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1 Introduction

During the last decade many changes in mathematical provision in the sixth form have been called for; some of them arise not directly from any developments within the subject, but from the reorganisation of secondary education, the greater diversity of pupils entering the sixth form and the increasing demands for a mathematical qualification. More emphasis is being placed on mathematical processes by other subjects in the curriculum such as geography, economics and biology. Industry and commerce have stimulated a growing appreciation of the uses that can be made of mathematics in management, planning, design, communication and data analysis. Within mathematics itself some traditional topics have been displaced by more modern material, although this has not brought about all the changes that those concerned with such developments might have expected.* Finally computers have become readily available to schools and the pocket calculator has very largely displaced logarithms as a calculating aid in the sixth form, although both of these technological innovations have yet to make any general impact upon the way mathematics is taught.

Schools have tried to respond to these new demands by making the adjustments that seemed feasible within the constraints of the existing framework of external examinations and established notions of the sixth form. This has sometimes made the task of teaching less straightforward and has also raised concern in higher education about the variety of A-level syllabuses and some students' lack of mastery of certain mathematical processes. The objectives of mathematical education in the sixth form remain largely undefined in practice, with the great majority of schools depending on examination syllabuses to provide a sense of direction.

The main purpose of this publication is to stimulate discussion about the role of mathematics in the sixth form, methods of teaching and learning and the way these might respond to changing circumstances. The observations and comments are based on a survey by HM Inspectors of Schools of 89 sixth forms during the period 1978-80, together with a number of other visits to sixth forms at about this time including some in Wales. It was not practicable to select the schools and colleges on a statistically representative basis but they were chosen to illustrate various types, sizes and locations. Further details are given in Appendix A. The survey involved about 6 per cent of sixth formers in England and over 500 visits to sixth form classrooms to see work of every kind and level. Since A-level work occurs much more frequently than work in other courses in a typical sixth form timetable, HMI had less opportunity to observe non-A-level mathematics classes. What follows concentrates on the A-level aspects, although much of what is said has wider application.

*For example see p7 of The School Mathematics Project report of the Stoke Rochford Conference, March 1980 on A-level Mathematics (SMP 1980)

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This relatively short document, which is directed towards a wider audience than specialist mathematicians does not attempt to enter into mathematical detail about the content of courses. Readers may like to refer to the suggestions for further reading given in Appendix D. Special attention is also to be drawn to Chapter 7 of Aspects of secondary education in England (HMSO 1979) and the associated booklet, Supplementary information on mathematics (HMSO 1980). These reports, which have their basis in a national survey of pupils in their fourth and fifth years, incorporate many passages relevant to the teaching of mathematics in the sixth form especially to those students who are not of A-level calibre.

The interim report of this survey was made available as evidence to the Committee of Enquiry into the Teaching of Mathematics in Schools under the chairmanship of Dr W H Cockcroft. Mathematics in the sixth form was being drafted at the same time as the Cockcroft Committee's report*, and so it has been possible to comment on the Committee's findings relating to the sixth form only by way of marginal notes.

HMI are particularly grateful to those schools and colleges which have contributed to the survey and for their patience in collating information from their records.

*Mathematics counts (HMSO 1982)

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2 Sixth form mathematics courses

Over 60 per cent of sixth formers in the schools* involved in the survey were engaged in some kind of mathematics course at the time of HMI visits. Of the remaining students, some would already have completed a mathematics course in the sixth form and others acquired suitable qualifications in the subject in their fifth year. There were wide variations from school to school, both in the provision of courses and in terms of the proportion of students involved (see Appendix C). A few schools required all sixth formers to follow some kind of mathematics course unless they had already reached a suitable minimum standard in the same subject. Very few courses. accounting for about 3 per cent of the students (see Figure 1), did not lead to external examinations.

The demand for formal qualifications in mathematics dominates the provision of courses and largely determines their content. Quite often requirements are set by parents, employers and further/higher education courses which schools do their best to meet. Sometimes, however, the school itself chooses to use an examination, and its syllabus, as an incentive for study. In this case examination aims may not coincide with the purpose a course is serving, for example it may need to support another A-level subject. Courses therefore become tailored to available examination syllabuses without regard for their suitability.

The size of a school was, as might be expected, a limiting factor in determining the range of courses available; nevertheless, some small sixth forms managed well in the number of mathematics courses they offered (see Appendix C). In contrast, some large sixth forms did not make the provision that might have been expected. Judging what is appropriate in terms of course provision must depend on local circumstances but there are some general questions which schools might ask themselves in determining their needs. For example, are there arrangements which:

• satisfy the demand for a basic qualification for employment/further education?
• provide mathematical support for other areas of the curriculum?
• contribute to the broadening of mathematical awareness of all students?
• meet the varied requirements of A-level students and their differing career interests?
• pay regard to those students with special mathematical talent? Not all these questions necessarily imply separate course provision.

*'Schools' is used throughout the text as an abbreviation for both schools and sixth form colleges. Where there is a need to distinguish between schools and colleges this is made clear.

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Figure 1
Distribution of sixth formers between different types of mathematics course in the schools and colleges included in the HMI survey

HMI studied the various possible factors influencing the proportion of girls taking up mathematics in the sixth form. Few generalisations emerged from the survey but it certainly appeared that girls were attracted to courses with a statistical bias; schools offering only the mechanics option usually had a lower proportion of girls studying A-level mathematics.

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3 Advanced level courses

Types of course

The description of A-level mathematics* courses has undergone some changes over the years so it is necessary to clarify terminology. There are broadly speaking two categories: single-subject and double-subject. Single subject leads to one A-level qualification. The double-subject course gives the student two separate A-level qualifications. Subjects commonly available from examination boards are given in Table 1.

Table 1 Title of A-level examination

Note: Statistics and Computer Science are full A-level subjects and they may be taken as components of the usual three-subject course. But more commonly they are taken as supplementary subjects, perhaps as a fourth A-level which mayor may not receive time in the timetable.

It was rare to find a student who was taking Applied Mathematics without also following the single-subject Pure Mathematics course. The main decisions are whether the student is going to study Mathematics with the mechanics or the statistics option (in some cases the examination syllabus may include both) and whether mathematics is to be offered as a single or double-subject. A school may also adopt a syllabus which contains a number of 'modern' topics or covers a wider range of mathematical applications. Syllabuses therefore have tended to become known as modern, compromise or traditional according to the content although distinctions have become increasingly blurred. Because of their size and in order to accommodate students with different mathematical backgrounds, most of the sixth form colleges in the survey offered both a modern/compromise syllabus and a traditional syllabus; schools usually restricted themselves to one type of syllabus. Examination boards are reducing the number of alternative syllabuses to avoid distinctions between 'traditional' and 'modern'.

A significant number of schools changed from a modern/compromise O-Ievel syllabus in the fifth year to a traditional A-level syllabus in the sixth form; there was little sign that the reverse

*Where reference is made to the titles of examinations, capital letters are used. School subjects and courses are referred to by lower case letters.

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took place. Some teachers took the view that traditional A-level courses were more acceptable in higher/further education, particularly for the physical sciences, and they provided a greater opportunity for the weaker candidate to succeed. Such arguments may or may not be valid but they draw attention to some of the factors that teachers have to consider when selecting an examination syllabus.

Although Pure Mathematics and Applied Mathematics were frequently partners as a double-subject combination, there were some schools where Pure Mathematics played a different role. For instance, Pure Mathematics (on its own) was sometime offered to students, often girls, as a way of avoiding mechanics; on other occasions it was thought to be a suitable course for students who were not able or willing to take the double-subject. It was also used to facilitate economical timetabling in a school with both single- and double-subject candidates since both groups could be combined for Pure Mathematics. The division of sixth form advanced courses into 'pure' and 'applied' has a long history and has its parallel in university mathematics courses, although in the latter context 'pure' involves a level of abstraction inappropriate to schools. A school course in mathematics needs to keep a balance between theory, technical facility and applications which promote an appreciation of the problems mathematics can solve. The use of the single-subject Pure Mathematics as a way round logistical difficulties may disadvantage some students later in their career when they require a broader appreciation of the subject*.

Six schools included in the survey ran courses in single-subject A-level Statistics and 14 schools arranged a course in A-level Computer Science, sometimes in conjunction with local colleges of further education. These courses, which affected about 3 per cent of A-level mathematics students, were mainly supplementary to other A-level mathematics courses; for a few students they constituted a third mathematics subject. The advent of the micro-computer is rapidly changing the role which computing can play in mathematical education at all levels, quite apart from its part in computer studies/science courses, and this will necessitate some reconsideration of existing arrangements. The development of the computer as an aid to learning and exploring mathematics may result in less emphasis being given by mathematics departments to organising courses aimed at computing as a vocational study. Some schools have already made the distinction and regard A-level Computer Science as a subject outside the province of the mathematics department. The linking of vocationally-biased computer studies courses to colleges of further education might be, at least in the short term, the direction which best serves the interest of mathematics. This could result in staff being released to play a greater part in the development of the microcomputer as an aid to the teaching of mathematics. The advantages of using the computer creatively with mathematics students was sufficiently evident in a number of schools to suggest that greater emphasis should be given to developments in this direction.

Mathematics or statistics or both?

The last 10 to 15 years have seen a considerable expansion in statistics as an alternative to mechanics or as an adjunct to it. At the same time there has been a marked change in the character and

*Mathematics counts: Para 565 expresses the view "that all A-level mathematics courses should contain some substantial element of 'applied mathematics'" and that the Committee "do not favour single-subject A-level courses which consist of pure mathematics only". Para 589 also refers to the distinction between pure and applied mathematics "which, as we have already pointed out, we consider to be undesirable at school level".

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composition of A-level mathematics classes. Students have been less inclined to see mathematics as closely allied to physics and are becoming increasingly aware of the value of an A-level qualification in the subject. Over half the girls taking A-level mathematics in the schools included in the survey were combining mathematics with at least one non-science subject (Figure 2). This was reflected in the higher proportion of girls compared with boys offering statistics rather than mechanics (Figure 3). A-level subjects such as economics, geography and biology were closely associated with the choice of the statistics alternative has clearly provided room for expansion without necessarily displacing the old partnership between mathematics and physics from its prime position.

Figure 2
Percentage of A-level mathematics students in the survey who combined mathematics with one/two (or more) non-science A-level subjects

Note A 'non-science' A-level was understood to mean a subject other than physics, chemistry or one of the biological sciences.

About half the students in the survey schools included statistics in their A-level mathematics course and two-thirds included mechanics (Figure 3). The balance might have been more even but for the fact that nearly a third of the schools did not provide a course which included statistics. In those schools not offering the statistics option there was a tendency for a lower than average proportion of the sixth form to take up A-level mathematics. There were exceptions to this in some of the schools using single-subject Pure Mathematics as the alternative course to Mathematics with the mechanics option. Attention has already been drawn to the disadvantages of this strategy for some students.

Destinations of A-Ievel mathematics students

The wider range of A-level subject combinations which include mathematics, and the availability of courses with the statistics option, result in A-level mathematics students making increasingly diverse

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choices of career and study after leaving school. Although not all school records made it possible to provide a full analysis of the destination of every student who had left school in the year prior to the survey, there was enough information to arrive at some worthwhile conclusions. The main features of Table 2 are confirmed by information taken from the 1978-79 leavers' survey conducted by the Department of Education and Science*.

Figure 3
Percentage of A-level mathematics candidates in the survey offering different options in the subject.

Note This table may contain some errors due to inadequate information about the options followed by double-subject candidates and uncertainty concerning those students taking a syllabus containing both mechanics and statistics. A student following a syllabus containing both mechanics and statistics may not necessarily offer both for examination.

The leavers' survey also provided some interesting information about the way A-level mathematics grades were distributed in relation to destinations (Table 3). Mathematics degree courses absorbed about a quarter of all A-level mathematics students who had achieved grade A: engineering took a similar proportion. Nearly a third of the students awarded grade A read for degrees in the biological sciences or the non-sciences. A quarter of all non-science undergraduates had been candidates for A-level mathematics (Table 4). Of those A-level mathematics students who had entered non-degree courses in HE/FE and employment, 46 per cent and 45 per cent respectively had achieved a pass grade (A-E).

Two-thirds of the students who have been prepared for A-level mathematics leave school to follow courses or careers outside the field of mathematics and physical science. The approach to A-level mathematics has been largely designed until now with higher education needs in these two main areas in mind. Many regard this to be the primary purpose and students choosing to study mathematics need to understand this situation. Furthermore this view would be

*The survey is made annually in October and is based upon a 10 per cent sample of pupils who have left school in the preceding academic year.

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strengthened if those who failed to achieve a pass grade in A-level mathematics were not included in the analysis. However, the demand for an A-level qualification in mathematics continues to grow and it becomes increasingly difficult to provide for all students without taking a more flexible attitude towards the aims of an A-level course.

Table 2 Destinations of students leaving school or college having completed an A-level course in mathematics

Perhaps, therefore, A-level mathematics should be available in a form that provides a distinct alternative to existing courses, using an approach that widens its appeal to students without entailing a loss of status as an A-level subject. For some students the proposed Intermediate-level may well provide the appropriate alternative. At present the statistics option which has a somewhat theoretical emphasis is effectively the only other route for students who need an A-level mathematics qualification. A course based on applications could be more empirical but not necessarily less intellectually demanding. Such a course might overcome some of the difficulties faced by students who study A-level subjects which use mathematical techniques without an adequate conceptual foundation in mathematics being provided. It was disappointing to find that some

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attempts to move in this direction through the introduction of project and practical work in statistics and by means of the Schools Council Sixth Form Mathematics Project, Mathematics applicable, have met with a limited response from schools. Lack of status and uncertainty of acceptance are no doubt contributory factors which have hindered this kind of development despite its educational merits.*

Table 3 Destinations of A-level mathematics candidates who obtained grade A (Source: DES leavers' survey 1978-79)

Table 4 Distribution of A-level mathematics grades of students entering degree courses (Source: DES leavers' survey 1978-79)

Conditions of entry to A-level mathematics courses

The criteria for selecting suitable students for A-level mathematics courses varied widely from one school to another. Schools with a high proportion of able pupils tended to impose stricter conditions than those concerned with finding sufficient numbers to make up a viable teaching group. Sometimes the requirements placed mathematics on a rather exclusive level compared with other A-level subjects. Conditions of entry usually, but not always, depended on O-Ievel

Mathematics counts: Para 596 makes similar suggestions in the context of the proposed I-level examination.

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grade; grades A or B were required in a fifth of the schools. About a third of the schools accepted CSE grade 1 in mathematics but considerable uncertainty surrounded the acceptability of both this level of attainment and O-level grade C. Thirteen schools were able to supply HMI with the examination results of A-level candidates who had started their sixth form course with grade C at O-level. It appeared that at least a third of the candidates had passed A-level mathematics and some had attained grade B. Although there was not enough evidence collected about those students with CSE grade I to draw any firm conclusions, it was clear that some had achieved a pass grade in A-level. Performance in O-level/CSE at 16 provides some guidance about possible chances of success at A-level but it cannot take account of the significant developments which may take place in a student's capacities between the ages of 16 and 18. On the other hand, leniency in admitting students to A-level courses can lead to teaching which is over-concerned with preparation for examinations and may result in the student acquiring a marginally acceptable qualification but an inadequate command of the subject upon which to build further studies. A balance must be found between being too demanding and too flexible which will give students the opportunity to improve without distorting the broad aims of the course.

Double-subject provision

It has become established practice for some students to offer mathematics as two A-level subjects. The traditional examination combination of Pure Mathematics and Applied Mathematics has to some extent been replaced by Mathematics and Further Mathematics, each component of the latter involving both pure and applied topics. Applied Mathematics or Further Mathematics is usually taken by abler students seeking entry to courses in higher education where double-subject qualifications might be an advantage although not necessarily a requirement. Two-thirds of the schools and colleges in the survey provided a double-subject course: of these a third used the traditional pure and applied approach. About one-eighth of A-level mathematics students were studying the double-subject.

In some ways the entry requirements for double-subject mathematics courses appeared more flexible, in other ways more rigid than for the single-subject. Ten schools provided an O-level additional mathematics course in the fifth year insisted on a pass grade in this subject and sometimes this was coupled with an A grade in O-Ievel mathematics. At the other extreme there were similar numbers of schools where O-Ievel grade C was a sufficient basis for students to be considered. Sometimes the decision was left until the end of the first year in the sixth form. Many schools felt that double-subject mathematics should be reserved for those students with a 'flair' for the subject. The selection of students could also be influenced by organisational considerations such as the timetabling of extra periods, combinations with other A-level subjects and the size of teaching groups. Some schools felt that they had so few students of this calibre that provision would be uneconomic; others expressed the view that the extra A-level in mathematics was an essential inducement to work.

Some schools argued that a single-subject A-level course in mathematics could provide adequate opportunity for all students without the need for an extra A-level to extend the more able. There is

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no reason why single-subject courses should not offer sufficient opportunity provided that the range of work is not limited to examination questions. The arrangements that mathematics departments made for the double-subject sometimes led to complications, such as the need for early identification and separate provision, which had adverse repercussions lower down the school as well as in the sixth form. There were inequalities in the opportunities available to students in sixth forms of different size and type; provision for the double subject was very varied and so were the conditions for entry to these courses. There is also the question of whether a course comprising three A-level subjects of which two are mathematical provides a sufficiently broad educational experience. Although many teachers regard double-subject mathematics as an advantage when students seek admission to some university departments, there is increasing evidence that a high proportion of students are accepted to read mathematical subjects who have only a single-subject A-level qualification. It would be unfortunate if the able student's interest in mathematics could only be satisfied by the challenge of more demanding examinations. Indeed a concentration on examination work to the exclusion of other mathematical activities could produce students less able to profit from the opportunities available in higher education* (see Learning and study skills, page 21) .

Little first-hand evidence emerged from the survey about the provision for A-level special papers in mathematics or arrangements made for those students intending to take university entrance scholarships. Some schools made a place for this work especially in the Autumn term both before and after students had taken A-level: others felt that they rarely had a suitable candidate. For the majority of schools this kind of provision does not appear to have as high a priority as the achievement of high grades in A-level. Although this policy may result in the students involved gaining admission into higher education at the end of two years in the sixth form, undue concern for high grades may exclude new work which would both challenge and extend the able student.

Girls and A-level mathematics**

In recent years there has been an upward trend in the proportion of girls studying A-level mathematics. However, both nationally and in the survey there were three boys for every girl taking A-level mathematics, and for the double subject the ratio was 4:1. There were striking differences between schools but no single general factor emerged to explain how these differences arose. The attitudes and expectations of parents and teachers, local circumstances, school traditions, A-level subject combinations. suitability of courses, presence of women mathematics teachers, careers guidance, unsuitable O-Ievel subjects selected in the fourth year and attitudes generated during the main school course are but some of the influences at work. Although boys and girls might have been on a roughly equal footing throughout the main school course. by the time they had reached O-Ievel the ratio of boys to girls achieving a pass grade (A-C) was 3:2 (confirmed by national samples***). In some schools where it was possible to follow pupils through into the sixth year, it was found that twice as many boys as girls who had passed O-Ievel in the fifth year took up an A-level course in mathematics. Where schools provided an

*Mathematics counts Para 586 refers to some of the problems that can arise in some schools providing double-subject courses. The Committee take the view that double-subject courses should continue where "necessary staffing can be made available to teach the course without disadvantage to those who are studying mathematics at lower levels". In Para 588 attention is drawn to the fact that in 1979 only 55 per cent of entrants to degree courses in mathematical studies had qualifications in the double-subject. The report expresses the hope "that those who select students for admission to higher education will recognise that there are sound educational as well as economic reasons for offering only single-subject mathematics at A-level".

**HMI Series: Matters for Discussion 13 Girls and Science contains material relevant to the issues raised in this paragraph.

***DES leavers' survey

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O-Ievel additional mathematics course in the fifth year, the presence of girls was further reduced compared with boys (national figures indicate a ratio of 1:5*).

It has already been suggested that the availability of alternatives to the mechanics option in A-level mathematics course and the opportunity to combine mathematics with non-science subjects help to make mathematics more appealing to girls. It is sometimes said that single sex schools encourage a higher proportion or girls to study A-level mathematics. The position is in fact more complex. Figure 4 displays information on those schools in the survey which had more than 50 students taking at least one A-level subject. 43 per cent of the mixed schools attracted an above average proportion of girls to study A-level mathematics. Although a higher percentage of the single-sex (or former single-sex) girls' schools exceeded the average it

Figure 4

Proportion of girls taking A-level mathematics in those survey schools with traditional sixth forms over 50

*DES leavers' survey

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was not by an amount that distinguished them from some of the mixed schools. Two of the mixed schools had the highest levels of participation for girls. It was also noticeable that some of the former boys' schools had lower than average proportions of girls taking A-level mathematics.

It was noticeable that girls who had studied A-level mathematics went into a wide range of careers and courses of study, some of which were not closely associated with mathematics. Obviously schools are not in a position to control all the influences determining subject choice but they can avoid creating additional hurdles such as the notion that mathematics is 'more difficult' than other subjects. Although mathematics may be different in many respects from other A-level studies the opportunity to be successful, judging from examination pass rates, appears to be very similar. As it happened, the girls who had been entered for A-level mathematics in the year previous to the HMI survey achieved a slightly higher percentage of pass grades (A-E) than the boys and this was confirmed by national samples*.

*DES leavers' survey

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4 Other sixth form mathematics courses

Most sixth form courses lead to an external examination but this can disguise the purpose of those courses whose main aim is to provide mathematical support for other sixth form studies. There are also courses which are not so much supportive or qualifying as intended to encourage a general appreciation of mathematical ideas. The latter type of course usually comes under the umbrella of general studies and in some schools can also lead to an external examination. Finally there are activities associated with computing and mathematical clubs which may be informal in nature but are highly influential in the contribution they make. In the survey 27 schools had clubs of this kind and 12 of these were involved with computing. The numbers taking part were sometimes small and confined to particular groups of pupils or students.

Alternative Ordinary (AO) Level and Additional Mathematics

Mathematics courses at AO-Ievel appear under a number of titles including that of Additional Mathematics. Some examination boards prefer to describe an O-Ievel examination which incorporates extra topics as Additional Mathematics, and although many schools entered candidates in their fifth year (see Examination pressures, page 23) others also used it in the sixth form. Six schools in the survey arranged sixth form AO-Ievel courses in computer studies or in statistics. Altogether less than a third of the schools used one or more of the AO-Ievel mathematical examinations (or O-level Additional Mathematics) and this involved only about 2 per cent of sixth formers. However, this does not include those students who were taking Additional Mathematics examinations (at both AO and O-Ievel) after one year of their A-level mathematics course. In their case the examination was being used as a measure of progress towards A-level although the syllabuses for Additional Mathematics are not usually intended for this purpose. In a few cases students who are subsequently awarded an F grade at A-level might gain a qualification which they would not otherwise achieve. However, with courses already strongly directed towards examination requirements there is a risk that taking an extra examination with no counterpart in other subjects may divert attention and even give a false impression of how much has been achieved.

There have been some important innovations in syllabuses at AO-level which have yet to find a substantial footing in schools. They demonstrate some of the difficulties which have to be faced by any attempt to move far away from familiar territory, however worthy the aim. Schools Council Sixth Form Mathematics Project materials designed to foster the study of how mathematics can be applied to the solution of a wide range of problems in science and technology have already been mentioned. Likewise the School Mathematics Project Additional Mathematics course, which provides a broader logical

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view of mathematics, has received little attention from schools in the survey although in many cases they were using other SMP courses.

Ordinary level

O-Ievel Mathematics (or its equivalent) is a necessary qualification for many areas of employment or subsequent study. Over 80 per cent of the schools in the survey considered it essential to provide a course in the sixth form for students who had not already passed O-Ievel mathematics. It was often in these courses that the limitations of time and staffing were most apparent. Quite frequently students could not attend for all of the periods arranged, because periods timetabled coincided with those for other subjects. On some occasions classes reached the lower and mid-twenties in size and catered for students taking CSE as well as for those aiming for O-Ievel; furthermore some of the students were taking an examination in November and some in the following June. Additional complications arose where students had previously followed different syllabuses, either GCE or CSE, of the same or different examination boards. In such circumstances the task for the teacher was daunting and it was not surprising to find that work became restricted to practising examples from previous examination papers with little being planned to enhance the understanding and appreciation of the subject. The evidence from the survey was that one third of these O-Ievel candidates subsequently succeeded in passing the examination. Making this very necessary provision is undoubtedly a strain on many schools, since they may find themselves providing a less-than-adequate course which neither enables students to overcome continuing difficulties with the subject, nor, in some cases, to rectify misunderstandings of essential details.

Not all O-Ievel work was remedial or aimed directly at achieving a qualification. Quite a number of schools provided new targets by means of O-Ievel courses in subjects such as statistics and computer studies. Some of these courses were supportive of other subjects or belonged to a programme of minority studies. Where these courses are providing support it is important to ensure that they fulfil this aim and this may well imply some extension of the ideas embodied in the examination question papers (see Non-examination courses, page 17).

Certificate of Extended Education

In about one quarter of the schools the CEE was used to provide a one-year course matching to the maturity of the students, and encouraging the feeling of making a fresh start. These courses attracted a wide variety of sixth formers in the hope that they could achieve the equivalent of an O-Ievel qualification. Several of the courses involved options such as money management, computing and statistics, although there was a tendency to select options with a strong arithmetical content. It was usual for some form of project work to be included and sometimes an element of continuous assessment. Experience in both of these was valuable for mathematics departments, especially as it would not have been readily available through other mathematical courses in the sixth form. The course material seen was not as exciting as it sounded since it tended to rely mainly on the practice of techniques in a way that students had already experienced in the main school. Opportunities to introduce a greater degree of realism had been exploited in only relatively few of the schools. For example, little use was being made of the pocket

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calculator to handle data taken directly from first-hand sources. Project work tended to be preoccupied with description at the expense of analysing the mathematical aspects of the topic. Nevertheless, CEE was providing a framework to plan a one-year course which allowed some choice of teaching and assessment methods as well as opportunity to match the subject content to students' interests and requirements.

Certificate of Secondary Education

The increasing number of students remaining at school after the age of 16 who were not naturally inclined to a full A-level course but who could benefit from post-16 provision has produced a variety of responses from schools. There were schools which regarded O-Ievel and CEE as providing the appropriate course structure but there were others where CSE was regarded as the most suitable examination objective. CSE courses which were simply repeating fifth year work suffered from much the same shortcomings as the O-Ievel provision. There were schools where CEE examinations were not used and where a mode 3 CSE course designed for use in the sixth form would have suited the situation. One school, for example, had devised a CSE Additional Mathematics course which included a variety of assessment procedures as well as a wide range of mathematical ideas. In particular, students undertook a project aimed at bridging the gap between school mathematics and its domestic and vocational applications. A small number of schools offered computer studies as a CSE subject in the sixth form.

Courses in arithmetic and numeracy

A sprinkling of other courses mainly of an arithmetical character was observed, sometimes associated with secretarial work but quite often aimed at sixth formers who required some mathematical qualification in the absence of suitable grades in O-Ievel and CSE. Some of these, such as the Foundation courses operating under the auspices of the City and Guilds of London Institute, provided schools with an opportunity to expand their approach to numeracy. However, little evidence emerged during the visits to schools to show how these courses would develop both to meet the needs of industry and commerce and to satisfy ordinary consumer demands. Quite often the emphasis remained focused on the traditional skills of paper calculation with little use being made of the opportunities to bring the students face to face with the situations where calculation actually arises in shops, offices and workshops.

Non-examination courses supporting other subjects

Whether courses leading to existing external examinations provide the right kind of mathematical support for other subjects in the curriculum such as biology, geography and economics, deserves close scrutiny by schools. It was certainly not easy to find work going on in the mathematics classroom which utilised data drawn from what the students were doing elsewhere in their studies. A non-examination course sometimes had a better chance of being related to other subjects, but such courses were evident only in a fifth of the schools visited. Schools argue that students need the incentive of an examination but the price of this approach might be high in terms of relevance in the supportive role. Self-instructional material* has been published with this kind of work in mind but it was in use in only three

*Continuing mathematics project. NCET/SC. Longman.

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schools. In one sixth form college this material formed part of the resources of a room where for two thirds of the week a member of the mathematics staff was available to provide tutorial help.

General studies

A mathematical course aimed at developing a student's general appreciation of the subject was a fairly rare event. Possibly less is done now than formerly although much of the effort may have been redirected into computer activities. Nevertheless a few individual schools have showed enterprise with courses entitled 'Mathematical Games', 'Money and Society', 'Mathematical Ideas' and 'Consumer Mathematics'. These topics reflect a change of emphasis away from the philosophy and history of mathematics towards issues which cater for a wider audience. There is a strong case for both extending this work and keeping it practical; this is not to say that more intellectual and abstract pursuits should be neglected but that they should develop from situations which are firmly rooted in experience.

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5 Methods of teaching and learning

Teaching

The general style of teaching in the sixth form closely resembles the approach adopted with fourth and fifth year pupils and was commented upon in Chapter 7 of Aspects of secondary education in England (HMSO). The teacher presents a topic on the blackboard, works through an example and while the students carry out exercises based on the topic the teacher helps individuals. Sometimes the blackboard presentation relates to homework and quite often examples are taken from previous examination papers instead of a textbook or duplicated sheet. It is common for the student to take down a verbatim record of what is on the blackboard.

Within the constraints imposed by this kind of lesson pattern the teacher can employ a variety of techniques to enhance the effectiveness of the teaching. Despite this the majority of lessons visited appeared to depend on very few of these techniques; they were mainly instructional in character, made only limited provision for the interchange of ideas with students and offered insufficient opportunity for students to gain more than a restricted view of the subject. Teaching in these circumstances was highly predictable and tailored to the needs of what had become a passive and uncritical audience. The following comments by HMI are typical of many:

'The staff are well qualified in mathematics and demonstrate their mastery of the subject in the classroom. Their teaching style is largely that of authoritative exposition of the subject; they have analysed the work with the utmost care and in great detail and put it across meticulously. Unfortunately the challenge has been removed from the learning process except in the sense of doing harder and harder examples; there is no opportunity for the students to do any extended study or even to learn something independently.'

(Sixth form college)

'Notes were dictated and examples copied from the board; then a sheet of examples was worked through to ensure the mastery of the technique. The students were putting together their own notebook which would contain all the material necessary for the examination. This was clinically efficient preparation for A-level but it left little scope for mathematical thinking, problem-solving strategies or opportunities for the student to surprise the teacher with a novel solution. All the work done appeared to be confined to the syllabus and students were never asked to learn anything for themselves.'

(Independent school)

'The merely instructional nature of the teaching seems to have driven the students into a state of quiet acceptance. Less monologue, more dialogue and a greater awareness of the

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students' need to express themselves would help the teachers to create a more stimulating atmosphere.'

(Comprehensive school)

A mathematics lesson should encourage discussion and even debate. One teacher, of an A-level class following up the homework, asked his students for their answers, challenged their hypotheses and persuaded the students to question their assumptions. The arguments that emerged went beyond the limitations of the textbook and the syllabus. In another school the teacher developed the topic of simple harmonic motion with his A-level students working from first principles using carefully worded questions. Responses from the students were skilfully turned into a supplementary question by the teacher.

Topics can be given relevance by reference to applications from within mathematics or outside the subject; in particular, cross-curricular links need to be established. An A-level group studying polar graphs (using all overhead projector) were shown a book on radio-wave propagation. Another teacher listed a number of important types of differential equation and discussed with the students how the equations were applied in science. The same teacher also asked his students to guess possible solutions before discussing formal procedures with them. In a general studies course students were working on various projects such as 'furnishing the kitchen', 'decorating the bedroom' and a 'best-buy shopping basket'. All the calculations were derived from these projects.

It is sometimes both challenging and beneficial to learning to explore alternative solutions to a problem. Different solutions provide an opportunity to contrast methods, exercise problem-solving skills, establish relationships with other aspects of mathematics and encourage discussion. This was evident in an A-level class where the teacher was pursuing methods of solution which, although not producing the most elegant results, nevertheless provided an opportunity to illustrate different facets of the problem which might otherwise have passed unnoticed.

An unusual problem can often be used to enliven interest, provoke argument or demonstrate a particular teaching point. For example in a double-subject mathematics class the teacher used a number of problems from recreational mathematics to show how all the solutions depended on the same pattern of proof and could be expressed in similar notation.

Occasionally it is possible to draw upon students' own experience to provide motivation and achieve deeper understanding. A group of students following a CEE course were able to discuss percentage profit using data they had collected running the school shop. This work also

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provided suitable situations in which to use a pocket calculator. In general studies a teacher had distributed copies of an income-tax form and many students were able to talk about their experience as employees. An A-level class, was engaged in a course in experimental statistics. They worked in pairs and the data from practical activities replaced some of the textbook exercises.

From time to time students' interests should be extended beyond the range of problems and topics implicit in particular textbooks or examinations. One teacher who was familiar with Open University mathematics courses recommended students to watch particular programmes. In another school a girl was seen reading Kasner and Newman's Mathematics and the imagination - a -very untypical but valuable experience.

Sometimes it is possible to add force to an argument or explain more vividly using a visual aid. This was evident when a teacher of an A-level class used the cardboard cylinder from a kitchen paper roll to demonstrate the construction of a helix. The microcomputer will extend existing facilities, as in the A-level class where it was used to provide a graphic display of the successive approximations to the solution of a differential equation.*

Learning and study skills

From the point of view of the student the effective learning of mathematics requires the development of some special study skills as well as the extension of more general ones. A list of study skills relevant to mathematics might include how to:

• use books as a resource for reference and study
• make notes which encourage concentration on essential arguments and are not a simple transcription
• organise study and deploy time effectively
• revise and retrieve information efficiently
• pose questions and participate in discussion
• cooperate and learn as a member of a group
• learn from mistakes and cope with failure
• make abstractions and develop hypotheses
• devise strategies to solve problems
• formulate a problem in response to given circumstances
• discover the underlying pattern and logical structure
• construct a logical argument and set it down on paper
• develop a concise style of written expression using symbols and notations
• perceive and cultivate 'elegance' in argument and analysis.

*Mathematics counts Para 243 summarises its discussion of classroom practice in six elements "which need to be present in successful mathematics teaching to pupils of all ages". They are: exposition by the teacher, discussion, practical work, consolidation and practice, problem solving and application to everyday situations and, finally, investigational work. Paras 561 and 562 extend the views of the Committee to teaching methods in the sixth form.

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learning in detail from the blackboard at the end of a sixth form course as at the beginning. Judging from the use that students made of the library, learning from books was far less common in mathematics than in other sixth form subjects.

Teaching and learning

The teacher teaching and the student learning are not of course independent isolated activities; furthermore there are times when teaching and learning roles are reversed and teachers learn from their students and students teach one another. Study skills are acquired as much by inference from the teacher's style of teaching/learning as from more direct instruction. It is difficult to see how some study skills could develop in the many classrooms where students sat passively accepting without comment every word written on the blackboard. Mathematics is not the most obvious subject to promote a heated argument but teaching techniques need to encourage an exchange of ideas. Not all questions have only one acceptable answer nor problems only one method of solution yet this was the impression often given to students. Questions might be more hypothetical: "What would happen if we changed the sign?" "What happens if we change the operation from multiplication to division?" Many mathematical situations start with accepted 'rules of play' but "What happens if we change the rules?" Unless teachers are able to engage their students in a dialogue, they miss opportunities to discover whether effective learning is taking place and, if it is not, to decide how an explanation might be modified.

Mathematics is a subject which can easily become separated from other forms of experience and this can reduce its effectiveness as a means of communicating ideas, recording information and bringing precision to judgements. To improve on the present perspectives requires among other factors an appreciation of commercial and industrial applications and also of the role mathematics can play in other sixth form subjects. Suitably selected outside visits and work experience are one way of helping students to bring greater coherence to their studies and of providing opportunity to widen discussion and tap students' personal interests. To treat mathematics as a study entirely apart runs the risk of its eventual rejection by the vast majority of students when the necessary qualifications in the subject have been achieved. It may also deprive them of some of the flexibility of thinking which a mathematical qualification should aim to provide.

Just as mathematics draws life and purpose from the ways in which it provides a service to other studies and experience, so must it also sustain and develop a vigorous life of its own. To a large extent the examination syllabus provides teachers with the opportunity to take a generous look at a wide range of mathematical ideas but these may well need to be extended beyond the confines set by the questions used in examination papers to test knowledge of the syllabus. In particular, a stimulus needs to be provided to encourage students to sample the general literature of the subject including books of recreational mathematics. They require the experience of creating (or re-creating) mathematics as well as imitating the work of others. The subject has special satisfaction for those who like to explore and think for themselves. This is of particular importance to the future mathematician and teacher of the subject.

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Examination pressures

A distinction has already been made between mathematics for its own sake and mathematics for the sake of examinations. Teachers were very concerned about their responsibilities towards students and the concentration of effort and time needed to ensure success in public examinations. This was reflected during the visits to classrooms where the overriding importance attached to examination preparation was abundantly evident in the choice of work and the teaching approach. Mathematics departments often argued that there was little time for any work not essential to examinations even though this meant a curtailment of what they thought was desirable. The members of one mathematics department made this dilemma very clear in a document they had written in response to questions from their headmaster: "While understanding that many students see the course as a means to a useful qualification rather than as an education in the broad sense, members of the department feel that the students would benefit from being weaned away from their reliance on coercion and spoon-feeding and that more could be done to promote real pleasure and intellectual activity".

It is not hard to appreciate the difficulties which face mathematics departments, often with large groups of students having diverse interests and abilities, but it is nevertheless necessary to consider whether there are alternative ways of teaching and learning which might make more effective use of the time available. A better balance might be struck between what students can learn for themselves and what needs to be taught formally. Teaching might concentrate more on the central issues rather than on detail which students can explore for themselves with the use of books. Notes might be in the form of a duplicated handout rather than dictated or transcribed from the board. Eventually, of course, students should be prompted to abstract their own notes of proceedings. Typical solutions to problems might be filed and made accessible to students after their own attempts have failed or are in need of amplification. These and other procedures need to be aimed at promoting the most effective learning skills, so helping students to develop an increasing responsibility for their studies. This is not so much a matter of withdrawing support as redirecting support towards methods which develop more adult attitudes to study and which free the teacher to work on a broader canvas.

Preparing for the sixth form

The acquisition of all the appropriate learning skills needed by a mature sixth former must be based on the preparations made in the previous phase of education. Actual observation of the form that preparation took was beyond the scope of the survey but other evidence* indicates that it was aimed mainly at overcoming gaps in manipulative skills, and providing an earlier introduction to A-level topics, rather than to develop appropriate attitudes towards study. About a third of the schools were engaged in some pre-A-Ievel work towards the end of the summer term in the fifth year when pupils had largely made up their minds about the future; sixth form colleges often provided a short introductory course at this stage. A similar proportion of the school, made specific provision for the more mathematically able pupils by completing the O-Ievel mathematics course in the fourth year and starting a course aimed at O-Ievel Additional Mathematics. Some teachers thought it was essential to

*Aspects of secondary education in England. HMSO.

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start A-level work in the fifth year especially if the student was to offer the double-subject.

Extra mathematics for the more able gives rise to a number of complications. Figures collected from eight comprehensive schools with a fifth year additional mathematics course suggested that such a course had been taken by only about half the number of students who eventually elected to do A-level mathematics. This means that a school has either to divide the A-Ievel mathematics students in each year into two groups or be prepared to organise classes in which students have mixed backgrounds. Furthermore a much lower proportion* of girls was attracted to additional mathematics in the fifth year than was the case for the more elementary O-Ievel course. An additional mathematics course can be important in persuading the best students to choose A-level mathematics but it can also be a barrier to those whose mathematical talents might blossom later but who have not had the opportunity to take an additional mathematics course in the fifth year.

Some teachers maintained that the main school course should provide enough material to satisfy all requirements without the need to undertake an extra examination. If, however, the work concentrates solely upon examination questions then the most able pupils lack challenge. This is because traditional mathematical questions, unlike many questions in other curriculum subjects, are rarely capable of being interpreted at more than one level of difficulty. It might also be argued that sixth form preparation is not simply a matter of accelerating progress towards the next examination but that it is also a time in which to take a wider view of mathematical studies. There are good reasons for reviewing present practices to see whether they provide the kind of preparation which is needed or whether they achieve only limited gains.

*DES leavers' survey (1979-80) data indicates that 19 per cent of fourth and fifth year entries for O-level Additional Mathematics are girls compared with 43 per cent for O-level Mathematics.

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6 Staffing, organisation and resources

Availability of qualified staff

The survey did not indicate any general shortage of suitably qualified staff for teaching A-level work as far as the majority of students was concerned. This of course does not imply that other areas of mathematical teaching were equally well staffed, even in schools where the provision for A-level work was satisfactory. What it means to be 'suitably qualified' is a matter for some discussion since clearly any full definition must include aspects of teaching experience as well as academic qualification. Even if the definition were limited to the latter, there would still be questions about qualifications in which mathematics formed a part but not the whole. To be more explicit about qualifications would have required much more information than the survey team could reasonably have requested and the measure of the situation must therefore remain fairly crude and possibly unjust to some teachers. For example, if a degree in mathematics as the only subject of study had been taken as a measure, then 80 per cent of A-level teaching in 60 per cent of the schools and colleges would have been included as taught by 'suitably qualified' teachers. If 'suitably qualified' had been extended to include all graduates both trained and untrained with mathematics as a main or subsidiary subject in their degree then this would have embraced all the A-level teaching in three quarters of the schools. Six schools depended on non-graduate specialist mathematics teachers for more than half of the A-level teaching. Two schools, both with small sixth forms, appeared to have no graduate specialist on the mathematics staff. The availability of graduate specialists for A-level work tended to be related to school size and the size of the sixth form. In sixth form colleges A-level work occupied about two thirds of graduates timetables, in grammar schools about half and in comprehensive schools about a quarter. These of course are overall estimates and do not represent the working loads of individual teachers who may in fact have had higher or lower proportions of A-level work.

Table 5 A-Ievel mathematics teaching undertaken by graduates (trained and untrained) with degrees in mathematics (only) or mathematics and physics

A sixth form mathematics teacher, particularly for A-level work, must have sufficient subject knowledge to be able to explore the

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subject with confidence beyond the boundaries that examination questions appear to imply. The teacher must be able to do more than re-echo the textbook and provide the conventional solution to a problem. Facility is needed in showing students that some methods of solution are unprofitable, demand too much manipulation or that they are too long-winded. Insight is required into the history and philosophy of the subject, and sufficient knowledge to extract key issues and relationships from a proliferation of detail. Teaching skills may require greater versatility than experience as a student may have suggested. The teacher who was a talented pupil, learning effectively from teaching which employed only a limited range of techniques, has to adjust his subsequent teaching to take account of the wide variety of learning problems experienced by his pupils. Understandably, in-service education has not given a high priority to the skills of sixth form teaching, possibly because teachers have rather taken for granted this aspect of their work when compared with that of the main school.

Departmental planning

Although it is usual to find a written scheme of work for younger pupils, it is relatively rare to find very much more than an edited version of the examination syllabus as a guide to planning work for the sixth form. Decisions, nevertheless, need to be made about emphasis, additional material, teaching methods, suitable reference books, sequence, applications, links with other subjects, homework tasks, monitoring progress, marking and presentation of work, practical work, projects, the use of calculators/computers and, by no means least, the aims and objectives of the course, all of which would be worth setting down on paper. Without a formal statement new teachers take very much longer to adjust, probationers are deprived of an essential professional tool and the department is less likely to submit itself to the discipline of evaluating its performance. Meetings of the mathematics department in many schools provided a forum for discussion, but too often these were not held frequently enough to deal with more than the essential administrative matters. In some schools there were no formal departmental meetings at all.

Allocating time

Fairly uniform views are held in schools about the amount of time needed for teaching mathematics, with the result that three quarters of the schools in the survey followed much the same pattern, allocating 8 periods per (40-period) week for A-level and usually half this amount to other courses. Isolated examples were found of 6 periods and 12 periods per (40-period) week for A-level mathematics as a single-subject. For the double-subject about half the schools doubled the time given to the single-subject; the remainder were less liberal and often restricted themselves to 4 extra periods. These allocations tend to be determined by 'across the board' decisions to fit into an option block scheme for the whole sixth form in which every subject is on the same footing, with double-subject mathematics being treated as two separate A-levels. Such arrangements have their advantages administratively but they may not always encourage a close examination of how different subjects use the time and resources available, and whether there are justifiable differences which need to be acknowledged. For example should the double-subject be treated as if its components were two unrelated subjects or should the

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components be seen as extensions of one another? Furthermore if the students concerned are the most able, do they require as large an allocation of teaching time as other A-level candidates? Has the number of students in the group an effect on the time needed? How much time is to be allocated to teaching rather than supervised study or individual tutoring? Does too much teaching time encourage too much teaching of material which the student could and should study for himself? A better balance between teaching time and private study of the subject needs to be considered if teaching resources are to be used effectively and economically*.

Size of teaching groups

Since mathematics involves a greater number of sixth formers than most other subjects, it might be expected that teaching groups would be somewhat larger than the average size for sixth forms. During the survey the largest A-level class visited contained 25 students (see Table 6). Schools with over 100 students taking A-levels in various subjects tended to produce single-subject mathematics A-level groups with more than 9 or 10 students. The size of double-subject teaching groups (see Table 7) did not necessarily reflect the size of the sixth form; a large sixth form could have group sizes below that of a small sixth form because of the different policies schools adopted in selecting students. A third of sixth forms offering the double-subject had group sizes of four or less on those occasions where students were taught separately from those taking the single-subject. For other sixth form mathematics courses there were wide variations in class sizes (see Table 8) from two or three to the mid-20s arising sometimes from difficulties in accommodating on the timetable the variety of subject combinations which students had chosen.

Table 6 Sizes of single-subject A-level teaching groups in the survey schools

Table 7 Sizes of all double-subject A-level teaching groups in the survey schools

*Mathematics counts Para 587 expresses similar views.

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Table 8 Sizes of O-level/CSE/CEE mathematics teaching groups in the survey schools

Several factors must be taken into account when determining the suitable size of a class. Many schools in the survey sought to achieve reasonably economical group sizes in a variety of ways, including joint provision with other schools. There is no reason why group sizes should remain constant whatever the work in hand; it is necessary to consider closely the form and purposes of the lessons intended and also the kind of learning they are expected to stimulate. Large groups can generate a great deal of written work for the teacher to mark but this disadvantage should be weighed against the economic use of teaching staff. Some lessons seen during the survey could have been conducted just as effectively with a much larger group drawn from similar classes. A few schools combined groups for some parts of the work (for example, the pure mathematics element of the single-subject course) and separated them for others, with benefit. Quite often schools separate their single and double-subject candidates only for the additional periods (usually 4 to 6 per week) which are attended only by those taking the double-subject. A small number of schools found it necessary to combine first and second year classes for at least part of the time, and this arrangement was useful when the pattern of work enabled the first year students to benefit from the experience of those in their second year.

Accommodation

Sixth form mathematics understandably does not claim a high priority in the assignment of working areas. Classes were frequently found scattered around the premises in rooms that did little in their furnishing or by way of visual display to enhance an interest in the subject. With so many commercially produced posters and prints available, it is disappointing to find mathematics teaching taking place in such an arid atmosphere. Rooms were sometimes inconveniently small and with blackboards so limited that no problem could be completed without rubbing off the early part of the solution. Only one school in seven was considered by HMI to provide 'good' accommodation for sixth form mathematics. Quite modest expenditure might well transform many rooms.

Use of books

Mathematics, in contrast to some other subjects, seems to have become accepted by students as an activity which can be pursued with the minimum of dependence on books, except as a source of examples. This situation may have come about as a result of teaching styles which encourage the student to rely heavily on the teacher as almost the sole source of knowledge. Textbooks are often written to provide an

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independent means of learning but they are less frequently used for this purpose than they might be. During the survey HMI observed many lessons where the treatment of a topic did not differ greatly from what could have been read in the adopted textbook. The general (non-textbook) reading material in the library was rarely consulted by students.

Much has been said in recent years about the development of language skills and their contribution to all aspects of the curriculum, including mathematics. To some extent it is a little strange to find an emphasis in the primary school and early secondary years on learning mathematics from course books and workcards rather than by means of oral explanation while in the case of the sixth form the situation is often reversed. With classroom time at a premium it is imperative that students should be encouraged to read widely, to sample a variety of textbooks which offer different insights, to acquire some knowledge of the historical background of the subject, to obtain a grasp of some of the developments which are currently taking place and to find enjoyment in books of recreational mathematics. Furthermore a library should aim to provide for both students and teachers a collection of journals, magazines, articles and other non-book material which updates and extends the more permanent collection of books and encourages a sense of immediacy.

Other resources including computers

There have been several attempts over the years to encourage work that draws upon practical experience but this seems to have proved difficult for teachers to organise in practice. At sixth form level statistics provides the most promising area for reappraisal and a few schools were found in the survey where students collected their own experimental data for analysis. Some CEE work had encouraged a closer look at first-hand material, but on the whole mathematics was taught from a position too remote from the way mathematical processes are actually employed. A school needs to be continually collecting material, both in three-dimensional and printed form, which utilises mathematical ideas or provides the raw data for investigation. Not only has visual and manipulative experience an important contribution to make to understanding, it also provides motivation and the kind of experience which students will need to develop if they are to make use of mathematics in adult life.

In A-level work there was a very widespread use of the pocket calculator as a substitute for mathematical tables. It was disappointing to find much less use being made of them in other examination courses in mathematics. The fact that some examination rubrics would not allow their use during the actual examination was often interpreted to mean that they cannot be used at any time during the course. Much valuable mathematical activity can be derived from the use of these devices beyond the more obvious purposes for which they are largely used at the present time. In particular they allow students to carry out arithmetical processes using real data which previously would have proved too difficult or tedious. Courses need to be modified to exploit more fully this inexpensive piece of technology.

Even before the pocket calculator has had a serious chance to influence mathematical teaching, it is being overtaken, although not displaced, by more sophisticated computing devices which are rapidly

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becoming less expensive and are capable of providing both a visual and printed display. The influence that the micro-computer was having upon mathematics at the time of the survey was still in the early stages of development. However, schools were visited where students were encouraged to exercise initiative and ingenuity in devising and running computer programmes. These are qualities which mathematics courses need to develop.

As microcomputers become more readily available, they will be capable of changing significantly the way mathematics is presented visually in the classroom. Programming procedures will influence the methods used for solving problems and there will be a greater emphasis on numerical techniques. The microcomputer's capacity to store and rapidly reorganise data makes it an invaluable tool to carry out complex statistical and combinatorial investigations. The consequences for mathematics teaching are of the greatest significance and all concerned need to consider carefully how this expensive resource can be used to the best effect.

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7 Some issues

Distinctive features of sixth form work in general

There are a number of general characteristics of the 16-18 phase of education which mathematical courses need to take into account. The mainly voluntary nature of post-16 education means that the motivation of students and their attitudes to learning are more positive and this in turn affects the choice of teaching methods. Even greater importance is attached to qualifications and this provides both an incentive and a focus for study. At this stage students are more mature and their interests are more adult. They are able to comprehend ideas at a deeper intellectual and emotional level and this makes possible the pursuit of ideas which have a greater impact on their future outlook. Sixth form studies can usually be undertaken in smaller teaching groups which permit teachers and students to have a closer understanding of one another, more open discussion and greater opportunity to use a variety of teaching methods. During this phase students need to exercise an increasing responsibility for their own learning. Finally, there is a greater awareness amongst students of how personal and practical experience needs to relate to their studies.

Some special features of mathematical work in the sixth form

The most dominant influence on sixth form mathematics courses observed by HMI arose from the demand for mathematical qualifications. Without underestimating the importance of this influence, it is reasonable to question the educational implications. The achievement of appropriate qualifications is only one of the purposes of sixth form work; mathematics teaching may be tending to exaggerate one purpose at the expense of others. Earlier paragraphs have tried to indicate how existing courses might develop a wider perspective aimed at improving students' understanding, and also provide them with learning skills which they will need if they are to profit from the qualifications they have so zealously acquired.

Mathematics courses now have to accommodate a wider range of student interests and abilities than ever before, yet the subject is often presented in a way which is unnecessarily insular and inflexible. There has been a substantial growth in courses incorporating statistics but teaching methods have depended on data taken from textbooks rather than sources that encouraged experimental approaches. It is difficult to see how mathematics can become a useful and stimulating activity if technical accomplishment is not closely related to real applications. Furthermore a close relationship between theoretical mastery and the problems which the theory can solve makes an important contribution to mathematical understanding.

Schools have a special responsibility towards those who continue to study mathematics in higher education, particularly those students who might enter the teaching profession. Such students need to be excited by the structural patterns within mathematics and its capacity to throw light on apparently diverse and unrelated problems. Their

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experience as students in the sixth form may have a lasting influence in determining how they see their role as teachers.

The need for adjustments

In recent years higher proportions of school pupils have stayed on in sixth forms. Greater numbers have studied mathematics and after taking A-level have moved on to an increasing variety of occupations. At the same time development projects, in this country and elsewhere, have tried to put into courses a more powerful range of abstract ideas and also a wider range of applications. Whilst these influences have brought about some changes in the content of syllabuses, they have not altered substantially the way in which the subject is taught. The teaching seen during the survey relied predominantly on training students in a sequence of techniques, each illustrated by a comparatively limited range of stereotyped exercises. The systematic cultivation of self-reliance and the discipline of independent study were less in evidence; very little time indeed was given to the history and social context of the subject or its distinguishing characteristics as a human achievement. A style of teaching was adopted which was directed to producing the best results in the examination at the end of the course - an understandable but in practice restrictive response to the pressures to which schools are subjected.

There should be a more balanced compromise between the short-term advantages of a qualification and the long term interests that such a qualification is intended to serve. It seems to be the hope that the long-term gains will somehow materialise without being consciously sought and planned. More could be done to help young people to adjust to adulthood and that mathematics can make a distinct contribution. In recent years when there have been more pressing matters lower down the school system, the sixth form may not have been in the forefront of teachers' minds. To some extent therefore the sixth form has been left to operate along lines that appeared to work well in the past. In many ways it should be easier to change approaches with the sixth form than with younger pupils since classes are smaller and there is for many students a high level of commitment.

Lessons observed over the years in a number of sixth forms show that teaching for broader interest and teaching to excite the pupils' imagination are completely compatible with examination success. Such teaching is to be found in a number of schools whose examination records are good by any standards; although evidence of this kind can never be completely conclusive as sceptics may always claim that any school which is able to teach in this way is advantaged in its intake. However, more than one piece of research (see Appendix D) has shown that while mathematics is regarded by students in schools as a highly desirable qualification, it is well down the list of subjects in order of popularity. If more young people are to enjoy mathematics, to respond to it as an exciting challenge, and to see it as more than an essential qualification for some other career, then teachers need reassurance that employers and higher education are seeking young people with greater accomplishment than a formal qualification alone can signify. At the same time the qualifications themselves must be derived from syllabuses which leave time for the teacher to pursue a more extensive range of objectives in addition to the acquisition of technical competence.

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Appendices

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Appendix A

General data relating to the HMI sixth form survey

*DES Statistics of education 1978 Vol. 1 HMSO 1980 Education for 16-19 year olds. Table 1 HMSO 1980.

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Appendix B

Students taking A-level mathematics in the survey

*DES Statistics of education 1978 Vol 1. HMSO 1980

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Appendix C

Table C1
Background and student numbers for schools and colleges in the survey

NOTES

Column

1: Sixth form in process of transition (T) from selective or non-selective to comprehensive.
2: Age of entry. Some sixth form colleges have younger pupils in the process of being phased out.
3: Schools located in an urban/industrial environment (U). Mainly large conurbations and inner city areas.
4: If the school or college has been reorganised since 1970 then previous type of school - grammar (G), modern (M) or mixture (GM). This may have influenced provision in the existing sixth form and if the school was also transitional would indicate that the sixth form contained former grammar or modern school pupils. Former technical schools have been regarded as grammar.
5: If the former status of the school had affected the composition of the sixth form in terms of boy/girl bias this is indicated (b or g).
6: If school was single-sex this is indicated (b or g).
7: Size of traditional sixth form (TS) rounded off to the nearest 10.
8: Percentage of girls in the traditional sixth form (g/TS).
9: Size of new sixth form (NS) rounded off to the nearest 10 except for very small numbers.
10: Percentage of students in the sixth form who were new sixth formers.
11: Percentage of traditional sixth formers who were taking A-level mathematics (M/TS). Double-subject candidates counted once.
12: Percentage of A-level mathematics candidates who were girls (g/M).
13: Percentage of the traditional sixth form offering A-level mathematics as a double subject (FM/TS).
14: Percentage of the traditional sixth form taking A-level mathematics and one non-science A-level (1nS/TS)
15: Percentage of the traditional sixth form taking A-level mathematics and two non-science A-levels (2nS/TS).

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16: Percentage of A-level mathematics candidates also taking two non-science A-levels (2nS/M).

Table C1

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Table C1 (cont)

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Table C1 (cont)

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Table C2
Mathematics courses available in the sixth forms included in the survey

NOTES

Column

1. A-level mathematics - mechanics option. See also column 4.
2: A-level mathematics - statistics option. See also column 4.
3: Traditional A-level. Marked (x) if there is an entry in columns 1, 2 or 5.
4: Modern A-level eg SMP, ME1 or one of the alternatives offered by examining boards. In many instances these syllabuses contain both mechanics and statistics.
5: A-level pure mathematics.
6: A-level applied mathematics (AM) or further mathematics (FM). An entry in this column will show whether the school or college was offering the double subject.
7: A-level statistics as a single subject (ST) or A-level computer science (CS). This includes provision for computer science even when tuition took place outside the school or college.
8: AO-Ievel/O-Ievel additional mathematics (Add or applicable mathematics (App) or statistics (St) or computer studies (Cs).
9: O-Ievel mathematics (m) or statistics (st) or computer studies (cs).
10: CSE mathematics/arithmetic (m) or computer studies (cs).
11: CEE mathematics courses
12: RSA arithmetic or in one or two instances Certificate of Proficiency in Arithmetic.
13: City & Guilds Foundation course numeracy
14: Mathematics support for students studying sciences, economics and geography but not externally examined. This column needs to be read in conjunction with some other examination courses particularly those in additional mathematics and statistics.
15: General Studies. Any course, other than computing, aimed at sixth form students in general or related to A-level general studies.
16: Computing, including any general studies component, that was not examined externally.

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Table C2

[page 42]

[page 43]

[page 44]

Appendix D

Some suggestions for further reading

The following list is provided as guidance to readers interested in pursuing some of the issues raised in this publication. The choice is not intended to be exhaustive and is limited mainly to recent published material. Items marked with an asterisk(*) might provide a more selective basis for reference.

*Sixth Form Education in the Schools of Wales. HMSO 1978

The Sixth Form and its Alternatives. J. Dean, K Bradley, B Choppin & D Vincent. NFER 1979

*Study Skills at 16 Plus. (Research in Progress 4.) R Tabberer & J Allman. NFER 1981

RESEARCH ARTICLES/REPORTS

*Report on an enquiry into attitude and performance among pupils taking advanced level mathematics. J Selkirk. University of Newcastle upon Tyne, School of Education. 1972

*The Training of Mathematicians. R R McLone. SSRC 1973

A study of attitudes towards mathematics in relation to selected school characteristics. R F Kempa and J M McGough. British Journal of Educational Psychology, Volume 37, p296-304. 1977

A study of the attitudes of pupils studying advanced level mathematical degree courses. K D C Stoodley. University of Bradford. 1978

Interest as a factor in attainment in GCE A-level. J K Backhouse. University of Oxford, Department of Educational Studies. 1979

*The mathematical education of engineers at school and university. T J Heard. University of Durham, Department of Engineering Science. 1978

*Choice of mathematics at school and university. Oxford Educational Research Group, University of Oxford, Department of Educational Studies. Started 1978 but not yet complete.

Examination O-Ievel grades and teachers' estimates as predictors of the A-level results of UCCA applicants. R J L Murphy. British Journal of Educational Psychology, Volume 51, p1-9. 1981

REPORTS/CONFERENCES

Proceeding of the Third International Congress of Mathematical Education. Organising Committee of the 3rd ICME. 1977

*Mathematics for Sixth Formers. Association of Teachers of Mathematics. 1978

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*Report on the teaching of mathematics to non-specialists in the sixth form. Headmaster's Conference. 1978

*Statistical Education and Training for 16-19 year olds. Centre for Statistical Education, Department of Probability and Statistics, University of Sheffield. 1979

Proceedings of the University Mathematics Teaching Conferences held at the University of Nottingham:

Adapting University Mathematics to Current and Future Educational Needs. 1975
University Mathematics Curricula and the Future. 1976
Teaching Methods for Undergraduate Mathematics. 1977
Assessment and Service Teaching. Two aspects of University Mathematics. 1978
Undergraduate Mathematics - Using and Learning. 1979

A minimal core of syllabus for A-level mathematics. Standing Conference for University Entrance & Council for National Academic Awards. 1978

*A-level Mathematics. A report of the Stoke Rochford Conference. The School Mathematics Project. 1980

*Mathematics Counts (Cockcroft Report). HMSO 1982

Mathematics at A-level. A Discussion Paper on the Applied Content. The Mathematical Association 1982.

ARTICLES IN MATHEMATICAL JOURNALS

Mathematics at the university. The Mathematical Gazette, Volume 59, No. 410. p221-228 (also pamphlet). 1975

A-level grades - a matter of degree? J Anderson. The Mathematical Gazette, Volume 63. No. 423, p7-10. 1979

Creativity. A R Tammadge. The Mathematical Gazette. Volume 63, No. 425, p145-162. 1979

*Taken an A-level question ... A Fitzgerald. The Mathematical Gazette, Volume 64, No. 427, p1-4. 1980

Core Syllabuses and A-level Mathematics. Mathematics Teaching No. 90. p40-43. 1980

Various articles. Journal of Mathematical Modelling for Teachers. Department of Mathematics. Cranfield Institute of Technology. 1978 et al

TEACHING RESOURCES

A short list of books and pamphlets, but excluding textbooks, which teachers might find useful as a source of ideas.

Mathematics across the Curriculum. J Ling. Schools Council Project - The Mathematics Curriculum: A Critical Review. Blackie 1977

*Topics in Mathematics: Ideas for the Secondary Classroom. (Reprints of problems from the Mathematical Gazette). Bell & Hyman. 1979

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Evaluation: Of what, by whom, for what purpose? The Mathematical Association. 1979

Pupils' Projects: Their use in Secondary School Mathematics. The Mathematical Association. 1980

Booklists for the Teaching of Mathematics in Schools. The Mathematical Association. 1980

*Mathematics Teacher Education Project. G T Wain & D Woodrow. Tutor's Guide and Students' Material. (A project funded by the Nuffield Foundation to assist with teacher education). Blackie. 1980

*Teaching Statistics 11-16. Schools Council Project on Statistical Education (11-16). Foulsham. 1980

Diversions in Modern Mathematics. B Lewis. Heinemann. 1981 (An illustration of an 'enrichment' course for the sixth form)

*The Real World and Mathematics. H Burkhardt. Blackie 1981

Mathematical Games. M Gardner. Scientific American. A series of monthly articles that have been appearing since 1956. Sets of these have been reprinted under various titles and publishers:

Mathematical Puzzles and Diversions. Penguin. 1965
More Mathematical Puzzles and Diversions. Penguin. 1966
New Mathematical Diversions. George Allen & Unwin 1969
Further Mathematical Diversions. George Allen & Unwin/Penguin. 1970
Martin Gardner's Sixth Book of Mathematical Games. Freeman. 1971
*Mathematical Carnival. George Allen & Unwin/Penguin. 1976
Mathematical Magic Show. George Allen & Unwin. 1977
Mathematical Circus. Allen Lane. 1981