individual differences in mathematical ability: genetic, cognitive and behavioural factors

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Journal of Research in Special Educational Needs

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Volume 7

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Number 2

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2007 97–103doi: 10.1111/j.1471-3802.2007.00085.x

Blackwell Publishing Ltd

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Individual differences in mathematical ability: genetic, cognitive and behavioural factors

John W. Adams

University of Durham

Key words

: mathematical ability, mathematical difficulties, SEN.

Identifying individuals with mathematical difficulties(MD) is becoming increasingly important in oureducation system. However, recognising MD is onlythe first stage in the provision of special educationalneeds (SEN). Although planning the effective remedialsupport is vital, there is little consensus on theinterventions that are appropriate. There are two mainreasons for this: first, MD has a variety of manifestationswhich appear to change with age; and second,there are many potential causes for the difficultiesindividuals experience. This paper addresses theseissues by reviewing research evidence from three‘domains’ of psychological research (genetic, cognitive,behavioural), all of which appear to offer insights intopotential influences on mathematical ability.

Mathematics, like reading, is an important skill in oureveryday lives. Mathematical reasoning is something weall do, from simple counting to complex calculations.However, the ability to use these skills varies greatly anddifficulties emerging in childhood can persist through toadulthood. The introduction of the National NumeracyStrategy (Department for Education and Employment(DfEE), 1999) in schools acknowledged this problem in thebelief that a solid grounding in basic numeracy helpschildren to succeed in other subject areas in the curriculumand develop more advanced mathematical skills which areessential for higher education and employment.

For many years it was recognised that there was asubstantial literature (from a psychological perspective)on reading problems and other disabilities associated withliteracy and a paucity of research concerning disabilitiesrelated to mathematics. Some suggested that this was becausepoor numeracy was seen as being more socially acceptablethan processing poor literacy skills (e.g., O’Hare, Brown& Aitken, 1991). However, in recent years this bias hasbeen addressed and there is now an extensive literature onmathematical difficulties. This large and diverse body ofresearch into mathematical cognition, development anddifficulties offers a number of explanations for individualdifferences in mathematical ability.

The aim of this paper is to provide a concise overviewof this literature and to consider how different researchfindings mediate the type of intervention likely to resolvemathematical difficulties. Throughout the paper the genericterm ‘mathematics difficulties’ (MD) is adopted to describemaths performance significantly below expectation andconsequently requiring special needs intervention. Using atheoretical framework first suggested by Morton and Frith(1995), three different (but not mutually exclusive) researchdomains will be reviewed that provide evidence of potentialsources of individual differences in mathematical ability:genetics, cognition and behaviour. The examples of recentresearch chosen to illustrate each of these domains are bynecessity selective; however, the aim is to demonstrate theuse of this psychological framework in conceptualising anddealing with MD.

Genetic

It is becoming increasingly recognised that our genetic inheritanceinfluences our lifespan in terms of our predisposition tocertain illnesses, both physical and mental. The observationthat some learning difficulties such as dyslexia (Francks,MacPhie & Monaco, 2002) tend to run in families supportsthis view. Developmental dyscalculia, a mathematical disordersynonymous with MD diagnosed in children and adults,has also been reported to have a strong genetic link (seeButterworth, 2005; Shalev & Gross-Tsur, 2001). Alarcon,DeFries, Gillis-Light and Pennington (1997) found that58% of monozygotic and 39% of dizygotic twins shared thediagnosis of a mathematics disability. Shalev et al. (2001) alsoreported that half of all siblings of children with developmentaldyscalculia also share the diagnosis, an estimated five- totenfold the risk compared to the general population.

In his book,

The Mathematical Brain

, Brian Butterworth(1999) proposes that we are born with a number modulewhich ‘categorize(s) the world in terms of numerosities’(p. 7). This innate mathematical ability is considered to be,at least in part, genetically determined. The notion thatchildren are born with a mathematical brain is supported byresearch which suggests that pre-verbal infants and somespecies of animals appear to have an innate number sense(see Dehaene, 1997 for a review of the evidence). StanislasDehaene (1997, 1999, 2001) proposes that we have a

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non-verbal ability to spatially represent magnitude akin to amental number line. Located bilaterally in the parietotemporalareas of the brain, the number line is believed to be anevolutionary developed ability which helps us to representthings in terms of relative magnitude.

It is proposed that this innate ability to represent magnitudeprovides the foundation on which our culturally derivedsystem of numbers and mathematical operations is built(e.g., Butterworth, 1999). This is a bold statement and we areleft to consider whether individual differences in mathematicalability are the result of nature (i.e., genes) or nurture (e.g.,teaching). It is worth noting that Dehaene (1999) placesmore weight on nurture, concluding that ‘... impact of educationis probably much greater than any initial difference ...’ innumber sense.

In order to examine individual differences in the functionof the number module, Butterworth (2003) developed the

Dyscalculia Screener

. Targeted at schools and educationprofessionals, the computer-based screener assesses learningdifficulties in mathematics for children aged 5 to 14 years.Identification of impaired mathematical function is basedon the child’s speed of response (reaction time, RT) to twonumerical tasks: dot enumeration (comparing an array ofdots and a numeral) and number comparison (numericalstroop, identifying the larger of two numbers), and a measureof arithmetic ability. Butterworth argues that reaction timesare more sensitive than simple yes/no responding as theyreflect the efficiency with which children access theirnumerical magnitude representations. This move away fromthe traditional IQ discrepant approach is to be welcomed.A number of researchers have expressed concerns with thislatter approach (Bishop, 1997; Stanovich, 1991; Sternberg& Grigorenko, 2002), arguing that there is no plausiblereason why having a low IQ should inoculate a childfrom having a developmental disorder with a constitutionalorigin.

Overall, the Dyscalculia Screener appears a useful tool forscreening children with mathematical difficulties across abroad age range, but it has its weaknesses. Using only twocore numerical tasks, the Screener can assess only a limitedrange of mathematical knowledge (i.e., counting and magnitudejudgement). Later acquired knowledge of algebra, geometryand fractions may or may not be supported by these earlynumeracy skills. Second, it is not diagnostic in the sensethat teachers have no clear indication what the underlyingimpairments causing poor performance are. This is due tothe Screener’s reliance on simple RT to infer impairment.For example, slow solution RTs of simple arithmetic arebelieved to reflect that the child is relying on an inefficientcounting strategy and has not used a memory retrievalstrategy (Butterworth, 2003).

Finally, the Screener’s value is to some extent dependent onother measures which could indicate other areas of difficulty.For example, MD is often associated with problems such asreading (e.g., dyslexia) and behaviour (e.g., attention deficithyperactivity disorder, ADHD). In order to target remedial

help appropriately, there is a clear need to ascertain whetherthe MD is a primary or secondary problem, that is, whetherthe difficulties are the result of impaired numerical magnituderepresentations, or some other cause. An early diagnosis ofMD or other learning difficulties is desirable and interventionsshould be established as soon as possible. Possible remedialstrategies include methods of reinforcing representationsof magnitude through the use of concrete examples (e.g.,number lines).

However, if we are to accept a genetic abnormality as thecause of MD then the best course of action may well bemanaging the disability. For the individual with MD thismeans learning to come to terms with the academic impairmentand building on other cognitive strengths, rather than providinga cure.

Cognitive

The study of mathematical cognition attempts to answer thefundamental questions of structure and process in mathematicalprocessing: how is numerical and mathematical knowledgerepresented in the brain, and how do we solve mathematicalproblems? Cognitive psychologists approach these kinds ofquestions by trying to understand the functional architectureof the brain. By modelling observed human performance(e.g., doing mathematics), models of mathematical cognitionincluding computer simulations (see Ashcraft, 1992, for areview) have advanced our understanding of our mathematicalbrain. This is because in order to successfully model humanperformance, one first need to fully understand theprocesses children and adults go through in order to solvemathematical problems.

This dual approach is exemplified by the work of Robert Siegler,who over a number of years has studied the development ofmathematical problem solving, and developed a functionalnetwork model of mathematical development (e.g., StrategyChoice and Discovery Simulation (SCADS) model, Shrager& Siegler, 1998). Siegler’s Distribution of Associationsmodel (Siegler & Jenkins, 1989; Siegler & Shrager, 1984)was the first model to implement the developmental ofchildren’s strategy choice in mathematical problem solving.The model posits that correct and incorrect answers arestored in an interconnected network of number facts inlong-term memory. Through a dynamic process, answersgenerated by a child when solving a problem (e.g.,3 + 4 = ?) influence their future response to that problem.

For example, the child will become more efficient at usingthe counting algorithm and will begin to compute thecorrect answer more times than the incorrect answers, thusbuilding its associative strength. The correct answer (7)associated with the problem (3 + 4) will display a ‘

peakedness

’amongst the possible solutions. When this ‘

peakedness

’exceeds a threshold (the confidence criterion), the child willretrieve rather than compute the answer. The model alsoprovides a mechanism to account for strategy selection.Backup strategies such as counting will be used when theanswer(s) generated fail to reach the confidence criterionand the search length parameter (number of retrieval attempts)


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exceeds a preset level. Siegler argues that this process isautomatic and not the result of a conscious choice.

So does this model help to predict the deficits that might befound in an individual with MD? First, the model suggeststhat early accuracy in counting is essential for building astrong link between a given problem and its answer.Children who count inaccurately will be less likely to buildan efficient network of number facts in their long-termmemory. Second, those children who do not develop anetwork will have to rely on backup strategies such ascounting, which may be appropriate for small problems,but inefficient for larger and more complex ones. In otherwords, the lack of an efficient network of number factsseverely restricts the strategies available for mathematicalproblem solving. This is particularly salient as Siegler (1987,1999) has found that children use a variety of strategies atall times when problem solving, with continuously changingfrequencies of existing strategies and the occasional discoveryof new strategies.

A number of studies conducted in the USA (e.g., Geary,Bow-Thomas & Yao, 1992; Geary, Brown & Samaranayake,1991) have found that MD children are delayed, or fail tomake the transition, from using slow procedural countingstrategies to efficient memory retrieval. The hypothesisGeary sets out is that as MD children are slow and inaccuratecounters, hence they have a lower working (short-term)memory span, a distinct functional deficit when performingmental arithmetic which requires the retention of operandsand a counting sequence. As a result frequent calculationerrors can occur due to information being lost throughdecay, thus inducing the formation of weak associationsbetween operands and their sums in long-term memory.This in turn decreases the probability of MD childrendeveloping an efficient math-fact retrieval strategy.

In recent years, the role of working memory in mathematicalproblem solving has also received a great deal of attention(see DeStefano & LeFevre, 2004), due in part to theobservation that low working (short-term) memory span isfrequently associated with learning disorders, includingMD. A widely researched model of working memory is oneproposed by Baddeley and Hitch (1974, revised Baddeley,1986, 2000) which consists of a number of separate butinterconnected components. The three components of theirearly tripartite model, the phonological loop, visuo-spatialsketchpad and the central executive have all been associatedwith mathematical problem solving (see Holmes & Adams,2006).

Sue Gathercole and colleagues have in recent years reportedsignificant associations between measures of working memorycapacity and National Curriculum attainment. For example,Gathercole and Pickering (2000b) reported that phonologicalloop and central executive scores were significantly associatedwith six- and seven-year-olds’ performance on standardisedattainment tests in arithmetic. Central executive scores havealso been related to mathematics attainment at 7 (Gathercole,Pickering, Knight & Stegmann, 2004), 11 (Holmes, 2005;

Jarvis & Gathercole, 2003) and 14 years (Jarvis & Gathercole,2003). Visuo-spatial sketchpad scores have been related toNational Curriculum mathematics attainment at 7 (Gathercole& Pickering, 2000a), 11 and 14 years (Jarvis & Gathercole,2003) and phonological loop scores have been related toNational Curriculum mathematics performance at 7 and11 years (Gathercole et al., 2004).

This wealth of evidence supports the idea that workingmemory plays a key role in children’s mathematicssupporting both processing (calculation) and storage(retaining information). Hence, children with low workingmemory capacities will be at a disadvantage when problemsolving. A simple example of this was demonstrated byAdams and Hitch (1997) when they found that children’sability to solve complex mental additions was enhanced bysimply making the addends visible during the calculationprocess. Further insights into why working memory constrainslearning have recently been noted by an observational studyof young children identified as having very poor verbalworking memory function at school entry (Gathercole,Lamont & Alloway, 2006). These children frequently failedin learning activities that placed heavy demands on workingmemory; for example, forgetting instructions, problemsfollowing sequences and place keeping errors. As mathematicsinvolves keeping track of temporary information and goingthrough a series of processing stages it is easy to see howlow-working-memory children often fail to meet thedemands of individual learning episodes.

So if you have a student who shows evidence of one ormore cognitive deficits what can be done? Maths difficultiesassociated with poor strategy use and poor counting abilitycan be targeted with tailored interventions. Theoretically,one-to-one instruction should improve counting (arithmetic)performance and increase the child’s confidence (e.g.,

Numeracy Recovery Program

, Dowker, 2001). This in turnshould lead the child to develop a more efficient network ofnumber facts and therefore increase the range of strategiesavailable. However, this approach is labour intensive, timeconsuming and not without problems. First, although trainingin the procedural skills associated with counting needs to beflexible to allow the child to apply his or her newly acquiredskills across different problems (Geary, 1994), there is noreliable mechanism to translate this newly acquired knowledgeinto an efficient fact-retrieval network. Second, massedpractice and drill in number facts do not necessarilytranslate into better problem solving without concomitantshifts in strategy use (Goldman, Pellegrino & Mertz, 1988).Indeed, it is possible that this inability to operate betweenprocedural and declarative knowledge is at the core ofsome types of MD (see Temple & Sherwood, 2002).

For maths difficulties associated with memory impairmentthe best approach is to minimise the problems by effectiveclassroom management of the memory-related failures.While an ideal solution would be to remediate the memoryimpairment directly, there is little evidence that trainingworking memory in students with low working memoryskills leads to substantial gains in academic attainments

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(e.g., Turley-Ames & Whitfield, 2003). Interventions includethe use of memory aids (e.g., number lines, blocks and otherconcrete aids to counting, calculators), the simplification ofverbal instructions into smaller steps to reduce complexity(i.e., WM load), and empowering children with poor workingmemory (WM) to recognise their limitations and adopt newstrategies (Gathercole & Alloway, 2006). These may includeasking for help, or developing organisational strategies(e.g., dictionary of definitions, memory crib-cards).

Behavioural

Within the classroom and beyond, children’s numeracyproblems present themselves in a number of ways. Asalready noted, these difficulties can be specific or related toother learning problems such as language (e.g., specificlanguage impairment (SLI), see Fazio, 1999) and literacy(e.g., dyslexia). Issues of comorbidity are often complex andpose difficulties in getting to the locus of the mathematicaldifficulty as MD can mask, or be masked by, other potentialcauses of low achievement. As only a careful and detailedassessment of an individual’s strengths and weaknesses canachieve this, it is unlikely that we will ever be able to relyon a single test to diagnose MD/dyscalculia. Furthermore,it is evident that the key markers (deficits) of MD changethroughout development. For example, number productiondeficits are more likely to be found in children beginningformal mathematics instruction than in older children whohave mastered basic arithmetic transcoding skills.

In a longitudinal study of MD and comorbid MD/RD(reading disabilities) subgroups, Geary and colleagues(Geary, Hamson & Hoard, 2000; Geary, Hoard & Hamson,1999) reported the results of a comprehensive battery ofcognitive test measures given to the children in first andsecond grades. The overall pattern of deficit found in boththe MD and MD/RD children included poor countingknowledge, lower working memory spans and memoryretrieval errors. However, at first grade MD children showeda more circumscribed pattern of deficit. These includeddifficulties in number production and comprehension, poorunderstanding of order-irrelevance, and frequent countingprocedure and retrieval errors.

Finally there are two further sources of behavioural variationthat interact with cognitive factors that may explain individualdifferences in mathematical ability. These are cognitivestyle and maths anxiety.

An individual’s cognitive style describes his or her preferredapproach to organising and representing information (Riding,2002). Cognitive style has two dimensions: wholist-analytic(whether people view the whole or see things in parts)and verbal-imagery (whether people prefer to representinformation verbally or as pictures and images). Ridingcontends that people’s cognitive styles differ along thesedimensions in various combinations (i.e., analytic-verbalisers,wholistic-imagers, etc.). An important consequence ofhaving a particular style profile is the impact it has on theindividual’s working memory. Analytic and verbal stylestend to be more resource demanding than wholistic and

imager styles. This is not necessarily problematic whenWM capacity is not exceeded, but in those students withpoor WM an analytic verbaliser may experience the samememory problems outlined earlier. Again, suggestions toaid students with these difficulties by reducing overallprocessing load include the use of external representation,slowing down presentation, and using frequent revisiontechniques (Riding, Grimley, Dahraei & Banner, 2003).

On the other hand wholistic imagers may experience theirown difficulties. Although they are less susceptible to exceedingtheir WM capacity, they experience problems learningmathematics at primary level where the teaching and learningof arithmetic facts and strategy-use are a predominantlyverbal domain. Furthermore, research by Riding and colleaguesreport that in terms of overall academic achievement, wholisticimagers appear to do less well, especially in topics that requiredetailed analysis (e.g., Science, Music, Mathematics; seeRiding, 2002). Hence, some adjustment of teaching fromverbal to pictorial may provide the basis of any intervention.

Mathematics has long been associated with increased levelsof anxiety, both in students suffering from general anxietyto those who have a subject-specific problem (Hembree,1990). Ashcraft and Kirk (2001) studied people duringproblem solving to evaluate how maths anxiety affectedmathematical ability. They carried out studies with collegestudents who cited low, moderate, or high levels of mathsanxiety on a questionnaire. In one experiment, students firstsaw a set of letters to be remembered. They were thentimed as they performed a mental addition problem. Aftersolving it, the students were asked to recall the letters theyhad seen previously. High-math-anxiety students scoredpoorly on both tasks, but especially on the mental addition.Their performance was especially bad on problems thatinvolved carrying numbers, such as 47 + 18. However,when allowed to use pencil and paper during the additiontrials, they did as well as students without maths worries,indicating an underlying mathematical competence.

Overall, their results reveal that increased maths anxietywas associated with decreased working memory capacity.Individuals with high maths anxiety showed a significantdecline in problem-solving performance. They had lessworking memory space to effectively deal with mathsproblems, because their maths anxiety was using workingmemory space that could be used to solve maths problems.Furthermore, Ashcraft and Kirk found that students witha high level of maths anxiety enrolled in fewer mathscourses, received lower maths grades, and scored worse onworking-memory tests involving numbers.

Reducing anxiety is consequently the key to any interventionstrategy with maths anxious students.

Synthesis

The three levels of explanation outlined above, genetic,cognitive and behavioural, are not necessarily mutuallyexclusive. It has already been claimed that genetic differencesimpact upon basic numeracy skills (Butterworth, 2003),


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and as maths anxiety illustrates, a behavioural problemappears to interact with cognitive ability (i.e., memory). Inthe future these links may become more explicit, providinga chain of causality across the three domains. For example,Uta Frith (1997) has used this framework model to goodeffect in presenting a consensus model of phonologicaldyslexia. However, given our current level of understandingof MD, interventions are likely to be most effective whentargeted at the domain in which the difficulties are present.

One important issue raised by this psychological modelapproach to individual differences in mathematics attainment,as opposed to an educational perspective of developmentaldelay, is the tacit and sometimes explicit proposition (e.g.,genetics) that MD have a pathological origin and aretherefore abnormal. An important question which needs tobe addressed in the future by the diverse but complementaryapproaches is how professional educators and psychologistsconceptualise MD. This has been recently illustrated by thecontentious debate on the existence of dyslexia as a usefuland necessary construct (Elliot, 2005).

Finally, there is one important domain proposed by Mortonand Frith (1995) that undoubtedly influences how we dealwith MD. Our environment has influenced student /teacherviews of mathematics learning for as long as it has beenstudied. Gender differences have a longstanding history. Untilrecently, boys traditionally out-performed girls at maths atall levels (see Hyde, Fennema & Lamon, 1990), and even todayboys are more likely to study post-16 qualifications inmathematics. While this difference may be due to sex-baseddifferences (Geary, 1996), the fact that the gender effectsize has been decreasing during the last 30 years points toattitude change as a significant factor.

Conclusion

An important question on which to conclude is in whatcontext any interventions for MD are to be carried out. Theimpact of the National Numeracy Strategy has led to agrowing debate on whole-class teaching versus setting byability. Whitburn (2001) alluded to this dilemma when notingthat one of the major challenges in teaching mathematicsarises from the diversity of pupils’ attainments. Whileoutside the scope of this paper, a tentative conclusion isthat if at least some MD are ‘abnormal’ (and not the resultof delayed development), then individuals with thesedifficulties would be unlikely to benefit from an inclusiveapproach.

Experimental psychology has made significant progressin explaining possible sources of individual differences inmathematical ability. This review has attempted to illustratehow these differences can be conceptualised and howthe different sources of variation are likely to impact onintervention.

Address for correspondence

John W. Adams,Department of Psychology,University of Durham,Queen’s Campus,University Boulevard,Thornaby,Stockton on Tees,TS17 6BH,UK.Email: [email protected].

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