Relations between balancing and arithmetic skills in children – Evidence of cerebellar...

Journal of Neurolinguistics 24 (2011) 592–601

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jneurol ing

Relations between balancing and arithmetic skills inchildren – Evidence of cerebellar involvement?

Jan Lonnemann a,b,1,*, Janosch Linkersdörfer a,b,1, Vera Heselhaus a,b,Marcus Hasselhorn a,b,c, Sven Lindberg b,c

a Institute for Psychology, Goethe-University, Frankfurt, GermanybCenter for Individual Development and Adaptive Education of Children at Risk (IDeA), Frankfurt, GermanycGerman Institute for International Educational Research (DIPF), Frankfurt, Germany

a r t i c l e i n f o

Article history:Received 1 July 2010Received in revised form 2 February 2011Accepted 14 February 2011

Keywords:ArithmeticBalancingCerebellum

* Corresponding author. IDeA, Schloßstr. 29, 60424708 216.

E-mail address: j.lonnemann@idea-frankfurt.eu1 Shared first author.

0911-6044/$ – see front matter � 2011 Elsevier Ltdoi:10.1016/j.jneuroling.2011.02.005

a b s t r a c t

Reading performance has been shown to be linked with balancingskills, possibly indicating an involvement of the cerebellum inreading-related tasks. In our study, we examined whether a similarconnection can be detected for arithmetic performance. Weassessed basic arithmetic skills of 8–10-year-old children (n ¼ 53)and asked them to balance on the left or right foot, with eyes openor closed. Results revealed substantial correlations betweenperformance in arithmetic tasks and in the balancing tasks withclosed eyes even when controlling for attentional and reasoningcapabilities. These findings are interpreted in terms of a cerebellarinvolvement in arithmetic tasks. We propose that verbally medi-ated arithmetic tasks like multiplication might be related tocerebellar functions in different ways than those arithmetic tasksthat require elaboration strategies and quantity manipulations,such as subtraction.

1. Introduction

In 1990, Nicolson and Fawcett claimed that difficulties in learning to read reflect a generalimpairment in automatizing skills. This assumption was derived from the finding that compared to an

86 Frankfurt am Main, Germany. Tel.: þ49 (0)69 24708 224; fax: þ49 (0)69

(J. Lonnemann).

J. Lonnemann et al. / Journal of Neurolinguistics 24 (2011) 592–601 593

age-matched control group, dyslexic children showed difficulties with balancing, but only when per-forming a secondary task (Nicolson & Fawcett, 1990). The authors therefore assumed that in addition toreading, the skill of motor balance is poorly automatized in dyslexic children. Later on, the hypothe-sized automatization deficit was ascribed to a dysfunction in the neuronal circuitries of the cerebellum(cerebellar deficit hypothesis; Nicolson, Fawcett, & Dean, 1995; Nicolson, Fawcett, & Dean, 2001). Theoriginal balancing deficit was interpreted as evidence of a cerebellar dysfunction, as such motordifficulties are characteristic of cerebellar damage (e.g. Holmes,1917). Classic clinical tests of ‘cerebellarsigns’ like dystonia (lowmuscle tone) and dysmetria (difficulty in precisely measuredmovements) alsorevealed impairments in dyslexic children (Fawcett & Nicolson, 1999; Fawcett, Nicolson, & Dean, 1996).In addition, cerebellar lesions can cause deficits in time but not in loudness estimation (Ivry & Keele,1989), in eye-blink conditioning (Maschke et al., 2003), and in adaption to visual-field displace-ments (Baizer, Kralj-Hans, & Glickstein, 1999). Interestingly, similar difficulties have also been detectedin dyslexics (Brookes, Nicolson, & Fawcett, 2007; Nicolson, Daum, Schugens, Fawcett, & Schulz, 2002;Nicolson et al., 1995).

Neuroimaging studies provided further evidence in support of the cerebellar deficit hypothesis:bilateral activations in the cerebellum were found during word reading (Fiez & Petersen, 1998;Turkeltaub, Eden, Jones, & Zeffiro, 2002), and structural cerebellar anomalies were detected indyslexics (Eckert et al., 2003; Leonard et al., 2001; Pernet, Poline, Demonet, & Rousselet, 2009; Raeet al., 1998; 2002). Moreover, it could be demonstrated that cerebellar brain activation was lower fordyslexic adults than for controls when being asked to perform tasks that involve cerebellar structures(i.e. executing a previously learned sequence of finger movements or learning a new sequence,Nicolson et al., 1999).

The cerebellar deficit hypothesis has, however, also been put into question. Denckla, Rudel,Chapman, and Krieger (1985) argued that motor impairments were predominantly found in dyslexicchildrenwho also had attentional difficulties. This claim could be substantiated by studies investigatingmotor skills in carefully controlled samples of dyslexic children. Wimmer, Mayringer, and Raberger(1999) excluded dyslexic children with additional attention-deficit/hyperactivity disorder (ADHD)symptoms from their sample and found no indication of balancing problems. Furthermore, studies inchildren with reading disability and/or ADHD found that balancing difficulties were only (Raberger &Wimmer, 2003) or at least to some extent associated with ADHD (Ramus, Pidgeon, & Frith, 2003).

Nicolson and Fawcett (2007) tried to accommodate these findings by presenting a ‘neural-systems’approach, which is based on a distinction between procedural and declarative learning systems. Whiledeficits in the declarative learning system are assumed to cause generalized learning difficulties,dyslexia and ADHD (at least the inattentive subtype) are both attributed to an impaired procedurallearning system. This system is involved in learning new rule-based procedures and in controllingestablished sensori-motor and cognitive habits. The procedural system assumingly includes thecerebellum as well as prefrontal language areas, basal ganglia, and parietal structures. Motor andlanguage-related components of this system might be affected separately or in combination, resultingin different forms of developmental disorders. Consequently, both dyslexia and ADHD can presumablycome along with or without motor problems.

Even though the neural systems approach by Nicolson and Fawcett (2007) aims at reunitingresearch in developmental disorders by emphasizing commonalities rather than differences betweendifferent disorders, the authors do not take mathematical disabilities into account. This is remarkablebecause disorders in arithmetic frequently co-occur with attentional (e.g. Lindsay, Tomazic, Levine, &Accardo, 1999) as well as with reading disorders (e.g. Landerl & Moll, 2010). The mechanisms under-lying these comorbidities are largely unknown. It has been claimed that dyslexia and dyscalculia haveseparable cognitive profiles, namely a phonological deficit and a deficient number module, and thatthese cognitive deficits are additive in children with comorbid learning disabilities (Landerl,Fussenegger, Moll, & Willburger, 2009). However, the co-occurrence of dyslexia and dyscalculiamight alternatively be traced back to an impaired procedural learning system (Nicolson & Fawcett,2007) possibly involving cerebellar dysfunctions. Indeed, cerebellar activations are found duringarithmetic processes (see Zamarian, Ischebeck, & Delazer, 2009 for an overview). Bilateral activity inthe cerebellum has been reported when contrasting trained versus untrained complex multiplication(Delazer et al., 2003), multi-digit versus single-digit multiplication (Grabner et al., 2007), or when

J. Lonnemann et al. / Journal of Neurolinguistics 24 (2011) 592–601594

skilled abacus users mentally perform one-digit and three-digit number additions (Hanakawa, Honda,Okada, Fukuyama, & Shibasaki, 2003). Moreover, it has been proposed that a cerebellar dysfunctionmay cause difficulties in the acquisition and automatization of basic articulatory and auditory skills aswell as of visual skills such as eye movement and letter recognition (Fawcett et al., 1996). An additionalindirect effect might be that articulation deficits lead to a reduced effectiveness of working memoryfunctions (Nicolson et al., 2001; Ravizza et al., 2006). This wide spectrum of difficulties might not onlyprovoke reading but also arithmetic disorders. Indeed, arithmetic skills seem to be associated with eyemovements (Loetscher, Bockisch, & Brugger, 2008), working memory (e.g. Wilson & Swanson, 2001) aswell as with phonological processing (De Smedt, Taylor, Archibald, & Ansari, 2010). These indications ofcerebellar involvement have, however, not been associated with disorders in arithmetic.

The goal of the present study was to provide behavioural evidence for a possible connectionbetween motor and mathematical skills. To achieve this goal, balancing and mathematical skills of8–10-year-old children were assessed. In order to control for potential confounds, attentional andreasoning capabilities were also assessed for each child. We expected to find significant positivecorrelations between balancing and mathematical skills, even when controlling for attentional andreasoning capabilities. This would underline the notion of a cerebellar involvement in arithmeticprocesses in children.

2. Method

2.1. Participants

Participants were 53 (21 female, 6 left-handed2) children (mean age 9.0, range 8–10 years) recruitedfrom a primary school in Mühlheim am Main (Germany). All participants had normal or corrected tonormal vision. Written and informed consent was obtained from all parents and teachers involved.Based on the measures applied in our study (DEMAT 2þ and SPM Plus, see Materials), none of thechildren was considered as dyscalculic according to the research guidelines of the InternationalClassification of Diseases (ICD-10; WHO, 2006).

2.2. Materials

In order to assess balancing skills, the participating children were asked to balance on the left orright foot, with eyes open or closed. Mathematical skills were examined by the German scholasticachievement test for mathematics (DEMAT 2þ; Krajewski, Liehm, & Schneider, 2004) and by sets ofaddition, subtraction, and multiplication problems. Attentional capabilities were assessed by using thed2 Test of Attention (Brickenkamp, 2002) and reasoning capabilities via Raven’s Standard ProgressiveMatrices Plus (SPM Plus; Horn, 2009). While the balancing and the arithmetic tasks were carried outindividually, the DEMAT 2þ, the d2, and the SPM Plus were conducted in groups of about 20 children.Individual and group testing took place on different days.

2.2.1. Balancing tasksIn the balancing task children had to balance on the right or the left foot with eyes open or closed.

Each of the four conditions (left foot, eyes open; right foot, eyes open; left foot, eyes closed; right foot,eyes closed) was repeated three times with a short break in between trials to ascertain that childrenwere ready to start the next trial. Children started in one of these four conditions, fulfilling the differenteye-conditions in a blockwise manner (i.e. six trials with open eyes followed by six trials with closedeyes or vice versa). The foot-condition was always changed after three trials. Placing the raised footdown ended the trial. Maximum trial duration was 45 s. The experimenter measured the balancingtime by using a stopwatch. The sum of the balancing time of the three trials per condition is reported ins for each child.

2 Since at least some of the children were not able to determine their dominant foot only handedness is reported here.

2.2.2. DEMAT 2þThe DEMAT 2þ is a German core-curriculum-based test to assess children’s school achievement in

mathematics (Krajewski et al., 2004) comprising subscales of arithmetic (e.g. addition, subtraction,multiplication), applied mathematics, and geometry. Total scores, ranging from 0 to 36 are reported foreach child.

2.2.3. Arithmetic tasksThe addition and subtraction problems consisted of nine blocks of ten arithmetical problems (see

Lonnemann, Krinzinger, Knops, & Willmes, 2008); five blocks were addition problems and four blockssubtraction problems. The addition problems were divided into two blocks, in which a single-digitnumber had to be added to a two-digit number with only one of these blocks requiring carrying.Moreover, three blocks contained addition problems, inwhich two two-digit numbers had to be added.In only one of these latter blocks, one of the addends was a decade number. Among the remaining twoblocks without decade numbers, again, only one block required carrying. The subtraction problemswere structured in a similar way: there were two blocks, in which a single-digit number had to besubtracted from a two-digit number and two blocks, which required subtraction of a two-digit numberfrom another two-digit number. In both cases one block required borrowing, while the other one didnot. Children were given 30 s to work on a single block. The multiplication problems consisted of 40problems involving two single-digit numbers. Maximum duration was 10 min. Total scores, rangingfrom 0 to 40 for the multiplication problems, from 0 to 30 for the addition problems without carrying,and from 0 to 20 for the addition problems with carrying, subtraction problems without borrowing, aswell as for the subtraction problems with borrowing were used to assess arithmetic skills. Problemswith and without carrying/borrowing were used because problems involving these processes are morecomplicated and less automatized (e.g. Ashcraft, 1992).

2.2.4. d2 test of attentionThe d2 is a timed test of selective attention, including 14 test lines with 47 characters in each line.

Each character consists of a letter ‘d’ or ‘p’ marked with one, two, three, or four small dashes. Partic-ipants are required to cross out all occurrences of the letter ‘d’with two dashes while ignoring all othercharacters. The concentration performance score (number of the correctly crossed out relevant itemsminus the errors of commission) is reported for each child.

2.2.5. SPM plusRaven’s SPM Plus (Horn, 2009) is an untimed test, consisting of 60 non-coloured diagrammatic

puzzles, each with a missing part, which has to be identified from 6 or 8 options. The SPM Plus containsmore difficult items than the Raven’s SPM. Total scores, ranging from 0 to 60 were used to estimatereasoning capabilities.

3. Results

Descriptive statistics are listed in the lower section of Table 1. Pearson correlation coefficients werecomputed to look for possible associations between the different variables. Intercorrelation matricesare shown in the upper section of Table 1. While age was not related to any of the variables, SPM Plusscores were found to be positively correlated with all of the variables except for the balancing tasksinvolving the right foot. The concentration performance (d2) was also positively correlated with nearlyall variables but not with the subtraction task with borrowing and the task of balancing on the rightfoot with closed eyes. As expected, performance in the different balancing tasks as well as performancein the different arithmetic tasks was significantly correlated and significant correlations were foundbetween the arithmetic tasks and the DEMAT 2þ scores. Most importantly, significant positivecorrelations between measures of mathematical skills and performance in the balancing tasks werefound: DEMAT 2þ scores were related to balancing on the left foot with open eyes and multiplicationtasks were correlated with all of the balancing tasks. Moreover, addition tasks without carrying werecorrelated with all balancing tasks except for balancing on the right foot with closed eyes, subtractiontasks with borrowing were correlated with both balancing tasks (left and right foot) with closed eyes,

Table 1Pearson correlation coefficients and descriptive statistics (minimum, maximum, theoretical range, mean (M), standard deviation (SD)) for the observed variables.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

1. Age –

2. SPM Plus �.17 –

3. d2 �.10 .44** –

4. DEMAT 2þ �.18 .64** .25* –

5. Addition without carrying .09 .38** .43** .57** –

6. Addition with carrying �.03 .49** .30* .60** .80** –

7. Subtraction without borrow. �.01 .35** .27* .53** .68** .66** –

8. Subtraction with borrowing �.08 .40** .21 .50** .56** .68** .69** –

9. Multiplication �.22 .38** .29* .60** .43** .50** .44** .47** –

10. Balancing left open �.04 .25* .28* .25* .24* .17 .08 .21 .31* –

11. Balancing right open �.06 .19 .31* .18 .23* .06 .24* .20 .25* .78** –

12. Balancing left closed .00 .25* .36** .17 .20* .17 .11 .34** .37** .51** .41** –

13. Balancing right closed �.07 .07 .14 .12 �.02 �.02 .09 .26* .36** .50** .46** .70** –

Minimum (empirical) 101 months 10 �46 3 1 0 0 0 24 18 s 16 s 6 s 6 sMaximum (empirical) 130 months 34 177 36 29 16 17 13 40 135 s 135 s 94 s 97 sTheoretical range – 0–60 – 0–36 0–30 0–20 0–20 0–20 0–40 0–135 0–135 0–135 0–135M 109 months 24 108 25 18 8 10 5 36 94 99 28 28SD 5.3 months 5.4 44.7 7.7 5.8 3.7 4.2 3.1 3.7 36.9 34.7 22.0 23.4

*p < 0.05, **p < 0.01 (one-sided); n ¼ 53.

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and subtraction tasks without borrowing were found to be related to balancing on the right foot withopen eyes.

In order to control for potential confounds, partial correlations were computed. Age, reasoning(SPM Plus), and concentration (d2) were used as control variables. First order partial correlations foreach of these variables as well as third order partial correlations involving all three variables arereported. In all cases, performance in the different balancing tasks as well as performance in thedifferent arithmetic tasks was still significantly correlated and correlations between the differentarithmetic tasks and the DEMAT 2 þ scores also remained significant. Controlling for age had nosubstantial effect on the correlations between the variables representing mathematical skills andperformance in the balancing tasks (see Table 2).

Controlling for reasoning (SPM Plus), however, changed the pattern of results. As can be seen inTable 3, the correlation between the DEMAT 2þ scores and balancing on the left foot with open eyes aswell as correlations between addition tasks without carrying and the different balancing tasks declinedand they were no longer significant. Additionally, none of the correlations involving the task ofbalancing on the right foot with open eyes reached the conventional level of significance (alpha ¼ 5%).Significant correlations remained for the multiplication tasks and the different balancing tasks,whereby the correlation between the multiplication tasks and balancing on the right foot with openeyes was only marginal (p ¼ 0.082, one-sided). Correlations between the subtraction tasks withborrowing and both balancing tasks (left and right foot) with closed eyes also remained significant. Asimilar pattern of results was found when using concentration (d2) as control variable (see Table 4).Finally, when controlling for age, reasoning (SPM Plus), and concentration (d2), substantial correlationsremained between the balancing tasks with closed eyes and multiplication as well as subtraction taskswith borrowing (see Table 5).3

4. Discussion

The present study aimed to deliver behavioural evidence for a relationship between balancing andmathematical skills in children. Positive correlations between different measures of mathematicalcompetence and balancing skills could be detected. Controlling for reasoning and concentrationrevealed the importance of differentiating between balancing tasks with eyes open or closed as well asbetween different arithmetic tasks. Balancing with open eyes was found to be related to the perfor-mance in a mathematical school achievement test (DEMAT 2þ), to multiplication performance, as wellas to performance in addition and subtraction tasks without carrying/borrowing. When controlling forreasoning and concentration capabilities, however, these correlations declined substantially. Bothreasoning and concentration capabilities were positively correlated with arithmetic and balancingskills, whereby reasoning was more strongly related to mathematical skills and concentration corre-spondedmore strongly to balancing skills. The relationship between concentration and balancing skillsis in line with findings showing that attentional difficulties and motor impairments can occur incombination (e.g. Denckla et al., 1985). As a result, balancing with open eyes seems to be related toattentional mechanisms rather than to mathematical skills.

Performance in the balancing tasks with closed eyes revealed different results. For these tasksconnections with performance in the multiplication and in the subtraction tasks with borrowing werefound, which did not substantially decline when controlling for reasoning and concentration capa-bilities. Hence, connections between balancing and mathematical tasks primarily occurred for ‘purer’balancing tasks that can scarcely be controlled by volition.

With respect to measures of mathematical competence, our study revealed that substantialcorrelations with balancing tasks can be found for performance in multiplication and in subtraction

3 In order to make the different sets of arithmetic tasks comparable with respect to the number of problems, only the first 20multiplication problems as well as 20 addition problems without carrying (skipping the block, in which two-digit numbers anddecade numbers had to be added as there was no comparable block for subtraction problems) were considered in a supple-mentary correlation analysis. Results mainly support the findings of the analysis using a different number of problems withinthe different sets (see Appendix).

Table 2Partial correlation coefficients for the mathematical school achievement test (DEMAT 2þ), the balancing tasks, and the arith-metic tasks with age as control variable.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

1. DEMAT 2þ –

2. Addition without carrying .60** –

3. Addition with carrying .61** .81** –

4. Subtraction without borrow .54** .69** .66** –

5. Subtraction with borrowing .49** .57** .68** .69** –

6. Multiplication .58** .46** .50** .45** .47** –

7. Balancing left open .24* .25* .17 .08 .21 .30* –

8. Balancing right open .17 .24* .06 .24* .20 .24* .78** –

9. Balancing left closed .18 .25* .17 .11 .34** .38** .51** .41** –

10. Balancing right closed .11 �.02 �.02 .09 .26* .35** .50** .46** .70** –

*p < 0.05, **p < 0.01 (one-sided); n ¼ 53.

tasks with borrowing but not for less complex addition and subtraction tasks. Interestingly, learning tosolve complex arithmetic problems has been shown to involve cerebellar structures in adults (e.g.Delazer et al., 2003), giving reason to assume that our findings represent evidence of a cerebellarinvolvement in arithmetic tasks. Assigning the ideas of the cerebellar deficit hypothesis of dyslexia(Nicolson et al., 2001) to the domain of arithmetic could mean that initial motor impairments not onlycause reading but also arithmetic difficulties. For instance, if cerebellar deficits provoke problems in theacquisition and automatization of basic articulatory and auditory skills, this might interfere with theacquisition of number words and with the verbal manipulation of numbers, which is presumablyimportant for arithmetic operations that require access to a rote verbal memory of arithmetic facts,such as multiplication (Dehaene, Piazza, Pinel, & Cohen, 2003). This might explain our finding ofa relationship between balancing and multiplication skills. Subtraction tasks that involve borrowing,however, are not assumed to place particularly strong demands on a verbal coding of numbers butseem to be solved by elaboration strategies requiring some form of quantity manipulation (Dehaeneet al., 2003). These kinds of arithmetic operations might be related to cerebellar functions inanother way, namely via visual skills such as eye movements: recent findings indicate that arithmeticprocesses are accompanied by saccadic eye movements (Loetscher et al., 2008). Indeed, neuralcircuitries that are involved in eye movements seem to be recruited for mental arithmetic. Whileaddition seems to be associated with rightward eye movements, subtraction seems to involve leftwardeye movements (Knops, Thirion, Hubbard, Michel, & Dehaene, 2009). If cerebellar deficits causedifficulties in the acquisition and automatization of basic visual skills such as eye movements (Fawcettet al., 1996), this might in turn complicate the development of procedures that are necessary to solvearithmetic problems and potentially explain our finding of a connection between balancing andsubtraction skills. Alternatively, this finding might also be explained by a reduced effectiveness of

Table 3Partial correlation coefficients for the mathematical school achievement test (DEMAT 2þ), the balancing tasks, and the arith-metic tasks with reasoning (SPM Plus) as control variable.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

1. DEMAT 2þ –

6. Multiplication .49** .33** .39** .36** .38** –

7. Balancing left open .12 .17 .06 �.01 .13 .24* –

8. Balancing right open .07 .17 �.04 .19 .14 .20 .77** –

9. Balancing left closed .02 .18 .06 .03 .27* .31* .48** .38** –

*p < 0.05, **p < 0.01 (one-sided); n ¼ 53.

Table 4Partial correlation coefficients for the mathematical school achievement test (DEMAT 2þ), the balancing tasks, and the arith-metic tasks with concentration (d2) as control variable.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

1. DEMAT 2þ –

6. Multiplication .56** .35** .45** .40** .44** –

7. Balancing left open .19 .14 .09 .01 .16 .25* –

9. Balancing left closed .09 .11 .07 .02 .29* .30* .46** .34** –

*p < 0.05, **p < 0.01 (one-sided); n ¼ 53.

working memory functions possibly provoked by cerebellar impairments (Nicolson et al., 2001;Ravizza et al., 2006), because working memory seems to play a key role in subtraction tasks thatrequire borrowing (e.g. Imbo, Vandierendonck, & Vergauwe, 2007). At the same time, it raises thequestion of why therewas no substantial relation between balancing and addition skills. This, however,seems due to the fact that addition tasks without carrying were related to attentional and additiontasks with carrying to reasoning capabilities rather than to balancing skills, possibly indicating thataddition procedures might have been already automatized in children in our sample. Future studiesshould therefore examine relations between balancing and arithmetic skills in different age groups inorder to check whether these connections are age-dependent.

In sum, the present study revealed a relationship between balancing and arithmetic skills thatmight be interpreted in terms of a cerebellar involvement in arithmetic tasks. Indeed, connectionsbetween balancing and arithmetic can be partly ascribed to reasoning and/or attentional mechanisms,but this does not seem to be the whole story. Even after controlling for these variables, substantialrelationships remained. Nevertheless, the test we used to measure attentional capabilities might nothave been sufficiently comprehensive. Attentional difficulties that have been associatedwith cerebellarimpairments (e.g. Denckla et al., 1985), can involve not only inattention but also hyperactivity, whichcannot be appropriately assessed by the d2 test of attention. The potential confound, which is based onthe comorbid occurrence of attentional and arithmetic difficulties, should have been eliminated,however, because children with arithmetic difficulties are predominantly found to be inattentive butdo not display hyperactivity (Lindsay et al., 1999). In order to substantiate our findings, future studiesshould investigate the relationship between balancing and arithmetic skills in carefully controlledsamples of dyscalculic children without ADHD.

Since we only report correlations here, strong conclusions cannot be drawn: balancing and arith-metic skills of children seem to be linked. Thismight be due to a cerebellar involvement in both of these

Table 5Partial correlation coefficients for the mathematical school achievement test (DEMAT 2þ), the balancing tasks, and the arith-metic tasks with age, reasoning (SPM Plus), and concentration (d2) as control variables.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

1. DEMAT 2þ –

6. Multiplication .50** .35** .40** .36** .38** –

7. Balancing left open .13 .11 .04 �.04 .12 .22 –

9. Balancing left closed .04 .08 .03 �.02 .27* .30* .45** .33** –

*p < 0.05, **p < 0.01 (one-sided); n ¼ 53.

proficiencies. However, verbally mediated arithmetic tasks like multiplication might be related tocerebellar functions in different ways than such arithmetic tasks as require elaboration strategies andquantity manipulations, like subtraction. Further evidence is needed to substantiate this claim. In orderto do so, future studies should include other cerebellar tasks possibly involving eye movements as wellas measures of potential mediating factors like verbal skills or working memory. If the findings of thepresent study are supported by future research, they may provide important implications for thediagnosis and the prevention of dyscalculia.

Acknowledgements

Wewould like to thank all the participating children and teachers. Moreover, the first author wouldlike to thank Ernst Prasse for his inspiration. This research was funded by the Hessian initiative for thedevelopment of scientific and economic excellence (LOEWE).

Appendix

Table A.1Partial correlation coefficients for the mathematical school achievement test (DEMAT 2þ), the balancing tasks, and the arith-metic tasks with age, reasoning (SPM Plus), and concentration (d2) as control variables. Only 20 problems were considered foreach of the different sets of arithmetic tasks.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

1. DEMAT 2þ –

6. Multiplication .43** .35** .33* .28* .28* –

7. Balancing left open .13 .04 .04 �.04 .12 .20 –

9. Balancing left closed .04 .04 .03 �.02 .27* .21 .45** .33** –

10. Balancing right closed .09 �.06 �.07 .06 .26* .26* .49** .44** .71** –

*p < 0.05, **p < 0.01 (one-sided); n ¼ 53.

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