Cockcroft (1982)

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

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

Part 2

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

Part 3

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

Appendices

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

Index

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

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

ISBN 0 11 270522 7

Chapter 1 Why teach mathematics?
[pages 1 - 4]

1 There can be no doubt that there is general agreement that every child should study mathematics at school; indeed, the study of mathematics, together with that of English, is regarded by most people as being essential. It might therefore be argued that there is no need to answer the question which we have used as our chapter heading. It would be very difficult - perhaps impossible - to live a normal life in very many parts of the world in the twentieth century without making use of mathematics of some kind. This fact in itself could be thought to provide a sufficient reason for teaching mathematics, and in one sense this is undoubtedly true. However, we believe that it is of value to try to provide a more detailed answer.

2 Mathematics is only one of many subjects which are included in the school curriculum, yet there is greater pressure for children to succeed at mathematics than, for example, at history or geography, even though it is generally accepted that these subjects should also form part of the curriculum. This suggests that mathematics is in some way thought to be of especial importance. If we ask why this should be so, one of the reasons which is frequently given is that mathematics is 'useful'; it is clear, too, that this usefulness is in some way seen to be of a different kind from that of many other subjects in the curriculum. The usefulness of mathematics is perceived in different ways. For many it is seen in terms of the arithmetic skills which are needed for use at home or in the office or workshop; some see mathematics as the basis of scientific development and modern technology; some emphasise the increasing use of mathematical techniques as a management tool in commerce and industry.

3 We believe that all these perceptions of the usefulness of mathematics arise from the fact that mathematics provides a means of communication which is powerful, concise and unambiguous. Even though many of those who consider mathematics to be useful would probably not express the reason in these terms, we believe that it is the fact that mathematics can be used as a powerful means of communication which provides the principal reason for teaching mathematics to all children.

4 Mathematics can be used to present information in many ways, not only by means of figures and letters but also through the use of tables, charts and diagrams as well as of graphs and geometrical or technical drawings. Furthermore, the figures and other symbols which are used in mathematics can be manipulated and combined in systematic ways so that it is often possible to deduce further information about the situation to which the mathematics relates. For example, if we are told that a car has travelled for 3 hours at an average speed of 20 miles per hour, we can deduce that it has covered a distance of 60 miles. In order to obtain this result we made use of the fact that:

However, this mathematical statement also represents the calculation required to find the cost of 20 articles each costing 3p, the area of carpet required to cover a corridor 20 metres long and 3 metres wide and many other things as well. This provides an illustration of the fact that the same mathematical statement can arise from and represent many different situations. This fact has important consequences. Because the same mathematical statement can relate to more than one situation, results which have been obtained in solving a problem arising from one situation can often be seen to apply to a different situation. In this way mathematics can be used not only to explain the outcome of an event which has already occurred but also, and perhaps more importantly, to predict the outcome of an event which has yet to take place. Such a prediction may be simple, for example the amount of petrol which will be needed for a journey, its cost and the time which the journey will take; or it may be complex, such as the path which will be taken by a rocket launched into space or the load which can be supported by a bridge of given design. Indeed, it is the ability of mathematics to predict which has made possible many of the technological advances of recent years.

5 A second important reason for teaching mathematics must be its importance and usefulness in many other fields. It is fundamental to the study of the physical sciences and of engineering of all kinds. It is increasingly being used in medicine and the biological sciences, in geography and economics, in business and management studies. It is essential to the operations of industry and commerce in both office and workshop.

6 It is often suggested that mathematics should be studied in order to develop powers of logical thinking, accuracy and spatial awareness. The study of mathematics can certainly contribute to these ends but the extent to which it does so depends on the way in which mathematics is taught. Nor is its contribution unique; many other activities and the study of a number of other subjects can develop these powers as well. We therefore believe that the need to develop these powers does not in itself constitute a sufficient reason for studying mathematics rather than other things. However, teachers should be aware of the contribution which mathematics can make.

7 The inherent interest of mathematics and the appeal which it can have for many children and adults provide yet another reason for teaching mathematics in schools. The fact that 'puzzle corners' of various kinds appear in so many papers and periodicals testifies to the fact that the appeal of relatively elementary problems and puzzles is widespread; attempts to solve them can both provide enjoyment and also, in many cases, lead to increased mathematical understanding. For some people, too, the appeal of mathematics can be even greater and more intense. For instance:

Anna and I had both seen that maths was more than just working out problems. It was a doorway to magic, mysterious, brain-cracking worlds, worlds where you had to tread carefully, worlds where you made up your own rules, worlds where you had to accept complete responsibility for your actions. But it was exciting and vast beyond understanding. (1)

Even though it may be given to relatively few to achieve the insight and sense of wonder of 7 year old Anna and of the young man who in later years wrote the book, we believe it to be important that opportunities to do so should not be denied to anyone. Indeed, we hope that all those who learn mathematics will be enabled to become aware of the 'view through the doorway' which many pieces of mathematics can provide and be encouraged to venture through this doorway. However, we have to recognise that there are some who, even though they may glimpse the view from time to time as they become interested in particular activities, see in it no lasting attraction and remain indifferent or in some cases actively hostile to mathematics.

8 There are other reasons for teaching mathematics besides those which we have put forward in this chapter. However, we believe that the reasons which we have given make a more than sufficient case for teaching mathematics to all boys and girls and that foremost among them is the fact that mathematics can be used as a powerful means of communication - to represent, to explain and to predict.

9 It is interesting to note two very different uses which have been made of mathematics in the current Voyager space programme. Not only has the predictive power of mathematics been used to plan the details of the journeys of the two Voyager spacecraft but examples of mathematics have been included in the information about life on Earth which was affixed to each of the spacecraft before they were launched in 1977 to explore the outer Solar System and then to become 'emissaries of Earth to the realm of the stars'. (2) The reason for including examples of mathematics is explained in these words:

So far as we can tell, mathematical relationships should be valid for all planets, biologies, cultures, philosophies. We can imagine a planet with uranium hexafluoride in the atmosphere or a life form that lives mostly off interstellar dust, even if these are extremely unlikely contingencies. But we cannot imagine a civilisation for which one and one does not equal two or for which there is an integer interposed between eight and nine. For this reason, simple mathematical relationships may be even better means of communication between diverse species than references to physics and astronomy. The early part of the pictorial information on the Voyager record is rich in arithmetic, which also provides a kind of dictionary for simple mathematical information contained in later pictures, such as the size of a human being.

11 The problems of learning to use mathematics as a means of communication are not the same as those of learning to use one's native language. Native language provides a means of communication which is in use all the time and which, for the great majority of people, 'comes naturally', even though command of language needs to be developed and extended in the classroom. Furthermore, mistakes of grammar or of spelling do not, in general, render unintelligible the message which is being conveyed. On the other hand, mathematics does not 'come naturally' to most people in the way which is true of native language. It is not constantly being used; it has to be learned and practised; mistakes are of greater consequence. Mathematics also conveys information in a much more precise and concentrated way than is usually the case with the spoken or written word. For these reasons many people take a long time not only to become familiar with mathematical skills and ideas but to develop confidence in making use of them. Those who have been able to develop such confidence with relative ease should not underestimate the difficulties which many others experience, nor the extent of the help which can be required in order to be able to understand and to use mathematics.

Implications for teachers

12 We conclude this chapter by drawing the attention of those who teach mathematics in schools to what we believe to the implications of the reasons for teaching mathematics which we have discussed. In our view the mathematics teacher has the task

• of enabling each pupil to develop, within his capabilities, the mathematical skills and understanding required for adult life, for employment and for further study and training, while remaining aware of the difficulties which some pupils will experience in trying to gain such an appropriate understanding;
• of providing each pupil with such mathematics as may be needed for his study of other subjects;
• of helping each pupil to develop so far as is possible his appreciation and enjoyment of mathematics itself and his realisation of the role which it has played and will continue to play both in the development of science and technology and of our civilisation;
• above all, of making each pupil aware that mathematics provides him with a powerful means of communication.

Footnotes

(1) Fynn Mister God, this is Anna Collins Fount Paperbacks 1974

(2) Carl Sagan Murmurs of Earth Hodder and Stoughton 1979