Furthermore, language and namely, inversion of number words, has been shown to influence the compatibility effects in different studies. For two-digit number pairs, it has been shown that the unit-decade compatibility effect is more pronounced for languages with number word inversion (Nuerk et al., 2005, see also Pixner et al., 2011a for children data), at least when numbers are read from left-to-right (Moeller et al., 2015). It has been argued that number word inversion influences the comparison process as the unit digit being named first in the respective number words (erroneously) implies a higher importance and activation of the unit digit, although it is actually irrelevant for the decision. The higher activation of unit digits elevates the compatibility effect, because it is actually a unit interference effect, where the automatic activation of irrelevant unit comparisons cannot be completely suppressed thus hindering or prolonging responses in incompatible trials.

In the current study we were interested in whether and – if so – how three-digit number comparison is influenced by inversion. In languages with non-inverted number words, the order of digits in a three-digit number is the same as the order of constituents of the corresponding number word (e.g., in English: 372 ↔ three-hundred-and-seventy-two). Contrarily, this is not the case in languages with inverted number words (e.g., in German: 372 ↔ three-hundred-two-and-seventy; see **Figure 1**, for illustration). Similar to two-digit numbers, the unit digit is named before the tens digit following the hundreds digit. This may increase interference due to the irrelevant unit digit. However, in contrast to two-digit numbers, the neighborhood of the constituents differs between number words and Arabic numbers. The number word corresponding to the *unit* digit is

the direct neighbor of the number word corresponding to the hundred digit (i.e., three and seven in above example) whereas the direct neighbor of the hundred digit in Arabic digital notation still is the ten*s* digit (i.e., three and two in above example). If linguistic number word structure influences three-digit number processing, interference due to the unit digit might not be restricted to the neighboring Arabic tens digit but might also extend to the verbally neighboring hundred digit. Consequently, interference due to the unit digit should be more pronounced for German-speaking participants whereas for English-speaking participants it was observed that interference due to the decade digit was more pronounced (Korvorst and Damian, 2008; see also **Figure 1** for an exemplary illustration).

Paralleling the unit-decade compatibility effect for two-digit number comparison, both hundred-decade and hundred-unit compatibility can be defined for three-digit number comparison. A three-digit number pair is hundred-decade compatible when separate comparisons of the hundred and the decade digits lead to the same decisions (e.g., 742\_896; hundreds 7 *<* 8 and decades 4 *<* 9) and hundred-decade-incompatible when separate comparisons of hundreds and tens lead to opposing decisions (e.g., 362\_517, hundreds 3 *<* 5, but 6 *>* 1). Analogously, a three-digit number pair is defined as hundred-unit compatible when separate comparisons of hundred and unit digits yield the same decision (e.g., 742\_896, hundreds 7 *<* 8 and units 2 *<* 6) and hundred-unit-incompatible when these separate comparisons lead to opposing decisions (e.g., 537\_692, hundreds 5 *<* 6, but units 7 *>* 2). It is important to note that hundreddecade compatibility and hundred-unit compatibility are both attributes of one single number pair, which can be manipulated independently from each other.

So far, there are only few studies investigating language influences in three-digit number comparison tasks. In line with the larger interference due to decade-digits for languages with a non-inverted number word structure, Korvorst and Damian (2008) observed that, for English-speaking adults, the hundreddecade compatibility effect was descriptively more pronounced as compared to the hundred-unit compatibility effect. The authors interpreted this to indicate a left-to-right processing gradient reflecting partially sequential processing. However, it is important to note that in the original stimulus set of Korvorst and Damian (2008) hundred-decade compatibility was confounded with overall distance: overall distance was larger for hundred-decade compatible number pairs. This may have led to an inflation of the hundred-decade compatibility effect and, therefore, questions the proposed interpretation of a left-to-right processing gradient.

The only direct between-language comparison for three-digit numbers was conducted with children in third grade. Klein et al. (2013) observed that only German-speaking children exhibited a reliable hundred-unit compatibility effect, whereas no such effect was found for Italian-speaking children (a language without inversion). Moreover, the hundred-unit compatibility effect was more pronounced as compared to the hundreddecade compatibility effect for German-speaking third and fourth graders (Mann et al., 2012). This corroborates the argument on more pronounced unit interference when units neighboring hundreds verbally due to the inverted structure of German number words. However, the direct comparisons between the language groups was not significant, so that these differential language influences need to be treated with caution.

It is, however, important to note that with any outcome of above study, children's processing of multi-digit numbers cannot simply be generalized to adults. Children seem to move from a more sequential to a more parallel processing mode for both two-digit (Nuerk et al., 2004; Mann et al., 2011) and three-digit (Mann et al., 2012) numbers. It is well conceivable that language influences children's more sequential processing of three-digit numbers, but does not influence adult's more parallel and more automatic (cf. Kallai and Tzelgov, 2012) processing. Therefore, the question remains whether or not a stronger influence of unit interference in a language with inverted number words may only be a transient developmental phenomenon.

The present study set off to investigate inversion-related language specificities in three-digit number processing in German- (a language with inverted) as well as English-speaking (a language with non-inverted number words) adults. Because hundred-decade compatibility was confounded with overall distance in the stimulus set of Korvorst and Damian (2008) we created a new better matched stimulus set avoiding such confounds. Nevertheless, in line with results of Korvorst and Damian (2008), we expect to find reliable effects of hundreddecade as well as hundred-unit compatibility for both language groups indicating that three-digit numbers are processed in a parallel-decomposed manner. Yet, for the English-speaking participants, one would expect no differences in the size of hundred-decade and hundred-unit compatibility effects when the larger hundred-decade compatibility effect were indeed driven by the confounded stimulus set of Korvorst and Damian (2008). On the other hand, we had a specific hypothesis regarding the influence of the inversion property of German number words: for our German-speaking participants the interference due to the verbally first named unit-digit should be more pronounced as compared to the English-speaking participants. This interference should result in a relatively larger hundred-unit compatibility effect as compared to the hundred-decade compatibility effect for the German-speaking sample.

The evaluation of the pattern of compatibility effects for the two language groups will provide first empirical evidence on the question of whether the specific influence of the inverted number word structure on numerical cognition generalizes from 2- to 3-digit number processing. In particular, the proposed differential pattern of compatibility effects would corroborate the notion of inversion-related influences in German to persist into adulthood not only for the processing of two- but also of three-digit numbers.

# Materials and Methods

#### Participants

Twenty-five native German speakers (three male, four lefthanded) and 28 native English speakers (six male, two lefthanded) participated in the study. In each group, one participant had to be excluded due to error rates exceeding 10%. Mean age of the resulting samples was *M* = 23.08 years (SD = 6.28 years) for German- and *M* = 20.11 years (SD = 2.34 years) for Englishspeaking participants. Participants were recruited via postings at either the University of Tuebingen or the University of York and received course credits or 5€/4£ for compensation. All participants reported normal or corrected to normal vision. The study was approved by the local ethics committee of the University of York.

## Stimuli and Design

In total, the stimulus set consisted of 640 three-digit number pairs. Numbers containing the same digit more than once (i.e., 545 or 555), multiples of hundred (i.e., 200) and/or multiples of ten (i.e., 420) were not included in the stimulus set. For 320 of these number pairs, all corresponding digits differed from each other. For these critical items, the factors hundreddecade compatibility (HDC) and hundred-unit compatibility (HUC; each compatible vs. incompatible), as well as hundred (HD), decade (DD), and unit distance [UD; for all, small (1–3) vs. large (4–8)] were manipulated orthogonally in a 2 × 2 × 2 × 2 × 2 within-subject design. Problem size (the sum of the two numbers of a number pair) was matched for all resulting 32 conditions and overall as well as decade and unit distance was matched for the respective item conditions. Hundred distance could not be held constant for all conditions, as hundred distance is necessarily smaller for hundred-decade-compatible than hundred-decade-incompatible trials when problem size is held constant across conditions [hundred distances: *MHDC*−*comp* = 3.7, *SDHDC*−*comp* = 2.1, *MHDC*−*incomp* = 4.4, *SDHDC*−*incomp* = 2.2; *t*(318) = 3.31, *p <* 0.001]. Descriptive characteristics for these 320 critical number pairs are given in the supplementary material.

Additionally, 320 within-hundred number filler pairs were included. Filler items should prevent participants from focusing on decision-relevant hundred digits only. For half of these filler items, hundred digits were held constant (e.g., 475\_421) whereas for the other half hundred and decade digits were identical (e.g., 425\_421).

Number pairs were presented above each other in Arabic notation in white against a black background (font: "Arial," font size: 24 pt, bold) with a viewing distance of ∼50 cm.

#### Task and Procedure

In a magnitude comparison task participants had to indicate the larger of two three-digit numbers as fast and accurately as possible. In case the upper number in the display was larger, participants were instructed to press the '↑' key of a standard keyboard with their right index finger. When the lower number was larger, participants had to press the '↓' key with their left index finger. Instructions were given in the respective native language of participants. The two to-be-compared numbers of each pair appeared simultaneously and remained visible until a response key was pressed. Trials were separated by an intertrial interval of 500 ms. Trial order was randomized separately for each participant. Participants did not receive feedback as to the correctness of their response. Prior to the critical trials, participants performed 10 practice items, which were not part of the stimulus set.

# Results

Only correct responses were considered for analyses [mean error rate was 3.7%, SD = 2.1%; German: 4.0%, SD = 2.3%; English: 3.4%, SD = 1.9%; *t*(49) = 0.86, *p* = 0.392]. All three-digit number pairs with RTs faster than 200 ms were excluded from further analyses. Additionally, all number pairs with RTs deviating more than ±3 standard deviation from the individual participant's mean RT were excluded. This procedure led to a total loss of 1.4% of the data [German: *M* = 1.4%, SD = 0.4%; English: *M* = 1.4%, SD = 0.7%; *t*(49) = −3.43, *p* = 0.733]. As error rates were very low, analyses focused on RT. Nevertheless, a highly significant positive correlations between error rates and reaction times (German: *r* = 0.788, *p <* 0.001; English: *r* = 0.688, *p <* 0.001) indicated a similar response pattern for errors and RTs disconfirming a speed accuracy trade off. Compatibility and distance effects were evaluated by a 2 × 2 × 2 × 2 × 2 × 2

*M, mean; SD, standard deviation.*

ANOVA discerning the within-subject factors HDC (compatible vs. incompatible), HUC (compatible vs. incompatible), HD (small vs. large), DD (small vs. large), UD (small vs. large) as well as the between-subject factor language (German vs. English). Mean reaction times and standard deviations for all stimulus categories are presented in **Table 1**. In the following we will first report results relevant for possible language differences in compatibility effects, before describing distance effects as well as further modulations of compatibility effects through distances between respective digit positions.

# Compatibility Effects and Language Differences

In line with our hypothesis, reliable main effects of HDC and HUC were observed indicating that three-digit numbers were processed in a parallel and decomposed fashion. Hundred-decade compatible number pairs (*M* = 765 ms, SD = 172 ms) were on average responded to 14 ms faster than hundred-decade incompatible number pairs [*M* = 779 ms, SD = 165 ms; *F*(1,49) = 19.65, *p <* 0.001, η2 p = 0.29]. Additionally, response latencies were on average 27 ms shorter for hundred-unit compatible number pairs (*M* = 759 ms, SD = 165 ms) as compared to hundred-unit incompatible pairs [*M* = 786 ms, SD = 171 ms; *F*(1,49) = 39.96, *p <* 0.001, η2 p = 0.47].

In contrast to our hypotheses and in contrast to the compatibility pattern observed by Korvorst and Damian (2008), the main effect of HUC was descriptively larger as compared to the effect of HDC for both language groups (see **Figure 2** for an illustration).

In addition to the main effects of HDC and HUC, the interaction between HDC and HUC was significant

[*F*(1,49) = 4.42, *p* = 0.041, η2 p = 0.08]. In contrast to Korvorst and Damian (2008), this interaction indicated that the effect of HUC is larger for hundred-decade *incompatible* as compared to compatible number pairs (33 and 21 ms, respectively).

In contrast to what we expected, neither the interaction between HDC and language [*F*(1,49) = 0.11, *p* = 0.745] nor that between HUC and language [*F*(1,49) = 0.01, *p <* 0.919] nor the three-way interaction of HDC, HUC and language was significant [*F*(1,49) = 0.10, *p* = 0.756]. Overall reaction times did not differ between groups [German; *M* = 750 ms, SD = 196 ms, English: *M* = 792 ms, SD = 169 ms; *F*(1,49) = 0.81, *p* = 0.372]. Importantly there wasn't any reliable interaction with the factor language at all. These findings indicate that language did not modulate three-digit number processing. This interpretation is further corroborated by Bayesian analyses. Using the method proposed by Masson (2011), graded evidence for the null hypothesis (given the obtained data) can be calculated (see Masson, 2011, for a detailed description of the method).With respect to the interaction of both HDC and HUC with language, Bayesian analyses revealed that the probability of the null hypotheses (no differences between language groups) was 0.87 and 0.88, respectively. For the three-way interaction of HDC, HUC, and language the probability was 0.87. Applying the criteria suggested by Masson, probabilities above 0.75 can be considered positive evidence for the null hypothesis.

# Distance Effects and Influences of Digit Distances on Compatibility Effects

A significant main effect of hundred distance was found indexing number pairs with a large HD (*M* = 728 ms, SD = 159 ms) to be responded to 88 ms faster than pairs with a small HD [*M* = 816 ms, SD = 176 ms; *F*(1,49) = 534.75, *p <* 0.001, η2 p = 0.92]. This clearly indicates that number magnitude was processed in the task at hand. Moreover, the effect of DD was marginally significant [*F*(1,49) = 3.33, *p* = 0.074, η2 p = 0.06]. In line with the results of Korvorst and Damian (2008) this reflected a tendency toward an inverted DD effect: number pairs with a small DD (*M* = 770 ms, SD = 167 ms) tended to be processed faster than pairs with a large DD (*M* = 774 ms, SD = 166 ms). In addition, the interaction of HD and DD was significant [*F*(1,49) = 21.00, *p <* 0.001, η2 p = 0.30] indicating that the reversed DD effect was only present for small hundred distances but not for large HD (−15 and 7 ms, respectively).

With respect to the HDC effect, a reliable two-way interaction of HDC and DD was observed [*F*(1,49) = 9.30, *p* = 0.004, η2 p = 0.16]. In line with previous findings on the influence of digit distances on compatibility effects for two-digit numbers (Nuerk et al., 2001) and three-digit numbers (Korvorst and Damian, 2008), this interaction indicated that the HDC effect was more pronounced for larger as compared to smaller DD. Moreover, a reliable interaction of HDC and UD was observed [*F*(1,49) = 4.22, *p <* 0.045, η2 p = 0.08] indicating that the effect of HDC is larger for large UD when compared to small UD. Additionally, three three-way interactions were observed involving HDC. Firstly, the interaction of HDC and DD was further qualified by the three-way interaction of HDC, DD, and HD [*F*(1,49) = 4.49, *p* = 0.039, η2 p = 0.08]. Breaking down this three-way interaction into its constituting two-way interactions revealed that the interaction of HDC and DD was significant for small HD [*F*(1,49) = 11.92, *p* = 0.001, η2 p = 0.19] but not for large HD. Secondly, the interaction of HDC and DD was further qualified by the reliable three-way interaction of HDC, DD, and HUC [*F*(1,49) = 24.16, *p <* 0.001, η2 p = 0.33]. Breaking down this three-way interaction in two-way interactions of HDC and DD for hundred-unit-compatible and – incompatible number pairs showed that the interaction was significant for hundred-unit compatible number pairs [*F*(1,49) = 28.92, *p <* 0.001, η2 p = 0.37] but not for hundred-unit incompatible ones [*F*(1,49) = 0.17, *p* = 0.682]. Lastly, the interaction of HDC, UD, and HD was reliable [*F*(1,49) = 19.48, *p <* 0.001, η2 p = 0.28]. Breaking down this three-way interaction showed that the two-way interaction of HDC and UD was reliable for large HD [*F*(1,49) = 10.49, *p* = 0.002, η2 p = 0.17] but not for small HD.

For the HUC effect, modulations due to distances between the respective digit positions were observed as well. First, the interaction between HUC and UD was significant [*F*(1,49) = 28.85, *p <* 0.001, η2 p = 0.37] indicating that the effect of HUC was more pronounced for larger as compared to smaller UD. This two-way interaction was further qualified by two three-way interactions including HD and DD, respectively. Breaking down the three-way interaction of HUC, UD, and HD [*F*(1,49) = 8.89, *p* = 0.004, η2 p = 0.15] revealed that the interaction of HUC and UD was reliable for both large [*F*(1,49) = 12.11, *p* = 0.001, η2 p = 0.20] and small HD [*F*(1,49) = 32.32, *p <* 0.001, η2 p = 0.39]. However, while the HUC effect was larger for large UD for both large and small HD, the difference between the HUC effects for large and small UD was more pronounced for small HD. Breaking down the three-way interaction of HUC, UD, and DD [*F*(1,49) = 26.11, *p <* 0.001, η2 p = 0.35] showed that the two-way interaction of HUC and UD was reliable for small DD [*F*(1,49) = 42.44, *p <* 0.001, η2 p = 0.15] but not for large DD.

To further investigate the contribution of inter-digit distances, we ran a regression analysis similar to that conducted by Nuerk et al. (2001) for both language groups separately. Importantly, the results mirrored those of the ANOVA. After checking for collinearity between predictors (e.g., hundred distance was highly correlated with overall distance as well as logarithmic hundred distance), we included the predictors absolute hundred distance, absolute decade distance, absolute unit distance, categorical predictors of HDC and HUC, respectively, continuous predictors of HDC and HUC {e.g., the continuous HDC index for a compatible number pair is positive [732\_896; index: +6 (9–3)] while the continuous HDC index for an incompatible number pair is negative [762\_851; index: −1 (5–6)]}, and problem size (operationalized as the mean of the two to-be-compared numbers). For both language groups, regression analysis was highly predictive [German: *R* = 0.700, adj. *R*2 = 0.482, *F*(5,314) = 60.42, *p <* 0.001; English: *R* = 0.718, adj. *R*2 = 0.507, *F*(5,314) = 66.64, *p <* 0.001] with the same five predictors incorporated in the final model: absolute hundred distance (G: *b* = −0.620; E: *b* = −0.660), HDC categorical (G: *b* = −0.228; E: *b* = −.219), HUC categorical (G: *b* = −0.196; E: *b* = −0.200), HDC continuous (G: *b* = −0.081; E: *b* = −0.105) and problem size (G: *b* = 0.251; E: *b* = 0.190). Thus, the predictors for both languages were identical. Directly contrasting standardized *b*-weights of significant predictors between both languages using the method suggested by Brame et al. (1998) revealed no significant differences for any predictor (all *Z <* 1.1, all *p >* 0.27).

In sum, the results of these regression analyses are in line with previous research: the predictor hundred distance was the strongest predictor of RT indicating that participants indeed performed a magnitude comparison task. In addition, results suggest that even for three-digit numbers, compatibility effects are not simply categorical, but are influenced by the magnitudes of the involved digits. This is reflected by the inclusion of the continuous HDC variable in the final regression model, which did not code compatibility categorically, but defined by magnitude.

# Discussion

The aim of the current study was to investigate whether three-digit number processing in adults is influenced by the specific language property of number word inversion. Although we observed reliable hundred-decade and hundredunit compatibility effects, these were not modulated by language in the present study. Bayesian analyses substantiated these null effects. In sum, these data argue against a reliable influence of language on three-digit number processing in adults.

This was in contrast to our expectations because such language influences have been found repeatedly for *two-digit numbers* over a variety of tasks and participant groups (e.g., Nuerk et al., 2005; Macizo et al., 2010; Helmreich et al., 2011; Pixner et al., 2011a; Göbel et al., 2013; Imbo et al., 2014). For threedigit number processing, there is first evidence from child data indicating language influences, which were, however, small and not observed consistently (Klein et al., 2013). However, in contrast to this, no language effects were found at all in the present study when evaluating the influence of number word inversion on the pattern of HDC and HUC effects in German- and English-speaking adults using a three-digit number comparison task. This indicates that language influences on multi-digit number processing in children cannot simply be generalized to adults (Nuerk et al., 2004; Mann et al., 2011, 2012).

Nonetheless, comparable and reliable effects of HDC and HUC were observed for both German- and English-speaking adults. This indicated three-digit numbers to be processed in a parallel and decomposed manner in both language groups. Additionally, in both language groups the pattern of compatibility effects differed from the one observed for Englishspeaking participants previously reported by Korvorst and Damian (2008). In particular, we observed the effect of HUC to be descriptively larger than the effect of HDC contradicting a sequential left-to-right processing gradient in three-digit number processing.

# Lack of Language Differences in Adults

We observed significant HDC and HUC for German and English-speaking adults in ANOVA and regression analyses. In line with previous results for multi-digit numbers (e.g., Nuerk et al., 2001; Ganor-Stern et al., 2007; Macizo et al., 2010; Meyerhoff et al., 2012; Macizo and Herrera, 2013; Moeller et al., 2013) three-digit numbers thus seemed to be processed in a parallel and decomposed manner: the correct decision in number magnitude comparison was not only influenced by the decisive digit (i.e., the hundred digits in the case of between-hundred number pairs) but also by the separate comparisons of decisionirrelevant tens and units. This provides further evidence for the argument that place-value information and the magnitudes of single digits are considered automatically when multi-digit numbers are processed.

Importantly, and in contrast to previous results for twodigit numbers, the present data indicate that three-digit number processing is *not* influenced by number word inversion. The decision-irrelevant tens and units thus exhibited a comparable influence on three-digit number processing for both Germanand English-speaking participants. Unlike for tens and units, the position of the hundred digit is not inverted in German number words as compared to its position within the digit string (e.g., *3*84: *drei*hundertvierundachtzig, literally: *three* hundred four and eighty). Considering this, the present pattern of results indicates that interference due to inverted digits does specifically affect the inverted digits but not those next to the inverted ones. In turn, this might mask potential language differences in multi-digit number processing beyond the twodigit number range. Therefore, the observed compatibility effect pattern indicated that language influences observed for twodigit number processing do not generalize to three-digit number processing. Our results thereby indicate that inversion effects seem to be restricted to the digits being inverted (i.e., tens and units) and do not generalize to the verbally neighboring hundreds.

Please note, however, that the samples investigated in this study comprised only 25 and 28 participants, respectively. Therefore, one might speculate that null effects observed in this study might be attributable to power problems and/or type-1 errors associated with small sample sizes. Yet, this seems unlikely for at least two reasons. First, the observed null effects were substantiated by Bayesian analyses indicating them to be reliable. Second, it needs to be considered that for two-digit numbers, influences of inversion on the compatibility effect have been observed with similar sample sizes (e.g., Nuerk et al., 2005). So, even if statistically significant inversion influences on three-digit number processing might be detected with larger sample sizes, this would provide further evidence for our conclusion that these influences are most likely smaller and/or less reliable than for two-digit number processing in adults.

# Differences between Three-Digit Number Processing in Children and Adults

Investigating an adult sample, the present data did not indicate language to influence three-digit number processing. However, when considering previous evidence from children, data indicate language-specific developmental shifts from more sequential to more parallel decomposed processing of multi-digit numbers (e.g., Klein et al., 2013). As previously observed for two-digit numbers, compatibility effects for three-digit numbers seem to become more pronounced with increasing age. However, for non-inverted languages, the effect of *HUC* was present in (English-speaking) adults (this study and Korvorst and Damian, 2008) but not in (Italian-speaking) children (Klein et al., 2013). For the inverted German language, the effect of *HDC* effect was present in adults but not in children of third and fourth grade. Furthermore, for both German-speaking adults (this study) and children (Mann et al., 2012), the effect of *HUC* effect was larger than the hundred-decade compatibility effect. Thus, there are language influences on the developmental shift from more sequential toward more parallel decomposed processing of multidigit numbers with age and experience.

While the shift in processing patterns can be provoked by changes on the visuo-spatial and/or verbal processing level in non-inverted languages (i.e., English, Italian), in the inverted German language, a shift in processing verbal information is more probable. Because the tens did not interfere with the comparison process in German-speaking elementary school children, it can be assumed that, at least for children up to fourth grade, interference caused by verbal number words is more pronounced as compared to interference caused by the directly neighboring digits in symbolic Arabic number notation. Thus, auditory-verbal neighborhood of spoken number word elements seems to be more important at this point than visuospatial neighborhood of written digits. This might be due to children's tendency to verbalize what they are cognitively engaged in – in the present case three-digit Arabic numbers – more strongly than adults might do. In line with this, research on so-called private speech indicates that children's use of a more externalized, overt verbal thinking in kindergarten reduces gradually to rather internalized, soundless inner speech over the course of early elementary school (e.g., Kohlberg et al., 1968; Berk, 1992; Winsler and Naglieri, 2003). Although internalizing with age, language-supported processing might still be more pronounced in elementary school children when compared to adults. In turn, visuo-spatial neighborhood seems to become more salient with increasing age, experience, and automaticity in number processing whereas interference due to automatic activation of corresponding number word properties seems to become less salient. Taken together, these findings are in line with the notion of a developmental shift from more *language modulated* sequential processing of three-digit numbers to a more decomposed and parallel processing mode, which gets more independent of language with increasing age and experience.

# Sequential and Parallel Processing and Differential Compatibility Patterns

Combined sequential and parallel processing has been postulated for multi-digit numbers beyond the two digit number range (Korvorst and Damian, 2008; Meyerhoff et al., 2012). For threedigit numbers, Korvorst and Damian (2008) accounted for the larger HDC effect (as compared to the HUC effect) by suggesting a sequential left-to-right processing gradient enhancing the interfering role of the tens. In such sequential processing, tens are assumed to be processed directly following the hundreds and therefore interfere more than the subsequently processed unit digits.

While this explanation is appealing, we were not able to replicate Korvorst and Damian's (2008) results in this respect. Instead, we observed the HUC effect to be descriptively larger than the HDC effect for both the present English- and German-speaking participants. These results are inconsistent with the assumption of a left-to-right processing gradient. As a consequence, the question why differing compatibility effect patterns were found between the present study and that of Korvorst and Damian (2008) is of theoretical importance for our understanding of multi-digit number processing.

To account for this inconsistency, one should first consider differences in the stimulus set. As already described above, overall distance was larger for hundred-decade-compatible as compared to hundred-decade incompatible number pairs in the original stimulus set used by Korvorst and Damian (2008). Therefore, the HDC effect may have been inflated, because overall numerical distance and HDC were confounded. This confound was eliminated in the present stimulus set, which might have led to the smaller hundred-decade-compatibility effect in the present study. Yet, these contrasting results highlight the importance of matching task, stimuli, and procedures when aiming at evaluating multi-digit number processing.

Nevertheless, differences in stimulus characteristics may not be sufficient to explain that – in the present study – the HUC effect was larger than the HDC effect for both language groups. On a first glance, this seems somewhat counterintuitive since hundred and unit digits are visually and conceptually further apart but caused larger inter-digitinterference. A possible explanation for this finding is the effect of lateral masking. Investigated extensively in reading research, this effect describes the interference letters have on the processing of their neighboring letters. When a target letter is flanked by other letters, parafoveal and peripheral vision decreases and, therefore, the probability of correctly identifying the target letter decreases as well (Wolford and Chambers, 1983; Huckauf et al., 1999). Paralleling letter strings or words, one obvious difference between two-digit and three-digit numbers is that only in a three-digit number the decade digit has two neighbors, whereas the hundred and the unit digit only have one. This means less inhibitory influences for the two lateral digits (e.g., hundreds and units) which might in turn add to a descriptively more pronounced HUC effect.

# Conclusion

Taken together, our data suggest that there are limitations to language influences on multi-digit number processing – at least in adults. We observed no influences of number word inversion on three-digit number magnitude processing. This seems counter intuitive because we have seen an increasing number of papers in recent years showing language influences for a wide variety of numerical task, stimulus sets and participant groups. Therefore, the current data constrain these findings: language influences may not be ubiquitous but seem to be specific to stimulus sets, age groups, and probably tasks. Additionally, our data suggest that perceptual determinants of processing multiple elements deserve attention in multi-digit number processing research. On a practical level, future studies might wish to evaluate the development of language differences with age more systematically. Thereby, possible associations with numerical/arithmetical competencies may be investigated (e.g., Göbel et al., 2013), which, in turn, would allow to better understand the influence of inverted number word systems on children's numerical development. On a theoretical level, it would be desirable to better the interplay of parallel and sequential processing of in multi-digit numbers. Therefore, future studies may use eye-tracking to evaluate online what is actually going on during the comparison process.

# Acknowledgments

JB was supported by the Leibniz-Competition Fund (SAW-2014- IWM-4) providing funding to Elise Klein. KM and H-CN were principal investigators at the LEAD Graduate School [GSC1028], a project of the Excellence Initiative of the German federal and

# References

state governments. We would like to thank Silke Göbel and Carolin Maier for their help in data collection.

# Supplementary Material

The Supplementary Material for this article can be found online at: http://journal*.*frontiersin*.*org/article/10*.*3389/fpsyg*.* 2015*.*01216

**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

*Copyright © 2015 Bahnmueller, Moeller, Mann and Nuerk. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*

# The relation between language and arithmetic in bilinguals: insights from different stages of language acquisition

Amandine Van Rinsveld1 \*, Martin Brunner 2 , Karin Landerl 3 , Christine Schiltz 1 † and Sonja Ugen4 †

1 Education, Culture, Cognition and Society, Institute of Cognitive Science and Assessment, University of Luxembourg, Walferdange, Luxembourg, 2 Berlin-Brandenburg Institute for School Quality, Free University of Berlin, Berlin, Germany, 3 Department of Psychology, University of Graz, Graz, Austria, 4 Luxembourg Center for Educational Testing, University of Luxembourg, Luxembourg, Luxembourg

#### Edited by:

Hans-Christoph Nuerk, University of Tübingen, Germany

#### Reviewed by:

Marc Brysbaert, Ghent University, Belgium Claudia K. Friedrich, University of Tübingen, Germany

#### \*Correspondence:

Amandine Van Rinsveld, Education, Culture, Cognition and Society, University of Luxembourg, Route de Diekirch, L-7201 Walferdange, Luxembourg<EMAIL_ADDRESS>
† These authors have contributed equally to this work.

#### Specialty section:

This article was submitted to Developmental Psychology, a section of the journal Frontiers in Psychology

> Received: 20 October 2014 Accepted: 23 February 2015 Published: 13 March 2015

#### Citation:

Van Rinsveld A, Brunner M, Landerl K, Schiltz C and Ugen S (2015) The relation between language and arithmetic in bilinguals: insights from different stages of language acquisition. Front. Psychol. 6:265. doi: 10.3389/fpsyg.2015.00265 Solving arithmetic problems is a cognitive task that heavily relies on language processing. One might thus wonder whether this language-reliance leads to qualitative differences (e.g., greater difficulties, error types, etc.) in arithmetic for bilingual individuals who frequently have to solve arithmetic problems in more than one language. The present study investigated how proficiency in two languages interacts with arithmetic problem solving throughout language acquisition in adolescents and young adults. Additionally, we examined whether the number word structure that is specific to a given language plays a role in number processing over and above bilingual proficiency. We addressed these issues in a German–French educational bilingual setting, where there is a progressive transition from German to French as teaching language. Importantly, German and French number naming structures differ clearly, as two-digit number names follow a unit-ten order in German, but a ten-unit order in French. We implemented a transversal developmental design in which bilingual pupils from grades 7, 8, 10, 11, and young adults were asked to solve simple and complex additions in both languages. The results confirmed that language proficiency is crucial especially for complex addition computation. Simple additions in contrast can be retrieved equally well in both languages after extended language practice. Additional analyses revealed that over and above language proficiency, language-specific number word structures (e.g., unit-ten vs. ten-unit) also induced significant modulations of bilinguals' arithmetic performances. Taken together, these findings support the view of a strong relation between language and arithmetic in bilinguals.

Keywords: numbers, language learning, bilingualism, arithmetic, addition

# Introduction

Although every human can manipulate approximate numerical quantities independently from language (Xu and Spelke, 2000), acquiring and mastering symbolic representations of exact quantities critically depends on language and instruction. Amazonian tribes who have restricted or no number words for quantities larger than five (or even two) impressively illustrate the importance of language for exact quantity representations. While their members can handle and manipulate large numerosities approximately, they are not able to process and represent them exactly (Gordon, 2004; Pica et al., 2004). Formal education enables the acquisition of exact number representations through labeling sets using distinct number names (Fuson et al., 1982). In other words, exact numerical quantities are learned through the use of language (Le Corre and Carey, 2007), and consequently exact number processing remains under the influence of language long after exact number representation acquisition. Recent studies demonstrated that basic processes such as number comparison are performed in slightly different ways depending on task language (Nuerk et al., 2005; Macizo et al., 2010; Van Rinsveld et al., 2012). Yet language plays an especially crucial role in more complex numerical computations, such as arithmetic problem solving. In the present study, we investigated whether and how the progressive acquisition of multiple languages modulates arithmetic problem solving in bilinguals.

# Language and Arithmetic

Several studies provide strong evidence for an involvement of language in exact arithmetic (Spelke and Tsivkin, 2001). Exact calculations, contrary to approximate number processing, is thought to be represented in a specific language-coded format. Neuropsychological studies highlighted that the preservation of language is in fact necessary for arithmetic problem solving, as many authors reported an association between acalculia and aphasia (e.g., Delazer et al., 1999; Basso et al., 2000, 2005, but see Rossor et al., 1995; Cappelletti et al., 2001; Baldo and Dronkers, 2007). In the same way, neuro-imaging studies have shown that exact calculation tasks systematically activate specific language areas, arguing for an exact language-dependent system as opposed to a language-independent approximate system for number representations (Dehaene et al., 1999; Cohen et al., 2000; Stanescu-Cosson et al., 2000; Gruber et al., 2001; Venkatraman et al., 2006, but see Pesenti et al., 2000; Zago and Tzourio-Mazoyer, 2002; Benn et al., 2012).

Language is undoubtedly needed to build exact quantity representations, yet it still has to be clarified for what specific aspect of calculation language plays a crucial role. Heterogeneous solving strategies and processes can be involved in calculation depending on task difficulty (Beishuizen, 1993). Language may consequently affect distinct calculation types differentially. For that matter, it is important to separately examine the specific role played by language in each of the two classically distinguished arithmetic solving strategies. On the one hand, we distinguish simple calculations that are generally composed of one-digit operands (i.e., operands <10). For these problems it is widely accepted that learning and practice lead to a direct retrieval of their solutions from memory, as so-called "arithmetic facts" (Ashcraft, 1992; McCloskey, 1992; Fayol and Thevenot, 2012). However, there is less agreement concerning the storage format of these arithmetic facts: they could be represented in an abstract semantic format (McCloskey et al., 1985), a verbal format (Campbell, 1994; Dehaene and Cohen, 1995) or in a format depending on individual subject's preferences (Noel and Seron, 1993). Moreover, the importance of language in arithmetic fact retrieval is modulated by operation-type: simple addition and multiplication, in comparison to subtraction and division, especially rely on verbally coded facts probably because addition and multiplication facts are more often learned and used in their verbal code than subtraction and division (Lemer et al., 2003).

More complex calculations (i.e., operands >10), on the other hand, cannot directly be retrieved from memory but they require mental computations to be solved. These computations mainly rely on working memory resources to execute solving strategies, keep intermediate solutions in memory and update the final solution (Hitch, 1978; Ashcraft, 1995). According to Logie et al. (1994), the phonological loop of Bladdeley's working memory model (Baddeley, 1992) is used in mental calculation to verbally repeat the numbers. Studies using articulatory suppression during complex calculation have shown that phonological mediation occurs, especially when some elements of the problems disappear after a short presentation time (Fürst and Hitch, 2000). Moreover, a study with English-Welsh bilinguals found longer response times and more errors when calculations were performed in Welsh than in English, due to longer number words in the former (Ellis and Hennelly, 1980). Klessinger et al. (2012) revealed that the impact of number word lengths on exact additions was especially prominent in less proficient calculators. Similarly, a neuro-imaging study confirmed the crucial role of working memory in complex calculation and suggested that the working memory components engaged in the (visual or verbal) solving process may depend on individual solving strategies (Delazer et al., 2003). Taken together, these findings suggest that language is important for arithmetic problem solving at different levels because arithmetic facts are potentially represented or retrieved from memory in a verbal format and complex arithmetic solving processes rely at least partially on verbal working memory components.

Language is crucial for exact representation of large numerosities and for exact arithmetic problem solving, at least during the acquisition of these abilities. The different number naming systems used in different languages can modulate numerical performances during acquisition stages of numerical cognition but also in adults, who have long acquired these abilities (Campbell and Xue, 2001; Chen et al., 2009). Specifically, the order of tens and units in two-digit number words is a characteristic of number naming systems that can directly affect arithmetic performances. Brysbaert et al. (1998) showed that additions presented in the format "21 + 4" were solved faster by French-speaking participants (21 is pronounced twenty and one since French number words follow the ten-unit order), whereas the same additions presented in the format "4 + 21" were solved faster for Dutchspeaking participants (21 is pronounced one and twenty since Dutch number words following the unit-ten order). Moreover, Göbel et al. (2014) reported that German-speaking children had a larger carry effect in additions compared to Italian-speaking children. They explained this result by the greater similarity between the Arabic digit notation and the order of tens and units in Italian number words than in German number words. Indeed, Arabic digit notation follows the same order (from left to right) than the

tens-units order of Italian number words (e.g., "24" = twentyfour) but they are inverted in comparison to the unit-ten order of German number words (e.g., "24" = four-and-twenty).

Another difference between number naming systems from different languages is the use of a base-20 structure instead of base-10 structure for two-digit number words (e.g., in French and in Basque). Seron and Fayol (1994) highlighted the specific difficulties encountered by French-speaking children for 70 and 90 number words following this base-20 structure used in France, in comparison to Belgian French-speaking children who use base-10 structure for 70 and 90. The former took longer and made more errors than the latter when writing down Arabic digits in a number dictation task. The base-20 system seems to have an impact not only during development but also later since a study by Colomé et al. (2010) showed that adult Basque speakers are influenced by the base-20 system of their language when they solve addition problems (see also Salillas and Carreiras, 2014). Taken together, the structure of the number words in the language in which numbers are acquired appears to affect arithmetic performances during childhood and some influence on arithmetic computation even persists in adulthood.

# Bilinguals and Arithmetic

Given the critical role of language in arithmetic problem solving, how people using several languages (e.g., bilinguals) calculate is a particularly intriguing question. Many models of language processing in bilinguals support the idea that bilinguals' languages are active in parallel at any time, occasioning mutual interferences between languages (e.g., Kroll et al., 2014). It is generally assumed that interferences of the dominant language during the use of the non-dominant language are more consequential than interferences in the opposite direction, and the relative asymmetry in this mutual influence is a function of bilingual proficiency (Bialystok, 2009; Kroll et al., 2013). Indeed, higher proficiency level in a language lessons potential interferences of other languages on it. From the reports in the literature about bilingual's arithmetic problem solving it appears that several elements concerning the relative mastery of languages (i.e., language proficiency) as well as the structure of the number words in the involved languages directly modulate bilinguals' performances in arithmetic. Relevant data concerning these two aspects are highlighted below.

Early studies in bilingual speakers provided first indications that arithmetic skills is related to language proficiency. They indeed observed systematic advantages in response time and accuracy when bilinguals calculated in their first compared to their second (and less proficiently mastered) language (Marsh and Maki, 1976; McClain and Huang, 1982; Geary et al., 1993). Frenck-Mestre and Vaid (1993) tested addition fact-verification tasks in bilinguals with correct-outcome problems but also falseoutcome problems that could be related or unrelated to multiplication facts (i.e., 2+3 = 6 was a false-outcome addition related to a multiplication fact). The authors observed associative confusion when problems were presented in bilinguals' first language and in Arabic digits but not in bilinguals' second language, so that they argued in favor of automatic arithmetical fact retrieval in the first but not the second language.

More recently, neuro-imaging studies on late Chinese-English bilinguals suggested that the verbal code of the first language is needed to retrieve arithmetic facts when the network of arithmetic facts in second language is not sufficiently developed. Wang et al. (2007) observed that performing complex calculations in first and second languages rely on a common activation network, but with higher activations during calculations in second language. This was interpreted as evidence for extra language processing needs in second language; potentially translation of input from second into first language (Lin et al., 2011). Taken together, these bilingual studies point toward an advantage for both retrieving arithmetic facts and computing complex arithmetic problems in the first language, i.e., the language in which most bilinguals learned to do arithmetic.

However, the bilinguals tested in the aforementioned studies were all late bilinguals or clearly unbalanced bilinguals, so the picture may be a bit different in more balanced bilinguals or bilinguals who did not acquire arithmetic in their first language. Indeed, one study reported that highly proficient bilinguals produced arithmetic facts equally well in their two languages (Campbell and Epp, 2004). Moreover, a study with Philipino-English bilinguals reported better arithmetic fact verification performances in English number word presentation, which was their second language but also the language in which they learned arithmetic at school and which they reported as their preferred language for doing arithmetic (Bernardo, 2001). Furthermore, Salillas and Wicha (2012) provided evidence for strong associative networks between terms and solutions for problems in the language in which participants learned arithmetic, which was not necessarily their first language. Participants seemed to maintain these early-established networks in adulthood, independently of language proficiency. These results were supported by a recent study where bilinguals showed switching costs when they had to retrieve arithmetic facts in their untrained- vs. trained-language (Saalbach et al., 2013). Hence, when bilinguals solve arithmetic problems, the language in which arithmetic was learned might be even more critical than the first language or the language in which they are currently most proficient.

In sum, bilinguals' arithmetic performances can be modulated by language proficiency levels, language of math acquisition and number word structure of the respective spoken languages. However, we are still lacking extensive studies, which investigate the relation between these different factors and arithmetic performances in bilingual participants. Such approaches are nevertheless necessary to understand in detail how language contributes to numerical computations. It is for instance currently unclear whether in highly proficient bilinguals performance levels in arithmetic become equivalent for their two languages or whether they maintain an advantage for retrieving and/or calculating in the language of arithmetic acquisition. It is also not known what increasing language proficiency implies for simple and complex calculations. Finally, it remains to be explored how language-related differences in number word structures affect arithmetic performance in bilinguals (e.g., German units-tens vs. the French tens-units; German base-10 vs. French base-20 for number words between 70 and 99).

# The Present Study

One of the major issues when studying bilinguals is that there often are as many different stories and profiles of languages acquisition as individuals. However, age of acquisition and proficiency levels of languages in bilinguals may drastically influence various ranges of cognitive processes (e.g., Altarriba and Basnight-Brown, 2007). In the present study we took advantage of the unique German-French bilingual school system of Luxembourg in order to address the aforementioned questions concerning to the relation between language and arithmetic by tracking the development of addition solving in bilingual adolescents and young adults at five different stages of bilingual proficiency. Bilingualism is a major attribute of the Luxembourgish educational system, as German and French are both teaching languages. In primary school teaching is held exclusively in German, but during secondary school, teaching language progressively switches to French, so that the pupils become highly proficient both in German and French through their education.

We composed four samples of German-French bilingual pupils at different levels of Luxembourgish secondary school (i.e., grades 7, 8, 10, and 11) and one sample of German-French bilingual young adults (who had also attended secondary school in Luxembourg). All participants thus mastered both German and French. Pupil participants from grades seven to 11 incrementally improved their mastery in German and French, with a relative emphasis on French as this language was becoming their predominant teaching language. The young adults achieved the level of excellence in both French and German. Altogether this yielded a design encompassing five distinct stages of German-French bilingualism.

For empirical research on the interplay of language and arithmetic the bilingual context in Luxembourg is characterized by a double advantage. (a) Firstly, all participants of a given ageclass have a similar exposure to each of the two languages, as they are all first taught in German and then in French. This allows composing large samples of bilingual participants that are homogenous in terms of duration and amount of exposure to each language. Moreover, although bilingual, all participants acquired arithmetic in German. (b) Secondly, German-French bilinguals are particularly interesting because German and French languages use inverted number word structures. Two-digit number words follow the units-tens order in German (e.g., "four-and-twenty") but the tens-units order in French (e.g., "twenty-four" like in English).

The experimental tasks consisted in addition problems that participants had to solve both in German and in French during two separate sessions. Additions were presented in two different formats (i.e., visual presentation of Arabic digits and auditory presentation of number words) and consisting of two difficulty levels (i.e., simple and complex additions). Throughout the entire experiment participants had to give their answers orally in the language of the session. Thus, task language permeated task instructions as well as presentation and solution of the additions in the auditory format, whereas only task instructions and solution production were imbued by task language in the visual format. Based on the literature reviewed above a series of predictions concerning the influence of language on addition solving in German-French bilinguals could be derived. Moreover, we also formulated detailed proposals on how these language effects might express at different stages of bilingual proficiency.

# Effects of Calculation Complexity on Bilinguals' Arithmetic Solving

In simple additions participants are thought to retrieve the solution from memory (Ashcraft, 1995). Previous studies with bilinguals have shown evidences for early-encoded arithmetic facts in one of the bilinguals' languages (e.g., Frenck-Mestre and Vaid, 1993; Spelke and Tsivkin, 2001; Wang et al., 2007) underlined by format-depending representations (Dehaene and Cohen, 1995). Nevertheless, other studies have highlighted evidences for transferable facts from one language to the other in very proficient bilinguals (e.g., Campbell and Epp, 2004) suggesting a possible representation of numbers independent from any format or language of encoding (McCloskey et al., 1985). Consequently, it can be expected that highly proficient bilinguals (i.e., adults and older adolescent participants of the present study) retrieve addition facts equally well in German and French. Indeed, these participants should be proficient enough in French and/or have been sufficiently exposed to numbers in French to be able to solve the simple additions similarly in French as in German.

Language-related performance differences ought to predominantly arise with complex additions. Compared to simple calculations, more complex arithmetic problems are thought to rely on computational procedures composed of multiple processing steps (e.g., Fayol and Thevenot, 2012), which can be modulated by language proficiency but also by specific number word structures. With respect to task presentation format, language effects should be larger for auditory than visual presentation formats because the operands have to been kept in memory in the former (LeFevre et al., 2001). In line with the prominent role of language, we expected that participants of all proficiency levels solve complex additions better and faster in German than in French. Indeed all participants had acquired German earlier than French and German was also their language of arithmetic acquisition (Bernardo, 2001; Salillas and Wicha, 2012). At the highest bilingual proficiency levels this benefit should be reduced, but we anticipated that it might never be resorbed completely if the early constellation of bilingual proficiency is critical.

# Effects of Number Word Structures on Bilinguals' Arithmetic Solving

Performance differences that arise when bilinguals solve additions might also be due to the specific number word structure of the respective languages. To gauge the impact of the different two-digit number-naming systems used in French vs. German on arithmetic performance in the five different bilingualism proficiency groups, we investigated two aspects of the number words.

Firstly, we explored whether the particular base-20 number word structure used in French (but not in German) for numbers from 70 to 99 might impact arithmetic performances differentially across age-groups. Indeed, the number words under 70 follow the classical base-10 structure in both task languages, while the number words over 70 follow the base-10 structure in German but not in French (where they follow the base-20 structure). We expected to find a general problem size effect in both languages because arithmetic problems with larger numbers are assumed to be more difficult to solve than arithmetic problems with smaller numbers (Groen and Parkman, 1972). But more interestingly, we also assumed that additions involving numbers over 70 would be specifically difficult in French because of the base-20 structure (Seron and Fayol, 1994). This specific difficulty should thus be especially pronounced at lower French proficiency levels.

Secondly, we aimed to understand whether and how the order of tens and units in number words (i.e., tens-units in French vs. units-tens in German) plays a role in bilinguals' addition performances. As Pixner et al. (2011) reported that the number naming system used in a two-digit number transcoding task modulated the type of errors, we analyzed which errors bilingual participants made on complex additions across the different presentation formats and languages. Given the contrasting positions of units and tens in German and French it is plausible that the same bilingual participant makes errors that predominantly pertain to distinct value positions depending on the language in which the calculation is performed.

Testing these predictions on our unique German-French bilingual sample will allow us to better understand the relation between language and arithmetic in bilinguals and how this relation evolves with increasing bilingual proficiency levels. To the best of our knowledge, there are currently no studies that systematically investigated how the influence of number word structure on arithmetical performance evolves as a function of language proficiency. Taken together these original data should also yield new insights into the role of language in arithmetic and number processing in general.

# Methods

#### Participants

A total of 193 bilingual participants were recruited for the present study. The sample was composed of 36 pupils from grade 7 (21 females; mean age of 12.2 years; SD = 0.36 years), 33 pupils from grade 8 (13 females; mean age of 13.2 years; SD = 0.58 years), 35 pupils from grade 10 (15 females; mean age of 15.5 years; SD = 0.66 years), 41 pupils from grade 11 (19 females; mean age of 16.4 years; SD = 0.72 years) and 48 young adults (34 females; mean age of 22.4 years; SD = 2.67 years).

All participants thus spoke Luxemburgish (an official language of Luxembourg which developed from a dialectal variant of German) or German as native language and attended the Luxembourgish school system in the highest academic track, which prepares for attending college and university. Moreover, all study participants (including the adults) had attended Luxembourgish primary school that starts with German as teaching language. From second grade of primary school on, all participants learned French as a second language. Importantly, students in grades 7 and 8 were taught mathematics in French, whereas students in grades 10 and 11 were not only taught mathematics but also all of their other courses in French (except the German and English language courses). Over the school years, relative exposure and proficiency in French thus progressively increased and tended toward bilingualism with high proficiency levels in both German and French in the highest grades. Consequently, the adult group was composed of young adult participants who had become highly proficient German-French bilinguals through their education.

Native language(s), the number of years spent in Luxembourgish schools and linguistic background (under the form of selfassessment of language proficiency) were checked in a short questionnaire before starting the experiment in order to ensure that all participants also had similar exposures to languages in these respects. Adults received 20e for their participation. Informed consent was obtained from all participants.

# Stimuli

Eighty-four two-operand addition problems were presented during the entire experiment. The set was composed of 28 one-digit simple additions (e.g., 4 + 2) and 56 two-digit complex additions (e.g., 56 + 32). This stimulus set was split in four blocks of additions to be allocated to both presentation formats of the problems and to both language sessions: visual and auditory presentation of the numbers in the German session and visual and auditory presentation of the numbers in the French session.

Simple additions were composed of two one-digit operands ranging from 2 to 9. We excluded +1 additions and additions between the same operands (e.g., 7 + 7), resulting in a range of solutions from 5 to 17. The simple additions with carry (additions with a solution of 10 or more) and without carry (additions with a solution below 10) were equally distributed across the four blocks of additions.

Complex additions were composed of two two-digit operands ranging from 12 to 86 in order to keep solutions below 100. We excluded all additions including a zero or ties. Furthermore, problems with a repetition of the same digit between the operands or between one of the operands and the solution were excluded, resulting in a range of solutions from 35 to 98. The requirement of a carry to be solved (with or without carry), the position of the larger operand (left vs. right in visual presentation; first vs. second in auditory presentation) and the problem size (small when the solution ranged between 30 and 69 or large when the solution ranged between 70 and 98) were taken into account in the repartition of complex additions in the four blocks. Indeed, each block contained seven problems with carry and seven problems without carry, and seven problems of small size and seven problems of large size. In other words, among the small problems, half of them contained a carry and half of them did not, and the same for the large problems, so that problems with and without carry were distributed equally among problems of different sizes within each block. The assignation of the blocks to a presentation format and a language was balanced through participants. For instance, block 1 was be assigned to visual presentation of the French session for the first eight participants but the same block 1 was assigned to visual presentation of the German session for the next eight participants.

#### Procedure

We ran the experiment on an Apple 13′ Macbook using Psyscope X B57 (Cohen et al., 1993) where voice onset times of responses

TABLE 1 | Mean duration of presentation of auditory additions in ms with standard deviation for each complexity level of the additions as a function of language.

Mean presentation time differences between languages were not significant neither for simple additions, t(54) = 0.239; p = 0.812, not for complex additions, t(108) = 0.148; p = 0.883.

were recorded with a voice key on the Iolab USB Button Box. As the voice key only recorded the response onset, the experimenter wrote the solutions down and pressed a key to start the next trial, which started after an inter-trial interval of 500 ms. The onset of the response time (RT)—measurement started when the stimulus presentation was completed.

In the visual presentation format, additions appeared on a white screen in black (Arial, font size 90) until participants responded. In the auditory presentation format, participants had to listen to the additions via headphones (in both ears). The length of auditory presentation was controlled between languages separately for simple and complex additions, so that the mean duration of auditory presentation did not differ between languages (see **Table 1**). In both presentation formats, participants had to respond orally by giving the solution in the microphone in the language of the task. This means that for auditory presentation of the additions, RT-measurement started at the offset of the second operand.

The testing was organized in two language sessions: participants performed both presentation formats first in one task language and then in the other. Order of presentation formats and task languages were counterbalanced between participants. Instructions and interaction with the experimenter remained in German or in French, according to the session. Participants were tested individually and were instructed to respond as accurately and as fast as possible. Seven training items preceded the 21 additions of each block. The entire experiment lasted about 50 min.

#### Data Processing

### Effects of Calculation Complexity (Simple vs. Complex Additions)

In order to track the development of arithmetic problem solving in bilingual children and adults, correct response times (RTs) and correct response rates (CRs) during experimental tasks were collected at five different stages of language proficiency. Training items were not included in the dataset, and we also excluded RTs of all trials below or above three standard deviations from the mean of each participant and from the group mean. We excluded 4% of the trials in this way before analyzing the RTs.

We ran a preliminary analysis of variance (ANOVA) on the RTs and the CRs including all additions participants had to solve with Complexity2 × Format2 × Task language2 as within-subject factors and Age-group5 as between-subject factor. The two levels of complexity were the simple one-digit operand vs. the complex two-digit operand addition problems; format referred to visual or auditory presentation of the additions; and task language was German or French (for instructions, presentation of the additions in the auditory format, and production of the answer). The age-group factor had the following levels: seventh graders, eighth graders, tenth graders, eleventh graders or young adults. The aim of this preliminary ANOVA was to see whether it was relevant to analyze both complexity levels (simple vs. complex addition) separately. Therefore, we only report results from the effect and interactions with the complexity factor. Then, we ran analyses of variance (ANOVA) on the RTs and the CRs separately for each type of additions: i.e., the simple one-digit additions and the complex two-digit additions. Within each ANOVA we used Format2 × Task language2 as within-subject factors and Age-group5 as between-subject factor.

# Effects of Number Word Structure

To investigate how arithmetic performance is influenced by the different structures of number words in German and French we conducted two additional analyses. Firstly we tracked the impact of the particular base-20 number word structure used in French but not in German for numbers from 70 to 99 on the arithmetic performances across age-groups. Therefore, we introduced one more factor in the ANOVA on complex additions: the problem size. We categorized the items in two levels of problem size according to whether problems involved or not a number over 70. Indeed the number words under 70 follow the classical base-10 structure in both task languages, whereas the number words over 70 follow the base-10 structure in German but the base-20 structure French. We thus ran an ANOVA with Problem size2 × Format2 × Task language2 as within-subject factors and with Age-group5 as between-subject factor.

Secondly, we focused on the impact of the order of tens and units (i.e., ten-unit in French vs. unit-ten in German) in two-digit number words on arithmetic performances. We analyzed the type of errors participants made across different presentation formats and languages when solving complex additions involving on twodigit numbers. Within each task language and format, we listed the rate of errors (%) for which only the ten-digit was false ("tenerror," i.e., 34 instead of 24) and inversely, the rate of errors for which only the unit-digit was false ("unit-error," i.e., 34 instead of 35). Other types of errors were not included in the analyses because we found less than 2% of each type. We ran an ANOVA on these error rates with Error type2 × Format2 × Task language2 as within-subject factors and with Age-group5 as between-subject factor. The two levels of the error type factor corresponded to "ten-error" and "unit-error" and the level of the other factors were the same as in the previous analyses.

# Results

# Effects of Calculation Complexity (Simple vs. Complex Additions)

Preliminary ANOVA showed a strong effect of complexity on both RTs [F(1, 184) = 893.961; p < 0.001; η 2 = 0.829] and CRs [F(1, 185) = 510.891; p < 0.001; η 2 = 0.734]. Both in RTs and CRs, complexity modulated effects of language [RTs: F(1, 184) = 177.873; p < 0.001; η 2 = 0.492, CRs: F(1, 185) = 64.294; p < 0.001; η 2 = 0.258], format [RTs: F(1, 184) = 43.034; p < 0.001; η 2 = 0.190, CRs: F(1, 185) = 235.634; p < 0.001; η 2 = 0.560] and age-group [RTs: F(4, 184) = 12.740; p < 0.001; η 2 = 0.217, CRs: F(1, 185) = 3.880; p = 0.005; η 2 = 0.077]. We also observed a triple interaction between complexity, language and format [RTs: F(1, 184) = 13.119; p < 0.001; η 2 = 0.067, CRs: F(1, 185) = 9.768; p = 0.002; η 2 = 0.050]. Only in RTs, there was also a significant triple interaction between complexity, language and age-group [RTs: F(4, 184) = 4.610; p = 0.001; η 2 = 0.091]. Since all factors of the preliminary ANOVA interacted with complexity, we will directly report below separate analyses and results for both complexity levels.

#### Simple Additions

For the simple additions, overall mean RT was 1309 ms (SE = 32 ms) and overall mean CR was 96.4% (SE = 0.3%). We found an age-group effect on RTs [F(4, 184) = 12.710; p < 0.001; η 2 = 0.216] and on CRs [F(4, 185) = 3.038; p = 0.019; η 2 = 0.062], as participants solved the simple additions faster and more accurately with increasing age-group (see **Table 2**). Furthermore, simple additions were performed faster when they were presented in auditory than in visual format, F(1, 184) = 171.992; p < 0.001; η 2 = 0.483, but no difference between formats was observed in terms of CRs, F(1, 185) = 0.737; p = 0.392; η 2 = 0.004 (see **Figures 1A,B**). Thus, simple auditory additions were solved faster than visually presented ones, but correct response rates were similar for both formats.

Moreover, simple additions were performed faster in German than in French [RT: F(1, 184) = 77.199; p < 0.001; η 2 = 0.296],

TABLE 2 | Means of reaction times (RT) in ms and correct response rates (CR) in % with standard errors for each complexity level of the additions (simple vs. complex) and the general mean performances as a function of age-group.

FIGURE 1 | Mean reaction times in ms (A) and mean correct response rates in percentages (B) with standard errors for the simple additions in each task language (black line for German and red line for French) as a function of presentation format.

see **Figure 2A**. Participants also made fewer errors in German than in French [F(1, 185) = 9.782; p = 0.002; η 2 = 0.050], but this language effect on CRs was marginally modulated by the age-group (language × age-group: F(4, 185) = 2.234; p = 0.067; η 2 = 0.046), see **Figure 2B**. We decomposed this interaction by separately running a Format2 × Task language2 ANOVA on CRs in each age-group. It appeared that only the seventh graders were less accurate in French than in German, F(1, 34) = 4.074; p = 0.050; η 2 = 0.092, while all other age-groups performed with equal accuracy in both languages (all F's < 1 and p's > 0.05), see **Figure 2B**. No other interaction reached significance (all F's < 1 and p's > 0.05). In sum, participants solved simple additions faster in German than in French, but in both languages they performed the task faster when additions were presented in auditory than in visual format. In terms of accuracy, additions presented in both languages and presentation formats were performed equally well, except that seventh graders were less accurate in French than in German.

# Complex Additions

For the complex additions, overall mean RT was 4294 ms (SE = 124 ms) and overall mean CR was 79.2% (SE = 0.8%). We found an age-group effect on RTs [F(4, 185) = 14.008; p < 0.001; η 2 = 0.232] and on CRs [F(4, 185) = 5.976; p < 0.001; η 2 = 0.114], as participants from the older age-groups solved the complex additions faster and more accurately (see **Table 2**).

Regardless of task language complex additions were performed faster [F(1, 185) = 4.997; p = 0.027; η 2 = 0.026], and

TABLE 3 | Means of reaction times (RT) in ms and correct response rates (CR) in % with standard errors for each presentation format of the simple and complex additions as a function of age-group.

more accurately [F(1, 185) = 245.736; p < 0.001; η 2 = 0.571], when presented visually than in auditory format, see **Table 3**. Moreover, for CRs the format effect was modulated by age-group [format × age-group interaction: F(4, 185) = 8.365; p < 0.001; η 2 = 0.153]. Decomposition of this interaction showed that participants became more accurate with age for auditory presented additions. However, CRs remained similar across age-group for visually presented additions. This led to a progressively smaller error rate difference between visual and auditory formats with age-group (see **Table 3**).

In general, complex additions were also performed faster and more accurately when the task language was German than when it was French, RT: F(1, 185) = 201.922; p < 0.001; η 2 = 0.522 and CR: F(1, 185) = 113.630; p < 0.001; η 2 = 0.381, see **Figures 2C,D**. However, this language effect was modulated by the presentation format, both in terms of RTs, F(1, 185) = 10.729; p = 0.001; η 2 = 0.055, and CRs, F(1, 185) = 19.657; p < 0.001; η 2 = 0.096. Firstly, results from the pairwise comparisons on the RTs showed that even if complex additions were always performed faster in German than in French, the effect of the format (i.e., visual vs. auditory) was only significant for French, F(1, 185) = 9.867; p = 0.002; η 2 = 0.051, but not for German, F(1, 185) = 0.105; p = 0.746; η 2 = 0.001. Hence, auditorypresented additions were performed slower than visually presented additions only in French, see **Figure 3A**. Secondly, results from the pairwise comparisons on the CRs showed that languagerelated accuracy differences were larger in auditory, F(1, 185) = 109.919; p < 0.001; η 2 = 0.373, than visual presentation format, F(1, 185) = 28.536; p < 0.001; η 2 = 0.134, see **Figure 3B**.

Thus, when task language was French, participants were slower for additions presented in auditory compared to visual format, but the presentation format did not modulate RTs in German. Additionally, additions of the German session were always solved more accurately than additions of the French session and this effect of task-language was more pronounced for additions presented in auditory format. Finally, regardless of presentation format, task language also interacted with the agegroup on the RTs, F(4, 187) = 5.317; p < 0.001; η 2 = 0.103, but not on the CRs, F(4, 187) = 0.194; p = 0.941; η 2 = 0.004. Indeed, response times of both language sessions became increasingly similar with age, see **Figure 2C**.

When considering the above analyses it appears that the variability was different across age groups. Levene's test for homogeneity of variances across groups was indeed significant, as the younger groups performances were more heterogeneous than the older groups' (see standard errors reported in **Table 2**). This characteristic of the data is typical for transversal developmental comparisons, but it might have impacted the above-mentioned results and masked some interactions between age groups and task-language and/or presentation format effects. To cancel any potential influences of variance heterogeneity we therefore reconducted the same analyses after a standardization of the data per age-group. The results of this additional analysis are detailed in the Annex 1 of Supplementary Material.

To sum up results on both raw and standardized data, bilingual participants of all five age groups solved simple additions faster in German than in French. Moreover auditory format simple additions were performed faster than visual format additions in both languages1 . In contrast age group impacted the accuracy of simple addition solving, as seventh graders were overall less accurate in French than in German. This finding was confirmed

1Direct differences of RTs between formats of presentation can only be interpreted as modality-related measurement differences because RT recording started at the end of stimulus presentation in auditory format and at the beginning of stimulus presentation in visual format.

by the z-score analyses, which revealed that only participants of the tenth grade onwards solved simple additions with equal accuracy in both languages, even if they remained always slightly faster in German than in French.

Furthermore, complex additions were performed faster and better in German than in French. Critically, age group impacted RT differences observed when bilingual participants solved complex additions in German compared to French. Nevertheless, additional results on z-scores showed that task-language effect on RTs did no longer interact with age-group after standardization of the data. Thus, the effect of task language on RTs and CRs remained constant proportionally across age-groups.

Concerning presentation format, even though the differences of RTs cannot be interpreted per se, results on CRs showed that participants made more errors in auditory format than in visual format, especially in French compared to German were the CRs difference between formats was smaller. And this effect interacted with age-group as participants became more accurate for auditory-presented additions with increasing age. This last interaction between format and age-group remained significant after standardization of the data, suggesting that participants' ability to solve auditory presented additions genuinely improves with age.

# Effects of Number Word Structure

#### Base-10 vs. Base-20 Tens

Here we only report effects and interactions involving the problem size factor because other effects and interactions were already explained in detail in section Effects of Calculation Complexity (Simple vs. Complex Additions). In general, we observed lower CRs [F(1, 185) = 85.196; p < 0.001; η 2 = 0.315] and slower RTs [F(1, 173) = 151.138; p < 0.001; η 2 = 0.466] with problems over 70 than with problems under 70. Moreover, problem size interacted with the task-language both in RTs, F(1, 173) = 16.327; p < 0.001; η 2 = 0.086, and CRs, F(1, 185) = 52.912; p < 0.001; η 2 = 0.222.

To decompose this interaction, we ran pairwise comparisons. The problem size effect on the RTs was larger when the task language was French [F(1, 173) = 91.565; p < 0.001; η 2 = 0.346] than when it was German [F(1, 173) = 64.836; p < 0.001; η 2 = 0.273], see **Figure 4A**. In terms of CRs, problem size effect was only significant in French [F(1, 185) = 105.613; p < 0.001; η 2 = 0.363] but not in German [F(1, 185) = 2.878; p = 0.091; η 2 = 0.015], see **Figure 4B**. Further, the difference in CRs between German and French was smaller in problems under 70 [F(1, 185) = 11.763; p = 0.001; η 2 = 0.060] than in problems over 70 [F(1, 185) = 140.200; p < 0.001; η 2 = 0.431]. Finally, the problem size factor did not interact with any other factor, not even the age-group, all Fs < 1 and ps > 0.1. Thus, task language strongly modulated the effect of problem size in the direction that problem size effects were more pronounced when the task was performed in French than in German.

When considering the above analyses it appears that participants generally responded slower in French, which was also their less mastered language. Thus, the greater problem size effect found in French could also be due to participants' weaker French proficiency, independently of the structure of number words in this language. To rule out this alternative explanation, we reconducted this analysis after a standardization of the data per language, see results in Annex 2 of Supplementary Material. In summary, interactions of language and problem size remained significant after standardization of the data per language, suggesting that differences of problem size effect observed between languages in raw data are not a consequence of bilinguals' differences between languages in terms of language mastery.

#### Units-Tens vs. Tens-Units

In general, more errors were made on tens than units, F(1, 110) = 10.283; p = 0.002; η 2 = 0.085. Moreover, the task language × error type interaction was significant, F(1, 110) = 56.194; p < 0.001; η 2 = 0.338, and pairwise comparisons showed that there were more errors on the tens than on the units when additions were presented in German, F(1, 110) = 50.108; p < 0.001;

η 2 = 0.313, but inversely, when additions were presented in French, there were more errors on the units than on the tens, F(1, 110) = 9.594; p = 0.002; η 2 = 0.080. Additionally, there were more errors on the tens in German than in French, F(1, 110) = 48.293; p < 0.001; η 2 = 0.305, and more errors on the units in French than in German, F(1, 110) = 46.711; p < 0.001; η 2 = 0.298, see **Figure 5**.

Furthermore, the format of presentation interacted with task language, F(1, 110) = 7.783; p = 0.006; η 2 = 0.066. Pairwise comparisons of German vs. French addition errors showed the same pattern of results in both presentation formation but more errors of both types were made in German than in French for the auditory presentation format, F(1, 110) = 8.002; p = 0.006; η 2 = 0.068. In contrast, more errors of both types were made in visual than in auditory presentation format in French, F(1, 110) = 12.791; p = 0.001; η 2 = 0.104. It should be noted that this last interaction does not change the conclusions yielded by the aforementioned complex additions results, as here the error rates only referred to percentage of errors on unit vs. 10 digits in the incorrect solutions.

In summary, in German more errors were produced on the tens (e.g., "twenty" in "four-and-twenty"), whereas errors concerned predominantly the units in French ("four" in twentyfour"). This pattern of results was present for both presentation formats of the additions but was even more prominent in auditory format when task was performed in German and in visual format when task was performed in French.

# Discussion

To provide new insights into the question of bilingual's arithmetic problem solving we tracked arithmetic performances in German-French bilinguals at five different stages of their bilingual development from adolescence to adulthood. Four agegroups of pupils attending secondary school and one group of young adults had to provide oral answers to simple (i.e., addends <10) and complex (i.e., addends >10) addition problems presented once in a visual format (Arabic digits) and once in an auditory format (spoken number words). Moreover, all participants performed experimental tasks both in German and French in two distinct language sessions. Task language had a direct influence on solving complex addition problems, whereas only much weaker language effects were observed when participants retrieved answers for simple additions. From adolescence to adulthood complex additions performance considerably improved in both German and French, with especially noteworthy gains of accuracy in auditory-presented calculations. Yet, for complex additions a substantial language-related advantage for German additions remained in highly proficient adult bilinguals both in accuracy and response times. In contrast, participants tended to retrieve simple additions comparably well in German and French with increasing bilingual proficiency. In addition, the specific number word structures of German and French also significantly impacted bilinguals' arithmetic performance. Due to the base-20 structure of large French two-digit words, calculations with large numbers over 70 were less well-succeeded in French than German. Furthermore, the tendency to make errors

involving the second position of the number word led bilingual participants to produce more errors on the units when calculating in French and more decade-related errors in German. Firstly, we will discuss how language globally affected task performance and then separately consider simple and complex addition solving. Secondly, we will debate upon the effect of number word structure on bilinguals' arithmetic skills.

# Effects of Calculation Complexity

Overall additions were performed faster and with fewer errors in German than in French. This task language effect seemed to persist even in highly proficient adult bilinguals. As German was learned first by all participants it can be considered as their predominant language. In addition, it was also their language of arithmetic acquisition. Our results are consequently in agreement with the fact that (a) relative language predominance is known to promote arithmetic performance in bilinguals (Marsh and Maki, 1976; McClain and Huang, 1982; Frenck-Mestre and Vaid, 1993; Geary et al., 1993) and (b) bilingual adults solve numerical problems more proficiently in the language in which arithmetic was learned (Bernardo, 2001; Salillas and Wicha, 2012). The results also fit with the idea of non-selective language activation in bilinguals. Thus, lower arithmetic performances in French might also—at least partially—be due to less efficient access for French (in general) than for the predominant German (Bialystok, 2009; Kroll et al., 2013, 2014).

Nevertheless, a more nuanced picture emerged when considering separately how performance in simple and complex additions varied between the increasing language proficiency levels. With simple additions (e.g., 4 + 3 = 7) seventh and eighth graders were still marginally less accurate in French than German. But all other participants from grade 10 and upward did not show any accuracy difference between German and French when solving simple additions. If arithmetic acquisition language alone would explain language-related differences in bilinguals' arithmetic performance, then we would have expected an advantage for simple additions in the German session persisting in all age groups. However, here we observed that after 3 years of math-classes in French (i.e., grade 10 and upwards) participants solved simple additions with equal accuracy levels in German and French. This suggests that in addition to the importance the language for arithmetic acquisition, the current language proficiency level modulated the ability to retrieve simple arithmetic facts. Once a certain proficiency level was reached in both of the bilinguals' languages, the initial advantage for solving simple additions in the language in which they had been acquired (i.e., German) no longer applied for accuracy rates, as participants attained ceiling performances for these very simple arithmetic problems. However, even in adults some response time differences between languages remained, though reduced in comparison to other age-groups.

In complex addition (e.g., 54 + 13 = 67), language-related performance differences were more prominent as responses remained slower and less accurate in French than in German in all groups, which is consistent with the idea that complex additions require more processing steps and are therefore more likely to be influenced by language (Beishuizen, 1993). At first sight, raw data analysis indicated that complex addition response times of both language sessions became increasingly similar with age. However, when group differences in variance were eliminated by data standardization it appeared that language-related performance differences in favor of German remained of similar importance across all age groups. Concerning German, even with mathematics taught in French during the entire secondary school years, we observed neither decrease nor stagnation of arithmetic performances in comparison to French. Thus, complex calculation proficiency in the first language (i.e., German) seems to pursue a continuous development independently of the language in which formal math education is taught.

Complex additions were also affected differentially by presentation format of the additions, whereas no substantial difference was observed in simple additions. Participants made always more mistakes with auditory-presented additions. But French still enhanced this effect, with participants making on average 34% (± 0.01% SE) errors when computing auditorypresented complex additions (vs. 22% (± 0.01 SE) errors in German). Over and above this interaction with task language, auditory-presented complex additions were succeeded less well than visually-presented ones. However, the auditory disadvantage gradually reduced with increasing age (even in standardization data). This relative improvement for auditory-presented additions that was specific to complex problems might be due to developmental trends in cognitive and verbal abilities combined with a prolonged exposure to complex addition solving and an increasing math expertise. Indeed, as attested by the ceiling performances observed in simple additions, all participants were perfectly skilled to retrieve arithmetic facts, coherently with the common observation that children usually achieve arithmetic fact retrieval around the age of 8 years onwards (Barrouillet and Fayol, 1998; Butterworth, 2005). Although, participants' performances on complex additions did not reach any ceiling and continued to improve across age-groups in both languages. These observations fit well with the idea that solutions for complex additions cannot be retrieved directly from memory, even in adults (Ashcraft, 1995).

For this type of complex arithmetic computation, factors such as procedural knowledge, planning and working memory are known to play critical roles (Fürst and Hitch, 2000). In auditory presentation format, the additional need to keep the heard addends in working memory may interfere with using the phonological loop in the computation process. Consequently, participants made more errors for auditory presented additions than visually presented additions. This format effect in complex additions was especially pronounced when performing the additions in French, i.e., a language that was relatively less proficient (LeFevre et al., 2001) and/or distinct from the language of arithmetic acquisition. These findings nicely highlight the involvement of language in the numerical processing underlying complex additions. If participants had simply computed the results in their first language (i.e., German) and then translated them to the output language (i.e., French) this would have affected performance similarly in both the visual and the auditory presentation formats. But contrary to this prediction performance specifically dropped when participants computed auditory-presented complex additions in French. This may be due to the fact that arithmetic was learned in German or to globally weaker proficiency level in French. Due to the specific differences between number word structure in French and German languages, the interaction might also (at least partially) result from differences between French and German number naming systems. In the following paragraphs, we will further discuss the latter effects and their relation to arithmetic in German-French bilinguals.

# Effects of Number Word Structure

Languages differ in the way they construct two-digit number words (Campbell and Xue, 2001). This may directly influence bilinguals' addition skills and/or interact with other factors such as bilingual proficiency level and arithmetic acquisition language. Evaluating how arithmetic problem solving is influenced by number word structure in German-French bilinguals is particularly interesting because those both languages encounter two major differences in their number naming systems. Firstly, twodigit number words follow a unit-ten order in German (e.g., "24" = four-and-twenty) but a ten-unit order in French (e.g., "24" = twenty-four). Secondly, the 10 words for the numbers over 70 follow a base-10 structure in German (e.g., "72" = twoand-seventy) but a base-20 structure in French (e.g., "72" = sixty-twelve).

To characterize the effect of number word differences between languages on arithmetic performances, we conducted additional analyses on complex additions. Firstly, we focused on the base-10 vs. base-20 structure of large two-digit number words. Additions involving numbers under and over 70 were analyzed separately, since number words under 70 follow a base-10 structure in both language but number words over 70 follow a base-ten structure in German and a base-20 structure in French (e.g., "72" is pronounced as "sixty-twelve"). Not surprisingly, additions over 70 were solved overall slower than additions under 70 in both languages, confirming the classical problem size effect (Groen and Parkman, 1972). Nevertheless, the response time difference between additions under and over 70 was larger in French than in German. Moreover, in terms of accuracy, participants made more errors for additions over 70 than additions under 70 in French, but not in German where errors rates in additions under and over 70 were similar. Interestingly, these results were observed regardless of additions' presentation formats and participant groups. The latter observation demonstrates that the base-10 vs. base-20 effect is not modulated by bilingual proficiency groups. Nevertheless, it remains to be empirically determined whether specific difficulties for number words also occur in French-German bilinguals with French as first language. These findings confirm the early reports by Ellis and Hennelly (1980) that bilinguals' arithmetic skills are inevitably marked and modulated by the number word structure of the language in which they are currently calculating. In line with the present results, recent behavioral and electrophysiological studies indicate that these language-related characteristics might even impregnate basic number representations (Pixner et al., 2011; Salillas and Carreiras, 2014).

Secondly, we analyzed the type of errors participants made, namely whether more errors were made on the tens or on the units across different languages, presentation formats, and bilingual proficiency groups. As noted above, the number word structure in French and German differs in terms of which digit is pronounced at first in two-digit number words (ten vs. unit). It appeared that participants systematically produced more errors on the ten digit (e.g., "2" in "24") when calculating in German and more errors on the unit digit (e.g., "4" in "24") in French. Again, the presentation format and the group of participants did not modify this result. Thus, independent of the calculation language, errors seem to predominantly concern digits holding the second position of the solution number. These findings elegantly show how a language-independent focus on the first segment of number words can lead to qualitatively distinct numerical outcomes within different language contexts. Taken at face value, they imply that making calculation errors while computing prices in the range between 18 and 100 will become more expensive for a German- than for a French-speaking person.

# General Considerations

Literature provides divergent conclusions about the level at which bilinguals' different languages are involved in number processing and about the language in which bilinguals actually solve arithmetic problems. Many factors such as age of acquisition of the second language, language of teaching during school years and currently used language seem to determine the use of the language during arithmetic problem solving in bilinguals (Bernardo, 2001; Campbell and Epp, 2004; Salillas and Wicha, 2012). Investigating arithmetic performance in Luxembourgish adolescents and young adults who become highly proficient German-French bilinguals through the school system offered the rare opportunity to study large groups of bilingual participants at different bilingual proficiency levels who are homogeneously composed with respect to the previous factors.

Our findings obtained with German-French bilinguals at five distinct levels of bilingual proficiency extend the current knowledge by confirming that language plays a critical role in the computations underlying complex addition (i.e., operands above "10") at all bilingual stages. Participants' skills in computing additions in both German and French improved steadily with increasing bilingual proficiency levels from grade 7 to young adulthood. Nevertheless, participants of all age groups solved complex German additions faster and more accurately than French ones. This German advantage remained although mathematics is taught in French during the entire secondary school years. It is probably due to the fact that German is participants' first school language and their arithmetic acquisition language and that complex additions are not automatized enough to be free of any language help along the solving process. In contrast, simple addition facts (i.e., operands below "10") were accessed more directly and similarly in both languages, especially at later stages of second language acquisition. Indeed accuracy levels for simple additions were similar in French and German from grade 10 upwards, while their response times got closer. Thus, highly proficient bilinguals tend to be able to retrieve addition facts similarly in both languages suggesting that bilinguals' arithmetic fact retrieval may become either independent from the verbal code or automatized enough in different languages' verbal codes to lead to similar performances (Campbell and Epp, 2004).

The second part of our study explored the role of number word structure in bilinguals' arithmetic performance. German-French bilinguals indeed speak two languages that are characterized by inverted ten-unit structures of two-digit number words (unit-ten vs. ten-unit number words) and with different constructions of tens over 70 (base-10 vs. base-20). Consequently the full effect of number word structure on arithmetic computation could be highlighted optimally in this type of bilingual population. When additions were computed in French, specific response-delays and error-increases were observed for calculations involving number words over 70. Moreover, results from error analyses showed that participants of all age groups always committed more errors related to the digit that occurred in second position in the number word, i.e., tens in German and units in French. Taken together, both differences in German vs. French number word structures (two-digit words with base 10 vs. 20 and direct vs. inverted digit order) seemed to play a role in arithmetic processing at all bilingual proficiency stages.

In conclusion, the present study demonstrates that both (a) language proficiency levels and (b) number word structure affect addition solving performances in bilinguals. This leads to

# References

the conclusion that arithmetic significantly relies on language processes, especially in complex computations. Further studies will be needed to generalize the present findings to other number processing tasks (e.g., magnitude comparison), other arithmetic operations (e.g., subtraction, multiplication,) and other tasks with number words (e.g., math word problems).

# Acknowledgments

This work was supported by the Langnum CORE project funded by the Luxembourgish Fund for Scientific Research (FNR, Luxembourg). The authors declare APA ethical standards were followed in the conduct of this study. Authors gratefully thank all the participants for their collaboration to the study, the school principals who gave the permission to recruit pupil participants and the students who helped to collect the data.

# Supplementary Material

The Supplementary Material for this article can be found online at: http://www.frontiersin.org/journal/10.3389/fpsyg. 2015.00265/abstract

**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2015 Van Rinsveld, Brunner, Landerl, Schiltz and Ugen. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Number word structure in first and second language influences arithmetic skills

#### *Anat Prior\*, Michal Katz , Islam Mahajna and Orly Rubinsten*

*Edmond J. Safra Brain Research Center for the Study of Learning Disabilities, University of Haifa, Haifa, Israel*

Languages differ in how they represent numerical information, and specifically whether the verbal notation of numbers follows the same order as the symbolic notation (in non-inverted languages, e.g., Hebrew, "25, twenty-five") or whether the two notations diverge (in inverted languages, e.g., Arabic, "25, five-and-twenty"). We examined how the structure of number–words affects how arithmetic operations are processed by bilingual speakers of an inverted and a non-inverted language. We examined Arabic–Hebrew bilinguals' performance in the first language, L1 (inverted) and in the second language, L2 (non-inverted). Their performance was compared to that of Hebrew L1 speakers, who do not speak an inverted language. Participants judged the accuracy of addition problems presented aurally in L1, aurally in L2 or in visual symbolic notation. Problems were presented such that they matched or did not match the structure of number words in the language. Arabic–Hebrew bilinguals demonstrated both flexibility in processing and adaptation to the language of aural– verbal presentation – they were more accurate for the inverted order of presentation in Arabic, but more accurate for non-inverted order of presentation in Hebrew, thus exhibiting the same pattern found for native Hebrew speakers. In addition, whereas native Hebrew speakers preferred the non-inverted order in visual symbolic presentation as well, the Arabic–Hebrew bilinguals showed enhanced flexibility, without a significant preference for one order over the other, in either speed or accuracy. These findings suggest that arithmetic processing is sensitive to the linguistic representations of number words. Moreover, bilinguals exposed to inverted and non-inverted languages showed influence of both systems, and enhanced flexibility in processing. Thus, the L1 does not seem to have exclusive power in shaping numerical mental representations, but rather the system remains open to influences from a later learned L2.

Keywords: L1, L2, bilingualism, number processing, addition

# Introduction

Bilingual speakers have control of two languages and hence raise important questions regarding language and cognitive representations and processing. Such questions include the degree to which two languages are represented or processed independently versus interactively (e.g., Kroll and Stewart, 1994; Costa, 2005), as well as the impact of language on cognitive representations more generally. The fact that languages differ in their structural properties provides an

#### *Edited by:*

*Yvette Renee Harris, Miami University, USA*

# *Reviewed by:*

*Silvia Pixner, UMIT-The Health and Life Science University, Austria Yukari Okamoto, University of California, Santa Barbara, USA*

#### *\*Correspondence:*

*Anat Prior, Edmond J. Safra Brain Research Center for the Study of Learning Disabilities, Faculty of Education, University of Haifa, Mount Carmel, Haifa 31905, Israel<EMAIL_ADDRESS>
#### *Specialty section:*

*This article was submitted to Developmental Psychology, a section of the journal Frontiers in Psychology*

> *Received: 13 September 2014 Accepted: 23 February 2015 Published: 17 March 2015*

#### *Citation:*

*Prior A, Katz M, Mahajna I and Rubinsten O (2015) Number word structure in first and second language influences arithmetic skills. Front. Psychol. 6:266. doi: 10.3389/fpsyg.2015.00266* elegant method of investigating to what degree cognitive representations, in the current case in the numerical domain, are uniquely shaped by the native language (L1) or rather remain open to influences from a later acquired second language (L2). This question has been previously addressed in the domain of semantic/conceptual representations. Thus, Jiang (2002, 2004) proposed a model according to which the conceptual system is mostly shaped by the L1, except in highly proficient bilinguals. In contrast, Degani et al. (2011) demonstrated sensitivity of semantic processing to L2 lexical properties, even in unbalanced moderately proficient bilinguals (see also Cook, 2003; Laufer, 2003; Wolff and Ventura, 2009). The role of the L1 vs. the L2 in shaping representations and processing has also been investigated in the numerical domain (Gelman and Butterworth, 2005), again leading to conflicting results.

For example, Macizo et al. (2010) showed that the processing of number words in one language was not modulated by the way bilinguals processed number words in their alternative language, which differed in the structure of number words. In contrast, it has been consistently shown that even proficient L2 speakers resort to their L1 to perform mathematical operations (e.g., Spelke and Tsivkin, 2001). In addition, Salillas and Carreiras (2014) recently demonstrated specific influences of the structure of the language in which early math instruction occurred on the processing of numerical information in highly proficient balanced bilinguals.

These conflicting results raise the question to what degree people can learn to process numerical information according to the structure of their L2, when it differs markedly from that of the L1, specifically when they are highly proficient L2 speakers. The current work extends this controversial line of research by investigation the impact of both the L1 and the L2 on numerical representation and calculation in highly proficient bilinguals, whose languages differ in the structure of number words.

This question is of central importance, because arithmetic processing in monolinguals is closely linked to language (Dehaene, 1992; McCloskey, 1992; Campbell, 1994; De Smedt et al., 2010; Archibald et al., 2013 ). Thus, it has been shown that the four basic operations (addition, subtraction, multiplication, and division) are learned in school with different emphasis on quantity manipulations and on linguistic skills (Dehaene and Cohen, 1995; Delazer et al., 2006; Ischebeck et al., 2006), with multiplication and addition being retrieved from verbal memory but subtraction and division requiring manipulation of quantities (e.g., Zhou et al., 2006). In general, it has been suggested that with advanced age and practice, counting and using quantity knowledge to achieve an outcome is replaced as the strategy of choice by memory retrieval, similar to the way words are retrieved from the verbal lexicon, at least in cases of addition and multiplication (e.g., Delazer et al., 2006; Ischebeck et al., 2006). However, it should be noted that Pesenti et al. (2000) and Venkatraman et al. (2005) did not find any language-related frontal activations for symbolic exact arithmetic involving simple addition problems, suggesting that different strategies, other than retrieval from verbal memory, may be in use.

These findings lead to fascinating questions concerning the cognitive mechanisms underlying mathematical operations in proficient bilinguals, especially when information is presented in the L2. For example, when doing arithmetic in the L2, do bilinguals rely on the linguistic structure of that language, and how do these processes interact with the L1? These questions address the fundamental issue of whether human cognitive capacities related to the L1 and the L2 employ a shared or independent cognitive system. Numerical knowledge acts as a natural and ecological laboratory for the study of L1/L2 interactions, as bilinguals have three sets of symbols to represent the same semantic concept: written or symbolic digits (3), L1 number words (e.g., *shalosh* in Hebrew or *talate* in Arabic) and L2 number words. This makes it possible not only to study translation from L1 to L2 and from L2 to L1 but also from a common semantic meaning to written or verbal forms in either language. In the current study we extend the examination of bilingual cross-language interaction by asking whether the structure of number words in one language is modulated by the way bilinguals process numbers in the alternative language.

Languages differ in the structure of number words and how they are used, and such differences can shape the way in which speakers of a certain language process numbers. Thus, several studies set out to examine the idea that variability in mathematics performance may be related to differences in the cognitive organization of numbers that is affected by number–word characteristics of a language. Thus, number words in Chinese, Japanese, and Korean are congruent with the traditional base 10 numeration system, such that the spoken number corresponds exactly to the implied quantity represented in the written form (i.e., the number 49 is written in character symbols as four-10s-nine). Number words in English, on the other hand, may lack the elements of 10s and ones that are contained in them (i.e., the number 12, twelve). Miura et al. (1988) found that whereas first grade native speakers of English preferred to use a collection of unit blocks to represent numbers, speakers of base-10 languages more frequently used a construction of 10s and ones, in correspondence with the linguistic structure (see also Fuson and Kwon, 1992; Miller et al., 1995; Geary et al., 1996).

In a more recent study, Colomé et al. (2010) compared Italian and Catalan speakers. Italian is a base-10 language while Catalan number-words are constructed by combining multiples of 20 with units or with teens (e.g., the verbal representation of 35 is "twenty and fifteen"). Their results showed a consistent difference between the two groups in their preference toward a certain number–word structure when solving problems verbally and when typing their answers in Arabic numerals. The researchers concluded that language differences in the structure of number–words play a role when solving addition problems.

The current study focuses on the property of inversion, coined by Zuber et al. (2009) to describe the situation when the order of the symbolic and verbal notation of a number are inverted. For example, the number "25" in inverted languages is pronounced as "five and twenty." The inversion property affects all two-digit numbers from 21 to 98, repeats for the 10,000s, and is a feature of various languages such as Arabic, Danish, Dutch, and German. There is evidence showing that children who speak languages with inversion have difficulty in basic numerical transcoding tasks, namely the ability to translate numerals from one form to another, such as the Arabic notation "*27"* to verbal notation "*seven and twenty"* (e.g., Pixner et al., 2011; Imbo et al., 2014). Difficulty in such tasks, probably due to the multiple inversions required in representing two-digit numbers, consequently leads to working memory overload (Zuber et al., 2009).

In adults, Brysbaert et al. (1998) tested the theory that numerical addition is based on language processes by comparing French and Dutch-speaking participants solving addition problems such as 20 + 4 and 4 + 20 (unit + decade) and 21 + 5 and 5 + 21 (unit + decade-unit) presented either as Arabic numerals or as number words. The French participants solved operations like "20 + 4" and "21 + 5" faster than their counterparts in the other order, both when presented with Arabic digits and with number words. The Dutch participants differed in their performance; operations like "20 + 4" were preformed faster in this order only when presented in the numeric format, but no differences between "20 + 4" and "4 + 20" were found for the verbal format. When the operation consisted of decade-unit + unit (21 + 5), they were faster to answer in the inverse order (5 + 21). The results demonstrated some differences in preference for order of operations based on language when the questions were presented in written verbal form and the participants were asked to respond verbally. On the other hand, the two groups did not differ significantly in their responses when asked to type the answers numerically. The authors concluded that the numerical system is largely autonomous of the language system.

However, a later study by Nuerk et al. (2005) compared English and German speakers' performance in magnitude comparison. Two numbers were presented above each other on a computer screen and participants were asked to determine which number was larger. This study showed influence of the inversion property in a unit-decade compatibility effect. This compatibility effect is found when two-digit Arabic numbers are compared, such that cases where separate decade and unit comparisons lead to the same decision (e.g., 32\_47; in this case 3 *<* 4 and 2 *<* 7) are processed faster than incompatible trials (e.g., 37\_52; in this case 3 *<* 5, but 7 *>* 2). According to McCloskey (1992), there may be separate mental number line representations for decades and units which, in turn, may be separately processed in two-digit number comparison. If this is true, comparing a pair of incompatible numbers could be a more difficult and lengthy process than comparing a pair of compatible numbers.

In the above mentioned study, Nuerk et al. (2005) investigated the generality of the compatibility effect by comparing English and German speakers. They found that while for native German speakers the compatibility effect is much larger for large unit distances than for small unit distances, for native English speakers the compatibility effect is larger for small decade distances than that of the German speakers. Moreover, large unit distances and small decade distances led to disproportionately more errors for English participants but not for German participants. The authors therefore concluded that decade distance seemed to determine responses in English speakers, while overall distance was the most important predictor for German speakers, particularly when dealing with written number words. Thus, the lexical representation in a language influences magnitude comparison even when numbers are presented in a non-linguistic format.

A recent study conducted by Macizo and Herrera (2010) strengthens this conclusion by testing Spanish speakers' number processing when presented with two-digit number words in reverse form (unit-decade order, e.g., five-and-twenty). In each trial, one number word was presented above the other in the center of the screen and the participants had to select the larger of the two numbers. Based on the effects of the decade distance and the compatibility effect, the results showed that only decade distance was a significant predictor for difference in reaction time (RT). The authors concluded that speakers of non-inverted languages have learned a language-dependent process for analyzing written numbers in which decades have a major role regardless of the position in which they are presented experimentally. These findings reinforce the theory that the spoken language does in fact affect the way in which numbers are processed when presented in both numeric and verbal form.

To date, there is only a handful of studies that have taken a close look at number processing in bilinguals who speak both an inverted language and a non-inverted language. These studies have suggested that bilinguals process two-digit number words selectively in their L1 and L2 and that they do not seem to transcode number words from their L2 into Arabic number format. In other words, most studies have found that the processing of number words in the L1 does not influence the way bilinguals process number words in their L2 (Macizo et al., 2010, 2011). Macizo et al. (2010) examined the way Italian/German bilinguals performed a number comparison task by presenting them with compatible and incompatible number–word pairs in their two languages. Participants were faster when presented with compatible pairs than incompatible pairs in German, while they were slower when presented with compatible pairs than incompatible pairs in Italian. The authors concluded that bilingual speakers are not bound to the number-structure of their L1 and the relative reliance on the decade and unit values differ depending on the language of presentation; when processing number–words in an inverted language, they rely on the unit values, and when processing number–words in a non-inverted language, they rely more on the decade values.

A more recent study by Macizo et al. (2011), also investigated between-language influences by comparing Spanish/English and German/English bilinguals' performance on a number comparison task. Their results show that both bilingual groups presented a reverse compatibility effect when performing the comparison task in the L2 (a non-inverted language) but differed in the way they processed L1 numbers. A reverse compatibility effect was observed in the L1 Spanish task for the Spanish/English bilinguals (an expected pattern for a non-inverted language), and a regular compatibility effect was observed in the L1 German task for the German/English bilinguals (an expected pattern for an inverted language). The finding that bilinguals processed two-digit number words selectively in their L1 and L2 means that bilinguals are influenced by the language of presentation and process numbers according to the expected pattern for each language.

Taking such recent findings into account, the question that remains unanswered is whether or not cross-language influences exist in other numerical processing tasks, namely in arithmetic calculation. We investigate this question by presenting Arabic/Hebrew bilinguals and native Hebrew speakers with addition problems composed both in visual–symbolic notation and in aural–verbal presentation. Similar to Brysbaert et al. (1998) we manipulated the order in which the elements of the addition problems were presented (20 + 5 vs. 5 + 20) such that they did or did not match the structure of number words in the language. To our knowledge, this was the first study to use number words in an aural–verbal format instead of presenting number words in a written format. Thus, the current study will test the influence of language on number processing by examining the effect of the structure of number words in a language on processing addition problems, as well as the susceptibility of speakers of inverted and non-inverted languages to decade and unit numerical values.

We are particularly interested in whether the organization of numerical processing is determined by one's L1, which in this case is also the language of math instruction, or whether it is open to influences from the L2 as well. If the former is true, the performance of Arabic speakers in both aural–verbal presentation and in visual–symbolic presentation should reflect the inversion property of their L1. However, if the latter is true there are two possible patterns: they might show different preferences depending on the language in which the problem is presented. The second option is that Arabic–Hebrew bilinguals in the current study might show enhanced flexibility in processing, such that they become less sensitive overall to differences between presentations that match inverted or non-inverted structures.

# Materials and Methods

#### Participants

Sixty three students from the University of Haifa participated in the study: 31 Arabic–Hebrew bilinguals (22 women, mean age 22) and 32 Hebrew–English bilinguals (20 women, mean age 26). Participants were recruited through flyers and online ads. Participants gave informed consent and were paid 30 NIS an hour (45–60 NIS in total). The study was approved by the research ethics committee of the University of Haifa. All participants included in the study reported no history of language and\or numerical disabilities.

#### Materials

## Language Experience and Proficiency Questionnaire (LEAP-Q)