to justify adding the scores to form a single 3DD score. The frequency distribu­ tion of the 3DD total score, shown in Table C.6, was nearly symmetrical. An item analysis is given in Table C.7.

The reliability (Cronbach alpha) of the test was calculated as 0.85, which was considered satisfactory.

The frequency distribution of HST scores is also shown in Table C.6; it was clearly positively skewed, with the mode not far away from the chance score of 3. To estimate the reliability of the test, five answer papers were selected at random from each school and the 100 item scores 272

transcribed and subjected to an item analysis. The results

(Table C.8) show that HST was hardly any improvement over

the trial version (Table A.7). The KR-20 reliability of the

test was 0.71. HST was clearly too difficult for the present

sample.

The frequency distribution of BOX scores, shown in

Table C.6, was nearly symmetrical. Since BOX was a speed

test, its reliability was estimated from the correlation

between the separately-timed halves. Using the Spearman-

Brown formula to correct for the halved length, the reliabil­

ity of BOX in this sample was estimated to be 0.83. This is within the range of reliabilities (0.76-0.85) obtained in

the West African standardization of this test (Schwartz,

1964b).

The correlations between the three scores are given in

Table 6.3. The covariance matrices were significantly dif­

ferent for the two sexes (Box's test, chi-square = 20.69, d.f. = 6, p < 0.01), the correlations being higher for boys

than girls. The pattern was nevertheless similar in both

sexes: The highest correlation was between 3DD and BOX,

there was a slightly lower one between HST and 3DD, and a much lower one between BOX and HST. 273

6.33 Method of analysis

Means and standard deviations of scores on the three spatial tests are given in Tables C.9-C.11.

Analysis of the spatial test scores was already compli­ cated by the vacant cells for village high and technical high schools, but it was further confused by the discovery of factors confounded with the city-town contrast in high and technical high schools.

Table 6.3

Correlations between spatial test scores, by sex^

Test 3DD HST BOX

3DD 52 58 HST 47 39 BOX 51 25

^Decimal points omitted. Correlations above the diagonal refer to boys (N = 277), below the diagonal to girls (N = 352); P(r ^ 0.15) 0 .01.

There are only four mixed high schools in Kingston, com­ pared with five boys' schools and eight girls' schools. When a child takes the 11+ examination, parents have to choose which two high schools they would prefer him or her to attend, but priority is given to the candidates with the highest 274

marks. As the single sex schools are for historical reasons

the most prestigious, the mixed high schools in Kingston

tend to get the poorer students. This is confirmed by

their later examination results (Table B.l). Single-sex

schools are also preferred in the country areas, but the

resulting differences are much less marked (Table C.I2), if

only because of the greater distances between the schools.

Thus, the necessity for testing students in mixed schools

meant that although the two town high schools included in

the survey were probably representative of all town high

schools, the two Kingston high schools were distinctly below

average. There was also the fact that both the town high

schools were using a mathematics textbook which incorporates

practical geometrical activities specifically designed to

improve spatial ability (Mitchelmore et al., 1970-75), where­

as neither of the city high schools was using such a course.

As a result, differences between the town and city high

schools surveyed could not be attributed to environmental differences.

The relation between city and town technical high

schools was confounded by the fact that most of the students

in the town schools had transferred from Grade 8 in all-age 275 schools and were in their second year, whereas most of the city students had transferred from Grade 9 in junior secondary schools and were in their first year. Also, the

Kingston students had started to specialize (in mechanical engineering or commerce in both of the classes tested) where­ as the country students had not.

It was therefore decided that the school type comparison should be made by combining the four town and city schools of each type and analyzing without regard to environment.

The environmental comparison and an estimate of the type- environment interaction would be obtained by an analysis of the results of the junior secondary and all-age schools in the three environments; in any case, it was better to exclude the high and technical high schools from the environmental comparison because they draw their pupils from a much wider area than the junior secondary and all-age schools. Since the two analyses were not strictly orthogonal, it was not appropriate to use a single pooled error estimate; instead, it was decided to carry out each analysis independently.

6.34 School type comparison

The following analysis is based on the scores of students in the sixteen city and town schools surveyed. It 276 is a two-way analysis by sex and school type, with four schools nested within each type.

The cell means are reported in Table C.13 and illus­ trated in Fig. 6.3. (Fig. 6.3 also shows the mean scores of a teacher's college sample; see Section 7.52 for a discus­ sion of these scores.) A multivariate analysis of variance is shown in Table C.14. The sex difference was highly significant (P< 0.001) for all tests, and the type effect was also significant (P< 0.001 for 3DD and HST, P <0.01 for

BOX). The sex-type interaction was significant only for

3DD (P<0.01). Combining the two sexes for each type of school, it was found using Tukey's post hoc procedure that:

1. high school students scored significantly higher (P<0.01) than students from each other type of school on all tests;

2. technical high school students scored significantly higher (P<0.01) than junior secondary and than all-age school students on all tests; and

3. junior secondary students scored signifi­ cantly higher than all-age school students only on 3DD (P<0.05).

The superiority of junior secondary over all-age school students on 3DD was presumably the result of the boys' having studied technical drawing. The significant sex- type interaction on 3DD appears to be due to the 277

Three-Dimensional Drawing test

Boys 30 Mean score Girls 23 + Men

20 Women -o 15

Hidden Shapes Test

10 Mean score + 8 o

6

Boxes test

25 Mean score 20

15

10

l - g Ta h r ’ihTechnical Junior All-age Teachers’High schools high secondary schools college schools schools

Fig, 6.5 Mean spatial test scores in city and town schools, by sex and school type. Crosses refer to males and circles to females. The lines connect data points from the Grade Nine survey; the unattached data points are from the spatial training experiment (Chapter 7). 278

greater superiority of boys over girls in the technical

high schools, another possible result of boys' extra

drawing experience.

These significant differences suggest that most of the

students of high spatial ability are in the high and

technical high schools; however, there may still be consid­

erable numbers of high-ability students in the junior

secondary and all-age schools. To examine this question

further, results of BOX (the 3-dimensional test which seemed

less susceptible to specific experience) were examined in more detail. The items on this test are of two types :

Half the items involve a single fold and are relatively

easy, the other half involve multiple folds and are much more difficult. A score of 50% can therefore be regarded

as showing a minimal level of ability to imagine transforma­

tions in space. A student who scored over 50% was called

a good visualizer. About 22% of the entire sample were good visualizers.

Table 6.4 shows the percentages of the boys and girls

in each type of school who were good visualizers. (Since

the analysis in the next section showed no significant dif­

ference by environment on BOX, village junior secondary and 279

Table 6.4

Percentages of students scoring over 50% on BOX test. by sex and school type

High Technical Junior All-age Sex schools high secondary schools Boys 63.0 40.3 23.1 16.9 Girls 26.7 14.5 7.5 11.0

all-age schools have been included in the analysis of number of good visualizers.) The results suggest that technical high schools may have a smaller percentage of good visualizers than they would like (yet all the boys were taking several industrial arts courses), and that junior secondary and even all-age schools still have a substantial number of students who would probably benefit from a more extended technical education.

Table 6.4 also shows that, although boys were clearly superior to girls, there was still a large number of girls who were good visualizers. In fact, because there were more girls than boys in this sample(as in the school population in general), there were even more female good visualizers than Table 6.4 suggests: No less than 36% of the good visualizers were girls. Yet only one girl out of the 62 280 tested in the four technical high schools, and none out of the 120 in the six junior secondary schools, was taking any technical subject apart from mathematics and general science.

It is interesting to compare the mean BOX scores with those obtained during the West African standardization of this test (Schwarz, 1954a). In 1962, samples of Liberian and Nigerian students with three or four years of secondary education obtained mean scores of 21.5 (boys) and 12.6

(girls). The Nigerians would have been selected for secondary education by an examination similar to the

Jamaican 11+, but it is not known how the Liberians were selected. Jamaican high school means were 25.0 and 18.0 respectively— considerably higher than the West African norms. However, it is quite likely that, after twelve years of industrial and educational development, West Africans would also exceed those norms today.

6.35 Environmental comparison

The following analysis is based on data from the twelve junior secondary and all-age schools surveyed. It is a three-way analysis by sex, environment and school type, with two schools nested within each combination of environment and type. 281

The previous analysis by school type was designed to

study differences between schools as they exist now, whereas

the environmental analysis was more concerned with forces

which had shaped a personfe life in the past. It was there­

fore decided to refine the data for the environmental compari­

son by eliminating students who had not spent most of their

lives in the region surrounding the school in an environment

similar to that of the school. At the same time, other

sources of variation were reduced by eliminating students

who were outside the normal age range for the grade, who

had less than the normal number of years of education (which

in this case usually meant that they had not attended school

regularly), or who had transferred to the school within the past year. Of the 384 junior secondary and all-age school

students tested, 127 were thus eliminated as untypical; details are given in Table C.15. All the students included in the analysis were aged between 13:11 and 15:10; had had

8 , 9 or 10 years of education; had spent at least two years in their present school; and had lived most of their lives within 10 km of the school (in Kingston for students in

Kingston schools, in the town or a nearby village for students in town schools, and in a village for students in village schools). 282

The mean scores for the refined data are given by sex, environment and school type in Table C.16 and illustrated in

Fig. 5.4. The analysis of variance is reported in Table

C.17. As in the previous analysis (Section 6.34), the sex difference was significant for all tests (P •< 0.001) and junior secondary school students were superior to all-age school students only on 3DD (P <1 0.01). In the present analysis, the sex-type interaction was significant for 3DD

(P < 0.01) and HST (P < 0.05), the difference between junior secondary and all-age school students being greater for boys than girls on both tests. The environmental effect was significant only for HST (P < 0.01) and none of the interactions with environment were significant. Com­ bining the sexes and school types within each environment, it was found using Tukey's procedure that city students scored significantly higher on HST than town students

(P < 0.05) and village students (P < 0.01), but town and village students were not significantly different. On 3DD and BOX, the tendency was for city students again to score higher than village students, but slightly lower than town students. 283

Three-Dimensional Drawing test

24 ___ Mean f Junior secondary score 22

20

18 All-age

16 o— -o Junior secondary All-age

Hidden Shapes Test

Mean score

4- Junior secondary

Q— — 5 + All-age —O Junior secondary 4 ^ All-age Boxes test

16 Mean score 14 All-age 12 Junior secondary 10

8 Junior secondary

6 All-age

_L City Town Village

Fig. 6.4 Mesin spatial test scores in refined junior secondary and all-age school data, by sex, environment and school type. Crosses and full lines refer to boys, circles and broken lines to girls. 284

6.36 Influence of background variables

By means of the PDQ, information was collected on the background variables of age, years of education, writing hand, place of upbringing, number of siblings, presence of an adult male in the home during childhood, occupation of parents or guardian, and ambition. For reservations on the use of the writing hand as a measure of handedness and on

Miller's (1967b) rating of social class, see Section 5.51.

Place of upbringing was in this case ranked in a presumed decreasing order of sophistication as follows: Europe or

North America (0), Kingston (1), small city or a village under the urban influence of Kingston (2), town (3), and village (4); definitions are given in Section 6.23.

The analyses of Sections 6.34 and 6.35 were repeated for the background variables. In the city and town school data, the sex difference was significant for length of education

(girls had been at school longer), number of siblings (girls came from larger families) and ambition (fewer girls had a spatial/technical ambition). The school type effect was significant for all background variables except handedness and place of upbringing: As expected, technical high school students were older and had been at school longer than 285

students at other types of school. High school students came from smaller families of higher social class than junior secondary and all-age school students, with technical high school students intermediate. There was a much lower proportion of students with spatial/technical ambitions in all-age schools than in other types of school; it is interesting that the largest proportion of girls with such ambitions was found in technical high schools although only one girl was taking technical courses. Statistics by school type are given in Table 6.5

In the refined junior secondary and all-age school data, the environmental effect was significant for writing hand, number of siblings and social class (as well as place of upbringing). There were more left-handers and smaller families amongst city students, with little differ­ ence between town and village students. The social class difference was probably an artifact of the classification system whereby, for example, small farmers were placed in

Category 5 but small tradesmen in Category 4. Statistics by environment are given in Table 6 .6 . 286

Table 6.5

Summary statistics on background variables in city and town schools, by school type

High Technical Junior All-age Statistic Sex schools high secondary schools

Mean age in years Both 15.14 16.00 15.27 15.00

Mean number of years of Boys 8.85 9.29 8.57 8.18 education Girls 8.90 9.26 9.01 8.55

Percentage of left­ handers Both 6.5 4.7 8.9 7.2

Mean number Boys 4.35 4.89 6.18 6.04 of siblings Girls 4.61 5.49 7.45 6.48

Percentage having adult Both 81.3 68.5 75.6 53.2 male in home

Mean social class Both 3.35 3.84 4.19 4.17

Percentage having spatial/ Boys 63.5 77.3 71.4 34.1 technical Girls 7.0 11.5 6.8 3.0 ambition 287

Table 6.5

Summary statistics on background variables in refined junior secondary and all-age school data, by environment

Statistics City Town Village Mean age in years 15.11 15.10 15.05

Mean number of years of education 8.78 8.94 8.86

Percentage of left-handers 13.9 4. 2 3.8

Mean number of siblings 5.68 7.72 6.76

Percentage having adult male in home 62.5 72.6 68.8

Mean social class 3.81 4.53 4.56

Percentage having spatial/technical ambition Boys 53.3 48.4 43.3 Girls 7.1 4.7 10.0

Statistics on background variables for the entire sample are given in Table C.18.

Correlations between the background variables, exclud­ ing age and length of education because of their narrow range, are given in Table 6.7. All available data from all 288

Table 6.7

Correlations between selected background variables. by sex^

Variable Higher 1 2 3 4 5 6 score

1. Writing hand Right 11 08 03 -06 -03

2. Place of upbringing Village 04 26 03 31 13

3. Number of siblings More 00 30 -13 34 15

4. Adult male in home Absent -00 -04 -04 01 13

5. Social class Lower -01 40 36 07 07

6 . Ambition Verbal/ personal —06 -01 -08 02 05

Decimal points omitted. Correlations above the diagonal refer to boys, below the diagonal to girls. Ns range from 218 to 274 for boys and from 309 to 362 for girls; P( I r):^ 0.13) = 0.05, P(|r|;S0.17) = 0.01.

20 schools were used in the calculation of these correlations,

The larger sample size and wider range of variation make the present results rather more reliable than those obtained in the developmental survey (Section 5.51). The significant correlation of place of upbringing with number of siblings generalizes a result of the analysis of the refined junior 289 secondary and all-age school data. The significant corre­ lation of social class with place of upbringing, and there­ fore that of social class with number of siblings, are probably artifacts of the classification system. Table 5.7 shows that handedness, presence of an adult male, and ambition were virtually unrelated to each other or to place of upbringing, number of siblings, or social class.

Table 6.8

Correlations of background variables with spatial test scores, by sex&

Background 3DD HST BOX variable Boys Writing hand -08 -00 -05 Place of upbringing -14 -19 -12 Number of siblings -18 — 18 -11 Adult male in home -21 -18 -14 Social class -18 -24 -16 Ambition -27 -13 -17

Girls Writing hand 01 04 -03 Place of upbringing -25 -21 -14 Number of siblings -22 -17 -14 Adult male in home -05 -06 -00 Social class -23 -21 -14 Ambition -01 -07 02

^Decimal points omitted. Ns ranged from 221 to 277 for boys and fromI 310 to 362 for girls; P({ r{ ^ 0.14) = 0.05 and P(|r|> 0.18) = 0.01. 290

Correlations of the background variables with the

spatial test scores are shown in Table 6 .8 . Handedness was again unrelated to spatial ability, and the presence

of an adult male tended to be associated with higher test

scores in boys but not girls. A rural upbringing, large

family and low social class all tended to be associated with lower spatial ability. However, all correlations were rather low, which suggests that environmental factors of the type investigated are not important determinants of spatial ability.

Table 6.8 also shows that boys with spatial/technical ambitions tended to have higher spatial ability, but this was not the case for girls. The low correlations for boys suggest that they need more vocational guidance; girls have clearly not been getting any effective guidance at all. CHAPTER SEVEN

THE SPATIAL TRAINING EXPERIMENT

The major purpose of the spatial training experiment was to test the effectiveness of a unit in practical, elementary 3D geometry for improving spatial ability

(research question 4, Section 1.42). The experiment was conducted with two groups of teachers' college students, people who would shortly be in a position to influence the perceptual development of a large number of Jamaican children. Other purposes of the experiment were to study further the interrelations of various facets of spatial ability (research question 1) and the relation of spatial ability to sex, handedness, educational and environmental experience, social class, family size, and family composi­ tion (research question 3) in older students.

7.1 Review of previous research

7.11 Technical programs

Several researchers have investigated the incidental effectiveness of various high school and college courses in drafting, descriptive geometry and similar technical

291 292

courses for improving spatial ability. Most studies suffer

serious design deficiencies and give only the barest

details of the courses taken, but the results are still

worth reviewing.

Faubian et al. (1942) compared two groups of 100

soldiers who had or had not taken a 40-hour course in

drafting, blueprint reading and metalwork. Subjects were

not randomly assigned, but the mean IQs of the two groups

were initially nearly equal. There was no significant dif­

ference between the two groups on Surface Development Test

scores at the end of the 6-week course.

Churchill et al. (1942) administered the same Surface

Development test to 66 soldiers before and after a 9-week,

400-hour drafting course and to 60 soldiers before and

after a Water Purification course of the same length.

The mean score of the first group improved from 71.9 to

93,1, whereas the mean score of the second group increased

from 52.5 to 64.1. Churchill et al. suggest that the

greater gains compared to the Faubian et al. (1942) study were due to the greater length and intensity of the draft­

ing course. They may, however, also have been due to non-

linearity in the test used as a measure of spatial ability; 293 with initially unequal groups, it is impossible to make inferences as to the effectiveness of the training course.

In the largest study of this type so far reported.

Blade and Watson (1955) administered the CEEB Spatial Test

Form 1 to a large number of college and non-engineering students in three institutions at the start of their course, after one year, and again at the end of the four years.

The engineering students had higher initial scores (60 com­ pared to 24 at one college, 56 compared to 39 at another) and showed a greater gain after one year (12 compared to 5 and 6 respectively), but did not show any further gains by the end of the course. Again, the initial non-equivalence of the groups makes inferences risky.

In an interesting post hoc study, Myers (1958) tested

425 midshipmen who had taken mechanical drawing in high school and 425 who had not, and found them not significantly different in general aptitude and spatial ability. However, those who had done mechanical drawing obtained significantly higher scores in an engineering drawing/descriptive geometry course than the other group. There was also a significant interaction: The previous experience of mechanical drawing made a bigger difference for subjects with higher spatial ability than for those with low spatial ability. 294

Mendicino (1958) drew 150 matched pairs from students in vocational and non-vocational high schools» The subjects in the vocational schools took a one-year machine shop/ drafting course, whereas the other group studied a general curriculum not including shop or drafting, but presumably including geometry. Mean scores of the two groups on the

DAT Space Relations test were equivalent at the beginning of the year and again at the end, both groups improving from

34 to 45. (These scores are close to the norms for Grades

9 and 10 respectively; see Bennett et al., 1965.) A simi­ lar result was found for scores on the DAT Mechanical

Reasoning test.

Stallings (1958) tested the effectiveness of a 5-week descriptive geometry course on the spatial ability of 255 college freshmen engineering students using the CEEB Spatial

Relations Tests. Gains of 5 point on Form 1 and 7 points on Form 2 were significant, but in the absence of a control group, no deductions were possible.

The only report of this nature from developing countries is due to McFie (1951). Twenty-six boys aged between 15 and

19 years on entering a technical school in Uganda were given seven IQ subtests at the beginning and end of their 295

2%-year training in the building, carpentry, and automotive trades. Significant gains were recorded for Picture De­

scription, Memory for Designs (P <, 0.01) and Block Designs

( P < 0.05), but not for Comprehension, Similarities,

Arithmetic, or Picture Arrangement. Once again, no control group was used, and the percentage gains recorded were actually well within the range of test-retest increases for control groups in other studies. It also seems likely that a multivariate analysis would show no significant overall gain.

The results quoted in this section are therefore incon­ clusive. Some studies show greater gains for students who take a technical course, but shortcomings in design make it impossible to make any causal deductions.

7.12 Mathematical programs

In this section, investigations of the incidental effectiveness of various mathematics programs will be reviewed.

Ranucci (1952) asked whether the Solid Geometry course then regularly taught in one semester of Grade 12 in U.S. high schools had any effect on spatial ability. It did not.

Cohen (1959) also investigated the Grade 12 Solid Geometry 296

course: Half the sample of 126 students constructed models

to illustrate the theorems and the other half did not, but

there was no difference in space perception scores aL the

end of the semester.

Brown (1954) obtained significant differences in DAT

Space Relations scores for three groups of students in

Grades 10 and 11: The group which studied a one-year inte­

grated plane/solid geometry course gained less than those

who studied only plane geometry for a year, and less than a

third group which completed a two-year sequence of plane

geometry, advanced algebra and solid geometry. It is not

clear what one should deduce from these results 1

Two researchers have investigated the effect of elemen­

tary geometry instruction on spatial ability. Turner (1967)

used crystal models to teach symmetry to students from

Grades 4 and 5 and found that there was significant transfer,

mostly to other tasks involving a plane of symmetry.

Williford (1970) studied a unit on transformational geometry:

Students in Grades 2 and 3 who were taught this 6-hour unit

showed significant gains in geometry achievement, but did

not score significantly higher than the control group on a

composite test of spatial ability. 297

One further report is of interest. Bishop (1973) found that children in the first form of several English secondary schools (approximately Grade 7) who had attended elementary schools where structural apparatus was used ex­ tensively for teaching arithmetic obtained higher scores on

NFER Spatial Test 1 than those who had attended more tradi­ tional schools. Although the analysis was post hoc and the statistical procedure used rather suspect, the fact that the same effect was observed in four independent comparisons suggests that it was real.

The general pattern of the above results is that the study of formal geometry late in secondary school has little effect on spatial ability, whereas more informal experiences in elementary school could have a cumulative beneficial effect.

7.13 Perceptual training programs

Most research on the training of 2D perception in early childhood aims to find effective procedures for im­ proving reading and writing rather than visualization. It will be reviewed here briefly for the sake of completeness.

The training of visual discrimination is related to reading. Covington (1952) found that merely projecting test 298 shapes to kindergarten pupils for short periods each day for a fortnight was sufficient to reduce to insignificance a wide initial social class-related difference in matching ability. Williams (1969) found that tracing and copying were not as effective as discrimination practice in teaching kindergarten children to discriminate letter-like forms, a result also obtained by Bee & Walker (1968).

Improvement in copying is related to writing. Success­ ful training programs are described by Strayer and Ames

(1972) who gave discrimination training to 5-year olds,

Kennegeiter (1968) who trained 3-year olds on the critical elements of shape, and Boguslavskaya (cited by Zaporozhets,

1965) who taught 3- to 7-year olds using modelling activi­ ties .

Marianne Prostig has developed an entire training program of perceptual-motor activities intended to help disadvantaged children learn to read (Frostig & Horne,

1964). Its initial promise has, however, not stood up to closer examination (Walsh & Dangelo, 1971); although students who follow the program gain in perceptual skills and obtain higher reading readiness scores, they do not necessarily do better on later reading tests (Braithwaite,

1972). 299

The above results clearly show that perceptual devel­ opment in early childhood can be accelerated by appropriate training. There is, however, considerable doubt as to whether the gains can be maintained after training and whether they can lead to improvements in related skills.

7.14 Spatial training programs

Programs which are specifically intended to improve 3D visualization or representation will be reviewed next.

Coordination of perspectives Stages in the development of children's ability to visualize a view of a scene from a different viewpoint were summarized in Section 2.42. Eliot

(1950, 1963) tried out several methods of teaching this skill to children in kindergarten and Grades 1 and 3, in­ cluding object assembly, picture identification, and sketching, but no training effect was found. Miller et al.

(1969) tested three training programs with 7-year old children; they found only one of them effective and then only with boys. A later elaboration of this method, in­ volving much equipment and considerable carefully-planned student activity, proved successful with both boys and girls in a small Grade 3 sample (Miller & Miller, 1970). The programs in the above studies all used between 8 and 12 300

half-hour training sessions.

Drawing instruction Several researchers have studied the

results of instruction in drawing, both on the subject's

representation of space and on other aspects of spatial

ability.

In Dubin's method (1946), the teacher discussed the

drawings of 2- to 4-year old children with them in terms of

their present level of spatial representation, trying to

advance the child to the next stage without giving specific

rules. Using matched pairs, a significantly greater im­ provement was found in the experimental group after one quarter's instruction.

Salome (1955) used a "critical feature" method of draw­

ing instruction, training Grade 4 and 5 students to attend

to contours and to salient features on the contour of a

shape. Significant improvements in the representational

quality of drawings were found after only eight lessons.

Kens1er (1965) hypothesized that field independent sub­

jects would be better able to learn perspective drawing than

field dependent subjects. Four classes of Grade 7 students were taught perspective drawing, two of them with "per­ ceptual training to help ^s to see in perspective," for five 301 periods. All four classes, and two other control classes,

were given the PMA and MAT space tests, an embedded figures

test to measure field independence, and a test of drawing

a cuboid from their own direction and from another direction.

Training produced the expected improvement in drawing from

one's own viewpoint but not from other's, and there was no

significant correlation between gain scores on the drawing

test and field independence or gain scores on the space tests.

Rennels (1970) also studied the relation of field inde­ pendence to the ability to learn from drawing instruction.

The study was unique in using inner-city Grade 8 students, but the experimental design and the quality of the research report are so poor that the only conclusion which can be drawn is that one class taught by a discussion method per­ formed better on the posttests than one taught by a lecture method, who were no better than a control class.

Silverman (1962) had the interesting idea of comparing the transfer effects of 2D and 3D art activities. Four matched groups of high school students received either 2D art instruction, 3D art instruction, both, or neither. At the end of one semester, there were no significant differ- 302

ences between the four groups on the MAT Space Relations

test. One more interesting idea goes to the wind I

Two researchers have investigated methods of teaching

orthographic projection. Clark (1971) compared a method

which emphasized the geometric principles of projection with

one which relied on visualization; the former was found more effective with Grade 5 students. Franchek (1971) found

in Grade 7 students no significant differences in effective­

ness between a treatment which dealt only with 3-view projections, a treatment which started with 2-view projec­ tions, and a treatment which started with 1-view projections.

Both researchers rated subjects for visual-haptic aptitude, but found no significant aptitude-treatment interaction.

Visualization training A few researchers have attempted to train spatial visualization directly. Activities similar to those described by Miller and Miller (1970) for training coordination of perspectives are advocated by Petitclerc

(1972) for teaching elementary schoolchildren visualization of solid shapes, but she gives no evaluative data. Con­ trolled studies are reported for one college and two high school studies:

Van Voorhis (1941) gave twelve lessons on the estimation of length, area and angles, parts of figures, rotation. 303 visualization, and perspective to college students of low

spatial ability. Included in his program were 3D noughts

and crosses (tic-tac-toe) and practice with a stereoscope.

Scores on Thurstone's Cards and Figures (tests of spatial orientation) improved significantly compared with a control group.

More recently. Carpenter et al. (1965) developed a programmed instruction package to teach spatial visualization to Grade 8 students. The ten units of this program, each contained in about 50 frames, were as follows:

1. Basic terminology, estimation of length 2. Additional terminology, discrimination training 3. Discrimination between plane figures 4. Integration of parts into a plane pattern 5. Introduction to solids and nets 6. Nets of cubes and pyramids 7. Nets of cones and cylinders; discrimination between solids 8 . Discrimination of cubes from nets 9. Discrimination of solids from nets; effects of rotation 10. Discrimination of solids from nets: projecting self within framework of solid

The program was accompanied by a book of geometric designs and cut-out nets and a box of wooden models, the pattern- folding and solid-object manipulation being intended to provide tactual-kinesthetic as well as visual feedback to the learner. Also prepared as optional extras were a 15 min 304 motivational film on the vocational applications of spatial

ability and an 81-frame filmstrip showing the surface development process in more detail.

The pilot-study is reported by Brinkmann (1966).

Twenty-seven matched pairs (matched on sex and score on DAT

Space Relations) were obtained from two Grade 8 classes in one school. The experimental subjects (all in one class) worked through the programmed package at their own rate while the control subjects continued their regular mathe­ matics program. The experimental group obtained signifi­ cantly higher gain scores on the DAT Space Relations test than the control group. In the experimental group, most of the gains were recorded in the middle half of the ability range (a mean gain of 23 raw score points from an initial mean of 36).

In the main study (Carpenter et al., 1965) various com­ binations of programmed package, film and filmstrip were used in seventeen Grade 8 classes and six Grade 9 classes in three other high schools. Although the hypothesis that the effectiveness of the different combinations would be in the same rank order as their estimated instructional load­ ings was not confirmed, all the combinations that included the programmed package were found more effective (in terms of 305

pooled means adjusted by covariance on initial scores) than

those combinations that did not include the programmed

package. Although once again plagued by design and inter­

pretation problems caused by the non-random assignment of

subjects to treatments, the overall results give impressive

support for the trainability of visualization at the Grade 8

level.

Wolfe (1970) adapted a visualization training program developed by Volker and Ranucci (University of the State of

New York, 1957) for television presentation, and administered the videotaped program to six classes of students, two classes in each of Grades 7, 8 and 9. The ten lessons, which were alternated with student activity periods, dealt with the following topics:

1. Symmetry of plane figures 2. a. Reflections, rotations and translations b. Two-figure plane intersections 3. Partition and assembly of plane figures 4. Transition to solid relationships 5. Cutting and counting solid figures 6. Intersection of planes and solids 7. Vertices, regions, and edges 8. Orthographic projection 9. Rotation of plane and solid figures about an axis 10. Topology

Many of the activities seemed to have been modelled on spatial tests such as Card Rotations (Lesson 2), Form Boards 306

and Hidden Figures (3), Cube Comparisons and Surface Develop­ ment (4), Cube Counting (5), Intersections (6), and Views

(8), and one of the criterion tests consisted of fifty mis­ cellaneous items of the same type. Significant differences in gain scores were obtained on that test, but generally not on the Guilford-Zimmerman Spatial Orientation and Spatial

Visualization tests, whose item types were not included in the training program. It seems safe to conclude that the training was task-specific and no transfer to related spatial tasks occurred.

Developing country studies Only two training studies have been reported from developing countries, both of them con­ cerned with teaching pictorial depth perception. A third report (Duncan et al. 1973) states that the authors have successfully taught pictorial depth perception using specified methods, but gives no data from controlled experi­ ments .

Dawson (1967a) formed two matched groups of 12 Sierra

Leonean males aged about 18 who were unable to detect depth cues in perspective drawings (their mean score was less than

4% on Dawson's test). One group was taught the standard depth cues by a method in which subjects looked at the out­ 307

side scenery through a small hole, copied the dominant

lines on the window pane, and gradually learned to sketch directly onto paper. Photographs were also taken to illustrate depth cues. Three months after the instruction

(six months after the pretest), the experimental group scored 42% compared to the control group's 5^g%, a highly significant difference. Dawson observed also that the rank order of final scores was highly correlated (rho =

0.88, P < 0.001) with pretest scores on Koh's Block Design test, which he interpreted as a measure of field inde­ pendence.

Serpell & Deregowski (1972) found eight subjects in each of four Grade 7 classes of a Zambian primary school who were judged to be 2-dimensional by Deregowski's construction test (Deregowski, 1958a), and gave each class a different treatment for four class periods. One class discussed depth cues in photographs; one class discussed depth cues in films; the third class had two lessons from each of the first two treatments; and the control class discussed only the content of the photographs and films without reference to depth cues. There was a slight suggestion that the first treatment was most effective, but many post hoc analyses had 308

to be performed before a significant difference was found.

The authors ascribed the rather inconclusive results to the

short treatment period and to administrative difficulties in running the experiment.

7.2 Theory and practice of spatial training

7.21 Principles of spatial training

The studies reviewed in Section 7.1 will now be examined with the aim of extracting some general principles to guide the design of instructional programs aimed at im­ proving spatial ability.

Despite all the contradictions and imperfections, the preponderance of the empirical evidence shows that various aspects of spatial ability can be taught, at least to some of the people some of the time. It is a rare study in which the posttest mean is less than the pretest mean, even in the control group; and given the long time-span of some of the studies, the practice effect must be due to spatial learning not memorization or familiarity. It may take a great deal of searching to find an effective teaching technique, and an elaborate program may result, but the point is that most people have not reached the upper limit of their genetic potential in spatial ability. Spatial ability 309

can be taught.

Why are some training programs effective and others

not? For one thing, it is clear that spatial training

must be intentional. None of the studies reviewed above

gave any support to the hypothesis that courses in drafting,

geometry, or art cause significant improvement in spatial

visualization. It may be that these subjects, as they

are usually taught, are loaded with other considerations

such as accuracy, logic, or self-expression, and develop

from a prerequisite level of spatial ability without leading

to any improvement beyond that level. This suggestion is

supported by the observation that students electing such

courses start with higher spatial ability (Blade & Watson,

1955; Churchill et al., 1942; Siemahkowski & MacKnight, 1971;

Silverman, 1962) as well as the fact that those who do

better in these courses also have higher spatial ability

to begin with (see Section 1.23).

There is a suggestion that the teaching method is also

important, even in programs whose main intention is improve­ ment in spatial visualization. The successful treatment

reported by Rennels (1970) and the training method of Dawson

(1967a) both required the subjects to reflect on their 310

Visual experience, to take up a subjective attitude, and to derive general principles which could then be used to make or interpret 2D representations of 3D objects or scenes.

Dubin (1946) used a similar method at an earlier level.

By contrast, the lecture method used by Rennels (1970) pro­ duced no gains over five weeks. Wolfe's program (1970) was also highly directive and specific, factors which could have contributed to its lack of success. Although dis­ covery methods have been rightly criticised in some quarters

(Ausubel, 1968), perceptual development is one field where the subject must come to see things for himself, to coin a phrase. Verbal instruction can be used to teach drawing techniques and even spatial operations, but if the subject cannot relate these to what he sees around him, they will remain isolated skills. Although he can help the subject to carry out this integration, the instructor cannot tell him what to see: Only the subject can change how he per­ ceives the world. This may be why a teaching method in which the subject looks at things and reflects on what he sees is superior to a method in which he merely follows instructions.

All the successful spatial training programs had students active with physical materials. The role of 311

activity in perceptual learning was discussed in Section

2.52. It was concluded that there is little evidence for

any direct influence (for example, through muscular feed­ back) , but that activity generally increases the amount of visual information available to the learner and heightens his attention to relevant properties of the stimulus materials, either of which could account for the superiority of an active over a passive method. A specific way in which activity might be effective in teaching is by giving the student constructive rather than récognitive exercises.

Consider, for example, teaching a subject how a cube is represented in a diagram. If he is asked to draw a cube, and draws it incorrectly, he is often aware that it is not correct, but unable to modify his drawing appropriately.

(This was frequently observed in the developmental survey; see Section 5.34.) By comparing his drawing with a correct drawing, some differences can be detected and related to properties of the cube; at some point, the subject comes to formulate a general principle and his spatial ability (not just his representational ability) improves accordingly.

Simply recognizing a correct drawing of a cube is far easier to teach (see Section 5.34 again) but also far less effective, 312

because the errors in the incorrect representation do not

have to be attended to (Olson, 1970). The greater diffi­

culty and teaching effectiveness of construction as opposed

to recognition is not, of course, confined to perceptual

tasks; writing a dissertation or solving mathematics

problems is much more difficult than reading textbooks or

learning theorems, and the old adage that the best way to

learn something is to teach it is still respected (Bausell

& Moody, 1974).

In summary, a successful spatial training program is

likely to be deliberate, reflective, and constructive.

7.22 The JSP spatial training program

The training unit tested in the experiment to be de­

scribed below is based on material first developed by the

Joint Schools Project (JSP) in Ghana (Mitchelmore, 1971).

The material was based on principles similar to those just

outlined, without, however, the background of the literature

reviews above.

The JSP mathematics series (Mitchelmore & Raynor, 1967-

75) was written by a group of teachers who felt that the mathematics curriculum then current in Ghana was not meeting

the needs of their students. A major problem lay in the 313 geometry curriculum. Students' difficulties, as revealed in public examinations (Section 1.31), convinced the JSP authors that students suffered serious deficiencies in the basic spatial concepts on which high school geometry was built, and that it would be more suitable if the high schools included early in the course some work which might help students develop their geometric intuition before they reached the more abstract work. It was assumed that the deficiencies were caused by lack of the appropriate experi­ ences— exploratory activities and interaction with the physical environment— in early childhood; and it was proposed to overcome these deficiencies by structuring practical 2D and 3D activities for the classroom situation. The teaching method was to be discussion based on discoveries arising out of these activities, followed by practical applications of the results. Thus the spatial training aspect of the JSP course satisfied the three criteria enunciated at the end of the last section: It was deliberately designed to improve spatial ability, students were to be continually constructing figures, and they were to learn by reflecting on the results of their activity.

Most of the spatial training program occurs in the first two books of the JSP series, intended for Grades 7 and 314

8 . A basic tenet in constructing this program was that, since plane shapes are abstracted from the real, physical,

3D world, the proper order of teaching in the absence of an adequate intuitional background is to start with 3D figures and progress to 2D figures. This principle, which is re­ garded as heresy by many mathematicians and mathematics educators, meant a complete reversal of the traditional sequence in which 3D geometry is delayed to the last possible moment. There are four components to the program:

1. Model-making Students make models of the basic solids (cube, cuboid, pyramid, prism, cylinder, cone) by designing their nets and cutting them out of cardboard.

After ensuring that students know the basic shapes, and giving them plenty of help in making a cube, students are then thrown on their own resources. Feedback from this constructive activity not only draws attention to the 2D shapes of the faces and the 3D relations between them, it also motivates accuracy in the use of drawing instruments.

For example, consider how a student makes a model of a right square pyramid. From a model or drawing, he sees that the sloping faces are triangles and that there is one triangle adjoining each side of the base. Suppose he then sees that the net consists of a square with four triangles: 315

He now has to decide what size to draw the triangles. Per­ haps he can work out that each must be isosceles, but omits to draw them congruent; he will find out his mistake when he tries to assemble the model. The actual process of trying to stick unequal edges together demonstrates an important relation between adjacent faces of a solid figure in a way which is more likely to be remembered than a statement or a demonstration coming from someone else. Of course, if the student can already visualize how the faces of a pyramid are related to each other, the exercise will have little value (unless his model does not fit well, in which case he might learn to be more careful with his draw­ ing) ; if he cannot, he will be helped to acquire the ability while at the same time learning a little more about triangles.

2. Plane metric geometry The first year of the JSP course contains many activities in drawing and measuring: length, angle, area, patterns, vectors, scale drawing, surveying, and symmetry.

A large part of this activity consists simply of getting ex­ perience with shapes and their more obvious properties.

3. Transformation geometry From the symmetry of solids, the JSP program progresses to symmetry of plane figures and repeating patterns and then to translation, reflection and rotation. These transformations are then used in the study of the properties of the common polygons. 316

Spatial ability was characterized in Chapter 1 as the visualization of the results of a transformation, and per­ ceptual development in Chapter 2 as the formation of a progressively more complex structure of transformation groups. Transformation geometry, taught by a constructive- reflective method in a sequence which compresses the natural order of development, may therefore be expected to be partic­ ularly effective in developing spatial ability.

4. 3D drawing In the second year, the drawing of a cuboid— which provides the rectangular frame­ work fundamental to the mathematical analysis of 3D space— is taught explicitly. Both ortho­ graphic and central projections are taught.

Students are guided to adopt a subjective attitude of view­ ing objects and to discover and apply the basic properties of faithful representation. The ability to sketch simple solids is useful in many areas of school studies.

The JSP course received much formative evaluation during its development, and extensive revisions were made on the basis of feedback from teachers who used the preliminary and trial editions. Although the course has subsequently been extensively adopted in schools and colleges in Africa and the Caribbean (Mitchelmore et al., 1970-75), it has received little summative evaluation (Mitchelmore, 1973a). 317

The experiment to be described was therefore seen as of wide

interest.

7.3 Method

There was no opportunity to arrange an experimental test of the effectiveness of the JSP spatial training program amongst students in Grades 7 and 8 of high school, the level for which it was designed. However, the principal of Mico College, Kingston, kindly agreed to allow the writer to conduct a test of part of the program in conjunction with the college's on-going program of curriculum develop­ ment. As will be seen below, the student teachers who thus made up the experimental sample were at approximately the same level of geometrical knowledge and spatial ability as the original target population. The following sections de­ scribe the procedures used in executing the experiment.

7.31 The setting

Mico College is the oldest and most prestigious of the teachers' colleges in Jamaica, and has recently expanded con­ siderably to supply the needs of the burgeoning school population. Of the 1973-74 enrollment of 650, 210 were junior secondary students (training to be junior secondary 318

school teachers) and 440 were primary students (prospective

primary school teachers); there were approximately equal

numbers of men and women in each group. The junior

secondary students specialize in two teaching subjects in

addition to English, Education and one elective subject; they

are also required to reach a minimum competency level in mathematics. The primary students, on the other hand, all

study a core curriculum of Mathematics, Science, English,

Social Studies and Education, to which a few electives may be added. All students spend two years in college,

followed by a one-year internship in a school. In view of the small and varied nature of the mathematics classes taken by the junior secondary students, the experimental population was defined as all the prospective primary school teachers in the college.

The impression gained from local educators (no firm data are available) is that no form of geometrical activity occurs in most primary schools, and that very few of the geometrical topics recommended for the junior secondary schools are actually taught. As a result, geometry has traditionally had no place in teacher education, and the students have remained as ignorant at the end of the course 319 as they were at the beginning. Recognizing this problem, the curriculum at Mico College was recently changed so that all junior secondary students would study some e],emen- tary geometry, including model-making activities similar to those provided in the JSP course. Furthermore, the college had also adopted the JSP course (Books 1-3) as the textbook for the primary students entering in September 1973, so they too would study some elementary geometry. This combin­ ation of fortunate circumstances meant that the JSP materials could be used for the experiment with only minor adaptations; that both first and second year primary students could participate in the experiment; and that the mathematics tutors were ready, willing and able to teach the experi­ mental program.

7.32 Development of instructional materials

Students enter teachers' colleges with a varied back­ ground in which the only common elements are a primary education and sufficiently high marks on the national

Teachers' College Entrance Examination. Most would have attended one or more of the different types of secondary school (Section 3.23), but many would also or only have attended evening classes to reach the required standard. 320

The college had recently determined to rearrange its offer­

ings to take greater account of this variety. As an interim

measure, the program for each subject area had been divided

into semester courses, some of which could be omitted by

the brighter students, and streaming (homogeneous grouping)

had been introduced for the first year students. The long

term aim was to break each course down further into modules

and to allow students to work through these at their own pace; college graduation would then depend on satisfactory

completion of a specified number of modules rather than residence for a specified period.

Individualization had also been promoted by UWl/UNESCO/

UNICEF/UNDP Project RLA/142 which was, through workshops attended by teachers' college tutors from several West Indian countries, developing "unit packages" in several curriculum area, including mathematics, for use with intending junior secondary school teachers (Collins, 1973). Besides workcards to guide students' activities, each unit package was to con­ tain all the textual, manipulative, audiovisual, and evalu­ ative materials needed for that unit.

It was therefore decided that the JSP materials should be adapted for the experiment by writing workcards and allow­ ing students to work individually to progress at their own 321 rate. As no previous attempts to do this at the college level in Jamaica were known, the experiment therefore became a test of individualization as much as of spatial training.

As a precaution, a pilot-test was planned in order to familiarize both students and tutors with the use of work­ cards and to uncover problems which could invalidate the main experiment. It was also decided to run separate experi­ ments with the first and second year students at different times in order to allow time for revision of the workcards or classroom procedures between one experiment and the other.

The author met with the five mathematics tutors at the college for two hours weekly between September and November

1973. After extensive discussions of the aims of the experi­ ment and the needs of the students, two teams were formed.

The author and two college tutors wrote the workcards for the experimental unit and a pilot unit, and the other three tutors developed a control unit on statistics and wrote the workcards for it. The workcards followed the style used by the UNESCO project mentioned above, each card giving the objectives, activities and evaluation for one topic or theme.

In the experimental unit, students needed to be able to use a ruler, set square and compasses to construct nets 322 of solids accurately. Since tutors' previous experience was that most students lacked these skills, it was decided that the pilot unit should be on the use of geometrical instruments. The workcards, text pages, and workbook pages for this unit are given in Appendix K.

The experimental training program, called the Solids unit, was formed by merging parts of the two 3D components of the JSP spatial training program (Section 7.22). After an initial discussion on the shapes present in the environ­ ment, students make model cubes and cuboids and learn how to make an adequate sketch of a cuboid by drawing parallels in three inclined directions. They then make models of simple pyramids and prisms and a cone and a cylinder and practice sketching each figure. Application exercises cover elementary properties of the solids, such as patterns in the numbers of vertices, edges and faces or the spatial relations between them. The workcards, text pages, and workbook pages for the Solids unit are given in Appendix L, which also includes the achievement test used for the final evaluation.

The topic for the control program. Statistics, was chosen to match the experimental program for novelty.

Statistics was another area of mathematics which the college 323

tutors wished to introduce into the curriculum for both

junior secondary and primary students, and which neither

first year nor second year primary students had so far

studied. This unit was also constructed by merging two

sections of the JSP course, one on measures of central ten­

dency and one on graphical representation; some original

manipulative exercises illustrating the use of an assumed

mean were added. The workcards, text pages, and workbook

pages for the Statistics unit are given in Appendix M, as

is the achievement test used with the unit.

Only first year students had the JSP textbooks. For

second year students, relevant textbook pages were dupli­

cated. Workbook pages were duplicated for all students.

7.33 Pilot-testing

The pilot unit on Use of Instruments (Appendix K) was

studied by all first and second year primary students in

the last 2^2 weeks of Christmas term (Fall quarter) 1973.

It was found that the material was of the appropriate

level and that the use of workcards, although accompanied by

some difficulties, was perfectly feasible. The following

points arose from direct observation and from discussion with the tutors. 324

Performance Progress was initially very slow, due partly

to students" unfamiliarity with the use of workcards and

partly to ignorance of the most basic geometric conventions

such as how to name and label points and lines. It had been

assumed that students would at least have learned basic

geometrical notation and terminology in school (the sections

on this in the JSP book had therefore been omitted), but

clearly this was not the case.

By the end of two weeks (8 periods), all students had

reached Cards 4, 5 or 5, with no noticeable difference

between the three ability groups in the first year. The

optional cards 4A and 5A, which had been included to accom­

modate those students who finished far ahead of the remainder,

were not needed for that purpose. Card 6A (activities 1 & 2) was set in the last half-week and, along with evaluation

item 2 on card 4, used to obtain a student grade for college

records. Performance was regarded as satisfactory.

Reading difficulties Many students clearly had difficulty

reading and following instructions. Students frequently

started on a workbook exercise without reading the assigned

textual material, did the exercise poorly, and had to be

sent back to do it again. This was so frustrating at first 325

that some tutors took to checking that students had read

the text material before allowing them to start on the

workbook exercise. Some comprehension difficulties were

observed (most of them due to previously unnoticed subtle­

ties in the textbook), but the major problem appeared to be

that students did not expect to find instructions in print

and were not prepared to act on them. The same problem was

observed during the development of group tests (Chapter 4),

where it caused a switch from printed to oral administra­

tion.

It was expected that students' ability to follow printed instructions would improve with practice. If this expecta­

tion is realized, then the introduction of the workcard method could bring educational benefits beyond those origin­

ally intended.

Independence Possibly related to the difficulties in follow­ ing printed instructions was an initial reluctance to work independently. At first, students brought every single answer to the tutor for checking instead of waiting till the whole assignment was complete. Answers were frequently wild and ridiculous; it appeared that students were trying to satisfy the tutor instead of satisfying themselves of the correctness of the response. An instruction to make a conjecture about 326 three points (they were collinear) and then check it, was completely lost because the ideas of guessing and checking were both foreign to most students.

A further symptom of student dependence was their reluctance to work independently of each other. At first, conjectures were shared around the class, destroying the aim of getting students to think for themselves, and students waited for their friends to finish each card, voiding the aim of getting students to work at their own pace. It may be that complete independence is a false goal, not only be­ cause the students in this type of sample are not used to working individually but because sharing of ideas is both educationally and socially desirable. It was therefore decided to allow students to help each other in reading textual materials, explaining ideas, practising techniques, and exploratory activities, but to ban sharing of answers and discourage complete dependence on one partner.

The growth of both independence and responsible, inter­ action is another result of individualized instruction which could have widespread educational effects. 327

Efficiency Since mathematics classes were held at four

different times, an attempt was made to economize on paper

by printing only about 150 copies of each workcard. As a

result, tutors frequently had to visit the next classroom to borrow cards or the office of the Head of the Mathematics

Department to obtain new ones, and students could not be allowed to take cards home to complete assignments in their own time. For the main experiment, it was therefore decided to supply each tutor with one copy of each workcard for each of his students and to pack all the workcards for each unit in a filing box.

A more difficult administrative problem to solve was the matter of evaluating students' work on each card. There was frequently a long queue of students, especially at the start of a period, waiting for the tutor to check their work before going onto the next card; as a result, quite a lot of class time was wasted and the tutor was unable to give all students the attention they may have needed. The problem was only eased when most students were on Card 5, where the activities took longer and could be checked very quickly. By the nature of the material in the Use of Instru­ ments unit, it was not possible to write answer sheets to enable students to check their own work; and although students 328

had been expected to do a certain amount of checking by

looking critically at their finished drawings, they usually

did not. When students come to work on several different

units in the same classroom, it is possible that most

students would be able to check their work from answer

sheets, leaving the tutor more time for tutoring.

Summary Despite the difficulties noted, it was considered by all concerned that use of workcards was potentially

superior to teacher-led instruction. Even with the delays and frustrations experienced in the pilot test, more students were active during mathematics lessons, since sitting back and barely listening had become impossible. And even if students did not progress as quickly as they might have under the previous system, there was growth in self- reliance, independence and integrity which could be far more valuable to a potential teacher than the mathematics he or she learned. It was therefore agreed to carry out the main experiment using workcards as in the pilot test.

7.34 Experimental designs

The division of the primary students for mathematics classes was different in the first and second years, neces- 329

sitating different designs and independent analyses.

First year The first year primary students were divided

into two groups, lA and IB, depending on their choice of

elective subjects. For mathematics, each group was divided

into three classes (A: advanced, N: normal, and S; slow) on

the basis of their scores on a test administered at the beginning of the year. All three classes in each group had mathematics lessons at the same time; lA had one 85-minute period and two 40-minute periods, whilst IB had two 85- minute periods. Neither group had more than one period in any of the three segments of the day (periods 1-3, 4-6, and

7-9, separated by morning and lunch breaks). There was thus no reason to expect the two groups to be different, but as a partial control each unit was studied by two classes from one group and one class from the other group. The six classes were taught by four tutors, to be called A, B, C and

D, who had to stay with their classes; as a partial control on the teacher factor, no tutor was given the same unit in two classes. Under these two restrictions, the two classes at each level (A, N and S) were assigned to treatments at random. The resulting assignment is shown in Table 7.1. 330

Table 7.1

Assignments of first year classes and tutors to treatments

Solids unit Statistics unit Level Group Tutor Group Tutor A lA C IB B N IB D lA D S lA B IB A

The major dependent variables were posttest scores on

the 3DD and BOX tests. Partly for reasons of intrinsic

interest and partly to reduce the problems of administering

these two tests to a hundred students at a time, a Solomon

Four Group design (Campbell & Stanley, 1956) was adopted

for 3DD and BOX. The analysis was therefore four-way; the

design is shown in Table 7.2, which also gives the maximum

number of students available for each cell. All factors

were fixed.

Other variables were scores on HST (which was given to

all students as a pretest for possible use as a covariate,

and again as a posttest to obtain a test-retest reliability

estimate and to check its sensitivity to spatial training)

and scores on the Solids and Statistics achievement tests

(which were given to all students as posttests to find how well the content of each unit had been learnt). 331

Table 7.2

Research design for first year experiment and numbers of students available in each cell

Solids unit Statistics unit Level Sex Pretest No pretest Pretest No pretest A M 6 6 8 9 F 8 8 6 6

NM 9 9 8 8 F 8 8 9 9

SM 6 5 3 3 F 12 12 11 11

Second year The second year primary students were also divided into two groups (2B and 2C) according to their choice of electives. However, there was no grouping by ability, each group being divided by alphabetical order into four classes. All four classes in each group had their mathe­ matics lessons at the same time, each group having two 85- minute periods which fell into different segments of the day.

The eight classes were taught by the same four tutors as taught the first year, each tutor teaching one class from each group.

A Solomon Four Group design was also adopted for 3DD and BOX in the second year experiment. The added control on group and tutor effects gave the five-way incomplete block 332 design shown in Table 7.3. The tutor effect was random, the others fixed.

Table 7.3

Research design for second year experiment and numbers of students available in each cell

Solids unit Statistics unit Tutor Group Sex Pretest No pretest Pretest No pretest

2B M 7 7 F 6 6 A 2C M 6 6 P 7 7

2B M 8 8 F 8 8 B 2C M 6 5 F 7 7

2B M 7 7 F 5 5 C 2C M 6 6 F 6 6

2B M 6 6 F 6 6 D 2C M 6 6 F 8 8 333

HST was given as a pretest (but not as a posttest) to all second year students, and the two achievement tests were given as posttests.

7.35 Hypotheses

The major hypothesis in both years was as follows:

Hj^; Students who study the Solids unit gain in 3D drawing and visualization ability.

This hypothesis was tested by comparing posttest scores on

3DD and BOX of students who had just studied the Solids unit with those who had just studied the Statistics unit.

This assumes that studying the Statistics unit has no effect on spatial ability, a reasonable assumption since the only spatial content is a little graph-drawing.

A second important hypothesis is the following:

H^: Student teachers can learn mathematics by the workcard method.

This hypothesis was tested by comparing scores of the two treatment groups on the two achievement tests. It was expected that the group which had just studied the Solids unit would score higher on the Solids achievement test and lower on the Statistics test than the group which had just studied the Statistics unit. 334

Two minor hypotheses were suggested by the literature;

Field independence is not affected by spatial training.

H : Gains in spatial ability due to training are not linearly related to initial levels of spatial ability.

Teachers who have used the JSP course with high school students have frequently observed that girls appeared to benefit more than boys from the practical activities in 3D geometry^ This suggested the following hypothesis for the present experiment:

Hg: Women gain more in spatial ability as a result of studying the Solids unit than men do.

These and several further general hypotheses and questions are tested in Section 7.4.

7.36 Procedures

Pretesting Pretests were administered to all first and second year primary students during the first four mathe­ matics periods scheduled for each group (lA, IB, 2B and 2C) in the first week of Easter Term, commencing 7 January 1974.

In the first period, the Head of the Mathematics Department introduced the writer to the students, gave out an informa­ tion sheet (Appendix N), explained the purpose and proce­ dures for the experiment, and asked for students' cooperation. 335

Copies of the PDQ were then given out and completed under the writer's direction. (The form used in the teachers' college was similar to that shown in Appendix F, except that, for obvious reasons, students were not asked to state their ambition.) For two groups, the initial orientation was carried out in their regular classrooms with the writer hopping from one room to the next; for the other two groups, a single large room was used.

In the second period for each group, HST was adminis­ tered in a single large room to all the students in that group. At the end of the period, the students who had been selected (at random) to take the 2DD and BOX pretests were so informed ; they took these tests during the third and fourth periods, the other students being released from classes at that time.

Individual testing After the pretesting, 24 students were selected at random from the largest of the cells in each year and asked to take three of the individual tests used in the developmental survey (HFT, SRT and GIM). The 16 who actually attended were from the classes shown in Table 7.4; their data were subsequently eliminated in the analysis of the training experiment. (Most of the students who did not 336

attend for individual testing were second year students who were busy preparing for teaching practice.) Testing was

completed before any subject had started on the Solids unit.

Table 7.4

Sources of students for individual testing

First year Secondyear Sex A N SAB C D lA IB lA IB lA IB 2B 2C 2B 2C 2B 2C 2B 2C M 2 1 2 1 1 F 1 1 2 2 1 1 1

First year experiment First year students started on the

Solids and Statistics units in the second week of term.

For each class, tutors were given a file box containing the workcards for the appropriate unit, and a sheet for record­ ing when each student completed each workcard. For the

Solids unit, each student was given one sheet of card 20 cm

X 50 cm, and paste and scissors were made available in the classroom. Apparatus for the Statistics experiments was made available as and when needed.

It was originally intended to allow 3 weeks (12 periods) for each unit. However, students progressed much more slowly 337 than expected, so that by the middle of the third week, most of the students studying the Solids unit had only reached

Card 5 and most of those studying the Statistics unit were on Card 4. It was therefore decided to extend the treatment period to four weeks to enable most students to complete the

Solids unit up to Card 7; some further activities (on dis­ persion) were written for students who completed Card 5 of the

Statistics unit before the end of the extra week.

At the end of the four weeks allotted to the units, all students took 3DD, BOX, HST, the two achievement tests, and a questionnaire on teaching methods (TMQ). The two achievement tests were given first, by the tutors in their regular class­ rooms. In order to avoid any interference caused by reaction to taking a test on material not yet covered (an unusual situ­ ation for these students), each class took the test on the unit just covered first. Each test took about one period.

The writer administered the 3DD and BOX next, in one large room. The full instructions were used, since half of each group had not taken the tests at the start of the term.

HST was administered in the fifth period allocated to post­ testing; since all students had taken this test previously, instructions were abbreviated and the test given by tutors in their classrooms (see Appendix N). The TMQ (also in 338

Appendix N) was given out at the end of this period, to be completed at home and returned the next day.

In order to ensure that all students would cover the same material during the year, the classes which had studied the Solids unit then studied the Statistics unit and vice versa. It was not considered appropriate to take up yet more time by readministering all the posttests, although they would have given useful information on retention, so at the end of the second unit students took only the achievement test for that unit. Apart from the single purpose of obtain­ ing estimates of test-retest reliabilities for the achievement tests (Section 7.42), the results of studying the second unit will be ignored in the analysis below.

Details of the first year experimental timetable are given in Table D.l.

Second year experiment Because of intervening teaching practice, the second year students did not start studying the

Solids and Statistics units until the eighth week of term.

To save time, only three weeks (12 periods) were allotted to each unit; this was feasible because of the greater maturity of the second year students and the experience gained by tutors in running the first year experiment. The only sub­ 339 stantive change in the units was the addition of one card to each unit summarizing the unit objectives; this was intended to help students in their revision for the final achievement test on that unit. (Students were not given the objectives of the unit which they had not studied.)

Because of the larger size of the second year groups, it was necessary to split them for the 3DD and BOX posttests.

In the first two periods for each group, the two classes which had just studied the Statistics unit took the posttests while the two classes which had just studied the Solids unit started on the Statistics unit; in the next two periods, the two classes which had just studied the Statistics unit started on the Solids unit while the other two classes took the posttests. Thus the experimental (Solids) group had two periods of the control treatment (Statistics) in addition to the experimental treatment, whereas the control group had only the control treatment; although this was somewhat unsat­ isfactory, this order was preferable to the reverse order in which the control group would have experienced two periods of the experimental treatment.

As in the first year, the achievement tests were given before 3DD and BOX; but to save time, no HST posttest was given to second year students. It was still necessary to run 340 over into the next term before all students had studied both units. The TMQ was administered in class at the end of the experiment.

Details of the second year experimental timetable are given in Table D.2.

Experimental conditions Testing conditions were not ideal.

It was usually impossible to determine until the last moment which room would be free to use for testing, so there was no time and frequently no space to arrange desks with adequate spacing. In many cases, no desks were available and arm-rests or knees had to be used instead. First year students were most cooperative, but second years less so

(it was suggested that they were impatient to proceed with preparations for teaching practice); however, although occasional stern words had to be used to get them to stop writing after time had been called, testing was completed satisfactorily.

Several students attended the pretest sesssions although their names had not been called, apparently thinking they would have a second change of "passing" the test. There was some absenteeism for the spatial posttests, especially amongst the second year students, but most students took 341

the achievement tests (which they know were to be used for

college grades). Except for the 3DD test, the procedures devised for Grade 9 were probably too lengthy; however, they were followed precisely in order to allow post hoc comparisons between college and school students.

Several tests were given eight times (twice to each of the two groups in the two years) but students appeared to have heeded the request not to discuss the contents with their colleagues. For example, tutors reported that no students showed any signs of recognition when they took the achievement test at the end of their second unit, even though they had taken the same test only four weeks before; also, class mean scores obtained by the second group to study each unit were not radically different from the scores obtained by the first group (no tests of the significance of the difference were made). These results give confidence that students also did not discuss the contents of the ex­ perimental and control units with colleagues in other classes.

There were only a few hitches in the administration of the units. Material for the statistics experiments was occasionally late in arriving, and second year students complained of having to use poorly duplicated sheets instead 342

of the textbook. The hold-ups while tutors checked student

work were still common, as were many of the other difficul­

ties mentioned in Section 7.33, but they were judged to be

far less severe.

All in all, it was felt that the experimental conditions

allowed a fair test of the hypotheses at issue.

7.4 Results

7.41 Numbers

The numbers of students who provided complete data on

the appropriate spatial tests (HST pretest and all three posttests in the first year, HST pretest and 3DD and BOX posttests in the second year) and the Solids achievement test, excluding those who were given the individual tests,

are reported in Tables 7.5 and 7.5. Most of the subsequent

analysis is based on the data from these 167 first year

and 148 second year students.

The attrition rates were about 16% in the first year

and 30% in the second year. Simple comparisons based on

HST pretest scores (Table D.3) indicate that, except for the small number of first year men, the students with missing data were not significantly different from those 343

Table 7.5

Numbers of first year students with full data, by level, sex, treatment and pretest condition

Solids unit Statistics unit Level Sex Pretest No pretest Pretest No pretest A M 5 6 7 9 F 7 7 6 5

M 6 9 11 4 N F 8 8 6 8

M 5 7 3 4 S F 10 9 7 10

Table 7.5

Numbers of second year students with full data, by tutor, sex, treatment, and pretest condition

Solids unit Statistics unit Tutor S Pretest No pretest Pretest No pretest

M 5 6 4 6 A F 1 5 6 3

g M 6 7 6 4 F 6 5 6 3

Q M 6 5 7 2 F 1 4 6 4

g M 5 2 7 2 F 5 5 4 4 344

with full data. It was therefore assumed that attrition

introduced negligible bias into the results.

Background characteristics of the students are dis­

cussed in Section 7.51.

7.42 Test characteristics

Frequency distributions of the three pretests are given in Table D.5. The 3DD and BOX distributions were symmetrical, as in the Grade Nine survey (Table C.5), and the HST dis­ tribution was also symmetrical in this sample. Item analyses for the 3DD pretests are presented in Tables D.7 and

D.8 ; item statistics were again quite similar to those ob­ tained in the Grade Nine survey (Table C.7).

Correlations between the test scores in the control classes (those which studied the Statistics unit) are given in Tables D .4 and D.5. Combining results for men and women, these give the test-retest reliability estimates shown in

Table 7.7. Other reliability estimates were calculated and are also shown. Most of the reliabilities are satisfactory, being only slightly lower than those obtained in the Grade

Nine survey (Section 6.32). The HST reliability was much

lower, but it may be that, because of the different methods 345 of administration of the HST pretest and posttest (see

Section 7.35), 0.51 is a gross underestimate of the relia­ bility of this test.

Table 7.7

Reliabilities of spatial test scores, by year

First year Second year Estimate Test N r N r

Test-retest 3DD 65 0.73 51 0.83 (control Poles 0.65 0.62 classes Bottles 0.48 0.42 only) Cubes 0.69 0.61 Edges 0.87 0.86 HST 127 0.51 - - BOX 65 0.75 52 0.74

Cronbach 3DD pretest 99 0.79 120 0.79 alpha Poles 0.47 0.37 (all Bottles 0.44 0.54 classes) Cubes 0.78 0.68 Edges 0.75 0.79 3DD posttest 191 0.81 167 0.80 Poles 0.31 0.43 Bottles 0.50 0.50 Cubes 0.78 0.73 Edges 0.79 0.75

Split BOX pretest 99 0.80 122 0.84 (all BOX posttest 192 0.80 172 0.82 classes) 346

To test the linearity of the relation of pre-post gains

to initial scores in the training group (H^ in Section 7.35),

a succession of polynomial models was fitted to the relation

between posttest and pretest scores on 3DD and BOX. In no

case was the error reduction due to adding quadratic or

cubic terms significant, so H4 was rejected. F-values are

given in Table D.9.

The correlations between the test scores in the experi­ mental classes (those which studied the Solids unit) are given in Tables D.IO and D.ll. Combining results for men

and women gives the test-retest correlations shown in

Table 7.8; the fact that they are in most cases only slightly smaller than the test-retest reliabilities in the control classes (Table 7.7) shows that the training program had a fairly uniform effect (if any).

The Solids achievement test was scored for four ob­ jectives: names; faces, edges and vertices; sketches; and nets. These gave Cronbach alphas of 0.69 in both years. The test-retest correlations (for the students who took the test first after studying the Statistics unit and again after the

Solids unit) were 0.58 in the first year and 0.65 in the second year; these are probably underestimates of the relia­ bility because of the intervention of the Solids unit. The 347

Table 7.8

Test-retest correlations for spatial tests in experimental classes, by year

First year Second year Test N r N r 3LD 31 0.80 39 0.74 Poles 0.66 0.47 Bottles 0.23 0.24 Cubes 0.53 0.58 Edges 0.71 0.72 HST 62 0.51 - - BOX 31 0.69 41 0.78

Table 7.9

Correlations between spatial pretests in both years combined, by sex^

Test 3DD HST BOX

3d d 31 46 HST 40 20 BOX 47 11

^Decimal points omitted. Correlations above the diagonal refer to men. below the diagonal to women. Ns range from 103 to 115; P(r:s 0.15) = 0.05, P(r:^ 0.23) = 0 .01. 348

reliability of the Solids achievement test thus appeared to

be adequate.

The reliability of the Statistics achievement test

could only be estimated from the test-retest correlations

for the students who took the test first after studying the

Solids unit and again after the Statistics unit; these were 0.23 in the first year and 0.44 in the second year.

Even though they are underestimates of the test reliability, these figures suggest that the test was rather unreliable.

Correlations between the 3DD, HST and BOX tests at their several administrations are given for various subgroups in Tables D.4, D.5, D.IO and D.ll. They are given for the pretests in both years combined in Table 7.9. The covariance matrices were significantly different for the two sexes

(Box's test, chi-square = 19.13, d.f. = 6, P .ci 0.01); the covariances were all higher for men than women, but the corre­ lations in the two sexes were not consistently related. As in the Grade Nine survey, the correlation between HST and BOX was much lower than the other two correlations within each sex (Section 5.32).

7.43 Treatment effects on 3-dimensional drawing and visualization

To test the major hypothesis (Section 7.35), the 349

posttest scores on 3DD and BOX were subjected to multivariate

analyses of variance. By virtue of the design, the signifi­ cances of the retest effect, the test-treatment interaction,

and the sex effect could also be tested.

The correlations of the HST pretest scores with the 3DD and BOX posttest scores were moderate, mostly about 0.36

(see Tables D.4, D.5, D.IO and D.ll), so HST was not used as a covariate. In view of the individualization of both treatments, it was considered admissable to take the student as unit of analysis in the first instance, although no causal deductions could be made because of the lack of randomization at the student level.

Cell means for the 3DD and BOX posttests are given in

Tables D.12 and D.13 and illustrated in Fig. 7.1 and 7.2.

An attempt to analyze the second year data using a restricted model to take account of the vacant cells failed through ill conditioning of the data matrix. As there was no reason to expect any difference between groups 2B and 2C, data were therefore collapsed over groups to give a complete factorial design for this and all subsequent analyses of second year data; the design is shown in Table 7.6. The analyses of variance are reported in Tables D.14 and D.15, and the 350

First year

Men Women

50 Mean score '• A

25

20

Solids Statistics Solids Statistics

Second year

Men Women V, 50 Mean score 4 : 25 '• c cf B

20 C* D

B

15

Solids Statistics Solids Statistics

Fig» 7.1 Mean scores on 3DD posttests by level/tutor, pretest condition, treatment, sex and year. Letters refer to levels in the first year and tutors in the second year. Full lines and left-hand labels refer to pretested students; broken lines and right-hand labels refer to non-pretested students. 351

First year

Men Women A Mean A score S

20 N S N

j. —I__ Solids Statistics Solids Statistics

Second year

Men Women C 50 Mean B score D 25 D

C 20 A A B 15

10

V,

Solids Statistics Solids Statistics

Fig. 7.2 Mean scores on BOX posttests by level/tutor, pretest condition, treatment, sex and year. Letters refer to levels in the first year and tutors in the second year. Full lines and left-hand labels refer to pretested students; broken lines and right-hand labels refer to non-pretested students. 352

significance levels of the main effects are given in

Table 7.10; none of the multivariate interactions was

significant at P < 0.10.

Table 7.10

Significance levels of main effects in analyses of variance of 3DD and BOX posttest scores, by year

Year Source Multivariate 3DD BOX

First Level 0.009 0.003 0.069 Sex 0.001 0.001 0.002 Pretest 0.069 0.177 0.026 Treatment 0.846 0.982 0.581

Second Sex 0.005 0.001 0.035 Tutor 0.698 0.746 0.547 Pretest 0.101 0. 158 0.093 Treatment 0. 230 0.056 0.628

The sex effect was significant in both years, with men consistently scoring higher than women, but the pretest effect was significant only on BOX in the first year. The level effect was in the expected direction in the first year but significant only for 3DD, and the tutor effect was not significant in the second year. More interesting is the fact that, in both years, neither the treatment effect nor the pretest-treatment interaction was significant. The 353

major hypothesis was therefore rejected: The unit on Solids

had no significant effect on the 3-dimensional drawing or

visualization ability of either first or second year

students.

Analysis of gains The use of crude gain scores has been

criticized by Cronbach and Furby (1970). To test the

hypothesis that women make greater gains than men as a

result of studying the Solids unit, analysis of covariance was therefore used instead. Separate analyses were performed

for 3DD and BOX posttests• (for those students who had taken the pretests), using the corresponding pretest as covariate.

Homogeneity of regression was not rejected for 3DD

(P> 0.44) or BOX (P 0.52) in the first year, but was not tested in the second year because of the small numbers in certain cells. The results of the analyses are given in detail in Tables D.16-D.19 and summarized in Fig. 7.3 and 7.4 and Table 7.11; the only significant interaction (P < 0.01) was of level with treatment for BOX in the first year. As expected, the level effect in the first year was removed by covariance, but the sex effect also became non-significant for both tests in both years. Hypothesis Hg was therefore rejected: Women did not make greater gains than men. How- 354

First year

Men Women S Mean score N N A A S

22

X X Solids Statistics Solids Statistics

Second year

Men Women

Mean score

24

22

Solids Statistics Solids Statistics

Fig. 7.3 Meain scores on 3DD posttests adjusted for covariance with 3DD pretests, by level/tutor, treatment, sex and year. Letters refer to levels In the first year and tutors In the second year. 355

First year

Men Women 30 Mean A score 28

S 26 S A 24

22

20 N

N

1 X Solids Statistics Solids Statistics

Second year

Men Women

30 Mean A score 28

26 C B 24 C B D 22 C A 20

X X Solids Statistics Solids Statistics

Fig. 7.4 Mefioi scores on BOX posttests adjusted for covariance with BOX pretests, by level/tutor, treatment, sex and year. Letters refer to levels in the first year and tutors in the second year. 356

Table 7.11

Significance levels of main effects in analysis of covariance of 3DD and BOX posttests using corresponding pretest as covariate, by year

Year Source 3DD BOX First Level 0.939 0.090 Sex 0.830 0.623 Treatment 0.153 0.774

Second Sex 0.186 0.177 Tutor 0.809 0. 296 Treatment 0 . 108 0.537

ever, since the treatment effect was significant only for

3DD in the second year (Table 7.11), it cannot be claimed that the general hypothesis was given a fair test.

7.44 Treatment effects on field independence

There was no reason to expect work on the Solids unit to improve the ability to extract figures from an embedding background. To check this hypothesis (H3 in Section 7.35), an analysis of covariance of the HST posttest scores of first year students was carried out, using HST pretest scores as the covariate.

The analysis is reported in Table D.20, and the ad­ justed cell means (omitting the pretest factor) are given in Table D.21 and illustrated in Fig. 7.5. None of the main 357

WomenMen

Mean A score 10

N s

S

N

A

X X Solids Statistics Solids Statistics

Fi%. 7.5 Mean scores on HST posttest in first year adjusted foi covariance with HST pretest, by level, sex and treatment.

effects and interactions was significant, supporting the hypothesis. However, since the treatment did not have any effect on 3DD and BOX scores either, and HST may have been of low reliability, this is rather weak support.

7.45 Treatment effects on achievement

To test the second hypothesis (H2 in Section 7.35), scores on the Solids and Statistics achievement tests were subjected to separate analyses of variance. The correlations of the

Solids achievement test scores with the HST pretest scores were fairly low, averaging 0-26 (see Tables D.4, D.5, D.IO and D.ll), so no covariate was used. 358

The cell mean scores (omitting the pretest factor) are given in Tables D.22 and D.23 and illustrated in Fig. 7.6 and

7.7. The analyses of variance are reported in Tables D.24 and D.25 and the significance levels of the main effects in

Table 7.12; there were significant interactions only for the

Solids test, level x treatment (P < 0.01) and the fourth- order interaction ( P < 0.05) in the first year and sex x treatment (P < 0.05) in the second year.

Table 7.12

Significance levels of main effects in analyses of variance of achievement test scores, by year

Year Source Solids Statistics First Level 0.004 0.001 Sex 0.033 0.244 Pretest 0.504 0.091 Treatment 0.001 0.001

Second Sex 0.049 0.984 Tutor 0.541 0.026 Pretest 0.280 0. 278 Treatment 0.001 0.005

As expected, the level effect was significant in the first year and the pretest effect was non-significant throughout. The treatment effects were highly significant, and in the expected direction in each case. It is inter- 359

First year

Men Women

A A 20

Mean N score S s 15 N

10

I Solids Statistics Solids Statistics

Second year

Men Women

20 Mean score

10

Solids Statistics Solids Statistics

Fig. 7*6 Mean scores on Solids achievement test, by level/tutor, treatment, sex and year. Letters refer to levels in the first year and tutors in the second year. 350

First year

Men Women

Mean score

20

IX Solids Statistics Solids Statistics

Second year

Men Women

30 Mean score

20

10

Solids Statistics Solids Statistics

FiK. 7.7 Mean scores on Statistics achievement test, by level/tutor, treatment, sex and year. Letters refer to levels in the first year and tutors in the second year. 361

esting that the sex effect was significant for the Solids but

not for the Statistics achievement test.

The analysis was repeated by level or tutor and treat­ ment using class mean scores. Since classes had been

assigned at random to treatment, significant treatment effects from such analyses could be interpreted causally.

The treatment effects were indeed significant (P < 0.05 on the Solids achievement test in the first year, P <0.01 in the other three cases), showing that the greater achievement was most likely the result of studying the appropriate unit

(see Table D.26 for details). Hypothesis Hg was therefore accepted: Student teachers can learn mathematics using the workcard method.

Although significant amounts had been learnt from the two workcard units, the mean scores on the achievement tests were low, about 40% for Solids and 55% for Statistics. It was judged that only about 19% of the students reached a satisfactory level on Solids and about 28% on Statistics

(satisfactory in the sense of sufficient mastery to proceed to dependent units). Questionnaire responses (Appendix N) and tutor observations (Section 7.33) showed that many students had difficulties working independently. Correla­ tions between rate of working and achievement were obtained 362

for the seven classes which studied the Solids unit (there

was too little spread in the rate of working the Statistics

unit); they averaged about 0.31 (see Table D.27). Thus

there was a slight tendency for the better students to work

faster and the slower workers to do less well. It is possible that individualization helped some students but hindered others.

The important educational question is whether the work­ card method is more or less efficient than the lecture method used previously. An attempt was made to compare the efficiencies of the two methods in May 1974: One first year group was assigned to study a three-week unit on Integers using workcards while the other group studied the same unit following a lecture method. No significant differences were found on the unit posttests, but there were so many design deficiencies that it would be hazardous to make any de­ ductions. However, the college tutors subsequently started writing further individualized units, convinced that the affective benefits would outweigh any possible cognitive shortcomings of the workcard method. 363

7.46 Conclusions

It may be safely concluded from the above analyses that

the Solids unit did not lead to the expected improvement in

spatial ability, although there was significant progress in

learning the geometrical objectives of the unit. It is important to consider why one was possible without the other.

A closer look at results on the Solids achievement test shows more clearly what was and was not learnt as a result of studying the Solids unit. The pattern of scores was similar in both sexes and both years: There was a clear gain in knowledge of the mathematical names of everyday shapes, from

2.2 to 4.8 correct out of 7, on average. Knowledge of con­ cepts of faces, edges and vertices also improved, but only from 3.8 to 5.6 out of 16; this small improvement could have been made simply by finding out what prisms and pyramids were.

The sketches made by the students who had studied the Solids unit were not very different in quality from the sketches made by those who had studied the Statistics unit; mean scores were 4.9 and 4.5 out of 10, respectively. Scores on the nets objective improved from 0.1 to 2.5 out of 10; this small gain could have been the result of learning the names pyramid, cone and cuboid. The general deduction is that. 364 as a result of studying the Solids unit, students learned the names of various solids but learned little about their spatial properties.

It is thus no longer surprising that students did not improve in spatial ability. The basic hypothesis of the training program, that learning how to draw and construct elementary solids leads to improvement in spatial ability, was not given a fair test because students simply did not learn those skills.

But why did students learn so little about the spatial properties of elementary solids? Why, when they had been instructed in how to draw cuboids and other regular solids, did their sketches not improve? Why, when they had been doing little else but design, cut out, and assemble nets of various solids for the previous three weeks, were they unable to sketch three of them in the final evaluation? As high school students in Grades 7 and 8 regularly learn elementary geometry from the same material used in the

Solids unit, using a traditional teaching method, the fault must lie either in the students or the method of instruction, or both. Two explanations suggest themselves. 365

1. The present students had such poor backgrounds that they were unable to catch up.

The student teachers were about 10 yr older than the

average high school Grade 7 student, yet they were at about

the same level both in their spatial ability and their

geometrical knowledge. If the foundations of intellectual

development are laid down in earlier years (say up to

adolescence), it may be too late to train spatial ability by the time students reach teachers' college. This pessi­ mistic hypothesis can only be properly tested using a much longer spatial training program than the one tested in the present experiment.

2. Students did not reflect on the results of their model-making activities.

Because of students' background of authoritarian teaching, it was suspected that many students merely followed in­

structions, did just what they were told, and begged

assistance where needed, solely in order to produce work to satisfy the tutor. It is possible that, for these students, the goal of producing an acceptable model or drawing was so

attractive that the objective of learning geometrical properties was overlooked.

Whatever the reasons, it is clear that the Solids unit, in its present form and when used alone, is not suitable for 366

training the spatial ability of teachers' college students.

The experiment cannot be considered a failure, however,

since it has pointed out two characteristics of student

teachers (their poor background and their lack of inde­

pendence) which could be causes of widespread difficulties

and which deserve more direct attention. The effectiveness

of the JSP spatial training program with young high school

students remains an open question.

7.5 Further analyses of teachers' college data

7.51 Influence of background variables

By means of the PDQ, information was collected from

all students on the background variables of age, number of years of full-time education, writing hand, type of secondary education, place of upbringing, number of siblings, presence of an adult male in the home during childhood, and occupation of parents or guardian. For reservations on the use of the writing hand as a measure of handedness and on

Miller's (1967b) rating of social class, see Section 5.51.

Because many students who reported no adult male present during childhood gave the mother's or grandmother's occupa­ tion as "housewife," it was decided to make social class ratings only when a father was present. So many students 367

were found to have attended extension or evening classes

for their secondary education that the number of years of

full-time education was regarded as irrelevant and omitted

from the analysis. Place of upbringing was classified

simply as Kingston or country. The analysis below covers

all prospective primary school teachers in both years at

the college.

Table 7.13 summarizes the data collected. The wide range of ages and previous school experience is evident.

Over 90% were from the country although the college is situated near the centre of Kingston. As in the country schools included in the Grade Nine survey (Table 6.6), the majority of fathers were small farmers (category 5); most of the remainder were manual workers.

Correlations between the ordered variables are given in

Table 7.14. They were again low, probably because of the narrow range of variation on most of the variables. The only consistent correlation shows that students who reported being looked after as a child by both parents (or some other male-female combination) tended to have more brothers and sisters than those looked after only by their mothers (or other female or females). 358

Table 7.13

Summary statistics on background variables, by year^

First Second Variable Category year year

Age Mean 22.21 23.41 S.D. 3.24 3.87

Writing Left 6 12 hand Right 190 201

Last school All-age 70 96 attended Junior secondary 20 17 full-time High 35 29 Technical high 9 9 Comprehensive 10 9 Continuation 3 1 Private high 24 29 Vocational 8 10 Commercial 14 10

Place of Kingston 14 15 upbringing Country 182 198

Number of Mean 6.11 6.25 siblings S.D. 2.91 1.26 Adult male Present 139 154 in home Absent 56 54

Social 1 1 0 class of 2 0 0 father 3 18 16 (where 4 49 64 present) 5 60 69 6 9 3

^Except for age and number of siblings, the table gives the frequency of each category. 369

Table 7.14

Correlations between background variables, by sex^

Higher Variable score 1 2 3 4 5 6

1. Age Older 02 06 -06 04 01 2. Writing hand Right -04 -05 06 -09 -10 3. Place of upbringing Country -06 03 04 10 15 4. Number of siblings More 04 15 22 -33 06 5, Adult male in home Absent -04 -07 -10 -26 - 6. Social class Lower -02 01 10 11 -04

^Decimal points omitted. Correlations above the diagonal refer to men, below the diagonal to women. Ns vary from 1421 to 216 ; P (1r 1 4 0 -16) = 0.05, P(lr| ^0. 21) = 0 .01.

Correlations of the ordered background variables with

the spatial pretest scores are given in Table 7.15. Only

one out of 36 was significant (probably a chance result),

showing again that the environmental factors considered are probably not important determinants of spatial ability.

To test the effect of previous education, schools were

first grouped into three categories using the results of

the Grade Nine survey to suggest similarities. The cate­ gories were: Public high schools; technical high, compre­ hensive high, and vocational schools; and others. Although

students from high schools generally had higher means than 370

Table 7.15

Correlations of background variables with spatial pretest scores, by sex^

Background Higher 3DD HST BOX variable score Men Age Older -08 -05 -18 Writing hand Right 06 10 -01 Place of upbringing Country -02 -13 09 Number of siblings More -20 -11 -08 Adult male in home Absent 23 -00 12 Social class Lower -02 02 -00

Women

Age Older 00 -10 14 Writing hand Right -06 -12 —06 Place of upbringing Country 05 -04 00 Number of siblings More -10 -10 -14 Adult male in home Absent 05 -05 08 Social class Lower 05 -08 07

Decimal points omitted, Ns range from 76 to 143; P(| r I ^0.22) = 0.05.

other students on the spatial pretests, a two-way analysis of variance (category by sex) showed a significant difference between the three categories only for BOX in the second year (Tables D.28 and D.29). It seems likely that teachers' colleges admit the poorer high school students and the brighter students from the other secondary schools, so that school type differences are all but eliminated. 371

7.52 Colleqe-school comparisons

It is interesting to compare scores of the teachers' college students on 3DD, HST and BOX with those obtained by students tested in the Grade Nine survey.

Overall mean scores of the teacher's college students on the spatial pretests are given in Table D.30 and illus­ trated in Fig. 5.3 (Section 6.34). There was no uniform pattern of relation to the school scores, but most of the college means were near to the technical high school means— the women were somewhat higher on average, the men somewhat lower.

Further college-school comparisons are discussed in the next section.

7.53 Results of individual testing

The mean scores on HFT, SRT drawing, and GIM of the seven men and nine women tested are given in Table D.31 and illustrated in Fig. 5.2, 5.9 and 5.10 in Section 5.3. The men did not score significantly higher than the women on either of the spatial tests, and the only significant dif­ ference in illusion susceptibility was for Horizontal-

Vertical 1 (Table D.31). 372

Scores may be compared with those obtained in the developmental survey. Compared with the Grade Nine high school students, the college students scored lower on SRT drawing and higher on HFT, but in no case did the difference reach significance (Table D.32). It could be that the similarity in scores was due to the age difference between the two samples (the men were on average 5.9 yr older, the women 9.2 yr older than the school students), suggesting that spatial ability continues to develop through late adolescence.

In view of the suspected low reliability of GIM, it is difficult to be confident in making comparisons between college and school students' susceptibility to geometric illusions. However, 8 of the 10 comparisons in Fig. 5.10 show lower illusion susceptibility in the older students, which is in agreement with previous results (Section 2.85).

The data from this small sample also provide some information on the relation between the group and individual tests. Correlations of HFT and SRT drawing scores with 3DD, HST and BOX pretests are given in Table 7.16; data for both sexes have been combined because of the small sample size. It is noticeable that the two embedded figures tests (HFT and HST) had the highest correlation, even though 373

they also had the lowest reliabilities; in fact, the inter-

correlation was higher than the reliabilities! Changing

the mode of administration did not seem to affect the

character of this test.

Table 7.15

Correlations between group and individual spatial tests^

Individual Group test test 3DD HSTBOX HFT 69 74 63

SRT drawing 64 50 58

^Decimal points omitted. N = 16; P( r 0.43) = 0.05, P( r ^ 0.57) = 0.01.

7.54 Validity of spatial ability tests

To obtain some idea about the relation of spatial ability in Jamaican students (as measured by 3DD, HST and

BOX) to their abilities in other directions, measures of general intelligence, mathematics achievement, and industrial arts achievement were obtained for as many students as possible. College records provided a Learning

Potential score for most students in both years; this was the score on a verbal intelligence test administered as part 374 of the college entrance examination, taken about one year before by the first year students and two years before by the second year students. The mathematics achievement score obtained for the first year students was their score on the placement test (mainly arithmetic) administered during their first week in college, about 4 months before. For the second year students, the mathematics grade on the first year examinations, taken about 7 months before, was used.

Industrial arts grades were obtained for most men in six of the eight second year classes by averaging grades on the metalwork and woodwork sections of the course, taken in the first semester of their second year.

Correlations are presented in Table 7.17. They are all low and many are not significantly different from zero. For men, the only consistently significant correla­ tions were those between Learning Potential and BOX and between Mathematics and 3DD. The size of the correlation between BOX and industrial arts scores in the second year was of the same order as previously reported validities

(see Section 4.21). There were no consistent correlations amongst the women. 375

Table 7.17

Correlations of spatial pretest scores and learning Potential, mathematics and industrial arts scores, by sex and year^

First year Second year Variable 3DD HST BOX 3DD HST BOX Men Learning Potential 26 17 35 36 37 43 Mathematics 52 14 36 29 17 22

Industrial arts — — - 23 -03 34

Women Learning Potential 17 25 23 45 13 31 Mathematics 27 07 17 28 31 23 a Decimal points omitted. Ns range from 36 to 89. P(r ^ 0.28) = 0.25, P(r ^ 0.38) = 0 .01.

The reliability of the Learning Potential score was probably quite respectable; but the spread of the placement examination scores was poor and of the college mathematics and industrial arts grades even poorer, so these scores are unlikely to have been of good reliability. The above results should therefore be treated with considerable reserve. CHAPTER EIGHT

SUMMARY AND CONCLUSIONS

8.1 Aims and procedures

This study arose from the author's interest in the difficulties which students in Ghana appeared to have in applying plane geometry and trigonometry to 3-dimensional problems. It was assumed that these difficulties were mainly due to inadequate spatial ability, particularly the ability to draw perspective diagrams of regular 3D figures.

The purpose of the present study was to find out more about the spatial ability of students in another, more advanced, developing country: Jamaica. Particular aims were to study the relation of 3D drawing ability to other facets of spatial ability and to investigate the influences of various educational, sociological and physical factors.

To measure spatial ability, two new tests were developed, two were extensively adapted, and two were used for the first time in Jamaica. The new tests were both concerned with 3D drawing: The Solid Representation Test

376 377

(SRT) was an individual test of the ability to draw four common mathematical shapes— the cuboid, cylinder, pyramid, and cube; and the Three-Dimensional Drawing test (3DD) was a group test which mainly measured the ability to represent the 3D relations of parallel and perpendicular in a drawing.

Two items of 3DD, Poles and Bottles, were also given as individual tests. Two embedded figures tests were specially adapted for this study: The Hidden Figures Test

(HFT) was adapted so that it could be administered to in­ dividuals over a wide age range, and the Hidden Shapes Test

(HST) so that it could be used with groups of students of poor reading ability. The other two tests used were the

Pacific Design Construction Test (DCT), a block design test first developed in Papua New Guinea, and the I-D Boxes test

(BOX), first developed in West Africa. All the group tests were administered using extensive visual aids and incorpor­ ating supervised practice tests.

Questionnaires and interviews were used to obtain data on sex, age, handedness, previous education, family size, family composition, social class, and ambition. During interviews with individuals, susceptibility to five illusions was found using the Geometric Illusion Measures (GIM), a test 378 which has been used all over the world, and eye dominance and skin color were also assessed.

The study consisted of three phases. In the first phase, the developmental survey, the individual tests were adminis­ tered to samples of high-ability students in the capital city,

Kingston. Eighty students (40 boys and 40 girls) aged 7-15 yr, selected from 8 high schools and the top streams of 12 of the better elementary schools, were tested. The group tests were used in the other two phases. In the Grade Nine survey, the tests were administered to 639 students (277 boys and

362 girls) from 20 secondary schools of four types chosen to represent three distinctive types of geographical/urban environment. In the spatial training experiment, conducted with 416 students (196 men and 220 women) in a teachers' college in Kingston, the effectiveness for improving spatial ability of an individualized unit in practical, elementary

3D geometry was assessed.

8.2 Results and conclusions

The aim of the present section is to relate the results of the three phases of this study to the five research ques­ tions formulated in Section 1.42. Psychometric characteris­ tics of the tests used are also reviewed. Results are 379

presented in greater detail in Chapters 5, 5 and 7; to avoid

tedious repetition, references to specific sections of

those chapters are omitted.

8.21 Reliability of the spatial tests

DCT and HFT Each of these tests showed a near-rectangular

distribution of scores, a reliability in excess of 0.90, and

a clear developmental pattern. They may therefore both be

regarded as perfectly satisfactory for the type of group

comparisons involved in the developmental survey.

SRT Comparison of scores under different exposure conditions,

a clear developmental pattern, and a high Cronbach alpha of

0.96, combine to show that the SRT drawing score was also perfectly satisfactory. Only the bimodality in its distribu­ tion warrants caution in the interpretation of results. The

SRT selection score, on the other hand, had a narrow range and a low reliability (0,53) and several of the alternatives were never selected; this score was therefore only useful for gross comparisons.

Poles This test was rather difficult to score as the slopes did not change in any clear relation to age. Its reliability was consequently low: Cronbach alphas were 0.71 in the 380 developmental survey, 0.69 in the Grade Nine survey, and

from 0.31 to 0.47 in the training experiment, while test- retest reliabilities of 0.62 and 0.65 were obtained in the training experiment. These figures indicate that Poles was unsuitable for making comparisons between small samples such as those in the developmental survey. The bimodal dis­ tribution of scores also warrants caution; it suggests that the test might be better scored for stages, but the modes were not distinct and no means of clearly separating them suggested itself.

Bottles This test showed a clear developmental pattern and had a satisfactory reliability of 0.91 in the developmental survey. The distribution of scores was bimodal: 30 subjects scored zero while the scores of the remaining 50 were more or less symmetrically distributed about a mean of 7.2 out of 10. Only the top part of this distribution appeared in the samples of the older students, and the narrower range reduced the reliability. Cronbach alphas were only 0.57 in the Grade Nine survey and 0.44-0.54 in the training experi­ ment, and the test-retest reliabilities were 0.42 and 0.48 for the college students. The severe bimodality suggests scoring this test for stages, as has been done by Dasen 381

(1974), but the distribution of scores makes it unlikely that second and third stages could be reliably differentiated.

3DD The reliability of 3DD was satisfactory for the compari­ son of large samples. Cronbach alphas were 0.85 in the Grade

Nine survey and 0.79-0.81 in the training experiment, and test-retest reliabilities for the college students were 0.73 and 0.83. The items Cubes and Edges were of considerably higher reliability than Poles and Bottles (see above);

Cronbach alphas varied from 0.58 to 0.78 for Cubes and from

0.75 to 0.80 for Edges, and test-retest reliabilities were

0.61 and 0.69 for Cubes and 0.86 and 0.87 for Edges. The high reliabilities of Edges are particularly remarkable, considering the comparative crudity of the scoring system.

Cubes and Edges were also more highly correlated with total

3DD scores than Poles and Bottles (0.77-0.85 compared to

0.55-0.75). It thus seems that this test was measuring the ability to represent parallels and perpendiculars in a drawing, and that the first two items contained some irrele­ vant variance due to the additional concepts of horizontal and vertical. 382

HST The reliability of this test was only estimated twice;

a Cronbach alpha of 0.71 was obtained in the Grade Nine

survey and a test-retest reliability of 0.51 in the spatial training experiment. The distribution of scores was positively skewed in the former case, but symmetrical in the latter, so it is suggested that the lower second estimate may have resulted from changes in administration procedures between the pretest and the posttest. The test is of doubtful reliability even for making comparisons between large samples. The item analysis showed that most of the items taken directly from Vz-2 (French et al., 1963) were poor discriminators in the Grade 9 sample.

BOX This test gave a symmetrical distribution in both the

Grade 9 and the teachers' college samples, and estimates of its reliability (0.80-0.84) matched those obtained during its standardization (Schwarz, 1964b). The only reservation to be expressed about this test is that there were many students who guessed at random— especially in the school sample, where 15% scored zero or less. There was no evi­ dence such as bimodality to suggest that the cause was failure to communicate the test task. It is likely that guessing was encouraged by the inevitable strain to work as 383

fast as possible on this test, the only one of the three group

tests in which the time factor was salient.

8.22 The structure of spatial ability

Too few tests were given in the present study toallow a very close analysis of the structure of intellectual abilities in Jamaican students, but some educated guesses can be made. In a poorly differentiated structure, in which spatial ability would not be a valid construct, the correla­ tions between all cognitive tests, and in particular between all "spatial" tests, would be high. If spatial ability were viable as a unitary construct, spatial inter­ correlations would be uniformly moderate. If spatial ability were better broken down into subfactors, some intercorrelations would be moderate, others low. Finally, if the intellectual structure were well differentiated, but no facet of spatial ability was a psychometrically valid construct, the correla­ tions between spatial tests would be uniformly low. It is only necessary to measure relatively few intercorrelations to obtain some clues as to which alternative might be the case.

In considering the intercorrelations obtained in the developmental survey, scores on Poles and SRT selection may be 384 ignored because of their low reliability. Intercorrela­ tions of the four remaining tests were uniformly high

(0.54-0.79 for boys and 0.79-0.91 for girls), but that was partly a result of the fact that performance improved linearly with age. When age was partialled out, the inter­ correlations became more moderate (0.30-0.53 for boys and

0.59-0.80 for girls). As the partial correlations represent an average taken over ages 7-15 yr, these results suggest that, during this period, spatial ability is a valid construct in boys but not in girls (at least amongst high- ability students) .

Results obtained in the Grade Nine survey suggest further differentiation by age 15-16 yr. In each sex, the correlations of 3DD with HST and BOX were approximately equal (mean 0.52) whilst the correlation between HST and

BOX was much lower (mean 0.32). The same pattern was ob­ served in the college students, who were some 7-8 yr older on average, but the correlations were lower (means 0.41 and

0.16, respectively). These results are consistent with the finding in Western setting that spatial visualization and field independence are separate factors of spatial ability

(Section 1.22), and suggest that 3D drawing might load on both factors. This structure was not shown by the pattern 385 of correlations between the group and individual tests

amongst college students, but since only a very small sample was tested, that evidence cannot be given so much weight.

8.23 Sex differences in the development of spatial ability

As expected, performance on all the spatial tests used in the developmental survey improved with age. In fact, statistical analysis showed that the relation to grade level did not deviate significantly from linearity on any of the tests.

Somewhat less expected was the difference in develop­ mental pattern amongst boys and girls. In the developmental survey, patterns on the four most reliable tests were similar: At Grade 1, boys and girls both made low scores; at Grade 3, boys obtained about double the girls' score; at Grade 5, boys scored about three times the girls' score; but at Grades 7 and 9, boys and girls scored about the same again. Looked at in another way, boys showed the greatest improvement between Grades 1 and 5, after which they did not change very much, whereas girls showed little improve­ ment between Grades 1 and 5, but improved to reach approxi­ mately the same level as boys by Grades 7 and 9. Results of the Grade Nine survey and the spatial training experiment. 386 using larger samples and different tests, showed that there was in fact still a significant difference in favor of males in adolescence and adulthood.

The fact that boys' clear superiority in Grade 5 was completely eliminated by Grade 7 suggests an artificial effect due to selection at 11+. Only about 7.5% of Grade 5 children eventually reach high school, and the sampling method for Grade 5 could not have completely compensated for this factor. (Indeed, in the Grade 5 check testing, it was found that only half of those selected had obtained passes to high school.) More important, in order to obtain equal numbers of passes for boys and girls, the pass mark is always lower for boys than for girls; this differential selection would automatically increase girls' scores relative to boys' on any test which is positively correlated with 11+ examination scores. However, the selection effect on spatial ability appears to be quite weak: In the Grade

Nine survey, the variance within each high school was still about 70% of the variance in the entire sample. It is also noticeable that the sex difference within the high schools was larger than that in the non-selective junior secondary and all-age schools. It thus seems that, amongst high- ability students generally, the gap between boys' and girls' 387

spatial test scores does become narrower between Grades 5 and

7, although possibly not to the full extent indicated by the present results.

The finding that the spatial intercorrelations were higher for girls than boys, even when age was partialled out, may thus be re-interpreted as showing that spatial ability evolves as a valid construct later in Jamaican girls (at ages 13-15 yr) than boys(ages 9-11 yr). This is consistent with the results of the Grade Nine survey, which suggested no sex difference in the overall structure of spatial ability at age 15-16 yr. Although there was a significant sex difference between the covariance matrices in. both the

Grade 9 and the teachers' college samples, the differences in correlations were quite small and rather inconsistent.

Taking the results of the three studies together, and making due allowance for sampling and measurement errors, the implication is that spatial ability in Jamaican students develops later in females than in males and that it does not develop to quite the same level in females as in males. The difference in spatial ability between the two sexes is probably highest in the upper primary school.

None of the proposed causes of the sex differences reviewed in Section 2.74 provides any explanation for this 388 pattern of development. If the sole cause were increase in hormonal output in males starting at about age 8, the sex difference would widen until puberty and remain fairly constant thereafter, but this is not what was observed to happen. The girls' spurt between ages 11 and 15 suggests some further physiological factor acting in girls only.

A possible explanation is that the output of female hormones in girls increases at the same time as the output of male hormones increases in boys (Dr a little earlier, since puberty is usually a year or two earlier in girls), thus inhibiting the development of spatial ability at a time when it is enhanced in boys; production of female hormones in girls then drops off at the same time or a little earlier than output of male hormones drops off in boys, so that development is allowed to proceed in girls at a time when the enhancement in boys begins to decrease.

In Section 3.34, it was noted that Jamaican women appeared to be far more independent than women in other countries. In the light of previous research (Section 2.74), it was expected that the sex difference in spatial ability, and especially in field independence, would be smaller in

Jamaica. In fact, although there are no comparative data for the tests used in this study, the sex difference was so 389 large as to make its relative size of purely academic interest. This forces a reconsideration of the situation:

At all except the very top levels of many organizations, females outnumber males, do most of the work, and yet defer authority to the males. It is suggested that, despite appearances, male dominance is the norm in Jamaican society:

Women's apparent independence may be the result of the men's tendency to take it easy and to rely on their status as men to make up for their deficiencies. Kerr's (1952) opinion cited in Section 3.33, that Jamaican males expect the world to look after them with maternal care, is relevant. In childhood, girls are also expected to conform to authority more than boys, although in home and school the authority is usually female. It thus seems that sex differences in spatial ability in Jamaica can still be at least partly as­ cribed to social differences in role expectations. It must be noted, however, that differences in the pattern of develop­ ment of spatial ability cannot be explained on this basis; even if it could be shown that girls change radically in conformity between Grade 5 and Grade 7, it is inconceivable that such a personality change could have such a great effect in such a short time. 390

8.24 Influence on background variables on spatial ability

The results of the school type and environmental analy­ ses were quite clear, but few positive results were obtained from the correlational analyses of personal data collected by interview and questionnaire. Except for age and sex, correlations with spatial test scores tended to be low and inconsistent between sexes and between samples. It is sus­ pected that most of the assessments of background variables were unreliable. Number of siblings was probably the most reliable measure. Hand dominance was reliably measured in the developmental survey but not in the other two phases of the study. The measures of place of upbringing and ambition were probably of adequate reliability, but it is doubtful whether presence of an adult male during childhood, social class, or skin color were reliably assessed.

The impression of unreliability is strengthened by the failure to replicate well-established relations. Thus, it was expected that higher social class would be associated with a smaller family size, a father present and a lighter skin color (Henriques, 1968; Miller, 1967b), but only 5 out of 24 correlations obtained between these four variables were significant, two of them in the wrong direction. By contrast, 391 significant correlations were obtained between the more reliable variables (number of siblings, handedness, and place of upbringing) for which plausible post hoc explanations could be constructed.

The following results were obtained. In view of the suspected unreliability, although significant results may be given the usual weight, inferences from non-significant results should be made with caution.

Sex and age Developmental patterns and sex differences in spatial ability were discussed in Section 8.23.

Skin color The color of a students' skin bore no significant relation to his performance on any of the spatial tests given in the developmental survey.

Handedness No significant relation of handedness to spatial ability was found, except in its interaction with eyedness

(see below).

Evedness In the first analysis of the data from the devel­ opmental survey, right-eyedness was associated only with higher Poles scores and only for girls; when age was con­ trolled, Bottles and SRT drawing scores were added, still only for girls. Further analysis implicated the interaction of eye-and hand dominance: For all three tests, right-handed 392

left-eyed students scored lower than others, as found by

previous researchers.

Social class In the developmental survey, correlations with

DCT, HFT and SRT drawing scores were significant for girls

only, higher social class being associated with higher

scores. In the Grade Nine survey, correlations with 3DD,

HST and BOX scores were significant in both sexes, but low

(0.14 to 0.24). No significant correlations were found in

the teachers' college sample.

Number of siblings Results for number of siblings were

almost the same as for social class (with only the SRT score changed from drawing to selection), large families being associated with lower scores. As number of siblings was probably reliably measured, and was only weakly related to social class, it may be deduced that family condi­ tions do have an influence on the development of spatial ability, but that the influence is rather weak.

Adult male in the home Most correlations with this variable were non-significant. Low but significant correlations (0.14 to 0.21) were found for boys in the Grade Nine survey, show­ ing that the presence of a father or father-substitute was associated with higher spatial test scores. It is possible 393

that the stability of the home does not have the effect on cognitive development which has been hypothesized, or that those who are badly affected drop out of school by Grade 9, but it seems more likely that the question was poorly phrased or that students deliberately or through ignorance mis- reported the conditions of their upbringing. The results sug­ gest that fathers have a greater influence on boys' than girls' spatial ability, but the evidence is not very strong.

Place of upbringing Evidence on the influence of the place of upbringing comes only from the Grade Nine survey, since the other two samples were too homogeneous on this variable.

The analysis of variance of refined data from the junior secondary and all-age schools showed a significant environ­ mental effect only for HST, with city students scoring higher than town and village students, who were not signifi­ cantly different. Correlational analysis of the entire data suggested a rather weak effect in the same direction for all three tests. school type There were clear differences in spatial ability between students in the four types of secondary school in­ cluded in the Grade Nine survey. On all three tests, high 394 school students scored significantly higher than technical high school students, who in turn scored significantly higher than junior secondary and all-age school students; junior secondary and all-age school students were only significantly different on 3DD. In general, the type of secondary school attended cannot be said to influence spatial ability; the gross differences are caused by the correlation of spatial ability with scores on the various selection examinations.

However, there was evidence that the drawing instruction which boys receive in technical high and junior secondary schools influenced their scores on 3DD (but not HST or BOX).

Ambition It was found in the Grade Nine survey that boys with spatial/technical ambitions tended to have slightly higher spatial ability than boys with verbal/personal ambitions. There was no relation between ambition and ability amongst the girls.

8.25 Training of spatial ability

It was only possible to examine the training of spatial ability amongst teachers' college students. Classes of students in their first and second year at one college were each randomly divided into two groups. The experimental groups studied a unit on Solids which involved making models 395 and sketches of elementary 3-dimensional geometrical shapes; the control groups studied a unit on Statistics. Both units were adapted for individualized study from material in the Joint Schools Project Mathematics series

(Mitchelmore et al., 1970-75); first year students studied the units for 4 weeks, second year students for 3 weeks.

The tests 3DD, HST and BOX were used to measure students' spatial ability before and after studying each unit. Analy­ ses of variance and covariance showed conclusively that there was no significant difference between the two treatment groups at the end of the experimental period: The spatial training program did not produce the hoped-for gains in spatial ability.

There were, however, large and significant differences between the two groups on the Solids unit achievement test.

From an examination of results on that test, it was deduced that what students had learned from the Solids unit was mostly the names of various shapes and not their spatial properties. There had therefore been inadequate opportunity for the occurrence of the generalizations which could have led to an improvement in students' spatial ability.

Two explanations were advanced to account for student teachers' failure to learn spatial properties from the 396

Solids unit. The more pessimistic was that they were too old to develop in spatial ability, no matter what activities they followed to make up for shortcomings in their earlier spatial experiences. The other explanation was that students' oft-noted dependence on their tutors had caused them to proceed mechanically without adequate reflection on the results of their activity. Both of these factors would have less influence on the younger students for whom the spatial training program was originally designed.

8.26 Geometric illusions

Tests of susceptibility to the Muller-Lyer, Horizontal-

Vertical inverted T and L, Sander Parallelogram, and Ponzo illusions yielded no significant age trend or sex difference.

The graphs showed considerable error and only a vague resemblance to results obtained by other researchers.

Susceptibilities to the five illusions tested were inde­ pendent of each other and of all the background variables measured.

Correlations of illusion susceptibilities with scores on the spatial tests were low and showed inconsistencies between tests and between sexes. The clearest relations were for the Horizontal-Vertical inverted T illusion for bo]YS and the Muller-Lyer illusion for girls; in both cases, 397 higher spatial ability was associated with lower illusion

susceptibility. It was also found that boys who reported

a 3D interpretation of the Ponzo diagram were significantly more susceptible to the illusion; but the difference was negligible amongst the girls. These results suggest that it might be profitable to look for the cause of illusion sus­ ceptibility not in the ability to interpret or make 2D diagrams of 3D objects, but in the tendency to put a 3D interpretation on line drawings. They also suggest that there may be complex sex differences in illusion suscepti­ bility.

8.3 Implications and recommendations

The results of this study have implications for present educational practice in Jamaica and suggest many directions for future research and development.

8.31 Technical education

Selection As long as the present system of selection for the different types of secondary education by competitive examination continues, it is important to try to place all those who appear to be capable of benefitting from a technical education in either a high school or a technical high school. 398

(Although junior secondary schools also provide some

technical education, it has so far been only at an elemen­

tary level.) The results of the Grade Nine survey indicate that there are at present significant numbers of students of high spatial ability still in junior secondary and all­ age schools, probably due in large part to low correlations of spatial test scores with scores obtained on the various selection examinations (lit, 13t and GNAT). The size of these correlations is not known, but the correlations between the spatial test scores in the Grade Nine survey

(0.25-0.58) contrast sharply with those obtained by Isaacs

(1974) in a similar Grade 9 sample between scores on mathe­ matics, English, and verbal and non-verbal intelligence tests similar to those used in the GNAT (0.53-0.81).

It is therefore recommended that renewed attempts be made to develop group tests of spatial ability which could be included in the present public examination batteries.

Such tests would have to be much simpler to administer than the tests included in this study for research purposes. As the pilot-testing in this study suggested a serious effect due to reading ability, an important first step is to investigate more closely the effects of differences in administrative procedures; any proposed test which uses an 399 abbreviated method of administration should be cross­ validated with the same test administered after the manner of 3DD, HST and BOX.

High schools The correlation between spatial test scores and selection examination scores is high enough to ensure that most students in high schools are of high spatial ability. Many of these no doubt find their vocation in science or mathematics, but there is likely to be a consid­ erable number who would do better in a technical or technological field. The second recommendation is that the trend for high schools to add technical subjects to their curriculum be encouraged.

Country areas The present study shows that junior secondary and all-age school students in the country areas have an average level of 3D visualization and drawing ability which is not significantly different from that in the towns and cities. Care should therefore be taken to see that country students have the same opportunities for technical education. It is not suggested that country areas have been neglected, but the lower population density makes it more difficult to provide as rich a variety of educational options as can be provided in urban centers, and the country 400 junior secondary schools are too far apart for many students to reach them. As it is impracticable to supply all all­ age schools with adequate equipment or staff, it is recom­ mended that technical education centers be established in country areas which all-age school students could attend on a part-time basis. For example, ten schools could share one center, the students from each school attending for half a day each week. Such a center need not be very large. In fact, the data suggest that there may be many students in the all-age schools in towns and cities who could benefit from a similar arrangement.

Vocational guidance Almost all boys in technical high schools at present follow a technical course of one sort or another. The present results show that there is a wide range of spatial ability amongst these students, indicating that some might be better advised to follow alternative courses. No such advice is commonly available to students, either in technical high schools or in other secondary schools where the need may rather be to restrict numbers because of limited resources, and the result is a rather weak relation between ambition and ability. It is recom­ mended that tests of spatial and other abilities be developed for use in vocational guidance in Jamaican secondary schools. 401

Tests similar to those developed for the public examinations could probably be used for vocational guidance, but they would need adapting to allow for different average levels of ability in each population. Considerable research on the predictive validity of spatial and other tests is needed to ensure maximum predictive efficiency for the vocational guidance battery.

The present policy of the Jamaican government, announced after this study was started, is for junior secondary schools to be extended into Grades 10 and 11, with an emphasis in these years on marketable vocational skills.

New procedures for assessing graduates of the new so-called secondary schools are to be developed, and it is likely that pressure will be put on the high schools to follow suit and drop their present emphasis on G.C.E. O- and A-levels.

From there it is a short step to the abolition of selec­ tion and the provision of a wide range of academic and vocational courses in all post-primary institutions. Such a step would remove the need for the inclusion of a spatial ability test in the public examinations, but it would make the development of adequate vocational guidance batteries even more urgent. 402

Sex bias Few Jamaican girls (about 1% in Grade 9) have a technical ambition, and even fewer (less than 1%) take any technical courses in secondary school apart from mathematics and natural and domestic science. Yet more than a third of the best visualizers in the Grade Nine survey were girls.

Clearly a lot of talent is being wasted. It is recommended that every effort should be made to encourage girls to con­ sider technical careers and to enroll in industrial arts courses. The major determinant of career choice is probably social norms, but schools do have considerable influence.

As a start, technical high and junior secondary schools could cease scheduling industrial arts and domestic science as alternatives on the timetable; girls who wished to study industrial arts would then not have to drop domestic science and boys who did not do well in industrial arts could be provided with a more congenial alternative.

The inclusion of spatial ability tests in the 11+ exam­ ination battery would raise boys' total scores (on which selection is based) relative to girls'. It would, however, have a serious disadvantage in that girls' spatial ability appears to develop most between 11 and 15 yr so that a test at 11-12 yr would be of poor validity for girls. (In fact, there is nothing to show that other intellectual abilities 403 do not develop in a similar uneven pattern which is dif­

ferent for the two sexes.) This is further support for the argument that present rigid selection procedures need to be modified.

8.32 Geometry in schools

It was noted in Section 1.31 that the standard of performance on geometry items on the GNAT was very poor.

Further evidence of poor geometrical knowledge in Jamaican students was collected at several points in this study. In the individual testing, students were asked if they knew the names of the five solids they had just drawn. The mean number correct was 0.04 in primary school, 1.5 in high school and 2.3 in teachers' college. The most common error was in naming a face instead of the solid (e.g., rectangle for cuboid, circle for cylinder, triangle for pyramid, and square for cube), which confirms that the only geometry taught is plane geometry. The clearest indicator of the low priority afforded geometry was the situation in the teachers' college: This was the first time that prospective primary school teachers had been taught any geometry— there had been no need to teach student-teachers geometry because they would never have to teach the subject themselves. The 404 geometric ignorance of the present student-teachers only

confirms this inference.

Thus one reason why geometrical knowledge is on the

average poor is that the subject is not taught. Another reason is that teachers are afraid of the subject; there

seems little doubt of this, in view of their own ignorance of the subject and its omission from their training.

Another reason is the relative unimportance of geometry in the selection tests ; The 11+ mathematics paper has no geometrical items and the GNAT only a few. In general, teachers do not waste time teaching subjects which are not going to be examined. A fourth reason is that geometry has not been seen to be of any relevance to elementary school children: Geometry is a subject which you study for O-level.

These factors are, of course, all bound up in a network of relations in which the poor level of performance is cause as well as effect. The vicious circles can only be broken by stress applied at several points simultaneously. The introduction of geometry into the teachers' college curriculum is a starting point, but it will come to nothing unless more geometry is taught in schools. The recently introduced "new math" project has brought some geometry into primary school 405

classrooms, but it seems to have been restricted to learning

the names of common plane figures. It is recommended that

the primary school geometry curriculum be expanded by intro­

ducing more practical activities such as drawing and model-

making and by encouraging a more exploratory and self-

expressive approach to the subject. Incidentally, since

this is likely to be one area where boys would excel girls,

such activities would probably improve boys' academic self-

concept at the same time as they increase the girls'

spatial ability.

Unfortunately, teachers are unlikely to teach more

geometry until examinations require it. It is recommended

that more geometrical items be included in selection

examinations and that teachers be informed in advance of the

changes. The addition of a spatial ability test might have

the same effect as the inclusion of geometrical items.

The present practice of attempting to keep teachers in

complete ignorance of the contents of the various selection

examinations is to be deprecated, since it ignores a potentially powerful lever on the school curriculum.

Another way in which the study of geometry in schools could be encouraged is to release it from its euclidean

straightjacket and to demonstrate its practical value in 406

understanding spatial relations. Throughout mathematics,

the appreciation of abstract patterns and the foretaste of

vocational applications is likely to be seen as more rele­

vant than the construction of logical sequences or the manip­

ulation of meaningless symbols. Much progress is being made

in this direction in the development of individualized units

for Grades 10 and 11 in the new secondary schools. It is

hoped that several units in practical elementary geometry will be included in the new curriculum for those grades.

The main aim of these units should be to bring out the

shapes and geometrical relations implicit in man-made

objects in the local environment.

8.33 Teacher education

The spatial training experiment showed that a 4-week unit on drawing and making simple solids taught more about their names than about their shapes. Clearly, much more needs to be done in a teachers' college mathematics course if teachers are going to make gains in geometric knowledge and spatial ability sufficient to enable them to teach geometry accurately and with confidence. One solution is simply to do more work on geometry, covering concepts such as plane figures, area, volume, tessellations, symmetry and 407 transformations. It is doubtful if this would lead to much more than an increased vocabulary, but it is possible that the accumulation of geometrical experience might precipitate a breakthrough in spatial ability. The second solution is to try to get students to reflect more extensively on their own actions and to do more to evaluate their own under­ standing. This latter action might be more difficult to implement but it is likely to have farther-reaching effects.

It is recommended that teachers' colleges set up mathematics laboratories and include geometrical activities structured in such a way as to encourage reflection and self-evalua­ tion. Individualized programs such as the Solids and

Statistics units should also be rewritten to include many more opportunities for the student to check up on his progress towards unit objectives. Research on the relative effectiveness of individualized, laboratory and lecture methods should be continued, not to obtain a final answer but to identify the strengths and weaknesses of each method; to fully achieve this objective, it will be neces­ sary to find ways of measuring affective gains, especially in the areas of integrity and self-reliance. 408

8.34 Test development

Individual tests DCT appears to be satisfactory for general use with high-ability students between about ages 9 and 13, but its characteristics amongst lower ability students need to be investigated. HFT was also of high reliability, but it could be improved by re-ordering the items in order of difficulty and replacing the last item by an easier design.

Like DCT, HFT appears to be insufficiently discriminating in younger children and therefore probably in students of lower ability than the ones included in the developmental survey.

Of the two HVT tests. Bottles appears satisfactory for developmental use, but would be more discriminating in older students if there were more titled bottles (say 60° to the vertical, one inverted). Further attempts should be made to score the test by stages, and results should be compared to those of Dasen (1974). On the other hand, Poles presents something of a conundrum. The slope of the poles simply did not show any clear developmental pattern, so it is unlikely that a revised scoring system would lead to any great improvement in reliability or validity. As it seems likely that ability to represent the vertical is as closely linked to the other spatial test scores as they are to each other. 409

the most conservative assessment is that Poles is a poor test

and should be set aside. It is still a fascinating question

as to what does influence the slope of the poles.

SRT appears to have the makings of a useful test. The

scoring system was developed on a post hoc basis, and needs cross-validation; in particular, inter-judge reliability needs to be estimated. The high correlation between scores under Conditions 1 and 2, and their high reliabilities, suggest that a useful drawing score could be obtained using only one condition. The cone needs to be replaced by a solid which gives scoreable drawings, perhaps a triangular prism; addition of this solid would probably compensate for the loss of reliability to be expected on cross-validation.

For best results, stages in drawing the fifth solid should be predicted in advance. The alternatives for Condition 3 should be replaced by more representative drawings, chosen from those collected in the present study; it is still an open question whether the selection task is too easy to pro­ duce a reliable score in schoolchildren. SRT would be of wider application if it could be given as a group test; investigations should be made to find if the point of view and method of display could be standardized sufficiently to produce a reliable score. 410

The evidence of this and other studies is that GIM is rather unreliable in its present form. A study should be made to estimate its reliability, and if it is as low as expected, efforts should be made to improve it. A control on exposure time is the obvious first step; the actual exposure time used must be carefully chosen because the effect is different at different ages (Piaget, 1969). Since repetition reduces the illusion effect, it might also be better to present the stimuli in a mixed order instead of by type of illusion, as was done by Taylor (1972). Taylor's alternative format (selecting the best match from three stimuli) should also be investigated. Finally, the instruc­ tions might be changed to hint that some illusion is to be expected and to instruct subjects to compare the two lines most carefully. Again, data could be collected more quickly if this test could be reliably administered to groups, perhaps using slide projection.

Group tests 3DD may be regarded as an encouraging start towards developing an accurate measure of the ability to represent rectangularity in three dimensions. The item analyses suggest several improvements. Cubes and Edges are the most reliable and valid of the four items, but each 411 could be improved; the less discriminating features in the scoring system for Cubes could be replaced, and the last block in Edges could be replaced by an easier shape.

Bottles is not much less reliable or valid than Cubes and

Edges, even though two of the four bottles were almost always drawn correctly in the group testing, so addition of two more sloping bottles should bring the item statistics up to an acceptable level. As stated above, it is doubtful whether

Poles could be improved; instead, it would probably be better to investigate alternative item types more similar to

Cubes and Edges. Even Bottles might be better replaced because of its poor face validity relative to Cubes and Edges.

Development of HST still has a long way to go before it can be used across all types of school in Jamaica. Half of the items are too hard; it might be better to start again with all sixteen of the NLSMA Grade 5 items plus four slightly more difficult items, with the same 15 minute time limit. There seems nothing wrong with the method of admin­ istration or the format of the test paper.

BOX appears to rely too much on the speed factor. An investigation should be made to find the effect of increasing the time limit on this test, in an effort to reduce the 412

number of students who apparently guess at random. If the

number of guessers is still large, clinical procedures

should be followed to investigate students' difficulties in

more detail and to distinguish communication failures from

visualization failures.

As previously recommended, attempts should be renewed

to develop abbreviated administration procedures for spatial

tests. During pilot testing, no difficulty was experienced

in administering the first trial version of 3DD and the NLSMA

Hidden Figures Test (both in a standard printed format) with

high school students, and the explanations in the final versions of 3DD, HST and BOX were excessively elaborate for

teachers' college students. The standard format is likely

to be far more suitable for verbally competent students.

8.35 Further research

In addition to the curriculum and test development proposed in previous sections, much further psychological research is needed to replicate and extend the results of the present study.

Tentative evidence has been found of spatial visualiza­ tion and field independence factors in Jamaican Grade 9 students. It is vital for educational prediction purposes 413

that the relation of these factors to verbal and numerical

ability be established as soon as possible. In particular,

the contribution of verbal ability to variance on the

standard printed type of test should be estimated. A full

scale factorial study of intelligence is probably premature

at this point.

The relation of 3D drawing ability to spatial visualiza­

tion and field independence should be further studied. The

relation to field independence found in this study is some­

what surprising, but could arise as follows: To understand what is a correct line drawing of an object, it is necessary

to be able to take up a subjective perceptual attitude; in

effect, this means disembedding the 2D information provided by the edges of the object from the 3D information provided by the faces. This reasoning suggests that both 3D drawing

ability and field independence should be correlated with

shape constancy; this should be investigated.

The hypothesis that spatial ability develops later in

Jamaican girls than boys, but does not reach quite the same level, was derived from two studies with some contradictory features. Required is a further study which would somehow get over the problem of selection at 11+ and clarify the 414 growth of spatial ability between ages 9 and 13 yr. This would probably require the testing of wider ability groups at these ages. Such testing could also show whether spatial ability develops later in lower ability groups, or whether it develops at the same time but not so much. Of course, a longitudinal study would be the best way of studying developmental influences, but this may not be practicable.

The SRT selection task was clearly quite easy even for beginning primary school children. The test should be given to younger children to investigate the earlier development of the ability to recognize drawings of regular objects. The model outlined in Section 2-54 suggests several hypotheses. In particular, children should be asked which of the alternatives presented could be a drawing of each solid, as well as which was the best drawing. More alternatives would have to be used, and probably Blumbaugh's (1971) method of "posting" each stimulus figure would be more suit­ able than the multiple-choice technique used in this study.

It was suggested in Section 8.26 that illusion sus­ ceptibility might be related to the tendency (rather than the ability) to interpret line drawings 3-dimensionally. Such a relation would explain why adding textural detail increased the potency of the Ponzo illusion in Ugandian college students 415 but not villagers (Leibowitz & Pick, 1972). In the develop­ mental survey, previous drawing of 3D figures did not change the tendency enough to affect illusion susceptibility, but the diagrams in the two activities were quite dissimilar and no attempt was made to draw subjects' attention to any possible relation between them. The influence could be very specific. The hypothesized relation could be investigated by extending the techniques used by Leibowitz & Pick (1972) and in the present study to other illusions, especially the

Muller-Lyer, Horizontal-Vertical and Sander Parallelogram illusions. It would still be necessary to find what causes one individual to prefer a 3D interpretation of a line draw­ ing and another a 2D interpretation; Deregowski (1971) has shown that there are marked cross-cultural differences in this tendency and in its development with age.

Although all correlations of spatial ability and illusion susceptibility with skin color were very small in this study, only a rather poor measure of skin color was used. It is recommended that efforts be made to develop a more objective measure of skin pigmentation, if only to allow the controversy to be put to rest. An extensive study relating skin pigmentation, opthalmoscopic and physical 416

assessments of macular pigmentation, spatial ability, and illusion susceptibility should then be mounted. Both eyes

should be examined (Bone & Sparrock (1971) used only the right eye), and hand and eye dominance should be controlled.

This study produced very little evidence about the in­ fluence of sociological factors on spatial ability. It is more likely that the factors were poorly measured than that they have no influence. Even if Miller's (1967b) occupational classification could be improved, it is doubtful whether students' information about their parents is sufficient to enable an accurate determination. It is hoped that Jamaican sociologists will continue their studies of the relation between occupational prestige, income, education and home conditions, but even then the reliable assessment of social class will require more than the simple administration of a questionnaire. It is recommended that students' reports of parental occupation be treated with extreme reserve, and used only for gross contrasts such as manual/non-manual.

The spatial training program proved ineffective as a remedial measure with teachers' college students. It still remains to test its effectiveness with high school students for whom the materials were originally intended and where 417 teachers have usually felt them to be effective. Grade 7 would be nearer the period of most rapid perceptual develop­ ment for high ability students of both sexes, so there would be no question of students being too old to benefit. It may also be expected that younger students would be more inde­ pendent and better able to work from an individualized program, especially if the workcards were revised as sug­ gested above by including more questions requiring reflection and self-evaluation.

The method of instruction used by Dubin (1945) suggests a clinical method of teaching 3D drawing which might also be tested at various levels: The tutor would discuss each student's drawings with him, bringing his attention particularly to the shortcomings which distinguish his present drawings from those to be expected in the next stage of development. The author's experience during his year of research in Jamaica convinces him that it would not be difficult to find schools that would be as whole­ heartedly cooperative in carrying out such training studies as was the teachers' college in the present study. BIBLIOGRAPHY

Allport, G. VI., & Pettigrew, T. F. Cultural influence on the perception of movement; The trapezoidal illusion among Zulus. Journal of Abnormal and Social Psychology, 1957, 55, 104-113.

American Institutes for Research. I-D examiner's manual. Pittsburgh, Pa.: The Institutes, no date.

Anastasi, A. Differential psychology. (3rd ed.) New York: Macmillan, 1958.

Annett, M. A model of the inheritance of handedness and cerebral dominance. Nature, 1964, 204, 59-60.

Annett, M. The binomial distribution of right, mixed and left handedness. Quarterly Journal of Experimental Psychology, 1967, 19, 327-333.

Annett, M. A classification of hand preference by associa­ tion analysis. British Journal of Psychology, 1970, 61, 303-321.

Annis, R. C., & Frost, B. Human visual ecology and orienta­ tion anisotropies in acuity. Science, 1973, 182, 729-731.

Armstrong, R. E., Rubin, E., Stewart, V. M . , & Kuntner, L. Susceptibility to the Muller-Lyer, Sander Parallelogram, and Ames Distorted Room illusions as a function of age, sex and retinal pigmentation among urban midwestern children. Unpublished paper. Northwestern University, 1970.

Araheim, R. Art and visual perception; A psychology of the creative eye. Berkeley, Calif.; University of California Press, 1954.

Ausubel, D. P. Educational psychology; A cognitive view. New York; Holt, Rinehart, & Winston, 1968.

418 419

Baker, S. R., & Lalley, L. The relation of visualization skills to achievement in freshman chemistry. Journal of Chemistry Education, 1972, 49, 775-776.

Barakat, M. K. A factorial study of mathematical abilities. British Journal of Psychology (Statistics Section), 1951, 4, 137-156.

Barnhard, E. N. Developmental stages in compositional con­ struction in children's drawings. Journal of Experi­ mental Education, 1942, 11, 156-184.

Barry, H., Bacon, M. K. & Child, I. D. A cross cultural survey of some sex differences in socialization. Journal of Abnormal and Social Psychology, 1957, 55, 327-332.

Bausell, R. B., & Moody, W. B. Learning through doing in teacher education: A proposal. The Arithmetic Teacher. 1974, 21, 436-438.

Beard, R. M. An investigation into mathematical concepts among Ghanaian children. Teacher Education in New Countries, 1968, 9, 3-14 & 132-145.

Bee, H. L., & Walker, R. S. Experimental modification of the lag between perceiving and performing. Psychonomic Science, 1968, 11, 127-128.

Bennett, G. K., Seashore, H. G., & Wesman, A. G. Differ­ ential Aptitude Tests: Manual. (4th ed.) New York : The Psychological Corporation, 1966.

Bennett, P. Survey of mathematical skills and associated processes in teachers' colleges. Unpublished research. School of Education, University of the West Indies, Mona, Jamaica.

Berry, J. W. Temne and Eskimo perceptual skills. Inter­ national Journal of Psychology, 1966, 1, 207-229.

Berry, J. W. Ecology, perceptual development, and the Muller-Lyer illusion. British Journal of Psychology. 1968, 59, 205-210. 420

Berry, J. W. On cross-cultural comparability. International Journal of Psychology, 1959, 4, 119-128.

Berry, J. W. Ecological and cultural factors in spatial perceptual development. Canadian Journal of Behavioral Science, 1971, 3, 324-336. (a)

Berry, J. W. Muller-Lyer susceptibility: Culture, ecology or race? International Journal of Psychology, 1971, 6, 193-197. (b)

Berry, J. W. An ecological approach to cross cultural psychology. Netherlands Journal of Psychology, 1974, in press.

Berry, J. W . , & Annis, R. C. Ecology, culture, and psycho­ logical differentiation. International Journal of Psychology, 1974, in press.

Beveridge, W. M. Racial differences in phenomenal regression. British Journal of Psychology, 1935, 26, 59-62.

Bhatia, C. M. Performance tests of intelligence under Indian conditions. Bombay: Oxford University Press, 1955.

Bieri, J. Parental identification, acceptance of authority, and within-sex differences in cognitive behavior. Journal of Abnormal and Social Psychology, 1960, 60, 76- 79.

Biesheuvel, S. African intelligence. Johannesburg: South African Institute of Race Relations, 1943.

Biesheuval, S. Psychological tests and their application to non-European peoples. In G. B. Jeffrey (Ed.), The year­ book of education. London: Evans, 1949.

Bishop, A. J. Use of structural apparatus and spatial ability: A possible relationship. Research in Education, 1973, 9, 43-49.

Blade, M. P., & Watson, W. S. Increase in spatial visuali­ zation test scores during engineering study. Psychologi­ cal Monographs: General and Applied, 1955, 69 (Whole No. 397). 421

Blake, R. B. Industrial application of tests developed for illiterate and semiliterate people. Paper presented at the NATO Conference on the Influence of Cultural Factors on Test Performance, Istanbul, 1971.

Blakemore, C., & Mitchell, D. E. Environmental modification of the visual cortex and the neural basis of learning and memory. Nature. 1973, 241, 467-468.

Blum, D. M., & Chagnon, J. G. The effect of age, sex, and language on rotation in a visual-motor task. Journal of Social Psychology, 1967, 71, 125-132.

Boe, B. L. A study of the ability of secondary school pupils to perceive plane sections of solid figures. The Mathematics Teacher, 1968, 61, 415-421.

Bone, R. A . , & Sparrock, J. M. B. Comparison of macular pigment densities in human eyes. Vision Research, 1971, 11, 1057-1064.

Borich, G. P., & Bauman, P. M. Convergent and discriminant validation of the French and Guilford-Zimmermann spatial orientation and spatial visualization factors. Educa­ tional and Psychological Measurement, 1972, 32, 1029-1033.

Bower, T. G. R. Discrimination of depth in premotor infants. Psychonomic Science, 1964, 1, 368.

Bower, T. G. R. Slant perception and shape constancy in infants. Science, 1966, 151, 832-834.

Box, G. E. P. A general distribution theory for a class of likelihood criteria. Biometrika, 1949, 36, 317-346.

Braithwaite, R. J. The Frostig test and training programme - How valuable? The Slow Learning Child, 1972, 19, 86-91.

Brault, H. Etude genetique de la constance des formes. Psychologie Française, 1962, 7, 270-282. 422

Brinkmann, E. Programmed instruction as a technique for improving spatial visualization. Journal of Applied Psychologv, 1966, 50, 179-184.

Broverman, D. M., Klaiber, E. L., Kobayashi, Y., & Vogel, W. Roles of activation and inhibition in sex differences in cognitive abilities. Psychological Review, 1968, 75, 23-50.

Broverman, D. N., Clarkson, F. E., Klaiber, E. L., & Vogel, V. The ability to automize: A function basic to learning and performance. In C. Kris (Ed.), Learning disabilities; Multidisciplinary approaches to identification, diagnosis and remedial education. New York: Macmillan, 1972.

Brown, F. R. The effect of an experimental course in geometry on the ability to visualize in three dimensions. Unpublished doctoral dissertation. University of Illinois, 1954. Dissertation Abstracts, 15, 83-84.

Brumbaugh, D. Isolation of factors that influence the abil­ ity of young children to associate a solid with a repre­ sentation of that solid. The Arithmetic Teacher, 1971, 18, 49-52.

Brunswik, E. Perception and the representative design of psychological experiments. Berkeley, California: University of California Press, 1956.

Burt, C. The structure of the mind: A review of the results of factor analysis. British Journal of Educa­ tional Psychology, 1949, 19, 110-111 & 176-199.

Caldwell, J. R. Structure-of-intellect factor abilities relating to performance in tenth grade modern geometry. Unpublished doctoral dissertation. University of Southern California, 1970. Dissertation Abstracts, 31A, 212-213.

Campbell, D. T. Distinguishing differences in perception from failures of communication in cross-cultural studies, In F. C. S. Northrop & H. H. Livingston (Eds.), Cross- 423

cultural understanding; Epistomology in anthropology. New York: Harper & Row, 1964.

Campbell, D. T. Parallelogram reproduction and the "carpentered world" hypothesis: Suggestions for cross- cultural research. Unpublished paper. Northwestern Uniyersity, 1969.

Campbell, D. T., & Stanley, J. C. Experimental and guasi- experimental designs for research. Chicago: Rand McNally, 1966.

Carlsmith, L. Effect of early father absence on scholastic aptitude. Haryard Educational Reyiew, 1964, 34, 3-21.

Carlson, J. A. The effect of instructions and perspectiye- drawing ability on perceptual constancies and geometrical illusions. Journal of Experimental Psychology, 1966, 72, 874-879.

Carpenter, F., Brinkraann, E. H. , & Lirones, D. S. Educa­ bility of students in the visualization of objects in space. Cooperatiye Research Project No. 1474, Uniyersity of Michigan, Ann Arbor, 1965.

Cassidy, F. G. Jamaica talk: Three hundred years of the English Language in Jamaica. (2nd ed.) New York: Macmillan, 1971.

Cattell, R. B. A culture-free test. Scale 3 . New York: The Psychological Corporation, 1944.

Chetyerukhin, N. F. An experimental investigation of pupils' spatial concepts and spatial imagination. In J. Kilpatrick & I. Wirszup (Eds.), Soviet studies in the psychology of learning and teaching mathematics. Vol. 5. Stanford, Calif.: School Mathematics Study Group, 1971.

Churchill, B. D ., Curtis, J. M. , Coombs, C. H., & Hassell, T. W. Effect of engineer school training on the surface development test. Educational and Psychological Measurement. 1942, 2, 279-280. 424

Clark, F. E. Effects of two learning treatments on the understanding of orthographic projection by students varying in visual-haptic aptitude. Unpublished doctoral dissertation, University of Missouri, 1971. Disserta­ tion Abstracts, 32, 5033A.

Clarke, E. Mv mother who fathered m e . London: Allen & Unwin, 1957.

Cleary, A. Jamaican proverbs. Kingston, Jamaica: Brainbuster Publications, 1971.

Cohen, L. An evaluation of a technique to improve space perception through the construction of models by students in a course in solid geometry. Unpublished doctoral dissertation, Yeshiva University, 1959. Dissertation Abstracts, 21, 1136.

Cole, M . , Gay, J., Click, J., & Sharp, D. The cultural context of learning and thinking: An exploration in experimental anthropology. New York: Basic Books, 1971.

Collins, A. J. (Ed.). Teachers' College Mathematics Project: Outline units. Workshop reports, UWl/UNESCO/ UNICEF/UNDP Project RLA/142, University of the West Indies, Mona, Jamaica, February & June 1973.

Combridge, J. T. The Mathematical Association reaches its first century. Part 2. Mathematics in Schools, 1972, 1 (2), 6-8 .

Cousins, D., & Abravanel, E. Some findings relevant to the hypothesis that topological features are differentiated prior to euclidean features during growth. British Journal of Psychology, 1971, 62, 475-479.

Covington, M. V. Some effects of stimulus familiarization on discrimination. Unpublished doctoral dissertation. University of California, 1962.

Cowley, J. J., & Murray, M. Some aspects of the development of spatial concepts in Zulu children. Journal for Social Research (Pretoria), 1962, 13, 1-28. 425

Cronbach, L. J., & Furby, L. How should we measure change - or should we? Psychological Bulletin, 1970, 74, 68-80.

Cryns, A. G. J. African intelligence: A critical survey of cross-cultural research in Africa South of the Sahara. Journal of Social Psychology, 1962, 57, 283-301.

Dasen, P. R. Cross-cultural Piagetian research: A summary. Journal of Cross-Cultural Psychology, 1972, 3, 23-29.

Dasen, P. R. The influence of ecology, culture and European contact on cognitiye development in Australian Aborigines. In J. W. Berry & P. R. Dasen (Eds.), Culture and cognition: Readings in cross-cultural psychology. London: Methuen, 1974.

Dastoor, D. P., & Emoyon, A. C. Performance of nine-year old children on block design test. West African Journal of Education, 1972, 16, 293-300.

Dayidson, H. P. A study of the confusing letters, B, D, P and Q. Journal of Genetic Psychology, 1935, 47, 458-468.

Dayis, C. M., & Carlson, J. A. A cross-cultural study of the strength of the Mueller-Lyer illusion as a function of attentional factors. Journal of Personality and Social Psychology, 1970, 16, 403-410.

Dayis, E. J. A study of the ability of selected school pupils to perceiye the plane sections of selected solid figures. Unpublished doctoral dissertation, Uniyersity of Florida, 1969. Dissertation Abstracts, 31A, 57.

Dawson, J. L. M. Kwashiorkor, gynaecomastia, and feminization processes. Journal of Tropical Medicine and Hygiene, 1966, 69, 175-179.

Dawson, J. L. M. Cultural and physiological influences upon spatial-perceptual processes in West Africa, Part I. International Journal of Psychology, 1967, 2, 115-125. (a) 426

Dawson, J. L. M. Cultural and physiological influences upon spatial-perceptual processes in West Africa. Part II. International Journal of Psychology, 1967, 2, 171- 185. (b)

Dawson, J. L. M. Theory and research in cross-cultural psychology. Bulletin of the British Psychological Association, 1971, 24, 291-306.

Dawson, J. L. M. Effects of sex hormones on cognitive styles in rats and men. Behavior Genetics, 1972, 2, 21-42. (a)

Dawson, J. L. M. Temne-Arunta hand/eye dominance and cognitive style. International Journal of Psychology, 1972, 7, 219-233. (b)

Dawson, J. L. M. Temne-Arunta hand/eye dominance and susceptibility to geometric illusions. Perceptual and Motor Skills, 1973, 37, 659-667.

Dawson, J. L. M., Young, B. M., & Choi, P. P. C, Developmental influences on geometric illusion sus­ ceptibility among Hong Kong Chinese children. Journal of Cross-Cultural Psychology, 1973, 4, 49-74.

Dawson, J. L. M., Young, B. M . , & Choi, P. P. C. Develcpmenal influences on pictorial perception among Hong Kong Chinese children. Journal of Cross-Cultural Psychology, 1974, 5, 3-22.

De Vries, R. Relationships among Piagetian, achievement, and intelligence assessments. Paper presented at the Annual Meeting of the American Educational Research Association, New Orleans, February 1973.

Denner, B., & Cashdan, S. Sensory processing and the recognition of forms in nursery school children. British Journal of Psychology, 1967, 58, 101-104.

Deregowski, J. B. Difficulties in pictorial perception in Africa. British Journal of Psychology, 1968, 59, 195- 204. (a) 427

Deregowski, J. B. Pictorial recognition in subjects from a relatively pictureless environment. African Social Re­ search, 1958, 5, 356-364. (b)

Deregowski, J. B. Preference for chain-type drawings in Zambian domestic servants and primary school children. Psycholoqia Africana, 1969, 12, 172-180.

Deregowski, J. B. Chain-type drawings: A further note. Perceptual and Motor Skills, 1970, 30, 102.

Deregowski, J. B. Orientation and perception of pictorial depth. International Journal of Psychology, 1971, 6, 111-114.

Deregowski, J. B. Drawing ability of Soli rural children: A note. Journal of Social Psychology, 1972, 86, 311-312. (a)

Deregowski, J. B. Pictorial perception and culture. Scientific American, 1972, 227 (5), 82-88. (b)

Deregowski, J. B. Reproduction of Kohs-type figures: A cross-cultural study. British Journal of Psychology, 1972, 63, 283-296. (c)

Dinnerstein, D. Preyious and concurrent visual experience as determinants of phenomenal shape. American Journal of Psychology, 1965, 78, 235-242.

Dodwell, P. C. Children's understanding of spatial con­ cepts. Canadian Journal of Psychology. 1963, 17, 141-161.

Du Toit, B. M. Pictorial depth perception and linguistic relativity. Psychologie Africans, 1966, 11, 51-53.

Dubin, E. The effect of training on the tempoof development of graphic representations in preschool children. Journal of Experimental Education, 1946, 15, 166-173.

Duncan, H. F., Gourlay, N., & Hudson, W. A study of pictorial perception among Bantu and white primary school children in South Africa. Johannesburg, S. Africa: Wit- watersrand University Press, 1973. 428

Dwyer, C. A. Sex differences in reading: An evaluation and a critique of current theories. Review of Educa­ tional Research, 1973, 43, 455-467.

Dyson-Hudson, R., & Dyson-Hudson, N. Subsistence herding in Uganda. Scientific American, 1969, 220(2), 76-89.

Eisner, E. W. The development of drawing characteristics of culturally advantaged and disadvantaged children. Project No. 3086, U. S. Department of Health, Education and Welfare, Office of Education Bureau of Research, 1967.

Eliot, J. Effects of age and training upon children's con­ ceptualization of space. Unpublished doctoral disserta­ tion, Stanford University, 1960. Dissertation Abstracts, 27, 2065A.

Eliot, J. Report on a spatial relations unit and per­ spective test. Council for Public Schools, Inc., Boston, Mass., 1963.

Eng, H. The psychologv of children's drawings. (2nd ed.) London: Routledge & Kegan Paul, 1954.

Esty, E. T. An investigation of children's concepts of certain aspects of topology. Unpublished doctoral dis­ sertation, Harvard University, 1970. Dissertation Abstracts, 3lA, 3773.

Ewart, J. P. The neural basis of visually guided behavior. Scientific American, 1974, 230(3), 34-42.

Faubian, R. W., Cleveland, E. A., & Hassell, T. W. The influence of training on mechanical aptitude test scores. Educational and Psychological Measurement, 1942, 2, 91-94.

Pestinger, L., Ono, H., Burnham, C. A., & Bamber, D. Efference and the conscious experience of perception. Journal of Experimental Psychology Monographs, 1967, 74 (4, Whole No. 637).

Figeroa, J. J. Society, schools and progress in the West Indies. Oxford: Pergamon, 1971. 429

Fisher, G. H, Developmental features of behavior and per­ ception: I. Visual and tactile-kinaesthetic shape perception. British Journal of Educational Psychology, 1965, 56, 69-78.

Fonseca, L., & Kearl, B. Comprehension of pictorial symbols: An experiment in rural Brazil. Bulletin No. 30, Department of Agricultural Journalism, University of Wisconsin College of Agriculture, 1960.

Forsdale, J. H., & Forsdale, L. Film literacy. Teacher's College Record. 1966, 67, 608-617.

Fraisse, P., & Vautrey, P. The influence of age, sex and specialized training on the vertical-horizontal illusion. Quarterly Journal of Experimental Psychology, 1956, 8, 114-120.

Franchek, S. J. Multiview orthographic projection concepts and the learner: Three instructional strategies. Unpub­ lished doctoral dissertation, Pennsylvania State Univer­ sity, 1971. Dissertation Abstracts, 32, 5100A.

Frederickson, W. A., & Geurin, J. Age difference in per­ ceptual judgment on the Mueller-Lyer illusion. Perceptual and Motor Skills, 1973, 36, 131-135.

Freeman, N. H., & Janikoun, R. Intellectual realism in children's drawings of a familiar object with distinctive features. Child Development, 1972, 43, 1116-1121.

French, J. W. The description of aptitude and achievement tests in terms of rotated factors. Psychometric Mono­ graphs, 1951, 5.

French, J. W . , Ekstrom, R. B., & Price, L. A. Kit of refer­ ence tests for cognitive factors. Princeton, N. J.: Educational Testing Service, 1963.

Frijda, N., & Jahoda, G. On the scope and methods of cross- cultural research. International Journal of Psychology, 1966, 1, 110-127.

Frostig, M., & Horne, D. The Frostig program for the development of visual perception. Chicago: Follett, 1964. 430

Gardner, R. W. Field-dependence as a determinant of sus­ ceptibility to certain illusions. American Psychologist, 1957, 12, 397.

Gardner, R. W. Cognitive controls of attention deployment as determinants of visual illusions. Journal of Abnormal and Social Psychologv. 1961, 62, 120-127.

Garron, D. C. Sex-linked, recessive inheritance of spatial and numerical abilities, and Turner's syndrome. Psychological Review, 1970, 77, 147-152.

Gay, J., & Cole, M. The new mathematics and an old culture. New York: Holt, Rinehart & Winston, 1967.

Ghiselli, E. E. The measurement of occupational aptitude. University of California Publications in Psychology, 1955, 8(2), 101-216.

Gibson, E. J. Principles of perceptual learning and development. New York: Appleton-Century-Crofts, 1969.

Gibson, E. J., Gibson, J. J., Pick, A. D., & Osser, H. A developmental study of the discrimination of letter­ like forms. Journal of Comparative Physiological Psychology, 1962, 55, 897-906.

Gibson, J. J The perception of the visual world. Boston: Houghton Mifflin, 1950.

Gibson, J. J. The senses considered as perceptual systems. Boston: Houghton Mifflin, 1966.

Goldstein, K., & Scheerer, M. Abstract and concrete be­ havior. An experimental study with special tests. Psychological Monographs, 1941, 53, 2 (Whole No. 239).

Gombrich, E. H. Art and illusion: A study in the psychology of pictorial representation. New York: Pantheon Books, 1960.

Gordon, S. C. A century of West Indian education: A source book. London: Longmans, Green, 1963. 431

Gough, H. G., & Olton, R. M. Field dependence as related to non-verbal measures of perceptual performance and cognitive ability. Journal of Consulting and Clinical Psychology. 1972, 38, 338-342.

Graham, F. K., Berman, P. W., & Ernhart, C. B. Develop­ ment in preschool children of the ability to copy forms. Child Development, 1960, 31, 339-359.

Grant, G. V. Spatial thinking: A dimension of African intellect. Psycholoqia Africana, 1970, 13, 222-239.

Grant, G. V. Conceptual reasoning: Another dimension of African intellect. Psycholoqia Africana, 1972, 14, 170-185.

Greenfield, P. M., & Bruner, J. S. Culture and cognitive growth. International Journal of Psychology, 1966, 1, 89-107.

Gregor, J. A., & McPherson, D. A. A study of susceptibility to geometric illusion among cultural subgroups of Australian aborigines. Psycholoqia Africana, 1965, 11, 1-13.

Gregory, R. L. Distortion of visual shape as inappropriate constancy scaling. Nature, 1963, 199, 678-680.

Gronwall, D. M. A., & Sampson, H. Ocular dominance. British Journal of Psychology, 1971, 62, 175-185.

Guignot, E., Mace, H., & Vurpillot, E. Influence de la consigne sur une mesure de constance de forme. Bulletin de Psychologie, 1963, 16, 11-12.

Guilford, J. P. The nature of human intelligence. New York: McGraw Hill, 1967.

Guilford, J. P., & Zimmermann, W. S. The Guilford-Zimmermann Aptitude Survey. Beverley Hills, Calif.: Sheridan, 1956.

Guthrie, G. Behavior and malnutrition. Paper presented at the Second International Conference of the International Association for Cross Cultural Psychology, Kingston, Ontario, August 1974. 432

Hanna, G. S. An investigation of selected ability, apti­ tude, interest, and personality characteristics relevant to success in high school geometry. Unpublished doctoral dissertation, University of Southern California, 1965. Dissertation Abstracts, 26, 3152-3153.

Harris, D. Children's drawings as measures of intellectual maturity. New York: Harcourt, Brace & World, 1963.

Harris, L. J. Discrimination of left and right, and the development of the logic of relations. Merri11-Palmer Quarterly. 1972, 18, 307-320.

Harris, L. J. Neurophysiological factors in spatial development. Paper presented to the Biennial Meetings of the Society for Research in Child Development, Philadelphia, March 1973.

Hathaway, W. E. The degree and nature of the relations between traditional psychometric and Piagetian develop­ mental measures of mental development. Paper presented to the Annual Meeting of the American Educational Research Association, New Orleans, February 1973.

Heiss, A. Zum problem der isolierenden abstraktion. Neue psychologie Studie, 1930, 4, 285-318.

Held, R. Plasticity in sensory motor systems. Scientific American, 1965, 213(5), 86-94.

Helmholtz, H. von. Handbook of physiological optics. Vol. 3. New York: Optical Society of America, 1925.

Henderson, K. B. (Ed.) Geometry in the mathematics curriculum. 36th Yearbook of the National Council of Teachers of Mathematics. Reston, Virginia: The Council, 1974.

Henriques, F. West Indian family organization. Caribbean Quarterly, 1967, 13(4), 31-40.

Henriques, F. Family and colour in Jamaica. (2nd ed.) London: MacGibbon & Kee, 1968. 433

Herskovits, M. J., Campbell, D. T., & Segall, M. H. A cross-cultural study of perception. (2nd ed.) Indianapolis: Bobbs-Merri11, 1969.

Hochberg, J. E., & Brooks, V. Pictorial recognition as an unlearned ability: A study of one child's performance. American Journal of Psychology, 1962, 75, 624-628.

Hoffman, W. C. The Lie algebra of visual perception. Journal of Mathematical Psychology, 1966, 3, 65-98.

Holmes, A. C. A study of understanding of visual symbols in Kenya. London: Overseas Visual Aids Centre, 1963.

Howard, I. P., & Templeton, W. B. Human spatial orienta­ tion. New York: John Wiley, 1966.

Hudson, W. Pictorial depth perception in subcultural groups in Africa. Journal of Social Psychology, 1960, 52, 183-208.

Hudson, W. Cultural problems in pictorial perception. South African Journal of Science, 1962, 58, 189-195. (a)

Hudson, W. Pictorial perception and educational adaptation in Africa. Psycholoqia Africana, 1962, 9, 226-239. (b)

Hudson, W. The study of the problem of pictorial perception among unacculturated groups. International Journal of Psychology. 1967, 2, 90-107.

Hurwitz, S. J., & Hurwitz, E. F. Jamaica; A historical portrait. New York: Praeger, 1971.

Husen, T. (Ed.) International study of achievement in mathematics. 2 vols. New York: Wiley, 1967.

Huttenlocher, J. Children's ability to order and orient objects. Child Development. 1967, 38, 1169-1176.

Irvine, S. H. Adapting tests to the cultural setting: A comment. Occupational Psychology. 1965, 39, 13-23.

Irvine, S. H. Contributions of ability testing in Africa to a general theory of intellect. Journal of Biosocial Science. Supplement, 1969, 1, 91-102. (a) 434

Irvine, S. H. Culture and mental ability. New Scientist, 1969, 42, 230-231. (b)

Irvine, S. H. The factor analysis of African abilities and attainments; Constructs across cultures. Psychological Bulletin. 1969, 71, 20-32. (c)

Irvine, S. H. Affect and construct: A cross-cultural check on theories of intelligence. Journal of Social Psychol­ ogv. 1970, 80, 23-30.

Isaacs, I. Some factors related to the performance in mathematics of third year students in Jamaican post­ primary schools. Unpublished Master's thesis. Univer­ sity of the West Indies, Mona, Jamaica, 1974.

Jaensch, E. R. Eidetic imagery. London: Kegan Paul, Trench & Trubner, 1930.

Jackson, D. N . , Messick, S., & Myers, C. T. Evaluation of group and individual forms of embedded-figures measures of field independence. Educational and Psychological Measurement. 1964, 24, 177-192.

Jahoda, G. Assessment of abstract behavior in a non- Western culture. Journal of Abnormal Social Psychology, 1956, 53, 237-243.

Jahoda, G. Geometric illusions and environment: A study in Ghana. British Journal of Psychology. 1966, 57, 193-199.

Jahoda, G. Cross-cultural use of the Perceptual Maze Test. British Journal of Educational Psychology. 1969, 39, 82-86.

Jahoda, G. Retinal pigmentation, illusion susceptibility and space perception. International Journal of Psychol­ ogy. 1971, 6, 199-208.

Jahoda, G., & Stacey, B. Susceptibility to geometrical illusions according to culture and professional train­ ing. Perception and Psychophysics, 1970, 7, 179-184. 435

Jeffrey, W. E. Variables in early discrimination learning: I. Motor responses in the training of left-right dis­ crimination. Child Development, 1958, 29, 269-275.

Jeffrey, W. E. Discrimination of oblique lines by children. Journal of Comparative and Physiological Psychology, 1966, 62, 154-156.

Jensen, A. R. Genetics and education. London: Methuen, 1972.

Jensen, A. R. Level I and level II abilities in three ethnic groups. American Educational Research Journal. 1973, 10, 263-276.

Kannegieter, R. B. The effects of a learning program in perceptual-motor activity upon the visual perception of shape. ERIC document No. ED 030 494, 1968.

Kellaghan, T. Abstraction and categorization in African children. International Journal of Psychology, 1968, 3, 115-120.

Kendall, I. M. The organization of mental abilities of a Pedi group in cultural transition. Johannesburg, S. Africa: National Institute for Personnel Research, 1971.

Kendall, I. M. A comparative study of the organization of mental abilities of two matched ethnic groups: The Venda and the Pedi. Johannesburg, S. Africa: National Insti­ tute for Personnel Research, 1972.

Kennedy, J. M. A psychology of picture perception: Images and information. San Francisco, Calif.: Jossey-Bass, 1973.

Kensler, G. L. The effects of perceptual training and modes of perceiving upon individual differences in ability to learn perspective drawing. Studies in Art Education, 1965, 34-41. 436

Kerr, M. Children's drawings of houses. British Journal of Medical Psychology. 1936, 16, 206-218.

Kerr, M. Personality and conflict in Jamaica. Liverpool: The University Press, 1952.

Kerschensteiner, G. Die Entwickelung der Zeichnerischen Begebunq. Munich: Gerbert, 1905.

Kershner, J. R. Children's spatial representation of directional movement and figure orientations along horizontal and vertical dimensions. Perceptual and Motor Skills. 1970, 31, 641-642.

Kidd, A. H., & Rivoire, J. L. The culture-fair aspects of the development of spatial perception. Journal of Genetic Psychology, 1965, 106, 101-111.

Kieren, T. E. Review of research on activity learning. Review of Educational Research, 1969, 39, 509-522.

Kilbride, P. L., & Robbins, M. C. Linear perspective, pictorial depth perception, and education among the Baganda. Perceptual and Motor Skills, 1968, 27, 601-602.

Kimura, D. The asymmetry of the human brain. Scientific American, 1973, 228(3), 70-78.

Klamkin, M. S. On the ideal role of an industrial mathe­ matician, and its educational implications. American Mathematical Monthly, 1971, 78, 53-76.

Kleinfeld, J. S. Intellectual strengths in culturally different groups: An Eskimo illustration. Review of Educational Research, 1973, 43, 341-359.

Kohs, S. C. Intelligence measurement. New York: MacMillan, 1923.

Laurendeau, M . , & Pinard, A. The development of the concept of space in the child. New York: International Univer­ sities Press, 1970. 437

Leibowitz, H., & Heisel, M. A. L'évolution de l'illusion de Ponzo on fonction del* âge. Archives de Psychologie, 1958, 36, 3 2 8 -3 3 1 .

Leibowitz, H. W . , Brislin, R., Perlmutter, L., & Hennessy, R. Ponzo perspective illusion as a manifesta­ tion of space perception. Science, 1969, 166, 1174-1176.

Leibowitz, H. W., & Pick, H. A. Cross-cultural and educa­ tional aspects of the Ponzo perspective illusion. Perception and Psychophysics. 1972, 12, 430-432.

Lester, S. Pictorial equivalents for spatial relations: A comparative analysis of drawings of American and Nigerian schoolchildren. Paper presented at the Second International Conference of the International Associa­ tion for Cross Cultural Psychology, Kingston, Ontario, August 1974.

Levy, J. Possible basis for the evolution of lateral specialization of the human brain. Nature, 1969, 224, 614-615.

Lewis, H. P. Developmental stages in children's representa­ tions of spatial relations in drawing. Studies in Art Education, 1962, 3(2), 69-76.

Lewis, H. Spatial representation in drawing as a corre­ late of development and a basis for picture preference. Journal of Genetic Psychology, 1963, 102, 95-107.

Ling, B.-C. Form discrimination as a learning cue in infants. Comparative Psychology Monographs, 1941, 17 2 (Whole No. 86).

Littlejohn, J. Temne space. Anthropological Quarterly, 1963, 63, 1-17.

Lloyd, B. B. Perception and cognition: A cross-cultural perspective. Harmondsworth, England: Penguin, 1972.

Loomis, W. F. Skin-pigment regulation of vitamin D bio­ synthesis in man. Science, 1967, 157, 510-516. 438

Lovell, K. A follow-up study of some aspects of the work of Piaget and Inhelder on the child's conception of space. British Journal of Educational Psychology, 1959, 29, 104-117.

Lowenfeld, V., & Brittain, W. L. Creative and mental growth. (4th ed.) New York: Macmillan, 1965.

Luquet, G.-H. Le dessin enfantin. Paris: Felix Alcan, 1927.

MacArthur, R. Sex differences in field dependence for the Eskimo: Replication of Berry's findings. International Journal of Psychology, 1967, 2, 139-140.

Maccoby, E. E. Sex differences in intellectual function­ ing. In E. E. Maccoby (Ed.), The development of sex differences. London: Tavistock, 1967.

Maccoby, E. E., & Bee, H. L. Some speculations concerning the lag between perceiving and performing. Child Development, 1965, 36, 367-378.

Maccoby, E. E., & Jacklin, C. N. Sex differences in in­ tellectual functionings. In Assessment in a pluralistic society. Proceedings of the 1972 Invitational Confer­ ence on Testing Problems. Princeton, N. J.: Educa­ tional Testing Service, 1973.

MacDonald, A. Selection of African personnel: Reports on the work of the Selection of Personnel Technical and Research Unit, M.E.F. London: War Office Archives, 1944-45.

Maistfiaux, R. La sous-evolution des noirs d'Afrique: Sa nature, ses causes, ses remedes. Revue Psychologique des Peuples. 1955, 10, 167-189 & 397-456.

Manley, D. R. Mental ability in Jamaica. Social and Economie Studies, 1963, 12, 51-71.

Manley, D. R. The School Certificate examination, Jamaica, 1962. Social and Economic Studies, 1969, 18, 54-71. 439

Martin, J. L. An analysis of Piaget's topological tasks from a mathematical point of view. Paper presented at the Annual Meeting of the National Council of Teachers of Mathematics, Houston, April 1973. (a)

Martin, J. L. An investigation of the development of selected topological properties in the representational space of young children. Paper presented at the Annual Meeting of the American Educational Research Association, New Orleans, February 1973. (b)

Matarazzo, J. D. Wechsler's Measurement and Appraisal of Adult Intelligence. (5th ed.) Baltimore: Williams & Wilkins, 1972.

McCallum, D. I. The relationship between psychological test-scores and mathematical attainment. Unpublished report. Inner London Education Authority Research Unit, 1970.

McClearn, G. E. Psychological research and behavioral phenotypes. In J. N. Spuhler (Ed.), Genetic diversity and human behavior. Chicago: Aldine, 1967.

McConnell, J. Abstract behavior among the Tepehuan. Journal of Abnormal Social Psychology, 1954, 49, 109-110.

McElroy, M. A. Aesthetic ranking tests with Arnheim Land aborigines. Bulletin of the British Psychological Society, 1955, 26, 44,

McFie, J. African performance on an intelligence test. Uganda Journal, 1954, 18, 34-43.

McFie, J. The effect of education on African performance in a group of intellectual tests. British Journal of Educational Psychology, 1961, 31, 232-240.

McGurk, E. Susceptibility to visual illusions. Journal of Psychology, 1965, 61, 127-143.

McGurk, H. Infant discrimination of orientation. Journal of Experimental Child Psychology, 1972, 14, 151-164. 440

Meizlik, F. Study of the effect of sex and cultural variables on field independence in a Jewish subculture. Unpublished Master's thesis. City University of New York, 1973.

Mendicino, L. Mechanical reasoning and space perception: Native capacity or experience? Personnel and Guidance Journal. 1958, 36, 335-338.

Meneghini, K. A., & Leibowitz, H. W. The effect of stimulus distance and age on shape constancy. Journal of Experimental Psychology, 1967, 74, 241-248.

Michael, W., Guilford, J., Fruchter, B. , & Zimmerman, W. The description of spatial-visualization abilities. Educational and psychological measurement, 1957, 17, 185-199.

Miller, E. Handedness and the pattern of human ability. British Journal of Psychology, 1971, 62, 111-112.

Miller, E. L. Ambitions of Jamaican adolescents and the school system. Caribbean Quarterly, 1967, 13(1), 29-33. (a)

Miller, E. L. A study of body image, its relationship to self concept, anxiety and certain social and physical variables in a selected group of Jamaican adolescents. Unpublished Master's thesis. University of the West Indies, Mona, Jamaica, 1967. (b)

Miller, E. L., Body image, physical beauty and colour among Jamaican adolescents. Social and Economic Studies, 1969, 18, 72-89.

Miller, E. L. Experimenter effect and the reports of Jamaican adolescents on beauty and body image. Social and Economic Studies, 1972, 21, 353-390.

Miller, J. W . , Boismier, J., & Hooks, J. Training in spatial conceptualization: Teacher-directed activities, automated and combination programs. Journal of Experimental Edu­ cation, 1969, 38, 87-92. 441

Miller, J. W., & Miller, H. G. A successful attempt to train children in coordination of projective space. George Peabody College for Teachers, Nashville, Tenn., 1970.

Ministry of Education. Annual report, 1972-'73. Kingston, Jamaica: The Ministry, 1973.

Mitchelmore, M. C. The Joint Schools Project. Mathematics Teaching. 1971, 54, 30-32.

Mitchelmore, M. C. Performance in a modern mathematics curriculum. West African Journal of Education, 1973, 17, 295-305. (a)

Mitchelmore, M. C. Spatial ability and three-dimensional drawing. Magazine of the Mathematical Association (Kingston, Jamaica), 1973, 5(2), 15-40. (b)

Mitchelmore, M. C., & Raynor, B. (Eds.) Joint Schools Project mathematics. 7 vols. London: Longman, 1967-75.

Mitchelmore, M. C., Raynor, B., & Isaacs, I. (Eds.) Joint Schools Project mathematics. Caribbean edition. 5 vols. London: Longman, 1970-1975.

Monroe, R. L., & Monroe, R. H. Effect of environmental experience on spatial ability in an East African society. Journal of Social Psychology, 1971, 83, 15-22.

Mundy-Castle, A. C. Pictorial depth perception in Ghanaian children. International Journal of Psychology, 1966, 1, 290-300.

Mundy-Castle, A. C., & Nelson, G. K. A neurophysiological study of the Knysua forest workers. Psychologia Africana, 1962, 9, 240-272.

Munro, T., Lark-Horoyitz, B., & Barnhart, E. N. Children's art abilities: Studies at the Cleveland Museum of Art. Journal of Experimental Education, 1942, 11, 97-155. 442

Mycock, R. W. The relationship between spatial ability and performance on some Piagetian tasks. (Summary) British Journal of Educational Psychology, 1969, 39, 196-197.

Myers, C. T. Some observations and opinions concerning spatial relations tests. Paper presented to the Annual Convention of the American Psychological Association, New York, 1957.

Myers, C. T. The effects of training in mechanical drawing on spatial relations test scores as predictors of engin­ eering drawing grades. Research and Development Report RB-58-4, Educational Testing Service, Princeton, N. J., 1958.

Nerlove, S. B., Monroe, R. H., & Monroe, R. L. Effect of environmental experience on spatial ability: A repli­ cation. Journal of Social Psychology, 1971, 84, 3-10.

Nissen, H. W . , Machover, S., & Kinder, E. F. A study of performance tests given to a group of native African negro children. British Journal of Psychology, 1935, 25, 308-355.

Oh, C. Y. A study of the development of children's ability to perceive depth in static two-dimensional pictures. Unpublished doctoral dissertation, Indiana University, 1969. Dissertation Abstracts 29B, 3470.

Ombredane, A., Bertelson, P., & Beniest-Noirot, E. Speed and accuracy of performance of an African native popula­ tion and of Belgian children on a paper-and-pencil per­ ceptual task. Journal of Social Psychology, 1958, 47, 327-337.

Okonji, M. O. Culture and children's understanding of geometry. International Journal of Psychology, 1971, 6, 121-128.

Olson, D. R. Cognitive development: The child's acouisi- tion of diaqonality. New York: Academic Press, 1970. 443

Oltman, P. K. & Capobianco, F. Field dependence and eye dominance. Perceptual and Motor Skills, 1967, 25, 645-646,

Ord, I. G. Manual for Pacific Design Construction Test. Hawthorn, Victoria, Australia: Australian Council for Educational Research, 1968.

Ord, I. G. Mental tests for pre-literates. London: Ginn, 1970.

Osborne, A. R., & Crosswhite, F. J. Forces and issues related to curriculum and instruction, 7-12. In P. S. Jones & A. F. Coxford (Eds.), A history of mathematics education in the United States and Canada. 32nd Year­ book of the National Council of Teachers of Mathematics. Washington B.C.: The Council, 1970.

Osborne, R. T., & Gregor, A. J. Racial differences in heritability estimates for tests of spatial ability. Perceptual and Motor Skills, 1968, 27, 735-739.

Over, R. Explanations of geometrical illusions. Psycho­ logical Bulletin, 1968, 70, 545-562.

Page, E. I. Haptic perception: A consideration of one of the investigations of Piaget and Inhelder. Educational Review, 1959, 11, 115-124.

Palow, W. P. A study of the ability of public school students to visualize particular perspectives of selected solid figures. Unpublished doctoral dissertation. University of Florida, 1959. Dissertation Abstracts, 31A, 78.

Peel, E. A. Experimental examination of some of Piaget's schemata concerning children's perception and thinking, and a discussion of their educational significance. British Journal of Educational Psychology, 1959, 29, 89-103.

Petitclerc, G. The 3-D test for visualization skill. San Raphael, California: Academic Therapy Publications, 1972. 444

Piaget, J. Judgment and reasoning in the child. New York: York: Harcourt Brace, 1924.

Piaget, J. The mechanisms of perception. London: Rout­ ledge & Kegan Paul, 1969.

Piaget, J., & Inhelder, B. The child's conception of space. New York: Norton, 1967.

Pollack, R. H. Contour detectibility thresholds as a function of chronological age. Perceptual and Motor Skills, 1963, 17, 411-417.

Pollack, R. H., & Silvar, S. D. Magnitude of the ML illu­ sion in children as a function of the pigmentation of the Fundus Oculi. Psychonomic Science. 1967, 8, 83-84.

Poole, H. E. The effect of urbanisation upon concept attainment among Hausa children in Northern Nigeria. British Journal of Educational Psychology, 1968, 38, 57-63.

Ranucci, E. The effect of the study of solid geometry on certain aspects of space perception abilities. Unpub­ lished doctoral dissertation. Teachers College, Columbia, 1952. Dissertation Abstracts, 12, 662-653.

Raven, J. C. Standard Progressive Matrices: Sets A, B, C, D and E . London: Lewis, 1958.

Rebelsky, F. Adult perception of the horizontal. Per­ ceptual and Motor Skills, 1964, 19, 371-374.

Rengstorff, R. H. The types and incidence of hand-eye preference and its relation to certain reading abilities American Journal of Optometry, 1967, 44, 233-238.

Rennels, M. R. The effects of instructional methodology in art education upon achievement on spatial tasks by disadvantaged Negro youths. Journal of Negro Education, 1970, 39, 116-123.

Reuning, H., & Wortley, W. Psychological studies of the Bushmen. Psychologia Africana Monograph Supplement, 1973, No. 7. 445

Rivoire, J. L. Development of reference systems in children. Unpublished doctoral dissertation. University of Arizona, 1961.

Rouma, G. Le langage graphique de 1*enfant (2nd ed.) Brussels: Misch & Tron, 1913.

Rudel, R. G., & Teuber, H. L. Discrimination of direction of line in children. Journal of Comparative Physiologi­ cal Psychology, 1963, 56, 892-898.

Ruscoe, G. C. Dysfunctionality in Jamaican education. Un­ published doctoral dissertation. University of Michigan, 1963. Dissertation Abstracts, 24, 2328.

Salome, R. H. The effects of perceptual training upon the two-dimensional drawings of children. Studies in Art Education, 1965, 7, 18-33.

School Mathematics Study Group. Minimum goals for mathe­ matics education. SMSG Newsletter no. 38, August 1972.

Schwarz, P. A. Aptitude tests for use in the developing nations. Pittsburgh, Pa.: American Institute for Research, 1961.

Schwarz, P. A. Adapting tests to the cultural setting. Educational and Psychological Measurement, 1963, 23, 673-685.

Schwarz, P. A. Development of manpower screening tests for the developing nations: A manual of scoring procedures and norms ... Pittsburgh, Pa.: American Institutes for Research in the Behavioral Sciences, 1964. (a)

Schwarz, P. A. Development of manpower screening tests for the developing nations: A technical manual and re­ port ... Pittsburgh, Pa.: American Institutes for Research in the Behavioral Sciences, 1964. (b)

Schwarz, P. A., & Krug, R. E. Ability testing in developing countries : A handbook of principles and techniques. New York: Praeger, 1972.

Schwitzgebel, R. The performance of Dutch and Zulu adults on selected perceptual tasks. Journal of Social Psychol­ ogy, 1962, 57, 73-77. 446

Segall, M. H. Nutrition and cognitive development: A design for research. Paper presented at the Second International Association for Cross Cultural Psychology, Kingston, Ontario, 1974.

Segall, M. H., Campbell, D. T., & Herskovits, M. J. The influence of culture on visual perception. Indianapolis, Indiana: Bobbs-Merri11, 1966.

Serpell, R. How perception differs among cultures. New Society. 1972, 20, 620-623.

Serpell, R., & Deregowski, J. B. Teaching pictorial depth perception: A classroom experiment. Human Development Research Unit Reports No. 21, University of Zambia, Lusaka, 1972.

Servais, W . , & Varga, T. (Eds.) Teaching school mathe­ matics . Harmondsworth, England: Penguin, 1971.

Shapiro, M. B. The rotation of drawings by illiterate Africana. Journal of Social Psychology, 1960, 52, 17-30.

Shaw, B. Visual symbols suryey. London: Centre for Educational Development Overseas, 1969.

Shemyakin, F. N. Orientation in space. In B. G. Anan'yev et al.. Psychological Science in the U.S.S.R., Vol. I. Washington, D.C.: U. S. Joint Publications Research Service, 1961.

Sherman, J. Problems of sex differences in space per­ ception and aspects of intellectual functioning. Psychological Review, 1967, 74, 290-299.

Siemankowski, F. T., & Macknight, F. C. Spatial cognition, a success prognosticator in college science courses. Paper presented to the Annual Meeting of the National Association for Research in Science Teaching, 1971.

Silvar, S. D., & Pollack, R. H. Racial differences in pigmentation of the Fundus Oculi. Psychonomic Science, 1967, 7, 159-160. 447

Silverman, R. Comparing the effects of two- versus three- dimensional art activity upon spatial visualization, aesthetic judgment and art interest. Unpublished doctoral dissertation, Stanford University, 1962. Dissertation Abstracts, 23, 2017.

Skemp, R. R. The psychology of learning mathematics. Harmondsworth, England: Penguin, 1971.

Smith, I. M. Spatial ability: Its education and social significance. London: University of London Press, 1964.

Smith, I. M. The use of diagnostic tests for assessing the abilities of overseas students attending institutions of further education. The Vocational Aspects of Educa­ tion, 1970, 22, 1-8.

Smith, I. M. The use of diagnostic tests for assessing the abilities of overseas students attending institu­ tions of further education: Part II. The Vocational Aspect of Education, 1971, 23, 39-48.

Smith, I. M- , & Lawes, J. S. Spatial Test 3 (Newcastle Spatial Test). Slough, England: National Foundation for Educational Research, 1959.

Smith, M. G. Education and occupational choice in rural Jamaica. Social and Economic Studies, 1960, 9, 343-354.

Smith, M. G. West Indian family structure. University of Washington Press, 1962.

Smith, M. G. The plural society in the British West Indies. Berkeley, California: University of California Press, 1965.

Snider, J. G. Aptitude tests for West Africa. West African Journal of Education, 1972, 16, 171-177.

Sowder, L. High versus low geometry achievers in the N.L.S.M.A. Y-population. Journal for Research in Mathematics Education, 1974, 5, 20-27. 448

Spaulding, S. Communication potential of pictorial illustrations. Audio-Visual Communication Review, 1956, 4, 31-46.

Stallings, W. M. The effects of teaching descriptive geometry in General Engineering 103 on spatial relations test scores. Research Report No. 280, Measurement and Research Division, Office of Instructional Research, University of Illinois, 1968.

Stafford, R. E. Sex differences in spatial visualization as evidence of sex-linked inheritance. Perceptual and Motor Skills. 1961, 13, 428.

Stern, W. Uber verlagte Raumformen. Zeitschrift fur anqewandte Psychologie, 1909, 2, 498-526.

Stern, W. Die zeichnerische Entwicklung eines Knaben vom 4. bis zum 7. Jahre. Zeitschrift fur angewandte Psychologie, 1910, 3, 1-31.

Stewart, V. M. A cross-cultural test of the "carpentered environment" hypothesis using three geometric illusions in Zambia. Unpublished doctoral dissertation. Northwestern University, 1971. Dissertation Abstracts, 32A, 5340.

Stewart, V. M. Tests of the "carpentered world" hypothesis by race and environment in America and Zambia. Inter­ national Journal of Psychology, 1973, 8, 83-94.

Stewart, V. M. A cross-cultural test of the "carpentered world" hypothesis using the Ames Distorted Room illusion. International Journal of Psychology, 1974, in press. (a)

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. (b)

Strayer, J., & Ames, E. W. Stimulus orientation and the apparent lag between perception and performance. Child Development, 1972, 43, 1345-1354. 449

Strodtbeck, F. Considerations of meta-method in cross- cultural studies. American Anthropologist, 1964, 66, 223-229.

Stuart-Mason, A. Health and hormones. Harmondsworth, England: Penguin, 1963.

Taylor, T. R. Optical illusions: Their relationship to field dependence, their theory, and their factor structure. Johannesburg, S. Africa: National Institute for Personnel Research, 1972.

Temp1er, A. J. The relationship between field dependence- independence and concept attainment. Psycholoqia Africana, 1972, 14, 121-129.

Terman, L. M., & Merrill, M. A. Measuring intelligence. Boston, Mass.: Houghton Mifflin, 1937.

Thouless, R. H. A racial difference in perception. Journal of Abnormal & Social Psychology, 1933, 4, 330- 339.

Thurstone, L. L. Primary mental abilities. Psychometric Monographs, 1938, 1.

Tuddenham, R. D. A "Piagetian" test of cognitive develop­ ment. In W. Bo Dockrell (Ed.), On intelligence. Toronto: Ontario Institute of Studies in Education, 1970.

Turner, M. L. The learning of symmetry principles and their transfer to tests of spatial ability. Unpublished doctoral dissertation. University of California, Berkeley, 1967. Dissertation Abstracts, 29, 1140A.

United Nations Educational, Scientific and Cultural Organization. East African Regional Workshop on Book Production, 1962: A report. Paris: The Organization, 1963.

United Nations Special Fund Project, "Assistance in Physical Development Planning." National atlas of Jamaica. Kingston, Jamaica: Department of Town Plan­ ning, Ministry of Finance and Planning, 1971. (a) 450

United Nations Special Fund Project, "Assistance in Physical Development Planning." Urban structure and policy. Kingston, Jamaica: Department of Town Planning, Ministry of Finance and Planning, 1971. (b)

University of the State of New York. Experimental unit on space visualization. Albany, N.Y.: The State Educa­ tion Department, 1967.

Van Voorhis, W. R. The improvement of space perception ability by training. Unpublished doctoral dissertation, Pennsylvania State University, 1941.

Vernon, P. E. The structure of human abilities. London: Methuen, 1961.

Vernon, P. E. Environmental handicaps and intellectual development. British Journal of Educational Psychology, 1965, 35, 9-20 & 117-126.

Vernon, P. E. Educational and intellectual development among Canadian Indians and Eskimos. Educational Review, 1966, 18, 79-91.

Vernon, P. E. Administration of group intelligence tests to East African pupils. British Journal of Educational Psychology, 1967, 31, 282-291. (a)

Vernon, P. E. Abilities and educational attainments in an East African environment. Journal of Special Education. 1967, 1, 335-345. (b)

Vernon, P. E. Intelligence and cultural environment. London: Methuen, 1969.

Walsh, J. F., & Dangelo, R. Effectiveness of Frostig Program for Visual Perceptual Training with Head Start children. Perceptual and Motor Skills, 1971, 32, 944-946.

Walters, A. A genetic study of geometrical-optical illusions. Genetic Psychology Monographs, 1942, 25, 101-155o 451

Wapner, S. & Werner, H. Perceptual development; An inves­ tigation within the framework of sensory-tonic field theory. Worcester, Mass.: Clark University Press, 1957.

Watts, A. F., Pidgeon, D. A., & Richards, M. K. B. Spatial Test 2 (Three-dimentional). Slough, England: National Foundation for Educational Research, 1951.

Wechsler, D. Wechsler Intelligence Scale for Children: Manual. New York: The Psychological Corporation, 1949.

Werdelin, I. Geometrical ability and the space factors in boys and girls. Lund, Sweden: C. W. K. Gleerups, 1961.

West Indian Population Census. Census of population of Jamaica. Kingston, Jamaica: Department of Statistics,1961.

Westbrook, H. R. Intellectual processes related to mathe­ matics achievement at grade levels 4, 5 and 6. Unpub­ lished doctoral dissertation. University of Georgia, 1965. Dissertation Abstracts 26, 6520.

Western New Mexico University. A study of visual per­ ceptions in early childhood. ERIC document No. ED 023 451, 1967.

White, A., Handler, P., & Smith, E. L. Principles of bio­ chemistry. (3rd ed.) New York: McGraw Hill, 1964.

Whorf, B. L. Science and linguistics. The Technical Review, 1940, 40, 229-231 & 247-248.

Whorf, B. L. Language, thought, and reality. Cambridge, Mass.: M.I.T. Press, 1956.

Williams, J. P. Effects of discrimination and reproduction training on ability to discriminate letter-like forms. American Educational Research Journal, 1969, 6, 501-514.

Williams, R. L. The supply of essential skills in less developed countries. Mona, Jamaica : Institute of Social and Economic Research, University of the West Indies, 1965. 452

Williford, H. J. A study of transformational geometry instruction in the primary grades. Unpublished doctoral dissertation. University of Georgia, 1970. Dissertation Abstracts. 31, 6462A.

Wilson, J. W . , Cahen, L. S., & Begle, E. G. N.L.S.M.A. Reports No. 1; X-population test batteries. Stanford, Calif.: School Mathematics Study Group, 1958.

Winer, B. J. Statistical principles in experimental design. (2nd ed.) New York: McGraw Hill, 1971.

Winter, W. The perception of safety posters by Bantu indus­ trial workers. Psycholoqia Africana, 1963, 10, 127-135.

Witkins, H. A. Individual differences in ease of perception of embedded figures. Journal of Personality, 1950, 19, 1-15.

Witkin, H. A., Dyk, R. B., Paterson, H. P., Goodenough, D. R., & Karp, S. A. Psychological differentiation: Studies of development. New York: Wiley, 1962.

Witkin, H. A., Oltman, P. K., Cox, P. W., Ehrlichman, E., Hamm, R. M. , & Ring1er, R. W. PieId-dependence- independence and psychological differentiation: A bib­ liography through 1972 with index. Princeton, N. J.: Educational Testing Service, 1973.

Wober, M. The meaning and stability of Raven's Matrices test among Africans. International Journal of Psychology, 1969, 4, 229-235.

Wober, M. Confrontation of the H-V illusion and a test of 3-dimensional perception in Nigeria. Perceptual and Motor Skills. 1970, 31, 105-106.

Wohlwill, J. P. Developmental studies of perception. Psychological Bulletin, 1960, 57, 249-288.

Wohlwill, J. P. The perspective illusion: Perceived size and distance in fields varying in suggested depth, in children and adults. Journal of Experimental Psychology, 1962, 64, 300-310. 453

Wolfe, L. R. The effects of space visualization training on spatial ability and arithemtic achievement of Junior High School students. Unpublished doctoral dissertation. State University of New York, Albany, 1970. Disserta­ tion Abstracts. 31A, 2801.

Wrigley, J. The factorial nature of ability in elementary mathematics. British Journal of Educational Psychology, 1958, 28, 61-78.

Wursten, H. Recherches sur le développement des perceptions, IX. Archives de Psychologie, 1947, 32, 1-144.

Zaporozhets, A. V. The development of perception in the preschool child. Monographs of the Society for Research in Child Development, 1965, 30, 82-101.

Zimmermann, W. S. The influence of item complexity upon the factor composition of a spatial visualization test. Educational and Psychological Measurement, 1954, 14, 106. (a).

Zimmermann, W. S. Hypotheses concerning the nature of the spatial factors. Educational and Psychological Measure­ ment, 1954, 14, 396-406. (b). THE PERCEPTUAL DEVELOPMENT OF JAMAICAN STUDENTS, WITH SPECIAL REFERENCE TO VISUALIZATION AND DRAWING OF THREE-DIMENSIONAL GEOMETRICAL FIGURES AND THE EFFECTS OF SPATIAL TRAINING

VOLUME II

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By

Michael Charles Mitchelmore, M.A., Cert. Ed.

*****

The Ohio State University 1974

Reading Committee: Approved by

F. Joe Crosswhite

Fred L. Damarin, Jr.

Dean H. Owen ______Adviser Richard J. Shumway ‘'College of Education CONTENTS

Page

APPENDIX A. SUPPLEMENTARY TABLES: TEST DEVELOPMENT...... 454

A.1 Pilot-testing timetable for group tests . . 455 A.2 Pilot-testing timetable for individual t e s t s ...... 455 A.3 Item analysis of scores on first trial version of 3DD...... 457 A.4 Means and standard deviations of scores on second and final versions of 3DD, by s c h o o l ...... 458 A.5 Item statistics for high and low scorers on 3DD...... 459 A .6 Item analysis of scores on NLSMA Hidden Figures Test...... 460 A.7 Item analysis of scores on trial version of HST...... 461 A .8 Social classification of occupations in J a m a i c a ...... 462 A.9 Occupations classified as spatial/ technical and verbal/personal ...... 463

APPENDIX B. SUPPLEMENTARY TABLES: DEVELOPMENTAL SURVEY...... 464

B.l Academic indexes of Kingston high schools . 465 B.2 Academic indexes of 25 highest-ranking Kingston mixed elementary schools ...... 466 B.3 Summary data on schools included in developmental survey...... 467 B.4 Numbers of untypical students tested in each school ...... 468 B.5 Frequency distribution of DOT scores. . . . 469 B .6 Item analysis of DOT scores ...... 469 B.7 Means and standard deviations of spatial test scores, by grade and s e x ...... 470 B .8 Trend analysis of spatial test scores, by grade and sex...... 472

xxviii CONTENTS (continued)

Page

APPENDIX B (continued)

B.9 Means and standard deviations of DOT and HVT scores in Grade 5 check testing, by s e x ...... 473 B.IO Frequency distribution of HFT scores. . . . 473 B.Il Item analysis of HFT scores ...... 474 B.12 Frequency distribution of Poles median inclinations, by section...... 474 B.13 Frequency distribution of Poles median inclinations, by grade and section...... 475 B.14 Frequency distributions of Poles and Bottles scores...... 476 B.15 Frequency distribution of Bottles inclinations, by grade and bottle ...... 477 B.16 Means and standard deviations of Poles and Bottles scores in Grade 9 high school samples, by s e x ...... 478 B.17 Cross-tabulation of SRT scores in Conditions 1 and 2, by solid ...... 479 B.IB Cross-tabulation of SRT scores in Conditions 2 and 3, by solid...... 480 B.19 Frequency distribution of SRT scores in Condition 2, by grade and solid ...... 481 B.20 Frequency distribution of SRT scores in Condition 3, by grade and solid ...... 482 B.21 Repeated measures analysis of SRT school mean scores by solid, grade, sex and condi­ tion (Conditions 1 and 2 ) ...... 483 B.22 Frequency distributions of SRT drawing and selection scores...... 484 B.23 Numbers of students with two or more Guttman errors on GIM illusions, by grade . 484 B.24 Trend analysis of GIM scores, by grade, sex and order ...... 485 B.25 Means and standard deviations of GIM scores, by grade and sex...... 487 B.26 Frequency distributions of background variables, by grade ...... 488

xxix CONTENTS (continued) Page

APPENDIX B (continued)

B.27 Analysis of variance of spatial test scores, by hand and eye dominance...... 489 B.28 Mean spatial test scores, by hand and eye dominance...... 490

APPENDIX C. SUPPLEMENTARY TABLES: GRADE NINE SURVEY. 491

C.l Grade Nine survey population ...... 492 C.2 Schools in survey area eliminated from survey population...... 495 C.3 Means and standard deviations of spatial test scores in eliminated village "high s c h o o l s " ...... 495 C.4 Summary data on schools included in Grade Nine survey...... 497 C.5 3DD inter-item correlations...... 498 C .6 Frequency distributions of spatial test s c o r e s ...... 498 C.l Item analysis of 3DD scores...... 499 0.8 Item analysis of HST scores...... 500 C.9 Means and standard deviations of 3DD scores, by school and sex...... 501 C.IO Means and standard deviations of HST scores, by school and sex...... 502 C.ll Means and standard deviations of BOX scores, by school and sex...... 503 C.12 Academic indexes of non-Kingston high schools...... 504 C.13 Mean spatial test scores in combined city and town school data, by sex and school type...... 505 C.14 Analysis of variance of spatial test scores in combined city and town school data, by sex and school t y p e ...... 506 C.15 Numbers of untypical students eliminated from junior secondary and all-age schools. . 507 C.16 Mean spatial test scores in refined junior secondary and all-age school data, by sex, environment and school t y p e ...... 508

XXX CONTENTS (continued)

Page

APPENDIX C (continued)

C.17 Analysis of variance of spatial test scores in refined junior secondary and all-age school data, by sex, environment and school type...... 508

C.18 Summary statistics on background variables for all Grade Nine students tested...... 510

APPENDIX D. SUPPLEMENTARY TABLES: SPATIAL TRAINING EXPERIMENT ...... 511

D.l Experimental timetable: First year ...... 512 D.2 Experimental timetable: Second year...... 514 D.3 Means and standard deviations of HST pre­ test scores of students with full data and with missing data, by year and sex...... 516 D.4 Correlations between spatial tests and Solids achievement test scores in first year Statistics group ...... 516 D.5 Correlations between spatial tests and Solids achievement test scores in second year Statistics group ...... 517 D .6 Frequency distributions of spatial pre­ test scores, by year...... 517 D.7 Item analysis of 3DD pretest scores in first year...... 518 D .8 Item analysis of 3DD pretest scores in second y e a r ...... 519 D.9 F-values for successive terms in step­ wise polynomial regression of 3DD and BOX posttest scores on corresponding pretest, by year and s e x ...... 520 D.IO Correlations between spatial tests and Solids achievement test scores in first year Solids group ...... 521 D.ll Correlations between spatial tests and Solids achievement test scores in second year Solids group ...... 521

xxxi CONTENTS (conti nued)

Page

APPENDIX D (continued)

D.12 Mean scores on 3DD and BOX posttests in first year, by level, sex, treatment and pretest condition ...... 5 22 D.13 Mean scores on 3DD and BOX posttests in second year, by tutor, sex, treatment and pretest condition...... 523 D.14 Analysis of variance of 3DD and BOX posttest scores in first year, by level, sex, treatment and pretest condition. . . . 524 D.15 Analysis of variance of 3DD and BOX posttest scores in second year, by tutor, sex, treatment and pretest condition. . . . 525 D.16 Mean scores on 3DD and BOX posttests in first year, adjusted for covariance with corresponding pretest, by level, sex and t r e a t m e n t ...... 526 D.17 Mean scores on 3DD and BOX posttests in second year, adjusted for covariance with corresponding pretest, by tutor, sex and treatment ...... 527 D.18 Analysis of covariance of 3DD and BOX posttest scores in first year, using corresponding pretest as covariate, by level, sex and treatment...... 528 D.19 Analysis of covariance of 3DD and BOX posttest scores in second year, using corresponding pretest as covariate, by tutor, sex and treatment...... 529 D.20 Analysis of covariance of HST posttest scores in first year, using HST pretest scores as covariate, by level, sex, treat­ ment and pretest condition...... 530 D.21 Mean scores on HST posttest in first year, adjusted for covariance with HST pretest, by level, sex and treatment. . . . 531 D.22 Mean scores on achievement tests in first year, by level, sex and treatment . . 531

X X X I 1 CONTENTS (continued)

Page APPENDIX D (continued)

D.22 Mean scores on achievement tests in first year, by level, sex and treatment. . 531 D.23 Mean scores on achievement tests in second year, by tutor, sex and treatment . 532 D.24 Analysis of variance of achievement test scores in first year, by level, sex, treatment and pretest condition ...... 5 33 D.25 Analysis of variance of achievement test scores in second year, by tutor, sex, treatment and pretest condition . . . 534 D.26 Analysis of variance of achievement test class mean scores, by level/tutor and treatment ...... 5 35 D.27 Correlations between rate of working and scores on Solids achievement test. . . 535 D.28 Analysis of variance and means of spatial pretest scores in first year, by sex and category of previous education...... 5 35 D.29 Analysis of variance and means of spatial pretest scores in second year, by sex and category of previous education . . 5 37 D.30 Means and standard deviations of spatial pretest scores, by sex and year...... 538 D.31 Means and standard deviations of HFT, SRT drawing, and GIM scores, by sex...... 539 D.32 Means and standard deviations of HFT and SRT drawing scores in Grade Nine and college samples, by sex...... 539

APPENDIX E. PRELIMINARY VERSIONS OF GROUP TEST INSTRUMENTS...... 540

Three-Dimensional Drawing test (Trial version 1). 541

Three-Dimensional Drawing test (Trial version 2). 55 3

NLSMA Hidden Figures T e s t ...... 559

Hidden Shapes Test (Trial version)...... 566

xxxiii CONTENTS (continued)

Page

APPENDIX F. GROUP TEST INSTRUMENTS...... 568

Three-Dimensional Drawing test ...... 569

Hidden Shapes T e s t ...... 574

I-D Boxes t e s t ...... 576

Personal Data Questionnaire...... 579

APPENDIX G. INSTRUCTIONS FOR ADMINISTERING GROUP TESTS...... 580

Opening statement...... 581

Three-Dimensional Drawing test ...... 582

Hidden Shapes T e s t ...... 586

I-D Boxes t e s t ...... 588

Personal Data Questionnaire...... 590

APPENDIX H. INDIVIDUAL TEST MATERIALS...... 593

Record form...... 594

Hidden Figures Test ...... 600

APPENDIX I. INSTRUCTIONS FOR ADMINISTERING INDIVIDUAL TESTS ...... 601

Opening statement...... 602

Eye-Hand Dominance Schedule...... 602

Design Construction Test ...... 604

Hidden Figures Test...... 604

Horizontal-Vertical Test ...... 605

xxxiv CONTENTS (continued)

Page

APPENDIX I (continued)

Solid Representation Test...... 605

Geometric Illusion Measures...... 607

Personal Data Interview...... 607

APPENDIX J. INSTRUCTIONS FOR SCORING DRAWING TESTS . 608

Three-Dimensional Drawing test ...... 609

Solid Representation T e s t ...... 615

APPENDIX K. MATERIALS FOR PILOT UNIT (SPATIAL TRAINING EXPERIMENT) ...... 619

Workcards...... 520

Text p a g e s ...... 624

Resource cards ...... 632

APPENDIX L. MATERIALS FOR SOLIDS UNIT (SPATIAL TRAINING EXPERIMENT) ...... 642

Workcards...... 643

Text p a g e s ...... 649

Resource cards ...... 665

Solids achievement test...... 670

APPENDIX M. MATERIALS FOR STATISTICS UNIT (SPATIAL TRAINING EXPERIMENT) ...... 675

Workcards...... 676

Text p a g e s ...... 679

XXXV CONTENTS (continued)

Page

APPENDIX M (continued)

Resource cards ...... 691

Statistics achievement test...... 694

APPENDIX N. MISCELLANEOUS ADMINISTRATIVE DOCUMENTS (SPATIAL TRAINING EXPERIMENT)...... 696

Student information sheet...... 697

Instructions for administering HST posttest. . . . 699

Teaching Methods Questionnaire ...... 700

xxxvx APPENDIX A

SUPPLEMENTARY TABLES; TEST DEVELOPMENT

454 455

Table A.l

Pilot-testing timetable for group tests

Date School Grade N Tests^ (1973)

May St. Mary Elementary School, Columbus 7 19 3DD

May Bishop Ready High School, Columbus 9 53 3DD

June Immaculate Conception High School, Columbus 7 46 3DD

November Windward Road All-Age School, Kingston 7 47 HFT, 3DD

November Immaculate Conception High School, Kingston 9 38 HFT, 3DD

November St. George's College, Kingston 9 37 HFT, 3DD

November St. Michael's All-Age School, Kingston 9 35 3DD, HST

November Ardenne High School, Kingston 9 35 HST, BOX

November Maverley All-Age School, Kingston 9 30 BOX, PDQ

November Tarrant Junior Secondary School, Kingston 9 38 3DD, HST

HFT = NLSMA Hidden Figures Test; for other abbreviations, see Section 4.35. 456

Table A.2

Pilot-testing timetable for individual tests

Date School______Grade N Tests a

May Devonshire Road Elementary 1-6 60 SRT School, Columbus

November Hope Valley Experimental 3&5 ' HVT, SRT, School, Kingston PDI, HFT, EDS, DAP

November Windward Road All-Age School, HVT, SRT, Kingston PDI, HFT, EDS, DAP, GIM

November St. George's College, HFT, SRT, Kingston PDI, DAP, EDS, GIM

November Immaculate Conception High 9 HFT, SRT, School, Kingston GIM

December Hope Valley Experimental 3&6 PDP, DCT, School, Kingston HFT

^DAP = Draw-a-Person test, PDP = Pictorial Depth Per­ ception test; for other abbreviations, see Section 4.49, Table A.3

Item analysis of scores on first trial version of 3DD

Item-test Item difficulty^ Subitem difficulty^ Item correlation^ number Grade 7 Grade 9 Grade 7 Grade 9 Grade 7 Grade 9 1&2^ 48 57 57,48,40 67,64,44 62 47 3d 26 30 42, 47, 20, 13, 10 55,40,28, 10,08 63 76 4d 39 51 64, 35,50, 31, 20 73,46,58,40,38 74 77 5 32 37 52, 23, 22 56,31,25 45 50 6 20 23 23,30,06 31, 34, 07 44 35 ?d 68 68 88,55,50,94 95,61,46,87 61 54 8^ 53 52 -- 62 66 9 46 48 46,33,58 49, 42,54 43 52

^Expressed as percentages of the maximum possible.

^Decimal points omitted. c Used in second trial version but subsequently dropped.

^Used in second trial and final versions (3 = Edges, 4 = Cubes, 7 = Bottles, 8 = Poles).

in '«J 458

Table A.4

Means and standard deviations of scores on second trial and final versions of 3DD, by school^

School Poles Bottles Cubes Edges Total

Second trial version^

Boys' high M 78.7 81.1 76.8 64.5 76.2 (N=37) SD 25.0 18.7 20.7 24.4 16.9

Girls' high M 70.4 73.2 68.2 37.8 62.7 (N=38) SD 23.8 26.9 20.1 25.4 14.9

Mixed all-age^ M 30.7 38.1 23.8 4.5 24.6 (N=47) SD 29.9 31.6 12.4 8.0 17.4

Final version

Mixed junior M 47.7 74.5 40.0 29.3 48.7 secondary (N=38) SD 29.2 18.1 20.7 30.4 15.4

Mixed all-age M 45.5 75.1 26.9 25.7 44.4 (N=35) SD 29.0 18.0 20.0 24.2 16.6

All scores are expressed as percentages of the maximum possible.

^Items ordered Cubes, Edges, Poles, Bottles.

Grade 7 class; all others were Grade 9 classes. 459

Table A.5

Item statistics for high and low scorers on 3DD^

Poles Bottles Cubes Edges Total

Highest scorers M 88.9 87.0 82.3 69.6 83.8 (N=44) SD 13.1 14.1 15.5 20.6 7.4

Lowest scorers M 24.6 64.1 25.4 14.9 32.1 (N=41) SD 21.8 21.3 16.9 16.6 8.4

^Grade 7 students omitted; remaining sample size 148. All scores expressed as percentages of the maximum possible. 460

Table A.6

Item analysis of scores on NLSMA Hidden Figures Test^

Item All-age school (N=47) High schools (N=75) number P P' PNT ULI P P* PNT ULI b 1 87 87 0 44 97 97 0 7 2, 36 36 0 22 85 90 5 37 3^ 47 47 0 22 92 93 1 19 4b 49 49 0 33 92 95 3 19 5b 49 51 4 22 81 90 9 44

6b 30 33 9 44 72 81 11 52 7 23 27 13 11 85 90 5 37 8b 43 49 13 67 88 92 4 33 9 21 28 23 33 87 90 4 33 10 23 38 30 44 95 92 0 11

lib 15 26 42 44 69 73 5 70 12b 30 56 47 56 85 93 8 41 13 9 16 47 11 75 84 11 60 14 6 12 51 22 72 83 13 63 15 11 21 51 44 73 82 11 63

16 2 5 53 11 61 75 19 82

P = percentage giving correct answer; P' = percentage of those attempting item giving correct answer; PNT = percent­ age not attempting item; ULI = discrimination index based on upper and lower 27% on total score distribution.

^Used in subsequent version (HST trial version). 461

Table A.7

Item analysis of scores on trial version of HST&

Item P P" ULI number PNT

1 76 85 10 52 2 70 84 18 37 3 54 67 20 67 4 33 72 54 74 5 40 61 34 89

6 75 84 10 44 7 42 69 40 59 8 73 89 18 26 9 37 65 43 55 10 12 36 67 44

11 20 60 66 48 12 18 40 55 30 13 15 52 71 26 14 18 49 62 22 15 10 46 78 19

^For abbreviations. see note to Table A.6 . Analysis based on data from 108 subj ects. 462

Table A.8

Social classification of occupations in Jamaica

Social class Examples of occupations

1. Higher Farmers and land proprietors of more professional than 500 acres, university professors and managerial and senior lecturers, doctors, lawyers. High Court judges, engineers, owners of large commercial and industrial enter­ prises, directors and managers of large enterprises. Chief of Police and army, heads and assistant heads of government departments.

2. Lower Senior civil servants, headteachers of professional large secondary schools, magistrates, and managerial farmers with 100-499 acres, superin­ tendents of police, senior officers of army, assistant managers of large es­ tablishments, managers and directors of medium size establishments, univer­ sity lecturers, heads of large denominations.

3. Highly skilled Teachers, nurses, druggists, salesmen, ministers of religion, junior officers in army, inspectors of police, other civil servants, stenographers, accountants, typists, owners of small enterprises, farmers with 50-99 acres, secretaries, clerks, highly skilled technicians.

Skilled Carpenters, plumbers, cabinet makers, drivers, bus conductors, policemen, corporals, private soldiers, farmers with 10-49 acres, dressmakers, tailors, masons, tilers, curio workers, etc. 463

Table A.8 (continued)

Social class Examples of occupations 5. Semi-skilled Factory workers, waitresses, waiters. bartenders, porters, office maids. postmen, machine operators. etc.

6 . Unskilled Domestic workers, watchmen. peddlers. casual workers, portworkers, fish- vendors, higglers, etc.

Table A.9

Occupations classified as spatial/technical and verbal/personal

Spatial/technical Verbal/personal Engineer, architect Lawyer, politician Surgeon, dentist Doctor, nurse Scientist, technician Veterinarian Draughtsman, artist Teacher Pilot Accountant Designer Secretary, typist Builder, contractor Telephonist, clerk Carpenter, plumber Policeman Mechanic, welder Printer Electrician Driver Tailor, dressmaker Housewife APPENDIX B

SUPPLEMENTARY TABLES: DEVELOPMENTAL SURVEY

464 465

Table B.I

Academic indexes of Kingston high schools

School Sex Academic index^ St. George's College Boys 28.2 Immaculate Conception H.S. Girls 22.9 St. Andrew's H.S. Girls 19.8 Wolmer's Boys' H.S. Boys 19.8 Kingston College Boys 19.0

Queen's H.S. Girls 18.8 Jamaica College Boys 18.8 St. Hugh's H.S. Girls 18.2 Holy Childhood H.S. Girls 17.2 Wolmer's Girls' H.S. Girls 15.5

Alpha Academy Girls 15.5 Camperdown H.S. Mixed 14.9 Merl Grove H.S. Girls 12.5 Excelsior H.S. Mixed 12.0 Mead owbrook H.S. Mixed 10.3

Ardenne H.S. Mixed 9.2 Calabar H.S. Boys 7.8

The total number of passes in Mathematics and English Language on the 1972 GCE 0-level examination, as a percentage of the total school enrollment. 466

Table B.2

Academic indexes of 25 highest-ranking Kingston mixed elementary schools^

School Number of passes Academic index^

St. Richard's 66 37.0 Harbour View 97 27.7 Swallowfield 29 26.6 Pembroke Hall 72 25.1 Rousseau 35 24.3

Alpha Junior 62 23.7 Providence 65 23.4 St. Jude's 27 22.9 Melrose 29 22.8 Windward Road 20 22.7

Duhaney Park 80 22.7 Dunrobin 36 21.7 St. Francis 48 21.4 Calabar 21 20.8 North St. Congregational 9 20.5

Chetolah Park 25 19.1 John Mills 19 18.4 Maverley 24 17.7 Shortwood Practising 35 17.0 Mona Heights 51 17.0

Mico Practising 25 16.4 Whitfield 18 14.9 Jones Town 35 14.5 New Day 10 13.5 Balmagie 10 13.5

^There are a further forty Kingston public schools. Three are single-sex schools; the remainder each entered less than 25 students for the 1973 Common Entrance examination and had academic indexes of less than 13.5.

^The number of passes on the 1973 Common Entrance examina­ tion, as a percentage of the number of entries. 467

Table B.3

Summary data on schools included in developmental survey

Academic Test Sex School Type^ Sex Grade index order 1 B 1 Harbour^View^ AAS Mixed 27.7 20 2 Melrose ' AAS Mixed 22.8 9

G 1 Rousseau ^ PS Mixed 24.3 19 2 Pembroke Hall PS Mixed 25.1 16

3 B 1 Alpha Junior PS Mixed 23.7 17 2 Duhaney Park PS Mixed 22.7 3

G 1 Mona Heights PS Mixed 17.0 8 2 St. Francis AAS Mixed 21.4 10

5 B 1 Providence^'^ PS Mixed 23.4 18 2 Dunrobin PS Mixed 21.7 1

G 1 St. Richard's PS Mixed 37.0 11 2 Shortwood AAS Mixed 17.0 6

7 B 1 Kingston College HS Boys 19.0 13 2 Camperdown HS Mixed 14.9 4

G 1 Queen's HS Girls 18.8 7 2 Wolmer's Girls' HS Girls 16.5 14

9 B 1 Ardenne^ HS Mixed 9.2 15 2 Jamaica College HS Boys 18.8 5

G 1 Meadowbrook HS Mixed 10.3 12 2 Holy Childhood? HS Girls 17.2 2

AAS = all-age school, PS = primary school, HS = high school. ^Unstreamed; students chosen from ten brightest in class. ^Alternate replacing school in severely depressed area. ^Two students tested on two consecutive school days. ^Students tested in afternoon. ^Alternate replacing school where students not selected at random.

^One abnormal student replaced by an alternate. 468

Table B.4

Numbers of untypical students tested in each school

Number of untypical students tested School Grade Sex Years of Place of number Age Total education upbringing 1 B 1 1 0 0 1 2 1 0 1 1

G 1 3 0 0 3 2 0 0 0 0

3 B 1 3 0 0 3 2 3 1 0 3

G 1 3 1 1 4 2 1 2 0 2

5 B 1 0 0 0 0 2 0 2 0 2

G 1 0 1 1 2 2 0 0 0 0

7 B 1 0 0 0 0 2 0 0 0 0

G 1 0 0 0 0 2 0 2 0 2

9 B 1 0 2 1 3 2 1 0 0 1

G 1 0 0 0 0 2 0 1 1 1 469

Table B.5

Frequency distribution of DCT scores

Score Frequency Score Frequency

0-4 16 20-24 9 5-9 9 25-29 8 11-14 14 30-34 11 15-19 4 35-39 9

Table B .6

Item analysis of DCT scores

Item Maximum Mean Number at­ Item-test number possible score tempting item correlation

1 2 1.66 80 0.41 2 2 1.55 80 0.65 3 2 1.61 80 0.63 4 2 1.38 79 0.68 5 2 1.02 78 0.66

6 2 1.31 75 0.79 7 2 1.16 66 0.82 8 3 1.61 61 0.89 9 4 1.08 61 0.81 10 4 1.06 58 0.77

11 4 1.12 54 0.80 12 5 1.90 44 0.89 13 5 1.51 37 0.90 470

Table B.7

Means and standard deviations of spatial test scoresi. by grade and sex

Sex Grade 1 3 5 7 9 DCT

Boys M 3.50 14.50 27.25 25.50 29.75 SD 2.39 6.55 7.19 9.38 5.90

Girls M 5.88 10.00 8.62 25.00 31.00 SD 4.22 9.56 4.10 10.18 4.28

HFT

Boys M 5.38 13.38 23.00 20.75 28.62 SD 7.13 6.68 8.37 8.83 4.93

Girls M 5.25 8.50 3.88 24.62 28.50 SD 6.04 7.76 6.03 6.19 8.11

Poles

Boys M 2.88 3.50 4.12 2.62 6.88 SD 1.64 2.33 1.96 1.51 1.55

Girls M 2.00 2.12 1.88 5.25 5.25 SD 2.20 2.36 1.55 2.05 2.76

Bottles

Boys M 0.75 5.00 6.50 7.12 7.38 SD 2.12 3.74 3.02 2.10 1.92

Girls M 0.00 1.88 2.38 6.12 7.75 SD 0.00 2.80 3.29 3.91 1.28 471

Table B.7 (continued)

Grade Sex 1 3 5 7 9

SRT drawing

Boys M 3.75 15.12 17.75 18.25 26.62 SD 3.41 4.05 9.00 8.55 2.62

Girls M 1.75 5.50 4.12 19.00 22.88 SD 2.05 6.87 3.52 6.50 3.31

SRT selection

Boys M 11.62 13.50 14.50 14.00 15.75 SD 4.17 1.20 1.51 1.77 0.46

Girls M 11.25 11.75 11.88 15.38 15.12 SD 3.28 2.25 3.00 0.92 0.84 Table B.S

Trend analysis of spatial test scores, by grade and sex

Multi­ HVT SRT Source variate DCT HFT Poles Bottles Drawing Selection

Grade (G) Linear 1 0.001 MS 6630.60 5428.89 112.22 493.50 5378.55 316.41 F 149.98 105.96 29.60 66.43 155.87 57.70 P< 0.001 0.001 0.001 0.001 0.001 0.001

Residual 3 0.309 MS 8.50 50.53 7.91 6.39 61.09 9.88 F 0.18 0.99 2.09 0.86 2.18 1.80 P< 0.907 0.405 0.112 0.467 0.100 0.156

Sex (X) 1 0.004 MS 320.00 332.11 9.80 59.51 638.44 12.80 F 7.24 6.48 2.58 8.01 22.73 2.33 PC 0.009 0.C13 0.113 0.006 0.001 0.132

Interaction 4 0.001 MS 294.59 321.55 14.80 13.61 137.29 9.18 (GX) F 6.66 6.28 3.90 1.83 4.89 1.67 P4 0.001 0.001 0.007 0.135 0.002 0.168

Between 10 0.583 MS 64.52 45.91 6.18 5.36 45.70 2.70 schools^ F 1.46 0.90 1.63 0.72 1.63 0.49 Pz. 0.177 0.542 0.120 0.701 0.121 0.888

Within 60 MS 44.21 51.24 3.79 7.43 28.09 5.48 schools

^Because the hypothesis of zero between-school variance was not rejected at P< 0.25, the within-schools error term was used in testing the significance of G, X and GX. to 473

Table B.9

Means and standard deviations of DCT and HVT scores in Grade 5 check testing, by sex

HVT Sex DCT Poles Bottles Boys M 21.12 5.25 7.00 (N=8) SD 6.71 2.25 1.60

Girls M 10.38 2.12 4.38 (N=8 ) SD 7.17 1.64 2.72

t 3.10 3.17 2.35 P< 0.008 0.007 0.034

Table B.IO

Frequency distribution of HFT scores

Score Frequency Score Frequency 0 9 21-25 13 1-5 12 26-30 11 6-10 10 31-35 5 11-15 7 36-40 3 16-20 9 41-50 1 474

Table B.ll

Item analysis of HFT scores

Item Mean Item-test Item Mean Item-test number score^ correlation number score^ correlation 1 2.72 0.75 6 1.62 0.82 2 2.01 0.85 7 1.10 0.70 3 1.79 0.81 8 1.64 0.88 4 2.28 0.80 9 0.51 0.63 5 2.10 0.84 10 0.45 0.36

Maximum possible = 5.

Table B.12

Frequency distribution of Poles median inclinations, by section

a Section Angle' Top Middle Bottom -170° to -50° 3 0 1 -49° to -30° 12 3 2 -29° to -10° 8 4 4

-9° to 10° 8 6 30 11° to 30° 13 5 24 31° to 50° 10 9 16 51° to 70° 21 6 3

71° to 90° 5 10 0 91° to 110° 0 14 0 111° to 130° 0 16 0 131° to 150° 0 7 0

Measured clockwise from the vertical position. 475

Table B.13

Frequency distribution of Poles median inclinations, by grade and section

Grade Angle ^ Score 1 3 5 7 9 Top section

Up to -38° 0 3 3 1 2 1 -37° to -25° 1 0 3 0 1 1 -24° to -9° 2 2 0 0 3 3 -8° to 8° 3 0 0 2 2 4 9° to 20° 2 1 1 2 0 5 21° to 47° 1 2 3 3 4 1 48° to 90° 0 8 8 8 4 1

Middle section

Up to -38° 1 0 1 0 1 0 -37° to -25° 2 0 0 0 0 1 -24° to -9° 3 0 1 0 2 1 -8° to 8° 4 0 1 0 0 4 9° to 20° 3 0 0 1 1 0 21° to 47° 2 5 4 1 0 3 48° to 90° 1 2 1 6 4 3 Beyond 91° 0 9 8 8 8 4

Bottom section

Up to -38° 0 1 1 0 0 0 -37° to -25° 1 0 1 0 0 0 -24° to -9° 2 1 2 0 3 1 -8° to 8° 3 2 3 2 4 12 9° to 20° 2 0 3 5 5 2 21° to 47° 1 11 3 7 4 1 48° to 90° 0 1 3 1 0 0

^Measured clockwise from the vertical position. 476

Table B.14

Frequency distributions of Poles and Bottles scores

Frequency Score Poles Bottles

0 5 30

1 19 0 2 7 0 3 8 2 4 8 2 5 14 4

6 10 9 7 4 11 8 1 9 9 3 8 10 1 5 477

Table B.15

Frequency distribution of Bottles inclinations, by grade and bottle

Illus- Grade Angle Score tration 1 3 5 7 9

Bottle 2

-179° to -24° 0 15 11 7 4 1 -23° to -14° 1 0 0 1 0 2 -13° to -7° 2 /S 0 0 1 1 1 - 6° to 6° 3 1 2 2 7 5 7° to 13° 2 0 2 2 1 4 14° to 35° 1 ▲ 0 0 2 3 3 35° to 180° 0 0 1 1 0 0 Bottle 3

-179° to -61° 0 ■O 15 8 6 3 1 -60° to - n o 1 0 0 0 1 1 -10° to 10° 2 a 0 7 10 12 14 11° to 60° 1 0 0 0 0 0 61° to 1800 0 cm 0 1 0 0 0 Bottle 4

-179° to -67° 0 15 7 6 2 1 -660 to -31° 1 % 0 0 2 0 1 -30° to -7° 2 0 0 1 3 1 -6° to 6° 3 0 1 2 3 5 7° to 23° 2 0 2 1 3 4 24° to 96° 1 1 6 4 5 4 97° to 180° 0 % 0 0 0 0 0 Bottle 5

190° to -97° 0 B 15 7 6 2 0 -96° to -11° 1 0 0 0 0 0 -10° to 10° 2 I 9 10 14 16 11° to 96° 1 y 0 0 0 0 0 97° to 170 ° 0 0 0 0 0 0

^Measured anticlockwise from the horizontal position. 478

Table B.16

Means and standard deviations of Poles and Bottles scores in Grade 9 high school samples, by sex

Poles Bottles Test Type Boys Girls Boys Girls HVT Individual N 8 8 8 8 M 6.88 5.25 7.38 7.75 SD 1.55 2.76 1.92 1.28

3DD Group N 54 73 54 73 M 7.87 6.49 8.83 8.10 SD 2.48 2.26 1.24 1.69

t 1.10 1.45 2.88 0.56 P > 0.20 > 0.10 < 0.01 >0.50 479

Table B.I7

Cross-tabulation of SRT scores in Conditions 1 and 2, by solid sonre under Score under Condition 2 Condition 1 0 1 2 3 4 Cuboid

0 27 7 2 1 1 1 8 2 2 5 5 3 3 1 2 8 6 4 1 1

Cylinder 0 17 2 1 14 1 2 1 1 3 4 1 3 1 12 6 4 2 15 Pyramid

0 18 4 1 2 1 1 9 5 2 2 2 1 4 5 9 3 2 11 4 4 0

Cube 0 25 2 1 1 1 4 2 10 5 3 4 17 3 4 1 2 4 480

Table B.18

Cross-tabulation of SRT scores in Conditions 2 and 3, by solid

Score under Score under Condition 3 Condition 2 0 1 2 3 4 Cuboid 0 4 1 5 17 1 1 1 1 12 2 12 3 1 14 4 1 10 Cylinder

0 1 2 3 13 1 6 11 2 1 2 3 1 18 4 2 20 Pyramid 0 10 1 3 2 4 1 3 2 5 6 1 2 5 5 3 3 2 9 13 4 6

Cube

0 4 2 1 19 1 2 1 3 2 1 2 13 3 1 24 4 1 6 481

Table B.19

Frequency distribution of SRT scores in Condition 2, by grade and solid

Grade ücore 1 3 5 7 9 Cuboid 0 12 6 9 1 4 7 1 4 2 1 3 5 2 3 2 1 6 6 4 2 1 8 Cylinder 0 9 4 4 2 1 6 3 5 3 2 1 2 0 0 3 6 2 5 6 4 1 5 6 10

Pyramid 0 10 5 3 2 1 5 3 7 2 2 1 4 2 3 3 3 4 3 8 9 4 1 1 4

Cube

11 5 9 1 1 4 2 0 0 2 1 3 3 7 2 3 6 3 8 8 4 1 0 5 482

Table B.20

Frequency distribution of SRT scores in Condition 3, by grade and solid

Grade Score 1 3 5 7 9 Cuboid 0 3 2 1 1 1 2 5 2 1 3 0 0 0 1 4 7 11 15 15 16 Cylinder 0 1 1 2 1 4 3 2 0 0 0 0 3 4 0 0 0 4 9 15 12 13 15

Pyramid 0 6 2 5 1 3 0 0 2 3 6 2 3 1 3 2 8 3 4 5 4 2 0 6 9 10 Cube

0 3 1 1 1 2 0 0 2 3 0 0 3 1 2 0 1 1 4 7 13 15 15 15 483

Table B.21

solid, grade, sex and condition (Conditions 1 and 2)

Multi­ Source d .f. variate Cuboid Cylinder Pyramid Cube p/- Between schools Grade (G) 4 0.002 MS 157.91 175.29 102.94 154.41 F . 41.28 14.31 13.63 19.80 P< 0.001 0.001 0.001 0.001

Sex (X) 1 0.005 MS 87.02 67.60 32.40 136.90 F 22.75 5.52 4.29 17.55 Pc 0.001 0.041 0.065 0.002

GX 4 0.244 MS 18.59 23.54 10.21 21.59 F 4.86 1.92 1.35 2.77 P-i 0.019 0.183 0.317 0.087

Error 10 MS 3.82 12.25 7.55 7.80

Within schools

Condition (C) 1 0.001 MS 24.02 1.60 25.60 0.40 F 25.97 2.91 34.13 0.29 P^ 0.001 0.119 0.001 0.505

GO 4 0.029 MS 3.46 ■ 0.79 2.41 2.45 F 3.74 1.43 3.22 1.76 P/L 0.041 0.293 0.061 0.214

XC 1 0.305 MS 5.62 0.40 0.10 0.40 F 6.08 0.73 0.13 0.29 P 4 0.033 0.414 0.723 0.605

GXC 4 0.500 MS 0.44 0.59 0.54 2.09 F 0.47 1.07 0.72 1.49 P c 0.755 0.422 0.599 0.277

Error 10 MS 0.96 0.55 0.75 1.40 484

Table B.22

Frequency distributions of SRT drawing and selection scores

Drawing Selection Score Frequency Score Frequency 0-4 24 0-2 0 5-8 8 3-4 1 9-12 6 5-6 2 13-16 5 7-8 6

17-20 12 9-10 7 21-24 11 11-12 10 25-28 10 13-14 20 29-32 4 15-16 34

Table B .23

Numbers of students with two or more Guttmann errors on GIM illusions, by grade

Grade Illusion 1 3 5 7 9 Muller--Lyer 2 1 2 0 1 Horizontal-Vertical 1 1 2 1 1 0 Horizontal-Vertical 2 1 0 0 1 0 Sander Parallelogram 0 5 6 2 1 Ponzo 3 4 0 2 1 Table B.24

Trend analysis of GIM scores, by grade, sex and order

Multi­ Source d.f. variate ML HVl HV2 SP Pz P< Grade (G) Linear 1 0.386 MS 4.22 3.60 0.90 0.90 0.31 F 1.28 2.32 0.39 0.91 0.25 0.284 0.158 0.548 0.362 0.626

Residual 3 0.381 MS 0.78 6.11 1.72 0.48 2.96 F 0.24 3.94 0.74 0.48 2.44 PZL 0.870 0.043 0.553 0.703 0.124

Sex (X) 1 0.699 MS 0.05 0.20 3.20 2.82 0.31 F 0.02 0.13 1.38 2.85 0.26 P < 0.904 0.727 0.268 0.122 0.623

Order (0) 1 0.451 MS 0.20 1.56 0.05 1.51 6.61 F 0.17 1.80 0.04 1.03 2.86 P< 0.679 0.218 0.846 0.315 0.099

GX 1 0.389 MS 4.36 0.73 1.26 1.91 4.38 F 1.32 0.47 0.54 1.90 3.61 P < 0.327 0.756 0.708 0.182 0.045

GO 1 0.931 MS 0.95 0.46 0.55 0.67 2.05 F 0.83 0.40 0.42 0.46 0.89 00 P< 0.516 0.810 0.791 0.767 0.481 Table B.24 (continued)

Multi­ Source d.f. variate ML HVI HV2 SP Pz P < XO 1 0.018 MS 5.00 2.45 5.00 3.77 0.65 F 4.35 2.13 3.85 5.51 1.51 PC 0.043 0.152 0.057 0.059 0.423

Schools 10 0.074 MS 3.30 1.55 2.32 0.99 1.21 (S(GX))B F 2.87 1.35 1,79 0.68 0.52 PC 0.009 0.240 0.095 0.740 0.863

GXO 4 0.235 MS 3.88 0.61 0.50 1.29 2.70 F 3.37 0.53 0.38 0.88 1.17 PC 0.018 0.716 0.818 0.482 0.340

SO(GX)^ 10 0.431 MS 2.40 0.25 1.28 1.24 2.96 F 2.09 0.22 0.98 0.85 1.28 PC 0.049 0.993 0.475 0.589 0.274

Within schools 40 MS 1.15 1.15 1.30 1.46 2.31

Because the hypothesis of zero between-school variance was re­ jected at 0.25, S(GX) was used as the error term for test­ ing the significance of G, X and GX.

^Because the hypothesis of zero school-order interaction vari­ ance was not rejected at P ^ 0.25, the within-schools error term 00 (D was used for testing the significance of O, GO, XO and GXC. 487

Table B.25

Means and standard deviations of GIM scores, by grade and sex

Grade Sex 1 3 5 7 9 Muller-Lyer Boys M 5.38 6.12 5.25 6.50 5.25 SD 0.74 1.25 1.28 2.39 1.16

Girls M 6.50 5.50 6.12 5.12 5.00 SD 0.93 0.54 1.55 1.73 0.93 Horizontal-Vertical 1 Boys M 6.00 5.38 5.75 4.50 5.25 SD 1.20 0.74 1.39 0.76 0.89

Girls M 5.50 5.38 6.38 4.50 5.62 SD 1.31 0.74 1.06 1.07 0.74 Horizontal-Vertical 2 Boys M 6.12 5.62 6.88 6.62 6.62 SD 0.99 1.30 0.64 1.41 1.30

Girls M 6.00 6.00 6.38 5.50 6.00 SD 1.20 1.51 0.92 0.54 1.51 Sander Parallelogram Boys M 5.75 5.00 5.50 5.12 6.00 SD 1.67 1.07 0.54 1.25 1.51

Girls M 5.38 5.38 5.25 5.00 4.50 SD 1.30 1.30 1.04 0.93 0.54 Ponzo

Boys M 5.25 5.62 6.00 5.38 6.62 SD 1.49 1.51 1.31 1.30 2.00

Girls M 6.25 5.62 6.50 5.00 4.88 SD 1.83 1.51 1.20 1.51 1.36 488

Table B.26

Frequency distributions of background variables, by grade

Grade Category 1 3 5 7 9 Hand dominance Left 0 1 0 0 0 Mixed 4 4 3 4 6 Right 12 11 13 12 10 Eye dominance

Left 10 5 9 5 10 Right 6 11 7 11 6

Number of siblings

Mean 4.25 3.75 4.88 3.75 3.25 Standard deviation 1.92 2.62 2.16 2.52 2.27

Adult male in home Present 10 11 8 8 13 Absent 6 5 8 8 3 Social Class

Higher professional 0 0 0 0 1 Lower professional 0 0 1 1 1 Highly skilled 2 7 5 7 9 Skilled 7 4 3 5 3 Semi-skilled 1 0 3 2 0 Unskilled 0 1 0 0 1

Ambition Spatial/technical 2 1 1 5 7 Verbal/personal 13 13 14 11 9

Skin color White 0 0 0 0 1 Light brown 0 3 4 1 0 Mid brown 5 0 3 3 5 Dark brown 5 8 3 6 2 Black 6 5 6 6 8 Table B.27

Analysis of variance of spatial test scores, by hand and eye dominance

Multi­ Source d.f. variate DCT HFT HVT SRT P < Poles Bottles Drawing Selection

Hand 1 0.223 MS 161.79 4.94 19.64 2.47 245.30 2.53 dominance F 1.14 0.04 3.43 0.19 2.59 0.26 (H) P < 0. 289 0.849 0.068 0.665 0.112 0.610

Eye 1 0.005 MS 47.63 284.54 25.02 79.97 351.25 0.08 dominance F 0.34 2.11 4.38 6.11 3.70 0.01 (E) P< 0.564 0.151 0.040 0.016 0.058 0.928

HE 1 0.293 MS 454.23 184.03 14.73 48.91 88.89 18.08 F 3.20 1.36 2.57 3.74 0.94 1.88 P< 0.078 0.247 0.113 0.057 0.336 0.174

Error 76 MS 141.94 135.00 5.72 13.09 94.82 9.62

00 LO 490

Table B.28

Mean spatial test scores, by hand and eye dominance

Mixed hand ed Right handed Test Left eyed Right eyed Left eyed Right eyed

DCT 22.79 16.25 14.68 19.15 HFT 17.07 15.75 12.80 18.48

Poles 4.57 4.25 2.40 4.06 Bottles 5.00 4.38 2.68 5.67

SRT drawing 24.07 24.75 17.24 22.79 SRT selection 17.93 16.38 16.60 17.24

N 14 8 25 33 APPENDIX C

SUPPLEMENTARY TABLES: GRADE NINE SURVEY

491 492

Table C.l

Grade Nine survey population

Academic Urban Environment School index^ index^ High schools

City Camperdown 14.9 60 Excelsior 12.0 60 Meadowbrook 10.3 60 Ard enne 9.2 60

Town St. Mary H.S. 15.4 30 Knox 14.6 28 Clarendon 9.7 30

Village Happy Grove c 10 Oberlin c 17 St. Mary's College 7.6 17

Technical high schools

City St. Andrew d 60 Kingston d 60

Town Dinthi11 d e Holmwood d e

Junior secondary schools

City Denham Town 46.2 60 Holy Trinity 28.0 60 Pembroke Hall 14.5 60 Vauxhall 11.8 60 Norman Manley 11.3 60 Penwood 10.9 60 Papine 10.2 60 Haile Selassie 9.7 60 Tarrant 9.3 60 Kingston 6.7 60 St. Anne's 1.7 60 493

Table C.l (continued)

Academic Urban Environment School index^ index^ Town Lionel Town 16.8 30 Spaldings 14.0 28 Yallas 13.9 32 Kellits 13.2 25 Richmond 9.9 25 Porus 7.6 29 Ewarton 4.5 34 Bog Walk 2.2 31 Oracabessa 2.2 34

Village Glengoffe 30.7 18 Alston 26.7 17 McGrath 14.8 10 Bellefield 10.1 10 Lennon 8.7 10 Thompson Town 7.4 18 Fair Prospect 3.3 17

All-age schools

City Harbour View 40.4 60 St. Francis 33.3 60 Swallowfield 32.3 60 Mico Practising 27.5 60 All Saints 23.0 60 John Mills 22.6 60 Norman Gardens 22.0 60 Whitfield 20.7 60 Calabar 17.5 60 Cockburn Gardens 17.4 60 Shortwood Practising 14.6 60 Central Branch 13.3 60 New Day 12.5 60 Windward Road 12.3 60 St. Michael's 11.8 60 Maverley 8.5 60 Melrose 3.8 60 Seaward 2.9 60 494

Table C.l (continued)

Academic Urban Environment School index^ index'

Town Hope Bay 15.8 24 Chapelton 7.1 34 Bath 6.7 29

Village Newstead 54.9 10 Harewood 38.9 10 Ginger Ridge 30.8 12 Kitson Town 28.1 12 Bryce 27.3 18 Seafieid 21.2 10 Boown's Hall 21.1 10 Water Valley 20.5 10 Rock Hall 20.0 10 Carron Hall 19.4 11 Johns Hall 18.8 18 Brandon Hill 17.6 10 Mile Gully 16.7 18 Pike 15.3 10 Goshen 11.9 10 Crooked River 11.6 10 Top Jackson 10.5 10 Craighead 7.9 10 Bethany 2.9 10

For the definitions of academic index, see Section 5.22 for high schools and Section 6.25 for junior secondary and all-age schools.

^For the definition of urban index, see Section 6.23.

^Academic indexes could not be calculated for these schools, which were operated in combination with junior secondary schools.

^The academic index was not defined for technical schools.

^The urban index of these schools was not assessed; see Section 6.23. 495 Table C.2

Schools in survey area eliminated from survey population

Reason Schools Eliminated High schools Single sex (boys) Calabar, Jamaica, Kingston, St. George's, Wolme's Boys

Single sex (girls) Alpha Academy, Holy Childhoo, Immaculate Conception, Merl Grove, Queen's, St. Andrew's, St. Catherine, St. Hugh's, Wolmer's Girls

In small cities Glenmuir, Manchester, Morant Bay, St. Jago, Titchfield Technical high school In sugar belt Vere Junior secondary schools Intermediate village/town Guy's Hill, Trinityville

In sugar belt Stokes Hall

In small cities Buff Bay, Christiana, Denbigh, May Pen, Morant Bay, Old Harbour, Port Antonio, Port Maria, St. Catherine, Spanish Town

Attached to high school Oberlin, Happy Grove

All-age schools

Entered less than 30 students for 1973 GNAT A total of 286 schools

Intermediate village/town Bull Bay, Race Course

In sugar belt Wood Hall, Hayes

In small cities Crescent, Linstead, Mandeville, Villa Road

Strong urban influence Gregory Park, White Marl 496

Table C.3

Means and standard deviations of spatial test scores in eliminated village "high schools"

School Sex 3DD HST BOX 1 Boys M 22.43 5.93 13.86 (N=14) SD 6.22 2.92 13.40

Girls M 15.86 4.34 10.83 (N=29) SD 4.85 2.41 10.19

2 Boys M 23.31 7.54 20.92 (N=13) SD 5.74 3.46 11.82

Girls M 16.76 5.33 13.10 (N=21) SD 4.23 2.58 10.98 497

Table C.4

Summary data on schools included in Grade Nine survey

Rank of Envir­ Urban Test school Academic class onment index index order tested^ High schools City 1 Meadowbrook 10.3 60 -/2 1 2 Camperdown^ 14.9 60 2/4 3

Town 1 Knox 14.6 28 2/3 4 2 St. Mary 15.4 30 1/2 5

Technical high schools

City 1 Kingston - 60 -/6 2 2 St. Andrew - 60 -/5 17

Town 1 Dinthill __ 2/4 12 2 Holmwood - - 3/5 6

Junior secondary schools City 1 Penwood 10.9 60 3/8 7 2 Norman Manley 11.3 60 3/9 8

Town 1 Porus 7.6 29 2/4 16 2 Richmond 9.9 25 2/4 9

Village 1 Bellefield*^ 10.1 10 3/6 13 2 Lennon 8.7 10 2&3/4 19

All-age schools

City 1 Whitfield*^ 20.7 60 -/3 15 2 Cockburn Gdns 17.4 60 1/3 11

Town 1 Hope Bay 15.8 24 -/I 10 2 Chapelton® 7.1 34 2/3 18

Village 1 Carron Hall 19.4 11 1/3 14 2 Johns Hall 18.8 18 -/I 20

^E.G. 2/4 means that the class tested was the second stream out of four; a dash indicates that the classes were un­ streamed . bAlternate replacing school where principal uncooperative. c Students tested in afternoon.

^Alternate replacing school with no Grade Nine classes.

Grade 7-9 divided into three groups by ability instead of age. 498

Table C.5

3DD inter-item correlations^

Item Poles Bottles Cubes Edges , Total Poles 32 55 55 80 Bottle s 32 36 38 58 Cubes 36 37 69 85 Edges 32 25 51 86 Total 72 67 76 73

^Decimal points omitted. Correlations above the diagonal refer to boys (N=277) and below the diagonal to girls (N=362). P( 1 rj^O.16) =0.01.

Table C .6

Frequency distributions of spatial test scores

3DD HST BOX Score Frequency Score Frequency Score Frequency 0-5 4 0-1 9 4"io 7 6-10 45 2-3 108 "9-0 87 11-15 98 4-5 187 1-10 175 16-20 173 6-7 146 11-20 147

21-25 138 8-9 93 21-30 159 26-30 83 10-11 62 31-40 59 31-35 62 12-13 22 41-48 5 36-40 36 14-15 12 499

Table C.l

Item analysis of 3DD scores

Maximum Mean Item-test Item possible score correlation Poles top 3 1.52 0.59 middle 4 1.65 0.63 bottom 3 2.12 0.58 Poles total 10 5.30 0.75 Bottles 2 3 1.93 0.48 3 2 1.84 0.38 4 3 1.97 0.53 5 2 1.92 0.30

Bottles total 10 7.67 0.64

Cubes 1 1 0.87 0.34 2 1 0.38 0.42 3 1 0.24 0.35 4 1 0.85 0.34 5 1 0.39 0.45 6 1 0.50 0.64 7 1 0.52 0.37 8 1 0.19 0.61 9 1 0.34 0.57 10 1 0.19 0.65

Cubes total 10 4.47 0.83

Edges 1 2 0.99 0.68 2 3 1.24 0.64 3 3 1.44 0.75 4 2 0.29 0.60

Edges total 10 3.96 0.84 500

. Table C .8

Item analysis of HST scores^

Item P ?' PNT ULI 1 80 86 7 37 2 68 78 13 44 3 42 52 19 65 4 53 77 31 58 5 50 65 23 61

6 66 84 21 64 7 42 64 34 88 8 34 39 13 25 9 39 50 22 59 10 22 42 47 56

11 21 39 46 55 12 18 33 45 30 13 11 24 54 22 14 26 45 42 44 15 14 30 53 25

^Based on random sample of five students from each of the twenty survey schools. For abbreviations, see note to Table A.5. Table C .9

Means and standard deviations of 3DD scores, by school and sex

High Technical Junior All-age Envir­ School schools high secondary schools onment number Boys Girls Boys Girls Boys Girls Boys Girls City 1 M 32.30 25.93 29.22 20.06 20.09 15.67 19.73 15.00 SD 4.52 6 . 21 5.44 5.12 7.16 7.62 6.26 5.70

2 M 28.38 23.83 29.88 20.60 25.29 19.44 18.82 18.23 SD 5.69 6.74 7.20 5.85 7.01 6.48 4.24 5 .60 Town 1 M 32.35 25.08 30.33 18.60 24.00 18.17 19.38 16.93 SD 7.24 6.93 5.81 5.15 7.02 5.89 6.61 5.15

2 M 33.18 23.09 25.25 19.73 23.85 15.58 19.19 16.30 SD 5.06 6.07 5.71 3.71 5.35 5.40 7.76 4.94

Village 1 M —— — — 24.71 17.54 19.00 14.41 SD 9.08 5.52 7.01 5.36

2 M - - - - 19.38 14.50 17.80 15.62 SD 6.90 4.78 5.44 4.33

in o Table C.IO

Means and standard deviations of HST scores, by school and sex

High Technical Junior All-age Envir­ School schools high secondary schools onment number Boys Girls Boys Girls Boys Girls Boys Girls

City 1 M 10.00 8.37 5.89 5.38 6.00 5.13 6.67 5.23 SD 3.33 3.27 2.40 2.31 2.49 2.53 2.13 2.49

2 M 8.25 7.25 8.50 7.53 7.47 4.89 6.27 4.68 SD 2.21 2.60 3.29 2.70 1.97 1.68 2.24 1.62 Town 1 M 10.94 8.25 8.76 6.00 6.15 3.83 4.12 5.36 SD 2.53 3.57 2.76 2.30 1.72 1.25 1.64 1.82

2 M 12.00 7.96 6.33 5.27 5.46 4.62 5.69 4,75 SD 1.95 2.38 2.96 2.94 3.15 2.30 2.41 1.83

Village 1 M - “■ - - 4.93 4.92 5.40 4.04 SD 2.16 1.98 2.22 1.83

2 M - — - - 5.08 4.75 5.07 3.85 SD 1.85 1.80 1.79 1.99

in o to Table C.ll

Means and standard deviations of BOX scores, by school and sex

High Technical Junior All-age Envir­ School schools high secondary schools onment number Bovs Girls Boys Girls Boys Girls Boys Girls City 1 M 21.60 19.33 25.33 16.19 7.82 7.60 9.07 3.46 SD 12.56 10.01 13.26 12.78 10.54 12.16 10.35 7.87

2 M 21.12 12.08 24.94 15.80 21.76 8.33 13.09 14.41 SD 9.04 10.82 11.36 10.21 11.99 10.56 10.62 11.86 Town 1 M 25.76 21.25 19.33 11.40 11.15 12.67 14.62 8.57 SD 14.69 7.86 11.08 11.42 11.27 10.49 11.15 12.70

2 M 32.36 18.09 15.58 15.93 19.54 9.17 16.25 12.00 SD 3.44 9.23 7.86 7.41 11.25 9.14 14.07 13.37

Village 1 M -- -- 13.57 8.33 8.80 6.00 SD 11.31 11.64 11.24 9.78

2 M --- - 7.85 6.85 14.73 8.69 sn 10.99 10.78 12.96 9.98

Ul o (jj 504

Table C .12

Academic indexes of non-Kingston high schools^

School Sex Academic index Munro College Boys 27.5 Montego Bay H.S. Girls 23.4 Hampton H.S. Girls 21.8 York Castle H.S. Mixed 19.1 St. Jago H.S. Mixed 15.7

St. Mary H.S. Mixed 15.4 Westwood H.S. Girls 15.4 Knox College Mixed 14.6 Mt. Alvernia H.S. Girls 12.9 St. Hilda’s H.S. Girls 11.3

Cornwall College Boys 10.6 Manchester H.S. Mixed 10.5 Glenmuir H.S. Mixed 9.9 Clarendon College Mixed 9.7 Ferncourt H.S. Mixed 9.2

Titchfield H.S. Mixed 8.9 St. Mary's College Mixed 7.6 Mannings H.S. Mixed 5.6 a Academic indexes could not be calculated for a further five high schools operated in combina- tion with junior secondary schools. 505

Table C.13

Mean spatial test scores in combined city and town school data, by sex and school type

School 3DD HSTBOX type Boys Girls Boys Girls Boys Girls

High 31.33 24.59 10.18 8.04 24.96 18.08

Technical high 29.02 19.75 7.49 6.03 21.61 14.85

Junior secondary 23.57 17.15 6.37 4.60 15.83 9.49

All-age 19.30 16.80 5.86 4.94 13.14 10.46 506

Table C.14

Analysis of variance of spatial test scores in combined city and town school data, by sex and school type

Multi­ Source d.f. variate 3DD HST BOX P <

Sex (X) 1 0.001 MS 5286.89 319.85 4448.72 F 166.50 34.22 20.73 P < 0.001 0.001 0.001

School type (T) 3 0.001 MS 2270.70 364.78 2575.69 F 40.51 16.91 8.31 P< 0.001 0.001 0.003

XT 3 0.056 MS 234.83 8.06 118.99 F 7.40 0.86 0.55 P< 0.005 0.487 0.655

Schools (S(T))S 12 0.001 MS 56.05 21.58 309.93 F 1.54 3.58 2.56 P < 0.106 0.001 0.003

SX(T)^ 12 0.109 MS 31.75 9.35 214.61 F 0.87 1.55 1.77 P < 0.575 0.103 0.050

Within schools 471 MS 36.39 6.03 121.06

^Used for testing the significance of T.

^Used for testing the significance of X and XT. 507

Table C.15

Numbers of untypical students eliminated from junior secondary and all-age schools

Number of untypical students eliminated Envir­ School Years of Place of Recent onment number Age education upbringing transfer Total Junior secondary schools

City 1 0 2 1 3 6 2 2 4 2 1 9

Town 1 3 2 6 0 11 2 2, 0 1 4 6

Village 1 2 8 5 3 21 2 3 5 6 5 14

All-age schools

City 1 2 11 2 1 13 2 7 8 3 4 17

Town 1 0 0 1 0 1 2 4 3 5 3 12

Village 1 7 10 2 0 13 2 2 1 1 0 4 508

Table C.16

Mean spatial test scores in refined junior secondary and all-age school data, by sex, environment and school type

Enviro- 3DD HST BOX onment Boys Girls Boys Girls Boys Girls Junior secondary schools

City 23.38 17.48 6.90 5.04 16.10 7.44 Town 23.88 16.29 5.81 4.26 12.69 10.80 Village 21.46 15.22 5.77 4.61 10.62 8.00 All-■age schools City 17.38 16.33 6.15 5.05 9.69 11.28 Town 19.06 16.72 4.75 5.17 15.81 8.72 Village 17.90 14.36 4.80 3.75 12.65 6.11

Table C.17

Analysis of variance of spatial test scores in refined junior secondary and all-age school data, by sex, environment and school type

Multi­ Source d.f. variate 3DD HSTBOX

Sex (X) 0.001 MS 1261.60 74.42 1254.43 F 34.23 18.36 10.85 0.001 0.001 0.001

School type (T) 0.014 MS 327.49 11.15 28.25 F 8.89 2.75 0.24 P ^ 0.003 0.099 0.622

Environ­ 0.020 MS 78.88 24.15 164.18 ment (E) F 2.14 5.96 1.42 P < 0.120 0.003 0.244 509

Table C.17 (continued)

Multi­ Source d.f. variate 3DD HST BOX P< XT 1 0.010 MS 269.49 16.90 0.02 F 7.31 4.17 0.00 P< 0.007 0.042 0.989

XE 2 0.773 MS 13.77 3.75 1.32 F 0.37 0.92 0.01 P< 0.689 0.398 0.989

TE 2 0.65 7 MS 12.33 6.73 0.43 F 0.33 1.66 0.00 P< 0.716 0.192 0.996

XTE 2 0.135 MS 9.29 4.58 347.57 F 0.25 1.13 3.00 P< 0.777 0.325 0.051

Schools 6 0.301 MS 45.29 0.61 218.58 (S(TE)) F 1.23 0.15 1.89 P< 0.292 0.989 0.083

XS(TE) 6 0.450 MS 13.05 4.93 215.06 F 0.35 1.22 1.86 P< 0.907 0.299 0.089

Within schools^ 233 MS 36.86 4.05 115.55

Because the hypotheses of zero between-school and sex- school interaction variances were not rejected at P<0.25, the within-school error term was used in testing the main effects and interactions. 510

Table C.18

Summary statistics on background variables for all Grade Nine students tested

School type Technical high Others Age Mean 16.00 15.09 S.D. 0.59 0.64

Years of education Mean 9.28 8.61 S.D. 1.12 1.06

Writing hand Left Right Frequency 38 594

Place of upbringing 0 1 2 3 4 Frequency 18 222 54 71 271 Ambition Spatial/technical Verbal/personal Frequency Boys 169 103 Girls 27 334

Number of Mean; 5.95 S.D. : siblings 3.17

Adult male in home Present Absent Frequency 310 202 Social class 1 2 3 4 5 6 Frequency 13 12 158 199 119 41 APPENDIX D

SUPPLEMENTARY TABLES: SPATIAL TRAINING EXPERIMENT

511 512

Table D.l

Experimental timetable: First year

Week Period Group number number lA IB Easter term 1 1 PDQ Last day 2 HST of vacation 3 3DD PDQ 4 BOXHST

2 1 3DD 2 BOX 3 4 First unit (15 periods): First unit 3 1-4 Periods 1 & 2 (16 periods) 4 1-4 of Week 5 5 1-4 lost to public holiday 6 1 2 3 Solids AT Solids AT Statistics AT& Statistics AT^

7 1 3DD 3DD 2 BOX BOX 3 HST/TMQ HST/TMQ 4 8 1-4 Second unit 9 1-4 (15 periods): Second unit 10 1-4 Period 3 of (15 periods) Week 8 lost to 11 1 public holiday 2 3 Solids/Stat. AT^ 4 Solids/Stat AT^ Continuation of 12 1-4 Continuation of regular course regular course 513

Table D.l (continued)

Period Group number lA IB Days and 1 Tuesdays 8.15 Mondays 14.20 times 2 Tuesdays 9.00 Mondays 15.05 for each 3 Wednesdays 14.20 Thursdays 11.05 period 4 Fridays 11.05 Thursdays 11.50

Solids achievement test given first in classes which studied the Solids unit first, Statistics achievement test first in classes which studied the Statistics unit first.

Solids achievement test given only in classes which studied the Solids unit second. Statistics achievement test only in classes which studied the Statistics unit second. 514

Table D.2

Experimental timetable: Second year

Week Period Group number number 2B 2C Easter term 1 1 PDQPDQ 2 HST HST 3 3DD 3DD 4 BOX BOX 2 1-4

3 1-4 Preparation Preparation

4 1-4 for teaching for teaching practice and practice 5 1-4

6 1-4 teaching teaching practice practice 7 1-4

8 1 Public holiday 2 3 First unit 4 (12 periods)

9 1—4 First unit 10 1-4 (12 periods)

11 1 Solids AT 2 Statistics AT^ 3 4 College holiday C1&C4: 3DD & BOX C2&C3: Statistics 12 1 Solids AT C1&C4: Solids 2 Statistics AT^ C2&C3: 3DD & BOX 3 B1&B4: 3DD & BOX 4 B2&B3: Statistics

13 1 B1&B4: Solids 2 B2&B3: 3DD & BOX Second unit 3 (12 periods) 4 Second unit (12 periods) 14 1 2 515

Table D.2 (continued)

Week Period GrouD number number 2B 20 Summer term 1 1-4 Second unit (continued) 2 1 Second unit Solids/Stat AT& 2 (continued) 3 4

3 1 Solids/Stat AT^ Continuation of 2 regular course 3 Continuation of 4 regular course

Days and 1 Wednesdays 8.15 Tuesdays 11.05 times 2 Wednesdays 9.00 Tuesdays 11.50 for each 3 Fridays 13.35 Thursdays 13.35 period 4 Fridays 14.20 Thursdays 14.20

Solids achievement test given first in classes which studied the Solids unit first. Statistics achievement test first in classes which studied the Statistics unit first.

Solids achievement test given only in classes which studied the Solids unit second. Statistics achievement test only in classes which studied the Statistics unit second. 516

Table D.3

Means and standard deviations of HST pretest scores of students with full data and with missing data, by year and sex

First year Second year Group Men Women Men Women Students N 76 91 80 68 with full M 7.71 6.84 8.55 7.72 data SD 3.00 2.75 3.25 3.19

Students N 9 19 27 35 with miss- M 9.89 6.95 8.44 7.43 ing data SD 3.76 2.27 2.59 2.51 t 2.01 0.17 0.15 0.47 P< 0.048 0.868 0.879 0.639

Table D.4

Correlations between spatial test and Solids Achievement test scores in first year Statistics groupé

Test 1 2 3 4 5 6 7 1. 3DD pretest 65 46 09 57 62 24 2. 3DD posttest 72 36 27 33 33 34 3. HST pretest 46 40 53 10 19 13 4. HST posttest 09 32 41 17 26 18

5. BOX pretest 53 72 09 13 81 06 6 . BOX posttest 24 44 15 16 62 19

7. Solids achieve­ ment test 28 32 22 06 07 15

Decimal points omitted. Correlations above the diagonal refer to men; Ns range from 31 to 61; P(r ^ 0.30) = 0.05, P(r ^ 0.41) = 0.01. Correlations below the diagonal refer to women; Ns range from 20 to 40; P(r 0.36) = 0.05, P(r ^ 0.49) = 0.01. 517 Table D.5

Correlations between spatial tests and Solids achievement test scores in second year Statistics groupé

Test 1 2 3 4 5 6 1. 3DD pretest 85 06 45 55 82 2. 3DD posttest 82 33 56 72 70

3. HST pretest 39 32 04 33 28

4. BOX pretest 55 61 -16 75 53 5. BOX posttest 60 64 13 73 58

5. Solids achieve­ ment test 45 51 14 36 32

^Decimal points omitted. Correlations above the diagonal refer to men; Ns range from 26 to 51. Correlations below the diagonal refer to women; Ns range from 25 tc• 51. P(r> 0. 32) = 0.05, P(r> 0 .44) = 0 .01.

Table D. 6

Frequency distributions of spatial pretest scores, by year

3DD HST BOX Frequency Frequency Frequency First Second First Second First Second Score Score Score year year year year year year 0-5 0 0 0-1 1 1 <-10 0 0 6-10 3 4 2-3 17 13 “9-0 5 5 11-15 13 11 4-5 41 32 1-10 19 22 16-20 31 41 6-7 43 40 11-20 33 41

21-25 21 32 8-9 45 63 21-30 30 39 26-30 17 20 10-11 32 30 31-40 11 14 31-35 7 6 12-13 13 22 41-48 1 1 36-40 7 6 14-15 3 9 518

Table D.7

Item analysis of 3DD pretest scores in first year

Maximum. Mean Item-test Item possible score correlation Poles top 3 1.68 0.47 middle 3 1.84 0.55 bottom 3 2.14 0.42

Poles total 10 5.66 0.69

Bottles 2 3 2.12 0.53 3 2 1.94 0.22 4 3 2.09 0.45 5 2 1.96 0.11 Bottles total 10 8.11 0.60

Cubes 1 1 0.95 0.18 2 1 0.46 0.37 3 1 0.35 0.48 4 1 0.90 0.26 5 1 0.32 0.56 6 1 0.46 0.56 7 1 0.38 0.35 8 1 0.19 0.52 9 1 0.32 0.58 10 1 0.17 0.62

Cubes total 10 4.52 0.79

Edges 2 2 0.98 0.64 3 3 1.07 0.71 4 3 1.37 0.80 5 2 0.24 0.57 Edges total 10 3.67 0.85 519

Table D.8

Item analysis of 3DD pretest scores in second year

Maximum Mean Item-test Item possible score correlation

Poles top 3 1.73 0.43 middle 4 1.60 0.46 bottom 3 2.26 0.48

Poles total 10 5.59 0.67 Bottles 2 3 1.94 0.41 3 2 1.88 0.26 4 3 2.09 0.51 5 2 1.97 0.25 Bottles total 10 7.88 0.55

Cubes 1 1 0.93 0.21 2 1 0.47 0.38 3 1 0.32 0.17 4 1 0.91 0.21 5 1 0.34 0.43 6 1 0.58 0.37 7 1 0.29 0.49 8 1 0.19 0.56 9 1 0.32 0.50 10 1 0.32 0.53

Cubes total 10 4.68 0.77

Edges 1 2 0.88 0.69 2 3 1.06 0.66 3 3 1.39 0.73 4 2 0.28 0.59

Edges total 10 3.61 0.84 520

Table D.9

F-value for successive terms in stepwise polynomial regression of 3DD and BOX posttest scores on corresponding pretest score, by year and sex

First year Secondyear Term Men Women Men Women (N=41) (N=52) (N=50) (N=38) 3DD

Linear 31.83 64.39 88.00 41.97 Quadratic 0.35 0.02 0.15 1.12 Cubic 0.01 1.84 0.08 4.08

BOX

Linear 55.20 37.03 19.46 46.75 Quadratic 0.25 0.18 0.05 0.17 Cubic 0.51 1.51 0.74 0.00 521

Table D.IO

Correlations between spatial tests and Solids achievement test scores in first year Solids groupé

Test 1 2 3 4 5 6 7 1. 3DD pretest 81 25 58 30 42 63 2. 3DD posttest 76 36 62 26 71 66

3. HST pretest 21 45 62 29 36 05 4. HST posttest 21 28 45 26 46 33

5. BOX pretest 37 37 29 01 52 39 6 . BOX posttest 28 39 32 03 75 60

7. Solids achieve­ ment test 57 69 55 25 45 49

^Decimal points omitted. Correlations above the diagonal refer to men; Ns range from 10 to 23; P(r^ 0.50) = 0.05, P ( r 0.66) = 0.01. Correlations below the diagonal refer to women; Ns range from 20 to 39; P(r^ 0.36) = 0.05, P(r 0.49) = 0.01.

Table D. 11

Correlations between spatial tests and Solids achievement test scores1 in second year Solids groupé

Test 1 2 3 4 5 6 1. 3DD pretest 75 52 46 64 68 2. 3DD posttest 56 57 65 46 52

3. HST pretest 43 31 49 34 56

4. BOX pretest 45 37 35 75 41 5. BOX pretest 21 12 28 79 49

6 . Solids achieve­ ment test 29 25 42 33 61

^Decimal points omitted. Correlations above the diagonal refer to men; Ns range from 24 to 51; P(r^ 0.33) = 0.05, P(r;;^ 0.45) = 0.01. Correlations below the diagonal refer to women; Ns range from 15 to 45; P (r^ 0.41) = 0.05, P(r;^ 0.56) = 0.01. 522

Table D.12

Mean scores on 3DD and BOX posttests in first year, by level, sex, treatment and pretest condition

Solids unit Statistics unit Level Sex Pretest No pretest Pretest No pretest 3DD

A Men 30.80 30.33 30.00 27.56 Women 24.43 21.00 22.83 23.40

N Men 24.83 27.89 25.82 30.25 Women 22.50 18.38 24.33 21.25

S Men 26.00 23.43 22.33 25.50 Women 22.50 18.00 19.57 18.90

BOX

A Men 28.20 29.00 29.29 23.00 Women 31.14 20.14 19.67 18.60

N Men 20.33 22.33 23.46 26.75 Women 17.88 14.20 27.67 18.12

S Men 28.00 21.43 20.33 26.20 Women 20.90 19.89 20.29 13.10 523

Table D.13

Mean scores on 3DD and BOX posttests in second year, by tutor, sex, treatment and pretest condition

Solids unit Statistics unit Tutor Sex Pretest No pretest Pretest No pretest

3DD

A Men 26.00 31.50 29.60 26.33 Women 22.00 22.33 23.17 23.50

B Men 27.57 31.60 23.50 27.14 Women 23.67 20.67 26.00 17.20

C Men 29.50 26.80 26.5 7 25.33 Women 23.00 24.60 24.17 15.75

D Men 28.20 24.67 28.71 24.00 Women 23.75 22.00 19.25 19.80

BOX

A Men 18.25 19.17 29.20 24.17 Women 18.00 23.67 20.50 22.00

B Men 28.86 22.40 12.17 25.14 Women 17.33 11.67 22.29 4.00

C Men 32.00 14.80 29.29 18.67 Women 22.00 17.40 24.33 12.25

D Men 27.60 15.33 26.14 14.50 Women 25.50 17.75 15.50 12.40 Table D.14

Analysis of variance of 3DD and BOX posttest scores in first year, by level, sex, treatment and pretest condition

Multi­ 3DD BOX Source d.f. variate MS P< MS F PC PC Level (L) 2 0.009 202.49 5.90 0.003 268.28 2.73 0.069 Sex (X) 1 0.001 1463.53 42.65 0.001 973.80 9.91 0.002 Pretest (?) 1 0.069 63.20 1.82 0.177 500.74 5.10 0.026 Treatment (T) 1 0.846 0.02 0.00 0.982 30.12 0.31 0-581

LX 2 0.85 3 13.59 0.40 0.674 14.18 0.14 0.866 LP 2 0.874 10.87 0.32 0.729 42.64 0.43 0.649 LT 2 0.152 25.15 0.73 0.482 329.75 3.36 0.038 XP 1 0.167 104.62 3.05 0.083 172.90 1.76 0.187 XT 1 0.708 14.78 0.43 0.513 6.64 0.09 0.795 PT 1 0.496 40.88 1.91 0.277 0.57 0.01 0.940

LXP 2 0.521 50.08 1.46 0.236 45.94 0.47 0.627 LXT 2 0.786 5.62 0.16 0.849 59.58 0.61 0.547 LPT 2 0.924 10.87 0.32 0.729 12.36 0.13 0.882 XPT 1 0.748 7.34 0.21 0.644 16.84 0.17 0.680

LXPT 2 0.279 13.46 0.39 0.676 253.81 2.58 0.079

Error 143 34.32 98.28

in to Table D.15

Analysis of variance of 3DD and BOX posttest scores in second year, by tutor, sex, treatment and pretest condition

Multi­ Source^ d.f. variate 3DD BOX P< MSF P< MS F P < sex (X) 1 0.005 1115.34 470.86 0.001 1095.75 13.38 0.035 Tutor (U) 3 0.698 15.90 0.41 0.746 97.84 0.71 0.547 Pretest (P) 1 0.101 59.23 3.50 0.158 1327.41 5.94 0.093 Treatment (T) 1 0.230 123.21 9.27 0.056 36.15 0.29 0.628

XU 3 0.814 2.37 0.06 0.980 81.91 0.60 0.619 XP 1 0.298 80.73 1.01 0.388 15.39 0.05 0.836 XT 1 0.751 0.00 0.00 0.992 81.81 0.65 0.479 UP 3 0.568 16.95 0.44 0.727 223.52 1.63 0.187 UT 3 0.660 13.29 0.34 0.795 124.79 0.91 0.439 PT 1 0.155 83.38 5.12 0.109 8.98 0.14 0.730

XUP 3 0.204 79.72 2.05 0.110 301.94 2.20 0.092 XUT 1 0.595 26.51 0.68 0.564 125.61 0.91 0.437 XPT 1 0.552 3.65 0.10 0.778 277.10 1.77 0.275 UPT 3 0.906 16.27 0.42 0.740 62.52 0.46 0.714

XUPT 3 0.466 38.36 0.99 0.401 156.20 1.14 0.337

Error 123 38.81 137.47

U 1 Factor U random. For all effects not involving U, the error term was the to corresponding interaction with U. For all effects involving U, the within- Ln cells error term was used. 526

Table D.16

Mean scores on 3DD and BOX posttests in first year, adjusted for covariance with corresponding pretest, by level, sex and treatment

Level Sex Solids unit Statistics unit

3DD

A Men 24.93 23.96 Women 24.94 26.06

N Men 26.19 22.91 Women 25.19 24.20

S Men 27.80 20.81 Women 24.50 24.47

BOX

A Men 25.22 25.52 Women 29.64 25.24

N Men 18.59 26.06 Women 20.63 24.46

S Men 26.76 18.42 Women 26.18 21.05 527

Table D.17

Mean scores on 3DD and BOX posttests in second year, adjusted for covariance with corresponding pretest, by tutor, sex and treatment

Tutor Sex Solids unit Statistics unit 3DD A Men 26.05 24.87 Women 32.16 25.38

B Men 27.45 25.29 Women 24.90 23.88

C Men 26.28 27.17 Women 28.98 25.54

D Men 27.87 25.50 Women 26.74 22.09

BOX

A Men 29.61 24.49 Women 21.15 23.79

B Men 24.99 18.45 Women 23.35 19.04

C Men 23.56 26.59 Women 22.09 24.97

D Men 25.40 25.09 Women 23.00 22.33 Table D.18

Analysis of covariance of 3DD and BOX posttest scores in first year, using corresponding pretest as covariate, by level, sex and treatment

3DD BOX Source d.f. MS F P< MS F P <

Regression 1 1630.88 73.03 0.001 3528.24 72.91 0.001

Level (L) 2 1.40 0.06 0.939 120.24 2.48 0.090 Sex (X) 1 1.04 0.05 0.830 11.80 0.24 0.623 Treatment (T) 1 46.55 2.08 0.153 4.02 0.08 0.774

LX 2 2.31 0.10 0.902 6.54 0.14 0.874 LT 2 10.81 0.48 0.618 292.55 6.04 0.004 XT 1 61.71 2.76 0.100 25.79 0.53 0.467

LXT 2 11.69 0.52 0.594 27.61 0.57 0.567

Error 83 22.33 48.39

Ul to 00 Table D.19

Analysis of covariance of 3DD and BOX posttest scores in second year, using corresponding pretest as covariate, by tutor. sex and treatment

3DD BOX Source^ d.f. MS F P< MS F P < Regression 1 1769.44 113.90 0.001 4855.21 83.05 0.001

Sex (X) 1 45.75 3.94 0.186 46.61 4.21 0.177 Tutor (U) 3 5.01 0.32 0.809 73.29 1.25 0.296 Treatment (T) 1 16.65 7.80 0.108 51.64 0.54 0.537

XU 3 13.44 0.86 0.463 7.40 0.13 0.944 XT 1 14.87 0.98 0.427 5.96 0.35 0.616 UT 3 7.61 0.49 0.690 77.64 1.33 0.271

XUT 3 10.17 0.66 0.583 11.56 0.20 0.898

Error 73 15.54 58.46

^Factor U random; see note to Table D.15.

U l N) VO 530

Table D.20

Analysis of covariance of HST posttest scores in first year, using HST pretest scores as covariate, by level, sex, treatment and pretest conditions

Source d.f. MS F P < Regression 1 282.11 41.25 0.001

Level (L) 2 3.74 0.55 0.580 Sex (X) 1 15.35 2.24 0.136 Pretest (P) 1 1.69 0.25 0.620 Treatment (T) 1 5.04 0.74 0.392

LX 2 15.60 2.28 0.106 LP 2 14.12 2.06 0.131 LT 2 4.75 0.69 0.501 XP 1 0.06 0.01 0.925 XT 1 23.50 3.44 0.066 PT 1 2.70 0.39 0.531

LXP 2 1.47 0.21 0.807 LXT 2 9.60 1.40 0.249 LPT 2 4.11 0.60 0.550 XPT 1 16.06 2.35 0.128

LXPT 2 9.94 4.97 0.485

Error 142 6.84 531 Table D.21

Mean scores on HST posttest in first year, adjusted for covariance with HST pretest, by level, sex and treatment

Level Sex Solids unit Statistics unit A Men 10.80 9.10 Women 7.30 8.92

B Men 9.44 8.12 Women 7.96 8.66

C Men 8.95 8.26 Women 9.51 8.49

Table D.22

Mean scores on achievement tests in first year, by level, sex and treatment

Level Sex Solids unit Statistics unit Solids achievement test A Men 22.10 11.62 Women 21.33 9.82

N Men 18.07 10.73 Women 14.00 9.15

S Men 16.46 11.14 Women 17.74 8.35

Statistics achievement test

A Men 12.10 26.94 Women 13.17 28.45

N Men 13.73 28.87 Women 14.06 31.46

S Men 11.55 22.14 Women 8.84 21.88 532

Table D.23

Mean scores on achievement tests in second year, by tutor, sex and treatment

Tutor Sex Solids unit Statistics unit Solids achievement test

A Men 20.64 12.54 Women 16.10 8.71

B Men 19.36 11.59 Women 14.35 9.75

C Men 17.07 10.64 Women 16.11 10.75

D Men 21.40 10.54 Women 16.58 10.25

Statistics achievement test

A Men 17.00 27.73 Women 17.70 25.79

B Men 16.91 25.12 Women 13.42 22.42

C Men 16.86 29.00 Women 14.00 33.50

D Men 13.70 28.54 Women 19.50 28.83 Table D.24

Analysis

Solids Statistics Source d.f. MS F P < MS F P < Level (L) 2 91.38 5.72 0.004 634.03 14.66 0.001 Sex (X) 1 74.41 4.66 0.033 59.07 1.37 0.244 Pretest (P) 1 7. 19 0.45 0.504 125.57 2.90 0.091 Treatment (T) 1 2688.99 168.26 0.001 8371.72 193.62 0.001

LX 2 24.55 1.54 0.219 39.46 0.91 0.404 LP 2 3.12 0.20 0.823 2.52 0.06 0.943 LT 2 79.36 4.97 0.008 80.72 1.87 0.159 XP 1 1.63 0.10 0.750 50. 35 1.15 0 . 282 XT 1 2.18 0.14 0.712 6.85 0.16 0.691 PT 1 8.08 0.51 0.478 166.71 3.86 0.052

LXP 2 43.63 2.73 0.069 22.72 0.52 0.592 LXT 2 32.78 2.05 0.133 3.47 0.08 0.923 LPT 2 4.64 0.29 0.748 78.82 1.82 0.165 XPT 1 0.60 0.04 0.847 2.12 0.05 0.825

LXPT 2 53.47 3.35 0.038 6.63 0.15 0.858

Errer 138 15.98 43.24

U l U) UJ Table D.25

Analysis of variance of achievement test scores in second year, by tutor, sex, treatment and pretest condition

Solids Statistics Source" d.f. MS F P< MS F P < Sex (X) 1 382.61 10.27 0.049 0.02 0.00 0.984 Tutor (U) 3 19.06 0.72 0.541 156.44 3.18 0.026 Pretest (?) 1 72.80 1.73 0.280 107.77 1.75 0.278 Treatment (T) 1 2408.68 167.36 0.001 5523.30 57.22 0.005

XU 3 37.26 1.41 0.242 45.91 0.93 0.426 XP 1 9.65 1.35 0.331 0.02 0.00 0.977 XT 1 64.25 6.39 0.086 12.58 0.21 0.675 UP 3 42.17 1.60 0.193 61.65 1.25 0.292 UT 3 14.39 0.54 0.653 96.53 1.96 0.121 PT 1 13.48 0.29 0.629 11.24 0.19 0.691

XUP 3 7.20 0.27 0.845 26.42 0.54 0.657 XUT 1 10.05 0.38 0.757 58.82 1.20 0.313 XPT 1 12.94 0.24 0.658 0.44 0.01 0.927 UPT 1 46.82 1.77 0.155 58.74 1.20 0.313

XUPT 3 53.96 2.04 0.110 45.36 0.92 0.431

Error 150 26.44 49.16

Factor U random; see note to Table D.15, Ul 0 0 535

Table D.26

Analysis of variance of achievement test class mean scores, by level/tutor and treatment

Solids Statistics Source d.f. MS FPMS F P P First year

Level 2 6.79 2.32 0.301 19.48 9.14 0.099 Treatment 1 103.01 35.24 0.027 310.57 145.66 0.008 Residual 2 2.92 2.13 Second year

Level 3 0.47 0.49 0.715 7.90 1.56 0.361 Treatment 1 102.56 105.87 0.002 235.76 46.65 0.006 Residual 3 0.97 5.05

Table D.27

Correlations between rate of working and scores on Solids achievement test

First year Second year Level N r Tutor N r A 27 0.45 A 17 0.00 N 36 0.08 B 19 0.33 S 36 0.41 C 22 0.18 D 19 0.73

^For N = 20, P( 1 r|;>0.42) = 0.05. 536

Table D.28

Analysis of variance and means of spatial pretest scores in first year, by sex and category of previous education

Multi­ Source d.f. variate 3DD HSTBOX

Sex (X) 1 0.001 MS 1264.75 26.93 447.45 F 33.57 3.06 4.46 P< 0.001 0.084 0.037

Previous edu­ 0.236 MS 90.26 25.24 73.83 cation (V) 2 F 2.40 2.87 0.74 P< 0.097 0.062 0.482

XV 2 0.496 MS 51.16 10.93 173.03 F 1.36 1.24 1.73 PüC 0.262 0.294 0.184

Error 91 MS 37.68 8.80 100.26

Means

High schools Men 30.71 10.71 27.00 Women 20.27 7.82 16.27

Technical schools Men 27.85 7.86 24.00 Women 17.50 5.83 15.17

Other schools Men 24.48 7.55 18.62 Women 18.54 7.03 16.84 537

Table D.29

Analysis of variance and means of spatial pretest scores in second year, by sex and category of previous education

Multi­ Source d.f. variate 3DD HST BOX P < Sex (X) 1 0.004 MS 538.81 8.60 596.88 F 13.52 0.95 6.01 P < 0.001 0.332 0.016

Previous edu­ cation (V) 2 0.084 MS 80.63 1.61 423.85 F 2.02 0 . 18 4.27 p<: 0.137 0.837 0.016

XV 2 0.548 MS 39.13 2.83 144.90 F 0.98 0.31 1.46 P< 0.378 0.732 0.237

Error 113 MS 39.85 9.05 99.30 Means

High schools Men 29.40 7.80 30.60 Women 21.21 8.36 18.36

Technical schools Men 25.00 8.50 16.00 Women 17.50 7.75 5.75

Other schools Men 23.20 8.29 20.29 Women 19.32 7.51 16.54 538

Table D.30

Means and standard deviations of spatial pretest scores, by sex and year

First year Second year Test Men Women Men Women

3DD N 45 56 61 59 M 26.05 18.80 23.80 19.64 SD 7.09 5.43 6.78 5.90

HSTN 85 110 107 103 M 7.94 6.85 8.52 7.62 SD 3.13 2.66 3.08 2.97

BOXN 43 56 63 59 M 20.86 16.14 20.52 16.24 SD 10.93 9.48 10.06 10.29 539

Table D.31

Means and standard deviations of HFT, SRT drawing and GIM scores, by sex

SRT GIM Sex HFT drawing ML HVI HV2 SP Pz

Men M 31.57 22.14 4.14 5.71 7.29 5.86 5.00 (N=7) SD 9.98 8.11 0.69 1.11 0.95 1.34 1.16

Women M 28.55 19.11 4.22 5.11 5.11 5.00 4.78 (N=9) SD 8.35 5.11 0.97 1.05 0.78 0.86 0.83

t 0 .66 0.92 0.18 1.11 5.03 1.55 0.45 P < 0.521 0.375 0.858 0.286 0.001 0.143 0.661

Table D .32

Means and standard deviations of HFT and SRT drawing scores in Grade Nine and college samples, by sex

HFT SRT drawing Sample Male Female Male Female

Grade Nine N 8 8 8 8 M 28.62 28.50 26.62 22.88 SD 4.93 8.11 2.62 3.31

College N 7 9 7 9 M 31.57 28.56 22.14 19.11 SD 9.98 8.35 8.11 5.11

t 0.74 0.01 1.48 1.77 P-C 0.472 0.989 0.162 0.096 APPENDIX E

PRELIMINARY VERSIONS OF GROUP TEST INSTRUMENTS

540 541

THREE-DIMENSIONAL DRAWING TEST

(Trial version 1)

This is a test of your ability to visualize and draw simple solid objects. This is part of a scientific research project. Your performance on this test will help show how well Columbus students can visualize three-dimensional objects. Thank you for helping in this research. Your score will not be used for grading purposes.

Please write your name and other details in the following spaces.

Name: Sex:

School: Grade:

Age: yr mo Period: _____

You have most of the period for this test. All you will need is a pencil. Do not use a ruler for the drawings.

You will be asked to write the time when you finish each page of this test. Just copy down the time from the clock in your classroom, to the nearest minute. The time taken will not be used in scoring, but will be used in improving this test.

When everyone is ready to start, write the time here: and turn over. 542

, The figure on the right below represents a pyramid,

(a) On the base of the pyramid, find two opposite corners. Make a mark (•) at these two corners. (b) Find an edge which does not pass through either of the marked corners. Hake a mark in the middle of that edge, (c) Draw a triangle which has its corners at the marks that you have made.

2, The figure on the right below represents a cube. (a) Find two opposite corners of the cube. Make a mark (©) at each of these corners. (b) Find two parallel edges of the cube which do not pass through the marked corners. Make a mark in the middle of these two edges. (c) Draw a four-sided figure which has its corners at the four marks that you have made.

WRITE THE TIME WHEN YOU FINISHED THIS PAGE: NOW turn over and do Ou. 3» 543

The following diagrams all represent prisms. Using broken lines, draw in the hidden edges of each prism. The first one is done for you.

TIME FINISHED: Now turn over and do Qu, 4. 544 if. The following objects are made from cubes, stuck together. Some of the edges of the cubes are shown. You are to draw the remaining edges, (Do not draw the hidden edges.) The first one is done for you.

TIME FINISHED: Turn over. 545

5* The following diagrams represent cubes. Hidden edges are shown by broken lines The dotted lines show where a cut is to be made in each solid. What does the cut surface look like? Draw its actual shape below each diagram. The first one has been done for you.

TIME FINISHED: Turn over. 546

6. The figures below represent the side view of a cone. The broken lines shown where a cut is to be made in the cone. What would the cut surface look like? Draw each cut surface below the diagram. The first one has been done for you. I

TIME FINISHED: Turn over, 547

7, The first of the drawings below shows a closed bottle which is half full of water. The other four drawings show the bottle tilted into different positions. Show how the water will appear in the tilted bottles, (Draw in the water level and shade the water.)

I

TIME FINISHED: Turn over. 548

8. The drawing below shows a roadway winding into the distance. Draw the telegraph poles on one side of the road. (The first one is drawn for you.)

TIME FINISHED: Turn over. 549

9. Draw simple diagrams to Illustrate the following situations.

(a) A ladder is placed against a vertical wall. It stands on horizontal ground.

(b) A boy stands on a horizontal floor. There is a light in the ceiling, to his right. The light casts a shadow of the boy on the ground.

(c) Sugarville is a town in the middle of a wide plain (i.e. a flat piece of land). Knox is due North of Sugarville, and Acton is due West of Sugarville.

TIME FINISHED: Turn over. 550

10. Draw simple diagrams to illustrate the following situations.

(a) A house is built on a rectangular base. The base measures 12 ft by 16 ft. The roof slopes one way only. The front wall is 6 ft high, and the back wall is 8 ft high.

(b) A courtyard is in the form of a square. There are two vertical poles at opposite corners of the yard. An electric wire runs from the top of one pole to the top of the other.

(c) An aircraft is coming in to land at Port Columbus airport. It is 2 mi iVest and 1 mi North of the airport, and flying at a height of 3,000 ft.

TIME FINISHED: Turn over. 551

11, Make simple drawings of the following objects,

(a) A cup (b) A sugar cube

(c) A soup tin (d) A suitcase

(e) A milk carton (f) An ice-cream cone

(g) A sharpened pencil (h) A church spire

TIME FINISHED: Check over your work on Qu 1-11. If there is still some time left, then turn over and do the bonus question. 552

Bonus Draw a picture of your school, from the front yard.

Comments Do you have any comments on this test? Did you like it? Was it too easy? too hard? Were any particular questions too easy? too hard? Would you like to learn to draw three- dimensional objects better? Have you done these exercises before, say in art classes or in shop drawing classes? Do you like drawing?

(Continue overleaf, if necessary.) 553

Name:

School;

THREE-DIMENSION/i

DRAWING TEST

(Trial version 2)

This tost contains five drawing exercises. The answers are to he drawn in this booklet.

Do the drawings in pencil. Do not use a ruler.

There is one exercise on each page. You will be told how much time to spend on each exorcise. Please do not turn over to the next exorcise until you are asked to do so.

DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO START 554

Exercise 1 (k minutes)

This diagram represents a solid block:

The block Is made from small cubes like this:

Some more blocks are shomi below. These blocks are also made from small cubes. Draw the cubes In each object.

(To help you, some lines are drawn already.)

DO NOT TURN OVER UNTIL THE SIGNAL IS GIVEN. 555

Exercise 2 (4 minutes)

This diagram represents a solid block:

The block has some odges that you cannot see. The hidden edges can be drawn like this:

In the same way, draw the hidden edges of the following blocks.

STOP DO NOT TURN OVER UNTIL THE SIGNAL IS GIVEN. 556

Exercise 3 (3 minutes)

A . The diagram below represents a pyramid. Two opposite corners of the base are marked (•),

(i) Find an edge that does not pass through either of the marked corners. Make another mark (•) in the middle of that edge.

(ii) Draw the triangle that has its corners at the three marks.

B. The diagram below represents a cube. Two opposite corners of the cube are marked (*)«

(i) Find two parallel edges which do not pass through either of the marked corners. Make a mark (•) in tho middle of each of these two edges.

(ii) Draw the four-sided figure that has its corners at the four marks.

STOP DO NOT TURN OVER UNTIL THE SIGNAL IS GIVEN. 557

Exercise 4 (3 minutes)

The drawing below shows a road winding into the distance.

Draw the telegraph poles along one side of the road. (The first two poles are drawn for you.)

DO NOT TURN OVER UNTIL THE SIGNAL IS GIVEN. 558

Exercise 5 (2 minutes)

The drawing on the right shows a bottle which Is half-full of fruit juice.

The four drawings below show the same bottle tilted Into different positions. The top Is screwed on tightly, so the juice cannot come out.

Show how the juice appesœs In each bottle. (Draw the level of the top of the juice, and shade the juice.)

v_/STOP PLEASE DO NOT TURN BACK TO PREVIOUS QUESTIONS. At author's request, pages 559-567 not reproduced.

Available for consultation at

the Ohio State University. APPENDIX F

GROUP TEST INSTRUMENTS

568 569

Class number: ______Student number:

THREE-DIMENSIONAL

DRAWING

Do not open this booklet until you are told to start. 570

Exercise A 571 572

Exercise C

2 . 3.

4. 5. 573

Exorcise D

1.

2

4, At author’s request, pages 574-578 not reproduced.

Available for consultation at

the Ohio State University. 579

PERSONAL DATA QUESTIONNAIRE

Class number: ______Student number:____ _

1* AjS:e: I a m ______years old.

2, Date of birth: I was b o m o n ______

3. Sex; I am a ______.

4, Handedness : I write I'd.th m y ______hand.

5. Education:

I attended______in

I attended in

I attended in

6, Repeats ; I was in grade ______for two years.

7« Place; Rost of my life, I have lived in ______

8. Pamily; I have ______brothers and ______sisters,

9. Home ; Host of my life, I have been looked after by my

10, Occupation; M y is a

My __ is a

11, Ambition; When I leave school, I would most like to be a APPENDIX G

Instructions for administering group tests

580 581

Opening Statement

Good morning, boys and girls. My name is Mr. Mitchel- more. I have come from the university at Kingston. How many of you have been there?

I work in the School Education. There, we try to find out more about how people learn things, then we try to help teachers teach better. I'm interested in how well students can draw things, how well they can work with diagrams, and how well they can imagine things. I don't mean imagine to write stories, I mean imagine solid objects. (Pick up a book or other object to hand.) If I put this book behind my back (do it), can you imagine what it looks like? If I stand it up, can you imagine what the book looks like, even though you cannot see it? (Bring book from behind back and show to class.) Can you draw a picture of this book? If I show you a picture of something, can you imagine what it looks like? . . . That's the sort of thing I'm interested in.

There are many activities where you have to imagine what things look like. Boys, how many of you want to be engineers, technicians, mechanics, electricians, anything like that? ... Well, an engineer has to look at a diagram of something, imagine what it looks like, and then make it. Girls, do you like sewing? ... There again, you have to look at diagrams and imagine what the dress will look like, and then make it. A designer has to imagine what something looks like, and then draw it for someone else to make. There are many other things like that. That's why being able to imagine things is important.

I'm visiting many schools all over Jamaica, asking students to do three tests which will tell me how well they can draw, use diagrams, and imagine things. Will you do these tests for me this morning? ... These tests are not like the usual tests you do in school: There are no words and there are no numbers; only pictures and diagrams. And your scores will not be used for any grades in this school. So I think you should enjoy them. 582

So that no-one will know your score, I am going to give you a number to use instead of your name. (Shuffle numeral cards and deal out face down.) Take one of these cards; it will be your number for today. Write your number in your exercise book so that you will remember it. When I get home today, I will mark these tests and send the scores back to your teacher; but the list will have your number on it, not your name. If you remember your number, you will be able to find out how well you have done, and no-one else will know unless you tell them your number. Is that the best thing to do?

There is also a number for this class. It is (write up on blackboard.)

All you will need for these tests is a pencil and an eraser. Please put everything else away. Who does not have a pencil?

Three-dimensional Drawing test

Materials

1. One test booklet for each student.

2. The following visuals drawn on card approximately 50 cm X 70 cm, and a device for displaying these at the front of the room.

Ex A An enlargement of the test page, on blue card.

Ex B An enlargement of bottle No. 1, on yellow card. Transparent plastic covering at least the inside of the bottle.

Ex C An enlargement of the worked example No. 1, on pink card.

Ex D An enlargement of the cube in No, 1, but without the broken lines, on white card. Transparent plastic covering at least the region enclosed by the diagram of the cube. 583

3. The following models.

Ex B A bottle half-filled with colored liquid.

Ex C A cardboard model of block No. 1, measuring approximately 20 cm x 16 cm x 12 cm, with the face ruled in squares as shown in the diagram in the test paper. At one of the corners, a removable cube.

Ex D Two cubes of side approximately 20 cm, one in white card and one in transparent plastic, with the edges of both accentuated with black lines.

4. A red felt-tip pen or chinagraph pencil.

Procedure

Introduction (Give out test papers.) The first test is on drawing. Please write your class number and your student number on the front cover. In this booklet, there are four drawing exercises. We will do them one at a time. I will tell you what to do for each and how much time you will have. Do not turn over the page until I tell you. If you finish before the end, just close your booklet and wait. Do not look back at exercises which are finished.

Exercise A Open the booklet to Exercise A. (Display visual.) It is a blue page, like this. This is a picture of a road. It is straight here, then it bends, then it is straight, then it bends again, and disappears into the distance. (Indicate on visual. Point to telegraph poles.) Can you tell me what these are? ... Yes, they are light posts, that's right. Now, what I want you to do is to draw some more posts along this side of the road. (Point to right side of the road.) Draw 3 or 4 posts here (tap finger at four points to the right of the lower section of the road), 3 or 4 posts here (tap middle section), and 3 or 4 posts here (tap top section). Just draw simple posts, like these. (Indicate.) Do you see what you have to do? ... Put up your hand if you do not understand. Right, you have 3 minutes. Start now: (Check that no-one is using any form of straight edge. After 3 minutes, say;) Stop: Put your pencil down, everybody. (Proceed to next exercise before end of 3 minutes if all students have finished by then.) 584

Exercise B Turn over to Exercise B, on the yellow page. (Display visual.) Do you see this picture at the top of your page? It is a picture of this bottle. (Hold up model, upright.) But this picture does not show the juice in the bottle. How can we show the top of the juice in this picture? ... Yes, that's right. We can draw a line across the bottle here. (Mark horizontal line halfway up the bottle on visual.) Is the juice here (indicate the region above the line) or here (below the line)? ... Yes, of course it is. I'm going to mark an X to show where the juice is. (Do so.) We could shade this space, but it's quicker to mark an X. Will you all do that for No. 1? Then put your pencil down. (Check that all students have understood to draw a line and an X. ) .

Now look at the other pictures. Nos. 2, 3, 4 and 5. They show this bottle turned around into different positions. (Hold up bottle upright. Indicate turning movement with other hand, but do not turn bottle.) I want you to imagine what the juice would like like in each position; I'm not going to show you! Then, on each diagram, draw a line and a cross to show where the juice is, like we did for No. 1 (indicate). Does everyone understand? ... Right, you have 2 minutes for this exercise. Start now I (Check no-one uses a straight edge. After 2 minutes, say:) Stopl Put your pencil down, everybody. ... Has anyone not marked an X to show where the juice is? (Allow any defectors to insert X's.)

Exercise C Turn over to Exercise C on the pink page. (Dis­ play visual.) Do you see this picture at the top of your page? It is a picture of this block. (Display model; hold near visual.) Do you see? This block is made of small cubes stuck together. Here is one. (Detach corner cube and display clearly.) There are many cubes like this stuck together to make this block. (Hold model near diagram again.) There are 4 along here (indicate on upper front edge of block in diagram, using the single cube), 5 along here (indicate upper side edge), and 3 down here (indicate front vertical edge). If your mathematics is good, you will be able to see that there are 60 cubes altogether in this block. These lines on the faces (indicate on model) show the edges of the cubes where they are stuck together. (Demonstrate by re-inserting the corner cube.) Does everyone see that? Put up your hand if you don't understand. (If necessary, repeat instructions underlined above.) 585

Good. Now look at Nos. 2, 3, 4 and 5. These are pictures of four more blocks. They are not the same shape as this block (hold up model), but they are all made of cubes exactly the same as this one (hold up small cube). Some of them are made from less than 50 cubes, some from more than 60 cubes. Can you imagine that all those blocks are made of small cubes stuck together? ... want you to draw the lines on the faces to show the edges where the cubes are stuck together, like these (indicate on model). The lines on some of the faces haye been drawn already. You only haye to draw the lines on the other faces. Does eyeryone understand what to do? Put up your hand if you do not understand. (If necessary, repeat instructions underlined above.)

Right, you have 7 minutes for this exercise. Start now I (After 6 minutes say:) You have one minute left. (After 7 minutes:) Stopl Put your pencil down, everybody. (Proceed to next exercise before end of 7 minutes if all students have finished by then.)

Exercise D Turn over to the last exercise. Exercise D, on the white page. (Display visual.) This is a picture of this block. (Hold up white cube.) Here are the four edges at the front (indicate on model and diagram), here are the three edges going back (indicate on model and diagram), and here are the two edges at the back (indicate on model and diagram). There are some other edges of the cube at the back (indicate), but you cannot see them. (Hold cube between tester and class.) I can see them, but you can't, because you cannot see through the block. These are called hidden edges. The hidden edges are not shown in this dia­ gram (indicate) because you cannot see them.

Nov; imagine that you could see through the block, like this. (Hold up transparent model.) Can you see the hidden edges now? (Indicate.) We sometimes want to show the hidden edges on this diagram. We do that with broken lines, like this. (Mark hidden edges on visual using broken lines.) This is the diagram at the top of your page. Do you see it? The full black lines (point) show the edges which you see when you cannot see through the block, like this (hold up white cube); the broken red lines (point) show the extra edges you see when you can see through the block, like this (hold up transparent block). Does eyeryone understand? 586

Good. Now look at Nos. 2, 3, 4 and 5. They show some more blocks of different shapes. You cannot see through those blocks, so the hidden edges are not shown. I want you to imagine that you could see through those blocks; then draw the hidden edges, using broken lines as I did for No. 1 (indicate). Does everyone understand what they have to do? ... Right, you have 6 minutes for this exercise. Start now I (After 4 minutes say:) You have 2 more minutes. Close your book when you have finished. (After 6 minutes:) Stopl Put your pencil down, everybody. Close your books, that is the end of the first test. (Finish earlier if all students finish before end of 6 minutes.)

Hidden Shapes Test

Materials

1. One practice paper and one test paper for each student.

2. The following visuals mounted on card measuring approximately 90 cm x 70 cm and covered in transparent plastic, with a device for displaying them at the front of the room.

(a) An enlargement of the top part of the practice sheet (the hidden shapes and item No. 1).

(b) An enlargement of the hidden shapes and the four practice examples (items No. 2-5).

3. Transparencies of the five hidden shapes, and a water- color felt-tip pen or chinagraph pencil.

Procedure

Introduction Do you remember those puzzles you used to do in children's magazines where you had to find some object hidden in a picture? The next test is like that, but harder. Instead of everyday objects being hidden, there are going to be hidden shapes, shapes you have probably never seen before. Because this is so different we are going to have some practice before you do the test. 587

Practice (Give out practice papers.) When you get one of these papers, put it face down on your desk and write your class number and your student number at the top of the back. Don't turn it over until we are all ready to start.

Does everyone have a practice paper? Then turn it over, (Display visual.) Do you see these shapes at the top of your page? These are the shapes which are going to be hidden. There will be one, and only one, of them hidden in each picture. Look at No. 1. Can you see which shape is hidden in this picture? (Accept all answers without comment.) All right, let us look at each shape to see whether it is hidden in this picture. Here is shape A. (Hold up transparency of shape A. Show that it fits on A.) Let us see if we can find A in this picture. (Move it over No. 1 and show that it does not fit.) No, some of the lines are here, but not all of them. A is not the right answer. Let us try B. (Demonstrate as before.) No, most of the lines are there, but not all. B is not correct. Let us try C. (Demonstrate.) No, C is not correct. Let us try D. (Demonstrate.) Ah yes, here it is. All the lines are in this picture. D is the correct answer. Does everyone see that? ... Let us just check that E is not there. (Demonstrate.) No, E is not there. There is only one answer, D.

To show that the correct answer is D, underline the letter D here. (Mark visual.) Everyone do that. Make sure you make a nice dark mark, so that I will be sure that is your answer. (Check that all students are marking answers correctly.) There is only one answer to each question, so you should only underline one letter. If you mark two letters, your answer will be scored "wrong," even if one of your answers is correct.

There is one more thing I have to tell you; listen carefully: When I found shape D hidden in this picture, I did not have to turn it around (rotate D ) , and I did not have to turn it over (flip D ) . It was in the same position as it is at the top of the page. (Demonstrate once more by sliding D from the top to No. 1.) It will be the same for all these pictures. The hidden shape will always be in the same position as it is at the top of the page. 588

Now, Nos. 2, 3, 4 and 5 are for practice. See you many you can do in one minute. Start now! (Check that all students know what to do and mark their answers clearly. Use stopwatch but vary time allowance so that almost all the class completes No. 4 but no more than half completes No. 5.) Stop! Put your pencil down, everybody. (Ensure that all obey signal. Growl at those who are slow to react.)

(Display second visual.) Right, let's see how many you got right. What is the answer to No. 2? (Accept answers from several students before saying;) The answer is C. Let me show you. (Use transparency to demonstrate.) You should have underlined C. (Mark visual.) (Repeat for Qu 3 - E; Qu 4 - A; and Qu 5 - B.) How many got all four right? How many got three right? ... That's good, you are going to do well on this test.

Put your practice papers away now, and I'll give out the test papers. I'll collect both papers at the end.

Test When you receive a test paper, place it face down on your desk and write your class number and your student number at the top of the back. Do not turn it over until I tell you to start. (Give out test papers. Ensure that no-one turns over.)

In this test, there are 15 pictures like this. (Indi­ cate on visual.) Just find the one shape (point) which is hidden in each picture. Remember that it does not have to be turned around or turned over; always look for the shape in the same position as it is at the top of the page. The first questions are quite easy, like these (point to Nos. 4 and 5 on the visual), but the ones at the end are very difficult. Just do as many as you can. You will have 15 minutes. Everyone ready? Then turn over and start, now!

(After 10 minutes, say:) You have 5 more minutes. If you finish before the end, check your answers carefully. (After 15 minutes;) Stop! Put your pencil down, everybody.

I-D Boxes test

Follow instructions in the Examiner's Manual (American Institutes for Research, n.d.) with the following amendments, 589

Pages 5-12 Fold each net twice only, unless a student asks for a repeat.

Page 11 Before introducing the practice items, say:

In this test, each picture can be folded to make one of these two boxes (indicate on visual), either the box with the line between the two black faces (point) or the box with the line at the ends of the two black faces (point). The two boxes are always in the same order, with this one first (run finger down column to indi­ cate "between" box always first) and this one second (point). You just have to underline the box which you think is correct.

Page 13 Before giving out the test papers, say:

Does everyone understand what to do in this test? Good. Then put your green papers away and I'll give out the test papers. I will collect both papers at the end.

(Hold up test paper.) In this test, there are questions on both sides of the paper. There are 24 questions on this side and 24 questions on this side (indicate). We shall do one side, then stop, and then do the other side. You will have 4 minutes for each side. That's quite a short time, and I don't expect that you will all finish all the questions. After 4 minutes, some of you will have finished down to No. 24, some of you will still be on No. 12. Don't worry: just do as many as you can. Notice that the questions go down the page: 1, 2, 3, (indicate on question paper), just like the practice sheet (indicate on visual). The first questions are easier than the last ones, so if you do them in the correct order, down the page (show again), you will do the easier ones first.

When you get a test paper, place it on your desk with side 1 up (indicate numeral "1" on test paper). At the top of the page, where it says "number" and 590

"numéro," write your class number and your student number. When you have done that, look up. Don't start the test until everyone is ready and I tell you to start. Is that clear?

Personal Data Questionnaire

Introduction I would like to ask you to give me some information about yourselves. This is to help me to analyize your results on the tests you have done this morning. They are just simple questions like "Are you a boy or a girl?" Will you help me by answering these questions? ... Good.

Here is a sheet to write the answers to my questions. (Give out PDQ papers.) I will read out the questions, you just fill in the answers on this paper.

Please fill in your class number and student number at the top of the page, then I'll read out the questions when everyone is ready.

1. Age How old are you? Just fill in the answer in the space. If your 16th birthday is next week, fill in "I am 15 years old."

2. Date of Birth When were you born? Please tell me the date, the month, and the year; for example, 29 February 1925.

3. Sex Are you a boy or a girl? Or if you prefer, are you male or female?

4. Handedness Which hand are you writing with now?

5. Education Please tell me which schools you have been to since Grade 1, and which grades you were in at each school. For example, if you attended St. Peter's Primary School in Grades 1-6 and this school in grades 7-9, fill in this: (Demonstrate on chalkboard.) If you have been to more than two schools, use the other lines there. 591

5. Repeats Have you ever had to stay in a grade for a second year, perhaps because you were too young? If so, tell me which grade. If not, write "None" in that space.

7. Place Where have you lived most of your life? Please tell me the town and the parish. If you have lived in the Kingston-St. Andrew Corporate Area most of your life, just write "Kingston." Only write "St. Andrew" if you have lived in upper St. Andrew most of your life. If you have lived overseas most of your life, tell me the name of the city and the country.

8 . Family How many brothers and sisters do you have?

9. Home Who has looked after you most of your life? If you have lived most of your life with both your parents, fill in "father and mother." If you have lived most of your life with your mother only, and your father has not been at home, write "mother." If you have been looked after by a guardian, write "guardian" and say what relation they are to you; for example, "guardian (grandmother)." (Write three alternatives on blackboard.)

10. Occupation For each person you wrote in for Question 9, tell me what work they do. For example, if you have written "mother and father" for No. 9 (point to this alternative on chalkboard), tell me what they both do to earn a living. Please tell me as much as you can. If you know where they work, tell me that.

Let me give you some examples. If your mother does not do any work, write, "My mother is a housewife." I'll know what that means. If your father is a farmer, write, "My father is a farmer. He owns 5 acres of land and grows yams and sugar cane," or whatever it is. Tell me if he employs others to work for him, or if he only works for others. Tell me as much as you know about your parents or guardian's occupation so that I can understand what it is they do for a living. Put up your hand if you want me to help you. 592

11. Ambition Last question: What do you want to be when you leave school? Imagine that everything goes well for you and you pass all your examinations, what would you most like to be? Just one thing, please. Put up your hand if you want me to help you with spelling. APPENDIX H

INDIVIDUAL TEST MATERIALS

593 594

RECORD FORM

Name: Grade:

School; Date:

Part 1 : Time started Time finished

Part 2: Time started Time finished

1. EDS

1. Unscrewed bottle with hand.

Looked into bottle with ____ _ eye,

Reported seeing ______dots.

2. Used scissors with ______hand.

Looked through holes with ______hand.

Tried hole first, then hole

5. Looked through magnifying glass with eye.

Reported seeing ______

Eye dominance: 595

2. DCT Time Scoring Score

1: Correct on first attempt 60=2 Incomplete ___ Refl. ___ Rot, ___ Corrected/demonstrated, 2nd attempt 60=1 2: Correct on first attempt 60=2 Incomplete ___ Refl. ___ Rot, ___ Corrected/demonstrated. 2nd attempt 60= 1 3: Work as quickly as you can. Tell me wllen yc>u have fini;shed. Refl, ___ Rotd, ___ (No 2nd chance) 60=1,30=2 Incomplete. Demonstrated, 2nd attempt 60= 1 4: Now make this one. Work as quickly as you :an, What pieces go in here? ___ 6o=1,30=2

5; Now make this one. 60=1,30=2

6: Now this one. 90= 1,45=2

7; Work as quickly as you can. 90=1,45=2 Demonstrated correct design ___

8; This time use nine pieces. 120=2 ,60=3 _____ corner pieces at wrong angle (each -1)

9: Here is the next one. 180=2 ,90=3

10: Work as quickly as you can. 180=2 ,90=3

11: We need another trsiy for this one. 180=2,90=3 Only centre piece wrong __ _ (-1)

12: This time you will need more pieces. 240=3,120=4 One piece wrong _____ (-1) Copy symmetrical, 2 pcs. wrong ___ (-2) 13: Here is the last one. 240=2,180=3, 120=4 One piece wrong ____ (-1) Copy symmetrical, 2 pcs. wrong ___ (-2)

3-7: demonstrate rot. or refl. no more than once. Total

12-13: allow score if only 1 or 2 pcs. remain at time limit. Check S if last pc. inserted incorrectly, no more than twice. Stop testing if 3 scores 0 on 3 consecutive designs or on #12. 597

4/7, GIM

This test was done the drawing tests HVT and SRT,

Comprehension check 1 Horizontal-Vertical 2 (J_) RB 1 B 29 30 RB 2 t R 31 RB 32 RB 3 L X 33 R B 4 L X 34 RB 35 R B Muller-Lyer illusion 36 R B 37 RB 5 L R 38 RB 6 LR 39 RB L R 7 Sander Parallelogram 8 L R 9 L R 40 L R 10 LR 41 LR 11 LR 42 LR 12 LR 43 LR 13 LR 44 LR 14 LR 45 LR 15 LR 46 L R 16 L R 47 L R 17 LR 48 L R 49 LR Horizontal- Vertical 1 (~j) Ponzo illusion 18 R B 19 R B 50 T B 20 R B 51 TB 21 RB 52 T B 22 R B 53 T B 23 R B 54 TB 24 R B 55 T B 25 RB 56 TB 26 RB 57 TB 27 R B 58 TB 28 R B 59 T B 60 TB

ML HV1 HV2 SP Pz No» of illusion-supported responses Scale order of highest response No. of Guttman scale errors 596

3._H-FT

Took ______seconds to trace Practice 2. Took ______seconds to trace Practice 3. Mean tracing time = ___ seconds. Time on Practice ______seconds.

Time to Item correct Search Scale number timet) finish^ score^

1

2

3

4

5

6

7

8

9

10

Total

®Mark N if not seen after 80 s I if not complete after 30 s plus tracing time X if incorrect figure drawn and not corrected before 80 s plus tracing time.

^Subtract mean tracing time from time to correct finish.

Search time (s) 0-7 8-11 12-20 21-40 hi—80 Fail Scale score 5 4 3 2 1 0

Enter time finished on front page 598

5/6. PDI

1. Age; _____

2. Date of birth: ______

3. Sex: ______

4. What schools have you attended? School Grades

5. Have you ever had to stay in a grade for a second year?

6. Where have you lived most of your life?

7* Ho?/ many brothers and sisters do you have? ______B

8. V/ho has looked after you at home most of your life?

9» What work do(es) he/she/they do?

______is a ______

is a

10. What do you want to be when you leave school?

11. Skin colour: Wh LB MB DB B1 599

6 A . HVT

Comments:

7/5. SRT

Comments:

Hand used for writing:

Hand used for drawing:

Hand dominance: ______600

HIDDEN FIGURES

Name:

Copyright © 1962 by Educational Testing Service. All rights reserved. Adanted and reproduced by permission.

HFT was in the form of a booklet 215 mm x 90 mm, printed on blue paper. Each page contained one item, for example:

Practice 1

Items were selected from HST (Appendix F) as follows:

Practice items Test items

HFT HST HFT HST

1 1 1 6 2 3 2 7 3 4 3 9 4 5 4 1 5 4

6 10 7 11 8 13 9 13 10 14 APPENDIX I

Instructions for administering individual tests

601 602

Opening statement

Hello, what's your name? ... Do you like drawing? ... Do you like doing puzzles? ... Good, I've got several puzzles and drawings here for you to do this morning.

My name's Mr. Mitchelmore. I'm from the university at Mona, have you been there? I'm trying to find out how well students can draw and imagine things and work with pictures. Will you help me by doing these puzzles and drawings? ... Good.

I've got two tests here to do now, and then some more after lunch. It'll take us about an hour altogether. O.K.?

Eye-hand Dominance Schedule

Materials

1. A small screw-top clear-glass bottle with six small dots painted on the bottom face and covered with trans­ lucent white paper; the remainder of the bottle painted opaque blue.

2. For each student, one piece of opaque paper 7 cm x 21 cm, divided by lines into three 7 cm squares numbered 1, 2, and 3. In the centre of the squares, holes made by a file punch, a compass point, and a pin, respectively. Three cards approximately 13 cm x 8 cm of different colors, with a different number 2 cm high on one side of each card. A pair of scissors.

3. A Jamaican 50* note. On the back, a segment of the balcony containing eight small pillars circled and arrowed in pencil. A small hand magnifying glass.

Procedure

Before we do these two puzzles, I want to see how good your eyes are. 603

1. (Give bottle to S_ and say:) Look into this bottle, hold it up to the light, and tell me what you see on the bottom. (Record hand used to unscrew bottle, eye used to look into it, and the number of dots reported, (if necessary, add:) Hold it up close to your eye.

2. That's right. Now look at these three holes. (Show S_ paper with holes.) Which is the smallest hole which you can see through? ... Do you think you could read through there? ... Which is the smallest one you could read through? ... All right, then cut out that square, and I'll give you a short test. (Rummage in kit while £ cuts out square. Record hand used for cutting.) Good. Now hold that square up to your eye and look through the hole; cover your other eye with your other hand. That's right. I'm going to show you a card ; you tell me what color it is. (Hold up first card with plain face toward S^. ) What color is this? ... (Turn card over.) What number is this? ... (Record eye and number of hole used. Repeat with another hole if necessary to keep up pretence of finding ^'s ability to see through small holes or if it is not clear which eye ^ is using.)

3. Right, here's the last of these three eye tests. You know what tnis is, don't you? (Lay 50* note on table in front of ) In this circle, there are some small white lines going up and down. (Indicate.) Can you tell how many white lines there are there? You can use this magnifying glass to help you. (If necessary add:) Hold the glass up close to your eye. It's usually better that way. (Record eye used and number of lines reported.) Yes, I believe that is correct. Some people have difficulty counting lines when they are so close together.

Good, your eyes are all right, so we can go on to these two puzzles now. 604

Design Construction Test

Follow instructions in Pacific Design Construction Test Manual (Ord, 1958), with one amendment: For item 8, if ^ omits white pieces, proceed as for Item 4. Note that designs have to be presented upside-down to 2 (Ord, 1970, p. 121).

Hidden Figures Test

Here is a different kind of puzzle. Write your name on here, will you? (Record hand used. Open to Practice 1.) On each page of this booklet, there are two pictures. The shape of this side (draw around left-hand figure with blue felt-tip pen) is hidden in the picture on this side: let me show you (draw around "hidden figure" in right-hand diagram). Do you see that this shape (point to outline in right-hand diagram) is the same as this shape (point to left-hand diagram)? ... It is exactly the same; it is not turned around (demonstrate by hand movement) and it is not turned over (demonstrate).

Well, that's the puzzle. You have to find where the shape on this side of each page is hidden in the picture on this side (indicate). I'm going to see how quickly you can find each shape. (Pick up stopwatch and start it.) Try this one. (Turn over to Practice 2.) Can you see where this shape is hidden in there? ... (Hand ^ felt-tip pen.) Then draw around it. (Record time taken to trace around figure. Use zero-reset button to avoid distracting £. ) Good. Turn over and try the next one. (Record time to trace Practice 3.) Good, turn over. (Record time to trace Practice 4.)

(In Practice 2-4, if S_ fails to find hidden figure in 80 s, give a clue to enable him to start or if necessary draw the entire figure. If gives up before 80 s, encour­ age him to keep trying. If S_ draws wrong figure, say:) No, that's not quite right; it must be exactly the same as this shape. (If adds lines to original figure, say:) No, you can only use the lines that are there already. (If first attempt is wrong, allow a second attempt. If ^ hesi­ tates on any of the practice examples, indicate such on 605

record form and obtain tracing time from one of the test items. If omits one line of hidden shape, say:) Draw all the way around it.

Good, now you know what to do, don't you? You can turn over each page yourself; do each one as quickly as you can. Start now. (For each item, record time taken from first viewing page until last line of hidden shape is drawn cor­ rectly. If S^ draws an incorrect shape, say:) No, that's not quite correct. Try again. (If S_ gives up before 80 s, say:) Keep trying. (If S_ has not completed correct figure at the end of 80 s plus tracing time, say:) You can't find it? Turn over then, and try the next one. (Do not demonstrate correct response to failed test items, except possibly the last one to be attempted.)

(Discontinue if ^ obtains three consecutive zeros, saying:) We'll stop there. The rest would be too difficult for you, they are for older students.

Horizontal-Vertical Test

Materials and procedure as for 3DD Exercises A and B (Appendix G), with obvious modifications for individual administration.

Solid Representation Test

Materials

1. Apparatus and chin rests as described in Section 4.45.

2. For each S_, a booklet consisting of ten blank pages measuring approximately 11 cm x 20 cm, numbered 2, 3, 3, 4, 5, 1, 2, 3, 4, 5 respectively, with a cover sheet.

Procedure

Introduction Here's the next drawing I would like you to do. In this box, there are five wooden blocks, of different 606

shapes. Each block is painted the same color as the number on the front; can you tell me what these colors are? (Record any gross errors.)

I'm going to show you these blocks one at a time. The first time, I will open the lid and then close it again very quickly, so you can draw the block from memory. The second time, I will leave the lid open so that you can draw the block while you are looking at it. Do you see?

Here's a booklet for you to do the drawings in. There is one page for each drawing. Will you please write your name on the front?

(Select appropriate chin rest.) Now, I want you to rest your chin on here while you do your drawings. Is that com­ fortable? This is to make sure that all students look at these blocks from the same direction. (Lift apparatus onto wedge at other end of chin rest.)

Condition 1 Are you ready? ... I'll show you the blue block first. I will show it to you for quite a short time, so look carefully. Then draw the block exactly as you see it here. Make your drawing look solid, like a photograph. Ready? (Open lid to full extent as quickly as possible, remove hand from lid to a distance of about 20 cm, and then close the lid quickly without slamming. Aim at an exposure time of 1 s.) ... Have you finished? Then turn over, and I'll show you the next block, the red block. Ready? (Repeat for Nos. 2-5. Ensure that keeps chin on rest for each exposure. If _S starts to shade a drawing, stop ^ by saying;) You don't need to shade it. Just draw the lines. (Similarly if ^ starts showing the box. The first time leaves a broken line in a drawing, say:) What does the broken line mean? (If S^ indicates that they show hidden edges, say:) No, I want you to draw only what you saw from there. Please erase those lines. (If S^ indicates that they are visible, say:) Then draw them with full lines, like the others. (Similarly, if ^ shows hidden edges with full lines, say:) Could you see all those edges? ... Draw only what you saw. (Do not correct subsequent drawings showing hidden edges.) 607

Condition 2 Good. Now I'm going to show you each block again, but this time you can look at it as long as you like. Turn over. Remember to make your drawing look solid, like a photograph, and to draw exactly what you see from there. (Expose blocks one at a time. Allow S_ as much time as he he wishes. Allow as many erasures as necessary. Ensure ^ keeps chin on rest at all times he is drawing.) Have you finished? Good. (Remove chin rest.)

Condition 3 Now have a look at these. These are copies of drawings which other students have made of these blocks. Tell me which of them is the best drawing of each block. (Open lid to No. 1.) Here is the first one. (Place stimulus card on table in front of S^. ) Which of these is the best drawing of this block? (Record response. Repeat for Nos. 2-5.)

Postscript Good, you did well on those drawings. Have you ever drawn those blocks before? ... Do you do a lot of draw­ ing? ... (Enquire into S_‘s previous experience of drawing and source of instruction, if any. Record.) Do you know what these shapes are called? (Open lids one by one. Record names given by S_. )

Geometric Illusion Measures

Follow instructions in manual (Herskovits et al., 1969), omitting Poggendorff illusion. After diagram 60, ask:

Have you seen any pictures like those before? (Enquire further to find if ^ recognized pictures as illusions. Record.) What does this picture look like to you? (Point to diagram 60. Question S_ until it is clear whether ^ is interpreting diagram two- or three-dimensionally. Record.)

Personal Data Interview

Follow instructions for Personal Data Questionnaire (Appendix G), omitting question 4, with obvious modifica­ tions for individual administration. APPENDIX J

INSTRUCTIONS FOR SCORING DRAWING TESTS

608 609

Instructions for scoring: 5DD

Exercise A

Scoring is based on the slopes of the poles alongside each of the three straight sections of the "road." For each section, select a tynical pole (probably the one with median slope) and score its slope according to the table below; then add the three scores. If the slope falls on the boundary between two ranges, score according to the modal range of the poles in that section. For the central range (-8° to 8^^) only, award the stated score only if at least 7^% of the poles drawn fall within or on the boundary; otherwise, deduct one point. (E.g., if slopes range from 0° to 12° in the middle section, with a median of 6°, score only 3 if more than Z3% of the noles have slopes in excess of 8°.) Maximum score 10,

Slope <“90 “8 - 8 9-20 range^ "90-"38 “57-“23 “24-“9 21-47 48-90 >90

Top and bottom 0 0 1 2 3 2 1 0 0 sections

Middle 0 1 2 section 3 4 3 2 1 0

'‘Angles measured in degrees, clockwise from the vertical position. 610

Exercise B

Scoring is based on the slope of the juice level in each bottle. Score each slope according to the table below, then add the four scores. If a line is curved or segmented, draw a line of average slope and score that. Maximum score 10,

Bottle 2 Bottle 3 Bottle 4 Bottle 5

Slope ^ Score Slope Score Slope Score Slope Score

< "35 0 <"60 0 < " 9 6 0 <"96 0

"33-"14 1 "60-"11 1 "96-"24 1 "96-"11 1

"13- "7 2 “ 10- 10 2 “23- "7 2 "10- 10 2

“6- 6 3 11— 60 1 "6- 6 3 11- 96 1

7- 13 2 > 6 0 0 7- 30 2 > 9 6 0

14- 23 1 31- 66 1

> 23 0 > 66 0

Angles measured in degrees, clockwise from the horizontal position. Note that all cases where the juice is marked above the line score zero. 611

Exercise C

Score 1 point for the presence of each of the following features. Examples of acceptable drawings are shown overleaf. Maximum score 10.

No. 2 Look at the front face:

(1) One and only one more or less horizontal line connecting with given line

(2) One and only one vertical line drawn near the middle

Look at the top face:

(3) Line(s) drawn from front to back edges (all) parallel to the side edges: difference between intercepts less than 2 mm

No. 3 Look at the front face:

(4) Horizontal line connecting corner to given line, as shown

Look at the lower horizontal face:

(5) One and only one line drawn from left to right edges, not through a corner

(6) No line from front to back edges drawn parallel to the vertical edges of the block: difference between right intercepts at least 2 mm

No. 4 Look at the top face only:

(7) Line drawn parallel to at least one of the back edges

(8) Back corner divided into four quadrangular regions

No. 5 Look at the top face:

(9) Face divided into four quadrangular regions

Look at the lower front corner:

(10) Cube at front corner completed by correct two lines drawn on lower horizontal surface: both intercepts no more than 1 mm from corner 612

Score 1 Score 0 Score 1 Score 0 (6 )

(2) (7)

(3)

mm

(4) (9)

(5) (10) 613

Exercise D

Score points for each item as indicated below. Examples of each score sire shown overleaf. Maximum score 10.

No. 2

Score 2 if three lines are drawn from the correct vertices more or less parallel to the edges of the cuboid (not more than 6° from psirallel)

1 if three lines are drawn from the correct vertices, but not all parallel to the edges of the cuboid

0 otherwise

No. 3

Score 3 for more or less correct drawing

2 if it is topologically correct but at least one of the hidden edges deviates more than 6° from correct direction

1 if only the long hidden edge drawn, or if only a triangular hidden face drawn at the back

0 otherwise

No. 4

Score 3 for more or less correct drav/ing

2 for one face correct but not all, whatever else is drawn

1 if some edges correct, whatever else is drawn

0 otherwise

No. 5

Score 2 if more or less correct (not more than 6^ from parallel)

1 if correct three edges are drawn, but not all parallel to the given edges or with extra lines added

0 otherwise 614

Score Item 0 1 2 3

2 —

5

7 3 7 3 7 7

4 7 ' 7 7 3 3 ' 3 7 3 7 ' " ’^

0 7 ^ 7 3

7 7 " 3 7 - 0 7 3

5 — 615

Instructions for scoring SRT

1. Cuboid

Stage Examples

A rectangle

E Several rect­ angular faces juxtaposed

----\ Primitive depth depiction; either wrong number of faces or correct faces with top ■ |\ face rectangular / \ B \

Advanced depth depiction: top face shown in perspective. Also drav/ing from wrong view or showing hidden edges

Correct within drawing error

Capital letters indicate alternatives used in Condition 3, 616

2. Cylinder

Stage Criterion Examples

Front or top O view or general outline 0

Showing both end D faces (either O O circles or ellipses; l O i

Top face shown by circle, bottom face by straight line

RR Either top face A ____ _ ,__ ^ shown by ellipse or bottom face shown by curve

Correct within drawing error (top and bottom same curvature) □ L/ 617

3« Pyramid.

Stage Criterion Examples

0 A triangle

Showing wrong number of faces or three tri­ angular faces with common base and no fore­ shortening

Primitive depth depiction: either foreshortening of side faces or at least one base edge sloping, not both cues

Advanced depth depiction: both side faces fore­ shortened and both base edges sloping

Correct within drawing error (angle between sloping base edges not more than 10°) 518

4. Cube

Stage Criterion Examples

A square or rectangle □

Several squares or rectangles juxtaposed

Primitive depth depiction: either wrong number of faces or correct faces without vertical edges drawn parallel era

Advanced depth O i depiction: correct A A number of faces and all vertical E edges drawn parallel. Also drawing from wrong view or showing hidden edges

Correct within drawing error APPENDIX K

MATERIALS FOR PILOT UNIT

(SPATIAL TRAINING EXPERIMENT)

619 620

Unit: USE OF INSTRUMENTS j Card: 1

Theme: Using a ruler

Objective: 1. To draw a line accurately through two given points.

Activities: 1. Read JSPM Bk 1 Section 4-5 Ip 30). 2. Do Workbook Exercise 9 on Card R1.

Evaluation: 1. Bring your work to the tutor for checking and assistance. 2. Repeat inaccurate work using your own diagrams.

Unit: USE OF INSTRUMENTS Card: 2

Theme: Perpendicular lines

Objectives: 1. To recognize perpendicular lines. 2. To draw accurately a line through a given point perpendicular to a given lino, using a ruler and a set square.

Activities: 1. Read the first part of JSPM Bk 1 Section 4-6 Ip 30). 2. Do Workbook Exercise 10 on Card R2-1. 3. Read the second pai’t of Section 4-6 (pp 30-31). Consult tutor if in difficulty. 4. Do Workbook Exercise 11 on Card R2-2.

Evaluotion: 1. Bring your work to the tutor for checking and assistance. 2. Repeat inaccurate work using your own diagrams.

. . . . . J 621

Unit: USE OF INSTRUMENTS Card: 3

Theme: Parallel lines

Objectives: 1. To recognize parallel lines. 2. To draw accurately a line through a given point parallel to a given line, using a ruler and a set square.

Activities: 1. Read JSPM Bk 1 Section 4-7 (pp 51-32). Consult tutor If In difficulty. 2. Do Workbook Exercise 12 on Card R3.

Evaluation: 1. Bring your work to the tutor for checking and assistance. 2. Repeat Inaccurate work using your own diagrams.

Unit: USE OF INSTRUMENTS Card: 4

Theme: Rectangles and squares Ob.lectlves: 1. To recognize rectangles and squares. 2. To draw accurately rectangles and squeo-es of given sizes.

Activities: 1. A rectangle Is a four-sided figure with adjacent sides perpendicular. Its opposite sides are parallel and equal In length. A pqusLre Is a rectangle In which all the sides are equal In length. (a) Make a list of five objects In this room which have the shape of a rectangle. (b) Name three objects which are squares. 2. Draw accurately a square of side 2 In. 3. Draw accurately a rectangle measuring 7 cm x 5 cm.

Evaluation: 1, Bring your work to the tutor for checking. 2. Do JSPM Bk 1 Ex 4e Qu 5 & 6 (p 32). 622

Unit; ÜSE OF INSTRUMENTS Card; 4A

Objective: To find out more about pareillel and perpendicular lines.

Activities: 1. Do JSPM Bk 1 Ex L Qu 8 (p 98).

2. Do JSPM Bk 1 Ex if0 Qu 1 8e 2 (p 32). Be careful to use your Instruments! 3. Do JSPM Bk 1 Ex 4e Qu 3 (p 32). How many letters of the alphabet consist of only parallel and perpendicular lines? 4. A domino is formed by Joining two equfll^quares, like this:I I I . These figures: I I I I & " J could be called trominoes. Figures made of four equal Squares are tetrominoes. and figures made of five equal squares are pentoalnoes. How many differently-shaped trominoes are there? Tetrominoes? Pentominoes? Draw them all accurately, using a square of side 2 cm. 5. Draw accurately a square of side 2 in. Divide the square into ^ in. squares. Draw the diagonals of the small squares (the lines joining opposite corners of each square). What do you notice?

Evaluaij.on: 1. Discuss your results with other students.

Unit: USE OF INSTRUMENTS Card: 5

Theme: Drawing circles

Objective: 1. To draw accurately circles and arcs of circles of given radius and given centre.

Activities: 1. Read JSPM Bk 1 Section 6-1 (p 46).

2. Do Workbook Exercise 16 on Card R5.

3. Do JSPM Bk 1 Ex 6a Qu 2 8e 3 (pp 46-47).

Evaluation: 1. Bring your work to the tutor for checking and assistance. 2. Repeat inaccurate work. 623

Unit: USE OF INSTRUMENTS Card: 6

Theme: Drawing triangles

Objective: 1. To draw a triangle accurately, given the length of its sides.

Activities: 1. Read JSPM Bk 1 Section 6-2 (p 47). 2. Do Ex 6b Qu 2 & 5. Check answer to Qu 5 before proceeding. Correct if necessary. 5. Work Section 6-3 (pp 48-49) in small groups of four or five students and your tutor. 4. Do Ex 6c Qu 1, 4, 5, 7, 10, 13 & 15 (p 49).

Evaluation: 1. Bring your work to the tutor for checking and assistance. 2. Repeat unsatisfactory work and do extra exercises if necessary to achieve objective.

Unit: USE OF INSTRUMENTS Card: 6A

Theme: Further drawing

Ob.iective; 1. To gain extra facility in the ueo of compaaees and other instrumenta.

Activities: 1. Do JSPM Bk 1 Ex 6c Qu 12 (p 49). 2. Do JSPM Bk 1 Ex K Qu 7 (p 98). 5. Do JSPM Bk 1 Ex 6a Qu 1, 4 & 5 (pp 46-47).

Evaluation; 1, Discuss answers with other students. In case of difficulty, consult the tutor. 624

4-5 Drawing We say that the edges AB and BC shown in the In the remainder of this chapter, you are going to figure are perpendicular. We can al.so say that AB learn something about accurate drawing. For this, and BC are perpendicular lines, or that AB is you w ill need a sharp pencil. perpendicular to BC. Find some perpendicular lines in your classroom. (a) (cl Which of the lines in this figure are perpendicular?

V

Why are (a) and (b) bad for accurate drawing? Check them using your folded paper. T ry th is test o f drawing. Draw a few lines in The instrument used to test if two lines are your exercise book using a ruler. Now rub them perpendicular is the set square. out. Can you see the place where you drew the We mark lines which are perpendicular like this: lines? I f you can, you pressed too hard. Can you sec them on the next page? If you can. you pressed much too hard.

Turn to Exercise 9 in yuiir Workbook.

Exercise 4d Turn to Exercise 10 in voiir Workbook. Repeat the questions in Fx 9 in the Workbook using your own lines, points and circle. Drawing perpentdicuiar lines 4—6 Perpendiculcir lines Suppose that we want to draw a line through a Tear a piece of paper out of a rough book and fold point P perpendicular to a line m. We can use a it in tw o . Then fold it again so that the first crease ruler and a set square to do this accurately as folds onto itself: follows. (It does not matter whether ? is on m or not. Both cases are shown here.)

Second crease

30

Pages 624-641 copyright 1967 by Longman Group Ltd. All rights reserved. Adapted and reproduced by permission. 625

A ll these lines point in the same direction as m. They will never meet m. however ftir they are e.xtended. These lines are called parallel lines; they are all parallel to m. Are there any parallel lines in your classroom? Here is a sketch o f a cuboid.

A

B

W hich edges are (a) parallel to AB? (b) parallel to AD? (c) perpendicular to DC? Id) perpendicular to EH? First place your ruler along the given line m. Then place your set square with one o f its shorter edges against the ruler. Hold the ruler, and slide the Drawing parallel lines set square along it u n til the other short edge goes Suppose that we want to draw a line through P through the point P. parallel to a line m. Use a ruler and a set square Hold the set square, and draw the required line. as follows. Check your drawing with your set square, placing it on both sides of the line you have drawn:

Slide to new position

Turn 10 Exercise 11 in rour iVorkhooE

First put one edge o f your set square against the 4-7 Parallel lines line. Put your ruler against another edge. The ruler then marks the position of the line n perpendicular Draw a straight line m in your e.xercisc book. to m, (You do not need to draw n.) Draw a straight line n perpendicular to m. Using Now hold your ruler lirmly, and slide the set your ruler and set square, draw several lines square along the ruler until the edge passes through perpendicular to n. the point P. 31 626

Which of I he lines in this figure do you think are (c) parallel? □

4 (a) Draw PQ 8 cm long. (h) Draw a line through P perpendicular to PQ. M ark S on it so that PS is 6 cm long. Check using your ruler and set square. (c) Draw a line through Q perpendicular to PQ. To show that two lines are parallel, we put M ark R on it so that QR is 6 cm long. arrowheads on them, like this: (d) Draw the line RS. le)* What shape have you drawn? If)* What can you say about RS? Check it. 5 (a) Draw AB 50 mm long. (b) Draw AC and BD perpendicular to AB. Make AC 25 mm long and draw CD perpen­ dicular to AC to cut BD at D. Turn in F.xen isr 12 in ymir Workbook. (c) What shape have you drawn? Id) W hat can you say about C D and BD? Exercise 4e Check it. 1 Are the lines/n and n parallel? 6 Draw a line EF 64 mm long. Draw EH perpen­ dicular to EF. and make EH 64 mm long too. Draw one line through H parallel to EF. Draw another line through F parallel to EH. Let these tw o lines meet at G. W hat figure is E F G H ? Check it.

2 Are the lines a and b perpendicular?

3 Draw the following words accurately. Use your ruler and set square to draw parallels and perpendiculars. Choose vour own sizes. (a) (b) J u cJ J 32 627

6 More Solids

6-1 Circles A segment is a region between a chord and the circumference. Look ai the surface of a milk tin or a penny. A semicircle is half a circle, cut off by a diameter. Examine the curved edges carefully. They have a very special shape. They arc called circles. Try to draw a few circles freehand. Are you Turn to Exercise 16 in your Workbook. satisfied w ith them? We use a pair o f compasses to draw better circles. Before using compasses, make Exercise 6a sure that the pencil is not loose. H old the compasses at the top and lean them slightly in the direction 1(D) Draw a circle on tracing paper and turn the you are turning them. Practise draw ing some circles paper around a little. Does the circle look any on a piece o f paper. M ake them different sizes. different'.’ Look at it through the back of the paper. Is it the same? Parts of a circle Repeal this with a large S and then a large C. The ancient philosophers called the circle the perfect curve'. 'vVhat is special about a circle? 2 Draw a circle of radius about 3 cm in the middle of the page. Carel'ully go over the circumference in red. Name the centre A. Choose any point on the circumference. Name .centre it P. Draw the circle, with this point P as centre.

Choose another point Q on the red circum­ ference. With centre Q, draw another circle sector passing through A. Your figure should now look like the sketch below.

The centre of a circle is where you put the point of the compasses. The circumference is the line drawn by the pencil. A radius (plural: radii) is a straight line drawn from the centre to a point on the circumference. A chord is a straight line joining two points on the circumference. A diameter is a chord pa.ssing through the centre. An arc is part of the circumference. A sector is a region between two radii and the Draw a number of circles like this. .Space the circumference. centres etenlv around the red circle. All the.se

46 628

circles must puss through A. This gives a point B such that AB = 68 mm. Whut is the outline of the pattern you hu\c drawn? 3 Draw a circle of radius 25 mm on the left half of the page. Carefully colour the circumference red. Name the point on the red circle nearest the left-hand margin A. Choose a point on the red circumference. Draw the circle, with this point as centre, passing through the point A. Choose another point on the red circumference. With this point as the centre, draw another circle to pass through A. Draw a number of circles like this with their You may have a pair of dividers in your maths centres evenly spaced out around the red circle. set. These may be used fo r measuring lengths, as in They must all pass through A. the diagram below. If you do not have a pair of The outline of this pattern is called a ccmiioid. dividers, a pair of compasses may be used instead. Find out what this word means and explain why it is used for this shape. 4 Draw a circle o f radius 2 cm in the centre of the page. Colour its circumference red. Mark a point in blue about li cm outside the circle on the left. Choose :i point on the red circle. Draw a circle with this point as centre and passing through the blue point. Draw a number of circles like this with their centres on the red circle and all passing through the blue point. The outline of this pattern is called a limaçon. Find out what the word means and explain why it is used for this shape. 5 Draw circles with their centres on a given circle and all touching a diameter of that circle. This o utline is called a ncphniid.

6-2 Drawing and Open your dividers or compasses to the given measuring lengths length. Then measure this length on your ruler. What is the above length, to the nearest millimetre? In Chapter .5, you learned to draw and measure lengths to the nearest millimetre. You can now use compasses to do this more accurately. F.xcrcise 6b To draw a line of length 6K mm accurately, lirst I Draw a straight line about 15 cm long. Mark a draw a line a link' loiuirr than (iK mm Isay about point .A near one end. 8 cm long). Mark a point A near one end. Open I'sing your compasses, mark points B. C. your compasses to 68 mm on your ruler, as shown D, I: and F on the line so that A B = 80 mm, in the diagram at the top of the page. Put the point AC = 110 mm, A D = 95 mm, A E = 56 mm of the compasses on A, and draw a small arc and A F = 121 mm. cutting the line: , From the given measurements, calciilaie the ^ ------4- lengths of BC. CF and DE. 47 629

N ow inecisiiH' ihc lengths o f BC. CF tirnJ DE on 6-3 Constructions your drawing. How accurtilc was your drawing? with compasses 2 Draw a straight line about 15 cm long. Mark a Qiu’stiiiit I point L near the left end. Using your compasses, mark points M. N. O Choose a point near the centre of a page of your and P on the line, as follows. exercise book. Call this point P. (a) M to the right of L. LM = 60 mm Use your compasses to mark at least ten points, (b) N to the right of M. MN = 45 mm each 8 cm from P. (c) O to the left o f N . N O = 53 mm If you drew all the points which are 8 cm from P. Id) P to the right of O. OP = 27 mm what would you obtain? From the given measurements, calculate the Now use your compasses to draw the complete length of LP. Now measure LP on your drawing. circle. How accurate was your drawing? Question 2 3 This question refers to the map of West .Africa Draw a line PQ of length 50 mm near the centre on the inside front cover of this book. o f your page. Using your dividers (or compasses), measure Use your compasses to draw all the points which the following lengths to the nearest millimetre. are 40 mm from P. (a)* Freetown-Bamako Use your compasses again to draw all the points (b)* Accra-Tamale which arc 25 mm from Q. (c) Ib a d a n -K a n o Name the points of intersection of the two circles Id) The distance between the lines of longitude R and S. (the lines running north-south) What is the distance of R from P? |c) The distance from Sokoto to the equator What is the distance of R from 0'-’ 4* Draw a line AB of length 5? mm. .Are there any other points wltich arc 40 mm from Using your set square, draw a straight line P and 25 mm from Q? through B perpendicular to AB. Mark a point C on it so that BC = 42 mm. Draw .AC. Measure AC to the nearest m illi­ Triangles metre. We can use the ideas in the above questions to 5 Draw a line HK of length 100 mm. help us draw a triangle when we know the lengths Mark a point I on HK. so that HI = 7H mm. of its sides. Draw a line through I perpendicular to HK. To construct a triangle PQR in which PQ = Mark a point J on this line, so that I.I is 53 mm 50 mm. PR = 40 mm and QR = 25 mm. start by long. drawing PQ. Now draw two arcs; the lirst arc with Measure HJ and JK to the nearest millimetre. centre P and radius 40 mm. and the second arc with centre Q and radius 25 min. R is the point 6 |a) Draw a line U V o f length 85 mm. where they cross. (It is not necessary to draw the Draw a line through U perpendicular to whole circles to find R.) Draw the arcs lightly, and UV. Mark a point X on this line, so that U.X do not rub them out. is 37 mm long. Draw a line through V perpendicular to UV. Mark a point W on this line, so that VW is also 37 mm long. |b) What length do you expect XW to be? Measure it. and check your guess. Ic) What do you expect to be true about the lengths of UW and XV? .Measure them, and check vour ttuess.

48 630

To complete the triangle, join P to R and join 7 Construe! a triangle H JK in which all the sides Q to R: are 80 mm long. A triangle whose sides are all equal is ealled an equilateral triangle,

8 Draw an equilateral triangle of side 62 mm.

9 Copy the circles on the front cover. (The centres of the circles are the vertices of an equilateral triangle.) Notice that we can also construct the triangle PQR with R below PQ. 10 Draw a circle of radius 5 cm. Always use compasses for this construction. You Mark a point A on its circumference. cannot do it accurately with a ruler only. Put the compass point at A and. with the compasses still open to 5 cm. mark a point B on the circumference. Exercise 6c Now put the compass point at B and. with 1 D raw AB = 70 m m . the same radius, mark C on the circumference. Using ^our compasses, find a point (' so that Continue in this way. w ith a new centre each the distance from A to C is 55 mm and the time. You should linish back at A. distance from B to C is 45 mm. Join AB. BC. C D . and so on to A. Join A to C. and join B to C. The figure you have drawn is called a regular What figure base you drawn ' hexagon. Find what these words mean.

2 Draw VZ = K5 mm. 11 Draw a regular hexagon o f side 44 mm. Using your compasses, lind a point X so that the distance from X to V is 6N mm and the 12 Draw any two lines .AB and AC meeting at A. distance from X to Z is 72 mm. Make sure they are not perpendicular. Make Draw XY and XZ. AC = 9 cm and AB = 5 cm. Find two points D so that BD = 9 cm and 3 D raw K L = 6 cm. CD = 5 cm. What do you notice about the two Using your compasses, find a point M so that figures ABDC? Check it. KM = 4 cna and LM = 5 cm. 13 Draw two perpendicular lines. HK and LM. Draw MK and ML. meeting at O . M ake H O = K O = 45 mm and 4 Draw DE = 4 cm. L O = M O = 65 mm. Using your compasses, tind a point F so that Complete the figure H LKM . EF = 3 cm and DF = 5 cm. The figure you have drawn is called a rhombus. The point which is exactly in the middle of Measure the sides of the rhombus. What do D F is called the mid-point of D F. Find it and you notice? name it O. Draw a circle w ith centre O to pass through 14 Draw a rhombus of side 5 cm. the point D. Compare your drawing with your neigh­ What do you notice? bour's drawing. Are they the same?

5 Draw PQ = 9 cm. Find a point R so that 15 Draw two perpendicular lines. KT and IE, PR = 7 cm and QR = 7 c.m. Draw PR and QR. meeting at A. Make KA = IK mm. AT = 53 Notice that this triangle has two sides equal mm and lA = AE = 38 mm. in length. A triangle with two sides equal is Complete the figure KITE. called an isosceles triangle. What is the name of the figure you have drawn ? 6 Draw an isoceles trianule in which two sides Measure the sides of the figure KITE. What are f>l mm lone and I he third side is 47 mm lone. do vou notice'.’ 49 631

2 A piece o f wood is I 35 m long. Tw o pieces, each 2 A man had to walk one kilometre to his home. of length 0-34 m. tire cut off. What length of wood He walked half the way and stopped to cat a few is left? bananas. He then walked on for a third of a kilometre, before stopping for a rest. How much 3 The circumference of a circle can be found farther had he to walk? Give the answer to the appro.ximately by multiplying the diameter of the circle bv 3f. nearest metre. (a) What is the circumference of a circle, if the 3 Write, in index form, the value of the 3 in each radius is 2 i m? of the following numerals. (b) What is the diameter of a circle, if the cir­ (a)322pi,, (b )1 3 6 ,„.„ (c) 223,^, cumference is I2> m? Two of the numerals above represent the same 4(a) Write the next two terms in the following number. W hat is the num ber? W rite your answer sequences. in base ten. (i) (2.3i5(.7i. ...) 4 Find the angles marked by the letters. (ii) 100. 10. I. 0 1....) (b) W rite the numbers in the follow ing sequence as fractions. (■i1

(c) Is the sequence I -)■* ■ finite or infinite? 5 A bottle holds S50 ml of concentrated sul­ phuric acid. One litre of concentrated sulphuric 40 acid weighs 1-83 kg. Find the weight of acid in the bottle, in kilogrammes to ! d.p. If the acid is divided into three equal parts, what is the weight of each part? Give the answer in kilogrammes to 2 d.p. 6 In a piece of wood, there are two holes. The first Write down: is a square o f side 6 cm. The second is a circle of J) the first 15 term s o f the sequence o f m u ltip les radius 3 cm. There is one common solid that will o f 6. fit both holes exactly. What is the solid, and what (b| the first 20 terms of the sequence of multiples size is it? o f 4. 7 Draw a circle of radius 54 mm. centre A. Join A (c) the first 10 terms of the sequence of numbers to a point B on the circumference. which are m ultiples o f both 6 and 4. Construct a line through B perpendicular to 6 ( a ) Factorise (i) 95. (ii) 3 800 into primes, writing AB. Extend this line in both directions. your answer in index form wherever possible. Does this line intersect the circle? What is (b) ' Divide 3 8 by 0 95. special about the line? 7 A piece of card is a rectangle 10 cm by 18 cm. 8 Draw a graph of the first six numbers in the A cuboid, which is 55 mm long. 30 mm wide and

sequence( I2„ I 3r,,,,r. 14r: 38 mm high, is made from the card. Make a sketch showing how the net is drawn. 8 Draw AB 96 mm long. Exercise L At A. draw a perpendicular AC. 72 mm long. 1 Add the followingbinary fractions, Join BC. Measure BC. (a) 101 101 + (b)IIIOfll Find D. the mid-point of BC. 10011 II 101 Measure AD. What do you notice?

98 632

Unit: USE OF INSTRUMENTS Card; R1

Page 1 of 2

Exercise 9

1. Mark any three points P, Q, R (in that order) on the line n on the right. Draw the lines AQ and PB. Call the point where they meet X. Draw the lines AR and CP. Call the point where they meet Y. Draw the lines BR and CO. Call the point where they meet Z, What do you notice about the points X, Y and Z?

Chock it.

2. Mark a point E on the line AD on the left. Find the intersection of the lines EC and BD. Call this F. Find the intersection of the lines EB and AF. Call this G. What do you notice about the

points D, G and H? ______

Check it. 633

Unit: USB OF INSTRUMENTS Card: R1

Page 2 of 2

5. Extend the lines AB and XY until they meet. Call their intersection L. Mark any two points C and Z on the line n. Extend the lines CB and ZY until they meet. Call their intersection M. Extend the lines CA and ZX until they meet. Call their intersection N. What do you notice atout the

points L, M and N? ______

Check it.

4. Mark six points A, B. C, D, E and F on the circle on the left . Draw the lines AE and FB. Call their intersection L. Draw the lines AD and CF. Call their intersection M. Draw the lines CE and BD. Call their intersection N. What do you notice about the

points L, M and N? ______

Check it. 634

Unit: USE OF INSTRUMENTS Card: R2-1

Pago 1 01 2

Exercise 10

Which of the following pairs of lines do you think are perpendicular? Mark the perpendicular lines in pencil.

Use your set square to check your answers to Qu. 1, 635

Unit: USE OF INSTRUMENTS Card: R2-1

Page 2 of 2

3. UelnR your ruler only, draw a line perpendicular to each line at the point marked. Then check your drawing with your eat square. Check each drawing before you go on to the next one. 636

Unit: USE OF INSTRUMENTS Card: R2-2

Page 1 of 3

Exercise 11

Using your ruler and set square, draw a line through each point P perpendicular to each line m.

(over) 637

Unit: USE OF INSTRUMENTS Card: R2-2

Page 2 of 3

(a) Draw a straight line through A perpendicular to AB

(b) Draw a straight line through 0 perpendicular to CB

(c) Find the point of intersection of the two lines you have drawn. Call this point D.

(d) The figure ABCD is a . 638

Unit: USE OF INSTRUMENTS Card: R2-2

Page 5 of 3

3.

(a) Draw a straight line through A perpendicular to m. Mark the point of intersection of p^ and m.

(b) Draw a straight line through perpendicular to n, Mark C^, the point of intersection of and n.

(c) Draw a straight line pg through perpendicular to m. Mark the point of intersection of p^ and m.

Id) Draw a straight line q^ through Bg perpendicular to n. Mark Cg, the point of intersection of qg and n.

(e) Continue drawing p y q^, p^, q^, ... as far as you can.

(f) What can you say about the zig-zag line you have drawn? Unit: ÜSE OF INSTRUMENTS Card: R3

Page 1 of 2

Exercise 12

Using your ruler and set square, draw lines through the given points parallel to each line.

2. (a) In the figure below, draw straight lines through A and C parallel to m.

(over) 640

Unit: USE OF INSTRUMENTS Card: R3

Page 2 of 2

(b) A four-sided figure with two opposite sides parallel is called a trapezium (trapezia). How many trapezia are there

in the figure you have drawn? _ ____

3.

(a) Draw two lines parallel to n.

(b) Draw three lines parallel to n.

(c) A four-sided figure in which both pairs of opposite sides are parallel is called a parallelogram. How many parallelograms

aro there in the figure you have drawn? _ _ _ _ _ 641

Unit: USE OF INSTRUMENTS Card: R5

Page 1 of 1 Exercise 16

1. Repeat the design shown three times in the squares provided. Then colour them.

2. Repeat each of the designs in the large squares directly below. Then colour them. (A

+ -f APPENDIX L

MATERIALS FOR SOLIDS UNIT

(SPATIAL TRAINING EXPERIMENT)

642 643

Unit: SOLIDS Card: 1

Theme: Everyday aolida

Objectives: 1. To recoRnize the shapes of a cube, cuboid, cylinder, prism, pyramid, cone and sphere. 2. To identify faces, edges and vertices of simple solids. 3. To distinguish planes, lines and points.

Activities: 1. Discuss in largo group JSPM Bk 1 Section 4-I to 4 -4 (pp 25 -29). Include Ex 4a Qu 1 & 2, Ex 4b Qu 1-3 and Ex 4c Qu I-5 .

2. Do Ex 4b Qu 5-10 and Ex 4c Qu 8.

Evaluation: 1. Bring your work to the tutor for checking and assistance.

Unit: SOLIDS Card:

Theme: Making a cube

Objectives: 1. To design the net of a cube of given side. 2. To draw accurately the net of a cube.

3 . To construct a cardboard model of a cube.

Activity: 1. Do JSPM Bk 1 Ex 4f Qu 1-3 (p 33)• Workbook page 69 is on Card R2.

Evaluation: 1. Bring your work to the tutor for checking and assistance. 2. Repeat inaccurate work until you are satisfied with the results. 644

Unit: SOLIDS Card:

Theme: Making a cuboid

Objectives; 1. To design the net of a cuboid of given dimensions. 2. To draw accurately the net of a cuboid. 3. To construct a cardboard model of a cuboid.

Activity: 1. Do JSPM Bk 1 Ex 4f Qu 4 (p 33). Bring your net design to the tutor before drawing it accurately on cardboard.

Evaluation: 1. Do JSPM Bk 1 Ex 4f Qu 5(a). 2. If you need to (or want to), do also Qu 5(b).

Unit: SOLIDS Card: 4

Theme: Drawing a cuboid

Objective: 1. To sketch a cuboid using parallel projection.

Activities: 1. Read carefully JSPM Bk 2 Sections 4-1 and 4-2 (pp 38-40).

2. Do Workbook Exercise 6 on Card R4. 3. Practise drawing cuboids. Work with a partner. Take it in turns to draw three edges of a cuboid, and let the other person complete the sketch. 4. Do Ex 4a Qu 2 & 3 (Bk 2 p 40).

Evaluation: 1. Bring your work to the tutor for checking and assistance. 645

ünit: SOLIDS Card: 4A

Theme: Further work on cuboide.

Objective: 1, To gain extra facility with cuboide.

Activities: 1. Do JSPM Bk 1 Ex 4f Qu 6 (p 33)» 2. Do JSPM Bk 2 Ex 4a Qu 4 (p 40).

Evaluation: 1. Show your work to the tutor.

Unit: SOLIDS Card: 5

Theme: Making and drawing pyramide.

Objectives: 1. To design the net of a pyramid given the lengths of its edges. ?.. To draw accurately the nets of triangular, rectangular and square pyramids. 3. To construct cardboard models of these pyramids. 4. To sketch simple pyramids.

Activities: 1. Read JSPM Bk 1 Section 6-4 (p 50). 2. Do Workbook Ex 1? Qu 1 & 2 on Card R5.

3. Do Ex 6 d Qu 1-4.

Evaluation; 1. Bring your work to the tutor for checking and assistance. 646

Unit: SOLIDS Card: 6

Theme: Making and drawing prisms

Objectives: 1. To design the net of a prism given the shape of the cross section.

2 . To draw accurately the nets of triangular, rectangular and regular hexagonal prisms. 5. To construct cardboard models of these prisms. 4. To sketch prisms given the cross section.

Activities: 1. Read JSPM Bk 1 Section 6-5 (p 50).

2 . Do Ex 6 e Qu 1-5« Bring your work to the tutor for checking before you proceed.

5. Do Ex 6 e Qu 4 & 5. 4. Make a sketch of a hexagonal prism.

Evaluation: 1. Bring your work to the tutor for checking and assistance.

Unit; SOLIDS Card: 7

Theme: Cones and cylinders

Objectives: 1. To design and draw accurately the net of an open cylinder given its height and its circumference. 2. To recognize the net of a cone.

Activities; 1. Do JSPM Bk 1 Ex 6 f Qu 1, 2 i 5» 2. Sketch the models you have made.

Evaluation: 1. Bring your work to the tutor for checking and assistance. 647

Unit: SOLIDS Card: 8

Theme: Cross sections

Ob.1ecti.ve: 1. To identify the cross sections of pyramids, prisms, cylinders and cones.

Activities: 1 « Divide the class Into four groups. Each group do one of the following exercises in JSPM Bk 1: (1) Ex 6 d Qu 5 (11) Ex 6 e Qu 6 (ill) Ex 6 f Qu 5 (Iv) Ex 6 f Qu 6

2. In a large group, discuss Ex 6 f Qu 4 & 7.

Evaluation: None.

Unit: SOLIDS Card: 9

Theme: Polyhedra

Oblectlves: 1. To make models of further polyhedra. 2. To discover the relation between the number of faces, edges and vertices of a polyhedron.

Activities: 1. A pol.vhedron Is a closed solid with plane faces. Make some more polyhedra of your choice. Sketch each one you make. (Rofer to JSPM Bk 1 Ex 6 g Qu 5~9 end anproprlate sections of other textbooks in the library, e.g. SMP Bk 1. See also Cundy & Rollett: Mathematical Models.)

2. Do JSPM Bk 1 Ex 6 g Qu 10 (p 54).

Evaluation: 1. Bring your work to the tutor for checking and assistance. 648

Unit: SOLIDS Card: 9A

Theme: Truncated solids

Objective: 1, To design and construct models of solids obtained by truncating pyramids and cubes.

Activities; 1. Do JSPM Bk 2 Ex ifb Qu 1 & 2 (pp 43-44). 2. Do any further questions from this exercise which Interest you.

Evaluation: 1. Show your work to the tutor and other students.

ünit: SOLIDS Card: SO

Summary of Objectives

1. To recognize the shapes of the cube, cuboid, cylinder, prism, pyramid, cone eind sphere.

2. To Identify faces, edges and vertices of solids.

3. To distinguish planes, lines and points.

4. To design the nets of cubes, cuboids, pyramids, prisms and cylinders of given dimensions, and to recognize the net of a cone.

5. To draw accurately the nets of cubes, cuboids, pyramids (only triangular, rectangular and square pyramids), prisms (only triangular, rectangular and regular hexagonal prisms), and open cylinders of given dimensions.

6. To construct cardboard models of cubes, cuboids, pyramids, prisms, cylinders and cones.

7. To sketch the above solids using parallel projection. 649

4 Solids (Book 1)

4 -1 Shape You should look at objects around you in this way. Try to handle them. Look at them carefully. In this chapter, we arc going to study the shapes of Find in what way they are like other objects in objects which we sec and use every day. We shall shape and how they are different. We call these see how the shapes are alike and how they are objects solids. different. A football and an orange aie very different Look at the picture below. Are any of the objects in size, colour and weight but they are alike in which you can see alike? Which arc the most shape. com m on shapes?

U'' -.h;

ftHHjs I

N. A U i \

25

Pages 649-669 copyright Q 1967, 1968 by Longmat: Group Ltd. All rights reserved. Adapted and reproduced by permission. 650

4-2 Some common solids Which objects pictured on page 25 are cylinders? Make a list of ten objects which arc cylinders. For the next few lessons you should have some of the following objects in front of you. The prism A chalk box An orange A cigarette tin A milk tin The cuboid and cylinder are special kinds of A table-tennis ball A cubc-sugar box prism. Here arc some more prisms: A packet of soap powder A tennis ball The shapes of these objects are very common. We often see the same shapes in other objects.

The cuboid

Many objects have the same shape as a chalk box. T his shape is called a cuboid. Which other objects pictured on page 25 are cuboids? Make a list of five objects which are prisms (but We shall want to draw sketches of the solids we not cuboids or cylinders). are studying. A solid can look quite different when we look at it from different places. Here are four sketches of a cuboid: The pyramid

The shapes sketched below are called pyramids. They arc not quite so comm.on as the cylinder, cuboid and sphere. They are usually seen as parts of objects. (b) (c)

(dt

Look at a chalk box in three ways so that it looks like the sketches (a), (b) and (cl. In sketch (d). the dotted lines represent hidden edges. These sketches are all useful. But (cl and (d) are The cone used most often, because they give us a good idea of the whole shape. The shape sketched below is a special sort of Make a list of ten objects which are cuboids. pyramid called a cone.

The cylinder

The shape of a milk tin is called a eyiindiT.Here arc four sketches of a cylinder. Look at a cylinder so that it looks like these sketches:

Which objects pictured on page 25 are pyramids and which arc cones?

26 651

The sphere 4-3 Surfaces of solids

Many objects have the same shape as a table- Often the surface of a solid is clearly divided into tennis ball. This shape is called a sphere. Here is a several regions. These regions are called faces. sketch of a sphere: An edge is where tw o faces meet. A corner, a pointed part, of a solid is called a vertex. iThe plural of vertex is reniccs.)

Edge Vertex

Face Edge W hich objects pictured on page 25 are spheres'.' Face Vertex

Face Exercise 4a ! What shape are the following solids? Vortex (a)* A cigarette (b)* A cannon ball A face is either a flat surface or a curved surface. (c)* A yam mound (d) A bar of soap An edge is either a straight line or a curved line. (e) An ant-lion's hole If) A heap of gravel 2 What shape are the following solids? (a) A board duster (b) A bamboo pole Exercise 4b ic) A plank of wood Id) A record 1 Write down the mathematical names of any (e) The end o f a If) A beaker solids you can find which have: sharpened pencil la l a Hat face, 3 M ake lists of: lb) a curved face. la l five objects which are shperes. Id * both a fiat face and a curved ftice, Ibl three objects which are pyramids. (d)* only one face. Is this face curved or flat? Id three objects which are cones. 2 Write down the mathematical names of any 4 The diagram below shows a familiar scene near solids you can find which have: Kano. Nigeria. la) only one edge, You can see a pile o f sacks of groundnuts. lb) no edge at all, What shape is this pile? Ic) a straight edge, (d ) a curved edge, le) both a curved edge and a straight edge. 3 Write down the mathematical names of any solids you can find which have: la)* only one vertex, lb)* no vertices, Ic) four or more vertices. Id) two vertices. Id three vertices.

4* W hich solids have a fiat face which is this shape'.’ 5 Make three quite different sketches of a cone. 6 Make a sketch of the Earth as seen from a spaeeship. 7 T urn a sphere round and look at it in many ways. What do you notice about it? This shape is called a rectangle. 27 652

Here are some more reelangles: Here are some more triangles:

12 Which solids have a flat circular face? 13 (a) H ow many faces has an egg? 5 One solid has rectangles for all its faces. Which (b) H ow many edges has an egg? solid is it? (c) H ow many vertices has an egg? 6 (a) H ow many faces has a cuboid? (b) Can you ever see them all at once? (c) When you look at a chalk bo.x in different positions you can see different numbers of faces. W hat is the greatest number you can see from one position'.’ w 7 A rc any o f the faces o f a cuboid the same? 8 (a) H ow many edges has a cuboid? (b) What is the greatest number you can see at once? (c) Use your ruler to find whether any are of the same length 14 (a I H ow many faces has an ear-ring? lb) How man) edges has an ear-ring? 9 (a) How many \ertices has a cuboid? Ic) How many vertices has an ear-ring? (b) What is the greatest number of vertices that you can see at once ? 15 (a) How many faces has yo ur desk? (b) H ow many edges has you r desk? 10(D) If all the edges ot a cuboid are equal it is (c) How manv vertices has \ our desk? called a cube. Name three objects which arc cubes. The faces o f a cube are special rectangles called squares: 4-4 Planes, lines and points

Look at your blackboard. It is a flat surface. The m athem atical name for a Hat surface is a plane. If you wanted a bigger blackboard, you could extend it until it covered the whole of the front wall of the classroom. In the same way. any plime can be extended in any direction, as far as you like. Which other solids have faces which are The top of your desk is a plane. If you extended squares? it. you would get a larger top to your desk. If you 11 W hich solids have flat faces o f this shape, called extended it far enough, it would join the top of the a triangle'' desk next to you in your row. The top of the next desk is therefore an extension of the plane of the top of your desk. Look for some more planes in your classroom. W hat would you get if you extended these planes? In vour classroom, where do the floor and the

28 553

from will) meet? Find some mon- nl.met in your Open the paper out. What is the relation classroom whicii meet. Can you lind any planes between the fold and 'he line you started with? that do not meet? 5 What is the intersectiv . ot two curved lines? If two planes meet, do vou alwavs get a straight 6 Name the intersection of the straight line m line? and the curved line n ir. he follow ing figure. The place where two things meet is called their intersection. The intersection of two planes which meet is a straight line. Look for some straight lines in your classroom. Are they all intersections of two planes? N ow look at one o f the vertical edges of a wall of your classroom. Where does it meet the floor? Find some more places in the room where a 7 Are the intersections of the following pairs of straight line meets a plane. Is the intersection of a objects points, straight lines or curved lines? straight line and a plane always a point? (a) The top of a milK tin and the curved surface In order to talk about lines and points, we use lb) The ceiling of your classroom and the front letters to name them. We give capital letters to w all points and small letters to lines, writing them close Ic) T w o o f the sloping edges o f a pyram id to the points or lines. Id) A flag pole and the ground tD |e) A desk and a pencil which is lying on it •B 10 A table and a table tennis ball which is lying on it Igl A table and a bottle which is standing on it 8 Below is a sketch of a cuboid in which the \ertices have been named. The straight line through two points D and E is called ‘the line DE'.

A Exercise 4c 1 (a) Find two straight lines in your classroom which do not meet. (b) Find another two straight lines in your E F classroom which do meet. What is their Name the intersections of the following lines intersection? and planes. If they do not intersect, write 'no 2 Look again at the solids shown on page 26. intersection'. (a) W hat is the intersection of tw o plane faces? la) AE and EF (b) What is the intersection of a plane face and lb) AE and FB a curved face? |c) AE and HE (c) What is the intersection of two straight Id) Plane liFGC and AE edges? le) Plane ABFE and plane CGFB â(D ) Flerc is a straight line; ------. lO C D and plane B FG C H ow can you cheek that it is straight'.’ Igl Plane EHGF and plane ADCB How can you make it into a curved line? 4 Tear a piece of paper out of a rough book. Draw a short line on it with a ruler. Fold the paper along the line. Extend the fold to the edge of the paper. W hat shape is the fold? W hy? 29 654

4 - 8 M a k in g a cuboid Is this the net of a cube? Find as many arrangements of the six squares Exercise 4f as you can which make the net of a cube. 1 Cut out the shape on page 69 of your Workbook. This shape is called the net of a cube, becau.se 3 M ake a cube o f side 25 mm. it can be folded to make a cube. 4(D ) Sketch the net of a cuboid which is 8 cm Fold the net along the dotted lines. Make it long. 6 cm wide and 4 cm high. (Put in the fold into a cube. (The shaded parts of the net are lines, but leave out the flaps for now.) flaps which you can use to stick the faces Which lines are parallel on the net? together.) Which lines are perpendicular on the net?

2(D) The net of the cube which you cut out in W hich faces w ill come together when the net is folded to make a cuboid? Sketch in some flaps Qu 1 was made up of six squares. Why? They were arranged like this: for sticking it together. W hich lines arc the same length on the net? Draw the net accurately (including the flaps). Cut it out and make the cuboid. 5 Make the following cuboids. (a) 75 mm x 50 mm x 25 mm (b) 35mm x 35mm x 100mm 6 Make a large matchbox. Here is another arrangement of the six squares: (Make the matchbox cover 150 mm x 90 mm x 44 mm. Make the matchbox tray' 150 mm x 86 mm X 40 mm.)

S um m ary

You have learnt about the follow ing shapes in this chapter. Use this list to look up any you have Is it the net of a cube? forgotten. Here is a th ird arrangement: Cube, cuboid (pages 26. 321 Cylinder, prism (page 26) P yram id, cone (page 26) Sphere (page 27) Rectangle, square, triangle (pages 27 28) Trapezium (W o rkb o o k, page 21) Parallelogram (Workbook, page 22)

33 655

6-4 Pyramids cuts out a square of side 40 mm. the second a square of side 39 m m . the th ird one of side 3X mm Below you sec some sketches of pyramids. and so on. .Make a hole through the centre of each square and thread them in the same order on to a straight piece of wire.

6 - 5 Prisms

When you cut through an object, the new face which you get is called a cross section. If you cut straight across a pencil, the cross (4) sections at various places are all the same, usually a circle or a regular hexagon as in the next column.

(3) 1 Cut here

(5)

Cross-section The vertex at the top o f each is called the apex. The face at the bottom is called the base. The base is shaded in each o f the sketches. A pyramid is named after its base Tlu's tlpure '3) .A solid like this is called a prism The following is a sketch of a pyramid on a square base' or a are all prisms. 'square pyramid'. Find the names of the other five. A pyramid may be made with any base. Cross section

Turn to Exercise 17 in vour Workbook. t.en g lh •

Exercise 6d 1 Sketch the net of a pyramid on a triangular base. A M ark in the same colour the edges which will Cross section come together. Sketch in some llaps for sticking it together. If all the edges o f the pyram id are to be S cm long, draw the r,et accurately using your com­ passes. Put in Haps for glueing, cut it out and make your own model. This shape is also called a regular tetrahedron. 2 Sketch a pyram id on a square base and a py ram id on a triangular base. 3 Is there a pyramid with only three faces'.’ 4 W hat shape are the bases of the famous Egyptian pyramids'.’ Find out how big they are. 5 A solid py ramid can be made as a class project using fairlv thick cardboard. The first student Prisms are named after their cross sections. 50 656

Exercise 6e 1(D) (il) What is another name for a rectangular prism? (b) What is another name for a circular prism? (c) Is a cone a prism? 2 Make a sketch of a triangular prism.

3 W hat shapes are the faces o f a prism? Sketch the net of a prism with a triangular cross section. M ark in the same colour the edges which will come together. If the triangle is to have sides ft cm and the The cross sections of a cone and a cylinder can prism is to be 10 cm long, mark these lengths on be circles, but they can be other shapes as well. your sketch. W hat shapes can they be? 4 Draw accurately the net for a triangular prism with the sides of the triangle 6 cm and the length 10 cm. Put in flaps, cut it out and stick it together. 5 Copy and complete the following table. Exercise 6f 1 Obtain a piece of paper in the shape of a rectangle. Fold it into the shape of a cylinder. Number of: Number of We require a tin whose height is to be b cm. Name of ------.sides o f the The length of the circumference of the base is solid Ver- _ , „ coss to he lOctn. Using your instruments, make an Edites Faces tices section accurate plan of the piece of metal you would cut out to form the curved face of the tin. Triangular prism 1 Now look at a tin and see how it is made. 2 Draw a circle of radius approximately 8 cm on Rectangular prism a plain piece of paper. Draw a sector and cut it out: Pentagonal prism

Hexagonal prism 1

Can you see any pattern in the numbers? .S\. 6 Obtain a piece of fairly thick cardboard and cut C ut out out any shape you like. Copy and cut out as many as you can exactly like the first shape. How can you fit them together to make a prism? W hat is its name? Fold tiie remainder into the curved surface of a cone. If you fold a smaller sector, how will the cone be dilTcrent? Check by lolding the sector you removed. 6-6 Some solids with curved 3*’ Obtain a piece of fairly thick cardboard. Cut out as many circles of radius 5 cm as you can. surfaces M ake a small hole through the centre o f each The sphere is interesting bectiuse its cross section and thread them on to a straight piece of wire. is alwavs a circle, wherever vou cut it. What hav e vou made? 51 r~) /

4(D) Hold the cylinder you made in Qu 3 by the Id) A pile o f exercise books wire. Now turn the cylinder round the wire. It le) A tennis ball will not look any different in its new position. If) A ruler So we say that the wire is an axis of the cylinder. lul A Fan' milk carton If you look at a wheel on a lorry or a bicycle Ihl A coin you will see that it turns round an axle. The axle li) A horizontal cross section of a Hag pole always lies along the axis o f the wheel. 2 Put a penny (or butut, cent, kobo or pesewa) Does a cone have an axis? on a table. W hat is the largest num ber of Does a sphere have an axis? pennies that can be placed around it so that each 5 Cut out a paper rectangle 12 cm by 6 cm. Mark one touches it? 2 cm intervals along the longer sides and draw What geometrical figure is this connected lines as shown below. with? 3 Draw a circle of radius 5 cm with centre O, A 2cm 2cm 2cm 2cm 2cm 2cm D Draw a diameter AC, Use your set square to draw another diameter DB perpendicular to AC, Join AB, BC, CD, DA, What shape is figure ABCD? Draw another circle, with centre O and radius 4 cm. What do you notice? By joining four points on the smaller circle, you can draw another square. Do it. L ___ 4 Draw a circle of radius 36 mm. Mark a point P 2cm C B 2cm 2cm 2cm 2cm on the circumference. Find a point Q on the circumference. 4! mm Fold it into a cylinder with the lines outside. from P. Side A D should meet side BC. Through Q. draw a straight line perpendicular Describe the curve you have obtained. It is to PQ. (Use your set square.) Let the line meet called a helix. Which articles are made in this the circumference at R. shape? Draw a straight line from P to R. Measure 6 A cone can be made as a class project. QR and PR. What do you notice about the chord PR? The first student cuts out a cardboard circle of radius 40 mm, the second a circle of radius 5 Draw this net accurately. All the sides are 59 mm, the third a circle of radius .3S mm, and 75 mm long. so on. Make a small hole through the centre of each and thread them in order on to a straight piece o f wire. 7* W hat can you say about the cross sections o f a given sphere through its centre? 8 Can you make a sphere out of a piece of paper?

75mm Exercise 6g (.iH.scellancoii.i) 1 What are the mathematical names for the shapes o f the follow ing? lal A table top 75mm (b| A face ol a coin Ic) A pencil 52 658

Cut nul the PCI. :iP(J make it into a solid. 104mm This shape is called a regular octahedron. What can you find out about it? 104mm 6 Draw the following net accurately.

147mm

75mm

75mm [~ 75mm 104mm

75mm 104mm 104mm 104mm

Draw the net accurately. Put in the flaps, cut it out and make the model. Can you fit two of these together to make a cube? Cut out the net. and make it into a pyramid 9 Make models of the following pyramids. w ith ou t a base. What can you lind out about this pyramid? Collect a number of these pyram ids. Try fitting two or more of them together. What shapes can you make? 104mm 7 M. fake a model of the following pyramid on a square base. A ll the sloping sides are the same length.

104mm

77mm

77mm

104mm

104mm

94mm

104mm Collect a number of these pyramids. Try fitting •Si.v of these (three of each) can be fitted tw o or more o f them together. W hat shapes ogether to make a cube. Can you do it? can you make? (Hint: two of one of the pyramids and one of the other can he lilted together to make the 8 What solid is this the net of? arism in Qu 8.) 53 659

10 Collect together as many solids with plane S um m ary faces as you can find. M ake a table like 'be following, and fill it in.

N um ber of: You have learnt about the following in this Name o f ! V a- F solid chapter. Use this list to look up any you have Vertices Edees | Faces forgotten. (V) 1 (K) 1 (F l Circle, circumference, radius, chord, diameter, Cube 8 1 12 1 6 14 arc. sector, segment, semicircle (page 46) Square i 1 Isosceles and equilateral triangle (page 49) pyram id 5 5 I 8 1 i '0 Regular he.vagon (page 49) Rhombus (page 49) Find a relation between V. E and F. Regular tetrahedron (page 50)

54 660

4 Three-Dimensional Drawing (Book 2)

4—1 Dimensions F ig ;

Here is a point: f How many measurements do you have to take in order to say exactly where P is on this page? Two horizontal edges meet at the bottom of this We have to measure how tar it is from the top edge. We can draw these in several ways— it o f the page, and also how far from one side. depends on how we are looking at the matchbox. Because tw o measurements arc required, we say Fig,2 shows three common ways. that this page is 2-dimensional. The word ’dimen­ sion' is a special word for 'measurement'. (a) A ll surfaces are 2 dimensional. How many measurements do you have to take in order to say exactly where P is in this book? As well as measuring the distance of P from the top and from the edge o f the book, we must also Co) measum how far it is liirow^h the book, how far it is from the front cover. Because three measure­ ments are required, we say that this book is l-dimensional. (cf All solids are 3-dimensional. This is why it is difficult to draw a solid object Fig 2 on paper. Solids are 3-dimensional: they have length, width and depth. But paper is 2-dimen­ Notice that the right angles arc drawn differently sional; it only has length and width. A ll we can do in each diagram . Fig 3 shows the right angles is to draw in such a way that our sketch looks as between the vertical edges and the horizontal edges. if it has depth as well. In this chapter, we are going to discuss one im­ portant method of draw ing 3-dimcnsional figures. la) It IS best to attempt all the exercises /ree/aW . A fter some practice, you will find that this is the quickest way. If you want a very neat drawing, use a ruler.

(b)

4-2 Parallel projection Suppose that we want to draw a matchbox which is lying on a horizontal table. The three edges of (el the matchbox which meet at each vertex arc per­ pendicular to each other. We draw a set o f three Fig 3 perpendicular edges first. First draw a line straight up the page to represent The horizontal edges are also perpendicular to a vertical edge (Fig 1). each other.

38 661

Fig 4 shows the approximate ciireelions of the Fig 6 shows how the middle sketch (hi was com­ lines, as they arc drawn on the paper. pleted. Notice that parallel lines were used at each stage. Be careful always lo use parallel lines in your sketches. (a) 457

(b)

Stage I Stage 2

(c)

F is 4

However, we are making a sketch, so it is not necessary to make the angles exactly these sizes. Stage 3 . St 4 Do not use a protractor. Copy the sketches in Fig 2 freehand, in the meiiiod of parallci projection, hues a/;/< /; arc parallel in 3 iiimcnsians are i/rawn /H i'dllel an llic paper. So the sketches of the matchbox are completed by drawing parallel lines (Fig 5).

I Stage 6 (a)

(b) Stage 7 Stage 8

Fig 6

Notice that liplu lines were used un til the construc­ tion was finished (Stage 7). Then, in Stage 8. the visible edges were drawn over w ith full lines. The hidden edges were shown by broken lines. .Mways draw liy lil lines u n lil you hare finished a (c) eonsiriierinn. Then go over the important lines again to make the figure stand out. For more complicated objects, it is often better Fig 5 not to show the hidden edees at all. .lust leave their

39 66:

construction lines as they are; do not erase them. (0 The full name for this projection is "oblique parallel projection'. In technical subjects, it is generally known as 'oblique projection'.

Turn lo E \cn i.si‘ (> in your H'orkhook.

Practise drawing cuboids. Work with a partner. Take it in turns to draw three edges o f a cuboid and to complete the sketch. Do not worry if your first attempts look wrong. Keep on practising until you can draw a cuboid quickly and neatly without using a ruler (or an eraser.'). Practise each o f the three methods shown above.

(d) Exercise 4a In this exercise, use purullel projection for all tlrmrint’s. 1 Choose one of your textbooks. Put it in the following positions. Sketch it in each position. (a) Lay it on the table, so that you can re;id the title on the cover. (b) Lay it on the table, so that you can read the title on the spine. (c) Stand it up, so that you can read the title on 3 On the matchbox you drew in question 2(a). the cover. print the trade mark, which is LITE-IT'. But (d) Stand it up so that you can read the title on remember that the letters should be drawn in ihe spine. parallel projection as well. 2 Copy the following drawings freehand. 4 Draw the following freehand. (a) (a) Your school trunk with the lid closed (b) Your school trunk with the ltd open (c) A table Id) A solid letter T 5 (D) The figure below is a sketch of a pyramid on a horizontal square base. So A B C D represents a square. But what shape is it on this paper?

(b)

40 663

(c) W hat shape are the sloping faces o f this solid? What shape is the top face'’ The original pyram id had a base with sides o f length 100 m m , and sloping edges o f length 115 mm. In the truncated pyramid, the sloping edges are only 90 mm long. W rite these lengths on your sketch. Make a sketch of the net of this truncated pyram id, and write in the lengths o f the edges. Construct the net accurately. Make the model. 2 (a) Sketch lig h tly a cube o f side 60 mm. Also (d ) sketch its net. (b)On your cube, choose one ol the vertices nearest you. Mark the three points on the adjacent edges which are 20 mm from this vertex. Join the three points in red.

14 (a) Sketch a pyramid on a square base. On your sketch, siiow the planes of syin- rnetry.of the pyramid. (b) Sketch a prism with cross section an isos­ celes triangle. Show its planes of symmetry. 15 N'îakc a careful drawing of a rectangular pyra­ mid. using a ruler to draw the lines. Make the same construction on vour net. Use colour to show the planes and a.xis of sym m etry.

4—3 Some truncated solids A truncated solid is a solid w ith certain corners cut off by plane cuts. Many interesting solids can be made by truncating the common solids.

Exercise 4b 1 This sketch represents a truncated square pyramid. Copy it.

Imagine that the cube has been truncated at this corner by a cut through these lines. How many faces has the new solid? W hat shapes are iiiey? W hat are the lengths o f the edges?

43 664

(c) Im agine that another corner has been cut olF This truncated solid is called a n'gu/ur o d a - in this way. Add the extra lines on your cube hvdron. How many faces does it have? What and on your net. using the same colour. shape are they? If the sides o f the tetrahedron are (d) Continue truncating each vertex like this. 100 mm long, what lengths are the edges ol the When you have truncated all the vertices, go octahedron? over the visible edges o f the resulting solid in Construct the net of the octahedron. (Un­ red. Show also the hidden edges. fortunately, this net cannot be constructed by (e) H o w m any face.s has the solid? W hat shapes truncating the net of the tetrahedron.) Makeghe and sizes are they? Write in the lengths on model. your net. How can you add the triangular 4 Sketch a cube in parallel projection. Mark the faces to your net ? mid-points of each edge, including the hidden ( f ) C onstruct the net accurately. M ake the model. edges. 3 Sketch lightly a figure to represent a regular Join the m id-points o f the fo ur edges o f the top tetrahedron. face. What shape does this represent?

Do the same on the front and side faces. Imagine you have done the same for the remain­ Mark the mid points of each edge, including ing faces. Then you have drawn another tru n ­ the hidden one. How many mid-points are there? cated cube. This one has a special name; it is Join the mid-points, including the hidden one, called a cubocuilwdron. H o w many faces has it? in such a way as to truncate each corner. Draw W hat shape arc they? your lines lightly. This sketch shows one corner If the sides o f the cube are 80 mm long, con­ truncated. struct the net of the cuboctahedron. Make the model.

5 Copy the following diagram on graph paper, using a scale o f 1 in. to 1 cm.

When you have finished, go over the visible edges of the resulting solid. Show also the hidden edges.

44 665

Dnit: SOLIDS Card; R2

Page 1 of 1

Workbook page 69 666

Dnit: SOLIDS Card: %

Page 1 of 2

Exercise 6

1. Using parallel projection, complete sketches of cuboids from the following edges.

(a) (b)

(c) (d)

(e) (f)

(over) 667

Unit: SOLIDS Card: R4

Page 2 of 2

(g) (h)

(i) (j)

(k) (1) 668

Unit: SOLIDS Card: R5

Page 1 of 2

Exercise 17

1. We are going to draw the net of a square pyramid.

Base

V/hat shape are the sloping faces of any pyramid (except a cone)?

Two faces of a square pyramid are drawn in the diagram above: the square base and one of the sloping faces. Use your compasses to help you draw the other three faces. Caution: Make sure they are the correct size! Colour the pairs of edges which will come together when the net is folded into a pyramid. Use a different colour for each pair of edges. Draw some flaps to help stick the model together. Then cut out the diagram and make the pyramid. If your diagram was wrong, try again on another sheet of paper. 669

Unit; SOLIDS Card: R5

Page 2 of 2

2. Fill in the following table.

Number of:

Name of pyramid Number of sides of the base Vertices Edges Faces

Triangular pyramid

Rectangular pyramid

Pentagonal pyramid

Hexagonal pyramid

Can you see any pattern in the numbers? 570

Solids achievement test

Time allowed; 30 min.

Answer all questions on this paper

Name the mathematical shapes of the following 3”dlmenslonal objects. (a) A soap packet ... (h; A football _ (d) (cj A church spire ______(dj A church roof ______(e) A glass louvre _

2. State whether the following statements are true or false.

(a) All edges of a cube have the same length. (b) All faces of a cuboid are squares. (c) All edges of a cube are parallel. (d) Every face of a cuboid is perpendicular to four other faces of the cuboid, (e) Every vertex of a cube is the intersection of three edges of the cube. (f; A cuboid has the same number of vertices as an octagonal u.vramid.

Complete the following sketches, Don't show the hidden edges, (a) A cube (b) A prism (c) A cylinder 671 if. Draw simple 3-dimensional sketches of the following objects. Then naune the mathematical shapes in each object, (a) A domino (b) A "Vim" carton

A domino has the shape of a ______A "Vim" carton has the shape of a

Sketch nets of the following solids, (a) A square pyramid (b) A cone

6, (a) Sketch a net of a cuboid which measures 5 cm x 8 cm X 10 cm. Don’t show the flaps.

(b) On your sketch above, mark the lengths (in centimetres) of all the lines. 672

7. (a) "Every pyramid has an even number of edges" is a true statement. (i) Illustrate this statement by choosing two types of pyramid and finding their number of edges.

A ______pyramid has ______edges.

A ______pyramid has ______edges.

(ii) Explain why the statement is true for all pyramids.

(b) "The number of edges of a prism is always divisible by 3" is another true statement. (i) Illustrate this statement by choosing two types of prism and finding their number of edges:

A prism with a ______cross-section has _____ edges.

A prism with a ______cross-section has ____ edges.

(ii) Explain why this statement is true for all prisms. 673

Solids achievement test: mark scheme

Qu Answers & marks Comments

(a) cuboid (a) Allow prism but not cube, (b) sphere' (c) Allow cone but not (e.g.) triangular pyramid, (c) pyramid^ (d) Allow prism, but not (e.g.) (d) triangular prism square prism. (e) cuboid^ (e) Do not allow rectangle or oblong. In all cases, allow correct © adjective.

(a) true . (b) false (c) false (d) true^ (e) true! (f) falsel©

Score 2 for good attempt to draw parallels to complete sketch (with or without a o ruler). Allcv; 1 if lines are in correct position but not parallel, or if hidden edges are shown.

(a) (b) Score 2 for a good sketch, Q 1 for a poor $D sketch, and 0 for any 2D sketch. Allow \[ cube or cuboid and the 2 appropriate sketch for (b),

1 1 as if "carton” mistaken for cuboid cylinder packet or box. ©

(a) (b) Allow 1 for good attempt which is not quite correct, even allowing for drawing error; e.g. (a) triangles too squat, (b) complete circumference drawn without instruction to © cut out sector 674

Qu Answers & marks Comments s (a) Score 2 for six rectangles s s correctly connected, not s s « necessarily ■ ; shown. (There 1 1 are several correct combin­ 1 1 1 ations of 5» 3 and 10.) 10 1 1 1 lo Ignore errors of scale. 1 1 1 1 Allow 1 for a good attempt 1 1 S' 5- g which is not quite correct. s 5 (b) Score from 14 lengths shown: for each error, subtract 1 from 4 (uo negatives!). Ignore (a) 2 marks (b) 4 marks omission of units and of © lengths on folds.

7 (a)li; e.g. square - o (a)(1) Allow 1 for each correct triangular - 6 combination of name and (ii) # of sloping edges = number. # of vertices of base (ii) Score 1 for each of the = # of edges of base. three ideas listed, however .% . total # of edges = expressed. Be generous if twice ir of edges of base. muddled explanations appear .*. number of edges of to be caused by language a pyramid is even. 5 difficulties. (b)(i) e.g. triangular-9 (b)(i) Score 1 for each correct rectangular - 12 2 combination. (ii) ir of. lateral edges (ii) Total 3» score as for = ir of vertices of base (a)(i). Note that = # of edges of base = "lateral" means parallel a of edges of top face. to the axis of the prism. . . totiîl ir o f edges = 3 X # of edges of base. .*. a of edges of a prism is divisible by 3. - 3

Maximum score: 4 5 |

Objective scores

"Names" ...... qu & Ou (names only) "Faces, edges and vertices" ... Qu & 1 "Sketches" ...... q u & Ou (sketches only) "Nets" ...... Qu & 6 APPENDIX M

MATERIALS FOR STATISTICS UNIT

(SPATIAL TRAINING EXPERIMENT)

675 676

U n it: STATISTICS Card: 1

Objectives: 1. To collect data. 2. To arrange data in intervals.

Activities: 1. Perform experiments 1, 2 & 3 (not necessarily in that order) on Card K1. 2. Answer the questions s e t. 3 . Read JSPM Bk 2 Section 14-1 (pp 156-157).

Evaluation: 1. Have you completed the required tables and answered the questions asked? I f in difficulty, seek the help of your tutor.

U nit: STATISTICS Card: 2

Objectives: 1. To read information from a column or bar graph. 2. To record inform ation using a column or bar graph.

Activities: 1. Read JSPM Bk 1 Section 13-2 (p 150). 2. Do Ex 15b Qu 1, 5 & 9. 5. Read JSPM Bk 1 Section 15-3 (p 154). 4 . Do Ex 15c Qu 1 & 4.

Evaluation: 1. Check Ex 15b against answer sheet. 2 . Show Ex 15b to tu to r. 3. Make the necessary corrections. 677

U n it: STATISTICS Card: 3

Objectives: 1. To read information presented by pictographs, piecharts and line graphs. 2 . To use the above graphs to display data.

A c tiv itie s : 1. Do JSPM Bk 1 Ex 15b Qu 2, 4 & 11. 2. Do JSPM Bk 1 Ex 15 c Qu 2, 5 & 5.

E valuation; 1. Check Ex 15b against answer sheet, 2 . Show Ex 15 c to tutor. 3 . You w i l l need to correct any errors.

U n it: STATISTICS Card: 4

Objectives: 1. To compute the arithmetic mean from a lis t or simple frequency table. 2. To determine the mode and median of a set of numbers.

Activities: 1. Read JSPM Bk 2 Section 14-2 (pp 158-159). 2. Use data collected in class to compute mean. 3 . Read JSPM Bk 2 Section 14-3 (pp 160-162). 4. Refer to questions you answered when doing Card 1. Id e n tify the mode and the mediain. 5. Do JSPM Bk 2 Ex 14f Qu 3, 5 & 6.

Evaluation: 1. Check your re s u lts against answer sheet. 678

U n it: STATISTICS Card: 5

O b jective: 1, To determine the mean of a set o f numbers using an assumed mean.

Activities: 1. Do one or more of the experiments on Card R5. 2. Read JSPM Bk 2 pp 159-160. 3 . Do Ex 14c Qu 1 & 2. 4* Use the assumed mean method to determine the mean of the data collected in one experiment from Card 1.

Evaluation: 1. Check against answer sheet.

U n it: STATISTICS Card: SO

Summary o f O bjectives

1. To collect data and arrange it into tables.

2. To read statistical information displayed graphically (as column graphs, bar ch arts, pictographs, p ie ch arts, and line graphs).

3. To record statistical data graphically, using an appropriate method,

4. To compute the arith m e tic mean, the mode, and the median of a l i s t of numbers.

5. To compute the arith m e tic mean, the mode, and the median o f a set o f numbers given by a simple frequency ta b le .

6. To compute the mean o f a set of numbers using an assumed mean. 679

(Book 1) Height of a Plant

16 The number of hours each day that you spend (a) working, (b) sleeping, (c) resting, (d) eating, ':rr. (e) playing. :ülx. 17 The length of time that your local radio station gives each week to (aI music, (b| news, (c) religion, (dI education. |e) women's programmes. 18 Collect some statistics on any other topic which interests you.

15-2 Graphic display

Many ways have been invented by artists and mathematicians for drawing pictures of statistical (a) When was the plant 15 cm high? mappings. These pictures help us to understand (b) When was the plant growing fastest? quickly what the information is about. There are (c) How high was the plant after 2( days? various ways of doing this, and the method of display depends on the type of information that 3* Groundnut Exports from West .Africa has been collected. Some of these methods are (1967-69 Average) shown in the next exercise. 100 000 200 000 tonnes I I Senegal Exercise 15b Nigeria

1* kainfall Recorded At Freetown in 1963 The Gambia

iger

(a) For whfch years ilo these figures apply? (b) Which country exported the greatest weight of groundnuts dur ng these years? (c) Can we tell easily what weight of ground­ nuts The Gambia exported? (d) How could we improve tl.e graph so that we could answer (c) more q lickly?

1 Proportion of Working Population Engaged in Agriculture (1965)

INDIA 70%

GHANA 6 6 JAMAICA 3 4 % ^, (a) What do J, F, M, A D stand for? 6 6 (b) In which .month did most rain fall? JAPAN 2 4 % ^ , (c) How much rain fell in October? 6 f When w;is the rainy season? (dl U.K. 4 % ^ (c) When was the dry season? 150

Pages 679-690 copyright 1967, 1968 by Longman Group Ltd. All rights reserved. Reproduced by permission. 580

(a) W hich country had the highest percentage Population and Energy Consumption of people working in agriculture? in Selected Regions (1970) (b) Which country had the lowest percentage of people working in agriculture? (c) Cover over the percentages given in the middle column of the graph. By looking at the drawings of the men. can you tell what proportion of the w orking population of Japan was engaged in agriculture ? I

i Protection against Malaria

Numbers Protecteil by W H O * Control Programme

North Western Far East Africa 1949 America Europe (ex China) 1950 □ Population nergy consumption 1951 (a) Which region had the greatest population? 1952 (b) Which region had the greatest energy 1953 consumption? (c) Which region had the greatest energy 1 2 3 4 5 consumption per head of population'.’ (m illio n s ; (d) Which region had th.e least energy con­ sumption per head o f population ? Numbers Protected by World W ide Eradication Programme 7 World Production of Cocoa (1969/70) 409 43 1958 Dominican 0 Republic 1959 200 Q ] Ecuador 1960 224 Cameroon 1r " I 50 100 150 200 250 300 Ivory Coast (millions)

(a) Which years does the first graph cover? Brazil (b) Which years does the second graph cover? (c) How many more people were protected L_y Nigeria between 1952 and 1953? Figures are in (d) How many more people were protected G hana thousand tonnes between 1958 and 1959? (c) W hich represents the larger increase? (a) Which country produced most cocoa in (0 The numbers protected against malaria 1969 70? increased more during the World Wide (b) Which country produced more cocoa. Eradication Programme than during the Brazil or Nigeria? WHO Control Programme. Yet it does (c) Did you look at the pictures or the numbers not appear like this on the graph. Why not? to answer (b)? ' World Health Organization Id) How could this graph be improved? 151 681

8 Average Composition of Selected Foods 10 M ark of Students on a Test 0 50% 100%

Meat Cereals TJ 5 Beans Greens

Fruit

Eggs Milk 20 40 60 80 100 Mark (%) □ Protein Fat In the above diagram, block A shows that 3 □ students obtained between 4 0 and 4 9 ''„. (a) How many students obtained marks in the I I Carbohydrates | | Water and Waste range 80",, to 89";,? (a) Which foods have the least amount of (b) In which range of marks are there 5 water and waste? students? (b) Which food contains no carbohydrates? (c) Which range of marks has the most (c) Which food has an equal amount of pro­ students? tein, fat and carbohydrates? (d) Can we say how many students scored 0 "„? (d) Which food contains most fat? (e) Can we say how many students scored (el Ciin you find a hotter way of displaying I00"„? these facts?

Distribution of Public Expenditure in a I Number of Students in Each Form n Seven Year Development Plan at Joinspro College

Industry

Agriculture Education jBoyîiin! Transport Health Housing etc. Electricity' Water, etc

'Misc.

(a) What is most money being spent on under f-TTHn+T-l the Seven Year Development Plan? (a) How many students are there In Form 2? (b) Approximately what fraction of the whole (b) How many girls are there in Form 2? is this? (c) H ow many boys tire there in Form 2? (c) W hich is receiving the larger share. A g ri­ (d) W hat disadvantage has this method o f culture or Education? display? Id) Can we say how much each is receiving? (e) Is the total number o f students at Joinspro (el What arc the advantages and disadvan- College increasing or decreasing? taees o f this method? 102 682

12 Temperature Change during a Day 14 Relatire Humidity at Arhimota School, Ghana (April 1965)

iviidOiJiJiil

(a) What was the hottest period of the day ? lb) What was the coldest period of the day? (c) What was the maximum temperature? |a) Find out what 'Relative Humidity' means. Id) What was the minimum temperattire? |b| On which days was it greatest? |e) When was the temperature about 20 C? |c) On which day would the least amount of water appear on the outside o f a glass o f cold water?

Î3 Average .Monthly Temperature Variation at Khartoum, Sudan 15 Number o f Hospital Beds in Nigeria, 1940-70

t-4- 'T '- r- - - *--•—j- • . — -i- -i-g — *~ 1 ...... " iz T . . r- 1 .... ■ i- r-T— 1 . . ...f.... -- r • - f r i - T — iïn’ , - r :| ■ u' 9200 17 100 30 500 : : ... ij beds in b-ds in beds in ::cr_ fJ .Kp;. ?:js: p 1950 1960 — J- |a) Cover over the bottom line of the diagram |the number o f beds in each year). (a) What is the hottest month of the year? Between 1950 and I960 did the number of (b) What is the coldest month of the year? beds (c) Whtit is the lowest average temperature for |i| increase by half again, the hcittcst month? lii) double, (d) What is the higitcst average temperature liii) treble? for the coldest month ? lb) Now uncover the numbers. Was your (e) W hich m onth has the least range of answer to la) correct? temperature? Ic) Explain. in In which month is the minimum tempera­ Id) What are the advantages and disadvan- ture hii’hest? taees of this method? 153 583

15—3 Choosing a method of Population of Ten West African display Countries ( 1968)

Population C ountry So far in this chapter, we have considered sets of (m illions) facts given in a table. We have seen how such facts can be displayed graphically. Elut when We are Dahomev 2 5 given a set o f facts, it is not alwtiys easy to decide Ghana 8-2 which method of graphical display would illustrate Guinea 3-8 the facts best. Ivorv Coast 4 6 In the exercise which follows, yoa will be drawing Liberia 1-3 graphs from the tables given in Ex 15a. For the first five questions the methods of display are Nigeria 62 suggested. But after that you w ill be left to decide Senegal 38 which method you think is best. In most statistical Sierra Leone 2 5 graphs, the use of colour can help to make the facts T ogo 18 stand out more clearlv. Upper Volta 51

Illustrate this table by a graph similar to that of Ex 15b Q u 3. Exercise 15c

1 Rainfall rccordeii during June, 1964 Distribution of Workers under a (Mfantsipim School, Ghana) Seven Year Development Plan

N um ber of Trade W orkers Date 15th 16th I7th 18th

Agriculture 300000 Rainfall (mm) 225 13 45 135 M in in g 25000 19th 20th 21st Manufacturing 225000 Construction 105000 <1 2 6 U tilitie s 10000 Services 165 000 T ransport 160000 Illustrate the information in this table by a Commerce 85000 graph sim ilar to that of Ex 15b Q u I .

Illustrate this table by a graph similar to that Height of a Boy o f Ex 15b Q u 4.

Age (yr) 10 I 11 ■ 12 1 13 14 15 5 Temperature of a Hospital Patient 1 1 Time 2amj6amjlOam 2pm Height (cm) 122 129 134 1 142 ,55 165 6 p ni 110 p ni

fcmp. rC) 3V0 ?90 36-5 370 38-5

Illustrate the information in this table by a Illustrate this table by a graph similar to that graph sim ilar to that o f Ex 15b Q u 2. o f Ex 15b Q u 14.

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14 Frequency (Book 2)

14-1 Frequency tables In the table above, the first live results have been tallied. In Book 1, we studied the ways in which statis­ tical data can be represented. This chapter will be The completed table is as follows: concerned with; (a) how the data is collected, (b) how it is grouped together in a suitable way tor voTt; RttSüLTS frequi;ncy graphical representation, (c) how it is interpreted. A statistician receives his information in many yes Uu., i-LPi l-j-fU i-j-Pi / / ■' / i 24 ways. He might have the observations that a tech­ nician has made. He might have the replies to no H~t: f-ri-r H -r‘ 20 certain questions that he has asked many people. The first thing he must do with any information is d.k. ■ ; 6 to sort it out. He does this by grouping the results together in some way. An important method of sorting out informa­ Note that when we come to tally the fifth one of tion is to form a frequency table. To do this, count a group, we do so by drawing a line across the other how frequently (how often) eacii possüilc result four. occurs in the data. I his wiii show the frequency of each result. We shall illustrate how to fonn a frequency table with two examples.

Exa/iip/c 1 In a certain school. 50 hoys were Example 2 T w o classes in a school were given a asked whether they liked mathematics or not. On test. The marks, out o f 40. were as follow s: a piece of paper, they had to write "yes', 'no' or 14 ‘don't know' ( d.k.'). The results were as follows: 26 . 15 19 17 27 35 7 36 23 25 22 21 31 29 33 yes: vest d.k.; no: no; yes; n o ; no; ye:,; yes; yes: 27 17 19 11 28 36 29 26 no; no; no; yes; d.k.; yes; yes; yes; yes; no; d.k.; 32 27 23 ->2 33 18 13 20 no; no; yes; no; yes; no: yes: yes; no: d.k.: no: 9 IS 37 32 27 27 33 19 yes: yes; d.k.; no; yes; no; no; yes; d.k. ; yes; yes; 26 21 29 28 22 20 11 29 no; yes; no; no ; yes; yes. 26 25 23 17 16 18 25 16 A frequency table is formed by tallying these Hind the frequency o f each mark. opinions: A simple frequency table does not help us sort v o n out the data in this ctise. W hy not'.' Instead, group the marks together, say 0 5. yes 6 10. II 15 ...... 20 40. The frequency o f these groups will tell us more about the test results than no the frequency of each m ark.

The frequency table will then be drawn up like this:

156 585

7 6 8 10 5 7 7 8 sc()Ri:s RLSULTS 1 TKIigiliNCY 4 9 10 6 7 9 5 5 3 8 6 6 9 10 7 6 0-5 9 8 5 II 7 12 9 4 10 8 7 2 II 12 10 8 6-10 6 5 4 8 5 4 7 6 8 5 9 5 7 9 6 5 11-15 1 4 8 8 6 10 7 7 3 8 II 2 7 9 10 6 II 16-20 I 1 4 8 6 II 7 8 6 9 8 7 10 5 9 3 4 21-25 I 1 8 6 7 5 4 10 II 3 7 6 5 9 26-30 ! Form a frequency table for this data. 31-35 ; Illustrate the results of the experiment by i drawing a suitable diagram. 36-40 : 3 In a Ghanaian family the following were the ages (to the nearest year) of the living members: This is called a groupai ficiitwiuy tabic. 3 21 II 48 6 15 1 34 36 26 45 0 17 4 36 9 Turn to Exercise 32 in your li'orkhook. 2 13 5 38 25 17 10 73 II 46 7 33 9 0 59 18 1 25 41 28 6') 53 94 A fiequcncy graph is called a histo<;rani if the 3 7 30 14 29 22 3 37 6 original dat;i consists of ninnbcrs or nicasurcmcnts. 8 27 The data is often grouped together in some way. The graph which you drew in Workbook E\ 32 Form a grouped frequency table for the ages Qu 2 is a histogram. The graph in Ex 32 Qu I is not O '), 10-19. 20-29. and so on. a h isto g ra m - it is a hur churl. Illustrate the result by drawing a suitable diagram.

4 In a survey conducted by a manufacturer of Exercise 14a men's sandals, the following results were ob­ The I'reiiuency lahies from Qu I and 2 w ill he tained. Thev are the size o f sandals worn bv needed laler in the chapler. ach n; ;an in a group o f about 100 men. 8 7 8 9 7 8 8 6 I ’ The following is a record of the number of goals 7 10 8 8 6 7 10 9 scored by the A ll Stars football tetim in each 8 8 8 7 9 8 8 9 match for one season: 6 9 8 7 7 9 5 6 3 2 2 4 0 I 2 3 9 8 9 7 10 9 7 8 6 5 3 1 2 7 3 9 11 8 11 8 8 8 9 2 3 4 2 0 4 1 3 8 12 9 10 7 5 7 8 2 4 1 5 2 3 4 9 10 7 8 10 9 8 12 Form a frequency tabk; for this data, and 7 13 6 8 8 7 8 10 illustrate it by a histogram. 9 8 8 9 7 10 8 9 7 9 II 8 6 7 9 8 2 In an experiment, two dice were thrown and the 8 8 7 6 10 7 9 8 numbers added. This was done 100 times. The Form a frequency ttible for this data and illus­ results were as follow s: trate it by a histogram.

157 686

In an examination the following percentages average. Collecting statistics can help us to answer were obtained bv a set of students: such questions as: 59 70 78 44 55 69 67 80 W hat is the average cocoa crop fo r a farm er in 47 36 51 76 57 37 82 90 Nigeria'?' 66 46 40 58 79 83 41 38 What is the average income of a family in 77 77 38 33 43 77 69 46 Sierra Leone'?' 30 35 55 49 61 58 41 34 ‘What is the average number of children in a 27 24 54 39 22 15 26 25 Gambian family?' 17 27 34 39 5! 53 93 80 ‘On average, which is the nost popular pro­ 71 63 61 68 69 65 59 54 gramme on Radio Ghana'.’' 47 75 89 85 60 84 82 52 ‘W hat is the average rainfall n the wet season'?' 92 88 75 69 65 59 42 90 ‘W hat was his average crickc score this season'?' 88 85 75 ‘W hat was the average me ‘k o f the class in Form a grouped frequency table fo r these Mathematics'/' percentages, grouping them from 0-9. 10-19. The meaning of ‘average' depends on the kind of 20-29. and so on. questions asked. In ordinary language, an average Illustrate the results by drawing a histogram. is a m iddle value o r a value typical o f a whole set o f 6 The passes o f ten boys in the School Certiticatc results. It is a single number which represents a examination are shown in the table below. Sub­ large collection o f numbers. There are several jects are abbreviated thus: Mathematics (M). methods o f choosing this value, so there are several Additional Mathematics (AM). Science (S). mathematical averages. The most im portant ones A d d itio n a l Science [AS), tngiish Language are the u rit/iiiw iic mean, the intnle and the median. (Lang). English Literature (Lit). French (F). Geography (G). History (Hi. .Music (Mu). .Art (A). Nigerian Languages (V). Latin (L). Arithm etic mean Bible Knowledge (BK). The arithm etic mean (or inei:n for short) of a set of numbers is the average you use when you work NAME PASSES out your average mark in an c.xamination. The mean value o f a set o f numbers is: J. K. Ajayi Lang. M. S. F. H. G. A. the sum o f all the numbers in the set A. Akpam BK. Lang. Lit. M. L. F. H. G. K. O. Ayeni M. AM. S. AS. Lang. G. Lit. the number o f members in the set ' P. J. Ayuba Lang. G, F. H. L it. The marks o f a class in a test were as follow s: B. K, Buba Mu. A. V. G. Lang. Lit. iM.

R. Chukwuka Lang. Lit. E. H. G. S. V. .MARK FREQUENCY J. Eze M . G . S. A M . Lang. F.

K. Olubunmi L, Mu. G, H. Lang. Lit. M. A. 0 0 D. Okonjo S. AS. M.AM.Lang. G. F. 1 J. K . Umaru Lang. Lit. M. S. AS. F. G, 2 ! 3 1 Form a frequency table showing passes in the 4 various subjects. Illustrate the results by diawing a suitable 5 7 diagram . 6 6 7 4 8 2 9 0 14-2 The arithmetic mean ,0 1 A very important idea in statistics is that of an

158 687

The m arks fo r the whole class must have boon : Use of an assumetJ mean

Working out an arithmetic mean can often be 5 very tedious. The work can sometimes be simpli­ 6 fied by using an assitnicd mean. 10 In this method, we first guess an approxim ation Copy and complete the following '.•. omonts: to the mean of the set of numbers. We then find T h e num ber o f students was . , , out how much each number d ille is (or i/eiiu/c.v) The sum o f all the marks was .. . from this assumed mean. By working out the T h e mean m ark fo r the class was . . , mean o f the deriaiians. we can find the actual mean. This method is illustrated in the two examples How can you find the number of students from below. the frequency table? How can you find the sum of all the marks from the frequency table?

E.\am/}/c 3 Find the mean of the follow ing set of numbers; Exercise 14b 1 Find the mean o f the follow ing sets o f numbers. 483 487 488 495 502 (a )*2 7 3 6 5 3 “ 9 503 505 508 509 515 (b )*2 8 9 6 10 7 20 I I 8 8 ( 0 2 9 9 9 9 9 9 9 Solution Let us assume that the mean is 500. (d) 4 5 4 6 7 5 7 4 6 5

2 In Ex 14a Qu 1. you made a frequency table for the number o f goals scored by tlie All Stars foot­ Nl.MHliR Oti'.IATION ball team in all their matches in one season. 1 Use your table to calculate the mean number 483 ' - 1 7 o f goals scored per match during that season. 487 - 1 3 WW 1 - 1 2 3 Calculate the mean value of the dice scores given 495 - 5 in Ex 14a Qu 2. 502 + 2 503 + 3 4 A man who kept chickens recorded how many 505 ' + 5 eggs they produced each day. The results for one 508 + 8 m on th were as follow s. 509 ; + 9 515 + 15

NUMBliR o r bGGS FRi;QtJf-;Nrv 1 T otals' - 4 7 + 42 35 1 • 3 6 ! 3 37 ! 2 Total deviation = -47 + 42 = -5 38 I 5 -5 Mean deviation - 0 5 39 I 5 10 4 0 i 8 Actual mean = 500 - 0 5 = 499-5 41 1 4 4 2

Calculate the mean value of the number of E.\ainplo 4 Find the mean of the set of marks eggs produced each day. given by the following frequency table.

159 688

14 MARK Mean deviation = = Ü 56 Actual mean = 5 + 0 56 = 5 56 The mean is 5 6 (to I d.p.).

Exercise 14c I Use an assumed mean to find the arithmetic mean of the following sets of numbers. (a) 3 > 4 4 5 6 7 7 8 10 lb ) 763 768 770 773 778 779 784 788 790 2*Calculate the mean of the set of numbers given in the following frequency table.

NUMBER EREOUENCY

47 ! 1 48 2 Solution 49 3 Assumed mean = 5. 50 1 3 51 1 2 — ...... DEVIATION rOEAI. DEVIA I II,N 52 0 .MARK EREVtENTY EUR VlAKK 53 4 54 1 0 0 - 5 0 3 Use an assumed mean to find the mean age o f 1 1 - 4 , - 4 your class to the nearest month. 4 Use an assumed mean to calculate the mean 2 1 - 3 - 3 number of eggs produced each day by the chickens in E.\ 14b Qu 4. 3 1 - 2 _ 2

4 2 - 1 _ 2 14-3 Mode and median 5 7 , 0 ’ 0 The mode (or iiukial uiliii’} o f a set o f numbers is 6 6 1 ‘ + 6 the number which occurs most often. The modal mark in E.xample 4 (p 159) is 5; this occurs seven

7 4 2 ! + X times, which is more frequently than any other mark. 8 2 3 + 6 There is a French expression you may have met -

Totals 25 : - I I + 2 5 Exercise I4d 1 Find the modal values of the following sets of Total deviation = — 11 + 25=14 numbers.

160 689

(a)*2 7 3 6 5 3 9 (a) 2 2 3 3 3 5 5 6 6 (b)*2 8 9 6 10 7 20 11 * 8 (b)*4 4 5 5 5 6 6 6 6 7 (c) 2 9 9 9 9 9 9 9 (c) 8 3 5 4 X 2 7 4 3 4 7 (d) 4 5 4 6 7 5 7 4 (d) 4 8 2 5 4 7 9 4 6 2 8 Wh;ii do you notice about (d)7 (A set of num­ 4 6 7 5 1 10 bers like this is called hiniothil.) 3 Find the median number of goals scored by the 2 Find the m odal number o f eggs produced each All Stars from the data of Ex 14a Qu I. day by the chickens in Fx 14b Qu 4. 4 Find the median score for the data o f Ex 14a Qu 2. 3 Find the modal number of goals scored by the A ll Stars from the data of Ex 14a Qu 1. Which average? 4 Find the modal score for the data o f Ex 14a Qu 2, In a survey conducted in 1965 in a Secondary School in G'^.ana. a sample of 150 students (about one-lifih of the school) were asked how many children there were in their families. All the The median o f a set o f numbers is the number children had to have the same father. which comes in the middle when the numbers are The results were as follow s: written in order. For example, the marks in a

Latin test are 1. 2. 2. 3, 4. 4. 5. 5, 6. There are nine NtlMULR OF FRFQL'KNrV students, so the middle one is the fifth: CHtLDRFN IN FAMILY 1 2 3 4 4 5 5 6

4 marks middle 4 marks mark The median m ark is therefore 4. If there is an even number of marks, there is no ‘middle number'. In this case, we find the median by taking the mean o f the m iddle pair. For example, i f the marks are 1 .2 .2 . 3. 4. 4. 5, 6. the middle pair is the fourth and fifth numbers: I 2 2 3 4 4 5 6

3 marks middle 3 marks pair The median is therefore 31 (the mean o f 3 and 4). W hat happens if the marks are I, 2. 4. 4. .s. 6?

Question How can you tind a median from a fre­ quency table?

Exercise 14e 1 Find the medians o f the follow ing sets o f numbers. .411 other value.s (a)*2 7 3 6 5 3 9 (b)*2 S 9 6 10 7 20 11 8 8 (c) 2 9 9 9 9 9 9 9 Calculate the mean, mode and median of the (d) 4 5 4 6 7 5 7 4 6 5 numbers o f child, en in these families. 2 Find the mode and median of the following sets W hich o f these averages best dc.scribes the size o f numbers. of the families?

161 590

Which is highest, ihc mean, the median or the 1 Nt'.VIIIIÎR OF WORDS mode? Which is smaiicsl? Why do you think this FRFQUF.NrV ' IN A LINE happens? W ill it always happen?

How would you answer the question: What is 1 I the average number of children in a Ghanaian 2 . 0 ia m ily ? 3 1 4 2 5 0

6 2 Exercise J4f 7 3 8 4 1 W ork out the mean, mode and median for the 9 9 data o f E.x 14a Qu 4. 10 17 A manufacturer wants to make just one size o f sandal. So he wants to know the average size i l 21 o f men's sandals. Which average should he use? 12 10 13 1 14 2 2 A school in Liberia was collecting for a new hall, and asked for contributions from old students. After contacting thirty old students, the follow­ (a) Calculate the three averages. ing contributions were received: (b) Which of these three numbers best repre­ sents the average number of words per line in the book? .vMOi'NTtNiun.t.AR:: int'O !t)0 SO 50 30 10 5 0

(i The table below shows the burning times of a FREUUKxrY 2 :2.2 3'4 6,8.3 number of candles of the same make, to the nearest j hour. (a) Calculate the three averages. TI.ME (h) 2 :2l;2U2^i3 3l:3t (b) What was the most usual contribution?

(c) Which average best represents this data? FREtyUKNrV 2 1 i 4 : 5 ! 7 3 0 : I W hy? (a) Calculate the three averages, 3 The total scores of each member of the school (b) W hich o f these three numbers best represents cricke t team in the scttson were 107. 93. 40. 45. the average burning time of this make of 346. 29. 50. 71. 53. 81. 112. 'vVhat average best candle? represents the score of an 'average' player and w hat score is this? 14-4 Dispersion

4 Is it possible to give an 'average colour pre­ Expcrinwnl 1 (D o nal use a ruler in this experi­ ference' for the group of people in Workbook ment.) Ex 32 Qu I ? (a) Let every member of the class estimate the width of this book to the nearest centimetre. 5*The following frequency table was obtained for (b) Collect the results u f the class, and form a fre­ the number of words per line in a book. These quency table. results came from two pages selected at random (c) Draw a histogram of the estimates. from the book. Now answer these questions:

162 691

Unit: STATISTICS Card: HI

Page 1 of 2

Experiment 1

1 .1 Each student is to go to the weighing station and have his/her weight recorded to the nearest pound. (Two students to be responsible for the station.;

1.2 When all weights have been recorded, student in charge will list them on the chalkboard. Each students makes a copy.

1.3 Arrange data in descending order.

1.4 Prepare a table as illustrated and fill in space.

110 & over Weight (lb) 111-120 171-180 under ISO

Number of persons

Make up another table with your own choice of intervals.

1.5 Answer the following: (a) What weight falls in the middle of the list? (b; In what weight range do most persons fall? 69:

Unit; STATISTICS Card: R1

Page 2 of 2

Experiment 2

2*1 Each student is to go to one of the height stations and have his/her height recorded to the nearest centimetre. (Two students responsible for station.)

2.2 When all heights have been taken, student in charge will list them on the chalkboard. Each student makes a copy.

2*3 Arrange data in ascending order.

2.4 Prepare a table as illustrated and fill in the spaces.

150 & Height (cm) 151-160 191-200 under Number of persons

Make up a new table with your own choice of intervals.

2.5 Answer the following: (a) What height falls in the centre of the list? (b) In which height range do most persons fall?

Experiment 3

3.1 One student will check with each member of class for day of the week on which their birthday falls, using a tally method to record results, as follows: Sun '//M II M o n ///

Sat

3.2 Results will be posted on the chalkboaird for others. 693

Unit; STATISTICS Card: R5

Page 1 of 1

1. You have been given three bottles containing different quantities of water and a funnel. Without the use of any other apparatus, adjust the levels of the water in all three bottles to the same height. Write down briefly the method you used. Use this method to find the mean of (a) 6, 15» 45; (b) 8, 12, 13, 3.

2. You have been given three rods made up of unifix cubes, Adjust them to the same length. The cubes must not be counted and the shortest rod must not be pulled apart. Ask your tutor to set up a similar problem using four rods. Write down briefly the method you used. Use this method to find the mean of (a) 6, 15» 45; Cb) 8, 12, 13» 3.

3. You have been given bags of sand, two other bags and a balance. Use this apparatus to divide the sand equally into three parts. Your experiment should suggest a way of finding the mean of 7, Î6, 46 or 7, 11, 12, 2. Describe how this might be done. Find the mean in each case. ly,

694

Statistics achievement test

Time allowed: 30 min

Answer all questions on this paper

A survey was done of all the girls (0 walking past a school between 3 p.m. r 4 and 4 p.m. on a Saturday afternoon. •H The colours of the dresses were bC - 'X ' noted and the following graph drawn. (H Answer the questions. O u (a) How many girls were wearing 0) # red dresses? i (b) How many girls were wearing orange dresses? (c) Which were the most popular colours? © 7 3 < u ® a © (dj How many girls were there bCi © H 3 © 4-> a W CU rH © altogether? ® p; *4 u ZD (e) Is it true that no girls were O cu wearing brown dresses on that Saturday?

When a driver of a car puts his brakes on, the car does not stop immediately. It takes a time to slow down and come to a halt. The table below shows stopping distances at different speeds. Choose a suitable method of displaying the facts.

SPEFD (m.p.h.) 10 20 30 40 50 60

APPROX. STOPPING 15 40 75 120 175 240 DISTANCE (ft.)

(Over) 695

3. The marks obtained by a class of students is shown below. (a) Prepare a simple frequency table of these marks. (b) Determine the mean, mode and median for these scores. 7 21 15 12 8 15 10 12 14 15 13 19 17 11 19 18 14 15 21 16

Mark scheme

Qu Answers Marks Comments

(a) 5 2 (b) 0 2 (c) Blue, green, red 2 (c) Allow 1 if "red" omitted (d) 24 2 (e) Allovf 2 for "don’t know", (e) Cannot tell 2 1 for "no" and 0 for "yes"

Any line graph 6 Allow 4 for column or bar graph Correct scales 4 Allow 2 if one scale correct Plotting of points 4 Deduct 1 for each error Labelling 6 Allow 3 if one label correct; feo deduct 1 for each unit missing

(a) Mark 7 8 S 10 11 12 13 14 15 16 17 18 19 20 21 Frequency 1 1 C 1 2 5 2 2 1 1 1 2 0 2

4 Deduct 1 for each error; do not penalize omission of 9 or 20 (b) Mean = - ^ = 14.4 6 Allow 2 for attempting to find total of scores, 2 for attemp­ ting to divide by number of students, 2 for correct answer Mode = 13 4 Allow 2 if correct by student's table Median = 14 6 Allow 4 for correct method, and (20) 2 for correct answer

Maximum score: 50 APPENDIX N

MISCELLANEOUS ADMINISTRATIVE DOCUMENTS

(SPATIAL TRAINING EXPERIMENT)

696 697

Information sheet

MICO COLLEGE MATHEMATICS CURRICULUM EXPERIMENT

During this term, the Mathematics Department will he running an educational experiment. This experiment will involve two units; Solids and Statistics, Each unit has been written on workcards like the unit on Use of Instruments which you studied at the end of last term.

There are two aims of this experiment: To find out (i) whether teachers* college students can learn effectively using the workcard method, and (ii) whether the study of solids improves visual imagination. Each year will be divided into two groups, and each group will study the two units at different times. Several tests will be given before and after each unit.

There will be two types of test. Tests on the content of each unit will be given by class tutors; scores on these tests will be used for assigning grades to each student. Tests of visual imagination will be given by Hr, Mitchelmors; scores on these tests will not be used in assigning grades.

These tests will tell us the effectiveness of each unit. For example, the classes who have just studied the Solids unit should score far higher than those who have only studied the Statistics unit. It is very important that the classes straying Solids do not discuss the units with the classes scunyin g S tatistics until the experiment is completed. Discuss the material with members of your own class only. It is also important that you do not discuss the tests with other students. Your cooperation is needed to malte this experiment a success.

The results of the experiment will be reported to you next term. The timetable for this term is shown overleaf. 698

Timetable

First Year students Second Year students Week Week No. beginning 3 classes 3 classes 4 classes 4 classes

1 Jan 7 First test sessions 1 First test sessions 1 2 Jan 14 j —... rreucuravion-'' Solids Statistics ! Jan 21 for teaching 5 unit unit 4 Jan 28 r j 5 Feb 4 Second test sessions i — .....M,,— , 1 ■ ------J 1 Teaching 6 Feb 11 practice Statistics Solids I ■■ - - Feb 18 7 unit unit 1 8 Feb 23 Solids Statistics! 9 Mar 4 Third test sessions j s unit unit 10 Mar 11 1 Lcav S^ooXUnb 1 11 Mar 18 vuHLZnu&uiOD 1 12 Mar 23 of mathematics Statistics! Solids ■ lUii t - ~ Uir-iv- 13 Apr 1

14 Apr 8 Third test sessions 699

Instructions for adminlstsring HST posttest

1. Ensure that everyone has a pencil and an eraser.

2. Remind students of the test content, as follows: "This is the same test which you did at the beginning of the term. There are five shapes at the top of the page (indicate on tutor's copy, but do not let students see the items) and you have to find which shape is hidden in each of these pictures (indicate test items). You show your answer by underlining one of the five letters below each picture, by filling in the space between these small lines (indicate). Remember, the shape is always in the same position as it is at the top of the page; it does not have to be turned around or turned over."

3* Give out the white practice sheets, face down. Ask students not to turn over until they are told to start.

4. Ask students to write their name and group (1AA, IAN, ec:.) and today's date at the top of the blank side of the paper*

3. Say: "These first five items are for practice. See how many you can do in one minute. Turn over and start now." Stop work after exactly one minute. Ensure that all students put their pencils down immediately.

6. Read out the correct answers: 1-D, 2-C, 3~S, 4"A and 3-B. Demonstrate No. 5 on the blackboard. Also check that everyone is marking their answers correctly, ^

7* Ask the students to put the white practice sheets out of the way somewhere.

8. Give out the blue test sheets, face down. Ask students not to turn over until they are told to start.

9. Ask students to write their name and group and today's date at the top of the blank side of the paper,

10. Say: "There are 15 items in this test. See how many you can do in 15 minutes. Turn over and start now,"

11. Stop work after exactly 15 min.* Ensure that all students put down their pencils immediately.

12. Ask students to put the white sheet on top of the blue sheet, and collect both sheets as quickly as possible,

♦If a student finishes well before time, give him a copy of the Teaching Methods Questionnaire to read through. 700

TEACHING METHODS QUESTIONNAIRE

Last week, you completed your second unit using workcards. We would like to find out whether this teaching method could be used for other units. From the tests you have just done, we can find out how much you have learnt using workcards. Now, we would like to find out how much you like this method. This questionnaire is anonymous. Please give us your honest opinion and express yourself freely on the questions asked.

Think about the unit which you completed last week, the unit

o # .

(a) How did you enjoy working on that unit?

I liked it very much 28 I liked it 121 I didn't like it 51 I didn't like itat all 32

(b) If this unit had been taught by the lecture method, do you think you v/ould have learnt more or less?

I would have learnt more by the lecture method 102 I would have learnt about the same amount 81 I would have learnt less by the lecture method 3^

(c) Which parts of the unit did you find most difficult?

Miscellaneous ; 183

(d) Were any parts so easy as to be a waste of time?

No: 123

Yes: 33

Miscellaneous: 42

^Responses for both units and both years combined, (Over) 701

(e) What were the major administrative difficulties in using this unit?

(Answers combined with Qu, 2(c), q.v.)

(f) Do you think we should use workcards for this unit with next year's students?

Yes, without any changes __ ^ Yes, but make some changes 124 No, don't use workcards 3^

(g) What changes would you recommend in this unit?

(Answers combined with Qu. 2(d), q.v.)

2. Next, think about the use of workcards as a method of teaching mathematics in this college.

(a) How do you like the v/orkcard method of studying mathematics?

I like it very much 29 I liko it 105 I don't like it 50 I don't like it at all 26

(b) In your opinion, what are the main advantages of using workcards for studying mathematics in college?

Develops independence, self-confidence, etc.; 69

Makes one think, find out for oneself, etc.: 55

Allows one to work at one's own pace, etc.: 59

Administratively more efficient: 32

Suggests practical teaching method: 6 (Over) 702

(c) What are the main disadvantages?

Not enough guidance from tutor, etc.: 82

Encourages copying, laziness, absenteeism, etc,; 59

Uninteresting, frustrating, etc,: 33

Administratively inefficient: 50

(d) How could the disadvantages be overcome? More guidance from tutor, more lectures, etc,: 143

Impose time limits, check all assignments, etc,: 24

Make more interesting, allow group work, etc,: 42

Make more efficient (many ways suggested): 42

(e) More of the mathematics courses in this college could be arranged on workcards. If this was done, mathematics classes would have to be completely reorganised* Say which of the following changes you would like and which you would dislike:

(i) Instead of working in a large group, the students in each class would work individually or in small groups. Like 124 Dislike 14 Don't mind 77 (ii) Instead of lecturing all the students at once, the tutor would spend most of his time helping individuals. Like 169 Dislike 36 Don't mind (iii) Students would have to check most of their answers from answer sheets, and consult the tutor only when in difficulty. Like 117 Dislike 43 Don't mind 50 (iv) In each class, students would be working on different parts of the same unit at the same time. Like 80 Dislike 51 Don't mind 24 (v) In each class, students would be working on different units at the same time. Like 63 Dislike 61 Don't mind GO

(Over) 703

(vi) A student would take a test as soon as he completes a unit. The students in each class fould therefore take the same test at different times. Like 64 Dislike 86 Don't mind 61 (vii) If a student's performance on a test was not satisfactory, then he or she would do some more work on that unit and later take another test on the same unit. Like 119 Dislike _i7__ Don't mind 77 (viii) In order to graduate from college, each student would have to complete a certain number of mathematics units satisfaci orily. The student could do them all in a short time or spread them out over several semesters If he wished. Like 94 Dislike _ .70 Don' t mind 62

(f) In view of your answers to part (e), how much use of workcards would you like to see in this college?

Use workcards for the whole i-aths. programme 11 Use workcards for most of th.. programme 52 Use workcards for some parts of some courses 137 Don't use workcards for any >:>f the courses 20

3, Finally, think about the use of workrards as a teaching method in primary schools.

(a) When you leave this college and start teaching, do you intend to use workcards in your classroom (if the principal agrees)?

Yes, all the time __6_ Yes, most of the time 34 Yes, some of the time 157 No, at no time 15

(b) Please give reasons for your answer to 3(a):

Miscellaneous: 188