Is the National Numeracy Strategy Research-based?

IS THE NATIONAL NUMERACY STRATEGYRESEARCH-BASED?

by MARGARET BROWN, MIKE ASKEW, DAVE BAKER, HAZEL DENVIR andALISON MILLETT, School of Education, King’s College London

ABSTRACT: The British Government has recently agreed proposals fora National Numeracy Strategy which claims to be based on evidenceconcerning ‘what works’. This article reviews the literature in each keyarea in which recommendations are made, and makes a judgement ofwhether the claim is justified. In some areas (e.g. calculators) the recom-mendations run counter to the evidence.

Key words: numeracy, national numeracy strategy, researchsummary, calculators

1. INTRODUCTION

It is the proud claim of the Numeracy Task Force, appointed by thegovernment to design a National Numeracy Strategy to be imple-mented in England in 1999/2000, that their recommendations arebased on research evidence.

We have aimed throughout our work to look at the evidence to findsolutions to any problems with mathematics achievement, and tomake practical recommendations based on methods that have beenshown to be effective in raising standards of primary mathematics.

(DfEE, 1988b, p. 7)

This paper explores the extent to which this research-based claimis supported. It is important to stress that if the claim is notsupported in one or more aspects this does not necessarily invalidatethe recommendations; there may be other compelling reasons whythey should be implemented. Indeed the one of us who was amember of the Task Force clearly has in general agreed the policy,in full knowledge of those areas in which the claim was weak. Inreviewing the literature on numeracy we will be brief and by no

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means comprehensive, but will focus on those issues in the NationalNumeracy Strategy which are most contentious, referring only to thekey literature that we believe to be most reliable and apposite.

2. WHAT IS NUMERACY?

The definition of numeracy is a shifting one, both historically andgeographically. It appears to have been coined in the CrowtherReport (DES, 1959) as meaning scientific literacy in a broad senseand on the way was memorably re-defined by an HMI as ‘sensible useof a 4-function calculator’ (Girling, 1977). Until recently numeracywas probably conceived of functionally, as for example:

‘…the ability to process, communicate, and interpret numericalinformation in a variety of contexts.’ (Askew et al., 1997a, p. 4)

Numeracy is still used in this sense in New Zealand, but the word isunknown in the United States, where ‘mathematical literacy’ is acommon phrase and includes more than number. However‘number sense’ (McIntosh, Reys and Reys, 1992) is used in the USto include conceptual aspects of numeracy.

The definition proposed for the National Numeracy Strategy(DfEE, 1998b, p. 11) is that of the National Numeracy Project, whichlike that used by the Basic Skills Unit, focuses on ‘proficiency’,regarding numeracy as a culturally neutral and value-free set ofautonomous ‘basic numerical skills’ (ALBSU 1993, p. 13), emphasis-ing mental and written calculation and knowledge of number factssuch as multiplication tables. Although reference is made to a ‘vari-ety of contexts’ in the preamble, the detailed bullet points whichfollow get no nearer real life than ‘make sense of number problems’suggesting that ‘contexts’ refers only to the artificial contexts usedin textbook ‘word-problems’. Noss (1997) connected the narrow-ness of this definition with the Secretary of State’s simplistic views asexpressed in the White Paper Excellence in Schools: ‘the first task of theeducation service is to ensure that every child is taught to read, writeand add up’ (DfEE, 1997, p. 9). Noss also stresses the danger ofidentifying mathematics with numeracy, a danger now becomingmore apparent as the National Numeracy Project’s detailed numer-acy framework has with few modifications become the Framework forTeaching Mathematics: Reception to Year 6 (National Numeracy Project,1998), and is currently driving the revision of the national curricu-lum in mathematics.

In contrast the social practice model (Baker and Street, 1993; Baker,1996), is based on an acceptance of the cultural and ideological

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nature of numeracy as the set of numeracy practices individualsengage in during their lives. This draws upon the socio-culturalmilieu of everyday or work settings and is consonant with the stud-ies of literacy practices (Heath, 1983; Street, 1996). It includes infor-mal mathematics sometimes characterised as ethnomathematics(D’Ambrosio, 1985; Bishop, 1988; Harris, 1997), and also considersformal education as a distinctive cultural setting (Walkerdine,1988).

Recent research on numeracy as social practice, mainly fromoutside the UK, highlights the gaps between everyday numeracymethods and formal school methods (Scribner, 1984; Saxe, 1988,1991; Nunes et al., 1993; Masingila et al., 1996). Lave (1988) hasprovided the theoretical approach of situated cognition, in whichthe role of context has been seen to be of major significance. Studiesin England have confirmed that even when given classroom numberproblems, children rarely use the methods taught in schools(Plunkett, 1979; Hart (ed.), 1981; Johnson (ed.), 1989; Gray, 1991;Thompson, 1997).

The divergence between school-bound and socio-cultural mean-ings of numeracy explains other differences in interpretation ofresearch evidence discussed in later sections.

3. IS THERE AN ENGLISH PROBLEM OF LOW ATTAINMENT IN NUMERACY?

The National Numeracy Strategy was born out of a view held by this(and the previous) government that standards of numeracy arecurrently low. However the evidence cited by Ministers often revealsmisunderstandings about the design and status of national curricu-lum levels and the technicalities of national assessment. McIntosh(1981) provides quotations which show that officials have deploredwhat they perceive as falling levels of numeracy since at least themiddle of the last century.

It is not clear to what extent there is a significant problem ofcommerce and industry related to weak numeracy skills in thegeneral population; most concern recently has come from numer-ate disciplines in higher education (Sutherland and Pozzi, 1995;London Mathematical Society et al., 1995) and recruiters of gradu-ates, and may therefore relate to the 16–19 phase. Although theCockcroft Committee was launched in response to a perceivedconcern among employers, further investigations showed this not tobe substantiated (DES/WO, 1982). The little workplace researchthat does exist suggests that requirements differ from the traditionalskills associated with school number (Harris, 1991). In particular,

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Noss’s recent evidence (1997) suggests that the widespread use ofcomputers requires a workforce which has a sophisticated under-standing of the mathematical basis of models incorporated in soft-ware, rather than traditional computational skills which are morequickly and accurately performed by machines.

While the Basic Skills Agency estimated that a quarter of thepopulation had skill levels that would make it difficult to completeeveryday tasks successfully, less than one in ten of those identifiedfelt they experienced any real problems with numeracy (Bynner andParsons, 1997). Again this discrepancy may reflect the divisionbetween the artificial nature of the pencil and paper test used in thesurvey and common informal numeracy practices, including use ofcalculators.

Turning to international comparisons, in spite of the real diffi-culty of drawing valid inferences from the results (Brown, 1998), itcannot be disputed that in pencil-and-paper testing of numberunderstanding and skills, England generally scores below average,whether at primary level (Reynolds and Farrell, 1996; Mullis et al.,1997; Harris, Keys and Fernandes, 1997), at secondary level(Reynolds and Farrell, 1996; Beaton et al., 1996; Keys, Harris andFernandes, 1996) or among adults (Basic Skills Agency, 1997).

However when the items require the use of numeracy in practicalor contextual problem-solving, English pupils are consistentlyamong the highest performers (Lapointe, Mead & Phillips, 1989;Harmon et al., 1997; Planel et al. (in press) ). A clue to Englishsuccess in problem-solving may be that English students tend to bemore confident about their mathematical ability than most others(Beaton et al., 1996; Mullis et al., 1997), and are more used to work-ing on their own (Lapointe, Mead and Phillips, 1989). They also areset problems which require more than procedural knowledge, inclass and in national tests, due to national differences in culturalbeliefs about the aims of primary education (Planel et al., in press).

Thus the answer to whether there is a real numeracy problemdepends on the relative weighting put on problem-solving, and onmore conventional classroom pencil and paper arithmetic.

4. HAVE WE GOT THE RIGHT CURRICULUM?

The National Numeracy Strategy does not itself set a new curricu-lum for primary mathematics, other than a recommendation that‘oral and mental work should be included in every lesson’, and astrong discouragement from using calculators. However the Strategyrequires the distribution to all schools of the Framework for Teaching

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Mathematics from Reception to Year 6 developed by the NationalNumeracy Project. While not statutory, the use of the document isstrongly urged, in particular by schools seen as less effective innumeracy, and it will form the focus of national training. Manyschools will adopt it so as to avoid criticism from Ofsted inspectorsand Local Education Authority (LEA) officials.

The Framework sets the curriculum in at least four different senses,which will be considered separately. First, it indicates the overallcontent and balance across the different curriculum areas in terms ofboth the detailed objectives and the number of lessons to bedevoted to each, including the emphasis on calculators and computers.It also indicates the sequencing of the curriculum by recommendingexactly what should be taught in each week of each year. Finally itsformat as a fixed curriculum to be taught to all pupils regardless ofattainment indicates that very little curricular differentiation isintended.

Content and balance

The Framework places greater emphasis on oral and mental strate-gies, and also, in older age groups, on traditional written skills, thanis perceived to have been the case, and it is thought to be ambitiouswhen compared with current content in most schools, and indeedwith other countries (Howson, Sutherland et al., 1998). TIMSSresults relating to the coverage of the items in the international testsdo suggest that the current English curriculum, while wide in termsof topic spread, is less extensive than most (Mullis et al., 1997).

A multivariate analysis of international comparative research(Burstein, 1992) has suggested that differential coverage of thecurriculum by different countries (‘opportunity to learn’) is a rela-tively significant cause of international differences, although itaccounted for only 10% of the variance. However some countrieswith broad coverage, like the United States which claimed that chil-dren would have covered all the items tested in TIMSS (Mullis et al.,1997), did not perform better than others, such as the Netherlands,which only claimed to cover about half of them. This is in line withother research that suggests that children do not necessarily learnthe mathematics that they are taught (Hart, ed., 1981; Johnson, ed.,1989; Denvir & Brown, 1986).

Certainly the importance of curriculum emphasis is supported bythe TIMSS primary results which show that although English pupilsperformed relatively poorly in number items, the same pupilsscored higher than all other countries in the geometry items. Other

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international studies (Travers and Westbury, 1989; Lapointe, Meadand Askew, 1992; Schmidt et al., 1997a) show that both aims andcurriculum emphasis differ substantially across countries. Thissuggests that a re-adjustment of the curriculum towards more prac-tice in mental and written calculation as incorporated in theFramework document will indeed improve English performance innumber items in international tests. But the price of improvedperformance in international number tests could well be a decreasein performance in the use of mathematics in practical problem-solv-ing, in other areas of mathematics and in other subject-areas, likescience. Although there is probably a consensus that improved oraland mental skills are essential to both numeracy practices in manyfields and to further study of mathematics, discussion of the detailedbalance of curricular priorities has not been overt.

Calculators and computers

The headlines which greeted the announcement of the NationalNumeracy Strategy focused on the statement in the governmentpress release that there would be:

‘a ban on the use of calculators by children up to the age of eightand restricted use throughout the remainder of primary school’.

It was not made clear that this ban was not a recommendation of theTask Force Report, which contained the carefully worded compromise:

‘Calculators are best used in primary schools in the later years ofKey Stage 2…teachers should teach pupils how to use themconstructively and efficiently…Used well, however, calculatorscan be an effective tool for learning about numbers and thenumber system…’(DfEE, 1998b, p. 4)

Research on the effect of calculators on attainment in numeracy iscarefully reviewed by Ruthven (1997), and will not be repeated here.His conclusion concerning primary schools is that:

‘the degree of calculator use remains modest in most schools andby most pupils…however tempting it might be to cast the calcula-tor as scapegoat for disappointing mathematical performance atprimary level, the available evidence provides scant support forthis position…’ (Ruthven, 1997, p. 18)

The research on the effect of computer use in raising standards issimilarly inconclusive (Watson, ed., 1988, BECTa, 1998). Theseresults on computers and calculators are also supported by the more

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recent multivariate analysis of the factors affecting mathematicalperformance in TIMSS among English primary schools (Keys et al.,1998).

Sequencing

The way the curriculum is sequenced also differs between countries.For example Bierhoff (1996) points out that some continentalschools often deal for the first two or three years with oral andmental methods and small numbers only, and introduce writtencalculation and larger numbers rather later than in England.Although there is an attempt to delay written calculation by the TaskForce, this recommendation has been made repeatedly on previousoccasions (e.g. DES/WO, 1982) but, for reasons which are not clear,never implemented in English classrooms.

The continental curriculum also contains more intensive work onfewer ideas, and assumes earlier work has been assimilated; Englishprimary textbooks tend instead to move around rapidly and toconstantly recapitulate (Bierhoff, 1996). The same criticism of ‘asplintered vision’ leading to a curriculum ‘a mile wide, an inchdeep’ has also been made about the United States, in contrast withJapan (Schmidt et al., 1997b). However the arguments made bythese authors are not completely convincing; for example Germanyhas a typical continental curriculum yet, when sampling differencesare taken into account, achieves less well in mathematics at age 13than either the United States or England (Beaton et al., 1996).

The scheme of work suggested in the Framework maintains therapid change and recapitulation model, with a tendency to fragmentthe curriculum into many detailed objectives to be taught separatelyin logical sequence (Noss, 1997). This style has the advantage ofdrawing to the teacher’s attention the steps involved; dependenceon textbooks has meant that teachers are not always aware of these(Aubrey, 1994; Millett and Johnson, 1996). However the stepwiseapproach is a poor match to the existing evidence about the waypupils learn. While many pupils supply connections for themselvesand have therefore already mastered the next step before it istaught, others are not ready for it because they have not yet graspedthe previous step (Johnson, ed., 1989; Denvir and Brown, 1986).Hence there may well be an increasing problem up the school ofchildren’s learning becoming seriously out of phase with theFramework. Although permission is given for teachers to depart fromthe sequence, it is not clear that, with minimal training, all teacherswill feel confident enough to do so.

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Curricular differentiation

International multivariate analyses also suggest that a key factor innational attainment is the degree to which the curriculum is differ-entiated (Burstein, 1992). On the whole successful countries, likethose on the Pacific rim, maintain an ambitious curriculum for allpupils, whereas countries which offer some children reduced curric-ula, like Britain and the United States, do less well.

However, the downside of this ambitious uniform curriculum isthat many pupils in Pacific rim countries feel that they are not ableto keep up, and hence that they are weak at mathematics (Robitailleand Garden, 1989; Beaton et al., 1996; Mullis et al., 1997). This islikely to partially account for the very high level of participation inprivate coaching in such countries, which in turn lifts attainmentlevels. In contrast these studies show that English pupils generallyhave very positive attitudes, which unusually pertain across thewhole attainment range, and may enhance problem-solvingperformance.

Lack of mathematical confidence, rather than lack of attainment,was the key problem identified by the Cockcroft Committee fromresearch findings in industrial settings and randomly sampled adultsin their homes (DES/WO, 1982). The Committee strongly endorsedcurricular differentiation as a solution, responding to evidence of awide spread of attainment (Hart, ed., 1991). Learning to usenumber ideas in different contexts is very complex (Nunes andBryant, 1996), and some children take much longer than others toacquire understanding and skills (Denvir and Brown, 1986; Brown,1996b). In setting work, primary teachers tended to consistentlyunderestimate the range of attainment in their class, leaving highand low attainers without appropriate tasks (Bennett et al., 1984).

There may be good reasons now to reverse this policy of curricu-lar differentiation, but we should not forget the reasons for intro-ducing it.

5. WHAT FORM OF CLASS ORGANISATION IS EFFECTIVE?

The strongest recommendation in the National Numeracy Strategyis that in the first bullet point:

‘…a daily mathematics lesson to all pupils, lasting between 45 and60 minutes depending on pupils’ ages. Teachers should teach thewhole class together for a high proportion of the lesson….

(DfEE, 1998b, p. 2)

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Small group collaborative work is also endorsed in preference toindividual work (DfEE, 1998a, p. 20). This raises many issues whichwill be dealt with under the separate sub-headings of Time, Classorganisation and Differentiation.

Time

Much has been made of the notion of the numeracy hour, but theTIMSS data suggest that English pupils in Years 4 and 5 in 1994already spent on average between 55 and 60 minutes daily on math-ematics, an allocation exceeded by only three countries, one ofwhom scored lower than England (Mullis et al., 1997). Howeverthere may be some advantage to ensuring that all schools devote thisamount of time.

Class organisation

The intention to implement a ‘high proportion’ of whole classteaching is probably the most radical proposal in the NumeracyStrategy. The data from TIMSS suggest a smaller proportion ofEnglish teachers of Year 5 pupils (11%) use whole class teaching inmost mathematics lessons than that of teachers in almost any othercountry (Mullis et al., 1997).

In spite of the claims of the Task Force that ‘there is support inthe research’ for ‘an association between more successful teachingof numeracy and a higher proportion of whole class teaching’(DfEE, 1998b, p. 19), the evidence is not unambiguous. It dependsmostly on large-scale correlational studies, which cannot easily estab-lish causation, and can only report variables that are measurable.Many medium and large-scale studies show no significant effect(Aitken, Bennett and Hesketh, 1981, which is a re-analysis, after crit-icism, of the original data from Bennett, 1976; Burstein, 1992; Askewet al., 1997a), although some of these depend on questionnairerather than observational data. In reviewing generic Dutch studies,Creemers (1997) noted that the proportion of whole class teachingappeared to have a significant correlation with attainment (positivein all cases) in only 3 out of the 29 studies.

In other large scale statistical studies there has been a positivecorrelation between whole-class teaching and attainment (Galton,Simon and Croll, 1980; Galton and Simon, eds, 1980; Good, Grouwsand Ebermeier, 1983; Brophy and Good, 1986). However, notingthat in individual cases particularly poor results have also been asso-ciated with whole class styles, investigators have cited evidence for

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the quality of teacher-pupil interaction being a much more impor-tant factor than class organisation (Good and Grouws, 1979; Goodand Biddle, 1988; Galton, 1995). These authors suggest that a wholeclass format may make better use of high quality teaching, but mayequally increase the negative effect of lower quality interaction.

Most of the studies quoted above have shown individualised formsof organisation as the least effective. However, a recent multi-levelanalysis of the factors contributing to high achievement in Year 5 inTIMSS in English primary schools found that use of individualisedworking from worksheets in most lessons was significantly associatedwith higher attainment, whereas some whole class teaching in mostlessons was not quite significantly associated with lower attainment.However these effects were not found in the Year 4 sample, wherethe factors if anything were reversed (Keys et al., 1998).

Peterson (1979), in a review of mathematical learning studies,found that with the more direct approaches of traditional wholeclass teaching, students tended to perform slightly better onachievement tests (although the effect sizes were small). Howeverthey performed worse on tests of more abstract thinking, such ascreativity and problem-solving.

Several authors have found that using small group collaborativelearning was related to high-level mathematics achievement (e.g.Peterson and Fennema, 1985; Slavin, 1989), although once againGood, Mulryan and McCaslin (1992) in reviewing this work feel thatit is the level of thinking generated rather than the format which isimportant.

It is perhaps worth noting that most of the studies quoted in thissection do not report differential effects on different groups ofpupils, although, interestingly in terms of current national priori-ties, Bennett reported in both analyses (Bennett, 1976; Aitken,Bennett and Hesketh, 1981) that a style avoiding class teachingappeared to favour low attaining pupils, especially boys.

Differentiation

The National Numeracy Strategy has stated an aim to reduce therange of attainment in a class, and hence:

‘we are concerned that children should not continue to work atmany different levels, with the teacher placing them in a widerange of differentiated groups’ (DfEE, 1998b, p. 54).

This advice is followed also by a general discouragement fromsetting across parallel classes (or across age groups).

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It is correct that the national and international literature showsthat differentiation by setting is usually found to be a neutral factorin attainment, and may if anything be slightly negative in its effects(Slavin, 1990; Burstein, 1992; Hallam and Toutounji, 1996;Creemers, 1997; Boaler, 1997; Askew et al., 1997a).

Similarly although the evidence is somewhat equivocal, it seemsthat differentiated grouping may sometimes slightly lower the aver-age attainment and widen the attainment spread (Good, Mulryan,et al., 1992). There is also some evidence from TIMSS of a statisticalassociation in England between whole class teaching and a smallerstandard deviation (Keys et al., 1998), although it could be thatmore homogeneity (as in a setted school) enables whole class teach-ing, rather than the reverse.

On the other hand, there is no evidence that unsetted classes anda common curriculum necessarily reduce the range. The countriesthat have the largest standard deviations are exactly those of thePacific rim, like Japan and Korea, which teach unsetted classes on anundifferentiated curriculum in the way that the National NumeracyStrategy recommends (Mullis et al., 1997; Beaton et al., 1996). Theabove average standard deviation in England does not necessarilyreflect on differentiated grouping policies as it is accounted for bythe fact that most European countries exclude from testing between10% and 30% of the lowest attaining pupils in the age cohort(Brown, 1998).

In fact where entire cohorts are sampled, the spread of mathemat-ical attainment is surprisingly uniform across very different cultures,education systems and forms of class organisation (Lapointe, Meadand Askew, 1992). To what extent this range is inherent, or is a prod-uct of a culture that believes it to be inherent, or is caused by differ-ent kinds of home or school cultures, is still unclear.

6. WHAT STYLE OF TEACHING IS EFFECTIVE?

The review of research on class organisation in the previous sectionconcluded that teaching quality was a far more salient factor thanclass organisation. This raises the problem of characterising teach-ing quality. The two sections below examine teaching style andassessment respectively.

Teaching style

The National Numeracy Strategy emphasises oral and mental work,and requires that ‘teachers provide clear instruction, use effective

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questioning techniques and make good use of pupils’ responses’(DfEE, 1998b, p. 14). The format of lessons is exemplified in apublication from the National Numeracy Project (1997).

In fact large scale surveys find only small effects on primarynumeracy attainment that are due to individual teachers andschools, typically less than 10% of the variance that remains amongpupils after the effects of the much more powerful pupil variableshave been removed (Mortimore et al., 1988; Creemers, 1997;Burstein, 1992). Creemers found that scores for ‘teaching quality’(judged on the basis of characteristics previously found to correlatewith attainment) were about 10 percentage points higher forTaiwan, Norway and the Netherlands than they were for the USA,UK and Hong Kong, yet Taiwan and Hong Kong were ahead of theother four countries on attainment.

Both international and English observational studies seem toshow some agreement however on some of the aspects of teacherquality which correlate with attainment. These include the use ofhigher order questions, statements and tasks which requirethought rather than practice; emphasis on establishing, throughdialogue, meanings and connections between different mathe-matical ideas and contexts; collaborative problem-solving in classand small group settings; more autonomy for students to developand discuss their own methods and ideas (Creemers, 1997; Bell,1993; Cobb and Bauersfeld, 1995; Yackel and Cobb, 1996; Wood,1996; Stigler and Hiebert 1997; Boaler, 1997; Askew et al.,1997a).

The description of the qualities of effective teaching in the TaskForce reports (DfEE, 1998a, p. 19–20; 1998b, p. 14) reflects thesefindings. However since the documents also freely use with approvalwords like ‘instruction’, ‘direct teaching’, ‘effective questioningtechniques’, and the Framework refers to large numbers of frag-mented procedures, there are risks that many teachers may perceivethe Numeracy Strategy as being supportive of a more traditional‘transmission’ style.

Askew et al. (1997a) found that low numeracy gains wereobtained by transmission approaches, where teachers demonstratedspecific procedures, often preceded by practical and/or diagram-matic justifications. They often included significant periods of wholeclass interaction in the form of question and answer dialogue, but itwas highly structured by teachers as a form of ‘cued elicitation’, withlow-level questions designed to funnel pupils’ responses towards therequired answer (Bauersfeld, 1988; Voigt, 1985, 1994; Edwards andMercer, 1987; Yang and Cobb, 1995).

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Earlier American literature (Brophy and Good, 1986) is morepositive about the effects of a well-structured transmission style. But,as noted previously, Peterson concluded that positive effects weremainly restricted to the achievement of learning repetitive basicskills; there was evidence that ‘higher order thinking may require aless direct instructional approach’ (Peterson, 1988, p. 5–6).

Assessment

The National Numeracy Strategy endorses ‘high quality formativeassessment’ (DfEE, 1998b, p. 59) interacting with pupils to makeclear what the objectives are, assessing how far they haveprogressed in achieving them, and using the information incurriculum planning.

The Strategy is in line with the findings of a recent researchreview across subjects and phases (Black and Wiliam, 1998), andwith findings that effective teachers of numeracy are characterisedby personal, complex and multi-dimensional systems of formativeassessment, and by a tendency to assess at the start of a new topic, soas to inform the teaching, instead of, or in addition to, testing at theend (Askew et al., 1997a).

However, the tension is not fully acknowledged between therequirement to plan using assessment information, which is likely toreveal a wide range of attainment, and the expectation that teacherswill keep the whole class together in following a Framework forTeaching that is presented in the form of week by week objectivesthroughout the primary school.

7. WHAT MAKES A TEACHER EFFECTIVE IN TEACHING NUMERACY?

The National Numeracy Strategy is about assisting all teachers toimplement certain recommended practices, and does not thereforeconsider teachers as individuals. Although Begle (1979) famouslyfound no salient characteristics which discriminated between lesseffective and more effective mathematics teachers, more qualitativestudies are producing different findings:

To understand why teachers behave as they do, it appears neces-sary to understand also their beliefs about mathematics and class-room instruction (Good and Biddle, 1988, p. 127).

We will examine separately teachers’ subject knowledge and theirbeliefs, pedagogical knowledge and practice.

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Teachers’ subject knowledgeMany research studies present a deficit model of primary teachers’own mathematical knowledge, suggesting that less effective teachershave weak subject knowledge (Wragg et al., 1989; Bennett andTurner-Bisset, 1993; Aubrey, 1994). Leinhardt et al. (1991)concluded that subject knowledge impacted in several ways. Forexample, teachers’ mental plans for lessons were dependent upontheir familiarity with the content to be taught (Borko et al., 1988) aswere the questions asked and explanations offered to pupils(Bennett et al., 1993). Feelings of inadequacy over subject knowl-edge has also been shown to lead to over-reliance on a commercialscheme (Millett and Johnson, 1996).

However much of this literature is based on judgements of prac-tice rather than learning gains, and it does not always include a rangeof teachers in the sample. Askew et al. (1997b) show that it is difficultto distinguish effective teachers from less effective in relation to qual-ifications or subject knowledge, except for increased fluency indiscussing conceptual connections, mainly in terms of classroompractice. However some less effective teachers demonstrated thattheir own subject knowledge was more procedural than conceptual.

Other studies (e.g. Carpenter et al., 1988) also suggest that it maybe more important to have a sound grasp of pedagogical contentknowledge than subject content knowledge.

Beliefs, pedagogical knowledge and practicesAlthough many studies, particularly in the USA, focus on effectiveclassroom practice in terms of teachers’ routines (Berliner, 1988),research demonstrates the difficulties that teachers experience inadopting new practices without an appreciation of, and belief in, theunderlying principles (Alexander, 1992). Moreover, some teachershave adopted the rhetoric of ‘good’ practice in teaching mathemat-ics without changes to their actual practices (Desforges andCockburn, 1987; Eraut, 1982).

Cobb (1986) raises the issue of the relationship between teach-ers’ belief systems and pupils’ learning. Beliefs about the nature ofthe subject have been explored in the literature (Ernest, 1989;Lerman, 1990; Thompson, 1984) but have not yet been shown to beas important as pedagogical beliefs. Askew et al., (1997a) demon-strate that effective teachers of numeracy tend to hold commonpedagogical beliefs about such aspects as the nature of numeracy,pupils’ potential, and ways of teaching, which are not shared by lesseffective teachers.

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Cobb, Yackel and Wood (1988) contend that neither beliefs norpractice have primacy, but that ‘beliefs and practices are dialecticallyrelated’ (p. 24). This suggests that teacher development is likely tobe an iterative and long-term process, since teachers may need toexamine and work on their belief systems as well as work at chang-ing practices.

8. WHAT IS THE BEST WAY OF INCREASING THE EFFECTIVENESS OF ALLTEACHERS?

The National Numeracy Strategy is intended to raise standards bychanging both the content emphasis and the methods used in theclassroom. The dissemination strategy is by running 3-day trainingsessions in each area attended by three teachers from each school,including the head and mathematics co-ordinator. They will then‘cascade’ the training to other teachers in the school during threewidely separated training days. Support will be available fromprinted materials, video-tapes, and opportunities to observe well-taught lessons, and from local consultants. Co-ordinators will receive5 days additional release. For schools identified as needing particu-lar help, additional training courses and consultant time will bemade available (DfEE, 1998b).

Askew et al. (1997a,b), both statistically and according to teachers’own reports, found that off-site 10 or 20-day courses of professionaldevelopment in mathematics run by higher education over extendedperiods seemed to be the most reliable way of changing beliefs andpractices so as to significantly improve effectiveness in numeracy.This supports evidence from teachers’ views reported in Halpin,Croll, and Redman (1990). It is perhaps ironic that the governmentchose to stop all funding for such courses in the year prior to the startof the numeracy strategy; it is by no means clear whether compacted3-day substitutes will be sufficient to make a real difference. Hopkins(1989) echoes the findings of Joyce and Showers (1980) with refer-ence to the ‘ubiquitous “one-shot” inservice workshops that haveproven to be so demonstrably inefficient’ (p. 86).

There is support in choosing the school as the unit for change formost teachers, since, in examining how teachers develop their knowl-edge, beliefs and practices, many writers cite collegiality as providinga reference group for support as an important factor (Eraut, 1982;Biggs, 1983; Critchley and Casey, 1984; Nias, 1985; Pinner andShuard, 1985; Miles, 1986; Pirie, 1987; Fullan, 1991; Fullan andHargreaves, 1992). Certainly much literature in the UK hassupported the need for school-centred or school-based professional

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development (e.g. Easen, 1985; Ball, 1991; Bolam, 1982). Hopkins(1989) notes the power of linking professional development toschool improvement. Studies on school effectiveness and improve-ment (Mortimore et al., 1988; Sammons et al., 1995) identifyinggeneric factors such as effective leadership, consistency of beliefsand practice, and staff collaboration as being important.

However, Smyth (1989) pointed out that in-school developmenthas its own problems. Wilcox and Gray (1996), examining inspec-tion recommendations, found those relating to areas of teachingand learning and curriculum delivery had low levels of implemen-tation up to 21 months after the original inspections. Millett (1996)identified the particular problems in mathematics where teacherscollectively lack the confidence to discuss issues in depth and to takerisks in changing practice. Askew et al. (1997b) found that it waspossible for schools to act as a powerful influence in improving theeffectiveness of their teachers in teaching numeracy, but that itappeared to require special circumstances, both some expert staff inleadership positions and a substantial investment of time overseveral years in sharing ideas, experiences and team teaching. Theissue of how knowledge can be transferred to schools, identified asin need of improvement, is a particular problem (Brown, Duffieldand Riddell, 1995; Hopkins, 1991).

It is good therefore to see investment in training and co-ordina-tor time provided in the Strategy, and additional training andsupport for weaker schools. Nevertheless, the doubts are whether itis enough on a one-year basis, and whether there will be enoughstaff who are sufficiently expert to provide appropriate leadership.Although previous initiatives in mathematics demonstrate the likeli-hood of misinterpretation by teachers of what is required (Askew,1996), the supplementary central exemplar material should assist inclarifying the aims (Fullan, 1991).

9. WHAT HELP CAN BE GIVEN AT HOME?

All national and international surveys of attainment in primarymathematics have demonstrated the overwhelming importance ofhome factors, which tend to swamp any factors related to schools(DES, 1967; Mortimore et al., 1988; Burstein, 1992; Keys et al, inpress). Hence the National Numeracy Strategy appears to be justi-fied in trying to incorporate a strong strand of family-focusedactivities into the programme (DfEE, 1998b, p. 74–85).

Certainly several researchers have found that many children start-ing school have more skills than teachers recognise (Carpenter et

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al., 1983; Munn, 1994; Aubrey, 1993), and that already there arelarge socio-economic differences in attainment (Hughes, 1986).

There is very little research in the area of increasing numeracyattainment through family intervention. The major evidence in thehome-school area relates to work in the IMPACT project (Merttensand Vass, eds, 1993), in which sending activities home for families towork on collaboratively was found to be successful.

The incorporation of more homework into the NationalNumeracy Strategy seems a plausible way of raising standards.International surveys (Beaton et al., 1996; Mullis et al., 1997) showEngland to currently require less homework than most other coun-tries. Brown (1996a) estimated the average total amount of timedevoted to mathematics per year, both in and out of school, by aJapanese child to be almost twice as much as that of an English child.

10. CONCLUSIONS

The discussion of whether the recommendations in the NationalNumeracy Strategy are based on research has shown varied results;sometimes recommendations are quite strongly underpinned,sometimes the evidence is ambiguous, sometimes there is little rele-vant literature, and sometimes the research is at odds with therecommendations. This demonstrates two points:

1. The research findings are sometimes equivocal and allow differences ofinterpretation.

In most cases the literature reveals that school, teachers, teachingorganisation and teaching methods have a relatively small effect onnumeracy attainment. Thus it is difficult to pick up consistentmessages which relate to improved effectiveness. The complexity ofthe findings and of the possible interpretations suggests that minis-terial desires for simply telling ‘what works’ are unrealistic.

2. There are always many practical constraints on policy which are likelyto over-ride empirical evidence.

As well as the ever-present financial constraints, which limit forexample the amount of training which is possible, there areconstraints such as that on calculator recommendations, imposed bythe more or less established positions of Task Force members, ofobservers representing national agencies (who in this case almostoutnumbered Task Force members), and of Ministers.

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The National Numeracy Strategy is itself a huge national researchproject; it is hoped that research funds will be made available toenable researchers to document it fully, so that next time we may allbe better informed.

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Correspondence:Professor Margaret BrownSchool of Education, King’s College LondonCornwall HouseWaterloo RoadLondon SE1 8WA

Received on: 28 September, 1998Accepted for publication on: 30 September, 1998

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Documents

Is the National Numeracy Strategy Research-based?