Cockcroft (1982)

Notes on the text
Preliminary pages Foreword, Membership, Contents, Introduction

Part 1
Chapter 1 Why teach mathematics?
Chapter 2 The mathematical needs of adult life
Chapter 3 The mathematical needs of employment
Chapter 4 The mathematical needs of further and higher education

Part 2

Chapter 5 Mathematics in schools
Chapter 6 Mathematics in the primary years
Chapter 7 Calculators and computers
Chapter 8 Assessment and continuity
Chapter 9 Mathematics in the secondary years
Chapter 10 Examinations at 16+
Chapter 11 Mathematics in the sixth form

Part 3

Chapter 12 Facilities for teaching mathematics
Chapter 13 The supply of mathematics teachers
Chapter 14 Initial training courses
Chapter 15 In-service support for teachers of mathematics
Chapter 16 Some other matters
Chapter 17 The way ahead

Appendices

Appendix 1 Statistical information
Appendix 2 Gender differences in mathematical performance
Appendix 3 List of those who made submissions
Appendix 4 Visits and meetings
Appendix 5 List of abbreviations

Index

Report of the Committee of Inquiry into the teaching of mathematics in schools under the chairmanship of Dr WH Cockcroft

London: Her Majesty's Stationery Office 1982
© Crown copyright material is reproduced with the permission of the Controller of HMSO and the Queen's Printer for Scotland.

ISBN 0 11 270522 7

Appendix 2 Differences in mathematical performance between girls and boys
By Miss HB Shuard
[pages 273 - 287]

B1 Pupils in schools are often classified according to their sex, and discussion of their educational programme may take their sex into account, either directly or because of social custom. Until fairly recently in Britain, it was commonplace to discuss separately the education of boys and girls, including the mathematics courses they should follow, and expectations of their mathematical attainment were different. In this appendix, differences in mathematical performance are described, and possible causes of these differences are discussed. Historical and statistical evidence, and the considerable volume of research on educational differences between the sexes, are drawn upon. Suggestions are made for measures which might help to improve the mathematical performance of girls.

Historical background

B2 Mathematics established itself in the curriculum of boys' public and secondary schools in the first half of the nineteenth century, and when girls' secondary schools began to be founded later in the century, the pioneers of girls' education wished to introduce its study into their schools. Lecturing in 1848, Professor FD Maurice discussed the curriculum of the newly opened Queen's College; his remarks about mathematics have often been quoted out of context as a belief 'that women students were unlikely to advance far in mathematics' (1). However, the belief which he actually expressed was that although 'we are aware that our pupils are not likely to advance far in Mathematics', there were positive benefits which girls would gain from its study, and 'the least bit of knowledge ... must be good' (2). At another new girls' school, Cheltenham Ladies' College, Miss Dorothea Beale acknowledged that although she wished to introduce mathematics into the curriculum, it would spell financial ruin for the school, because parents did not wish their daughters to study it. Even arithmetic was suspect; around 1860, a father wrote to Miss Beale, on deciding to send his daughters to another school, 'My dear lady, if my daughters were going to be bankers, it would be very well to teach arithmetic as you do, but really there is no need' (3). However, the opening of the Cambridge Local Examinations to girls in 1863 gave impetus to the teaching of arithmetic and mathematics in girls' secondary schools. Of the first 25 candidates from the North London Collegiate School, 10 failed in arithmetic; the headmistress, Miss Buss, was horrified, and the teaching of arithmetic at once became a matter of extreme importance in her school (3). Only three years later, girls were doing as well in arithmetic as in other subjects in the Cambridge Local Examinations, and when substantial numbers of girls' secondary schools were founded after 1873, mathematics became a regular subject of the curriculum. At first there were great difficulties because of the inadequate supply of teachers; this slowly improved as more women graduated from universities, but in 1912 it was estimated that, out of some 900 women teaching mathematics in secondary schools, only about one third had themselves studied as far as the calculus. However, the boys' schools were no better off, for about the same proportion of men teachers of mathematics in secondary schools had studied calculus (1).

B3 In the public elementary schools, both boys and girls learned arithmetic, but as the Royal Commission on the Elementary Education Acts (4) reported in 1888, 'as the time of the girls is largely taken up with needlework, the time they can give to arithmetic is less than that which can be given by boys'. They therefore recommended that the arithmetical requirements of the curriculum should be modified in the case of girls.

B4 In the 1912 Report on the teaching of mathematics in Britain (1), Miss ER Gwatkin, headmistress of Queen Mary High School, Liverpool, noted that mathematics occupied a good position in the curriculum of most girls' secondary schools, but the pressure caused by the introduction of a wider range of subjects into the curriculum was causing this position to be questioned. Among the objections to mathematics as an important subject for girls were that the subject was uninteresting to most girls, that its utilitarian value to them was negligible, which could explain the lack of interest, and that its difficulty put a strain on pupils out of all proportion to the benefit received. Miss Gwatkin presented a clear case against these objections, using such arguments as the following:

All girls ought to grow up reasonable beings, and many of them do not; all girls ought to acquire a knowledge of the meaning of language, and a power of using it accurately, and many of them do not. Mathematics, properly taught, will help to both ends. In all branches of Mathematics, though perhaps more especially in Geometry, it is necessary to be clear-headed.

Women are said to be inconsistent, and, moreover, to be quite unable to recognise their inconsistency. There is no place where the penalties of inconsistency are more striking than in the mathematical classroom.

Mathematics ... offers unique opportunities to the teacher for recognising and encouraging independent thought. ... Girls, perhaps, need a greater stimulus to independent thought than boys do.

Much harm is done by rating too highly a girl's power to produce - somehow, anyhow - the correct answer to a given sum, and more attention should be paid to style, and more credit given for connected and intelligible explanations, both oral and written.

A teacher of girls is, perhaps, too easily satisfied when her pupils are working steadily and conscientiously along the lines which she has laid down for them; a boy is almost certain to go off at a tangent ... routine does him less harm, because he is less susceptible to its influence. Probably one of the weaknesses of girls is that they will submit to so much dullness without resentment. ... Many girls who are apparently good workers are really mentally lazy, they reproduce, but they do not produce. A teacher needs to be alive to this danger, and to realise that it is her business to stimulate intellectual curiosity, the desire to know, and not only to know, but to find out at first hand; she has every opportunity in Mathematics, where hardly any of the work should be reproductive.

B5 In 1923, when the Consultative Committee of the Board of Education reported on differentiation of the curriculum for boys and girls (5), the Regulations for Secondary Schools provided that:

an approved course in a combination of these (domestic) subjects may for girls over 15 years of age be substituted partially or wholly for Science and for Mathematics other than Arithmetic.

The present degree of girls' inferiority in this subject should not be regarded as permanent, being due partly to unskilful teaching of an old-fashioned kind and partly to an impression among parents, which has influenced the timetable, that Mathematics is unsuitable for girls.

the fact that the girls miss the scale-drawing and 'plan and elevation' work of the handicraft course should be a reason not for doing less but for doing more of such work in mathematics, and the same remarks apply to the practical measurement which girls miss through not studying elementary physics.

In mental capacity and intellectual interests they have much in common, the range of difference in either sex being greater than the difference between the sexes. But in early adolescence the thoughts of boys and girls are turning so strongly towards their future roles as men and women that it would be entirely inappropriate to base their education solely on their intellectual similarity.

Present-day differences in the examination performance of boys and girls

B7 Today, almost all girls as well as boys study mathematics up to the age of 16, but girls are still not as successful in mathematics examinations as are boys. The proportion of entries for public examinations in mathematics by girls decreases as the level of the examination increases. At CSE, almost equal numbers of boys and girls enter (7), although the same was not true in earlier years. In O Level mathematics, only 44 per cent of the 1979 entry was from girls, while in A Level, 26 per cent of the 1979 entry was from girls. However, the position is improving; in 1968, 37 per cent of the mathematics O Level entry was from girls, and only 17 per cent of the mathematics A Level entry.

B8 Information about the relative performance of boys and girls has also been obtained from the DES Survey of 10 per cent of school leavers, discussed in Appendix 1. This data relates to pupils who left school in 1979, so that not all their examination results were obtained in the same year. At O Level, 24.5 per cent of boys who left school in 1979 had a pass in mathematics at grade A, B or C, but only 17.6 per cent of girls, and at the highest grade, grade A, more than twice as many boys (5.5 per cent) as girls (2.6 per cent) obtained this grade. This is significant for their future careers, as amongst the pupils who went on to take A Level mathematics, 91 per cent of both boys and girls who had obtained a grade A pass at O Level passed at A Level, while only 48 per cent of boys and 36 per cent of girls who had a grade C O Level pass obtained an A Level pass. However, a smaller proportion of the girls who had a grade A pass in O Level mathematics continued to A Level than did boys; the percentages were 48.6 per cent of girls as against 66.9 per cent of boys. In summary, a smaller proportion of girls than boys enter for O Level mathematics; of those who do enter, a smaller proportion of girls achieve high grades, and of those with high grades, a smaller proportion of girls than boys proceed to A Level. The result of this is that nearly three times as many boys as girls entered for A Level mathematics in 1979: 12,350 girls as against 34,670 boys. Although the overall pass rates for boys and girls were very similar (boys 73.4 per cent, girls 73.0 per cent), at the highest grade boys were more successful; 10.1 per cent of girls obtained a grade A pass, as against 15.4 per cent of boys.

B9 Although boys are more successful than girls in public examinations in mathematics, the opposite holds in English, as Appendix 1 shows. The picture which emerges is that girls underachieve in public examinations in mathematics compared with boys, while in English the position is reversed. This pattern continues into the teaching careers of men and women, so that in 1978, among the 258,811 women teachers and 180,060 men teachers in maintained schools, there were 2,484 women mathematics graduates, as against 5,264 men mathematics graduates (7), The same pattern holds among students who follow the BEd route into teaching; the percentages of 1980 entrants to subject-oriented BEd courses who took mathematics either as a single subject or in combination were: men 17.4 per cent, women 8 per cent.

Details of differences in mathematical performance

B10 In order to see how girls' performance in mathematics might be improved, it is necessary to look at details of the differences. This information can be gathered from a number of sources at different age levels. More work has been done in the USA than in Britain on sex differences in mathematics; the position in the two countries seems to be roughly comparable, and American as well as British work is therefore referred to.

B11 In the project reported in Mathematics and the 10 year old, Ward (8) tested 2,296 children in England and Wales; each item was administered to more than 550 children. Girls performed significantly better than boys on 11 items out of 91. These items were on computation with whole numbers and money, and on entirely verbal items involving naming geometric shapes and making a deduction from given verbal (non-numerical) information. Girls also did significantly better on one logic item involving mappings. Boys performed significantly better on 14 items out of 91. Four of these items were on place value, and of the other five place value items, boys did better on four. The other items on which boys did significantly better involved measurement and visual items, word problems and reversing an operation, as in 105 ÷ ? = 21. The items on which girls did significantly better were easier, with average success rate 64 per cent, as against an average success rate of 49 per cent for the items on which boys did significantly better. The items on which girls did better were also thought to be more important by their teachers, in a survey carried out as part of the project. The types of items on which boys did better at the age of 10 become more important as children get older. An understanding of place value, an ability to reason about a word problem and to reverse an operation become more important as pupils proceed towards O Level, when routine computation is less important than problem-solving ability.

B12 The first APU Primary Survey, in 1978 (9), found some differences between boys and girls in the results of its written tests at the age of eleven:

The girls' mean score is significantly higher statistically in computation (whole numbers and decimals). The boys' mean score is significantly higher statistically in three sub-categories: length, area, volume and capacity; applications of number; and rate and ratio.

B13 The first APU Secondary Survey (11) found that by the age of 15/16, the composition of the top 10 per cent of achievers was: boys 61.5 per cent and girls 38.5 per cent. However, the difference between proportions of boys and girls in the middle 10 per cent of achievers and in the bottom 10 per cent was small. Over all the written tests, boys had higher scores than girls in every sub-category, and were furthest ahead in descriptive geometry (a success rate of 7 per cent more than that for girls), rate and ratio (6 per cent) and mensuration (5 per cent). There were also considerable differences between boys and girls in the practical tests, particularly in the mass/weight topic, in which 19 per cent more boys than girls succeeded in finding the mass of one peg from a bag of equal small pegs when given only a 20 gram mass and a balance. This topic was an extension of the task on weighing used in the Primary Surveys.

B14 In 1973 and 1974, Wood (12) analysed performance on the London Board O Level Syllabus C papers. On the multiple-choice papers he found that, in addition to the fact that girls generally performed worse than boys, they were particularly weak on items on scale, pie-charts and probability, and had a poor grasp of size and of distance-time graphs. Girls were strong on sets, Venn diagrams, matrices, the real number-line or line-segments, and on straightforward vector addition. In the free-response questions, probability and geometry were much more popular among boys, and girls' relative performance was worst on a question about size and estimation. Girls in single-sex schools did rather better than girls in mixed schools, but the items on which girls from single-sex schools out-performed girls from mixed schools were on substitution, intersection of sets, reflection and matrix definition. The performance on the 'boys' items' remained low. The comparison between single-sex and mixed schools needs to be treated with some caution, as in 1973-4 many of the single-sex schools were selective, and it is therefore not clear how far like was compared with like. Wood summarises

None of the items on which girls out-performed boys required what could be termed problem solving behaviour; instead they call for recognition or classification, the supplying of definitions, application of techniques, substitution of numbers into an algebraic expression and so forth, just the kind of operations which are most susceptible to drilling.

B15 A number of similar studies have been carried out in the USA. Many of those which dealt with children of ages between 10 and 14 were surveyed by Fennema in 1974 (14).

She concluded that

Girls performed slightly better than did boys in the least complex skill (computation) ... In the 77 tests of more complex cognitive skills (comprehension, application and analysis) five tests had results that favoured girls, while 54 tests showed significant differences in favour of boys. The conclusion is inescapable that the boys of these populations learned the mathematics measured by these tests better than did the girls ...

... in overall performance on tests measuring mathematics learning ... there are no significant differences that consistently appear between the learning of boys and girls in the fourth to ninth grade ... if a difference does exist, girls tend to perform better in tests of mathematical computation, and boys tend to perform better in tests of mathematical reasoning.

B17 The testing carried out in 1964 as part of the International Study of Achievement in Mathematics (16) showed a similar pattern. In all twelve developed countries which took part in the study, the performance of boys was higher than that of girls at the age of 13, and the performance of boys was further ahead on verbal mathematical problems than on computational problems. However, there was a considerable difference between countries, the sex differences in performance being greatest in Belgium and Japan, and least in the USA and Sweden.

B18 There has been much research on the subject of differential performance between the sexes on non-mathematical tests of various types, and it is well established that males tend to perform better than females on tests of spatial visualisation, which include the ability to rotate objects in the mind and to orient oneself or other objects in space (17). Evidently, some aspects of mathematics make use of this ability, but it is not yet clear how far spatial visualisation is directly related to the learning of mathematics. In tests of verbal ability, on the other hand, the average ability of girls is greater than that of boys, but it is not known how important this ability is in the learning of mathematics. On all tests, however, the overlap between the sexes is very large, and it would be a gross distortion to expect that most boys would be better than most girls at, for example, tasks involving spatial visualisation.

Reasons suggested for differences in performance between boys and girls

B19 A number of biological theories have been put forward to explain differences between the sexes in spatial visualisation. Three of these theories concern a recessive gene on the x-chromosome, the role of sex hormones, and differences in brain lateralisation (18). The question of biological differences in either spatial visualisation or mathematical ability is not yet fully understood, but in view of the fact that differences between the sexes in mathematical attainment are more marked in some countries than in others (16), there would seem to be factors other than differences in ability in spatial visualisation which influence differences in mathematical attainment. These factors may be classified as follows: first, patterns of socialisation may be produced by child-rearing practices and peer group pressures; secondly, the expectations of schools and individual teachers may affect pupils' performance; finally, the pupils' own motivation may have significant effects on their attainment. All these factors may be expected to interact with one another in influencing the different mathematical attainments of boys and girls. These factors are now surveyed, and reference is made to research studies concerned with them.

Socialisation patterns

B20 A child-rearing practice which may have an effect on mathematical attainment is the fact that boys are given significantly more spatial and scientific toys, rather than the dolls which girls receive (17). A recent British study (19) found significant differences between the spontaneous play of boys and girls in a nursery school; girls engaged in more fantasy and creative play, while boys chose more construction play and play with sand and water. Throughout childhood, boys play more with constructional toys and take part in more physical games, both of which promote spatial awareness and problem solving activity. Boys are encouraged to be more independent, a valuable characteristic for problem solving, while girls are expected to be more passive and conformist, and to spend time helping mother around the house, rather than helping father with 'do-it-yourself' and with the car, both of which are more directly related to measurement, shape and calculation than are washing up and simple cooking and sewing. Boys seem to receive more attention, more punishment and more praise from adults, and adults respond to boys as if they find them more interesting and more attention-provoking than girls (20). Thus, boys' ideas seem to be more valued by adults, and so boys put their ideas forward. This may have some consequences for later mathematics learning in school.

B21 Peer group pressures centre on sex stereotyping, and on the view of mathematics taken by children's contemporaries. Peer group pressure increases in adolescence, and is enhanced by the influence of pop culture, the media and the teenage magazines, some of which still put forward stereotypes which confirm the restricted image of the girl who is engaged at the age of 16 or 17 and married at 20 (20). Thus, the study of mathematics may seem pointless to girls who are influenced by these pressures.

B22 Recent studies suggest that the sex-typing of mathematics is decreasing, but that male prejudice against girls' involvement in mathematics still exists, or at least girls believe it does (21). It has also been shown that mathematically gifted girls fear that their achievement will have negative consequences for their relationships with boys (22) and that girls who underachieve in mathematics see intellectual achievement as only appropriate for men (21). The evidence as to whether boys do in fact stereotype mathematics as a male domain is somewhat conflicting (23, 21). In providing opportunities for mathematically gifted pupils in Baltimore, Fox (24) found that more 13 year old boys than girls were eager to enrol in a special mathematics course, and that many girls dropped out of the course. They appeared to be afraid that their participation would have negative social consequences for them.

Factors within the school

B23 It is widely believed in the USA that the most important influence on learning and achievement in mathematics is how great an opportunity the pupil has to learn mathematics, and that boys take more advanced mathematics courses than girls do, so that they have more opportunity to learn mathematics. However, the 1978 National Assessment of Educational Progress in the USA (15) shows only very small differences in course-taking in mathematics between the sexes. This is not the case in England and Wales, where more boys than girls take O Level as against CSE courses, and where this pattern is greatly intensified at A Level. Hence, in the APU's testing of mathematical attainment in England and Wales at the age of 16, boys might be expected to score rather higher than girls, because more boys than girls take O Level courses.

B24 Mathematics is not only learned in mathematics lessons; a great deal of mathematics is used and learned in science, especially in the physical sciences, and in technical drawing. In these subjects, not only is mathematics learned and practised in a practical context, but the message is conveyed that mathematics is useful in the technical, scientific and employment worlds, particularly in the traditional worlds of boy's employment and interests. In the physical sciences and technical drawing there is a great difference between numbers of boys and girls who take these subjects.

B25 In mixed schools, in groups in which boys and girls are following the same course, there is some evidence that boys still have more opportunity to learn that do girls. Secondary school teachers have been shown to interact more with boys than they do with girls (25) and to give more serious consideration to boys' ideas (20); they also give boys more opportunity to respond to higher cognitive level questions (26). High-achieving girls have been found to receive significantly less attention in mathematics classes than do high-achieving boys (27).

B26 Even in the primary school, there are differences between the treatment of boys and girls. In a 1973 survey (28) HMI found that

... Boys engage in a wider range of crafts involving the use of a range of tools and materials leading to three-dimensional modelling and construction and the use of measurement the boys' experience further helps to familiarise them with geometrical ideas ...

B27 In primary schools, too, children are expected to behave in ways regarded as appropriate to their sex. Boys and girls are asked to help in different ways in the classroom; boys move PE apparatus and milk crates, while girls tidy up and arrange displays (29). Primary teachers see girls as sensible, obedient, hard-working and cooperative, while boys are excitable and talkative, and need more supervision and attention (20). Thus, no doubt, teachers find themselves interacting more with boys than with girls.

B28 Dweck has investigated a phenomenon she calls 'learned helplessness' (30). 'Learned helplessness' exists when a person believes that failure at a task is insurmountable, and it is accompanied by a deterioration in performance. Dweck found that some children become incompetent following a failure, while others rise to the challenge, persist and improve their performance. She found that the child's interpretation of the failure, what he thinks caused it and whether he thinks he can overcome it, predicts his response. She was able to train children to attribute their failures to insufficient effort rather than to innate inability; most of them then showed improvement in a problem solving task following a failure. In contrast, children whom she arranged to have experiences only of successful problem solving during her experiment did not improve their ability to cope with failure when it occurred. She then investigated by classroom observation (in the USA), the ways in which elementary school teachers commented on pupils' failures. She found that more of the negative feedback received by boys referred to conduct than to intellectual ability, and even feedback about their work was largely about neatness, instruction-following or style of presentation. On the other hand, much of the negative feedback girls received for their work related to its intellectual aspects. However, Fennema (31) made similar observations of pupils aged 12 in mathematics classes, and found relatively few instances in which the teacher provided feedback to either boys or girls which attributed success or failure to such causes. Fennema, however, noted that boys who were high in confidence received a great deal more attention from their mathematics teachers than did any of the girls, whether they were confident or not, or than the less confident boys. In another recent study (32), Fennema has found significant differences between boys and girls in the ways in which they themselves attribute their successes and failures in mathematics learning. People may attribute their successes or failures to relatively stable and unchanging causes such as their own ability (or lack of it), or to relatively unstable and changeable causes such as lack of effort, the difficulty or ease of the task, the unfairness of the teacher, or luck. Fennema's work confirmed, for the learning of mathematics, findings obtained by workers in other areas, that boys are more likely to attribute their successes to stable causes such as ability and their failures to unstable causes such as lack of effort, while girls attribute their successes to unstable causes such as the effort they put into their work, and their failures to stable causes such as their lack of ability.

Motivation and attitudes towards mathematics

B29 There is some evidence that boys see more clearly than do girls that mathematics will be useful in their future lives and work. Fox, surveying American research in 1977 (22), found many studies which suggested that girls are less oriented towards careers outside the home than are boys, and that the usefulness of mathematics in the traditional women's careers in business, nursing, teaching and the social services is less plain than is its usefulness in traditional men's careers. Fennema, however, has reported that in the 1978 NAEP survey (15), no significant differences were found between boys and girls in the perceived usefulness of mathematics. This suggests that American attitudes may be changing. The recent British evidence on this point is inconclusive. A survey of the attitudes of 13 year old Sheffield schoolchildren (33), found a small but significant difference in the mean scores for boys and girls on the attitude scales, with boys showing a greater liking for mathematics. It was also found that the most common reason for liking mathematics is its perceived usefulness in getting a job, or in doing a job, or in general everyday use; one boy is quoted as saying 'It is helping you to get a better job with good pay even though the lessons may be boring and confusing'.

B30 The APU has surveyed the attitudes of 11 year old children to mathematics. In the first survey (9), it was found that most children agreed with statements about the usefulness of mathematics, and that there were no sex differences. However, primary mathematics is fairly clearly related to everyday life, while the mathematics of the secondary school is less clearly oriented towards daily living, and more towards qualifications, further study, and use in science and technology. In the second survey, in 1979 (10), it was found that significantly more boys than girls believed that they usually understood a new mathematical idea quickly, that they were usually correct in their work and that mathematics was one of their better subjects. However, significantly more girls than boys confirmed that they often got into difficulty with mathematics and were surprised when they succeeded. Thus, 11 year old girls were already showing a tendency to attribute failure to stable causes such as lack of ability, while boys showed greater self-confidence in their mathematics. In their comments on individual mathematical topics, significantly more boys than girls liked and found easy topics such as measurement and geometry, while girls preferred numerical topics such as factors and multiplication tables.

B31 Many people also regard mathematics as a male domain. Weiner (20) maintains that

... mathematics is regarded by pupils of all ages both primary and secondary (and by teachers) as a subject at which boys excel.

B32 Children acquire many of their attitudes from their parents. Parents still often hold lower educational aspirations for girls than for boys, and it has been found in the USA that low levels of mathematical achievement are more easily accepted by parents of girls than by parents of boys (34). Teachers, too, may unconsciously play a part in the sex role stereotyping which reinforces children's attitudes to mathematics. It is well-known that children's books often reinforce traditional pictures of sex roles; mathematics textbooks and examination papers sometimes do the same. It has been said that in one well-known British secondary mathematics series, all the shopping is done by the female sex, while in another example, the girls knit while the boys make concrete or play football (35).

B33 Pupils' self-confidence and self-concept may affect their achievement in mathematics, and sex differences in self-confidence in mathematics have been found in several studies (21). In one study (36) most students gave lack of effort as the reason why they failed to be successful in most subjects of the curriculum. In mathematics, however, 26 per cent of girls gave their lack of ability as the reason, but only 15 per cent of boys. Fennema (32) states that females often feel inadequate about intellectual, problem solving activities, and underestimate their ability to solve mathematical problems. The APU Primary Surveys confirm that there is a comparative lack of mathematical self-confidence among girls in England at an age as early as 11. High anxiety is also associated with lower achievement in mathematics, and as schooling progresses, American girls have been found to display greater anxiety of a debilitating type. Recent work by Buxton (37) as well as the report of the study undertaken by ACACE for our inquiry, suggest that anxiety about mathematics also occurs frequently in Britain.

Evidence submitted to the Committee

B34 In view of the evidence cited earlier which shows the differences in examination success in mathematics between boys and girls in England and Wales, and the fact that there is a large volume of research evidence on sex differences in mathematics, it is surprising that little evidence was submitted to the Committee relating to this topic. However, the Committee for Girls and Mathematics pointed out to us that

The debate on standards, particularly in relation to the mathematical needs of industry, has often seemed to be directed very largely towards boys. Schools, the careers service and industry appear to have shown very little initiative in encouraging or attracting girls with ability in mathematics into some of the fields where there are shortfalls of good applicants.

Strategies for improvement

B35 The evidence reviewed in this appendix suggests a number of strategies which might improve the attitudes and achievement of girls in mathematics. Girls should be helped to realise that mathematics is as important for their daily lives, and in their future careers, as it is for boys. Investigations into the uses of mathematics in employment have shown the importance of mathematical knowledge in, for instance, the traditional women's career of nursing, while failure to obtain O Level mathematics at grade C or higher now provides a barrier to entry to teaching. In less traditional women's careers, public awareness of equal opportunities for women is increasing, and more girls are entering these careers. It will be a consequence of this development that greater demands will be made on girls' mathematics. Boys as well as girls need to be aware of the importance of mathematics for women's careers and daily lives, so that boys do not unconsciously emphasise outdated stereotypes in their expectations of girls.

B36 Teachers should ensure that girls receive additional help and encouragement in the areas of measuring, spatial and diagrammatic work and problem solving, and should ensure that girls attain a good grasp of the principles of place value. Girls should be encouraged to tackle higher cognitive level tasks, and not be content with success at low level tasks such as routine computation.

B37 Teachers need to become aware of the fact that they may unconsciously give cues to both boys and girls, and that these cues may affect not only attitudes to mathematics but also the learning of mathematics. If a teacher responds to pupils in a way which conveys the message to a boy that mathematics is important for him, that he is expected to succeed and that lack of success is due to his lack of effort, while a girl receives the message that her lack of success is due to lack of ability and that lack of mathematical ability is common and unimportant in girls, then it is not surprising if the girl gives up trying while the boy tries harder. Thus, teachers need to be consciously aware of the importance of helping girls to see their successes in mathematics as the result of their good mathematical ability, and not solely due to their hard work. Teachers also need to be well informed of the specific areas of mathematics in which girls may need additional experience and help if they are to achieve well, and of the importance of ensuring that girls are given the necessary opportunities.

B38 Authors, publishers, examiners and teachers should ensure that written material in mathematics does not reinforce the stereotyping of boys as active, exploratory problem solvers while girls are portrayed as passive helpers whose interests do not extend beyond fashion and the home. Applications of mathematics should encompass those with which girls as well as boys can identify. Teachers also need to ensure that mathematics is not presented as a male domain in the daily oral work of the classroom, as well as in written materials.

B39 In choosing their options in the secondary school, girls should be encouraged to take more subjects in which the uses of mathematics are made plain. As well as the traditional subjects of physics and technical drawing, the newer Craft, Design and Technology and computer studies can encourage a problem solving approach which is relevant to today's world.

B40 Careers guidance should make plain to girls, early in the secondary school, the qualifications which they will need for entry to various occupations, and the importance of mathematics among those qualifications. Mathematics often acts as a 'filter', whose absence as a qualification can exclude girls from many fields of employment, training and further education.

B41 Research is needed in Britain into the causes of girls' comparative underachievement in mathematics. A good deal of the research quoted above was undertaken in the USA, and it is not clear how accurate is its application in the British educational system, and in a society whose expectations are not exactly the same as those of society in the USA. It is clear, however, from the statistics of public examinations in England and Wales that, even in the last few years, many fewer girls than boys were studying mathematics at the higher levels, and that those who continued the study of mathematics did not perform as well as did boys at the highest grades.

References

(1) Special Reports on Educational Subjects: Volume 26 The teaching of mathematics in the United Kingdom HMSO, 1912.

(2) Maurice FD Queen's College, London: its objects and methods London, 1848.

(3) Kamm J How different from us Bodley Head, 1958.

(4) Final Report of the Commissioners appointed to Inquire into the Elementary Education Acts, England and Wales (the Cross Report). HMSO, 1880.

(5) Board of Education Report of the Consultative Committee on Differentiation of the curriculum for boys and girls respectively in secondary schools HMSO, 1923.

(6) Board of Education Suggestions for the consideration of teachers and others concerned in the work of public elementary schools HMSO. First issue 1905. The handbook was reissued in 1909, 1912, 1915, 1918, 1922, 1923, 1927 and 1937. There were substantial changes to the section on mathematics in 1915, 1927 and 1937.

(7) DES Statistics of education HMSO, 1978.

(8) Ward M Mathematics and the 10 year old Schools Council Working Paper 61, Evans/Methuen Educational, 1979.

(9) Assessment of Performance Unit Mathematical development. Primary survey report No. 1 HMSO, 1980.

(10) Assessment of Performance Unit Mathematical development. Primary survey report No. 2 HMSO, 1981.

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