Reform of mathematical education in primary schools: the experiment in Barking & Dagenham.

Prais, S.J.

The need to raise the vocational skills of the broad cross-section of Britain's workforce - rather than concentrating on the number of university graduates - has formed a principal conclusion of the series of international comparisons of productivity in matched samples of industrial plants carried out by the National Institute during the past fifteen years.(1) Discussion of the precise measures to secure improvements in skills have now risen close to the top of public policy debate, with some recent re-assuring signs of convergence between the main political parties. The country's most important gaps in skills are now recognised as being in technical and vocational qualifications - at the 'craft' or 'intermediate level' - such as the long-established and internationally respected City and Guilds awards at part II, and the hoped-for corresponding Level 3 of the newer, but much-criticised, NVQs.

At university graduate level the numbers qualifying have more than doubled in the past decade. But at craft-level the numbers qualifying remain low, with only about a quarter of each year-group qualifying in Britain, compared with about two-thirds in Germany or Switzerland (these proportions are to be taken as rough orders of magnitude; precise figures depend on which grades of craftsman and technician are included). The main difficulty in increasing numbers qualifying at craft-level in technical occupations in Britain lies in low school-leaving standards, particularly in mathematics, for those in the lower half of the attainment range.

Problems in school mathematics are of long-standing in England; they were recognised as long ago as the early nineteenth century by Kay-Shuttleworth (who became the equivalent of the present-day Secretary of State for Education), and later in that century by Matthew Arnold (the equivalent of our HM Chief Inspector of Schools). Both were much concerned with comparisons of our pupils' schooling attainments with the Continent. Nearer our own days, in 1982 the all-embracing wisdom at that time on problems of mathematics teaching in English schools was embodied in the lengthy official Cockcroft report, Mathematics Counts. But that report was too all-embracing and not sufficiently precise and discriminating in its recommendations: it set out lists of mathematical topics to be covered at school according to three levels of final attainment but, by not setting out sufficiently clearly the priorities in mathematical topics to be mastered at each age, it condoned much variability in pupils' attainments; and by not indicating sufficiently clearly the appropriate balance of teaching styles, free rein was given to methods of teaching in which, to give extreme examples, pupils in primary schools sit in groups around tables, many having their backs to the blackboard, and for much of the time are left to progress at their own pace in their own individualised ways, with inadequate guidance from the teacher. A considerable variability amongst pupils' attainments is a hardly surprising consequence of such teaching methods; that variability is considerably greater than in Continental countries, and makes teaching a much harder task for our teachers here than abroad. These problems have not diminished under the regime of specified 'attainment targets' and associated specified 'levels of attainment' for obligatory subjects instituted after 1988 by the National Curriculum. It is the weaker pupils who have continued to suffer more seriously from this approach.(2)

Wide and varied worries continue to be expressed in Britain on the state of mathematical teaching. Professors of leading universities' mathematics departments last year voiced their strong doubts as to the adequacy of the standards attained by students now entering university to study mathematics, and even stronger doubts on the mathematical standards of the much increased number of students now accepted for engineering degree courses. Their concern is with, say, the top quarter of the attainment range: they recognise that something has gone seriously wrong at secondary school.(3) Employers face a broader section of the ability range, and raise persistent doubts about basic numeracy levels of school leavers. Economic and social commentators worry about the adequacy of the country's skill-base in an era of worldwide rapid technological progress and point to the need to match - and be in advance of - increasing international competition; they worry about the growth of unemployment among the young and low-skilled, particularly as a result of ever-advancing automation; and many worry about social tensions resulting from the increasing income-gap between those who are skilled and able to secure employment, and those who are unskilled and more frequently unable to secure employment. Unemployment is undoubtedly the worry closest to the hearts of average citizens, and to the minds of politicians who rely on their votes.

Following the National Institute's earlier visits to Continental industrial training establishments and vocational colleges, a systematic series of visits to Continental primary and secondary schools has been carried out in recent years by National Institute researchers jointly with school inspectors and senior teachers mainly from the London Borough of Barking and Dagenham; the visits focused on schooling standards and the associated teaching methods in mathematics and related vocational subjects. The visits amplified what could be deduced, at least in outline, from earlier international tests that had been set to large samples of pupils since 1964 at roughly decennial intervals by the IEA and IAEP.(4) Those earlier tests were not without their limitations; participation in the randomly-selected samples was not satisfactory in Britain, with a suspicion that low-attaining schools and low-attaining pupils were under-represented (for example, only 47 per cent of pupils selected for the English sample in the 1990-91 IAEP survey participated, compared with 80 per cent in Switzerland). It was consequently difficult to obtain an entirely reliable indication of how our low-attaining pupils - say, those at the lower quartile - compared with those of other countries. What could be learnt from these earlier surveys of the main pedagogic contrasts between Britain and other countries was also very restricted. Those earlier surveys were nevertheless valuable in pointing to an important 'paradox' in Britain's educational achievements: while Britain's top pupils used to perform at least as well as, and perhaps better, than top pupils of other countries - and Britain was right to pride itself on that achievement - yet its average and below-average pupils were trailing behind pupils of similar age in leading European industrial countries and Japan.

The latter deficiencies were evident to the eyes of our visiting teams to Continental schools (which included France, Germany, the Netherlands and Switzerland); and they were evident throughout secondary-school ages. The deficiencies could be traced to the primary phase of schooling, more particularly to the great variability of British pupils' attainments in upper primary school classes (the 'junior school', or 'Key Stage 2'). On receiving pupils from their feeder primary schools, British secondary schools clearly have to face a seriously greater task in coping with pupils' variable attainments than their Continental counterparts; and that task is greater in mathematics because of its more 'linear development': what can be learnt by a pupil today depends more strongly than in most other subjects on what has been mastered yesterday. Under-attainment tends to persist, and initial variability tends to increase more rapidly as pupils progress through school. Whilst some remediation of primary school failings is undoubtedly possible at secondary school, most of the problem - in the judgement of our visiting teams - could be put right only at its source, that is to say, at the primary stage. This highly important conclusion led to the current experimental reforms in six junior schools in Barking and Dagenham, beginning with 15 classes in Year 4 (ages 8-9) in January 1995, rising to 30 classes in Years 4 and 5 in September 1995.

In general terms the better use of teaching time and the enthusiasm with which the changes have been welcomed by teachers and parents warrant a brief written account at this stage. However, the benefits to be derived from such an experiment need to be allowed to cumulate over a number of years, and perhaps another two years need to elapse before any quantitative assessment will prove possible.

What has been done so far, in the light of what our visiting teams recognised as effective in Continental classes and judged relevant to our problems in this country, is summarised below under the headings of new teaching styles and new teaching materials. These two aspects are interdependent; to use one without the other must be expected to be ineffective. And both aspects need considerable advance preparation and heavy investment of resources. A third heading under which reforms still need to be considered - on the basis of our Continental observations - relates to various organisational matters (class groupings, etc.) as will be discussed in the final section.

Teaching styles

The pithy - but potentially misleading - way of describing the changes in teaching styles adopted in our experimental classes is to say that they involve more 'whole-class teaching'. That is far from being an unambiguous concept, as pointed out by Professor Robin Alexander in a recent lecture.(5) It would be more correct - if a quick description is required - to describe the new style in the negative terms of what is sought to be avoided: the object is to move away from currently prevalent methods of primary teaching that lead to undue variability in pupils' attainments. In many classes pupils are now typically divided into separate groups working at different levels, usually sitting in groups of 4-6 round their own tables, with the teacher carrying a very heavy burden in moving from table to table to keep each group moving forward, while pursued by questions from individual pupils who have not understood something and are stuck, or have finished and want to know what to do next. The researches of Professor Galton (Leicester; the ORACLE project) have shown that this is no mere caricature, but a weakness of classroom methods that have become widespread in Britain in the past generation. These methods inevitably lead to the familiar 'fanning out' of pupils' attainments as they move to higher classes.

The object of the Barking and Dagenham reforms is to teach in such a way that the whole class is kept more closely together in its attainments. The greater part of each lesson is devoted to a teacher-led question-and-answer session directed to the whole class. With experience, the teacher learns to set the pace and difficulty of successive questions so that the greater part of the class is fully engaged mentally (is 'on task') for the greater part of the time. As far as possible every child participates; a child is invited to answer, not because he has put his hand up first, or even at all, but chosen in such a way as to elucidate the range of potential difficulties, and to maintain the lesson's rhythm, challenge and interest. Oral enunciation by the pupil promotes the mental fixing of new concepts, precise expression and 'good English'.(6) The development of competence in mental processes is given priority over written work, especially at younger ages (say, till age 9). Pupils come to the front, enunciate their answer in whole sentences, and write their answer on the board. No less important from a wider educational stance, this approach develops a higher quality of socialisation. The pupil coming to the front benefits in becoming not unduly bashful nor tongue-tied; and the rest of the class learns to be helpful and sympathetic, and never to mock or make fun of others' mistakes. The experience of collaborative learning - involving the whole class - promotes mutual consideration and socialisation in a wider sense. The latter aspects are given much emphasis in Swiss classes that our teams have visited, and are now equally emphasised in Barking and Dagenham.

Pupils' participation on these lines leads them to discover for themselves - with their own minds - the import of the lesson. Emphasis on discovery as an ingredient of learning seems no lower on this approach, and in practice seems to be more effectively delivered, than in the individualistic styles of teaching advocated by 'progressive educationists', in which lack of guidance by the teacher permits the child to pursue too many time-consuming fruitless culs-de-sac.

While the major part of each lesson is conducted in this way, there is systematic alternation of deskwork and various forms of group-work or educational 'games'. A variety of styles within each lesson maintains interest and rhythm, and that variety is integral to the new approach. During deskwork the teacher takes the opportunity to provide additional individual help for pupils who have difficulties and need to catch up; and extension worksheets are provided for those who work quickly.

This approach to teaching is some distance from the lecturing style that so often springs to the mind of English teachers when 'whole-class teaching' is mentioned. I am also not sure whether it is sufficiently similar to 'exploratory discourse', as discussed by Professor Alexander in his recent lecture, to be categorised under that heading.(7) To suggest that our 'new' methods in Barking and Dagenham are otherwise now never to be seen in England would be absurd; but it is an approach which has been well developed and is now widely and systematically used in certain parts of the Continent (particularly Switzerland and southern Germany); to judge from the writings of a number of nineteenth century English observers, it seems to have been a distinguishing feature of teaching methods there for a very long time.

Teaching materials

Improved teaching materials form an essential ingredient of our new approach. The main distinctive features are (a) more pupils' exercises to ensure better consolidation; (b) a finer gradation of difficulties in approaching a new topic, with adequate exercises after each small step; (c) a detailed teachers' manual including a suggested allocation of the year's work to each week, and suggested lesson-plans incorporating, for example, 'starter activities' (such as the first five minutes on a mental arithmetic game), OHP transparencies relating to the main oral part of each lesson, and worksheets of examples for deskwork in the second part of the lesson. Guidance for the main oral part includes question-and-answer procedures to introduce and consolidate concepts and operations that are the focus of that lesson.

The development of existing British teaching material to incorporate these features seemed desirable to our visiting teams on observing teaching in progress in Continental classrooms. A subsequent specimen analysis of the main mathematics textbooks used in schools at ages 8-9 in England, Switzerland and Germany (based on the most important textbooks used by at least half the pupils in each country) showed, for example, about 'three times as many examples for pupils to work on' in Continental than corresponding English textbooks; there is greater continuity on the Continent in the treatment of a new topic with 'Continental pupils having available some six times as many exercises on a topic as English pupils before moving on to the next topic'; and more distinct gradation in the Continental textbook, so that difficult concepts are not introduced till pupils have been given full opportunity to master simpler underlying concepts.(8)

There is nothing intuitively obvious, nor derivable from first principles, as to the right number of exercises at each step, nor the right number of intermediate steps, nor the best sequence of steps; it is a matter of teaching experience and objectives. So long as we are content that only, say, the top third of pupils need become sufficiently proficient in mathematics to attain standards in mathematics at the end of compulsory schooling suitable for entering a technical traineeship, then perhaps existing English teaching methods are adequate; but if we are to raise that proportion substantially, say, to two-thirds, to meet the requirements of advanced workplace technologies, then careful attention needs to be given to such details of how teaching is carried out in countries which are more successful with their middle and low-attaining pupils.

The above are the principles on which new teaching materials are being developed and trialled for this project, taking into account the experience embodied in recent Continental pupils' textbooks and teachers' manuals. Materials have been developed so far only for Years 4 and 5, starting from what they have learnt from current materials and methods in earlier Years. As new materials for younger children are gradually developed (Year 2 should be introduced later this year), and children learn more successfully with their help in their earlier years, so our materials developed for later years will need to be revised (some of that material originally intended for later years will be used at earlier stages of schooling). Much developmental work thus lies ahead.

Organisational reforms

In the fullness of time the changes described above may be expected to bridge much of the gap between the mathematical attainments of primary school pupils in Britain and leading Continental countries. But there are grounds for suspecting that in many schools the gap will not be completely bridged without adopting some additional organisational reforms with a view to further narrowing the range of pupils' attainments facing a teacher in each class. The greater the homogeneity, the greater the possible rate of progress: that is the central underlying principle. Continental primary schools manage to teach on a 'mixed-ability basis', and reach high attainments for all pupils, without setting or streaming into separate classes according to pupils' levels of attainment as is sometimes thought necessary in Britain. That greater homogeneity on the Continent undoubtedly depends in part on organisational aspects that still need to be considered, or reconsidered, in this country. No great resources would be required by these organisational changes - more a change in frame of mind as to how schools can best serve the interests of all their pupils (I have to emphasise that the following are no more than suggestions for discussion: their potential contribution to higher attainments warrant further research, and priorities for implementation in this country are still very much a matter of personal judgement).

More flexibility in age of school-entry In general, it is clear that children develop at very different and very varying rates; the question that needs to be asked in Britain is whether greater flexibility in date of entry, say, 3-4 months either side of the normal twelve months period, would be an advantage. Some form of 'school-readiness' testing could be made available to advise parents, the final decision usually resting with them. The issue of principle is whether a child should be put into a class to which his attainments best fit him; or whether his future is to be determined solely by his calendar age since birth (rather than his 'gestational age'). No perfect solution to such problems is possible: the pragmatic issue is whether an arrangement more on current flexible Continental lines is better than current rigid practice here. Benefits from a flexible age-range must be expected to arise not only to the child concerned, but also to the teacher who finds a class more manageable when a slow-developing child enters when he is more mature and closer in capability to the rest of the class; and there are consequential benefits to the rest of the class if less of the teacher's time has to be absorbed by children with difficulties.(9)

Half-classes for some lessons Another organisational arrangement to promote the learning of younger children (also observed on the Continent) is to split the class into two halves for, say, a quarter of the pupils' lessons in the core subjects of mathematics and reading; the teacher has only about a dozen pupils before her in such lessons, and is able to concentrate to a greater extent on difficulties faced by individual pupils. The other half of the class could be looked after by one of the many teaching support staff now employed in our primary schools (in 1995 there was one full-time equivalent educational support staff for every four qualified teachers on our primary schools).(10) That other half of the class might be engaged, for example, on individual deskwork or craftwork (on the Continent school-times are more flexible, and on some days half the class comes in later or leaves later). Any additional resources involved would in any event not be great, since the complete class is taught together for the great majority of the time.

In distinguishing this arrangement from general pressures for smaller classes, the important point to notice is that any additional resources needed by this arrangement are more carefully targeted. The arrangement is similar in intention to, but more effective in its application than, that current in English schools where a teacher occasionally spends some minutes with a group of a half dozen pupils sitting round a particular table, while other pupils in that room proceed with their separate activities watched out of the corner of the teacher's eye.

Class teachers for more than one year It seems to be agreed among Continental teachers that it takes perhaps a whole term, or longer, till a teacher becomes sufficiently acquainted with a new class of pupils - understands their individual learning problems, including any originating in circumstances at home - to know how best to motivate each of them in the most effective way.(11) For this reason Continental primary schools often arrange for teachers to take the same class of pupils for 2-4 years at a time. The risk of a child falling into the hands of a less than adequate teacher for such a long period seems less of a worry on the Continent than the benefits arising from continuity; perhaps this is because teaching is less strenuous, more manageable and less variable on the Continent, having been developed on the basis of the features outlined above. It will take some time in our experimental schools before a sufficiency of those features become satisfactorily developed here to make a period longer than a single year with a class the normal rule for teachers; but already at this early stage some teachers here have appreciated the likely benefits, and would be glad of such an opportunity.

The features listed above seem to have contributed substantially to reducing variability among pupils' attainments, and to reducing pressures in Continental schools for class-repetition. They also seem to have reduced the proportion of pupils having 'special educational needs'; the latter proportion has been put in England as being in the region of 20 per cent ever since the Warnock Report (1981). Children of that level of attainment were to be seen, as a broad impression, only half as frequently in Swiss schools.

Research and the tasks ahead

Finally, a general remark may be offered on the role of international comparative research and the role of systematic observation by experienced practising teachers. The new features and changes introduced so far in our experimental classes in Barking and Dagenham - relating to the precise types and mix of teaching styles, and the precise content of textbooks and teachers' manuals (degree of consolidation, gradation of difficulty) - were decided on after repeated systematic observation of Continental practice by our teams of teachers and researchers; but these features are rarely described and analysed with adequate clarity and in requisite detail in academic international educational research, such as the IEA or IAEP studies; the same holds for the organisational aspects just outlined. What success the present project may claim is very much the result of experienced practising English teachers and inspectors forming the core of our observation teams, and looking for classroom methods that might contribute to a much shorter 'tail of under-achieving pupils' than we have become accustomed to in England. Earlier international research undoubtedly provided background and starting points: but the drawing of relevant empirical conclusions, and the setting of priorities, depended on their experienced eyes.

The weight of the tasks ahead must not be underestimated. Successive revisions of draft teaching materials will be necessary in the light of cumulating experience for some years to come. Teacher training will need to be adapted with the help of videos planned to illustrate the distinctive components of the new teaching styles with which English trainee-teachers are unfamiliar (Continental trainees were themselves taught in that way and need less instruction in these styles). Contact with interested teacher-training colleges should lead to more critical analysis of these approaches and their more efficient propagation. Methods of assessment of pupils' progress will need to be developed, based on the objective that the vast majority of each class need to reach specified minimum standards in each year of schooling. And the whole of that approach developed for the primary years of schooling will need to be extended to the secondary years, and the associated research will need to be carried out.

Once the initial investments have been made to cover the transition, delivering an improved quality of education should not need vastly increased resources in terms of recurrent costs: that is the central theme of this project. Much foresight, patience and devotion will be required by teachers and researchers directly active in those transitional developments.

This Note has been prepared in response to the Editorial Board's invitation to provide a personal commentary on developments in the Barking and Dagenham project. Whilst remaining responsible for errors and misinterpretations, I should like to thank all concerned for their co-operation at Barking and Dagenham - particularly Mr Roger Luxton (Principal Inspector) and Mr Graham Last (Senior Inspector, Primary); on the Continent, I wish particularly to thank Professor Aurin (Freiburg i. B.); Professor HJ Streiff (Head of the Teacher Training College, Zurich, when we made our first visits there) and Swiss Federal and Cantonal educational authorities.

NOTES

(1) In 1981 the Institute's study Productivity and Industrial Structure (by SJ Prais in collaboration with A Daly, DT Jones and K Wagner; Cambridge UP) compared Britain with Germany and the United States; the gap in vocational training formed the subject of ch. 4, and the need to use it, was one of the main 'lessons' drawn in the final chapter 19. A series of subsequent comparative investigations of particular manufacturing industries and of one service sector (hotels) were surveyed in Productivity, Education and Training (by SJ Prais, Cambridge UP, 1995); the detailed underlying investigations were issued in two supplementary volumes of reprints (vol. 1, 1990; vol. 2, 1995; available from the Institute). A more detailed analysis of the transition From School to Productive Work: Britain and Switzerland Compared is in press (CUP, end 1996).

(2) Details are in ch. 4 of my 1995 book, Productivity, Education and Training; see also my paper, Improving school mathematics in practice, in Proceedings of a Seminar on Mathematics Education (Gatsby Charitable Foundation, London, November 1995), esp. pp. 5-6, nn. 12 and 14 on historical aspects; and p. 7, n. 17 on current difficulties on English primary teaching. On the difficulties associated with the teaching styles that have become current in English schools, see the well-known official report by R Alexander, J Rose and C Woodhead, Curriculum Organisation and Classroom Practice in Primary Schools (DES, 1992); and two valuable books published in the past year: R Alexander et al., Versions of Primary Education (Routledge, 1995), esp. ch. 6; and P Croll and N Hastings (eds), Effective Primary Teaching: Research-based Classroom Strategies (Fulton, 1996).

(3) AG Howson (chairman of a working party of the London Mathematical Society, The Institute of Mathematics and its Application, and the Royal Statistical Society), Tackling the Mathematics Problem (London Mathematics Society, October 1995).

(4) The International Association for the Evaluation of Educational Achievement (IEA) carried out mathematical tests in 1964, 1981 and (results still awaited) in March 1995; a dozen countries participated in the first round, rising to about 45 in the latest round. Attainments in other subjects were tested by the IEA in intervening years. A less detailed study of pupils' attainments in mathematics and science was carried out by the US-based International Assessment of Educational Progress (IAEP) in 1990-91 covering some twenty countries (a preliminary 'feasibility' study was carried out by the IAEP in 1988 in six countries; Spain was the only European participant apart from the UK). In all these surveys, usually 1-3000 pupils at each age in each country were given the same tests, each test lasting one school-period (about 45-minutes; in mathematical tests no calculators were permitted).

(5) Robin Alexander, Other primary schools and ours: hazards of international comparison. Delivered at the University of Warwick on 18 June 1996 (available as an Occasional Paper from the Centre for Research in Elementary and Primary Education, U. Warwick, Coventry CV4 7AL).

(6) On precision of expression, cf. also the Swiss emphasis in the teaching of practical subjects on clean working, precision, perseverance, reliability and responsibility (often put under the umbrella term of Arbeitscharakter or 'good work habits'; see p. 91 of my Productivity, Education and Training).

(7) p. 22.

(8) H Bierhoff, Laying the Foundations of Numeracy (National Institute Discussion Paper no. 90, January 1996), pp. 42-3. The finer gradation is illustrated in that study in relation to learning to add two-digit numbers, such as 56 + 37, which advances from the simpler 50 + 30 in six distinct intermediate steps (ibid. p. 27). English texts tend not to distinguish as many intermediate steps, and provide fewer consolidation exercises after each step.

(9) Children born pre-term present a particularly clear anomaly in relation to our current school-entry requirements based on birth within a precise twelve-months period: a child born, for example, two weeks' pre-term and just before the critical date determining the required year of school-entry, enters a year earlier than a child born at full term two weeks later; the former child is more likely to have problems in keeping pace with his class because of his early entry, and those problems often persist. Twins born either side of midnight of the determining date were recently reported as being directed by the local educational authority to enter school in successive years. In that case, an appeal led to a sensible solution; but anything less extreme leads to the literal application of the birthday rule irrespective of a child's state of development.

(10) Ofsted, Class-size and the Quality of Education (1995), p. 41. Present support staff often do not have the qualified status legally permitting them to take charge of children outside the surveillance of a qualified class-teacher; this anomaly is one that deserves addressing from many points of view (including whether a child with SEN is being properly catered for even under present arrangements; see next sub-section).

(11) These issues are discussed further in App. C of our forthcoming book (1996) on transition from school to work in Switzerland and England.