⁽¹⁾ The present articte follows one in the immediately preceding issue of this Review which examined the relation between produc tivity and workforce qualifications in a matched sample of British and German metalworking firms (‘Productivity, machinery and skills in a sample of British and German manufacturing plants: results of a pilot inquiry’ by A. Daly. D.M.W.N. Hitchens and K. Wagner, National Institute Economic Review, no. 111, Febru ary 1985). An account of the varied aspects of the German schooling system as observed by an English teacher may be found in Europe at School, by N. Newcombe (Methuen, 1977). See also the valuable surveys by M. Wilkinson, Lessons from Europe (Centre for Policy Studies, London, 1977), and by T.J. Russell, The Educational System of the Federal Republic of Germany (Combe Lodge information Bank Paper 1310, Further Education Staff College, Blagdon, Bristol, 1977). Much useful information on comparative studies in education is summarised by T. Husen, The School in Question (Oxford, 1979); but this has to be read critically (see Appendix B below for an example of the kind of care that is necessary). Previous comparisons in the present series referred to above are: ‘Vocational qualifications of the labourforce in Britain and Germany’ by S.J. Prais, in National Institute Economic Review, no. 98, November 1981; ‘Some practical aspects of human capital investment: training standards in five occupations in Britain and Germany’ by S.J. Prais and K. Wagner, National Institute Economic Review, no. 105, August 1983. Our immense debt is acknowledged on pp. 71-2 to the many who assisted us in the preparation of the present article, and by commenting on earlier versions. The present version includes the results of further comparisons of mathematical attainments between England and Baden-Württemberg.

⁽¹⁾ See Bundesminster für Bildung und Wissenscliafl, Grund-und Strukturdaten 1982/3, pp. 60-3, and table 1 below. The rapidly changing balance of pupil numbers amongst the three types of German school may be expected to lead to changes in relative school standards—a matter of the greatest scientific interest and likely to involve further policy changes. As explained further below (p. 56), there was previously also a ‘middle type’ in England, the Technical and Central (commercial) school, but it never attained the characteristic importance that the Realschule attained in Germany.

⁽²⁾ The age of transfer varies according to the Land; the tendency has been to move selection to the later age.

⁽³⁾ Only 4 per cent of German school-leavers in 1982 were from Comprehensive schoofs; even in Berlin, where these schools are most frequent, they account for only one in four of secondary school pupils.

⁽⁴⁾ Berlin is also somewhat less conservative than other parts of Germany in its approach to educational reform, and a little ctoser to the English approach.

⁽⁵⁾ School timetables for the three types of school in Germany are compared with that of the average English in Appendix C of National Institute Discussion Paper no. 60. The statistics that follow are taken for the relevant years from Allgemeines Schul wesen and Berufliches Schulwesen (for vocational schools), published by the German Federal Statistical Office in their series Bildung und Kultur 11/1 and 11/2; summaries are conveniently provided in the German Statistisches Jahrbuch (for example, the issue for 1981, p.346), and in the statistical handbook of the Ministry of Education (Grund- und Strukturdaten, Bundesminister für Bildung und Wissenschaft).

⁽⁶⁾ For the benefit of German readers it needs to be said that sixth forms in English secondary schools normally last two years.

⁽¹⁾ It is of interest that Greek remains oblgatory in the ‘sixth forms’ of classical (‘Altsprächliche’) Gymnasien in certain Länder.

⁽¹⁾ The adjustments made to the published figures in the above paragraphs to avoid double counting, and to yield the numbers attaining their highest certificates at the end of full-time schooling, are inevitably approximate. The official publication Grund- und Strukturdaten, 1981/2 (pp. 60-3), shows unadjusted totals for each particular school-leaving certificate, the grand total of which exceeds (by about a tenth) the number of pupils in each age- cohort. The proportionate distribution is however much the same as shown here.

⁽²⁾ 91 per cent in England in 1983, of whom 7 per cent were in ‘middle schools deemed secondary’ which are usually grouped by the Department of Education together with Comprehensives. Statistics are not always available for the whole UK on schooling, and it is necessary here, and below, to refer occasionally to England alone, or to England and Wales. The sources are Statis tics of Education (vol. 1) Schools, (vol. 2) School Leavers CSE and GCE, and Education Statistics for the United Kingdom (HMSO). Since 1980, summary statistics are available in the DES Statistical Bulletin (approximately monthly).

⁽³⁾ Based on Statistics of Education, Schools, 1978, pp.8-10. The statistics on Technical and similar schools have to be regarded as approximate. The official source shows only 3 per cent in Techni cal schools: but a further 7 per cent were in ‘other secondary’ schools, which included ‘selective central’ schools, together with ‘bilateral modern/technical’ schools and multilateral schools. We have here allocated the latter 7 per cent as 2 per cent with Techni cal schools, and 5 per cent with Comprehensives.

⁽¹⁾ School Leavers, 1981, table C12. Statistics collected by the DES on school-leavers are based on a 10 per cent sample of pupils in all schools, including independent schools (but excluding special schools; before 1977-8 independent schools not ‘recognised as efficient’ were also excluded).

⁽²⁾ Ibid., table C22; this indirect method of estimation has to be used here in the absence of a published total for pupils in establish ments of further education who attained a ‘higher grade- combination’.

⁽³⁾ Statistical Supplement to the Eighteenth Report 1979-80, Universities Central Council on Admissions, pp.16-7 (compare tables E7 and E8); more detailed information, relating to the acceptance rate for those with two points, is unfortunately not published.

⁽⁴⁾ For a detailed distribution of UCCA scores we relied on the results of the National Child Development Study, an unpublished report (section 15.5) kindly made available by Richard Ives of the National Children‘s Bureau, London (for a published summary of some of the results, see K. Fogelman's paper in Publishing School Examination Results: A Discussion, Bedford Way Papers 5, University of London institute of Education, 1981, pp.28-9).

⁽⁵⁾ See vol. II with that title, ed. T. Husén (Almquist & Wiksell, Stockholm, 1967), especially pp. 24-5, 69 and 86. Much the same ground is covered by N. Postlethwaite, School Organisation and Student Achievement: A Study Based On Student Achievement in Mathematics in Twelve Countries (Wiley, 1967). No one should read these studies without taking a pinch of the salt provided in an exceptionally extensive, critical—but enlightening—review by H. Freudenthal, ‘Pupils’ achievements internationally compared', published on pp. 127-186 of the journal he edits, Educational Studies in Mathematics, vol. 6, no. 2, July 1975 (Reidel, Dordrecht- Holland); however, the critical aspects are of greater relevance later in this article than they are here.

⁽¹⁾ ‘Population 3a’ in England included those in ‘science sixth forms’ who took mathematics as one of their subjects (see Husen, op. cit., pp.46 and 172). About a quarter of all A-level leavers were prob ably covered by this definition (see DES, Statistical Bulletin 11/84, table 11).

⁽²⁾ The test was not, however, exactly the same (Husén, vol. 1, p.105). Whilst it may seem obvious, in the above context, that the British system of specialisation was the substantial source of the higher marks obtained by sixth-form mathematicians, the ex perience of some other countries included in the sample study shows that equally high marks in mathematics can be achieved while taking nine subjects (Israel, 36; Belgium, 35; France, 33), and at much the same age as in Britain (Husén, p.86; Postleth waite, p.110). A notional adjustment (Husén, p.116 et seq.) for the different fractions of the age-group at school at higher ages in the different countries does not change the above conclusions relating to Britain and Germany; the adjustment is based on assuming that the best pupils in the country with a higher fraction can fairly be compared with those in another country with a lower fraction staying on to higher ages. The adjustment becomes more prob lematical in relation to countries such as the US and Sweden where very much larger fractions are at school at higher ages; in these, the hypothetical ‘cream’ of pupils in the survey may well yield a sample of significantly higher ability than in countries where selection is carried out by the schooling system at a lower age.

⁽³⁾ By way of example, some details may be given of present requirements in the ‘sixth forms’ of Berlin Gymnasien. The period of study is 2½ years, with a possible further year if required to achieve the requisite minimum of 28 ‘credits’ (each successfully completed course taken for a half year counts as a ‘credit’; three school periods a week are required for most subjects, but six periods are required in the two major subjects of specialisation). The full curriculum of 13 subjects is obligatory in the first half year, which is introductory (and does not reach credit levels); thereafter, the following are the minimum requirements: four credits each (which normally implies instruction throughout the remainder of schooling) in German, sciences, and in geography/political history; two credits each in mathematics, a foreign language and music/art. Specialised major fields of study are to be chosen from: language/literature/arts, geography/history/politics, and science/ mathematics.

⁽⁴⁾ Grund- und Strukturdaten, 1983/84, p. 108. It must be noted that because some students change their subject of study, or continue for a second degree, the number registered for the first year of a university course was a quarter higher than the number who were in their first year at university (ibid., p. 124, et seq.).

⁽⁵⁾ Education Statistics for the United Kingdom 1983, p.24; less an allowance of 10 per cent for overseas students (see DES Statisti cal Bulletin, 13/80, table 3).

⁽⁶⁾ DES Statistical Bulletin, 8/84, table 4(i).

⁽¹⁾ In Germany in 1981, there were 78,000 graduates in all subjects, excluding teaching qualifications and foreign students; in the UK in 1981-2, there were 77,000 graduates, excluding those going on to teacher training and other further education, and excluding over seas graduates who returned home (Grund. und Strukturdaten 1983/84, p. 154; Education Statistics for the UK 1983, p. 25).

⁽²⁾ See Prais (1981), pp.53-4, for comparisons between Britain and Germany of numbers of first degrees by subject of study. Professor Dahrendorf, in his address to the annual Conference of German University Rektors (Vice-Chancellors) in 1982 (reported in the Berlin Tagesspiegel of 4 May), strongly criticised the Ger man university system as unduly expensive per graduate produced.

⁽³⁾ The German school marking system, curiously enough, runs from a top mark of 1 to a bottom mark of 6; the pass-mark is usually 4.

⁽¹⁾ DES, School leavers 1981, table C8.

⁽²⁾ Ibid., table C11.

⁽³⁾ This indirect estimation procedure is necessary since the stati stics used here are based on the highest achievements of pupils leaving school, measured in the year they left (see the introduction to the source just quoted). The alternative is to rely on statistics based on all examination results in a particular year; but this is not satisfactory, since pupils may spread their examinations over a number of years. For the sake of clarity it may be added that the pupils who attain university entrance standards (with 2-3 A-levels) are excluded here, since they have been covered in Section 3 above.

⁽⁴⁾ The DES makes no estimate for this group (see DES, School Leavers 1977, pp.xxv, para. 43). An approximate indirect estimate can be obtained from the recently published results of the National Child Development Study (see p. 57 above, footnote 4, for references), which show that 10.1 per cent of 19-year olds have achieved five or more O-level passes, but no A-levels, compared with 9.2 per cent shown by the DES sample of school-leavers. The difference of one per cent is presumably largely due to this group; sampling errors inevitably imply that this estimate is approximate.

⁽⁵⁾ The comparisons of French examinations were based mainly on London CSE and O-level, and on the Bavarian Realschule leaving-examination. It perhaps needs to be said again here that standards vary slightly amongst the various examination boards, and the above limited comparisons should not be pressed too closely. It is sufficient for the present purposes that a broad corre spondence was found between the standards considered here for the two countries. French is normally the first foreign language in English schools, but in German schools it is usually the second foreign language (after English: the main exception is the Saar land, where French has priority over English); however, this difference seems unlikely to affect the above comparisons to any considerable extent.

⁽¹⁾ See the report produced under the chairmanship of Dr J. H. Cockroft (Vice-Chancellor of the New University of Ulster), Mathematics Counts (311 pp., HMSO, 1982); the Committee had been set up following a recommendation by a House of Commons Select Committee inquiring into the attainment of school leavers in 1977. Only four pages (pp. 236-9) dealt with mathematical education in other countries. No mention was made of the Intemational Study of Achievement in Mathematics (see p. 57 above). The role of tests is at one point virtually dismissed on the basis that ‘no one has ever grown taller as a result of being measured’ (p. 123). That tests encourage pupils to revise, and help parents and teachers to check progress, seems to have carried too little weight with this Committee.

⁽²⁾ The examinations compared were the London GCE O-level mathematics paper (syllabus D), the London CSE papers in arithmetic and mathematics, and the Realschule leaving examination as set in Baden-Wurttemberg. Papers for Saarland and Bavaria were also considered and appeared of similar standard.

⁽³⁾ Statistics on this are scarce. In the 1960s, the IEA sample inquiry showed 21 per cent of Grammar school pupils did not sit an O-level in mathematics (D. A. Pidgeon, ed., Achievement in Mathematics: A National Study in Secondary Schools, National Foundation for Educational Research, 1967, p.251). In 1981-2, 80 per cent of all Grammar school leavers passed in mathematics at O-level (unpublished information kindly made available by DES).

⁽⁴⁾ See K. Fogeiman, op.cit., pp.29-30; DES, Statistics of Education 1977, vol.2, School Leaders, pp.19, 44-9; DES, Statistical Bulletin, 11/84, table 8; Cockroft, op. cit., pp. 248-50.

⁽¹⁾ V. Dundas-Grant, ‘The education of the adolescent: recent developments in secondary education in France’, Cornparative Education (vol. 10, no.2, 1982), p. 33; her contrast is however over-stated, since it needs to bring into account A-level weavers.

⁽²⁾ Prais and Wagner, op. cit., 1983.

⁽³⁾ Baden-Württemberg has made an important start in centrally- organised examinations for this sector, as discussed below.

⁽⁴⁾ A fuller investigation might usefully look at attainments in writing, sciences and foreign languages.

⁽¹⁾ These points are well recognised; cf., generally, the survey by HM Inspectors of Schools, Aspects of Secondary Education in England (DES; HMSO, 1979); and Cockroft, Report, pp. 128 et seq.

⁽²⁾ DES, School Leavers 1977, p. viii, and Statistical Bulletin 11/84, table 3.

⁽³⁾ Cockroft, Report p. 131. The term ‘other grades’ is used in the official statistics to describe what are here called lower grades'.

⁽⁴⁾ The use of ‘Mode 3 syllabuses’, where the school sets and marks its own examinations, subject only to ‘monitoring’ by the CSE board, has exacerbated the problems faced by employers in recognising the value of these certificates, the School Council Working Paper no. 68 by D. Bird and M. Hiscox, Mathematics in School and Employment: A Study of Liaison Activities (Methuen Educational, 1981), especially pp. 17-8 and 110.

⁽⁵⁾ Cockroft, p.250; and School Leavers 1981, lable C25.

⁽⁶⁾ DES Press Notice, 15 November 1982. The numbers passing at CSE-level in a core of, for example, three subjects is not ascertain able from published statistics since the main table on that subject (for example, table C8, 1981; similarly in previous issues) is sub stantially in error—so we were informed by DES on enquiring into certain inconsistencies. We are grateful to the DES for information from an unpublished table which suggests that those passing CSE mathematics usually also pass CSE English and science. It is of course indicative of the general lack of concern with pupils' attainments in a range of core subjects that errors in these basic statistics have not been previously noted.

⁽⁷⁾ Allgemeines Schulwesen 1982, p.22. As many as 40 per cent of those leaving Berlin Hauptschulen do so without their Abschluss; this is partly because of the high proportion of children of recent immigrants. Another reason is that Berlin is one of the few places in Germany where Comprehensive schools have become important; these have tended to take the better pupils away from Hauptschulen, leaving the latter schools with pupils of tower ability.

⁽¹⁾ See Appendix B for details of the adjustments.

⁽²⁾ In fact, the German average for all pupils was a little higher than that attained by the lower half of pupils in England; but bearing in mind sampling variability, it is safer to note the similarity than the 2.3 points (10 per cent) higher German achievement. The present Secretary of State for Education and Science has adopted as a ‘goal…. bringing 80-90 per cent of 16 year old pupils up to the standard of performance now expected of the average pupil’; this was first announced in his Sheffield policy statement in January 1984 (shortly after an early version of this paper was circulated), and appears again in his most recent policy statement Science 5-16 (HMSO, March 1985), p. 14.

⁽¹⁾ See Husén, vol. I, especially pp. 92, 97 and 98.

⁽²⁾ Aspects of Secondary Education in England: A Survey by HM Inspectors of Schools (Department of Education and Science; HMSO, 1979), p.156.

⁽³⁾ ‘Strong recommendations’, in the parlance of HMI, are stronger than ordinary ‘recommendations’.

⁽⁴⁾ These very worrying figures were relegated to a subsequently published supplement: Aspects of Secondary Education in England: Supplementary Information on Mathematics (DES, HMSO, 1980), p.44.

⁽⁵⁾ Cockroft Report (1982), p. 236; as mentioned above (p.00), the few paragraphs devoted to mathematical education in other coun tries dealt only with very general matters. The HC Select Commit tee was very much aware of the lack of detailed knowledge of what is happening in other countries; it noted the DES was not able to provide much help on this, and recommended that the ‘DES either initiate or encourage select further inquiries …’ (The Attainments of the School Leaver, Tenth Report from the Expenditure Commit tee, HMSO, 1977; vol. I, para. 157, pp. xlix-I).

⁽⁶⁾ ‘It is clear that at the present time it would not be possible to require all those who teach mathematics at secondary level to hold a minimum mathematical qualification’ said the Cockroft Report (p. 232); and a further investigation was recommended!

⁽⁷⁾ This is not the place to go further into this sad history. The interested reader may consult the report on the School Council‘s ‘Mathematics for the Majority Project’ which was specifically designed to cater for those of below average ability. It has left very little impact (see P. Kaner, chapter 10, Evaluation in Curriculum Development, Schools Council, 1973; and the review by R. Munro in J. Curriculum Studies, 1974, p. 175). The consequences of so-called ‘modern mathematics’ are examined by G. Howson, C. Keitel and J. Kilpatrick in Curriculum Development in Mathematics (Cambridge University Press, 1981) who concluded (p.238): It is almost thirty years since the ‘modern math’ reform began … the practical results of such an enormous expenditure of labour and commitment have been relatively insignificant. The problem remains—many Johnnys still cannot add!'

⁽¹⁾ See Professor N. Postiethwaite‘s special tabulations of the IEA results, ‘The bottom half in lower secondary schooling’, in Education and Economic Performance (ed. G. D. N. Worswick, Gower Publishing, 1985). That article contains some highly interesting tabulations of IEA studies of attainments in science and reading comprehension for bottom-half pupils in a number of countries; these confirm that the problem of poor bottom-half attainments in England is not confined to mathematics, but extends to other subjects as well.

⁽²⁾ The examples below are taken from the Rahmenplan für Unter richt und Erziehung in der Berliner Schule (1981), especially pp. 33, 35-6; and ditto, Fach Arbeitslehre (Sonderdruck, 1983), pp. 59-114. For lack of space, the full details on the various options cannot be given here, but it should be noted that cross- options are encouraged between the industrial and domestic courses. Much exemplary material is contained in the detailed teachers' guides on Arbeitslehre produced by the Pädagogisches Zentrum Berlin (Uhlandstrasse 97); several duplicated textbooks have been produced by groups of teachers for each year and each section of the course.

⁽¹⁾ Aspects of Secondary Education in England, op. cit. (1979). The quotations that follow are from pp. 14, 24, 40, 230-1, 236, 266-7.

⁽¹⁾ DES, Technology in Schools (1982). This survey was based on 90 schools which represented ‘roughly 10 per cent of secondary schools which were running technology courses at that time’ (p.4). Out of a total of some 5,000 secondary schools, the grossed-up total of 900 schools accounts for just under 20 per cent. Only four of the 90 schools in the sample had obligatory courses (p. 10).

⁽²⁾ DES School Leavers 1981, table C30; for more detail see earlier issues such as 1977, tables 24-5. The table for the most recent year is difficult to interpret since, for example, passes in woodwork and metalwork are taken together, leading to double counting of the number of pupils involved. It needs to be stressed that this kind of information has not been made available by DES for its sample of school-leavers; we are thus compelled to rely on information from examination boards, based on subjects passed. Some of these passes are from colleges of further education rather than from schools.

⁽³⁾ See DES, Craft, Design and Technology in Schools; Some Suc cessful Examples (HMSO, 1980).

⁽⁴⁾ DES, HM Inspectorate, Curriculum 11-16 (1977), p. 1. In Ger many it is taken as self-evident that pre-vocational school insaruction is of value subsequently at work and in the daily business of organising one's life, and is also immediately valuable in stimuiating the motivation at school of many pupils who otherwise find ‘book-learning’ lacking in relevance. Much applied research is carried out there on the links between school and work, as evident from the thousand-page bibliography (Forschungs-Dokumentation) issued three times a year by the Nürnberg lnstitut für Ar beitsmarkt und Berufsforschung. An interesting British study of the initial impact of schooling on industrial performance was carried out by the Engineering Industry Training Board; it noted the signifi cant positive benefits on subsequent work-planning and budgeting, especially for techician trainees (D. Mathews, The Relevance of School Learning Experiences to Performance in Industry, EITB, 1977). There is much scope for further research on these lines.(5) See Unesco Statistical Yearbooks. The lower proportion of GDP may be related to fewer ancillary staff which, in turn, is related to a shorter school day; there is scope for a detailed comparative costing exercise on these matters.

⁽¹⁾ By Professor Sir James Dewar in his presidential address to the British Association in 1902! (British Association, 1903: quoted by L. F. Haber, The Chemical Industry 1900-1930, Oxford, 1971, p.53). The same view was also extensively advanced by Alfred Marshall in his writings (cf. his well-known statement: ‘Thus all the world has much to learn from German methods of education’, Industry and Trade, Macmillan, 4th edn., 1923, p.130).

⁽²⁾ R. W. Phillips, ‘Some preliminary results from the Second IEA mathematics study’ (paper presented to the IEA General Assem bly, August 1983), table 4.

⁽³⁾ ‘Implications of the IEA studies of curriculum and instruction’, chapter 3 in Educational Policy and International Assessment (ed. A.C. Purves, and D.V. Levine, McCutcham, Berkeley, Cal., 1975), pp.78-9.

⁽⁴⁾ At the time of writing no details are available on this, beyond DES statements that work is in progress and will be internally assessed.

⁽¹⁾ Durkheim's book was based on lectures delivered at the begin ning of this century primarily to French students of education; he devoted a highly interesting chapter to the development of the Realist stream in continental schooling (see E. Durkheim, The Evolution of Educational Thought, translated by P. Collins from the French edition of 1938; Routledge, London, 1977), chapter 23.

⁽²⁾ Curriculum 11-16, p.4.

⁽³⁾ DES, A Framework for the School Curriculum (January 1980), pp. 6-8.

⁽⁴⁾ DES, The School Curriculum (March 1981).

⁽⁵⁾ See J. White et al., No, Minister: A Critique of the DES Paper ‘The School Curriculum’ (Bedford Way Papers 4, University of London Institute of Education, 1981), especially the devastating first essay, ‘Enigmatic Guidelines’ by J. White.

⁽¹⁾ GCSE, The National Criteria (a series of 21 booklets issued by DES), HMSO, 1985.

⁽²⁾ Para. 80.

⁽³⁾ See paras. 67, 69, 109; para 37 states: ‘The government does not propose to introduce legislation affecting the powers of the Secretaries of State in relation to the curriculum’.

⁽¹⁾ Department of Education and Science, Assessment of Perfofm ance Unit, Mathematical Development: Secondary Survey Report No, 1, by D.D. Foxman, R.M. Martini, J.A. Tuson and M.J. Cresswell (HMSO, 1980) and subsequent issues.

⁽²⁾ Initial tests were applied to 4,500 pupils in 70 schools in 1977/8; see J. Wöppel, ‘Das neue Abschlussverfahren der Hauptschule’, Lehren und Lernen, 1979, Heft 10, p. 2 and pp. 24-36 (see also ibid. 1978, Heft 6, pp. 1-54; 1982, Heft 3, pp. 64-80). For statistics of schooNeavers with Abschluss, see Allge meines Schulwesen, 1982, p. 27.

⁽³⁾ In Baden-Würtiemberg in 1982, 41 per cent of all school-leavers came from Hauptschulen.

⁽⁴⁾ APU2, p. 44, question 20H.

⁽¹⁾ Unpublished APU questions made available to us for the present comparisons.

⁽²⁾ Interested readers may obtain a translation into English of the 1979 Baden-Württemberg Hauptschule mathematics tests on writing to the authors at the National Institute.

⁽¹⁾ As explained in the main article (p. 57), the results of the IEA study were reported in two volumes edited by Husén, and in a parallel volume by Postlethwaite; for brevity, these are referred to in this appendix as HI, HII and P.

⁽²⁾ HI, p. 158; Hli, p. 23. Separate samples were taken for Scotland which yielded a similar picture; for simplicity the exposition here is confined to the English samples (in reality: to England and Wales). The lower German variability is undoubtedly an important factor in promoting higher average attainments in successive classes; lower variability within German classes is clearly related to the widespread practice of grade-repeating (advancement to a higher class depends on mastering material in the lower class).

⁽³⁾ HI, p. 271; P, p. 116.

⁽⁴⁾ Fortunately, a whole chapter (HI, pp. 163-83) was devoted to a description of the administration of the inquiry in England; only Sweden was honoured at similar length. We do not know what problems arose in other countries; further details for Finland, Japan, the Netherlands and Scotland were published in Inter national Review of Education, 15 1969, no. 2.

⁽⁵⁾ HI, p. 171, lower half of page. The exposition given there of the error is not easy to follow. As described there, group 1 consisted of 13-year olds in the second year; group 2 consisted of 13-year olds in the third year; and group 3 consisted of the remainder of those in the third year. The relevance of these ‘groups’ to the IEA ‘Populations 1 a and 1 b’ is not immediately obvious: Population 1 a is close to ‘group 1’ plus ‘group 2’; and Population 1 b ought to consist of ‘group 1’ plus the remainder in the second year (an undefined ‘group’ in this context!). There seems no point in pursuing here a word-by-word analysis of this paragraph; it is sufficient for the present purposes that there was an admission of an error.

The chapter is attributed jointly to four authors (see p. 163). In a later chapter by a different author (p. 220), it is stated that in ‘England: the percentage of the 13-year olds in the 1 b grade group is approximately 84 per cent’ (p. 235). This is virtually impossible if the average age of 1 b pupils is 14:4 (the remaining 16 per cent would have to have an average age of over 16½). Presumably a further mistake was made here in the labelling of the various groups. A more detailed report devoted to the English study makes it clear that third-form pupils were chosen for Population 1 b, but without explaining how this choice is to be reconciled with that of other countries (D. A. Pidgeon, ed., Achievement in Mathematics: A National Study in Secondary Schools, NFER, 1967, p. 8; see also p. 310 which explicitly and mistakenly says that ‘combining groups 2 and 3 gives the agreed international Population 1 b … containing the majority of thirteen year olds’).

⁽¹⁾ For other countries in which there an age between samples 1a and 1b, differences in the direction in and ages can be detected in the tables shown in HII, p. 72 and P, pp. 116-7. The condusion in the text above would not really be altered if, using techniques, we the age adjust ment on the experience of all countries in the IEA study (as in HII, p. 71). The report on the English contribution (Pidgeon, op. cit., p. 99) gives slightly different values for the averaoe scores of the samples, but the differences arise only in the decimal place and do not affect the substance of the above argument. There is a suggestion in a more detailed analysis of IEA data (Population 1b) by the late Professor Choppin that the relation between achieve ment and month of birth is slightly non-linear; this is probably the result of more older pupils in the age-group 13:0 to 13:11 being in a higher class, and subject to a step-jump in their educational experience, as compared with younger pupits. Since we are here concerned with the average advance of a whole age-group, a linear approximation is not likely to be far out B. H. Choppin, ‘The relationship achievement and age’, Educational Research, 1969, especially p. 24).

⁽²⁾ HI, pp. 148, 153 and 260, table 14.1B.

⁽³⁾ See the graphs, HI, p. 225 and P, p. 55 (identical).

⁽⁴⁾ HII, p. 81, table 3.15.

⁽⁵⁾ W. Schuttze and L. Riemenschneider, ‘Eine vergleichende Studie über die Ergelonisse des Mathematikunterrichts in zwölf Ländern’, Deutsches Institut für Internationale Pädagogische Forschung, 34 (1969), p. 4. The numbers in the two sources do not correspond exactly, since those published in the international surveys have re-weighted to allow for differential response in the sub-samples (there seems to have been some over sampling of Realschulen).

⁽⁶⁾ HII, p. 81.

⁽⁷⁾ Pidgeon, op. cit., pp. 53, 60, 99.

⁽⁸⁾ Schuitze and Riemenschneider, op. cit., table 8.

⁽⁹⁾ We were not able to make an exact conventional adjustment for guessing, since the number not answering each of the questions was not published; accordingly we adjusted by linear interpolation between the published figures (unadjusted, adjusted) for Gymna sien (38.2, 33.7) and ‘Remainder’ schools (29.9, 24.4).

⁽¹⁰⁾ For England, Pidgeon, op. cit., p. 99, interpolated to provide an estimate at age 13:8; for Germany, Schuttze and Riemenschneider, loc. cit., and HII, p. 81.