Symbolic and non-symbolic distance effects in children and their connection with arithmetic skills

Journal of Neurolinguistics 24 (2011) 583–591

Contents lists available at ScienceDirect

Journal of Neurolinguisticsjournal homepage: www.elsevier .com/locate/

jneurol ing

Symbolic and non-symbolic distance effects in children andtheir connection with arithmetic skills

Jan Lonnemann a,b,1,*, Janosch Linkersdörfer a,b,1, 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:Numerical cognitionMagnitude comparisonDistance effectSymbolic number representationNon-symbolic number representationArithmetic skills

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

E-mail address: [email protected] Shared first author.

0911-6044/$ – see front matter � 2011 Elsevier Ltdoi:10.1016/j.jneuroling.2011.02.004

a b s t r a c t

The ability to compare numerical magnitudes is assuminglyrelated to children’s arithmetic skills. The role of symbolic andnon-symbolic number representations in this relationship is,however, still a matter of debate. To address this issue we assessedaddition and subtraction skills of 8–10-year-old children (n ¼ 35)and asked them to compare numerical magnitudes of dot patternsand Arabic digits in different numerical ranges. Results revealedthat the relationship between numerical magnitude comparisonsand arithmetic skills is not restricted to symbolic stimuli, but thatit can also be detected for non-symbolic dot patterns. The range ofnumerosities for which this relationship was found and themanner in which the magnitude comparison was related toarithmetic skills differed regarding the dots and digits. Thesefindings highlight the role of both symbolic and non-symbolicnumber representations in the development of arithmetic skillsand strengthen the view of different developmental trajectoriesunderlying these representations.

� 2011 Elsevier Ltd. All rights reserved.

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

(J. Lonnemann).

d. All rights reserved.

J. Lonnemann et al. / Journal of Neurolinguistics 24 (2011) 583–591584

1. Introduction

Magnitude comparison is a crucial ability for everyday life that we share with other species (e.g.Brannon, 2005).Human infants can compare sets representedbydot arrayson thebasis of number. At sixmonths of age children seem to be sensitive to the difference between 8 and 16, but fail to discriminatebetween 8 and 12 dots (Xu & Spelke, 2000). Similar to our performance in discriminating other physicaldimensions like line length or pitch (e.g. Henmon,1906) comparing numerical magnitudes depends onthe proportion by which magnitudes differ. We are faster and more accurate in comparing dot arrayswith respect to their magnitude the further apart they are (e.g. van Oeffelen & Vos,1982). This so-calleddistance effect (DE) is also found for symbolic number magnitude representations like Arabic numerals(Moyer& Landauer,1967),which is ascribed to amappingof symbolic representations ontoapproximatenon-symbolic number magnitude representations (Verguts & Fias, 2004). In contrast, it has beenproposed that non-symbolic and symbolic number representations might draw on different processes(Zorzi & Butterworth, 1999). According to this view, the DE emerges as a consequence of a nonlineardecision process rather than from an approximate representation of numerical magnitude.

Sekuler and Mierkiewicz (1977) reported a DE for Arabic numerals in children. The strength of thiseffect was found to decreasewith age (see also Holloway & Ansari, 2008).While the effect’s strength forolder children (grades four and seven) was comparable to that of adults, it was more pronounced foryounger children (kindergarten and first grade). The decreasing size of the DE may represent anincrease in the precision of children’s numerical representations (Holloway & Ansari, 2009). Thesenumerical magnitude representations assumingly serve as a foundation on which mathematicalcompetences like arithmetic skills are built (Butterworth, 2005). In three recent studies, the associationbetween symbolic as well as non-symbolic DE’s and mathematical skills was explored (Holloway &Ansari, 2009; Mussolin, Mejias, & Noël, 2010; Rousselle & Noel, 2007). Rousselle and Noel (2007)demonstrated that second graders with mathematical disabilities are impaired regarding symbolicbut not with regard to non-symbolic numerical magnitude comparison. The authors claimed thatchildren with mathematical disabilities do not have difficulties in processing numerical magnitudesper se but rather in accessing the meaning of symbolic numerals. These findings are similar to the onesby Holloway and Ansari (2009), reflecting a relationship between symbolic, but not non-symbolicnumerical magnitude comparison and mathematical skills in six to eight year old children. It isimportant to note a difference between these two studies with respect to the manner in which the DEwas related to performance on tests of mathematical competence. While children with mathematicaldisabilities had smaller distance effects than typically developing children (Rousselle & Noel, 2007),children with lower mathematical skills were shown to have larger DE’s in the study by Holloway andAnsari (2009). Rousselle and Noel (2007) assumed that children with mathematical disabilities usedpeculiar strategies for comparing Arabic digits in order to compensate for their impaired ability inextracting the meaning from Arabic numerals. On the other hand, Holloway and Ansari (2009) claimedthat children with less efficient, but not impaired, strategies to access the meaning of numericalsymbols show relatively lower mathematical skills as well as larger DE’s. In sum, both findings seem toconverge on the access deficit hypothesis by Rousselle and Noel (2007), implying that the efficiencywith which children access and use the meaning of symbolic numerals is related to their mathematicalcompetence. Objection to this interpretation has been raised by Cohen Kadosh andWalsh (2009), whoclaimed that a better mapping between a numerical symbol and its meaning can explain overall fasterreaction times in childrenwith better mathematical achievement, but cannot explain differences in theDE, as the symbolic DE occurs at a point in time (e.g. at the level of representation or during responseselection) when such a mapping should already have taken place. In a recent study, however, Mussolinet al. (2010) revealed that childrenwith mathematical disabilities showed a greater numerical distanceeffect than control children, irrespective of the number format, favoring the idea of a representationrather than an access deficit in children with dyscalculia.

An important issue that has to be addressed when comparing symbolic and non-symbolicmagnitude comparison tasks is that qualitatively different mechanisms seem to be involved inapprehending dot patterns of different quantities. While estimation or counting processes are involvedin processing large sets of dots, quantities from 1 to 3 are supposedly apprehended by automaticquantification processes called ‘subitizing’ (e.g. Trick & Pylyshyn,1994). The performance pattern in the

J. Lonnemann et al. / Journal of Neurolinguistics 24 (2011) 583–591 585

subitizing range differs from that observed with large numerosities, because it does not depend on thenumerical ratio but on the absolute difference between different magnitudes (Feigenson, Dehaene, &Spelke, 2004). It has been argued that subitizing may be crucial for the acquisition of an exactrepresentation of number (Le Corre & Carey, 2007) and may underlie dyscalculia (Landerl, Bevan, &Butterworth, 2004). Indeed, Koontz and Berch (1996) found that children with dyscalculia appearedto use time-consuming counting strategies rather than subitizing in a dot-matching task, suggestingthat these basic capacities could be tied to the understanding of numerosity. As a result, when seekingto investigate associations between non-symbolic DE’s and mathematical skills, different subsets ofquantities have to be used to avoid interference between subitizing and other quantification processes.This issuewas not addressed by either of the three recent studies examining these kinds of associations(Holloway & Ansari, 2009; Mussolin et al. (2010); Rousselle & Noel, 2007). While Rousselle and Noel(2007) only used numerosities above the subitizing range, Holloway and Ansari (2009) as well asMussolin et al. (2010) used an intermixed stimulus set that did not allow for running set-specificanalyses. For this reason the present study aims to investigate the relationship between arithmeticskills and magnitude comparisons within and beyond the subitizing range.

2. Method

2.1. Participants

Participants were 35 (18 female) Chinese children (mean age 8.8, range 8–10 years) recruited froma primary school in Shanghai (China). All participants had normal or corrected to normal vision.

2.2. Materials

Magnitude comparison tasks were used to assess DE’s in children and arithmetic skills wereexamined by sets of addition and subtraction problems. All tasks were carried out individually, whichtook about 1 h per child.

In the magnitude comparison tasks either two Arabic digits or two dot patterns were presented onscreen. The two stimuli were arranged in a horizontal fashion. Children had to indicate the side withthe larger numerical magnitude by answering with the left index finger when it was larger on the lefthand side and by using the right index finger when it was larger on the right hand side. Responses weregiven via the ‘S’ and ‘L’ buttons of a notebook keyboard. Comparison pairs varied along four numericaldistances (whole range of distances). In order to account for the involvement of qualitatively differentmechanisms in processing small and large numerosities, subsets of comparison pairs including onlynumerical magnitudes from 1 to 3 (range 1–3) and pairs ranging only from 4 to 6 (range 4–6) wereused. The dot patterns were displayed with a black frame and only one configuration per magnitudewas used to assure comparability with the digits. The different comparison pairs are shown in Table 1and the dot patterns are illustrated in Fig. 1.

Each of the 12 comparison pairs presented in Table 1 appeared eight times, four times in ascendingand four times in descending order. Arabic digits were presented in Times 60-point font, according towhich dot patterns were approximately matched in size (visual angle of 1.7� in height from a viewingdistance of about 60 cm). Reaction times (RT) and error rates (ER) were recorded and the instruction

Table 1Comparison pairs for the different numerical distances with pairs ranging only from 1 to 3 (bold font) andpairs ranging only from 4 to 6 (italics font).

Distance

1 2 3 4

1–2 1–3 2–5 1–52–3 2–4 3–6 2–64–5 3–5 – –

5–6 4–6 – –

Fig. 1. Illustration of the different dot patterns.

J. Lonnemann et al. / Journal of Neurolinguistics 24 (2011) 583–591586

stressed both speed and accuracy. The trials were pseudo-randomized so that there were no consec-utive identical comparison pairs and numerical distance was not identical on more than threeconsecutive trials. The experiment was preceded by six warm-up trials to familiarize participants withthe task (data not recorded) and controlled by a notebook with Presentation� software (Neuro-behavioral Systems, Inc.). Black-colored targets were presented on a 17" colour screen against a whitebackground. A target stimulus appeared until the response, but only up to a maximum duration of4000 ms, and was followed by a black screen for 700 ms.

The addition and subtraction problems consisted of nine blocks of ten arithmetical problems (seeLonnemann, 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 (e.g. 82 þ 5) with only one of these blocks requiringcarrying (e.g. 43 þ 9). Moreover, three blocks contained addition problems, in which two two-digitnumbers had to be added (e.g. 24 þ 65). In only one of these latter blocks, one of the addends wasa decade number (e.g. 68 þ 30). Among the remaining two blocks without decade numbers, again,only one block required carrying (e.g. 13 þ 88). The subtraction problems were structured in a similarway: there were two blocks, in which a single-digit number had to be subtracted from a two-digitnumber (e.g. 25�3) and two blocks, which required subtraction of a two-digit number from anothertwo-digit number (e.g. 76 � 23). In both cases one block required borrowing (e.g. 54 � 7 or 82 � 45),while the other one did not. Children were given 30 s to work on a single block. Total scores, rangingfrom 0 to 50 for the addition problems and from 0 to 40 for subtraction problems, were used toestimate arithmetic skills.

3. Results

To evaluate data from the administration of the magnitude comparison tasks, we conductedrepeated measures analyses of variance (ANOVAs) and post-hoc tests for further RT investigation. ERwas low (2.2%) and therefore not analyzed. The Huynh–Feldt epsilon (Huynh & Feldt, 1976) wascomputed to correct the degrees of freedom of the F-statistics in case of significant (alpha ¼ 10%) non-sphericity. Only correct responses were used for calculating mean RT. Trials in which no responseoccurred were classified as errors. Responses below 200 ms were excluded from further analysis, aswell as responses outside an interval of�3 standard deviations around the individual mean. Trimmingresulted in 1.7% of response exclusions. Two participants were excluded from further analyses becausethey had given only incorrect answers in one of the different conditions.

Pearson correlation coefficients were employed to look for possible relations between the size ofthe DE and arithmetic skills. To assess the DE, mean difference values were computed for eachparticipant with positive values indexing the DE. RT for comparisons with large numerical distances(mean RT for distances 3 and 4 for thewhole range of distances/mean RT for distance 2 for range 1–3 or4–6) was subtracted from those with small numerical distances (distance 1) separately for the twodifferent magnitude comparison tasks. These difference values were then divided by RT for largedistance comparisons (mean RT for distances 3 and 4 for the whole range of distances/mean RT fordistance 2 for range 1–3 or 4–6) for each child to yield a measure that accounts for individual differ-ences in RT (see Holloway & Ansari, 2009). All effects were tested by using a significance level ofalpha ¼ 5%.

J. Lonnemann et al. / Journal of Neurolinguistics 24 (2011) 583–591 587

3.1. Magnitude comparison

An ANOVA including the within-subject factors distance (1, 2, 3, 4)2 and stimulus set (dots, digits)revealed significant main effects for distance (F(3, 96) ¼ 79.81, p < .001, partial eta-squared ¼ .71,epsilon¼ .83) and for stimulus set (F(1, 32)¼ 62.65,p< .001, partial eta-squared¼ .66). Themaineffect fordistance was characterized by faster RT for large compared to small distances, while the main effect forstimulus set showed a faster mean RT for digits compared to dots. Additionally, a significant interactionwas found between the distance and stimulus set (F(3, 96) ¼ 27.99, p < .001, partial eta-squared ¼ .47,epsilon ¼ .87). Separate ANOVAs for the two different magnitude comparison tasks revealed significantmain effects for distance in both stimulus set conditions (dots: F(3, 96) ¼ 63.87, p < .001, partial eta-squared ¼ .67, epsilon ¼ .82; digits: F(3, 96) ¼ 21.86, p < .001, partial eta-squared ¼ .41, epsilon ¼ .77).Significant main effects for stimulus set were also found for all different distances (distance 1: F(1,32) ¼ 90.77, p < .001, partial eta-squared ¼ .74; distance 2: F(1, 32) ¼ 56.76, p < .001, partial eta-squared¼ .64; distance 3: F(1, 32)¼ 43.82, p< .001, partial eta-squared¼ .58; distance 4: F(1, 32)¼ 16.08,p< .001, partial eta-squared¼ .33). Hence, distance effectsweremore pronounced for dots and the effectsof the stimulus set increased with decreasing distance (see Fig. 2a). To illustrate the distribution of thestimulus-specific DE’s over the whole range of latencies, plots with the cumulative density function fordeciles (Ridderinkhof, 2002) are provided in Fig.1d/e. Visual inspection of the graphs reveals that the DE’smainly arose between the fifth and the ninth response deciles for both stimulus formats.

In order to look for the involvement of different mechanisms in processing small and large numer-osities, anotherANOVA including thewithin-subject factors distance (1, 2)3, stimulus set (dots, digits), andrange (1–3, 4–6) was conducted. Significant main effects for distance (F(1, 32) ¼ 125.88, p < .001, partialeta-squared ¼ .80), for stimulus set (F(2, 64) ¼ 96.37, p < .001, partial eta-squared ¼ .75), and for range(F(1, 32) ¼ 57.88, p < .001, partial eta-squared ¼ .64) were found. The main effect for distance was char-acterized by faster RT for large distances, themain effect for stimulus set showed fastermeanRT for digits,and themaineffect for rangewascharacterizedby fasterRT for range1–3.Significant interactionsbetweendistance and stimulus set (F(1, 32) ¼ 36.66, p < .001, partial eta-squared ¼ .53), distance and range(F(1, 32) ¼ 21.57, p < .001, partial eta-squared ¼ .40), stimulus set and range (F(1, 32) ¼ 29.78, p < .001,partial eta-squared ¼ .48), and distance, stimulus set, and range (F(1, 32) ¼ 14.20, p ¼ .001, partial eta-squared¼ .31)were found. Further analyses attributed thefindings to the fact that in the caseof large (¼2)distances the main effect for range was significant for dots (F(1, 32) ¼ 33.21, p < .001, partial eta-squared ¼ .51), but only marginal for digits (F(1, 32) ¼ 4.10, p ¼ .051, partial eta-squared ¼ .11). All othermain effects reached significance for the different conditions, see Fig. 2b/c for a depiction of these results.

3.2. Associations between the DE and arithmetic skills

Intercorrelations of the different variables are shown in Table 2. For both magnitude comparisontasks the DE for the whole range of distances was positively correlated with the DE for the range 4–6,but not for the range 1–3. A negative correlationwas found between the DE (whole range) for dots andthe DE (range 1–3) for digits. In line with our expectations, we found significant correlations betweenthe performance in the addition and in the subtraction task. Associations between the DE and arith-metic skills were reflected in a negative correlation between the subtraction task and the DE (range4–6) for digits on the one hand, and in a positive correlation between the subtraction task and the DE(range 1–3) for dots on the other hand (see Fig. 3).

4. Discussion

We used dot patterns and Arabic digits in the same numerical range to assess non-symbolic andsymbolic numerical magnitude representations in children. DE’s were found for both of the different

2 Note that out of the 96 trials, 32 trials included the distances 1 or 2 while only 16 trials included the distances 3 or 4, seeTable 1.

3 Note that 16 trials included the distance 1 while only 8 trials included the distance 2 for each of the two ranges, see Table 1.

Fig. 2. Distance effects and cumulative density functions for dots and digits. (a) Mean reaction times for correct responses in msseparately for the different stimulus sets (dots, digits) as a function of the factor distance (1, 2, 3, 4). (b and c) Mean reaction times forcorrect responses in ms separately for dots (b) and digits (c) as a function of the factors distance (1, 2) and range (1–3, 4–6). (d and e)Cumulative density functions for dots (d) and digits (e) as a function of the factor distance (1 versus the mean of 3 and 4). Thecumulative probability of responding is plotted as a function of mean reaction times (only correct responses) for each of ten responsespeed deciles.

J. Lonnemann et al. / Journal of Neurolinguistics 24 (2011) 583–591588

stimulus sets. Comparing dot stimuli took significantly longer than the comparison of digits. This mightbe due to the fact that representations of digits are more distinct than those of dot stimuli, making theprocess of magnitude comparison easier (Holloway & Ansari, 2009). To look for the involvement ofdifferent mechanisms in processing dot patterns of different quantities, comparison pairs that onlyincluded numerical magnitudes within a small (1–3) and a large (4–6) range were analyzed separately.Effects of these different ranges were predominantly found for dots. While small subsets of dots mighthave been apprehended by subitizing processes (e.g. Trick & Pylyshyn, 1994), more time-consumingcounting processes might have been used to process larger subsets of dots. It is important to note thatthe use of non-numerical perceptual cues would have been sufficient to make correct decisions whencomparing dot patterns, since numerical magnitude was correlated with density in the stimulus set ofthe present study. In this case, however, latencies should have been comparatively fast, because thenumerical magnitudes represented by the different dot patterns would not have been processed.Significantly longer latencies for dots as compared to digits, therefore, rule out the possibility that theparticipants relied on these cues.

Associations between the DE’s for the different stimulus sets and arithmetic skills were reflected ina negative correlation between the subtraction task and the DE (range 4–6) for digits as well as ina positive correlation between the subtraction task and the DE (range 1–3) for dots. The formercorrelation is in line with previous findings (Holloway & Ansari, 2009), showing a connection betweensymbolic distance effects and mathematical skills. In our study, we only found this association for thelarger range (4–6) of Arabic digits. Using numerosities ranging from 1–9, Holloway and Ansari (2009)demonstrated that this kind of relationship primarily exists in 6-year-olds and is diminished at the ageof 8 years. They therefore suggested that it would be more apt to use two-digit numbers in order toassess DE’s to reveal connections with mathematical skills in older children. Our results show that thisassociation can still be captured in 8-10-year-olds but only when small numerosities (1–3) are notconsidered.

In addition, we also found a connection between arithmetic skills and the DE in the non-symboliccomparison task with dots. Interestingly, this relationship was only found in the subitizing range (1–3)

Table 2Pearson correlation coefficients between the different DE’s and performance in the arithmetic tasks.

Measure 1 2 3 4 5 6 7 8

1. DE digits (whole range) – .21 .40* �.01 �.02 .12 �.27 �.122. DE digits (range 1–3) – .18 �.38* �.13 �.02 �.10 .103. DE digits (range 4–6) – �.14 �.23 �.12 �.23 �.35*4. DE dots (whole range) – .21 .73** �.04 .015. DE dots (range 1–3) – .04 .13 .47**6. DE dots (range 4–6) – .03 �.047. Addition

(theoretical range: 0–50; empirical range: 32–50; M: 45; SD: 4.3)– .57**

8. Subtraction(theoretical range: 0–40; empirical range: 23–40; M: 32; SD: 5.1)

–

*p < .05, **p < .01 (two-sided); n ¼ 33.

J. Lonnemann et al. / Journal of Neurolinguistics 24 (2011) 583–591 589

and, moreover, children showing a better performance in the subtraction task tended to be those withlarger DE’s. Based on the notion that the size of the DE decreases with increasing precision of theunderlying representations (Holloway & Ansari, 2009), it seems likely that children with bettersubtraction skills represented dots in this range in a less distinct way. This in turn might imply thatthese children more strongly relied on subitizing when asked to compare dots in the range of 1–3.Children with relatively low subtraction skills, on the other hand, might have used more precisestrategies like verbalizing the quantities. If so, then children with better subtraction skills should havebeen faster when asked to compare dot patterns in the subitizing range. Indeed, the correlationbetween subtraction skills and RT for the comparison of dot stimuli (range 1–3) was negative but notsignificant (r ¼ �.10; p ¼ .57, two-sided). Seemingly, the use of more precise strategies did not requiremuch more time than subitizing one to three dots. All in all, this line of interpretation agrees with theidea that basic perceptual skills like subitizing may be tied to arithmetic skills (e.g. Landerl et al., 2004).

While differences in the ability to subitize 1–3 dotsmay lie at the bottom of the connection betweennon-symbolic DE’s and subtraction skills, the mechanism underlying the relationship betweensymbolic DE’s and subtraction skills is not clear. It might be due to a difference in the efficiency withwhich children map between a numerical symbol and its meaning (see Rousselle & Noel, 2007).However, as Cohen Kadosh and Walsh (2009) argued, the symbolic DE occurs at a point in time whensuch a mapping should already have taken place. In support of this view, cumulative density functionplots (Ridderinkhof, 2002) demonstrated that the DE’s for digits as well as for dots mainly arose forrelatively late responses. One might assume that differences in the magnitude representation per semight play a role. This, however, can only be true when magnitude representations for digits and dotsare independent from each other because otherwise effects for symbolic and for non-symbolic DE’sshould be comparable. As symbolic and non-symbolic DE’s were found to be uncorrelated (or evennegatively correlated) in the present study, we suggest that our findings support the view that differentdevelopmental trajectories underlie the representation of symbolic and non-symbolic numericalmagnitude (Cohen Kadosh & Walsh, 2009). Evidence for this idea comes from neuroimaging studiesindicating that numerical representations for dots and digits are coded by separate neural circuits(notation-specific effects in left intraparietal areas; Piazza, Pinel, Le Bihan, & Dehaene, 2007).Furthermore, single cell recordings in monkeys also showed notation-dependent activity for dots anddigits in the parietal cortex (Diester & Nieder, 2007).

Fig. 3. Correlation between the performance in the subtraction task (raw score totals, theoretical range: 0–40) and the DE separatelyfor dots (range 1–3) and for digits (range 4–6).

J. Lonnemann et al. / Journal of Neurolinguistics 24 (2011) 583–591590

We were able to capture the aforementioned findings by analyzing connections between differentsubsets of magnitudes and different arithmetic types. As the present study reveals, it is important todifferentiate the stimulus set when assessing arithmetic skills. Although the performance in theaddition and in the subtraction task was positively correlated, connections with the DE’s were onlyfound for the subtraction task. This might be due to the fact that the subtraction task was better suitedto separate good and bad calculators because it is less automated and requires more numeracy skillsthan addition. Moreover, differentiating between subsets of magnitudes was also effective. For bothmagnitude comparison tasks the DE for thewhole range of distances was positively correlatedwith theDE for the range 4–6 while it was never associated with the small range (1–3). In the case of dot stimulithis pattern of results is conceivable because qualitatively different processes seem to be involvedwhen apprehending stimuli in the different ranges. For Arabic digits, on the contrary, these effects ofdifferent numerical ranges might be caused by the fact that numbers from 1 to 3 are much morefrequent and therefore easier to process than numbers beyond this range.

In conclusion, results from our study revealed that numerical magnitude comparisons with dotsand digits are related to arithmetic skills in different ways. The underlying representations seem tofollow different developmental trajectories. As these findings could only be captured by using differentsubsets of magnitudes and different arithmetic types, this should be incorporated in future studies.Moreover, children’s subitizing skills and their connection with arithmetic skills should be studiedmore carefully. If it is true that these basic perceptual capacities are related to arithmetic skills, theirassessment might be helpful in the early identification of arithmetic deficits.

Acknowledgments

This research was funded by the Hessian initiative for the development of scientific and economicexcellence (LOEWE). We would like to thank all the participating children, Guopeng Chen (East ChinaNormal University, Shanghai), Gerd Lüer, Uta Lass, Markus Reitt (Georg-August-University, Göttingen),and Song Yan (Jacobs-University, Bremen) for their support.

References

Brannon, E. M. (2005). What animals know about numbers. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition(pp. 235–249). Hove, UK: Psychology Press.

Butterworth, B. (2005). The development of arithmetical skills. Journal of Child Psychology and Psychiatry, 46, 3–18.Cohen Kadosh, R., & Walsh, V. (2009). Numerical representation in the parietal lobes: abstract or not abstract? Behavioral and

Brain Sciences, 32(3–4), 313–328.Diester, I., & Nieder, A. (2007). Semantic associations between signs and numerical categories in the prefrontal cortex. PLoS

Biology, 5(11). e294.Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8(7), 307–314.Henmon, V. A. C. (1906). The time of perception as a measure of differences in sensation. Archives of Philosophy, Psychology and

Scientific Methods, 8, 5–75.Holloway, I. D., & Ansari, D. (2008). Domain-specific and domain-general changes in children’s development of number

comparison. Developmental Science, 11(5), 644–649.Holloway, I. D., & Ansari, D. (2009). Mapping numerical magnitudes onto symbols: the numerical distance effect and individual

differences in children’s mathematics achievement. Journal of Experimental Child Psychology, 103(1), 17–29.Huynh, H., & Feldt, L. (1976). Estimation of the box correction for degrees of freedom from sample data in randomized block and

split-plot designs. Journal of Educational Statistics, 1, 69–92.Koontz, K. L., & Berch, D. B. (1996). Identifying simple numerical stimuli: processing inefficiencies exhibited by arithmetic

learning disabled children. Mathematical Cognition, 2, 1–23.Landerl, K., Bevan, A., & Butterworth, B. (2004). Developmental dyscalculia and basic numerical capacities: a study of 8–9-year-

old students. Cognition, 93(2), 99–125.Le Corre, M., & Carey, S. (2007). One, two, three, four, nothing more: an investigation of the conceptual sources of the verbal

counting principles. Cognition, 105(2), 395–438.Lonnemann, J., Krinzinger, H., Knops, A., & Willmes, K. (2008). Spatial representations of numbers in children and their

connection with calculation skills. Cortex, 44(4), 420–428.Moyer, R. S., & Landauer, T. K. (1967). Time required for judgements of numerical inequality. Nature, 215(5109), 1519–1520.Mussolin, C., Mejias, S., & Noël, M.-P. (2010). Symbolic and nonsymbolic number comparison in children with and without

dyscalculia. Cognition, 115(1), 10–25.Piazza, M., Pinel, P., Le Bihan, D., & Dehaene, S. (2007). A magnitude code common to numerosities and number symbols in

human intraparietal cortex. Neuron, 53(2), 293–305.

J. Lonnemann et al. / Journal of Neurolinguistics 24 (2011) 583–591 591

Ridderinkhof, K. R. (2002). Acitvation and suppression in conflict tasks: empirical clarification through distributionalanalyses. In W. Prinz, & B. Hommel (Eds.), Common mechanisms in perception and action. Attention & performance,Vol. XIX (pp. 494–5119). Oxford: Oxford University Press.

Rousselle, L., & Noel, M. (2007). Basic numerical skills in childrenwith mathematics learning disskills: a comparison of symbolicvs non-symbolic number magnitude processing. Cognition, 102(3), 361–395.

Sekuler, R., & Mierkiewicz, D. (1977). Children’s judgments of numerical inequality. Child Development, 48, 630–633.Trick, L. M., & Pylyshyn, Z. W. (1994). Why are small and large numbers enumerated differently. A limited-capacity preattentive

stage in vision. Psychological Review, 101, 80–102.van Oeffelen, M. P., & Vos, P. G. (1982). A probabilistic model for the discrimination of visual number. Perception & Psychophysics,

32(2), 163–170.Verguts, T., & Fias, W. (2004). Representation of number in animals and humans: a neural model. Journal of Cognitive Neuro-

science, 16(9), 1493–1504.Xu, F., & Spelke, E. S. (2000). Large number discrimination in 6-month-old infants. Cognition, 74(1), B1–B11.Zorzi, M., & Butterworth, B. (1999). A computational model of number comparison. In M. Hahn, & S. C. Stoness (Eds.),

Proceedings of the Twenty First Annual Conference of the Cognitive Science Society. Mahwah, NJ: Lawrence Erlbaum.