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Mathematical attainments in primary schooling: Raising standards and reducing diversity.

Julia Whitburn (*)

Concern over poor standards in mathematics among English school leavers has led to a number of government initiatives in recent years. Without a secure foundation of mathematical understanding and competence during the primary school years, later learning in mathematics is problematic. This paper examines recent major initiatives at the primary stage of schooling and their effect on raising standards, including the National Numeracy Strategy and the Improving Primary Mathematics (IPM) project. The latter project, influenced by successful Continental approaches to teaching mathematics, aimed both to raise average standards of attainment and to reduce the large variation in attainment that has, in the past, characterised the performance of English pupils.

Although the new teaching approaches, and the innovatively detailed teaching materials, developed by the IPM project have enabled significant improvements to be effected, concern remains over the low attainment in England of an unduly large proportion of pupils (as compared with Continental schools). It is suggested that serious consideration needs to be given to adopting arrangements that are the norm in several other countries -- namely, to introduce some flexibility in age of entry to schooling (at present in England this is governed strictly by date of birth). Such a change would, it is suggested, significantly reduce the number of low attainers and range of attainment within a class, and make a teacher's task of successful interactive whole-class teaching more manageable.

Background

The purpose of this paper is to assess the progress made during the past decade in raising standards of mathematical attainment among primary school pupils in England, and the principal tools that have been effective and, in conclusion, to examine how the further progress needed for English pupils to rival continental standards may best be achieved.

The large-scale international studies of achievement in 1995 have most recently drawn public attention to the poor average standards of mathematical attainment -- though evident already in the earlier international studies of 1964 and 1981. Since then, there have been a number of governmental initiatives designed to raise levels of attainment. Improving standards of mathematics, however, is a long process: mathematical learning is developed layer upon layer and, like building a wall, is as secure only as the foundations on which it stands. The government is naturally anxious to demonstrate improved standards among school leavers, yet securing improvement depends on establishing firmer foundations of understanding and competence in the primary years of schooling. Results of initiatives will be apparent among school leavers only after several years, and we should not rush to prejudge the effects of the reforms. It has, however, become very apparent that in the London Borough of Barking and Dagenham (LBBD), wher e pupils have been participating for the past five years in a mathematics initiative based on continental methods of teaching -- entitled 'Improving Primary Mathematics' (IPM) -- standards of average attainment have improved dramatically faster than so far at the national level. The IPM initiative has been especially effective in reducing the proportion of lower-attaining pupils and the ways in which this has been achieved are discussed below. We shall refer to some newly-available comparative data in respect of attainment of samples of primary age pupils in Switzerland to demonstrate that the average standard achieved there still remains ahead of even the improved standard in Barking and Dagenham; moreover, the Swiss schooling attainments are more consistent (less variable) in terms of individual pupils' attainments. Reasons for this are discussed, especially in relation to the narrower attainment-range of pupils in classes in Switzerland and its contribution to making teachers' tasks more manageable.

For further significant progress to be made here, external (organisational) changes are required that are largely beyond the control of individual schools. Changes to the structure of the education system are needed that require an Education Minister with boldness of vision and determination.

We begin with a brief consideration of the evidence regarding attainment at the national level, before examining in greater detail the evidence for progress made by pupils participating in the IPM project.

The national picture

A major source of evidence regarding levels of attainment in mathematics in England remains the Third International Mathematics and Science Study (TIMSS) that was conducted in 1995 and involved pupils in 41 countries (Beaton et al., 1996; Mullis et al., 1997). At that time, the mean scores of primary pupils in Year 5 (at about 9+) in England (corresponding in age to fourth grade in other countries) were ranked 17th out of 29 participating countries, and secondary pupils aged fourteen (Year 9 in England, eighth grade in other countries) were ranked 25th out of the 41 countries. (1)

It is in the crucial years of primary schooling that the foundations for later learning in mathematics need to be laid, since success in the later years of schooling depends on a secure understanding of number structure. Indeed, many countries achieving higher average standards in mathematics have shown a greater tendency than England to concentrate on mastery of numeracy during the primary stage of schooling, and to leave algebra, geometry and probability until skills in basic calculations have been secured. Their curricula for the first few years of schooling reflect an appreciation that success in mathematics at later stages requires a sound understanding and facility with a range of basic calculating strategies together with the ability to apply these in contexts appropriately and accurately.

The weaknesses of English primary pupils in basic numeracy were clearly revealed by the results of the TIMSS study; the nature of these weaknesses were convincingly illustrated by responses to specific questions in that study. As noticed at the time, there was an extraordinary inability by 9-year-old English pupils to select the correct answer (from a choice of four) to the four-digit subtraction question:

6000

-2369

This was answered correctly by a mere 36 per cent of English children aged nine, compared with 91 per cent, 86 per cent and 89 per cent of children the same age in Hungary, Netherlands and Japan respectively; this demonstrated all too clearly the lack of a secure knowledge in relation to basic calculations. If allowance is made for the fact that a multiple choice question may be correctly answered through guesswork, then the percentage of English pupils 'knowing' the correct answer falls to an estimated 15 per cent, compared with about 85 percent of pupils in the three other countries mentioned. Another question -- this time open-ended -- required the addition fact 4 + 4 + 4 + 4 + 4 + = 20 to be written as a multiplication fact: this was answered correctly by only 53 per cent of English children compared with 80 per cent in Hungary, 84 per cent in the Netherlands an 92 per cent in Japan. Poor knowledge of multiplication facts among English pupils was again evident in responses to a (multiple choice) question requiring the selection of a number to go in the number sentence 4 x? < 17; this was answered correctly by only 56 per cent of English pupils compared with 79 per cent, 70 per cent and 89 per cent of pupils the same age in Hungary, the Netherlands and Japan.

Similarly, poor understanding of the concept of fractions was more common among English children, or example, in a diagram showing nine squares of which five had been shaded, only 48 per cent of English children were able to identify the correct fraction (5/9) from a choice of four answers, compared with 58 percent of children in Hungary, 67 per cent in the Netherlands and 89 per cent in Japan.

For many concerned with mathematics standard in England, however, it was the poor performance of the weaker pupils that caused greatest anxiety. The scores achieved by the lowest 5 per cent and lowest 25 percent of primary pupils are shown in table 1. The 25th percentile score of English primary pupils (452) is, for example, below the 5th percentile score achieved by pupil of the same age in the Netherlands (462), suggesting that there are proportionately more than five times as many very low achieving children in England as there are in the Netherlands; similarly in relation to the other countries shown here. The international studies compared the mathematical attainment of pupils of similar ages, regardless of the number of years spent in formal schooling. Pupils in England, where the age of starting school is considerably earlier than that of most other countries, had had the benefit of more years of schooling than pupils in other countries.

A similar pattern is apparent in table 2, which gives the comparable percentile scores for secondary school pupils at age fourteen (Year 9). More Western European countries took part in this study, and so it is possible to include figures relating to a wider range of countries. Again, we see that the scores achieved by the weakest S per cent and 25 per cent of pupils in England are lower than those in other countries (Scotland achieved much the same as England). If pupils in the German-speaking part of Switzerland are considered, it can be seen that proportionately there are at least five times as many pupils in England who scored below the lowest 5 per cent score in that part of Switzerland.

The unremarkable (disappointing!) performance of both primary and secondary English pupils in TIMSS supported the messages conveyed by the earlier (if methodologically less sophisticated) First and Second International Mathematics and Science Studies of 1964 and 1981 (FIMS and SIMS). Further, by the time that TIMSS had been conducted, awareness of the 'long tail of underachievement' in mathematics among English pupils had been brought to the nation's attention (by, among others, researchers at NIESR; Prais, 1981, 1987). However, it had often been argued that although average standards might be disappointing, and there might be an over-large proportion of low attaining pupils, at the top end of the distribution pupils in England outperformed their peers in other countries, as had been shown by the first IEA study in 1964. The results of TIMSS (a generation later in 1995), however, dispelled that idea in respect of both primary and secondary pupils. At the primary level, only 7 per cent of English pupils in Yea r 5 achieved the scores of the top 10 per cent of pupils internationally, compared with 11 per cent, 13 per cent and 23 per cent of similarly-aged pupils in Hungary, Netherlands and Japan respectively.

At age fourteen (our Year 9), the scores of the top 5 per cent and 25 per cent of pupils in England remained below the standards achieved by pupils in competitor countries. Table 3 shows also that the score of the top 5 per cent of English pupils was only a few points above that achieved by the top 25 per cent of pupils in German-speaking Switzerland. The revelation that the standard achieved by the best of English pupils had been comfortably overtaken by high attaining pupils in several other countries served to focus the government's attention on the seriousness of the problem. As a result, decisions for change were taken quickly - some would say with too little consideration of the full implications.

Action taken by the successive governments in an attempt to improve mathematics attainment

Although the years since 1995 have seen a plethora of new initiatives designed to raise achievement in mathematics, concern over standards goes back much further. The appointment of the committee of enquiry into the teaching of mathematics under the chairmanship of Sir William Cockroft in 1978 reflected awareness of, inter alia, 'the apparent lack of basic computation skills in many children, the increasing mathematical demands made on adults, the lack of qualified maths teachers... '(Cockroft, 1982, p. ix). The Report recognised many of the difficulties that teachers, especially in secondary schools, faced in their attempts to motivate pupils in studying a subject that was perceived as both 'dull and difficult'. Many of the recommendations reflected the need to relate the study of mathematics to the experiences of adult life, and the Report served to influence secondary teaching of mathematics in this regard. Many of the recommendations made then still need emphasis today - for example, the recommendation no t to expect written recording of mathematics at too early an age (i.e. greater early emphasis on mental calculation).

Until 1989, however, primary teachers in England enjoyed an unusual degree of autonomy in their classrooms since, unlike many other countries, there was no nationally-agreed curriculum, and schools were relatively free to make their own decisions regarding curricula and the mix of subjects taught. (GCSE and its predecessors influenced to a great extent the curricula in secondary schools.) This changed in 1989 with the introduction of a National Curriculum for the teaching of all subjects in state schools. Since then the National Curriculum has been amended several times but many critics have regarded the way in which each National Curriculum for mathematics has been stated as having two major shortcomings: first, far more detailed specific guidance (not just principles) is needed on the teaching on each topic; and, second, a curriculum should be stated for each year of schooling (rather than each Key Stage - a group of 2-4 years of schooling) in order that teachers, pupils and parents may all be clear as to t he topics to be covered and the standard to be achieved by the end of each school year.

A subsequent far-reaching initiative introduced by the government since 1995 - reflecting its continued concern over the low average standards in mathematics - has been to remedy the shortcomings of the National Curriculum for primary mathematics through the publication of a National Numeracy Framework that specifies in a detailed way and on a year-by-year basis the topics to be taught. (2) There is no doubt that this was significantly influenced by an initiative in primary mathematics that began in the London Borough of Barking & Dagenham (LBBD). This initiative, Improving Primary Mathematics (IPM), was the result of a collaboration between inspectors in LBBD LEA and researchers at the National Institute of Economic & Social Research (NIESR), together with the educational authorities in Zurich, Switzerland. (3) This initiative - its methods and its success - are examined in a later section. (4)

The introduction of the National Numeracy Strategy (NNS) into all state primary schools from September 1999 represented a significant change of direction by the government from the earlier approach to the teaching of mathematics in primary schools. (5) Following the IPM approach in providing much finer detail on the topics to be taught each year, the NNS also provided exemplar lesson plans and advice on the implementation of a three-part lesson structure that was an essential part of the NNS approach to teaching mathematics. (6) By requiring a daily 'numeracy hour' (perhaps its most widely publicised recommendation), the NNS not only regularised but increased the amount of time spent on mathematics in primary schools. Representing a major shift from earlier recommended pedagogy, the NNS also encouraged teachers to move away from teaching to several small groups of pupils - an individualised approach to learning mathematics had dominated primary classrooms for the previous thirty years since the Plowden Report - and to adopt a whole-class teaching approach for a crucial part of every lesson. The radical nature of the introduction of the NNS should not be underestimated; it is an indication of the major shift in attitudes to the teaching of primary mathematics that has taken place during recent years.

Other less fundamental government initiatives since 1995 to raise standards of primary mathematics have included the publication of 'league tables' of result of national tests, the identification of 'leading teachers', and special consideration of children identified as 'gifted and talented'. In respect of low attaining children, funding has been provided for summer school to enable pupils achieving Level 3 at the end of Key Stage 2 to reach the 'expected' Level 4 before beginning secondary schooling.

Achievements so far

What has been the effect of the numerous initiatives so far? Has there been 'value for money' in the sense that standards in mathematics have improved in real terms? We know that officially-published results of the national tests have improved year by year, with increased proportions of pupils in successive years gaining the so-called 'expected' level. At the end of Key Stage 2, for example, the percentage of pupils gaining the 'expected' Level 4 has increased from 59 per cent in 1998 to 72 per cent in 2000, with a government pledge that this will be raised to 75 per cent by 2002. Yet for this to be regarded as an objective measure of improvement, we need to be certain that the standard required for a Level 4 has been invariant over time. We have to rely on government assurances for this - with only a slight doubt being raised in our minds by the fact that the percentage of the marks required to achieve a Level 4 has fallen slightly over the years. Were we inclined to be suspicious, we might wonder at the fa ct that the decision as to the number of marks required to gain Level 4 each year is set only after the test papers have been marked; on the other hand, we appreciate the need for the examiners to he confident that they have set the levels at the 'correct' standard -- which, questionable as it may seem, can only he done through examining the pupils' answers. It is a pity that the arguments and the detailed consideration governing these decisions have not been made public.

It is perhaps too soon, however, for any significant improvement in school-leaving standards to be evident. As stated at the outset, improving mathematics attainments of school leavers is a long-term process, and we need to wait until the better foundations laid in primary schooling can be built upon during Key Stage 3 more successfully than previously.

The need for well-qualified teachers

As Cockroft pointed out in 1982,

'No efforts to improve the quality of mathematics teaching are likely to succeed unless there is an adequate supply of suitably qualified mathematics teachers.' (ibid., p. 244)

Yet at the present time, the shortage of teachers well-qualified to teach mathematics is possibly more acute than ever before. Recruitment on a large scale of teachers from overseas is currently taking place in order to meet the shortage of primary teachers; at the secondary level we are told that the number of maths teachers at KS3 who have no qualification in the subject has quadrupled in the last three years (Henry, 2001a). A headline in The Times recently claimed, 'One in five new teachers not suitable for job', and the Chief Inspector for Schools has expressed his concern over the lack of well-qualified teachers.

The supply of primary teachers well-qualified to teach mathematics, however, is related to the mathematics qualifications of school leavers. At present, the current requirement is for all entrants to teacher training to have achieved a grade C (or equivalent) in GCSE mathematics, and, from informal samples of B.Ed/BA (QTS) students training to become primary teachers, it would appear that very few of the students have more than the minimum qualification in mathematics. (7) Since it is estimated that 94 per cent of C grades are awarded to candidates taking the Intermediate Tier of papers in GCSE examinations (DfEE, 1999b), it seems unlikely that many students becoming primary teachers will have studied the syllabus for the Higher Tier papers in GCSE mathematics. This is a contributory factor to the insecure levels of subject knowledge apparent among many teacher trainees and their clear lack of self-confidence in mathematics. The introduction of a compulsory Numeracy Test for all qualifying teachers from 2000 onwards may improve levels of subject knowledge but may also undermine self-confidence yet further.

Very few intending primary teachers continue with mathematics beyond the compulsory years of schooling; pupils taking A-level mathematics (only about 9 per cent) will, in the main, be drawn from those who have taken the Higher Tier of GCSE examinations. The recent introduction of AS levels has apparently had little, if any, effect on the numbers continuing with mathematics.

This contrasts with the pattern in many continental countries (and indeed many countries around the world) where the expectation is that pupils continue with the study of mathematics throughout all their years of schooling, and may well need to pass an examination in mathematics in order to proceed to any form of further/higher education or training. In France, Germany and Switzerland, for example, a good mark in the mathematics paper that forms a compulsory part of the Baccalaureate, Abitur or Maturitat is a prerequisite for university entrance regardless of the subject or discipline for future study. In Japan some 80 per cent of upper secondary school pupils choose to continue with mathematics into the 11th grade (Year 12). (8) There is no doubt that this improves the subject knowledge of entrants to teacher training, and ensures that primary teachers are competent and confident regarding their knowledge of mathematics. From my observations of levels of subject knowledge of Japanese and English trainee prim ary teachers, it is clear that those in England need to be improved if pupils are not to suffer in the future from inadequately-prepared teachers.

The Improving Primary Mathematics Project

For more than ten years, a team of researchers at NIESR has been involved in making international comparisons of mathematics standards at different levels of schooling, with many of the comparisons being carried out in collaboration with inspectors and advisory teachers in LBBD. Over one hundred classes in Switzerland, Germany and Holland have been visited, in order to understand better the relationship between teaching methods, teaching materials and successful teaching in mathematics. The consistently high quality of the mathematics teaching observed in German-speaking Switzerland (the highest achieving European region in mathematics in the most recent international tests) was especially impressive. One key feature was the level of teachers' skills in whole-class interactive teaching, and the way in which all pupils participated on an equal basis. During the past five years, we have taken several teams of primary head teachers and classroom teachers to classrooms in Switzerland for extensive observation of teaching of mathematics. They were impressed that teachers were assisted in achieving a consistently high standard of teaching through the excellence of the teachers' manuals and pupils' materials. In their judgement, these materials were superior to anything available in the commercially published schemes in England. F or the past five years, we have been working together to develop teaching materials of a quality, detail and depth comparable to those we have seen in Switzerland (Bierhoff, 1996). A series of detailed teachers' manuals for Years 1-6 of schooling, accompanied by pupils' workbooks and textbooks, have been produced (of course fully meeting the needs of the National Numeracy Strategy and Curriculum 2000) incorporating the carefully sequenced approach of the Swiss teaching materials.

The materials assume the use of a highly interactive whole-class teaching approach, reflecting the successful practice observed in Swiss primary schools. In respect of the layout of classrooms -- to which much importance is attributed in Switzerland -- the horseshoe shape used in Swiss primary schools (or sometimes double-horseshoe, depending on a classroom's dimensions) has been adopted, replacing the earlier typical arrangement of pupils seated in groups of 4-6. (9) This arrangement improves sight-lines between teacher and pupils, pupils and other pupils, and pupils have a clearer view to the board or OHP screen. The horseshoe shape also encourages pupils to address the whole class when answering questions, so promoting pupils' participation and concentration, and improving a feeling of community and co-operation among the class as a whole unit.

The new teaching materials consist of pupils' materials in the form of either workbooks, textbooks and -- more important -- substantial teachers' manuals, which provide detailed lesson-by-lesson plans including OHTs, pupils' worksheets and other resources. The influence of the early work of the IPM project can be seen in many of the NNS ideas and strategies. Yet there are certain important distinctive features of the IPM. These include:

* more attention to securing a sounder foundation in children's understanding of mathematics, through more detailed, sequenced, step-by-step teaching in the very early years;

* greater emphasis on mental work, including developing high standards of oracy;

* the provision of detailed plans for each mathematics lesson (not merely example lessons as for NNS) throughout each school year, together with the necessary resources;

* providing more guidance on the pedagogy needed for successful whole-class teaching -- for example, the levels and types of questioning, maintaining the pace of lessons and encouraging the active participation of all children.

The IPM project perhaps differs most noticeably from the NNS in its provision of the last two items; both of these are regarded as joint prerequisites for maintaining high quality teaching across topics and between schools. IPM has made an important contribution to developing teacher quality by the regular and detailed in-service training that has been provided; the detailed discussions of pedagogy have helped classroom teachers to understand the complexities of successful questioning of pupils and ways of developing dialogue for taking learning forward, including increasing pupil participation.

The IPM materials are designed to be used with pupils of all attainment levels in a mixed ability setting. All pupils are involved in the 'starter' activity (provided as a mental warm-up to each lesson) and in the 'development' section of each lesson, through answering questions demonstrating to the class and giving clear oral explanations using precise mathematical vocabulary. Children's contributions in taking the lesson forward are valued, and they learn to listen to each other. Written work is provided for consolidation and practice only after understanding has been established, and generally accounts for less than one-third of lesson time.

Teaching is systematic, and each new concept is broken down into several very small stages, which are mastered individually and progressively. Once children have mastered conceptual understanding, they are expected to develop the quick recall of number facts which is essential for the application of calculating strategies. Additional pupil materials are provided for 'quick finishers'; these provide questions of a more challenging nature on the lesson topic rather than expecting children to progress on to new concepts without the assistance of the teachers.

The new teaching materials and new teaching methods were introduced gradually, beginning in 1995 with six pilot schools in Barking and Dagenham, and are now being used by more than 19,000 children in over fifty schools in five boroughs.

Monitoring the effectiveness of the project

Independent evidence regarding the effectiveness of the project is provided by results in national tests (SATs). When the IPM project began, Barking & Dagenham was among the lowest achieving boroughs in the country. There has been a steady improvement since then, and in 2000, for the first time, the results in mathematics at age seven (the end of Key Stage 1) for pupils in Barking & Dagenham were just above the average standard nationally, with 91 per cent of all children achieving Level 2 or better, compared with 90 per cent in England as a whole. It took slightly longer for the benefit of the IPM project to be visible in the Key Stage 2 results, since it was only in July 2001 that the first cohort of pupils in the project reached the end of Year 6. Provisional figures for 2001 indicate that, whereas at the national level the percentage of pupils achieving the 'expected' Level 4 or better fell by 1 percent from the 2000 figure to 71 per cent, in LBBD the figure increased to 72 per cent (see table 4). Moreove r, provisional figures for LBBD indicate that during the period 1997-2001 the percentage of pupils gaining Level 5 in mathematics at the end of Key Stage 2 (the level 'expected' after two further years of schooling) increased from 11 per cent to 23 per cent, showing that higher-attaining pupils have benefited from the IPM project.

A key objective of the project, however, was to reduce levels of low attainment; the percentages of pupils failing to achieve the 'expected' Level 2 in mathematics at the end of Key Stage 1 are, therefore, shown in table 5.

Reducing low attainment

Among the national initiatives to raise standards of mathematics attainment, it is difficult to discern adequate emphasis on efforts focused on the problems faced by low-attainers. While at Key Stage 1 the proportion of pupils failing to achieve Level 2 has fallen from 15 per cent in 1998 to 10 per cent in 2000, the number of children recorded as having identified special educational needs continues to increase, with a corresponding increase in the percentage of children with more serious problems having 'statements' of educational need. The main focus of the support programmes is, quite correctly, for these children to improve their skills in literacy; there is relatively little specific provision, however, aimed at raising mathematical standards among low-attaining pupils or, more fundamentally, for preventing pupils from becoming low attainers. Of course, as a nation we are not alone in having low-attaining pupils in mathematics, but it seems that other countries are more successful in that they have, roug hly, only a fifth of our fraction of low attainers (see tables 1 and 2). We have already mentioned the criterion in some countries of a required standard of attainment for progression to a subsequent grade. Children in France or Germany are likely to receive supplementary weekend or holiday instruction, or ultimately be required to repeat a year of schooling if they fall too far behind; in England that approach is regarded as potentially too damaging in other respects, although studies have not been able to produce evidence to support this (Robinson et al., 1992).

In Switzerland, the beginning of schooling is likely to be delayed for a year for the minority of children (one or two per class) who are clearly not ready for formal learning. Alternatively, slower-learning children may go to a separate class which takes two years over the work normally covered in the first year of schooling, in order to obtain a sound and secure foundation before joining mainstream classes and proceeding in the usual way. These examples serve to illustrate that other countries acknowledge the existence of slower learning children and have developed ways of accommodating them that maximise their chances of returning to the mainstream and succeeding educationally.

In England, the tradition is for pupils to progress strictly according to the twelve-month span of ages in which they are grouped, regardless of their level of attainment (our educational system has cynically been described as the only one which requires a pupil to attend for eleven years but does not require anything to be learned). There will, additionally, be a small minority of children -- perhaps 1-2 per cent -- with specific disabilities, who benefit from the support of a teacher with specialist knowledge and expertise in the particular disability. The proportion of children who benefit from being in a special class -- perhaps within a mainstream school -- continues to be subject to debate; policy and practice across the world differ widely in this respect.

If, however, we require all pupils to progress through the years of schooling at the same rate, then arguably we should make greater efforts to reduce the incidence of low-attainment and provide greater support for slower learners. To this end, the Improving Primary Mathematics project has developed special materials, to be used in addition to the normal programme with children who, for one reason or another fall behind. This has contributed to the success in Barking and Dagenham of reducing the incidence of low achievement; between 1998 and 2000 the percentage of children failing to achieve Level 2 at the end of Key Stage 1 fell from 18 per cent to 9 per cent -- faster than the national average (see table 5).

Clearly it is important to reduce low attainment yet further, but this is very easy to say and enormously difficult to achieve, especially when we know one of the causes of low-attainment is social deprivation. The complexities of social deprivation are readily acknowledged and we know that even children from economically secure homes may experience severe deprivation in terms of poor parental interaction and difficult social relationships. It is only in terms of economic affluence, however, that we have an indicator -- albeit imperfect -- that as long been recognised as related to academic performance: this is the percentage of children eligible for free school meals, widely used at the school level as an indicator of deprivation. If we look at the figures for national achievement on the KS1 tests in mathematics we see that there is a clear relationship between the percentage of eligibility for free school meals and low attainment.

It is clear that, at the national level, a child in a school where over 50 per cent of children receive FSM is about six times as likely to be a low attainer in mathematics than a child in a school where under 8 per cent of children are eligible for FSM. Clearly more than economic affluence is at issue here: homes with low incomes may often be those with only a single parent who, if at work, may not be at home when the child returns from school. What can be done to reduce this level of inequality? What can schools or teachers do to compensate for the level of disadvantage indicated by free school meals? This question deserves very serious consideration, in the light of the government's recently published plans to impose 'educational inclusion' to ensure that all pupils get a fair deal at school' (Ofsted, 2000a). (10)

The data in table 6 for schools in LBBD using IPM materials show that in each category of school the percentage of pupils failing to achieve Level 2 is better than the national figure, and the difference is especially apparent among schools with more than 50 per cent of pupils eligible for FSM. However -- and this is a highly significant point in making overall comparisons -- these comments take no account of the fact that the distribution of schools by FSM category is very different in the IPM schools from those in England as a whole. We can see the difference between the two distributions clearly in chart 1.

When we compare the KS1 results for all IPM schools together in Barking & Dagenham with those achieved at the national level, and note that the performance of IPM pupils was better than the national average in 2000, we need to take account of the distribution of disadvantage.

If the distribution of disadvantage among schools in Barking and Dagenham was similar to that in the nation as a whole, then (calculating a weighted average, or an 'index number') the estimated total proportion of pupils failing to reach Level 2 would have been only 5 per cent in 2000, compared with a national figure of 10 per cent - only half the national average of low attainers.

Pupils in the IPM project have thus made progress in the following ways:

* standards of average attainment at the ends of KS1 and KS2 are now as good as (or better than) those in England as a whole;

* the improvement in percentages of IPM pupils achieving 'expected' levels has been faster than in England as a whole;

* the proportion of low-attaining pupils has been reduced - faster than in England as a whole;

* if levels of social deprivation are considered, the reduction in low attainment in IPM schools has been even more remarkable.

Season of birth

A further factor contributing to low attainment in England, however, is the distinctive, very strict application of a twelve-month spread of birthdays for admitting children to school and thereafter to each successive class. This fails to allow for the different developmental rates of children prior to schooling or to provide for additional compensatory support that would allow previously slow-developing children to 'catch up'. Differences in development and the consequential need to allow for this in school admissions policy are acknowledged in many countries with high standards in mathematics attainment. In parts of Hungary, for example, admission to school is based on a consideration of children's ages within a three-year-span and an evaluation of children's developmental levels (Nagy, 1989), and similar considerations apply in Switzerland.

We have analysed the IPM test results of a three-year cohort of Year 2 pupils according to low attainment and season of birth (defining low attainment for the purposes of this analysis as achieving a score of less than 40 per cent); the variation in the proportion of low attaining pupils is shown in table 7.

The pattern of children born in the summer months achieving less at school than children born earlier in the school calendar is of course well documented (Sharp et al., 1994; Sharp and Benefeld, 1995). Using the IPM test data, however, we have been able to analyse the effects on children with the double disadvantage of (a) attending schools with high eligibility for free school meals and (b) who also happen to be born in the summer months. For example, children born in June-August, and attending schools with over 50 per cent FSM, have a 23 per cent chance of being low-attainers, whereas for those from schools with under 8 per cent FSM and who happen to be born in the autumn quarter the chance is less than 4 per cent.

It might seem that as little can be done to mitigate the effects of season of birth as of home background; but at least as far as far as general stow maturation or readiness for school is concerned, the Swiss/Continental approach is to allow 4-6 months flexibility in relation to the nominal twelve months of birthdays for entry to primary school. This flexible arrangement cannot eliminate underachievement entirely, but the proportion of low attaining children -- and hence the diversity of attainment within a class -- is reduced, thus making whole-class teaching more manageable.

Comparisons of standards achieved using IPM materials with those in Switzerland

It is important, however, to continue to make comparisons with the standards achieved by primary school pupils in higher-achieving regions and countries and not to become complacent at the improvements so far achieved. Since no large-scale recent data are available regarding the attainments of primary aged pupils, we have conducted our own small-scale study. The detailed results of the tests to pupils aged 8 to 10 years (using identical mathematics questions) are provided in Appendix A, with the main points as follows:

* when Swiss and English pupils were matched by age, with Swiss pupils having had twenty months less schooling, the mean scores of the Swiss pupils were at least 10 per cent above those achieved by English pupils;

* the variation within a class of Swiss pupils was much smaller -- less than half as great as in England -- with a major factor being the fewer lower-attainers;

* the age range in each class was narrower for the English classes, reflecting the stricter policy of proceeding according to birth date.

We have also analysed the results of the Swiss pupils by age and found that the underachievement by younger pupils commented on earlier in relation to the performance of English pupils is much reduced (see table A2).

But we need to consider precisely why the Swiss pupils were able to achieve better results after twenty months less formal schooling. We know that the standard of teaching in the IPM project is high -- inspectors from LBBD and researchers from NIESR have carefully monitored this. We also know that the teaching materials used in IPM classrooms contain a carefully sequenced progression, broken down into small steps, and that the teaching manuals provide support on pedagogy and classroom organisation. We appreciate that the IPM teachers master the complex demands of successful whole-class teaching gradually, building and developing their skills year by year, but it has to be acknowledged, however, that -- in spite of our teachers' efforts -- the IPM pupils are nor yet achieving the standards that would be regarded as the norm in Swiss classrooms. We need to consider what other factors are both restricting average attainment and encouraging diversity in English classrooms.

From extensive visits to Swiss primary classrooms, we have become aware of a number of factors that, in our view, affect pupils' standards of attainment but are beyond the control of the individual teacher. These are identified below; they are put forward only as meriting further examination for their effect on teaching standards.

* Pre-schooling in Swiss classes concentrates on general preparation for formal schooling, in terms of developing pupils' social and emotional readiness for learning. Problems in these areas are identified and subject to a finely-grained diagnosis; additional help and support is provided as required.

* Pupils identified in pre-schooling as not being ready for formal learning at the same rate as other pupils are encouraged either to spend an extra year in pre-schooling or to join classes which take two years over the first year of formal schooling.

* In the normal course of events, primary school teachers retain a class for three years, thus saving valuable time at the beginning of each year in 'getting to know the pupils' and also providing sufficient time for teachers to address the problems of an individual pupil rather than passing them on to the next teacher.

The first two factors mentioned reflect official policy; both, however, assist children to participate on more equal terms once in the classroom. In England, with the current emphasis given during the Foundation Stage of schooling (Nursery and Reception years) to developing skills of reading, writing and recording of number, it may be difficult for teachers to pay sufficient attention to the social and emotional preparation for schooling.

The second factor requires only a change in attitude to enable a greater flexibility in terms of the age of children in a particular class. Until now, some difficulty would have been caused to the administration of SATs if flexibility of age were to be introduced, since the SATs have been designed to be taken by children at a particular age. It has now been suggested, however, that pupils might take SATs according to their 'readiness' rather than strictly by age (Henry, 2001b); this would smooth the way for introducing some flexibility in schooling age.

The third factor does not reflect official policy; allocation of staff to classes is often the result of negotiation between heads and their teachers. Many heads and teachers have privately acknowledged (to the present writer) the obvious benefits of teacher continuity for pupils' learning and emotional security, yet there appears to be a reluctance to put the obvious into practice. The common objection made concerns what would happen if a teacher failed to establish a positive relationship with a particular child, yet a teacher's professionalism should enable him/her to overcome potential difficulty in this respect. If we regard English teachers as being as professionally competent as Swiss teachers - for whom this does not appear to present any problems, and who react with surprise to the idea that they should 'get to know' a new set of pupils every year - then we would expect the benefits to considerably outweigh the disadvantages.

Summary

Improving standards of mathematics attainment is a long-term process, and it will take more years for the effects of the government initiatives to be evident among secondary pupils and school-leavers. At the more local level, the teaching approaches and lesson materials of the IPM project have had considerable effect in raising average standards of attainment and reducing levels of low attainment. Classroom teachers using IPM materials have successfully developed their skills of interactive whole-class teaching, having been much influenced by the teaching styles and classroom organisation in German-speaking Switzerland. Yet the gap between standards of attainment among IPM pupils and those in Zurich remains. A greater diversity of attainment among IPM pupils also continues, and increases the difficulty of whole-class teaching. To close the gap in standards and to reduce the diversity of attainment requires, it is proposed here, changes in the organisation of schooling that are beyond the control of the indivi dual classroom teacher and needs to be tackled at the national level.

(*.) National Institute of Economic and Social Research. [e-mail: j.whitburn@niesr.ac.uk]

NOTES

(1.) Although some still argue over the validity of international tests and the soundness of the methodology employed, it would appear likely that deficiencies in sampling and response rates - with under-representation of weaker pupils - operated to the apparent benefit of the performance of English pupils.

(2.) The National Numeracy Framework initially related to Years 1-6 of schooling but has now been extended to cover pupils in Years 7, 8 and 9.

(3.) Funding for the pilot study was provided by the Gatsby Foundation.

(4.) The extent of its influence has been noted by Professor Alexander (Alexander, 2000, p. 597).

(5.) The NNS was the outcome of the National Numeracy Report.

(6.) The NNS specified that a typical lesson in Years 1-6 would contain three parts:

* oral work and mental calculation (about 5 to 10 minutes)

* the main teaching activity (about 30 to 40 minutes)

* a plenary to round off the lesson (about 10 to 15 minutes) (DfEE, 1999a, p. 13).

(7.) Students without a GCSE pass in mathematics at C grade or higher may either take an Access course or sit a mathematics paper set by the provider of the initial teacher training. Little information is publicly available regarding the content or rig-our of these papers which are set and marked internally.

(8.) 97 per cent of pupils in Japan continue with upper secondary schooling beyond the compulsory years of schooling (Monbusho, 2000).

(9.) McNamara and Waugh (1993) discussed the effectiveness of a 'horseshoe' arrangement in achieving whole-class intereaction.

(10.) A further recent report (Ofsted, 2000b) commented in the context of Key Stage 2 results (p. II): 'The general pattern is that as disadvantage increases the achievement of pupils reduces.'

REFERENCES

Bierhoff, H. (1996), Laying the Foundations of Numeracy, Lon on, National Institute of Economic and Social Research.

Alexander, R. (2000), Culture and Pedagogy, Oxford, Blackwell.

Beaton, A. E., Mullis, I.V.S., Martin, M.O., Gonzalez, E.J., Kelly, D.L. and Smith, T.A. (1996), Mathematics Achievement in the Middle School Years: IEA's Third International Mathematics and Science Study (TIMSS). Chestnut Hill, TIMSS International Study Center.

Cockroft, W.H. (1982), Mathematics counts: Report of the Committee of Inquiry into the Teaching of Mathematics in Schools, London, HMSO.

Department for Education and Employment (DfEE) 1998), Autumn Package, London, DfEE.

--(1999a). The National Numeracy Strategy, London, DfEE.

--(1999b). Statistics of Education, London, DfEE.

--(2000). Autumn Package, London, DfEE.

Department for Education and Skills (DfES) (2001), First Release; National Assessments of 7,11 and 14 year olds (Provisional), London, DfES.

Henry, J. (2001 a), 'Short route to plug gaps', Times Educational Supplement, 17 August, p.6.

--(2001 b), 'Teachers may decide when to test pupils', Times Educational Supplement, 7 September, p.1.

McNamara. D.R. and Waugh, D.G. (1993), 'Classroom Organisation: a discussion of grouping strategies in the light of the "Three Wise Men's" report', School Organisation, 13 (I), pp. 41-50.

Nagy, J. (1989). Articulation of Pre-school with Primary School in Hungary: An Alternative Entry Model, Hamburg, UNESCO Institute for Education.

Monbusho (2000). Education in Japan 2000, Tokyo, Monbusho.

Mullis, I.V.S., Martin, M.O., Beaton, A.E., Gonzalez, E.J., Kelly, D.L. and Smith, T.A. (1997), Mathematics Achievement in the Primary School Years: lEA's Third International Mathematics and Science Study (TIMSS). Chestnut Hill, TIMSS International Study Center.

Office for Standards in Education (2000a), Strategies to Pro ate Educational Inclusion: Improving City Schools, London, Ofsted.

--(1987), 'Educating for productivity: Comparisons of Japanese and English schooling and vocational preparation', Nationa Institute Economic Review, 99, February.

--(2000b). Raising the Attainment of Minority Ethnic Pupils, Lon on, Ofsted.

Prais, S.J. (1981), 'Vocational qualifications of the labour force in Britain and Germany', National Institute Economic Review, 98, November.

--(1997), 'Whole-class Teaching, School-readiness and Pu ils' Mathematical Attainments', Oxford Review of Education, 23 , 3, pp. 275-90.

Robinson, W.P., Tayler, C.S. and Piolet, M. (1992), 'Redoublment in relation to self-perception and self-evaluation; France', Research in Education, 47, pp. 64-75.

Sharp, C. and Benefeld, P. (1995), Research into Season of Birth and School Achievement A Select Annotated Bibliography, Slough, National Foundation for Educational Research.

Sharp, C., Hutchison, D. and Whetton, C. (1994), 'How do season of birth and length of schooling affect children's attainment at key stage I?, Educational Research, 36, 2, pp. 107-21.

[Graph omitted]
Table 1.

Percentile scores in mathematics: lower attaining pupils in Year 5
(fourth grade)

Country 5th percentile score 25th percentile score

England 366 452
Hungary 404 488
Netherlands 462 528
Japan 458 545

Source: Mullis et al., 1997, Table C.I.

Note: The scores of the international tests were standardised by IEA to
an international mean of 500 and a standard deviation of 100.
Table 2.

Percentile score in mathematics: lower attaining pupils in Year 9
(eighth grade)

Country Score of lowest 5% Score of lowest 25%

England 361 443
Hungary 391 471
Netherlands 387 477
France 415 484
Germany 368 448
Switzerland 401 485
German-speaking CH 446 528
Japan 435 536

Source: Beaton et al., 1996, Tables D.3 an E.I.
Table 3.

Percentile score in mathematics: higher attaining pupils in Year 9
(eighth grade)

Country Score of highest 25% Score of highest 5%

England 570 665
Hungary 602 693
Netherlands 604 688
France 591 666
Germany 572 661
Switzerland 607 685
German-speaking CH 658 740
Japan 676 771

Source: Beaton et al., 1996, Tables D. 3 an E.I.
Table 4.

Percentages of pupils gaining Level 4 or better in mathematics at the
end of Key Stage 2: England and LBBD

Year % of pupils ganining Level 4 or better
 England LBBD

1995 44 28
1996 54 41
1997 62 54
1998 59 54
1999 69 68
2000 72 71
2001 provisional 71 72

Sources: DfEE, 1998; 2000; DfES, 2001; LBBD Statistics Unit.
Table 5.

Percentages of pupils failing to gain Level 2 in mathematics at the end
of Key Stage 1: 1998 and 2000

 % of pupils not gaining Level 2
Year England LBBD

1998 15 18
2000 10 9

Source: DfEE, 2000; LBBD Statistics Unit.
Table 6.

Percentage of pupils failing to achieve Level 2 in mathematics test at
Key Stage 1 by school percentage of free school meals: 2000

% FSM % pupils without Level 2
 England IPM LBBD

up to 8% 3 0
8-20% 7 3
20-35% 11 9
35-50% 15 12
More than 50% l8 12

Sources: DfEE, 2000. Tables 3.1, 3.2, 3.3, 3.4, 3.5; LBBD Statistical
Unit (unpublished data).
Table 7.

Season of birth and low attainment

Month of birth % of pupils gaining less than 40%

September-November 11
December-February 17
March-May 20
June-August 24

All pupils 18
No. of pupils tested 4095

Source: IPM test data.


Appendix A: Recent comparisons of mathematics attainment of pupils using IPM materials and those in Switzerland

Valid international comparisons of attainment are difficult to make. One major difficulty - acknowledged by the large-scale international studies - relates to different ages of starting school in different countries, and the consequent problem as to whether to match pupils by age or years of schooling. In Switzerland, the average age at which pupils begin the first class of formal schooling is 6 years 11 months, compared with an average age of 5 years 6 months in England for pupils entering Year 1. Pupils in three classes in Zurich completed a test designed for IPM pupils and thus not necessarily compatible with their curriculum. The pupils in Zurich were, on average, a similar age to the LBBD pupils but had twenty months less experience of formal schooling. Clearly, the samples of Zurich pupils are too small to be more than illustrative but results are consistent with those of an earlier study (Prais, 1997).

It was also possible to analyse the test results by age for the sample of pupils in Zurich. The lower achievement by younger pupils commented on in respect of English pupils is much reduced. Interestingly, the score of the more 'advanced' pupils who are up to six months younger than the expected age for their classes and have been admitted to school earlier than the norm is higher than the average of the other pupils, although it must he remembered that the numbers involved and very small. The score of the more slowly developing pupils who were admitted to school later than their contemporaries is only 4 percentage points behind the score of the other pupils - thus they appear to have been progressing reasonably satisfactorily within the mainstream class.
Table A1.

Comparison of standards of mathemat ics attainment: in LBBD & Zurich

 Primary schools pupils aged
 9-10 years 8-9 years
 LBBD Zurich LBBD Zurich

No. of pupils 1312 44 1571 24
Age: Mean(y/m) 9 9 9 9 8 9 8 9
SD (m) 3.5 4.6 3.5 4.4
Years of schooling 4 1/3 2 2/3 3 1/3 1 2/3
Scores: Mean % 77 87 69 84
SD % 21 8 27 10
Table A2.

Mean scores of Swiss pupils aged 9-11 years according to age

Quarter of birth Mean percentage % of pupils
 score

Up to 6 months younger 94 9
First quarter 83 13
Second quarter 85 20
Third quarter 90 14
Fourth quarter 86 18
Up to six months older 82 13
All pupils 86 100
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Date:Jan 1, 2002
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