Arithmetical Difficulties Developmental and Instructional Perspectives

Arithmetical Difficulties:Developmental and Instructional

Perspectives

EDUCATIONAL AND CHILD PSYCHOLOGYVOLUME 24, NUMBER 2

Guest EditorsPenelope Munn

Rea Reason

2 Educational & Child Psychology Vol 24 No 2

Contents3 About the contributors

5 Arithmetical difficulties: Developmental and instructional perspectives (extended editorial)Penny Munn & Rea Reason

16 Strategy flexibility in children with low achievement in mathematicsLieven Verschaffel, Joke Torbeyns, Bert DeSmedt, Koen Luwel & Wim Van Doreen

28 Early markers for arithmetic difficultiesPieter Stock, Annemie Desoete & Herbert Roeyers

40 Assessing and teaching children who have difficulty learning arithmeticAnn Gervasoni & Peter Sullivan

54 Assessing pupil knowledge of the sequential structure of numbersDavid Ellemor-Collins & Robert Wright

64 What can intervention tell us about the development of arithmetic?Ann Dowker

83 Investigating variability in classroom performance amongst childrenexhibiting difficulties with early arithmeticJenny Houssart

98 Language-based retrieval difficulties in arithmetic: A single case interventionstudy comparing two children with specific language impairmentTuire Koponen, Tuija Aro, Pekka Räsänen & Timo Ahonen

108 Achieving new heights in Cumbria: Raising standards in early numeracythrough mathematics recoveryRuth Willey, Amanda Holliday & Jim Martland

119 Educational psychologists’ assessment of children’s arithmetic skillsSusie Mackenzie

137 Linking children’s home and school mathematicsMartin Hughes, Pamela Greenhough, Wan Ching Yee, Jane Andrews,Jan Winter & Leida Salway

146 Supporting children with gaps in their mathematical understanding: The impactof the national numeracy strategy on children who find arithmetic difficultJean Gross

Educational & Child Psychology Vol 24 No 2 3

About the contributorsTimo Ahonen is a Professor of Developmental Psychology in the Department of Psychology, University

of Jyväskylä, Finland.

Jane Andrews is a Senior Lecturer in Education, University of the West of England, UK.

Tuija Aro is a Senior Researcher in the Niilo Mäki Institute, Jyväskylä, Finland.

Bert De Smedt is a Postdoctoral Fellow of the Department of Educational Sciences of the Katholieke

Universiteit Leuven, Belgium.

Annemie Desoete is a Professor in the Department of Clinical Psychology, University of Ghent, Belgium

and at the Artevelde Hogeschool, Belgium.

Ann Dowker is a University Research Lecturer in the Department of Experimental Psychology, University

of Oxford, UK.

David Ellemor-Collins is a Researcher in the School of Education, Southern Cross University, Australia.

Ann Gervasoni is a Senior Lecturer in the Faculty of Education, Australian Catholic University, Ballarat

Campus, Australia.

Pamela Greenhough is a Research Fellow in the Graduate School of Education, University of

Bristol, UK.

Jean Gross is the Director of Every Child a Reader, KPMG Foundation, UK.

Amanda Holliday is a Numeracy Consultant with Cumbria County Council Children’s Services, UK.

Jenny Houssart is a Senior Lecturer in the Institute of Education, University of London, UK.

Martin Hughes is a Professor in the Graduate School of Education, University of Bristol, UK.

Tuire Koponen is a Researcher in the Niilo Mäki Institute, Jyväskylä, Finland.

Koen Luwel is a Postdoctoral Fellow of the Fund for Scientific Research, Flanders (Belgium) associated

to the Department of Educational Sciences of the Katholieke Universiteit Leuven, Belgium.

Susie Mackenzie is an Educational Psychologist with Education Leeds, UK.

Jim Martland is a member of the International Board of Mathematics Recovery and Director in the

Mathematics Recovery Programme, UK.

Penny Munn is a Reader in Education at Strathclyde University, UK.

Pekka Räsänen is a Senior Researcher in the Niilo Mäki Institute, Jyväskylä, Finland.

Rea Reason is a Co-Director in the Doctoral programme for Practising Educational Psychologists,

School of Education, University of Manchester, UK.


About the contributors

Herbert Roeyers is a Professor of Developmental Disorders in the Department of Clinical Psychology,

University of Ghent Belgium.

Leida Salway was a Teacher-Researcher in the Graduate School of Education, University of Bristol, UK.

Pieter Stock is a Researcher in the Department of Clinical Psychology, University of Ghent, Belgium.

Peter Sullivan is a Professor of Science, Mathematics and Technology Education, Monash University,

Clayton, Australia.

Joke Torbeyns is a Postdoctoral Fellow of the Fund for Scientific Research, Flanders (Belgium)

associated to the Department of Educational Sciences of the Katholieke Universiteit Leuven, Belgium.

Wim van Dooren is a Postdoctoral Fellow of the Fund for Scientific Research, Flanders (Belgium)

associated to the Department of Educational Sciences of the Katholieke Universiteit Leuven, Belgium.

Lieven Verschaffel is a Professor of Educational Sciences in the Centre for Instructional Psychology and

Technology, Katholieke Universiteit Leuven, Belgium.

Wang Ching Yee is a Research Fellow in the Graduate School of Education, University of Bristol, UK.

Ruth Willey is a Specialist Educational Psychologist with Cumbria County Council Children’s

Services, UK.

Jan Winter is a Senior Lecturer in the Graduate School of Education, University of Bristol, UK.

Robert Wright is a Professor of Education in the School of Education, Southern Cross University,

Australia.

EDUCATIONAL PSYCHOLOGISTSwork in the area between psychologicaltheory and its applications in a range of

contexts. They are responsible both for itsdirect application and for common under-standing of psychology as it relates tochildren’s learning and development(Cameron, 2006; Farrell et al., 2006). Becauseof this it is both timely and relevant that weshould present this collection of papers onarithmetical difficulties reflecting currentresearch in the psychology of mathematicseducation.

There are a range of reasons for thegrowing interest in arithmetical difficultiesby educational psychologists, all related tochanging social and educational attitudestowards numeracy problems. We can identifythese changes as advances in psychologicalknowledge and research, the arrival in edu-cation of the term ‘dyscalculia’, changes ineducation due to the implementation of theNational Numeracy Strategy and increaseddemands of employers for numeracy.

First and foremost, there is a critical massof psychological and educational research thatcan inform knowledge of the development ofnumeracy. As outlined in this article and illus-trated by the international contributions tothis special issue, the combination of tworesearch traditions – cognitive-developmentalwork and the psychology of maths education –provides the basis for examining children’sconcepts and strategies from a constructiviststandpoint.

The migration of the term ‘dyscalculia’from neuropsychology to education providesa second reason for educational psychologiststo be interested. Indeed, the initial impetusfor the present issue was the publication ofPrimary National Strategy information thataddressed the concept of ‘dyscalculia’ (DfES,2001). A comprehensive DfES commissionedreport, entitled ‘What works for children withmathematical difficulties’ (Dowker, 2004),argued that arithmetical ability was notunitary and there were vast individualdifferences. Dyscalculia – major difficulties

Educational & Child Psychology Vol 24 No 2 5© The British Psychological Society 2007

Extended editorial

Arithmetical difficulties: Developmentaland instructional perspectivesPenny Munn & Rea Reason

AbstractThis introductory paper sets the scene for the special issue by outlining the research that informs our understanding of the way children learn arithmetic, and the difficulties that they may encounter. Overviewsof the papers contributed for this special issue reveal converging views on children’s arithmetical ability, theresearch uses of interventions, the influence of the social context, mathematics assessment and curriculumdesign, and the most effective forms of interventions for children who are struggling to learn. We draw sev-eral conclusions from our overview of the research and the contributions to this issue. Our main conclu-sions are that arithmetical difficulties are best understood within the framework of mathematicaldevelopment, and that discussion of dyscalculia should take place as part of this broader context. Secondaryconclusions are that educational psychologists need an understanding of current theories in mathematicseducation and that they can usefully contribute to a growing field in the individual and social psycholo-gies of mathematics education.

with basic number concepts – was also men-tioned in this report as applicable to a smallminority of learners. Nevertheless, as in thepresent journal issue, the main thrust of thereport was on understanding of the range ofareas involved in children’s difficulties withnumber.

The National Numeracy Strategy in England and Wales has had considerableimpact on the way number is taught in primary schools, and on teachers’ percep-tions and expectations of young childrensnumeracy learning (Millet, Brown & Askew2004). Although there is argument in somecircles about its effect on mathematics edu-cation (Brown, 2005; see also Hughes et al. inthis issue) there is no doubt that the NNSemphasis on computational aspects of arith-metic has had an effect on teachers andchildren, and has in some respects increasedchildren’s abilities. It has also brought intothe spotlight those children who have prob-lems with numeracy and whose difficultiesneed to be understood and addressed (seethe contribution of Gross in this issue).

Education systems around the world havebeen changing in the direction of increasingtheir emphasis on numeracy, but they havenot been doing this in a vacuum; they havebeen responding to a co-ordinated interna-tional demand for numeracy skills (see, forinstance, the ‘history’ page of the Interna-tional Statistical Literacy Project 2007). Lit-eracy problems have long been known todisadvantage adults in the workplace. Nowwe also know that adults with numeracyproblems are seriously disadvantaged in theworkplace (Bynner & Parsons, 1997). Mathsand numeracy are basic life skills that make amajor contribution to the five outcomes ofeducation as set out in ‘Every Child Matters’(DfES, 2004): to be safe, healthy, enjoyingand achieving, have economic security andmake a positive contribution to the commu-nity. Similarly, numeracy education makesmajor contributions to the four major aimsof the Scottish ‘Curriculum for Excellence’(to produce successful learners, confidentindividuals, responsible citizens and effective

contributors) (SEED Curriculum ReviewGroup, 2004).

For all these reasons, educational psy-chologists will find the papers in this specialissue both important and informative. Weshall also argue that the time is now ripe fortheir active professional engagement in thepractical issues that the papers raise.

Background in educational anddevelopmental psychologyEducational psychologists already have consid-erable understanding of research. This willenable them to get involved now with the prac-tical and theoretical issues that have beenraised over the past three decades in mathseducation and the psychology of numeracy.There have been two strands in the history ofresearch in arithmetical difficulties, whichhave evolved separately from each other. Onestrand began with Skemp’s (1973) work on thepsychology of maths education, which culmi-nated in the development of an internationalforum for all involved in maths educationresearch. This strand has been concerned withthe link between theory and education prac-tice across all sectors, and has been very pro-ductive (Gutierrez & Boero, 2006). Thesecond strand follows on from Piaget’s workon number (Piaget, 1952). Piaget was not aneductionalist, but his extensive work onnumber development was given educationalinterpretations by many, and was influentialuntil it was challenged by Gelman & Gallistel’s(1978) work on counting that showed thatvery young children do have some grasp of theprinciples of number. This apparent over-throw of Piagetian theory was followed by acomprehensive research programme thatmapped the understanding of number inbabies, young children and animals in waysthat were not defined by Piagetian theory (foran overview of this research see Sophian, 1996;for an overview of its theoretical relation toPiagetian and Educational theory see Bryant,1995). The cognitive strand of research isembedded in cognitive-developmentalpsychology and is not overly concerned withpractices within maths education. Some


Penny Munn & Rea Reason

researchers have worked on developing theapplications to maths education implicit in thecognitive-developmental tradition (for exam-ple, Resnick, 1989). However, the applicationsof experimental cognitive research to educa-tion are almost a research field in their ownright, since the high reliability and low validityinvolved in experimentation make it hard toapply such research directly to educationalpractice.

Now, the volume of research in both ofthese traditions of maths research hasreached a critical mass where psychologicaland educational theories can interact, andtheory and practice can inform each other. Itis at this point that educational psycholo-gists, who have considerable research skills,can usefully begin to get involved in thepoint at which bridges between theory andpractice can form. While most educationalpsychologists will know the cognitive-devel-opmental work on number, they may not beso familiar with the work on the psychologyof maths education. It may help here to sum-marise the main ways in which the field ofmaths education research (as it applies toyoung children) has developed since theearly days of Skemp (1973).(i) There has been a decisive shift away from

behaviourist accounts and a wholesaleadoption in the psychology of mathseducation community of constructivisttheories to explain development andlearning in mathematics.

(ii)Current research emphasises thesimultaneous development of additiveand multiplicative structures.

(iii)The future direction of research in thepsychology of maths education istheoretically driven. It emphasises themodelling and reasoning that underliesarithmetical performance, rather thanfocusing on the computation proceduresthemselves. (Mulligan & Vergnaud 2006)

These characteristics mean that maths edu-cation research can offer a coherenttheoretical framework that links to practice.The framework linking maths educationstudies is one that takes practical utility as its

first priority; the uniting factor is that theimplication of findings for educationalpractice is much clearer than is the case forfindings derived from cognitive developmen-tal research. The converse of this is that thelatter type of findings usually have muchclearer implications for theoretical develop-ments. This has resulted in the two strands ofresearch having markedly different charac-teristics. The cognitive developmentalresearch produces information that adds toour knowledge of psychological models ofnumber development, but it is not alwaysclear what relevance these models have tocurriculum development, teacher education,or classroom practice. The maths educationresearch, by contrast, produces informationthat adds to our understanding of curricu-lum development, teacher education andclassroom practice, but does not always con-tribute directly to a psychological model ofchildren’s number development.

We now know a great deal about arith-metical development in childhood, but thisknowledge needs to be integrated into aframework that is concerned with extendingboth psychological models and educationalunderstanding. One thing that is clear fromboth educational and psychological frame-works is that arithmetical development inchildren is non-linear and multidimensional.Young children draw together threads ofquantitative experience and link these withgrowing verbal number strings as they slowlyconstruct for themselves a rich understand-ing of number (for examples, see Fuson,1988). The non-linearity of developmentsare evidenced by their slow pace and by theway that children construct their mathemati-cal understanding across multiple domains.The complexity of mathematical under-standing in childhood contrasts starkly withvernacular accounts of arithmetical develop-ment. By ‘vernacular’ we mean the accountthat is implicit in the layout of many guidesfor (non maths specialist) teachers. (see forinstance Biggs & Sutton, 1983; Busbridge &Womack, 1991). Such vernacular accountscan describe children’s difficulties (as in ‘she


Editorial

cannot remember number bonds withinten’) but they are not helpful in remedyingsuch difficulties because they are devoid ofboth theory and explanation. Many inspiredmaths educators have struggled to upgradethe vernacular account by contributing a discourse of mathematics education to theprimary sector (see for instance Clemson &Clemson, 1994; Hughes, Desforges &Mitchell, 2000). Others have directed theirattempts at the social organisation of mathsteaching across the school (see for instanceAtkinson, 1996). The success of such dis-course and organisation will be essential ifteachers are to replace their intuitivedescriptions of children’s arithmetical diffi-culties with analyses that reflect the complex-ity of mathematical development and therecent growth in psycho-educational theories.

Contributions to this issueFrom an initial call for papers we haveselected 11 papers that represent excellencein the application of psychological theory tothe field of arithmetical difficulty. One verypractical aspect of most of these papers is thatthe concrete examples in the papers providesuggestions for educational psychologists asto the kinds of assessments and interventionsthey can apply within their daily practice. Forexample, Ellemor-Collins and Wright givedetailed illustrations of their four groups ofassessment tasks that one of the editors wasable to use immediately in her own assess-ments and suggestions for intervention. Ger-vasoni and Sullivan have illustrations withintheir framework of growth points thatinclude creative use of concrete materials.Ann Dowker outlines the nine componentswithin the Numeracy Recovery Scheme andher appendices include much detail that canbe of immediate relevance to practice. Mostof the other papers also contain informationand appendices with illustrations of the mate-rials being used.

The papers that were contributed to thisspecial edition form a wealth of descriptionthat shows both how much psychologicalapproaches have achieved in the teaching

and learning of arithmetic, and how muchthey still have to contribute. To illustrate this,we have grouped the papers into three thematic groups. The first group of papersillustrates the deep psychological under-standing of the complexity of children’sarithmetical ability that is current in the liter-ature. The second group of papers showshow we can use intervention to increaseknowledge of arithmetical difficulties. Thethird group of papers shows how psychologycan contribute to our understanding of thesocial contexts of arithmetical difficulties.

Papers illustrating the inherent complexity ofchildren’s arithmetical abilityFour of the papers describe the complexity ofarithmetical learning in the primary school,and articulate a profound understanding ofthe ways in which children’s arithmetical‘ability’ can be understood. Verschaffel et al.,Stock, Desoete and Roeyers; Gervasoni andSullivan; and Ellemor-Collins and Wright allfocus in very different ways on the complexityof children’s arithmetical ability, and on theimplications of this for approaches to arith-metical difficulties.

Vershaffel et al. discuss the importance ofchildren’s adaptivity, or strategy flexibility,for the full development of their mathemati-cal ability. They argue that such adaptivitymight arise from children’s perceptions ofclassroom demands, and that we shouldtherefore make sure that classrooms makecomplex and challenging demands from theearliest school years even of children whoseem to be low in ability. They warn that suchchallenge cannot be added in to the curri-culum once children have learned ‘thebasics’. By then it will be too late, andchildren will already have lost the opportu-nity to develop adaptivity.

Stock, Desoete and Roeyers’ work isaimed at discovering whether we can findprecursors of arithmetical difficulties whilechildren are in preschool. They comparethe predictive power of variations in mathe-matical reasoning with variations in subitis-ing ability. The findings show that it is



children’s number logic (their mathematicalreasoning) that gives the better predictionfrom preschool number knowledge acrossthe transition into school. This result sup-ports the view that there is not an elementary‘building block’ from which arithmeticalability is fashioned. Rather, it is complexeven at the beginning of education.

Gervasoni & Sullivan conclude that thereis no single ‘formula’ for describing childrenwho have difficulty learning arithmetic or fordescribing the instructional needs of thisdiverse group of children. One should notassume that, because a child has difficultywith one aspect of number learning, otherareas will also be problematic. They argueexplicitly that programmes need to includeall number domains in tandem (addition,subtraction, multiplication and division) aschildren’s construction of number knowl-edge in a specific domain is not dependenton prerequisite knowledge in anotherdomain, but dependent on being able totake advantage of a range of experiences inthat domain.

Ellemor-Collins & Wright examine thestrategies of ‘lower-attaining’ learners agedeight to ten years. They find that manypupils have not developed the strategy ofjumping by tens and may therefore not usesequence-based understanding such asrecognising that number sequences 8, 18,28, 38 are always ten steps apart. The paperrecommends that these pupils’ numberlearning should include a focus on numberword sequences, skip counting and incre-menting by tens and locating numbers in therange of 1–100 on a number line. The prac-tical relevance of their study supports thetheoretical message that complexity innumber learning should be respected evenwhen intervening in the upper primary age groups.

Papers illustrating how intervention can increase our understandingIn this group of papers, we have three contri-butions that offer incisive analysis ofchildren’s responses to mathematical situa-

tions, conclusions about how we understanddevelopmental progressions, and reasonedprogrammes of intervention for the lowestachieving children. The thread that unitesall these papers is their use of intervention to increase our understanding of arithmeti-cal difficulties by adding to psychologicaltheory.

Dowker’s account of ‘numeracy recovery’gives finely analysed detail of which aspects ofher intervention were effective. Her findingssupport the view that arithmetic is made upof multiple components that are relativelyindependent of each other. She shows howarithmetical difficulties can be amelioratedand then uses the results of the interventionto analyse for predictors of improvementamong the multiple components. She showshow her results add to our understanding ofarithmetical difficulties.

Houssart uses her position as researcher toreinterpret an aspect of behaviour that is trou-bling to teachers and often ignored by theo-reticians. Her account of low ability children’sdaily struggle with number highlights the day-to-day variability in what they appear able todo. Houssart comments that this variability isseen as a problem in teaching, since it chal-lenges basic teaching principles. However, sheshows how this variabilty can be seen as evi-dence of complexity in learning, and bringsthis neglected aspect of classroom learninginto theoretical perspective.

The contribution of Koponen et al. has amore clinical flavour in that they compare theperformances of two children considered tohave specific language impairment. However,what makes this contribution particularlyrelevant in this issue is that the focus is not ontesting but on the response of the children tointervention over a period of some twomonths. The authors use this response toelaborate theories of the relation betweenlanguage and arithmetic development.

Willey, Holliday & Martland’s account ofMaths Recovery in Cumbria documents theeffects of this programme on children’s con-ceptual development. They also to chart theeffect of the intervention on teaching and


Editorial

support staff, concluding that because itinspires the teachers to work from principlesrather than prescription, the programme hasan impact on the teachers’ beliefs aboutmaths teaching and learning as well as ontheir confidence in teaching number. Thisinclusion of the changes in teachers in a storyabout cognitive change in children is a valu-able contribution to theory and extends ourunderstanding of the role of maths teachingdiscourse in children’s development.

All of these papers demonstrate theunique strengths that psychological theorieshave in helping teachers and other profes-sionals to understand and intervene in arith-metical difficulties. They also demonstratethat the interventions themselves can in turncontribute to theory.

Papers illustrating the social psychological contexts of arithmetic difficultiesIn this group of papers there is a commonthread that illustrates the wider social psycho-logical perspective on arithmetic difficulties.This perspective includes children’s beliefsand experiences concerning arithmetic, aswell as the beliefs, guiding principles, andeducational principles that education profes-sionals bring to the class. These papers eachtell an interesting story of their own, but theyalso indicate that there are ways in which educational psychology can be used to betterunderstand and manage these out-of-classcontexts.

MacKenzie reviews the assessment toolsand approaches that are available to educa-tional psychologists, and notes the dearth oftools compared with those available for read-ing. However, her views on how educationalpsychologists might use the available tools inan interactive way, using their skills in obser-vation, emphasises the role that the practi-tioner’s constructs play in their view ofchildren’s abilities.

Hughes et al.’s account of out-of-schoolmathematics highlights the issue of ‘authentic-ity’ in mathematical activity – a characteristicthat is extremely difficult to build into schoolmaths activities, yet is often found in children’s

spontaneous activities at home. The authorssuggest that assessment of maths ability shouldroutinely include these out-of-school activities.They show that out-of-school maths has anemotional significance that is lacking in schoolmaths, and makes an important contributionto the attitudes that children bring into class.Their account brings out-of-school settingsinto a story about how children establish thepurposes of their learning. In particular, theirdiscussion of parental dilemmas over styles ofcalculation produces a pragmatic solution thatshould become part of every school’s numer-acy curriculum.

Gross’s account of how lower attainingpupils have not benefitted from the NationalNumeracy Strategy includes an analysis of theuse of diagnostic labels such as ‘dyscalculia’within the classroom ecology. She arguesstrongly against the effect that widespreaduse of such a term would have of takingresponsiblity for low attaining children out ofthe teacher’s day-to-day classroom manage-ment. Her analyses inform our understand-ing of how policy contexts can shape the experiences of children with arithmeticaldifficulties.

We have made three broad groupings ofthese papers for our own convenience inexplaining how these authors can collec-tively take us forward in our understandingof psychology’s role in maths education.However, there are common themes thatcross the boundaries of these groups. One such theme is the importance of non-linearity in curriculum design. That is, thecurriculum should not be designed in aheavy-handed ‘lock-step’ principle, in whichsupposedly simple aspects of maths are fol-lowed by supposedly complex aspects. Oneshould also not assume that, because a childhas difficulty with one aspect of numberlearning, other areas will also be problem-atic. We also find a thread running throughmany of the papers that urges us not to takeprecipitate decisions that limit the scope ofearly maths teaching, not to take poorly per-forming children out of the classroomteacher’s remit and make them somebody



else’s responsibility, and not to allowmechanical use of assessments to overrideour intuitions and direct experiences ofchildren’s maths learning. All of thesethreads are drawn from current psychologi-cal understanding of arithmetic learning as apart of mathematics education. There is afurther theme that echoes longstanding ten-sions over the differences between ‘mathe-matics’ and ‘computation’. All of the papersaddress this theme in one way or anotherand all conclude that while computationmay be important, it is not everything. Manyof the papers implicitly point out that if educators behave as though computation isall that matters, we should not be surprised if children respond by only learning how tocalculate.

Developmental progressionsin numberPsychological models of number under-standing are a combination of behaviouristand cognitivist description that somehowmanage to exclude children’s own subjectiveexperience of learning, and being taught,about number. Thus, there are manydescriptions of changes in strategy aschildren age but no real account of the sub-jective experiences that must drive thissequence of development. Maths educationresearch, by contrast, has used clinical inter-viewing techniques to produce a construc-tivist account of children’s typical journeythrough phases of number understanding(see, for instance, Steffe & Cobb, 1986).

Many of the papers in this issue draw onSteffe’s research, which has outlined a clearprogression in number concepts as theyrelate to mathematical activity. This progres-sion has a ‘step’ function in that movementfrom one stage to the next involves cognitiverestructuring rather than a slow accretion ofknowledge. (Steffe et al., 1983; Wright et al.,2000). Between the ages of roughly threeand seven, most children pass through thefollowing five broad stages in understandingarithmetical operations:● Emergent number stage : At this stage,

children are just beginning to constructthe counting sequence and are stilldeveloping one-to-one correspondencein counting.

● Perceptual number stage: At this stage,children can only deal with addingtogether quantities that are visible. In theclassroom, they look as though theyunderstand early maths operations,because they can work with visible quan-tities.

● Figurative number stage: At this stagechildren seem to be able to deal withscreened addition, but they are using a‘number sequence’ logic to achieve this. They can’t yet operate with cardinalnumbers. In the classroom they look as though they understand early maths operations because they can useordinal strategies to work with hiddenquantities.

● Initial number stage: At this stage childrenhave acquired an adult-like understandingof number, and are able to comprehendnumber symbols and operations such as addition and subtraction. Children need to reach this cognitive stage beforethey can grasp the elements of the formalnumber curriculum – cardinal number,written number symbols and numberoperations.

● Facile number stage: At this stage childrenare developing higher-order number con-cepts, and become able to focus on therelation between numbers, and on therelation between operations. Childrenneed to reach this cognitive stage beforethey can grasp elements of the moreadvanced primary curriculum such as frac-tions, ratio and percentages.It takes children several years to grow

from the emergent to the facile stage. Untilchildren have reached the Initial Numberstage they cannot grasp the mathematicalnature of operations such as addition or sub-traction and the formal written curriculum islost on them. There is much variability in thelength of time that children take to reachthis stage, and there are a number of sources


Editorial

of this variability (Ridler-Williams, 2003;Munn, 2006).

While educational theorists have beendescribing the delicate and complex stagesthat children pass through to understandnumber, they have also created a rich understanding of the ways in which childrenconstruct the verbal string of number words.Piaget long ago pointed out that children’snumber word string constructions do not nec-essarily co-incide with their number concepts,and present day research has given us a goodunderstanding of why this is so. Fuson (1988)and Wright (1994) have done the mostdetailed research on this aspect of develop-ment, and both describe an active construc-tion of the learned number word sequence.That is, children do not passively replicate thenumber word sequence and store it in mem-ory; they actively construct it and reconstructit as they learn it. This active construction hastheoretical and practical implications. On thetheoretical level, children’s number wordsequences provide a window onto the devel-opmental pathways they are taking towardsthe Initial Number Stage, at which numberwords become associated with cardinal mean-ing. On a practical level, it provides theunderstanding that children’s errors in thenumber word are never accidental, and canextend the practitioner’s understanding ofthe individual path that a particular child withdifficulties is treading towards cardinal under-standing or points beyond it. The papers by Willey et al. and by Wright et al. illustrate this approach.

Implications of developmentalprogressions in number for theconcept of dyscalculiaThe term ‘dyscalculia’ implies that there canbe problems of a neurological origin affect-ing very basic capacity for understandingnumbers (Butterworth, 2005). As stated in adocument published by the Department forEducation and Skills (2001), dyscalculiclearners may have difficulty understandingsimple number concepts, lack an intuitivegrasp of numbers and have problems learn-

ing number facts and procedures. Withinthe progression outlined above, dyscalculiamay be theorised as some kind of ‘blockage’at the perceptual and figurative stages ofnumber construction that prevents childrenfrom moving on to operate within the laterconceptual stages.

However, contributions in this volume,such as those by Verschaffel et al.; Gervasoniand Sullivan; and Dowker, show us the dan-gers inherent in assuming that such a ‘block-age’ is an insurmountable barrier tochildren’s progression to higher order mathe-matical thinking. If arithmetical understand-ing is made up of multiple components, andprogress is not linear, then it is possible forchildren to develop conceptual understand-ing in some domains that compensate forslower progress in others. Difficulties with oneaspect of number learning do not necessarilyimply that other areas will also be problem-atic. There will also be more than one routeto mathematical understanding. This viewsuggests that psychologists should assessexactly what children can or cannot do andplan appropriate interventions according tothese assessments. Such an approach wouldlook very different from one in which psychol-ogists use a diagnostic label that has poorlyarticulated assumptions about the implica-tions for intervention.

In her extensive review of the concept ofdyscalculia, Gifford (2005) concluded that‘an innate ability to apprehend precise num-bers is unconfirmed and unlikely’ (p. 37).The true picture is probably much morecomplex. Visualisation, motor control,language and symbolic representations ofnumbers develop interdependently, suggest-ing that various areas of the brain worktogether to develop number understanding.One area of evidence points to problems withshort term or working memory (Geary &Hoard, 2001) that might indeed explain whychildren struggling with basic literacy mayalso struggle with basic arithmetic. In thepresent issue, Koponen et al. argue thatspecific difficulty in retrieving verbal materialand in forming verbal associations seems



to be connected in the ability to learn arith-metical facts.

However, the multiple sensory and sym-bolic pathways that children use to encodemathematical experiences mean that there isprobably considerable plasticity in the neu-rological developments that underpin thegrowth of number understanding. This fac-tor means that screening tests claiming toreflect the neurological basis of arithmeticaldifficulties should be interpreted in the con-text of all other aspects of the child’s sym-bolic activity. MacKenzie (in this issue)argues that screening tests focussing on anyparticular cognitive area such as subitisingdo not have ecological validity in relation tothe way a child uses number in everyday lifeand do not necessarily provide clues as to thenature of the child’s actual computationaldifficulties, or gaps in understanding, that can lead to advice about appropriateinterventions. Gross also critiques medical-diagnostic models and argues convincinglyfor the importance of developing effectivestrategies that involve a teaching and learn-ing response rather than debates about cognitive causation.

A focus on learning opportunitiesThere may be purely experiential reasons forchildren becoming ‘stuck’ in their numberunderstanding and resembling those at ear-lier stages of development. It can be argued,for example, that because the traditional UKnumber curriculum moves rapidly intonumber operations in the first year, it is devel-opmentally inappropriate for a significantproportion of children in the first year ofschool. Ridler-Wiliams (2003), for instance,found that a large proportion of Scottishschoolchildren had not progressed to the‘Initial Number Sequence’ by the end oftheir first year at school and were thereforenot truly comprehending the number opera-tions they had been taught. Follow-up studyof these same children (Munn, 2006) high-lights the effect that their experiences of thenumber curriculum can have on their beliefsabout the nature of mathematics. Children

whose number development is not well syn-chronised with classroom demands candevelop a range of beliefs that locate arith-metical logic beyond the range of their ownthought processes. Often, their beliefs aboutnumber lessons as a species of guessing gameor rote-learned procedures clearly functionto protect their concepts of themselves aslearners. Without appropriate intervention,such children will continue to be left behindbecause they are constructing the arithmeti-cal problems presented to them in radicallydifferent ways from their more successfulclassmates. Kelly (2004) outlined three dif-ferent approaches to the curriculum. The‘curriculum as content’ approach privilegescertain areas of knowledge and views educa-tion as the transmission of cultural knowl-edge. The ‘curriculum as product’ approachis derived from early twentieth century scien-tific managerialism, and views education asthe effective delivery of specified objectives.The ‘curriculum as process’ approach isderived from child-centred education andviews education as the development of theindividual. The papers in this issue providestrong support for a curriculum that sets outto develop the individual rather than to deliver specified content or deliver mathematical ‘products’. Even if it were pos-sible to specify objectively maths curriculathat ‘delivered’ specific outcomes, thesewould be experienced very differently bychildren at different stages of mathematicalunderstanding.

ConclusionThe emphasis of this journal issue is oninforming educational psychologists andtheir colleagues about current researchregarding children’s number learning, thevariability of that learning and theapproaches available to develop the strate-gies and understandings of those childrenthat struggle with the learning. In our view,educational psychologists are well placed tobecome involved in these areas both aspractitioners and as theoreticians. We feelstrongly that they should grasp this opportu-


Editorial

nity and avoid a precipitate narrowing oftheir focus to arguments about labels forarithmetical difficulties. Current research inthe psychology of mathematical and arith-metical development is rightly concernedwith computational processes, but there isconsiderable scope for the development of asocial psychology of maths education. Thiswould encompass the psychological study ofthe communicative and discursive contexts

(at home and at school) in which mathemat-ical thinking develops. Here too we thinkthat educational psychologists could con-tribute worthwhile practitioner research onthe social and individual psychology of arith-metical difficulties.

Address for [email protected]



ReferencesAtkinson, S. (1996). Developing a Scheme of Work for

Primary Mathematics. Avon: Hodder & Stoughton.Biggs, E. & Sutton, J. (1983). Teaching Mathematics

5–9, A classroom guide. London: McGraw Hill.Brown, M. (2005). The role of mathematics educa-

tion research in influencing educational policy.(Closing Plenary) Proceedings of the Fourth Congressof the European Society for Research in MathematicsEducation. Sant Feliu de Guíxols, Spain, 2005.

Butterworth, B. (2005). Developmental dyscalculia.In J.I.D. Campbell (Ed.), Handbook of mathemati-cal cognition. New York: Psychology Press.

Busbridge, J. & Womack, D. (1991). Effective MathsTeaching. Cheltenham: Stanley Thornes.

Bryant, P. (1995). Children and arithmetic. Journal ofChild Psychology and Psychiatry. 36(1), 3–32.

Bynner, J. & Parsons, S. (1997). Does numeracy matter?London: The Basic Skills Agency.

Cameron, R.J. (2006). Educational psychology: Thedistinctive contribution. Educational Psychology inPractice, 22(4), 289–304.

Clemson, D. & Clemson, W. (1994). Mathematics inthe Early Years. London: Routledge.

Department for Education and Skills (2001). Guidanceto support pupils with dyslexia and dyscalculia (Ref:DfES 0512/2001). Nottingham: DfES Publications.

Department for Education and Skills (2004). EveryChild Matters: Change for Children. Nottingham:DfES Publications.

Dowker, A. (2004). What works for children withmathematical difficulties? Research Report 554,Department for Education and Skills. Nottingham:DfES Publications.

Farrell, P., Woods, K., Lewis, S., Rooney, S., Squires,G. & O’Connors, M. (2006). A review of the func-tions and contribution of educational psycholo-gists in England and Wales in the light of ‘EveryChild Matters: Change for Children’. ResearchReport 792, Department for Education and Skills. Nottingham: DfES Publications.

Fuson, K (1988). Children’s Counting and Concepts ofNumber. Chapter 1: Introduction and Overview of

Different Uses of Number Words. NY: Springer-Verlag.

Geary, D.C. & Hoard, M.K. (2001). Numerical andarithmetical deficits in learning-disabledchildren: Relation to Dyscalculia and dyslexia.Aphasiology, 15(7), 635–647.

Gelman & Gallistel (1978). The Child’s Understanding ofNumber. Cambridge, MA: Harvard University Press.

Gifford, S. (2005). Young children’s difficulties in learn-ing mathematics: Review of research in relation todyscalculia. Qualifications and CurriculumAuthority: www.qca.org.uk/13809.html.

Gutierrez, A. & Boero, P. (2006). Introduction. In A. Gutierrez, & P. Boero (Eds.), Handbook ofResearch on the Psychology of Mathematics Education: PME 1976–2006. (pp. 117–146) Rotterdam: Sense Publishers.

Hughes, M., Desforges, C. & Mitchell, C., with Carre, C.(2000). Numeracy and Beyond. Buckingham: OU Press.

International Statistical Literacy Project (2007). Historyof the ISLP and the World Numeracy Project 1994–2006.http://www.stat.aukland.ac.nz/~iase/islp/hist.

Kelly, A.V. (2004). Curriculum: Theory and Practice.London: PCP Sage.

Millet, A., Brown, M. & Askew, M. (Eds.), (2004). Pri-mary Mathematics and the Developing Professional:Multiple Perspectives on Attainment in Numeracy.Kluwer Academic Publishers.

Mulligan, J. & Vergnaud, G. (2006). Research onChildren’s Early Mathematical Development. In A. Gutierrez, & P. Boero (Eds.), Handbook ofResearch on the Psychology of MathematicsEducation: PME 1976–2006. (pp 117–146)Rotterdam: Sense Publishers.

Munn, P. (2006). Mathematics in early childhood –the early years maths curriculum in the UK andchildren’s numerical development. InternationalJournal of Early Childhood, 38(1), 99–112.

Piaget, J. (1952). The child’s conception of number.London: Routledge & Kegan Paul.

Resnick, L.B. (1989). Developing MathematicsKnowledge. American Psychologist, 44(2), 162–169.

Ridler-Williams, C. (2003). Is Children’s NumeracyDevelopment related to Communication in the Class-room? Unpublished PhD Thesis, University ofStrathclyde.

SEED Curriculum Review Group (2004). Purposes and Principles for the Curriculum 3–18. SEED Publications.

Skemp (1973). The Psychology of Learning Mathe-matics. Penguin.

Sophian, C. (1996). Children’s Numbers. Harper-Collins.

Steffe, L. & Cobb, P. (with E. von Glaserfeld) (1988).Construction of Arithmetic Meanings and Strategies.New York: Springer-Verlag.

Steffe, L., von Glaserfeld, E., Richards, J. & Cobb, P.(1983). Children’s Counting Types: Philosophy, Theoryand Application. New York: Praeger Scientific.

Wright, R.J. (1994). A Study of the Numerical devel-opment of 5 year olds and 6 year olds. EducationalStudies in Mathematics, 26, 25–44.

Wright, R., Martland, J. & Stafford, A. (2000). EarlyNumeracy: Assessment for Teaching and Intervention.London: Paul Chapman Publishing.


Editorial

16 Educational & Child Psychology Vol 24 No 2© The British Psychological Society 2007

INSTRUCTIONAL PSYCHOLOGISTSand mathematics educators have for along time emphasised the educational

importance of recognising and stimulatingflexibility in children’s self-constructedstrategies as a major pillar of their innova-tive approaches to (elementary) mathemat-ics education and have designed andimplemented instructional materials andinterventions aimed at the development ofsuch flexibility (see e.g. Brownell, 1945;Freudenthal, 1991; Thompson, 1999;Wittmann & Müller, 1990–1992). Especiallyin many curriculum reform documentsfrom the end of the previous century, suchas the Curriculum and evaluation standards forschool mathematics of the National Council ofTeachers of Mathematics in the US (1989,2000), the Numeracy Strategy in the UK(DfEE, 1999), the Proeve van een NationaalProgramma voor het Reken/wiskundeonderwijs

in The Netherlands (Treffers, De Moor &Feijs, 1990), the Handbuch ProduktiverRechenübungen in Germany (Wittmann &Müller, 1990), and the Ontwikkelingsdoelen enEindtermen for elementary education inFlanders (1998), as well as in many innova-tive curricula, textbooks, software, andother instructional materials based on thesereform documents, there is a basic belief inthe feasibility and educational value of striv-ing for strategy flexibility, also for theyounger and mathematically weakerchildren (Baroody et al., 2003; Kilpatrick,Swafford & Findell, 2001; Verschaffel, Greer& De Corte, in press). However, systematicand scrutinised research that convincinglysupports these basic claims is still ratherscarce. In this contribution, we reflect onthe flexible or adaptive1 choice and use ofsolution strategies in elementary schoolarithmetic. First, we present and discuss dif-

Strategy flexibility in children withlow achievement in mathematicsLieven Verschaffel, Joke Torbeyns, Bert De Smedt,

Koen Luwel & Wim Van Dooren

AbstractInstructional psychologists and mathematics educators have for a long time emphasised the educationalimportance of recognising and stimulating flexibility in children’s self-constructed strategies as a majorpillar of their innovative approaches to (elementary) mathematics education and have designed and imple-mented instructional materials and interventions aimed at the development of such flexibility. Especiallyin many curriculum reform documents from the last two decades as well as in many innovative curricula,textbooks, software, etc., there is a basic belief in the feasibility and educational value of striving for strategyflexibility, also for the younger and mathematically weaker children. However, systematic and scrutinisedresearch that convincingly supports these basic claims is still rather scarce. In this contribution, we reflecton the flexible or adaptive choice and use of solution strategies in elementary school arithmetic. In the firstpart of this article we give some conceptual and methodological reflections on the adaptivity issue. Morespecifically, we critically review definitions and operationalisations of strategy adaptivity that only take intoaccount task and subject characteristics and we argue for a concept and an approach that also involve thesociocultural context. Second, we address the question whether strategic flexibility is a parameter of strategiccompetence that differentiates mathematically strong and weak children. Finally, we discuss whether aim-ing for strategy flexibility is a feasible and valuable goal for all children, including the younger and math-ematically weaker and weakest ones.


Strategy flexibility in children with low achievement in mathematics

ferent conceptions and operationalisationsof strategy flexibility. Second, we address thequestion whether strategic flexibility is aparameter of strategic competence thatdifferentiates mathematically strong andweak children. Finally, we discuss whetheraiming for strategy flexibility is a feasibleand valuable goal for all children, includingthe younger and mathematically weakerand weakest ones.

Conceptions and operationalisationsof strategy flexibility/adaptivityStrategy flexibility or adaptivity is sometimesdefined and operationalised as using a varietyof solution strategies, without any further qualification (e.g. Heirdsfield & Cooper,2002). Although the availability of a variety ofstrategies and the ability to switch smoothlybetween these strategies can be consideredan essential stepping-stone towards flexibility,the mere use of different solution strategieson a series of similar mathematical items orproblems – without any attention to the efficiency of these strategy choices – can hardlybe considered as evidence of adaptivity. Afterall, on the one hand, one can switchsmoothly between different strategies in acompletely arbitrary and random way while,on the other hand, the consistent use of onesingle strategy for a whole series of arithmetictasks may sometimes be more adaptive thanswitching between a diversity of strategiesavailable in one’s repertoire (Verschaffelet al., submitted).

More frequently, strategy flexibility oradaptivity has been defined and opera-tionalised in relation to certain task characteris-

tics. For example, Van der Heijden (1993, p. 80) defines it as follows: ‘Flexibility instrategy use involves the flexible adaptationof one’s solution procedures to task character-istics’. He operationalised strategy flexibilityby analysing whether children systematicallyuse the 1010-procedure and the G10-proce-dure for, respectively, additions and subtrac-tions in the number domain up to 1002.Exactly the same definition is used by BlöteVan der Burg, and Klein (2001). In a similarway, Thompson (1999, p. 147) concurs in hisplea for developing flexibility in elementaryschool arithmetic that ‘mental calculationplaces great emphasis on the need to selectan appropriate computational strategy forthe actual numbers in the problem’. So,these authors first distinguish differentstrategies for doing additions and subtrac-tions, and, based on an analysis of thestrengths and weaknesses of these differentstrategies vis-à-vis certain types of problems3,they then define certain ‘problem type �

strategy type combinations’ as flexible andothers as not. As we have illustratedelsewhere (see Verschaffel et al., submitted),this view on flexible or adaptive strategy useis also found in many (so-called) reform-based textbooks, such as the first year’spupils’ book of a Flemish textbook seriesNieuwe Reken Raak (Bourdeaud’hui et al.,2002). In the textbook for Grade 1 childrenare taught three different strategies fordoing additions with sums between 10 and20: (a) the retrieval strategy (e.g. knowing byheart that 6 � 6 � 12), (b) the tie strategy(e.g. solving 6 � 7 by doing 6 � 6 � 1), and(c) the decomposition-to-ten strategy (e.g.

1For the sake of simplicity, we will use both terms as synonyms, although, as we have explained elsewhere

(Verschaffel, Luwel, Torbeyns & Van Dooren, submitted), some authors define both terms (slightly) differently.2The 1010 procedure involves splitting off the tens and the units in both integers and handling them separately

(e.g. 47 � 15 � .; 40 � 10 � 50; 7 � 5 � 12; 50 � 12 � 62). The application of the G10 procedure, on the

other hand, requires the child to add or subtract the tens and the units of the second integer to/from the first

unsplit integer (e.g. 47 � 15 � .; 47 � 10 � 57; 57 � 5 � 62).3This analysis consists of a purely rational task analysis (wherein the procedural and conceptual complexities of

the different strategies are compared), of an empirical analysis of the accuracy data from large groups of

students for the different (types of) problems, or of a combination of both kinds of analysis.


Lieven Verschaffel et al.

solving 6 � 7 by doing 6 � 4 � 3). Simulta-neously, children learn to link each strategyto a particular type of sum over ten for whichthat strategy is considered most efficient: (a)the retrieval strategy for tie sums (e.g.6 � 6 �), (b) the tie strategy for near-tiesums (e.g. 6 � 7 � or 8 � 7 �) and (c) thedecomposition-to-ten strategy for all othersums over ten (e.g. 6 � 8 � or 3 � 9 �).Although this conception and operationali-sation of flexibility/adaptivity can already beconsidered as more sophisticated than thefirst one, wherein it is simply identified withthe (random) use of multiple strategies, itremains, in our view, highly problematic todefine and operationalise strategy flexibility/adaptivity in terms of task characteristicsalone. Indeed, it is possible that, for a partic-ular subject and/or under particular circum-stances, the strategy choice process that theabove-mentioned authors call flexible/adaptive may become inflexible/inadaptive,and vice versa. Hereafter, we consider twoother groups of factors that need to be incor-porated into a more genuine concept offlexibility/adaptivity, besides task variables,namely subject and context variables.

A first additional set of factors, which has been intensively and systematically inves-tigated and modeled by cognitive psycholo-gists such as Siegler and his associates(Shrager & Siegler, 1998; Siegler, 1996,2000; see also Torbeyns, Arnaud, Lemaire &Verschaffel, 2004), are subject variables.Siegler’s computer model of how children’smastery of simple arithmetic sums develops,the Strategy Choice and Discovery Simulation(SCADS), indicates that whether a particularstrategy (e.g. retrieval or an extensivecounting strategy or a shortcut countingstrategy) is chosen to solve a particular itemby a particular child depends basically onhow accurately and how fast that strategy is

executed for that particular item and by thatparticular child, in comparison to otherconcurrent strategies available in the child’srepertoire. In other words, SCADS alwaystends to select and apply the strategy thatproduces the most beneficial combination ofspeed and accuracy for a particular sum (onthe basis of the accumulated data regardingspeed and accuracy available in the system’sdata base)4. Irrefutably, the adaptivityconcept underlying this computer model –which claims to be an accurate simulation ofhow ‘real’ children make strategy choicesand develop new strategies in the domain ofelementary arithmetic – reflects a morecomplex and more subtle view on the strat-egy choice process, wherein affordancesinherent in the task have to be considered inrelation to, and balanced with, features of the individual who is solving the task,especially how well (s)he masters the distinctstrategies.

More recent theoretical developments,especially the rise of the socioculturalperspective, have revealed that the issue offlexibility/adaptivity is even more compli-cated than is suggested by cognitive (com-puter) models such as the one by Siegler andhis colleagues (Shrager & Siegler, 1998;Siegler, 1996, 2000). More particularly, studieswithin the sociocultural view suggest that peo-ple may switch flexibly/adaptively betweenarithmetic strategies depending on not onlytask and subject variables, but also on variablesrelated to the (sociocultural) context. Relyingon the scarce theoretical and empiricalsociocultural literature, Ellis (1997, p. 492)shows that, with age and experience, childrendevelop (implicit) knowledge regarding ‘whata given culture defines as appropriate,adaptive, and wise’. This knowledge may alsoguide one’s strategy choices at an explicit orimplicit level. The idea of classroom situations

4Actually, the selection mechanism is somewhat more sophisticated, as the system also awards ‘novely points’ to

newly discovered strategies, which increase the chance that a new strategy will be attempted (regardless of its effi-

ciency in terms of speed and accuracy). This could be interpreted as a (first) attempt to include also a context

variable into the (computer) model.

and psychological experiments ‘as situationscharacterised by both social and task goals’(Ellis, 1997, p. 508) – or, to state it differently,as situations determined by, respectively, a‘didactical contract’ (Brousseau, 1997) or an‘experimental contract’ (Greer, 1997) – cancontribute, according to Ellis (1997), to ourunderstanding of strategy choice, particularlyto those choices that seem less than optimal atfirst glance (at least in terms of accuracy andspeed, which dominate cognitive-psychologi-cal research). So, children’s strategy choicesin elementary arithmetic are co-determinedby characteristics of the sociocultural contextin which they have to demonstrate their arith-metic skills: for example, what aspects of theirstrategic behaviour – instead of, or in additionto, the salient aspects of speed and accuracy –seem (most) valued in the classroom and/ortesting context, such as simplicity, elegance,formality, generality, intelligibility, certitude,originality, etc. of the solution strategy(see also Ellis, 1997; Lave & Wenger, 1991;Rogoff, 1990).

Based on the previous brief overview, wepropose the following working definition ofwhat it means to be adaptive in one’s strategychoices: By an adaptive choice of a strategy wemean the conscious or unconscious selection anduse of the most appropriate solution strategy on agiven mathematical item or problem, for a givenindividual, in a given sociocultural context. Bythe phrase ‘the most appropriate strategy’we simply do not mean ‘the strategy thatleads most quickly to the correct answer’ (asin the strictly cognitive-psychological senseof the term), although we do not excludethat in a particular instructional or testingsetting it may have that meaning (see alsoEllis, 1997; Verschaffel et al., submitted).

Are mathematically weak children less flexible thanmathematically strong ones?Over the past few years, researchers haveproposed and used several models to analyseand compare children’s strategic behaviour.An influential analytic model has beenproposed by Siegler and associates (Shrager& Siegler, 1998; Lemaire & Siegler, 1995;

Siegler, 1998, 2000). In their ‘model of strat-egy change’ Lemaire and Siegler (1995)distinguish four dimensions of strategic com-petence: (1) the repertoire of strategies thatpeople use to solve a set of items in a giventask domain, (2) the relative frequency withwhich the different strategies are applied tosolve this set of items, (3) the efficiency withwhich each strategy is executed (typicallymeasured in terms of speed and/or accuracyof strategy use), and (4) the adaptivity withwhich the various strategies are chosen andapplied on a given set of items, or stateddifferently, the efficiency of the strategychoices taking into account the relevant taskand subject parameters. So, Siegler’s theorydepicts cognitive development as charac-terised by a continuously changing reper-toire of coexisting strategies, which areapplied with continuously changing fre-quencies and proficiencies and which are exe-cuted in an increasingly adaptive way.

Whereas most research done within thisframework has looked at strategy choice and strategy change processes in childrenwith a normal arithmetic development, someresearchers have started to use this model(or similar models) to analyse low achievers’acquisition and use of arithmetic strategies incomparison to the strategic performance of their normally developing peers. Gray,Pitta and Tall (1997); Torbeyns, Verschaffeland Ghesquiere (2004a), and several othersconducted detailed analyses of the strategiesthat mathematically weak children andchildren with mathematical difficulties (MD)apply on single-digit additions and subtrac-tions like 5 � 8 and 12 � 5. Generally speak-ing, these analyses revealed that, comparedto their normally achieving peers, low achiev-ers and MD children use basically the sametypes of strategies. However, they rely moreoften on immature counting strategies andless often on more efficient mental strategies;in particular, they execute more advancedstrategies, like the decomposition-to-ten strat-egy, less accurately and slower than typicallydeveloping children. Most of these compara-tive studies have, however, not addressed the



fourth strategy parameter, namely flexibility,in their analysis of children’s strategy choices.Interestingly, some recent studies from ourcentre (Torbeyns et al., 2004a; Torbeyns,Verschaffel & Ghesquière, 2005) tried toaddress the flexibility parameter, by using theso-called choice/no-choice method (Siegler& Lemaire, 1997).

This method requires testing each partic-ipant under two types of conditions. In thechoice condition, participants can freelychoose which strategy they use to solve eachproblem. In the no-choice condition, theymust use one particular strategy to solve allproblems. Ideally, the number of no-choiceconditions equals the number of strategiesavailable in the choice condition. The oblig-atory use of one particular strategy on allproblems in the no-choice condition by eachparticipant allows the researcher to obtainunbiased estimates of the speed and accu-racy of the strategy (Siegler & Lemaire, 1997;Torbeyns et al., 2004b). The comparison ofthe accuracy and speed data of the differentstrategies gathered in the no-choice condi-tions, with the strategy choices made in thechoice condition, allows the researcher toassess the adaptivity of individual strategychoices in the choice condition in a scientif-ically valid way: Does the subject (in thechoice condition) solve each problem bymeans of the strategy that yields the bestperformance – in terms of accuracy andspeed – on this problem, as evidenced by the information obtained in the no-choiceconditions? Hereafter, we briefly and exem-plarily review one study from our own centrewherein we have applied this method tocompare the strategic performance, andespecially the strategy flexibility/adaptivity,of the strategy choices of children of differ-ent mathematical ability in the domain ofsingle-digit elementary arithmetic.

Torbeyns et al. (2005) investigated thestrategic performance of first graders ofdifferent mathematical achievement levelson arithmetic sums over ten by means of thechoice/no-choice method. They character-ized children’s strategy use with the four

above-mentioned parameters of Lemaire andSiegler’s (1995) model of strategy change.Participants were 83 first graders with high,moderate or low mathematical abilities(hereafter abbreviated as HA, MA, and LA)and who had been taught two mental cal-culation strategies as part of their regularinstruction – namely the above-mentioneddecomposition-to-10 strategy and the tiestrategy (7 � 8 � 7 � 7 � 1 � 14 � 1 � 15).All children were administered a series offive near tie sums over ten (like 7 � 8 � and7 � 6 � ) in different conditions. In the choicecondition, children could choose betweenthe decomposition-to-10 and the tie strategyon each near-tie sum. Afterwards they had tosolve, in two no-choice conditions, the samesums with the decomposition-to-10 and withthe tie strategy. Instructions and visualisationswere used to force children to choosebetween the two strategies (in the choice con-dition) or to apply only the intended strategy(in the no-choice conditions). To assesswhether children were able to solve the prob-lems by means of retrieval (i.e., by knowingthe answer to a sum by heart), a third no-choice condition was added to the design, inwhich the children had to solve the samenear-tie items (together with some extraretrieval items) by means of retrieval. First,with respect to strategy repertoire, the resultsfrom the choice condition revealed thatabout half of the children applied both typesof strategies to solve the near-tie items (57%),whereas the others either solved all sums by the decomposition-to-10 (25%) or the tiestrategy (18%). There were many morechildren who applied both strategies in theHA (77%) and MA group (65%) than amongthe LA children (31%). Second, childrenused the tie strategy (50%) with the samefrequency as the decomposition-to-10 strategy(50%) to solve the five near-tie sums in the choice condition, but the HA and MAchildren applied the tie strategy much moreoften than their LA peers. Third, theaccuracy and speed data from the no-choiceconditions showed that all children werequite efficient in performing both strategies





(with accuracies � .90), and that the HA andMA children applied these strategies moreefficiently (in terms of speed) than the LApupils. Finally, and for the topic of this papermost importantly, the comparison of theresults of the choice and no-choice condi-tions did not reveal any group differences instrategy adaptivity. More specifically, in thechoice condition mathematically HA and MA pupils did not solve more frequently thenear tie sums with the most efficient strategy(according to a comparative analysis of theefficiency data from the two no-choice con-ditions) compared with the LA pupils.

Torbeyns et al. (2005) provide severalpossible causes for this last unexpectedfinding. One of these explanations is ofparticular interest for the present article, asit reflects the growing awareness amongresearchers of the sociocultural (i.e., instruc-tional) factors that co-determine children’sstrategy choices in elementary arithmetic.According to the authors, the HA and MApupils might have based their strategy selec-tions more than the LA ones on (their inter-pretation of) the socio-mathematical normsand practices in their mathematics class-room (Yackel & Cobb, 1996) than on thestrategy efficiency characteristics. The text-book that was used in these classrooms wasNieuwe Reken Raak. As explained above, thistextbook strongly favours, or even imposes,one particular strategy for doing near tie sums, namely the tie strategy (e.g.6 � 7 � 6 � 6 � 1 � 13), above the otherstrategy, namely decomposition-to-10 (e.g.6 � 7 � 6 � 4 � 3 � 10 � 3 � 13). Inter-views with the classroom teachers revealedthat they had followed this textbook rigor-ously, at least for that part of the arithmeticcurriculum. So, arguably, these norms andpractices in these mathematics classes were strongly in favour of always solving near-tie sums with the tie strategy. Stated differ-ently, these HA and MA children may havedemonstrated a level of adaptivity that wassuboptimal for them (in the cognitive-psy-chological sense of the word), because theymade strategy selections in line with (their

interpretation of) the classroom norms andpractices about what it means to behave flex-ibly with regards to this particular kind ofarithmetic exercises. And these norms andpractices reflected a notion of strategy flexi-bility that was purely defined in terms of taskcharacteristics. So, if these pupils had beeninstructed in a mathematics classroomfavoring a more sophisticated concept offlexibility (that also allowed subjectiveconsiderations besides task-related oneswhen making strategy choices), one mighthave found the expected ability-relateddifference in strategy flexibility. But thisremains, of course, a hypothesis, whichshould be tested in future research.

Engendering strategy flexibility: From when on and for whom?As stated before, many current reform-baseddocuments and materials depart from thebasic belief that striving for strategy flexibil-ity is feasible and educationally valuable, alsofor the younger and mathematically weakerchildren (Baroody et al., 2003; Kilpatricket al., 2001; Verschaffel et al., in press). How-ever, systematic and scrutinised research thatconvincingly supports these basic claims is problematically scarce (Geary, 2003;Verschaffel et al., in press). Hereafter webriefly discuss the above-mentioned claimsconcerning the optimal age and the optimalpublic to strive for flexibility.

First, there is the issue of the optimalmoment for starting to strive for adaptivity/flexibility. Several authors argue that it isbetter first, and above all, to teach for routinemastery of a given arithmetic skill, andafterwards to change one’s aims and peda-gogy in the direction of flexible or adaptivestrategy use. This argument is supported bythe widespread belief that without long-term memory of previously learned facts,procedures, models, and representationaltools, there can be no flexible or adaptivethinking and by the observation that many children do not succeed in masteringfluency and/or automaticity in a givenarithmetic strategy, even after numerous



hours of regular classroom instruction and additional specific training (see e.g.Geary, 2003; Milo & Ruijssenaars, 2002;Warner, Davis, Alcock & Coppolo, 2002).This argument against a premature strivefor flexibility is opposed by many advocatesof the reform-based approach to mathemat-ics education, who conjecture that thedevelopment of adaptive expertise is notsomething that simply happens after pupilshave developed routine expertise in a givendomain. On the contrary, these authorsargue that education for flexibility shouldalready be present from the very beginningof the teaching/learning process (see e.g.Baroody, 2003; Gravemeijer, 1994; Selter,1998; Wittmann & Müller, 1990–1992). Thisidea is nicely expressed (albeit in more gen-eral terms) in the following quote fromBransford (2001, p. 3):

You don’t develop it in a ‘capstone course’ at the endof students’ senior year. Instead the path towardadaptive expertise is probably different from the pathtoward routine expertise. Adaptive expertise involveshabits of mind, attitudes, and ways of thinking andorganising one’s knowledge that are different fromroutine expertise and that take time to develop. Idon’t mean to imply that ‘you can’t teach an old rou-tine expert new tricks’. But it’s probably harder to dothis than to start people down an ‘adaptive expert-ise’ path to begin with – at least for most people.Some authors go even further and warn

against the rigidifying effects of years of dili-gent practice in routine expertise. Accordingto Feltovich, Spiro & Coulson, (1997, p. 126), ‘there are effects on cognition thatcome with such an extended practice thatcould lead to reduction in cognitive flexibility –to conditions of relative rigidity in thinkingand acting (while, we have noted, affectingother, more desirable goals, such as in effi-ciency and speed)’. Given the decrease inflexibility that might accompany increasedroutine experience in a certain domain, itseems very risky to design teaching/learningenvironments wherein one strives (only) for routine expertise first and postponesengendering flexibility/adaptivity until themoment routine expertise has been estab-

lished. Applied to the field of elementarymathematics education, an initial exclusivefocus on procedural fluency of some expli-citly taught procedures for particular prob-lem types may not only be unhelpful, butalso counterproductive for the developmentof adaptive expertise.

Closely related to the issue when to startstriving for strategy flexibility/adaptivity, isthe question whether promoting flexibleand adaptive strategy use is feasible and valu-able across different levels of mathematicalachievement, including the average and mathe-matically weaker ones. Threlfall (2002, p. 40)refers to the frequently heard claim that‘(because) only the more “mathematicallyminded” children will be capable of learninghow to make good choices, (. . .) flexibilityshould be abandoned as an objective for the“average” and “below average”’. The claimthat working for flexibility is unfeasible forthese latter groups of children is supportedby cognitive psychological research that hasdocumented lower working memory capac-ity in children with low achievement in math-ematics. In particular, the central executivecomponent of working memory, which isresponsible for the control, regulation andmonitoring of complex cognitive processes,appears to be related to individual differ-ences in math ability (e.g. Bull & Scerif,2001; McLean & Hitch, 1999; Swanson &Beebe-Frankenberger, 2004). This centralexecutive component has been fractionatedinto separate, though overlapping, func-tions, one of which includes the ability toshift between tasks or strategies (Baddeley,1996). Interestingly, some studies havereported that children with low achievementin mathematics have general difficulties inflexible shifting between solution strategies,as measured by standard cognitive shiftingtasks (Bull, Johnston & Roy, 1999; Bull &Scerif, 2001; McLean & Hitch, 1999). There-fore, the goal of developing strategy flexibil-ity might be unfeasible or even dangerousfor these children.

The claim that working for flexibility is unfeasible for these children is also

supported by the results of several interven-tion studies (e.g. Baxter, Woodward & Olson,2001; Geary, 2003; Milo & Ruijssenaars, 2002;Sowder, Philipp, Armstrong, & Schappelle,1998; Woodward & Baxter, 1997; Woodward,Monroe, & Baxter, 2001) indicating thatespecially mathematically weaker childrenand/or children with MD profit more frominstruction that pays major, if not exclusive,attention to developing efficiency (i.e., accu-racy and speed) in one arithmetic strategyand that considers flexibility as a strategicquality of only second-rate importance. How-ever, the results and conclusions of theseintervention studies are contradicted byother studies that do support the claim that not only mathematically able children but also lower-achievingchildren benefit more from reform-basedinstruction (which strongly aims at the devel-opment of strategy variety and flexibility)than traditionally-oriented direct instruction(which aims at the development of one particular strategy) (Baroody, 1996; Bottge,1999; Bottge, Heinrichs, Chan & Serlin, 2001;Bottge, Heinrichs, Mehta & Hung, 2002;Cichon & Ellis, 2003; Klein, Beishuizen &Treffers, 1998; Menne, 2001; Moser Opitz,2001; Van den Heuvel-Panhuizen, 2001).Given the large differences in (a) the arith-metic tasks and solution strategies beingaddressed in these intervention studies, (b)the age and characteristics of the pupilsbeing involved (and especially of themathematically weaker ones), (c) the natureof the intervention (and especially what kindof strategy flexibility/adaptivity is aimed atand how it is realised through instruction),and (d) how the effects are measured (andespecially how adequately flexibility/adaptivity is assessed), the contradictorynature of the results and conclusions is notsurprising and is an issue that imploresfurther research.

Anyhow, taking into account the theo-retical issues discussed in the first part ofthis article, we believe that providingchildren with a ‘(quasi-)algorithmic’ rulefor linking problem types to solution strate-

gies and with systematic training in the flu-ent application of that rule (as in theabove-mentioned Flemish method NieuweReken Raak) is not the kind of approachthat will yield flexible or adaptive expertiseas we have conceived and defined it. Thelatter kind of instruction, which is basedupon a notion of flexibility that merelylooks at task variables – without considera-tion of individual or contextual factors –misjudges the quintessence of the notionof ‘adaptivity’. Indeed, adaptivity involves apersonal and insightful choice based onweighing different kinds of affordances,not only task-related variables, but alsosubject- and context-related variables. Themore one dismisses a notion of strategyflexibility that merely looks at linking well-defined strategies to task characteristics (asin Nieuwe Reken Raak), the more one willagree that there is no easy and direct short-cut to becoming adaptive, and that this isnot something that can be trained ortaught but rather something that has to bepromoted or cultivated in a long-termperspective. Clearly, the more one adheresthe latter view on adaptivity, the more diffi-cult and challenging it might become tostrive for it, especially with younger andmathematically weak children. Whetherthis will be feasible and educationally valuable remains an open issue requiringfurther empirical research.

It is important that further interventionresearch about the feasibility and the opti-mal form of flexibility oriented instructionfor younger and mathematically weakerchildren is clear with respect to its focus. If the major instructional aim is to obtain inthe short term better gains in computationalfacility (operationalised as being able tosolve familiar sums quickly and correctly), itmight be more efficient to teach one singlestrategy for each arithmetic operation, whichis used with all such problems, or to providethe children with a ‘quasi-algorithmic’ ruleto associate certain problem types with cer-tain solution strategies and to train them inthe (routine) application of that rule, with-





out any concern for subject or context vari-ables. If, however, the instructional goals arebroader and more long-term, and comprisegenuine strategy flexibility – combined withgood understanding of mathematical con-cepts and principles, pattern recognitionskills, and appropriate beliefs, attitudes, andemotions towards mathematics and mathe-matics learning and teaching – then other,less routine-based instructional approachesmight be more appropriate, also for theyounger and mathematically weaker children.So, the question which instructional approachis the best for children, and for children withmathematical difficulties in particular, is notpurely an empirical one, but also, to asignificant degree, dependent on the valuesystem underlying one’s view of the purposesof mathematics education. Furthermore, thisquestion also has an ethical dimension: Is itequitable to expose children of average orlow ability (as assessed by traditional criteria)to a mathematics education that is intellec-tually less stimulating (Greer, personalcommunication)?

ConclusionIn this article, we have argued that forchildren of all ages and all ability levels, themost valuable instructional approach will benot the one in which children receive drill-and-practice in (quasi-)algorithmic tech-niques for selecting the most efficientprocedures for particular types of problems.Instead, we favour an instructional approachwherein children are cultivated in develop-ing their own preferences based on a per-sonal reflection on task, subject, and contextcharacteristics. However, we acknowledgethat these ambitious pleas to strive for such akind of adaptive expertise and the accompa-nying claims about the optimal road forreaching this ambitious goal, are still toomuch based on ‘rhetoric’ and anecdotal evi-

dence and too little on convincing evidencefrom empirical research. Especially theapplication of these ideas to mathematicallyweaker children and children with mathe-matical difficulties remains an open ques-tion, as supporting empirical evidence isproblematically scarce (as is the counterevi-dence). If we want to make progress in ourtheoretical understanding and practicalenhancement of strategy flexibility in ele-mentary arithmetic of children with lowachievement in mathematics (includingchildren with MD), there is a great need forcontinued research efforts, especially todesign experiments involving children fromdifferent ability ranges. Only when suchstudies have convincingly and repeatedlyshown that the reform-based approach,which strives for varied and flexible strategyuse effectively leads to the intended out-comes (without resulting in significant lossin computational accuracy and fluency),researchers will be in a good position toconvince policy-makers, teachers, andparents of the feasibility and value ofstriving for adaptive expertise for children ofall ages and all ability levels, rather thanreserving it as a ‘pinnacle’ for those whofirst have developed routine computationalexpertise.

AcknowledgementsThis research was partially supported byGrant GOA 2006/01 ‘Developing adaptiveexpertise in mathematics education’ fromthe Research Fund K.U. Leuven, Belgium

Address for correspondenceLieven Verschaffel, Centre for InstructionalPsychology and Technology, KatholiekeUniversiteit Leuven, Vesaliusstraat 2, 3000Leuven, Belgium. Tel: �32 16/32.62.58,Fax: �32 16/32.62.74E-mail: lieven.verschaffel@ ped.kuleuven.be

ReferencesBaddeley, A.J. (1996). Exploring the central executive.

Quarterly Journal of Experimental Psychology, 49, 5–28.Baroody, A.J. (1996). An investigative approach to

the mathematics instruction of children classi-fied as learning disabled. In D.K. Reid, W.P. Hresko et al. (Eds.), Cognitive approaches tolearning disabilities (third edition) (pp. 545–615).Austin, TX: PRO-ED.

Baroody, A.J. (2003). The development of adaptiveexpertise and flexibility: The integration ofconceptual and procedural knowledge. In A.J. Baroody & A. Dowker (Eds.), The developmentof arithmetic concepts and skills (pp. 1–34).Mahwah, NJ: Lawrence Erlbaum Associates.

Baroody, A.J., Wilkins, J.L.M. & Tiilikainen, S.H.(2003). The development of children’s under-standing of additive commutativity: From proto-quantitive concept to general concept? In A.J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills (pp. 127–160).Mahwah, NJ: Lawrence Erlbaum Associates.

Baxter, J.A., Woodward, J. & Olson, D. (2001). Effectsof reform-based mathematics instruction on lowachievers in five third-grade classrooms. ElementarySchool Journal, 101, 529–547.

Blöte, A.W., Van der Burg, E. & Klein, A.S. (2001).Students’ flexibility in solving two-digit additionand subtraction problems: Instruction effects.Journal of Educational Psychology, 93, 627–638.

Bottge, B.A. (1999). Effects of contextualized mathinstruction on problem solving of average andbelow-average achieving students. The Journal ofSpecial Education, 33, 81–92.

Bottge, B.A., Heinrichs, M., Chan, S. & Serlin, R.C.(2001). Anchoring adolescents’ understandingof math concepts in rich problem solving envi-ronments. Remedial and Special Education, 22,299–314.

Bottge, B.A., Heinrichs, M., Mehta, Z. & Hung, Y.(2002). Weighing the benefits of anchored mathinstruction for students with disabilities ingeneral education classes. The Journal of SpecialEducation, 35, 186–200.

Bourdeaud’hui, G., Caymax, J., Deboyser, G., Haepers,F., Jacobs, H., Matthys, L., Melis, J., Scheurweghs,M., Van den Eynde, R., Van der Avert, A., Vandeweyer, W., Verlinden, S., Westerlinck, W. &Wiecke, C. (2002). Nieuwe Reken Raak. Invulboek 1.Leuven, Belgium: Wolters.

Bransford, J. (2001). Thoughts on adaptive expertise.(Unpublished manuscript); Available at http://www.vanth.org/docs/AdaptiveExpertise.pdf.

Brousseau, G. (1997). Theory of didactical situations inmathematics. In N. Balacheff, M. Cooper, R. Sutherland & V. Warfield (Eds. & Trans.).Dordrecht, The Netherlands: Kluwer.

Brownell, W.A. (1945). When is arithmetic meaning-ful? Journal of Educational Research, 38, 481–498.

Bull, R., Johnston, R.S. & Roy, J.A. (1999). Exploringthe roles of the visual-spatial sketch pad andcentral executive in children’s arithmetical skills:Views from cognition and developmentalneuropsychology. Developmental Neuropsychology,15, 421–442.

Bull, R. & Scerif, G. (2001). Executive functioning asa predictor of children’s mathematics ability:Inhibition, switching, and working memory.Developmental Neuropsychology, 19, 273–293.

Cichon, D. & Ellis, J.G. (2003). The effects of MATHConnections on student achievement, confi-dence, and perception. In S.L. Senk & D.R. Thompson (Eds.), Standards-based SchoolMathematics Curricula: What are They? What doStudents Learn? (pp. 345–374). Mahwah, N.J.:Lawrence Erlbaum Associates.

Department for Education and Employment (1999).The National Numeracy Strategy Framework forTeaching Mathematics from Reception to Year 6.Londen: DfEE.

Ellis, S. (1997). Strategy choice in socioculturalcontext. Developmental Review, 17, 490–524.

Feltovich, P.J., Spiro, R.J. & Coulson, R.L. (1997). Issuesof expert flexibility in contexts characterized by complexity and change. In P.J. Feltovich, K.M. Ford & R.R. Hoffman (Eds.), Expertise in con-text: Human and machine (pp. 125–146). MenloPark, CA: AAAI Press.

Freudenthal, H. (1991). Revisiting mathematics educa-tion. Dordrecht: Reidel.

Geary, D.C. (2003). Arithmetical development: Com-mentary on Chapters 9 through 15 and futuredirections. In A.J. Baroody & A. Dowker (Eds.),The Development of Arithmetic Concepts and Skills:Constructing Adaptive Expertise (pp. 453–464).Mahwah, N.J.: Lawrence Erlbaum Associates.

Gray, E., Pitta, D. & Tall, D. (1997). The nature of theobject as an integral component of numericalprocesses. In E. Pehkonen (Ed.), Proceedings of the21st PME International Conference, 1, 115–130.

Gravemeijer, K. (1994). Developing realistic mathematicseducation. Utrecht, The Netherlands: FreudenthalInstitute, University of Utrecht.

Greer, B. (1997). Modelling reality in mathematicsclassrooms: The case of word problems. Learningand Instruction, 7, 293–307.

Heirdsfield, A.M., & Cooper, T.J. (2002). Flexibilityand inflexibility in accurate mental addition andsubtraction: Two case studies. Journal of Mathemat-ical Behavior, 21, 57–74.

Kilpatrick, J., Swafford, J. & Findell, B. (2001). Addingit up. Helping children learn mathematics. Washington,D.C.: National Academy Press.





Klein, A.S., Beishuizen, M. & Treffers, A. (1998). Theempty number line in Dutch second grades:Realistic versus gradual program design. Journalfor Research in Mathematics Education, 29, 443–464.

Lave, J. & Wenger, E. (1991). Situated learning:Legitimate, peripheral participation. Cambridge:Cambridge University Press.

Lemaire, P. & Siegler, R.S. (1995). Four aspects ofstrategic change: Contributions to children’slearning of multiplication. Journal for ExperimentalPsychology: General, 124, 83–97.

McLean, J.F. & Hitch, G.J. (1999). Working memoryimpairments in children with specific arithmeticlearning difficulties. Journal of Experimental ChildPsychology, 74, 240–260.

Menne, J.J.M. (2001). Met sprongen vooruit. Een productiefoefenprogramma voor zwakke rekenaars in het getal-lengebied tot 100 – een onderwijsexperiment. [Aproductive training program for mathematicallyweak children in the number domain up to 100 –a design study]. Utrecht, The Netherlands: CD-beta Press.

Milo, B. & Ruijssenaars, A.J.J.M. (2002). Strategiege-bruik van leerlingen in het speciaal basisonder-wijs: begeleiden of sturen? [Strategy instruction inspecial education: Guided of direct instruction?]Pedagogische Studiën, 79, 117–129.

Moser Opitz, E. (2001). Mathematical knowledgeand progress in the mathematical learning ofchildren with special needs in their first year ofschool. In MATHE 2000. Selected papers (pp.85–88). Dortmund, Germany: University ofDortmund, Department of Mathematics.

National Council of Teachers of Mathematics. (1989).Principles and standards for school mathematics.Reston, VA: National Council of Teachers ofMathematics.

National Council of Teachers of Mathematics. (2000).Principles and standards for school mathematics.[http://standards.nctm.org/document/index.htm].

Ontwikkelingsdoelen en eindtermen. Informatiemap voor deonderwijspraktijk. Gewoon basisonderwijs (1998).[Standards. Documentation for practitioners.Elementary education.] Brussel: Ministerievan de Vlaamse Gemeenschap, DepartementOnderwijs, Afdeling Informatie en Documentatie.

Rogoff, B. (1990). Apprenticeship in thinking. New York: Oxford University Press.

Swanson, H.L. & Beebe-Frankenberger, M. (2004).The relationship between working memory andmathematical problem solving in children at risk and not at risk for serious math difficulties.Journal of Educational Psychology, 96, 471–491.

Selter, C. (1998). Building on children’s mathe-matics. A teaching experiment in grade three.Educational Studies in Mathematics, 36, 1–27.

Shrager, J. & Siegler, R.S. (1998). SCADS: A model ofchildren’s strategy choices and strategy discoveries.Psychological Science, 9, 405–410.

Siegler, R.S. (1996). Emerging minds: the process of change in children’s thinking. Oxford: OxfordUniversity Press.

Siegler, R.S. (2000). The rebirth of children’s learn-ing. Child Development, 71, 26–35.

Siegler, R.S., & Lemaire, P. (1997). Older andyounger adults’ strategy choices in multiplica-tion: Testing predictions of ASCM using thechoice/no-choice method. Journal of ExperimentalPsychology: General, 126, 71–92.

Sowder, J., Philipp, R., Armstrong, B. & Schappelle, B.(1998). Middle-grade teachers’ mathematical knowl-edge and its relationship to instruction. Albany, NY: SUNY.

Thompson, I. (1999). Getting your head aroundmental calculation. In I. Thompson (Ed.), Issues inTeaching Numeracy in Primary Schools (pp. 145–156).Buckingham; U.K.: Open University Press.

Threlfall, J. (2002). Flexible mental calculation.Educational Studies in Mathematics, 50, 29–47.

Torbeyns, J., Arnaud, L., Lemaire, P. & Verschaffel, L.(2004a). Cognitive change as strategic change. InA. Demetriou & A. Raftopoulos (Eds.), CognitiveDevelopmental Change: Theories, Models andMeasurement (pp. 186–216). Cambridge, UK:Cambridge University Press.

Torbeyns, J., Verschaffel, L. & Ghesquière, P. (2004b).Strategy development in children with mathemat-ical disabilities: Insights from the choice/no-choice method and the chronological-age/ability-level-match design. Journal of LearningDisabilities, 37, 119–131.

Torbeyns, J., Verschaffel, L. & Ghesquière, P. (2005).Simple addition strategies in a first-grade classwith multiple strategy instruction. Cognition andInstruction, 23, 1–21.

Treffers, A., De Moor, E. & Feijs, E. (1990). Proeve van een national programma voor het reken/wiksun-deonderwijs op de basisschool. Deel 1. Overzicht eind-doelen. [Towards a national curriculum formathematics education in the elementary school.part 1. Overview of the goals.] Tilburg, TheNetherlands: Zwijsen.

Van den Heuvel-Panhuizen, M. (Ed.) (2001). Childrenlearn mathematics. A learning-teaching trajectory withintermediate attainment targets for calculation withwhole numbers in primary school. Groningen, TheNetherlands: Wolters Noordhoff.

Van der Heijden, M.K. (1993). Consistentie van aan-pakgedrag. [Consistency in solution behavior.]Lisse, The Netherlands: Swets & Zeitlinger.

Verschaffel, L., Greer, B. & De Corte, E. (in press).Whole number concepts and operations. In F.Lester (Ed.), Handbook of Research on Mathematics

Teaching and Learning (second version). New York:MacMillan.

Verschaffel, L., Luwel, K., Torbeyns, J. & Van Dooren, W. (submitted). Conceptualizing, investi-gating, and enhancing adaptive expertise in elemen-tary mathematics education. Manuscript submittedfor publication.

Warner, L.B., Davis, G.E., Alcock, L.J. & Coppolo, J.(2002). Flexible mathematical thinking and multi-ple representations in middle school mathemat-ics. Mediterranean Journal for Research in MathematicsEducation, 1(2), 37–61.

Wittmann, E.Ch., & Müller, G.N. (1990–1992). Handbuch produktiver rechenübungen. Vols. 1 & 2

[Handbook of productive arithmetic exercises.Volume 1 & 2]. Düsseldorf und Stuttgart, Germany: Klett Verlag.

Woodward, J. & Baxter, J. (1997). The effects of aninnovative approach to mathematics on academ-ically low achieving students in inclusive settings.Exceptional Children, 63(3), 373–388.

Woodward, J., Monroe, K. & Baxter, J. (2001).Enhancing student achievement on perform-ance assessments in mathematics. LearningDisabilities Quarterly, 24 (Winter), 33–46.

Yackel, E. & Cobb, P. (1996). Classroom sociomathe-matical norms and intellectual autonomy. Journalfor Research in Mathematics Education, 27, 458–477.




OVER THE past few years, mountingempirical evidence suggests that theearlier we recognise vulnerable young

children, the more likely we will be tosupport their subsequent development andprevent learning difficulties from occurringlater on (e.g. Coleman, Buysse & Neitzel,2006). However, especially children at-risk forreading disabilities have been at the forefrontof research. Surprisingly, few studies havebeen conducted to explore initial learningand the development of children’s arithmeticknowledge in the transition from preschoolto the primary grades (Aunola, Leskinen,Lerkkannen & Nurmi, 2004; LeFevre et al.,2005). In this article, we focus on preschoolpredictors, to add to our psychologicalunderstanding of initial development arith-metic skills and to help teachers respond toyoung children who may be at-risk for arith-metic learning disabilities as early as possible.

Several cognitive antecedents have beensuggested as factors that play a role in thedevelopment of initial arithmetic perform-ance and eventually as early marker for arith-

metic difficulties. In 1941, Piaget postulatedthat logical abilities, namely ‘seriation’, classi-fication’, and ‘conservation’ are conditionalto the development of arithmetic (Piaget &Szeminska, 1941). However, up till now, thedebate on the value of the Piagetian abilitiesfor arithmetic remains unsolved (for a review,see Lourenço & Machado, 1996). Besides thelogical abilities, reseachers focused on theimportance of ‘procedural’ and ‘conditionalcounting abilities’ in the development ofarithmetic performance. Finally, since Lan-derl, Bevan & Butterworth (2004) suggestedthat the core problem of arithmetic disabili-ties is a subitising deficit, it may be interestingto explore if subitising or ‘magnitude com-parison’ can be used as early markers forarithmetic difficulties.

Cognitive antecedents of initialarithmetic performanceLogical knowledge: seriation, classification andconservationPiaget and Szeminska (1941) specified thelogical abilities that children progressively

Early markers for arithmeticdifficultiesPieter Stock, Annemie Desoete & Herbert Roeyers

AbstractThe development of six prenumerical abilities and the relationship with numerical facility and arithmeticreasoning was explored in a longitudinal study (n � 108) for children in preschool (age 5 to 6) and Grade 1. ‘Procedural knowledge about counting’ was assessed using accuracy in counting objects. ‘Concep-tual knowledge about counting’ was assessed by asking children to make judgments about unusual right-to-left counts and reflections on why a procedure works. To assess the logical knowledge children completed‘seriation’, ‘conservation’ and ‘classification’ tasks. The ‘representation of number size’ was assessed bysubitising or dot and number comparison tasks. The results showed that about half of the variance in math-ematical reasoning (as measured by a standardised test at age 6 to 7) was associated with the prenumeri-cal skills of children in preschool. Especially conceptual counting knowledge and the seriation performancesin preschool, and the procedural counting knowledge in grade 1 were important for early arithmetic reason-ing. Numerical facility in grade 1 (as measured by a standardised test) was associated for a smaller pro-portion with the prenumerical skills of children. Moreover, it was possible to correctly classify in almostninety percent performance in grade 1 based on the six prenumerical skills in preschool. Again especiallyconceptual knowledge about counting and seriation were suitable early markers for arithmetic difficulties.


Early markers for arithmetic difficulties

acquire to master the concept of number.‘Seriation’ is the ability to sort a number of

objects based on differences on one or moredimensions while ignoring the similarities.In contrast, ‘classification’ is the ability to sortthose objects based on their similarities onone of more dimension, making abstractionof the differences. Once a child masters seri-ation and classification, it develops theknowledge that the number of objects in acollection only changes when one or moreobjects are added or removed. This conceptin logical thinking is called ‘conservation’(Piaget & Szeminska, 1941).

Since the publication of the work ofPiaget, there has been a substantial literaturedealing with Grade 1 (age 6 to 7) predictorsof subsequent development of arithmeticskills. For example, Arlin (1981) found thatwhether a child has reached the stage of con-crete operations or not was an importantcomponent of a child’s academic readiness.According to Kingma (1983), the combina-tion of conservation and seriation was a pre-dictor for number-language. Moreover,seriation tasks in preschool predicted thenumber-line comprehension in first-gradechildren (Kingma, 1984), whereas classifica-tion was found not suitable as predictor ofcomputational skills in grade 1 (Kingma,1983).

Since the publication of the work ofPiaget, researchers have criticized his theo-ries (Donaldson, 1978; for an overview, seeLourenço & Machado, 1996). Although thework of Piaget remains an essential refer-ence for practitioners working with childrenwith arithmetic problems (Grégoire, 2005),the Piagetian skills are no longer consideredas ‘conditional’ but rather as ‘precursors’ ofarithmetic development. Nevertheless, in ini-tial arithmetic, classification was found rele-vant to know the cardinal of a set (e.g. howmany balls can you see on this picture?),whereas seriation was needed when dealingwith ordinal numbers (e.g. circle the thirdball from the beginning). In addition, achild’s acquisition of conservation reflectedhis ability to think in a reversible way. This

way of thinking was especially beneficial forsolving reversal addition and subtractiontasks (e.g. 2 � � 5).

Counting knowledgeDuring the 1980s there was considerableinterest in exploring procedural and concep-tual knowledge in preschoolers’ counting(Le Corre, Van de Walle, Brannon, & Carey,2006; LeFevre et al., 2006).

Procedural knowledge is defined aschildren’s ability to perform an arithmetictask, for example, when a child can success-fully determine that there are five objects inan array (LeFevre et al., 2006). Proceduralknowledge can be assessed using accuracy incounting objects.

Conceptual knowledge reflects a child’sunderstanding of why a procedure works orwhether a procedure is legitimate (LeFevreet al., 2006). For counting, conceptual knowl-edge includes understanding five principles(Gallistel & Gelman,1992): (a) ‘stable-orderprinciple’ according to which the order ofnumber words must be invariant acrosscounted sets; (b) ‘one-one principle’ accord-ing to which every number word can only beattributed to one counted object; (c) ‘cardi-nality principle’ according to which the final number word pronounced in a count represents the numerosity of the set; (d) ‘abstraction principle’ according towhich any kind of object can be counted; (e) ‘order-irrelevance principle’ accordingto which objects can be counted in anyorder. Conceptual knowledge can beassessed by asking children to make judg-ments about types of counts they made (as inthis study) or types of counts modeled by ananimated frog (LeFevre et al., 2006).

Some advocates of the ‘continuity hypoth-esis’ (e.g. Gallistel & Gelman, 1992) claimedthat children have conceptual knowledgebefore their procedural counting skills arewell developed. Other researchers reportedthe opposite (e.g. Frye, Braisby, Lowe,Maroudas & Nicholls, 1989). The timing ofthe two types of knowledge may, however,largely depend on the particular task or the


Pieter Stock et al.

development may be iterative (Rittle-John-son, Siegler & Wagner, 2001).

It is obvious that early arithmetic strate-gies for addition and subtraction involvecounting in the ‘count all’ or ‘sum’ strategyin which the child first counts each collectionand then counts the combination of two collections starting from one (i.e.,2 � 5 � 1,2 . . . 1, 2, 3, 4, 5 . . . 1, 2, 3, 4, 5, 6, 7).As practice increases, older children usemore effective back-up strategies, such as the‘count-on’ strategy where they count up fromthe first addend the number of times indicated by the second addend (i.e.,2 � 5 � (2), 3, 4, 5, 6, 7) or the ‘min’ strategywhere they count up from the larger addendthe number of times indicated by the smalleraddend (i.e., “2 � 5 � 5 � 2 � (5), 6, 7”). Itis assumed that the retrieval strategy(“2 � 5 � 7, I know this by heart”) is madepossible by the learning and progressivestrengthening of memory associationsbetween problems and answers as a result ofthe repeated use of algorithms (Barrouillet &Lepine, 2005; Torbeyns, Verschaffel &Ghesquière, 2004). Geary and colleagues(Geary, Hoard, Byrd-Craven & DeSoto, 2004)found that children with specific arithmeticdifficulties had problems in counting.

Moreover, it has been suggested thatchildren’s basic conceptual understandingof how to count objects and their knowledgeof the order of numbers play an importantrole in arithmetic performance because theypromote the automatic use of arithmeticrelated information, allowing attentionalresources to be devoted to more complexarithmetic problem solving (Aunola, 2004).

Magnitude comparisonRepresentation of number size is alsoinvolved in numerical competence ( Jordan,Kaplan, Olah & Locuniak, 2006). Thisnumerical skill is involved in subitising (rapidapprehension of small numerosity) and inmagnitude comparison (i.e., knowing whichdigit in a pair is larger). There are some argu-ments that problems encountered by pupilswith arithmetic learning disabilities may be

due to a deficit in this skill (Butterworth,2003; Gersten, Jordan & Flojo, 2005; Stock,Desoete & Roeyers, 2006).

Initial arithmetic and arithmeticdifficultiesInitial arithmetic can be seen as a broaddomain of various computational skills.Dowker (2005) differentiated between twodomains: ‘mathematical reasoning’ and‘numerical facility’.

Children might have problems with arith-metic due to an insufficient knowledge base. Inaddition, by executing arithmetic problemsrepetitively basic number facts (e.g. 5 � 2 � _)become ‘automatic’.

Children with arithmetic disabilitiesoften lack numerical facility and do not knowbasic number facts by heart.

MethodAim and research questionsThe present study was designed to examineif we can predict the level of children’s arith-metic from their performance in preschool(age 5 to 6). The second purpose of thestudy was to test if children at-risk for arith-metic learning disabilities in grade 1 (age 6to 7) can be detected by their prenumericalskills in preschool.

ParticipantsIn a nonselected population a total of 108(54 girls, 54 boys) children from seven ran-domly selected preschools in the Dutchspeaking part of Belgium participated.Informed consent was obtained from all theparents. All children were tested in May ofpreschool (M � 5.9 years, SD � 4.0 months)and one year later in grade 1.

The original ‘nonselected’ sample con-sisted of all the children (n � 108). For someanalyses out of the original sample a smallersample (n � 67) of ‘low achieving’ (LA) and‘at least moderate achieving’ (MA) Caucasian,native Dutch-speaking children, without ahistory of ADHD, sensory impairment, braindamage, chronic poor health, seriousemotional or behavioural problems, or a

poor educational background were selected.The analysis was not run on all of thechildren, because the difference in groupsizes was too big. The LA-children (10 boysand 11 girls) performed below the 10th per-centile on at least one standardised arith-metic test and the low arithmeticperformance level was confirmed by theform teacher of the child. The MA-children(22 boys and 24 girls) scored above the 50thpercentile on both arithmetic tests, had anage-appropriate performance level (at leastlevel B; 60%) according to the form teacherand no signs of any learning disability.

All children and parents were fluent nativeDutch-speakers. A socio-economic status wasderived from the total number years of schol-arship of the parents (starting from the begin-ning of elementary school), with a mean of14.64 months (SD � 2.90) for mothers and14.62 months (SD � 2.69) for fathers.

MeasurementThe Kortrijk Arithmetic Test Revision(Kortrijkse Rekentest Revision, KRT-R)(Baudonck et al., 2006) is a Belgian test ofarithmetic reasoning which requires thatchildren solve mental arithmetic (e.g.19 � 7 � . . .) and number knowledge tasks(e.g. one less than eight is . . .). The psycho-metric value of the KRT-R has been demon-strated on a sample of 3,246 Dutch-speakingchildren from grade 1 till 6. In the study the standardised total percentile based onFlanders norms was used.

The Arithmetic Number Facts test(Tempo Test Rekenen, TTR; de Vos, 1992) isa numerical facility test which requires thatchildren in grade 1 solve as many numberfact problems as possible within two minutes(e.g. 3 � 2 � . . .). The psychometric valuehas been demonstrated for Flanders on asample of 10,059 children (Ghesquière &Ruijssenaars, 1994).

TEDI-MATH (Grégoire et al., 2004 Flem-ish adaptation) is a test designed for theassessment of arithmetic disabilities frompreschool till grade 3. The psychometricvalue has been demonstrated on a sample of

550 Dutch speaking Belgian children andhas been proven to be a well validated andreliable instrument (Desoete, 2006).

Procedural knowledge of counting (seeAppendix A subtest 1) was assessed using accuracy in counting numbers, counting forward to an upper bound (i.e., up to 6),counting forward from a lower bound (i.e.,from 3), counting forward with an upper andlower bound (i.e., from 5 up to 9), countingforward by number (e.g. what number youget when you count five numbers on fromeight), counting backward given a startingnumber (i.e., 7), and counting by step (i.e.,by 2) from it. One point was given for a cor-rect answer. A sum score was constructed(maximum: 14 points). Cronbach’s alpha was.73. Standardised percentiles were used.

Conceptual knowledge of counting (seeAppendix A subtest 2) was assessed with judg-ments about the validity of counting proce-dures. Children had to judge the counting oflinear and random patterns of drawings andcounters. To assess the abstraction principle,children had to count different kind ofobjects who were presented in a heap. Fur-thermore, a child who counted a set ofobjects was asked ‘how many objects arethere in total?’, or ‘how many objects arethere if you start counting with the leftmostobject in the array’. When children had tocount again to answer they did not gain anypoints, as this was considered to representgood procedural knowledge but a lack ofunderstanding of the counting principles ofGelman & Gallistel (1978). One point wasgiven for a correct answer with a correct moti-vation. A sum score was constructed (maxi-mum: 13 points). Cronbach’s alpha was .85. Standardised percentiles were used.

Three logical operations were assessed (seeAppendix A subtest 3). Children had to seriatenumbers (e.g. ‘Sort the cards from the onewith fewer trees to the one with the mosttrees’; maximum: 3 points). Cronbach’s alphafor the subtest was .68. Children had to makegroups of cards in order to assess the classifica-tion of numbers (e.g. ‘Make groups with thecards that go together’; maximum: 3 points).



Cronbach’s alpha was .73. Counters were usedto test the conservation of numbers (e.g. “Doyou have more counters than me? Do I havemore counters than you? Or do we have the same number of counters?And why is this?; maximum: 4 points). Onepoint was given for a correct answer with a cor-rect logical motivation. Cronbach’s alpha was.85. In the study standardised total percentilesbased on Flanders norms were used.

Magnitude comparison was assessed bycomparison dot sets (6 items) and numbers(12 items). Preschool children and firstgraders were asked were they saw most dots.The first graders were also asked whichnumber was closed to a certain targetnumber (see Appendix subtest 4). Onepoint was given for a correct answer. A sumscore was constructed (maximum: 6 pointsin preschool and 18 points in Grade 1).Cronbach’s alpha was .79. Standardised per-centiles based on Flanders norms were used.

Design and procedureAll preschool children were assessed individ-ually with the TEDI-MATH (Grégoire, VanNieuwenhoven & Noël, 2004) in a separateand quiet room by a trained tester. In addi-tion regular preschool teachers completed ateacher survey in the same period.

One year later, in grade 1 of elementaryschool the children completed the KRT-R(Baudonck et al., 2006) and the TTR (de Vos,1992) on the same day for about one hour intotal, supervised by a trained tester. Thechildren were also assessed individually inthe same week by the same trained testerwith TEDI-MATH. In addition regular elementary school teachers completed anelementary school teacher survey in thesame period. The testers, all psychologists ortherapists skilled in learning disabilities,received training in the assessment andinterpretation of arithmetic difficulties.

ResultsProspective assessmentSince all variables were normally distributedand did meet the assumptions for multiple

regression, two regression analyses were con-ducted in the nonselected sample to evaluatehow well the prenumerical arithmetic abilitiespredicted arithmetic reasoning and numericalfacility in grade 1. Six prenumerical abilities atage five to six were included simultaneously aspredictor variables: procedural countingknowledge, conceptual counting knowledge,seriation, classification, conservation and mag-nitude comparison. The univariate F-testswere Bonferroni-adjusted to control for thenumber of comparisons. With six comparisonsan adjusted alpha for each comparison isp � .008.

The linear combination of the prenumeri-cal abilities was significantly related to arith-metic reasoning assessed in grade 1 (at age 6to 7) with KRT, F (6, 107) � 18.659, p � .0005.R2 was .489. Conceptual counting knowledgeand seriation were especially beneficial forbeginning arithmetic reasoning (see Table 1).

The second multiple regression analysespointed out that the linear combination ofprenumerical abilities at age five to six wasalso significantly related to numerical facilityone year later (at age six to seven) assessedwith TTR, F (6, 107) � 9.655, p .0005. R2

was .327.Conceptual knowledge and seriation in

preschool seem to be especially beneficialfor both arithmetic reasoning and numericalfacility in grade l (see Table 1).

Concurrent assessmentTwo additional regression analyses were con-ducted to evaluate how well the concurrentprenumerical abilities in grade 1 (age 6 to 7)predicted arithmetic reasoning and numeri-cal facility in the same grade. The same sixprenumerical abilities were included simul-taneously as predictor variables. The linearcombination of prenumerical abilities wassignificantly related to mathematical reason-ing in the same grade, F(6, 107) � 6.067,p � .0005. R2 was .221 (see Table 2).

The linear combination of prenumericalabilities was also significantly related tonumerical facility in the same grade, and


Pieter Stock et al.



F(6, 107) � 3.056, p � .009. R2 was .103 (seeTable 2).

To answer the second research question,and to investigate if children at-risk for arith-metic learning disabilities in grade 1 (age 6to 7) can be detected by their prenumericalskills in preschool one year earlier a discrim-inant analysis on the sample (n � 67) of ‘lowachieving’ (LA) and ‘at least moderateachieving’ (MA) Caucasian, native Dutch-

speaking children. Wilks’ lambda was signi-ficant, � .51, �2 (6, N � 67) � 41.54,p � .0005, indicating that overall the predic-tors differentiated among the low achievingand the at least moderate achieving group.In Table 3 the standardised weights of thepredictors are presented. Based on thesecoefficients, conceptual knowledge of count-ing and seriation demonstrate the strongestrelationships with initial arithmetic.

Arithmetical reasoning (at age 6 to 7) Numerical facility (at age 6 to 7)

Prenumerical Unstandardised � t p Unstandardised � t pskills at age coefficients coefficients5 to 6

Constant �1.95 �.21 .83 .60 .09 .93

Proc. Knowledge .09 .18 1.89 .06 �.01 �.03 �.26 .80

Conc. Knowledge .14 .31 3.64 .008* .11 .39 3.97 .008*

Seriation .22 .35 4.39 .008* .10 .27 2.86 .008*

Classification .07 .13 1.74 .08 .05 .16 1.81 .08

Conservation .13 .08 1.08 .29 �.05 �.06 �.64 .53

Subitising .02 .02 .29 .77 .05 .12 1.51 .13

Table 1: Prediction of arithmetical reasoning and numerical facility in grade 1 (age 6 to 7) from prenumerical skills in preschool (age 5 to 6). Note: Proc. Knowledge � procedural knowledge,Conc. Knowledge � conceptual knowledge *p � .008

Arithmetical reasoning (at age 6 to 7) Numerical facility (at age 6 to 7)

Prenumerical Unstandardised � t p Unstandardised � t pskills at age coefficients coefficients5 to 6

Constant 9.33 1.47 .15 �.50 �.12 .90

Proc. knowledge .19 .32 3.39 .008* .07 .19 1.85 .07

Conc. knowledge .02 .03 .33 .74 �.01 �.02 �.25 .81

Seriation .06 .11 1.10 .27 .04 .11 1.02 .21

Classification .09 .19 2.13 .04 .06 .20 2.09 .04

Conservation .01 .02 .19 .85 �.02 �.07 �.65 .51

Subitising .16 .17 1.78 .08 .10 .18 1.82 .07

Table 2: Prediction of arithmetic reasoning skills and numerical facility from prenumerical skills at age 6 to 7. Note: Proc. Knowledge � procedural knowledge, Conc. Knowledge � conceptualknowledge *p � .008


Pieter Stock et al.

The mean prenumerical scores on thediscriminant function were consistent withthis interpretation. The at least moderateachieving group did better on conditionalknowledge and seriation than the low achiev-ing group. Based on the scores for these sixpredictors, 86.6 per cent was classified cor-rectly into the low achieving or at least mod-erate achieving group, whereas 83.6 per centof the cross-validated grouped cases wereclassified correctly. Based on the six prenu-merical scores 95.7 per cent (or 44 out of the46 children ) of the at least moderate achiev-ers and 66.7 per cent of low achievingchildren (or 14 out of 21 children) were clas-sified correctly.

Finally, correlations were computedbetween the preschool results and the resultsin grade 1 (see Table 4).

From Table 4 we can conclude a signifi-cant relationship between procedural andconceptual knowledge in preschool but not inGrade 1. Significant correlations were foundbetween preschool and grade 1 scores for procedural counting skills, seriation and conservation.

Discussion and conclusionsOver the past few years, increasing attentionhas been paid by policymakers to the idea ofearly childhood programs as important deter-

minants of children’s cognitive outcomes andschool readiness skills. Nevertheless, currentlymost arithmetic learning disabilities are notdetected until Grade 3 (Desoete, Roeyers &De Clercq, 2004). In this article, we focussedon the prediction of the level of children’sarithmetic in grade 1 from their performancein preschool, to add to our understanding ofinitial development arithmetic skills and tohelp teachers focus on early markers for arith-metic difficulties.

About half of the variance in arithmeticreasoning skills in first grade can be pre-dicted by assessing six prenumerical skills inpreschool. Only about one fifth of the vari-ance in arithmetic reasoning skills can bepredicted by assessing the same skills ingrade 1. Three markers showed significantcontributions: conceptual counting knowl-edge and seriation in preschool and proce-dural counting knowledge in grade 1.

The current results also suggest that morethen one third of the variance in numericalfacility in grade 1 can be predicted by assess-ing the prenumerical skills in preschool.Only about one tenth of the variance in factretrieval skills can be predicted by assessingthe same skills in grade 1, although no indi-vidual markers were significant.

The analysis of low and at least averageperforming first graders was consistent with

Prenumerical skills Standardised Mean Meanat age 5 to 6 canonical prenumerical prenumerical

discriminant score (SD) score (SD)function for LA for MA

coefficients (n � 21) (n � 46)Procedural counting knowledge .20 39.19 (19.89) 69.28 (24.52)

Conceptual counting knowledge .62 24.67 (23.60) 63.78 (29.68)

Classification .08 56.86 (29.79) 75.59 (25.42)

Seriation .64 62.55 (24.83) 89.65 (16.66)

Conservation .06 75.49 (6.24) 77.97 (9.66)

Magnitude comparison �.02 55.91 (21.23) 62.91 (24.82)

Table 3: Standardised canonical discriminant function coefficients of the predictors at age 5 to 6 forarithmetic and the mean values for LA and MA at age 6 to 7. Note: LA � low achievers at age 6 to 7, MA � at least average achievers at age 6 to 7

the previous analysis. Two thirds of the lower-elementary school children classified as ‘atrisk’ were classified correctly based on theirprenumerical performances in preschool atthe age five to six. Especially conceptualknowledge and seriation were suitable predic-tors of at-risk arithmetic performance. In linewith research of Kingma (1983) the Piagetianmodel had some value added since children-atrisk in grade 1 had lower scores seriation tasks,compared with at least average performingpeers. In line with Geary and Hoard (2005)children at-risk also had less developed count-ing knowledge and especially lacked concep-tual counting knowledge. Moreover, in linewith the idea of Rittle-Johnson and colleagues

(2001) arguing that procedural knowledgeand conceptual knowledge develop itera-tively, we found weak connections betweenthose components.

These results have as implication that inyoung children we should not only assesshow accurately children can count (proce-dural knowledge) but also how they masterthe counting principles of Gallistel &Gelman (1992). It might be interesting tosee if a controlled intervention focusing onconceptual counting knowledge and seri-ation skills in children at-risk can preventlearning difficulties from occurring later inthese vulnerable children.

In our dataset we found no significant con-

Preschool skills (age 5 to 6)

Preschool skills PK CK Cl Se Co Su

CK .60* – – – – –

Cl .35* .13 – – – –

Se .34* .33* .28* – – –

Co .07 �.05 .09 .36* – –

Su .26* .05 .07 .15 �.02 –

Grade 1 skills (age 6 to 7)

Grade 1 skills PK CK Cl Se Co Su

CK .16 – – – – –

Cl .17 .16 – – – –

Se .28* .17 .24 – – –

Co .14 .12 .08 .19 – –

Su .27* .19 .02 .17 .07 –

Preschool skills (age 5 to 6)

Grade 1 skills PK CK Cl Se Co Su

PK .37* .32* .32* .25 .11 .07

CK .16 .19 .31* .00 .09 .03

Cl .17 .17 .23 .05 �.15 . 34*

Se .29* .16 .24 .26* .12 .14

Co .14 .12 .08 .19 .28* .02

Su .27* .19 .02 .17 .07 .00

Table 4: Intercorrelation matrix. Note : PK � procedural knowledge of counting, CK � conceptualknowledge of counting, Cl � classification, Se � seriation, Co � conservation, Su � subitizing*p � .008




Pieter Stock et al.

tribution of subitising skills in the predictionof early arithmetic scores. Nevertheless, in linewith Landerl, Bevan & Butterworth (2004)children at-risk did worse on magnitude com-parison tasks than at least average performingpeers. It should be acknowledged that samplesize is a limitation of the present study. Obvi-ously sample size is not a problem for signifi-cant correlations or regressions. However,when analyses were not significant, a risk oftype 2- or �-error (concluding from the cohortthat there were no differences although inreality there were differences in the popula-tion) can not be excluded.

These results should be interpreted withcare since the analyses are correlational innature and numerosity skills might involve dif-ferent cognitive skills and might be age-, andintelligence-dependent and still maturing.Also more research on other variables such ashome and school environment, parentalinvolvement but also the facets of numericalcompetence (such as the knowledge of thenumerical system and the arithmetical opera-tions) is needed. Such longitudinal studieswith a multilevel design are currently beingprepared. In addition, in our sample manychildren (about 20 per cent) could be classi-fied, based on their test scores as pupils at-risk.All of those children had below critical cut-offscores (they scored � pc 10 on at least onearithmetics test) in grade 1. It would be inter-esting to investigate if these deficits persist

across two successive grades and if all thesechildren will demonstrate a severe and resist-ant learning disability or if some of them willbecome mildly delayed arithmetic problemsolvers having difficulties that are not relatedto learning disabilities. Research with largergroups of poor arithmetic performers andchildren with learning disabilities followed forlonger periode of time during elementaryschool seems therefore indicated, to replicatethe results of this study. In addition, not onlythe prenumerical but also the facets of numer-ical competence (such as the knowledge ofthe numerical system and the arithmeticaloperations) still need a full explanation. Suchstudies are currently being conducted.

Summarising, our dataset support thatwe should assess procedural counting knowl-edge, the ability to seriate in preschool, andalso the ability to distinguish essential forminessential counting characteristics (or theconceptual counting knowledge of youngchildren). Especially the inability to do suchthings in preschool (at the age 5 to 6) maybe a marker for later arithmetic disabilities.

Address for correspondenceDepartment of Clinical Psychology, ResearchGroup Developmental Disorders, Universityof Ghent, Henri Dunantlaan 2, B-9000 Gent,Belgium E-mail: [email protected]

ReferencesArlin, P.K. (1981). Piagetian tasks as predictors of

reading and math readiness in grades-K-1. Journalof Educational Psychology, 73, 712–721.

Aunola, K., Leskinen, E., Lerkkanen, M. & Nurmi, J.(2004). Developmental dynamics of math per-formance to Grade 2. Journal of Educational Psychology, 96, 699–713.

Barrouillet, P. & Lepine, R. (2005). Working memoryand children’s use of retrieval to solve additionproblems. Journal of Experimental Child Psychology,91, 183–204.

Baudonck, M., Debusschere, A., Dewulf, B., Samyn, F.,Vercaemst, V. & Desoete, A. (2006). De KortrijkseRekentest Revision KRT-R. [The Kortrijk ArithmeticTest Revision KRT-R]. Kortrijk: CAR Overleie.

Butterworth, B. (2003). Dyscalculia Screener. London:NFER Nelson Publishing Company Ltd.

Coleman, M.R., Buysse, V. & Neitzel, J. (2006). Recog-nition and response. An Early Intervening system for Young children at-risk for Learning Disabilities.Research synthesis and recommendations. UNIC FPGChild Development Institute.

Desoete, A. (2006). Validiteitsonderzoek met de TEDI-MATH9 [Validity research on the TEDI-MATH].Diagnostiekwijzer], 9.

Desoete, A., Roeyers, H. & De Clercq, A. (2004).Children with mathematics learning disablities inBelgium. Journal of learning disabilities, 37, 50–61.

De Vos, T. (1992). TTR. Tempotest rekenen [Arithmeticnumber fact test]. Lisse: Swets & Zeitlinger.

Donaldson, M.C. (1978). Children’s Minds. London:Fontana.

Dowker, A. (2005). Individual Differences in Arithmetic.Implications for Psychology, Neuroscience and Educa-tion. Hove, UK: Psychology Press.

Frye, D., Braisby, N., Lowe, J. Maroudas, C. & Nicholls,J. (1989). Young children’s understanding ofcounting and cardinality. Child Development, 60,1158–1171.

Gallistel, C. & Gelman, R. (1992). Preverbal and verbalcounting and computation. Cognition, 44, 43–74.

Geary, D.C. & Hoard, M.K. (2005). Learning disabili-ties in arithmetic and mathematics: Theoreticaland empirical perspectives. In J.I.D. Campbell(Ed.), Handbook of Mathematical Cognition (pp.253–268). New York: Psychology Press.

Geary, D.C, Hoard, M.K., Byrd-Craven, J. & DeSoto,M.C. (2004). Strategy choices in simple and com-plex addition: Contributions of working memoryand counting knowledge for children with math-ematical disability. Journal of Experimental ChildPsychology, 88, 121–151.

Gelman, R. & Galistel, C.R. (1978). The child’s under-standing of number. Cambridge, MA: Harvard University Press.

Gersten, R., Jordan, N.C. & Flojo, J.E. (2005). Earlyidentification and intervention for students withmathematics difficulties. Journal of Learning Dis-abilities, 38, 293–304.

Ghesquière, P. & Ruijssenaars, A. (1994). Vlaamse normen voor studietoetsen Rekenen en technisch lezenlager onderwijs. [Flemisch norms for schooltests onMathematics and Technical Reading in elementaryschool]. Leuven: K.U.L.–C.S.B.O.

Grégoire, J. (2005). Développement logique et com-pétences arithmétiques. Le modèle piagétien est-il toujours actuel? In M. Crahay, L. Verschaffel, E.De Corte, & J. Grégoire (Eds.) Enseignement etApprentissage des Mathématiques (pp. 57–77). Brussels: De Boeck.

Grégoire, J., Noel, M. & Van Nieuwenhoven (2004).TEDI-MATH. TEMA (Flamish adaptation: A. Desoete,H. Roeyers & M. Schittekatte): Brussel/ Harcourt:Antwerpen.

Kingma, J. (1983). Seriation, Correspondence, andtransitivity. Journal of Educational Psychology, 75,763–771.

Kingma, J. (1984). Traditional intelligence, Piagetiantasks, and initial arithemtic in kindergarten andprimary-school grade-one. Jounal of Genetic Psy-chology, 145, 49–60.

Landerl, K., Bevan, A. & Butterworth, B. (2004).Developmental dyscalculia and basic numericalcapacities: a study of 8–9-year-old students. Cogni-tion, 93, 99–125.

Le Corre, M., Van de Walle, G., Brannon, E.M. &Carey, S. (2006). Re-visiting the competence/ per-formance debate in the acquisition of the count-ing principle. Cognitive Psychology, 52, 130–169.

LeFevre, J.A., Smith-Chant, B.L., Fast, L., Skwarchuk,S.L., Sargla, E., Arnup, J.S., Penner-Wilger, M.,Bisanz, J. & Kamawar, D. (2006). What counts asknowing? The development of conceptual andprocedural knowledge of counting from kinder-garten through Grade 2. Journal of ExperimentalChild Psychology, 93, 285–303.

Lourenço, O. & Machado, A. (1996). In defense ofPiaget’s theory: A reply to 10 common criticisms.Psychological Review, 103, 143–164.

Jordan, N.C., Kaplan, D., Olah, L.N. & Locuniak, M.N.(2006). Numer sense growth in kindergarten: Alongitudinal investigation of children at risk for mathematics difficulties. Child Development, 77,153–175.

Piaget, J. & Szeminska, A. (1941). La genèse du nombrechez l’enfant. Neuchâtel: Delachaux et Niestlé. (7e

édition, 1991).Rittle-Johnson, R., Siegler, R.S. & Wagner, R.S.

(2001). Developing conceptual understandingand procedural skill in mathematics: An iterativeprocess. Journal of Educational Psychology, 93,346–362.

Stock, P., Desoete, A. & Roeyers, H. (2006). Focussingon mathematical disabilities: a search for definition, classisfication and assessment (pp. 29–62). In Soren V. Randall (Ed.) LearningDisabilities New Research Nova Science: Hauppage, NY.

Torbeyns, J., Verschaffel, L. & Ghesquière, P. (2004). Strategic aspects of simple addition andsubtraction: the influence of mathematical ability. Learning and Instruction, 14, 177–195.




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Subtest Content and example of item

1. Procedural —Counting as far as possible (without help � 2 points; with startingknowledge help � 1 point)(13 items, max — Counting forward to an upper bound (“up to 9 ”)14 points) —Counting forward to an upper bound (“up to 6 ”)

—Counting forward from a lower bound (“from 3 ”)—Counting forward from a lower bound (“from 7 ”)—Counting forward from a lower bound to an upper bound (“from 5 up to 9”)—Counting forward from a lower bound to an upper bound (“from 4 up to 8”)—Counting forward (“5 steps starting at 8 ”)—Counting forward (“6 steps starting at 9 ”)—Count backward (“from 7 ”)—Count backward (“from 15 ”)—Count by step (by 2)—Count by step (by 10)

2. Conceptual —Counting linear pattern of items (“how many rabbits are there? Howknowledge many rabbits are there in total?”, and “how many rabbits are there if you (7 items, max start counting with this one. Why?” ). Max 3 points13 points) —Counting linear pattern of items (“how many lions are there? How many

lions are there in total?”, and “how many lions have I hidden. Why? ”). Max 3 point

—Counting random pattern of items (“how many turtles are there? How manyturtles are there in total? ”). Max 2 points

—Counting random pattern of items (“how many sharks are there? How manysharks are there in total? ”). Max 2 points

—Counting a heterogeneous set of items (“how many animals are there intotal?”). Max 1 point

—Understanding of the cardinal (“Can you put as much counters as there areon this paper? ”) Max 1 point

—Understanding of the cardinal (“How many hat do I have in my hand, whenall the snowman had a hat on this picture?”) Max 1 point

3. Logical —Seriation: “Sort the cards form the one with fewer trees to the one with the operations on most trees. forgot this card” and “Can you put this card in the correct order?”numbers and “I give you carts with numbers now. Do the same as with the trees. (9 items, Start with the cart with smallest number and go on with the other carts”max 9 points) Max 3 points

—Classification: “Make groups with the cards that go together. Can you putthem together in another way?” and “Make groups with these cards thatgo together” Max 3 points

—Conservation: 4 items: Do you have more counters than me? Do I havemore counters than you? Or do we have the same number of counters?Why? Max 4 points

Appendix A: Subtsests and examples of test-items of the TEDI-MATH

(Continued )

Subtest Content and example of item

4. Estimation of Comparison of dot sets (subitising): in preschool and grade 1the size(18 items,18 points)

Where do you have most dots? Here or here? Show me.1 dot versus 3 dots3 dots versus 2 dots4 dots versus 6 dots7 dots versus 2 dots7 dots versus 12 dots15 dots versus 8 dots (see example)

max 6 pointsEstimation of size: Comparison of distance between numbers (in grade 1).

—Target number is 4. What number is closed to this (5 or 9)?—Target number is 2. What number is closed to this (7 or 4)?—Target number is 8. What number is closed to this (7 or 3)?—Target number is 9. What number is closed to this (5 or 7)?—Target number is 7. What number is closed to this (3 or 9)?—Target number is 3. What number is closed to this (8 or 2)? (see ex. above)—Target number is 5. What number is closed to this (3 or 9)?—Target number is 6. What number is closed to this (8 or 1)?—Target number is 32. What number is closed to this (59 or 24)?—Target number is 48. What number is closed to this (57 or 15)?—Target number is 61. What number is closed to this (53 or 99)?—Target number is 79. What number is closed to this (48 or 86)?

max 12 points.




ASSISTING CHILDREN who have difficulty learning arithmetic is not straightforward. This conclusion

became apparent to us after we analysed thenumber knowledge of several thousand Australian students during the Early NumeracyResearch Project (ENRP, Clarke, Cheeseman,Gervasoni, Gronn, Horne, McDonough,Montgomery, Roche, Sullivan, Clarke & Rowley, 2002). What we discovered is that thenumber knowledge of children who are vul-nerable is much more diverse that we hadanticipated. This finding has importantimplications for the way classroom teachersand specialist teachers assess students andplan and implement curriculum and instruc-tion that accelerates vulnerable children’snumber learning. First, teachers need assess-ment tools that are responsive to the diversityof children’s number knowledge and thatenable teachers to determine the extent ofchildren’s current mathematical knowledge

in the number domains. Second, teachersneed to provide effective learning experi-ences that can be easily customised torespond to children’s individual learningneeds. This article explores these aspects withthe aim of providing advice for teachers. Inso doing, we draw on findings from the ENRPthat researched mathematics learning in thefirst three years of schooling (ages 5, 6 and 7-years). Arithmetic difficulties may not bestraightforward, but once we recognise thisfact, then the way forward to helping studentsis clearer.

Vulnerability in arithmetic learningThe argument presented in this paper isbased on the assumption that it is importantfor school communities to identify childrenwho, as emerging school mathematicians andafter one year at school, have not thrived inthe school environment, and to provide thesechildren with the type of learning opportuni-

Assessing and teaching children whohave difficulty learning arithmeticAnn Gervasoni & Peter Sullivan

AbstractArithmetic difficulties have long captured the attention of teachers and researchers, but intervention programs for assisting children are seldom successful for all. Recent Australian research suggests that thisis because we have failed to recognise the complexity of arithmetic difficulties. Analysis of some 30,000 one-on-one clinical interviews conducted over three years during the Early Numeracy Research Project providedrich data for charting the pathway of young children’s number learning in four domains (Counting, PlaceValue, Addition and Subtraction Strategies, and Multiplication and Division Strategies), and for identi-fying children who were having difficulty. We describe such children as being vulnerable or at risk of notbeing able to take advantage of everyday classroom experiences. The data show that the combinations ofdomains in which children were vulnerable were diverse, and suggest that there is no single ‘formula’ fordescribing children who are vulnerable in number learning, or for describing the instructional needs of stu-dents. Indeed, children have learning needs that call for teachers to make individual decisions about theinstructional approach for each child. Further, the diversity of children’s mathematical knowledge in thefour domains suggests that knowledge in any one domain is not necessarily prerequisite for knowledge con-struction in another domain. This finding has implications for both intervention programs and for the wayin which school mathematics is introduced to children. It seems likely that children may benefit from con-current learning opportunities in all number domains, and that experiences in one domain should not bedelayed until a level of mathematical knowledge is constructed in another domain.


Assessing and teaching children who have difficulty learning arithmetic

ties and experiences that will enable them tothrive and extend their mathematical under-standing. Thus, the notion of interventionearly in schooling is important. The perspec-tive that underpinned our research was thatthose children who have not thrived have notyet received the type of experiences andopportunities necessary for them to con-struct the mathematical understandingsneeded to successfully engage with theschool mathematics curriculum, or to makesense of this curriculum. As a result, thesechildren are vulnerable and possibly at riskof poor learning outcomes. The term vulner-able is widely used in population studies (e.g.Hart, Brinkman & Blackmore, 2003), andrefers to children whose environmentsinclude risk factors that may lead to poordevelopmental outcomes. The challengeremains for teachers and school communi-ties to create learning environments anddesign mathematics instruction that enablesvulnerable children’s mathematics learningto flourish.

Using growth points to describechildren’s number learningThe data presented in this article werecollected from 1999 to 2001 as part of theENRP (Clarke et al., 2002), the fundamentalbuilding block of which was a set of growthpoints (see Appendix A) that describe keystages in the learning of various domains ofmathematics. The principles underlying theconstruction of the growth points were thatthey would:● describe the development of children’s

mathematical knowledge and under-standing in the first three years of school,through highlighting important ideas inearly mathematics understanding in aform and language that is useful forteachers;

● reflect the findings of relevant internationaland local research in mathematics education(e.g. Steffe, von Glasersfeld, Richards &Cobb, 1983; Steffe, Cobb & von Glasersfeld,1988; Fuson, 1992; Boulton-Lewis, 1996;Mulligan & Mitchelmore, 1996; Wright,

1998; Bobis & Gould, 1999);● reflect, where possible, the structure of

mathematics;● allow the mathematical knowledge and

understanding of individuals and groupsto be described; and

● enable identification of those students whomay benefit from additional assistance.The growth points developed formed a

framework for describing children’s develop-ment in four Number domains (Counting, PlaceValue, Addition and Subtraction Strategies, andMultiplication and Division Strategies), three Mea-surement domains (Length, Mass and Time),and two Space/Geometry domains (Properties ofShape, and Visualisation and Orientation). Theprocesses for validating the growth points, the interview items and the comparativeachievement of students in project and references schools are described in full inClarke et al. (2002).

To illustrate the nature of the growthpoints, the following are the points forthe Addition and Subtraction Strategiesdomain:1. Counts all to find the total of two

collections.2. Counts on from one number to find the

total of two collections.3. Given subtraction situations, chooses

appropriately from strategies includingcount back, count down to & count upfrom.

4. Uses basic strategies for solving addi-tion and subtraction problems (doubles,commutativity, adding 10, tens facts,other known facts).

5. Uses derived strategies for solving addi-tion and subtraction problems (near dou-bles, adding 9, build to next ten, factfamilies, intuitive strategies).

6. Given a range of tasks (including multi-digit numbers), can solve them mentally,using the appropriate strategies and aclear understanding of key concepts.Each growth point represents substantial

expansion in mathematical knowledge. In dis-cussions with teachers, growth points aredescribed as key “stepping stones” along paths


Ann Gervasoni & Peter Sullivan

to mathematical understanding (Clarke, 2001).It is not claimed that every student passes allgrowth points along the way, nor should thegrowth points be regarded as discrete. However, the order of the growth points pro-vides a guide to the possible trajectory (Cobb& McClain, 1999) of children’s learning. In asimilar way to that described by Owens &Gould (1999) in the Count Me In Too project:“the order is more or less the order in whichstrategies are likely to emerge and be used bychildren” (p. 4).

In summary, the framework of growthpoints help teachers to:● understand a possible trajectory for

describing children’s learning;● describe (following assessment) the

mathematics achievements of each child;● identify any children who may be vulner-

able in a given domain;● describe a zone of proximal development

in each domain so as to customise plan-ning and instruction; and

● identify the diversity of mathematicalknowledge in a class.

The clinical assessment interviewbased on the growth pointsThe framework of growth points formed thestructure for the creation of the assessmentitems used as the basis of a clinical interview,known as the Early Numeracy Interview(Department of Education Employment andTraining, 2001). Clinical interviews enablethe teacher to observe children as they solveproblems to determine the strategies usedand any misconceptions, and to probechildren’s mathematical understandingthrough thoughtful questioning (Ginsburg,1997; Wright, Martland & Stafford, 2000).The insights gained through this form ofassessment inform teachers about the partic-ular instructional needs of each studentmore powerfully than scores from traditionalpencil and paper tests.

Both the Early Numeracy Interview andthe framework of growth points were refinedthroughout the first two years of the ENRP inresponse to data collected from more

than 20,000 assessment interviews. In theENRP, the assessment interview was con-ducted with every child in project schools atthe beginning and end of each school year (March and November respectively). TheMarch interviews enabled teachers to deter-mine the growth points reached by children atthe beginning of the school year, to customisecurriculum planning and instructionaccordingly, and to identify any childrenwho were vulnerable in a particular domain.The November interview enabled growth overthe year to be determined for each child.

The Early Numeracy Interview takesbetween 30 to 40 minutes per student andduring the ENRP was conducted by the reg-ular classroom teacher. The full text of theinterview involves around 60 tasks, althoughno child is presented with all of these. Givensuccess with a particular task, the interviewercontinues with the next task in the givenmathematical domain (e.g. Addition andSubtraction Strategies) for as long as thechild is successful. If the child cannot per-form a particular task correctly, the inter-viewer moves on to the next domain ormoves into a detour, designed to elaboratemore clearly any difficulty a child might behaving in a particular area. Figure 1 showsthe first two questions from the Addition andSubtraction Strategies section of the inter-view. Words in italics are instructions for theinterviewer, and the symbols and arrows indicate which question to ask next.

Teachers gain insights into children’smathematical knowledge from their res-ponses to the interview tasks. For example, achild’s response to Question 18 enables theteacher to determine whether a child cancount-on or use reasoning strategies or knownfacts to solve the problem. Children who aresuccessful move on to Question 19, while theothers continue with 18(d) to determinewhether they can use a count-all strategy tosolve the problem when all items may be seen.

Question 19 provides information aboutchildren’s arithmetic strategies in a simplesubtraction context in which models are notinitially provided. Children who are not suc-

cessful with this task are provided with aprompt to use their fingers as a model. Ifchildren are still unable to use a successfulstrategy, the interviewer moves to the ques-tions on Multiplication and Division Strategies.Successful students move on to a series ofmore complex tasks in the Addition andSubtraction Strategies domain, with the finaltask involving the subtraction of 3-digitnumbers.

The Early Numeracy Interview providedteachers participating in the ENRP withinsights about children’s mathematicalknowledge that they reported might other-wise not have been forthcoming (Clarke,2001). The project found that teachers wereable to use this information to plan instruc-tion that would provide students with thebest possible opportunities to extend theirmathematical understanding.

Using growth points to identifyvulnerability in number learning for Grade 1 and 2 children (6- and 7 year-olds)Participants in the Early Numeracy Research Pro-ject in the year 2000 included 1497 Grade 1children (6-year-olds) and 1538 Grade 2children (7-year-olds) from 34 ENRP trialschools. These schools included Government,Catholic and Independent schools fromacross the State of Victoria in Australia thatwere widely representative of the Victorianpopulation.

In order to identify children who are vul-nerable in their number learning, some-times a line is drawn across a distribution oftest scores, and children ‘below’ the line aredeemed at risk (e.g. Ginsburg, 1997;Woodward & Baxter, 1997). In the ENRP thedecision on where to draw the line was madeon the basis of on the way growth points (Gervasoni, 2000). The on the way growth pointin any domain is used to identify children whohave constructed the mathematical knowledge

Figure 1: Excerpt from the addition and subtraction strategies section of the ENRP assessment interview

SECTION C: STRATEGIES FOR ADDITION & SUBTRACTION Equipment: green teddies, ice-cream lid, white, blue and yellow problem cards.

18) Counting On Place 9 green teddies on the table. a) Please get four green teddies for me. b) I have nine green teddies here (show the child the nine teddies, and then screen the nine teddies with the ice-cream lid).That’s nine teddies hiding here and four teddies here (point to the groups).c) Tell me how many teddies we have altogether. . . . Please explain how you worked it out.

Q19 If the child does not say 13, please go to part (d)

d) (Remove the lid). Please tell me how many there are altogether.

19) Counting Back For this question you need to listen to a story.a) Imagine you have 8 little biscuits in your play lunch and you eat 3. How many do you have left? . . . How did you work that out?

Q20 part (b)

b) Could you use your fingers to help you to work it out? (it’s fine to repeat the question, but no further prompts please).

Section D



that underpins the initial mathematicscurriculum in a particular domain and gradelevel, and who are likely to continue to learnsuccessfully. Not yet reaching the on the waygrowth point in a particular domain is anindicator that children may be vulnerable inthat domain and may benefit from support tohelp them reach the on the way growth point.

The development of appropriate on theway growth points for Grade 1 (the secondyear of school) and Grade 2 children wasguided by three data sources: the ENRPgrowth point distributions for Grade 1 andGrade 2 children in March 2000; the Victorian Curriculum and Standards Frame-work II (Board of Studies, 2000) for Grade 1and Grade 2; and the opinions of ENRPGrade 1 and Grade 2 classroom teachers (seeGervasoni, 2004, for elaboration of this). Theanalyses and synthesis of these data resultedin the following on the way growth pointsbeing established for children at the begin-ning of Grade 1 in Counting, Place Value,Addition and Subtraction Strategies, and Mul-tiplication and Division Strategies respec-tively:● can count collections of at least 20 items;● can read, write, order and interpret one

digit numbers;● counts all to find the total of two collections;● counts group items individually in multi-

plicative tasks.In other words, being able to count a

collection of 20 objects is one piece of evi-dence to suggest that children are ready forconventional Grade 1 experiences. If theycannot count 20 objects, they may be vulner-able and unable to take advantage of theactivities they will experience during the year.

The on the way growth points establishedfor children at the start of Grade 2 were:● counts forwards and backwards beyond

109 from any number;● can read, write, order and interpret two

digit numbers;● counts on from one number to find the

total of two collections;● uses grouping to solve multiplicative tasks.

In other words, a student who can read,write, order and interpret two digit numberswill benefit from the usual Grade 2 numeracyexperiences in the Place Value domain. Stu-dents who cannot do so are vulnerable andmay not be able to take advantage of thosesame experiences. Even though these respec-tive on the way growth points are to someextend arbitrary, they are based on consid-ered judgments and they form the basis ofthe data reported in the following section.

The domains and combinations of domains forwhich children were vulnerableWhen considering the domains and combi-nations of domains for which children werevulnerable, there are three aspects of inter-est: the number of students who are vulner-able in the respective domains; the extent towhich children are vulnerable in multipledomains; and the nature of vulnerable stu-dents’ achievements in domains in whichthey are not vulnerable.

The analysis of ENRP participants’ growthpoint data found that there were 576 (out of1497) Grade 1 children in 2000, or 38 per centof the group, who were vulnerable in at leastone number domain. The number of Grade 2children in 2000 who were vulnerable in atleast one number domain was 659 (n � 1538),or 43 per cent of the Grade 2 children. Thenumber of children who were vulnerable ineach number domain is shown in Table 1.

In Grade 1 (6-year-olds), the number ofstudents vulnerable in any domain accordingto the earlier definition is small but largeenough to require considered attention. Thenumber of students vulnerable in Multiplica-tion and Division Strategies is larger. These stu-dents were unable to solve a multiplicative taskeven by counting the items individually. Whilethis might be a characteristic of a curriculumin the first year of school that avoided thisdomain, such students still require support.The number of vulnerable students in Grade 2is larger, although it can be noted that this isperhaps a function of the definition.

To illustrate the implications for teach-ing, these data suggest that in a typically





sized Grade 1 class of 24 students, about 14children (62 per cent) will have reached theon the way points in all domains, while 10children (38 per cent) will be vulnerable inone or more number domains. About threechildren will be vulnerable in each of Count-ing, Place Value, and Addition and SubtractionStrategies, and seven children will be vulnera-ble in the Multiplication and Division Strategiesdomain. It would be usually assumed thatthere was considerable overlap in these vul-nerabilities.

In a typical Grade 2 class (7-year-olds),the number of students who have reachedthe on the way points in all domains is similar,but the higher percentages in the respectivedomains indicate that more students are vulnerable in multiple domains.

To explore the overlaps between thedomains for Grade 1 children, Figure 2provides a diagrammatic representation ofthe intersecting domains for which theGrade 1 children were vulnerable. Therewere 576 Grade 1 children who were vulner-able in at least one domain.

By way of explanation, there are 33 students (top right hand side) who are vul-nerable in Place Value but in no otherdomain, while the 36 students (nearby) arevulnerable in both Place Value and Multipli-cation and Division Strategies.

The most striking inference is that therewere only 23 children (out of a total of 1497)who were vulnerable in all four domains, andonly another 57 who were vulnerable in anythree domains. On one hand, this suggeststhat vulnerability is a function of absence ofspecific experiences. This may occur ifchildren do not have enough experiences in a

domain or are not able to take advantage ofthese so as to construct knowledge. On theother hand, it can be anticipated that specificinterventions can address the respectivevulnerabilities. We can expect that onceteachers recognise that any child is vulnerablein a given domain, then specific interventionscan be expected to redress this situation.

A similar diagram indicating the intersect-ing domains for which Grade 2 children werevulnerable is shown in Figure 3 (N � 1538).There were 659 Grade 2 children who werevulnerable in at least one domain.

While there is still a remarkable spread ofvulnerabilities in the respective domains,there are 219 (out of 1538) of the Grade 2swho were vulnerable in three or fourdomains. This is a higher proportion thanfor Grade 1 children. However, on the otherhand, most Grade 2 children who were vul-nerable were vulnerable in either one or twodomains, and these domains varied.

Overall, the diversity of domains andcombinations of domains in which Grade 2children were vulnerable is striking. Therewas a spread of vulnerability across alldomains, and there were no combinations ofdomains that were common for children whowere vulnerable.

Making inferences about vulnerabilityThe above figures indicate a surprising diver-sity of knowledge across the various numberdomains. To explore this further, we exam-ined, for students who were vulnerable inone domain, their performance in the otherdomains. Indeed, while a shortcoming of theEarly Numeracy Interview was the limitednumber of items associated with a particular

Domain Grade 1 (n � 1497) Grade 2 (n � 1538)

Counting 163 (11%) 371 (24%)

Place Value 149 (10%) 417 (27%)

Addition and Subtraction Strategies 176 (12%) 294 (19%)

Multiplication and Division Strategies 423 (28%) 280 (18%)

Table 1: Number of children in year 2000 who were vulnerable in each number domain



growth point in any one domain, its strengthwas its capacity to facilitate comparisons ofperformance across domains.

The following data refer to the studentsin Grades 1 and 2 who were vulnerable inCounting, and examines their achievedgrowth points in the other domains.

Of the 163 Grade 1 children who werevulnerable in Counting (which meant thatthey did not count a collection of about 20teddies accurately):● 64 per cent were at the level of reading,

writing, ordering and interpreting onedigit numbers;

● 48 per cent could solve the addition prob-lem (9 � 4 teddies) by counting-all theitems, and a further 14 per cent couldsolve the solve that problem by counting-on, that is without having to see the origi-nal nine teddies;

● 30 per cent could solve multiplicative tasksby counting objects one by one (actuallymaking and quantifying 4 groups of 2 ted-dies, and sharing 12 teddies among 4mats), and 14 per cent could use groupingstrategies (such as skip counting) to solvethose tasks;

● 61 per cent could accurately compare the

length of a string and a straw, and a further26 per cent could use paper clips as a unitto quantify the length of the straw.Clearly, just because a student is vulnerable

in Counting this does not mean that theyneed particular support in other domains.It is possible that some of these studentsmade a mistake with the counting of the set of 20 teddies. However, in the analysisacross six separate interviews for each child,we found extraordinary consistency inresponses from the students implying therewere few careless errors. In the projectschools, the interviews were administeredone by one by class teachers. This couldaccount for the low number of aberrantresults. Likewise the results are unlikely to bea result of fatigue, given that these itemswere asked at the start of the interview. Fur-ther the items were asked on the same dayreducing variations due to external circum-stances. A logical conclusion is that thecapacity of students in particular domains isa function of their prior experiences withwhatever are the necessary pre-requisiteknowledge and skills for that domain, and that the respective domains (e.g. Count-ing, Place Value, Addition and Subtraction

Figure 2: Diagrammatic representation of theintersecting domains for which Grade 1 children(6-year-olds) were vulnerable (n � 576)

33

15 8

23

18 16

636

37

12

48231 43

14 36

Figure 3: Diagrammatic representation of theintersecting domains for which Grade 2 children(7-year-olds) were vulnerable (n � 659)

10

44 54

93

8 20

3221

65 8367

60 58

19 24

Strategies, and Multiplication and DivisionStrategies) are less connected to each otherthan we had anticipated. The main result isthat teachers should be advised to evaluatestudents’ performance on particular tasks,and to take care when inferring fromperformance on one task to performance onanother, or from one domain to another.

There was less variability in the profiles ofthe Grade 2 children, but similar conclu-sions can be drawn nevertheless. Of the 371Grade 2 children who were vulnerable inCounting (meaning that they could not continue a counting sequence):● 26 per cent were at the level of reading,

writing, ordering and interpreting twodigit numbers, and a further 4 per centwere doing this for numbers up to 1000;

● 45 per cent could solve the additionproblem (9 � 4) by counting-on, and afurther 7 per cent could solve subtractiontasks like 8 – 3 and 12 – 9;

● 54 per cent could use grouping (evidentby skip counting) to solve multiplicativetasks, and a further 2 per cent couldabstract multiplicative thinking (mean-ing they could solve a task like 15 teddiesseated in 3 equal rows, without usingmodels);

● 51 per cent could accurately compare thelength of a string and a straw and a fur-ther 40 per cent could use paper clips asa unit to quantify the length of the straw.In other words, close to half of the stu-

dents who were vulnerable in the Countingdomain were performing at least up toexpectations in other number domains.

We had anticipated that Counting wouldbe fundamental to a capacity to respond toitems in Place Value, Addition and Subtrac-tion Strategies, and Multiplication and Divi-sion Strategies, but it seems that this is notthe case. This has implications both for cur-riculum and for teaching.

Discussion and implicationsThe findings presented in the previous sections highlight the diversity of mathemat-ical understandings amongst the group of

Grade 1 and Grade 2 children identified asvulnerable in aspects of learning schoolmathematics. This group is far from being ahomogeneous one. Indeed, there were nopatterns in the domains in which childrenwere vulnerable, or in any combinations ofdomains for which children were vulnerable.Vulnerability was widely distributed across allfour domains and combinations of domainsin both grade levels. However, one feature ofthe findings for Grade 1 children is worthnoting. Twice as many Grade 1 children werevulnerable in Multiplication and Divisionthan for any other domain, but this level ofvulnerability was not maintained for Grade 2children. It is likely that this finding is anartefact of the mathematics curriculum inVictoria that does not recommend Multipli-cation and Division experiences for childrenin the first year of school. It is thereforelikely that some teachers may not providesuch learning opportunities for their stu-dents. The fact that so many Grade 1children did reach the on the way growthpoint for Multiplication and Division indi-cates that children will benefit from oppor-tunities to enhance their construction ofknowledge in the Multiplication and Divi-sion domain throughout the first year ofschool. If this were to occur, then perhapsfewer Grade 1 children would be identifiedas vulnerable in the Multiplication andDivision domain.

A feature of the findings for Grade 2children worth noting is that a higher pro-portion of students were vulnerable in threeor four domains. It is likely that this findingis an artefact of the assessment interview andgrowth points. However, this situation alsoreflects the increasing complexity of themathematics curriculum that Grade 2children typically experience. An implica-tion of this finding may be that it is impor-tant to provide intervention programs forchildren who are vulnerable as early in theirschooling as possible, before their difficul-ties increase in complexity.

The findings have several other implica-tions for the instructional needs of children.





Most importantly, the results indicate thatchildren who are vulnerable in aspects oflearning school mathematics have diverselearning needs, and this calls for particularcustomised instructional responses fromteachers. It is likely that teachers needto make individual decisions about theinstructional approach for each child, andthat there is no ‘formula’ that will meet allchildren’s instructional needs. Further, thediversity of children’s mathematical knowl-edge in the four domains suggests thatknowledge in any one domain is not neces-sarily prerequisite for knowledge construc-tion in another domain. For example, someteachers may assume that children need tobe on the way in Counting before they areready for learning opportunities in the Addi-tion and Subtraction Strategies domain. Onthe contrary, the findings presented in Figures 1 and 2 indicate that some childrenwho are not on the way in Counting arealready on the way in Addition and Subtrac-tion, and this pattern is maintained for theother domains also. This finding has implica-tions for the way in which the school mathe-matics curriculum is introduced to children.It seems likely that children will benefit fromlearning opportunities in all four numberdomains, provided in tandem with oneanother, and that learning opportunities inone domain should not be delayed until alevel of mathematical knowledge is con-structed in another domain.

Responsive instruction and intervention forvulnerable learners of school mathematicsThe data presented earlier suggest thataddressing arithmetic difficulties is notstraight forward because of the diversity ofthe domains and combinations of domainsin which children are vulnerable. A themeemerging in the literature is the need forinstruction and learning experiences toclosely match children’s individual learningneeds (e.g. Ginsburg, 1997; Greaves, 2000;Wright et al., 2000). Rivera (1997) believesthat instruction is a critical variable in effec-tive programming for children with mathe-

matics learning difficulties, and that instruc-tion must be tailored to address individual needs, modified accordingly, andevaluated to ensure that learning is occur-ring. The need for ‘tailored’ instruction is because diversity among and within subgroups of children who have difficultywith learning mathematics is as great as forthat across the general population (Shon-koff & Phillips, 2000). Certainly the numberknowledge of the vulnerable learnersexplored in this paper was characterised byits diversity. Therefore, it is important to con-sider intervention for those with arithmeticdifficulties as a concept rather than a pro-gram, and for programs to respond to diver-sity, as target groups are heterogeneousrather than homogeneous.

Ginsburg (1997) articulated a process forusing the zone of proximal development(Vygotsky, 1978) for enhancing children’slearning, believing that this is an importantidea for assisting teachers develop suitablelearning opportunities for children. Thezone of proximal development is describedas ‘the distance between the actual develop-mental level, as determined by independentproblem solving, and the level of poten-tial development, as determined throughproblem-solving under adult guidance, or incollaboration with more capable peers’(Vygotsky, 1978, p. 86), and defines thosefunctions that have not yet matured but ‘are in the process of maturation’ (Vygotsky,1978, p. 86). Ginsburg’s process is that first, the teacher analyses children’s currentmathematical understandings and identifiestheir learning potential within the zone ofproximal development. Next, the teacherpresents a problem in an area with which thechild had difficulty, and provides hints toassist the problem solving process. Thesehints range from general metacognitivehints to those specific to the mathematicaldemands of the task. The amount of helpeach child needs is an estimate of his or herlearning efficiency within the domain. Theteacher continues presenting problems tothe student of a similar nature, providing as

much help as necessary, until the student isable to solve the problems independently.Finally, the teacher presents near, far andvery distant transfer problems and studentsare given assistance, as needed, to solvethem. Apart from enhancing learning, Ginsburg (1997) argues that this type oftechnique is required to establish the extentto which cognitive difficulties persist despiteconstant efforts to remove them.

Sullivan, Mousley and Zevenbergen(2006) also highlighted the importance ofteachers adjusting tasks to enable childrenwho experience difficulty to engage success-fully in learning opportunities. For example,they claim that student difficulties with a par-ticular task might be a result of: the number of steps; the number of variables; the modesof communicating responses; the number ofelements in recording; the degree of abstrac-tion or visualisation required; the size of thenumbers to be manipulated; the languagebeing used; or psychomotor considerations.They argued that a teacher could anticipatethese difficulties, and prepare prompts andassociated resources to: reduce the requirednumber of steps; reduce the requirednumber of variables; simplify the modes of representing results; reduce the writtenelements in recording; make the task moreconcrete; reduce the size of the numbersinvolved; simplify the language; or reducethe physical demand of any manipulatives.Thus, we are recommending that teachersbe oriented towards responding to individ-ual children’s learning needs by adjustingtasks to increase engagement, an importantaspect of assisting vulnerable learners.Wright et al. (2000) also advocated thatteachers should routinely make adjustmentsto planned activities on the basis ofchildren’s responses. Further, they arguethat tasks should be genuine problems forchildren, and that children should be chal-lenged to bring about reorganisations intheir thinking (Wright et al., 2000).

Each of the strategies outlined above isimportant to consider when teachingchildren with arithmetic difficulties. However,

fundamental to meeting children’s individuallearning needs is the notion of a framework of growth points or stages of development,such as that developed for the ENRP. Growth points help teachers to identifychildren’s zone of proximal development inmathematics so as to create appropriate learn-ing opportunities, and in order to adjust activ-ities to increase engagement and removefeatures that are creating barriers to learning.Thus, reference to a framework of growthpoints helps to ensure that instruction for vulnerable children is closely aligned tochildren’s initial and ongoing assessment, andis at the ‘cutting edge’ of each child’s knowl-edge (Wright et al., 2000).

In summary, we believe the followinginstructional practices are important forenhancing mathematics learning for childrenwho have arithmetic difficulties:1. targeting instruction within a child’s zone

of proximal development in eachdomain, based on current assessmentinformation about the child’s mathemati-cal understandings and the probablecourse of the child’s learning;

2. making adjustments to planned activitieson the basis of children’s responses;

3. presenting rich, challenging problemsthat promote ‘hard thinking’ within achild’s zone of proximal developmentand in an area with which the child haddifficulty; providing hints to assist theproblem solving process, ranging fromgeneral metacognitive hints to those spe-cific to the mathematical demands of thetask; continually presenting problems of asimilar nature, providing as much help asnecessary, until the student is able to solvethe problems independently.These approaches are important to con-

sider when providing intervention programsfor children with arithmetic difficulties, andwhen examining the effectiveness of anintervention program.

ConclusionThe findings presented in this paper suggestthat there is no single ‘formula’ for describing



children who have difficulty learning arith-metic or for describing the instructional needsof this diverse group of students. Further, it isnot possible to assume that because a child isvulnerable in one aspect of number learning,then he or she will be vulnerable in another.This finding may surprise some teachers andhighlights why assisting children with arith-metic difficulties is not straight forward. Meet-ing the diverse learning needs of children is achallenge, and requires teachers to be knowl-edgeable about how to identify each child’slearning needs and customise instructionaccordingly. This calls for rich assessment toolscapable of mapping the extent of children’sknowledge in a range of domains, and an asso-ciated framework of growth points capable ofguiding teachers curriculum and instructionaldecision-making.

It follows that intervention programs for children who are vulnerable need to beflexible in structure in order to meet thediverse learning needs of each participatingchild. Intervention teachers need to provideinstruction and feedback that is customisedfor the particular learning needs of eachchild, and based on knowledge of children’scurrent mathematical knowledge. Further,teachers need to ensure that children takeadvantage of learning experiences by draw-ing children’s attention to the salient featuresto facilitate the construction of knowledgeand understanding. Another issue for Inter-vention programs is that the diversity we dis-

covered in children’s number knowledge andabilities in a range of number domains suggests that programs need to focus on allnumber domains in tandem. It is not appro-priate to wait until children reach a certainlevel of knowledge in one domain beforeexperiences in another domain are intro-duced. Children’s construction of numberknowledge in a specific domain is notdependent on prerequisite knowledge inanother domain, but is dependent on beingable to take advantage of a range of experi-ences in a given domain.

Assisting children with arithmetic difficul-ties is complex, but teachers who areequipped with the tools necessary forresponding to the diverse needs of individualswho have not previously thrived when learn-ing arithmetic, are able to provide childrenwith the type of learning opportunities andexperiences that will enable them to thriveand extend their mathematical understandingfurther.

Address for CorrespondenceDr Ann Gervasoni, Australian Catholic University – Ballarat Campus, 1200 Mair St,Ballarat, VIC 3350, Australia E-mail: [email protected] Peter Sullivan, Monash University,Clayton VIC 3800, Australia E-mail: [email protected]



ReferencesBoard of Studies (2000). Curriculum and Standards

Framework II: Mathematics. Carlton, Victoria: Author.Bobis, J. & Gould, P. (1999). The mathematical

achievement of children in the Count Me In Tooprogram. In J.M. Truran & K.M. Truran (Eds.),Making the difference (Proceedings of the 22nd AnnualConference of the Mathematics Research Group of Australasia (pp. 84–90). Adelaide: MERGA.

Boulton-Lewis, G. (1996). Representations of placevalue knowledge and implications for teachingaddition and subtraction. In J. Mulligan & M. Mitchelmore (Eds.), Children’s number learn-ing: A research monograph of MERGA/AAMT (pp.

75–88). Adelaide: Australian Association of Math-ematics Teachers.

Clarke, D. (2001). Understanding, assessing and devel-oping young children’s mathematical thinking:Research as powerful tool for professional growth.In J. Bobis B. Perry & M. Mitchelmore (Eds.),Numeracy and beyond: Proceedings of the 24th AnnualConference of the Mathematics Education ResearchGroup of Australasia (Vol. 1, pp. 9–26). Sydney:MERGA.

Clarke, D., Cheeseman, J., Gervasoni, A., Gronn, D.,Horne, M., McDonough, A., Montgomery, P.,Roche, A., Sullivan, P., Clarke, B. & Rowley, G.

(2002). Early Numeracy Research Project Final Report.Melbourne: Australian Catholic University.

Cobb, P. & McClain, K. (1999). Supporting teachers’learning in social and institutional contexts. Paperpresented at the International Conference onMathematics Teacher Education, Taipei, Taiwan.

Department of Education Employment and Training(2001). Early numeracy interview booklet. Melbourne:Department of Education, Employment andTraining.

Gervasoni, A. (2000). Using growth-points profiles toidentify Grade 1 students who are at risk of notlearning school mathematics successfully. In J. Bana & A. Chapman (Eds.), Mathematics educationbeyond 2000. Proceedings of the 23rd Annual Conferenceof the Mathematics Education Research Group of Aus-tralasia. (pp. 275–282). Fremantle, WA: MERGA.

Gervasoni, A. (2004). Exploring an intervention strategyfor six ans seven year old children who are vulnerablein learning school mathematics. Unpublished PhDthesis, La Trobe University, Bundoora.

Ginsburg, H.P. (1997). Mathematical learning disabilities: A view from developmental psychol-ogy. Journal of Learning Disabilities, 30(1), 20–33.

Greaves, D. (2000). Private provider services for stu-dents with learning difficulties. In W. Louden,L.K.S. Chan, J. Elkins, D. Greaves, H. House, M. Milton, S. Nichols, J. Rivalland, M. Rohl & C. Van Kraayenoord (Eds.), Mapping the territory:Primary Students with Learning Difficulties: Literacyand Numeracy (Vol. 1, pp. 135–156). Canberra:Commonwealth of Australia.

Fuson, K. (1992). Research on whole number addi-tion and subtraction. In D.A. Grouws (Ed.),Handbook of Research on Mathematics Teaching andLearning (pp. 243–275). New York: Macmillan.

Hart, B., Brinkman, S. & Blackmore, S. (2003). Howwell are we raising our children in the North Metropol-itan Region? Perth: Population Health Program,North Metropolitan Heath Service.

Mulligan, J. & Mitchelmore, M. (1996). Children’srepresentations of multiplication and divisionword problems. In J. Mulligan & M. Mitchelmore(Eds.), Children’s Number Learning: A ResearchMonograph of MERGA/AAMT (pp. 163–184). Adelaide: Australian Association of MathematicsTeachers.

Owens, K. & Gould, P. (1999). Framework for elementary

school space mathematics (discussion paper).

Rivera, D.P. (1997). Mathematics education and stu-

dents with learning disabilities: Introduction to

the special series. Journal of Learning Disabilities,

30(1), 2–19.

Shonkoff, J. & Phillips, D. (2000). From neurons to

neighborhoods: The science of early childhood develop-

ment. Washington: National Academy Press.

Steffe, L., Cobb, P., & von Glasersfeld, E. (1988).

Construction of arithmetical meanings and strategies.

New York: Springer-Verlag.

Steffe, L., von Glasersfeld, E., Richards, J. & Cobb, P.

(1983). Children’s counting types: Philosophy, theory,

and application. New York: Praeger.

Sullivan, P., Mousley, J. & Zevenbergen, R. (2006).

Developing Guidelines for Teachers Helping

Students Experiencing Difficulty in Learning Math-

ematics. In P. Grootenboer, R. Zevenbergen & M.

Chinnappan, (Eds.), Proceedings of the29th Annual

Conference of the Mathematics Education Research Group

of Australasia (pp. 496–503), Canberra, July.

Vygotsky, L.S. (1978). Mind in society: The development

of higher mental processes. Cambridge, MA: Harvard

University Press.

Woodward, J. & Baxter, J. (1997). The effects of an

innovative approach to mathematics on academ-

ically low-achieving students in mainstreamed

settings. Exceptional Children, 63, 373–388.

Wright, R. (1998). An overview of a research-based

framework for assessing and teaching early

number learning. In C. Kanes, M. Goos & E.

Warren (Eds.), Teaching Mathematics in New

Times: Proceedings of the 21st Annual Conference of

the Mathematics Education Research Group of

Australasia (pp. 701–708). Brisbane, Queensland:

Mathematics Education Research Group of

Australasia.

Wright, R., Martland, J., & Stafford, A. (2000). Early

Numeracy: Assessment for teaching and intervention.

London: Paul Chapman Publishing



Growth Points for the number domainsNotes

● Growth points are not necessarily hierarchical,but involve increasingly complex reasoningand understanding.

● It must be emphasised that conclusions drawnin relation to placing students at levels withinthis framework are based on a 30-minute(approx.) interview only. Ongoing assessmentby the teacher during class will provide impor-tant further information for this purpose.

● Student understanding may be reported as a“0”. This should not be taken as an indicationof “no knowledge” or “no understanding”, butrather as an indication of a lack of evidenceof “1”.

CountingNot apparent. Not yet able to say sequenceof number names to 20.Rote counting. Rote counts the numbersequence to at least 20, but is not yet able toreliably count a collection of that size.Counting collections. Confidently counts acollection of around 20 objects.Counting by 1s (forward/backward, includingvariable starting points; before/after). A countforward and backwards from various startingpoints between 1 and 100; knows numbersbefore and after a given number.Counting from 0 by 2s, 5s, and 10s. Can countfrom 0 by 2s, 5s, and 10s to a given target.Counting from x (where x �0) by 2s, 5s, and10s. Given non-zero starting points, countsby 2s, 5s,10s to given target.Extending and applying counting skills. Countsfrom non-zero starting points by any 1-digitnumber, and applies counting skills in practi-cal tasks.

Place ValueNot apparent. Not yet able to read, write,interpret and order single digit numbers.Reading, writing, interpreting, and orderingsingle digit numbers. Can read, write, inter-pret and order single digit numbers.

Reading, writing, interpreting, and ordering two-digit numbers. Can read, write, interpretand order two-digit numbers.Reading, writing, interpreting, and ordering three-digit numbers. Can read, write, interpretand order 3-digit numbers.Reading, writing, interpreting, and ordering num-bers beyond 1000. Can read, write, interpret& order numbers beyond 1000.Extending and applying place value knowledge.Can extend and apply knowledge of placevalue in solving problems.

Strategies for addition and subtractionNot apparent. Not yet able to combine &count 2 collections of objects.Count all (two collections). Counts all to findthe total of two collections.Count on. Counts on from one number tofind the total of two collections.Count back/count down to/count up from. Insubtraction contexts, chooses suitably fromcount-back, count-down-to & count-up-fromstrategies.Basic strategies (doubles, commutativity, adding 10,tens facts, other known facts). Given an additionor subtraction problem, strategies such as dou-bles, commutativity, adding 10, tens facts, andother known facts are evident.Derived strategies (near doubles, adding 9, buildto next ten, fact families, intuitive strategies).Given an addition or subtraction problem,strategies such as near doubles, adding 9,build to next ten, fact families and intuitivestrategies are evident.Extending and applying addition and subtractionusing basic, derived and intuitive strategies.Given a range of tasks (including multi-digitnumbers), can solve them mentally, usingthe appropriate strategies and a clear under-standing of key concepts.Strategies for Multiplication and DivisionNot apparent. Not yet able to create andcount the total of several small groups.Counting group items as ones. To find thetotal in a multiple group situation, refers toindividual items only.



Appendix A: Early numeracy research project assessment framework

Modelling multiplication and division (all objectsperceived). Models all objects to solve multi-plicative and sharing situations.Abstracting multiplication and division. Solvesmultiplication and division problemswhere objects are not all modelled orperceived.Basic, derived and intuitive strategies for multipli-cation. Can solve a range of multiplicationproblems using strategies such as commuta-tivity, skip counting and building up fromknown facts.

Basic, derived and intuitive strategies for division.Can solve a range of division problems usingstrategies such as fact families and buildingup from known facts.Extending and applying multiplication anddivision. Can solve a range of multiplica-tion and division problems (including multi-digit numbers) in practical contexts.




MULTI-DIGIT ADDITION and sub-traction is an important aspect ofnumber sense and mental compu-

tation (Anghileri, 2000; Thompson & Smith,1999). As well, this topic is importantbecause it provides a basis for moreadvanced arithmetic. This article drawsfrom a current three-year project focusingon the development of pedagogical tools tosupport intervention in the number learn-ing of low-attaining third- and fourth-graders (8- to 10-year-olds). These toolsinclude schedules of diagnostic assessmenttasks and instructional procedures. Thispaper focuses on some of the assessmenttasks which enable assessment of knowledgeof the sequential structure of numbers.Developing significant knowledge of thesequential structure of numbers provides animportant basis for multi-digit addition andsubtraction (Beishuizen & Anghileri, 1998).The paper will:

1. elaborate the term ‘sequential structureof numbers’;

2. review literature relevant to the sequen-tial structure of numbers;

3. set out relevant diagnostic assessmenttasks; and

4. describe the range of low-attainingpupils’ responses to those tasks.

Sequential structure of numbersThis paper discusses low-attaining pupils’knowledge and use of what we call the‘sequential structure of numbers’. By‘sequential structure of numbers’ we refer tothe decade-based structures in the linearsequence 1, 2, 3, . . . 99, 100, 101. . . . Specifically,this number sequence consists of a sequenceof decades, which can be further organisedin a sequence of hundreds. The decadenumbers (10, 20, 30, . . .) are referencepoints in the sequence, at even intervals of

Assessing pupil knowledge of thesequential structure of numbersDavid Ellemor-Collins & Robert Wright

AbstractResearch on children’s mental strategies for multidigit addition and subtraction identifies two categories ofstrategy. Collections-based strategies involve partitioning numbers into tens and ones, and can be modeledwith base-ten materials. Sequence-based strategies involve keeping one number whole, and using the sequen-tial structure of numbers. They can be modelled as jumps on an empty number line. Studies have foundsequence-based strategies to be more successful, and to correlate with more robust arithmetic knowledge, par-ticularly among low-attaining pupils. Studies also suggest that sequence-based strategies and sequentialstructure are not explicitly developed in many primary mathematics classrooms. This report draws on resultsfrom a three-year project which has the goal of developing pedagogical tools for intervention in the numberlearning of low-attaining third- and fourth-graders (8- to 10-year-olds). These tools include assessmenttasks to inform intervention. The report focuses on four groups of assessment tasks that collectively enabledetailed documenting of pupils’ knowledge of the sequential structure of numbers. Tasks and pupils’responses are described in detail. Some examples follow. When asked to count back from 52, pupils said,‘52, 51, 40, 49, 48, and so on’. When asked to count back by tens from 336, pupils had difficulty contin-uing after 326. Thus teen numbers in the hundreds (316) presented particular difficulties. Pupils had dif-ficulty saying the number that is ten less than 306. Pupils had difficulty with locating the numbers 50,25, 62 and 98 on a number line on which zero and 100 were marked. The report provides insight intoassessing knowledge of sequential structure and argues that this is important basic number knowledge.


Assessing pupil knowledge of the sequential structure of numbers

ten. Each decade follows the same patternas, for example, ‘20, 21, 22, . . . 28, 29, 30’. Bythe neat symmetry in this sequence, a pair ofnumbers such as 18 and 28, or 71 and 81, isalways ten steps apart. Referring to thesequential structure of numbers, 57 can beregarded as one after 56, seven after 50,three before 60, or 10 after 47.

We can also describe ‘collections-based’structures in multidigit numbers. Theseinvolve thinking of numbers in terms ofcollections of ones, tens, hundreds and soon. For example, 57 can be constructed asfifty and seven, or as 7 ones and 5 tens.

Literature reviewEmphasis on mental computationIn the last 15 years, research and curriculumreforms in a range of countries highlight arenewed emphasis on mental computationwith multidigit numbers (Beishuizen &Anghileri, 1998; Thompson, 1997). An earlyemphasis on mental strategies, rather thanformal written algorithms, may bettersupport number sense and conceptualunderstanding of multidigit numbers, andsupport development of important connec-tions to related knowledge (Askew, Brown,Rhodes, Wiliam & Johnson, 1997; Hiebert &Wearne, 1996; McIntosh, Reys & Reys, 1992;Sowder, 1992; Yackel, 2001). Mental compu-tation can also stimulate the development ofnumerical reasoning and flexible, efficientcomputation (Anghileri, 2001; Treffers,1991).

Mental strategies: ‘sequence-based’ and ‘collections-based’In response to the emphasis on mental com-putation, research projects in several coun-tries focused on pupils’ informal mentalstrategies for multi-digit addition and subtrac-tion (Beishuizen, Van Putten & Van Mulken,1997; Cobb et al., 1997; Cooper, Heirdsfield, &Irons, 1995; Foxman & Beishuizen, 1999;Fuson et al., 1997; Ruthven, 1998; Thompson& Smith, 1999). Several studies described twomain categories of strategies – sequence-basedand collections-based (e.g. Beishuizen &

Anghileri, 1998; Cobb et al., 1997; Foxman &Beishuizen, 2002; Thompson & Smith, 1999).

The standard example of a sequence-based strategy is the ‘jump’ strategy. Jumpinvolves keeping the first number whole andadding (or subtracting) the second via aseries of jumps. For example, a pupil mightadd 57 and 26 using jump by reasoning asfollows: ‘57 and ten is 67, and ten more is 77;three more is 78, 79, 80; and three moremake 83.’ Researchers note that suchsequence-based strategies depend on knowl-edge of sequential structures to jump by ten,and to make steps and hops in the numbersequence (Fuson et al., 1997; Treffers & Buys,2001; Yackel, 2001). Classroom use of set-tings such as a number line that highlightsthe decades or a bead string with thedecades demarked by colour (1–10 is blue,11–20 is red, 21–30 is blue etc.) are linked topupil use of sequential structure andsequence-based strategies (Klein, Beishuizen& Treffers, 1998).

The standard example of a collections-based strategy is the ‘split’ strategy. Splitinvolves partitioning both numbers into tensand ones, adding (or subtracting) separatelywith the tens and the ones, and finally recom-bining the tens and ones subtotals. A pupilmight add 57 and 26 using split by reasoningas follows: ‘50 and 20 are 70, 7 and 6 are 13, 70and 13 make 83’. Collections-based strategiesuse collections-based structures (Fuson et al.,1997; Treffers & Buys, 2001; Yackel, 2001).Classroom settings such as base-ten blocks arelinked to the use of collections-based structuresand strategies (Beishuizen, 1993).

Fuson et al. (1997) suggest that anadvanced understanding of multi-digit addi-tion and subtraction requires an integrationof sequence-based and collections-basedstrategies. For example, an advanced pupilasked to add 5 doughnuts to 58 doughnutsmight use a sequence-based strategy, jump-ing through 60 to 63, which is more efficientthan split in this case; but when then askedhow many boxes of ten she could fill, use herknowledge of collections-based structure torecognize 6 tens in 63.


David Ellemor-Collins & Robert Wright

Infrequency of sequence-based strategies amonglow-attaining pupilsResearchers have found that low-attainingpupils tend to use split strategies, indicatingthe development of knowledge of collections-based structure (Beishuizen, 1993; Foxman& Beishuizen, 2002). Research also suggeststhat many low-attaining pupils do notdevelop the strategy of jumping by tens andthus may not develop sequence-based struc-tures (Beishuizen, 1993; Beishuizen et al.,1997; Menne, 2001). Thus it is unlikely thatthese pupils can advance to integratedsequence-collections-based strategies which,we would argue, is important for numbersense and mental computation.

Advantages of sequence-based strategiesJump strategies can develop as abbreviationsof pupils’ informal counting strategies(Beishuizen & Anghileri, 1998; Olive, 2001).Following the view that pupils’ knowledgeshould build on their informal strategies(Anghileri, 2001; Resnick, 1989), some resear-chers recommend teaching jump strategies(Klein et al., 1998). A common difficulty withmulti-digit addition and subtraction arises forpupils when they separate the digits in thetens place from the digits in the ones placeand do not adequately regroup. For example,57 � 26 is found to be ‘73’ or even ‘713’.These difficulties arise in the case of splitstrategies but do not arise in the case of jumpstrategies (Beishuizen & Anghileri, 1998;Cobb, 1991; Fuson et al., 1997). Beishuizenand colleagues found that pupils made signif-icantly more errors when using split strategiesthan when using jump strategies. Importantly,even within a group of pupils identified aslow-attaining, jump strategies were muchmore successful (Klein et al., 1998). Theseresults were confirmed by Foxman (2002).Studies comparing the use of split and jumpstrategies found that split led to more difficultydeveloping independence from concretematerials (Beishuizen, 1993), more proce-dural and conceptual confusion (Klein et al.,1998) and slower response times, suggesting aheavier load on working memory (Wolters,

Beishuizen, Broers & Knoppert, 1990). Sub-traction tasks are a source of particular diffi-culties in multidigit arithmetic, and thepotential confusions of subtraction using asplit strategy are well documented. Confusedresponses using split suggest the collections-based structure offers a problematic represen-tation of subtraction tasks (Fuson et al., 1997).Success with split requires strong numbersense and subtle insight into the procedureitself, whereas success with jump mainlyrequires knowing how to jump ten from anynumber (Beishuizen, 1993).

Developing flexibility with strategiesAn important goal in improving multidigitnumber sense is flexibility with strategies,including recognising efficient short-cuts andmaking adaptations for unfamiliar problems(McIntosh et al., 1992). Studies indicate thatpupils more readily adapt the jump strategyto make efficient computation choices.According to Beishuizen et al., this is due to‘the underlying mental representation of thenumber row up to 100’ (1997). That is, usingthe sequential structure of number readilysupports strategic insight into computationtasks.

In summary, pupils with arithmetic diffi-culties tend not to develop sequential struc-ture and sequence-based strategies such asjump. It is likely that this denies them anintegrated approach to multi-digit additionand subtraction, and access to the preferredstrategies of arithmetically successful pupils.Further, development of sequential structureand strategies might resolve a number oftypical multi-digit difficulties prevalent withcollections-based strategies.

Assessment task groups and responsesAs part of the project (referred to earlier inthis article), 204 low-attaining pupils wereinterviewed twice during the school year toassess their number knowledge. The pupilswere in third and fourth grades (8- to 10-year-olds) from a broad demographic rangeacross the state of Victoria, and were selectedfor the study based on low results in screen-

ing tests. A method involving one to one,dynamic interview was used, in which thepupil is posed number tasks, and the inter-viewer pays close attention to the pupil’sthinking process (Wright, Martland &Stafford, 2006). Interview assessments wererecorded on videotape for later analysis.

We use the term ‘task group’ to refer to agroup of closely related tasks used to investi-gate pupils’ knowledge of a specific topic. Inthis paper, we discuss four task groups wefound particularly valuable in assessing pupilknowledge of sequential structure:1. Number word sequences by ones.2. Number word sequences by tens.3. Incrementing and decrementing by ten.4. Locating numbers.

For each, we describe the range of low-attaining pupils’ responses and difficulties,evident from analysis of the videotaped inter-views.

Task group 1: Number word sequencesby onesFocusShort sequences of number words, backwardsand forwards; number word before or after;and bridging decades and hundreds.

Examples‘Count from 97. I’ll tell you when to stop.’

Stop at 113.‘Count backwards from 103.’ Stop at 95.‘Say the number that comes just after 109’.‘Say the number that comes just before 100’.Similarly for bridging 40, 210, 300, 990,

1000, 1100 forwards and backwards.

Low-attaining pupils’ difficultiesTable 1 sets out examples of pupils’ errorswith number word sequences. Errors bridging50 backwards indicate that the pupils have notfully constructed the number word sequence.Rather, they are aware of separated chainssuch as 41–49 and 51–59, and link thesechains incorrectly when going backwards(Skwarchuk & Anglin, 2002). In the range100 to 1000, number word sequence errorswere common (Table 1). In many of the cases

where pupils responded correctly to thesetasks, their responses indicated a lack of certi-tude, particularly when bridging decade orhundred numbers. All of our low-attainingpupils made errors with number wordsequences bridging 1000. Younger children’sdifficulties in establishing the number wordsequence are well documented (e.g. Fuson,Richards & Briars, 1982; Wright, 1994). Wehave found the persistent errors and uncer-tainties of these older children striking. Ourconclusion is that the assessment tasksdescribed above are indicative of areas ofknowledge that should be explicitly taught, atleast in the case of low-attaining pupils.

Task group 2: Number wordsequences by tensFocusNumber word sequences by tens, forwardsand backwards, on and off the decade.

Examples‘Count by tens.’ Stop at 120.‘Count by tens from 24. I’ll tell you when to

stop.’ Stop at 104.

Bridging 50 or 40 backwards‘52, 51, 40, 49, 48 . . .’‘52, 51, ^ 49, 48 . . .’‘52, 51, 50, 89, 88 . . .’‘42, 41, 40, 49, 48 . . .’

Bridging 100‘98, 99, 100, ten hundred’‘102, 101, ^ 99, 98 . . .’

Bridging 110 forwards‘108, 109, 1000, 1001 . . .’‘108, 109, 200, 201, 202 . . .’Number word after 109: ‘1000’

Bridging 200‘198, 199. That’s all I know.’‘198, 199, 1000, 1001 . . .’‘198, 199, ^ 201, 202 . . .’‘202, 201, ^ 199, 198 . . .’

Table 1: Pupil’s errors in oral number wordsequences. Note: Errors are marked in bold,omissions are marked with ‘�’.



‘Count by tens back from 52.’ Stop at 2.‘Count by tens from 167.’ Stop at 237.

Low-attaining pupils’ responses and difficultiesThe patterns of number word sequences bytens are inherent in the sequential structureof the base-ten number system. Jump strate-gies are derived from these patterns.Researchers have suggested that pupils canhave difficulty producing number wordsequences by tens off the decade, and hencebe unable to develop a jump strategy (e.g.Beishuizen, 1993).

Skip counting by tens on the decade. All pupilsinterviewed could produce the sequence ofdecade numbers ‘10, 20, 30, . . .’, althoughsome had difficulty in continuing beyond 90.

Cannot count by tens from 24. Some pupilscould not count by tens from 24. Responsesincluded: (a) ‘24, 25, 20’ and again ‘24, 25,20?’; and (b) ‘24, 30, 34, 40’. One pupil couldnot count by tens from 24, but could count bytens from 25 – ‘25, 35, 45…’. It seems thatthese pupils’ inability to make sense of the taskarises from an unfamiliarity with sequences oftens off the decade compared with sequencesof fives and of tens on the decade. Indeed,some of these pupils could count by tens onthe decade up to 1000.

Counting by ones. When asked to count bytens from 24, some pupils counted each tenby ones. This could be laborious and some-times unsuccessful. Sometimes, a pupilwould seemed to become aware of the patternthey were producing, and their sequencewould become more fluent perhaps curtail-ing the counting by ones.

Difficulties with teen numbers. Many pupilscould not coordinate the teen numbers witha larger number sequence. When countingby tens back from 52 (52, 42, 32, . . .), somepupils had difficulty after 22. Responsesincluded: (a) ‘… 22, 2’; (b) ‘…22, 14, 4’; and(c) ‘… 22, 10, 1’. Some were successful buttheir response involved counting back by

ones after 22. Many pupils had difficultieswith teen numbers in the hundreds, forexample saying ‘336, 326, 316, 314, 306,304.’ A few pupils had difficulties with teenswhen counting back by tens on the decade:(a) ‘70, 60, . . . 30, 20, 15, 10.’; (b) ‘70, 60,. . . 30, 12, 10.’; and (c) ‘70, 60, 50, 40, 12, no,20, 0? or 10?’. Irregularities in the names ofteens mask their ten-structure (Fuson et al.,1997) and this results in significant difficul-ties in saying sequences by tens.

Difficulties in the range 100 to 1000. Pupilswho could skip count by ten off the decade inthe range 1 to 100 experienced difficultieswith bridging one hundred or higher hun-dred numbers. One pupil said ‘177, 187, 197,one hundred and-’ then ‘297,’ then ‘207,217. . .’ . When skip counting back by ten onthe decade some pupils produced a sequencesuch as ‘430, 420, 410, 300, 390, 380. . .’. Thisis analogous to a common error amongyounger pupils at decade numbers whencounting backward by ones, for example,when counting back from 45, the pupil says,‘45, 44, 43, 42, 41, 30, 39, 38 . . .’ . Pupils whoerred when skip counting by saying 300 as thenumber ten less than 410, did not make acorresponding error when skip countingback by ten off the decade but there were dif-ficulties at hundred numbers such as ‘336,326, 316, 297’ corrected to ‘296’, and thusomitting 306. Another difficulty was discrimi-nating the new hundreds number from thetens number, for example, an attempt to skipcount by ten from 167 was ‘267, 367, 467’.When counting back by tens from 336, oneresponse was: ‘326, 316, 312’ (pause), ‘326,226, 206, two hundred and zero, 196, 186,176’. This sequence illustrates a persistentdifficulty with the teen numbers within thehundreds, and confusion when bridging 200.It is clear that knowledge of sequences of tensbeyond 100 is a significant extension ofknowledge of sequences of tens up to 100.

The responses described above areindicative of weaknesses in pupils’ knowl-edge of the sequential structure of numbers.





We believe it is important to address theseweaknesses through intervention.

Task group 3: Incrementing anddecrementing by 10FocusTen more and ten less, on and off the decade,and bridging decades and hundreds.

ExamplesShow ‘20’ on a card. Ask ‘Which number is

ten more than this?’Similarly for 79, 356, 195, 999.Show ‘30’ on a card. Ask ‘Which number is

ten less than this?’Similarly for 79, 356, 306, 1005.

Relationship to knowledge of the tens structure ofthe number sequenceWe were interested in whether pupils couldsolve the tasks in Task Group 3 withoutcounting by ones or trying to use an algo-rithm for addition or subtraction. We wouldregard such pupils as having knowledge ofthe tens structure of the number sequence.

The tasks of (a) incrementing and decre-menting by ten and (b) skip counting by tenseemed to be linked in the sense that pupilsshowed similar levels of advancement intheir responses to these task groups. Never-theless, the tasks involving incrementing ordecrementing by ten are distinct from tasksof skip counting by ten. There was incongru-ence in pupils’ responses on these two taskgroups. For example, one pupil skip countedby tens from 167 successfully: ‘177, 187, 197,297’ self-corrected to ‘207, 217. . .’, but couldnot solve ten more than 195: ‘one hundredand . . .’ changed to ‘220?’ changed to ‘225’.

Pupils might construe the incrementingtask as an addition task rather than a task basedon a number sequence with increments of ten.Thus they are unable to regard ten more asone increment of a sequence with incrementsof ten. Even if they can regard ten more as oneincrement in a sequence with increments often, they are apparently unable to incrementthe sequence from a standing start, when theincrement involves bridging a hundred

number. Alternatively, we could say that a pupilwho can increment by ten to bridge a hundrednumber has constructed a sequence-basedstrategy for the operation of adding ten.

Progressions in incrementing and decrementingby tenPupils’ success with tasks involving incre-menting or decrementing by ten tended toprogress as follows:● 2-digit off-the-decade: ‘ten more/less

than 79’,● 3-digit off-the-decade: ‘ten more/less

than 356’,● forward across a hundred number: ‘ten

more than 195’,● backward across a hundred number: ‘ten

less than 306’,● forward across 1000: ‘ten more than 999’,● backward across 1000: ‘ten less than

1005’.Thus a pupil who was successful at the

third progression (starting from the upper-most progression) was likely to succeed withthe tasks at the first two progressions and notsucceed with the tasks from the fourth pro-gression onward. Some pupils could notincrement by ten off the decade at all. Diffi-culties with the teen number sequence werealso evident in these tasks, for example, apupil could find ten more than 356, but notfind ten more than 306. Tasks involving1000, that is ten more than 999 and ten lessthan 1005, were especially difficult for virtu-ally all of the pupils.

‘Which number is ten less than 306?’The task of finding ten less than 306 wasparticularly difficult for many pupils. Pupils’responses included:● ‘I don’t know.’● Incorrect counting: ‘210?’, ‘299’, ‘300’.● Counting back by ones with an incorrect

sequence, and using fingers to keep trackof the ten counts: ‘305, 304, 303, 302,301, 330, 329, 328, 327, 326!’.

● Counting back by ones and answering ‘295’.● Counting back by ones successfully.● Jumping back ten successfully.



To solve this task by jumping back tenrequires knowing the decade before 301 isthe 290s. Many of these pupils did not knowthis or could not apply this knowledge tosolve the task, that is, they lacked knowledgeof the sequential structure of numbers.Further, many of the pupils did not countback by ones to solve this task. Apparentlythese pupils could not construct a represen-tation of this problem that was embedded inthe number sequence. Those pupils whoattempted to count back on this task wereconsistently more successful on other tasksinvolving incrementing by ten than pupilswho did not attempt to count back.

Task group 4: Locating numberson a number lineFocusLocating numbers on a linear representationof the number sequence from 0 to 100.

ExamplePupil is given a pen, and a line on paper withonly the endpoints 0 and 100 labeled (seeFigure 1). Pupil is asked to

‘Mark where 50 is on the line,’ and then‘Label that as 50.’ Similarly mark and label25; 98; 62.

Low-attaining pupils’ responses and difficultiesA locating number task requires knowledgeof the number sequence. To locate the num-bers efficiently requires using ideas such as:● 50 at half-way;● 25 at half-way to 50;● 98 at two steps before 100; and● 62 just after 60, which is ten after 50.

To do the task well also requires knowl-edge of linear measure and proportionwhich may be somewhat distinct from numbersequence knowledge. Pupils’ difficulties withthis task indicate a lack of knowledge of

sequential structure and also a lack ofknowledge of linear measure. We have foundpupils’ responses to be interesting, andrevealing of their number sequence knowl-edge. Four examples are discussed below.

Renee’s response, shown in Figure 2, istypical. It would seem there is some sense ofglobal location: 50 is placed at half-way; 25placed perhaps from a sense of decades, orfrom half of 50; 98 is probably located to benear 100, but with a weak sense of the meas-ure of the 2-step gap.

A weaker response can be very revealing.Helen (see Figure 2) does find 50 as ‘halfway’. But to locate 25, she marks all the onesfrom 0 to 25. She does not count in tens,though she does emphasise her ‘20’ point.Her 25 ends up almost at 50, and it is notclear whether she regards this as problem-atic. She locates 98 two steps back from 100,but the steps are too big. Helen is using anaspect of the number structure, but is notchecking against another aspect, that is, 98 as‘8 more than 90’. She does not seem to havea global or embedded sense of the structuresof the sequence. To locate 62, Helen againcounts by ones. She does count on from 50,and she emphasises the ‘60’ point along theway, showing some appreciation of how thenumber structure can support her solution.But she does not curtail counting by ones.

Nate finds each of his numbers by count-ing and marking fives (see Figure 2). Helocates 98 after counting to 95, and locates62 just past 60. He does not count by ones,but doesn’t regard the decades as referencepoints. Rather, he counts by fives. Perhapsmore striking is that, in contrast to Helen,Nate finds both 62 and 98 by countingby fives from five. He marks 98 just short of100 but does not use 100 as the reference tolocate 98. He does seem concerned to lineup the successive counts to 50 at the same 50mark, but he does not simply count-on from50—he begins afresh from five each time.His approach is analogous to counting-allrather than counting-on to solve an additiontask. Both Helen and Nate frequently countedby ones on their fingers to solve addition

0 100

Figure 1: Blank number line for ‘locating numbers’ task

tasks, and neither could skip count by tensoff the decade. Their responses on this TaskGroup indicate a lack of knowledge of struc-tures in the number sequence.

Both of these pupils received intensiveindividual intervention instruction afterthese assessments. The instruction did notfocus on tasks of locating numbers butincluded a significant focus on counting byones and tens, and on recording on an emptynumber line, additions using a jump strategy.When assessed with the locating number taskafter the intervention period, Helen was nolonger marking ones, and she located 25appropriately. Nate’s post-assessment responseis shown in Figure 2. His pen moves in jumpsof ten and one. This learning that Natedemonstrated in his post-assessment can beattributed to the instruction that focused onrecording jump strategy additions on anumber line. He shows a clear use of decadestructure, and he is no longer working fromone. Interestingly, his knowledge of linearmeasure and proportion has also advanced.

Thus, without explicit instruction on locatingnumbers, his broad development of numbersequence knowledge has made significantdifferences to his responses on the task groupof locating numbers.

This Task Group is very useful because itcan reveal knowledge of number sequencestructure, can differentiate levels of under-standing, and can enable learning over timeto be documented.

ConclusionWe claim that the sequential structure ofnumbers is important basic number knowl-edge. We advocate that pupils’ number learn-ing should include a focus on number wordsequences up to 1000, skip counting andincrementing by tens off the decade, andlocating numbers in the range 1 to 100. It isstriking that many third and fourth gradepupils (aged 8 to 10 years) are not successfulon the assessment tasks described in thisreport. In our view, a focus on sequentialstructure exemplifies an informed approach

Figure 2: Locating numbers on a number line



Renee

Helen, counting by ones

Nate, counting by fives from zero

Nate on post-assessment



ReferencesAnghileri, J. (2000). Teaching number sense. London:

Continuum.Anghileri, J. (2001). Intuitive approaches, mental

strategies and standard algorithms. In J. Anghileri(Ed.), Principles and practices in arithmetic teaching –Innovative approaches for the primary classroom.Buckingham: Open University Press.

Askew, M., Brown, M., Rhodes, V., Wiliam, D. &Johnson, D. (1997). Effective teachers of numeracy:Report of a study carried out for the Teacher TrainingAgency. London: King’s College, University ofLondon.

Beishuizen, M. (1993). Mental strategies and materi-als or models for addition and subtraction up to100 in Dutch Second Grades. Journal for Researchin Mathematics Education, 24(4), 294–323.

Beishuizen, M. (2001). Different approaches tomastering mental calculation strategies. In J.Anghileri (Ed.), Principles and practices in arith-

metic teaching (pp. 119–130). Buckingham: OpenUniversity Press.

Beishuizen, M. & Anghileri, J. (1998). Which mentalstrategies in the early number curriculum? A com-parison of British ideas and Dutch views. BritishEducational Research Journal, 24(3), 519–538.

Beishuizen, M., Van Putten, C. M. & Van Mulken, F.(1997). Mental arithmetic and strategy use withindirect number problems up to one hundred.Learning and Instruction, 7(1), 87–106.

Cobb, P. (1991). Reconstructing elementary schoolmathematics. Focus on Learning Problems in Mathe-matics, 13(2), 3–22.

Cobb, P., Gravemeijer, K.P.E., Yackel, E., McClain, K.& Whitenack, J. (1997). Mathematizing and sym-bolizing: the emergence of chains of significationin one first-grade classroom. In D. Kirshner &J.A. Whitson (Eds.), Situated Cognition Theory:

to tackling numeracy difficulties (Dowker,2005).

Studies suggest that weakness in thesesequence-based tasks is characteristic of low-attaining pupils (Beishuizen et al., 1997;Menne, 2001). Our study accords with this.We recommend that low-attaining pupils beassessed for knowledge of sequential struc-ture, and that intervention include explicitattention to development of this knowledge.The four assessment task groups discussed inthis report can inform detailed assessment ofpupils’ number sequence knowledge. We aredeveloping instructional activities for thistopic in our current research project with low-attaining pupils, trialling, for example, flexi-ble incrementing and decrementing by tensand ones (Wright, Martland, Stafford &Stanger, 2002) and jumping on an emptynumber line (Menne, 2001). We are alsodeveloping activities targeting the pupils’development of the related sequence-basedmental strategies for addition and subtrac-tion.

There has been considerable discussionof pupil and curriculum choices betweencollections-based and sequence-based strategiesfor addition and subtraction (Beishuizen,2001). Studies suggest that low-attaining

pupils can have more success with sequence-based addition strategies, such as jump, thanwith collection-based strategies, such as split(Beishuizen, 1993). Importantly, if teacherschoose to emphasise jump, pupils willrequire a co-development of knowledge ofsequential structure (Menne, 2001). Further,regardless of choice of arithmetic strategy(jump or split) our curriculum should recog-nise the importance of sequential structure asa basic aspect of number.

AcknowledgementsThe authors gratefully acknowledge thesupport for this project from the AustralianResearch Council under grant LP0348932and from the Catholic Education Commis-sion of Victoria. As well, the authors acknowl-edge the contributions of Gerard Lewis andCath Pearn (partner investigators) to thisproject. Finally, the authors express theirsincere thanks to the teachers, pupils andschools participating in the project.

Address for correspondenceSchool of Education, Southern CrossUniversity, PO Box 157, Lismore NSW 2480,AustraliaE-mail: [email protected]

Social, semiotic and neurobiological perspectives (pp.151–233). Mahwah, NJ: Lawrence Erlbaum.

Cooper, T.J., Heirdsfield, A. & Irons, C.J. (1995).Years 2 and 3 children’s strategies for mentaladdition and subtraction. In B. Atweh & S. Flavel(Eds.), Proceedings of the eighteenth annual conferenceof the Mathematics Education Research Group of Aus-tralasia (pp. 195–202). Darwin: MathematicsEducation Research Group of Australasia.

Dowker, A. (2005). Individual differences in arithmetic:Implications for psychology, neuroscience and educa-tion. Hove, UK: Psychology Press.

Foxman, D. & Beishuizen, M. (1999). Untaught men-tal calculation methods used by 11-year-olds.Mathematics in School, 5–7.

Foxman, D. & Beishuizen, M. (2002). Mental calcula-tion methods used by 11-year-olds in differentattainment bands: A reanalysis of data from the1987 APU Survey in the UK. Educational Studies inMathematics, 51(1–2), 41–69.

Fuson, K.C., Richards, J. & Briars, D. (1982). Theacquisition and elaboration of the number wordsequence. In C.J. Brainerd (Ed.), Progress in Cog-nitive development: Vol. 1 Children’s logical and math-ematical cognition (pp. 33–92). New York:Springer-Verlag.

Fuson, K.C., Wearne, D., Hiebert, J., Murray, H.,Human, P., Olivier, A., Carpenter, T.P. & Fen-nema, E. (1997). Children’s conceptual struc-tures for multidigit numbers and methods ofmultidigit addition and subtraction. Journal forResearch in Mathematics Education, 28, 130–162.

Hiebert, J. & Wearne, D. (1996). Instruction, under-standing, and skill in multidigit addition andsubtraction. Cognition and Instruction, 14(3),251–283.

Klein, A.S., Beishuizen, M. & Treffers, A. (1998). Theempty number line in Dutch second grades:Realistic versus gradual program design. Journalfor Research in Mathematics Education, 29(4),443–464.

McIntosh, A., Reys, B.J. & Reys, R.E. (1992). A proposed framework for examining basic numbersense. For the Learning of Mathematics, 12, 2–44.

Menne, J. (2001). Jumping ahead: an innovativeteaching program. In J. Anghileri (Ed.), Principlesand practices in arithmetic teaching – Innovativeapproaches for the primary classroom (pp. 95–106).Buckingham: Open University Press.

Olive, J. (2001). Children’s number sequences:An explanation of Steffe’s constructs and anextrapolation to rational number of arithmetic.The Mathematics Educator, 11(1), 4–9.

Resnick, L.B. (1989). Developing mathematicalknowledge. American Psychologist, 44, 162–169.

Ruthven, K. (1998). The use of mental, written, andcalculator strategies of numerical computationby upper primary students within a ‘calculatoraware’ number curriculum. British EducationalResearch Journal, 24(1), 21–42.

Skwarchuk, S. & Anglin, J.M. (2002). Children’saquisition of the English cardinal number words:A special case of vocabulary development. Journalof Educational Psychology, 94(1), 107–125.

Sowder, J.T. (1992). Making sense of numbers. InG. Leinhardt, R. Putnam, & R.A. Hattrup (Eds.),Analysis of arithmetic for mathematics teaching (pp.1–51). Hillsdale, NJ: Lawrence Erlbaum.

Thompson, I. (Ed.) (1997). Teaching and learning earlynumber. Buckingham: Open University Press.

Thompson, I. & Smith, F. (1999). Mental calculationstrategies for the addition and subtraction of 2-digitnumbers (Report for the Nuffield Foundation).Newcastle upon Tyne: University of Newcastleupon Tyne.

Treffers, A. (1991). Didactical background of amathematics program for primary education. InL. Streefland (Ed.), Realistic mathematics educationin primary school (pp. 21–56). Utrecht: Freuden-thal Institute.

Treffers, A. & Buys, K. (2001). Grade 2 and 3– Calcu-lation up to 100. In M. van den Heuvel-Pan-huizen (Ed.), Children learn mathematics. TheNetherlands: Freudenthal Institute, UtrechtUniversity.

Wolters, G., Beishuizen, M., Broers, G. & Knoppert, W.(1990). Mental arithmetic: Effects of calculationprocedure and problem difficulty on solutionlatency. Journal of Experimental Child Psychology, 49,20–30.

Wright, R.J. (1994). A study of the numerical devel-opment of 5-year-olds and 6-year-olds. EducationalStudies in Mathematics, 26, 25–44.

Wright, R.J., Martland, J. & Stafford, A.K. (2006).Early numeracy: Assessment for teaching andintervention (2nd ed.). London: Paul ChapmanPublishing.

Wright, R.J., Martland, J. Stafford, A.K. & Stanger, G.(2002). Teaching number: Advancing children’s skillsand strategies. London: Paul ChapmanPublishing.

Yackel, E. (2001). Perspectives on arithmetic fromclassroom-based research in the United States ofAmerica. In J. Anghileri (Ed.), Principles andpractices in arithmetic teaching – Innovative approachesfor the primary classroom (pp. 15–31). Buckingham:Open University Press.




T HERE IS a much smaller research baseon mathematical development and dif-ficulties than on some other areas of

development, such as language and literacy.However, there has recently been an increasedemphasis on mathematics in cognitive devel-opmental research (e.g. Baroody & Dowker,2003; Campbell, 2005; Royer, 2003); inneuroscience (Ansari, Garcia, Lucas, Hamon& Dhilil, 2005; Butterworth, 1999; Dehaene,1997); and in educational policy and practicein the UK and abroad (Askew & Brown,2001; Kilpatrick, Swafford & Fundell, 2001).

In particular, there is by now overwhelm-ing evidence from experimental, educa-tional and factor analytic studies of typicallydeveloping children and adults (e.g.Dowker, 1998, 2005; Geary & Widaman,1992; Ginsburg, 1977; Lefevre & Kulak,1994; Siegler, 1988); studies of children witharithmetical deficits (Butterworth, 2005;Dowker, 2005; Geary & Hoard, 2005; Gins-burg, 1977; Jordan & Hanich, 2000; Russell& Ginsburg, 1984; Shalev, Gross-Tsur &Manor, 1997); studies of patients (Butter-worth, 1999; Dehaene, 1997; Delazer, 2003);and functional brain imaging studies

(Castelli, Glaser & Butterworth, 2006;Dehaene, Spelke, Pinel, Stanescu & Tsivkin,1999; Gruber, Indefrey, Steinmetz & Klein-schmidt. 2001; Rickard, Romero, Basso,Wharton, Flitman & Grafman, 2000) thatarithmetical ability is not unitary. Its broadcomponents include counting, memory forarithmetical facts, the understanding of con-cepts, and the ability to follow procedures.Each of these broad components has, inturn, a number of narrower components:for example, counting includes knowledgeof the counting sequence, ability to followcounting procedures in counting sets ofobjects, and understanding of the principlesof counting: for example, that the lastnumber in a count sequence represents thenumber of objects in the set, and that count-ing a set of objects in different orders willgive the same answer (Greeno, Riley & Gel-man, 1984; Munn, 1997).

Moreover, though the different compo-nents often correlate with one another,weaknesses in any one of them can occur rel-atively independently of weaknesses in theothers. Weakness in even one componentcan ultimately take its toll on performance in

What can intervention tell us aboutarithmetical difficulties?Ann Dowker

Abstract146 children (mean age 6 years 10 months) were included in the Numeracy Recovery interventionprogramme. The programme involved working with children who have been identified by their teachers ashaving problems with arithmetic. These children were assessed on nine components of early numeracy, andreceived weekly individual intervention (half an hour a week for approximately 30 weeks) in the particu-lar components with which they have been found to have difficulty. Six months after the start of interven-tion, the children showed significant improvement on three standardised tests. Scores on three components(estimation; derived fact strategies; translation between concrete, numerical and verbal presentations ofarithmetic problems) were analysed in relation to one another; to general level of addition performance; tostandardised test scores; and to improvements in these scores. Regressions showed few relationships betweenthe components, addition level, and standardised test scores. However, addition level predicted both the initial standardised test scores and improvement in these scores; and derived fact strategy use showed a negative relationship to test improvement. Implications of these findings are discussed.


What can intervention tell us about arithmetical difficulties?

other components, partly because difficultywith one component may increase the risk ofthe child relying exclusively on another com-ponent, and failing to perceive and use rela-tionships between different arithmeticalprocesses and problems. In addition, whenchildren fail at certain tasks, they may cometo perceive themselves as ‘no good at maths’and develop a negative attitude to the sub-ject. However, the components describedhere are not seen as a hierarchy. A child mayperform well at an apparently difficult task(e.g. word problem solving) while perform-ing poorly at an apparently easier compo-nent (e.g. remembering the counting wordsequence). Though certain components mayfrequently form the basis for learning othercomponents, they need not always be prereq-uisites. Several studies (e.g. Denvir & Brown,1986) have suggested that it is not possible toestablish a strict hierarchy whereby any onecomponent invariably precedes anothercomponent.

Many children have difficulties withsome or most aspects of arithmetic. It ishard to estimate the proportion who havedifficulties, since this depends on the crite-ria that are used to define ‘difficulty’. More-over, as arithmetical thinking involves sucha wide variety of components, there aremany forms and causes of arithmetical diffi-culty, which may assume different degreesof importance in different tasks and situa-tions. It is likely that at least 15–20 per centof the population have difficulties with cer-tain aspects of arithmetic, sufficient tocause significant practical and educationalproblems for the individual (Bynner & Par-sons, 1997) though the proportion thatmight be described as dyscalculic is muchlower than this.

The present paper describes a study ofthe relationships between components ofarithmetic in a group of children selected forhaving arithmetical difficulties. The researchis based on the Numeracy Recovery Programme (Dowker, 2001, 2005), an inter-vention that takes a ‘multiple components’view of arithmetic.

The components were selected on thebasis of earlier research (Dowker, 1998) anddiscussions with teachers about what theyconsidered to be important components ofarithmetic, which were sources of difficultyfor children.

Since children’s performance in thesecomponents was assessed in depth beforethe interventions took place, the study canbe used to analyse relationships between thecomponents, and to investigate whether theyshould indeed be seen as separate multiplecomponents, they are just specific tests of anoverarching general arithmetical ability. Thiscan be studied by investigating the initialrelationships between components; by inves-tigating whether they predict standardisedtest scores similarly or differently; and byinvestigating whether they have a differentialrole in predicting test improvements.

Thus, the first aim of this study is toinvestigate whether there is a relationshipbetween scores on the different componentsof the Numeracy Recovery Programme, orwhether they are relatively independent. Previous research, as summarised above, hassuggested some functional independencebetween these components. It appears possi-ble, however, that different components canbe more closely linked in children withmathematical difficulties, due to difficultiesin one having an adverse effect on develop-ment of others, or due to the difficultiessharing a common cause.

The second aim of the study is to inves-tigate the extent to which different compo-nents might contribute independently notonly to standardised test scores, but to thelevel of improvement in these scores followingintervention. This is an important questionfor theoretical reasons, in as much as it canprovide information about the specificcomponents of arithmetic that have thegreatest effect on overall growth, and forpractical reasons, in helping us to predictwhich children are most likely to benefitfrom a relatively non-intensive intervention,and which may need more or differenthelp.

MethodSample146 children who had been included in theNumeracy Recovery programme. Theyincluded 64 boys and 82 girls. Ages rangedfrom five years six months to eight yearsthree months (mean: six years ten months;standard deviation 6.1 months).

Measures of arithmetical componentsArithmetical components were measuredusing assesments of three selected compo-nents of arithmetical ability (derived factstrategy use, estimation and translation)derived from the programme.The threecomponents (described more fully inAppendix 2) were selected because they hadalready been studied in some detail withunselected groups of children (Dowker,1997, 1998, 2005).

Outcome measuresThe subsample of 145 were assessed beforeand after intervention on the British AbilitiesScales Basic Number Skills subtest (Elliott,1997), the WOND Numerical Operations test(Wechsler & Rust, 1996), and the WISCArithmetic subtest (Weschler, 1991). The firsttwo place greatest emphasis on computationabilities and the latter on arithmetical reason-ing, and so the BAS and WOND were takenas measures of computation and the WISC asa measure of arithmetical reasoning.

AnalysesTo establish whether the arithmetical com-ponents were interrelated, three separatemultiple regression analyses were run witheach of the three selected components asoutcome measures in turn.

To establish whether the arithmeticalcomponents contributed to the improve-ment level, multiple regression analyses wererun using the improvement scores on BAS,WOND and WISC as outcome measures.

The numeracy recovery programmeNumeracy Recovery is an intervention basedon a ‘multiple components’ view of arithmetic

and has been described previously (Dowker,2001, 2005). The programme has involvedworking with children who have been identi-fied by their teachers as having problems witharithmetic. These children are assessed onnine components of early numeracy, whichare summarised in Appendices 1 and 2.

The children receive weekly individualintervention (half an hour a week) in theparticular components with which they havebeen found to have difficulty. The interven-tions are carried out by the classroom teach-ers, using techniques proposed by theresearcher. The teachers are released (eachteacher for half a day weekly) for the inter-vention. Each child remains in the programfor 30 weeks, or until their teachers feel they no longer need intervention, whicheveris shorter. New children join the project periodically.

The initial scores on the standardisedtests and retest scores after six months of thefirst 146 children to take part in the projectare described here. The median standardscores on the BAS Basic Number Skills sub-test were 96 initially and 100 after approxi-mately six months. The median standardscores on the WOND Numerical Operationstest were 91 initially and 94 after six months.The median standard scores on the WISCArithmetic subtest were 7 initially, and 8 aftersix months (the means were 6.8 initially and8.45 after six months). Wilcoxon testsshowed that all these improvements were sig-nificant at the 0.01 level. 101 of the 146children have been retested over periods ofat least a year and have maintained theirimprovement.

Using the intervention to examine the structure of arithmeticAs outlined in the introduction, there weretwo aims. The first was to investigate whetherthere were relationships between scores onthe different components or whether theywere relatively independent. The second wasto examine the extent to which differentcomponents might have contributed inde-


Ann Dowker

pendently to the level of improvement in stan-dardised scores following intervention.

The performance of a subsample ofchildren in the intervention group was exam-ined on three selected components of arith-metical ability (derived fact strategy use,estimation and translation) against outcomemeasures of the standardised tests. The threecomponents (described more fully in Appen-dix 2) were selected because they had alreadybeen studied in some detail with unselectedgroups of children (Dowker, 1997, 1998, 2005).

The children were divided into five levelsaccording to their performance on a mentalcalculation pre-test. Table 1 gives brief descrip-tions of the levels, and examples of theproblems that could and could not be solvedat these levels. In practice, only the first threelevels were represented in the present group.

ResultsTable 2 gives the descriptive statistics for all the variables for 146 children in the subsample. Before the intervention, 37 ofthese children were at Addition PerformanceLevel 1 (Beginning Arithmetic); 86 at Level 2(Facts to 10) and 23 at Level 3 (Facts to 25).

Table 3 shows the mean number of thethree kinds of additive strategies used bychildren from the intervention group at level1, 2 and 3 respectively. The table also shows,for comparison purposes, data on derived fact

and estimate strategies from an unselectedsample at the same levels (Dowker, 1998).

The figures show that these strategies areused more frequently by children at additionlevels 2 and 3 than by children at Additionlevel 1. Comparison shows that childrenfrom the 1998 unselected sample at all threelevels used derived fact strategies moreoften. The mean translation score of theintervention group as a whole was 22.58 (s.d.9.81). The unselected children studied byDowker, Gent & Tate (2000) were not dividedby performance levels in this way; but theoverall average score obtained by 6-year-oldswas 32, so on he whole these scores too werehigher in the unselected sample.

Relationships among differentmeasuresIn order to investigate the independent con-tributions that different factors made to tar-get measures, entry level multiple regressionswere carried out on each of the measures asdependent variable, with the other measuresas the predictors. Table 4 shows the results ofthe regression analyses.

Addition Level was included in the analy-ses because it was a determinant of the precisecontent of the derived fact strategy and estima-tion tasks, and it seemed desirable to controlfor it. As it is an ordinal rather than cardinalscore, there could however be doubts as to theappropriateness of its inclusion in a regres-sion. To check for this, the same analyses werecarried out with this variable omitted, andresults were identical as regards the signifi-cance of the other predictors, except that Esti-mation became a definitely rather thanborderline significant predictor of Transla-tion, (beta � 0.25; t � 2.23; p 0.05) and ofcourse vice versa (beta � 0.28; t � 2.23;p 0.05).

Factors contributing to standardised test scores,and to levels of improvement in these scores.Entry method linear multiple regressionswere carried out [1] with each standardisedscore at t1 as the dependent variable andAge, Addition Level, Addition Derived Fact

Table 1: Levels of arithmetical performance inaddition

Level Problem Problemjust within outsiderange range

Beginning 2 � 2 5 � 3arithmetic

Facts to 10 5 � 3 8 � 6

Simple facts 8 � 6 23 � 44

2-Digit 23 � 44 52 � 39(no carry)

2-Digit (carry) 52 � 39 523 � 168




Ann Dowker

n Mean Standard Rangedeviation

WOND (first score) 175 90.32 10.99 62 to 123

WOND (second score) 146 92.79 12.34 68 to126

WOND improvement 146 2.45 10.9 �23 to 33

BAS (first score) 175 95.19 11.76 67 to 123

BAS (second score) 146 100.38 12.45 62 to 133

BAS improvement 146 5.11 11.87 �25 to 49

WISC (first score) 175 6.86 2.87 2 to 17

WISC (second score) 146 8.33 2.5 2 to 15

WISC improvement 146 1.51 3.13 �8 to 11

Age at start (months) 175 80.7 6.05 66 to 97

Addition Level 175 1.94 0.67 1 to 4

Addition derived fact strategies 175 1.05 1.13 0 to 5

Estimation 175 4.08 2.21 0 to 9

Translation 175 21.9 9.93 2 to 47

Table 2: Means and standard deviations of test scores

Level 1 Level 2 Level 3

Intervention: Mean Sd Mean Sd Mean Sd

Number of 0.3 0.6 1.3 1.2 1.2 1.1Derived factstrategies

Number of 3 1.7 4.5 2.3 4.3 2.2reasonableestimates(out of 9)

Translation 15.3 7.2 25.0 9.5 25.9 8.4score

Unselected:

Number of 0.7 1.8 3.0derived factstrategies

Number of 2.4 5.4 5.0reasonableestimates

Table 3: Means and standard deviations of additive strategies by addition performance level for intervention and previous (unselected) samples

Strategies, Estimation, and Translation as thepredictors and [2] the improvement in score(t2 – t1) as the dependent variable and Age,t1 standardised scores, Addition Level, Addi-tion Derived Fact Strategies, Estimation, andTranslation as the predictors.

DiscussionThe children with arithmetical difficultiesappeared in general to show some weaknessesin the components investigated: derived factstrategies, estimation and translation, as com-pared with unselected children in other studies.The groups may not be directly comparable,due to the time lapse and changes in the edu-cational system since the studies of the unse-lected children; but the figures suggest thatthe children in the intervention group usedon average somewhat fewer derived factstrategies and make fewer reasonableestimates than the unselected children, andthat this was especially true of those at thehigher addition performance levels. Perhaps

children at the higher addition performancelevels were only regarded by their teachers asarithmetically weak and needing interventionif they did have additional weaknesses inaspects of arithmetical reasoning. The studyof unselected children’s translation (Dowker,2005) did not assess the children’s additionperformance level; but their translation per-formance as a group appeared to be some-what better than that of the children witharithmetical difficulties.

It should be noted, however, that not allchildren in the latter group performedpoorly in the components investigated; thatderived fact strategy use and estimation wereoften quite good; and that their translationperformance in particular seemed betterthan that which would have been predictedby Hughes (1986), who found extreme translation difficulties even in unselected 9-year-olds.

Standardised test scores are more relatedto some specific components of arithmetic

Predictor Beta t Significance

Derived facts

Addition level 0.17 1.32 n.s.

Estimation 0.1 0.78 n.s.

Translation 0.06 0.43 n.s

Age in months 0.8 �0.6 n.s.

Estimation


Derived facts 0.09 0.78 n.s.

Translation 0.26 1.93 P � 0.06(*)

Age in months 0.04 0.32 n.s.

Translation

Addition level 0.25 2.2 P 0.05*


Estimation 0.21 1.94 P � 0.06 (*)

Age in months 0.27 2.47 P 0.05*

Table 4: Results of regressions on task scores



Table 5: Results of regressions on standardised test scores and improvements in these scores (Continued on next page)


Ann Dowker


WOND (first score)

Addition level 0.6 5.71 P 0.01**



Translation �0.08 �0.74 n.s.

Age in months �0.41 �4.03 P 0.01**

WOND (improvement)


Derived facts �0.27 �2.33 P 0.01**

Estimation 0.21 1.85 P � 0.07 (*)

Translation 0.03 0.22 n.s.

Age in months �0.4 �2.9 P 0.05*

WOND (first score) �0.68 �4.81 P 0.01**

BAS (first score)




Translation 0.05 �0.43 n.s.

Age in months �0.14 �1.19 n.s.

BAS (improvement)

Addition level 0.25 2.2 P 0.05*

Derived facts �0.33 �3 P 0.01**



Age in months 0.25 �2.22 P 0.05*

BAS (first score) �0.66 �5.6 P 0.01**

WISC (first score)


Derived facts �0.07 �0.6 n.s.


Translation 0.27 2.24 P 0.5*


than to others. Addition Level was a significantindependent predictor of initial scores in alltests: not surprisingly, as all the standardisedtests emphasised competence at calculation.Neither Estimation nor Derived Fact Strate-gies was an independent predictor of any testscores. Translation predicted performance inthe WISC Arithmetic test, but not in the othertasks. This may be due to the fact that theWISC Arithmetic test places an emphasis onword problem solving, whereas the other testsplace greater emphasis on calculation and onreading and writing numbers.

The pattern was more complex withregard to the factors affecting improvementsin performance. Initial scores were negativepredictors of improvement in all three tests.This is not surprising, as the lower the initialscore, the more room for improvement; andthe initial scores had indeed only been included in the regressions so as to controlfor them, while investigating the effects ofthe scores on different components ofarithmetic.

None of these scores did in fact predictimprovement in Arithmetic. With regard to improvements in the WOND and BAStests, there was no significant effect of eitherEstimation or Translation. Addition Levelwas, however a positive independent predic-tor of improvement (even after partiallingout the initial score on the standardised test)

and Derived Fact Strategy score was a nega-tive independent predictor.

It is intriguing that different componentsof arithmetic appear to play different rolesin predicting improvement; but a lot moreresearch needs to be done before we candraw strong conclusions about the nature ofthese roles.

The finding that derived fact strategyscores appear to act as a negative predictorof improvement is puzzling, especially asmost studies (Dowker, 1998, 2005) show apositive relationship between derived factstrategy use and other aspects of arithmeticalperformance.

There are at least three possible explana-tions. One is that, in the case of childrenwith arithmetical difficulties, good derivedfact strategy use actually does make them lesslikely to show overall improvement in arith-metic at least in the short term, perhapsbecause the existence of useful compensa-tory strategies reduces the need and motiva-tion to acquire conventional strategies. Asecond possible explanation is that the asso-ciation is secondary to some other character-istics that are associated with relativestrengths in derived fact strategy use.Dowker (1998) found that children who aresignificantly better at derived fact strategyuse than at exact calculation (whether or notthey have actual calculation difficulties) are


WISC (improvement)



Estimation 0 0 n.s.



WISC (first score) �0.82 �8.0 P 0.01**

Table 5: (continued) Results of regressions on standardised test scores and improvements in these scores



more likely than others to show largediscrepancies, in either direction, betweenVerbal and Performance IQ. It may be thatsuch discrepancies are associated with lowerimprovements, at least in the short term, andespecially in a group selected for mathemat-ical difficulties, which may include somechildren with co-morbid disorders associatedwith Verbal/Performance IQ discrepancies.This hypothesis should be investigated in thefuture by giving the children IQ and perhapsother cognitive tests, and investigating rela-tionships to improvements in arithmetic.

A third possible explanation is that theeffects on improvement in performance arenot linked to the child’s arithmetical or cogni-tive characteristics, but to the form of inter-vention that was given. Children who wereweak at derived fact strategies were giveninterventions that involved derived fact strat-egy training; those who already performedwell at derived fact strategies, did not receivesuch intervention. Perhaps derived fact strat-egy training has a particularly beneficialeffect. It may be desirable to investigate theeffect of giving derived fact strategy trainingto all children in an intervention program, inaddition to the more specific interventionsfor components in which particular childrendemonstrate weaknesses.

Investigations revealed some general cor-relations between specific components ofarithmetic, but there were few significant inde-pendent relationships between these compo-nents. It is possible that more suchrelationships would be found if a larger sam-ple were studied. The results are, however,consistent with the view that arithmetic ismade up of many components; that ‘there isno such thing as arithmetical ability; onlyarithmetical abilities’ (Dowker, 2005). Indeed,they show rather less relationship between dif-ferent components than was found inDowker’s (1998) study of an unselected groupof children. That study (which did not investi-gate translation) showed significant independ-ent effects of addition level on both estimationand derived fact strategies, and a particularlystrong independent relationship between

derived fact strategy use and estimation. Bycontrast, in the present study, derived factstrategy use and estimation not only did notshow an independent relationship; they werenot even correlated before other factors werepartialled out.

It may be that in a group of children witharithmetical difficulties, there is even lessrelationship between different arithmeticalcomponents than in a typical sample. Per-haps in completely typical mathematicaldevelopment, different components, thoughperhaps functionally separable, do informand reinforce one another in the course ofdevelopment (as Baroody & Ginsburg (1986)propose for the development of principlesand procedures in younger children, in their‘mutual development’ theory). In childrenwith arithmetical difficulties, this integrationmay not occur to the same extent, eitherbecause it is impeded by marked weaknessesin individual components, or because of afailure in the integrative process itself.

However, another possible explanation forany differences found between Dowker’s(1998) study and the present one is that thefindings are linked to educational changes.There were crucial changes in British mathe-matics education in 1998–1999, with theintroduction of the National Numeracy Strat-egy (Department for Education and Employ-ment, 1998). There is no transparentlyobvious reason why the changes in mathemat-ics education at that time should have led togreater dissociation between different compo-nents of arithmetic. If anything, one mighthave expected that the more explicit struc-ture of the mathematics curriculum, and theinclusion of derived fact strategies and estima-tion in primary mathematics instruction,might have led to the components becomingmore integrated with one another. However,curriculum changes sometimes have effectsother than the predictable or intended ones.In any case, it is risky to assume that differ-ences in findings between groups are entirelythe result of group characteristics, when thereare also differences in the instruction thatthey have received. It would be desirable to


Ann Dowker

Askew, M. & Brown, M. (2001, eds.). Teaching andLearning Primary Numeracy: Policy, Practice andEffectiveness. Notts: British Educational ResearchAssociation.

Ansari, D., Garcia, N., Lucas, E., Hamon, K. & Dhilil, B.(2005). Neural correlates of symbolic numberprocessing in children and adults. Neuroreport, 16,1769–1775.

Baroody, A.J. & Ginsburg, H.P. (1986). The relation-ship between initial meaningful and mechanicalknowledge of arithmetic. In J. Hiebert (Ed.),Conceptual and Procedural Knowledge: The Case ofMathematics (pp. 75–112). Hillsdale, NJ: Erlbaum.

Baroody, A.J., Ginsburg, H.P. & Waxman, B. (1983).Children’s use of mathematical structure. Journalfor Research in Mathematics Education, 14, 156–168.

Baroody, A. and Dowker, A. (2003). The Developmentof Arithmetical Concepts and Skills. Mahwah, N.J.:Erlbaum.

Butterworth, B. (1999). The Mathematical Brain.London: Macmillan.

Butterworth, B. (2005). Developmental dyscalculia. InJ.I.D. Campbell (Ed.), Handbook of MathematicalCognition (pp. 455–467). Hove: Psychology Press.

Bynner, J. and Parsons, S. (1997). Does NumeracyMatter? London: Basic Skills Agency.

Campbell. J.I.D. (Ed.), (2005) Handbook of Mathemat-ical Cognition. Hove: Psychology Press.

Canobi, K.H., Reeve, R.A. & Pattison, P.E. (1998).The role of conceptual understanding inchildren’s addition problem solving. Developmen-tal Psychology, 34, 882–891.

Canobi, K.H., Reeve, R.A. & Pattison, P.E. (2003).Young children’s understanding of additionconcepts. Educational Psychology, 22, 513–532.

Castelli, F., Glaser, D.E. & Butterworth, B. (2006).Discrete and analogue quantity processing in theparietal lobe: a functional MRI study. Proceedingsof the National Academy of Sciences, 103, 4693–4698.

Cowan, R., Bailey, S., Christakis, A. & Dowker, A.D.(1996). Even more precisely understanding theorder irrelevance principle. Journal of Experi-mental Child Psychology, 61, 84–101.

compare the children in this group with uns-elected children undergoing similar mathe-matics instruction. Such a study is currentlyunderway.

ConclusionThe findings discussed in this paper stronglysupport the view that arithmetic is made upof multiple components rather than beingunitary; though further research is necessaryto establish how the relationships betweencomponents vary with ability level and witheducational factors.They are also consistentwith the view that arithmetical difficultiescan be significantly ameliorated by interven-tions targeting specific weaknesses, thoughmore research comparing different forms ofintervention would be needed to confirmthis view. There is more research to be doneon exactly how such interventions lead toimprovement. Indeed, the project is under-going further development and evaluation.

Further investigations are of course neces-sary to show whether and to what extent spe-cific interventions in mathematics are moreeffective in improving children’s mathemat-ics than other interventions which

provide children with individual attention:for example, interventions in literacy. It isalso desirable to investigate whether differentapproaches to such intervention (e.g. agewhen intervention starts; degree of intensive-ness; degree of individualization; the particu-lar components emphasised) may differ ingeneral effectiveness and/or differentiallyappropriate to different groups of children.

The present study also demonstrates thepossibilities for bidirectional relationshipsbetween research and intervention. Theproject integrates the implementation andevaluation of the intervention scheme withthe investigation of individual differences in,and relationships between, certain selectedcomponents of arithmetic. Thus, the inter-vention project, which was inspired by myearlier research and conclusions about thecomponents of arithmetic, also serves to testtheories about these components.

Address for correspondenceAnn Dowker, Department of ExperimentalPsychology, University of Oxford, SouthParks Road, Oxford OX1 3UD.E-mail: [email protected]

References



Decorte, E. & Verschaffel, L. (1987). The effect ofsemantic structure on first-graders’ strategies forsolving addition and subtraction word problems.Journal for Research in Mathematics Education, 18,363–381.

Dehaene, S. (1997). The Number Sense. London:Macmillan.

Dehaene, S., Spelke, E., Pinel, P., Stanescu, R. & Tsivkin, S.(1999). Sources of mathematical thinking: behav-ioural and brain-imaging evidence. Science, 5416,970–974.

Delazer, M. (2003). Neuropsychological findings on conceptual knowledge of arithmetic. In A. Baroody & A. Dowker (Eds.), The development ofarithmetical concepts and skills (pp. 385–407).Mahwah, N.J. Erlbaum.

Denvir, B. & Brown, M. (1986). Understandingnumber concepts in low-attaining 7-to-9-year-olds. Part 1: Development of descriptive frame-works and diagnostic instruments. EducationalStudies in Mathematics, 17, 15–36.

DfEE (1999). The National Numeracy Strategy: Frameworkfor Teaching Mathematics, Reception to Year 6. London:Department for Education and Employment.

Dowker, A.D. (1997). Young children’s addition esti-mates. Mathematical Cognition, 3, 141–154.

Dowker, A.D. (1998). Individual differences in nor-mal arithmetical development. In C. Donlan(Ed.), The Development of Mathematical Skills (pp.275–302). Hove: Psychology Press.

Dowker, A.D. (2001). Numeracy recovery: a pilotscheme for early intervention with youngchildren with numeracy difficulties. Support forLearning, 16, 6–10.

Dowker, A.D. (2005). Individual Differences in Arith-metic: Implications for Psychology, Neuroscience andEducation. Hove: Psychology Press.

Dowker, A., Gent, M. & Tate, L. (2000). Is arithmetica foreign language? Young children’s translationsbetween arithmetical problems in numerical,concrete and word problem formats. Paper pre-sented at the European Congress of Psychology,London.

Elliott, C.D., Smith, P. & McCullouch, K. (1997). BritishAbility Scales, 2nd Edition. Windsor: NFER.

Fuson, K. (1992). Research on learning and teachingaddition and subtraction of whole numbers. InG. Leinhardt, R. Putnam & R.A. Hattrop (Eds.),Analysis of Arithmetic for Mathematics Teaching.Hillsdale, N.J.: Erlbaum.

Geary, D.C. & Hoard, M.K. (2005). Learning disabili-ties in arithmetic and mathematics: theoreticaland empirical perspectives. In J.I.D. Campbell(Ed.), Handbook of Mathematical Cognition (pp.253–267). Hove: Psychology Press.

Geary, D.C. & Widaman, K.F. (1992). On the conver-gence of componential and psychometric mod-els. Intelligence, 16, 47–80.

Ginsburg, H.P. (1977). Children’s Arithmetic: How TheyLearn It and How You Teach It. New York, N.Y.:Van Norstrand.

Gray, E. & Tall, D. (1994). Duality, ambiguity and flex-ibility: a proceptual view of simple arithmetic.Journal for Research in Mathematics Education, 25,115–144.

Greeno, J., Riley, M. & Gelman, R. (1984). Concep-tual competence and children’s counting. Cogni-tive Psychology, 16, 94–134.

Griffin, S., Case, R. & Siegler, R. (1994). Rightstart:providing the central conceptual prerequisitesfor first formal learning of arithmetic to studentsat risk for school failure. In K. McGilly (Ed.),Classroom Learning: Integrating Cognitive Theory andClassroom Practice (pp. 25–50). Boston: M.I.T. Press.

Gruber, O., Indefrey, P., Steinmetz, H. & Kleinschmidt,A. (2001). Dissociating neural correlates of cog-nitive components in mental calculation, CerebralCortex, 11, 350–369.

Holt, J. (1966). How Children Fail. New York: Pitman.Hughes, M. (1986). Children and Number: Difficulties in

Learning Mathematics. Oxford; Blackwell.Jordan, N. & Hanich, L.B. (2000). Mathematical

thinking in second grade children with differentforms of LD. Journal of Learning Disabilities, 33,567–578.

Kilpatrick, J., Swafford, J. & Fundell, B. Adding It Up:Helping Children Learn Mathematics (Eds.) (2001).Washington, D.C.: National Academy Press.

Lefevre, J.A. & Kulak, A.G. (1994). Individual differ-ences in the obligatory activation of additionfacts. Memory and Cognition, 22, 188–200.

Lefevre, J.A., Greenham, S.L. & Waheed, N. (1993).The development of procedural and conceptualknowledge in computational estimation. Cogni-tion and Instruction, 11, 95–132.

Mayer, R. (2003). Mathematical problem solving. In J.M. Royer (Ed.), Mathematical Cognition. (pp. 69–92). Greenwich, CT: Information AgePublishing.

Mazzocco, M. & Myers, G.F. (2003). Complexities inidentifying and defining mathematics learningdisability in the primary school years. Annals ofDyslexia, 53, 218–253.

Munn, P. (1997). Children’s beliefs about counting.In I. Thompson (Ed.), Teaching and Learning EarlyNumber (pp. 9–19). Buckingham: Open UniversityPress.

Ostad, S. (1998). Developmental differences in solv-ing simple arithmetic word problems and simplenumber-fact problems: a comparison of mathe-matically normal and mathematically disabledchildren. Mathematical Cognition, 4, 1–19.

Rickard, T.C., Romero, S.G., Basso, G., Wharton, C.,Flitman, S. & Grafman, J. (2000) The calculatingbrain: an fMRI study. Neuropsychologia, 38,325–335.


Ann Dowker

Riley, M.S., Greeno, J.G. & Heller, J.I. (1983). Devel-opment of children’s problem-solving ability inarithmetic. In H.P. Ginsburg (Ed.). The Develop-ment of Mathematical Thinking (pp. 153–196). NewYork: Academic Press.

Royer, J.M. (Ed.), (2003) Mathematical Cognition.Greenwich, CT: Information Age Publishing.

Russell, R. & Ginsburg, H.P. (1984). Cognitive analy-sis of children’s mathematical difficulties. Cogni-tion and Instruction, 1, 217–244.

Shalev, R.S., Gross-Tsur, V. & Manor, O. (1997). Neu-ropsychological aspects of developmental dyscal-culia. Mathematical Cognition, 3, 105–170.

Siegler, R.S. (1988). Individual differences in strategychoice: good students, not-so-good students andperfectionists. Child Development, 59, 833–851.

Siegler, R.S. & Booth, J.L. (2005). Development ofnumerical estimation: a review. In J.I.D. Campbell(Ed.), Handbook of Mathematical Cognition (pp.197–212). Hove: Psychology Press.

Sowder, J.T. & Wheeler, M.M. (1989). The develop-ment of concepts and strategies use in computa-tional estimation. Journal for Research inMathematics Education, 20, 130–146.

Straker, A. (1996). Mental Maths for Ages 5 to 7: Teach-ers’ Book. Cambridge: Cambridge University Press.

Straker. A. (1999). The National Numeracy Project,1996–1999. In I. Thompson (Ed.), Issues in Teach-ing Numeracy in Primary Schools (pp. 39–48). Buck-ingham: Open University Press.

Temple. C.M. (1994). The cognitive neuropsy-chology of the developmental dyscalculias.Cahiers de Psychologie Cognitive/Current Psychology ofCognition, 13, 351–370.

Tronsky, L.T. & Royer, J.M. (2003). Relationshipsamong basic computational automaticity, workingmemory and complex problem solving: what weknow and what we need to know. In J.M. Royer(Ed.), Mathematical Cognition (pp. 117–145).Greenwich, CT: Information Age Publishing.

Warrington, E.K. (1982). The fractionation of arith-metical skills: a single case study. Quarterly Journalof Experimental Psychology, 34A, 31–51.

Wynn, K. (1990). Children’s understanding of count-ing. Cognition, 36, 155–193.

Yeo, D. (2003). Dyslexia, Dyspraxia and Mathematics.London: Whurr.

Wechsler, D. (1991). Wechsler Intelligence Scale forChildren, 3rd Edition. London: Harcourt.

Wechsler, D. & Rust, J. (1996). Wechsler ObjectiveNumerical Dimensions. London: Harcourt.




Ann Dowker

Aspect Assessment Intervention

Counting (i) accuracy of counting sets Children are given practice inprocedures of 5, 8, 10, 12 and 21 objects; counting sets of objects, ranging

(ii) rote verbal counting to in number from 5 to 25.10 and to 20

Counting principles The children watch an adult count Children practice counting and(i) Order irrelevance a set of objects, and are then asked answering order-irrelevance(ii) Repeated to predict the result of further counts: questions about very smalladdition by 1 and (a) in the reverse order (b) after the numbers of counters (up to 4).repeated addition of an object and (c) after He adult makes statements suchsubtraction by 1 the subtraction of an object. as, ‘It’s four this way, and four

Children are shown a set of 5 items, that way – it’s four whicheverand then shown one more item being way you count it!’ The child isadded, and asked to say, without given practice with increasinglycounting, how many there are now. large sets.This is repeated up to 15. Practice in observing andChildren are shown a set of 10 items, predicting the results of such and then shown one item being repeated additions and subtracted, and asked to say, without subtractions with counters (up counting, how many there are now. to 20). Verbal ‘number after’ This is repeated down to zero. and ‘number before’ problems:

‘What is the number before 8?’, ‘What is the number after 14?’, etc. Worksheets devised for the project, including repeated addition and subtraction by 1 from a set of circles. ‘Number After Dominoes’ and ‘Number Before Dominoes’ which are played like dominoes except thatthe added domino must be the number after (or before) the end item, rather than the same number.

Written Children are asked to read aloud Practice in reading and writingsymbolism a set of single-digit and two-digit numbers, sorting objects intofor numbers numbers. A similar set of numbers is groups of ten, and recording

dictated to them for writing. them as ‘20’, ‘30’, etc. andsorting and recording taskswhere there are extra units aswell as the groups of ten.

(Continued )

Appendix 1


Place value Children are asked to add 10s to Addition of tens to units andunits (20 � 3), to add 10s to 10s the tens to tens in several(20 � 30) and to combine the two different forms: Written numerals,into one operation (20 � 33). They Number line or block, Handsare also asked to point to the larger and fingers in pictures, 10-pence

number in pairs of 2-digit numbers, pieces and pennies or Anythat vary in the units (23 vs 26)’; apparatus (Multilink, Unifix)in the 10s (41 vs 51); or in with which the child is familiar. both tens and units in conflicting The fact that these give the directions (27 vs 31; 52 vs 48). same answers is emphasised.

Practice with arithmetical patterns such as: ‘20 � 10; 20 � 11; 20 � 12’, etc; being encouraged to use apparatus when necessary.

Word problem Word problems test Short addition and subtraction solving (Griffin et al., 1995) word problems of ‘Change’,

‘Compare’ and ‘Combine’ types are discussed with them: ‘What are the numbers that we have to work with?’ ‘What do we have to do with the numbers?’ ‘Do you think that we have to do an adding sum or a taking-away sum?’ ‘Do you think that John has more sweets or fewer sweets than he used to have?’, etc. They are encouraged to use counters to represent the operationsin the word problems, as well aswriting the sums numerically.

Translation Children are asked to read aloud a Children practice reading and between concrete, set of single-digit and two-digit writing numbers. Children with verbal and numbers. A similar set of numbers difficulties in reading or writing written number is dictated to them for writing. two-digit numbers (tens and

Children are presented with sums and units) are given practice in sortingare invited to ‘show how to do this objects into groups of ten, andsum with the counters’ in a rage of recording them as ‘20’, ‘30’, etc.translation contexts: numerical to They are then given such concrete, concrete to numerical, sorting and recording tasks verbal to concrete, verbal to where there are extra units as numerical, numerical to verbal well as the groups of ten.and concrete to verbal. The children are shown the The children’s performance on this same problems in different pretest is looked at in the context of forms and are shown that they their performance on the Written give the same resultsSymbolism and Word Problem pretests.

(Continued )




Ann Dowker


Derived fact Children are given the Addition and Training in the use and applicationstrategies Subtraction Principles Test (Dowker, of derived fact strategies

1998). They are given the answer to a (specifically commutativity, the problem and then asked them to solve n � 1 principle, and the inverse another problem that could be solved principle).quickly by the appropriate use of an arithmetical principle.

Arithmetical Children are presented with a series Children are given additional estimation of problems with estimates made by ‘Tom and Mary’ evaluation tasks,

imaginary characters (Tom and Mary). and are asked to give reasons The children are asked to evaluate the for their answers; and further estimates on a five-point ‘smiley faces’ practice in producing their own scale and to suggest ‘good guesses’ estimates. They also play for these problems themselves. ‘Twenty Questions’-type

number-guessing games (cf. Holt, 1966), which involve focussing on the range within which a number lies.

Number fact Russell and Ginsburg’s (1984) Number The child is asked to do the sameretrieval Facts Test has been expanded to sums repeatedly in the hope that

include some subtraction facts. the repetition will lead to retention of the facts involved. They also play ‘number games’ (e.g. Straker, 1996) that reinforcenumber fact knowledge.

1. Counting procedures: Arguably the mostbasic component of arithmetic is the abil-ity to make appropriate use of counting.While most six-year-olds have achievedrelatively effortless counting, a significantnumber have not (Griffin, Case & Siegler,1994; Yeo, 2003). This may seriouslyimpede their development of arithmetic,both because of the intrinsic logicalrelationships between counting andarithmetic, and because the effort ofcounting may distract attention fromother aspect of arithmetic (Gray & Tall,1994; Yeo, 2003).

2. Counting-related principles and theirapplication: Most counting principles areacquired before the age of five or six,even in children with some mathematicaldifficulties. However, the order irrele-vance principle (that counting the sameset of items in different orders will resultin the same number) is usually the latestof the main counting principles to beacquired (Cowan, Dowker, Christakis &Bailey, 1996); and is sometimes weakeven in six-year-olds. Evidence suggeststhat understanding the order irrelevanceprinciple is closely related to the ability topredict the result of adding or subtract-ing an item from a set (Cowan, Dowker,Christakis, & Bailey, 1996; Dowker, 2005).

3. Written symbolism for numbers: Here is muchevidence that children often experiencedifficulties with written arithmetical sym-bolism of all sorts, and in particular withrepre-senting quantities as numerals(Ginsburg, 1977; Fuson, 1992). Withregard to this component, children areasked to read aloud a set of single-digitand two-digit numbers. A similar set ofnumbers is dictated to them for writing.

4. Understanding the role of place value innumber operations and arithmetic : Thisinvolves the ability to add 10s to units (20 � 3 � 23); the ability to add 10s to10s (20 � 30 � 50); and the ability tocombine the two into one operation(20 � 33 � 53). A related task involves

pointing to the larger number in pairs of2-digit numbers, that vary either just withregard to the units (e.g. 23 versus 26)’;just with regard to the 10s (e.g. 41 versus51); or where both tens and units vary inconflicting directions (e.g. 27 versus 31;52 versus 48).

5. Word problem solving: There is a consider-able body of evidence (Hughes, 1986;Mayer, 2003; Riley, Greeno & Heller,1983) that young children often experi-ence difficulty with word problems inarithmetic, even when they are capable ofperforming the necessary calculations.Indeed, some studies have suggested(e.g. Russell & Ginsburg, 1984) that per-formance on word problems is one of thetasks that most strongly defines the differ-ence between mathematically normaland mathematically ‘disabled’ school-children. It is important to take intoaccount the nature of the problems, astheir semantic nature has a strong influ-ence on how easily they are solved. Forexample children tend to find problemsinvolving changes in quantity (‘Change’problems) easier than those involvingcomparisons between quantities (‘Com-pare’ problems) (Riley et al., 1983;DeCorte & Verschaffel, 1987).

6. Translation between arithmetical problems pre-sented in concrete, verbal and numerical for-mats: Several authors have suggested thattranslation between concrete, verbal andnumerical formats is a crucial area of dif-ficulty in children’s arithmetical develop-ment. For example, Hughes (1986)reported that many primary schoolchildren demonstrate difficulty in trans-lating between concrete and numericalformats (in either direction), even whenthey are reasonably proficient at doingsums in either one of these formats andhas suggested that this difficulty in trans-lation may be an important hindrance tochildren’s understanding of arithmetic.The children’s performance on theassessment described in the table is

Appendix 2: The nine components of the numeracy recovery programme



looked at in the context of their perform-ance on the Written Symbolism andWord Problem pretests. If the childrenperform particularly badly on transla-tions that involve numerical material,and also perform poorly on the WrittenSymbolism pretest, then it is likely thattheir main problem is with written sym-bolism. If the children perform particu-larly badly on translations that involveverbal material, and also perform poorlyon the Word Problems pretest, then it islikely that their main problem is withword problem comprehension. However,if they perform uniformly poorly on allparts of the translation pretest, and/or iftheir performance on the translationsinvolving numerical material is dispro-portionately worse than their perform-ance on the Written Symbolism pretest,and/or if their performance on thetranslations involving the verbal materialis disproportionately worse than theirperformance on the Word Problempretest, then it is likely that the problemis with translation as such.

7. Derived fact strategies in addition and subtrac-tion: One crucial aspect of arithmeticalreasoning is the ability to derive and pre-

dict unknown arithmetical facts fromknown facts, for example by using arith-metical principles such as commutativity,associativity, the addition/subtractioninverse principle (Baroody, Ginsburg &Waxman, 1983; Canobi, Reeve & Pattison,1998, 2004; Dowker, 1998). For example,if we know that 29 � 13 � 42, we can usethe commutativity principle to derive thefact that 13 � 29 is also 42.

8. Arithmetical estimation: The ability to esti-mate an approximate answer to an arith-metic problem, and to evaluate thereasonableness of an arithmetical esti-mate, are important aspects of arithmeti-cal reasoning (LeFevre et al., 1993; Siegler& Booth, 2005; Sowder & Wheeler, 1989).

9. Number fact retrieval : Although most psychologists, educators and mathemati-cians agree that memorisation of facts isnot the essence of arithmetic, knowledgeof number facts does contribute to effi-ciency in calculation (Tronsky & Royer,2003), and is a significant factor in distin-guishing between mathematically normaland mathematically ‘disabled’ children(Geary & Hoard, 2005; Ostad, 1998; Jordan & Hanich, 2000; Russell & Ginsburg, 2004).


Ann Dowker

Derived fact strategy task (use ofarithmetical principles)Children are given the answer to a problemand then asked to solve another problemthat can be solved quickly by using thisanswer, together with the principle underconsideration. The exact arithmetic prob-lems vary according to the previouslyassessed calculation ability of the child andare selected to be just a little too difficult forthe child to solve unaided. The principlesinvestigated are as follows, in order of theirdifficulty for the children:1. The identity principle (e.g. if one is told

that 8 � 6 � 14, then one can automati-cally give the answer ‘14’, without calcu-lating, if asked ‘What is 8 � 6?’).

2. The commutativity principle (e.g. if9 � 4 � 13, 4 � 9 must also be 13).

3. The n � 1 principle (e.g. if 23 � 44 � 67,23 � 45 must be 68).

4. The n – 1 principle (e.g. if 9 � 8 � 17,9 � 7 must be 17 – 1 or 16).

5. The addition/subtraction inverse principle(e.g. if 46 � 27 � 73, then 73 – 27 mustbe 46).A child is deemed to be able to use a prin-

ciple if (s)he can explain it and/or used it toderive at least 2 out of 3 unknown arithmeti-cal facts, while being unable to calculate anysums of similar difficulty when there is noopportunity to use the principle. TheDerived Facts score is the total number ofderived facts used.

Estimation taskThe addition estimation task has been usedin previous studies (Dowker, 1997, 1998,2003). In the present study, each child is pre-sented with an set of addition problemswithin their base correspondence as definedabove. Each set includes a group of ninesums to which a pair of imaginary characters(‘Tom & Mary’) estimate answers. Each set of‘Tom & Mary’s’ estimates includes three

good estimates (e.g. ‘7 � 2 � 10’,‘71 � 18 � 90’); three that are too small;and three that are too large. The childrenare asked to evaluate each guess on a five-point scale from ‘very good’ to ‘very silly’,represented by a set of schematic faces rang-ing from very smiling to very frowning, andwere themselves asked to suggest ‘goodguesses’ to the sums. The Estimation score isthe number of reasonable estimates, out of amaximum score of 9. Reasonable estimatesare defined as those that are within 30 percent of the correct answer, and are alsolarger than each of the addends.

Translation taskThe tasks involve translations between wordproblem, concrete and numerical formats foradditions and subtractions. The concrete for-mats involve the use of counters. Word prob-lems included Change, Compare andCombine problems for addition and subtrac-tion. All six combinations of presentationand response domain are given, as demon-strated below. No sum in any of these transla-tion tasks includes a number greater than 10.1. Translation from numerical to concrete (2

items): Children are presented withwritten sums (‘2 � 5 � 7’; ‘6 � � 4’);and invited to ‘show me how to do thissum with the counters’.

2. Translation from concrete to numerical(2 items): Children watch the researcherperform arithmetical operations withcounters (adding 7 counters to 2 coun-ters; subtracting 6 counters from 9 coun-ters) and are then asked to ‘write downthe sum that goes with what I did’.

3. Translation from verbal to concrete (5 items):Children are presented with word prob-lems and asked to ‘show me this storywith the counters’ e.g. ‘Paul had 4sweets; his mother gave him 3 more; sonow he has 7 sweets.’ (Addition: ‘Change’semantic category).

Appendix 3: A description of the three tasks used in the study



4. Translation from verbal to numerical (5items): Children were presented withword problems (similar but not identicalto those above), and asked to ‘writedown the sum that goes with the story’.

5. Translation from numerical to verbal (2 items):Children are presented with written sums(‘3 � 6 � 9’, ‘8 � 6 � 2’), and invited to‘tell me a story that goes with this sum’.

6. Translation from concrete to verbal (2 items):Children watch the researcher perform arithmetical operations with counters(e.g. adding 5 counters to 3 counters;

subtracting 6 counters from 9 counters)and then asked to ‘tell a story to go withwhat I just did with the counters.’The Translation score is calculated by giv-

ing 3 for every fully complete response, 2 forevery complete response, which involvesinverting the operation (e.g. representingthe story ‘Peter had 5 buns; he ate 3 buns; sonow he has 2 chocolates,’ as ‘3 � 2 � 5’rather than ‘5 – 2 � 3’); 1 for every incompleteresponse and 0 for every incorrect response. Asthe total number of items is 18, the maxi-mum possible score is 54.


Ann Dowker


THE ARTICLE uses detailed observa-tional data gathered in a natural class-room situation, to compare children’s

responses to similar tasks over a period oftime. While some variability may beexpected, observation shows major variabil-ity, with children apparently demonstratingspecific capacities only to ‘lose’ them later.The research questions are whether thereis evidence for variability of the perform-ance of individuals and whether it is possi-ble to explain any variability. Variability cantake many forms, for example betweenchildren or for individual children betweendifferent aspects of mathematics, differentaspects of number or tasks presented in dif-ferent ways. Another form of variability isbetween similar or identical tasks carriedout by the same child on different occa-sions. This final type of variability is the

main focus of this research although othertypes are relevant.

The research arises from a wider ethno-graphic study carried out in various settings.This particular research question arose inthis setting from the concern of the adultswho worked there. They expressed concernat the difficulty of assessing children duringclassroom activities due to perceived variabilityof their responses. The purpose of my inves-tigation, therefore, was firstly to interrogatemy existing data to see if there was evidencefor this variability. A further question waswhether variability really occurred over iden-tical or similar tasks or whether it could infact be explained by task differences.

The children concerned were all consid-ered to have learning difficulties in mathe-matics and the article ends by consideringwhether variability is a particular issue for

Investigating variability in classroomperformance amongst childrenexhibiting difficulties withearly arithmeticJenny Houssart

AbstractResearchers in both psychology and mathematics education acknowledge that children’s mathematicalperformance can vary inexplicably from day to day, though there has been little detailed investigation of theform of variability discussed in this paper. The paper builds on research suggesting this might be aparticular issue for children considered to have learning difficulties in mathematics. The childrenconcerned were seven- to nine-year-olds taught together for mathematics in a small group with high levelsof adult help in assessment, planning and teaching. Observational research was conducted, with theresearcher making weekly visits over the course of a year. Findings synthesise a range of evidence for eachchild gathered both during planned assessment tasks and as part of routine classroom activity. The dataare used to chart the performance of individuals over this period. Findings suggest that arithmetical capac-ities were not fixed and easily assessed, but varied from day to day. This variability is considered in somedetail with the aim of offering explanations for perceived differences. Elements such as task presentationand subtle mathematical differences between tasks provide partial explanations. Many differences remainunexplained, and it is argued that variability is in fact a feature of learning. Finally, implications for prac-tice in assessing children and planning for their mathematical development, are considered.

such children or is in fact a wider phenome-non. The paper’s contribution is to showthat classroom research confirms clinicalpsychological experiments on variability, sug-gesting that this may be integral to the veryprocess of learning itself rather than an aber-ration requiring alternative explanations.

BackgroundBoth psychologists and educational research-ers note, frequently as an aside, that children’smathematical capacities are not fixed andeasy to assess but vary markedly from day today or even between similar tasks on thesame day. The issue is not currently fore-grounded in English schools, possiblybecause it conflicts with contemporary initia-tives. For example, the National NumeracyProject (DfEE 1999) focuses on detailedlearning objectives, often to be shared withchildren with the hope that they will beachieved within the lesson. It is stated explic-itly (page 33) that assessment during everylesson should check that children havegrasped the main teaching points and deter-mine whether they can move on, or whethermisunderstandings need to be addressed.The implication is that such decisions canreasonably be made in a lesson and that, inthe medium term, records can be kept con-firming which key objectives have been met.The current mathematics curriculum inEngland puts an emphasis on teachingnumber, though other aspects of mathemat-ics are also included. Much of the detaileddiscussion about children with apparent dif-ficulties in mathematics focuses on number(e.g. Dowker, 2004, Wright et al., 2002).Other writers, (e.g. Gabb, 2005) assert thatpupils with special educational needs shouldhave an appropriate diet of mathematics, notjust restricted to basic number work. My ownresearch in other settings suggests that somepupils who are apparently low attainers inmathematics can respond well to non-numbertasks such as measuring or shape (Houssart,2004). However, most detailed studies ofchildren considered to be low attainers con-centrate on aspects of number.

A recent detailed discussion of individualdifferences in arithmetic (Dowker, 2005)brings together findings confirming thatmany children show variability across differ-ent aspects of number. A central theme ofDowker’s book is that it is not appropriate totalk about arithmetical ability, but ratherarithmetical abilities which can be groupedin to several categories. It is suggested thatthere can be strong discrepancies in eitherdirection between almost any two compo-nents. Although the focus is mainly on vari-ability between aspects of arithmetic, somepoints are also made relevant to variabilitybetween occasions. For example, in relationto her studies of estimation, Dowker suggeststhat the ‘know’ or ‘not know’ dichotomy inrelation to particular types of arithmetic isinadequate, and she refers to a ‘zone ofpartial knowledge and understanding’. Sheprovides many examples of individualsdemonstrating uneven performance acrossdifferent aspects of mathematics and usesthe phrase ‘cognitively uneven’ for thosewho have verbal reasoning which is eithermuch better or much worse than theirspatial reasoning. Despite detailed discus-sion of the performance of individuals onparticular aspects of arithmetic, the authorstresses the difficulty in trying to break downarithmetic into components for the purposesof assessment and intervention.

A major contributor to discussions of vari-ability in arithmetic is the noted psychologistSiegler (1996) who focuses on children’sstrategies and how changes occur in theirstrategies and ways of thinking. He suggeststhat evidence for variability is present in thedetail of much research but that, for severalreasons, this remains peripheral and under-emphasised. A key reason he advances is thatwithin Developmental Psychology variabilitybetween age groups is the main focus of attention;hence variability within groups is minimised.Although there is little work aimed directly atstudying variability, clues to variability can befound by looking at the findings of apparentlycontradictory studies. In particular, work chal-lenging Piaget (e.g. 1952, 1953) highlighted


Jenny Houssart

young children able to cope with conceptssuch as class inclusion and conservation at anearlier age than previously thought (e.g.McGarrigle & Donaldson, 1974, McGarrigle,Grieve & Hughes, 1978). Such findings arediscussed by Siegler (1996) who suggests thatrecognising variability of thinking is impor-tant in trying to reconcile evidence of youngchildren’s competence with evidence of theirincompetence.

Within Education some writers touch onpossible variability of performance, often inan assessment context. Black (1998) considerswhether a pupil might perform differently ondifferent days when discussing the reliabilityand validity of formal tests. He argues thatthis has received far less attention fromresearchers than other issues of reliabilityand validity. It could be argued that if vari-ability was widely recognised this wouldundermine the system of formal assessmentcurrently in use, and it is therefore perhapsnot surprising that this is not a high profileissue. Discussions held with teachers (Watson,2000) indicate their strong belief that pupilperformance varies from day to day. Theseteachers taught mathematics to childrenaged between ten and twelve years. Askedabout how they reached judgments abouttheir pupils’ mathematics, roughly half of thethirty teachers said it was possible for pupilsto be able to do some mathematics on oneday but not on the next. Related work by thesame author (Watson, 2001, 2006) suggeststhat the whole issue of teacher assessment isproblematic and that it is not possible to sayfor certain what a pupil knows.

Work aimed specifically at assessingpupils with learning difficulties in arithmeticalso accepts that there is some variability(Wright, Martland & Stafford, 2000). In thiscase the emphasis is again on variability ofstrategy, with the authors saying thatchildren frequently use strategies that areless sophisticated than those of which theyare capable. They give possible reasons forthis, including the facts that a less sophisti-cated strategy may be easier or that some fea-ture of the child’s thinking prior to solving

the current task may focus them on a lesssophisticated strategy.

A related, currently more prominentissue in education is that pupils may performdifferently according to how a task is pre-sented. Clausen-May (2005) discusses pupilswith different mathematics learning styles.She uses the outline VAK model, incorporat-ing Visual, Auditory and Kinesthetic styles ina discussion of how pupils may exhibitlearning differences in mathematics withpreferences for tasks involving seeing, hear-ing or doing. She also suggests that classroommathematics tasks may have a strong literacybase, disadvantaging some pupils. Differentthinking styles in mathematics are alsodiscussed by Chinn (2004) who uses the ideain the context of pupils considered to havelearning difficulties in mathematics. Bothwriters’ work is relevant to the current studybecause the data are drawn from a range ofclassroom tasks and it is possible that variabil-ity might be explained by pupils being askedto work on tasks presented in different ways.

There is therefore some support amongstboth writers and teachers for the idea of vari-able performance. However, research on theissue is relatively sparse for methodologicalreasons; such studies are time-consumingand problematic. Most of the research evi-dence is provided by Siegler and his col-leagues (Siegler & Jenkins, 1989; Siegler,1996; Siegler & Stern, 1998). These studiesuse micro genetic methods where individualsundertake the same task on several occa-sions, methods described in detail by Siegler& Crowley (1991) who argue that the con-cept of micro genetic methods and therationale for using them go back for overeighty years.

This paper uses data gathered in aclassroom whilst children undertake theirnormal activities. It is similar in some ways tomicro genetic studies since it is longitudi-nal, with children frequently returning tosimilar tasks. It differs from them in that theresearcher can not control the type ornumber of tasks carried out and because it isconducted in a classroom context rather


Investigating variability in classroom performance amongst children

than as a clinical experiment. This carrieslimitations in the number of times a childmay work on each calculation but has theadvantage of providing an opportunity to seewhether Siegler and his colleagues’ results inexperimental situations can be replicated ina ‘natural’ classroom context.

Context and methodData are drawn from a long-term researchproject carried out with four groups ofchildren using ethnographic methods. Thedata considered were gathered from onegroup of children aged seven to nine yearsold. The twelve children in the group, whowere drawn from different classes, were allconsidered to have learning difficulties as faras mathematics was concerned. The groupwas similar to other groups considered to below attainers in mathematics (Denvir, Stolz &Brown, 1982; Haylock, 1991; Robbins, 2000)in that they exhibited a range of apparentdifficulties with a corresponding range ofpossible reasons.

The teacher was joined in mathematicslessons by two classroom assistants. Whilstresearching, I visited the group for onemathematics lesson each week for a year,adopting a role similar to that of the class-room assistants and observed the children’sresponses to activities carried out with thewhole group. While children worked aloneor in smaller groups, I worked alongsidethem as requested by the teacher. I also occa-sionally conducted assessment activities withindividual children as requested by theteacher. Detailed notes of children’s responsesto tasks were kept.

The data was analysed by examining fieldnotes and extracting all those incidentswhich named individuals. These were reor-ganised to obtain a ‘personal record’ for eachchild providing information drawn over ayear for each individual, arranged chronolog-ically. Incidents were coded on each personalrecord according to the aspect of mathemat-ics concerned. The next step was to focus onexamples where there were a large numberof incidents for an individual featuring the

same aspect of mathematics. These incidentswere extracted and compared in order toexamine variations across the year. Thisprocess is exemplified in the sections belowwith particular reference to one child, Claire,and with examples drawn from otherchildren also shown for comparison.

FindingsOverview of activities. Initial examination ofthe personal records for each child gavesome indication of the curriculum coveredby this group of children and in particularwhich aspects of mathematics were revisitedseveral times. To illustrate this, the informa-tion from Claire’s personal record has beentabulated to show which aspects of mathe-matics she was observed studying across eachof the twenty-five weeks for which there wereobservations for her (see Table 1). Eachmark on the table indicates aspects of math-ematics covered in the notes, though thesevary from brief mentions to records of wholelessons containing several activities on thesame aspect. It is clear from the chart thatthe majority of time was spent on numberrather than on other aspects of mathematics.There are only six entries covering data han-dling, shape and space and measures andmost of these are very brief. This means it isimpossible to draw any conclusions aboutwhether Claire’s performance on numbertasks varied from her performance on otheraspects. The shape and space observation forWeek 11 related to Claire’s use of a com-puter program concerned with tables facts. Ithad been noted that Claire negotiated themaze with apparent ease. This raises a slightpossibility that Claire may be happier work-ing with spatial tasks but there is insufficientdata to examine this possibility. This was thesame for all children, with the vast majorityof observations being related to number. Forthis reason, the focus for the rest of thisarticle will be on variability within number.It is worth noting, however, that any childrenin this group with strengths in other aspectsof mathematics had few opportunities todemonstrate those strengths.


Jenny Houssart


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Focusing on number. Table 1 indicates alarge amount of data concerning Claire’sresponses to aspects of mathematics whichwere revisited frequently throughout theyear, such as addition, subtraction andcounting in twos, fives and tens. The nextstep in analysis was to look separately ataspects of mathematics for which there wassufficient data. Observations for chosenaspects were tabulated with brief detailsincluded of the task involved. At this pointtentative coding was used to indicatewhether the task was completed correctly(�), incorrectly (X), or whether theresponse was mixed (�). Because this repre-sents a fairly crude categorisation, somecomments were added to give details of theoutcome. Where responses were almostentirely correct a tick was used, but anysmall errors were noted. Similarly, ifresponses were almost entirely incorrect, across was used and any correct responsenoted. A cross was also used when noresponse was given, but this was noted. Thesymbol for mixed response was used for amixture of correct and incorrect answersand also for cases where work was com-pleted with adult help. Table 2 showsClaire’s response to addition tasks tabulatedin this way and Table 3 shows her responseto tasks involving counting in twos, fives andtens and multiples of two, five and ten.

Some observations can be made fromthese two tables. Firstly, it is clear that onseveral occasions, Claire did not provideanswers to questions or did not participate ingames or joint counting activities. It is notpossible to say for certain whether Clairecould have answered the questions or notand this presented a difficulty for the staffworking with her. Towards the beginning ofthe year, for example, when she did notanswer questions about the total number ofspots on a domino in Week 4, the staff wereconcerned that she did not understand thesevery basic ideas and was unable to answer.Later in the year, they felt that Claire some-times chose not to participate in tasks thatshe might be able to complete. There were

also a few occasions when Claire arrived atlessons distressed about matters outside ofmathematics and this may have been a factorin her lack of participation. The tables alsoshow some general tendencies. For example,Table 2 suggests that Claire achieved moresuccess in written addition tasks than in sim-ilar tasks presented in games or practical for-mats. Table 3 suggests that her counting andrecitation of tables was better when she waspicked to recite in front of the class or to anadult as an assessment activity. However, nei-ther of these patterns applies entirely. Gener-ally, both charts show a mixed performanceacross the weeks with Claire often failing toanswer or answering incorrectly, sometimeson tasks similar to those she had completedpreviously. This will be considered in moredetail later.

A similar process was carried out forother children and information is shown forSeth in Tables 4 and 5. Seth was one of theolder children in the group, and one of themost successful at mathematics, althoughthe adults remarked that he was not consis-tently successful. The charts for Seth differfrom those for Claire in several ways. Firstly,it was very rare for Seth not to provideanswers or participate in activities, so, whenhe did not answer correctly, incorrectanswers were given which sometimes shedlight on his difficulties. The chart for addi-tion also indicates that Seth usually com-pleted tasks correctly and sometimes didmore than was expected, for example withhis comments about addition of zero inWeek 4 and his systematic recording of possi-bilities in Week 7. Although Seth’s difficul-ties in Weeks 4, 14 and 17 have no obviousexplanation, it is not surprising that heneeded help with the second task in Week 20as it was harder than those carried out previ-ously. Seth also differs from Claire in that thetwo tables show a different picture. The tablefor place value suggests that he was lessconfident with this aspect and had some dif-ficulty with tens and units. This observationis supported by other entries on Seth’s per-sonal record. For example, Seth had difficul-


Jenny Houssart

ties in identifying multiples of two, five andten in Week 18. He ringed the number 205,suggesting it was a multiple of two, and whenchallenged about this said that it did end intwo. When asked to ring multiples of ten, heringed almost everything. Apparently, Seth’sperformance across aspects of number

was somewhat uneven, with his confidence in addition not being matched with hisunderstanding of place value. Care needs tobe taken here as it is acknowledged thatplace value is often more problematic thanaddition (Cockburn, 1999). However, Seth’sperformance on place value tasks was also



Week Task details Outcome Outcome details

Week 2 Addition to 10 X Does not answer

Week 4 Adding numerals from cards X Does not answer

Week 4 Adding spots on dominoes X Does not answer

Week 4 Totals to 10 X Does not answer

Week 5 Number walls (written addition) � Had adult help withfirst few examples,then worked alone

Week 6 Pairs to 10 (oral activity) – Mixture of correctand incorrect answers

Week 6 Making 10 (written activity) � 19 out of 20 calculationscorrect with no adult help

Week 8 Adding money (oral activity) � Correct answers, no help

Week 15 Addition to 20 (game format) X Not participating

Week 15 Number card addition to 20 – Initially incorrect butcorrected after adult help

Week 15 Addition dominoes X Not participating

Week 16 Make 10 worksheet (no help) X Mostly incorrect, heavyrubbing out

Week 16 Make 20 worksheet (no help) X Mostly incorrect, apparently tried to make use of a pattern

Week 16 Pairs to 20 (mental, then check X Incorrect answerwith calculator)

Week 16 Addition to 20, dice game X Not participating

Week 17 Make 10 and make 20 worksheet – Completed withintensive adult help

Week 20 Addition to 20 using number cards � Mixture of instantcorrect responses andcorrect responses aftercounting on fingers

Week 24 Dice addition – Task interrupted

Week 25 Addition of money (oral activity) � All answers correct

Table 2: Claire’s response to addition tasks. Key to outcomes: � � Mostly or entirely correct, – � Mixedresponse, X � Mostly or entirely incorrect or answers not given


Jenny Houssart


Week 5 Counting in 5s (game) X Incorrect answers

Week 5 Counting in 5s (missing – Correct answer with adult helpnumber game)

Week 7 Tape for 2x table – No response initially, joins in later, numbers are correctbut out of step with the questions

Week 7 Counting in 2s � Claire counts correctly up to 30s (joint oral activity) then misses out 36

Week 8 Counting in 5s – Intermittent participation(joint activity)

Week 8 5x table tape � Joins in enthusiastically andcorrectly volunteers to recite italone (not chosen)

Week 9 10x table tape � Joins in enthusiastically and correctly volunteers to say it alone

Week 11 2x, 5x and 10x tables as part – Needs extensive adult help to startof computer game with but later completes correctly

by counting in 2s, 5s and 10s

Week 12 2x table tape � Recited correctly alone asassessment task

Week 13 Counting in 2s (joint – Intermittent participationoral activity)

Week 13 Counting in 2s worksheet � Mostly correct, a few small errors

Week 14 Counting in 5s – Intermittent participation(oral activity)

Week 14 Counting down in 10s X Incorrect answers

Week 14 Counting in 10s, missing X Incorrect answersnumber game

Week 15 10x table � Correctly recites alone as assessment activity

Week 18 Counting in 5s (joint – Does not participate to startoral activity) with, then joins in correctly

Week 18 Multiples of 5, card activity – Intermittent participation

Week 18 Multiples of 5 (writing on board) � Completed correctly

Week 18 Multiples of 2, 5 and – Does not stay on task10 (worksheet)

Week 19 Counting in 2s, 5s and 10s � Completed correctly as individualassessment activity

Week 19 Counting in 2s oral activity – Initially incorrect, completes(high-starting numbers) with adult help

Week 23 Counting in 2s using coins X Incorrect answers, lots of adultencouragement

Table 3: Claire’s response to tasks involving counting in 2s, 5s and 10s and multiples of 2s, 5s and 10s

compared with that of others in this groupwho completed many of them correctlyyet had greater difficulties than Seth withaddition tasks.

Analysis of similar tasks. Further analysis wascarried out by extracting activities from eachaspect of mathematics which were as similaras possible. This led to a series of shorter

tables which tended to contain between fiveand ten activities. Carrying out this processfor Claire revealed a series of tables whichshowed mixed progress. Two of these tablesare shown as Table 6 and 7. Table 6 shows

those addition tasks which required Claire toadd up numbers rather than the harder taskswhich involved finding numbers that addedto a given total. This chart suggests that


Week 4 Adding spots on dominoes X Incorrect answers

Week 4 Domino addition � Discusses addition of zero

Week 5 Oral addition � Answers quickly andcorrectly in plenary

Week 7 Addition to twelve (written task) � All possibilities recordedsystematically

Week 8 Finding coins for given totals � Correctly answered fortotals of 10p, £1, 50p

Week 14 Addition to 100 – Some errors initially,corrected with adult help

Week 15 Make twenty (number cards) � Completed correctly,comments on connectionbetween 5 � 15 and 15 � 5

Week 16 Make ten worksheet (no help) � Completed without mistakes

Week 16 Make twenty worksheet (no help) � Completed without mistakes

Week 16 Make ten and make � Appeared to complete easilytwenty worksheet

Week 16 Adding three numbers � Completed correctly

Week 17 Make ten and make – Make ten part correctedtwenty worksheet easily, slows down and

makes some mistakes onmake twenty part

Week 20 Addition program on computer � Completed correctly and helped another child

Week 20 Two-digit addition using – Correct on 16 � 15, neededhundred square help with 32 � 33

Week 24 Adding four numbers � Completed correctly

Week 24 Dice addition � Completed correctly,number lines were availablebut Seth appeared to makeuse of known facts sometimes

Week 25 Oral addition � Seth appeared to find thesequestions (e.g. 3 � 2) easy

Table 4: Seth’s response to addition tasks



Claire’s performance improved over theyear, though with some variation across theweeks. It also shows, as mentioned before,that she was more likely to leave questionsunanswered than to answer incorrectly. Shedid also sometimes need adult help to startwith. For example, in the number walls activ-ity in Week 5 she used a number line to add,and although she was secure about startingat the first number rather than zero, sheneeded reminding to count one on the firstjump rather than on the starting number.After this initial help, Claire completed this

written activity correctly. The tables forClaire suggest that she tended to participateless enthusiastically in practical and gameactivities. The detailed observations forClaire contained comments that supportthis. For example, in Week 4, Claire andsome other children struggled with theintroduction to addition via spots on domi-noes. For this reason, the adults decided to

abandon the planned worksheet and carryon with the practical introduction. At thispoint, Claire said ‘Are we doing games all


Jenny Houssart


Week 2 Putting tiles on hundred square – Initially made mistakesbut completed correctlyafter adult explanation

Week 2 Making numbers with arrow cards X Confused tens and units

Week 3 Putting tiles on hundred square – Made initial mistakes,corrected after adult help

Week 3 Questions about hundred square – Mixture of correct andincorrect answers

Week 7 Representing numbers with � Completed correctlytens and units pieces

Week 7 Hundred square jigsaw � Completed quickly andcorrectly

Week 8 Counting stick X Answered questionsincorrectly

Week 16 Making numbers with cards � Could make and readthree-digit numbers

Table 5: Seth’s response to place value tasks

Week 4 Adding numerals from cards X Does not answer

Week 4 Adding spots on dominoes X Does not answer

Week 5 Number walls (written addition) � Had adult help with first fewexamples, then worked alone

Week 8 Adding money (oral activity) � Correct answers, no help

Week 15 Addition dominoes X Not participating

Week 24 Dice addition – Task interrupted

Week 25 Addition of money (oral activity) � All answers correct

Table 6: Claire adds single-digit numbers

day?’ Much later in the year, when work wasbeing discussed at the beginning of a lesson,Claire made the comment ‘I love sums, I lovewriting in my book’. It appears that Claireconsidered calculations written in books oron the whiteboard as ‘real maths’. AlthoughClaire’s attitude was extreme, it is echoed ina more moderate form in the findings ofGregory, Snell & Dowker, (1999) who carriedout an international study about attitudes tomathematics and suggested that childrenmay see written sums as a core aspect of thesubject.

Table 7 deals with activities in whichClaire was asked to count in twos or identifymultiples of two. It is not surprising thatClaire found the activity in Week 11 harder,as this computer game required her to mul-tiply given numbers by two, whereas the nor-mal activity was to chant multiplication factsin order only. The second task carried out inWeek 19 was also harder than some of theothers, as Claire was asked to count in twos

starting from 74. It is less easy to explainClaire’s difficulties in Week 23, when askedto count in twos as the teacher dropped 2pcoins in a tin. 12p was dropped in the tin andClaire said the amount was 50p. The activitywas repeated with intensive adult help.Looking at these activities for Claire suggestsvariability between occasions which cansometimes but not always be explained bylooking at differences in the way tasks werepresented or in the mathematics involved. Inmany cases, Claire’s incorrect answers couldnot be explained by considering commonerrors or misconceptions and it is possiblethat she sometimes gave answers based onthe idea of arithmetic as an arbitrary game asoutlined by Ginsburg (1977). For somechildren, very similar activities were identi-fied, often over consecutive weeks, to see ifthere was still variability.

There were several such cases wherechildren were recorded successfully complet-ing a mathematical task one week and then



Week 7 Tape for 2x table – No response initially, joins inlater, numbers are correct butout of step with the questions

Week 7 Counting in 2s (joint oral activity) � Claire counts correctly up to 30s then misses out 36

Week 11 2x, 5x and 10x tables as part – Needs extensive adult help toof computer game start with but later completes

correctly by counting in 2s, 5sand 10s

Week 12 2x table tape � Recited correctly alone as assessment task

Week 13 Counting in 2s (joint oral activity) – Intermittent participation

Week 13 Counting in 2s worksheet � Mostly correct, a few small errors

Week 19 Counting in 2s, 5s and 10s � Completed correctly as individual assessment activity

Week 19 Counting in 2s oral activity – Initially incorrect, completes (high-starting numbers) with adult help

Week 23 Counting in 2s using coins X Incorrect answers, lots of adultencouragement

Table 7: Claire counting in twos and multiples of two

experiencing difficulty with a very similartask the following week. It was also some-times recorded that children who did notappear to understand a piece of mathemat-ics however it was presented in one lessonwere able to cope with it in the next lesson.The following example concerns the per-formance of one child, Douglas, acrosssimilar tasks on three consecutive weeks.

Douglas was amongst those in the grouphaving greatest difficulty with mathematicsand his record reflected this. In an activityconcerning addition pairs which made 20,Douglas did not answer when asked whatshould be added to 15. In a later activity inthe same lesson, each child was given anumber tile and asked to find anothernumber tile to go with their tile to make 20.Douglas was given 19 but didn’t answer. Theteacher asked ‘How many do you need tomake 20?’ There was no answer, so theteacher said ‘Count on’. Douglas said, ‘20’.The teacher asked, ‘How many?’ and he said‘20’. A number line was found and used todemonstrate. Eventually, Douglas gave therequired answer: 1. This and similar inci-dents led me to conclude that Douglas couldnot provide the missing number in additionpairs. However, the following week I wasproved wrong. The children were given anumber and asked ‘What has to be added toit to make 20?’ They had calculators whichthey were allowed to check the addition withafter the numbers had been suggested. 15was given as the first number in the activityand Douglas correctly suggested 5 for thesecond number. The calculator was not usedto give Douglas the answer, merely to check,though this incident was in keeping with oth-ers which suggested he did better in an activ-ity involving technology even when thetechnology did not actually do the mathe-matics for him. This activity suggested to methat contrary to the evidence of the previousweek, Douglas was able to understand theidea of pairs of numbers with a given total.The following week the idea of pairs to 20was introduced in a different way. Thechildren had a worksheet on which they had

to ring and join pairs of numbers to total 20.Douglas made little progress with the sheetso I explained what he had to do and pickedsome individual numbers asking him for thepairs. The sheet was eventually completedwith a very high level of help.

Although it is not possible to be certainwhy this variability occurred, it seems likelythat part of the explanation lies in the waythe task was presented. Perhaps Douglas wasmotivated by calculator use and presentingthe task in that context maximised his poten-tial. Perhaps Douglas understood the under-lying idea but declined to cooperate whenthe task was presented in other ways, ormaybe he was genuinely unable to under-stand the format of the worksheet. Douglaswas similar to Claire in responding differ-ently when tasks were presented in differentways. This was true for others in the group.Some, in contrast to Claire, succeeded inmental calculation but had more difficultywhen the same calculations were presentedin written form. However, task presentationdid not explain all or even most of the casesof variability. There were many exampleswhere children were successful on a task oneweek but failed to carry out a similar taskpresented in a similar way the followingweek. Variability also occurred withinextended tasks carried out on one occasionas shown in the examples which follow.

Analysis of single tasks. Activities in thisclassroom commonly involved counting intwos, fives or tens. Often this counting wascarried out as a group but occasionallychildren were asked to count alone with theadults and other children listening. On oneoccasion Neil was asked to count alone infives to 100s. He started slowly and deliber-ately. He started 5, 10, 15, 20, 13. Asked to tryagain, he said 20, 25, 40 and was then correctto 70. He was unsure whether 74 or 75 wasnext and was helped by the adults. He endedwith 75, 80, 85, 100.

It was common for the teacher to com-plete a checklist concerning counting whenchildren were asked to count alone in this


Jenny Houssart

way. However, it was not clear whether Neilcould be recorded as able to count in fives to100. There are several possible reasons forhis slight difficulties. His slow start suggestedthat he may have been mentally adding onfive every time. The 13 is harder to explain,but could have been a combination ofskipping the 25 and confusing 13 with 30.Since Neil managed to count from 20 to 70correctly, he may have been aware of thenumber pattern involved. However, heconsidered 74 as a possibility suggesting thathe had not seen the pattern. Omitting 90and 95 may have been a mathematical error,an accidental slip or a desire to get to 100 asrequested.

On another occasion, the group wasworking on multiples of five and childrenhad been invited to write multiples of five onthe board. The numbers 55 and 80 were writ-ten and the teacher said one of those wasalso a multiple of ten. Michael quickly puthis hand up and answered 80. The teacherpraised him and asked him to write anothermultiple of 10 on the board. He wrote 56.

Michael’s enthusiasm for answering theteacher’s initial question and his correctanswer of 80 had suggested to the adults thathe could recognise multiples of ten. Theteacher’s intervention was important and waspresumably designed to confirm Michael’understanding. However, it had the oppositeeffect, leaving us wondering if he understoodmultiples at all. His correct answer couldhave been a guess, especially since he had twonumbers to choose from, though he seemedconfident and was not obliged to answer. The56 is harder to explain, though childrenoccasionally did activities related to hundredsquares where they were asked to identify thenext number and writing 56 on the boardnear to 55 would have been correct if thathad been the activity.

Summary of findings. The above incidentsare selected to show examples of differenttypes of variability. Variability across lessons, asillustrated by Douglas, was common, andcould sometimes be explained by factors such

as task presentation. In many cases, more thanone explanation was possible. Variabilitywithin individual incidents, as shown by Neil &Michael is harder to explain, suggesting thatreal difficulty exists in trying to determinefrom such an incident whether or not a child‘can do’ a piece of mathematics or evenwhether they did it successfully on that occa-sion. In other words, in general there was noexplanation available in terms of task presen-tation or indeed any other obvious factor formost cases of variability.

The adults who worked in the classroomwere well aware of the issue of variability, espe-cially for children such as Douglas & Claire,for whom this was a marked issue. My analysisof personal records indicates that variabilitywas actually present for all children, thoughin some cases it was much less frequent andwas not necessarily evident as part of normalclassroom assessment. Discussions amongstthe adults often attempted to explain variabil-ity in terms of factors such as task preference,lack of concentration and other personal orsocial factors. Such discussions often led tothe crucial issue of what should be done nextas far as planning with this group of childrenwas concerned; a tension existed betweenreinforcement and repetition or moving on,with perceived pressure to move on to harderaspects of mathematics.

Issues of mastery and progression arerelated to this dilemma. Some believe it desir-able to ‘master’ a piece of mathematics, thatis, by performing consistently. For some, ‘mas-tery’ is a pre-requisite for moving on. The ideaof progression in mathematics is based formany on the belief that any new piece ofmathematics can only successfully be learnedwhen those preceding it have been under-stood thoroughly. However, my findings sug-gest that this may be an unrealistic aspiration.

ImplicationsThese findings have clear implicationsfor practitioners by casting doubt on theusefulness of assessments conducted on sin-gle occasions, especially if they are based ononly one item only of each type. There are



dangers in extrapolating such assessments tomake statements about what a child ‘can’ or‘cannot’ do. It could rather be argued thatcontradictory assessment can be more usefulin diagnosing difficulties and in planning byindicating that children respond better tocertain types of activity, or have difficultieswith specific aspects of arithmetic or withparticular methods. This information can beused to structure appropriate activities, andto inform adults about those activities theyare likely to need particular help with. How-ever, my findings suggest that it is unrealisticto expect busy classroom practitioners tocompile detailed pictures of all children andreach appropriate conclusions. This is anextremely time-consuming task, and inter-pretation is problematic. Thus, althoughdetailed assessment can inform teaching itcan not be relied upon.

A further issue is whether the variabilityobserved applies particularly to groups ofchildren considered to have learning difficul-ties or whether it applies more widely. It is notpossible to answer this question from this data,but it is useful to speculate about why variabil-ity in performance was apparently so markedin this classroom. One possible reason is thatvariability is more common amongst childrenwho experience difficulties in learning arith-metic. Another is that the adults concernedwere in the unusual position of being able toobserve children closely as they worked on sim-ilar tasks over a long period of time and there-fore more able to observe variability, whichcould be present but less noticeable in othersituations. It is interesting to note that myanalysis of personal records detected variabilityamongst all children, even those for whom itwas not evident from normal classroom obser-vation. It appears possible that variability is anatural part of learning and is present in class-rooms, for all children.

Perhaps the key issue for practitioners ishow to proceed in situations similar to theone described. Teachers need to make deci-sions about when to move on to harderaspects of mathematics and when to repeator reinforce ideas. Some views of mathe-matics and learning point to the conclusionthat individual aspects should be mastered tothe point where performance is accurateand automatic before moving on. Perhapsthe fact that the children concerned areconsidered to be low attainers makes thisoption more tempting. However, my findingssuggest that aiming for mastery before mov-ing on is unrealistic and likely to be demor-alising for all concerned. In moving forward,however, teachers need to be aware thatreminders are often required. This could beseen as a positive step with reinforcementtaking place as required in order to enableprogress to be made rather than being seenas an end in itself. Frequent repetition oftasks can also produce a reaction in somechildren. Using micro genetic methods witha group of ten children, Siegler & Jenkins(1989) had to stop working with two of theten children because of the way they reactedto the repetition of tasks. One child becameover-anxious about succeeding on the tasks,whereas the other apparently became boredand gave evidence of not trying. It is possiblethat the situation in our classrooms in whichwork is frequently repeated brings about asimilar reaction in children. Ironically, thisrepetition is often carried out in a search formastery and automaticity.

Address for correspondenceDr J. Houssart, Institute of Education, 20 Bedford Way, London WC1H 0Al.E-mail: [email protected]


Jenny Houssart

Black, P. (1998). Testing: Friend or foe? Theory and practiceof assessment and testing. London: Falmer Press.

Chinn, S. (2004). The trouble with maths. London:Routledge-Falmer.

Clausen-May, T. (2005). Teaching maths to pupils withdifferent learning styles. London: Paul ChapmanPublishing.

Cockburn, A. (1999). Teaching mathematics with insight:The identification, diagnosis and remediation of youngchildren’s mathematical errors. London: FalmerPress.

Denvir, B., Stolz, C. & Brown, M. (1982). Low attainersin mathematics, 5–16: Policies and practices in schools.London: Methuen Educational.

DfEE (1999). The National Numeracy Strategy, frame-work for teaching mathematics from reception to Year 6.Sudbury: DfEE Publications.

Dowker, A. (2004). What works for children with mathe-matical difficulties? (DfES Research Report 554). Not-tingham: DfEE Publications.

Dowker, A. (2005). Individual differences in arithmetic:Implications for psychology, neuroscience and educa-tion. Hove and New York: Psychology Press.

Gabb, J. (2005). Equals at BCME 6. Equals, 11(2),22–23.

Ginsburg, H. (1977). Children’s arithmetic: The learningprocess. New York: D. Van Nostrand Company.

Gregory, A., Snell, J. & Dowker, A. (1999). Youngchildren’s attitudes to mathematics: A cross-culturalstudy. Paper presented at the Conference onLanguage, Reasoning and Early MathematicalDevelopment, University College London.

Haylock, D. (1991). Teaching mathematics to low attain-ers, 8–12. London: Paul Chapman Publishing.

Houssart, J. (2004). Low attainers in primary mathe-matics: The whisperers and the maths fairy. London:Routledge Falmer.

McGarrigle, J. & Donaldson, M. (1974). Conservationaccidents. Cognition, 3, 341–350.

McGarrigle, J., Grieve, R. & Hughes, M. (1978). Inter-preting inclusion: a contribution to the study ofthe child’s cognitive and linguistic development.Journal of Experimental Child Psychology, 26,528–550.

Piaget, J. (1952). The child’s conception of number. Lon-don: Routledge and Kegan Paul.

Piaget, J. (1953). How children form mathematicalconcepts. Scientific American, 189(5), 74–79.

Robbins, B. (2000). Inclusive mathematics 5–11.London: Continuum.

Siegler, R. (1996). Emerging minds: The process of changein children’s thinking. New York: Oxford UniversityPress.

Siegler, R. & Crowley, K. (1991). The microgeneticmethod: A direct means for studying cognitivedevelopment. American Psychologist, 46(6), 606–620.

Siegler, R. & Jenkins, E. (1989). How children discovernew strategies. Hillsdale, New Jersey: Lawrence Erl-baum Associates.

Siegler, R. & Stern, E. (1998). Conscious and uncon-scious strategy discoveries: A microgenetic analy-sis. Journal of Experimental Psychology, 127(4),377–397.

Watson, A. (2000). Mathematics teachers acting asinformal assessors: Practices, problems andrecommendations. Educational Studies in Mathe-matics, 41, 69–91.

Watson, A. (2001). Making judgments about pupils’mathematics. In P. Gates (Ed.), Issues in mathemat-ics teaching. London: Routledge Falmer.

Watson, A. (2006). Raising achievement in secondary math-ematics. Maidenhead: Open University Press.

Wright, R., Martland, J. & Stafford, A. (2000). Earlynumeracy: Assessment for teaching and intervention.London: Paul Chapman Publishing.

Wright, R., Martland, J., Stafford, A. & Stanger, G.(2002). Teaching number: Advancing children’s skillsand Strategies. London: Paul Chapman Publishing.



References


FLUENT CALCULATIONS are impor-tant cornerstones in mathematicallearning and are also essential in many

everyday situations. The learning developsgradually. First, when learning to solve simplearithmetic problems, children use calculationstrategies based on counting such as fingercounting or verbal counting (Siegler &Shrager, 1984). Initially, children count bothnumbers presented in the addition – counting-all procedure (Fuson, 1982; Geary, Hamson& Hoard, 2000). Later, children shift tocounting on from the cardinal value of thefirst (counting-on max) or larger number(counting-on min) presented, which is amore efficient strategy (Geary et al., 2000).Frequent successful use of counting strate-gies increases memory representations ofarithmetical facts and leads to a strategy ofretrieving arithmetical facts from long-term

memory (Barrouillet & Fayol, 1998; Siegler &Shrager, 1984). In age-appropriate develop-ment of arithmetical skill children usuallystart using fact retrieval as the main strategyby the age of nine years. However, somechildren have difficulties in acquiring theskill of fluent calculation. For example,children with specific language impairment(SLI) have been found to have difficulties inshifting from counting-based strategies to afact retrieval strategy (Fazio, 1999; Koponen,Mononen, Räsänen & Ahonen, 2006). Thesefindings suggest that learning arithmeticalfacts relies on linguistic processing. This ideahas also been presented in models of numberprocessing (e.g. Dehaene & Cohen, 1995).

Models of number processing state thatarithmetical facts are stored in an associativenetwork (e.g. Ashcraft, 1992; Campbell &Clark, 1988; Siegler & Shrager, 1984). Ashcraft’s

Language-based retrieval difficultiesin arithmetic: A single caseintervention study comparingtwo children with SLITuire Koponen, Tuija Aro, Pekka Räsänen & Timo Ahonen

AbstractThe aim of this single-case intervention study was to examine whether difficulties in fluent language-basedretrieval are related to learning to retrieve arithmetical facts from long-term memory. Two 10-year-oldFinnish-speaking children considered to have Specific Language Impairment (SLI) were trained individu-ally twice a week for two months using computerised game-like addition tasks. The participants werematched for non-verbal reasoning and non-verbal numerical skills as well as linguistic skills (verbal short-term memory, comprehension and vocabulary). The key cognitive difference between the participants wasnaming fluency. Child A had difficulties in fluent language-based retrieval while child B’s performance onthe same task was close to age-mean. A multiple baseline across-subjects design was used, with three base-line assessments and three follow-up assessments. Before the intervention both children used finger-count-ing strategies only. During the intervention child B progressed from finger-counting strategies to factretrieval, while child A continued to use finger counting only. The results suggest that the benefit of anintervention programme, focusing on teaching fluent calculation skills with simple additions, is related tothe specific features of the child’s language competencies. It is proposed that an inability to shift from a fin-ger counting to a fact retrieval strategy is connected to difficulties in fluent language-based retrieval fromlong-term memory.


Laguage-based retrieval difficulties in arithmetic

(1992) network retrieval model proposes thatarithmetical facts are represented in memoryin an organized network from which accessand retrieval occur via a process of spreadactivation. The strength with which the nodesare stored and interconnected is a functionof the frequency of occurrence and practice.How these nodes are represented remains anopen question. However, there is theoreticaljustification as well as empirical evidence forboth modular (e.g. McCloskey, Caramazza &Basil, 1985; Butterworth, 1999; Temple &Sherwood, 2002; Landerl, Bevan & Butter-worth, 2004) and non-modular (Campbell &Clark, 1988; Dehaene & Cohen, 1995) repre-sentations of arithmetical facts. Butterworth& colleagues (Butterworth, 1999; Landerlet al., 2004) propose that the numerical,rather than other cognitive domains, is cen-tral in arithmetical fact retrieval deficit. Incontrast, Campbell & Clark (1988) take theview that the associative network of arithmeti-cal facts includes multiple numerical codes,(e.g. phonological, semantic and visual)which are interconnected. Dehaene andCohen (1995) go even further by proposingthat arithmetical facts are stored in verbalform. The goal of the present study was totest the theory that simple calculation is a lan-guage-connected skill. Of particular interestwas whether problems in fluent language-based retrieval are related to learning toretrieve arithmetical facts from long-termmemory. This question was explored in a sin-gle case intervention study with two partici-pants considered to have specific languageimpairment (SLI).

Only a few studies have focused on themathematical skills of children with specificlanguage impairment (SLI). These studiesshow that children with SLI lag significantlybehind their age peers in arithmetical skills(e.g. Arvedson, 2002; Fazio, 1999; TiecheChristinat, Conne & Gaillard, 1995). Fluentcalculation skills, in particular, are one ofthose basic numerical skills that seem to behard to acquire if the child has a languageimpairment (Fazio, 1999; Koponen, et al.,2006). Fazio (1999) reported that compared

to their age peers, 9- to 10-year-old childrenwith SLI had more problems when fast arith-metical fact retrieval was demanded. Koponenand colleagues (2006) found that only 31per cent of 9- to 11-year-old children with SLIused fact retrieval as the main strategy in sin-gle-digit calculations. Most of the childrenwith SLI used counting-based strategies,despite having practised simple calculationfor several years. Rapid serial naming ofobjects and colours was the only one of thevariables that explained the differencesbetween the children who mainly usedretrieval and those who used counting-basedstrategies in single-digit calculations. Thechildren who calculated simple additionsand subtractions slowly also named coloursand objects more slowly than the childrenwho calculated fluently. Also, Temple andSherwood (2002) found that a group ofchildren with arithmetical difficulties wereslower at colour and object naming thancontrol children, although the authors didnot claim a causal relationship betweenrapid naming and difficulties in factretrieval. These findings of language-basedretrieval difficulties are in line with thetheory of Geary (1993), who has suggestedthat representing and retrieving phonologicalinformation from long-term memory mayunderlie problems in learning arithmeticalfacts as well as coexistence with readingdifficulty. The results of his later study(Geary, Hamson & Hoard, 2000) supportthis theory while suggesting that poor inhibi-tion of irrelevant associations might alsocontribute to fact retrieval difficulties (seeBarrouillet & Fayol, 1998).

The present study was concerned with theimpact of language-based retrieval difficultieson arithmetical acquisition. To better under-stand this connection, which so far has mainlybeen investigated via group studies, an inter-vention study design was used. Two SLIchildren with different abilities in language-based retrieval received an arithmetical inter-vention in order to examine whether thecapacity to learn arithmetical facts is relatedto language-based retrieving ability.

MethodParticipantsThe participants were two 10-year-oldFinnish speaking children (1 boy, 10; 10years and 1 girl, 10; 5 years) with specific lan-guage impairment comprising both recep-tive and expressive language. They had beendiagnosed by a phoniatrist using the ICD-10criteria (WHO, 1993). They had participatedin an extended compulsory education pro-gramme and were attending a state schoolfor children with SLI. They were studyingmathematics according to their individualeducational plan (IEP), using special educa-tion textbooks (concise versions of the stan-dard books) for second graders (aged 8 to 9in Finland). The children were drawn from alarger sample collected for a group assess-ment study conducted in spring 2002 (seeKoponen et al., 2006). These two participantswere chosen on the grounds that, while theyboth had calculation difficulties (calculationfluency was at the level of 6-year-oldpreschoolers without any formal educationin arithmetic), only one of them had diffi-culty in fluent language-based retrieval. Theywere matched in non-verbal reasoning skill,linguistic skills (digit span, sentence compre-hension and vocabulary), and non-verbalnumerical skills (comprehension of num-bers presented in Arabic numerals and asplay-money and estimation). In addition, thePaired-Associate Word Learning Test (WMS-R, Wechsler, 1995) was administered inorder to determine the participants’ abilityto learn verbal associations. Their raw andstandard scores are presented in Table1.

DesignThe intervention study was run in threephases: pretests, intervention and posttests(see Figure 1). A multiple baseline across-subjects design was used (Kazdin, 1982) toexamine the effects of this computer-aidedintervention on the calculation skill acquisi-tion of two children with SLI. Child A (male)had a one-week interval between the first andsecond and a three-week interval betweenthe second and third pre-intervention assess-

ments, while Child B (female) had threebaseline pre-intervention assessment ses-sions in eight days, after which the interven-tion was applied. The differences in theintervals between the baseline assessmentswere designed to enhance the validity ofinterpretation of the intervention effect,thus enabling it to be more powerfullyargued that the improvements found in thechildren’s calculation performances weredue mainly to the intervention and not, forexample, to teaching at school or matura-tion during the intervention.

The effects of the intervention wereassessed at three follow-up assessments. Bothparticipants had their first post-interventionsession three days after their last interven-tion and their second and third assessmentsone week and one-month, respectively, afterthe first post-intervention assessment.

MeasuresAt each of the six assessments the childrenwere presented with 31 simple additions tobe done using a computer. The childrenheard instructions through headphones andviewed examples of the task on the screen.The answers and time taken for each trial ofevery task were recorded on the computer.The computer-administered tasks wereconstructed and carried out using theNEURE program, a graphical experimentgeneration tool (see http://www.edu.fi/oppimateriaalit/neure). Each testing sessionwas also recorded on video.Addition task. Single-digit additions usingpermutations from one to nine were pre-sented one at a time in a horizontal formaton a computer screen. The additions were inrandom order. The children were instructedto answer as quickly as possible by pressingany key on the keyboard. After pressing a keythe addition disappeared and a box for theanswer appeared with the cursor in it. Thechildren were instructed to write the answerand to mouse-click an on-screen “OK” button.After giving the answer the children wereasked how they calculated this addition. Thedifferent strategies (fact retrieval, counting,


Tuire Koponen et al.

counting on the fingers, decomposition)were introduced and gone through with thechildren before the testing. The testing ses-

sion was also recorded on video and thechildren’s strategies were confirmed after-wards. The task consisted of two training

ChildrenChild A Child B

Task Raw score Standard score Raw score Standard score(M � 10, Sd � 3) (M � 10, Sd � 3)

Language skill

Digit Spana 6 3 6 3Comprehensionb 13 4 14 6Vocabulary c 38 NA 33 NANaming colors d 57 5 43 10Naming objects d 72 1 52 9Paired-Associate 9/22 N/A 14/36 N/AWord Learning Teste

RCPM (IQ)f 24 74 25 78Non-verbal number skillComparison of numbersg

Pre 11 6Post 12 7

Comprehension 10 10of numbersh

Estimationi 5 6

Table 1: Background measures. Note : a � Digit Span task (Wisc-III; Wechsler, 1991), raw score.b � Sentence comprehension (NEPSY; Korkman, Kirk & Kemp, 1997), raw score, max. 21.c � Word finding vocabulary test (Renfrew, 2001), raw score, max. 50. d � Rapid automatisednaming (colors and objects, RAN; Denkla & Rudel, 1974; the Finnish version Ahonen,Tuovinen & Leppäsaari, 1999), time in seconds. e � Paired-Associate Word Learning Test(WMS-R), raw score of 3 and 6 trials, max. 24/48. f � Raven’s Coloured Progressive Matrices(Raven, 1993), raw score. g � Number comparison, raw score, max. 15. h � Moneybag, rawscore, max. 20. i � Estimation, raw score, max. 20.

Figure 1: Multiple baseline design



items and 31 assessment items. There werethree parallel versions, each with 16 identicaland 15 different additions. The 15 differentadditions were selected so that each was ofthe same average level of difficulty accordingto Wheelers ranking (see Wheeler, 1939).Three scores were derived: calculationspeed, accuracy and fact retrieval strategy.

Intervention. The intervention was con-ducted twice weekly for a period of abouttwo months, each session lasting 45 minutes.The first author conducted all the interven-tion sessions during class time in a secludedroom at the school.

Previous studies (e.g. Tournaki, 2003)have shown that children with mathematicallearning difficulties benefit more fromstrategy instruction than from instructionthrough drill and practice. In the presentintervention study the children receivedinstruction and training in the use of moreefficient calculation strategies, in accor-dance with the typical development of calcu-lation strategies such as learning to use thecounting-on min instead of counting-on maxstrategy. A faster counting strategy increasesthe possibility of having all the terms presentat the same time in working memory inorder to form and store the associationsbetween the terms. Faster counting may alsoincrease calculation accuracy by shorteningthe counting process – a rather error-proneprocedure. The counting-on min strategywas introduced by using two kinds of prac-tices: first, the order-irrelevance principle inaddition was introduced. This was done withconcrete material by demonstrating thatwhen two sets containing items x and y arecombined, the final amount is always thesame, despite the arrangement or order ofthe objects (e.g. 2 red and 3 blue or 2 blueand 3 red balls; you have 2 balls and you get3 more or you have 3 balls and you gettwo more). After that the same idea was pre-sented in the addition context with numbersand the symbol for the operation: the sum ofa certain two numbers is always the same,despite the order in which they are pre-

sented (e.g. 2 � 3 � 3 � 2 � 5). Second, thechild was asked to calculate the addition 2 � 9and write down the answer. Next, the childwas asked to calculate 9 � 2 and write theanswer. After that it was discussed with thechild whether the answer was the same ordifferent, why it was the same (relating thispractice with the earlier one), which way itwas faster and easier to calculate and howthey could use this information when calcu-lating other additions like 3 � 9 or 4 � 7.During the computer games the childrenwere observed and reminded to use thecounting-on min strategy if they used thecounting-on max strategy.

Direct retrieval was trained both withcomputer tasks and board games. The taskswere of three kinds. In the first, mathemati-cal problems were presented in visual quan-tities without any arithmetical symbols. Thechildren were encouraged to identify thequantities without counting, name themand say the sum aloud. After the childrenhad solved the problem correctly, positivefeedback was given and the same problemwas presented with arithmetical symbols andnumbers. In the second, mathematical prob-lems were presented both in visual quanti-ties and in arithmetical symbols. Thechildren had to solve the problem and thenwrite the correct answer by pressing thenumber key. In the third, mathematicalproblems were presented in arithmeticaland number symbols without the correspon-ding visual quantities. Moreover, there waspractice in such tasks as memorization ofpairs of numbers which equal a givennumerical outcome (e.g. ‘try to find allnumber pairs which equal 4’). In these exer-cises counting-on strategies did not adaptwell; hence the aim of this task was tostrengthen the memory-associationsbetween the addends and the answers withthe support of the relevant conceptualknowledge. All the training tasks were sin-gle-digit additions using permutations fromone to nine.

Throughout the intervention thechildren were encouraged to retrieve



answers from their memory. The researchergave additional positive feedback when thechildren retrieved a fact from memory whenplaying the computer and board games.However, strong pressure or prohibitionfrom using counting-based strategies wasavoided. Both of the children were wellmotivated and co-operative and participatedwillingly in each intervention session.

Data AnalysisThe analysis was performed in three steps.The first examined whether there are anysignificant differences on the children’scalculation performances (on calculationspeed, accuracy and fact retrieval strategy)between the six assessment sessions. Thesecond examined whether the children’scalculation performances within the base-line and follow-up were stable. The thirdcompared the children’s pre and postintervention performances. For this pur-pose the baseline data were combined aswere the follow-up data. The baseline vari-able thus included all three baseline assess-ments (93 additions) and the follow-upvariable all three follow-up assessments (93additions).

Raw scores for calculation speed weresubjected to a non-parametric Friedman Test,which is a nonparametric equivalent of aone-sample repeated measures design. Boththe number of assessments (k) and samplesize (N) exceeded five and so the value ofstatistic (denote as Fr) was distributedapproximately as �2 with df � k � 1 (Siegel &Castellan, 1988). The raw scores for calcula-tion accuracy and fact retrieval strategy weresubjected to the non-parametric Cochrantest, which is identical to the Friedman testbut is applicable where all the variables arebinary. The sample size (N) exceeded fourand the product Nk was greater than 24 andso the value of statistic (denote as Q) wasdistributed approximately as �2 with df � k � 1(Siegel & Castellan, 1988). The non-parametric tests were used because thevariables were not normally distributed andthe database was small. The medians for

calculation speed as well as percentage ofaccuracy and use of the fact retrieval strategyat each baseline and follow-up assessmentsession are presented in Figure 2.

ResultsThe test for the possible effect of the assess-ment sessions on calculation speed showed asignificant Fr value for both children (childA Fr � 36,27, p .001 and child B Fr � 23,72, p .001), while for the effects on calcu-lation accuracy and fact retrieval strategysignificant differences were found for childB (Q � 29,48 exact p .001 and Q � 44,79,exact p .001), but not for child A (exactp � .05).

Calculation performance during the baselineand follow-upIn calculation speed both children had astable baseline (exact p � .05), with nosignificant differences in the children’scalculation speed between the three preassessments. Both children also had a stablefollow-up, with no significant differences intheir calculation speed between the threepost assessments.

In calculation accuracy child B had botha stable baseline (exact p � .05) and stablefollow-up (exact p � .05), with no signifi-cant differences in her calculation accuracybetween either the three pre assessments orbetween the three post assessments.

During the baseline assessments child Bdid not use the fact retrieval strategy at all,and thus her baseline score was not sub-jected to analysis. During the follow-up Bshowed significant differences in the use ofthe fact retrieval strategy in the first and sec-ond compared to the third post assessmentsession (exact p .05 and exact p .01),retrieving fewer facts from her memory inthe first (23 per cent) and second (10 percent) than in the third (42 per cent) assess-ment. At the time of the second post assess-ment session, child B’s teacher reported thatshe had had a ‘bad day’, and been upset andtearful, which most likely influenced her per-formance during the session. Nevertheless,



the assessment was done, as reschedulingwas not possible.

Comparison of calculation performance beforeand after the intervention. Child A showedsignificant differences in calculation speedbetween the baseline assessment and follow-up (Fr �25,04, exact p .001), being fasterafter the intervention was applied. As before,there were no significant differences in accu-racy or the use of the fact retrieval strategy.

Child B showed significant differences incalculation speed, accuracy and usage of thefact retrieval strategy (Fr � 21,77, exact p

.001, Q � 24,14, exact p .001, Q � 23,00,exact p .001), being faster, more accurateand using significantly more fact retrievalafter than before the intervention.

At the baseline both children usedcounting-based strategies, particularly thecounting-on max strategy. They took the firstaddend by lifting an equivalent number offingers without counting and then joined itto the second addend by lifting one finger ata time while counting (e.g. 3 � 5 � 3, 1, 2, 3,4, 5) until they got the answer (8).

The intervention did not help child A toshift from counting to fact retrieval. He didnot use the fact retrieval strategy in the firstor second post assessment at all. At the thirdpost assessment he remembered only oneanswer (8 � 1 � 9); using a ‘followingnumber’ rule rather than fact retrieval tosolve it. However, he was able to learn thecounting-on min strategy (instead of count-ing-on max strategy), although he used it



Figure 2: Calculation performance of child A and child B

Calculation speed, accuracy and fact retrieval strategy of child A

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only in the first and second but not in thethird post assessment.

During the intervention child B was ableto shift from finger-counting strategies to factretrieval. During the three follow-up sessionsshe used the fact retrieval strategy to solve 23per cent, 10 per cent and 42 per cent of allitems. In addition she was able to learn thecounting-on min strategy (instead of count-ing-on max strategy) and used it in all the postassessment sessions.

DiscussionPrevious studies have shown that childrenwith SLI have difficulties in retrieving arith-metical facts from memory (Fazio, 1999;Koponen et al., 2006). Koponen and others(2006) found that rapid serial naming ofobjects and colours differentiated thosechildren with SLI who after several years ofpractice still depended on slow calculationstrategies in single-digit additions andsubtractions from those who had learnt toretrieve those same calculations from mem-ory (Koponen et al., 2006). The aim of thisstudy was to explore further the questionwhether the development of calculation flu-ency shares some of the underlying process-ing abilities required in fluent retrieval ofthe names of objects or colours. The primaryfocus of the study was to examine the effectof an intensive intervention on the cal-culation skill of two participants: one haddifficulties in both fluent calculation andlanguage-based retrieval (child A) while theother had difficulties only in calculation(child B). The question to be considered waswhether difficulties in fluent language-basedretrieval had an impact on learning toretrieve arithmetical facts from long-termmemory.

Before the intervention both participantsused a finger-counting strategy (counting-onmax) only. Although both of them calculatedsignificantly faster after the intervention,only child B was able to progress partiallyfrom using finger counting to fact retrieval. Itwas of interest that child A, who was unable toretrieve facts from memory, had difficulties in

fluently retrieving the names of familiarobjects presented serially in the RAN. Like-wise, he was unable to retrieve eight associa-tive pairs of words during the six trials in theWMS-R.

In contrast, both children performedalike in the other linguistic and cognitivetasks, such as the digit span, in which bothwere able to recall three digits forward(short-term memory) and two digits back-ward (working memory). Because the partic-ipants were matched in all other linguisticand non-verbal reasoning skills, the differ-ence in their ability to benefit from the inter-vention suggests that a specific languagedifficulty in retrieving could be one aspectcausing the fact retrieval difficulty. In addi-tion, it should be considered whether childA’s weak performance in learning associativepairs of words is an indication of a storingdifficulty. At least, it can be said that he didnot form representations of arithmeticalfacts which were retrievable later. Thesepropositions are in line with the theory ofGeary (1993), who has suggested that factretrieval problems are a reflection of a moregeneral problem in the ability to representand retrieve phonetic information fromlong-term memory. This in turn couldexplain the coexistence of arithmetical andreading difficulties.

Some of the more recent theories pro-pose that a number-specific module could beresponsible for calculation deficit (Butter-worth, 1999). In order to see whether thesetheories offered a potential explanation ofthe present findings, the number skills of theparticipants were analysed. These analysesindicated that child B had severe difficultiesin understanding the structure of Arabicnumbers and their connections to magni-tude. For instance, she had difficulties inselecting the largest 2-digit number out ofthree items. In contrast, child A’s perform-ance in number comparison was closer tothat of educational age controls and he didnot commit errors with 2-digit numbers.Their ability at matching Arabic numberswith play money as well as estimating the dis-



tance between numbers was very similar.Thus, it can be concluded that a differencein number skills at the outset did not explainthe differences found in the effects of theintervention.

Another possible explanation for theresults could be that the motivation tochange strategies may not have been samefor both children. Child A was rather accu-rate from the start and thus the rewardsobtained by changing his strategy may nothave been as high for him as for Child B,who made more errors in the beginning butshowed a good increase in accuracy duringthe intervention. However, this is an unlikelyexplanation in view of the fact that Child Awas not resistant to the intervention, butstarted to use the counting-on min strategyduring the intervention. Moreover, both ofthe children were co-operative, motivatedand participated willingly in each interven-tion session.

The results also raise questions as to howit is possible that child B had not learnedsimple arithmetical facts during her 4-yearsschooling, and yet learned them during thetwo months’ intervention. There are severalpossible explanations. First, before the inter-vention child B committed many errors ofcalculation owing to her use of an error-prone counting strategy. During the inter-vention her counting strategy progressed

considerably, enhancing her accuracy andspeed, which in turn enabled her to formcorrect associations between the terms of theproblem and the answer. Second, comparedto the teaching in the classroom, the inter-vention was restricted, focusing only on sim-ple addition through very intensive practicewith immediate feedback and a largeamounts of repetition. Third, despite herSLI, child B did not present language diffi-culties of a type, which, according to ourinterpretation, might have an impact on thelearning of arithmetical facts.

The results suggest that the particularnature of the language impairment has amajor impact. That is, a specific difficulty inverbal retrieval and in forming verbal associa-tions seems to be connected to the ability tolearn arithmetical facts. In addition to lend-ing support to earlier findings, indicating thatchildren with language impairment have aspecific problem in fact retrieval, the presentstudy shows the importance of defining indetail the particular SLI phenotype when try-ing to understand the role played by languagein arithmetical difficulties.

ReferencesTuire Koponen, Nilo Maki Institute,Jyvaskyla, Finland.E-mail: [email protected]



References

Ahonen, T., Tuovinen, S. & Leppäsaari, T. (1999).Nopean sarjallisen nimeämisen testi [The test ofrapid serial naming. Research reports.].Jyväskylä: Haukkarannan koulun julkaisusarjat.Tutkimusraportit.

Ashcraft, M.H. (1992). Cognitive arithmetic: A reviewof data and theory. Cognition, 44, 75–106.

Arvedson, P.J. (2002). Young children with specificlanguage impairment and their numerical cogni-tion. Journal of Speech, Language, and HearingResearch, 45, 970–982.

Barrouillet, P. & Fayol, M. (1998). From algorithmiccomputing to direct retrieval: Evidence fromnumber and alphabetic arithmetic in childrenand adults. Memory & Cognition, 26, 355–368.

Butterworth, B. (1999). What counts: How every brain ishardwired for math. New York.

Campbell, J.I.D. & Clark, J.M. (1988). An encoding-complex view of cognitive number processing:Comment on McCloskey, Sokol & Goodman(1986). Journal of Experimental Psychology: General,117, 204–214.

Dehaene, S. & Cohen, L. (1995). Towards an anatom-ical and functional model of number processing.Mathematical Cognition, 1, 83–120.

Denckla, M.B. & Rudel, R. (1974). Rapid ‘automa-tized’ naming of pictured objects, colors, lettersand numbers by normal children. Cortex, 10,186–202.

Fazio, B.B. (1999). Arithmetic calculation, short-termmemory and language performance in children

with specific language impairment: A 5-yearfollow-up. Journal of Speech, Language and HearingResearch, 42, 420–421.

Fuson, K.C. (1982). An analysis of the counting onsolution procedure in addition. In T.P. Carpenter,J.M. Moser & T.A. Romberg (Eds.), Additionand subtraction: A cognitive perspective (pp. 67–81).Hillsdale NJ: Lawrence Erlbaum Associates, Inc.

Geary, D.C. (1993). Mathematical disabilities: Cogni-tive, neuropsychological, and genetic compo-nents. Psychological Bulletin, 114, 345–362.

Geary, D.C., Hamson, C.O. & Hoard, M.K. (2000).Numerical and arithmetical cognition: A longitu-dinal study of process and concept deficits inchildren with learning disability. Journal of Experi-mental Child Psychology, 77, 236–263.

Kazdin, A.E. (1982). Single-case research designs: Meth-ods for clinical and applied settings. New York:Oxford University Press.

Koponen, T., Mononen, R., Räsänen, P. & Ahonen, T.(2006). Basic numeracy in children with specificlanguage impairment: Heterogeneity and con-nections to language. Journal of Speech, Language,and Hearing Research, 46, 1–16.

Korkman, M., Kirk., U. & Kemp, S.L. (1997). NEPSY.Lasten neuropsykologinen tutkimus [NEPSY: Adevelopmental neuropsychological assessment].Helsinki, Finland: Psykologien kustannus.

Landerl, K., Bevan, A. & Butterworth, B. (2004).Developmental dyscalculia and basic numericalcapacities: A study of 8–9-year--old students.Cognition, 93, 99–125.

McCloskey, M., Caramazza, A. & Basili, A. (1985).Cognitive mechanisms in number processingand calculation: Evidence from dyscalculia. Brainand Cognition, 4, 171–196.

Niilo Mäki Instituutti 2002. NEURE-programme:http://www.nmi.jyu.fi/neure/.1.2.2003.

Raven, J.C. (1993). Coloured progressive matrices.Revised and reprinted. Oxford: Oxford Psycholo-gists Press Ltd.

Renfrew, C. (2001). Word finding vocabulary test (4th ed). Bicester: Speechmark Publishing Ltd.

Siegel, S. & Castellan N.J (1988). Nonparametricstatistics: for the Behavioral Sciences (2nd edn.)(pp. 174–183). Singapore: McGraw-Hill, Inc.

Siegler, R.S. & Shrager, J. (1984). Strategy choices inaddition and subtraction: How do children knowwhat to do? In C. Sophian (Ed.), Origins of cogni-tive skills (pp. 229–293). Hillsdale, NJ:LawrenceErlbaum Associates, Inc.

Temple, C.M. & Sherwood, S. (2002). Representationand retrieval of arithmetic facts: Developmentaldifficulties. Quarterly Journal of ExperimentalPsychology, 55, 733–752.

Tieche Christinat, C., Conne, F. & Gaillard, F. (1995).Number processing in language impaired school-children. Approche Neuropsychologique des Apprentis-sages chez l’enfant, hors série, 52–57.

Tournaki, N. (2003). The differential effects of teach-ing addition through strategy instruction versusdrill and practice to students with and withoutlearning disabilities. Journal of Learning Disabili-ties, 36, 449–458.

Wechsler, D. (1991). WISC-III. Wechslerin lastenälykkyysasteikko. [Wechsler Intelligence Scale ForChildren]. Helsinki, Finland: Psykologien Kus-tannus Oy.

Wechsler, D. (1995). WMS-R. Wechsler Memory Scale – revised. Helsinki, Finland: Psykologien Kus-tannus Oy.

Wheeler, L.R. (1939). A comparative study of thedifficulty of the 100 addition combinations.Journal of Genetic Psychology, 54, 295–312.

World Health Organization (1993). The ICD-10 classi-fication of mental and behavioural Disorders: Diagnos-tic criteria for research (pp. 142–143). Geneva.




Support for numeracy difficultiesWithin the British educational system, therehas traditionally been more emphasis placedupon addressing children’s literacy difficul-ties, than on addressing their numeracydifficulties. This is reflected in the resourceswhich Local Authorities have allocated tothe two areas: in many Authorities there willbe a team of specialist literacy teachers toaddress Special Needs in literacy, but no cor-responding team for such needs in numer-acy. However, we do have evidence both ofthe existence of children’s difficulties withnumeracy, and of the disadvantage whichpeople with poor numeracy skills suffer inadult life. Since the introduction of theNational Numeracy Strategy’s ‘Frameworkfor teaching mathematics’ in 1999, thenumber of children achieving the targetlevel for their age in mathematics at the endof Key Stage 2 has increased by 16 per cent

(DfES, 2004 and 2005a). But there is still along tail of children who do not achieve thetarget level, with a steady proportion ofchildren achieving below level 3, between2001 and 2004 (DfES, 2005b). The long-term consequences of numeracy difficultiesare serious: research suggests that, amongstadults, poor numeracy is more disadvanta-geous in the labour market than is poor liter-acy (Basic Skills Agency, 1997).

There is, then, a pressing need to addressthis area. The Primary National Strategy hasresponded through its model of ‘waves’ ofintervention: Wave 1 being high qualitylearning and teaching for all in daily lessons;Wave 2 being targeted, short term smallgroup interventions; and Wave 3 being amore individualised, short term interventionto address ‘fundamental errors and miscon-ceptions’ (DfES, 2005b). Wave 3 is intendedfor pupils in Key Stage 2. Thus, whilst Wave 2

Achieving new heights in Cumbria:Raising standards in early numeracythrough mathematics recoveryRuth Willey, Amanda Holliday & Jim Martland

AbstractThis article describes how standards in early numeracy were raised within Cumbria by the application ofthe Mathematics Recovery Programme. It reports data showing how children’s numeracy improved as aresult of the programme, and describes effective elements of the in-service teacher training programme whichwas implemented. This work is an example of how teachers and educational psychologists can work togetherto develop and disseminate good practice in teaching, which is based on a sound theoretical and evidencebase. Mathematics Recovery (MR) is an evolving, research-based programme which was first developed inthe 1990s, in order to meet the needs of children who were not reaching age-related expectations for numeracyskills. There is an underlying model of how children acquire strategies and numerical knowledge, and anexplicit set of principles of good teaching. The MR materials include short-term, intensive, individualteaching programmes, as well as group and class teaching. The paper reflects on the nature of ‘best practicein assessment for learning’ (DFES 2005b), which we argue is dynamic in character, in that the assessmentis embedded in the teaching, with the assessor playing a mediating role, supporting the learner to constructand elaborate their own model of number. We show how Mathematics Recovery implements this approachto assessment and teaching, through the design, implementation and evaluation of programmes of short-term intervention. The paper concludes with some evidence of how the above has impacted on teachers’ profes-sional development and changed classroom practice in Cumbria.


Achieving new heights in Cumbria

might be regarded as being early interven-tion to prevent children from falling behind,Wave 3 is targeted at children who havemade unsatisfactory progress across thethree years of infant-aged schooling.

In 2002 the Cumbria LEA pilotedand then implemented the MathematicsRecovery programme which complementsall three waves of intervention. In its originalform, it was a short term, intensive individualprogramme for pupils in Year 1 of Key Stage1, and thus constituted intervention to pre-vent failure, earlier than Wave 3 and moreintensive than Wave 2. There are not manysuch documented intervention programmesavailable for the Key Stage 1 age group. Inher recent review ‘What Works for Childrenwith Mathematical Difficulties’, Ann Dowkerreviews MR very positively, and cites it as oneof the two available large-scale, individu-alised, componential programmes based oncognitive theories of arithmetic (Dowker,2004). As will be seen below, the MRprogramme has also been developed so thatit can be applied more widely than just as anindividual programme, and it can be used aspart of Wave 1, Wave 2 and Wave 3.

The key features and origins of themathematics recovery programmeThe key features of MR can be summed upunder four headings – Early Intervention,Assessment, Teaching and Professional Devel-opment. The assessment and teaching strandsuse a strong underpinning theory of youngchildren’s mathematical learning which leadsto a comprehensive and integrated frame-work for both assessment and teaching. Theprogramme has a detailed approach to, andspecific diagnostic tools for, the assessment ofchildren’s early number strategies and knowl-edge. Following the assessment, teachers canemploy an especially developed instructionalapproach and distinctive instructional activi-ties which can be applied to individuals insmall-group or class situations. The pro-gramme also has an intensive, short-termteaching intervention for low-attaining 5–8-year-old children by specialist teachers. The

entire programme provides an extensiveprofessional development course to preparethe specialist teachers, and ongoing collegialand leader support for these teachers.

MR was originally developed in NewSouth Wales, between 1992 and 1995. Itemerged from detailed research studies ofhow children’s number knowledge develops(Wright, 1991; Aubrey, 1993; Young-Loveridge, 1989, 1991). From this, a modelof the usual course of this learning was con-structed, and assessment tools and tech-niques were developed, to enable individualchildren’s knowledge to be described in theterms of the model. Wright and his col-leagues went on to design an individualteaching approach and materials, intendedto move children on through the model, byworking in a very detailed way within thechild’s Zone of Proximal Development(ZPD), that is, planning instruction which isfocussed just beyond the child’s currentlevels of knowledge. These individual teach-ing programmes were evaluated, and shownto be very successful in moving children onthrough the stages and levels of the model.(Wright et al., 1994; Wright et al., 1998).

The approach has been further developedinto its current, published form (Wright,Martland & Stafford, 2006; Wright, Stanger,Stafford & Martland, 2006; Wright, Martland,Stafford & Stanger, 2002). Materials nowinclude assessment tools, teaching pro-grammes for individual children and a bookon using the approach in classroom teach-ing. MR is now in wide, international use, inAustralia, the USA, New Zealand, Canada,the UK and Ireland.

Assessment in the mathematicsrecovery programmeMathematics Recovery involves a distinctiveapproach to assessing young children’snumerical knowledge. The origins of thismethod are in research projects conductedin the 1980s and 1990s that focused onunderstanding children’s numerical strate-gies for addition and subtraction and themodifications children make to their strategies


Ruth Willey et al.

over time (e.g. Cobb & Steffe, 1982; Steffe &Cobb, 1988; Steffe et al., 1983; Wright, 1989,1991a). In 1998 the approach was extendedto include a focus on children’s early multi-plication and division knowledge. This workdrew on an extensive range of research(Steffe, 1992b; Steffe & Cobb, 1988; Steffe,1994; Mulligan, 1998) and the Count Me InToo project (NSW Department of Educationand Training, 1998).

The assessment in MR is distinctive ontwo counts. First, it is interview based andsecond, the assessment interview is video-taped so that the teacher does not need torecord the child’s responses during thecourse of the interview. The benefit of nothaving to make notes is that the assessor isfree to observe, listen and engage in ques-tions with the child in order to detect themost sophisticated strategy the child uses.

Underlying the development of MR is abelief that in early number learning it is veryimportant to understand, observe and takeaccount of children’s knowledge and strate-gies when solving tasks. Children’s earlynumerical knowledge varies greatly and theirstrategies are multifarious. Thus, acrosschildren, early numerical knowledge is char-acterized by both commonalities and diver-sity. As indicated by the research of Denvir &Brown (1986a, 1986b), it is insufficient tothink that every child’s early numericalknowledge develops along a common devel-opmental path. For example, one importantfactor in a particular child’s developmentalpath, it is believed, relates to the nature ofthe settings in which the child’s prior learn-ing has occurred. Also, children who mayappear to an observer to be in the same set-ting, or learning situation, will construct thesituation idiosyncratically and thus differentkinds of learning are likely to occur.

The child’s process of constructingnumerical knowledge can be thought of interms of progression or advancement.Children reconstruct or modify their currentstrategies and doing so is nothing more or lessthan progression, advancement or learning.Given this, it is useful to consider the notion

of the relative sophistication of children’sstrategies. For example, the child who has nomeans of working out nine plus three otherthan counting out nine counters from one,counting out three counters from one, andthen counting all of the counters from 1 to12, is using a far less sophisticated strategythan the child who ignores the counters andsays nine plus three is the same as ten plustwo, and I know that is 12 without counting.Understanding the progression of the strate-gies which children use in early number situa-tions is the key to advancing teaching staffs’professional knowledge and learning. Werefer to the progression as SEAL (Stages ofEarly Arithmetical Learning). They are:

Assessment tasks as a source of instructionalactivitiesVirtually all of the assessment tasks areideally suited for adaptation to instructionalactivities. Further, because the assessmenttasks are organised into task groups, thetasks within a task group or across severalgroups typically constitute an implied,instructional sequence. Again, although thetasks are presented in a format for one-to-one interaction, they are easily adapted tosituations involving small or large groupinstruction.

Implementation of mathematicsrecovery in CumbriaCumbria began its involvement with MR ona small scale in 2002 with a group of eightteachers, two Numeracy Consultants andone Educational Psychologist. Subsequently,a MR Team consisting of a Numeracy Con-sultant (two days per week), an Educational

Stage Significant tasks

Stage 0: Emergent Counting

Stage 1: Perceptual Counting

Stage 2: Figurative Counting

Stage 3: Initial Number Sequence

Stage 4: Intermediate Number Sequence

Stage 5: Facile Number Sequence

Psychologist (half a day per week) and threeteachers (one day per week each) hasworked to support and develop the use ofMR within the County.

A major focus is the running of an annualcourse to train teachers and teaching assis-tants. The course takes place over two terms,with assessment being covered in the firstterm, and the teaching programme in thesecond term. There is a total of seven centre-based training days, with two or more tutorvisits to participants’ schools. During thecourse, participants engage in video-tapedpractice assessments, and design and run ateaching programme with an individualchild. So far, 97 schools have undertaken thetraining, which represents almost one thirdof the primary schools in Cumbria. (Thetraining has also been found useful by somespecial school and secondary school teach-ing staff). Schools have been encouraged tosend a teacher and teaching assistant on thecourse together, in order to promote the useof the programme later in school at both theclassroom and individual child level. Theteachers and assistants have worked closelytogether on the programmes for children,and have found this particularly helpful inthe development of their skills.

Staff who successfully complete the MRtraining are able to apply for funding to runindividual MR programmes, for pupils whomthey have assessed as functioning well belowthe expected levels on the MR assessments.The effectiveness of these programmes isevaluated, through analysis of pupils’ results

on the MR assessments before and after theprogramme. Most pupils make gains of twoSEAL stages (e.g. they move from having tosee and count concrete objects in order toadd two sets (Stage 1 on SEAL) to being ableto work without visible objects and to ‘count-up-from’ and ‘count-down-from’ to solveaddition and subtraction problems, includ-ing missing addends and missing subtra-hends (Stage 3 on SEAL)). They alsoincrease their ability in other aspects ofnumber: saying forward and backwardnumber word sequences, to identifyingnumerals and recognising spatial patterns.Indeed, so far the small number of pupilswho have not made a gain of at least oneSEAL stage during their MR individual pro-gramme have all made measurable gains inthese other aspects. See Table 1 below, for asummary of the gains in SEAL stages made by2 1 0Cumbrian pupils who received individualprogrammes between April 04 and March 06.

These increases are similar to thosereported in the Australian MR research,although the Cumbrian programmes areshorter in duration (about 20 sessions, halfan hour each, taking place three or fourtimes a week – whereas the Australian pro-grammes were more than twice this length).As Ann Dowker says, ‘Relatively small amountsof individual intervention may make it possi-ble for a child to benefit far more fully fromwhole class teaching’ (Dowker, 2004). Evalu-ating how well the children have generalisedand adapted the learning is more difficult.

No stages 1 stage 2 stages 3 stages 4 stagesgained gained gained gained gained

April 04 – March 9 26 44 19 205 100 pupils

April 05 – March 4 31 57 12 606 110 pupils

% of total 6.2 27.1 48.1 14.8 0.04

Table 1: Gains in SEAL stages made by 210 Cumbrian pupils who received individual programmes(Holliday, 2005; Holliday, 2006)



Informal teacher reports, collected on tuto-rial and other visits to schools, consistentlysay that the pupils are performing better inclass, and that they have gained in confi-dence and independence. It will soon be pos-sible to analyse the mathematics SATs resultsfrom the end of Key Stage 1, tracking thosepupils who received a funded programme,and comparing results with comparablepupils who did not.

In addition to staff training and monitor-ing of individual programmes, the MR teamare developing other ways of supporting theuse of MR in schools. These include the pub-lication of guidance for schools on the use ofMR for group work and for work in theFoundation Stage; work to develop ICTmaterials for whole class use; work withteaching staff on using MR in the daily math-ematics lesson; establishing regular supportmeetings to update staff and for them toshare developments.

The development of the MR work withinCumbria would not have been possible with-out dedicated funding for the project. Thiswas not initially available. However, followingthe evaluation of the success of the firstcohort of training, it was possible to arguethe case for some of the existing SpecialNeeds resources to be directed towardsnumeracy. Currently, there is an annualbudget which pays for the salaries of the MRTeam, and for the delivery of some individ-ual programmes to pupils in schools.

Why does MR work so well?Our ongoing evaluation of the work suggeststhat MR is highly effective, including withchildren who have already received Wave 2interventions, in the form of small groupwork based on National Numeracy StrategySpringboards, or on National NumeracyStrategy objectives which had been trackedback to earlier curriculum stages. Yet manyof the MR teaching activities resemble thosein other programmes, and cover groundwhich is also in the Numeracy Strategy. Sowhy is it so successful, especially when used

with pupils who have some history of mathe-matics difficulty?

A large part of the answer to this, webelieve, lies in the way in which the assess-ment and teaching are used together, within aframework which is constructivist in itsnature. The assessment is not seen just as ameasure of what has been learned, but as anintegral and ongoing facet of the teaching,which will inform both what is taught next,and the approach and materials which areused in the teaching. Although the initialassessment does give summative informationabout the levels and stage at which the child isfunctioning, its central purpose is to allow aqualitative, detailed analysis of the strategiesthe child is using. The assessor presents thechild with a problem, observes the child work-ing, and explores the child’s responses(through questioning and judicious presenta-tion of new problems), to find out how thechild thought whilst solving the problem. Thisinformation is recorded (after the assessmentinterview) in a Pupil Profile, which highlightsthe child’s present strategies, strengths andweaknesses, and possible next steps for devel-opment. This profile is then used to designthe teaching programme for the pupil, draw-ing on the range of available teaching activi-ties within the MR materials.

This approach to assessment continuesthroughout the teaching programme, as thechild’s responses during teaching areobserved closely, and used to guide the nextteaching steps. The aim is always to be work-ing within the child’s Zone of ProximalDevelopment (Lunt, 1993; Lidz, 1995), sothat the child succeeds, with small but well-targeted prompts from the MR teacher. TheMR teacher role is critical here: it is to selectappropriate problems for the child, presentthem in a suitable setting, support the childsuccessfully to find their own solution to theproblem and help the child to reflect onwhat they are doing. This is not didacticteaching: modelling of solutions rarely hap-pens, and when it does is usually associatedwith the learning of basic facts (such as thewords in the forward number word


Ruth Willey et al.

sequence). This is a mediating role, and issimilar to the teacher role in the Dutch‘Realistic Mathematics’ work, which refers topupils as engaging in ‘guided reinvention’,and stresses the importance of knowledgebeing constructed by the child. (Gravemeijer,1994; Milo, Ruijssnaars & Seegers, 2005).

The problem-centred approach is used,within MR, as a very important tool for ensur-ing that the teaching remains constructivistin its orientation. A central aim of the MRprogramme is for children to develop theirown, increasingly powerful concepts ofnumber, which they will be able to use as abasis for subsequent learning (Cobb &Merkel, 1989). Because they have been devel-oped by the children elaborating their con-cepts in the course of their ownproblem-solving, these constructs will be fullyunderstood by the children, in a relational,rather than only an instrumental, manner(Skemp, 1976). Thus, the children will notmerely be following a learned ‘recipe’ forsolving a particular, familiar type of problem(showing instrumental understanding), butwill be able to devise their own strategies andalgorithms for solving novel types of problem(showing relational understanding). Such a‘shift from procedures to reasoning’(Wheatly & Reynolds, 1999) is essential, ifchildren are to become confident and inde-pendent learners who will be able to gener-alise and extend their knowledge in newcontexts. The problem-centred approach fos-ters this relational understanding, through‘developing a setting in which children caninvent and discuss their own strategies’(Cobb & Merkel, op cit). In an individual MRsession, this will be done by presenting thechild with a problem which is slightly moredifficult than those which the child has previ-ously solved, and allowing as much time as isnecessary for the child to work on the prob-lem. The teacher will observe closely, and usetheir knowledge of what strategies and con-cepts that child already possesses, to offerprompts that will lead the child towardsdeveloping more sophisticated strategies.Importantly, the child will be encouraged to

check their solutions, by using less sophisti-cated strategies. This will enable them tobuild links between their developing con-cepts, so that they are continuously elaborat-ing their mental model of the numbersystem, through solving the problems. AsWright expresses it, ‘for the constructivistteacher, advances in the children’s knowl-edge occur when the children modify theircurrent ways of operating in response to aproblematic situation.’ (Wright, 1990).

During the teaching sessions, the MRteacher continues to observe and assess thechild’s responses, with a strong focus on howthe child makes use of the support which theteacher offers. The teacher is continuouslymaking and testing hypotheses about whatexperiences will now help the child todevelop further their models of number. Thus,throughout the programme, the teacher canbe regarded as engaging in Dynamic Assess-ment (Lidz, 1995; Elliott, 2003). The role ofthe teacher is to mediate the child’s learningexperience, by locating the child’s Zone ofProximal Development (ZPD) (i.e., theregion where the child can only succeed withsome support), and working with the childin this Zone. An example from a teachingprogramme may serve to illustrate the styleof the teaching, showing how the teachermediates the child’s learning, choosing dif-ferent prompts and settings, in order to helpthe child to construct her responses. Thesection of dialogue and commentary belowcomes from an early session in a programmewith Gertie, a girl aged 6 years and 5 months.

Programme extract: Session 2:work on forward number wordsequence to 20

Gertie has been doing FNWS successfully, althoughshe hesitated at 12. The teacher presents the NumberWord After task, and Gertie succeeds with 5, 15, 24,29 and 7. The teacher then presents 12, and thefollowing dialogue happens:T: What number comes straight after 12?G: 11T: we’re going forwards, so it’s the

number that comes just after 12.



G: 14T: 12. . . . . .? (re-presents task, and waits)G: . . . . . .13T: Good. How did you do that?G: I just thought in my head it was.T: Did you count? (G shakes her head.)G then succeeds with NWA 3 and 12T: 19G: 18T: 19, what comes next, the number just

after 19? (re-presents task)G: . . . . . . . . . . . . . . .17At this point, G’s body language is interesting. Sheis slumped in her chair, with her hands up aroundher mouth, beginning to squirm. She seems bored ortired, and uncomfortable. She seems to be signallingthat she has had enough of this hard task, but Tpersists :T: You’re going backwards, you’re doing

the numbers before. If you were countingforwards, what would come just after 19?

No reply from G, after a long pause. T decides to tryto get G to count forwards, and listen to her ownvoice saying the number after 19 :T: Can we do some counting for-

wards . . . we’ll start at 18. 18 . . .G: 19T: 20G: 21T: OK, so what comes just after 19?G: 21T: 19. . .?G: 19… … … … … . .This has not worked. T brings out a numeral track,from 11 to 20. T points to each number, and G readsthem out correctly.T: Where’s 19?G points to 19T: What comes just after it?G: 20 (Points to it.)T puts numeral track away.T: What comes just after 19?G: 20T: Good. What comes just before 20?G: … … … … … . … .T: If you were counting, what would

come just before 20?G: … … … … … … … … . .19T: Well done!

In this extract, Gertie eventually succeeds with NWAfor 12 and 19, but with a lot of difficulty, and withexternal prompting. Future sessions will show thather counting skills vary a lot, from day to day. Sheis also worse in the afternoons, when she seems tired,and has more difficulty in concentrating. Severalinteresting points emerge, from this extract.Although Gertie can produce the Forward NumberWord Sequence beyond 20, she has not connectedthis knowledge to the Number Word After task. Theteacher attempts to help her make that connection.First, this is done verbally, by encouraging her tocount. (She uses a series of graduated prompts: Didyou count?… . If you were counting forwards, whatwould come just after 19?. . .do some counting for-wards. . . .) The teacher observes carefully, anddecides to fine tune the teaching, through the choiceof problems. (Having discovered a difficulty with12, she works on this, then asks a different question,before revisiting 12 to check. The difficulty with 19is then worked on, until this is resolved. At thispoint, Gertie has had enough of this task, and theteacher moves on to a different Key Topic.) Thisteaching is in Gertie’s ZPD: she cannot do it inde-pendently, but eventually gets there, with teachermediation.Perhaps the most powerful and positively

evaluated session in the teaching course isone which is run as a class tutorial, whereteaching staff bring scenarios from theirongoing teaching programmes and ask thegroup to help them with teaching ideas (i.e.,ideas for mediating the child’s experience).This leads to exchange of ideas and rich dis-cussion, focussed on how to set up an expe-rience which will support that particularchild to solve the problem in question. Atthis point in the course, it becomes evidentthat not only are the teaching staff workingwithin the children’s ZPD, but that they arealso developing their teaching skills throughworking within their own ZPD, using thetutors and each other for support.

Beyond individual programmesIn its original form, MR was delivered to chil-dren as short term individual programmes,targeted at Year 1 aged children who werebeginning to fall behind their peers in


Ruth Willey et al.



numeracy. However, it has developed consid-erably beyond this. For example, in NewSouth Wales and in New Zealand theapproach is used across the first three yearsof schooling, as a framework for the teachingof numeracy to all pupils. This implementa-tion is called Count Me In Too (CMIT), anddoes not deliver individual programmes tochildren. It has been evaluated (through preand post assessments of pupils, as well asquestionnaires and case studies with teach-ing staff and facilitators involved in the pro-gramme) as highly successful, both inpromoting pupil progress and in increasingteacher knowledge and understanding(Thomas & Ward, 2001). Key elements inthe success of this group approach werefound to be: increased teacher understand-ing of how children learn number; increasedfocus on the strategies which individualchildren actually use; the availability ofassessment tools which can be used to groupchildren appropriately for working onparticular learning objectives (Thomas &Ward, op cit).

The Cumbrian experience also shows thatthe effectiveness of MR goes well beyond theindividual programmes. Examples of thisinclude:● Teaching staff who attend the training

often respond to the assessment course byspontaneously implementing changes inthe way they deliver their class teaching.They come to the second and subsequenttraining sessions keen to talk aboutchanges they have already made. Manyfocus initially on the ‘mental and oralstarter’ part of their lessons, noticing thatmost children are working in their ZPDfor only a small part of this activity. Theythen find different ways to organise andpresent the activity, so that it is better dif-ferentiated to match the children’s needs.

● Teaching staff raise their expectations ofwhat children can achieve.

● About half of the teaching staff who com-plete the course apply for funding to runindividual programmes. (In the financialyear 2005–2006, funding was approved

for 111 programmes, and 93 per cent ofthose programmes have so far been com-pleted.) However, many of the remainingteaching staff use the approach to makechanges in their classroom teaching, toorganise groups of children for teaching,or to deliver teaching to small groups.

● Some teaching staff are using the assess-ment materials to track children’s progressthroughout the infant school, and topinpoint the need for specific interven-tions with particular children, or for stafftraining in particular areas of numeracyteaching.

● Teaching staff feel that their expertise inearly numeracy is recognised within theirschool and feel that their skills are usedto advantage, as colleagues consult themregarding children’s progress and theconstruction of teaching strategies.

● The teaching staff report that they canreadily use existing classroom materialsbut now in a more effective way: they tryto promote mental strategies throughproviding a set of integrated activities inmultiple settings.Although MR can be very effectively

applied in whole class and small groupcontexts, the MR Team take the view that, ininitially learning to apply the principles ofMR, it is extremely effective to work with an individual child. This allows the teacherto work continuously at finding where thechild is, and developing ways of supportingthem to move on. This cannot be done soprecisely when working with a group, wherethere is often the need to make a compro-mise, or move on before one child is reallyready.

The MR Team have formed the impres-sion that many teaching staff develop theirknowledge, understanding and practice ofnumeracy teaching considerably, throughusing MR. A research project is attemptingto explore these changes, through in-depthPersonal Construct Psychology interviewswith staff (Willey, ongoing). Results so farsuggest that staff constructs about teachingof number change markedly, following


Ruth Willey et al.

training and use of MR. Changes to constructsare in line with the underlying principles ofMR, and include the following:● Trying to take pupils back to first princi-

ples so they can build understanding,rather than trying to plug gaps in theirprocedural knowledge

● Having an understanding of howchildren develop number knowledge

● A growing commitment to promotingpupils’ independent learning

● A belief that good teaching will be enjoy-able and motivating for pupils

● Willingness to wait whilst pupils think,and to observe closely what they do

● A belief that children will learn effec-tively, if they are given tasks within theirZPD and a small amount of support

● A view of the teacher as facilitator andguide, rather than transmitter of knowl-edge.The staff who were interviewed were

asked to rate themselves on the construct‘teaches numeracy very well’, for both beforeand after the MR training. All but one ofthem felt that they had improved. Even staffwith many years of experience felt thatengaging with MR had moved their teachingon significantly.

It seems, then, that teaching staff dodevelop skills and knowledge through imple-menting MR, and that they do put this intopractice in their subsequent work withpupils. This is because MR provides a struc-ture within which staff feel safe to experi-ment with a more constructivist approach.The teaching activities given within theprogramme function as examples. Althoughit would be possible to run individual pro-grammes using only the teaching activitiesgiven, this does not generally happen: theteaching staff adapt, tailor and extend theactivities, to address more exactly the needsand interests of each child. The teachingstaff are able to do this because they havelearned to use the principles of MR, togenerate their own solutions to new situa-

tions. The teaching staff have gained confi-dence in pupils’ abilities to learn in a con-structivist way, and in their own abilities toguide such learning.

ConclusionWe have shown how one Local authority hasimplemented Mathematics Recovery andevaluated its impact. Individual pupils whoreceive MR programmes make good prog-ress in basic numeracy skills. Teachers andteaching assistants develop their knowledge,skills and confidence to teach numeracy.The Maths Recovery principles, assessmenttools and activities work well at a number oflevels: in individual programmes, in groupwork and in informing good classroomteaching.

The greatest power of MathematicsRecovery lies in its use as a tool for the pro-fessional development of teaching staff.Staff who engage with Mathematics Recov-ery develop an enhanced faith in pupils’ability to learn and to solve problems forthemselves. Alongside this, they becomemore confident in their own ability to assesswhere pupils are, and to offer appropriatesupport to help pupils learn. For many staff,this results in a significant shift in theirteaching style, away from the didactic andtowards a more pupil-focused, constructivistoutlook.

Address for correspondenceRuth Willey, Cumbria County Council,Children’s Services, Newbeck Centre, Wig-ton Rd, Carlisle, CA2 6LBE-mail: [email protected]

A. Holliday, Cumbria County Council,Children’s Services, Kendal Area Office,Busher Walk, Kendal, LA9 4RQ E-mail: [email protected]

J. Martland, Mathematics Recoveryprogramme (UK) Ltd, 11 Station Lane,Mickle Trafford, Chester, Cheshire, CH24EHE-mail: [email protected]

ReferencesAubrey, C. (1993). An investigation of the mathemat-

ical knowledge and competencies which youngchildren bring into school. British EducationalResearch Journal, 19(1), 27–41.

Basic Skills Agency (1997). Does numeracy matter?Evidence from the National Child Development Studyon the impact of poor numeracy on adult life. BasicSkills Agency.

Cobb, P. & Merkel, G. (1989) Thinking strategies:teaching arithmetic through problem-solvingNCTM Yearbook.

Denvir, B. & Brown, M. (1986a) Understanding ofnumber concepts in low attaining 7–9 year olds:Part 1. Development of descriptive frameworkand diagnostic instrument. Educational Studies inMathematics, 17, 15–36.

Denvir, B. & Brown, M. (1986b) Understanding ofnumber concepts in low attaining 7–9 year olds:Part II. The teaching studies. Educational Studiesin Mathematics, 17, 143–164.

Department for Education and Skills (DfES)(2005a). Reviewing the frameworks for teachingliteracy and mathematics. Primary National Strat-egy, 1786–2005 DOC-EN. London: DfES.

Department for Education and Skills (DfES)(2005b). Targeting support: implementing interven-tions for children with significant difficulties in mathe-matics. Primary National Strategy, 1083-2005.London: DfES.

Department for Education and Skills (DfES) (2004).Raising standards in mathematics – Achievingchildren’s targets: Primary National Strategy,1075–2004. London: DfES.

Dowker, A. (2004). What works for children withmathematical difficulties? Nottingham: DfES Publi-cations.

Elliott, J. (2003). Dynamic assessment in educationalsettings: Realising potential. Educational Review,55(1)15–32.

Gravemeijer, K.P.E. (1994). Developing realistic mathe-matics Education. Utrecht, The Netherlands: CD-BPress.

Holliday, A. (2005). Mathematics recovery evaluation offunding April 2004 to March 2005. Unpublishedreport of Cumbria Children’s Services.

Holliday, A. (2006). Mathematics recovery evaluationApril 2005 to March 2006. Unpublished report ofCumbria Children’s Services.

Lidz, C.S. (1991). Practitioners’ guide to dynamic assess-ment. New York: Guilford.

Lidz, C.S. (1995). Dynamic assessment and the legacyof L. S. Vygotsky. School Psychology International,16, 143–153.

Lunt, I. (1993). The practice of assessment. In H.Daniels (Ed.) Charting the agenda: educationalactivity after Vygotsky. London: Routledge.

Milo, B.F., Ruijssenaars, A.J.J.M. & Seeger, G. (2005).Math instruction for students with special educa-tional needs: effects of guiding versus directinginstruction. Educational and Child Psychology,22(4), 68–80.

Mulligan, J.T. (1998). A research-based frameworkfor assessing early multiplication and division. InC. Kanes, M. Goos & E. Warren (Eds.), Proceedingsof the 21st Annual Conference of the Mathematics Edu-cation Research Group of Australasia (Vol. 2, pp.404–411). Brisbane: Griffith University.

NSW Department of Education and Training (1998).Count me in too: A professional development package.Sydney: NSWDET.

Skemp, R. (1976). Relational understanding andinstrumental understanding. Mathematics Teach-ing, 77, 20–26.

Steffe, L.P., von Glasersfeld, E., Richards, J. & Cobb, P.(1983). Children’s counting types: Philosophy, theory,and application. New York: Praeger.

Steffe, L.R. & Cobb, P. (with E. von Glasersfeld)(1988). Construction of arithmetic meanings andstrategies. New York: Springer-Verlag.

Steffe, L.P. (1992b). Schemes of action and operationinvolving composite units. Learning and Individ-ual Differences, 4, 259–309.

Steffe, L.R. (1994). Children’s multiplying schemes,In G. Harel & J. Confrey (Eds.), The Developmentof Multiplicative Reasoning in the Learning of Mathe-matics (pp. 3–41). Albany, NY: State University ofNew York Press.

Thomas, G. & Ward, J. (2001). An evaluation of theCount Me In Too pilot project. Exploring issues inmathematics education. Wellington: New ZealandMinistry of Education.

Wheatley, G. & Reynolds, A. M. (1999). Part 1: Help-ing children learn mathematics. Coming to knownumbers. Mathematics Learning, 6–25.

Wright, R.J. (1989). Numerical development in thekindergarten year: A teaching experiment. Doc-toral Dissertation, University of Georgia.

Wright, R. (1991). What number knowledge is pos-sessed by children entering the kindergartenyear of school? Mathematics Education ResearchJournal, 3(1), 1–16.

Wright, R.J. (1990). Interactive communication: Con-straints and possibilities, In L.P. Steffe & T.Wood(Eds.), Transforming children’s mathematics educa-tion: International perspectives. Hillsdale NJ:Lawrence Erlbaum.

Wright, R.J. (1991a). An application of the epistemol-ogy of radical constructivism to the study oflearning. Australian Educational Researcher, 18(1),75–95.

Wright, R., Cowper, M., Stafford, A., Stanger, G. &Stewart, R. (1994). The mathematics recovery



project – A progress report: Specialist teachersworking with low-attaining first graders.Proceedings of the 17th annual conference of the Math-ematics Education Research Group of Australia, 2,709–716.

Wright, R., Stewart, R., Stafford, A. & Cain, R. (1998).Early mathematics. Proceedings of the 20th annualmeeting of the North American chapter of the Interna-tional Group for the Psychology of Mathematics Educa-tion, 1, 211–216.

Wright, R., Martland, J., Stafford, A. & Stanger, G.(2002). Teaching number: advancing children’sskills and strategies. London: Paul ChapmanPublishing/Sage.

Wright, R., Martland, J. & Stafford, A. (2006a). Earlynumeracy: Assessment for teaching and intervention.London: Paul Chapman Publishing/Sage.

Wright, R., Stanger, G., Stafford, A. & Martland, J.(2006b). Teaching number: Advancing children’sskills and strategies. London: Paul ChapmanPublishing/Sage.

Young-Loveridge, J. (1989). The development ofchildren’s number concepts: The first year ofschool. New Zealand Journal of Educational Studies,24(1), 47–64.

Young-Loveridge, J. (1991). The development of children’snumber concepts from ages five to nine, Volumes 1 and2. Hamilton, NZ: University of Waikato.

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T HIS ARTICLE seeks to address theneeds of educational psychologists currently working in the UK. There are

a number of reasons for this article at thistime: increasing understanding of the natureof mathematical skills, changes in the deliv-ery of the maths curriculum and the intro-duction of particular interventions toaddress children’s difficulties with mathe-matical skills. The purpose of this article is toconsider practical aspects of educationalpsychology work in the context of the othercontributions to this special issue providinginformation about areas of theory, researchand practice.

This article will focus on assessment ofindividual children as this continues to be asignificant part of the work of educationalpsychologists, though at times there will bereference to needs of specific groups orwhole school issues. Assessment is frequentlydiscussed in professional and academic

literature (Fredrickson, Webster & Wright,1991; Miller & Leyden, 1999; Freeman &Miller, 2001; Elliott, 2003: Tymms & Elliott,2006). This article does not seek to revisitarguments about relative merits of specificforms of assessment, or the question oflabels, which frequently arises in conversa-tions with parents, and others, when we areasked for an assessment. The aim here is toprovide a relevant summary of researchrelating to maths learning, and assessmenttools, to guide applied psychologists intheir thinking when asked to do an assess-ment in relation to problems with numbers.It is intended to promote assessment thatcan lead to a good understanding of thechild’s difficulties, strategies and miscon-ceptions and so that educational psycholo-gists can provide appropriate advice andintervention.


Educational psychologists’ assessmentof children’s arithmetic skillsSubmission to special issue educationaland child psychology september 2006Susie Mackenzie

AbstractThis paper explores a range of tools and approaches available to Educational Psychologists asked to assessmathematical skills and understanding. The main assessment tools and techniques commonly used byEPs are standardised tests, curriculum based assessment, observation and dynamic assessment. Comparedwith assessment of reading skills there are fewer standardised tests available for the assessment of mathe-matical skills and the validity of these is questioned. It is suggested that when an EP has a good under-standing of how mathematical skills develop and of the key cognitive skills, social experiences, attitudesand strategies that underpin progress then an interactive assessment style can emerge. I describe how obser-vation combined with interactive questioning using a detailed guide developed in Leeds provides a usefuland flexible adjunct to standardised tests and curriculum based assessment. This more wide-rangingapproach may be more time consuming but can give a better understanding of the gaps in the pupil’sunderstanding, the strategies they bring to mathematical tasks and how they may, or may not, be able touse these skills in everyday life.

The purpose of assessment ofmaths skillsFreeman & Miller (2001) provided a ‘taxo-nomy’ of assessment with three main para-digms: norm-referenced (psychometrics),criterion-referenced (curriculum based) anddynamic assessment or analysis of the child’sresponses in a learning situation.

Freeman & Miller (2001) repeated anassertion made originally by Fredrickson,Webster & Wright (1991) that assessment bya psychologist needs to go beyond a descrip-tion of what a child can or can’t do, and sug-gest that the aim of an assessment should beto help those around the child, or teachingthe child, to have a better understanding ofWHY the child has this pattern of strengthsor difficulties.

In the context of maths teaching andlearning Wiliam (2006) provided a thoroughanalysis of the role of assessment of mathslearning for teachers, and asked some ques-tions that are equally relevant to psycholo-gists involved in the assessment of mathsskills. Do we want an assessment that tells usthe child’s existing state of knowledge, orone that enables us to play a role in improv-ing or supporting learning? Do we want todo an assessment that gives us clues as to whythe child is failing or one which gives us ascore and tells us that the child is perform-ing below average for their age? For educa-tional psychologists one main purpose of anassessment of maths skills would be to formu-late and test hypotheses about why a particu-lar child is failing to progress or fallingbehind peers.

Another issue raised by Wiliam (2006) isthat most maths assessment tools measurethe children’s ability to compute, which isimportant, but this is not the same as thechildren’s numeracy: their ability to solveproblems using mathematical knowledge inthe real world. While it is important to assesswhat computations a child can and cannotdo, we may also want to assess children’s abil-ity to say whether an answer is correct or rea-sonable or to know when they have beenshort changed in a shop. At its most basic we

might find that a child can give correctanswers to test questions such as 3 � 2 � 5,or even 50 � 20 � 30, but a more important‘assessment question’ might be ‘Can thischild go into a shop with a 50 pence piece,buy a packet of crisps and come out with thecorrect change?’ Therefore another keyelement or purpose behind an assessment ofmaths skills could be to find out how thechild manages number problems in everydaylife (Hughes, Desforges & Mitchell, 2002).

Typical maths developmentand key skillsEducational psychologists need to have someidea of what it would be reasonable to expecta child to know, or do, at a particular age orstage of development. What maths tasksshould an average 3, 5, 7, 10 or 15 year oldbe able to do? When the National NumeracyStrategy was introduced in England andWales (DfES, 1999) specific targets were setfor each year group, though these did notnecessarily relate to the available evidencerelating to maths attainment (Hughes,2000), or take account of the evidence thatthere are wide variations in children’s levelsand rates of development of maths under-standing pre-school (Young-Loveridge,1989) and in first years of formal mathsteaching (Hughes, 2000; Munn, 2004).Research on the development of children’smathematical skills shows that maths devel-opment is far from a simple linear progres-sion (Nunes and Bryant, 1996; Donlan, 1998;Munn, 2004; Dowker, 2005), it may be betterto think in term of a set of skills rather thana unitary skill (Munn, 2004; Dowker, 2005).

Recent reviews by Munn (2004) andDowker (2005), looking at teaching mathe-matics and development of arithmeticand mathematical skills, conclude that thereare multiple components that make up eachperson’s ability, and that there are individualdifferences in the development of each ofthese components (see Dowker and Gervasoniin this issue). Therefore our assessment willneed to consider as many of these compo-nents as possible, and acknowledge that it is


Susie Mackenzie

not necessarily helpful to make simplisticstatements about what maths knowledge or skills an individual should have at a partic-ular age.

When thinking about factors underlyingdevelopment of maths skills we can start withcognitive factors such as perception of quan-tity, number sense and ‘subitising’– the abil-ity to recognise the number of objects in avisual display, for example, * * * * is a set offour (Deheane, 1997; Butterworth, 2003),working memory (Baddeley & Hitch, 1974;Adams & Hitch, 1998), knowledge aboutnumber (Dowker, 2005), and how this is dif-ferent to knowledge about how to carry outspecific procedures with numbers (Rittle-Johnson & Seigler, 1998), fluency of countingand speed of processing (Bull & Johnston,1997), strategies, application of rules, pro-cedures and holding numbers ‘in their head’while calculating (Wright, Martland &Stafford, 2000; Dowker, 2005).

Then there are another set of factors thatcould be described as interactive or environ-mental such as early everyday experiencesrelating to number, classroom experienceand willingness to solve real life problemsusing mathematical concepts and computa-tion strategies. For example, experienceswith number and maths concepts (such assize, quantity, shape) in the home (Young-Loveridge, 1989; Sammons, Sylva et al.,2002), the quality of teaching, class size, set-ting policies (Askew, 2001; Boaler & Wiliam,2001), nature of school curriculum andwhether it connects with the child’s homeexperiences with number and maths con-cepts (Hughes, 2000), myths and emotionsabout maths learning and how childrendevelop self-perception as a learner of maths (Walkerdine, 1998).

Anxiety about learning and doing maths(Buxton, 1981; Ashcroft & Faust, 1994) is aparticularly important area. Applied psychol-ogists will be aware of the role emotions playin all aspects of learning and assessment andthere is an extensive research literature doc-umenting negative emotions and problemswith children’s self-perception as a learner of

maths, from UK children as young as eightyears old (Newstead, 1998), to secondary age pupils (Walden & Walkerdine, 1985;Walkerdine, 1998) and even among studentsin Further and Higher sector of Education inUSA (Betz, 1978; Hembree, 1990) and in theUK (Mackenzie, 2002). There is evidence ofthe effects of anxiety on maths test perform-ance (Ashcraft & Faust, 1994) and advice onhow to cope with or overcome fears thatinterfere with performance of maths skills(Tobias, 1980; Buxton, 1981; Hembree,1990).

Mathematical skills and their develop-ment are complex, therefore a range of keyskills or components of maths understand-ing need to be considered when doing anassessment. To make an analogy with theassessment of reading difficulties most prac-titioners would agree that at least basic phon-ics, word decoding and understanding oftexts are key ‘ingredients’ of a reading assess-ment. An assessment of maths and numeracyskills may need to include the following(depending on the age of the child and thedescription of the problem provided by theirteacher or parent): understanding of basicconcepts of size and sets (at perceptual andsymbolic levels); knowledge of the numbersystem and counting skills; understanding ofthe language of maths; memory for specificnumber facts such as times tables or numberbonds; computation strategies and proce-dures; attitudes to learning maths and self-perception as a learner of maths; and finally,ability to solve problems using mathematicalconcepts and numbers in everyday life.

Problems that can occur in thedevelopment of arithmeticand maths skillsGifford (2005) provided a comprehensivereview of the literature and research evidencerelating to maths difficulties in children froma range of countries (including USA, Europeand the Middle East). Her summary of theevidence states that about five percent ofchildren may experience difficulties learningnumber skills, with boys and girls in equal


Educational psychologists assessment of children’s arithmetic skills

numbers. Within this context, the onlyagreed defining characteristic for ‘dyscalculia’is poor arithmetic skills (in particular poornumber fact recall) that persists despiteappropriate teaching. She also noted anotherfinding, common across a range of studiesand cultures, that difficulties with mathslearning can co-occur with a range of otherdifficulties to do with reading, language,spatial awareness, coordination, attentionand memory.

In a very small number of children, suchas those with global cognitive deficits andcomplex learning difficulties, there maybe a failure to develop even a perceptualunderstanding of number (Dehearne, 1997;Staves, 2001), or inability to develop frominitial iconic numerosity to understanding asymbolic and language based number system(Gifford, 2005). For other children, withoutcomplex difficulties, there are a range ofproblems. Some can rote count withoutunderstanding the principles underlyingcounting: count each object once and onlyonce, fixed order, the final number definesthe set, the number for the set is independ-ent of other qualities such as the size of theobjects and that the number of the set is thesame no matter which order you count indi-vidual items (Gelman & Galistel, 1978).Some children have good ‘situated mathe-matics’ for dealing with number in the realworld, but cannot connect this to the mathstaught in the classroom (Nunes & Bryant,1997), for example their understanding ofnumber relating to real objects may not ‘con-nect’ with the mathematical language andsymbolic number system used in classroomlearning (Sammons, Sylva et al., 2002; Munn,2004). For some children language deficitsmay contribute to difficulties understandingmaths (Donlan, 1998). Other difficultiescontributing to problems with maths includeuse of inefficient computation strategies ormisapplication of a learned strategy (Wrightet al., 2000), inappropriate targets for yeargroups driven by the National NumeracyStrategy and gaps in knowledge not pickedup by teachers (Hughes, 2000; Munn, 2004),

slow speed of processing (Bull & Johnston,1997) or working memory problems (Adams& Hitch, 1998; Henry & MacLean, 2003;McKenzie, Bull & Gray, 2003).

We should not be looking only at cognitiveprocessing difficulties but also consideringemotional and environmental/situationalfactors (Walkerdine, 1998; Gifford, 2005). Wealso need to be aware of groups that areknown to have specific difficulties learningmaths skills, for example, specific conditionssuch as Down’s syndrome (Bird & Buckley,2000) specific language disorders (Donlan,1998) and hearing impairments (Nunes &Moreno, 1998).

Equally we need to be wary of unhelpfulmyths and generalisations such as ‘girls can’tdo maths’, ‘you need to be brainy to domaths’, ‘Asian kids are good at maths’ or you‘won’t need maths if you are going to be aprofessional footballer, mechanic, rock staror mother’ that may be affecting assumptionsabout children’ learning (Walkerdine, 1998).

Psychometric assessmentPsychometric or ‘norm-referenced tests’compare an individual with a large group ofchildren, and children of a range of ages.Performance on these tests can be an indica-tion of capacities, but is not an explanationof those capacities (Howe, 1997), and inter-pretations of test scores will generally needto take notice of contextual factors, par-ticularly where the child being tested haslimited communication skills or where spoken English is not their first language.

Table 1 compares the content of some UKstandardised tests commonly used by educa-tional psychologists to look at mathematicalunderstanding, understanding of numbersand computation skills. When using theseassessment tools practitioners need to beaware of, and consider the implications fortest performance of, a number of issues.Where a test includes written calculations it isimportant to check that the format and sym-bols used accord with those familiar to thechild. For example, children at Key Stage 1 (5to 7 years old) who are following the National


Susie Mackenzie



Brit

ish

Abili

ty S

cale

s BA

S II

(Elli

ott,

Smit

h &

McC

ullo

ch, 1

996)

Sub

test

Cont

ent

Age

rang

eCo

mm

ent

Early

num

ber

conc

epts

Basi

c co

ncep

ts, s

ize,

mor

e th

an;

2:6–

7:11

Stan

dard

ised

prio

r to

impl

emen

tati

on o

f th

e N

NS.

Basi

c nu

mbe

r (z

ero

to 2

dig

its,

Mat

hem

atic

al la

ngua

ge u

sed

inte

ns a

nd u

nits

) and

cou

ntin

g.qu

esti

onin

g ev

en f

or v

ery

firs

t it

ems.

Qua

ntit

ativ

e re

ason

ing

Patt

ern,

mat

chin

g an

d ba

sic

num

ber

5:0–

17:1

1St

anda

rdis

ed p

rior

to im

plem

enta

tion

of

the

NN

S.co

ncep

ts; m

ore

than

, les

s th

an, d

oubl

e,

Early

item

s al

l vis

ual r

athe

r th

an n

eedi

ng

incr

easi

ng t

o m

ore

com

plex

rul

es s

uch

unde

rsta

ndin

g of

mat

hem

atic

al la

ngua

ge.

as d

ivid

e by

2 t

hen

take

aw

ay 1

.La

ter

item

s re

quire

chi

ld t

o ap

ply

know

ledg

eab

out

com

puta

tion

and

num

ber

syst

em r

equi

ring

hold

ing

inte

rim r

esul

ts in

WM

.

Num

ber

atta

inm

ent

Reco

gnit

ion

of n

umbe

rs, s

ymbo

lic

5:0–

17:1

1St

anda

rdis

ed p

rior

to im

plem

enta

tion

of

the

NN

S.co

mpu

tati

on (�

, �, x

and

);

frac

tion

s, Ve

rtic

al la

yout

of

com

puta

tion

and

sym

bols

use

dpe

rcen

tage

s, de

cim

al n

otat

ion,

pow

ers,

for

divi

sion

such

as

3)96

not

com

patib

le w

ith N

NS.

som

e ap

plic

atio

n of

num

ber

such

as

No

wor

d pr

oble

ms

or c

once

ptua

l con

tent

di

scou

nted

pric

es a

nd f

uel c

onsu

mpt

ion.

at e

arly

leve

ls.

Wec

hsle

r In

telli

genc

e Sc

ale

for

Child

ren

WIS

C IV

UK

(Wec

hsle

r, 20

04)

Sub

test

Cont

ent

Age

rang

eCo

mm

ent

Arit

hmet

ic T

est

Basi

c co

unti

ng, s

impl

e co

mpu

tati

on

6:0–

16:1

1St

anda

rdis

ed p

ost

impl

emen

tati

on o

f th

e N

NS.

(�, �

, x a

nd

) and

app

licat

ion

ofFi

rst

5 it

ems

have

vis

ual c

ues,

all o

ther

item

s nu

mbe

r an

d co

mpu

tati

on in

wor

d pr

oble

ms

rely

on

unde

rsta

ndin

g of

a v

erba

l que

stio

n an

don

hol

ding

info

rmat

ion

in W

M.

(Con

tinu

ed)


Susie Mackenzie

Tabl

e 1:

(Con

tinu

ed) C

ogni

tive

bat

terie

s us

ed b

y ed

ucat

iona

l psy

chol

ogis

ts in

the

ass

essm

ent

of a

rithm

etic

ski

lls

Wec

hsle

r In

divi

dual

Ach

ieve

men

t Te

st W

IAT

II U

K(W

echs

ler,

2005

)

Sub

test

Cont

ent

Age

rang

eCo

mm

ent

Mat

hem

atic

al R

easo

ning

Rang

es f

rom

bas

ic c

ount

ing,

mor

e th

an,

4:0–

16:1

1St

anda

rdis

ed p

ost

impl

emen

tati

on o

f th

e N

NS.

size

, sha

pe, p

atte

rn, k

now

ledg

e of

the

Co

mpl

ex m

athe

mat

ical

lang

uage

is u

sed

innu

mbe

r sy

stem

, mon

ey, c

lock

s an

d ca

lend

ars,

ques

tion

ing,

but

eve

ry t

est

item

has

vis

ual

to g

eom

etry

, fra

ctio

ns a

nd p

roba

bilit

y th

eory

info

rmat

ion

that

red

uces

load

on

WM

.Re

spon

se b

ookl

et t

ells

exa

min

er w

hat

skill

s ar

e be

ing

test

ed in

eac

h it

em a

nd in

clud

es a

grid

for

no

ting

qua

litat

ive

obse

rvat

ions

.

Num

ber

Ope

rati

ons

Reco

gnit

ion

and

writ

ing

of n

umbe

r sy

mbo

ls,

4:0–

16:1

1St

anda

rdis

ed p

ost

impl

emen

tati

on o

f th

e N

NS.

coun

ting

, and

com

puta

tion

(�, �

, x a

nd

), H

oriz

onta

l lay

out

of c

ompu

tati

ons

and

nota

tion

fr

acti

ons,

perc

enta

ges,

squa

re r

oots

and

co

mpa

tibl

e w

ith

NN

S.hi

gher

pow

ers,

geom

etry

Resp

onse

boo

klet

tel

ls e

xam

iner

wha

t sk

ills

are

bein

g te

sted

in e

ach

item

and

incl

udes

a g

rid f

or

noti

ng q

ualit

ativ

e ob

serv

atio

ns.

No

wor

d pr

oble

ms.

Numeracy Strategy will not be familiar withthe ‘vertical format’ used on the BAS IInumber skills attainment test, as they haveonly been introduced to calculation with ahorizontal format (e.g. 3 � 2 � 5 and4 � 6 � 24). Similarly some test forms will usethe ‘old fashioned’) rather than the sym-bol for division.

Large steps between test items make ithard to determine actual competence or spe-cific gaps or misconceptions and the contentof the tests tends to have a very narrow focuson symbolic computation and not to includeapplication of number (Wiliam, 2006), or totest ability to manage computation with visualor concrete support (number lines, structuredapparatus etc). Children may fail some testitems due to any one of many problems: lackof knowledge of the number system or mathsconcepts, limited language understanding,the demands made on working memory orheightened anxiety due to performing in atest situation with an unfamiliar adult. Thefocus when using these tools needs to be notonly on the actual scores and what an individ-ual got ‘right’ or ‘wrong’, but also on how the child approached the task, what strategiesthe child used to attempt to answer test itemsand careful scrutiny of ‘wrong’ answers to geta feel for an underlying gap in knowledge,misconception or mis-applied strategy.

Teachers will be familiar with test mat-erials for assessment relating to the NationalCurriculum such as Standard AttainmentTests (SATs) for each Key Stage of Educa-tion, NFER tests and such tests as the BasicNumber Screening Test (Gillham, 2001).The Basic Number Screening Test (BNS) hasthe advantage of being easy to administerand ‘child friendly’ and has two parallelforms allowing for test/re-test comparison(e.g. before and after an intervention). Stan-dardisation of the BNS is from 7 to 12 yearsbut there are not enough test items to allowdiscrimination at the bottom end of the scaleto get a clear picture of specific gaps inunderstanding for many primary agechildren with serious weaknesses in maths.

One additional assessment tool on the market is the Dyscalculia Screener (Butterworth, 2003). It is standardised forchildren aged 6 to 14 years, and as its namesuggests, it is supposed to identify children‘at risk’ by providing information to teachersabout skill levels on a set of specific tasks pre-sented on computer, with information aboutresponse times as well as numbers of correctand incorrect answers. Though responsetimes may be of interest, the computer taskshave little ecological validity (how the childuses number skills in everyday life), andwould not be advisable for any child experi-encing ‘maths anxiety’. The screener is amixture of tests based on assumptions aboutunderlying processes such as subitising, andstraight forward knowledge about thenumber system and computation. Profilesproduced for individuals on the screenerwould not necessarily provide clues as to thenature of child’s difficulties, or gaps in theirunderstanding, and do not lead to specificadvice for teachers about appropriate inter-ventions. The Dyscalulia Screener does nottest language understanding, visual memory,working memory or what strategies are beingused in computation.

At present we do not have a specific stan-dardised test designed to highlight strengthsand weaknesses of underlying skills when achild fails to make progress with maths, or togive us detailed information that could beused for the design of an individual interven-tion programme. The Working Memory TestBattery (Gathercole & Pickering, 2001)could be used to explore components ofworking memory that might be affecting per-formance of computation, but this would notidentify gaps in basic knowledge, problemswith the language of maths or inappropriateprocedures used in calculations.

If assessing a child that has been experi-encing failure with maths learning for sometime evidence relating to maths anxiety isworth noting and bearing in mind wheninterpreting test results. Ashcroft & Faust(1994) took groups of young adults who hadexperienced failure with maths at school and



gave them computations to do in ‘test’ andinformal conditions. Performance in testconditions was markedly worse than that ininformal conditions. Performance on testswhere participants knew their performancewas being monitored did not reflect theiractual ability to perform computations inless stressful conditions or in everyday life.

Standardised tests provide scores thatteachers or parents find useful as a guide tothe extent of the problem, or confirmationby an independent ‘expert’ of their assertionthat this child has significant difficulties(Freeman & Miller, 2001). Test performancemay give some indication as to whether ornot there is cause for concern or a need forfurther investigation or intervention. How-ever, scores on these tests on their own do little in terms of furthering understanding ofthe nature of the difficulties of an individualchild, tell us nothing of how the child canuse number skills in real life and do not nec-essarily provide clues for designing an appro-priate intervention.

Keeping detailed notes relating to thechild’s performance while administering astandardised test will be helpful when formu-lating hypotheses about the nature of theirdifficulties: such as strategies being used,need for visual aids or writing down workings,misapplying procedures or gaps in knowl-edge, time taken to retrieve information,what skills are fluent and which requireattentional resources. The WIAT (see Table1) does provide a useful guide in the recordbooklet for recording some observations.

Curriculum-based assessmentAt its most basic a ‘curriculum based assess-ment’ is a descriptive assessment giving aclear picture of what the child can do, with-out reference to what other children can orcan’t do, or what is expected at a given age.In terms of maths skills it might be statementssuch as ‘can add two single digits’ or ‘knowsnumber bonds to 10’. As with test scores suchstatements may be useful descriptors, orindications that there is a problem, but dolittle to answer ‘why’ questions or allow for

testing of a hypothesis about the nature ofthe problem.

When an educational psychologist isasked to look at progress of an individual orsmall number of children falling signifi-cantly behind peers National Curriculum(NC) levels routinely used by schools tomonitor progress are not detailed enough,or the child may not have reached NC level1 (i.e., the child does not have the skillsexpected of a five year old). In these cases wecan suggest use of ‘Performance Indicatorsfor Value Added Target Setting or PIVATS(Lancashire, 2002). PIVATS cover NC levels1 to 5, but in addition they provide ‘indica-tors’ for levels P1 to P8 leading up to NClevel 1. PIVATS categorise maths learning inthree components: ‘Number,’ ‘Applicationof Number’ and ‘Shape, Space and Measures’with detailed descriptive statements aboutwhat the child can do. In addition each level(P1 to P8) is broken down into five parti-cular skills. For example P5 level for Numberhas five ‘indicators’ if the child has achievedone of these indicators then they could besaid to be at NC level P5i, when they achieveanother of the indicators they are at levelP5ii and so on. These detailed descriptorscan be particularly useful when holding aconversation with a teacher regarding tar-gets for an Individual Education Plan (IEP),personalisation of learning and for monitor-ing learning over a term or year.

Informal, qualitative and interactive assessmentEducational psychologists can adopt appro-aches known as ‘dynamic assessment’ to lookat the skills the child brings to learning situ-ations in general (Elliott, 1993; Elliott, 2003;Resing, 2006) such as adaptive thinkingskills, mediation of learning, ability to learnfrom demonstration of a rule.

This approach can be applied to under-standing of maths constructs, ability to carryout particular mental maths tasks or usemaths in real life. Given the lack of diagnosticpsychometric tools discussed earlier, thesemay be more helpful in making sure an assess-ment can test a range of hypotheses or pro-


Susie Mackenzie

vide evidence for recommending a particularintervention, but it will take considerablymore time and by its nature it is not standard-ised and could be seen by some as biased orsubjective (Elliott, 1993; Dowker, 2005).

In an informal assessment of maths skillsthe educational psychologist can engage thechild in maths activities and games in arelaxed way (minimising test anxiety, Ashcraft& Faust, 1994). We can use observation,questioning and mediated learning to lookfor zones of proximal development and togather information relating to the child’sunderstanding of quantity, numbers, lan-guage of maths, counting fluency, strategies,recall of number facts and ability to applyknowledge in real, practical ways.

The assessment can become a conversa-tion: asking the child to do a range of mathsrelated activities, taking into account the ageof the child, expected targets for their age andinformation from school about what the childcan and can’t do, or how the child approachesactivities relating to maths and arithmetic. As‘prompts’ for questions and concepts to use toidentify specific maths learning problemsWright et al. (2000) provide a very usefulassessment technique (p39–62) and Denvir &Bibby (2001) have produced a very clear hier-archy of skills in their ‘Diagnostic Interviews inNumber Sense’.

Education Leeds School Support Servicehas devised their own assessment sheetsbased on principles set out by Denvir &Bibby (2001). A copy of the assessment guideis included as an appendix. The focus is onkey skills and both asking questions of thechild and asking the child to set challengesfor you, with hints as to what to recommendif a child cannot complete a particular task.It is very flexible, particular skills to check indetail can be related to the age and generaldevelopmental level of the child and whatthe class teacher or parent has already toldyou. This might include:● Does this child count with one to one

correspondence?● Does this child understand specific math-

ematical language?

● Is this child operating at an iconic or sym-bolic level of understanding of number?(Can the child count or compute usingsymbols or do they need objects, fingersetc.)

● How sound is their long-term memoryfor the number system/number bonds/multiplication facts/place value?

● What skills are fluent (retrieving numberfacts from long term memory) and whichrequire conscious effort (and workingmemory resources)?

● What strategies is the child using for com-putation (�, �, � and )?Items I have in my maths assessment kit,

used alongside this guide, include a range ofbeads for sorting, counting and estimating,large dice, playing cards, a small clock, realcoins (£1 to 1p), Numicon plates, place value(PV) arrow cards, a 100 square and whiteboard and marker. With children in the lastyear of Primary School and at Secondary levelI am interested in not only number but alsoapplication of knowledge especially relating toclocks and money (hence the small clock andset of coins in my kit), and their experiences asa ‘maths learner’ to check beliefs about maths,motivation and attitudes to learning aboutmaths. Most EPs would include in an assess-ment questions about likes and dislikes,favourite activities etc., and these can be easilyapplied to an assessment of affect, motivationand attitudes to learning about numbers ask-ing not only obvious questions such as ‘do youlike learning about numbers? And ‘do you feelconfident in maths lessons’, but also exceptionquestions such as ‘which bits of numeracylessons do you like best?’ or ‘which bits do youfind easy?’ or ‘how does Mrs X help you?’.

Results of assessments: advice and interventionIf the purpose of an assessment were to iden-tify what the nature of the problem is, thenone of the results would be to give advicetailored to help the individual. Educationalpsychologists may also be asked for theiradvice if a whole class appears not to be mak-ing progress or when tracking for the school



shows that ‘value added’ for maths is not asgood as that for literacy or science.

A useful place to start for both individualsolutions and class or school level interven-tions would be to discuss with staff theirwhole approach to maths teaching and sug-gest they look at publications from the DfESsuch as ‘Supporting Pupils with gaps in theirmathematical understanding’ (DfES, 2005;see also Gross in this issue), and encourageteachers to implement specific teaching toaddress gaps in understanding (Dowker,2005; Gifford, 2005; see also Dowker in thisissue). Sometimes a child struggling withmaths may be expected to practice skills theyfind difficult, rather than have specific teach-ing to address gaps in their understanding,but this should be avoided as it could becounter-productive, leading to frustration,boredom and lack of motivation (Boaler &Wiliam, 2001).

If assessment of the difficulties of an indi-vidual or group identifies a specific problem,then approaches can be recommended toaddress that problem. For example, if it is aproblem with working memory then meth-ods can be introduced to reduce memoryload (such as use of visual and concreteaids); if it is a problem of misapplication of aset of strategies then this can be addressedthrough specific teaching and explanation.There are now intensive structured pro-grammes available such as the MathematicsRecovery Programme (Wright et al., 2000; seealso Willey et al., this issue), the NumeracyRecovery Programme (Dowker, 2001;Dowker, 2004 see also Dowker this issue) andapproaches that make use of visual supportfor maths (Wing and Tacon, 1999; DfES,2005; Brighton & Hove, 2006). Older pupilswill benefit from developing strategy flexibil-ity or improving knowledge of the sequentialstructure of numbers (see Verschaffel et al; Elle-mor-Collins and Wright, this issue). Schoolsmight also want to consider peer tutoringand group collaboration, re-thinking settingarrangements and other ‘in house’ initiatives(Topping & Bamford, 1998; Boaler &Wiliam, 2001; Topping, Campbell, Douglas

& Smith, 2003; Dowker, 2005). Interventionfor older pupils and young adults mightalso need to address self-perceptions andconstructs relating to learning maths (Hembree, 1990).

Structured recovery and catch up pro-grammes for pupils with problems need to bedelivered by staff with a good understandingof mathematics, how skills develop and thenature of the child’s specific problems.Teaching assistants (TAs) are increasinglylikely to be the main providers of support forindividual children who need individuallearning plans, and good quality training isessential for TAs (Farrell, Balshaw & Polat,1999). Teaching assistants supportingchildren struggling with maths may need par-ticular assistance with understanding how toteach number concepts, or particular skillsand strategies, in which case the best ‘inter-vention’ might be to provide specific trainingfor TAs regarding the nature of mathematicaldevelopment: key concepts, language, pro-cessing required and so on. Alternatively wemight advise the class teacher, who we wouldexpect to know more about teaching mathe-matics, to let the TA oversee more capablestudents while she teaches the child with dif-ficulties or devises an individual programmeto be delivered by a TA.

Sure Start and other community initia-tives provide another potential ‘interven-tion’ through pre-school community basedschemes, such as the BigMath programme(Ginsburg, Balfanz & Greenes, 1999) whichwas designed to raise maths understandingin young children through games and other‘fun’ activities. Workshops can be run forfamilies with ideas for games and activities todo at home: everyday activities from themundane such as setting the table for ameal; counting songs with classics such as‘five speckled frogs’; card, dice and boardgames. These activities provide practice for arange of concepts, counting and computa-tion skills in fun and engaging ways, and canbuild confidence for parents as well aschildren (Warren & Westmoreland, 2000).


Susie Mackenzie

Adams, W. & Hitch, G.J. (1998). Children’s mentalarithmetic and working memory. Chapter 7 inC. Donlan (Ed.), The development of mathematicalskills. Hove: Psychology Press.

Ashcraft, M.H. & Faust, M.W. (1994). Mathematicsanxiety and mental arithmetic performance: anexploratory investigation. Cognition and Emotion,8, 97–125.

Askew, M. (2001). Policy, practices and principles inteaching numeracy: What makes a difference? InP. Gates (Ed.), Issues in mathematics teaching (chap-ter 8). London: Routledge.

Baddeley, A.D. & Hitch, G.J. (1974). WorkingMemory. In G.H. Bower (Ed.), The psychology oflearning and motivation, 8, 47–89.

Betz, N.E. (1978). Prevalence, distribution and correlates of maths anxiety in college students,Journal of Counselling Psychology, 25, 441–448.

Boaler, J. & Wiliam, D. (2001). ‘We’ve still got tolearn!’ Students’ perspectives on ability groupingand mathematics achievement. In P. Gates (Ed.),Issues in mathematics teaching. London: RoutledgePress.

Bird, G. & Buckley, S. (2001). Number skills of individ-uals with Down Syndrome – an overview. Portsmouth:Down Syndrome Educational Trust.

Brighton & Hove (2006). Visual Models and Imagessupported by Signs and Symbols: A ‘Wave 3’programme designed by Brighton and Hove LEA.

Bull, R & Johnston, R.S. (1997). Children’s arithmeticdifficulties: Contributions from processing speed,item identification and short-term memory. Jour-nal of Experimental and Children Psychology, 64, 1–24.

Butterworth, B. (2003). Dyscalculia screener: Highlight-ing pupils with specific learning difficulties in maths.Windsor: nferNelson.

Buxton, L. (1981). Do you panic about maths? Copingwith maths anxiety. London: Heinemann.

Dehaene, S. (1997). The number sense: How the mindcreates mathematics. London: Allen Lane.

Denvir, H. & Bibby, T. (2001). Diagnostic interviews innumber sense. London: King’s College.

DfEE (1999). The national numeracy strategy. Suffolk:Cambridge University Press.

DfES (2005). Primary national strategy: Using modelsand images to support mathematics teaching and learn-ing in Years 1 to 3. Norwich: HMSO.



ConclusionThis article aims to lead practitioners to amore thorough understanding and approachto the assessment of basic arithmetic (ability tocompute with numbers), numeracy (ability touse numbers in real life) and identifying gapsin mathematical understanding and knowl-edge. Educational psychologists will be awareof the five outcomes of education as set out inthe Children’s Act (2004): to be happy, safe,enjoying learning, become economicallysecure and able to make a positive contribu-tion to society. Basic numeracy and mathsskills contribute to all these, especially eco-nomic security, and helping children tobecome ‘mathematically literate citizens in society’(Wiliam, 2006).

Standardised tests and curriculum-basedassessments have their place in the psycholo-gist’s toolkit. However, if we want to answer‘why?’ questions, to consider differenthypotheses about the nature of the problemand to be in a position to recommend well-targeted interventions, then these approachesare best used with caution and in conjunc-

tion with more searching observation andinteractive questioning along the lines sug-gested in this article.

What is really important is that practi-tioners doing an assessment of maths diffi-culties have a good understanding of thecomplexity of maths learning and what dif-ferent factors might be contributing to thedifficulties of an individual. Educational psy-chologists are in an important position toinform and support those working directlywith the child, so we need to be wellinformed. Where children have problemslearning maths our efforts to make a differ-ence will be depend on the quality of ourown knowledge and understanding regard-ing maths learning in general and mathsproblems in particular.

Address for correspondenceSusie Mackenzie, Psychology and AssessmentService, Education Leeds Blenheim CentreCrowther Place Leeds LS6 2STE-mail: [email protected]

References

DfES (2005). Primary national strategy: Supportingchildren with gaps in their maths understanding: Wave3 mathematics. Norwich: HMSO.

Donlan, C. (Ed.) (1998). The development of mathe-matical skills. Hove: Psychology Press.

Dowker, A. (2001). Numeracy recovery: a pilotscheme for early intervention with youngchildren with numeracy difficulties. Support forLearning, 16(1), 6–10.

Dowker, A, (2004) What works for children with mathe-matical difficulties. London: DfES ResearchReport 554.

Dowker, A. (2005). Individual Differences in Arithmetic:implications for psychology, neuroscience and educa-tion. Hove: Psychology Press.

Elliott, C., Smith, P. & McCulloch, K. (1996). Britishability scales second sdition (BASII). Windsor:nferNelson.

Elliott, J. (1993). Assisted Assessment: if it is dynamicwhy is it so rarely employed? Educational and ChildPsychology, 10(4), 48–58.

Elliott, J. (2003). Dynamic Assessment in educationalsettings: Realising potential. Educational Review,55(1), 15–32.

Farrell, P., Balshaw, M. & Polat, F. (1999). The manage-ment, role and training of learning support assistants.London: Department for Education and Science.

Fredrickson, N., Webster, A. & Wright, A. (1991). Psy-chological assessment: A change of emphasis.Educational Psychology in Practice, 7, 20–29.

Freeman, L. & Miller, A. (2001). Norm-referenced,Criterion-referenced and dynamic assessment:What exactly is the point? Educational Psychologyin Practice, 17(1), 3–16.

Gathercole, S. & Pickering, S. (2001). The workingmemory test battery for children. London: HarcourtAssessment.

Gelman, R. & Gallistel, C.R. (1978). The child’s under-standing of number. Cambridge MA: HarvardUniversity Press.

Gifford, S. (2005). Young children’s difficulties in learn-ing mathematics: Review of research in relation todyscalculia. London: Qualifications and Curricu-lum Authority.

Gillham, B. (2001). Basic number screening test. Lon-don: Hodder & Stoughton.

Ginsburg, H.P., Balfanz, R. & Greene, C. (1999).Challenging mathematics for young children. In A. Costa (Ed.), Teaching for intelligence, II: A col-lection of articles. Arlington Heights: Skylight.

Hembree, R. (1990). The nature, effects and relief ofmaths anxiety. Journal for Research in Maths Educa-tion, 21, 33–46.

Henry, L.A. & MacLean, M. (2003). Relationshipsbetween working memory, expressive vocabularyand arithmetical reasoning in children with and

without intellectual disabilities. Educational andChild Psychology, 20(3), 51–64.

Howe, M.J.A. (1997). IQ in question: The truth aboutintelligence. London: Sage.

Hughes, M. (2000). The national numeracy strategy:Are we getting it right? The Psychology of EducationReview, 24, 3–11.

Hughes, M., Desforges, C. & Mitchell, C. (2002)Numeracy and beyond. Buckingham: Open Univer-sity Press.

Lancashire County Council (2002). Performance indi-cators for value added target setting: Assessment of learning, performance monitoring and effective target setting for all pupils revised edition. Preston:Lancashire Professional Development Service.

Mackenzie, S. (2002). Can we make maths count inHE? Journal of Further and Higher Education, 26(2),159–172.

McKenzie, B., Bull, R. & Gray, C. (2003). The effectsof phonological and visuospatial interference onchildren’s arithmetical performance. Educationaland Child Psychology, 20(3), 93–108.

Miller, A. & Leyden, G. (1999). A coherent frame-work for the application of psychology in schools.British Educational Research Journal, 25, 389–400.

Munn, P. (2004). The psychology of maths educationin the early primary years. The Psychology of Educa-tion Review, 28(1), 4–8.

Newstead, K. (1998). Aspects of children’s mathemat-ics anxiety. Educational Studies in Mathematics, 36,53–71.

Nunes, T. & Bryant, P. (1996). Children doing mathe-matics. Oxford: Blackwell.

Nunes, T. & Bryant, P. (Eds.) (1997), Learning andteaching mathematics: An international perspective.Hove: Psychology Press.

Nunes, T. & Moreno, C. (1998). Is hearing impairmenta cause of difficulties in learning mathematics? InC. Donlan (Ed.), The development of mathematicalskills (chapter 10). Hove: Psychology Press.

Rittle-Johnson, B. & Seigler, R.S. (1998). The rela-tionship between conceptual and proceduralknowledge in learning mathematics: A review. InC. Donlan (Ed.) The development of mathematicalskills (chapter 4). Hove: Psychology Press.

Resing, W.C.M. (2006). Using children’s ‘ability tolearn’ in diagnosis and assessment. Educationaland Child Psychology, 23(3), 6–11.

Sammons, P., Sylva, K., Melhuish, E. C., Siraj-Blatch-ford, I., Taggart, B., & Elliot, K. (2002). The effec-tive provision of pre-school education project. TechnicalPaper 8a: Measuring the impact on children’s cognitivedevelopment over the pre-school years. London: Insti-tute of Education/DfES.

Staves, L. (2001). Mathematics for children with severeand profound learning difficulties. London: DavidFulton.


Susie Mackenzie

Tobias, S. (1980). Maths anxiety: What you can doabout it. Today’s Education, 69(3), 26–29.

Topping, K. & Bamford, J. (1998). Parental involve-ment and peer tutoring in mathematics and science.London: David Fulton.

Topping, K.J., Campbell, J., Douglas, W. & Smith, A.(2003). Cross-age peer-tutoring in mathematicswith seven and eleven year olds: Influence onmathematical vocabulary, strategic dialogue andself-concept. Educational Research, 45(3), 287–308.

Tymms, P. & Elliott, J. (2006). Difficulties and dilem-mas in the assessment of special educationalneeds. Educational and Child Psychology, 23(3),25–34.

Walden, R. & Walkerdine, V. (1985). Girls andmathematics: From primary to secondary schooling.Bedford Way Papers 24 London: Heinemann.

Walkerdine, V. (1998). Counting girls out: Girls andmathematics (2nd edn). Hove: Falmer Press.

Warren, V. & Westmoreland, S. (2000). Number andplay in everyday life. In Drury, R., Miller, L. &

Campbell, R. (Eds.), Looking at early years educa-tion and care (chapter 16). London: David Fulton.

Wechsler, D. (2004). Wechsler Intelligence Scale for children(UK 4th edn). London: Harcourt Assessment.

Wechsler, D. (2005). Wechsler Individual AchievementTest (UK 2nd edn)London: Harcourt Assessment.

Wiliam, D. (2006). Keeping learning on track: Class-room assessment and the regulation of learning.In F.K. Lester (Ed.), NCTM handbook of research onmathematics education.

Wing, T. & Tacon, R. (1999). A visual approach to men-tal arithmetic. Paper presented at the BCME-4University College, Northampton.

Wright, R.J., Martland, J. & Stafford, A.K. (2000).Early numeracy: Assessment for teaching and interven-tion. London: Paul Chapman.

Young-Loveridge, J. (1989). The relationship betweenchildren’s home experiences and their mathe-matical skills on entry to school. Early ChildDevelopment and Care, 43, 43–59.




Susie Mackenzie

Appe

ndix

: Ed

ucat

ion

Leed

s m

aths

ass

essm

ent

guid

eTa

skQ

uest

ions

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erva

tion

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art

at 1

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pace

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n nu

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es n

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afte

r *

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nt t

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xt n

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xt 2

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ture

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nt t

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num

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ps

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num

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st b

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num

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line

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l giv

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num

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2 nu

mbe

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efor

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ll m

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e 2

num

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t co

me

e.g.

14

15 1

6 _

just

bef

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ctis

e w

ith

100

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lash

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1 to

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ary

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aske

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tim

e to

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ualis

e’

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ing

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sub

trac

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use

ow

n 10

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and

link

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ract

1 f

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me

wha

t su

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eans

● d

oes

this

men

tally

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. rel

ates

to

‘nex

t nu

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d ‘su

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onu

mbe

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oing

to

say

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‘num

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trac

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ses

fing

ers

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equi

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us●

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se

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ary

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stio

nsO

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ons

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mm

enda

tion

s

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num

ber

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ll m

e th

e nu

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1 m

ore

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nder

stan

ds k

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ular

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wor

k pr

acti

cally

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l lif

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oble

ms

is 1

mor

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an *

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an d

o w

hen

give

n a

real

life

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uild

in t

ime

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n 10

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ll m

e th

e nu

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1 le

ss●

use

fla

sh c

ards

/voc

abul

ary

book

that

is 1

less

than

*

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s pa

irs o

f nu

mbe

rsYo

u di

d re

ally

wel

l wit

h th

e ad

ding

up

● c

an d

o th

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ando

mly

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ck r

ecal

l●

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vis

ual p

rom

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whi

ch m

ake

10W

hat

does

thi

s nu

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y? (1

0)●

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ds t

o gi

ve a

nsw

ers

in a

lear

nt(s

choo

l and

hom

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ll m

e 2

num

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tha

t ad

d up

to

10, e

tc.

patt

ern/

sequ

ence

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omin

o ac

tivi

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le f

lips

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eeds

to

be g

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a n

umbe

r ●

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w p

roce

ssin

g ti

me

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s pa

irs o

f nu

mbe

rsAs

abo

vepr

ompt

whi

ch m

ake

20 a

nd●

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s no

t re

cogn

ise

com

mut

ativ

ity

rela

te t

o hi

gher

num

bers

● a

ll ab

ove

appl

y to

hig

her

num

bers

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2 nu

mbe

rs

Tell

me

wha

t *

add

* is

/mak

es/e

qual

s●

cou

nts

all

● w

ork

prac

tica

lly/r

eal l

ife

prob

lem

unde

r 10

/20

● c

ount

s on

fro

m f

irst

num

ber

● t

each

to

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t al

l●

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nts

on f

rom

larg

est

num

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t on

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kSu

btra

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ow c

ard,

e.g

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ead

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me.

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ount

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lyco

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n fr

om la

rges

t num

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r 10

/20

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e m

e th

e an

swer

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use

s kn

own

fact

s to

der

ive

use

num

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bond

sTe

ll m

e w

hat

* su

btra

ct /

take

aw

ay *

is/

use

doub

les

mak

es/e

qual

sSh

ow c

ard,

e.g

. 10

– 4.

Rea

d th

is t

o m

e.

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e m

e th

e an

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t fo

rwar

ds in

Co

unt

on in

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for

me

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nder

stan

ds in

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ctio

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pra

ctis

e w

ith

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to 1

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get

s to

90

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ps o

r co

nfus

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row

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s to

20

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rev

erts

to

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vis

ual p

rom

ptCo

unt

back

war

ds in

Co

unt

back

war

ds in

10s

fro

m 1

00●

get

s to

10

then

sto

ps●

rela

te t

o 10

0 sq

uare

10s

from

100

●do

es n

ot s

ay z

ero

whe

n co

unti

ngba

ck


Susie MackenzieTa

skQ

uest

ions

Obs

erva

tion

sRe

com

men

dati

ons

Add/

subt

ract

10

(the

nTe

ll m

e w

hat

10 a

dd 1

0 is

/mak

es●

doe

s no

t re

late

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s to

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ntin

g ●

use

PV/

Arro

w c

ards

20

/30,

etc

. )

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t ab

out

20 a

dd 1

0, e

tc.

on/b

ack

in 1

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writ

e ou

t seq

uenc

esW

hat

is 2

0 su

btra

ct/t

ake

away

10,

etc

.●

reve

rts

to f

inge

rs●

dev

elop

vis

ualis

atio

n of

mov

ing

card

sSa

y ho

w m

any

10s

Wha

t do

es t

his

num

ber

say?

(e.g

. 40)

● c

an r

ead/

writ

e m

ulti

ples

of

10●

use

fla

sh-c

ards

and

rel

ate

addi

ng t

oin

40,

etc

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ow m

any

10s

mak

e/ar

e th

ere

in 4

0?●

will

say

tha

t 40

is 4

ten

sco

untin

g on

/subt

ract

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to c

ount

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back

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ill s

ay t

hat

40 is

40

tens

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y ou

t PV

car

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rd if

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dig

it n

umbe

rs if

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ropr

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appr

opria

te a

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ount

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g. 1

, 2, 3

ten

s eq

uals

30,

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rela

te t

o 10

0 sq

uare

● u

se ‘t

y’ v

isua

l pro

mpt

Coun

t fo

rwar

ds

This

tim

e I w

ant

you

to c

ount

in●

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ble

to d

o th

is●

pra

ctis

e w

ith

100

squa

re a

nd P

V ca

rds

in 1

0 fr

om a

10

s st

arti

ng a

t th

e nu

mbe

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(e.g

. 27)

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an d

o th

is f

ollo

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card

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onst

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n pr

ompt

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e ou

t se

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ces

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g PV

car

ds/1

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bey

ond

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if a

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unt

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tim

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ant

you

to c

ount

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kwar

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fro

m g

iven

num

ber

in 1

0s s

tart

ing

at t

he n

umbe

r *

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. 52)

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10 t

o a

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sta

rt w

ith

the

num

ber

* (e

.g. 4

,6,8

,9)

● a

s ab

ove

● a

s ab

ove

give

n nu

mbe

rN

ow a

dd 1

0 (a

nd 1

0 an

d 10

, etc

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not

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g co

unti

ng o

n to

add

/●

use

fla

sh c

ards

and

rel

ate

to c

ount

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ting

bac

k to

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trac

ting

on/b

ack

Subt

ract

10

from

Le

t’s s

tart

wit

h th

e nu

mbe

r *

(e.g

. 96,

98,9

9)a

give

n nu

mbe

rN

ow s

ubtr

act/

take

aw

ay 1

0Ta

sk(t

ake

away

10

take

aw

ay 1

0, e

tc)

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Que

stio

nsO

bser

vati

ons

Reco

mm

enda

tion

s

Plac

e va

lue

Poin

t to

the

num

ber

**/W

rite

the

● c

an r

ead/

writ

e 2

digi

t nu

mbe

rs●

pra

ctis

e w

ith

PV c

ards

:Sa

y ho

w m

any

num

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** f

or m

e (o

r **

*)(3

dig

it if

app

ropr

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)–

put

the

10 c

ard

on t

he t

able

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ay10

tens

/uni

ts in

a

Show

me

the

tens

?●

reve

rses

the

ord

er o

f th

e di

gits

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t th

e 1

card

on

top

– sa

y 10

2

digi

t nu

mbe

rSh

ow m

e th

e un

its?

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s )

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an id

entif

y th

e te

ns d

igit

and

say

and

1 eq

uals

11

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man

y te

ns a

re t

here

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s Ts

)th

at 4

ten

s ar

e 40

, etc

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fit

the

2 ca

rd o

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p –

say

10 a

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do

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ng.

● c

an id

enti

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ens

digi

t bu

t un

able

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ls 1

2, e

tc.

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t is

20

�3

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�40

�3,

etc

.)to

giv

e th

e va

lue

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he 4

i.e.

, say

s●

use

the

‘tee

n’ v

isua

l pro

mpt

that

it is

40

tens

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that

it is

4●

writ

e 10

and

(�) 1

�10

etc

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part

itio

n,Le

t’s u

se P

V Ar

row

Car

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dd 2

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d 3.

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h 3

digi

t nu

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ld in

tim

e to

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ards

i.e. 2

3�

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hat

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titi

on 2

/3 d

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num

bers

usi

ng14

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100

�40

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hat

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his?

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the

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THE NATIONAL Numeracy Strategy(DfEE, 1999) was introduced into Eng-lish primary schools in 1999. Its overall

aim was to raise standards of attainmentin mathematics, and in particular to meetthe government’s target that by 2002 three-quarters of 11-year-olds should reach level 4in their mathematics SATs. Yet by 2006 it wasbeing accepted that although the Strategyhad been widely welcomed by teachers, it hadnot fully achieved its targets. A new approachwas required, one which was less prescriptivethan what had gone before and whichallowed teachers to be more creative andinnovative in their practice (DfES, 2006).

While the 2006 Framework is lessprescriptive in its approach, it shares withthe original National Numeracy Strategy anumber of key assumptions about the teach-ing and learning of mathematics. In par-ticular, both documents appear to regardmathematics as a set of skills and compe-tences which are acquired and assessedwithin the classroom setting, under the

supervision of a trained professional teacher.Despite general encouragement within the2006 Framework to build good home-schoollinks, there is little or no recognition thatmathematics may be acquired and used out-side the classroom, or that parents and otherfamily members might play an importantrole in supporting children’s mathematicallearning.

Such an approach within the primarymathematics curriculum is surprising, giventhat much recent research on mathematicslearning is taking a very different perspective.Many psychologists and mathematics educa-tors are paying increasing attention to theways in which mathematics is actuallyacquired and used in a wide range of real-lifecontexts. In a classic study by Nunes and hercolleagues (Carraher et al., 1985), theauthors looked at the ways in which youngBrazilian street traders carried out calcula-tions while conducting their businesses.Carraher et al., found that the traders notonly used methods which were very different

Linking children’s home and school mathematicsMartin Hughes, Pamela Greenhough, Wan Ching Yee,

Jane Andrews, Jan Winter & Leida Salway

AbstractCurrent approaches to teaching mathematics in English primary schools pay little attention to the kind ofmathematics which children engage in outside of school. This paper attempts to redress the balance bydescribing the nature and characteristics of children’s out-of-school mathematics, and looking at howconnections might be made between in-school and out-of-school mathematics. At home, mathematics isfrequently encountered during play and games, and in authentic household activities such as cooking andshopping. There are also more school-like mathematical activities such as homework and commerciallyavailable maths schemes. The paper argues that it is important for connections to be made between homeand school mathematics, but this is often impaired by teachers’ lack of knowledge about home mathematicsand by parents’ lack of knowledge about school mathematics. One solution to this problem lies in knowledgeexchange activities, and examples are provided of activities which operate in both the school-to-home andhome-to-school directions. The main implications for teachers and educational psychologists are to paymuch greater attention to children’s out-of-school mathematics, and to develop further ways of linking homeand school mathematics.

from those they had been taught in school,but they were also more successful with theirown methods than when the same problemswere presented as ‘school maths’. Morerecently, a number of studies have shown thatdiverse groups such as farmers (Abreu,1999), nurses (Hoyles et al., 2001) andfishermen (Nunes et al., 1993) typicallydevelop and use methods for mathematicalcalculation which are different from thoseconventionally taught in schools. Such find-ings provide support for Lave’s (1988) claimthat mathematics – like other cognitiveprocesses – is essentially ‘situated’, and thatdifferent cultural practices may generate andsupport different ways of doing mathematics.

A similar picture is starting to emergefrom the more limited research on homemathematics. Street et al. (2005) argued thatthe numeracy practices in which children par-ticipate at home and at school differ on anumber of dimensions. For example, homenumeracy practices usually centre on solvinga specific local problem, while school numer-acy practices are about learning skills or con-cepts determined by the teacher. Street et al .also argued that there is considerable varia-tion across home numeracy practices. Whilethey were at pains to point out that they arenot operating from a ‘deficit’ position, theyclaimed that a key component in children’smathematical attainment at school lies inthe degree of ‘consonance’ or ‘dissonance’between children’s home and school numer-acy practices.

In this paper we will look more closely atthe nature of home mathematics, and itsrelation to school mathematics. In particular,we will argue that the possibility of makingmeaningful connections between home andschool mathematics may be limited by thefact that teachers and parents often knowvery little about what is going on at homeand at school respectively. We will furtherargue that this problem can be addressed by‘knowledge exchange activities’ betweenhome and school.

The home school knowledgeexchange projectThis paper draws on the work of theHome School Knowledge Exchange Project,which took place between 2001 and 2005. Theoverall aim of the project was to develop andimplement programmes of home-schoolknowledge exchange activities and look attheir impact on children, teachers and parents.The project consisted of three main strands,focusing in turn on literacy at key stage 1,numeracy at key stage 2, and primary/second-ary transfer. The data to be presented in thispaper comes from the numeracy strand of theproject (see Winter et al., 2004; Winter et al.,2007 for more details of this strand).

The numeracy strand involved childrenin Years four and five from four contrastingprimary schools in Bristol and Cardiff. Ineach city one school had a relatively highproportion of pupils eligible for free schoolmeals, and one school had a relatively lowproportion of such pupils. In addition, theethnic composition of the sample attemptedto match the ethnic diversity to be found inthe two cities. Within each school, a teacher-researcher seconded part-time to the projectworked with teachers, parents and childrento develop and implement a range of home-school knowledge exchange activities,designed to communicate knowledge bet-ween home and school.

The impact of the HSKE activities wasevaluated using a range of measures. Theseincluded standardised assessments of attain-ment and attitude undertaken by all thechildren in the cohort (whose performancewas then compared with children from simi-lar schools where HSKE activities did nottake place). In addition, in each school sixchildren were chosen for more intensivestudy, on the basis of gender and attainment,and in-depth interviews were carried outwith these children, their teachers and theirparents. The data presented in this papercomes primarily from these ‘target’ childrenand their families. Findings related to otheraspects of this strand will be reported insubsequent papers.


Martin Hughes et al.


Linking children’s home and school mathematics

As part of the involvement in the project,each target family was loaned a video cameraand asked to record mathematics eventswhich took place in the home. This requestwas made after a long interview in which thekinds of mathematics taking place at homehad been explored. Where detailed interac-tions are reported below, they are based onthe transcriptions of these videos.

The nature of home mathematicsThe children in our project were engaged ina substantial number of activities whichinvolved mathematics in some way. Theseactivities fell into three main groups – playand games, authentic household activities, andschool-like activities.

Many of the games which the childrenplayed involved some kind of mathematics,and some of the games involved a great deal.In Monopoly, for example, the participantsneed to add together the numbers shown ontwo dice, move their counters the appropriatenumber of spaces around the board, and giveand receive change as they buy property andcollect rent from other players. They alsohave to make more strategic financial deci-sions about whether or not to buy a particularproperty or to pay a fine to get out of jail.

Some of the children also played gameswhich they had designed or developed them-selves. Ryan, for example, was filmed by hismother playing outside on the street with hisyounger brother and some friends. Thestreet was in a quiet housing estate andthere was little traffic. The game – which thechildren called ‘Kerbs’ – involved takingturns to throw a football from one side ofthe street to the other. The aim was to hitthe opposing kerb as near to the edge as pos-sible, and this counted as 20 points A nearhit meant the thrower could get a secondthrow, this time taken from the middle of thestreet, and a successful throw here scored 10points. The game proceeded at a high paceand it was clear that Ryan and his friendswere very familiar with it.

Several of the games involving mathe-matics had clear cultural roots outside the

UK. For example, Dhanu and his olderbrother made a video of themselves play-ing ‘Carrom’. This game, which has beendescribed as ‘a combination of pool, marblesand air hockey’, is extremely popular on theIndian subcontinent and in other countrieswith a substantial south-Asian population. Itis essentially a board game, and sinking one’spieces down various holes around the boardscores points. Dhanu and his brother wereclearly experienced and skilful players, witha strong shared understanding of the gamewhich meant that conversation was kept to aminimum.

It should be noted that mathematics wasoften present in the children’s play, includ-ing their fantasy or role-play. In the followingextract from an interview with Chloe, shetalks about how she likes to play at estate agents. She describes how inher play she recreates some of the mathematics that adults would need to engage in if they were really sellinghouses:Interviewer: So when you’re playing mort-gages, what does that involve, what do you dothere?Chloe: Like I pretend to talk to people and Ilike say how much money do you want tospend and .. what’s the total money you wantto spend, and they’ll say like 300,000, some-thing like that, and then I have to try andfind them a house, like how many roomsthey want – if they want like a three-bed-roomed house I have to try and look in …pretend to look in books for a three-bed-roomed house. And if they … like if theyfind … if they want … one in like say [name](local area where she lives), but then theywant one in say Newport, I’ll pretend togo …. I’ll go on like the laptop and I’ll lookand see what one’s the best quality and theyhave to choose and something like that.Then I write it all down, like where they’removing and how much they really want tospend, and then how much it costs, and thenthey have to write me a cheque out, and thenI’ll pretend to fax the cheques off, and thenI’ll do other stuff.



Authentic household activitiesMany everyday activities which take place inand around the home involve mathematics.Cooking, for example, can involve weighingquantities, using multiplication or division toamend a recipe, calculating the time neededfor different parts of the process, and switch-ing between different units of measurement.Planning a journey or holiday can involveconsulting bus or train timetables, settinga budget, and working out the most costeffective means of travel. Programming thevideo can involve calculating how long a pro-gramme will last, working out whether thereis enough blank tape on the video, and grap-pling with the 24-hour clock.

Most of the children in our research wereregularly engaged in these kind of house-hold activities. We termed them ‘authentic’(cf Brown et al ., 1989) because the purposesunderlying the activities are an integral part ofhousehold life. Amongst other things, thismeans that it is important for all thoseinvolved that the mathematics is carried outcorrectly. If a mistake is made, then someone’sfavourite TV programme will not be recorded,or their dinner will be ruined. At the sametime, it should be noted that these activitiesare not fundamentally about mathematics orlearning mathematics, despite the importantrole which mathematics plays in their success-ful completion.

Many of these authentic household activi-ties involve money. Nadia, for example,accompanied her mother on shopping trips asher mother’s English was relatively modest.Nadia’s role was to read out the prices on thegoods in the shops, check what was boughtagainst the shopping list, and make sure hermother received the correct change. Otherchildren encountered money through assist-ing in their parents’ work. Olivia’s motherwas the part-time manager of a local centrefor learning disabled people, and Olivia fre-quently helped her at the centre, running thecoffee-bar and taking responsibility for the tak-ings. Aaqil’s father was a taxi-driver, and Aaqilwould collect spare change from his fatherwhich he would then give to charity.

One girl in the study, Ellie, made a videodescribing how she used mathematics to solvean authentic household problem. Ellie’s fam-ily were going away on holiday for two weeks,and during this time their cat would belooked after by neighbours. Ellie had to workout how much cat food to leave them. The catfood was in the form of granules and the dailyamount depended on the weight of the cat.So, Ellie needed to weigh her cat. She placedthe bathroom scales in front of the camerabut the cat did not want to stay on the scalesby itself. Ellie told the camera how she wasgoing to solve the problem:

I’m going to weigh my cat. First of all I’m going tostand on the scales and tell you my weight, then I’mgoing to stand on the scales with the cat in my armsand take that weight and tell you. Then I’m goingto take away the first weight I tell you from the sec-ond weight, and the weight left will be my cat’sweight. From that I’ll be able to work out how muchcat food my cat’s going to need.In the next scene Ellie is holding up a

piece of card on which she has written:My weight � cat � 53kgMy weight � 461/2kg

6.5 kg

School maths at homeIn the two kinds of activities which we havedescribed above, the mathematics is embed-ded in play and games or authentic house-hold activities. Both these kinds of activitiesshare certain characteristics – they are basedaround purposes and intentions which areindependent of the mathematics involved.Indeed, they are not primarily about learn-ing mathematics at all.

In contrast, the third category of homemathematics activity is rather different. Herethe activity is essentially focused on learningmathematics, and its main purpose is todevelop new mathematical knowledge andskills or to practice and rehearse existingknowledge and skills. In this respect, theseactivities closely resemble mathematics activi-ties undertaken in school, and we will referto them here as ‘school maths at home’. Interms of the distinction made by Street et al .

(2005) they are activities belonging to the‘school domain’ but taking place on the‘home site’.

School maths took various forms at home.Perhaps the most common was homeworkwhich had actually been set by the child’steacher. In addition, some of the childrenattended maths classes given by private tutors,and had to carry out homework given by theirtutor. Other children were set maths problemsor calculations by their parents or siblings.Sometimes these were set in the context of agame in which the children would play‘schools’, and this would involve one child – inthe role of teacher – setting some maths prob-lems for the other one – in the role of pupil.For example, the video made by Nadia’s fam-ily shows her working through a sheet ofmaths problems drawn up by her older sister.In addition, some of the parents had boughtcommercially available maths schemes orsoftware packages, and the children workedthrough these at home.

While the distinctions between these dif-ferent types of home maths activities are easyto make in theory, in practice the boundariesbetween them can become blurred. The fol-lowing conversation took place while Mollywas playing a game of darts with her youngerbrother Stephen. However their mother’sinterventions gave the game a strong ‘school-maths’ feel.Mother: Stephen, who’s going first? Stephen?

(Stephen has picked up the darts and starts to throw)

Mother: OK then(Stephen throws 15, then treble 3,then hits the board outside the ring)

Mother: Right, Molly, how much did he get?(Molly looks at the board, says nothing)

Mother: Right we’ll count them up. One isfifteen

Molly: FifteenMother: and treble three, what’s three

threes?Molly: SixMother: No, treble three, three threes?Molly: Nine

Mother: Yes, nine plus fifteenMolly: (pauses while she works it out)

Twenty-fourMother: Yes. Right put Stephen’s score

down as twenty-four thenThe conversation continues in this veinthroughout the game.

Difficulties in linking home andschool mathematicsAs we have seen, children are engaged in arange of different activities at home whichinvolve mathematics in some way. It is clearlyimportant that ways should be found of link-ing this home mathematics to the mathe-matics taking place in school – indeed, wewould argue that children’s potential to learnmathematics will be substantially limitedunless this is done. Yet there seem to be anumber of reasons why such links are notmade as frequently as they might be.

First, there is a general lack of awarenessabout the existence and nature of home math-ematics. Teachers frequently assume thatthere are few mathematical activities takingplace at home, and that those activities whichare going on are of little value for school learn-ing – or indeed, may actually serve to under-mine it. Even if teachers are convinced of thevalue of home mathematics and the desirabil-ity of making links with it, they may not neces-sarily know how to go about it. Few teachersare trained in techniques for eliciting informa-tion from parents or other family members,and attempting to do so with a class of 30 ormore children from contrasting backgroundsmay seem a daunting task.

It is also possible that parents and childrenmay not wish to disclose information aboutfamily life, for whatever reason. Nadia, forexample, was growing up in a family of Bengaliorigin who still maintained strong links withtheir extended family in Bangladesh. Likemany families of South Asian origin, Nadia’sfamily use a system of finger counting in whichthe three sections on each finger are countedrather than just a single finger. This systemallows counting up to 30 on the two handscompared with up to 10 using the Western





system. Nadia was good at mathematics andwas strongly supported in this by her family.However, she did not use her finger countingmethod at school, and did not want herteacher to know about it. In this case, thebarrier between home and school was beingerected by the child herself (Incidentally,Abreu, 1999, reports that children often donot report on or value their home mathe-matics practices, which she discusses in termsof valorisation).

Just as teachers may not know very muchabout home mathematics, so parents andother family members often know very little –or think they know very little – about what ishappening in school. This has been particu-larly evident in the last few years in Englishprimary schools, with the introduction of theNational Numeracy and Literacy Strategies.Some parents may have noticed that workwhich their children do in school or bringhome as homework employs different termi-nologies and procedures from ones theymay be familiar with. Other parents have sim-ply have a vague awareness that ‘things aredifferent now’.

If parents feel that there is a differencebetween their approach to mathematics andthose used by the school, then they may feelinhibited from providing help when this isneeded, possibly on the grounds that thismight cause confusion for their children.Alternatively, they may go ahead and offerhelp anyway, even if the outcome is unsatis-factory. As Lucy’s mother pointed out:

What confuses me is that they do their calculationsslightly different to how we were taught to do them . . .I try and show her my way and she says ‘oh you don’tknow what you’re doing’ (laughs)While another mother who had been

educated in India commented that ‘I wish Iwent to school here but I didn’t . . .’

A further example comes from Ryan’smother, who grew up in Scotland and wastaught to use methods which were differentfrom those which Ryan was currently beingtaught. Ryan’s mother described how herattempts to help him often ended in conflictand lack of communication:

I can read it out to him but he always says I’m wrongbecause I’m not doing it properly . . . so . . .and weend up at loggerheads and I just think. . .I think wellyou need to just take it back to your teacher and sayyou can’t do it, ‘oh’ she says, ‘I’ve showed him andI’ve showed him and I’ve showed him, but he justdoesn’t seem to take it in’.On the video which Ryan’s mother made

for the project there is a long section inwhich Ryan is attempting a homework sheetof subtraction calculations. Much of this sec-tion shows Ryan’s mother attempting to helphim and Ryan resisting her help. At onepoint the following interchange takes place:Mother: To take em … to be able to take

five away frae three you have to putone unit off the four and put itonto the three, do you not?

Ryan: NoMother: Well why … You have toRyan: You don’t. Not in my school we

don’t. We do it a different way.Many parents were clearly feeling ‘des-

killed’ by the changes in mathematics teach-ing and their lack of knowledge of currentmethods being used at school. This feelingwas often compounded with their lack ofconfidence in their own mathematical abil-ity, as a result of their own experiences atschool and afterwards.

Exchanging knowledge between home and schoolIn the previous section we argued that bothteachers and parents may lack sufficientknowledge about the mathematics takingplace at home and school respectively, andthat this may severely limit the opportunitiesfor making links between the mathematicstaking place in the two contexts. In thissection we describe what we have termedhome-school knowledge exchange activities,which are designed to increase the exchangeof knowledge about mathematics betweenhome and school.

The activities developed and imple-mented on the Home School KnowledgeExchange Project fell into two main groups.First, there were activities which we called



school-to-home, where the primary aim wasto inform parents and/or other familymembers about the school mathematicscurriculum and the teaching methods beingused. Examples of such school-to-homeactivities included regular newsletters,home-school folders which travelled back-wards and forwards between home andschool, allowing space for both teachers andparents to make comments, and videos madeabout school maths lessons which were madeavailable for parents to see.

At one school parents were invited intothe classroom in small groups to be shownthe procedures being used by the teachers.One mother commented as follows:

They’re obviously teaching maths a lot different tothe way I learnt maths, which was the main prob-lem, communication. Her form teacher, they had anafternoon where some of the mums went in and theyactually taught us for an hour how they teachchildren. And it helped so much, we got all thesesheets and we came home and once I had it in myhead, this is how she’s got to do it . . . I mean, theanswer came out the same, whether I did it my wayor her way, but it was nice to know how they’re beingtaught, how they break it all down.This mother added that this had been

‘probably the best hour I’ve spent at theschool, actually’.

The second main type of knowledgeexchange activity were those we termed home-to-school. Here the overall aim was to bringknowledge about the children’s out-of-schoollives into the classroom so that connectionscould be made with the children’s schoolmathematics. These activities included pupilprofile sheets, which provided informationabout the children’s out-of-school interestsand aptitudes; bringing maths games fromhome into school and playing them duringlessons; and developing home maths trails,where the children sought out and tabulatedinformation about their home lives, such asthe average age of the family members.

In one particularly successful activityall the children were supplied with dispos-able cameras and asked to take photographsof ‘everyday maths’ events taking place

outside-of-school. The children returned thecameras to the school where the photographswere developed and put on displays, togetherwith pieces of writing produced by thechildren which explained what mathematicswas involved in the activity. The ‘everydaymaths’ activities fell mostly into the categoriesof ‘play and games’ and ‘authentic householdactivities’ described above, although therewere some more idiosyncratic ones such asworking out how long their grandparents hadbeen married!

While some children did this activity ontheir own, others involved their parents orother family members. In the followingquote Adam’s mother talks about how theyworked on the activity together:

He enjoyed doing it. And it made him think aboutwhat we actually do in the house that involves num-bers. Like the clock and telling the time, and goingto the shops, and change, and money. Because that’sanother thing, him and (his sister) were changingtheir English money into euros so that was moremaths trying to work that out. And also trying towork out when we were on holiday, if somethingwas 8 euros how much that was in Englishmoney . . . when he went to the shops I’d say ‘we cantake a picture of this’ and we laid it all out and thechange and the money, and then he took the picture’.Further details of the knowledge exchange

activities and their impact on children, teach-ers and parents can be found in Winter et al.(2007).

Discussion and implicationsIn this paper we have argued that currentapproaches to teaching mathematics in Eng-lish primary schools do not pay adequateattention to the kinds of mathematics inwhich children are engaging outside ofschool. We have attempted to redress thisimbalance by providing an account of someof the main features of home mathematics,based on data recently collected with thehelp of children and their families. Ouraccount makes clear that mathematics canbe found in a wide range of home activities,but that it usually looks rather different fromschool mathematics. For example, the kind

of mathematics involved in play and gamesand authentic household activities is charac-terised by being heavily embedded (to useDonaldson’s, 1978 term) in the immediateintentions and purposes of everyday life.Indeed, these purposes are usually notprimarily about learning mathematics, butare about winning the game, or cooking din-ner, or ensuring that the shopkeeper givesyou the right change. In this respect theyshow a marked contrast to the mathematicsactivities found in school (and the school-like activities which we also found at home),where the primary purpose of the activities isthe learning of mathematics.

We have also argued in this paper that it isimportant that children are able to makemeaningful connections between the twokinds of mathematics they encounter. Theseconnections might take different forms. Forexample, teachers might treat children’s out-of-school mathematical knowledge as aresource which they could draw on in theclassroom to help with the introduction of anew concept or procedure. Alternatively, theymight treat the contexts in which children usemathematics as sites for the application of themore abstract (or disembedded) knowledgeacquired in school (see Hughes et al ., 2000,for further development of these ideas).There are also potential roles for parents asmediators or ‘brokers’ (see Wenger, 1998)between school mathematics and children’sindividual out-of-school understandings, onthe grounds that parents are exceptionallywell placed for playing this role.

One factor which seems to be workingagainst such connections being made is thelack of knowledge by teachers and parents ofwhat is happening at home and schoolrespectively. In this paper we have describeda number of knowledge exchange activitieswhich have been successfully used in schoolson the Home School Knowledge ExchangeProject. What is more important than thespecific activities, however, is the recognitionthat such activities are desirable and canenhance children’s mathematics learning.The exact nature of the activities is less

important, and they can readily be modifiedto suit local circumstances.

One issue which arose in the paper wasthat of different calculation methods beingused by parents and children, and theconflict and frustration – or withdrawal ofsupport – this may lead to. While it is impor-tant that parents are made aware of themethods being taught at their children’sschools, it is also important that they come torecognise that there is no single correctmethod. Indeed, one of the strengths of the2006 Framework is that it emphasises thatchildren should leave primary school havinga range of methods – both written andmental – for carrying out calculations, andthat some may be more appropriate forsome kinds of problems than others. If thiscould be successfully communicated to parents, then many of the problems wenoted above might be eased. At the sametime, a home-to-school activity in whichchildren brought in examples of differentmethods used by different family members,which were then openly discussed and com-pared within the classroom, could well helpthe children to appreciate that, in mathe-matics, there are many ways to accomplishthe same purpose.

The observations reported here are alsorelevant for all those – including teachersand educational psychologists – who need tomake assessments of children’s ongoingmathematical progress. The present papersuggests that traditional school-based assess-ment procedures are likely to engage withonly one part of children’s mathematicalexperiences. If we want to obtain a fuller pic-ture of children’s competence in mathemat-ics then it might be valuable to widen therange of contexts, tasks and materials used toassess children, and to make these closer tothe kinds of meaningful and authentic math-ematical activities in which they engage athome. This in turn further emphasises andreinforces the need to draw on the knowl-edge of those who know the children in envi-ronments other than school – primarily butnot exclusively parents.





To conclude, one of the main implica-tions of this work is the need for all those con-cerned with children’s mathematics learningin school – whether they be policy-makers,teachers or educational psychologists – to beaware of and take account of the mathemat-ics which children are engaged in outside ofschool, and to look for ways of making mean-ingful links between in-school and out-of-school mathematics. The activities describedabove suggest some ways in which this mightbe done, but they represent only the start ofwhat needs to be a major shift in perspectiveabout the nature of mathematics learningand how it can be enhanced.

AcknowledgementsThis paper draws on the work of the HomeSchool Knowledge Exchange Project (HSKE),which was funded by the Economic and SocialResearch Council (reference number L139 25

1078) as part of the Teaching and LearningResearch Programme (TLRP). More informa-tion about the HSKE project and TLRP can befound at http://www.home-school-learning.org.uk and http://www.tlrp.org. We are verygrateful to the children, parents and teacherswho participated in the project and to theLEAs of Cardiff and Bristol for their support.The HSKE project team consisted of: MartinHughes (project director), Andrew Pollard(who is also director of TLRP), Jane Andrews,Anthony Feiler, Pamela Greenhough, DavidJohnson, Elizabeth McNess, Marilyn Osborn,Mary Scanlan, Leida Salway, Vicki Stinch-combe, Jan Winter and Wan Ching Yee.

Address for correspondenceGraduate School of Education, University ofBristol, 35 Berkeley Square, Bristol, BS8 1 JAE-mail: [email protected]

ReferencesAbreu, G. (1999). Learning mathematics in and out-

side school: Two views on situated learning. InJ. Bliss, R. Saljo & P. Light (Eds.), Learning Sites:Social and Technological Resources for Learning.Oxford: Pergamon, 17–31.

Brown, J.S., Collins, A. & Duguid, P. (1989). Situatedcognition and the culture of learning. Educa-tional Researcher, 18(1), 32–42.

Carraher, T. N., Carraher, D.W. & Schliemann, A.D.(1985). Mathematics in the streets and in schools,British Journal of Developmental Psychology, 3, 21–29.

Department for Education & Employment (1999).The National Numeracy Strategy: Framework for teach-ing mathematics from Reception to Year 6London: DfEE.

Department for Education and Skills (2006). Primaryframework for literacy and mathematics London:DfES.

Donaldson, M. (1978). Children’s minds London:Fontana.

Hoyles, C., Noss, R. & Pozzi, S. (2001). Proportionalreasoning in nursing practice. Journal for Researchin Mathematics Education 32(1), 4–27.

Hughes, M., Desforges, C. & Mitchell, C. (2000).Numeracy and beyond. Buckingham: Open Univer-sity Press.

Lave, J. (1988). Cognition in practice: Mind, mathematicsand culture in everyday life. Cambridge: Cambridge University Press.

Nunes, T., Schliemann, A. & Carraher, D. (1993). Streetmathematics and school mathematics. Cambridge:Cambridge University Press.

Street, B., Baker, D. & Tomlin, A. (2005). Navigatingnumeracies: Home/school numeracy practicesDordrecht: Springer.

Wenger, E. (1998). Communities of practice: Learning,meaning and identity. Cambridge: Cambridge Uni-versity Press.

Winter, J., Andrews, J., Greenhough, P., Hughes, M.,Salway, L. & Yee, W. C. (in press). Improving pri-mary mathematics: Linking home and school. Lon-don: Routledge.

Winter, J., Salway, L., Yee, W. & Hughes, M. (2004).Linking home and school mathematics: TheHome School Knowledge Exchange Project.Research In Mathematics Education, 6, 59–75.


IN THE context of this special journalissue, where other papers considerresearch and interventions addressing

children’s difficulties with mathematics orarithmetic (see for example Dowker; Gerva-soni & Sullivan; Willey, Holliday & Mart-land, this issue), it is important also tooutline and evaluate developments thathave taken place within the English educa-tional system. This paper sets out to dothat. It starts with the rationale of theNational Numeracy Strategy (NNS) andprovides information about the impact ofthe strategy on lower attaining pupils. Itthen considers responses by the NNS toconcerns about children with significantdifficulties in mathematics and the ration-ale, development and evaluation of addi-tional teaching materials described asWave 3 intervention. Finally, the papertakes a pragmatic stance in relation to theterm ‘dyscalculia’ and suggests how thisstance might inform educational psychol-ogy practice.

From its inception in 1999, the NNS hashad the explicit aim of narrowing the gapbetween higher and lower attaining children,identified in international studies as a partic-ular feature of the English system (Reynolds& Farrell, 1996). Resources have deliberatelybeen targeted according to levels of disad-vantage, and teachers have been encouragedto develop whole-class interactive teachingfollowed by carefully differentiated groupwork as an alternative to the lonely journeythrough graded textbooks that formedchildren’s main mathematical experiencebefore the introduction of the NNS.

The potential benefits for children whofind mathematics difficult have included theopportunity for them to learn from othersrather than being thrown back on their ownsometimes limited resources, the opportu-nity for teachers to pinpoint appropriatelearning objectives from a clearly definedprogression, the focus on oral work, the useof visual models to develop mental imagery,explicit teaching of key mathematical vocab-

Supporting children with gaps in theirmathematical understanding: Theimpact of the National NumeracyStrategy (NNS) on children who findmathematics difficultJean Gross

AbstractThis paper outlines and evaluates developments within the NNS that have addressed the needs of lowerattaining pupils. Data is presented to show that these pupils have not benefited from the NNS as much asother pupils. The rationale and form of the additional Wave 3 intervention materials are then described.Evaluation of impact has so far been qualitative, involving comments made by teachers and the childrenthemselves. Quantitative information is still awaited and will include statistics of on the number of pupilscontinuing to attain below National Curriculum Level 3 at the end of Key Stage 2 (at the age of 11 years).Implications for the term ‘dyscalculia’ are discussed. It is concluded that there is an opportunity, in therelatively new field of mathematical difficulties, to develop from the start effective teaching strategies thatwork for all rather than distinct routes based on diagnostic categories.


Supporting children with gaps in their mathematical understanding

ulary, and the emphasis on metacognition –children identifying and explaining thestrategies they use to solve problems.

To what extent have NNS goals been achieved?Overt goals, as stated above, often translateimperfectly into classroom practice andthere is some evidence (Hardman, 2003;McSherry & Ollerton, 2002; Blatchford et al.,2004), summarised in Table 1, to suggest thatthe interpretation and implementation ofNNS guidance often falls short of the ideal.

The gap between aspiration and reality mayhelp to explain a number of findings that raiseconcerns about the impact of the NNSon lower attainers. For example, severalresearchers (Landerl et al., 2004; Anderson,2000) have noted that public questioningduring whole-class teaching can lead to raisedanxiety levels for children who find mathemat-ics difficult. The Gatsby numeracy project(Muijs, 2003) found that low attaining childrenin Years 1 and 2 (age range five to seven years),who were supported by a teaching assistant, didnot make greater progress than controlchildren who were not supported.

Evidence on the impact of NNS on thestandards achieved by lower attaining pupilshas often been contradictory. The 2003

TIMSS (Trends in International Mathematicsand Science Study) found that the progressmade in mathematics since 1995 was larger inEngland than in any other country and thatthese improvements applied equally acrossthe range of attainment, including the mostand least able. However, findings from theLeverhulme numeracy research programme(Brown & Millett, 2003), suggested that therange of variation in results between lowerand higher attainers between 1998 and 2002had actually increased.

Evidence from national datasets showsthat, while the numbers of children achievingnationally expected levels at the end of KeyStage 2 have grown significantly (Figure 1),the reduction in numbers achieving very lowlevels (below National Curriculum Level 3)has been much less dramatic (Table 2).

The profile of children attaining belowLevel 3 in mathematics is of interest. It doessuggest some degree of specificity in mathe-matical difficulties: 5.9 per cent of the 2005Year 6 cohort were below Level 3 in mathe-matics only, compared to 6.3 per cent in Eng-lish only and 3.9 per cent in both subjects.And while 68 per cent of those below Level 3in English are boys, the corresponding figurefor mathematics is only 55 per cent. In math-ematics as in English, children experiencing

Stated goals of NNS Interpretation in schools

Whole class interactive strategies Questioning used to funnel pupils’ responses towards arather than didactic teaching required answer, rather than promote discussion and

cognitive engagement

Differentiation by ability Increased use of setting as a response to difficulty in copinggrouping within a class with diversity

Teachers work with all Lower attaining groups generally work with teachingability groups assitants rather than the teacher

Well-trained Teaching Teachers and TAs have little liaison time and TAs may lack Assistants (TAs) work with subject knowledge. Their role is often to support children in groups on appropriately completing inadequately differentiated tasksdifferentiated tasks underthe direction of the teacherand with frequentopportunities for liaison

Table 1: Interpretations of NNS goals


Jean Gross

social disadvantage are overrepresented inthe low attaining group (by a factor of two),as are summer born children (three summerborn for every two autumn born childrenachieving below Level 3).

Response by the NNS to concernsabout children with significantdifficulties in mathematicsConcern about the largely static numbers ofvery low attainers in mathematics led the NNSto develop a number of initiatives to raise stan-dards for these children. Professional develop-ment materials were produced aimed atimproving the quality of inclusive classroomteaching (DfES, 2002; DfES, 2004) and help-ing teachers work more effectively with teach-ing assistants (DfES, 2005a). The Strategiesalso developed a model based on three‘Waves’ of intervention, with Wave 1 represent-

ing high quality everyday inclusive classroomteaching, Wave 2 the provision of short peri-ods of assistance in small groups for childrenwho are not making satisfactory progress(about 20 per cent), and Wave 3additional targeted interventions for themuch smaller proportion (approximately 5per cent) for whom the additional small groupteaching is insufficient.

The first step in developing a strategy forWave 3 intervention was to map existingprovision in schools. In the spring of 2002all local authorities (LAs) were asked toprovide information on literacy and mathe-matics interventions used in their area. Fiftyper cent responded. All named at least oneWave 3 literacy intervention and 21 differentliteracy interventions were identified as infairly widespread use.

Figure 1: The percentage of children achieving Level 4 � at the end of Key Stage 2, projected towardsa 2008 target

8585

79

75

50

55

60

65

70

75

80

85

90

1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

+4level

gniv ei

hcasl i

pu

p%

English Maths

1998 1999 2000 2001 2002 2003 2004 2005

7 6 6 5 5 6.3 6.1 5.8

Table 2: Percentage of pupils achieving below Level 3 in mathematics at the end of Key Stage 2 (age around 11 years). Note: Figures before 2003 rounded to the nearest whole number

Only seven of the LAs mentionedany Wave 3 maths interventions. The mostwidely used programme, mentioned by threerespondents, was Mathematics Recovery(Wright et al., 2000), a one to one intensive(daily) teaching system for children in Year 1.The remaining responses described locallydevised interventions, including the use ofNumicon materials in a structured teachingprogramme (Horner, 2002). Other pro-grammes known to be in use at the time butnot mentioned in survey returns were FamilyNumeracy (Brooks & Hutchison, 2002),Barking and Dagenham’s Group EducationPlans (Whitburn, 1997), a Numeracy Recov-ery scheme for six year olds (Dowker, 2001)and Paired Maths(Topping et al., 2003), inwhich peers work together on a tutoringprogramme.The conclusions from the survey were thataddressing mathematical difficulties was a lowpriority in many schools and local authorities.In contrast to literacy, few programmes wereavailable. Those that existed were confined toparticular geographical areas. Consequently,the National Strategies took the decision notto develop or publish any new literacy pro-grammes, since a plethora was already avail-able, but rather to guide schools’ choice ofprogrammes by providing information ontheir evidence base. In mathematics the posi-tion was clearly different. There was a need tostimulate greater activity and there was aparticular gap in materials for Key Stage 2(children aged 7 to 11 years).

For Key Stage 1 (age five to seven years)the NNS developed support sessions builtinto published teaching plans (DfES, 2003).These describe twenty-minute sessions foruse with a group of children who are strug-gling with a key concept in the unit of work,to be taught in addition to the main lesson.To inform the development of new Key Stage2 materials, the DfES commissioned a reviewof research to identify what works forchildren with mathematical difficulties. Thereview (Dowker, 2004) suggested that:● mathematical difficulties are common

and often quite specific;

● they are equally common in boys andgirls, in contrast to language and literacydifficulties which are more common inboys;

● they are varied and heterogeneous, asarithmetical ability is not a single entity,but is made up of many components;

● they represent one end of a continuumrather than a discrete ‘disorder’;

● their causes are varied and include,for example, individual characteristics,inadequate or inappropriate teaching,absence from school resulting in gaps inmathematics learning, lack of preschoolhome experience with mathematicalactivities and language;

● children with mathematical difficultiestypically combine significant strengthswith specific weaknesses;

● some children have particular difficultieswith the language of mathematics;

● difficulty in remembering number facts isa very common component of arithmeticaldifficulties, often associated with dyslexia;

● other common areas of difficulty includeword problem solving, representation ofplace value and the ability to solve multi-step arithmetic problems.The review concluded that mathematical

difficulties can be addressed through appro-priate intervention, which ‘can take place suc-cessfully at any time and can make an impact . . .it is not the case that a large number of childrenare simply “bad at maths”, and that nothing canbe done about it’ (Dowker, 2004, p 42). Thereview drew out some general principles foreffective intervention:● It should be individualised, but the

amount of time given to such individu-alised work does not, in many cases, needto be very large to be effective.

● It should be provided as early as possible,partly because mathematical difficultiescan affect performance in other areas ofthe curriculum, and partly to prevent thedevelopment of negative attitudes to andanxiety about mathematics.

● Interventions that focus on the particularcomponents of mathematics with which



the child has difficulty are more likely tobe successful than those which follow aset ‘programme’.

● Distinguishing between ‘specific’ mathe-matical difficulties and those associatedwith generally low cognitive ability doesnot seem to be helpful in planning inter-vention.

Developing the materialsThe principles set out in Dowker’s reportwere used to develop Supporting childrenwith gaps in their mathematical understanding(DfES, 2005b), a set of materials for childrenin Key Stage 2. The materials take the formof assessment tools that allow class teachersto identify the particular errors and miscon-ceptions that are limiting a child’s abilityto progress. A series of booklets each addressa particular error or misconception, anddescribe teaching activities to undertakewith the child on a one-to-one basis.

The design of the materials draws on keypsychological understandings about how allchildren learn, and on recent investigationsof the pedagogies that are appropriate forchildren with special educational needs – the‘intensification of what is needed by all children’,and the ‘common pedagogies delivered underspecial conditions’ described by Lewis andNorwich (2005). These include:● Distributed practice – short, focused

periods of one-to-one work● Overlearning – continuing to practise new

learning beyond the stage of apparentmastery

● Review and practice at progressivelyincreasing intervals, to support recall

● An emphasis on metacognition – helpingthe child make explicit the strategies theyare using so that they can achieve inde-pendence in their learning

● Multisensory learning, with maximum useof structured equipment and everydaymaterials to model mathematical concepts

● Linking learning to familiar and relevantcontexts

● Attention to language – highlighting andmodelling key vocabulary throughout

● Experiences of success to build children’sconfidence in themselves as learners

● A clear link to whole-class activities● Involvement of parents and carers in

their children’s learningThe materials reflect best practice in

assessment for learning (Black & Wiliam,2001) through the use of questions to elicitinformation about children’s understand-ing, sharing the purpose of the activity withthe learners and encouraging children’sreflection on their learning so that theyidentify for themselves possible next steps.

Learning from the pilotTwenty-seven local authorities volunteeredto pilot the materials in 2003–2005. As aresult of feedback, improvements were madeto the assessment for learning opportunitiesthroughout the materials, further links towhole class work were added and activitiesin the form of games both for use duringteaching sessions and to involve parents andcarers incorporated.

The resource proved popular with schoolstaff and children. Children in the pilotnoted with pleasure the lack of worksheets.Schools reported that the pilot had helpedthem move on from a sole focus on literacyin their special needs support. Some schools,however, found it difficult to resource andimplement the suggested model of detailedassessment and tailored one-to-one interven-tion. Some simply dovetailed the activities toobjectives in the medium term planning foreach year group and used the activities withlower sets/less able pupils as and when theWave 3 activities and objectives coincided(Hurt, 2005).

The more effective local authoritiesresponded to issues like these by target-ing support from central services. In theLondon Borough of Bromley, for example,specialist teachers familiarised senior man-agers with the aims of the intervention,worked alongside class teachers to identifygaps in children’s knowledge, modelleduse of the teaching materials for teach-ing assistants, then gradually handed over


Jean Gross



more and more of the teaching to them.The outcome, as reported anecdotally, wasgreatly increased confidence amongstteaching assistants (many of whom hadbeen reluctant to be involved in a mathe-matics intervention) and a sustained andmore strategic use of the interventionmaterials in schools.

Evaluation of the impact of the materialsEvaluation of the Wave 3 intervention atnational level has so far been mainly in theform of qualitative comments with quantita-tive information on numbers of childrenattaining below Level 3 still awaited. Thereseems to have been much change at theattitudinal level. Schools have mentionedchildren’s raised self esteem and more activeparticipation in the daily mathematics lesson.For example: ‘We often hear “I can do it bymyself ” whereas before it was more of “I can’tdo it”’. ‘Anything that gets kids this excitedabout maths has to be worth putting lots oftime and effort into’. Typical comments frompupils have been: ‘I don’t feel sick any morebecause I can do this work’; ‘It gave me awarm feeling in my tummy because I coulddo it’; ‘I now feel confident to do my SATsnext summer’.

Some local authorities have undertakentheir own evaluations, generally initiated byeducational psychologists. In Norfolk, forexample, educational psychologists testedchildren in fourteen schools at the begin-ning and end of a six month interventionperiod using BEAM diagnostic interviews innumber sense (Denvir and Bibby, 2001)along with tests of cognitive ability. Theresults showed that almost all of the pupilshad made greater progress in the time theWave 3 materials had been used than inprevious years. Many had increased theirNational Curriculum levels by at least onesub-level. Not surprisingly the greatestgains were in schools that had used the mate-rials in a planned, systematic way and hadeffective support from senior management( Johnson, 2005).

Local findings like these are helpful, butwait to be replicated nationally. Certainly thequalitative findings from the pilot need to betreated with caution, given the history ofadherence to unproven Wave 3 literacy pro-grammes based only on perceptions thatchildren ‘love doing them’ and ‘grow in con-fidence’. Warm feelings like these need totranslate into hard results if we are to takethem seriously.

There is, then, need for further researchinto the impact of the NNS materials,but also much to be learned from the impactof a messy but ‘real’ activity in the field.Controlled experimental studies, robust asthey might be, are often hard to replicatewhen they move beyond the conditions inwhich they were originally devised. TheNational Strategies’ traditional means ofevaluation (combining feedback from userswith impact on national attainment pat-terns), whilst less rigorous, at least has theadvantage of a firm grounding in an appliedcontext.

Pragmatic stance on the emergingdyscalculia labelEarly guidance from the NNS (DfES, 2001)made explicit use of the term dyscalculia,defining it as ‘a condition that affects the abil-ity to acquire arithmetical skills’. As muchas anything this represented a wish to givechildren’s mathematical difficulties a promi-nence in teachers’ minds similar to thatengendered by the dyslexia label.

Later NNS publications, however, includ-ing the Wave 3 mathematics interventionmaterials, are more cautious in the use of theterm. The reasons are those that will be famil-iar to educational psychologists in relation todyslexia: the evidence of a continuum of dif-ficulties, rather than a discrete cut-off point,and the consequent risks of medical modelsthat suggest children might either ‘have’, or‘not have’ a particular learning difficulty, justas they ‘have’ or ‘don’t have’ measles. Thereis also a perceived risk of inadvertentlyencouraging varying levels of provision forchildren with different cognitive abilities,

based on the inappropriate but pervasive useof a discrepancy model. Finally, note hasbeen taken of the Dowker research overview(op.cit) that failed to give support to the idea of a discrete teaching methodology for any one group of learners with arithmeticdifficulties.

Another risk is an over-investment of timeand energy in diagnosis and labelling in localauthority support services. It is salutary toreflect on the response of one special needssupport service to the National Strategies’Wave 3 materials, which was to use NFER-Nelson dyscalculia tests to identify childrenrequiring intervention. The tests revealedthree children who were ‘dyscalculic’, butthere were another twenty who could not domaths and whose support needs were, as far asanyone could see, identical. In educationalterms, the support service’s time might betterhave been spent working with teachers tohelp them overcome perceived barriers toappropriate, teacher-led use of the Wave 3intervention.

This is an example of how a medical-diagnostic model based on assumed pathol-ogy can obstruct the development ofappropriate curricular responses in the class-room. It is not unique. We know that teach-ers are identifying ever more children withspecial needs (DfES, 2006) and that the rateof so-called special educational needs (SEN)amongst boys of primary age is reaching epi-demic proportions (almost one third ofeight year old boys are now identified bytheir teachers as having special educationalneeds). It would seem that the concept hasbecome little more than an excuse: a socialconstruction that identifies a large numberof children as not teachable unless they areprovided with additional resources andsupport. Yet the provision of such additionalresources may not be helpful: the number ofteaching assistants has doubled in Englandsince 1998 (Blatchford et al., 2004), most ofthem working with children who find learn-ing difficult, but the percentage of very lowattainers has hardly shifted over the sameperiod.

It is important to avoid this pattern repeat-ing as we develop our knowledge aboutmathematical difficulties. The focus needs tobe on holding such children in the teacher’ssphere of responsibility, rather than placingthem outside it. A medical-diagnostic modelis unlikely to achieve this. Diagnosis andlabelling can have negative effects onchildren’s attainment as illustrated by Tymms& Merrell (2004). According to theirresearch, information for teachers on strate-gies to help inattentive and hyperactivechildren boosted the pupils’ attainments butscreening and identifying specific pupils ashaving ADHD had the opposite effect. Theexplanation suggested by the researchers wasthat, once ‘labelled’, teachers focused onkeeping the pupils happy and calm ratherthan encouraging them to achieve.

Yet diagnosis and labelling can also havebenefits. For some conditions such as autisticspectrum disorders (ASD) there is evidencethat early diagnosis is important in reducingfamily stress and for the later mental healthof the children involved, because it leadsadults to respond in ways that minimise thepotentially disabling effects of autism (Gross,1994; Papps & Dyson, 2004). Many childrenand adults would equally attest to the impor-tance of having a label such as dyslexia,because of its power in eliminating the dis-abling effects of how others construe them –most usually as stupid or lazy.

A social model of disability requires usto listen carefully to the perspectives ofthe learner, perhaps more carefully than tosterile debates about whether certain condi-tions exist or do not exist. For the moment,then, educational psychologists might wantto adopt the scientifically less interesting buteducationally more useful approach oftaking dyscalculia by its literal meaning (aninability to calculate). They can then startfrom the assumption that all children whostruggle with numbers and the numbersystem are to some extent dyscalculic andproceed, in their work with teachers, to themuch more important question ‘So what arewe going to do to about it in the classroom?’


Jean Gross



ConclusionIn their analysis of pedagogies for low attain-ing children, Dyson & Hick (2005) con-cluded that successful literacy interventionshave ‘a basis in a universally applicable model ofreading development that leads them to play downan aetiological approach to understandingchildren’s difficulties in favour of a functional one.Put simply, whatever the underlying causes ofchildren’s falling behind, the reading task remainsthe same’ (p.195).

The parallels between mathematics andliteracy are evident, and the history of dyslexiaresearch and debate illuminating. We havethe opportunity, in the relatively new field ofmathematical difficulties, to develop from thestart effective teaching strategies that workfor all rather than distinct routes based ondiagnostic categories.

Implications for educational psychologypractice are clear. In their consultations withindividual teachers, they need to encouragea teaching and learning response ratherthan one based on the probing of cognitivecausation. The ‘diagnosis’ should be aboutwhat the children know and what they needto learn: teachers still need support with this,and educational psychologists can help bymodelling the use of assessment tools such as

those provided in the NNS materials.The skills of educational psychologists,

moreover, can be applied at systems level,helping the school develop effective strategicmanagement of additional provision (Gross& White, 2003). Effective provision forchildren with mathematical difficulties invol-ves a coherent whole-school approach. Itrequires systematic, targeted and time lim-ited support informed by data and evidenceon what works. There is a need for good sys-tems for tracking and regular review of pupilprogress, close connections between theintervention and the work of the class asa whole, the positive engagement of par-ents and carers, and rigorous evaluation.Educational psychologists have traditionallyhad the skills to help schools develop suchsystems. It is to be hoped that theirbackground and training in the future willcontinue to provide them with the expertisethey need to provide this type of consultativesupport.

Address for CorrespondenceJean Gross, Every Child a Reader, KPMGFoundation, Salisbury Square, Londn EC4Y8BB.E-mail: [email protected].

ReferencesAnderson, J. (2000). Teacher questioning and pupil anxiety

in the primary classroom. Paper presented to theBritish Educational Research Association confer-ence, Research Student Symposium, Cardiff Uni-versity, September 2000.

Black, P. & Wiliam, D. (2001). Inside the Black Box: rais-ing standards through classroom assessment. London:King’s College London School of Education.

Blatchford, P., Russell, A., Bassett, P., Brown, P. &Martin, C. (2004). The role and effects of teachingassistants in English primary schools (Years 4 to 6).London: DfES.

Brooks, G. & Hutchison, D. (2002). Family numeracyadds on. London: Basic Skills Agency.

Brown, M. & Millett, A. (2003). Has the NationalNumeracy Strategy raised standards? In I. Thompson(Ed.), Enhancing primary mathematics teaching. Buck-ingham: Open University Press.

Denvir, H. & Bibby, T. (2001). Diagnostic interviews innumber sense. London: BEAM Education.

DfES (2001). Guidance to support pupils with dyslexiaand dyscalculia. London: DfES.

DfES (2002). Including all children in the literacy hourand daily mathematics lesson. London: DfES.

DfES (2003). Models and images. London: DfES.DfES (2004). Learning and teaching for children with

SEN in the primary years. London: DfES.DfES (2005a). The effective management of teaching assis-

tants to raise standards in literacy and mathematics.London: DfES.

DfES (2005b). Supporting children with gaps in theirmathematical understanding. London: DfES.

DfES (2006). Special educational needs in England,January 2006. London: DfES.

Dowker, A. (2001). Interventions in numeracy: thedevelopment of a numeracy recovery projectfor young children with arithmetical difficulties.Support for Learning, 16, 6–10.

Dowker, A. (2004). What works for children with mathe-matical difficulties. London: DfES.


Jean Gross

Dyson, A. & Hick, P. (2005). Low attainment,. In A.Lewis & B. Norwich (Eds.), Special teaching for spe-cial children? Buckingham: Open University Press.

Gross, J. (1994). Asperger syndrome: A labelworth having. Educational Psychology in Practice,19(2), 104–10.

Gross, J. & White, A. (2003). Special needs and schoolimprovement. London: David Fulton.

Hardman, F. (2003). Interactive whole class teaching in thenational literacy and numeracy strategies. Paper pre-sented at the Learning Conference 2003: WhatLearning Means. University of London, Institute ofEducation.

Horner, V. (2002). Counting on it. Special, Autumn2002.

Hurt, L. (2005). Personal communication.Johnson, J. (2005). Norfolk Wave 3. Personal commu-

nication.Landerl, K., Bevan, A. & Butterworth, B. (2004). Devel-

opmental dyscalculia and basic numerical capaci-ties: A study of 8–9 year old students. Cognition, 93,99–125.

Lewis, A. & Norwich, B. (Eds.) (2005). Special teachingfor special children? Buckingham: Open UniversityPress.

McSherry, K. & Ollerton, M. (2002). Grouping patternsin primary schools. Mathematics in school, 31, 2–7.

Muijs, D. (2003). The effectiveness of the use of learn-ing support assistants in improving the mathemat-ics achievement of low achieving pupils in primaryschool. Educational Research, 45(3), 219–230.

Papps, I. & Dyson, A. (2004). The costs and benefits ofearlier identification and effective intervention. Lon-don: DfES.

Reynolds, D. & Farrell, S. (1996). Worlds apart: A reviewof the international surveys of educational achievementinvolving England. London: HMSO.

Topping, K., Campbell, J., Douglas, W. & Smith, A.(2003). Cross-age peer tutoring in mathematicswith seven and 11 year–olds. Educational Research,45(3), 231–240.

Tymms, P. & Merrell, C. (2004). Screening and inter-ventions for inattentive, hyperactive and impulsivechildren. University of Durham: http://pips.cem.dur.uk/PDFs/ESRCReport.pdf.

Whitburn, J. (1997). Improving mathematics attainment:lessons from abroad? Paper presented to ScottishEducational Research Association annual confer-ence, September 18th–20th, University of Dundee.

Wright, R., Martland, J. & Stafford, A. (2000). Earlynumeracy: Assessment for teaching and intervention.London: Chapman.

Supporting children with gaps in their mathemati-cal understanding focuses on addition/subtrac-tion and multiplication/division objectives inthe English National Numeracy Strategymathematics framework.The materials consist of a series of teachingactivities, each addressing one of the follow-ing common errors and misconceptions.

Addition and subtractionDifficulty in counting – can only begincounting at one, lacks systematic approachesMisunderstanding the meaning of one more and one less, and not being able to iden-tify the number before or after a given numberFailing to relate the combining of groups ofobjects to additionNot being confident about when to stopcounting when subtracting in answer to thequestion ‘how many are left?’Making counting mistakes when using teennumbers and/or crossing boundariesDifficulty in remembering number pairstotalling between 10 and 20Counting up unreliably – still counting thesmaller number to get one too many in theanswerFailing to relate finding a difference and complementary addition to the operation ofsubtractionNot making links between addition and subtraction and/or recognising inversesNot readily using number patterns tosupport calculatingInsecure understanding of the structure ofthe number systemDifficulty in partitioningDifficulty in deciding when to use calcula-tions laid out in columnsDifficulty in adding three numbers in a columnInefficient counting strategies for large numbersRounding inaccurately, particularly with dec-imals, and having little sense of the size ofthe numbers involved

Difficulty in partitioning numbers withzero place holders and/or numbers lessthan oneDifficulty in choosing suitable methods forcalculations that cross boundaries.

Multiplication and divisionConfusing numbers when counting in twosDifficulty with identifying doubles andadding a small number to itselfMaking unequal groups when grouping, andbeing unable to compare the groupsDifficulty when sharing in dealing with anyleft over after making equal groupsDifficulty in counting reliably in tens from amultiple of tenNot understand vocabulary such as ‘groupsof’, ‘multiplied by’Failing to link counting up in equal steps tothe operation of multiplicationSeeing an array as a collection of onesrather than focusing on ‘rows of’ or‘columns of ’Failure to use partitioning when doublingDifficulty in relating multiplying by two toknown facts about doublesFailure to use knowledge of doubles to findhalf of a numberNot being systematic when sharing intoequal groups or using the language of divi-sion to describe the processNot understanding that ‘sets of’ or‘groups of’ need to be subtracted to solve aproblemLack of confident recall of multiplicationfactsNot understanding the relationship betweenmultiplication and division factsNot understanding the operation of multi-plying by tenFailure to apply partitioning and recombin-ing when multiplyingAssuming that the commutative law holds fordivisionWriting a remainder that is larger than thedivisor

Appendix




Jean Gross

Not understanding the significance of aremainderNot understanding rules about multiplyingand dividing by powers of ten and the associative law

Difficulty in interpreting a remainder as afractionInterpreting division as sharing but not asgrouping (repeated subtraction)Difficulty in making reasonable estimates.

Arithmetical Difficulties Developmental and Instructional Perspectives

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