Transparency of power

Transparency of power refers to the existence of different degrees of morpho-syntactic language transparencies in verbal numbers. In many Asian languages (i.e. Mandarin Chinese, Vietnamese) the morpho-syntactic structure of the verbal number system is highly consistent with the Arabic number system, and in general with the base-10 (i.e. 10*x + y, where x indexes the base and y the unit, see [21]). This linguistic characteristic, also termed transparency of power, facilitates learning to count [22–24] and solving arithmetic problems [25–27]. Another morpho-syntactic difference concerns the inversion between tens and ones as for example in Dutch and in German (i.e. y + 10*x; in German 42 is “zwei-und-vierzig”, literally “two-and-forty”). This linguistic characteristic, called transparency of order [28], explains some transcoding error patterns and reaction times [17, 18, 29–33], gives rise to specific pattern of compatibility effects [34–36] and complicates the solution of certain arithmetic problems [37–39].

Transparency of power can also vary with a change of base, for example a base-20 system (20*x + y), also called vigesimal. The use of the vigesimal system is found for example in French, Basque, or in Diola-Fogny (see [40]). French, to be precise, has a mixed system since it uses base-10 and base-20 systems. Up to the `60s in French, like in English, the counting system follows the base-10 rule. However, `80s to `90s follow a base-20 system, e.g. 87 = 4*20 + 7, “quatre-vingt-sept”, literally “four-twenty-seven”. Moreover, the teens (11 to 16) are lexical primitives, like in English and in stark contrast to the more transparent Asian languages. Note that the transparency contrast is additionally increased for 71 to 76 and 91 to 96, which are composed with a base-20 and teens, e.g. 96 = 4*20 + 16, “quatre-vingt-seize”, literally “four-twenty-sixteen”. Furthermore, the vigesimal system is subject to regional variances, for example in Belgian Wallonia `70s and ‘90s are not vigesimal (i.e. “septante” for “seventy” and “nonante” for “ninety”, see [41]) and in some Swiss-French cantons the vigesimal system is entirely absent (i.e. including for 80, “huitante”, literally “eighty”). In a study comparing French-speaking 1^st graders from Wallonia and France with a number dictation task, more mistakes were committed in the `70s and `90s by the latter [41]. In another study comparing English-speaking to French-speaking 5^th graders, numbers above 60 were slower to transcode, revealing a cost for base-20 numbers [42]. Further indicating an interaction between the number structure and number processing, Basque-speaking adults solved additions faster when the operand was in the base-20 structure. For example, the solutions for 20 + 15 is facilitated over 25 + 10, since the result 35, is said “hogeita-hamabost”, literally “twenty-fifteen” [43]. Strikingly, for Basque-speakers the distance effect also leads to different event-related potential brain responses for vigesimal compared to decimal Arabic digits [44]. While providing interesting insights into the neuro-cognitive mechanisms of vigesimal number processing, these studies might be limited by potential cultural and educational confounds associated with group comparisons [25] or curricular differences [45]. Hence, between-subject comparisons should be complemented by within-subject designs with multilingual participants. Furthermore, they do not provide information about the developmental trajectories of the acquisition of a language with a different number system since they focus on single age groups. They consequently need to be completed by designs comparing different age groups of multilinguals.

To sum up, the transparency of power refers to the morpho-syntactic structure of a language’s number word system. Results with different populations and using various tasks consistently indicate an advantage for processing numbers in more transparent languages. While some studies started to explore the impact of vigesimal number structures, only a few studies focused on within-subject designs with bilinguals. And to the best of our knowledge, studies on the developmental trajectory in bilingual learners are still entirely missing.

Language of mathematics acquisition

In a questionnaire asking multilinguals which language they preferentially use for mental calculations, the majority reported a preference for their first language L1, supposedly also their language of math acquisition (LM+) [46]. In contrast, solving arithmetic problems in the non-preferred language resulted in cognitive costs for vocal answers to arithmetic problems presented as Arabic digits [47, 48], auditory [49], or written number words [50]. While highlighting the impact of language dominance when doing math, these designs confound general L1 benefits with potential domain-specific benefits from the language in which mathematic was first learned [51].

Training experiments with bilinguals doing arithmetic in both languages indeed indicate a benefit for solving arithmetic problems in the language in which they were trained. Spelke and Tviskin [52] investigated Russian-English bilinguals, who were trained to solve arithmetic either in the dominant language, L1 (Russian) or their L2 (English). The results showed a cost for solving arithmetical problems when switching to the untrained language, independently if the testing was in the L1 or L2, indicating a Language Switching Cost (LSC). LSC in the context of math training was replicated in 9^th and 11^th graders attending German-French bilingual education curricula, who were trained to solve arithmetic in German or French [53]. Similar LSC was found in German-French [54] and German-English bilingual adults [55]. Hence, independently from language dominance, arithmetic and mathematical problem solving are facilitated when tested in the same language as they are learned, and they are accompanied by a cost in the untrained language. These findings underline the importance of how multilingual education school curricula are designed.

The LSC generalizes to more ecological learning settings, showing that mathematical problems are solved more accurately in LM+, even if it is not the dominant language (i.e. the LM+ not being the same language as the L1). For example, Bernardo [56] investigated arithmetic among high school Filipino-English bilinguals who have Filipino as L1 but specifically learned mathematics in English (LM+). The results indicate a cost for arithmetical problems written in number words in Filipino (i.e. being the L1 but LM-) compared to English (i.e. being an L2 but LM+), which in turn showed comparable results than to problems presented as Arabic digits. The critical role of LM+ or LM1 is also confirmed by studies on highly proficient bilinguals. Note that here we make a distinction between the LM+, defined as the (only) language of learning mathematics, and LM1 defined as the first language of learning mathematics (which is followed by other languages used later in the learning process). For example in school curricula where the first language for learning mathematics is German (LM1) and math classes later switch to a second language (French, LM2), systematic costs have been found for LM2 arithmetic problem solving despite an equal number of years training the LM2 [57]. A recent meta-analysis found an advantage for solving arithmetic and naming numbers (but not for magnitude comparison tasks) in the L1 [58].

These behavioural findings are confirmed by recent neuroimaging studies on arithmetic and number comparison. Recording electroencephalogram during a true or false judgment of simple multiplications, Salillas and Wicha [59] studied fluent Spanish-English bilingual adults who had learned arithmetic in either English or Spanish, the LM+ respectively. The problems presented in the LM+ were solved faster and the corresponding event-related potentials showed a larger N400 response than for problems in LM-, independently of the language representing LM+ and LM-. Assessing two-digit number comparison in Spanish-Basque bilinguals [44], could even show that the numerical distance effect was modulated by the language of math acquisition. Functional magnetic resonance imaging studies revealed that the LM1 recruited more temporal regions, supposedly related to direct semantic retrieval, than the LM2 for simple additions. In turn, the LM2 recruited a network of regions indicating the need for more generic cognitive resources [60]. On the contrary, Cerda et al., [61] recently investigated Spanish-English bilingual children’s performance in a multiplication verification task and observed similar ERP responses in both of their languages. Although the language of math acquisition largely impacts bilingual’s arithmetic skills, under certain conditions, such as early learning stages, the bilingual brain might consequently reveal some flexibility.

These results tend to support the above-mentioned TCM stating that precise numbers are encoded in a language-dependent format [62]. They fit less with the classical view from bilingualism research stipulating that representations are independent of languages (e.g. [63, 64]) and that both bilingual’s languages are active, even in situations when only one language is needed (e.g. [65, 66]). However, they agree with recent reports from the bilingualism literature that academic knowledge acquired in one language is retrieved more efficiently in the learning language compared to another language [54]. Further research on numerical cognition is consequently needed to fully understand this complex bilingual situation. Studying how language of math acquisition influences number transcoding is especially interesting since the cognitive mechanisms of transcoding are closely related to word retrieval. Yet, such studies are still missing, both in adults and in children.

Present study

The present study aimed to better understand the interaction between language and numbers by investigating number transcoding of German-French bilinguals from four different age groups. We targeted 5^th, 8^th and 11^th graders, as well as adults to assess performances before and after the switch in the language of mathematical learning implemented in 7^th grade in the Luxembourgish education system.

In the Luxembourgish multilingual school system, pre-schools (3 to 5 y.o.) are in Luxembourgish, which is linguistically close to German. The teaching language in primary school (1^st to 6^th grade, 6 to 12 y.o.) is German, except for French lessons. From 7^th grade to 10^th grade, the majority of subjects are taught in German, except for mathematics and French lessons, which are taught in French. Then from 11^th grade until the end of obligatory school, all topics are thought in French. This multilingual education system aims to render students highly proficient in both German and French. In sum, critically for number transcoding, the language for teaching and learning mathematics switches from German to French at 7^th grade [67]. Hence students’ first language of math acquisition (LM1) is German, while their second language of math learning (LM2) is French. Therefore allowing within-subject designs from a sample with the same educational background. Furthermore, German-French bilingualism is interesting concerning number word structures since both languages differ in transparency of order and their transparency of power. Previous studies on this specific population have reported language effects in magnitude comparison tasks, showing comparable compatibility effects to monolingual German [68]. While arithmetic problems were solved faster in German than in French [48], at least for complex additions [57]. Since arithmetic correlates with transcoding [17], we expected similar findings with transcoding tasks. The present study investigates the question of the role language proficiency has on number transcoding. Using a cross-sectional design allowed us to study whether and how (a) number word transparency of power and (b) language of math acquisition influence two-digit number transcoding at different stages of bilingualism.

We explored the effect of transparency of power by comparing transcoding of French numbers above 70 (`70s `80s `90s) and below 60 (`30s `40s `50s), following respectively a base-20 and a base-10 structure. We expected that transcoding performances in both bilinguals’ languages would improve with age. We hypothesized that independently of bilinguals’ age, language would influence number processing such that non vigesimal French numbers (following a base-10 structure) would be transcoded better than vigesimal numbers over 70 (following a base-20 structure), revealing an effect of transparency of power.

For the impact of language of math acquisition we compared the performances in both of the bilinguals’ languages in the four age groups. Transcoding requires to access and retrieve lexical information on numbers stored in long-term memory. Therefore, we predicted that transcoding would be better in German (LM1) than French (LM2). To capture the different facets of this retrieval process [69], we deployed two complementary transcoding tasks. In the reading aloud task participants had to name Arabic digits, while the verbal-visual matching task required the matching of spoken number words with the corresponding Arabic digit. Both tasks are assumed to tap into distinct retrieval mechanisms, i.e. free recall and recognition respectively. The two tasks might thus reveal a somewhat different result pattern such as more marked linguistic influences in the reading aloud free recall task than on the verbal-visual recognition task, as already observed in Van Rinsveld et al. [68].

Participants

From the initial full sample of 125, we first excluded participants who reported having French as their mother tongue, as these participants might have acquired French number words outside the context of formal schooling. This led to the exclusion of 25 participants. One additional adult was removed because of an otherwise missing measure in one of the crossed factors. Therefore we excluded a total of 26 participants, leading to a final sample of N = 99. This final sample consisted of four age groups: 5^th graders n = 26, age = 10.69(0.55), 8th graders n = 28, age = 13.46(0.56), 11^th graders n = 25, age = 16.48(0.59), adults n = 20, age = 23.58(5.12). The language profiles varied among the participants, 66 reported Luxembourgish as mother tongue, 5 Portuguese, 4 German, 4 English, 4 Italian, 3 Polish, and the 13 remaining all reported a different mother tongue.

Despite the different language backgrounds, all participants were currently living in Luxembourg, where Luxembourgish, German and French are official languages that can be encountered in daily life. Importantly the language support for the formal acquisition of mathematics through school curricula is first in German (LM1) from 1^st to 6^th grade and switches to French (LM2) in 7^th grade. Therefore young pupils start the school curriculum with different degrees of proficiency in French and German to progressively become proficient bilingual adults throughout their education. The four age groups of German-French bilinguals were therefore sampled at different ages of LM2 (French) acquisition: 5^th graders being before the language switch in mathematics, while the three older groups have gradually increasing experience and familiarity with practicing mathematics in French.

All pupils were enrolled and tested in schools, while the adults were enrolled and tested at the university. The Review Panel of the University of Luxembourg revised and approved the study. Informed written consent was obtained from all the children’s parents and adult participants. Monetary compensation after study completion was given to both pupils and adults.

Material and stimuli

Headsets connected to the Iolab USB Button Box were used to both (a) record the voice onsets of the reading aloud task and (b) play the auditory stimuli for the verbal-visual matching task. The experiments were run with Psyscope XB7 [70] on an Apple 13’ MacBook. Reaction times were measured with the Iolab USB Button Box from the end of the auditory recording until button press. Stimuli for each task were 28 different two-digit numbers ranging from 31 to 98. These were further subdivided into two sets of 14 distinct stimuli each, one for the German and the other for the French blocks. The experiment was part of a larger study, which lasted 45 minutes, at the end of which the participants were compensated with a gift voucher. Ties (e.g. 44) and tens (e.g. 30) were excluded from the sets. The list of all the stimuli used can be found in Table 1 in S3 File. The order in which the sets were presented was randomized. In the reading aloud task, the Arabic digits were presented visually on a computer screen. They appeared in the centre of a white screen in black (Arial, font size 90) and remained visible until participant responses. For the verbal-visual matching task, German and French recordings of two female native German and French speakers were presented as auditory stimuli (as in [42]). In the verbal-visual matching task, there were four possible visual Arabic numbers: one target and three distractors, which positions varied randomly. The three distractor-stimuli consisted of: unit distractors in which one unit was randomly added or subtracted to the heard number (e.g. for 42 distractors were 43 or 41). Ten distractors in which a ten unit was randomly added or subtracted to the heard number (e.g. for 42 the distractor was 32 or 52). Unit and ten distractors in which 11 were randomly added or subtracted from the target number.

Procedure

All participants were tested individually in a quiet room, children in the schools, and adults at the University of Luxembourg. The participants passed both the reading aloud and verbal-visual matching tasks first in German or French in a counter-balanced order. Both tasks started with a five-stimuli training followed by 14 test stimuli. Participants were instructed to respond as accurately and fast as possible.

In the reading aloud task after the participant named the visually presented number, the answers were written down by the experimenter, who started the following trial by button press. Reaction times were measured with voice key from stimuli screen presentation until first vocal onset. In the verbal-visual matching task, the auditory stimuli were first presented via the headsets. Then, four numbers were visually presented on the screen. Reaction time recording started when the auditory stimulus presentation was completed and ended when one of the response buttons was pressed. Stimuli were presented with an inter-trial interval of 500 ms. At the end of the experiment the participants were compensated with a gift voucher.

Design and statistical analyses

The transcoding of two-digit numbers was compared in a transversal developmental design by Linear Mixed Models (LMM) on the dependent variables reaction time and correct responses. To control for age related variabilities analogous analyses were conducted on z-score transformed RT, separately for each group. Furthermore, to control for word length (see [71]), we replicated the same analyses taking into account only numbers where the corresponding number words were constituted of four syllables (see Table 1 in S3 File for which stimuli were included in these analyses). Data were analysed with R [72] using the afex package [73], while graphs were drawn with ggplot2 [74].

Analyses

Both tasks were analysed with Linear Mixed Models (LMM) in R using the mixed function from the afex package’s mixed function [73], which relies on lme4 [76]. Follow-up analyses were computed with emmeans [77]. When our initially designed full model failed to converge, we reduced the model complexity by gradually simplifying the random effect structure until convergence was reached.

The analyses aimed at testing for all age groups the two main hypotheses: (1) the effect of transparency of power in French and (2) the effect of language of math acquisition. First, the effect of transparency of power in French, corresponding to the cost for vigesimal numbers in the base-20 system was tested by comparing small numbers (i.e. ‘30s, ‘40s, and ‘50s) to large numbers (i.e. ‘70s, ‘80s, and ‘90s) in French only (see A in Fig 1). For this hypothesis, we predicted that both tasks would lead to worse performances, henceforth a cost, for large compared to small numbers. Developmentally, the cost might either be re-absorbed with increasing training using LM2 number words or remain stable across age groups. Additional control for the potential confounder of a number size effect was implemented by applying the same comparison within German (A.1 in Fig 1). Second, the hypothesis concerning the language of math acquisition was tested by comparing only small numbers (i.e. ‘30s, ‘40s, and ‘50s) in French and German (B in Fig 1). We predicted that both tasks would lead to a cost in French compared to German. Developmentally, this cost could remain stable throughout adolescence and adulthood, or gradually diminish with age, potentially even resorbing in adults.

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Fig 1. Venn diagram of the design rationale.

Design rationale of the comparisons on the different hypotheses of number word transparency (A) and age of math acquisition (B).

https://doi.org/10.1371/journal.pone.0273391.g001

Reading aloud task

Reaction times.

The maximal linear mixed model was defined with main effects and interactions between the fixed between-group factor Age (levels: 5^th, 8^th, 11^th grade and adults) and two fixed within-group factors: Language (German, French), and Number Size (Large, Small), all levels being defined as categorical. As random factors, we modelled individual differences (i.e. Subject) and item-related variability (i.e. Item). The maximum model was defined taking into account individual differences by using random slopes and intercepts per Subject for the interaction between Language and Number Size. Moreover, item-related variability was modelled using random intercepts and slopes per Item for the interaction between Language and Age. This led to the model with the following R syntax form (A0): (A0)

However, since the maximal model led to a singular fit (due to high correlations between the random parts of Number Size per Subject and Age per Item) we reduced the complexity of the model by removing those problematic random terms [78]. Therefore, the final model takes the following R syntax form (A): (A)

P-values and degrees of freedom were obtained with the Satterthwaite approximation method by comparing the full model against the model without the effect [73]. Follow-ups were calculated by comparing estimated marginal means (EMM) obtained with the emmeans package [77], and p-values were adjusted with the Bonferroni method. All contrasts were set to sum.

The model for the reading aloud task RTs resulted in three significant main effects and three two-way interactions (see Table 1). The main effect of Age was decomposed with follow-up analyses showing that the 5^th graders (978(555) ms) were slower than the older groups, which had similar RTs (8^th 788(282) ms, 11^th 777(246) ms and adults 822(320) ms; standard deviations in parenthesis). Furthermore, the main effect of Language indicated overall slower naming in French than in German, and the main effect of Number Size indicated slower naming for bigger than smaller numbers. Significant two-way interactions were found between Age and Language, Age and Number Size, as well as Language and Number Size. Finally the three-way interaction was also significant, see Table 1 and Fig 2.

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Fig 2. Reaction times of the reading aloud task.

For each age group as a function of “Language” and “Number Size”. Large numbers correspond to ‘70s, ‘80s and ‘90s, while small numbers correspond to ‘30s, ‘40s and ‘50s. Ribbons represent standard error.

https://doi.org/10.1371/journal.pone.0273391.g002

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Table 1. Results of the reading aloud task’s RT’s linear mixed model.

https://doi.org/10.1371/journal.pone.0273391.t001

Follow-up analyses were performed on the model’s estimated marginal means (EMMs) of the interaction’s terms [see 79] with Satterthwaite estimation of degrees of freedom. Then EMM were compared with two-tailed pairwise tests with Bonferroni adjusted p-values. This confirmed the first hypothesis on number word transparency (cf. A in Fig 1), since in French the vigesimal ‘70s, ‘80s and ‘90s numbers were named slower than ‘30s, ‘40s and ‘50s numbers by all age groups: 282 ms slower for 5^th graders, 219 ms for 8^th, 188 ms or 11^th and 236 ms for adults (all t>5.13, p<.001). While as control (cf. A.1 in Fig 1), there was no significant difference in German (max difference 9 ms) between naming large or small numbers (all t<0.30, p = 1.0.).

Secondly, the hypothesis on language of math acquisition was confirmed by the two youngest age groups (cf. B in Fig 1). Naming in French (LM2) compared to German (LM1) was significantly slower by 510 ms for 5^th graders (t(126.17) = 9.09, p<.001) and by 177 ms for 8^th graders (t(117.4) = 2.83, p<.05). Although positive, the difference was not significant for the two older age groups: 161 ms for 11^th grades (t(117.9) = 2.42, p = .10) and 158 ms for adults (t(117.17) = 1.94, p = .32). To further establish the robustness of these results, we replicated the analyses using the same model (A), but on RTs transformed into z-scores per age-group (see Fig 3) and on a sub-sampled dataset including only four-syllable long number words (see S2 File).

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Fig 3. Z-score reaction times of the reading aloud task.

Standardized reaction times for each age group as a function of “Language” and “Number Size”. Large numbers correspond to ‘70s, ‘80s and ‘90s, while small numbers correspond to ‘30s, ‘40s and ‘50s.

https://doi.org/10.1371/journal.pone.0273391.g003

The z-score transformation aimed to reduce the variability of RT among age groups and confirmed the main effects and the interaction between Language and Number Size, while the triple interaction was not significant anymore (see Table 1 in S2 File). Using the dataset with number words of 4 syllables (see Table 3 in S2 File) to control for the possible word length effect confounder ([see 71]) also replicated the main effects and the two-way interactions, but again the three-way interactions between Age, Language and Number Size was not significant anymore.

Correct responses.

Correct responses (CR) were analysed using a binomial approach and p-values estimated by the likelihood ratio test. Since applying the same model as for RT (see (A)) did not converge, we had to drop the random per-Item factor and the fixed factor Age. In sum the final model had the following syntax (B): (B)

The failure of convergence with the more complex model might be due to ceiling effects, as the task was very simple, particularly for older age groups, see Fig 4 and Table 2 in S1 File.

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Fig 4. Correct Responses in the reading aloud task.

Percent correct for each age group as a function of “Language” and “Number Size”. Large numbers correspond to ‘70s, ‘80s and ‘90s, while small numbers correspond to ‘30s, ‘40s and ‘50s. Ribbons represent standard error.

https://doi.org/10.1371/journal.pone.0273391.g004

The binomial model (see (B) and Table 2) indicated a significant main effect of Language and in contradiction to the RT analyses, the main effect of Number Size was not significant. However, critically, like for RT, the interaction between Language and Number Size persisted with CR.

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Table 2. Results of the reading aloud task’s CR’s linear mixed model.

https://doi.org/10.1371/journal.pone.0273391.t002

Follow-up analyses confirmed with CR the pattern observed with RT: in French, 4.8% more errors were made with ‘70s, ‘80s and ‘90s than ‘30s, ‘40s and ‘50s numbers (z = −3.37, p<.01), indicating a cost for vigesimal number transcoding. The same contrast in German did not lead to any significant differences (z = 0.63, n.s.). Secondly, comparing ‘30s, ‘40s and ‘50s numbers in both languages revealed 2.6% more errors were made in French than in German (z = −2.89, p<.05), pointing towards a disadvantage for transcoding numbers in French (LM2).

In summary, the analyses on RT and CR confirmed both hypotheses which predicted effects of (1) transparency of power penalizing vigesimal number words and (2) language of math acquisition benefitting LM1 (i.e. German). The effect of transparency of power was robust across age groups since it could be replicated on z-score transformed data and by limiting the analyses to four-syllable long number words. The effect of language of math acquisition was found only in the two youngest age groups of the initial analyses, but persisted across age groups in the two additional analyses.

Verbal-visual matching task

Reaction times.

For coherence of interpretation and comparison with the previous verbal-visual matching task, the same model (A) and parameters for p-values, degrees of freedom and follow-up’s were applied here on RT. That is the factors Language, Age and Number Size were modelled as fixed effects. Additionally the model included random factors to consider item-related variability as a function of language and random intercepts to account for individual differences per subject.

As a result, all main effects, two-way and three-way interaction were significant (all F > 11.74, p < .001, see Table 3). Overall a similar pattern than for the reading aloud task was found: the main effect of Age was driven by the slow responses measured in 5^th graders (as shown by the follow-up analyses), the main effect of Language indicated slower responses in French than in German, and the effect of Number Size revealed slower responses for large (i.e. `70s, `80s and `90s) than small numbers (i.e. `30s, `40s and `50s) All two-way interactions were significant (see Table 3) and we found a three-way interaction between Age, Language and Number Size.

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Table 3. Results of the verbal-visual matching task’s RT’s linear mixed model.

https://doi.org/10.1371/journal.pone.0273391.t003

Follow-up analyses were applied comparing the estimated marginal means with paired comparisons on Satterthwaite corrected degrees of freedom and Bonferroni adjusted p-values. Follow-up analyses confirm the hypothesis relating to transparency of power, given a cost in French for ‘70s, ‘80s and ‘90s compared to ‘30s, ‘40s and ‘50s numbers for all age groups: 5^th graders with a 682 ms cost (t(97.97) = 10.84, p<.001), 8^th graders 339 ms cost (t(53.31) = 5.27, p<.001), 11^th graders a 202 ms cost (t(31.66) = 3.35, p<.01), and adults with a 218 ms cost (t(77.60) = 3.30, p<.01). In contrast, the same comparison in German did not result in any significant differences (all t<1.05, n.s., max difference = 46 ms). In sum, the cost for vigesimal numbers observed in the reading aloud task is replicated with the verbal-visual matching task for all age groups, see Fig 5.

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Fig 5. Reaction times in the verbal-visual matching task.

For each age group as a function of “Language” and “Number Size”. Large numbers correspond to ‘70s, ‘80s and ‘90s, while small numbers correspond to ‘30s, ‘40s and ‘50s. Ribbons represent standard error.

https://doi.org/10.1371/journal.pone.0273391.g005

However, in comparison to the reading aloud task, the hypothesis on language of math acquisition tested on small numbers (i.e. `30s, `40s and `50s; see B in Fig 1) was less supported, despite all differences in all four age-groups being positive. Indeed, performance advantages for German (LM1) were observed only in the youngest age group: while 5^th graders were 682 ms slower in French than in German (t(121.9) = 5.55, p<.001), the same comparison between older age groups was not significant, 340 ms for 8^th graders (t(106.54) = 1.25, p = 1.0), 202 ms for 11^th graders (t(107.1) = 1.34, p = 1.0) and 218 ms for adults (t(104.8) = 0.82, p = 1.0). As for the previous reading aloud task, we compared with z-scores transformed and on a subsampled dataset to test the robustness of these results, see Fig 6.

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Fig 6. Z-score reaction times of the verbal-visual matching task.

Standardized reaction times for each age group as a function of “Language” and “number Size”. Large numbers correspond to ‘70s, ‘80s and ‘90s, while small numbers correspond to ‘30s, ‘40s and ‘50s.

https://doi.org/10.1371/journal.pone.0273391.g006

The linear model applied on z-scores (see Fig 6) replicated the results with raw data described above, except, the three-way interaction with age, suggesting that the age differences observed with raw RTs, might be caused by differences between age-group variability (see Table 3 in S2 File). The analyses on the subset with four syllable-long number words replicated the effects mainly for the younger age groups, as the number size effect revealed that a cost associated with vigesimal numbers was present for 5^th and 8^th graders. Moreover, the effect of language in favour of the first language of math acquisition (LM1, i.e. German) was significant only in 5^th graders (See Table 3 in S2 File).

Correct responses.

For the matching task, the correct responses were analysed with the same method and model as for the reading aloud task, namely a binomial approach and likelihood ratio test, see Formula (B). Thus for the verbal-visual matching task, we modelled again the main effects and interactions between Language and Number Size and added random intercept per subject to consider individual differences.

Similarly to the reading aloud task, the correct response rates showed a main effect of Language indicating in average 5.45% more errors were done in French than in German, see Table 4. The main effect of Number Size was marginally significant, potentially indicating more errors for large numbers. The two-way interaction between Language and Number Size was also significant again, see Fig 7.

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Fig 7. Correct Responses in the reading aloud task.

Percent correct for each age group as a function of “Language” and “Number Size”. Large numbers correspond to ‘70s, ‘80s and ‘90s, while small numbers correspond to ‘30s, ‘40s and ‘50s. Ribbons represent standard error.

https://doi.org/10.1371/journal.pone.0273391.g007

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Table 4. Results of the verbal-visual matching task’s CR’s linear mixed model.

https://doi.org/10.1371/journal.pone.0273391.t004

Follow-up analyses indicate that 8.5% more errors were made when matching the French vigesimal ‘70s, ‘80s and ‘90s numbers than ‘30s, ‘40s and ‘50s numbers (z = −4.75, p<.001), as would be expected from the effect of transparency of power. No such difference was found in German (z = 1.37, p = 1.0). Secondly, comparing small numbers in both languages revealed no differences between French (LM2) and German (LM1) (z = −0.46, p = 1.0), thus again failing to reveal effects of language of math acquisition in the verbal-visual matching task.

In summary, for the verbal-visual matching task the (1) transparency of power hypothesis could be confirmed, as transcoding performances were overall lower for French ‘70s, ‘80s and ‘90s numbers having a vigesimal structure, stably across age groups. Concerning the hypothesis on (2) language of math acquisition, the advantage for transcoding in German (LM1) was only stable across age groups, when considering standardized RTs (see Fig 6).

Transparency of power

In French, `70s, `80s and `90s numbers differ in their transparency of power [28] from `30s, `40s, `50s numbers, with the former being in base-20. Comparing these numbers across both tasks, we found a cost for French `70s, `80s and `90s compared to `30s, `40s and `50s numbers across age groups and task. We interpret this as an effect of transparency of power that is independent of participant’s age and bilingual proficiency level. An advantage for larger compared to smaller numbers could also be explained by a number size effect [80]. However, this interpretation can be excluded, because the same comparison in German did not reveal any differences which, under the untested assumption that 5^th graders do not have mature magnitude representations, would speak against the prediction of semantic models. The effect was not explained by word length either (see [71]), since it was replicated with all age groups except adults when analysing only number words having the same length of four syllables. Comparable results were observed in terms of correct responses, although the different age groups could not be compared due to insufficient model fit. Since the number of correct responses displayed ceiling effects for older participants in both tasks and languages, correct responses might lack sensitivity and is therefore not an ideal dependent variable for measuring language differences in the present study [29].

The cost for less transparent number words structures is in line with previous results. Investigating participants from the same German-French bilingual population than the present study, Van Rinsveld and colleagues reported that arithmetical problems with numbers over 70 in French led to slower, less accurate solutions than in German both in visual and auditory modes [57]. Furthermore, our results replicate and extend previous transcoding studies, finding more errors in France-French (less transparent) than Wallonia-French (more transparent) speaking seven years old children [41]. Costs for base-20 numbers were also found for number reading and matching tasks when comparing 5^th graders speaking French or English (i.e. base-10 numbers) [42]. The present study extends these findings by comparing different age groups of increasingly proficient German-French bilinguals performing the same transcoding tasks. Notably, the cost was found for both reading aloud and number matching tasks that rely on transcoding from visual to verbal processes and vice-versa. The results from the z-scores standardization suggest that the cost explained by a difference in transparency of power was generally and stably persisted from 5^th grade up to adulthood. In other words the cost maintains across age groups, tasks and the associated degree of language proficiency. In the following, we give two possible accounts for the cost for transcoding French base-20 numbers.

From the perspective of cognitive models presented in the introduction, the cost for transparency of power in French would fit with Power and Dal Martello’s semantic model if rules reflecting the vigesimal form are added, since more rules would mean slower production [9]. Taking into account development, the asemantic ADAPT model (A Developmental Asemantic and Procedural model for Transcoding [6]), would partly fit with our results. ADAPT was proposed for number dictation (i.e. verbal to visual Arabic transcoding), hence similar processes could have taken place in the verbal to visual matching and reading aloud tasks (i.e. visual Arabic to verbal) of the present study design. In ADAPT, French `70s, `80s and `90s require more rules and therefore also more processing time. However, the model further proposes a lexicalization through repetitions over development, i.e. leading to faster transcoding of `70s, `80s and `90s. Yet, when standardizing age groups variability through z-score transformations, we noted that numbers above 70 were similarly slower to transcode in all age groups, arguing against a lexicalization with training. Also, in answer to an interesting comment in the review process, we conducted additional analyses on reaction times (see S4 File), by adding all decades as levels for the Number Size factor (hence 8 levels from ‘30s to ‘90s). Those additional analyses confirm that for both tasks in French, reaction times become slower from the ‘70s decade onwards. The non-lexicalization of `70s, `80s and `90s vigesimal numbers in French might interact with their late formal acquisition (i.e. LM2 acquired at 7<sup>th</sup> grade, around 11–12 years old). Finally, the cost for `70s, `80s and `90s numbers also fits with Dotan and Freidman’s recent transcoding model which suggests that reading `70s and `90s French number words requires additional irregular rules. When reading 75 for example, a number word frame would be structured by a decade and a teen instead of ten, then the ten frame filling would be changed from “7” to “6” (the filled frame would be: [6:tens] [15:teens] to give “soixante-quinze”, literally “sixty-fifteen”) [12].

Speculatively, it could be argued that for French morpho-syntactically complex number words that are not lexicalized, multiple additional morphemes need to be retrieved (see i.e. [81]). This could be the origin of the cognitive slow-down as it requires to retrieve more lexical morphemes (i.e. four for 97: “quatre”, “vingt”, “dix”, “sept”). Eventually, these morphemes, which are made of other numbers might interfere among each other and slow down RTs (i.e. proactive interference [82, 83]).

Please note that all three models would also generally predict slower responses for German number words above 12 due to the additional inversion rule in German. Such effects of transparency of order have been found in previous studies [17, 29, 33], but it is possible these were masked in the present study by the effect of language of math acquisition described in the following section.

To sum up, the effect of transparency of power was confirmed by costs for French `70s, `80s and `90s vigesimal numbers, independently from age or task. The origin of this cost could be explained by the non-automatized, hence not lexicalized, transcoding process for these non-transparent numbers in French. However, since French is the second language of mathematical acquisition (LM2) of the current sample, it remains an empirical question whether this interpretation can be generalized to early learners of French.

Language of mathematical acquisition: LM1 vs. LM2

To assess whether and how language of math acquisition impacted transcoding in the different age groups, we compared performances in German (LM1) and French (LM2). To avoid the potentially confounding effect of transparency of power described above, only `30s, `40s and `50s in both languages were compared. We can also exclude that differences in transparency of order between German and French explain LM2 costs since this would have meant the opposite effect, that is slower responses in German (LM1, having an inverted number word structure, e.g. [17, 31–33]), rather than in French (LM2, having a non-inverted number word structure). In line with our expectations, we observed costs during the reading aloud task in LM2 (French) compared to the LM1 (German); these costs were observed in the two youngest age groups in the analyses on RT, but they appeared at all ages when considering standardized response speed and four-syllable words. In the verbal-visual matching task, costs were consistently visible only in the analyses of standardized response times.

Our findings are in line with studies reporting qualitatively different arithmetic performance with LM+ compared to LM- in bilingual Filipino-English and Spanish-Basque participants, even if LM- corresponds to participants’ mother language [56, 59]. Finally, they also match and extend the finding that solving addition problems was slower in LM2 than LM1 in participants coming from the same education system than in the present study [57].

Interestingly, while the language of math acquisition impacted the number reading task, this was considerably less the case in the verbal-visual matching task. Lexically, the different pattern of results for the LM2 could be explained by different memory retrieval mechanisms [69] since the number reading task can be considered as a form of free recall while the matching task is more similar to a familiarity judgment. During free recall all possible number words can interfere with the retrieval of the correct number word, entailing a kind of lexical competition among the different verbal codes causing a cost for the less dominant language (LM2). In contrast, visual familiarity of the target number might underlie participants’ responses during matching, weakening the role of language code(s) activation during this task. Similar task differences were indeed reported by Vander Beken and Brysbaert [69] in a study investigating the recall of text in university students’ first and second languages. An additional explanation might rely on the nature of the stimuli. In the matching task, the number word input is already language-specific and therefore might be less susceptible to between-language lexical competition. While for the number reading task the identical visual Arabic digits might lead to a lexical competition between the LM1 and LM2.

To interpret the LM2 cost different theoretical perspectives can be taken. A possible interpretation can be derived from the ADAPT model of number transcoding [6]. LM1 (German) could be lexicalized (i.e. directly retrieved from long-term memory), while the LM2 (French), could rely on slower procedural rules, even for numbers under 60. In line with this view, weaker fMRI temporal lobe activation was observed when solving simple additions in LM2, proposedly reflecting less verbal retrieval than for LM1 additions [60]. Furthermore, since in ADAPT algorithmic rules are enacted by the short-term memory, it could potentially impact its capacity by using more resources [84, 85] and in turn explain parts of the LM2 costs observed in the same bilingual population for exact arithmetic [57]. It is worthwhile noticing that this disadvantage for storing and accessing LM2 numbers in verbal short-term memory might also impact the results obtained in neuro-developmental diagnostic tests. Indeed, using LM2 numbers for tests such as the number span test of the Wechsler intelligence scale or for different tasks in dyscalculia batteries might hamper performance and lead to an underestimation of children’s cognitive abilities.

The LM2 cost is also explainable from a psycho-linguistic perspective: here we present a syntactic and a lexical interpretation. Syntactically, the LM2 cost might result from an over-generalization of the LM1 syntactic inversion rule (see for example [29]): transcoding number words in French would require inhibition of the inversion rule (over)learned from German. A lexical explanations can be found in more general bilingual models such as the bilingual interactive activation model (BIA+), predicting a response competition between both languages, with faster activation for the more frequently used language [86]. In this framework, the slower lexical retrieval of LM2 would also have detrimental impacts on short-term memory [71, 87] and hence arithmetic [88].

In summary, the performance was overall better for LM1 (German) compared to LM2 (French) `30s, `40s, `50s numbers, confirming the importance of language of math acquisition for two-digit number transcoding in bilinguals. These effects were stronger in the number reading task than in the verbal-visual matching task and could be interpreted from bilingual lexical or syntactic perspectives.

A limitation of the present study is that it did not allow to disentangle the role played by the language of math acquisition from the familiarity with number words in the two languages. Since the language used for early learning (LM1) is probably also the one used more frequently, both factors might be confounded. Yet, both processes are supposed to rely on different neuronal substrates [15]. It is, however, noteworthy that math education extended for one year longer in LM2 (7 years of secondary education) than in LM1 (6 years of primary education), which might help to balance the frequencies in both languages.

Concerning the effect of transparency of power which we observed with `70s, `80s and `90s French numbers, we cannot exclude that part of the effect comes from the mixed nature of French number words. Since French contains both base-10 and base-20 numbers, it remains indeed to be determined whether similar effects would be observed with a language containing exclusively base-20 numbers (see for example [89]). Likewise, further interpretation of the LM2 costs would benefit from a study assessing the effects of LM1 on transcoding with a language having simpler number word structures (e.g. English or Asian languages). Finally, it is still debated whether transcoding requires access to semantics or not (see [6]). Further studies should investigate LM2 costs during number processing tasks systematically activating number semantics (e.g. in masked priming designs as in [90]).

Taken together the present results confirm and extend the interactions between language and number processing observed in more complex tasks such as arithmetic problem solving, corroborating the role of language in numerical cognition.