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Sex-related differences in mathematics achievement scores in Grade 4 and Grade 8 in Kerala, India Nair, Leila Karunakaran 1984

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SEX-RELATED DIFFERENCES IN MATHEMATICS ACHIEVEMENT SCORES IN GRADE 4 AND GRADE 8 IN KERALA, INDIA By LEILA KARUNAKARAN NAIR M.A. Uni v e r s i t y of Madras, India, 1957 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in THE FACULTY OF GRADUATE STUDIES (Department of Mathematics and Science Education) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1984 © L e i l a Karunakaran Nair, 1984 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t copying o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f f^c^Li^c^ACu, C W c / $IASU<^B <^cplu^<zoJ^cy^ The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date DE-6 (3/81) ABSTRACT The p u r p o s e o f t h i s s t u d y was t o i n v e s t i g a t e whether s e x -r e l a t e d d i f f e r e n c e s i n m a t h e m a t i c s a c h i e v e m e n t s c o r e s e x i s t e d i n a sample from t h e u n d e r - d e v e l o p e d w o r l d ; and i f so, what b e a r i n g t h e f i n d i n g s had on t h e c u r r e n t " n a t u r e " and " n u r t u r e " p o s i t i o n s on s u c h d i f f e r e n c e s . A t o t a l of 1 3 7 7 s t u d e n t s a t two g r a d e l e v e l s , G r a d e 4 and Grade 8, drawn from t h r e e d i s t r i c t s i n t h e s t a t e o f K e r a l a , I n d i a , were t e s t e d . Items f o r t h e i n s t r u m e n t s o f t e s t i n g were drawn from t h e B r i t i s h C o l u m b i a L e a r n i n g A s s e s s m e n t S t u d y i n M a t h e m a t i c s of 1981 f o r t h e Grade 4 l e v e l and f r o m t h e Second I n t e r n a t i o n a l M a t h e m a t i c s S t u d y c o n d u c t e d i n B r i t i s h C o l u m b i a i n 1982 f o r t h e G r a d e 8 l e v e l . The r e s u l t s showed t h a t s e x - r e l a t e d d i f f e r e n c e s i n m a t h e m a t i c s a c h i e v e m e n t s c o r e s e x i s t e d a t t h e two g r a d e l e v e l s i n v a r y i n g d e g r e e s d e p e n d i n g on t h e u r b a n - r u r a l l o c a t i o n o f t h e sample, and n o t c o n s i s t e n t l y i n f a v o u r o f t h e same s e x . In t h e u r b a n sample, boys o u t p e r f o r m e d g i r l s by 7% a t t h e Grade 4 l e v e l a nd g i r l s o u t p e r f o r m e d boys by 3% a t t h e Gra d e 8 l e v e l . In t h e r u r a l sample, g i r l s o u t p e r f o r m e d boys by 2% a t t h e Grade 4 l e v e l and boys o u t p e r f o r m e d g i r l s by 8% a t t h e G r a d e 8 l e v e l . B a s e d on t h e r e s u l t s of t h i s s t u d y , i t i s h y p o t h e s i z e d t h a t i f t h e " n a t u r e - p o s i t i o n " o f g e n e t i c male s u p e r i o r i t y i n m a t h e m a t i c a l a b i l i t y i s t h e r e a s o n f o r t h e i n s t a n c e s o f s u p e r i o r male scores in thi s study, data at the same grade levels that show marginally superior female achievement scores in a di f f e r e n t locale indicate that "nurture" of some sort can l i k e l y 'remedy any de f i c i e n c i e s in mathematical a b i l i t y that may be imposed by "nature". Research Supervisor: Dr. D. F. R o b i t a i l l e . iv CONTENTS Page LIST OF TABLES v i i ACKNOWLEDGEMENT v i i i Chapter 1 . INTRODUCTION 1 Theories to Explain Sex-related Differences 3 The Nature-Position 4 The Nurture-Position 5 The Project 6 Major Questions Posed in the Study 7 Reasons for the Separate Study of Urban and Rural Samples 8 Reasons for the Choice of Kerala 11 2. REVIEW OF RELATED LITERATURE 14 B r i t i s h Columbia Learning Assessment Studies ..15 Alberta Assessment of School Mathematics 17 Case for Sex-related Differences in Mathematical Reasoning A b i l i t y 18 H i s t o r i c a l Perspective 23 Are Males Superior in Spatial V i s u a l i z a t i o n A b i l i t y ? 24 V Is Mathematical Achievement Dependent on Spatial V i s u a l i z a t i o n A b i l i t y ? 25 Present Status of Genetic-Based Theories 29 Literature on the Nurture-Position 30 Summary 33 Some Studies in Other Cultures 35 Some Recent Studies in India 37 3. METHOD 40 Sample 41 School Structure in Kerala 42 Selection of Sample 43 Instruments of Testing 46 Test A for Grade 4 Level 46 Test B for Grade 8 Level 46 Procedure 47 4. RESULTS 49 Description of Sample 49 Comparison of Mean Scores 49 Urban Sample 50 Grade 4 50 Grade 8 50 Rural Sample 50 Grade 4 50 v i Grade 8 51 5. SUMMARY OF RESULTS and IMPLICATIONS 57 Findings 57 Bearing of Findings on Nature-Nurture Positions 58 Alternative Inference from Findings 59 Limitations of the Study 59 Suggestions for Further Research 60 REFERENCES ..62 APPENDIX I. Map of Kerala 74 II. Syllabus in Mathematics for Kerala Schools 76 Standard IV 77 Standard VIII 78 II I . Instruments of Testing 79 Test A 80 Test B 93 IV. Tables of Results by Item 107 v i i L I S T OF TABLES TABLE Page 1. D i s t r i b u t i o n o f S u b j e c t s i n Sample 52 2. G r a d e 4 L e v e l : Urban S c o r e s i n P e r c e n t a g e s by Domain .53 3. G r a d e 8 L e v e l : Urban S c o r e s i n P e r c e n t a g e s by Domain .54 4. G r a d e 4 L e v e l : R u r a l S c o r e s i n P e r c e n t a g e s by Domain .55 5. G r a d e 8 L e v e l : R u r a l S c o r e s i n P e r c e n t a g e s by Domain .56 6. I t e m w i s e p - v a l u e s on T e s t A a t G r a d e 4 L e v e l 108 7. I t e m w i s e p - v a l u e s on T e s t B a t G r a d e 8 L e v e l 109 v i i i A C K N O W L E D G E M E N T S I am m o s t g r a t e f u l t o t h e G o v e r n m e n t o f I n d i a f o r p e r m i t t i n g me t o c o n d u c t t h i s s t u d y , a n d t o t h e G o v e r n m e n t o f K e r a l a f o r m a k i n g i t p o s s i b l e . My s p e c i a l t h a n k s a r e d u e t o S r i . J o h n M a t h a i , D i r e c t o r o f P u b l i c I n s t r u c t i o n , K e r a l a , a n d D r . V e d a m a n i M a n u e l , H e a d o f t h e D e p a r t m e n t o f E d u c a t i o n , U n i v e r s i t y o f K e r a l a , w h o i n t h e i r d i f f e r e n t c a p a c i t i e s g a v e t h e p r o j e c t t h e i m p e t u s i t n e e d e d . I n i t s f i e l d w o r k , t h i s p r o j e c t i s s o i n d e b t e d t o t h e S t a t e I n s t i t u t e o f E d u c a t i o n , K e r a l a , t h a t i t w o u l d b e v a i n t o a t t e m p t t o d i s c h a r g e t h a t d e b t . I c a n o n l y m e n t i o n w i t h g r a t i t u d e t h e a c t i v e p a r t i c i p a t i o n o f S r i . T h o m a s M a t h e w s , E v a l u a t i o n O f f i c e r , S r i . N . B h a s k a r a n , D e p u t y D i r e c t o r ( M a t h e m a t i c s ) a n d S m t . P . V . M e e n a k s h i A m m a l , T e x t b o o k s R e s e a r c h O f f i c e r . My t h a n k s a r e a l s o d u e t o t h e h e a d s , t e a c h e r s a n d s t u d e n t s o f t h e v a r i o u s s c h o o l s w h o w i l l i n g l y p a r t i c i p a t e d i n t h e p r o j e c t . T o my c o m m i t t e e o f a d v i s o r s a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a w i t h o u t w h o s e g u i d a n c e t h i s s t u d y c o u l d n o t h a v e b e e n b r o u g h t t o a c o n c l u s i o n , I o f f e r my g r a t e f u l t h a n k s . T o my s u p e r v i s o r , D r . D . F . R o b i t a i l l e , I m u s t r e m a i n e v e r i n d e b t e d , a s w i t h o u t h i s e n c o u r a g e m e n t , s u p p o r t a n d c o n f i d e n c e t h i s s t u d y w o u l d n o t h a v e k n o w n a b e g i n n i n g . F i n a l l y , I w o u l d l i k e t o t h a n k t h e C a n a d i a n F e d e r a t i o n o f U n i v e r s i t y W o m e n , w h o b y g r a n t i n g me t h e A l i c e W i l s o n A w a r d f o r 1 9 8 3 - 8 4 , e a s e d t h e w e i g h t o f my f i n a n c i a l c o m m i t m e n t t o t h i s s t u d y . 1 Chapter 1 INTRODUCTION There i s a consensus in educational research that males outperform females on school mathematics achievement tests. While theories of male i n t e l l e c t u a l superiority have not been lacking through history, the e a r l i e s t evidence of extensive research data came with the F i r s t International Mathematics Study conducted across twelve countries by the International Association for the Evaluation of Educational Achievement (IEA), in 1964. A limited p i l o t project conducted as a forerunner to this study in Belgium, England, Finland, France, Germany, Is r a e l , Poland, Scotland, Sweden, Switzerland, the United States and Yugoslavia had shown a marginal difference in favour of g i r l s in numerical a b i l i t y at the 13 year old age l e v e l (Foshay, 1962). Consequently, when the data from the F i r s t International Mathematics Study was analyzed, i t was with some surprise that the n u l l hypothesis on sex-related differences in mathematics achievement scores at each of the four population lev e l s (9 years, 13 years, 17 years and Pre-University) was rejected (Husen, 1967). In every country studied males had outperformed females. Since then, a large number of studies on sex-related 2 differences in mathematics achievement scores have been implemented in many of these countries, most of a l l in the United States. By the early seventies, interest in this issue in the United States had been "spurred by p o l i t i c a l groups who were fig h t i n g the occupational segregation ex i s t i n g in their country and who had i d e n t i f i e d mathematics as the c r i t i c a l f i l t e r preventing access to many jobs and careers" (Mura, 1982, p.16). As a res u l t , a series of intervention measures to promote female achievement in mathematics gained impetus (Blum & Givant, 1982; Fennema, Wolleat, Pedro, Becker, & DeVancy, 1981; Fox, 1976b; S e l l s , 1982) and a narrowing of sex-related differences has been observed (Armstrong, 1981; Levine & Ornstein, 1983; Tobias, 1982). A l l the same, differences s t i l l exist in favour of males above the junior high school l e v e l (Kapoor, 1983; NAEP, 1980). In the United Kingdom and A u s t r a l i a , there i s some evidence of sex-related differences in mathematics achievement scores in favour of g i r l s in the lower age group of 11 years and under (Byrne, 1978; Shelley, 1982); but male superiority seems to be the rule in older age groups (Cornelius & Cockburn, 1978; Harding, 1977; Kelly, 1976, 1981; Murphy, 1978; Preece, 1979). That the situation i s much the same in the countries of Western Europe i s evident from a statement of concern on "education and equality of opportunity for g i r l s and women" (Council of Europe, 1982, p. 1) expressed at the 11th session of the Standing Conference of European Ministers of Education at 3 The Hague in 1979. Consequently, a workshop was held early in May, 1981 at Honefoss where thirteen of the countries were represented. At the workshop, each of the countries -- Austria, Belgium, Finland, the Federal Republic of Germany, Ireland, I t a l y , the Netherlands, Norway, Portugal, Spain, Sweden, Switzerland and the United Kingdom — c i t e d male advantage in career and occupation as a result of male superiority in mathematics and the physical sciences (Council of Europe, 1982). Sex-related differences in mathematics achievement scores was a major topic at the Fourth International Congress on Mathematical Education held at Berkeley in 1980. As a result of the proceedings, an international survey was undertaken by Dr. Erika Schildkamp-Kundiger, in cooperation with the Second International Mathematics Study (IEA), and published under the t i t l e "An International Review of Gender and Mathematics" in 1982. The contributions from almost every country c i t e d male superiority in achievement scores in mathematics and the physical sciences (Schildkamp-Kundiger, 1982). Theories To Explain Sex-related Differences It would appear that evidence of sex-related differences in favour of males in mathematics achievement scores in the developed world i s unequivocal at the present time. But what is controversial i s whether these differences are to be interpreted as evidence of innate superior mathematical a b i l i t y in the male, 4 or as evidence of the extent of inequality of opportunity in re l a t i o n to mathematical achievement that is b u i l t into human soc ie t y . The Nature-Position Those who would favour the former interpretation, or what has been c a l l e d the "nature-position", support i t on grounds of b i o l o g i c a l or genetic differences between the male and the female (Benbow & Stanley, 1980, 1981, 1982; Broverman, I. K., Vogel, Broverman, D. M., Clarkson and Rosenkrantz, 1972; Buffery & Gray, 1972; Corah, 1965; Garron, 1970; Stafford, 1961, 1963, 1972). While such a theory dates far back into history, the rationale for i t has changed p e r i o d i c a l l y . At one point superior male i n t e l l i g e n c e was attributed to superior brain size, then to superior brain weight, or even superior muscle strength ( E l l i s , 1908; Lourbet, 1896; Mozans, 1913). The most persuasive theory at the present time seems to be developed along the following l i n e s : (1) Males have superior s p a t i a l v i s u a l i z a t i o n a b i l i t y . (2) Mathematical a b i l i t y i s p o s i t i v e l y correlated with s p a t i a l v i s u a l i z a t i o n . (3) Therefore males are superior in mathematical a b i l i t y . That males have superior s p a t i a l a b i l i t y i s explained on the basis of (a) differences in l a t e r a l brain development, (b) 5 the X-linked chromosome inheritance theory, or (c) the theory of d i f f e r e n t i a l hormonal infuences. The Nurture-Position Those who support the alternate interpretation for lesser achievement scores in females, or subscribe to what has been c a l l e d the "nurture-position" attribute i t to socio-psychological-cultural factors such as: (1) Sex-role stereotyping. (2) Mathematics viewed as belonging to the male domain. (3) Fear of ostracism by male peer group. (4) Mathematics anxiety. (5) Lack of mathematical a c t i v i t y in childhood. (6) Lack of encouragement from parents and/or teachers and counsellors. (7) D i f f e r e n t i a l course-taking. While neither position can be "proved" in the s t r i c t sense of the word, a basic tenet of g e n e r a l i z a b i l i t y in experimental research d i f f e r e n t i a t e s between the c r e d i b i l i t y of the two hypotheses generated. The nurture theory, which i s based on factors extraneous to the ind i v i d u a l , generalizes over a population to which those factors apply. That such factors apply to most of human society as i t exists today is in c i d e n t a l . The nature theory on the other hand, generalizes over the human 6 race. Quantitative data that have generated t h i s genetic theory have been gathered almost e n t i r e l y from the developed world (Lancy, 1983). Empirical studies have been centred largely in the North American continent, Europe, A u s t r a l i a , New Zealand and Japan. The Soviet Union has consistently avoided considering sex as a determining factor in research studies in mathematics education ( K r u t e t s k i i , 1976; Menchinskaya, 1946, 1969). Hence, i t would appear that a hypothesis for the human race i s generated on the basis of evidence from a stratum of less than 25% of the world population. If any theory that speaks for humanity as a whole i s to be considered, such evidence as i s c i t e d , should be manifest at a l l l e v e l s and strata (Lancy, 1983). The Second International Mathematics Study of the IEA currently in progress may provide such a p r o b a b i l i t y sample. But empirical studies in diverse geographical, c u l t u r a l and economic strata would seem to be necessary before theories of d i f f e r e n t i a l b i o l o g i c a l embedding of mathematical a b i l i t y or a b i l i t i e s are entertained. The Project This study therefore, investigated whether sex-related differences in mathematics achievement scores exist in a sample from a d i f f e r e n t part of the world, in a d i f f e r e n t economic stratum, in a d i f f e r e n t culture and with a d i f f e r e n t educational history, v i z . the State of Kerala in India. Limitations of time 7 and funds 1 r e s t r i c t e d the study to two grade levels in three d i s t r i c t s in the State. Major Questions Posed in the Study The two major questions posed in the study were: (1) Are there sex-related differences in mathematics achievement scores at Grade 4 and Grade 8 levels in the urban schools tested? (2) Are there sex-related differences in mathematics achievement scores at Grade 4 and 'Grade 8 leve l s in the rural schools tested? A secondary objective was to determine i f the findings in (1) and (2) had any bearing on the "nature" and "nurture" positions on sex-related differences in mathematics achievement scores. 1The Project was partly supported by the A l i c e Wilson Award for 1983-84 from the Canadian Federation for University Women. 8 Reasons for the Separate Study of Urban and Rural Samples The reasons for the separate study of urban and rural samples are best i l l u s t r a t e d by the study of another investigation in India which did not draw any such urban-rural d i s t i n c t i o n s . In the second phase of the F i r s t International Study conducted by the IEA, a "Six Subject Survey", science among them, was conducted across 22 nations. Three of these countries, Chile, India and Thailand were, from the under-developed world. But the mean scores for these countries were so low in a l l subjects that they were excluded from most of the more detailed analyses (Comber & Keeves, 1973). In a later analysis of the data from India where the metropolitan and rural schools were separated, i t was found that the mean for the c i t y schools at the population I I , III and IV levels (14 year olds, 17 year olds and Pre-University) exceeded the median for the developed countries while that for the population I l e v e l remained below (Shukla, 1974). The analyses showed that differences were small between Indian and international scores in urban areas, the difference decreasing with higher levels of population; but that differences were large in rural areas, the difference increasing with higher levels of population. A sharp drop in mathematics achievement scores with higher grade levels from the primary stage has also been reported from another recent study in an under-developed area, v i z . The Indigenous Mathematics Project in Papua, New 9 Guinea (Lancy, 1983). In India, any study of academic achievement is confounded by the large i l l i t e r a c y figures that combine with caste, class and economic status to make within-country differences between rur a l and urban areas almost as large as the differences between India and the developed world (Jayaraman, 1981; Shukla, 1974). That these conditions confuse findings from any investigation of a conventionally representative sample of a reasonable size is evident from some other facts surrounding the Six Subject Survey in India. At the time of the study, 80.1% of India's population l i v e d in v i l l a g e s (the figure for 1981 was nearly the same, 80%), 70.65% of the population was i l l i t e r a t e and an estimated 30% or more l i v e d below the poverty l i n e (Iyer, 1970). In the Six Subject Survey, India took part in science and reading, and samples were drawn from the Hindi-speaking States as the largest one-language region in the country. There are 14 other o f f i c i a l languages in India and 1652 mother-tongues of which only about 400 are non-literate (Pattanayak, 1981). Even i f only those languages that are spoken by 50 000 people or over are counted, there are s t i l l 25 d i f f e r e n t languages in the country, with educational standards varying widely among the various language groups (Jayaraman, 1981). On the variable of l i t e r a c y , the relevant figures were: Bihar - 19.97%, Madhya Pradesh - 22.03%, Haryana - 26.69%, Rajasthan - 18.79%, Uttar Pradesh - 21.64% and Delhi - 56.65%. 10 Delhi alone, a metropolis with a population of 4 m i l l i o n , raised the l i t e r a c y figure of the sample to that for a l l of India, 29.35. The rest of the states represented a t o t a l population of 222 m i l l ion (Shukla, 1974). A representative sample of schools in these states could not, and did not, include any of the schools teaching in the medium of English (which are attended largely by middle and upper class children in urban areas) as less than 4% of the Indian population i s acquainted with the English language even today (Pattanayak, 1981). Yet, the ef f e c t i v e administration of the country in every aspect draws largely from t h i s section of the population (Lewandowski, 1980). As has been mentioned, the result of the International Study was that the scores, both in science and in reading, were so low that no worthwhile analyses could be made. That any small sample randomly selected, even i f proportionately s t r a t i f i e d , cannot present an accurate picture of the country may be evident from a few other s t a t i s t i c s : (1) that 450 m i l l i o n i l l i t e r a t e people, an estimated 50% of the world's i l l i t e r a t e population, l i v e in India (Manorama Yearbook, 1983). (2) that the t h i r d largest pool of s c i e n t i s t s and technical personnel in the world l i v e in India (Prasad, 1983). (3) that India i s the 9th most i n d u s t r i a l i z e d country in the world (Narayanan, 1982). (4) that India provides technological know-how to 37 other nations, including some in the developed world (Ummat, 1982). 11 This study therefore, takes into cognizance the fact that the middle classes of India, l i v i n g in i t s urban areas, whose to t a l numbers exceed the combined populations of the United Kingdom, Canada, Japan and France, but who are yet outnumbered 5 to 1 by the less p r i v i l e g e d , l i v i n g mostly in the rural areas (Census Commissioner of India, 1981) merit study in their own right. As t h i s investigation i s a limited individual project on a sample of about 1400 subjects, no probability sampling was attempted. Instead, the study was confined to three d i s t r i c t s in a single state, Kerala, and samples were drawn from i t s rural and urban areas and studied separately. Reasons for the choice of Kerala The choice of Kerala was partly one of convenience ( i t is the investigator's home state), and partly due to the fact that the state i s d i s t i n c t i v e in India in the area of public education. Ever since census studies were i n i t i a t e d in 1881, Kerala has consistently registered the highest l i t e r a c y figures in India (Lewandowski, 1980). At present the l i t e r a c y figure for Kerala i s 69.17% and that for a l l of India, 36.17% (Census Commissioner of India, 1981). The female l i t e r a c y figure for Kerala i s 64.48%, the highest in India. The state of Maharashtra follows next in rank with a female l i t e r a c y figure of 35.08%. In the past six years the state has almost achieved what 1 2 remains a target for the rest of the country, 100% enrollment of i t s 6-11 year olds in school (Government of Kerala, 1981). But t h i s rapid advance in public education on an economy that is the poorest among a l l the states, has not been without i t s anomalies. One quarter of Kerala's schools lack washrooms, one t h i r d lack drinking water, one half have no e l e c t r i c i t y (Government of Kerala, 1978). These facts must speak for what educational f a c i l i t i e s in the nature of books, laboratories, gymnasiums, arts and c r a f t s etc. the schools are l i k e l y to have. The state spends 55% of i t s revenue on education. Yet the expenditure per pupil per school day works out to 17 cents in elementary schools and 25 cents in the high schools (Government of Kerala, 1979). The average expenditure for elementary through secondary school in B r i t i s h Columbia for the year 1980-81 works out to $21 per pupil per school day despite the fact that the province spends no more than 15.5% of i t s budget on education ( S t a t i s t i c s Canada, 1983). In many ways Kerala epitomizes the contradictions that p r e v a i l in India. It i s the smallest state (area: 38 863 km2), with the highest density of population, 654 persons to the square kilometer (the average in other states i s 387). The state also has the lowest death rate in India, and the lowest figures for infant mortality, which i s attributed to i t s higher l e v e l of female l i t e r a c y (Chattopadhyay, 1983). With Education, Family Planning and Health Care together accounting for three quarters 1 3 of i t s expenditure, the State has also succeeded in bringing down the population growth rate to 1.80 per thousand per annum, the lowest growth rate in India (Narayanan, 1982). The female-male r a t i o in India had progressively declined to 935 females per 1000 males over the past century, which has been associated with d i f f e r e n t i a l treatment meted out to g i r l s in food and n u t r i t i o n within the family. For Kerala alone, the r a t i o i s about the same as for the rest of the world, 1035 females to 1000 males (Manorama Yearbook, 1983). But for every departure from the norm can also be c i t e d a point of conformity with the rest of India. For instance, while female teachers outnumber male teachers by 9% in Kerala, the highest lev e l s in schools, be i t primary, upper primary or secondary, are taught more by males than females (Government of Kerala, 1981, see Table 15). The sex-related hierarchy i s unmistakable. 14 Chapter 2 REVIEW OF RELATED LITERATURE In a review of l i t e r a t u r e on sex-related differences in mathematics achievement in the United States, Elizabeth Fennema (1977) draws a dividing l i n e around 1974. Studies reported before that date show differences in favour of males while those reported after that date are more ambivalent. The reasons behind th i s s h i f t in trends can be seen in two major factors. F i r s t of a l l , there has been a reduction in sex-related differences in mathematics achievement scores (Blum & Givant, 1982; Dees, 1982; Levine & Ornstein, 1983; Schonberger, 1976; S e l l s , 1982; Tobias, 1982; Usiskin, 1982). Secondly, the less disparate achievement scores between the sexes in combination with the more sensitive climate of controversy as a result of the active feminist and equal rights movements in the country (Mura, 1982), i s leading to more cautious conclusions being drawn from the data (Gray & Schafer, 1981) . A l l the same, i t must be mentioned that there are some notable exceptions to t h i s trend (Benbow & Stanley, 1980, 1981, 1982) . Canada and the United States are not always d i f f e r e n t i a t e d in educational research as there are, undeniably, a great deal in common between the two countries. Yet, the differences that 1 5 exist in educational and s o c i a l environments could be considerable in certain areas (Mura, 1982). It is evident in that neither of the factors described above apply wholly to Canada. Sex-related differences in mathematics achievement scores were f i r s t mentioned in Canada with the B r i t i s h Columbia Learning Assessment Studies of 1977 (Mura, 1982). Such research as has been reported since then'cite larger and not smaller sex-related differences ( R o b i t a i l l e , 1981; Sawada, Olson and Sigurdson, 1981). B r i t i s h Columbia Learning Assessment Studies In the f i r s t Learning Assessment Study in B r i t i s h Columbia (1977), the test content was divided into three domains: Computation and Knowledge, Comprehension, and Applications. In the nine comparisons possible in these domains at the grades 4, 8 and 12 l e v e l s , g i r l s outperformed boys in the computation domain in grades 8 and 12. In the seven other comparisons the boys were superior although a l l differences were small. Nevertheless, the differences must be considered educationally s i g n i f i c a n t as the study was based on a census and not a sample. The study also drew attention to the fact that g i r l s did not elect to take mathematics subjects in the same proportion as boys. While the male-female r a t i o was 51 to 49 at the Grade 4 and Grade 8 l e v e l s , in the optional Algebra 12 group i t was 58 to 42 ( R o b i t a i l l e & S h e r r i l l , 1977). 16 In the second Learning Assessment Study (1981) conducted four years l a t e r , the test was divided into five content domains: Number and Operation, Geometry, Measurement, Algebraic Topics and Computer Literacy. In mean scores, at the Grade 4 l e v e l , there were s i g n i f i c a n t differences only in two domains: in Geometry in favour of g i r l s and in Measurement in favour of boys, both differences being small. At the Grade 8 l e v e l , there were s i g n i f i c a n t but small differences in three areas: Geometry, Measurement and Computer Literacy; a l l three were in favour of boys. At the Grade 12 l e v e l there were s i g n i f i c a n t differences in a l l domains ranging between 7% and 17%, a l l in favour of ma1e s. In the Learning Assessment Study of 1981, a further analysis was done by dividing the Grade 12 population into sub-populations of those whose la s t or current mathematics course was Algebra 12, Algebra 11 or Mathematics 10. It was found that males outperformed females in each of the sub-populations, in each of the domains. It was only on a single objective within the domain of Algebraic Topics that there was no sex-related difference. The male-female r a t i o in Algebra 12 had increased to 60 to 40 in the i n t e r v a l . The r a t i o was s i m i l a r l y high in Computing Science, Geometry and Trade Mathematics ( R o b i t a i l l e , 1981)'. As demonstrated by scores in the two Learning Assessment Studies of 1977 and 1981, where domains were c l a s s i f i e d 1 7 d i f f e r e n t l y but overlapped considerably in content, sex-related differences in favour of males in mathematics achievement had widened in B r i t i s h Columbia. Females also continued to be a minority in upper level optional courses in secondary school mathemat i c s . Alberta Assessment Of School Mathematics In the spring of 1978, the province of Alberta conducted an assessment study in mathematics at the grades 3, 6, 9 and 12 level s (Olson, Sawada and Sigurdson, 1979; Sawada et a l , 1981). The content areas were: Number, Algebra, Geometry, Measurement and S t a t i s t i c s over the three cognitive levels of Knowledge, Comprehension and Application. An analysis was made of the number of instances where the mean score for one sex was s i g n i f i c a n t l y superior to the mean score for the other sex. On th i s c r i t e r i o n , comparison by grade-level indicated that males outperformed females 17 to 11 at Grade 3, 26 to 10 at Grade 6, 28 to 6 at Grade 9 and 43 to 3 at Grade 12. By mathematical content area, in the 20 comparisons possible, only in the area of Number at the Grade 3 l e v e l did females do better. A l l other comparisons favoured boys. By the three cognitive l e v e l s , males did better than females 15 to 9 in Knowledge, 30 to 6 in Comprehension and 33 to 2 at Applications. At the Knowledge lev e l alone g i r l s were ahead at grades 3 and 6, but the boys were ahead by Grade 9, and considerably so by Grade 12. 18 The Alberta study seems remarkable not only for i t s consistent findings of male superiority in mathematics achievement scores by grade, by content, and by cognitive l e v e l s , but also for i t s extreme caution in drawing conclusions from such unequivocal data. The authors recommend examining school systems, modes of instruction and choice of test items for sex bias. They suggest that as 90% of the test items were selected from studies in the United States such as the National Assessment for Educational Progress (NAEP) which have been found to be sex biased ( T i t t l e , McCarthy and Steckler, 1974), i t i s possible that such a bias pervades th i s study as well. The report concludes with a recommendation that Canada investigate what sort of socio-educational-cultural variables give r i s e to male superiority in mathematics so that changes may be introduced to give females a f a i r opportunity in an area of such sig n i f i c a n c e . Case for Sex-related Differences in Mathematical Reasoning  A b i l i t y In d i r e c t contrast to the inferences drawn from the Alberta Assessment of School Mathematics (Olson et a l . , 1979; Sawada et a l . , 1981) i s that done in the Study of Mathematically Precocious Youth (SMPY) at Johns Hopkins (Benbow & Stanley, 1982). Between 1972 and 1974 the SMPY i d e n t i f i e d , through a 19 talent search, over 2000 seventh and eighth grade students who had performed as well as the national sample of eleventh and twelfth grade females at the Mathematics (SAT-M) and Verbal (SAT-V) sections of the Scholastic Aptitude Test. Benbow and Stanley pursued the mathematical careers of these students through the following fiv e years. It was found that in rel a t i o n to successive SAT-M and SAT-V scores, scores on the Advance Placement Program Examination in Calculus, and on the Mathematics Achievement Test, Levels I & II, males scored higher than females. Therefore, i t was hypothesized, that males may have superior mathematical a b i l i t y for genetic reasons of either d i f f e r i n g l a t e r a l development of the brain, or prenatal hormonal influences (Benbow & Stanley, 1982). However, the data revealed: (1) that males took more mathematics courses than females especially in College Algebra, Analytic Geometry and Calculus, although a l l subjects took the same mathematics courses up to Grade 11 l e v e l (p.604); ( 2 ) that males took the courses one semester e a r l i e r on an average, than did the females (p. 604); (3) that the proportion of females taking the upper l e v e l optional courses in mathematics was smaller than for boys (p. 608). When females did take these courses, i t was reported that their 20 mean course grades were marginally higher than those for males. This fact was explained by the observation: The mathematics course-grade differences can probably be explained by the sex differences favouring g i r l s that have been found in conduct and demeanor in school (Baker, 1981; see Entwisle & Hayduk, 1981, in press). G i r l s have better conduct and demeanor. This p o s s i b i l i t y i s consistent with the stronger relationship between  mathematics reasoning a b i l i t y and mathematics  course grades for boys than g i r l s . Unfortunately, we could not control for conduct or demeanor, (p.617, emphasis added) What can be inferred from the reported data i s that females did not take mathematics courses semester after semester as did the males but only with breaks (and losses?). The "mathematics reasoning a b i l i t y " referred to i s the measure of the SAT-M score at the time of the talent search, and the "stronger relati o n s h i p . . . " translates to the fact that boys' scores were high or low in the courses depending on whether they were high or low in the SAT-M, but g i r l s ' scores in the courses did not follow the pattern of their scores in the SAT-M. The inference could well be that one needs to explain away the higher scoring of g i r l s in mathematics courses, as the authors have done; or that the performance of g i r l s in the SAT-M tended to be e r r a t i c . When one considers that scores on the SAT-M were supposed to predict aptitude for mathematics, and that course grades are supposed to indicate achievement in mathematics, i t appears to be inconsistent that mathematics reasoning a b i l i t y should be i d e n t i f i e d with the prediction c r i t e r i o n , but should be 21 summarily divorced from mathematics achievement to make way for "conduct and demeanor". Another questionable aspect of the Benbow & Stanley (1982) study is the persistent use of "mathematical reasoning a b i l i t y " interchangably with SAT-M scores. Apart from the instance c i t e d e a r l i e r , to c i t e a few others: (1) "when mathemaical reasoning a b i l i t y was controlled f o r . . . " (p. 611), when what was meant was that SAT-M scores were controlled for. (2) "the less well developed reasoning a b i l i t y of g i r l s . . . " (p. 618), when what the authors had access to were the lower SAT-M scores of g i r l s . (3) The conclusion: "We conclude that sex differences in mathematical reasoning a b i l i t y and achievement are widely noted in t h i s highly able group of students, they persist over several years, and they are better accounted for by the sex difference in mathematical reasoning a b i l i t y than by sex differences in expressed attitudes toward mathematics and mathematics course taking in junior and senior high school" (p. 619). There appears to be no evidence in the research l i t e r a t u r e to indicate that the quest to ide n t i f y mathematical reasoning a b i l i t y which began with Hadamard (1954) and Poincare (1963) concluded with the SAT-M. On the contrary, i t appears that the quest goes on (Luchins & Luchins, 1980). In fact, the a b i l i t y of the SAT-M even to assess mathematical reasoning a b i l i t y has not 22 gone unquestioned (Aiken, 1982; Gray & Schafer, 1981). In a study commissioned by the College Entrance Examination Board, i t was found that while SAT-V was impervious to coaching and d r i l l i n g , SAT-M scores could be raised by as much as one standard deviation by such preparation (Pike & Evans, 1972). As a further conclusion, Benbow and Stanley extend the i d e n t i f i c a t i o n of SAT-M scores from mathematical reasoning a b i l i t y to general reasoning a b i l i t y . To quote the authors: "Moreover, why boys tend to reason better than g i r l s from at least as early as second grade (Dougherty et a l . , Note 5) onward i s also, of course, not clea r " (p. 620). On the basis of the data presented in the la s t two studies (Benbow & Stanley, 1982; Sawada et a l , 1981), i t would appear that i t is Sawada et a l . , rather than Benbow & Stanley, who have uncontroversial evidence of male superiority in mathematics achievement scores. While i t i s Benbow and Stanley, and not Sawada et a l . , who claim genetic male superiority in mathematical reasoning a b i l i t y . These two studies have been dealt with at some length as they demonstrate dramatically that: (1) superior male performance on mathematics achievement tests i s s t i l l a fact, and (2) that interpretations of related factors and circumstances are highly subjective, and may be used to lend support to any plausible theory for sex-related differences. 23 H i s t o r i c a l Perspective No assessment of theories of inherent or genetic or gender-based male i n t e l l e c t u a l superiority can be placed in perspective unless i t i s considered in i t s h i s t o r i c a l context. A r i s t o t l e i s an oft-quoted authority on the theory, and no doubt he had innumerable unrecorded forbears. The preoccupation of trying to "prove" such theories l o g i c a l l y seem to have originated with the spurt in mathematical thought in the Renaissance period in Europe. With Newton and Leibnitz began not only the calculus, but an " i f - t h e n " approach to s o c i o l o g i c a l problems (Davis & Hersh, 1981). This search to give l o g i c a l backing to blindly-accepted theories can be seen in the measuring and weighing of craniums and grey matter to provide s c i e n t i f i c proof for the e f f e c t i v e superiority of male over female, white over black, west over east (Gould, 1981; LeBon, 1879; Vogt, 1864). But even i f such proof has proved elusive (Chorover, 1979), the means have become progressively more sophisticated. The current theory i s one of superior male s p a t i a l a b i l i t y due to d i f f e r e n t i a l brain l a t e r a l i z a t i o n (Bufferey & Grey, 1972; Harshman & Remington, 1976; Levy, 1972, 1974; McGee, 1979; McGlone & Kertesz, 1973). Granting that b i o l o g i c a l differences in the human anatomy and related functions must exist between the male and the female, i t i s of interest to investigate the succeeding two 24 steps in the l o g i c a l sequence: (1) that males are superior in sp a t i a l v i s u a l i z a t i o n , (2) that mathematics achievement i s dependent on spa t i a l v i s u a l i z a t i o n . Are Males Superior in Spatial V i s u a l i z a t i o n A b i l i t y ? Up to the mid-seventies, superior s p a t i a l v i s u a l i z a t i o n a b i l i t y was almost uniformally associated with males (Carey, 1915; Fennema, 1974; Maccoby & J a c k l i n , 1974; Schonberger, 1976). Since then inferences from data have become less conclusive. In the F a l l of 1978, the Education Commission of the States (ECS) conducted the Women in Mathematics Project, a survey of 13 year olds and high school seniors in the United States. At about the same time the ECS also conducted the second National Assessment of Educational Progress (NAEP). In the Women in Mathematics Project the test content was divided into the four domains of Computation, Algebra, Problem Solving and Spatial V i s u a l i z a t i o n and in the NAEP i t was divided into the three domains of Computation, Algebra and Applications. In Computation and Algebra, neither study showed any s i g n i f i c a n t difference between males and females at the 13 year l e v e l . There was no si g n i f i c a n t difference in the problem-solving domain, but there was a s i g n i f i c a n t difference in favour of males in Applications 25 and a similar s i g n i f i c a n t difference in favour of females in sp a t i a l v i s u a l i z a t i o n (Armstrong, 1981; NAEP, 1980). It can be concluded that evidence of superior s p a t i a l v i s u a l i z a t i o n a b i l i t y i s ambivalent between the sexes. Is Mathematics Achievement Dependent On Spatial V i s u a l i z a t i o n  A b i l i t y ? In a study sponsored by the National Science Foundation in the United States, a pr o b a b i l i t y sample of 1330 students in grades 6-8 in four school areas were studied (Fennema & Sherman, 1977). There were s i g n i f i c a n t sex-related differences in achievement in favour of males in one of the areas but i t was not accompanied by any difference in sp a t i a l v i s u a l i z a t i o n . In the same study in grades 9-12 (N=1233), two of four schools showed s i g n i f i c a n t differences in favour of males in mathematics achievement but only one of them showed a male advantage in sp a t i a l v i s u a l i z a t i o n . A causal rela t i o n s h i p between s p a t i a l v i s u a l i z a t i o n and mathematics achievement has been questioned by Werdelin despite what apparantly i s a "strong pedagogical reason to believe in a connection between the a b i l i t y to v i s u a l i z e and geometric a b i l i t y " (Werdelin, 1971). Logical as such a connection appears to be, empirical research has f a i l e d to establish i t s existence (Aiken 1973; Fruchter, 1954; Lean & Clements, 1981; Murray, 1949; Radatz, 1979; Smith, 1964; Very, 1967; Werdelin, 1971). 26 On the cognitive l e v e l , Twyman (1972) distinguishes between "memory images" and "abstract images" and the "use" of imagery. The irrelevant d e t a i l s in vi s u a l images, he said, could d i s t r a c t the " o r i g i n a l stimulus" from making the necessary abstractions. A similar d i s t i n c t i o n in cognitive a b i l i t i e s i s made by Menchinskaya (1946) and Kru t e t s k i i (1976). Kr u t e t s k i i categorizes problem solvers into three types: analytic, geometric and harmonic, where those in the last category s h i f t between the other two modes with ease. However, the geometric mode was found to be more r e s t r i c t i v e than the analytic mode. Lean and Clements (1981) in reviewing l i t e r a t u r e related to sp a t i a l v i s u a l i z a t i o n and mathematics achievement make the statement that mathematics educators may need further research done to c l a r i f y the implications of information-processing theories. They quote the experience of educational psychologists who had conducted research for decades on the assumption that "auditory" and " v i s u a l " learners could be i d e n t i f i e d , only to find that evidence of their separate existence was ambiguous (DeBoth & Dominowski, 1978; Jensen, 1971). Lean and Clements (1981) in their study of 116 foundation year engineering students at Papua, New Guinea, administered a battery of fi v e s p a t i a l a b i l i t y tests, a test in Pure Mathematics and a test in Applied Mathematics. Multiple Regression analysis of data revealed that a l l of the s p a t i a l a b i l i t y tests together accounted for 10% of the variance in Pure 27 Mathematics, and after the effects of Pure Mathematics had been p a r t i a l l e d out, i t accounted for 2% of the variance in Applied Mathematics. One inference drawn from the analysis was that "the tendency towards superior performance on mathematics tests by students who preferred a v e r b a l - l o g i c a l mode of processing mathematical information might be due to a developed a b i l i t y to abstract readily, and therefore avoid the formation of unnecessary images" (Lean & Clements, 1981, p. 296). This was further supported by a study on seventh grade students in V i c t o r i a , A u s t r a l i a , by Suwarsono (Clements, 1981, 1982). The greater significance of a n a l y t i c a l s k i l l s over s p a t i a l s k i l l s was the finding in another study done in New Delhi, India in 1963-67 (Sharma, 1973a, 1973b). Students from standards 8-11, where standard 11 was the f i n a l year of secondary school, in the 12+ to 19+ age group (N=2628), were the subjects in a combined cross-sectional and longitudinal study on achievement scores in the six areas: verbal, numerical, mechanical and s p a t i a l a b i l i t i e s , deductive reasoning, and c l e r i c a l speed and accuracy. The students were divided into fi v e a b i l i t y l e v e l s on the basis of t h e i r mean comprehension scores on the tests and growth curves were traced for each domain by clas s , by modal age in each class and by a b i l i t y levels in each c l a s s . The growth curves showed that: (1) verbal and reasoning a b i l i t i e s were very high for the upper levels in standard 8, but that those at lower lev e l s grew at a faster rate so that the gap 28 was narrowed by standard 11; (2) Numerical a b i l i t y , mechanical a b i l i t y , c l e r i c a l speed and accuracy grew at p a r a l l e l rates so that whatever differences there had been between the various leve l s at standard 8 was maintained through to standard 11; (3) In s p a t i a l a b i l i t y alone, the abler groups not only started at a higher l e v e l in standard 8, but progressed at a faster rate so that the gap widened considerably by standard 11. These results more or less conform with the Benbow and Stanley (1982) finding that males who were lower than females on the SAT-V scores in the Talent Search almost wiped out the difference by the end of high school, while growth in SAT-M scores were only s l i g h t l y higher for males. But conformity ends with the data. Benbow and Stanley concluded that males grew at a faster rate than females. Sharma inferred that there was a plateauing in verbal and reasoning a b i l i t i e s so that those who reached high levels e a r l i e r grew at a slower pace. No sex-related differences were studied although the sample was a mixed one. Another interesting result from the Sharma study was that when the six a b i l i t i e s were rank ordered for each of the fiv e l e v e l s in each class, for the top l e v e l , s p a t i a l a b i l i t y ranked the l a s t in standard 8, 4th in standard 9, and 5th in standard 10 and standard 11. Verbal, Numerical and Reasoning a b i l i t i e s outranked s p a t i a l a b i l i t y in each of the classes with a single exception of verbal a b i l i t y following s p a t i a l a b i l i t y in 29 standard 9. The question arises as to how much of the variance in achievement scores for the high scorers would have been accounted for by s p a t i a l a b i l i t y after the effects of verbal, numerical and reasoning a b i l i t i e s had been p a r t i a l l e d out. Even on the basis of rank ordering, the results seem to be in conformity with that of Lean and Clements (1981). In an e a r l i e r study of the SMPY (Benbow & Stanley, 1980) i t was reported that between 1972 and 1979, high scoring males far outnumbered high scoring females in SAT-M. The authors therefore "favoured the hypothesis that sex differences in achievement in and attitude toward mathematics result from superior male a b i l i t y , which may in turn be related to greater male a b i l i t y in s p a t i a l tasks." (Benbow & Stanley, 1980, p.1264) But as yet, while there has been f a i r evidence that high levels of s p a t i a l a b i l i t y are found among high achievers in mathematics, and low achievers in mathematics are low in s p a t i a l a b i l i t y , there has been no evidence that i t i s the superior s p a t i a l a b i l i t y that contributes to the superior mathematics achievement. In fact there i s evidence that mathematics achievement can be high with or without superior s p a t i a l v i s u a l i z a t i o n a b i l i t y (Fennema & Sherman, 1977; Lean & Clements, 1981; Roach,,1979; Robinson & Gray, 1974; Satterly, 1968, 1976) Present Status Of Genetic-based Theories If the genetic-based theories for sex-related differences 30 in mathematics achievement scores are based on superior male s p a t i a l v i s u a l i z a t i o n a b i l i t y , i t would appear that recent findings disrupt the sequence of l o g i c a l deduction described on page 4, though they do not d i s c r e d i t the theories per se. D i f f e r e n t i a l brain l a t e r a l i z a t i o n , or any of the other factors, could bring to l i g h t some other discriminating a b i l i t y which could account for the d i f f e r e n t i a l in achievement scores. Considering the r e l a t i v e l y short span in which theories have sh i f t e d from brain size to brain side, such a discovery could not be considered u n l i k e l y . If Benbow and Stanley (1980, 1981, 1982) are representative of the school of thought that supports genetic, sex-related differences in mathematical a b i l i t y , i t i s a point of note that i t was data from the same study that generated a hypothesis of male superiority based on s p a t i a l a b i l i t y in 1980, and one based on brain l a t e r a l i z a t i o n in 1982. In the interval s p a t i a l a b i l i t y had become increasingly controversial as a discriminating factor, and the more comprehensive c r i t e r i o n of brain l a t e r a l i z a t i o n was needed. Literature on the Nurture-Position While neither nature, nor nurture, can be contrived, accident victims and war veterans who had sustained brain i n j u r i e s provided some data for experimental studies on brain l a t e r a l i z a t i o n (McGlone & Kertesz, 1973). Even though no such opportunity exists for the study of induced changes in nurture, 31 e x p e r i m e n t a l s t u d i e s i n t h e t h e o r y have been d e s i g n e d t o d e m o n s t r a t e t h e s e n s i t i v i t y o f f e m a l e m a t h e m a t i c s a c h i e v e m e n t s c o r e s t o : s e x - r o l e s t e r e o t y p i n g ( C a s s e r l y , 1982; Fox, 1976a; Kagan, 1964; K e l l y , 1981; S m i t h e r s & C o l l i n g s , 1981), p e r c e p t i o n o f m a t h e m a t i c s as a male domain ( E r n e s t , 1976; Fennema, 1977; Fox, 1977), a n x i e t y and l a c k of c o n f i d e n c e ( C r a n d a l l , K a t k o v s k y and P r e s t o n , 1962; C r o s s w h i t e , 1975; E r n e s t , 1978; Kagan, 1964; Maccoby & J a c k l i n , 1973; S e p p i e & K e e l i n g , 1978; T o b i a s , 1976), l a c k of c h i l d h o o d m a t h e m a t i c a l a c t i v i t i e s ( Fox, 1976b; L u c h i n s & L u c h i n s , 1980; Osen, 1974), f e a r o f s u c c e s s ( F i n k , 1969; H o r n e r , 1968; Humphreys, 1982), l a c k of encour a g e m e n t from p a r e n t s a n d / o r t e a c h e r s and c o u n s e l l o r s (Osen, 1974; T o b i a s , 1982), and v a r i o u s c o m b i n a t i o n s o f t h e s e and o t h e r exogenous f a c t o r s ( C a r e y , 1958; McMahon, 1971; Ormerod, 1971, 1973, 1975; S c h o n b e r g e r , 1978; S e l l s , 1982; S p e n d e r , 1982). The s i n g l e most s i g n i f i c a n t r e s e a r c h e v i d e n c e i n s u p p o r t o f t h e e x ogenous t h e o r y i s t h e c o n s i d e r a b l e r a n g e o f o v e r l a p between male and f e m a l e a c h i e v e m e n t s c o r e s i n m a t h e m a t i c s and t h e p h y s i c a l s c i e n c e s (Fennema, 1977; K e l l y , 1981; T o b i a s , 1982). T h i s was e v i d e n t i n t h e F i r s t I n t e r n a t i o n a l M a t h e m a t i c s S t u d y (Husen, 1967), i n t h e SMPY (Benbow & S t a n l e y , 1980, 1982) i n t h e f i r s t , s e c o n d and t h i r d NAEP s t u d i e s ( K a p o o r , 1983; M u l l i s , 1975; NAEP, 1975, 1980), i n t h e f i r s t and s e c o n d B r i t i s h C o l u m b i a L e a r n i n g A s s e s s m e n t S t u d i e s ( R o b i t a i l l e & S h e r r i l l , 1977; R o b i t a i l l e , 1981) and i n t h e Women i n M a t h e m a t i c s P r o j e c t 3 2 ( A r m s t r o n g , 1 9 8 1 ) . I n e a c h o f t h e s e i n v e s t i g a t i o n s , a l a r g e p r o p o r t i o n o f f e m a l e s s c o r e d a s h i g h a s , o r h i g h e r t h a n a l a r g e p r o p o r t i o n o f m a l e s . I n f a c t , S h a r m a a n d M e i g h a n ( 1 9 8 0 ) c i t e t h i s o b s e r v a t i o n a s r e a s o n f o r t h e i r i n v e s t i g a t i o n o f t h e G C E ' 0 ' l e v e l m a t h e m a t i c s s c o r e s i n E n g l a n d f o r a m o r e c o n s i s t e n t r e a s o n f o r s e x - r e l a t e d d i f f e r e n c e s . T h e S h a r m a & M e i g h a n s t u d y r e s u l t e d i n f i n d i n g a s i g n i f i c a n t c o r r e l a t i o n b e t w e e n m a t h e m a t i c s a c h i e v e m e n t s c o r e s a n d w h e t h e r o r n o t t h e s u b j e c t w a s a s t u d e n t o f p h y s i c s a s w e l l , r a t h e r t h a n t h e s e x o f t h e s u b j e c t . I n t h e F i r s t I n t e r n a t i o n a l M a t h e m a t i c s S t u d y , c r o s s -n a t i o n a l d i f f e r e n c e s f a r e x c e e d e d t h e s e x - r e l a t e d w i t h i n -n a t i o n a l d i f f e r e n c e s . F o r i n s t a n c e , s e x d i f f e r e n c e s a t t h e 1 3 -y e a r - o l d l e v e l w e r e g r e a t e s t i n T h e N e t h e r l a n d s , B e l g i u m , J a p a n a n d E n g l a n d . B u t t h e " l o w - a c h i e v i n g " g i r l s o f t h e s e c o u n t r i e s o u t p e r f o r m e d t h e " h i g h - a c h i e v i n g " b o y s o f m a n y o t h e r c o u n t r i e s i n c l u d i n g t h e U n i t e d S t a t e s . A n o t h e r r e s u l t o f t h e s t u d y w a s t h a t I s r a e l h a d t h e h i g h e s t m e a n s c o r e f o r t h e 13 y e a r o l d p o p u l a t i o n a n d I s r a e l i g i r l s s c o r e d h i g h e r t h a n I s r a e l i b o y s b o t h i n v e r b a l a s w e l l a s c o m p u t a t i o n a l s u b t e s t s ( H u s e n , 1 9 6 7 ) . B u t o v e r a l l , i n t h e 4 2 s e x - r e l a t e d c o m p a r i s o n s t h a t w e r e m a d e , b o y s h a d t h e a d v a n t a g e i n 4 0 o f t h e m . B e n j a m i n B l o o m , s u m m a r i z i n g t h e r e s u l t s o n s e x - r e l a t e d d i f f e r e n c e s c o n c l u d e d w i t h t h i s s t a t e m e n t : We h a d o r i g i n a l l y a t t e m p t e d t o u n d e r s t a n d t h e s e d i f f e r e n c e s i n m a t h e m a t i c s a s f u n c t i o n s o f t h e 33 d i f f e r e n t i a l roles of males and females in the d i f f e r e n t countries. However, we f i n d so few exceptions to the rule of male superiority in mathematics that we are led to believe that variation in the roles of the sexes in these  countries w i l l not be helpful in understanding mathematics achievement differences between the sexes (Husen, 1967, p.259, emphasis added). In the two decades that have elapsed since that study, although there are indications that sex-related differences in some of the countries have narrowed (Brush, 1980; Levine & Ornstein, 1983), and studies in d i f f e r i n g sex-roles p a r t i a l l y accounted for the differences to the conviction of some, essential sex-related differences and the attendant controversy over i t s implications s t i l l p e r s i s t in these countries. Studies in other countries for an added dimension on the issues appear to be necessary. Summary Research evidence therefore appears to indicate that: 1. Sex-related differences in mathematics achievement scores exist in the developed world (Husen, 1967; Kapoor, 1983; Kelly, 1981; NAEP, 1975, 1980; R o b i t a i l l e , 1981). 2. The difference in scores in favour of males, widens with age and grade l e v e l (Kapoor, 1983; NAEP, 1975, 1980; R o b i t a i l l e & S h e r r i l l , 1977; R o b i t a i l l e , 1981; Sawada et a l , 1981). 3. In the past decade these differences have tended to 34 decrease in the United States, but they s t i l l exist (Benbow & Stanley, 1980; Blum & Givant, 1982; Levine & Ornstein, 1983; S e l l s , 1982; Tobias, 1982). The rationale for male superiority in mathematics achievement scores s t i l l evident regardless of observed trends is mainly divided into two schools: I. The Nature-Theory, which is based primarily on the sequence of deductions that (i) mathematics achievement scores are dependent on s p a t i a l v i s u a l i z a t i o n a b i l i t y , and ( i i ) males are superior in s p a t i a l v i s u a l i z a t i o n a b i l i t y (Carey, 1915; Fennema, 1974; Maccoby & J a c k l i n , 1974). Recent studies, however, make the assumption of male superiority in s p a t i a l v i s u a l i z a t i o n a b i l i t y ' questionable (Armstrong, 1981; Fennema & Sherman, 1977). Even i f such superior s p a t i a l a b i l i t y should ex i s t , whether achievement in mathematics i s dependent on s p a t i a l a b i l i t y has become increasingly dubious (Aiken, 1973; Clements, 1982; Fruchter, 1954; Lean & Clements, 1981; Murray, 1949; Radatz, 1979; Smith, 1964; Very, 1967; Werdelin, 1971). That s p a t i a l v i s u a l i z a t i o n a b i l i t y contributes l i t t l e to superior mathematics achievement was the conclusion in many such studies ( K r u t e t s k i i , 1976; Lean & Clements, 1981; Menchinskaya, 1946; Roach, 1979; Robinson & Gray, 1974; Satterly, 1976; Sharma, 1973a, 1973b). II. The Nurture-Theory, attributes sex-related differences in 35 mathematics achievement scores to s o c i o l o g i c a l , p s y c h o l o g i c a l and environmental f a c t o r s . While c o n s i d e r a b l e research has been done in t h i s area of i n v e s t i g a t i o n , evidence that s e x - r e l a t e d d i f f e r e n c e s i n mathematics achievement scores e x i s t s o l e l y because of such f a c t o r s has yet to be demonstrated. However, mathematics achievement scores of females have been shown to be s e n s i t i v e to s o c i o - p s y c h o l o g i c a l and environmental f a c t o r s (Crosswhite, 1975; E r n e s t , 1976; Fennema, 1977; Fox, 1977; Humphreys, 1982; Luchins & Luchins, 1980; Newton, 1981). Some St u d i e s In Other C u l t u r e s Recent s t u d i e s i n the West Indies and on West Indian and A s i a n c h i l d r e n i n B r i t a i n have produced r e s u l t s at v a r i a n c e with many that have been c i t e d . Roach (1979) t e s t e d 206 boys and 212 g i r l s randomly chosen from Grade 6 c l a s s e s i n 5 Jamaican schools on mathematics achievement, conceptual s t y l e and i n t e l l i g e n c e . I t was found that mathematics achievement had s i g n i f i c a n t p o s i t i v e c o r r e l a t i o n s with a n a l y t i c a l c onceptual s t y l e and i n t e l l i g e n c e and that the mean scores f o r g i r l s were s i g n i f i c a n t l y higher than that f o r boys (p<.0l). The v a r i a n c e among g i r l s was a l s o found to be l a r g e r than that f o r boys. Most s t u d i e s up to now have recorded not only higher scores f o r males, but a l s o a higher v a r i a n c e among males. No s i g n i f i c a n t 36 sex-related differences had been found in Jamican schools in two e a r l i e r studies as well (Vernon, 1961; Isaacs, 1974). Driver (1980) studied achievement in mathematics, science and English of 2300 West Indian, Asian and English sudents of the 16+ age group from 5 m u l t i r a c i a l schools in B r i t a i n . In mathematics and science, Asian boys and g i r l s (no differences are mentioned) did best of a l l , and the ranking among the others was, West Indian g i r l s , English boys, West Indian boys and English g i r l s , in that order. As larger numbers of English g i r l s dropped out of higher l e v e l mathematics courses than others further analysis was done c o n t r o l l i n g for course background. The West Indian g i r l s s t i l l kept their rank. Jahoda (1979) compared 72 Ghanian children in grades 2, 4 and 6 (in their last three weeks of school) with a p a r a l l e l group in Scotland in grades 3, 5 and 7 (in their f i r s t three weeks of school) in s p a t i a l perception tasks. A s i g n i f i c a n t sex-related difference in block construction in favour of males was the same in the two samples. There was no difference in either group in mental rotation or pattern assembling. Tanner and Trown (1979) investigated the e f f e c t s of c u l t u r a l change on mathematical thought by studying samples of immigrant Asian school children in two age groups, 10/11 years and 12/13 years, from 22 schools in 11 towns in the i n d u s t r i a l northwest of England. Among children who had had a l l their primary education in the country, Asian g i r l s were markedly 37 superior to their male counterparts at age 10/11, and marginally so at 12/13. Some Recent Studies in India Sex-related differences in mathematics achievement scores have not been an issue in educational research in India as the preoccupation in the country is s t i l l mainly with the problem of i l l i t e r a c y (Sunder, 1982). However, a few studies that have a tangential bearing on this investigation are cited below. Sinha (1980) compared the competence of students from four typical schools, a government boys' school, a government g i r l s ' school, a private boys' school and a private g i r l s ' school, with a total of 5200 students. Competence was operationally defined to cover components such as acquisition of s k i l l s , self confidence, positive self concept, internal control of reinforcement, moderate and/or high level of achievement and positive leadership qualities among others. The private g i r l s ' school ranked f i r s t on leadership qual i t ies , intelligence, extroversion and self concept. That the result does not extend to a l l g i r l s is indicated by other studies of status and personality of women across India (Gaur, 1980; Sharma, 1979), which found that while women of the upper classes held themselves responsible for their success, the middle and lower classes held God or Fate responsible. Upper class g i r l s in India are to be found almost entirely in private g i r l s ' schools 38 (Jayaraman,1981). Chauhan and Singh (1982) in an investigation into the study habits of 500 10-12 year olds in the Simla d i s t r i c t found no sex-related differences in either urban or rural populations, but the mean scores, according to the manual of the Study Habits Inventory used, were average in urban areas and below normal in rur a l areas. The effect of parental profession on study habits was also found to be s i g n i f i c a n t at the .05 l e v e l . The order of mean scores by parental profession were: Teaching-171, Government Services-163, Defence Services-161, Business-160, and Agriculture-156. Another study investigated the differences in the role of the secondary school teacher in urban and rur a l areas (Shah, 1971). The data revealed that: (1) heads of urban i n s t i t u t i o n s were generally better q u a l i f i e d than heads of rur a l i n s t i t u t i o n s ; (2) heads of i n s t i t u t i o n s , teachers and parents in urban areas set subject-training as the objective of top p r i o r i t y while in rural areas the p r i o r i t y was on character-building, good-citizenship and subject-training, in that order. Bhargava (1982) studied the effects of prolonged deprivation on academic achievement. The independent variables covered 15 material, s o c i a l and emotional dimensions such as food, clothing, housing, motivational experiences and childhood experiences. 13 of these dimensions were found to be s i g n i f i c a n t , 8 of them at the .01 l e v e l . A c r u c i a l result was 39 t h a t these f a c t o r s a f f e c t e d the p o t e n t i a l h i g h a c h i e v e r s , male and female, f a r more than the low a c h i e v e r s . However, no st u d y was made as t o whether the e f f e c t s v a r i e d i n e x t e n t w i t h sex. 40 Chapter 3 METHOD In I n d i a , s t a n d a r d i z e d achievement t e s t s i n mathematics have yet to be used i n e d u c a t i o n a l s t u d i e s . In the mathematics papers of p u b l i c examinations conducted at the end of the high school program throughout the country, g i r l s have been found to do j u s t as w e l l , or b e t t e r than boys (Hate, 1969). But a general comment on the s u p e r i o r performance of g i r l s i n e x t e r n a l examinations i n India has been that the q u e s t i o n s are l a r g e l y designed to t e s t t e x t u a l knowledge and t h e r e f o r e more l i k e l y to favour females than males (Hate, 1969). Hence, i n t h i s study i t was c o n s i d e r e d that the r e s u l t s would be more d e f i n i t i v e i f t e s t items that had a l r e a d y been used on known samples i n the developed world were chosen r a t h e r than indigenous ones. The items were t h e r e f o r e s e l e c t e d from the B r i t i s h Columbia L e a r n i n g Assessment Study of 1981 ( R o b i t a i l l e , 1981), and the Second I n t e r n a t i o n a l Mathematics Study conducted i n B r i t i s h Columbia i n 1982 ( R o b i t a i l l e , O'Shea, & D i r k s , 1982). C o n s i d e r i n g the l a r g e d i s p a r i t y i n t e c h n o l o g i c a l development, economic s u f f i c i e n c y , standard of l i v i n g and e d u c a t i o n a l o b j e c t i v e s , a d i r e c t comparison of mean scores between K e r a l a and B r i t i s h Columbia would be n e i t h e r j u s t i f i a b l e (Shukla, 1974; I n k e l e s , 1979; N i l e s , 1981a, 1981b), nor 41 p r o f i t a b l e . But i t was hoped that an a n a l y s i s of s e x - r e l a t e d d i f f e r e n c e s i n a s e c t i o n of the p o p u l a t i o n of I n d i a , on the b a s i s of t e s t s used i n a developed region such as B r i t i s h Columbia, would extend the f r o n t i e r s w i t h i n which such i n v e s t i g a t i o n s have been made. A d e s c r i p t i o n of the method and procedure employed in the study i s presented i n t h i s chapter. Sample A t o t a l of 1377 students i n 37 c l a s s e s from 18 schools i n 3 d i s t r i c t s of K e r a l a were t e s t e d . The d i s t r i b u t i o n was: 781 students from standard 9 (384 boys and 397 g i r l s ) and 596 students from standard 5 (300 boys and 296 g i r l s ) . The t o t a l p o p u l a t i o n i n these grades i n the schools i n the sample were: 3798 in standard 9 (1789 boys and 2009 g i r l s ) and 2107 i n standard 5 (1124 boys and 983 g i r l s ) , f o r a t o t a l of 5905 students. The three d i s t r i c t s i n K e r a l a were s e l e c t e d on the b a s i s of l o c a t i o n and e d u c a t i o n a l h i s t o r y : Cannanore at the northern end, Ernakulam i n c e n t r a l K e r a l a and Trivandrum at the southern t i p (see map i n Appendix I ) . H i s t o r i c a l l y , they represent the three d i f f e r e n t s e c t i o n s of B r i t i s h Malabar and the P r i n c e l y S t a t e s of Cochin and Travancore, each of which had a d i f f e r e n t e d u c a t i o n a l system up to 1956. As an index of the d i f f e r e n c e s i n t h e i r e d u c a t i o n a l h i s t o r y , the female l i t e r a c y f i g u r e s f o r 1891 f o r the three areas were, 0.66% f o r B r i t i s h Malabar, 3.76% f o r 42 Cochin State and 2.69% f o r Travancore State when the f i g u r e f o r a l l of Ind i a was 1% (Lewandowski, 1980). The three regions were merged i n 1956, on the b a s i s of t h e i r common language Malayalam, to form the s t a t e of K e r a l a . Since then, the education system' has been uniform throughout the S t a t e . School S t r u c t u r e i n K e r a l a At present, there are two d i f f e r e n t school s t r u c t u r e s in K e r a l a : (1) The Higher Secondary Schools run by the C e n t r a l Goverment with a uniform system a l l over I n d i a . These schools are f o r the c h i l d r e n of Government servants and members of other p u b l i c s e r v i c e s who are s u b j e c t to t r a n s f e r from p l a c e to p l a c e , so that t h e i r s t u d i e s do not get i n t e r r u p t e d . These schools, which are a l l l o c a t e d i n urban areas, run through 11 standards; standard 10 and standard 11 c o v e r i n g the s e n i o r secondary stage of most American schools i n the A r t s and S c i e n c e s . Graduates from these schools are admitted d i r e c t l y to a degree program i n the u n i v e r s i t i e s ; (2) The Secondary School Leaving C e r t i f i c a t e (SSLC) s t r u c t u r e that i s f o l l o w e d by the l a r g e m a j o r i t y of government and p r i v a t e s c h o o l s i n the S t a t e . The schools run through 10 standards; the f i r s t four standards c o n s t i t u t e the primary s e c t i o n , the next three standards, 5-7, c o n s t i t u t e the upper primary s e c t i o n , and the l a s t three standards, 8-10, c o n s t i t u t e the high school s e c t i o n . The l a s t two years of high school allow f o r some s e l e c t i o n of courses, but mathematics i s a 43 compulsory s u b j e c t . Graduates of t h i s school system undergo two years of p r e - u n i v e r s i t y s t u d i e s before they are admitted to a degree program. S e l e c t i o n of Sample As the m a j o r i t y of schools i n K e r a l a f o l l o w the l a t t e r system -- ,over 500 000 c a n d i d a t e s appeared f o r the SSLC examination of 1983 -- the samples were drawn e n t i r e l y from t h i s s e c t i o n of the s c h o o l p o p u l a t i o n . The t o t a l number of such schools i n the three d i s t r i c t s s e l e c t e d were 3565, and the t o t a l number of students at the two grade l e v e l s under study (standards 9 and 5), were 350 000 (Govt. Of K e r a l a , 1981). As mentioned i n Chapter 1, with the l i m i t e d s i z e of sample envisaged i n the study and the wide d i s p a r i t y i n standards between the v a r i o u s schools i n q u a l i t y of i n s t r u c t i o n , c l a s s , c a s t e , economic s t a t u s and e d u c a t i o n a l h i s t o r y of p o p u l a t i o n , no sampling on a r e p r e s e n t a t i v e b a s i s was attempted. Instead, a l i s t of s c h o o l s that represented the mode i n e d u c a t i o n a l standards was drawn up by the E v a l u a t i o n Department of the S t a t e I n s i t u t e of E d u c a t i o n , K e r a l a , l o c a t e d at Trivandrum. T h i s l i s t was drawn mainly on the b a s i s of the percentage of passes from the v a r i o u s s c h o o l s i n the SSLC examination at the end of standard 10. The schools i n the sample were then s e l e c t e d from t h i s l i s t . In the r u r a l areas, there was one c o e d u c a t i o n a l high school 44 per v i l l a g e . There were no s i n g l e - s e x s c h o o l s . The s e l e c t i o n c o u l d t h e r e f o r e be made randomly from the comprehensive l i s t of r u r a l s c h o o l s . The urban s e l e c t i o n was made so that at l e a s t one Boys' s c h o o l , one G i r l s ' school and one c o e d u c a t i o n a l school was s e l e c t e d from each d i s t r i c t . The s i n g l e - s e x schools were s e l e c t e d on the b a s i s of p a r i t y of s t a n d i n g . To c i t e an example from B r i t i s h Columbia, i f S t . Thomas More C o l l e g i a t e (a Boys' school i n Burnaby) was s e l e c t e d from the Boys' s c h o o l s , i t would be p a i r e d not with C r o f t o n House School f o r G i r l s which i s more " e l i t e " , but with Marian High School (a G i r l s ' s chool i n Burnaby run by the same r e l i g i o u s order) which i s fed by the same f a m i l i e s as the Boys' s c h o o l . Or, i t might be s a i d that the Boys' schools and G i r l s ' schools were s e l e c t e d i n matched p a i r s . The c l a s s e s at the v a r i o u s standard l e v e l s were chosen by the heads of schools i n the sample. No system of streaming was f o l l o w e d i n any of the s c h o o l s . The c l a s s e s were d i v i d e d almost always by a l p h a b e t i c a l order of names; except i n the case of g i r l s , which i s d i s c u s s e d i n the f o l l o w i n g paragraph. The same teacher, or at most two teachers, taught mathematics to a l l the c l a s s e s i n Standard 8. At the primary l e v e l s , mathematics was taught by the c l a s s teacher i n almost a l l s c h o o l s . In terms of numbers i n a c l a s s , Standard 8 f o r g i r l s i n r u r a l areas tended to be the more numerous; the l a r g e s t number encountered was 62. In g e n e r a l , a l l - f e m a l e c l a s s e s tended to be l a r g e r than a l l - m a l e c l a s s e s . An e x p l a n a t i o n given by one of the headmasters f o r the 45 d i s p a r i t y i n c l a s s - s i z e was that i t was e a s i e r to maintain d i s c i p l i n e i n a l a r g e c l a s s of g i r l s than boys. The s e l e c t i o n was made with the i n t e n t i o n of d i s t i n g u i s h i n g between c o e d u c a t i o n a l schools and s i n g l e - s e x s c h o o l s . But, at the time of t e s t i n g , i t was d i s c o v e r e d that with the exception of two urban s c h o o l s , every high s c h o o l had d i v i d e d i t s students i n t o c l a s s e s by sex. So i n e f f e c t , almost every high school student i n K e r a l a i s i n a s i n g l e sex environment i n the classroom, i f not always i n the s c h o o l . Although t h i s de f a c t o s i n g l e - s e x s t a t u s of r u r a l schools c o u l d have i t s i m p l i c a t i o n s f o r s e x - r e l a t e d d i f f e r e n c e s i n achievement, f o r the purposes of t h i s study, the d i s t i n c t i o n between s i n g l e - s e x and c o e d u c a t i o n a l schools was c o l l a p s e d . Schools i n K e r a l a , l i k e those i n other p a r t s of I n d i a , have the c h o i c e of E n g l i s h or the n a t i v e language as the medium of i n s t r u c t i o n . But t h i s c h o i c e i s o p e r a t i v e almost e n t i r e l y i n urban ar e a s . In the sample, a l l r u r a l schools f u n c t i o n e d i n the medium of Malayalam, and a l a r g e number (80%) of the urban schools f u n c t i o n e d i n E n g l i s h . The mean ages of the students at the d i f f e r e n t standard l e v e l s were: 13 years, 6 months f o r standard 9; and 9 years, 5 months f o r standard 5. There were no s i g n i f i c a n t d i f f e r e n c e s i n age between the urban and r u r a l samples. 46 Instruments of T e s t i n g Test A f o r Grade 4 L e v e l Form 4C of the B r i t i s h Columbia Learning Assessment Study ( R o b i t a i l l e , 1981), with ten items d e l e t e d to make t o t a l of 36 items, was chosen f o r t h i s l e v e l . The items d e l e t e d were: item 46 in Number and Operations, item 18 i n Measurement of time, item 34 i n Measurement of temperature, item 31 i n monetary measure, items 19, 44 and 45 i n P r o b a b i l i t y and items 5 and 24 in Computer L i t e r a c y . Where p o s s i b l e , names and contexts were changed to s u i t the l o c a l l i f e s t y l e (see Appendix I I I ) . T e s t i n g time was 35 minutes. Test B f o r Grade 8 L e v e l A s e l e c t i o n of 38 items from the Core Test and the four Rotated Forms i n the Second I n t e r n a t i o n a l Mathematics Study of B r i t i s h Columbia of 1982 ( R o b i t a i l l e et a l . , 1982) was made. Names and con t e x t s were changed where p o s s i b l e , as f o r Test A. The domains of t e s t i n g were Algebr a , Geometry, Measurement, A r i t h m e t i c , R a t i o and P r o b a b i l i t y & S t a t i s t i c s (Appendix I I I ) . T e s t i n g time was 35 minutes. No attempt was made to match the items on Test A or Test B with the Ke r a l a s y l l a b u s i n the su b j e c t (see Appendix I I ) , as the primary o b j e c t i v e was not a l e a r n i n g assessment of K e r a l a 47 s c h o o l s , but an i n v e s t i g a t a t i o n of s e x - r e l a t e d d i f f e r e n c e s when boys and g i r l s from K e r a l a were faced with mathematical q u e s t i o n s , f a m i l i a r or otherwise. The t e s t s were t r a n s l a t e d i n t o Malayalam with the help of the Textbook Department of the State I n s t i t u t e of Education. They were then t r a n s l a t e d back i n t o E n g l i s h . The E n g l i s h v e r s i o n s of the t e s t s are given i n Appendix I I I . Procedure Schools i n K e r a l a reopened a f t e r the summer v a c a t i o n of two and a h a l f months on June 16, 1983. 1 The t e s t s were administered to a l l i n the t h i r d week of c l a s s e s , J u l y 4-8. Two members of the S t a t e I n s t i t u t e of Education and' the i n v e s t i g a t o r took charge of one d i s t r i c t each. Each of the schools i n the sample was v i s i t e d by the member concerned and a meeting h e l d with the head of the school and the teachers of the c l a s s e s t a k i n g the t e s t s . I t was decided that no member should enter the classroom before or du r i n g t e s t i n g , so that c o n d i t i o n s remained normal. The t e s t s were given, and c o l l e c t e d by the c l a s s t e a c h e r s . The i n v e s t i g a t o r scored a l l the t e s t s . 1 S c h o o l s o r d i n a r i l y reopen a f t e r the summer break on June 1. The delayed monsoon and the attendant drought caused the date to be postponed. As i t happened, the S.W.Monsoon which normally reaches the K e r a l a coast around June 1, d i d f i n a l l y a r r i v e — on June 16. 48 Poor p r i n t i n g had caused some of the diagrams to be smudged and u n c l e a r . 1 As a r e s u l t , item 29 of Test A, and items 13 and 35 of Test B were omitted. The mean percentage scores f o r the samples, with the omissions, by sex and l o c a t i o n , were found to i n c r e a s e almost uniformly by about 1% l e a v i n g the d i f f e r e n c e i n scores u n a f f e c t e d . The data presented i n the f o l l o w i n g chapter are with the above mentioned items d e l e t e d . 1Due to a combination of s t r i k e s i n v a r i o u s s e c t i o n s of i n d u s t r y and a statewide cut on i n d u s t r i a l use of power d u r i n g the drought, the three major p r i n t i n g presses i n the c a p i t a l were c l o s e d i n d e f i n i t e l y . As a r e s u l t a lesser-known and l e s s e f f i c i e n t p r i v a t e press had to be p a t r o n i z e d . 49 Chapter 4 RESULTS The scores on Test A and Test B were recorded by sex and l o c a t i o n , and the percentage of s u b j e c t s who scored c o r r e c t l y on each item (p-values) were c a l c u l a t e d f o r each group. From the p-value s on items obtained f o r each l o c a t i o n , the percentage mean scores in each of the domains of t e s t i n g were c a l c u l a t e d by sex and compared. The r e s u l t s obtained are presented i n t h i s c h a p ter. D e s c r i p t i o n of Sample The d i s t r i b u t i o n of s u b j e c t s by sex i n the urban and r u r a l samples at the two grade l e v e l s , Grade 4 and Grade 8, are presented i n Table 1. Comparison of Mean Scores Mean scores i n the v a r i o u s domains of t e s t i n g f o r the urban and r u r a l samples at the two grade l e v e l s are presented i n Tables 2-5. The percentage of s u b j e c t s who scored c o r r e c t l y on each item i n the two t e s t s (p-values) are given i n Appendix IV, Tables 6 & 7. 50 Urban Sample Grade 4 At the Grade 4 l e v e l , boys outperformed g i r l s i n every domain of t e s t i n g (Table 2). The l a r g e s t d i f f e r e n c e of 15% was in Measurement, followed by 8% i n Algebra and 5% each i n Number and Operations, and in Geometry. In o v e r a l l mean s c o r e s , boys outperformed g i r l s by 7%. Grade 8 At the Grade 8 l e v e l , boys outperformed g i r l s i n two domains: i n P r o b a b i l i t y and S t a t i s t i c s by 4%, and i n R a t i o by 1% (Table 3). The performance of boys and g i r l s were on a par i n Geometry. In the remaining four domains, g i r l s outperformed boys: by 11% i n F r a c t i o n s , 4% i n Measurement, 4% i n A r i t h m e t i c and 2% i n A l g e b r a . In o v e r a l l mean s c o r e s , g i r l s outperformed boys by 3%. Rural Sample Grade 4 At the Grade 4 l e v e l , boys and g i r l s performed on a par i n Measurement (Table 4). In the remaining three domains g i r l s outperformed boys. The d i f f e r e n c e was 3% each i n Number and Operations, and i n Geometry, and 2% i n A l g e b r a . In o v e r a l l mean scores g i r l s outperformed boys by 2%. 51 Grade 8 At the Grade 8 l e v e l , both boys and g i r l s performed equally poorly (28%) in Arithmetic, and in Ratio (Table 5). In the remaining five domains boys were ahead of the g i r l s : by 16% in Algebra, 15% in Fractions, 10% in Measurement, 6% in Probability and S t a t i s t i c s , and 2% in Geometry. In overall mean scores, boys outperformed g i r l s by 8%. 52 Table 1 D i s t r i b u t i o n of Subjects i n Sample Grade L e v e l Urban R u r a l T o t a l Boys G i r l s Boys G i r l s Grade 4 149 161 151 135 596 Grade 8 205 191 179 206 781 T o t a l 354 352 330 341 1377 Table 2 Grade 4 L e v e l : Urban Scores i n Percentages by Domain Domain Boys G i r l s D i f f e r e n c e Col.2 - Col.3 No. & Op. (17) 1 64 59 5 Algebra (4) 37 29 8 Geometry (7) 49 44 5 Measurement (7) 54 39 15 Weighted Mean 56 49 7 Numbers i n parentheses i n column 1 i n d i c a t e the number of items i n each domain. 54 Table 3 Grade 8 L e v e l : Urban Scores i n Percentages by Domain Domain Boys G i r l s D i f f e r e n c e Col.2 - Col.3 Algebra ( 9 ) 1 55 57 -2 Geometry (10) 44 44 0 F r a c t i o n s (5) 44 55 -1 1 Measurement (4) 49 . 53 -4 A r i t h m e t i c (3) 46 50 -4 P r o b a b i l i t y & S t a t i s t i c s (3) 50 46 4 Ra t i o (2) 46 45 1 Weighted Mean 48 51 -3 Numbers i n parentheses i n column 1 i n d i c a t e the number of items i n each domain. 55 Table 4 Grade 4 L e v e l : Rural Scores i n Percentages by Domain Domain Boys G i r l s Di f ference Col.2 - Col.3 No. & Op. ( 1 7 ) 1 60 63 -3 Algebra (4) 43 45 -2 Geometry (7) 45 48 -3 Measurement (7) 53 53 0 Weighted Mean 54 56 -2 1 Numbers i n parentheses i n column 1 i n d i c a t e the number of items i n each domain. 56 Table 5 Grade 8 L e v e l : Rural Scores i n Percentages by Domain Domain Boys G i r l s Di f ference Col.2 - Col.3 Algebra ( 9 ) 1 46 30 16 Geometry (10) 34 32 2 F r a c t i o n s (5) 41 26 1 5 Measurement (4) 41 31 10 A r i t h m e t i c (3) 28 28 0 P r o b a b i l i t y & S t a t i s t i c s (3) 36 30 6 R a t i o (2) 28 28 0 Weighted Mean 38 30 8 1 Numbers i n parentheses i n column 1 i n d i c a t e the number of items i n each domain. 57 Chapter 5 SUMMARY of RESULTS and IMPLICATIONS A summary of the r e s u l t s of the study, i t s bearing on the "nature" and "nuture" p o s i t i o n s on s e x - r e l a t e d d i f f e r e n c e s in mathematics achievement sc o r e s , and an a l t e r n a t i v e i n f e r e n c e drawn from a post hoc a n a l y s i s of the f i n d i n g s are presented i n t h i s c h a p ter. F i n d i n g s The f i n d i n g s in response to the s t a t e d o b j e c t i v e s of t h i s study were: ( 1 ) In the urban sample, s e x - r e l a t e d d i f f e r e n c e s e x i s t e d at each of the grade l e v e l s s t u d i e d . At the Grade 4 l e v e l the d i f f e r e n c e was i n favour of the boys by 7%; at the Grade 8 l e v e l the d i f f e r e n c e was i n favour of the g i r l s by 3%. (2) In the r u r a l sample, s e x - r e l a t e d d i f f e r e n c e s e x i s t e d at each of the grade l e v e l s s t u d i e d . At the Grade 4 l e v e l the d i f f e r e n c e was i n favour of the g i r l s by 2%; 58 at the Grade 8 l e v e l the d i f f e r e n c e was i n favour of the boys by 8%. Bearing of F i n d i n g s on the "Nature-Nurture" P o s i t i o n s At the Grade 4 l e v e l (9 y e a r - o l d s ) , boys appear to have an a p p r e c i a b l e advantage over g i r l s i n the urban s e t t i n g while g i r l s d i s p l a y a marginal advantage i n the r u r a l s e t t i n g . If the former i s c o n s i d e r e d to support a genetic theory of male s u p e r i o r i t y i n mathematical a b i l i t y , then g i r l s i n the r u r a l s e t t i n g must be assumed to f u n c t i o n under some advantages, r e l a t i v e to g i r l s i n urban areas, that counteract the r e s u l t s of t h e i r g e n e t i c i n f e r i o r i t y . At the Grade 8 l e v e l (13 y e a r - o l d s ) , boys appear to have an a p p r e c i a b l e advantage over g i r l s i n the r u r a l s e t t i n g , and g i r l s d i s p l a y a marginal advantage over boys in the urban s e t t i n g . I f the male s u p e r i o r i t y in- r u r a l areas i s due to genetic s u p e r i o r i t y i n male mathematical a b i l i t y , then g i r l s i n urban areas must f u n c t i o n under some advantages, r e l a t i v e to r u r a l g i r l s , that more than count e r a c t t h e i r g e n e t i c i n f e r i o r i t y . As t h i s study recorded no s o c i o - p s y c h o l o g i c a l v a r i a b l e s , the only unequivocal c o n c l u s i o n that can be a r r i v e d at on the b a s i s of evidence from the samples c o n s i d e r e d i s : I f the "nature" p o s i t i o n f o r s e x - r e l a t e d d i f f e r e n c e s i n mathematics achievement scores holds, i n the age group of 9-13 years, i t i s not beyond being remedied by f a c t o r s that must c o n s t i t u t e the 59 "nurture" p o s i t i o n . A l t e r n a t i v e Inference from F i n d i n g s An a l t e r n a t i v e i n t e r p r e t a t i o n of the f i n d i n g s suggests i t s e l f when u r b a n - r u r a l d i f f e r e n c e s i n mean scores by sex are examined although such an a n a l y s i s was not a s t a t e d o b j e c t i v e of the study. At the Grade 4 l e v e l , the u r b a n - r u r a l d i f f e r e n c e i n scores f o r boys i s 2%, while that f o r g i r l s i s 7% (see Tables 2 & 4). At the Grade 8 l e v e l , the u r b a n - r u r a l d i f f e r e n c e f o r boys i s 10%, while that f o r g i r l s i s 21% (see Tables 3 & 5). Together with the f i n d i n g that g i r l s achieve both the highest and the lowest mean scores at each grade l e v e l (at the Grade 4 l e v e l urban boys and r u r a l g i r l s share the highest s t a n d i n g ) , an app a r e n t l y l e g i t i m a t e i n f e r e n c e c o u l d be: (1) that g i r l s are not i n f e r i o r to boys i n mathematical a b i l i t y , but (2) that such exogenous f a c t o r s as tend to i n h i b i t , or depress, mathematics achievement s c o r e s , as i s evident i n the u r b a n - r u r a l d i f f e r e n c e s , have a gr e a t e r e f f e c t on g i r l s than boys. L i m i t a t i o n s of the Study T h i s study was designed l a r g e l y as an e x p l o r a t o r y study i n s e x - r e l a t e d d i f f e r e n c e s i n mathematics achievement scores i n an area where no such study had been undertaken. The e x i s t e n c e of 60 s e x - r e l a t e d d i f f e r e n c e s i n favour of boys i n one sample, and in favour of g i r l s i n another, both of the same age and grade l e v e l , d i f f e r i n g only by l o c a l e , i n d i c a t e that exogeneous data on socio-economic, c u l t u r a l and i n s t r u c t i o n a l v a r i a b l e s need to be c o n s i d e r e d , and may be expected to shed l i g h t on the i s s u e . Suggestions f o r Further Research R e p l i c a t i o n of the study i n other d i s t r i c t s of K e r a l a , and in other p a r t s of India may serve to e s t a b l i s h a more c o n s i s t e n t p a t t e r n of s e x - r e l a t e d d i f f e r e n c e s i n mathematics achievement scores between urban and r u r a l l o c a t i o n s . I f , on the other hand, the r e s u l t s of t h i s study are repeated, i t may . i n d i c a t e that reasons f o r s e x - r e l a t e d d i f f e r e n c e s i n mathematics achievement scores are to be found, not i n g e n e t i c d i f f e r e n c e s i n a b i l i t y , but i n : ( 1 ) D i f f e r e n c e s i n e d u c a t i o n a l v a r i a b l e s such as c l a s s - s i z e , teacher q u a l i f i c a t i o n s , e d u c a t i o n a l o b j e c t i v e s , study h a b i t s of c h i l d r e n or medium of i n s t r u c t i o n , and t h e i r i n t e r a c t i o n with sex. (2) D i f f e r e n c e s i n s o c i o - p s y c h o l o g i c a l f a c t o r s between urban and r u r a l areas such as m o t i v a t i o n , d e s i r e f o r upward s o c i a l m o b i l i t y i n g i r l s , e roding of t r a d i t i o n a l v a l u e s / i n h i b i t i o n s i n r e l a t i o n t o g i r l s i n urban areas. 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D i f f e r e n t i a l F a c t o r S t r u c t u r e i n Mathematical A b i l i t y . Genetic Psychology Monographs, 1967, 75, pp.169-207. Vogt, C. L e c t u r e s on Man. London: Longman, Green, Longman & Roberts, 1864. Werdelin, I. Geometric A b i l i t y and the Space F a c t o r i n Boys and  G i r l s . Lund, Sweden: U n i v e r s i t y of Lund, 1971. 74 APPENDIX I Map of K e r a l a 75 76 APPENDIX II S y l l a b u s i n Mathematics f o r K e r a l a Schools 77 Standard IV No. Of p e r i o d s . 1. D i a g n o s t i c t e s t i n g and remedial t e a c h i n g . 55 2. Numbers up to those having nine d i g i t s . 15 3. M u l t i p l i c a t i o n . 35 4. D i v i s i o n . 35 5. Problems i n v o l v i n g the four fundamental o p e r a t i o n s . 20 6. Number-line. 10 7. F r a c t i o n s . 25 8. Decimal F r a c t i o n s . 45 9. Measurement of Time. 15 10. P i c t o r i a l Diagrams. 10 11. R e c o g n i t i o n of Geometrical shapes and f i g u r e s . 10 12. D i s c o v e r i n g P a t t e r n s . 5 Total...280 Standard VIII Sets, Venn diagrams, o p e r a t i o n s . Real Numbers. Approximate Numbers, t h e i r p r e c i s i o n and accuracy. Always-true sentences. Open sentences i n one v a r i a b l e . Open sentences i n two v r i a b l e s . B a s i c s concepts i n Geometry. T r i a n g l e s and P e r p e n d i c u l a r s . T r i a n g l e s ( c o n t i n u e d ) . D i s c e r n i n g P a t t e r n s . No. Of hours. 20 10 1 5 20 22 22 26 1 5 1 5 10 T o t a l . . . 175 APPENDIX III Instruments of T e s t i n g 80 TEST A (Grade 4 l e v e l ) 1. S u b t r a c t : 86- 64 n 22 68 n 54 n 118. n I don ' t know . . n 2. Add: 678+ 901. n 9+ 991 n 34 621 n 721 n I don ' t know n 3. I f you c o n n e c t e d t h e s e t h r e e d o t s w i t h s t r a i g h t l i n e s what shape would you g e t ? • Square n R e c t a n g l e n • T r i a n g l e n • C i r c l e n I don't know n 81 4. W h i c h shows 1 / 4 shaded? I d o n ' t know n 6. I f one c h i l d needs two b o o k s , how .many books do t h r e e c h i l d r e n need? 3 . n 82 6 n 8 n 9 n I don ' t know n 7. 10 t e n s "= One mi 1 1 i on. . . n t h o u s a n d . . .n t e n . . . n I d o n ' t know..n 8. A box c o n t a i n s 2 b l u e p e n s , 4 r e d pens and 1 b l a c k p en. V i n o d c o u l d n o t d e c i d e w h e t h e r he wanted a r e d pen o r a b l u e p e n . He c l o s e d h i s e y e s and p i c k e d one f r o m t h e box. Which c o l o u r i s he more l i k e l y t o p i c k ? b l u e o r b l a c k .....n r e d n b l u e n b l a c k n I don ' t know n 9. Which u n i t s h o u l d be u s e d t o measure t h e l e n g t h of a hou s e ? m i l l i m e t r e s n c e n t i m e t r e s n 83 metres n k i l o m e t r e s ' n I don't know n 10. Jane f i n i s h e d her homework a t 7145. Which c l o c k shows t h i s t ime? I d o n ' t know....n 11. What i s t h e remainder i n the f o l l o w i n g d i v 9 28 3) 28 9 27 3 1 1 I don't know s i o n problem? . . n . . n . . n . . n 12. E i g h t c a r s a r e p a r k e d on a r o a d . 1/4 of the c a r s a r e new. How many c a r s a r e new? 1 . . . n 2 . . . n 3 . . . n 4 . . .n I don ' t know '. . n 13. C o u n t i n g by 10s, t h e n e x t t h r e e numbers a r e : 780, 700, n, n, n, 791, 792, 793....n 720 , 730 , 740 n 800, 810, 820 n 800, 801 , 802. . . .n I d o n ' t know........ n 14. Round 1368 t o t h e n e a r e s t h u n d r e d : 1300...n 1400...n 3 000...n 4000...n I don ' t know n 15. I t was a c o l d r a i n y d ay. When J o h n l o o k e d t h e r m o m e t r e , i t showed: 23 C.. . n 3 4" C . . . n 3*' C. . .n 1 6° C . . . n I don ' t know n 85-16. W h i c h number s a y s f o u r t h o u s a n d two h u n d r e d s i x t y f i v e ? 42065...n 5624...n 40265. . .n 4265...n I don ' t know n 17. A h i p p o p o t a m u s w e i g h s 1153 k g . , an e l e p h a n t w e i g h s 1127 k g , a b u f f a l o w e i g h s 1196 kg,and a g i r a f f e w e i g h s 1 1 8 3 kg. R o u n d i n g t o t h e n e a r e s t 100 k g , w h i c h w e i g h s c l o s e s t t o 1100 kg? h i p p o . . . n e l e p h a n t . . . n buf f a l o . . . n g i r a f f e . . . n I don ' t know n 1 8 . I n w h i c h f i g u r e a r e a l l a n g l e s t h e same s i z e ? 12 86 I don ' t know n 19. C h a n d r a n t o o k t w e l v e r u p e e s t o buy s t a m p s . He bought 6 stamps a t 8 p a i s e e a c h , 8 stamps a t 3 p a i s e e a c h and 5 stamps a t 25 p a i s e e a c h . How much d i d he s p e n d on stamps? Rs. 10.03. . .n Rs. 0 . 55 . ..n Rs. 1 .97 . . .n Rs. 13.97. . .n I don ' t know n 20. Which number o r numbers c a n go i n t o t h e b l a n k to- make t h i s number s e n t e n c e TRUE? 5 + ( ) <12 7 . . . n any number l e s s t h a n 7...n any number g r e a t e r t h a n 7...n no number.....n I don ' t know n 21. Which i s t h e s m a l l e s t number t h a t can be made u s i n g a l l t h e d i g i t s 4, 3 , 9 , 1 ? 1934. . . n 1439. . .n 1349. . .n 1943...n I don ' t know n 22. W h i c h box i s -1/5 ( o n e - f i f t h ) s h a d e d ? I don ' t know n 23. How many l i n e s of symmetry does t h i s shape have? 1 . . . n 2 . . . n 3 . . . n ,4 . . .n I don ' t know n 88 24. 156 r o u n d e d t o t h e n e a r e s t 10 i s : 160. . .n 170. . .n 150. . .n 140. . .n I d o n ' t know n 25. J o h n p u t a w i r e f e n c e r o u n d h i s r e c t a n g u l a r g a r d e n . The g a r d e n i s 10 m e t r e s l o n g and 6 m e t r e s w i d e . How many m e t r e s of f e n c i n g d i d he u s e ? 16...n 32... n-36...n 6 0 . . . n I d o n ' t know n 26. How many c e n t i m e t r e s a r e t h e r e i n one m e t r e ? 1 . . . n 10 . . .n 100. . .n 1 000. . .n I d o n ' t know....n 27. F i g u r e s t h a t a r e t h e same s i z e and shape a r e c o n g r u e n t f i g u r e s . Which of the f o l l o w i n g a r e c o n g r u e n t ? A A and B...n A and D...n C and D...n B and C...n I don't know...n 28. About how l o n g i s t h i s p e n c i l ? 1 c e n t i m e t r e . . . . n 5 c e n t i m e t r e s . . . n 1 metre...n .10 m e t r e s . . . n I don't know...n 29. What i s the a r e a of t h i s shape i n s q u a r e . c e n t i m e t r e s C L J 31. A b o t t l e o f m i l k i s l i k e l y t o h o l d : 1 mi 1 1 i 1 i t r e . . . n 10 mi 1 1 i 1 i t r e s . . . n 1 l i t r e . . .n 100 l i t r e s . . .• ' I d o n ' t know . . . n 91 32. One f a c e i s shaded on t h i s cube. How many f a c e s does the cube have? 33. Which i s t r u e ? 35 = 30 + 5 n 35 > 30 + 5 n 35 > 5 + 30 n 30 + 5 < 35 n I d on't know....n 34. S a n t h o s h was t e s t i n g h i s model p l a n e . F o u r of h i s f r i e n d s g i v e n below, g u e s s e d as t o how l o n g i t would s t a y i n the a i r . The p l a n e s t a y e d up f o r 17 m i n u t e s . Who g u e s s e d c l o s e s t t o the c o r r e c t t i me? S u r e s h G i r i j a P r a k a s h R a v i . i i •j / 31! - j i -Time ( i n minutes) 92 S u r e s h . . . n P r a k a s h . . . n Gi r i j a . . . n R a v i . . . n I don't know ...n 35. Madhu has 51 soda b o t t l e s and 8 wooden boxes. Each box h o l d s 6 b o t t l e s . I f Madhu f i l l s a l l the boxes, how many b o t t l e s w i l l t h e r e be l e f t o v e r ? 6 . . . n 8 . . . n 3 . . . n 1 4 ... n I don 1 t know...n ' 36. Madhu d i s t r i b u t e d 51 s o d a - f i l l e d b o t t l e s t o the shops. He c o l l e c t e d back 30 of the empty b o t t l e s . H i s s i s t e r L e e l a c o l l e c t e d t h e r e s t . How many b o t t l e s d i d L e e l a c o l l e c t ? 18. . .n 14... n 2 1 ... n 44 ... n I don't know....n 9 3 T E S T B ( G r a d e 8 l e v e l ) 1 . W h i c h o f t h e f o l l o w i n g s e q u e n c e s o f n u m b e r s i s i n t h e o r d e r i n w h i c h t h e y o c c u r f r o m l e f t t o r i g h t o n t h e n u m b e r l i n e ? ( a ) { 0 , 1 / 2 , - 1 } ; ( b ) { 0 , - 1 , 1 / 2 } ; ( c ) { - 1 , - 1 / 2 , 0} '(d) {-1 , 0 , 1 / 2 } ; ( e ) { - 1 / 2 , -1 , 0} A n s 2 . W h a t i s t h e v a l u e o f * s ' ? ( a ) 7 ; ( b ) 1 3 ; ( c ) 15 ( d ) 1 7 ; ( e ) N o n e o f t h e s e . A n s : 3 . Suma w a l k e d f r o m h e r h o u s e t o t h e r a i l w a y s t a t i o n w h i c h i s 3 . 1 k i l o m e t r e s a w a y . D u r i n g h e r w a l k s h e l o s t h e r w a t c h , w e n t b a c k 1 . 7 k i l o m e t r e s t o f i n d i t , a n d t h e n c o n t i n u e d i n t h e o r i g i n a l d i r e c t i o n u n t i l s h e r e a c h e d t h e r a i l w a y s t a t i o n . How many k i l o m e t r e s h a d Suma w a l k e d a l t o g e t h e r w h e n s h e a r r i v e d a t t h e r a i l w a y s t a t i o n ? 9^ (a) 1.4; (b) 4.8; ( c ) 6.5 (d) 8.2; (e) None of t h e s e . Ans : 4. (-2) x (-3) i s e q u a l t o : (a) -6; (b) - 5 ; (d) 5; (e) 6 ( c ) - 1 ; Ans 5. I n w h i c h d i a g r a m below i s t h e s e c o n d f i g u r e t h e image of t h e f i r s t f i g u r e u n d e r a r e f l e c t i o n i n a l i n e ? W (?) L CP) Ans : 6. The t r i a n g l e s shown above a r e c o n g r u e n t . The m e a s u r e s of some 9 5 s i d e s and a n g l e s a r e shown. What i s x? (a) 52; (b) 55; (c) 65; (d) 73; (e) 75. Ans 7. The f o l l o w i n g t a b l e shows the number of t r e e s p l a n t e d a l o n g a highway i n a week. Day Mon. T u e s . Wed, T h u r s . F r i . No. of T r e e s 80 50 60 90 75 On the d i a g r a m below t h e g r a p h f o r t h e f i r s t two d a y s ' p l a n t i n g has been drawn. I f the g r a p h was c o m p l e t e d , which p o i n t would i n d i c a t e the t o p of the g r a p h on T h u r s d a y ? 100 U 4 3 60 il i 20 f\ p W C A n 9 V A r '/////// i %& D -/ft A y/////A '///////,. v/////& ////////, //////A Mon Tue Wed Thurs F r i 9 6 (a) A; (b) B; (c) C; (d) D; (e) E, Ans cm k cm 4 90° "I 90° 0 C - T n • . —.-to.-There is a brass plate of the shape and diensions shown in the figure above. What is i t s area in square centimetres? (a) 16; (b) 24; (c) 32; (d) 64; (e) 9.6. Ans : C 9. B -AB, CD and EF are concurrent. The measures of certain angles are shown. What is the value of x? (a) 54; (b) 62; (c) 64; (d) 126; (e) 128; Ans : 10. Simplify: 5x + 3y + 2x - 4y (a) 7x + 7y; (b) 8x - 2y; (c) 6xy; 97 (d) 7x - y; (e) 7x + y. Ans : 11. What i s t h e volume of a r e c t a n g u l a r box w i t h i n t e r i o r d i m e n s i o n s 10 cm l o n g , 10 cm w i d e , and 7 cm h i g h ? (a) 27 c c ; (b) 70 c c ; ( c ) 140 c c ; (d) 280 c c ; (e) 700 c c . Ans .: 12. I f P=LW and i f P=12 and L=3, t h e n W i s e q u a l t o (a) 3/4; (b) 3; ( c ) 4; (d) 12; (e) 36. Ans : 13. - -™ - • P Q - R i U V M N 0 "7 X' ' w The d i a g r a m shows a c a r d b o a r d cube t h a t has been c u t a l o n g some edges a nd f l a t t e n e d o u t . I f i t i s f o l d e d up a g a i n i n t o a c u b e , w h i c h two c o r n e r s w i l l t o u c h a t c o r n e r P? (a) Q and S; (b) T and Y; ( c ) W and Y; (d) T and V; (e) U and Y. Ans. : 98 14. Which of the following is a pair of equivalent fractions? (A) 5/8, 2/3; (b) 5/6, 2/3; (c) 4/5, 14/15; (d) 3/5, 9/15; (e) 1/2, 14/24. Ans 15. Which of these is a TRUE statement about the information shown on the graph? tn -p c v 3 -P CO o o S3 22 20 18 16 1U 12 10 8 6 It 2 0 STUDENTS IN GRADES 1,2,3 AND 1* 1 1 1 P // 7/, 1 // " p-1 1 1 P 1 2 3 Grades Boys G i r l s (a) Standard 2 is the smallest class. (b) Standards 2 and 4 have the same number of students. (c) Standard3 has twice as many boys as g i r l s . (d) Standard 4 has more g i r l s than boys. 99 (e) S t a n d a r d 1 has as many b o y s as t h e r e a r e g i r l s i n S t a n d a r d 4. Ans : 16. Q={ 1 , 2 , 3, 4, 5, 6, 7, 8, 9 } R={ 3, 5, 7, 9, 11, 13 } S=Q f] R T h e r e a r e 9 e l e m e n t s i n s e t Q and 6 i n s e t R. How many e l e m e n t s a r e t h e r e i n s e t S? (a) 16; (b) 11; ( c ) 7; (d) 4; (e) 2. Ans : 17. 2/5 + 3/8 i s e q u a l t o (a) 5/13; (b) 5/40; ( c ) 6/40; (d) 16/15; (e) 31/40. 18. 0.40 x 6.38 i s e q u a l t o (a) .2552; (b) 2.452; ( c ) 2.552; (d) 24.52; (e) 25.52. Ans : 1.9. On l e v e l g r o u n d , a boy 5 u n i t s t a l l c a s t s a shadow 3 u n i t s l o n g . A t t h e same t i m e , a n e a r b y t e l e p h o n e p o l e 45 u n i t s h i g h c a s t s a shadow, t h e l e n g t h o f w h i c h i n t h e same u n i t s . i s (a) 24; (b) 27; (c) 30; (d) 60; (e) 75. 100 Ans 20. If 6x 3 = 1 5 then 6x 15 - 3 (i) and 6x 1 2 ( i i ) and x 12/6 ( i i i ) x 2 (iv) If there is an error in the above reasoning, i t f i r s t occurs in (a) ( i ) ; (b) ( i i ) ; (c) ( i i i ) ; (d) ( i v ) ; (e) None of these. Ans : 21. The value of 2 3 + 3'2 is (a) 30; (b) 36; (c) 64; (d) 72; (e) None of these. The tot a l area of the two triangles i s , in square centimetres (a) 6 x 8; (b) (6 x 8)/2; (c) (10 x 6)/2; (d) (16 x 12)/2; (e) (20 x 12)/2. Ans 22 . Ans 101 23. A b o t t l e of soda, i n c l u d i n g the p r i c e of the b o t t l e , c o s t s a p a i s e , but t h e r e i s a r e f u n d of b p a i s e on each empty b o t t l e r e t u r n e d . How much w i l l G opi have t o pay f o r x b o t t l e s i f he b r i n g s back y_ empty b o t t l e s . (a) ax + by p a i s e ; (b) ax - by p a i s e ; (c) (a - b)x p a i s e ; (d) (a+x) - (b+y) p a i s e ; (e) None of t h e s e . Ans : 24. In a s c h o o l of 800 p u p i l s , 300 a r e boys. The r a t i o of the number of boys t o the number of g i r l s i s (a) 3 : 8; (b) 5 : 8; (c) 3 : 11; (d) 5 : 3; (e) 3 : 5. Ans : 25. The a r i t h m e t i c mean ( a v e r a g e ) of 1.50, 2.40, 3.75, i s e q u a l t o (a) 2.40; (b) 2.55; (c) 3.75; (d) 7.65; (e) None of t h e s e . Ans : 26. A q u a d r i l a t e r a l MUST be a p a r a l l e l o g r a m i f i t has (a) One p a i r of a d j a c e n t s i d e s e q u a l . 102 (b) One pair of p a r a l l e l sides. (c) A diagonal as axis of symmetry. (d) Two adjacent-angles equal. (e) Two pairs of .parallel sides. Ans : 27. One of the following points can be joined to the point (-3, 4) by a line segment which cuts NEITHER the x, NOR the y axis. Which one? (a) (-2, 3); (b) (2,-3); (c) (2, 3); (d) (-2,-3); (e) (4,-3). Ans : 28. Which of the following is the most l i k e l y to be nearest to the weight of a normal man? (a) 8.5 kg; (b) 85 kg; (c) 185 kg; (d) 850 kg; (e) 1850kg. Ans : 29. Matchsticks are arranged as follows: If the pattern continued, how many matchsticks are used in making the 10th figure? 1 0 3 (a) 30; (b) 3 3 ; ( c ) 36; (d) 3 9 ; (e) 42. Ans : 30. The l e n g t h of t h e c i r c u m f r e n c e of t h e c i r c l e w i t h c e n t r e a t 0 i s 24, and t h e l e n g t h o f t h e a r c RS i s 4. What i s t h e c e n t r a l a n g l e ROS t o t h e n e a r e s t d e g r e e ? (a) 24; (b) 30; ( c ) 45; (d) 60; (e) 90. Ans : 31. 30 i s 75% o f what number? (a) 40; (b) 90; ( c ) 105; (d) 225; (e) 2250. Ans : 32. What i s t h e s q u a r e r o o t o f 12 x 75? (a9 6.25; (b) 30; ( c ) 87; (d) 625; (e) 900. Ans : 3 3 . I n t h e number 847.36, t h e d i g i t 6 r e p r e s e n t s (a) 6 x1/100; (b) 6 x 1 / 1 0 ; ' ( c ) 6 x 1 ; (d) 6x10; (e) 6x100 Ans : 104 34. I f the segment PQ were drawn f o r each f i g u r e shown below, i t would d i v i d e one of the f i g u r e s i n t o two c o n g r u e n t t r i a n g l e s . Which f i g u r e ? (a) P (d) 7 Ans 35. s o 1 0 0 On t h e s c a l e t h e r e a d i n g i n d i c a t e d by the arrow i s between (a) 51 and 52; (b) 57 and 58; (c) 60 and 62; (d) 62 and 64; (e) 64 and 66. Ans 36. 1 0 5 What a r e t h e c o o r d i n a t e s o f P? y 4 -3 2-- t -3 -2 . ! i ! | * : 1 2 3 i+ -2 -3 O P -k -(a) (-3, 4 ) ; (b) (-4,-3); (c) ( 3, 4 ) ; (d) ( 4,-3); (e) ( - 4 , 3 ) . Ans : 37. The t a b l e below g i v e s the r e l a t i o n between the h e i g h t from which a b a l l i s dropp'ed (d) , and t h e h e i g h t t o which i t bounces ( b ) . d 50 80 100 150 b 25 40 50 75 Which f o r m u l a d e s c r i b e s t h i s r e l a t i o n ? (a) b = d 2 ; (b) b = 2d; (c) b = d/2; (d) b = d + 25; (e) b = d - 25. Ans : 38. The a i r t e m p e r a t u r e a t the f o o t of a mountain i s 31 d e g r e e s . On t o p of t h e mountain the t e m p e r a t u r e i s -7 d e g r e e s . How much warmer i n d e g r e e s i s t h e a i r a t t h e f o o t o f t h e m o u n t a i n ? (a) -38; (b) -24; ( c ) 7; (d) 24; (e) 38. And : A p p e n d i x I V T a b l e s of R e s u l t s by I t e m 108 Table 6 Itemwise p-values on Test A at Grade 4 L e v e l . I tern Urban K e r a l a Rural K e r a l a No. Boys G i r l s D i f f Boys G i r l s D i f f . 1 . 96 96 0 85 90 -5 2. 93 89 4 66 79 -13 3. 79 57 22 79 72 7 4. 70 68 2 72 64 8 5. 37 29 8 43 55 -12 6. 90 84 6 73 90 -17 7. 77 81 -4 80 80 0 8. 50 40 10 64 64 0 9. 55 34 21 48 57 -9 10. 69 46 23 57 50 7 1 1 . 94 78 16 74 95 -21 12. 30 19 1 1 52 40 12 13. 84 68 16 72 77 -5 14. 36 35 1 29 22 7 15.. 44 38 6 58 41 17 16. 59 • 56 3 42 51 . -9 17. 48 25 23 62 57 5 18. 50 55 -5 60 64 -4 19. 56 28 28 58 47 1 1 20. 35 1 7 18 29 30 -1 21 . 64 65 -1 63 64 -1 22. 64 68 -4 52 66 -14 23. 26 21 5 25 25 0 24. 42 37 5 37 40 -3 25. 25 5 20 39 42 -3 26. 60 51 9 62 57 5 27. 63 55 8 36 41 -5 28. 60 30 30 50 50 0 30. 30 34 -4 27 23 4 31 . 66 71 -5 58 73 -15 32. 61 58 3 47 59 -12 33. 47 43 4 41 38 3 34. 16 16 0 37 46 -9 35. 35 34 1 40 55 -15 36. 53 61 -8 64 51 1 3 Note. D i f f . = p-value for Boys - p-value f o r G i r l s 109 Table 7 Itemwise p-values on Test B at Grade 8 L e v e l I tern Urban K e r a l a Rural K e r a l a No. Boys G i r l s D i f f . Boys G i r l s D i f f . 1 . 84 77 7 82 49 33 2. 31 39 -8 29 27 2 3. 43 34 9 59 22 37 4. 83 83 0 63 33 30 5. 69 45 24 34 37 -3 6. 49 47 2 55 34 21 7. 76 67 9 37 41 -4 8. 28 22 6 44 25 19 9. 69 68 1 40 37 3 10. 45 64 -19 35 21 1 4 1 1 . 75 84 -9 56 36 20 12. 79 83 -4 38 35 3 14. 67 84 -17 53 34 19 15. 55 47 8 38 33 5 16. 74 71 - 3 58 37 21 17. 39 51 ' -12 19 19 0 18. 48 57 -9 34 33 1 19. 51 28 23 39 26 1 3 20. 39 39 0 35 30 5 21 . 75 66 9 21 27 -6 22. 18 24 -6 20 21 -1 23. 31 41 -10 32 22 1 0 24. 50 47 3 1 5 25 -10 25. 18 24 -6 32 16 1 6 26. 50 62 -12 35 46 -1 1 27. 1 3 19 -6 1 7 22 -5 28. 73 80 -7 43 41 2 29. 36 48 -12 32 34 -2 30. 35 34 1 30 21 9 31 . 42 42 0 40 30 10 32. 27 36 -9 30 22 8 33. 23 47 -24- 39 24 1 5 34. 39 61 -22 32 32 0 36. 31 39 -8 30 36 -6 37. 28 37 -9 47 19 28 38. 30 15 1 5 20 25 -5 

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