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Cockcroft (1982) Notes on the text
Part 1
Part 2 Chapter 5 Mathematics in schools
Part 3 Chapter 12 Facilities for teaching mathematics
Appendices Appendix 1 Statistical information
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The Cockcroft Report (1982)
Mathematics counts 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
ISBN 0 11 270522 7
Chapter 8 Assessment and continuity
Assessment 414 Assessment is an essential part of the work of all teachers; it needs to be carried out in a variety of ways and for a variety of purposes. Much assessment is based on the marking of written work and on information which the teacher gains as a result of comments made and questions asked during discussion with an individual pupil, a group or a class. Assessment can also be rapid and informal - perhaps a brief conversation to discover whether a point has been understood or a glance at a piece of work which is in progress in order to see whether a difficulty has been overcome. When pupils are engaged in practical activity, it is necessary for the teacher to watch them at work in order to discover whether skills such as counting, measurement or the use of drawing instruments are being mastered. Assessment in all of these ways is necessary in order to enable the teacher to monitor the development and progress of pupils. Marking 415 The form of assessment which is most immediately apparent to a pupil is the marking of written work; this may be routine class work or a more formal test. Such marking needs to be both diagnostic and supportive. A cross is of little assistance to a pupil unless it is accompanied by an indication of where the mistake has occurred, together with some explanation of what is wrong or a request to consult the teacher when the work has been returned. This style of marking also enables the teacher to become aware of the kinds of mistakes which are being made and to prepare later lessons in the light of this knowledge. 416 In the course of our visits to both primary and secondary schools we found a very wide variation in the amount of time and care given by teachers to the marking of pupils' work. In some cases marking was of the diagnostic and supportive type which we have already described. On the other hand, we found classrooms in which, although every piece of work had been marked, the result was merely page after page of ticks or crosses with little or no indication of where an error had occurred or what was wrong. There were classrooms in which marking had been so infrequent that pupils had continued to repeat the same mistake because a mathematical concept or routine had not been understood. We were also aware of instances in which, in our view, too little work was being set for pupils in secondary schools to do out of school because the teacher was not able to find the time to mark it. 417 Work in mathematics generates a great deal of marking and it is not usually possible nor, in general, desirable for a teacher to mark every piece of work which is done. A proper balance needs to be maintained between time spent on marking and time spent on preparation of lessons; however, work which has been marked should be returned promptly. Guidance about methods of marking and ways of resolving this problem should form part of a school's scheme of work in mathematics. In a well run class it should not be necessary for a teacher to mark every piece of work in detail. Exercises involving routine practice or applications can well be marked by pupils themselves, either from a list of answers or, in the case of computational exercises, by checking with a calculator. Pupils can then seek help if they are unable to locate and correct their mistakes. We believe that there is value in pupils learning to do this, though it is necessary for the teacher to maintain a general oversight in order to ensure that such marking is being carried out accurately. On the other hand, work involving problem solving or the application of more than a single skill will need appraisal by the teacher. As with so many other things, it is a sensible balance which is required. Recording of progress 418 Assessment needs to be accompanied by appropriate recording of progress. All teachers carry in their heads a wealth of information about the pupils they teach and it is not always easy to record this concisely on paper. Nevertheless we believe that in both primary and secondary schools it is necessary to maintain a written record of a pupil's progress, not only as a reminder to his teacher and for the information of the head or head of department but also to enable continuity to be maintained when the pupil changes teacher or school. We discuss this further in paragraph 431. 419 It is not easy to decide on the most suitable form in which a record should be kept. Many LEAs provide a standard record card which schools are required to complete and which accompanies a pupil on transfer to another school. However, this may need to be supplemented by a more extensive record. In primary schools it can be helpful for this record to be related to LEA guidelines, though it is necessary to be aware that a checklist of the kind which records the topics which a child has attempted is open to abuse if children and teacher race to reach the end of the list. However, such a list can help a teacher to ensure that all children receive a balanced coverage of topics. In secondary schools the record can be related to the departmental scheme of work. Whatever form of recording is used, some effort should be made to record those qualities which can only be assessed by the professional judgement of the teacher, such as a pupil's persistence in working at a problem, his ability to use his knowledge and his ability to discuss mathematics orally. Testing 420 Testing, whether written, oral or practical, should never be an end in itself but should be a means of providing information which can form the basis of future action. When tests are used in class they should be matched to the level of the pupils who are taking them. In a class which contains pupils whose attainment varies widely, it may not always be appropriate to give the same test to everyone. It is very easy to undermine confidence by using a test which is too difficult and a test which is too easy for those who are taking it serves little useful purpose, although there can from time to time be a place for a straightforward test designed to encourage pupils to develop mastery of a specific skill. It is important that tests used in the classroom should be followed by opportunity for pupils to explain the thinking, whether correct or not, which has led to the answers they have given and for discussion of difficulties and misunderstandings which may be revealed. Teachers need also to be aware of the difference between 'learning something for the test' and learning in such a way that what has been learned is assimilated and retained for use in the future. Learning of this latter kind is not necessarily revealed by the result of a single test. Standardised tests 421 Some testing has a wider purpose than the teacher's day to day assessment. The head teacher or the LEA may wish all pupils to be tested using nationally validated tests. Results of these tests give some guidance about the ability range and level of attainment of the pupils compared with national norms and should always be made available to the teacher. We are aware of increasing pressure in recent years from some members of the public and from some local councillors for the introduction of 'blanket' testing in mathematics for all pupils. We have been told that at the present time about one third of all local authorities administer some form of mathematics test to their pupils, in some cases at more than one age. However, a remark which has been drawn to our attention - 'no one has ever grown taller as a result of being measured' - underlines the fact that testing does not of itself lead to learning or to the raising of standards. 422 If LEA testing of this kind is carried out, the tests which are used need to be selected carefully so that they will provide as much information as possible; they should not concentrate exclusively on particular aspects of the mathematics curriculum. The results of the tests should be capable of use within each school to help teachers to diagnose, and if possible remedy, deficiencies and weaknesses. They should also be capable of use by the LEA as an aid in the identification both of successful practice and of schools which may be in need of additional support. 423 It is important to realise that standardised tests measure only some aspects of mathematical attainment. They do not test such aspects as attitude and perseverance nor, very often, the ability to apply mathematics in unfamiliar situations. We therefore recommend that, when the results of standardised tests are used as part of a transfer procedure between the different stages of schooling, these results should not be used by themselves but in conjunction with an assessment of these wider aspects provided by the pupil's previous school. Evaluation 424 The results of assessment and testing should be used for the purpose of evaluation. The teacher should use the records of individual pupils to evaluate the progress of the class and compare this with the aims and objectives set out in the school's scheme of work. Evaluation of this kind also provides opportunity for the teacher to appraise the results of his or her own teaching, and perhaps, in collaboration with the mathematics coordinator or head of department, to modify the scheme of work in some way. For example, if a topic has caused difficulty it may be necessary to reconsider the context in which it was introduced or the teaching approach which was used. In some cases, it may be felt that in a future year the topic should be deferred for comparable groups of pupils; on the other hand, there may be other topics which it is decided to introduce at an earlier stage. Mathematics coordinators, heads of department and head teachers need to evaluate the work of groups of classes and of the school as a whole. We believe that evaluation of this kind is not always carried out on a sufficient scale nor on a sufficiently regular basis. It is only as a result of such evaluation that it is possible both to identify successful practice and also to identify areas in which help is required. Assessment of Performance Unit 425 The mathematics testing of pupils aged 11 and 15 which is being carried out by the Assessment of Performance Unit (APU) has yet another purpose. Its aim is to develop methods of assessing and monitoring the achievement of children at school and to seek to identify the incidence of under-achievement. It makes use of a sampling technique which does not provide information about the performance of individual pupils, classes or schools. The information which is being assembled by APU seems to us to be a potential source of very valuable information for those who plan mathematics courses at all levels, produce textbooks and other classroom materials and draw up examination syllabuses, as well as for all who teach mathematics. We hope that ways will be found to enable maximum use to be made of this information. We recommend that in the near future an overall appraisal should be prepared of the educational implications of the mathematics testing which has been carried out so far. 426 A novel aspect of the work of APU is the practical testing which is being undertaken. The examples of this which have been released show clearly the potential of work of this kind in the classroom and we believe that all teachers of mathematics should be made aware of this aspect of APU's work.
Continuity 427 We have already referred to the need for schemes of work to be prepared which will give adequate guidance to teachers; we have also drawn attention to the necessity of maintaining a record of each pupil's progress. If the scheme of work is properly used and the record maintained, it should be possible within a single school to ensure consistency of teaching method and continuity of syllabus as a pupil moves from class to class and from teacher to teacher. The pupil's exercise book will also provide a record of recent work and, unless there have been staff changes, it is possible for a pupil's new teacher to obtain information from the teacher who taught him previously. 428 At the time of transfer to a new school, continuity is more difficult to achieve but no less important to maintain. Transfer is most often to a larger school and many children can be apprehensive both of their new surroundings and also of the prospect of work with new and unfamiliar teachers. In recent years all schools have, we believe, become increasingly aware of the need to ease problems of transfer. It is common for children to pay a preliminary visit to the school to which they will transfer and for staff of the receiving school to visit the schools from which their pupils will come. However, liaison activities of this kind, while making a major contribution to ease of transfer from a pastoral and social point of view, do not always pay as much attention as we would wish to ensuring continuity of mathematical development. Yet if a pupil is suddenly expected to attempt work which is beyond his capacity or finds himself bored at the outset by having to repeat work which has already been mastered, not only is his mathematical development interrupted but the whole process of transfer is subjected to unnecessary strain. During the first year after transfer, some schools send a report to their contributory schools about the progress of the pupils whom they have received. This is a practice which we commend; if the reports include reference to progress in mathematics, they can contribute towards the development of continuity in mathematics for pupils who will transfer in the future. 429 Problems of continuity on transfer from infant to junior school or from first to middle school are often less great than those which exist on entry to secondary or upper school. This is because infant and junior or first and middle schools are often closer together, their methods of working more similar and the range of attainment which exists among those who transfer less wide than is the case with older pupils. Nevertheless, problems can and do exist even in the case of transfer from infant to junior schools in adjoining premises. However, we believe that the greatest problems exist on transfer to secondary or upper school. These schools often receive pupils from a large number of contributory schools; in some areas, too, children from a single primary school can transfer to two or more secondary schools. As we have pointed out in paragraph 342, the spread in the mathematical attainment of these pupils can be very great and in these circumstances it is not very easy to make sure that pupils continue their mathematical education at a level and a speed which is appropriate. For convenience we refer in the paragraphs which follow to transfer between primary and secondary school; our remarks apply equally to transfer to middle school and upper school. 430 It is essential that there should be discussion between those who teach in primary and secondary schools in the same area and that such discussion should take place in an atmosphere of mutual professional respect. Both primary and secondary teachers need to take steps to acquaint themselves with the methods and materials which each uses. We believe that there are many secondary teachers who are unfamiliar with the approach to mathematics which their pupils have been using at the primary stage and also many primary teachers who have not taken steps to discover the type and range of work which those who leave them will undertake during their first term at secondary school. The outcome of such discussions should be overall agreement about the central topics which will have been tackled at the primary stage by pupils of different levels of attainment; where LEA guidelines exist it should be related to these guidelines. We stress that agreement of this kind should take account of the differences in attainment which will exist at the time of transfer. It should therefore be formulated in terms of a 'progression' of work and not of an agreed list of topics with which all children are expected to be familiar at the time of transfer. However 'reasonable' such a list may seem to be, it cannot be suitable for all pupils. Furthermore, the existence of such a list can produce undesirable pressure on teachers in primary schools to cover all that is in the list even though some pupils will not be ready for some of the work which it includes. 431 We referred in paragraph 419 to the fact that many LEAs provide a record card which accompanies each pupil from school to school, and also to the fact that it is not easy to produce a record card which is sufficiently concise to be completed easily and sufficiently detailed to provide adequate information. However, provided that both primary and secondary schools are committed to the use of record cards, much valuable information about a pupil's range of understanding and skill can be passed on in a way which will help to ensure continuity. It can also be of help to attach to the record card two or three examples of recent work selected to indicate a pupil's general level and style of working. 432 In some cases pupils are given a mathematics test at the time of transfer to secondary school. Such a test is sometimes drawn up by the secondary school in consultation with its contributory schools and sometimes carried out on an LEA basis. When tests are used, we recommend that they should be conducted before transfer in the primary school in which the pupil feels at home, rather than during the first few days at secondary school in an environment in which the pupil may not yet have settled. We believe that pupils entering secondary school should not be tested in any formal way until they have had time to settle in their new school and preferably also had time to complete some work on a new topic. Where a transfer test is used it should be related to LEA guidelines or to the 'progression' which has been agreed between primary and secondary schools. This should enable those sections of the test to be used which are suited to a pupil's attainment and avoid the necessity of requiring a pupil to attempt questions on work which has not yet been covered. 433 Some teachers in secondary schools have told us that they take little or no notice of information which they receive from their contributory schools about the mathematical attainment of pupils because they prefer to give all their pupils a 'fresh start'. We cannot accept that it can be justifiable to ignore information provided by schools in which pupils may have spent as long as seven years. It can, of course, happen that a pupil who has not made good progress at his previous school is able at a new school to make much better progress as a result of encountering a different teacher and perhaps a different approach to the subject; the same thing can happen as a result of change of teacher within the same school. It is important that schools should be aware of this possibility and that at all stages the arrangements for teaching mathematics are such as to encourage and support progress of this kind, whatever the age of the pupil. We would also point out that such improved progress is in any case more likely to occur if the teaching takes account of information which is available about a pupil's previous difficulties and sets out to resolve them. 434 Whatever steps may be taken to avoid categorising pupils prematurely in terms of their mathematical attainment, and to allow for the fact that a change of school may result in greatly improved progress, we can see no justification at all in ignoring and failing to act upon a report from a contributory school which indicates that a particular pupil is good at mathematics. To ignore such information may result in a pupil being given work which is far below the level of which he is capable. This will not only interrupt the pupil's progress but may also lead to the development of undesirable attitudes to mathematics because the pupil feels that his teachers are failing to recognise his ability. 435 Problems of continuity can also arise when pupils transfer to the sixth form. In 11-18 schools the change should present few problems provided that sixth form teachers recognise that some students will need guidance concerning the different pattern of study which is required of them and especially the need to organise private study time effectively. Problems of continuity are likely to be greater when pupils transfer at 16+ to the sixth form of another school or to a sixth form, further education or tertiary college. The need for consultation and transfer of information about attainment in mathematics is just as great at this stage as at the time of transfer from primary to secondary school and those who receive students at 16+ will need to take account of the variety of different backgrounds from which the students have come. Close liaison will be needed to ensure that choices of courses for sixth form study are realistic; pupils should neither embark upon courses for which they are insufficiently qualified nor be debarred from courses which they are likely to be able to undertake successfully. 
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