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 7 Calculators and computers
[pages 109 - 120]

372 We devote a separate chapter to electronic calculators and computers because we believe that their increasing availability at low cost is of the greatest significance for the teaching of mathematics. As a result of the fact that in the mid 1970s the price of electronic calculators started to fall very rapidly, very many adults now possess calculators of their own. So, too, do considerable numbers of school pupils; and many pupils of all ages and abilities who do not have a calculator of their own have access at home to a calculator belonging to some other member of the family.

373 At the present time a similarly rapid drop is taking place in the price of small computers, now more usually referred to as microcomputers, and a government programme has been announced which is intended to ensure that at least one microcomputer will be available in every secondary school by the end of 1982. Most microcomputers display their input and output on a television-type screen; some also have a printer which will provide a permanent record if one is required. As yet the number of families who possess a microcomputer is still relatively small but it seems clear that this number is likely to increase very considerably within the next few years. We are therefore in a situation in which increasing numbers of children will grow up in homes in which calculators and microcomputers are readily available, in which there is access to a variety of information services displayed on domestic television sets and in which the playing of 'interactive' games, either on microcomputers or by means of special attachments to television sets, is commonplace.

374 These developments have very great implications for the teaching of many subjects in schools. So far as the teaching of mathematics is concerned, we believe that there are two fundamental matters which need to be considered. The first concerns the ways in which calculators and microcomputers can be used to assist and improve the teaching of mathematics in the classroom. The second concerns the extent to which the availability of calculators and microprocessors should change the content of what is taught or the relative stress which is placed on different topics within the mathematics syllabus. We have not listed separately the question of the relationship between the use of calculators and the development of fluency in mental and written calculation because we believe that, although important, it arises as one aspect of these fundamental matters and needs to be considered in that context.

Calculators

375 Among the submissions which we have received there are many which make reference to the use of electronic calculators either in school or at work and the range of views which is expressed is very great indeed. For instance:

Exercise of the basic skills should not depend upon use of calculators: these should be limited to higher education.

Mathematics must be a compulsory subject, taught to a reasonable standard, using one's loaf and not a calculator.

The members were agreed on the tremendous part electronic calculators and machines now play in employment. Accordingly, it would seem sensible for the teaching of mathematics to introduce pupils to machines and to familiarise them with their capabilities and use.

Calculators have revolutionised computation and barely numerate students can overcome their weaknesses with these.

377 It is also clear that there is widespread public concern about the use of calculators by children who have not yet mastered the traditional pencil and paper methods of computation. It is feared that children who use calculators too early will not acquire fluency in computation nor confident recall of basic number facts. These fears are understandable and should not be ignored. However, such research evidence as is at present available suggests that there may be advantages which more than compensate for any possible disadvantages. In recent years a considerable number of research studies carried out in the United States of America have compared the computational performance of groups of pupils who have used calculators with that of groups of pupils who have not. Some of these studies have reported improvements among those who have used calculators in attitudes towards mathematics, in personal computational skills, in understanding of concepts and in problem solving; other studies have found no differences which are statistically significant between the performance of those who have used calculators and those who have not. From all the studies the weight of evidence is strong that the use of calculators has not produced any adverse effect on basic computational ability. We believe that this is important and should be better known both to teachers and to the public at large. Nevertheless, it remains incumbent upon those who teach mathematics to ensure that the development of appropriate skills of mental and written calculation is not neglected. Nor should a school overlook the need to make parents aware of its policy in regard to the use of calculators by pupils.

378 We wish to stress that the availability of a calculator in no way reduces the need for mathematical understanding on the part of the person who is using it. We have already explained that, for example, knowing how to multiply and knowing when to multiply involve distinct aspects of teaching and learning. A calculator can be of no use until a decision has been made as to the mathematical operation which needs to be carried out and experience shows that children (and also adults) whose mathematical understanding is weak are very often reluctant to make use of a calculator. We believe that this is a crucial point which is not always appreciated, especially by those who criticise the use of calculators in schools.

379 There can be little doubt of the motivating effect which calculators have for very many children, even at an early age. This is illustrated by the following extract from one of the submissions we have received from parents.

Following professional advice of mathematical colleagues we kept calculators away from our children until their late teens. But the youngest at age 6 got hold of a calculator to help out his 'tables' and found it such fun that he has been much more mathematically inclined since. So perhaps it would be wise to introduce simple calculators at an early age.

381 It is, of course, easy to make mistakes when using a calculator. These can occasionally arise from faulty functioning of the calculator itself but are much more likely to be the result of faulty operation by the person who is using it. For this reason it is essential that pupils should be enabled to acquire good habits in the use of calculators so as to guard against mistakes. Pupils must learn that calculations should be repeated, if possible by entering the numbers in a different order; that subtraction can be checked by addition and division by multiplication. It is also necessary to stress the importance of checking answers by means of suitable estimation and approximation. Discussion of all these matters in the classroom provides opportunity for increasing pupils' understanding of mathematical concepts and routines and so contributes to their progress.

382 It is also important that pupils should realise that not all calculators operate in exactly the same way; and that they should be able to 'explore' a calculator which is unfamiliar in order to discover whether it has particular characteristics which need to be taken into account when using it. It follows that we cannot regard the fact that the pupils in a class may possess different models of calculator as being a reason for avoiding their use. Comparison of the routines needed to perform the same task on different models of calculator can be very instructive.

383 Before we discuss more specifically some of the ways in which calculators can be used in the primary and secondary years we wish to draw attention to one further matter which we believe to be of importance. This is the use of calculators for the purpose of encouraging mathematical investigation. This can start at the primary stage, as we describe in the following paragraph. At the secondary stage, pupils should be encouraged to undertake exploration of the capabilities of the calculator itself; for example, of the largest and smallest numbers which can be entered, of what happens when the result of a calculation becomes too large or too small for the calculator to accommodate, of the use which can be made of memories and constant facilities and of the most economical methods of carrying out a variety of calculations. Investigations of this kind are appropriate for pupils of all levels of attainment and assist in developing awareness of the range of facilities which calculators provide. We suspect that many adults who possess calculators and use them regularly are not fully aware of the capabilities of the machines which they own; for example, they may not be able to say what would happen if, in given circumstances, they pressed the 'equals' key twice in succession. Calculators can also be used for more general investigations of many kinds. Work of this kind is especially valuable as a means of extending the mathematical insight of pupils whose attainment is high.

The use of calculators in the primary years

384 Many primary teachers feel some uncertainty about introducing calculators into their classrooms until more guidance is available than is the case at present. However, the increasing availability of calculators in many homes means that many children are likely to have access to a calculator at an early age and, by the age of 6, will have started to experiment to see what they can do. We therefore believe that primary teachers need to be able to use a calculator themselves, and that they should ensure that some calculators are available in the classroom for children to use. These can be used as an aid in discovery and investigational work. For example, a number of additions of the form 4 + 6, 14 + 6, 24 + 6, ..., 4 + 16, 14 + 16, 24 + 16, ... can be carried out with a calculator to enable a child to find out the result of adding numbers ending in 4 and 6. This can be followed up in subsequent mental work and other similar number patterns explored.

385 There are other ways in which a calculator can be used as an aid to teaching. For example, by entering the number 572 and then asking a child to change the display to 502, to 5720 or to 57.2, in each case by means of a single arithmetical operation, it is possible to reinforce understanding of place value. If the child cannot carry out the necessary operations the teacher is enabled to locate the area in which lack of understanding exists and to provide appropriate additional experience. Again, many children find difficulty in writing down the number which is, for example, one more than 6399 or one less than 6500. The calculator can assist with problems of this kind and provide a visual display which can be reconciled with an approach to the same question by means of the use of structural apparatus.

386 Children who make use of calculators are likely to meet decimals and negative numbers earlier than is usual at present. This provides the teacher with new opportunities of discussing these topics and the context in which they arise; the fact that some calculators record whole numbers as, for example, 5. or, less commonly, 5.000000 may also lead to questions. Because calculators differ in their capabilities and in the format in which they display their results, teachers need to be aware that some types of calculator are more suitable than others for use at the primary stage. Calculators also enable children to deal with larger numbers than would otherwise be possible. This means that investigations of such things as number patterns can be extended further. A calculator also makes it possible to deal with 'real life situations'; for example, it is possible to find the average height or weight of a whole class rather than of only a small number of children.

387 There is as yet little evidence about the extent to which a calculator should be used instead of pencil and paper for purposes of calculation in the primary years; nor is there evidence about the eventual balance to be obtained at the primary stage between calculations carried out mentally, on paper, or with a calculator. However, it is clear that the arithmetical aspects of the primary curriculum cannot but be affected by the increasing availability of calculators. In our view, it is right that primary teachers should allow children to make use of calculators for appropriate purposes, while remembering that, for the reasons which we have discussed in Chapter 6, it will remain essential that children acquire a secure grasp of the 'number facts' up to 10 + 10 and 10 x 10 and the ability to carry out, both mentally and on paper, straightforward calculations which make use of these facts.

388 It is likely that the use of calculators may bring about some change in the order in which different parts of the primary mathematics syllabus are introduced. We have already referred to the earlier introduction of the concepts of decimals and negative numbers. Decimals will assume greater importance in relation to fractions than is the case at present. Some development work on the use of calculators in the primary years is going on at the present time. In our view, more is needed both to consider the use of calculators as an aid to teaching and learning within the primary mathematics curriculum as a whole and also the extent to which the arithmetical aspects of the curriculum may need to be modified. We believe that priority should be given to this work and to providing associated in-service training for teachers.

The use of calculators in the secondary years

389 We believe that there is one overriding reason why all secondary pupils should, as part of their mathematics course, be taught and allowed to use a calculator. This arises from the increasing use which is being made of calculators both in employment and in adult life. We believe that calculators are also likely to be used increasingly in other curricular areas. We have already drawn attention to the necessity of establishing good habits when using a calculator so that mistakes can be avoided. The prime responsibility for doing this must lie with the mathematics department. If the necessary instruction is not given as part of the mathematics course, it is unlikely to be given at all. Furthermore, the necessary skills cannot be established in a few short lessons; like all other skills, they need to be practised and to have time to develop. As we pointed out in paragraph 381, the development of these skills provides opportunity for developing understanding in other ways as well.

390 Although training in the use of calculators must not be allowed to interfere with the acquisition of appropriate skills of mental and written computation, we believe that their availability should influence the complexity of the calculations which pupils are expected to carry out with pencil and paper and also the time which is spent in practising such calculations. We believe that it is reasonable to expect that most secondary pupils should be able, without using a calculator, to multiply by whole numbers up to 100, even thought they may often use a calculator to carry out calculations of this kind. Some pupils will, without difficulty, be able to multiply by larger numbers and they should not be dissuaded from acquiring this skill. Pupils should also be able to divide by numbers up to 10. However, 'long division' has always presented greater difficulty than 'long multiplication', and we suspect that many adults who were taught this process whilst at school would be unable to explain why the method works. We believe that it is not profitable for pupils to spend time practising the traditional method of setting out long division on paper, but that they should normally use a calculator. Once again, pupils who are interested in acquiring the ability to perform this calculation 'long-hand' should not be dissuaded. We believe that lower-attaining secondary pupils should use a calculator for all except the most straightforward calculations. For these pupils emphasis should be placed on using the calculator correctly and employing suitable checking procedures.

391 The availability of calculators makes it necessary to consider also the extent to which mathematical tables will continue to be used in the future. The purpose of using logarithm tables is to avoid having to carry out heavy calculations with pencil and paper by making use of a method which is easier and quicker; and also, in some cases, to perform calculations, such as those involving fractional indices, which it is very difficult to carry out with pencil and paper. A calculator provides a means of carrying out such calculations in a still more straightforward way and we believe there can be no doubt that calculators should replace logarithm tables as the everyday aid to calculation. The more sophisticated 'scientific' calculators enable the values of trigonometrical functions such as sine, cosine and tangent to be obtained without recourse to tables. We have been told that many pupils already possess calculators of this kind. As the availability of these calculators increases we believe that the use of trigonometrical tables will decline and eventually cease; we would encourage this.

392 Calculators can be used to assist in the teaching of a number of topics, including work with operations, functions, exponents, polynomials, square roots and problem solving. However, relatively little published material is yet available which illustrates ways in which calculators can be used as a means of developing mathematical understanding; much of the material which does exist is contained in the publications of the professional mathematical associations. We suspect that very few of those who teach mathematics at secondary level are at present making use of calculators as an aid to teaching; and that most of those who encourage their pupils to use calculators regard them solely as a way of avoiding tiresome computation. This can, of course, be very helpful when teaching a topic such as trigonometry, which usually involves a considerable amount of computation in its early stages. A reduction in the time spent on computation means that there is greater opportunity to develop the underlying concepts. Examples involving the use of Pythagoras' theorem to calculate lengths are also much easier if a calculator is available. However, these are not examples of the use of calculators in the way to which we referred at the beginning of this paragraph. There is an urgent need for an increase in the limited amount of work which is at present being undertaken to develop classroom materials designed to develop understanding of fundamental principles.

The availability of calculators

393 Throughout the preceding paragraphs we have assumed that calculators will be available for pupils to use. We conclude our discussion of the use of calculators in the secondary years by considering two matters which are related and which are likely influence the policy about the use of calculators in the classroom which is adopted by individual schools. The first is the extent to which calculators are available for the use of pupils in secondary schools, the second the question of their use in public examinations. At present there is no common policy among examination boards concerning the use of calculators in mathematics examinations. Some boards do not permit calculators to be used at all, some allow them in certain examinations or certain papers but not in others, a few permit calculators to be used freely.

394 One reason which is often given for not allowing calculators to be used in public examinations is that candidates who do not possess calculators will be at a disadvantage. This argument assumes that a school is unlikely to be able to provide a calculator for the use of each pupil who is attempting a mathematics examination and who does not have a calculator of his own. We do not accept that this should be the case. At the present time the cost of providing some 200 calculators of a type suitable for use by pupils in the 11-16 age range would not exceed £2,000 at a generous estimate. This number of calculators would enable all the pupils in a 6-form entry 11-16 school (approximately 180 pupils in each year group) who were likely to be studying mathematics at anyone time to have the use of a calculator, even if no pupil possessed his own; it would also provide some spares. This number would also enable all those likely to be attempting a public examination in mathematics on any one day to be provided with a calculator. We have been told by one LEA that the approximate initial cost of equipping a specialist teaching area for science is £17,000, for woodwork £14,000, for metalwork £25,000, for cookery and housecraft £16,000, for needlework £7,500. This means that the total cost of equipping specialist areas for the teaching of science, craft and home economics in a 6-form entry school is unlikely to be less then £150,000. We have also been told that the total cost of equipping six rooms of the kind usually provided for the teaching of mathematics is about £10,000. We do not believe that it is unreasonable to expect that one seventieth of the difference between these sums should be spent on providing calculators and we recommend that, when equipping new secondary schools, sufficient calculators should be provided to enable each pupil to have the use of one during mathematics lessons. We recognise, however, that at a time of increasing financial pressure some of those schools which do not yet have sufficient calculators may find difficulty in providing money from their capitation allowances for the purchase of additional calculators. We therefore recommend that steps should be taken, perhaps by means of a scheme similar to that for providing microcomputers in secondary schools, to ensure that the necessary calculators are available in secondary schools as soon as possible and in any case not later than 1985.

395 It follows from this that we believe that examination boards should design their syllabuses and examinations on the assumption that all candidates will have access to a calculator by 1985. Because it will still be necessary to ensure that candidates are able to carry out straightforward calculations without recourse to a calculator, it may well be appropriate that the use of calculators should not be permitted in certain papers. It will, however, be equally important that candidates should also demonstrate their ability to use a calculator effectively.

Computer studies

396 Computer studies was first introduced into the curriculum of some secondary schools during the 1960s. It was developed largely by teachers of mathematics and in many schools there would at the present time be no computer studies in the curriculum at all if mathematicians did not undertake the teaching. For this reason, many people both within the education system and outside it have assumed that computer studies should be regarded as part of the responsibility of mathematics departments.

397 At an early stage of our work we sought advice from a number of people whom we knew to be expert in the field of computer studies in schools, so that we might decide whether computer studies (as opposed to the use of a computer as an aid to teaching mathematics) should be considered as part of the 'mathematics in schools' to which our terms of reference require us to give attention. Their view was unanimous that computer studies should not be regarded as part of mathematics but should ideally exist within a separate department. It was pointed out that the teaching of computer studies within the mathematics department was liable to give a misleading view of the subject, with too great a stress on programming and numerical operations and too little on the much wider fields of data processing and social implications; and that training as a mathematics teacher was not of itself sufficient to teach computer studies adequately.

398 However, at the present time the number of specialist teachers of computer studies is very small indeed and, although teachers of subjects other than mathematics are gradually becoming involved, the process is a slow one. It therefore seems inevitable that for the time being, and probably for some time to come, a significant proportion of the teaching of computer studies will be undertaken by mathematics teachers, who will in consequence have less time available to teach mathematics. It is probably also the case that mathematics teachers who have displayed the initiative to develop computer studies courses are likely to be among the more enterprising and effective teachers and so the loss to mathematics teaching is correspondingly greater.

399 It is not easy to quantify the amount of time which mathematics teachers devote to the teaching of computer studies. At the present time some 36,000 pupils each year attempt public examinations of one kind or another in computer studies or computer science, but not all of those who follow courses in computer studies necessarily attempt a public examination in this subject. However, it seems likely that the time which mathematics teachers devote to the teaching of computer studies is already equivalent to the normal full-time teaching commitment of at least 600 mathematics teachers, and probably of rather more than this number. We believe that this raises the question of the extent to which computer studies should be taught as a separate subject, if it is done at the expense of good mathematics teaching.

400 There is an associated demand on the time of mathematics teachers of which it is necessary to be aware, because it is a demand which is likely to increase as microcomputers become increasingly available in schools: This is the amount of time which can be taken up in responding to the requests of colleagues on the staff to write computer programs for use in the teaching of other subjects. It can take many hours to write even a short program and ensure that it runs satisfactorily. This means that although the purpose of a program may be clearly defined, its preparation is likely to involve a great deal of work, perhaps to the detriment of the mathematics teaching of staff who are already hard pressed.

401 Computing also makes demands on the time of mathematics advisers. Although a few LEAs now have advisers or advisory teachers who are responsible for computing, it is more usual for this responsibility to be assigned to mathematics advisers. Some mathematics advisers have told us that work in this field already occupies a disproportionate amount of their time. There is clear danger that, unless LEAs make suitable additional provision, not only will the time which mathematics advisers devote to the teaching of mathematics be eroded still further as more schools acquire microcomputers, but also insufficient help will be available for the development of the effective use of microcomputers themselves.

Computers as an aid to teaching mathematics

402 There can be no doubt that the increasing availability of microcomputers in schools offers considerable opportunity to teachers of mathematics both to enhance their existing practice and also to work in ways which have not hitherto been possible. In particular, the availability of a visual display offers many possibilities for the imaginative pictorial presentation of mathematical work of many kinds.

403 Nevertheless, we feel it right to point out at the outset that, although these possibilities exist and are at the present time being exploited by a very small number of teachers, we are still at a very early stage in the development of their use as an aid to teaching mathematics. The amount of work which needs to be done before microcomputers are likely to have any major effect on mathematics teaching is very great indeed and, in the paragraphs which follow, we can do no more than make suggestions based on the limited amount of information which is available to us.

404 The fact that a school possesses one, or several, microcomputers will not of itself improve the teaching of mathematics or of any other subject. It does no more than make available an aid to teaching which, if it is to be properly exploited, requires teachers who have the necessary knowledge and skill and who have been supplied with, or have had time to prepare, suitable teaching programs. It does, however, also provide a valuable resource of which individual pupils can make use and from which some are likely to derive considerable benefit; we discuss this further in paragraph 412.

405 We have already drawn attention to the fact that relatively little advantage is yet being taken of the possibilities which electronic calculators offer as an aid to mathematics teaching. Even in schools in which computer studies courses are well established and are taught by those who teach mathematics, the use which is made in the mathematics classroom of the computer facilities which are available seems often to be very limited. It is, of course, sometimes the case that programs have to be sent away to be processed and that interactive working with the computer is not possible; nor may printout in graphical form be available. Nevertheless, even in these circumstances opportunities exist, especially in respect of work at higher levels, for using the computer to assist mathematics teaching.

406 We mention this matter not because we wish in any way to decry or discourage the provision of microcomputers in schools but in order to underline the extent of the changes in classroom practice which their successful use will require and the small amount of progress in the use of other aids which has so far been made by many teachers of mathematics. For example, experience shows that teachers of mathematics do not always use the graphical and pictorial potentialities of overhead projectors to maximum advantage. We welcome the Micro-Electronics Education Programme which the government have set up. If good use is to be made of the materials which will be produced, a very extensive programme of in-service training and of follow-up support for teachers in schools and teacher training establishments will be required.

407 There is at present relatively little software (ie prepared programs) available which can be used to assist mathematics teaching. Furthermore, it has been pointed out to us that much of the limited amount of software which is available is of poor quality, with programs which are badly written and documented, sometimes inaccurate and sometimes merely 'gimmicks'. There can be no point in producing software to teach a topic which can be taught more effectively in some other way. The fundamental criterion at all stages must be the extent to which any piece of software offers opportunity to enhance and improve work in the classroom. Even though the price of microcomputers may continue to fall, the cost of producing software will not. Those who purchase software for use in the classroom need therefore to make sure that it is of good quality.

Use in the primary years

408 Members of our Committee visited two primary schools which were known to possess microcomputers. The software which was in use had been prepared by teachers in these schools and had been designed to suit the needs of children of different ages. In a reception class of 4 and 5 year olds, a program in the form of a game was in use which was designed to develop recognition of figures and letters. With 9 year old juniors an interactive program, which required a group of children to locate a 'hidden' star, was being used to teach the use of coordinates. This generated valuable discussion as the group worked out the best strategy to use and also provided opportunity for the development of logical thinking. A third program provided practice in arithmetic skills for 7 and 8 year olds. In every case, when using these and other programs, the children were working with great concentration and the motivation which use of the microcomputer provided was very evident.

409 We believe that it is by means of programs of this kind that microcomputers are likely to make their greatest contribution to mathematics teaching in the primary years. The programs can be stored on cassettes and used by children singly or in groups, as well as by the teacher. There is clearly an urgent need for programs to be written for use by children in the primary years and we hope that some of the money allotted to the Micro-Electronics Education Programme will be used for this purpose. We believe that special attention should be paid to the development of programs for mathematical activities which will encourage problem solving and logical thinking in a mathematical context.

Use in the secondary years

410 Among the secondary schools which we visited were two which were making regular use of computers in their mathematics courses. In one school a computer-assisted learning scheme was in use for first and second year pupils. In the other school the computer was used as a means of introducing various algebraic concepts to first year pupils of above-average attainment.

411 At the secondary stage we believe that there is a special need to develop the potentiality of the high-resolution graphical display which is now available on many microcomputers. This enables work to be done on graph plotting and, at a higher level, can be used to provide a visual presentation of basic ideas in calculus and of the use of iterative methods to solve equations. Many geometrical properties can also be demonstrated in ways which have hitherto only been possible by using cine-films. Moreover, the interactive nature of work with a microcomputer offers opportunity for pupils to develop greater understanding of many of the mathematical concepts which they will meet. Once again, however, we wish to emphasise the need to produce programs which are not just 'extras' but which can contribute to the main-stream mathematical work of the school.

Individual use by pupils

412 We suggest that, in very many secondary schools, the most fruitful results of the availability of microcomputers are likely in the first instance to arise from their use by individual pupils. The motivation provided by access to a microcomputer can be even greater than that provided by a calculator. Experience shows that pupils are happy to 'investigate' computer systems to which they are given access and it is important that their ready acceptance of technological innovation should be fully exploited. Many examples already exist of individual pupils who, often largely self-taught, have developed extensive and effective programs of many kinds. We therefore urge that all secondary schools which possess microcomputers should make them available for use by individual pupils to the greatest extent which is possible. Such access should be afforded both during lesson times, if the machine is not being used for other purposes, and by organising computer clubs which operate outside normal classroom hours. Pupils frequently learn to write programs from one another, and we are aware of schools in which older pupils give time to teaching younger ones. Even teachers who are experienced in the use of microcomputers may find that they are able to learn a considerable amount by observing the activities of their pupils.

413 Although children of primary age will not normally reach the stage of writing their own programs, teachers need to be aware that there are a few children of high attainment whose interest and understanding are such that they wish to attempt work of this kind. In the schools which we visited, one 10 year old had started to write programs in BASIC as a result of studying programs published in magazines and watching his own teacher working on the microcomputer. An 8 year old, although not yet at this stage, was beginning to ask questions about programming. Children who show interest of this kind should be encouraged.