Norwood (1943)

Notes on the text
Preliminary pages Membership, Contents, Introduction

Part I Secondary education
Chapter 1 Nature of secondary education
Chapter 2 Secondary education as it is
Chapter 3 Secondary education as it might be

Part II Examinations
Chapter 4 Existing examinations
Chapter 5 The reorganisation of examinations (i)
Chapter 6 The reorganisation of examinations (ii)
Chapter 7 The Inspectorate

Part III Curriculum
Chapter 8 The curriculum in general
Chapter 9 Physical education
Chapter 10 Religious education
Chapter 11 English
Chapter 12 History
Chapter 13 Geography
Chapter 14 Mathematics
Chapter 15 Natural science
Chapter 16 Modern languages
Chapter 17 Classics
Chapter 18 Art, music, handicrafts
Chapter 19 Domestic subjects
Chapter 20 Education for commerce
Chapter 21 Wales and the teaching of Welsh

Conclusions and recommendations
Appendix A Note on the Central Welsh Board
Appendix B List of witnesses
Index

Report of the Committee of the Secondary School Examinations Council appointed by the President of the Board of Education in 1941

London: HM Stationery Office 1943

Chapter 14 Mathematics
[pages 104 - 108. In the printed version this is Part III Chapter VII]

Amid the changes in curriculum which have taken place in the last forty years mathematics perhaps more than any other subject has retained the position assigned to it by tradition. At the present time it is very rare indeed for any boy to give up mathematics during the main school course or to omit it from the subjects which he offers for the School Certificate examination. With girls the situation is different; in girls' schools the place assigned to mathematics varies far more than in boys' schools according to the attitude of the head mistress and the strength of the mathematical staff. Thus in some schools very few girls drop the subject; in others a third or even more omit it from the School Certificate examination.

As regards the examination itself mathematics is taken by about 90 per cent of the total number of candidates and thus ranks next to English and French. The position of these three subjects has, of course, been influenced to great extent by the 'group' requirements. At the same time it is sometimes felt, particularly in girls' schools, that there are a number of pupils, who, being weak in most subjects or having a blind spot for mathematics, find the standard of the examination in mathematics hard, and often great difficulty is experienced in bringing them up to the required level.

The position of mathematics, however, is not to be explained only by tradition or by examination regulations. Even if the value of the subject on other grounds is excluded, there yet remains the practical consideration that for many pupils mathematics is necessary for their careers; for others it is necessary in order to gain admittance to courses of advanced work in other subjects, whether at the university or elsewhere.

Mathematics then occupies a position in the schools strongly entrenched by tradition, the influence of examinations and practical considerations of career or utility. Yet it would be a mistake to imagine that as a result of the security so enjoyed methods and aims have become stereotyped. In this subject no less than in other traditional subjects there has been much rethinking of aim and method. Compared with the theory and practice of thirty years ago, from which standpoint we suspect that some of the criticism made on mathematics is directed, mathematics has reformed itself; the various branches of the subject have coalesced, dead matter has been pruned away; the course has gained in unity and embraces content which some years ago was reserved only for advanced students; progress has been accelerated and laborious formal proof and rigidly logical sequence have been replaced by shorter methods and by the demonstration of mathematical principles and their practical application. Many teachers would be the first to agree that in many respects revision of syllabus might with advantage be carried even further, particularly in the lower forms of schools; but they would point to the changes already made as an earnest of others to come.

The point which concerns us first in the above review of the situation is the place of mathematics in the curriculum. In practice, the subject occupies more time throughout the main school than any other single subject; admittedly English in the broad sense in which it runs through many subjects engages more time, but this is natural and desirable, and it might equally be said that mathematics is inherent in branches of natural science teaching. For pupils who show ability in mathematics or who need or may need that it should be carried to an advanced stage for purposes of a career we are not disposed to suggest a reduction in the content of the mathematical course or in the time given to it. On the other hand we are not convinced by the evidence put before us that pupils, whose abilities lean most markedly in other directions or whose disability in mathematics is established beyond reasonable doubt, should devote the same attention to mathematics or be expected to cover the same ground in the same way as those whose interests and needs justify mathematics as an important part of their school course. It may be objected that at present the situation is met by the two courses 'elementary' and 'additional mathematics' required for the School Certificate examination. Yet reference to recent figures shows that, whereas 91 per cent of the candidates took elementary mathematics, only 5 per cent took additional mathematics, and we understand that the great majority of that 5 per cent consisted of boys. In short, we should recommend that differentiation should be carried further than these figures suggest, that while a high standard, such as that conveniently indicated by 'additional mathematics', should be retained for a proportion of pupils, and reached by a larger proportion than at present, it should also be possible for some pupils to undertake a course in mathematics different from, and less exacting in content than the normal course.

We believe the establishment of this principle to be right not merely as a concession to those who show marked disability in mathematics, but also in the interest of those whose interest and ability have been shown to lie elsewhere.

As regards the pupil whose tastes are 'all round', who is equally able in mathematics and in languages, for example, and in art, we think it impossible to make any general observations; it may be to his advantage, or it may not, to carry mathematics to as advanced a stage as he is capable of carrying it; this is a problem which can be decided only at the school in the light of such considerations as probable career, special promise in particular ways, general cast of mind, nature of the teaching, total load carried by the pupil.

We are thus faced with a set of needs which demand satisfaction and which we may summarise thus:

(i) to discover aptitude, mathematical or other,

(ii) to establish at least a groundwork of mathematics necessary for everyday affairs,

(iii) to furnish opportunity for the ablest boys and also for those who, while not among the ablest, need or may need mathematics for a career,

(iv) to provide for those who through lack of ability or interest in mathematics make little progress,

(v) to provide for those who, while having ability for mathematics, yet on account of special abilities and tastes in other subjects prefer and are well advised to undertake a lightened course in it.

(a) In the first three years of the secondary grammar school mathematics should in our opinion be taken by all pupils. Opinion is on the whole in favour of mathematics being taught during these years, as well as later, in 'sets' arranged to allow for varying ability. We regard three years as a minimum for all pupils for these reasons; first, it is essential that all pupils should gain at least a knowledge of such mathematics as is necessary for everyday affairs and some acquaintance with the most elementary mathematical principles; secondly, full opportunity must be given for mathematical ability or disability to declare itself after experience of the personality and the teaching methods of more than one teacher. We stress this latter point because we regard the early stages in mathematics as most important, a pupil's liking and appreciation of the subject often depending on the way in which it was presented to him then.

(b) We contemplate that for the succeeding two years the majority of pupils should continue a course of mathematics which would be appropriate to those who need mathematics for their career; some of them would certainly carry the subject to a higher level than the rest.

This 'normal' course from 11+ to 16+ should include, we believe, parts of arithmetic, algebra, graphs, geometry and trigonometry, treated not as isolated subjects but with the fullest measure of coordination. For the treatment of mathematics now stresses frequent numerical illustration, and this change has affected not only trigonometry, making it more suitable for less able pupils, but also algebra and graphs. Graphs would not be taught as a separate topic, but rather to illustrate and help forward the subject in hand. In geometry formal proofs of geometrical propositions would give way to informal explanations and the use of algebraical and trigonometrical methods where appropriate. Solid geometry would be touched upon in the first lessons and references to three dimensions would recur throughout the course. On the ground that they stultify the teaching, few rules would be learnt, and pupils would be taught to rely upon a few fundamental principles and their own power to use them. The process, already advanced, of purging away old-fashioned topics from algebra, arithmetic and geometry would be carried still further. The method of treatment would naturally vary according to the abilities of pupils; we do not suggest that the same treatment would be suitable for all.

For many pupils it would be desirable and possible to cover more ground. The nature of an 'additional' syllabus would depend very much upon the choice of the individual school. Sometimes mechanics is included in the syllabus of physics, and when this is done the additional course in mathematics might reasonably consist of an introduction on unambitious lines to the differential and integral calculus with such parts of numerical coordinate geometry and algebra and trigonometry as are necessary. If mechanics is not part of the physics course, it would reasonably be treated in mathematics. Another suitable topic would be the use of the globes, involving emphasis on certain parts of spherical geometry and trigonometry. The mathematical aspect of the terrestrial globe has gained added importance and interest from the development of flying. The celestial globe has a double function, the purely astronomical and its application to navigation. Both astronomy and navigation would be suitable parts of the additional course and would make great appeal. But in the choice of content for this additional syllabus the individual schools should have complete freedom.

(c) For pupils who during the three years instruction from 11-14 showed little ability or were little attracted by the subject, we should recommend a lighter course, different in content and less exacting in time than the normal course outlined above. Throughout it practical applications and illustrations would be stressed. It would include arithmetic extended to include elementary trigonometry, the elements of algebra, that is the formulation of arithmetical problems in algebraic shape, equations such as arise from and are useful in arithmetic: such parts of geometry as are needed for mensuration and trigonometry, and for the construction of simple solids; formal proofs in geometry would be omitted, but geometrical questions in the form 'Is it true?' would be used. Possibly some elementary mechanics including experimental work would be attempted.

In the sixth form mathematics would become, under the arrangement for examinations suggested in an earlier chapter, (i) a subject for scholarship examinations either alone or in conjunction with natural science, (ii) a subject for the school leaving examination taken at 18+ for entry to universities and professions and for other purposes. On this we make no observations. But we would suggest that statistics is a suitable study for boys in the sixth form who are likely to go into posts in business or to read economics at a university. For we think that in statistics, geography, descriptive economics and aspects of public administration, together with a foreign language, an excellent introduction to their later work and studies could be provided.