MARGARET MORTON, BARBARA REILLY, ELIZABETH ROBINSON AND SHARLEEN FORBES

A COMPARATIVE STUDY OF TWO NATIONWIDE EXAMINATIONS: MATHS WITH CALCULUS AND MATHS WITH STATISTICS

ABSTRACT. In recent years a group at Auckland University has made an analysis of the results in the nationwide Year 12 calculus examination and the 'Equity in Mathematics Education' team in Wellington has done a similar investigation of the corresponding statistics paper. Both groups initially analysed the raw (unscaled) marks from the candidates for the 1987 and 1988 examinations on the basis of gender, school authority and school type. Of these three effects the first two were significant for the statistics examination and the last two for the calculus paper. The candidates were then divided into cohorts depending on whether they took both or just one of these mathematics papers. The effect of the number of mathematics papers a student took was highly significant, consequently the three-way analysis was repeated for each cohort. Gender was then no longer significant in either study but type and authority were significant for both analyses. An overview of the effect of gender in a question by question analysis of each examination has been included.

PURPOSE AND METHOD

This paper investigates the effect of gender on the results of two nationwide (Bursary) mathematics examinations, calculus and statistics, taken by Year 12 students. Three additional factors are also considered: whether a student sat both mathematics papers or only one, the type of school which each student attended (single sex or co-educational) and the authority of the school (private, state or integrated). By controlling for these variables, an attempt is made to determine the extent to which the gender difference found can be accounted for by the other three effects. No consideration is taken of the influence of the particular form of assessment used (long written questions in a formal nationwide examination). The results of all candidates for whom full data was available were considered. Since the scripts were marked anonymously, there was no indication to the marker of the name, gender, school or residential location of any candidate. Originally the results of the calculus and statistics examinations were analysed independently by two different research groups. This paper combines these findings in order to present this joint comparative study. Analysis of variance (ANOVA) via the standard Type III General Linear Model in the statistical computing package SAS was utilised. Unless otherwise stated the 1 percent level of significance is used. The data was initially analysed considering just three effects: school type, school authority and gender. Subsequently information was obtained as to whether candidates also sat the second possible mathematics paper. Students taking the statistics examination were then divided into two groups (those taking only the Educational Studies in Mathematics 26: 367-387, 1994. (~) 1994 Kluwer Academic Publishers. Printed in the Netherlands.

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M. MORTON ET AL.

statistics examination versus those taking both mathematics papers) and the threeway analysis of variance repeated on each subgroup. Students taking the calculus paper were subdivided in the same fashion and their results analysed similarly.

THE NEW ZEALAND BURSARIES EXAMINATION

At the end of their secondary schooling (Year 12) most New Zealand students who plan to continue to tertiary education sit a nationwide (Bursary) examination in five subjects. Prior to 1990 the Bursary examination was administered nationally by the Universities Entrance Board (UEB). This body has now been replaced by the New Zealand Qualifications Authority (NZQA). The Bursary examination is not compulsory but tertiary institutions take students' results into consideration if they apply for admission into a faculty with limited enrolment (for example, law, medicine, commerce). Students entering the Bursary examination are charged a fee and are examined in a given subject at the same time throughout the country in specially organised locations under strictly supervised conditions. Two of the subjects students may choose are 'Mathematics with Calculus' (MWC) and 'Mathematics with Statistics' (MWS). Both examinations are three hour written tests, tables of formulae are provided and calculators are permitted. For most of the candidates the material examined will have been taught and learned in the classroom. Although there is a definite syllabus for each subject, details such as difficulty, number of questions to be answered and emphasis on particular topics for each relevant paper are left to the current national examiners. New subject examiners, from different parts of New Zealand, are chosen every few years. For the years covered in this study, the multiple choice format for questions was not used. There is no attempt in this paper to measure the influence on performance of these external conditions of the examination process. The prescriptions for the MWC and MWS papers were first introduced in their current form in 1986 and there is some overlap in their pure mathematics content. Previously the two Bursary mathematics papers were known as Pure Mathematics and Applied Mathematics. The MWC paper is intended for those students who want to continue with mathematics, engineering or the physical sciences at the tertiary level whereas the MWS paper is less specialized and intended for those students wishing to pursue careers which use mathematics or statistics. The number of students entering the Bursary mathematics examinations over recent years is given in Table I. In 1989, whilst English continued to be the most popular paper, MWS ranked second followed by MWC in third position. In New Zealand the minimum school leaving age was fifteen (about Year 10), until 1992 when it was raised to sixteen, so the final year(s) of high school are optional. During the time of this study only approximately 30 percent of the students entering secondary school continued through to the final years. Consequently a large amount of 'self selection' had taken place before a student became a Bursary candidate. There appeared to be further self selection by gender with regard to choosing to sit the mathematics papers. Although there were almost

369

MATHEMATICSEXAMINATIONS

TABLE I Bursary entries by subject (UEB figures) Subject

1986

1987

1988

1989

% Change 1986-89

Maths with Statistics Maths with Calculus All subjects

6845 5682 50216

8446 (23.4) 6543 (15.2) 62320 (24.1)

10152(20.2) 7372 (12.7) 73482 (17.9)

11175(10.I) 7830 (6.2) 82431 (12.2)

63% 38% 64%

Changes in the number of entries compared to the previous year, expressed as a percent, are given in parentheses.

equal numbers of males and females in this final year of secondary schooling, only about 40 percent of B.ursary mathematics candidates were female. In 1989 the percentage of females sitting MWS compared to the total sitting Bursary over all subjects was 57 percent, the corresponding figure for MWC was 35 percent; for males the analogous percentages were 79 percent and 60 percent. This paper analyses the results for females who have made the same mathematics subject choice as males to take either one or both of these examinations. Gender differences in mathematics relating to participation, attitudes and achievement by New Zealand school children between 1970 and 1979 were summarised by Stewart (1981). Included in her paper is a sample of the results from the 1979 Bursary Pure and Applied Mathematics examinations which indicated that 33 percent of the pure mathematics and 17 percent of the applied mathematics candidates were female. Males dominated the high marks but the mean marks were not too dissimilar, favouring males by 2 percent in Pure Mathematics and 4 percent in Applied Mathematics. The 1985 Pure Mathematics paper was analysed for gender and school type effects (Reilly et al., 1987), however because of the substantial change in the syllabi for both Bursary mathematics examinations from 1986 onwards these previous results are not included in detail in this paper.

OVERVIEW OF THE 1987 AND 1988 PAPERS

The MWC results are presented for 6363 candidates in 1987 and 7159 candidates in 1988. In 1987 the mean mark was 48.0 (sd = 24.3) and in 1988 it increased to 50.4 (sd = 22.5). The overall distribution of marks is shown in the histograms in Fig. 1. Comparing the spread of the marks for these two years it appears that the 1988 exam achieved a possibly more desirable result. In 1987 there were 13 percent very low scoring (mark < 20) and 6 percent very high scoring (mark > 90) students. These percentages dropped in 1988 to 9 percent and 5 percent respectively. The 1987 MWS information represents 8070 students (mean = 52.5, sd = 19.9); in 1988 the candidate numbers increased substantially to 9961 (mean = 49.4, sd = 21.9). The histograms in Fig. 2 indicate the range of

370

M. MORTONETAL.

MAKIK3

~0

MAKK3

Fig. 1. Totalmarksfor 1987 (left) and 1988 (right) MWC examinations.

10°

e,

o

,.=

o

2D

40

60

so

ioo

o

~o

o

1o

do

~o0

MARK5

MARKS

Fig. 2. Total marks for 1987 (left) and 1988 (right) MWS examinations.

total marks. A comparison of the distribution of marks over the two years shows that although there seems to be the same proportion of candidates scoring more than 75 marks the proportion of students in the low scoring (mark < 25) group has increased and the proportion scoring a mark between 50 and 75 has decreased. The lower quartile has dropped from 35 marks in 1987 to 30 in 1988 whereas the upper quartiles in the two years are similar (65 and 64). Although any examination will vary in difficulty over different years it does appear that, for the MWS paper, the increase in number of candidates has been in the lower scoring categories.

ANALYSISBY GENDER Research in the United States by Fennema and Carpenter (1981), Armstrong (1985), Senk and Usisken (1983), Smith and Walker (1988) and Hyde et al. (1990), British studies by Wood (1976), Shuard (1982) and Bradberry (1989), studies in Singapore reviewed and discussed by Kaur (1992), research by Stewart (1981),

MATHEMATICS EXAMINATIONS

371

Pattison and Grieve (1984), Forbes et al. (1990) and Reilly and Morton (1991) in Australia and New Zealand, and comparative international studies (Garden, 1989; Robitaille, 1989; Hanna et al., 1990; Ethington, 1990) indicate a similar pattern of gender-related differences in mathematics performance. Prior to high school few, if any, differences are found. At high school and later small differences in favour of males often appeared. In particular, as the student sample became more selective the magnitude of the gender difference favouring males became larger. However it needs to be kept in mind that these differences are not consistent across countries. A number of the above mentioned authors have suggested that gender differences in mathematics performance vary as a function of the content areas tested and the nature of the task being undertaken. Consequently, because of similarities in educational systems, it is particularly interesting to compare the MWC portion of this study to some specific results of an Australian (Pattison et al., 1984) study and a New Zealand (Wily, 1986) investigation which consider examination results on similar content material. The Australian study gave an analysis of gender differences in three consecutive years (1979, 1980 and 1981) of the Victorian Higher School Certificate Pure Mathematics examination which is taken by students in their last year of secondary school. A question-by-question analysis of performance differences is included. Questions showing a gender difference in favour of females involved calculus (instructions to differentiate or integrate expressions), sketching a rational function, inverse functions, and complex numbers. Problems in analytic geometry and the application of calculus to maximum and minimum problems showed the strongest gender difference in favour of males. Wily explored some of the gender differences in the New Zealand component of the 1981 IEA Mathematics Survey. She found that at the seventh form (Year 12) level, females achieved better than males on questions which were familiar, wellpracticed, followed an easily recalled sequence of steps to obtain the solution, required algebraic or numerical accuracy, and generally did not depend on the use of a diagram. Questions where males performed better than the females had the following general characteristics; requiring a sound understanding of concepts which are developed in the upper secondary school such as 'limits' and 'integration', understanding a word problem, unfamiliar content or method necessitating a confident attack, use and interpretation of a given diagram, drawing of and subsequent use of a diagram to assist in obtaining the solution, and spatial visualization.

Mathematics with Calculus - The Gender Effect

Summary details for the 1987 examination are 3959 (62%) male enrolments (mean = 48.5, sd = 24.8) and 2404 (38%) females (mean = 47.0, sd = 23.3). For 1988, the corresponding statistics are 4538 (63%) males (mean = 51.0, sd -- 22.9) and 2621 (37%) females (mean = 49.4, sd = 21.9). In neither year is the difference in means significant. Therefore the gender difference may be

372

M. MORTON ET AL.

TABLEII Question by question analysisof the 1987 MWC examination 1987 Question

Topics

1 2 3 4 5

Algebra 100 Complexnumbers 68 Trigonometry 58 Conics 47 Trapezoidal rule, DE's 77 Derivatives, related rates 94 Integrals, area 83 Maximum/minimum 64

6 7 8

Percentage answering Male Female

Mean m a r k Significant Male Female difference?

100 73 65 45

9.6 6.8 7.6 7.2

9.1 6.2 7.2 6.6

Yes Yes Yes Yes

77

8.0

7.8

No

97 91 46

8.7 7.3 9.7

9.1 7.5 8.7

Yes* No Yes

* Difference is in favourof females.

explained largely by the other two factors, school type and school authority. While, in 1987 and 1988 (and 1985), there is a small but consistent difference in mean performance over the whole paper in favour of males this is not always the case when a question-by-question analysis is made. The full details, along with a listing of the questions, are available in the papers by Reilly et al. (1987), Morton et al. (1988 and 1989). A summary of the 1987 and 1988 question-by-question analyses is given in Tables II and III. The 'Significant difference?' column indicates whether the difference in mean scores is significant when the effect of gender alone is considered. An asterisk (*) indicates that the difference is in favour of the females, otherwise it is in favour of the males. In 1987 Question 1 was compulsory (20 marks) and students answered five of the remaining seven questions (16 marks each) to complete the paper. There was a change of both format and examiners for the 1988 paper. Whereas in previous years the compulsory question essentially covered topics in algebra, in 1988 it was increased to 35 marks and covered basic questions from the whole syllabus. Students answered five of the remaining seven questions (13 marks each) to complete the paper. The studies mentioned earlier (Pattison et al., 1984; Wily, 1986; Reilly et al., 1987) showed there is generally no significant gender difference in achievement in questions on complex numbers and straightforward differentiation. The 1987 and 1988 MWC results indicate that topics where females do as well as or better than males tend to involve complex numbers or calculus. It is interesting to note that in New Zealand these topics are first introduced in the sixth form (Year 11). Patterns of question choice by females and males were found to be similar. A comparison of percentages of students within gender groups in the 'high' and 'low' scoring categories is given in Table IV. In the 1987 MWC paper, 38 percent

373

MATHEMATICSEXAMINATIONS

TABLEIII Question by questionanalysisof the 1988 MWC examination 1988 Question 1-7 8 9 10 11 12 13 14

Topics

Percentage answering Male Female

Mean m a r k Significant Male Female difference?

All topics Algebra, complex numbers Binomialexpansion, sequences, series Conics Trigonometry Derivatives, related rates, max/rain Area, volume DE's, related rates, isoclines

100

100

21.5

20.9

Yes

77

83

7.1

7.2

No

81 74 57

82 70 58

6.2 6.2 5.2

5.7 5.6 4.9

Yes Yes Yes

86 71

88 70

5.8 5.8

5.8 5.8

No No

43

4l

5.2

5.0

No

TABLEIV Percentages of MWC students(by gender)in the 'low' and 'high' scoring categories Year 1987 MWC 1988 MWC

Low (_<20) Male

Female

High (>90) Male

Female

13.8 9.1

12.3 8.9

6.8 5.5

5.5 3.7

of the candidates and nearly 33 percent of those scoring more than 90 marks were female. In 1988, 37 percent of the candidates were female but only 28 percent scored in the 'high' category.

M a t h e m a t i c s with Statistics - The Gender Effect In marked contrast to the M W C examination there is a significant difference (1987 p < 0.02; 1988 p < 0.009) between the mean marks of the males and females in both years studied even when the factors school type and school authority are included in the analysis. Summary statistics for 1987 are 4796 (59%) males (mean = 54.0, sd = 20.5) and 3274 (41%) females (mean --- 50.4, sd = 18.9); for 1988, 5755 (58%) males (mean --- 51.1, sd = 22.4) and 4206 (42%) females (mean = 47.0, sd = 21.0). Both the examiners and the format remained the same for the 1987 and 1988

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M. MORTON ET AL.

TABLE V Question by question analysis of the 1987 MWS examination 1987 Question 1 2 3 4 5 6 7 8 9 10 11

Topics

Percentage answering Male Female

Mean mark Male Female

Significant difference?

Compulsory Algorithms Series, exponential functions Linear programming Graph, iterative method Logarithmic graphs Probability, expectation Standard deviation, Poisson Sampling, conf. int., hypothesis test Graph, hypothesis test Binomial distributions

100 42

100 32

14.9 8.9

14.4 7.3

Yes Yes

53 45

57 41

7.3 6.1

6,6 5.2

Yes Yes

46 37

45 36

9.5 9.8

8.2 8.7

Yes Yes

66

63

8.2

7.4

Yes

71

74

7.9

7.7

Yes

48

97

7.7

7.8

No

21

28

4.6

5.0

No

47

47

8.0

7.7

Yes

M W S papers. Question 1 (25 marks) was compulsory and consisted of short basic questions over all topics; the remainder of the paper was divided into two sections, one containing the statistics topics and the other the mathematics topics. Students answered five questions (15 marks each), choosing no more than three questions from each section. In both years more candidates chose to do three statistics question than three mathematics questions, this choice was even more pronounced for females. Question-by-question analyses, see Tables V and VI, show that as a group males scored significantly higher than females in almost every question. The only exception over the two years being two statistics questions in 1987 and one in 1988 where there is no significant difference and one essay question on statistics in 1988 where females scored higher than males. The lack o f females in the 'high' scoring category for the M W C paper is also evident in the M W S paper. Table VII shows the percentages of 'high' and ' l o w ' scoring candidates within each gender and year group. In the 1987 M W S examination 41 percent of candidates were female but only 26 percent of those scoring more than 90 marks were female. In 1988, 42 percent o f candidates were female while 32 percent scored in the 'high' category.

375

MATHEMATICS EXAMINATIONS

TABLE VI Question by question analysis of the 1988 MWS examination 1988 Question 1 2 3 4 5 6 7 8 9 10 11

Topics

Percentage answering Male Female

Mean mark Male Female

Significant difference?

Compulsory Series, errors, proof Algorithms, errors Linear programming Curve fitting, exponential DE Graph, maximum, iteration Poisson, binomial, hypothesis test Normal distribution, sampling Probability Descriptive statistics Essay on project

100 26 20 90

100 31 13 91

13.5 5.0 6.7 7.6

12.5 3.7 4.3 7.0

Yes Yes Yes Yes

29

24

7.9

6.7

Yes

44

41

8.1

6.7

Yes

63

62

8.7

8.3

Yes

73 32 74 34

74 32 80 37

9.4 6.8 8.2 5.1

8.8 5.5 8.1 5.5

Yes Yes No Yes*

* Difference is in favour of females.

TABLE VII Percentages of MWS students (by gender) in the 'low' and 'high' scoring categories Year

1987 MWS 1988 MWS

Low (_<20) Male

Female

High (>90) Male

Female

4.2 8.9

3.7 9.6

3.9 3.6

2.1 2.3

ANALYSIS BY TYPE OF SCHOOL In recent years, there h a v e been a n u m b e r of reports and studies c o m p a r i n g the attitudes and a c h i e v e m e n t s of students educated in single sex and co-educational schools. R e s e a r c h in Britain (Wood, 1976; Steedman, 1980) indicated that the m a t h e m a t i c a l attainment level of f e m a l e s f r o m single sex schools was higher than that o f f e m a l e s f r o m c o - e d u c a t i o n a l schools, but cautioned that their results needed to be interpreted with care. In c o m m o n with m u c h o f the research c o m p a r i n g school types, their studies w e r e hindered by the difficulty of n o n - e q u i v a l e n t group c o m p a r i s o n s due to the possibility o f a m o r e selective student intake by the single

376

M. MORTON ET AL.

sex schools. In the USA Lee and Bryk (1986), considering pupils within a private Catholic school system in both single sex and co-educational schools, were able to control for some pre-existing characteristics; they suggested that single sex schools deliver some specific advantages to their students, especially female students. Two studies of local interest considered the reorganisation of two single sex schools in the same neighbourhood, one for boys and one for girls, into two coeducational schools. The first, commissioned by a New Zealand school board in Marlborough (Battersby et al., 1983), recommended that, in view of many factors including financial costs, effect on teachers' positions, the ramifications of 'zoning' and ambivalence about the effect on the academic and social development of the students, the schools remain single sex. The second, an Australian study (Marsh et al., 1989), monitored such a reorganisation over a five year period, noting possible changes in student self-concepts, academic achievement and teachers' perceptions of advantages and disadvantages. They found that the size of the gender differences in english and mathematics performance were unaffected by the transition and for both sexes there was a clear increase in self-concepts. Prior to the transition teachers, while in favour of the idea, had suggested the advantages could be stronger for the boys. Afterwards some of them commented that they might have been influenced by the media publicity which had suggested the girls would be disadvantaged in some areas related to self-concepts and mathematics. To make a fair comparison of the relative merits of single sex and mixed classes, some researchers have looked at both types within the same school, for in this way many variables can be eliminated. A British investigation by Smith (1986), comparing groups of high school males and females who were taught in segregated and co-educational classes for five years, gave mixed results. He concluded that a great deal can be done to improve the performance of females in mathematics without recourse to segregated classes but that single sex classes were worth encouraging in the first and second years, particularly if a special scheme of work designed to meet the needs of females in mathematics were developed. Willis and Kenway (1986) argued that the evidence that single sex instruction for females as a strategy to improve their confidence, attitude and achievement in subject areas traditionally regarded as male domains is of questionable value. They claimed "the single sex strategy in its most popular manifestion is unlikely to change the educational opportunities of girls in any fundamental way because it focuses almost exclusively on changing the behaviour of girls. It neglects the much more difficult problem of changing the attitudes and behaviour of boys and teachers, and the nature of the curriculum itself". Mathematics with Calculus - The School Type Effect

In 1987 there were 3706 (58%) students (mean = 47.2, sd = 24.1) from coeducational schools and 2657 (42%) students (mean = 49.0, sd = 24.4) from single sex schools, equivalent figures for 1988 are 4223 (59%) students (mean = 49.3, sd = 22.3) from co-educational schools and 2936 (41%) students (mean = 52.0, sd = 22.7) from single sex schools. For both years this difference in means

377

MATHEMATICSEXAMINATIONS

TABLE VIII Population sizes (N), means and standard deviations (sd) for the MWC candidates grouped by gender and school type

Year

Male

Female

Co-ed

SingleSex

Co-ed

SingleSex

2269 48.I 24.5

1690 49.1 25.3

1437 45.7 23.5

967 48.9 22.9

2631 50.2 22.7

1907 52.2 23.2

1592 48.0 21.8

1029 51.7 21.8

1987 MWC

N mean sd 1988 MWC

N mean sd

is highly significant (1987 p < 0.0001; 1988 p < 0.0001). In view of the continuing debate on the relative merits of single sex versus coeducational schooling the relevant student numbers, means and standard deviations for the four groupings by gender and type are given in Table VIII. For both years females from single sex schools average at least three more marks than those at co-educational schools, the corresponding difference in male means is less.

Mathematics with Statistics - The School Type Effect

From co-educational schools 4618 (57%) students (mean = 53.0, sd = 19.6) sat MWS in 1987, from single sex schools there were 3452 (43%) students (mean = 52.0, sd = 20.4). Matching figures for 1988 are 5744 (58%) candidates (mean = 49.3, sd = 21.6) from co-educational schools and 4156 (42%) pupils (mean = 49.6, sd = 22.3) from single sex schools. Unlike the MWC results there is no significant difference between school types in the marks from the MWS candidates and the disparity in performance between females from single sex and co-educational schools is not evident. The gender by school type split, in common with the MWC results, only shows that females are not achieving as well as males in both co-educational and single sex schools. For comparison purposes the appropriate student numbers, means and standard deviations are given in Table IX.

ANALYSIS BY SCHOOL AUTHORITY

In New Zealand there are three different types of secondary school authority; state, integrated and private. The majority of schools are state schools, controlled and financed by the state, and include both single sex and co-educational schools.

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M. MORTONET AL.

TABLE IX Population sizes (N), means and standard deviations (sd) for the MWS candidates grouped by gender and school type Year

Male

Female

Co-ed

SingleSex

Co-ed

SingleSex

2635

2161

1983

1291

1987MWS

N mean sd

55.1 19.9

52.8 21.0

50.2 18.8

50.6 19.2

1988 MWS

N

3205

2511

2539

1645

mean

51.4

51.0

46.6

47.6

sd

21.9

23.0

21.0

21.0

During the time of this study most of the students in state schools came from a zoned region about the school, but a limited number of out-of-zone enrolments were allowed. Private schools receive some funding from the state and are attended by students whose parents can afford to pay the fees. Integrated schools are schools of special character, usually religious (the majority are Roman Catholic) or philosophical, that were once private and are now state funded. Many parents choose to send their children to these schools because of the time and emphasis placed on religious instruction. It is difficult to describe the socio-economic characteristics of typical schools from each of the three authority groups. All that can be said is that the majority of pupils at private schools will be from middle to high socio-economic levels. A recent study in Australia by Williams and Carpenter (1990) discussed the apparent public/private differences in educational outcomes in Australia. This is particularly relevant to the NZ situation because of the similarity between the two education systems. In Australia the three types of school authority may be identified as government (public, corresponding to state schools in NZ), Catholic (private, affiliated with the Roman Catholic church) and independent (private, affiliated with Protestant churches). Two contending explanations, 'quality education' versus 'selective-socio-economic-recruitment' for the apparent public/private differences in educational outcomes were examined. Their evidence suggested that in Australia a good part of the observed between-system difference in performance was attributable to between-system differences in student attributes, that is due to a family advantage rather than a school advantage. Possibly the same explanatory vector, 'selective-socio-economic-recruitment' versus 'quality education', may underpin the NZ results. At this stage the relevant socio-economic data is not available.

379

MATHEMATICSEXAMINATIONS

TABLE X Population sizes (N), means and standard deviations (sd) for the MWC candidates grouped by gender by school authority Year

State

Integrated

Private

5040 (79%) 48.5 24.3

713 (11%) 40.8 23.2

610 (10%) 51.5 23.9

5842 (82%) 50.8 22.7

726 (10%) 43.9 21.1

591 (8%) 54.8 21.2

1987 MWC

N mean sd 1988 MWC

N mean sd

Mathematics with Calculus - The School Authority Effect

An overview of the results for the three types of school authority is given in Table X. In both years the difference in means is highly significant (1987 p < 0.0001; 1988 p < 0.0001). Not unexpectedly students from private schools achieve the best results, those from state schools are reasonably close but there is a distinct gap between them and the results of candidates from integrated schools. For interest, and later comparison with the MWS results, candidate numbers, means and standard deviations for the six divisions grouped on the basis of authority and gender are given in Table XI. Means for males and females at integrated schools are always lower than those for students at state and private schools. The difference in means between private and state schools is more pronounced for females than for males. In fact females at private schools have the highest mean mark of all groupings.

Mathematics with Statistics - The School Authority Effect

For the MWS paper, as in the case of the MWC paper, the effect of authority is significant (1987 p < 0.0001; 1988 p < 0.0001). The number of students in each group together with the means and standard deviations are given in Table XII. The mean for students from integrated schools is always lower than those for other authorities but there are smaller differences between the means for students from state and private schools. Compared to the MWC results the means for the three authorities are much closer. Students in state schools appear to be a less homogeneous group usually having a larger variation in their scores. The two-way interaction between gender and authority is significant (1987 p < 0.01; 1988 p < 0.002). Table XIII contains the relevant numbers, means and standard deviations for each of the six groupings by gender and school authority. An examination of this interaction shows the difference in means between the three

380

M. MORTONET AL.

TABLE Xl Population sizes (N), means and standard deviations (sd) for the MWC candidates grouped by school authority and gender Year

Male State

Integrated

Private

Female State

Integrated

Private

3185 49.2 24.7

409 41.2 24.7

365 51.0 24.6

1855 47.4 23.5

304 40.1 20.9

245 52.2 22.9

3757 51.5 23.0

421 44.8 21.8

360 52.7 21.7

2085 49.5 22.0

305 42.6 20.0

231 58.0 19.9

1987MWC

N mean sd 1988 M W C N

mean sd

TABLE XlI Population sizes (N), means and standard deviations (sd) for the MWS candidates grouped by school authority Year 1987 M W S N

mean sd 1988 M W S N mean

sd

State

Integrated

Private

6387 (89%) 53.0 20.0

842 (10%) 48.5 19.4

841 (10%) 53.1 19.7

7970 (80%) 49.7 22.1

971 (10%) 45.6 21.2

959 (10%) 51.3 20.9

authorities is not always the same for males and females. As for MWC, males in state and private schools perform at a similar level, whereas females at private schools have a higher mean mark than females in other schools. Unlike MWC, there is little difference in the mean marks of females from state and integrated schools. These results may be confounded by the effect of school type. Private schools are mainly single sex schools and the above comparison is not with single sex state schools.

ANALYSIS BY NUMBER OF MATHEMATICS PAPERS TAKEN T r a d i t i o n a l l y f e m a l e s h a v e n o t c h o s e n to s t u d y m a t h e m a t i c s as m u c h as h a v e m a l e s in a d v a n c e d s e c o n d a r y s c h o o l classes, thus o n e o f t e n c o m p a r e s a p o p u l a t i o n o f

MATHEMATICSEXAMINATIONS

3 81

TABLEXIII Population sizes (N), means and standard deviations (sd) for the MWS candidates grouped by school authority and gender Year

Male State

Integrated Private

Female State

Integrated Private

3807 54.9 20.4

496 48.0 20.1

497 53.1 20.4

2584 50.2 19.0

346 49.1 I8.4

344 53.0 18.7

4601 51.9 22.4

529 45.8 22.3

586 50.7 21.5

3369 46.6 21.2

442 45.3 19.8

373 52.2 19.9

1987 MWS

N mean sd 1988 MWS

N mean sd

males who have spent more time studying mathematics with a population of females who have studied less. Comments on the different amounts of mathematics studied are included in Fennema (1979), Fennema and Carpenter (1981), Wily (1986), and the British Royal Society Report, 'Girls and Mathematics' (1986). In New Zealand all students who elect to study Bursary mathematics will have studied the same amount of mathematics until the end of the previous year (Year 11). Splitting the Bursary candidates into divisions according to how many mathematics papers they took offered a good opportunity to study the impact of an additional mathematics paper at Year 12. The outcome of analysing their 1988 marks on this basis in the M W C study gives a mean of 43.9 (sd = 22.2, N = 1630) for the group who took only MWC, and a mean of 52.4 (sd = 22.3, N = 5529) for those students who took two mathematics papers. Analogous results for the M W S group show an even more striking difference between the one and two paper groups. The mean for MWS students who took one paper was 37.3 (sd -- 17.6, N = 4346) and for those who took two papers 58.8 (sd = 20.3, N -- 5615). (The discrepancy in the numbers given of students taking both papers is because each research group acted independently in omitting a student's record on the basis of incomplete data.) Clearly the effect of a student taking both papers instead of just one is highly significant (MWS p < 0.001; M W C p < 0.0001). Investigation of the 1987 MWS and MWC results gave similar findings to those of 1988 and are not presented. Students sitting both papers have an advantage over those sitting only one because of the overlap in the pure mathematics section and the greater depth studied in these topics in MWC. This is especially true if the MWS paper has a large calculus content in the statistics questions. There is the further possibility that it is those students who really like mathematics who chose to do both papers, consequently one would expect them to do well.

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As mentioned earlier, the MWS paper is intended for a broader range of pupils than MWC. Students of medium to high ability who intend to continue with mathematics at the tertiary level are more inclined to take just MWC or both the mathematics papers. Consequently the group taking just MWS contains a disproportionally larger number of students of lower ability. This is reflected in the fact that the difference in means between the one paper and two paper groups for MWS is double that for MWC. Since the difference in achievement between the candidates taking both mathematics papers and those taking just one was so great, particularly in the MWS paper, it was felt appropriate to consider the two groups separately within each examination. Consequently the MWC candidates were divided into two groups, those sitting just MWC and those sitting both MWC and MWS, and the effect of gender, school type and school authority reconsidered within each of these two cohorts. The MWS candidates were similarly divided into two groups and the same analysis repeated. The outcome of this further analysis, allowing for number of Bursary mathematics papers sat, is now presented.

Analysis by Gender In each of the two possible groupings of students by gender, within the one paper and two paper bands, there is no significant difference by gender in the means. Whereas one would expect this for the MWC candidates, since gender is not significant when school type and authority are controlled for, this shows that if number of mathematics papers is also allowed for, the gender differences in the MWS paper are no longer significant. Specific details for the MWC examinees are: one paper (females - mean = 43.2, sd = 21.6, N = 730: males - mean = 44.5, sd = 22.7, N = 900), two papers (females- mean = 51.9, sd = 21.5, N = 1891: males - mean = 52.6, sd = 22.7, N = 3638). Corresponding numbers for MWS are: one paper (females - mean = 37.7, sd = 16.9, N = 2296: males mean = 36.9, sd = 18.4, N = 2050), two papers (females - mean = 58.3, sd = 19.9, N = 1910: m a l e s - m e a n = 59.0, sd = 20.5, N = 3705). For both males and females, within each examination, the difference in performance between the one and two paper groups is much greater than that within each gender group. If we examine the percentages of males and females doing both MWS and MWC or one paper only, it is apparent that a smaller group of females is opting to do both mathematics papers in the seventh form (Year 12). In 1988 nearly 11,500 candidates attempted one or both of the Bursary mathematics papers, while 55 percent of the males sat both papers only 38 percent of the females chose to do this. For males the percentage sitting both papers is much higher than the percentage sitting just MWS, the reverse situation is true for females. Possibly this explains the apparent gender differences in performance in the MWS examination when only school type and authority are controlled for.

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Analysis by School Type The effect of school type is significant in both the M W C study (one paper, p < 0.0001; two papers, p < 0.001) and in the M W S groups (one paper, p < 0.05; two papers, p < 0.01) with students from single sex schools doing significantly better than those from co-educational schools. Relevant information for the M W C study is: one paper (co-educational - mean = 40.3, sd = 21.3, N = 766: single sex - mean = 47.1, sd = 22.5, N = 864), two papers (co-educational mean = 51.3, sd = 22.1, N = 3457: single sex - mean = 54.0, sd = 22.5, N = 2072). F o r M W S the details are: one paper ( c o - e d u c a t i o n a l - mean = 36.3, sd = 17.0, N = 2264: single sex - mean = 38.5, sd = 18.3, N = 2047), two papers (co-educational - mean = 57.7, sd = 20.2, N = 3480: single sex mean = 60.5, sd = 20.3, N = 2109). For the M W S study, this is the first time that type o f school is significant but the differences are small compared to the authority effect. In both single sex and co-educational schools a lower percentage of females than males is opting to take both Bursary mathematics papers. In 1988 approximately 5,000 candidates from single sex schools and 6,500 from co-educational schools took one or both Bursary mathematics papers. In single sex schools 34 percent o f the female candidates attempted both papers compared to 47 percent o f the males. The corresponding numbers for co-educational schools are 42 percent o f the females and 62 percent of the males. It does need to be noted that higher percentages o f both females and males from co-educational schools do two papers. This m a y imply that the significant difference in means in favour of single sex schools is an artifact of a different selection of examination candidates.

Analysis by School Authority The effect of school authority is significant in both the M W C study (one paper p < 0.0001; two papers p < 0.0001) and in the M W S groups (one paper p < 0.01; two papers p < 0.01) with the candidates from integrated schools scoring consistently lower. For both the one paper and two paper groups candidates from private schools achieved significantly better than those from state schools. Pertinent numbers for the M W C students are: one paper (private - m e a n . = 49.9, sd = 21.4, N = 173: state - mean = 44.4, sd = 22.4, N = 1248: integrated mean = 36.2, sd = 19.8, N = 209), two papers (private - mean -- 56.8, sd = 20.8, N = 418: state - mean = 52.6, sd = 22.5, N = 4594: integrated - mean = 47.0, sd = 20.8, N --- 517). For M W S the details are: one paper (private - mean = 42.7, sd = 18.5, N --- 562: state - mean = 36.8, sd --- 17.5, N = 3298: integrated - mean = 34.3, sd = 16.3, N = 451), two papers (private - mean = 63.5, sd ---- 17.9, N = 397: state - mean = 58.7, sd = 20.4, N = 4672: integrated - mean = 55.4, sd = 20.1, N = 520). These differences are usually larger than those due to type o f school (viz. single sex versus co-educational). Unlike the gender and school type effects, which were moderated when students were further divided according to the number of papers they took, the school -

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authority effect remains strong. This seems to either add more evidence to the 'selective-socio-economic-recruitment' argument mentioned earlier or may indicate the schools own influence in the paper selection process. It is however necessary to keep in mind the very small sample numbers from both private and integrated schools compared to those from state schools.

SUMMARY

Since the introduction of the new Bursary mathematics syllabii in 1986 the growth of students entering MWS has kept pace with the increase in entries in the Bursary examination overall; but the rise in MWC entries has been less. Theoretically students have free choice about which of these mathematics papers they take, but not all schools are able to offer both and others dictate to those students wanting to select a single paper which one it must be. The increased numbers in MWS have been substantially greater for females than for males but still, in 1989, females were only 42 percent of the MWS candidates; the percentage of females taking MWC has remained more constant at about 37 percent. For the two years investigated, in both the MWC and MWS papers, the mean mark of the male candidates is higher. It is only for the MWS candidates that this gender difference is significant. A question by question analysis of the MWC paper shows some types of problems where females do as well as males for both years, for the MWS paper no such trend emerges. In all the examinations studied the top 25 percent of candidates contains a higher proportion of males than females. However, while the bottom 25 percent of candidates generally contains a higher proportion of females, males dominate the very low (mark < 10) range. This greater variation in males' marks may reflect different pressures for males and females to continue in mathematics. The mean performance of students from integrated schools is always lower than that of students from either state or private schools; students from private schools usually achieve better than those in state schools. Possibly this is a reflection of the differing socio-economic background of students in different school authorities. The effect of school type on the two examinations is inconsistent in that it is significant for the MWC students but not for the MWS students. Whether or not a candidate also took the second Bursary mathematics paper seems to be the major factor affecting performance. For the MWS candidates, when the one paper and two paper cohorts are analysed separately, gender differences in performance almost disappear. School authority continues to be significant and for the first time school type becomes significant. In contrast, within the MWC examination, school type and school authority remain significant. Regardless of the effect being considered the group doing both papers performed significantly better than either of the groups doing only one paper.

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ACKNOWLEDGEMENTS All four authors are indebted ot their co-researchers Alan Lee, John Pemberton and Ivan Reilly at Auckland University and Thora Blithe and Megan Clarke at Victoria University in Wellington. Both research groups acknowledge the assistance of the Secretary and staff of the Universities Entrance Board (now the New Zealand Qualifications Authority) in giving them access to Bursary candidate information and their results. The Auckland group acknowledges the University of Auckland Research Committee for a succession of grants enabling them to analyse the data for the M W C paper. The Wellington team are grateful to the Department of Education for a grant which enabled them to obtain both the results presented here and also carry out a number o f other research projects relevant to mathematics education (Forbes et al., 1990). Opinions expressed in this paper are those of the authors, and do not necessarily represent the views of any other group or person. The authors wish to acknowledge the constructive comments o f the referee which have enhanced the presentation of the results. Space considerations have required the omission of a series of box plots and tables. The interested reader can write to one of the first two authors for a copy of the fuller version o f the paper.

REFERENCES Armstrong, J. M.: 1985, 'A national assessment of participation and achievement of women in mathematics', in S. Chipman, L. Brush and D. Wilson (eds.), Women and Mathematics: Balancing the Equation, Lawrence Erlbaum Associates, New Jersey, pp. 59-94. Battersby, D., Cocklin, B. and Vincent, C.: 1983, Beyond the Crossroads: An Investigation into Single-sex and Co-educational Secondary Schooling in Blenheim, New Zealand, Department of Education, Massey University, Palmerston North, New Zealand. Bradberry, J.: 1989, 'Gender differences in mathematical attainment at 16+', Educational Studies 15(3), 301-314. British Royal Society: 1986, Girls and Mathematics, A Report by the Joint Mathematical Education Committee, The Royal Society and The Institute of Mathematics and Its Applications,London. Ethington, C.: 1990, 'Gender differences in mathematics: An international perspective', Journal for Research in Mathematics Education 21(1), 74-80. Fennema, E.: 1979, 'Women and females in mathematics', Educational Studies in Mathematics 10, 389-401. Fennema, E. and Carpenter, T.: 1981, 'Sex-related differences in mathematics: Results from national assessment', Mathematics Teacher 74(7), 554-559. Forbes, S., Blithe, T., Clark, M. and Robinson, E.: 1990, Mathematics for AU? Summary of a Study of Participation, Performance, Gender and Ethnic Differences in Mathematics in New Zealand Secondary Schools, A Report to the Ministry of Education, Wellington,New Zealand. Garden, R.: 1989, 'Students' Achievements: Population B', in D. Robitaille and R. Garden (eds.), The lEA Study of Mathematics IL' Contexts and Outcomes of School Mathematics, Pergamon Press,

NSW, Australia, pp. 126-152. Hanna, G., Kundiger, E. and Larouche, C.: 1990, 'Mathematical achievement of grade 12 girls in fifteen countries', in L. Burton (ed.), Gender and Mathematics: An International Perspective, Cassell, London, pp. 87-97. Hyde, J., Fennema, E. and Lamon, S.: 1990, 'Gender differences in mathematics performance: A meta-analysis', Psychological Bulletin 107(2), 139-155.

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Kaur, B.: 1992, 'Gender and mathematics: The Singapore perspective', paper presented at the Seventh International Congress on Mathematical Education, Quebec City, Canada. Lee, V. and Bryk, A.: 1986, 'Effects of single-sex secondary schools on student achievement and attitudes', Journal of Educational Psychology 78(5), 381-395. Marsh, H. W., Owens, L., Marsh, M. R. and Smith, I. D.: 1989, 'From single-sex to coed schools', in G. C. Leder and S. N. Sampson (eds.), Educating Girls: Practice and Research, Allen and Unwin, Australia, pp. 144-157. Morton, M., Pemberton, J., Reilly, B., Reilly, I. and Lee, A.: 1988, 'Is the gender gap in bursaries mathematics with calculus narrowing?', NZ Mathematics Magazine 25(4), 3-21. Morton, M., Pemberton, J., Reilly, B., Reilly, I. and Lee, A.: 1989, 'Bursaries mathematics with calculus revisited', NZ Mathematics Magazine 26(3), 17--40. Pattison, P. and Grieve, N.: 1984, 'Do spatial skills contribute to sex differences in different types of mathematical problems', Journal of Educational Psychology 76(4), 678-689. Pattison, P., Holton, D. and Gordon, R.: 1984, 'Describing sex differences in mathematics', preprint. ReiUy, B., Reilly, I., Pemberton, J. and Lee, A.: 1987, 'Gender differences in performance in bursaries pure mathematics', NZMathematics Magazine 24(1), 19-33. Reilly, B. and Morton, M.: 1991, 'Performance in a nationwide mathematics examination at teriary entrance level', Contributions to the Sixth International Conference on Gender and Science and Technology 1, 301-309. Robitaille, D.: 1989, 'Students' achievements: Population A', in D. Robitaille and R. Garden (eds.), The lEA Study of Mathematics II: Contexts and Outcomes of School Mathematics, Pergamon Press, NSW, Australia, pp. 102-125. SAS Institute Inc.: 1989, SAS/STAT User's Guide, Version 6, Fourth Edition, Volume 1, Cary, NC: SAS Institute Inc., 943 pp. Senk, S.andUsiskin, Z.: 1983, 'Geometryproofwriting: A new view ofsex differences in mathematics ability', American Journal of Education 91, 187-201. Shuard, H.B.: 1982, 'Differences in mathematical performance between girls and boys', in W. H. Cockcroft (chairman), Mathematics Counts, Her Majesty's Stationery Office, London, pp. 273-287. Smith, S.: 1986, Separate Tables?: An Investigation into Single Sex Setting in Mathematics, Manchester Equal Opportunities Commission. Smith, S. and Walker, W.: 1988, 'Sex differences in New York State Regents Examination: Support for the differential course taking hypothesis' , Journalfor Research in Mathematics Education 19, 81-85. Steedman, J.: 1980, Progress in Secondary Schools: Findingsfrom the National Child Development Study, National Children's Bureau. Stewart, C.: 1981, Sex Differences in Mathematics, Research Report Series No. 4, Research and Statistics Division, Department of Education, Wellington, New Zealand. Williams, T. and Carpenter, P.: 1990, 'Private schooling and public achievement', Australian Journal of Education 34(1), 3-24. Willis, S. and Kenway, J.: 1986, 'On overcoming sexism in schooling: To marginalize or mainslream', Australian Journal of Education 30, 132--49. Wily, H.: 1986, 'Women in mathematics: Some gender differences from the lEA survey in New Zealand', NZ Mathematics Magazine 23(2), 39-49. Wood, R.: 1976, 'Sex differences in mathematics achievement at GCE ordinary level', Educational Studies 2(2), 141-160.

Margaret J. Morton Department o f Mathematics and Statistics University o f Auckland Private Bag 92019 Auckland N e w Zealand

MATHEMATICS EXAMINATIONS

Barbara J. ReiUy Student Learning Unit~Departmentof Mathematics and Statistics University of Auckland Private Bag 92019 Auckland New Zealand Elizabeth Robinson Community Health Department, Medical School University of Auckland Private Bag 92019 Auckland New Zealand Sharleen Forbes Monitoring and Evaluation Ministry of Maori Development Private Bag Wellington New Zealand

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