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 1 Statistical information
[pages 246 - 272]

A1 This appendix contains a selection of the considerable amount of statistical information which has been made available to the Committee from a number of sources. We are most grateful to all who have helped us. We acknowledge in particular the assistance we have received from DES Statistics Branch and the Universities Statistical Record; both have, at our request, devised and carried out additional investigations. Some reordering, reclassification and amalgamation of the data has been undertaken by members of the Committee and its secretariat.

List of tables

1 School population
2 Numbers of school leavers
3 O Level grades in Mathematics: as percentages of all leavers
4 CSE grades in Mathematics: as percentages of all leavers
5 'Best' grades in Mathematics: as percentages of all leavers
6 O Level grades in Additional Mathematics: by age
7 Sixth form pupils studying Mathematics at A Level
8 A Level grades in Mathematics: sixth form pupils
9 A Level Mathematics grades related to O Level grades of the same pupils
10 A Level passes in Mathematics: as percentages of all leavers
11 Comparison of O Level and CSE results in Mathematics and English: as percentages of all leavers
12 Comparison of O Level and CSE results in Mathematics and English: cumulative percentages
13 Comparison of grades in Mathematics and English: by sex, as percentages of all leavers
14 Average class sizes: selected subjects
15 DES 1977 Survey of secondary school staffing: composition of sample
16 Mathematics teaching in maintained secondary schools: by levels of qualification of teachers
17 Estimated percentages of mathematics curriculum 'suitably' staffed
18 Percentage of mathematics curriculum 'suitably' staffed: schools included in the sample
19 Deployment of 'unsuitably' qualified teachers in comprehensive schools: by year groups
20 Full-time teachers in maintained nursery, primary and secondary schools
21 Full-time teachers in maintained schools who are mathematics graduates: by sex and phase
22 Rates of leaving teaching: by age and sex
23 Trained and untrained mathematics graduates: entry and wastage, by age
24 Entry to Postgraduate Certificate in Education (PGCE) courses
25 Qualifications of students entering PGCE courses who are taking main method courses in mathematics
26 Entry to BEd courses in 1980
27 Entrants to first degree or first degree and first diploma courses: numbers with A Level mathematics
28 Entrants to first degree or first degree and first diploma courses: qualifications in mathematics
29 Entrants to first degree or first degree and first diploma courses: as percentages of all entrants with stated qualification
30. Entrants to first degree or first degree and first diploma courses: as percentages of entrants to each subject group with stated qualification
31 Destinations of graduates in mathematical studies
32 Type of permanent home employment: graduates in mathematical studies
33 Services to management and financial work: graduates in mathematical studies
34 Graduate employment in computer programming: by subject group

Pupil numbers

A2 The figures in Table 1 are based on age in September. In broad terms, 'primary' refers to pupils aged 5 to 11 and 'secondary' to pupils aged 11 to 16, together with those who elect to remain in full-time education in schools after the age of 16. The secondary projections therefore make assumptions about staying-on rates. Schools include sixth form colleges but not tertiary colleges, which operate under FE regulations.

Table 1 School population

Examination performance of pupils in schools

A3 The only information which is available about the examination performance of pupils in schools on a national scale comes from the annual 10 per cent survey of school leavers in both maintained and independent schools which is carried out by the DES and Welsh Office. This survey collects information about all school leavers whose birthdays fall on the 5th, 15th or 25th day of each calendar month and includes details of their examination entries and results.

A4 Because the information relates to pupils who leave school in any one year, the examination results which are recorded may have been obtained over a period of years before leaving school. The figures do not therefore relate to a complete year group. However, the relativities between the number of leavers in any year, the number of examination entries in that year and the size of the year group have remained reasonably constant for the years 1977 to 1979. We therefore believe that the picture displayed in the tables which follow is unlikely to differ significantly from the picture which would have emerged if it had been possible to obtain comparable information for a complete year group.

A5 The information for all three years relates to both maintained and independent schools in England and Wales. It refers to pupils in schools (including sixth form colleges) but not to students in FE or tertiary colleges. The examination results of this latter group are therefore included only up to the time at which they left school. In very general terms this means that most of their O Level and CSE results are included but that their A Level results are not. In cases in which a school leaver has entered for the same examination more than once he is credited with the highest grade obtained. 'Mathematics' includes Statistics and Computer Studies when these are taken as separate subjects.

A6 The information displayed is for the whole population, derived by multiplying up from the 10 per cent sample surveyed, and is therefore subject to sampling error. In cases where the numbers are large the true figures are unlikely to differ from those given by more than half of one percentage point. In tables which relate to smaller numbers, the error may be greater, up to 3 per cent.

Table 2 Numbers of school leavers

Table 3 O Level grades in Mathematics: as percentages of all leavers

Table 4 CSE grades in Mathematics: as percentages of all leavers

A7 Some pupils attempt both O Level and CSE examinations in mathematics, either in the same year or in different years. An aggregation of the O Level results given in Table 3 and the CSE results given in Table 4 does not therefore give a picture of the overall situation. Table 5 amalgamates the O Level and CSE results obtained by school leavers in such a way that each pupil appears only once, according to his 'best' grade. In the case of leavers whose highest grades included both O Level grade A, B, C or O Level pass and CSE grade 1 we have taken account of the O Level result and discounted the CSE result. We have grouped together those whose 'best' grade was either O Level grade D or CSE grade 2 and those whose 'best' grade was either O Level grade E or CSE grade 3. 'Best' grades have been defined in the order presented in Table 5; for example, if a pupil obtained O Level grade E and CSE grade 2 it is the CSE grade which has been included, whereas if a pupil obtained O Level grade E and CSE grade 4 it is the O Level grade which has been included. We emphasise that these steps have been taken solely for the purpose of drawing up the table and should not be interpreted in any other sense. In the table, 'higher grades' include O Level grades A, B or C, O Level pass and CSE grade 1; 'lower grades' include O Level grades D and E and CSE grades 2 to 5.

Table 5 'Best' grades in mathematics: as percentages of all leavers

A8 The table which follows analyses examination results in Additional Mathematics at O Level or AO Level according to the age at which the best result was obtained. 'By age 16' means during or before the school year in which the pupil reached the age of 16; 'at age 17', during the school year in which the pupil reached the age of 17; 'at age 18 or over', during or after the school year in which the pupil reached the age of 18. Thus, in general, 'by age 16' refers to results obtained in the fifth form, or earlier; 'at age 17' to results obtained in the first-year sixth; 'at age 18 or over' to results obtained in the second year sixth or later. The proportion of leavers who had attempted the examination at some stage increased from 3.5 per cent in 1977 to 3.7 per cent in 1979 but the overall pattern of entry and success was similar for all three years. We therefore give the figures for leavers in 1979 only.

Table 6 O Level grades in Additional Mathematics: by age

Table 7 Sixth form pupils studying Mathematics at A Level

Table 8 A Level grades in Mathematics: sixth form pupils

Table 9 A Level Mathematics grades related to O Level grades of the same pupils

Table 10 A Level passes in Mathematics: as percentages of all leavers

Comparison of examination results in mathematics and English

A9 Tables 11, 12 and 13 compare the O Level and CSE results obtained by pupils in mathematics/arithmetic with those obtained in English. The 'best' grades in English have been defined in the way which is explained in paragraph A7. If more than one English subject was attempted, the overall 'best' grade has been used.

Table 11 Comparison of O Level and CSE results in Mathematics and English: as percentages of all leavers

Table 12 Comparison of O Level and CSE results in Mathematics and English: cumulative percentages

Table 13 Comparison of grades in Mathematics and English: by sex, as percentages of all leavers

Table 14 Average class sizes: selected subjects

Qualifications of those who teach mathematics

A10 Tables 15 to 18 are based on information obtained from the Survey of Secondary School Staffing carried out by the DES in 1977.

Table 15 DES 1977 Survey of Secondary School Staffing: composition of sample

A11 Some of the analysis displayed in this section is restricted to the schools in the sample. In other cases, estimated national totals have been derived by using 'weighted multipliers' which take account of the percentage of schools of each type in the sample in relation to the total numbers of such schools in the country. In categorising the 11/12 to 18 schools by size, those with up to 800 pupils were regarded as small, those with 801 to 1200 pupils as medium and those with more than 1200 pupils as large. Information about the staffing of middle schools is given in paragraph A15 and is not included in Tables 16 to 18.

A12 The four categories of qualification of teachers to which we referred in paragraph 625 [in chapter 13] were defined as follows:

'Good'

• Trained graduates, or equivalent, with mathematics as the first, main or only subject of a degree course.
• Bachelors of Education (BEd) with mathematics as a main specialist subject.
• Teachers whose general qualifications were of either of these types with mathematics as a subsidiary subject provided their main specialism was in a related subject (see Note below).

• Trained graduates, graduate equivalents, or BEd with mathematics as a second or subsidiary specialism if their first subject was not related.
• Untrained graduates with mathematics as first, main or only subject.
• Teachers holding the Certificate in Education, having followed a secondary course in which mathematics was their first, main or only specialism.
• Teachers with no initial mathematical qualifications who had a further qualification resulting from a course of at least one year in which mathematics was the main subject.

• Teachers holding the Certificate in Education having followed a secondary course with mathematics as a second or subsidiary subject provided their first or main subject was related.
• Teachers holding the Certificate in Education having followed a Junior or Junior/Secondary course with mathematics as their first or main subject.
• Teachers in the immediately preceding category with subsidiary mathematics provided their main subject was related.
• Graduates in any subject provided their course included a related subject.

• Qualified teachers without any recorded mathematics and not covered by any previous specification.
• Teachers holding the Certificate in Education with mathematics subsidiary to an unrelated subject.
• Teachers without any initial qualification who possessed a further qualification which did not lead to graduate status and in which mathematics was not the main subject.

A13 Tables 16 to 19 examine the distribution of all mathematics teaching between teachers in the qualification categories defined above. Numbers of periods have been standardised to a 40 period week.

Table 16 Mathematics teaching in maintained secondary schools: by levels of qualification of teachers

Table 17 Estimated percentages of mathematics curriculum 'suitably' staffed

A14 The percentages and totals in Table 16 conceal the extensive variation between individual schools of the same type. One aspect of this is shown in Table 18.

Table 18 Percentage of mathematics curriculum 'suitably' staffed: schools included in the sample

Table 19 Deployment of 'unsuitably' qualified teachers in comprehensive schools: by year groups

A15 Analysis of the qualifications of those teaching mathematics to pupils aged 11 or over in the middle schools for pupils aged 9 to 13 which were included in the sample shows that, according to the categories set out in paragraph A12, 6 per cent of the teaching was by teachers whose qualification was 'good', 17 per cent by teachers whose qualification was 'acceptable', 15 per cent by teachers whose qualification was 'weak' and 62 per cent by teachers whose qualification was 'nil'. As we point out in paragraph 629 [in chapter 13], it could be argued that, for teachers in middle schools, some of the qualifications classified as 'weak' on the ground that training had been for a younger age group should be regarded as 'acceptable'. If all the qualifications classified as 'weak' were to be regarded as 'acceptable', it would be the case that 38 per cent of the teaching of mathematics to pupils aged 11 or more would have been by 'suitably' qualified teachers, and 62 per cent by teachers whose qualification was 'unsuitable' .

Supply of teachers in maintained schools

A16 Precise definition of the term 'mathematics graduate' is not possible; for example, the Universities Statistical Record lists 63 different degree courses under the main heading of 'mathematics'. (It is this group of courses which we have called 'mathematical studies' in the main body of the report; see note to paragraph 150 [in chapter 4].) Published information about numbers of teachers who are mathematics graduates is therefore difficult to reconcile because of the differing definitions which have been used to compile the information. For this reason Tables 20 to 23 which follow do not all show the same total of mathematics graduates. When recording the qualifications of graduates, DES records show education as the first subject of qualification of all holders of the BEd degree. For holders of BEd who have taken mathematics as a main subject, mathematics appears as the second subject of qualification. In Tables 20 to 23, 'all graduates' include holders of the BEd degree, but references to graduates who have mathematics as the only or first subject of their degree do not include holders of BEd.

Table 20 Full-time teachers in maintained nursery, primary and secondary schools

A17 Table 21 analyses mathematics graduates by the sector in which they are teaching and displays the number of men and women who make up the total. The figures include those who have mathematics named as the only subject of their degree or who have mathematics named as the first of two subjects in their degree. Consequently the numbers shown in Table 21 are somewhat larger than those in Table 20.

Table 21 Full-time teachers in maintained schools who are mathematics graduates: by sex and phase

A18 Table 22 gives details by age and sex of the rates at which mathematics graduates leave teaching, and compares these with the rates of leaving for all graduate teachers. Table 23 compares entry and wastage rates for trained and untrained mathematics graduates. Both tables relate to graduates who have mathematics as the only subject of their degree or as the first subject of a degree which does not include science. The tables have been compiled from information which is supplied to the DES each year by local education authorities.

Table 22 Rates of leaving teaching: by age and sex

Table 23 Trained and untrained mathematics graduates: entry and wastage, by age

Entry to initial training

A19 Tables 24 to 26 provide information about entry to initial training courses.

Table 24 Entry to Postgraduate Certificate in Education (PGCE) courses

Table 25 Qualifications of students entering PGCE courses who are taking main method courses in mathematics

Table 26 Entry to BEd courses in 1980

Entrants to universities in England and Wales

A20 Tables 27 to 30 which follow are derived from information supplied by the Universities Statistical Record. They refer to entrants to full-time and sandwich first degree or first degree and first diploma courses at universities in England and Wales who had home fee-paying status and whose entry qualification was based on A Levels. Over the years 1973-1979 these constituted about 92 per cent of all entrants to these courses at universities in England and Wales.

Table 27 Entrants to first degree or first degree and first diploma courses: numbers with A Level mathematics

A21 Tables 28 to 30 provide details of the distribution of the entrants defined in paragraph A20 between categories of degree courses, and also details of their mathematical qualifications. The categories used are

• Engineering and technology, including the various types of engineering, mining, metallurgy, surveying, other technologies and combinations of these.
• Physical sciences, including mathematics/physics, physics, chemistry, general and combined physical sciences.
• Mathematical studies, including mathematics, statistics, computer science and combinations of these; and combinations of mathematics with other subjects.
• Medical and dental, including pre-clinical medicine and dentistry, and paramedical courses such as pharmacy.
• Biological sciences, including veterinary and agricultural studies and forestry.
• Other sciences, including combinations of physical and biological sciences and of various sciences with other subjects.
• Business studies, including business and management studies, economics and accountancy.
• Geography.
• All other subjects.

A Level qualification in Mathematics denotes a pass in at least one of the subjects Pure Mathematics, Applied Mathematics, Pure and Applied Mathematics (including Mathematics with Statistics), Further Mathematics; double-subject Mathematics denotes a pass in Applied Mathematics or Further Mathematics.

Table 28 Entrants to first degree or first degree and first diploma courses: qualifications in mathematics

Table 29 Entrants to first degree or first degree and first diploma courses: as percentages of all entrants with stated qualifications

Table 30 Entrants to first degree or first degree and first diploma courses: as percentages of entrants to each subject group with stated qualification

Destinations of graduates in mathematical studies

A22 Tables 31 to 33 provide information about the destination of those obtaining first degrees in mathematical studies (see paragraph A16 and note to paragraph 150 [in chapter 4]) from universities in England and Wales in the years 1977, 1978 and 1979. They are not restricted to those with home fee-paying status. In each of these years the overall proportion of those obtaining first degrees who had home fee-paying status was more than 90 per cent of men and more than 95 per cent of women.

Table 31 Destinations of graduates in mathematical studies

Table 32 Type of permanent home employment: Graduates in mathematical studies

Table 33 Services to management and financial work: Graduates in mathematical studies

A23 Table 34 compares the number of graduates in mathematical studies who gained employment in computer programming with the number of graduates from other subject groups who were similarly employed.

Table 34 Graduate employment in computer programming: by subject group