Trends in Neuroscience and Education 2 (2013) 65–73

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Trends in Neuroscience and Education

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Research in Perspective

Contributions of longitudinal studies to evolving definitionsand knowledge of developmental dyscalculia

Michèle M.M. Mazzocco a,b,n, Pekka Räsänen c

a Institute of Child Development, University of Minnesota, 51 East River Parkway, Minneapolis, MN 55455, USAb Johns Hopkins University Schools of Education and Medicine, 2800 N. Charles Street, Baltimore, MD 21218, USAc Niilo Mäki Institute, P.O. Box 35 (Asemakatu 4), 40014 University of Jyväskylä, Finland

a r t i c l e i n f o

Article history:Received 26 February 2013Accepted 27 May 2013

Keywords:DyscalculiaMathematics learning disabilityLongitudinal studiesMath anxietyArithmetic disorders

93/$ - see front matter & 2013 Elsevier GmbHx.doi.org/10.1016/j.tine.2013.05.001

esponding author. Tel.: +1 612 614 2982; fax:ail addresses: Mazzocco@umn.edu (M.M.M. Msanen@nmi.fi (P. Räsänen).

a b s t r a c t

In the last 20 years, longitudinal studies have demonstrated that it is important to attend to the stability ofmathematical performance over time as a facet of dyscalculia, that the manifestation of mathematicsdifficulties changes with development, and that individual differences in cognitive profiles and learningtrajectories observed in children with mathematics difficulties implicate differences between dyscalculic andnon-dyscalculic subgroups. Intra-individual differences over time, and external factors related to children'slearning environments, also contribute to performance trajectories; moreover, these factors may explain theinconsistent performance profiles observed among many students whose difficulty with mathematicsemerges later or diminishes over time. Longitudinal studies on DD are also an important tool to elucidatewhy some children are more responsive to mathematics intervention than others.

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Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652. DD vs. other forms of mathematics difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653. Diagnostic definitions of DD require a longitudinal perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664. Stability over time—one component of DD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695. Stability of dyscalculia may be enduring, but not constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696. Cognitive underpinnings of dyscalculia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707. Potential socio-affective factors in DD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

1. Introduction

Longitudinal studies make a unique contribution to our under-standing of developmental dyscalculia (DD, or mathematics learningdisability (MLD), terms we herein consider synonymous). Althoughboth cross-sectional and longitudinal approaches are useful fordescribing concurrent cognitive profiles and correlates of DD withinor across age groups, only longitudinal studies can reveal the trajec-tories of mathematics and related skills acquisition without potentialconfounds of cohort effects. Accordingly, longitudinal studies candelineate the timing of evolving relationships between associatedskills at different periods of development, and whether children with

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+1 612 625 2093.azzocco),

vs. without DD experience alternative pathways to mathematicsachievement or delayed but shared pathways. Intervention studies,which are longitudinal by design, may show how students' learningenvironments affect these trajectories and which cognitive, social, orenvironmental determinants interact with intervention effects. Thus,longitudinal studies inform the development of theories of change inmathematics learning. Here we review some of the primary contribu-tions longitudinal studies have made to recent efforts to define DD andhow those contributions inform best practices for prevention andremediation of DD.

2. DD vs. other forms of mathematics difficulties

Defining DD is challenging (Box 1). When identifying if studentsqualify for special education services for mathematics instruction,knowledge of whether they have DD or another form of mathematicsdifficulty may be unnecessary; but this knowledge is essential for

Box 1–Why is developmental dyscalculia so difficult to define?

� The term “developmental dyscalculia” (DD) does not refer

to all forms of mathematics difficulty seen in childhood.

� Some children phenotypically show features of DD at

some point of development, but their difficulties are not

linked to a DD genotype; this is common among children

with inadequate home or school learning environments

linked to poverty.

� DD is considered a mathematics disorder, and mathe-

matics encompasses a very broad range of cognitive

abilities, skills, and strategies influenced further by innate,

environmental, cognitive, and social factors.

� DD or some components of DD are likely to represent an

extreme on a continuum of skills and abilities; therefore, it

may be difficult to establish boundaries between typical

development and DD, and knowledge of typical mathe-

matics development and function can inform studies of

DD. However, DD or some components of DD appear

qualitatively distinct from other forms of low mathematics

achievement, limiting the extent to which we can general-

ize findings from studies of typical mathematics develop-

ment to the study of DD.

� Research on DD is increasing, but remains limited

compared to research on other learning disabilities, so

the knowledge base on which current definitions are

based is still emerging.

� Existing research on DD has been fragmented. In view of

the lack of universally accepted screening tools for DD or

validated “core deficits”, researchers develop and use a

range of measures in their studies. These measures vary

even when addressing the same construct (such as

“counting” or “magnitude comparison”); even standar-

dized test norms vary across countries. Studies replicating

previous findings using the same measures, and espe-

cially analyzing intervention effectiveness using the same

educational programs, have been rare exceptions.

� Across research studies, educational media, and govern-

ment reports, the terminology used when referring to DD

is inconsistent. Math learning disability (MLD) has been

used as synonymous with DD (as we do in this article), but

also as distinct from DD when MLD is used to refer to the

larger category of mathematics difficulties (MD), it is

intentionally referring to all children who struggle with

math. The emphasis on MD is understandable, given that

all such children need our research and educational

attention. However, not all these children have the severe,

specific disability in math that we refer to herein as DD.

M.M.M. Mazzocco, P. Räsänen / Trends in Neuroscience and Education 2 (2013) 65–7366

developing theories of why children struggle with mathematics andfor establishing and testing treatment priorities. A constraint on thedepth of our knowledge in this area stems from the paucity ofresearch on DD, particularly relative to research on other learningdisorders [12]. A related obstacle is the lack of universal classificationcriteria for DD, leading to inconsistent composition of DD samplesacross studies (as reviewed by Butterworth [9,10,11] and Murphyet al. [66]) that sometimes include children with milder or moretransient forms of mathematics difficulties. Until recently,assessment-based cut-off scores used to define DD samples werealso highly variable and nearly always applied dichotomously (unlesscontrasted with and without comorbidities; e.g., [75,40]; see Landeret al., this issue). A proposed trichotomous approach [61,26,66]compares children with persistently deficient (DD) or moderatelylow mathematics achievement (LA) in mathematics to each other

and to typically achieving (TA) peers. Group differences in cognitiveprofiles to emerge from this approach are not solely quantitative[13,17], and support the notion that DD and mathematics difficulties(MD) are both heterogeneous but are not synonymous [56,76].However, the boundaries between DD and other forms of MD remainfuzzy, in part because their differences from each other are incon-sistent across studies, absent in some studies or in some areas offunction, and dependent on competencies being assessed and theages at which assessments occur (Table 1). Longitudinal studies haveenhanced awareness that multiple pathways may lead to DD [49],that DD is just one subset of MD [56], that MD linked to poverty andother poor learning environments may manifest as DD, and that highthreshold cut-points can mask DD characteristics in research sam-ples. The implication for clinical and educational practices is thatindividual and developmental variation should be considered whenattempting to diagnose or rule out DD.

3. Diagnostic definitions of DD require a longitudinalperspective

Primary classification systems of developmental disorders, theICD-10 and the DSM-IV, describe disorders of arithmetic skills interms of a discrepancy between low arithmetical abilities andoverall intelligence level and chronological age, and in parti-cular focus on difficulty acquiring formal arithmetic operations.Although not stated explicitly, these diagnostic criteria require thatlearning difficulties are evident over a period of time. In view ofempirically validated limitations of discrepancy based criteria [20],response-to-intervention (RtI, [38]) models have become favoredfor confirmatory diagnostics (e.g. [12,37]), but only a few studieshave addressed the effects of educational interventions on thepersistence of DD (Fig.1 A). Fuchs and colleagues [21] found thatabout one third of children initially meeting discrepancy criteria forDD no longer met the criteria after 16 weeks of intensive tutoring.However, prevalence rates of DD varied significantly depending onthe test used to ascertain their mathematical achievement. Thisteam has since demonstrated that response to intervention varies asa function of whether DD co-occurs with reading disability (RD)[22], depending on the type of intervention employed. For instance,among 3rd graders (8-year-olds) who received tutoring for numbercombination skills, those with DD but no RD showed greater gainson number combinations when tutoring included word problems,whereas children with DD and RD showed greater gains whennumber combination tutoring was not presented through wordproblems. Children with DD and RD showed greater gains when astrategy-use lesson was followed by daily practice, relative tostudents with DD only, for whom regular practice did not altereffect sizes. These and other findings from Fuchs' lab support thenotion of DD7RD as distinct subtypes of DD, as was proposedearlier by Geary [24].

Iuculano [39] showed different rates of response to an earlyintervention program developed in the UK for 2nd graders(6-year-olds) with significantly delayed numerical development.In addition to curriculum based monitoring assessments, thesechildren completed a number sense battery designed to differ-entiate DD from LA [8]. Only children with LA benefitted from thestandard intervention, making age-appropriate gains. However,their performance returned to the level observed in children withDD within three months following the end of the intervention.This means that even children with LA (that is, with less severeMD than in DD) fall behind without additional support andattention, and that children with DD are more resistant than thosewith LA to responding to at least some standard interventions.

Table 1Examplesa of longitudinal studies of the characteristics of developmental dyscalculia (DDb), published between 2001 and 2013.

Study Age span inyears

Primary variables of interest Primary findings relevant to DD (typically referredto as MLD or MD in the original papers)

Anderson [1] 9–13 Comorbidity between DD and reading disability (RD) Children with DD and those with DD+RD displayed severearithmetic weaknesses in factual knowledge, conceptualknowledge, and procedural and problem-solving skills during all3 time points studied. Both DD groups displayed a smallweakness related to visual–spatial working memory. Childrenidentified as having DD at age 9 did not catch up with theirnormally achieving peers.

Auerbach et al. [3] 10–17 Behavioral problems in children with DD Children diagnosed with persistent or non-persistent DD at age10–11 years were followed for 6 years via parent and self-reportson the child behavior checklist, which was completed at each ofthree two-year intervals. Group differences emerged at time 3, inthe direction of children with persistent DD having significantlymore attention and externalizing problems, with some scores inthe clinical range; yet most scores were within normal limits forboth groups at all points of assessment.

Aunola et al. [5] 6–8 Cognitive antecedents of math performance,developmental trajectories

Evaluating math performance at 6 time points, the authors foundhigh stability and increasing variance in math performance overtime, and that counting skills in kindergarten predict 2nd grademath performance and growth.

Boets et al. [6] 5–8 Indicators of visual dorsal stream such as motionsensitivity as a predictor of arithmetic skills

Coherent motion sensitivity at age 5 years predicts subtraction,but not multiplication performance accuracy, at age 8 years.

Chong and Siegel [13] 7–9 Persistence of math fluency and procedural skills,DD vs. LA

Children with DD had more persistent retrieval deficits thanchildren with low math achievement (LA), whereas proceduraldeficits were less persistent.

D'Amico andPassolunghi [15]

9–11 Rate of access to long term memory and inhibitionin children with DD

Children with DD showed deficits in both lexical access (asmeasured by Rapid Automatized Naming, or RAN) and oneffortful inhibition as measured by a negative priming task.

De Smedt et al. [16] 6–7 Magnitude judgment and later math achievement Magnitude judgment speed at age 6 predicts math achievementat age 7

Desoete et al. [17] 5–7 Non-symbolic and symbolic number skills as predictors offormal arithmetic procedures and fluency

The authors evaluated DD, LA, and TA profiles. They found thatdifferences in the profiles of children DD vs. LA suggest morepersistent and distinct number deficits in DD than in LA overtime and that children with DD are less accurate on non-symbolic number comparison at age 5; and on symboliccomparison at ages 5 and 7.

Desoete and Jacques[18]

6–7 Early number skills and later math achievement Numerosity skills at age 7 predict math achievement at age 8.

Fayol et al. [19] 5–6 Domain-general neuropsychological tests as predictorsof arithmetic skills

The performances in neuropsychological tests evaluating thesomato-sensory skills measured at 5 years of age predicted asignificant proportion (420%) of the variance in the arithmetictests administered to the same children one year later.

Garrett et al. [23] 7–9 Metacognitive skills of prediction and evaluation ofmath problem solving accuracy in children with DD or TA

When asked to predict which math problems they can solvecorrectly; or to provide a post-hoc evaluation of whether theirsolution to a math problem is correct, children with DD were lessaccurate than children without DD. However, children with DDwere as likely as their peers to fail a math problem that they hadrated as one they could not solve correctly.

Geary et al. [26] 6–7 Cognitive profiles of DD (MLD), LA, and TA This study was the first to use the terms MLD and LA to differentiatestrict vs. lenient criteria to define math disabilities. Childrenwith DDshowed impairment on a wider range of math assessments than theLA group, and their performance on several of these tasks wasmediated by WM or processing speed tasks. Children with LA hadpoorer math fluency than their typically achieving peers.

Geary et al. [25] 6–9 Math achievement and skills focused primarily onforced fact retrieval; WM, processing speed (RAN)

The nature and persistence of fact retrieval difficulties differbetween DD and one LA subgroup, with DD children showinggreater deficits. This support the notion that these are differentetiologies. However, severe fact retrieval deficits were also seenin a subgroup of students with LA.

Hanich et al. [31] 7 (4 evaluationsduring 2ndgrade)

Math and numerical skills were examined amongchildren with math or comorbid math and reading deficits

Children with math+reading difficulties were more impaired onlinguistic (exact) tasks than were children with only MD; bothgroups were impaired on approximate arithmetic tasks.

Hannula andLehtinen [32]

3.5–6 Stability of Spontaneous Focusing on Numerosity (SFON), itsconnections to counting skills

There were individual differences in SFON, as well as stability inchildren's SFON across tasks during the follow-up. Path analysesindicated a reciprocal relationship between SFON and countingdevelopment.

Hannula et al. [33] 6–8 SFON as a predictor of arithmetic and reading skills The Spontaneous Focusing on Numerosity (SFON) tendency inkindergarten was a significant domain-specific predictor ofarithmetical skills, but not reading skills, assessed at the end ofgrade 2.

Hannula et al. [34] 4–5 Reciprocal developmental connections betweensubitizing, SFON, and counting

The number sequence production skills and SFON were directlyassociated with each other while the connections between SFONand object counting skills were mediated by subitizing-basedenumeration suggesting multiple pathways to enumerationskills.

M.M.M. Mazzocco, P. Räsänen / Trends in Neuroscience and Education 2 (2013) 65–73 67

Table 1 (continued )

Study Age span inyears

Primary variables of interest Primary findings relevant to DD (typically referredto as MLD or MD in the original papers)

Hecht et al. [35] 7–10 The contribution of phonological processing to mathcomputation skills

There are links between phonological processing and growth inmath computation; these vary within the time span examined.Phonological awareness (PA), phonological memory, and rapidnaming (RAN) at age 7 account for variation in rate of growth inmath computation by age 10; at 8 and 9 years only, PA accountsfor variation in growth by age 10.Arithmetic fluency (controlling for processing speed per se)accounts for growth in math computational skills.

Jordan et al. [40] 7–8 Arithmetic, fluency, and counting; later math achievementgrowth, in children with comorbid math and readingdifficulties

Children with math and reading difficulties had more severedifficulties with math than did children with only mathdifficulties, and were more impaired on linguistic (exact) tasks;both MD groups were impaired on approximate tasks.

Jordan et al. [41] 5–6 Number sense as a predictor of later mathematicsperformance

Growth in number sense predicted the arithmetic performancein grade 1; gender and SES also predicted math performance andgrowth.

Judge andWatson [43]

5–11 Individually administered mathematics tests subject to itemresponse theory (IRT) analysis

Using longitudinal data from the U.S. Early ChildhoodLongitudinal Study-Kindergarten Cohort (ECLS-K), the authorsexamined mathematics achievement and growth trajectories tofind that mathematics LD is persistent, that indicators of poormath outcomes are evident at kindergarten entry, and yet manychildren with mathematics LD remain unidentified until lateelementary school.

Koponen et al. [45] 5–10 Counting skills and processing speed (RAN) as predictors ofsingle-digit calculation and reading skills

Counting skills and RAN predicted fluency in arithmetic andreading. Procedural calculation skills at 4th grade were predictedby number concept skills measured at kindergarten and single-digit calculation fluency measured at 4th grade.

Krajewski andSchneider [46]

6–9 Early number skills and later math achievement A wide range of early number knowledge skills and numbernaming fluency predict math achievement at grade 4.

Krinzinger et al. [47] 7–9 Association between math anxiety, children's evaluation ofmathematics, and their calculation performance

Both mathematics anxiety and calculation were associated withchildren's self reported evaluation of mathematics, but not witheach other.

LeFevre et al. [49] 4–7 Linguistic, spatial, attention, and math skills Math competencies are independent from one another: thequantitative, linguistic, and spatial attention skills made uniquecontributions to preschool number skills, and their relationshipto later math skills depended on math competencies measured.

LeFevre et al. [50] 5–7 Home numeracy experiences as predictors of math skills Math, but not reading-related activities at home, explained 4% ofthe variance in math skills after SES and general cognitive skillswere controlled.

Mazzocco andGrimm [60]

5–13 Growth in Rapid Automatized Naming (RAN) speedfor Letters, Numbers, and Colors subtests

Despite similar levels of stability of RAN performance over timeacross groups, there is between-child variability in RAN speed atage 5 and 13, with executive functions accounting for some of thatvariation. More pronounced and global slowing on RANwas seen inthe DD group vs. LA group, both relative to TA. There were differentrelations across RAN subtests for MLD vs. LA subgroups.

Mazzocco and Myers[61]

5–8 Profile differences between children with strict (DD) andlenient (LA) criteria for DD on mathematics achievement anddomain general predictor variables

DD and LA groups have similar frequency of comorbid RD andvisuospatial difficulties; about 66% of children meeting criteriafor DD or LA continue to meet those criteria from grades K-3;stability in math performance levels is an importantcharacteristic of DD. Classification of DD varies depending on themeasures used to classify it. One-time single assessments of DDshould be avoided in the absence of a gold standard assessment.

Mazzocco andThompson [63]

5–8 Kindergarten predictors of 3rd grade DD status Kindergarten performance on “number sense” items (symbolicskills; number constancy, numeral reading; magnitudejudgments, mental addition of sumso9) correctly classified�88% of children as meeting or failing to meet strict andpersistent criteria for DD at grades 2 and 3; no gender effectsobserved.

Mazzocco andDevlin [57]

11–13 Rank ordering fractions/decimals, in children with DD,LA, and TA

There was a high frequency of persistent deficits in fractionsknowledge (through age 13) among children who failed thefractions knowledge task at age 11, in both DD and LA groups, butmore so in the DD group. Children with DD made moreconceptual errors than children with LA (or TA). At age 11, a goodpredictor of whether poor fractions knowledge would persist toage 13 years was children's accuracy of labeling (naming)decimals.

Mazzocco et al. [58] 11–13 Timed addition and multiplication in children withDD or LA

Children with DD or LA made more fact retrieval errors than TAchildren on “hard” items (e.g., sums420, multiples of 6, 7 or 8),but children with DD made more and different kinds of factretrieval errors than did children with LA (or TA) on “easy” items(e.g., sumso20, multiples of 2, 5 or 10).

Morgan et al. [64] 5–10 Stability of MLD Using an existing national database (ECLS-K), children werecategorized as having MLD if they were at the lowest 10percentiles in two assessments in kindergarten Fall and Springterms; and 65% of these children were still categorized as MLD at5th grade.

M.M.M. Mazzocco, P. Räsänen / Trends in Neuroscience and Education 2 (2013) 65–7368

Table 1 (continued )

Study Age span inyears

Primary variables of interest Primary findings relevant to DD (typically referredto as MLD or MD in the original papers)

Murphy et al. [66] 5–8 Strict vs. lenient criteria for DD This was the first study to directly test and demonstrate thatcognitive profile differences exist in subgroups of childrenmeeting strict vs. lenient criteria for MLD; the labelscorresponding to these criteria were later designated as MLD andLA, by Geary et al. [26] (see this table).

Reeve et al. [74] 6–10 (7measurements)

Stability and predictive validity of group membershipbased on development and performance in dot enumerationand number comparison tasks

Dot enumeration (DE) and number comparison (NC) abilitieswere measured 7 times over 6 years. 69% of children wereassigned to the same subgroup across age. None of the childrenfell from higher performing groups to the lowest performinggroup. Subgroups did not differ in processing speed or nonverbalreasoning, suggesting that DE and NC reflect individualdifferences specific to the domain of numbers. Both measurespredicted computation ability at 6 years, 9.5 years, and 10 years.

Shalev et al. [78] 10–16 (5th, 8th,& 11th grades)

Persistence of DD over time Most students diagnosed with DD at 5th grade continued to havemath difficulties at grade 8 (95%), and 40% continued to meetcriteria for DD at grade 11. DD persistence was associated withseverity of DD at grade 5, inattention, and writing problems.

Stock et al. [79] 6–10 The contribution of consistently meeting DDcriteria to DD classification accuracy

Children meeting criteria for persistent DD, LA, and TA wereevaluated; as were children with inconsistent DD or LA.Significant profile led researchers to conclude that children withDD can be identified early, but should be based on their scoresfrom 2 consecutive years.

Swanson [80] 6–8 Working memory and arithmetic word problemcomputation

Central executive and phonological memory at age 6 predictedword problem computations at age 8; growth in problem solvingwas not related to phonological or visuospatial memory.

Van der Ven et al.[82]

7–8 Measures of inhibition, cognitive shifting, and updatingsubject to confirmatory factor analysis and to correlates ofmathematics

Central executive skills at age 7 predict math achievement at age8, but this association is evident for updating skills only (notinhibition or shifting).

Vukovic et al. [84] 7–8 The role of math anxiety and WM as a predictor ofdevelopment of math skills

Higher levels of mathematics anxiety in second grade predictedlower gains in children's mathematical applications betweensecond and third grade, but only for children with higher levelsof working memory.

Zhang et al. [86] 5–8 Counting, written and spoken language, and visuospatialskills; math growth/achievement

Early letter knowledge and visuospatial relations at kindergartenpredicted later math performance and growth in math throughgrade 4; this relationship was mediated by early counting skills.

a These studies collectively exemplify the range of predictor or mediating variables of interest in the study of DD, and the range of research subtopics relevant to DD. Thisis not a comprehensive list of longitudinal studies of DD.

b Here we use the term "DD" although in many of these studies the terms MLD or MD were used.

M.M.M. Mazzocco, P. Räsänen / Trends in Neuroscience and Education 2 (2013) 65–73 69

4. Stability over time—one component of DD

Mathematics related abilities vary, but many center on earlyemerging number skills for which individual differences areobserved throughout the lifespan (e.g., [30]). The stability ofnumber skills has been observed during infancy [51], from pre-school to early school age years [52], and from primary schoolthrough elementary [74] to high school [28]. Whether and howrobustly mathematics difficulties persist depends on the type ofskills assessed. For instance, both fluency (timed arithmetic) andprocedural (untimed calculation) difficulties characterize youngchildren with DD, but while procedural difficulties diminish overtime, fluency deficits persist [13,27,31,85] even to 8th grade [27].These discrepant trajectories illustrate the independence of arith-metic or mathematics skills, a notion supported by data from atwin study showing that variance in timed mathematics fluency isindependent not only from untimed mathematics performancebut also from timed reading fluency [70]. It is important toconsider that persistent mathematics deficits are also observedin a subset of the general population, including some childrenwithLA and children at risk for poor outcomes due to poverty, who donot have DD—especially when no remedial instruction is offered.Thus stability of mathematics difficulties may characterize DD, butis not limited to DD.

Since most standardized mathematics assessments typically covera broad range of mathematics principles or problem types, assess-ment of performance specificity is limited by their use, and relianceon performance on these measures at a single point in time leads to

false positive cases of DD when students occasionally underperform[61] and to false negative cases when over-learned rote skills masklack of genuine mastery (e.g., [65]). Likewise, false positive cases arecommon when opportunities to learn are not equally distributedacross, for instance, socioeconomic strata (reviewed by Jordan andLevine [42]) or because of children's different linguistic background[4,67,85]. Importantly, these “false positives” do not diminish theneed to attend to children's mathematics difficulty, but rather reflectthat the etiology may not be DD and that the nature of educationalsupport may need to vary.

5. Stability of dyscalculia may be enduring, but not constant

Like many forms of development, stability over time does notimply linear changes nor constancy across all time points; indeed,longitudinal studies illustrate how overall persistence of mathe-matics difficulties over time are punctuated by occasional indica-tions of mastery, or at least higher than deficient performance.First, while reports that �66% of childrenwith DD (or MD) continueto meet DD (MD) criteria over time suggest stability, the remaining33% obviously have inconsistent performance. Regardless ofwhether (or how many) “inconsistent” performers have DD, theseparticipants are routinely omitted from studies of DD for failing tomeet consistency criteria. Second, many children who eventuallymeet DD criteria—and do so consistently—did not meet thesecriteria until 2nd grade or later [43,61], reflecting a developmentalcomponent of DD stability. Finally, “mastery” of a skill may diminish

Box 2–Key questions for future longitudinal studies of develop-mental dyscalculia.

Etiology of DD

� What are the “core deficit(s)” of developmental dyscalcu-

lia (DD), and how are these deficits manifested across the

lifespan?

� How does manifestation of DD in late adulthood differ

from typical changes in numerical skills associated

with aging?

� Are there distinct subtypes of DD? If yes, do these subtypes

exist at the level of a DD genotype and/or phenotype?

� What is the interplay between genetic, social, and

educational factors in the development of DD?

Manifestation and classification of DD

� What components of number knowledge and number

skills (i.e., “number sense”) are distinct from one another?

How can these components be measured both mean-

ingfully and efficiently, and which components play a

meaningful role in the development of numeracy, mathe-

matics skills, mathematics achievement, and DD?

� What alternative pathways to low numeracy exist among

domain-general and domain-specific factors at genetic,

neural, cognitive and behavioral levels, and how do these

contribute to differential diagnosis of DD?

� Which pathways to numeracy and math achievement are

malleable, and which pathways inform efforts to support

compensatory mechanisms for students with severe DD?

� If any skills underlying DD are malleable, when during

development are they most malleable?

Prevention, education, and intervention

� How do home learning environments, early education,

and interventions influence the phenotype of DD? What

efforts, and at what doses, help to prevent children with

DD from failing to learn school mathematics?

� What is the effectiveness of DD prevention and early

interventions prior to school entry? Is early intervention

more effective and enduring than intervention delivered in

later childhood?

� To what extent do adults with DD respond differently to

intervention than children?

� What are the effects of comorbidity on responsiveness to

intervention?

� How can educational support and therapeutic interven-

tions lessen the anxiety and emotional stress caused by

DD and other types of math difficulties, including LA?

Replication of research findings on DD

� Replication studies are needed in all areas of DD research,

given the short history of the field. What constitutes a

replication, considering the existing variation across

studies in our classification of DD samples, the range of

inclusive or exclusive focus on DD relative to other

mathematics difficulties in participant samples, and the

discrepancy with which constructs targeted across studies

(such as number sense) are measured?

� What core set of shared measures can researchers agree

to include in their research batteries in order to advance

the field and increase replication?

0

20

40

60

80

100

120

140

2001-2006 2007-2012

Non-InterventionIntervention

0

10

20

30

40

50

60

70

2001-2006 2007-2012

NumberAttentionWorking MemoryExecutive FunctionVisual MemorySpatial Memory

Fig. 1. The total number of PsychInfo hits on studies of DD/MLD (since 2001)focused on select areas of interest, and the change in frequency of publications inthose areas over two six-year periods. (A) The total number of publications of DD/MLD, and the proportion of those focused on intervention, during two six-yearperiods. (B) The total number of publications focused on select cognitive correlatesof MLD/DD; the totals per variable category are not mutually exclusive becausemany publications focused on multiple variables.

M.M.M. Mazzocco, P. Räsänen / Trends in Neuroscience and Education 2 (2013) 65–7370

once rehearsal is no longer a focus of instruction—in other words,when regular practice stops (as per [39]), which may account forinstability of a particular skills or set of skills over time.

Inconsistency in performance also applies to variable perfor-mance across or within mathematics domains or problem types.For instance, in recent work on fractions knowledge, 9 year oldswith dyscalculia were as accurate as their peers (�90% accurate)when comparing relative magnitude of two fractions, but only if theitem pairs shared common numerators [62]. Relative to their peers,performance was impaired on other types of fraction pairs both atthe same point in time and four years later. Their peers achievedceiling level performance by about 11 years of age. Despite beingindistinguishable from their LA peers on knowledge of fractions atage 9, by age 10, the DD and LD groups differed, the gap between theLA and TA groups narrowed, and the gap between the DD and TAgroups widened. Such variations in trajectories over time are notlimited to higher order mathematics skills like fractions. Noël andRousselle [68] suggest that even basic tasks like symbolic and non-symbolic magnitude comparison show different developmentaltime-frames in their sensitivity to differentiating DD from LA and TA.

6. Cognitive underpinnings of dyscalculia

Whereas earlier debates regarding cognitive domains of dyscal-culia DD focused on “domain specific” (i.e., numerical processing)skills (e.g., [48]) vs. “domain general” skills (e.g., [24]), currentdebates have changed in two important ways. First, each of these

M.M.M. Mazzocco, P. Räsänen / Trends in Neuroscience and Education 2 (2013) 65–73 71

broad subsets of cognition is being differentiated further. It is no longersufficient to refer to “number sense” except in general terms, in viewof emerging delineation of symbolic vs. non-symbolic number skills(see Ansari et al., this issue), the facility with which young childrenspontaneously pay attention to quantitative aspects in their envir-onment [33,34], the protracted development of automaticity withwhich symbolic representations are linked to number words andnumerals (e.g., [7,29]), and the gradually emerging acceptance thatnumber sense has multiple components [72] some of whichmay notyet be identified. Longitudinal studies are needed to uncover whatthese components are, how and when components including non-symbolic and symbolic number skills become intertwined, thisappears to occur prior to and during school entry, based on cross-sectional analysis of 4 to 6-year-olds (e.g., [44]); and what role theseskills play in DD. There is some evidence that non-symbolic numberskills deficits underlie DD [71] but not LA [59], although thesefindings have not been replicated consistently (see De Smedt et al.,this issue), and deficits in symbolic number skills also differentiatethese groups [74]. Evidence for an association between “numbersense” and formal mathematics skills may depend on the measuresused.

Second, the debate is no longer between domain general vs.specific, but rather concerns the relative unique contributions ofthese skills and their relationship with each other, as potentialmediators of the relationships between cognitive skills andmathematics achievement (e.g., [14,49,86]). It is clear that workingmemory (WM, particularly the central executive), processingspeed, and other cognitive skills have a significant role in mathe-matics learning and that these roles vary with development andsometimes across DD, LA, and TA groups. The role of these skills isalso task and strategy dependent. This emerging knowledgeremains incomplete (as elaborated by Fias et al., this issue),particularly given the heightened attention to number and WMskills at the expense of even more limited research on visual andspatial skills (Fig. 1B).

7. Potential socio-affective factors in DD

Children with DD experience repeated failure through thedemands of their school curriculum, persistent low exam scores,and feelings of incompetence compared to their classmates.Children with DD show stronger negative emotional reactions tonumerical tasks than their typically performing peers [77]. Simi-larly, math anxiety causes an “affective drop”, a decline in math-ematical performance on tasks with high working memory loads[2], linked to physiological reactions in anticipation of mathema-tical activities being similar to physical pain [53]. Beyond theimmediate avoidance of mathematics tasks, individuals with mathanxiety avoid coursework, college majors, or career paths thatinvolve mathematics [54].

In a study spanning kindergarten to 4th grade, Hirvonen et al.[36] showed that mathematics performance and task-avoidantbehavior develop in synchrony: low performance in mathematicsincreases math task-avoidant behavior and vice versa, andan increase in mathematics task-avoidant behavior leads tofewer opportunities to improve mathematics performance. Thiscross-over interaction between development of skills and task-motivation is evident already in kindergarten—for mathematics,not for literacy [83].

Teachers may unintentionally enhance this process [81] byattributing mathematics success to ability and effort for childrenwith high mathematics task-motivation, which in turn contributesto an increase in the child's interest in mathematics. Whenteachers attribute children's success and failure to external causes,children show a decrease in mathematics-related interest. It is

encouraging that a teacher's instructional support in the classroomreduces task-avoidant behavior [69]. Low achievement in mathe-matics and insufficient working memory are risk factors for mathanxiety. However, longitudinal studies propose that first throughthird graders' mathematics anxiety does not relate to mathema-tical performance, but to self-image [47]—a notion supported byBeilock's findings that the negative influence of teachers' mathanxiety on their students' mathematics achievement is mediatedby students' acceptance of gender stereotypes about mathematicsability. Moreover, mathematics anxiety may primarily affect chil-dren with higher levels of working memory [73,84].

Note that these aforementioned findings do not pertain to DD;the nature of mathematics anxiety in young children with DD and itsrelation to mathematical performance over time remains unspeci-fied, but should be studied, in view of recent evidence that mathe-matics anxiety may be linked to low numeracy [55].

8. Summary

Dyscalculia is a biologically influenced disorder characterized bydifficulty learning and performing mathematics across the lifespan.DD reflects both numerical and non-numerical sources of individualdifferences in function and development. Although distinct from otherforms of mathematics difficulties (MD), it shares features of therelationships between cognitive, socio-emotional, and mathematicsspecific skills seen in the general population. Additional longitudinalstudies are needed to identify the nature and timing of these rela-tionships (Box 2). In the meantime, diagnosticians and educators mustconsider multiple possible sources of children's mathematics difficul-ties and alternative educational supports, and should not expect allchildren with DD to conform to a highly specified behavioral orlearning profile.

References

[1] Andersson U. Skill development in different components of arithmetic andbasic cognitive functions: findings from a 3-year longitudinal study of childrenwith different types of learning difficulties. Journal of Educational Psychology2010;102(1):115–34, http://dx.doi.org/10.1037/a0016838.

[2] Ashcraft MH, Moore AM. Mathematics anxiety and the affective drop inperformance. Journal of Psychoeducational Assessment 2009;27:197–205,http://dx.doi.org/10.1177/0734282908330580.

[3] Auerbach JG, Gross-Tsur V, Manor O, Shalev RS. Emotional and behavioralcharacteristics over a six-year period in youths with persistent and nonpersis-tent dyscalculia. Journal of Learning Disabilities 2008;41:263–73, http://dx.doi.org/10.1177/0022219408315637.

[4] Aunio P, Hautamäki J, Sajaniemi N, Van Luit JEH. Early numeracy in low-performing young children. British Educational Research Journal 2009;35:25–46, http://dx.doi.org/10.1080/01411920802041822.

[5] Aunola K, Leskinen E, Lerkkanen M-K, Nurmi J-E. Developmental dynamics ofmath performance from preschool to grade 2. Journal of Educational Psychol-ogy 2004;96(4):699–713, http://dx.doi.org/10.1037/0022-0663.96.4.699.

[6] Boets B, De Smedt B, Ghesquiere P. Coherent motion sensitivity predictsindividual differences in subtraction. Research in Developmental Disabilities2011;32:1075–80, http://dx.doi.org/10.1016/j.ridd.2011.01.024.

[7] Bugden S, Ansari D. Individual differences in children's mathematicalcompetence are related to the intentional but not automatic processing ofArabic numerals. Cognition 2011;118:32–44 doi: http://dx.doi.org/10.1016/j.cognition.2010.09.005.

[8] Butterworth B. Dyscalculia screener. London: Nelson Publishing Company Ltd.;2003.

[9] Butterworth B. Developmental dyscalculia. In: Campbell JID, editor. Handbookof mathematical cognition. New York and Hove, East Sussex: Psychology Press;2005 p. 455–67.

[10] Butterworth B. Foundational numerical capacities and the origins of dyscal-culia. Trends in Cognitive Sciences 2010;14:534–41, http://dx.doi.org/10.1016/j.tics.2010.09.007.

[11] Butterworth B, Kovas Y. Understanding neurocognitive developmental dis-orders can improve education for all. Science 2013;340(6130):300–5, http://dx.doi.org/10.1126/science.1231022.

[12] Cavendish W. Identification of learning disabilities: implications of proposedDSM-5 criteria for school-based assessment. Journal of Learning Disabilities2013;46(1):52–7, http://dx.doi.org/10.1177/0022219412464352.

M.M.M. Mazzocco, P. Räsänen / Trends in Neuroscience and Education 2 (2013) 65–7372

[13] Chong SL, Siegel LS. Stability of computational deficits in math learningdisability from second through fifth grades. Developmental Neuropsychology2008;33:300–17, http://dx.doi.org/10.1080/87565640801982387.

[14] Cirino PT. The interrelationships of mathematical precursors in kindergarten.Journal of Experimental Child Psychology 2011;108(4):713–33, http://dx.doi.org/10.1016/j.jecp.2010.11.004.

[15] D'Amico A, Passolunghi MC. Naming speed and effortful and automaticinhibition in children with arithmetic learning disabilities. Learning andIndividual Differences 2009;19:170–80, http://dx.doi.org/10.1016/j.lindif.2009.01.001.

[16] De Smedt B, Verschaffel L, Ghesquière P. The predictive value of numericalmagnitude comparison for individual differences in mathematics achieve-ment. Journal of Experimental Child Psychology 2009;103:469–79, http://dx.doi.org/10.1016/j.jecp.2009.01.010.

[17] Desoete A, Ceulemans A, De Weerdt F, Pieters S. Can we predict mathematicallearning disabilities from symbolic and non-symbolic comparison tasks inkindergarten? Findings from a longitudinal study British Journal ofEducational Psychology 2012;82(1):64–81, http://dx.doi.org/10.1348/2044-8279.002002.

[18] Desoete A, Gregoire J. Numerical competence in young children and inchildren with mathematics learning disabilities. Learning and IndividualDifferences 2006;16:351–67, http://dx.doi.org/10.1016/j.lindif.2006.12.006.

[19] Fayol M, Barrouillet P, Marinthe C. Predicting achievement from neuropsy-chological performance: a longitudinal study. Cognition 1998;68:B63–70.

[20] Frances DJ, Fletcher JM, Stuebing KK, Lyon GR, Shaywitz BA, Shaywitz SE. Psycho-metric approaches to the identification of learning disabilities: IQ and achievementscores are not sufficient. Journal of Learning Disabilities 2005;38:98–110, http://dx.doi.org/10.1177/00222194050380020101.

[21] Fuchs LS, Compton DL, Fuchs D, Paulsen K, Bryant JD, Hamlett CL. Theprevention, identification, and cognitive determinants of math difficulty.Journal of Educational Psychology 2005;97(3):493–513, http://dx.doi.org/10.1037/0022-0663.97.3.493.

[22] Fuchs LS, Fuchs D, Compton DL. Intervention effects for students withcomorbid forms of learning disability: understanding the needs ofnonresponders. Journal of Learning Disabilities 2012. http://dx.doi.org/10.1177/0022219412468889.

[23] Garrett AJ, Mazzocco MMM, Baker L. Development of the metacognitive skillsof prediction and evaluation in children with or without math disability.Learning Disabilities Research and Practice 2006;21:77–88, http://dx.doi.org/10.1111/j.1540-5826.2006.00208.x.

[24] Geary DC. Mathematical disabilities: cognitive, neuropsychological, andgenetic components. Psychological Bulletin 1993;114(2):345–62, http://dx.doi.org/10.1037/0033-2909.114.2.345.

[25] Geary DC, Hoard MK, Bailey DH. Fact retrieval deficits in low achievingchildren and children with mathematical learning disability. Journal ofLearning Disabilities 2012;45:291–307, http://dx.doi.org/10.1177/0022219410392046.

[26] Geary DC, Hoard MK, Byrd-Craven J, Nugent L, Numtee C. Cognitive mechanismsunderlying achievement deficits in children with mathematical learning disabil-ity. Child Development 2007;78:1343–59, http://dx.doi.org/10.1111/j.1467-8624.2007.01069.x.

[27] Geary DC, Hoard MK, Nugent L, Bailey DH. Mathematical cognition deficits inchildren with learning disabilities and persistent low achievement: a five-yearprospective study. Journal of Educational Psychology 2012;104:206–23, http://dx.doi.org/10.1037/a0025398.

[28] Geary DC, Hoard MK, Nugent L, Bailey DH. Adolescents' functional numeracy ispredicted by their school entry number system knowledge. PLoS One 2013;8(1):e54651, http://dx.doi.org/10.1371/journal.pone.0054651.

[29] Girelli L, Lucangeli D, Butterworth B. The development of automaticity inaccessing number magnitude. Journal of Experimental Child Psychology2000;76(2):104–22, http://dx.doi.org/10.1006/jecp.2000.2564.

[30] Halberda J, Ly R, Wilmer JB, Naiman DQ, Germine L. Number sense across thelifespan as revealed by a massive internet-based sample. Proceedings of theNational Academy of Sciences 2012;109:11116–20, http://dx.doi.org/10.1073/pnas.1200196109.

[31] Hanich LB, Jordan NC, Kaplan D, Dick J. Performance across different areas ofmathematical cognition in children with learning difficulties. Journal ofEducational Psychology 2001;93:615–26, http://dx.doi.org/10.1037/0022-0663.93.3.615.

[32] Hannula MM, Lehtinen E. Spontaneous focusing on numerosity and mathe-matical skills of young children. Learning and Instruction 2005;15(3)237–56, http://dx.doi.org/10.1016/j.learninstruc.2005.04.005.

[33] Hannula MM, Lepola J, Lehtinen E. Spontaneous focusing on numerosity as adomain-specific predictor of arithmetical skills. Journal of Experimental ChildPsychology 2010;107(4):394–406, http://dx.doi.org/10.1016/j.jecp.2010.06.00.

[34] Hannula MM, Räsänen P, Lehtinen E. Development of counting skills: role ofspontaneous focusing on numerosity and subitizing-based enumeration.Mathematical Thinking and Learning 2007;9(1):51–7, http://dx.doi.org/10.1207/s15327833mtl0901_4.

[35] Hecht SA, Torgesen JK, Wagner RK, Rashotte CA. The relations betweenphonological processing abilities and emerging individual differences inmathematical computation skills: a longitudinal study from second to fifthgrades. Journal of Experimental Child Psychology 2001;79:192–227, http://dx.doi.org/10.1006/jecp.2000.2586.

[36] Hirvonen R, Tolvanen A, Aunola K, Nurmi JE. The developmental dynamics oftask-avoidant behavior and math performance in kindergarten and

elementary school. Learning and Individual Differences 2012;22(6):715–23http://dx.doi.org/10.1016/j.lindif.2012.05.014.

[37] Hughes S. Comprehensive assessment must play a role in RTI. In: Fletcher-Janzen E, Reynolds C, editors. Neuropsychological perspectives on learningdisabilities in the era of RTI: recommendations for diagnosis and intervention.Hoboken, NJ: John Wiley; 2008 p. 115–30.

[38] Individuals with Disabilities Education Improvement Act (IDEA). Pub. L. No.108-446, 118 Stat. 2647: 2004.

[39] Iuculano T. Good and bad at numbers: typical and atypical developmentof number processing and arithmetic. Unpublished doctoral dissertation.London, UK: University College London; 2012 Available from: http://discov-ery.ucl.ac.uk/1355958/1/5952.pdf.

[40] Jordan NC, Hanich LB, Kaplan D. A longitudinal study of mathematicalcompetencies in children with specific mathematics difficulties versus chil-dren with comorbid mathematics and reading difficulties. Child Development2003;74:834–50, http://dx.doi.org/10.1111/1467-8624.00571.

[41] Jordan NC, Kaplan D, Locuniak MN, Ramineni C. Predicting first-grade mathachievement from developmental number sense trajectories. Learning Dis-abilities Research & Practice 2007;22(1):36–46, http://dx.doi.org/10.1111/j.1540-5826.2007.00229.x.

[42] Jordan NC, Levine SC. Socioeconomic variation, number competence, andmathematics learning difficulties in young children. Developmental Disabil-ities Research Reviews 2009;15(1):60–8, http://dx.doi.org/10.1002/ddrr.46.

[43] Judge S, Watson SMR. Longitudinal outcomes for mathematics achievementfor students with learning disabilities. Journal of Educational Research2011;104:147–57, http://dx.doi.org/10.1080/00220671003636729.

[44] Kolkman ME, Kroesbergen EH, Leseman PPM. Early numerical developmentand the role of non-symbolic and symbolic skills. Learning and Instruction2013;25:95–103, http://dx.doi.org/10.1016/j.learninstruc.2012.12.001.

[45] Koponen T, Aunola K, Ahonen T, Nurmi JE. Cognitive predictors of single-digitand procedural calculation skills and their covariation with reading skill.Journal of Experimental Child Psychology 2007;97(3):220–41, http://dx.doi.org/10.1016/j.jecp.2007.03.001.

[46] Krajewski K, Schneider W. Early development of quantity to number-wordlinkage as a precursor of mathematical school achievement and mathematicaldifficulties: findings from a four-year longitudinal study. Learning and Instruction2009;19:513–26, http://dx.doi.org/10.1016/j.learninstruc.2008.10.002.

[47] Krinzinger H, Kaufmann L, Willmes K. Math anxiety and math ability inearly primary school years. Journal of Psychoeducational Assessment2009;27:206–25 http://dx.doi.org/10.1177/0734282908330583.

[48] Landerl K, Bevan A, Butterworth B. Developmental dyscalculia and basicnumerical capacities: a study of 8–9-year-old students. Cognition 2004;93(2):99–125, http://dx.doi.org/10.1016/j.cognition.2003.11.004.

[49] LeFevre J-A, Fast L, Skwarchuk S-L, Smith-Chant BL, Bisanz J, Kamawar D, et al.Pathways to mathematics: longitudinal predictors of performance. ChildDevelopment 2010;81(6):1753–67, http://dx.doi.org/10.1111/j.1467-8624.2010.01508.x.

[50] LeFevre J-A, Skwarchuk S-L, Smith-Chant BL, Fast L, Kamawar D, Bisanz J. Homenumeracy experiences and children's math performance in the earlyschool years. Canadian Journal of Behavioural Science/Revue Canadienne desSciences du Comportement 2009;41(2):55–66, http://dx.doi.org/10.1037/a0014532.

[51] Libertus ME, Brannon EM. Stable individual differences in number discrimina-tion in infancy. Developmental Science 2010;13(6):900–6, http://dx.doi.org/10.1111/j.1467-7687.2009.00948.x.

[52] Libertus ME, Feigenson L, Halberda J. Preschool acuity of the approximatenumber system correlates with school math ability. Developmental Science2011;14:1292–300, http://dx.doi.org/10.1111/j.1467-7687.2011.01080.x.

[53] Lyons IM, Beilock SL. When math hurts: math anxiety predicts pain networkactivation in anticipation of doing math. PloS One 2012;7(10):e48076http://dx.doi.org/10.1371/journal.pone.0048076.

[54] Ma X. A meta-analysis of the relationship between anxiety toward mathe-matics and achievement in mathematics. Journal for Research in MathematicsEducation 1999;30:520–40, http://dx.doi.org/10.2307/749772.

[55] Maloney EA, Risko EF, Ansari D, Fugelsang J. Mathematics anxiety affects countingbut not subitizing during visual enumeration. Cognition 2010;114:293–7, http://dx.doi.org/10.1016/j.cognition.2009.09.013.

[56] Mazzocco MMM. Defining and differentiating mathematical learning dis-abilities and difficulties. In: Berch DB, Mazzocco MMM, editors. Why ismath so hard for some children: the nature and origins of mathematicslearning difficulties and disabilities. Baltimore: Paul H. Brookes; 2007p. 29–47.

[57] Mazzocco MMM, Devlin KT. Parts and ‘holes’: gaps in rational number sense inchildren with vs. without mathematical learning disability. DevelopmentalScience 2008;11:681–91, http://dx.doi.org/10.1111/j.1467-7687.2008.00717.x.

[58] Mazzocco MMM, Devlin KT, McKenney JL. Is it a fact? Timed arithmeticperformance of children with mathematical learning disabilities (MLD) variesas a function of how MLD is defined Developmental Neuropsychology2008;33:318–44, http://dx.doi.org/10.1080/87565640801982403.

[59] Mazzocco MMM, Feigenson L, Halberda J. Impaired acuity of theapproximate number system underlies mathematical learning disability(dyscalculia). Child Development 2011;82:1224–37, http://dx.doi.org/10.1111/j.1467-8624.2011.01608.x.

[60] Mazzocco MMM, Grimm K. Growth in rapid automatized naming from gradesK-8 in children with math or reading disabilities. Journal of LearningDisabilities 2013. http://dx.doi.org/10.1177/0022219413477475, in press.

M.M.M. Mazzocco, P. Räsänen / Trends in Neuroscience and Education 2 (2013) 65–73 73

[61] Mazzocco MMM, Myers GF. Complexities in identifying and defining mathe-matics learning disability in the primary school age years. Annals of Dyslexia2003;53:218–53, http://dx.doi.org/10.1007/s11881-003-0011-7.

[62] Mazzocco MMM, Myers GF, Lewis KE, Hanich LB, Murphy MM. Limitedknowledge of fraction representations differentiates middle school studentswith mathematics learning disability (dyscalculia) vs. low mathematicsachievement. Journal of Experimental Child Psychology 2013;115:371–87,http://dx.doi.org/10.1016/j.jecp.2013.01.005.

[63] Mazzocco MMM, Thompson RE. Kindergarten predictors of mathlearning disability. Learning Disabilities Research and Practice 2005;20:142–55, http://dx.doi.org/10.1111/j.1540-5826.2005.00129.x.

[64] Morgan PL, Farkas G, Wu Qiong. Five-year growth trajectories of kindergartenchildren with learning difficulties in mathematics. Journal of Learning Dis-abilities 2009;42(4):306–21, http://dx.doi.org/10.1177/0022219408331037.

[65] Murphy MM, Mazzocco MMM. Rote numeric skills may mask underlyingmathematical disabilities in girls with fragile X syndrome. Developmental Neu-ropsychology 2008;33:345–64, http://dx.doi.org/10.1080/87565640801982429.

[66] Murphy MM, Mazzocco MMM, Hanich LB, Early MC. Cognitive characteristicsof children with mathematics learning disability (MLD) vary as a function ofthe cutoff criterion used to define MLD. Journal of Learning Disabilities2007;40:458–78, http://dx.doi.org/10.1177/00222194070400050901.

[67] Niklas F, Segerer R, Schmiedeler S, Schneider W. Findet sich ein Matthäus–Effekt in der Kompetenzentwicklung von jungen Kindern mit oder ohneMigrationshintergrund? Frühe Bildung 2012;1(1):26–33, http://dx.doi.org/10.1026/2191-9186/a000022.

[68] Noël M-P, Rousselle L. Developmental changes in the profiles of dyscalculia:an explanation based on a double exact-and-approximate number represen-tation model. Frontiers in Human Neuroscience 2011;5:165, http://dx.doi.org/10.3389/fnhum.2011.00165.

[69] Pakarinen E, Kiuru N, Lerkkanen M-K, Poikkeus A-M, Ahonen T, Nurmi J-E.Instructional support predicts children's task avoidance in kindergarten. EarlyChildhood Research Quarterly 2011;26(3):376–86 doi:http://dx.doi.org/10.1016/j.ecresq.2010.11.003.

[70] Petrill S, Logan J, Hart S, Vincent P, Thompson L, Kovas Y, et al. Math fluency isetiologically distinct from untimed math performance, decoding fluency, anduntimed reading performance: evidence from a twin study. Journal of LearningDisabilities 2012;45(4):371–81, http://dx.doi.org/10.1177/0022219411407926.

[71] Piazza M, Facoetti A, Trussardi AN, Berteletti I, Conte S, Lucangeli D, et al.Developmental trajectory of number acuity reveals a severe impairment indevelopmental dyscalculia. Cognition 2010;116(1):33–41, http://dx.doi.org/10.1016/j.cognition.2010.03.012.

[72] Purpura DJ, Lonigan CJ. Informal numeracy skills: the structure and relationsamong numbering, relations, and arithmetic operations in preschool. Amer-ican Educational Research Journal 2013;50(1):178–209, http://dx.doi.org/10.3102/0002831212465332.

[73] Ramirez G, Gunderson EA, Levine SC, Beilock SL. Math anxiety, working memoryand math achievement in early elementary school. Journal of Cognition andDevelopment 2013;14:187–202, http://dx.doi.org/10.1080/15248372.2012.664593.

[74] Reeve R, Reynolds F, Humberstone J, Butterworth B. Stability and change inmarkers of core numerical competencies. Journal of Experimental Psychology:General 2012;141:649–66, http://dx.doi.org/10.1037/a0027520.

[75] Rourke BP, Conway JA. Disabilities of arithmetic and mathematical reasoning:perspectives from neurology and neuropsychology. Journal of Learning Dis-abilities 1997;30(1):34–46, http://dx.doi.org/10.1177/002221949703000103.

[76] Rubinsten O, Henik A. Developmental dyscalculia: heterogeneity might notmean different mechanisms. Trends in Cognitive Science 2009;13:92–9, http://dx.doi.org/10.1016/j.tics.2008.11.002.

[77] Rubinsten O, Tannock R. Mathematics anxiety in children with developmentaldyscalculia. Behavioral and Brain Functions 2010;6:46, http://dx.doi.org/10.1186/1744-9081-6-46.

[78] Shalev RS, Manor O, Gross-Tsur V. Developmental dyscalculia: a prospectivesix-year follow up. Developmental Medicine & Child Neurology 2005;47:121–5, http://dx.doi.org/10.1017/S0012162205000216.

[79] Stock P, Desoete A, Roeyers H. Detecting children with arithmetic disabilitiesfrom kindergarten: evidence from a 3-year longitudinal study on the role ofpreparatory arithmetic abilities. Journal of Learning Disabilities 2010;43(3):250–68, http://dx.doi.org/10.1177/0022219409345011.

[80] Swanson HL. Working memory, attention, and mathematical problem solving:a longitudinal study of elementary school children. Journal of EducationalPsychology 2011;103:821–37, http://dx.doi.org/10.1037/a0025114.

[81] Upadyaya K, Viljaranta J, Lerkkanen M-K, Poikkeus A-M, Nurmi J-E. Cross-lagged relations between kindergarten teachers' causal attributions, andchildren's interest value and performance in mathematics. Social Psychologyof Education 2012;15:181–206, http://dx.doi.org/10.1007/s11218-011-9171-1.

[82] Van der Ven SH, Kroesbergen EH, Boom J, Leseman PP. The development ofexecutive functions and early mathematics: a dynamic relationship. BritishJournal of Educational Psychology 2012;82:100–19, http://dx.doi.org/10.1111/j.2044-8279.2011.02035.x.

[83] Viljaranta J, Lerkkanen M-K, Poikkeus A-M, Aunola K, Nurmi J-E. Cross-laggedrelations between task motivation and performance in arithmetic and literacyin kindergarten. Learning and Instruction 2009;19(4):335–44 10.1016/j.learninstruc.2008.06.011.

[84] Vukovic RK, Kieffer MJ, Bailey SP, Harari RR. Mathematics anxiety in youngchildren: concurrent and longitudinal associations with mathematical perfor-mance. Contemporary Educational Psychology 2013;38(1):1–10 10.1016/j.cedpsych.2012.09.001.

[85] Vukovic RK, Siegel LS. Academic and cognitive characteristics of persistentmathematics difficulty form first through fourth grade. Learning DisabilitiesResearch & Practice 2010;25:25–38, http://dx.doi.org/10.1111/j.1540-5826.2009.00298.x.

[86] Zhang X, Koponen T, Räsänen P, Aunola K, Lerkkanen M-L, Nurmi JE. Linguisticand spatial skills predict early arithmetic development via counting sequenceknowledge. Child Development 2013 in press.