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Waddell (1978) Notes on the text Part I Preliminary pages Contents, Membership
Part II Preliminary pages Contents, Membership Report of the Education Study Group (ESG) Glossary, Introduction
Report of the Cost Study Group (CSG) Chapter 1 Introduction
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The Waddell Report (1978)
School examinations Report of the Steering Committee established to consider proposals for replacing the General Certificate of Education Ordinary Level and Certificate of Secondary Education examinations by a common system of examining Chairman: Sir James Waddell CB Presented to Parliament by the Secretary of State for Education and Science and the Secretary of State for Wales by Command of Her Majesty July 1978 London: Her Majesty's Stationery Office 1978
Part I: Cmnd 7281-I
Part II - ESG Report Chapter 3 Mathematics
Introduction 59. The study of mathematics includes a wide variety of activities and understanding, ranging from the acquisition of basic skills of numeracy to the consideration of abstract concepts, and involves their use in a large number of practical applications. There is much current debate about which aspects are most important to pupils of different abilities and at different stages. There is a wide diversity of approaches, often polarised between 'traditional' and 'modern' content; this complicates the discussion of feasibility, since an examination can test only a selected number of aspects. It was no part of our brief to adjudicate between these approaches, but it was necessary to recognise their existence in the joint examinations. 60. The task of assessing, within a common system, the whole group of pupils currently covered by CSE and GCE O Level examinations in mathematics thus presents a considerable challenge. The most able (some, but not all of whom, will wish to continue with mathematics as a study) will need to confront difficulties when a problem has to be analysed, a number of concepts and techniques marshalled, and a solution accurately produced. The less able will not only find the subject intrinsically difficult but will also have to contend with poor reading and comprehension, and difficulty in expressing their ideas and showing their skills on paper. The problem for a common system is to confront this range of pupils appropriately and positively. An extra dimension is added by the fact that, partly because of the importance of mathematics as a subject and partly because employers and higher education often demand a qualification in it, a mathematics examination is taken by a high proportion of pupils. Research suggests that in 1974 about 72 per cent of the age group entered for this subject. The evidence 61. The main body of evidence available was the set of joint examinations. Discussions were held with two of the consortia (1 and 5) and with subject interests. The numbers of candidates involved in the examinations in the different years are tabulated below. Further reference to the examinations is made by number as indicated. The ability range 62. The whole range of ability under consideration was adequately represented in four of the studies (1-4) but not necessarily in proportion to the distribution in the school population. This is borne out by the evidence of the NFER Test 100 of scholastic aptitude, which also confirmed that in 3 and 4 the range extended well below the 40th percentile, a fact further supported by the percentage of low grades awarded in 4. There is evidence that in recent years only borderline candidates have been entered in 5 and, from some schools, in 1 in 1977. Scheme 6 did not recruit a sufficient number of less able pupils. Those schemes which were basically 'traditional' (1 and 3) or basically 'modern' (5) may not have attracted a representative range of schools. The syllabus and objectives 63. All the syllabuses included a broad statement of aims and a list of content to be covered, but teachers had to rely on specimen and past examination papers for an indication of the expected depth of treatment of the topics listed in the syllabus. No syllabus included a detailed statement of objectives, although 2 attempted a clear statement of aims and these were well balanced between applications and study of the subject for its own sake. No syllabus referred to or encouraged open-ended or investigational work; on the other hand none precluded these or a discovery approach/practical bias to the teaching. 64. The schemes included little syllabus development but in 2 a good attempt was made to merge 'traditional' and 'modern' content. Scheme 1 was basically 'traditional' and 5 was basically 'modern' in content; it could be argued that these lack balance because certain aspects were omitted. Two other schemes (3 and 4) included both 'modern' and 'traditional' content in optional sections, and in the core of 4. Choice within a syllabus should perhaps be between meaningful options and not between 'traditional' and 'modern' content as in 4 and to some extent in 3. Scheme 3 probably developed as it did because the 'modern' content was part of the CSE syllabuses; an unfortunate by-product might have been the identification of the 'modern' option with the less able. In that scheme, also, the options were limited to two; a more wide-scale use of the examination would require more options to be made available to schools. In a similar way alternative parallel schemes would have to be provided for 1 and 5. 65. One scheme (5) probably did not provide an adequate basis for A Level work because of the 'modern' content of the syllabus, its lack of depth and examination questions which were insufficiently demanding. Schemes 1, 3 (only the syllabus with alternative A) and 4 provided a basis of further study which was probably as good as that of the O Levels of the parent GCE boards. It is easy to see why some teachers thought that 2 provided an improved basis for A Level; the syllabus was broader and the examination more demanding than for O Level. Calculus was included as an option in 2, 3 and 4. 66. Basic skills of calculation were tested in all the examinations but sometimes in isolation from applications or in contrived situations (criticisms which can of course be made of O Level and CSE examinations as well). Basic skills and applications were given insufficient attention in 5 in which certain other topics were rather over-valued. Some of the 'modern' content was tested in some examination papers by rather trivial and meaningless examples. Estimation, approximation, limits of accuracy and significant figures found a place on most syllabuses and were occasionally tested. 67. Looked at from the point of view of less able pupils, for whom the examination was terminal, some of the syllabuses were rather full and heavy. The most unsatisfactory were those in which all pupils had to cover the whole syllabus - 1 (Scheme I) and 4. Some difficult topics, which would lack meaning and value for less able pupils, were included in 5 - for example the study of transformations for their own sake. Each of the syllabuses of 1 (Scheme II), 2 and 3 would have provided the basis for a course which was as worthwhile as that based on the CSE examinations of the parent boards. 68. Some syllabuses emphasised understanding and applications rather than memorisation of facts and lengthy manipulation. The issue to candidates of a sheet listing required formulae and standard symbols was a helpful feature of 3; the formulae required in certain questions were also quoted on some papers in 2 and 4. Although some examinations required skills and techniques to be applied only to very straightforward problems (1 and 5), they were no more deficient than many CSE examinations in this respect. In 4, only one school (with 50 to 70 candidates each year) made use of the positive feature that a Mode III element could replace the third paper, over the period 1975 to 1977. In 3, 102 candidates submitted course work in 1976 as an alternative to the third paper, and in 1977 454 candidates did so. 69. Taken together, the syllabuses and objectives of the joint examinations were sufficiently varied and well-considered to make the examinations useful as evidence on feasibility, and to support the view that, if a common system is introduced, appropriate syllabuses can be devised in the period leading up to its introduction. The syllabus, the examination and the ability range 70. The examinations devised by the consortia to cover their syllabuses and objectives give some indications about the kind of common system which would be suitable for mathematics. In particular, there are clear signs that examinations based on common papers taken by all pupils have run into considerable difficulties. Indeed, it is significant that those consortia which ran joint examinations in the form of a common examination - 1 (Scheme I), 4 and 5 - have switched or are switching to some form of papers differentiated according to difficulty. 71. The pattern of two common papers in 1 (Scheme I) was never popular with schools and will be discontinued after 1977. It was not considered appropriate to face less able candidates with material with which they could not cope and to make able candidates spend time on simple tasks. The three written papers of 4 included a difficult third paper based on one of three parallel, alternative syllabuses; a Mode III option was also available as an alternative to one of these three, although this has had a poor take-up. This arrangement turned out to be unsatisfactory for less able pupils; in the last two years 50 per cent to 70 per cent of the candidates scored less than 25 per cent on paper 3 alternatives. Here again, a change to differentiated papers is envisaged. 72. The pattern of 5 was also unsatisfactory. A large number of structured questions made great demands on the less able; in the early years a large number of candidates achieved little on the second paper, and the absence of difficult problems meant that insufficient demands were made on the most able. Those involved with this study have conceded that, in catering for the needs of the average, they neglected the needs of both the least and the most able groups. The scheme has recently been developed (non-operationally) to introduce new approaches more suitable for the extremes of the ability range; a pattern of differentiated papers, comprising four papers, will probably be tried in 1979. 73. The other three main studies provided different possible models for a common system from which a range of other models can be derived; all included a hard paper based on an extended syllabus. In 1 (Scheme II) there were three papers on an incline of difficulty, with candidates taking two. In 2 there were two papers and an optional further harder paper, with course work assessment for all. In 3 there were two papers, and either a third paper or internal assessment of course work. In the first two of these schemes a restricted range of grades was available on the results of the easier combination of papers. These three schemes all worked sufficiently well to indicate a possible way towards a satisfactory common system, and we believe that it is in this direction, rather than towards a common examination, that progress could be made in catering for the extremes of the ability range in mathematics. 74. There was a large range of types of questions in the joint examinations. Indeed, a wider range of assessment techniques was used in some of the experimental examinations than is to be found in any single O Level or CSE examination. This was certainly true of 2, and also of 3 and 4 for some pupils. This should have meant that a wider range of objectives could be tested. Externally-set examination papers for all pupils occurred only in 1 and 5. The hours pupils spent in the examination room varied from 2¾ (plus teacher assessment) to 4 or 5¼. but the examiners tried to reduce the examination load on the less able pupils in 3 and 4. 75. Objective tests were included in 2, 3 and 4 and were machine-marked in 2 and 3; these gave good syllabus coverage and tested basic skills in an efficient manner. They, and the short-answer questions which were also well used, ensured that those poor at reading and writing were given an opportunity to tackle some mathematical questions. Moreover, all candidates were tested on the core syllabus by both objective/short-answer questions and more open-ended/problem questions which required greater skill and deeper understanding. 76. Course work was included in 3, with detailed guidance being given to the teachers covering possible study topics, the supervision of pupils, time allocation, presentation of work, records and assessment. Such guidance is essential though it may make great demands on the teacher. Project work was tried in 6 and 7 and topic work was a part of 6 and of the course work option in 3. This kind of work can be educationally beneficial if well conceived and properly supervised; but it might make little contribution to discrimination as was found in 7. 77. Teacher assessment was an important part of 2. Some of the teachers involved in this study experienced internal assessment for the first time; the majority of teachers favoured its inclusion and none doubted that internal assessment had a place in mathematics. Guidance was given by the boards and some criteria were listed but these were described in terms which, apparently, the teachers found difficult to interpret with precision. 78. No choice of paper, or of question within a paper, was allowed the candidates in 1 (Scheme I) and 5, and this contributed to their unsatisfactory features; the demands were too great for many and not great enough for some. However, although there was similarly no choice of question on the first two papers of 2, 3 and 4 or on any of the papers of 1 (Scheme I), this did not appear to have disadvantages, owing to the different nature of these papers. Less able pupils probably have difficulty in making wise choices, and the absence of choice ensures that all the candidates do the same examination and that teachers cover the whole syllabus. The mark allocation for different questions was declared on some papers in 1 and 4 and was an integral part of 3. This was probably only satisfactory on the hardest paper taken by the more able pupils, because it required many judgements and choices to be made; the candidates were required to enter the examination room aware of which parts of the paper they would attempt to answer. 79. Some of the weaker candidates were discouraged by the hard paper in 2 for which they were inappropriately entered; by the equivalent one in 3 in years when there was no alternative; and by that in 4 because it had to be taken. This was inevitable; questions chosen to discriminate well were likely to discourage many pupils. In 1974 too many pupils worked the hard paper of 2; fewer lower ability candidates were entered in 1975. Teachers and parents in consultation with pupils appeared to be reluctant to decide which combination of papers candidates should take. This was also true, to some extent, of 1 (Scheme II). It is interesting to note that the revised and only scheme of WJEC for 1978 requires teachers to make this decision: either papers 1 and 2 or papers 2 and 3 for each pupil. 80. In general, the studies provided evidence to indicate that the least able must be spared the ordeal of facing a very hard written paper, but that a difficult paper must be included in the examination. The most able performed adequately on the easier papers and found enough interest in them. The take-up of the Mode III option of 4 and the course work option of 3 were disappointing, but it can be argued that it is desirable that every scheme should include this kind of option for the sake of both teachers and pupils. The teacher assessment imposed on the schools in 2 appeared to work satisfactorily. Marking and grading 81. The marking schemes appeared to be generally satisfactory with one exception (5), in which too few marks rewarded method/approach. Working to an agreed marking scheme examiners produced a single order of merit for each examination, and in general discrimination was good. There may have been some bunching in the distribution of scores on some papers in 3 for example, and amongst weaker candidates in 5; but the examination boards said that no great difficulties were experienced in fixing the various grade boundaries. 82. The predictable problems of grading were experienced in the schemes of 'take 2 papers from 3' in 1 and 8. There was the difficulty of awarding an appropriate grade to a pupil who performed badly on the harder combination of papers intended to discriminate across the upper ability range. There was also the problem of equating the standard of work required for the same grade on different combinations of papers. In these studies, no sophisticated statistical techniques were used: the two populations were graded independently and comparisons made through the common paper. This was also the procedure adopted in 2. 83. No problems should be experienced in the large-scale use of the written papers in any major extension of the examinations, but more precise guidance concerning teacher assessment would be needed, together with further consideration of moderation procedures. Conclusion 84. Although no single joint examination in mathematics offers a complete solution to the problems of assessing the performance of pupils over the wide range of ability envisaged, taken together they have indicated the way forward to a feasible common system of examining. They have demonstrated a number of possibilities and made use of a wide range of assessment techniques. There is a need for different forms of question for the able pupils which are not appropriate for the majority, and this suggests that in mathematics a common examination comprising only written papers taken by all the candidates cannot be satisfactory for both extremes of the ability range. The most satisfactory of the examinations adopted an approach through papers differentiated according to ability, including a paper based on an extended syllabus for the more able. 
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