Bilingual children access multiplication facts from semantic memory equivalently across languages: Evidence from the N400

Brain and Language 198 (2019) 104679

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Bilingual children access multiplication facts from semantic memory equivalently across languages: Evidence from the N400

T

Vanessa R. Cerdaa, , Amandine E. Greniera, Nicole Y.Y. Wichaa,b ⁎

a b

University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249, USA University of Texas Health San Antonio, 7703 Floyd Curl Dr., San Antonio, TX 78229, USA

ARTICLE INFO

ABSTRACT

Keywords: Bilingualism Arithmetic ERPs Children Spoken number words N400 Multiplication

Typically, bilinguals learn multiplication facts in only one instruction language. Consequently, these facts may be represented and/or accessed as language-specific memories, requiring a qualitatively different retrieval process in their other language. Indeed, behavioral studies reveal that bilinguals verify arithmetic facts faster and better in the language of learning. Here, event-related potentials (ERPs) were used as a window into the neurocognitive processes underlying this language bias in children. ERPs were recorded while bilingual children verified the correctness of multiplication solutions. Operands were presented as spoken number words in Spanish and English, separately. Although a language bias was revealed in behavior, both languages elicited the same ERP correctness effect, an N400, reflecting similar cognitive processes in both languages. This suggests that the source of the behavioral difference is not at the level of semantic access. Our findings highlight the flexibility of the bilingual brain, especially when both languages are learned early.

1. Introduction In typical American classrooms, children learn multiplication tables through verbal rehearsal until the facts become overlearned verbal expressions (e.g., “two times four equals eight”). In turn, it is thought that these arithmetic facts have language-based representations in long-term memory (Campbell & Epp, 2004; Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999). If a person speaks more than one language, however, it is less clear how these facts are represented or how they are accessed when using different languages. Bilinguals typically report that they prefer to do math in one language (Dewaele, 2007). Several studies have demonstrated that bilinguals indeed process arithmetic more efficiently and more accurately in their native language or the language used to learn arithmetic (LA+) compared to the other language (LA−) (Bernardo & Calleja, 2005; Bernardo, 2002; Frenck-Mestre & Vaid, 1993; Marsh & Maki, 1976; Salillas & Wicha, 2012; Spelke & Tsivkin, 2001). However, the neurocognitive processes that account for this difference in performance have yet to be determined. This study is the first to examine in bilingual children the neurocognitive processes engaged when performing arithmetic in each of their languages. Given that bilinguals are the norm around the world, not the exception, it is critical to understand how the bilingual brain handles foundational cognitive skills that can be affected by language experience. This is relevant for bilingual arithmetic given that, if a bias ⁎

toward the language of learning exists, bilingual children may be at a developmental disadvantage compared to their monolingual peers when they are tested in their other language. Using a less efficient language could cause a domino effect for more complex mathematical operations that depend on these core building blocks. Research shows that monolingual children who use retrieval processes for arithmetic problems more often than less efficient procedural processes (i.e., counting or recursive addition) achieve higher levels of math performance through high school (Geary, Hoard, Nugent, & Bailey, 2013; Price, Mazzocco, & Ansari, 2013; Siegler et al., 2012). Therefore, it is critical to understand if the differences in performance across languages in bilinguals index differences in accessing math facts from memory in each language. Behavioral studies investigating a potential language bias for arithmetic have shown that bilingual children and adults are faster and more accurate at retrieving and verifying arithmetic facts in LA+ than in LA− (e.g., Marsh & Maki, 1976; Salillas & Wicha, 2012; Spelke & Tsivkin, 2001). Although some studies assume that the native language is the language of learning (e.g., Frenck-Mestre & Vaid, 1993), other studies have shown that it is the language of learning arithmetic and not the native language, per se, that shows improved performance (for example, in native speakers of Tagalog attending school in English) (e.g., Bernardo, 2002; Bernardo & Calleja, 2005). These performance differences across languages are thought to arise from encoding

Corresponding author. E-mail addresses: [email protected] (V.R. Cerda), [email protected] (A.E. Grenier), [email protected] (N.Y.Y. Wicha).

https://doi.org/10.1016/j.bandl.2019.104679 Received 4 February 2019; Received in revised form 8 August 2019; Accepted 9 August 2019 Available online 21 August 2019 0093-934X/ © 2019 Elsevier Inc. All rights reserved.

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differences (i.e., in which language the representation of these facts is stored; Spelke & Tsivkin, 2001), and/or differences in efficiency of retrieving the multiplication facts from the arithmetic memory networks in each language (e.g., Campbell & Epp, 2004). There are several models that address how arithmetic facts might be structured and accessed from memory in a monolingual system, but only one that explicitly addresses bilinguals. Notably, these models are based on data from adults, and do not address if arithmetic should be processed differently in children. However, in an attempt to provide a theoretical context for our research with bilingual children, we focus here on the two most prominent math models in the literature, the Triple Code Model, which does not explicitly address bilinguals, and the Encoding Complex Model, which does (Campbell & Epp, 2004; Dehaene, 1992). Both of these models propose that simple arithmetic problems are stored in memory as memorized arithmetic facts. However, they differ on the specifics of these representations, and in turn, the implications for the underlying neurocognitive mechanisms that support them. The Triple Code Model (TCM) proposes that “multiplication tables are just part of a learned lexicon of verbal associations” (Dehaene, 1992, pg. 34). In this view, arithmetic facts are solely represented in a “verbal code”, which for monolinguals is by default the language in which math concepts were learned. TCM proposes that these facts would require transcoding from a digit format (2 × 4 = 6) to the verbal code (”two times four equals six”) in order to access these facts from memory. That is, although number words and number digits have separate representations, arithmetic facts are only represented in the verbal code (Dehaene & Cohen, 1995, 1997; Dehaene, 1992). Dehaene argues that TCM “is deeply constrained by (1) limiting the mental codes for numbers to three, and (2) assigning each process to only one prespecified code.” (Dehaene, 1992, p. 33). This implies that there can be no other mental code for the second language. He and others have inferred from this that bilinguals might use qualitatively different cognitive processes, for example, direct retrieval from LA+ versus translation from LA− to LA+, and, by inference engage qualitatively different brain processes for arithmetic facts in each language (see Dehaene, Piazza, Pinel, & Cohen, 2005, p. 442 and also Dehaene et al., 1999; Spelke & Tsivkin, 2001; Van Rinsveld, Dricot, Guillaume, Rossion, & Schiltz, 2017). In contrast, the Encoding Complex Model (ECM), the only math model to explicitly address bilingual arithmetic, proposes that arithmetic facts are represented in memory separately in each of the formats in which they can appear (Campbell & Epp, 2004). For example, Spanish-English bilinguals would have three representations for a given problem: “2 × 4 = 8”, “two times four equals eight”, or “dos por cuatro son ocho”. The arithmetic facts are retrieved directly from these separate format-specific stores, but with more efficient access in one language/format relative to the others based on experience. In turn, retrieving arithmetic facts from memory is most efficient when they are presented as visual numerals, followed by the stronger language and then the weaker language. ECM does not explicitly propose brain mechanisms to support these separate representations. However, it can be inferred that arithmetic facts should engage similar neurocognitive processes in both languages, namely retrieving the facts from memory, given that equivalent representations exist in each language. Therefore, bilinguals should show only quantitative differences in the speed and accuracy of retrieval based on the strength of the representations in each language, but no qualitative difference in the neurocognitive process engaged. Neither model clearly defines if arithmetic facts share space with the rest of the language system, but both models argue that these formatspecific representations are separate from the “magnitude code” that represents numerosity (or what the magnitude of the number actually is). Additionally, ECM proposes that the separate language representations are not directly connected to each other (Campbell & Epp,

2004) and instead are indirectly connected through the digit representation or through the “magnitude” code. Notably, both of these models are inconsistent with research that suggests that bilinguals maintain both languages simultaneously active even when only one is in use, reflecting interconnected and interactive language networks (Dijkstra & van Heuven, 2002; Hermans, Bongaerts, De Bot, & Schreuder, 1998; Thierry & Wu, 2007). Moreover, bilinguals who learn both languages early in life show little to no differences in brain activation patterns when processing words in each of their languages (Hernandez, Martinez, & Kohnert, 2000; Kim, Relkin, Lee, & Hirsch, 1997). Thus, bilingualism research suggests that a closer examination is needed to determine if arithmetic facts are treated differently across languages. A challenge in addressing this question is finding a measure that provides a window into the neurocognitive processes underlying the speed and accuracy differences in behavior across languages. Behavioral measures have provided critical findings that have led to the development of the above-mentioned models (Bernardo, 2002; Campbell, Kanz, & Xue, 1999; Dehaene et al., 1999; Frenck-Mestre & Vaid, 1993; Marsh & Maki, 1976; Spelke & Tsivkin, 2001; see Wicha, Dickson, & Martinez-Lincoln, 2018 for review). However, these measures provide a single data point that reflects the culmination of sensory and cognitive processes leading up to the behavioral response. For example, in judging the correctness of a solution to a multiplication problem, a button press can be faster or slower for a variety of underlying causes. In turn, the specific processes must be inferred by careful experimental design, such as comparing across solutions that differ in difficulty or type of arithmetic problem. Event related potentials (ERPs) have been used as a complimentary measure to these behavioral indices to help elucidate the cause of the effects more directly. ERPs provide a continuous measure of brain activity. In turn, changes in brain activity can be measured in the period of time leading up to a behavioral response (Dickson et al., 2018; Jost, Hennighausen, & Rösler, 2004; Martinez-Lincoln, Cortinas, & Wicha, 2015; Niedeggen & Rösler, 1999; Niedeggen, Rösler, & Jost, 1999; Prieto-Corona et al., 2010; Salillas & Wicha, 2012). Moreover, ERPs are a multidimensional measure of brain activity that can reveal both quantitative differences across conditions, as indexed by the amplitude or timing of a brain component, as well as qualitative differences in the type of brain response elicited, as indexed by the polarity or scalp distribution of the ERP. In adult ERP studies, verification tasks have been used in which participants judge the correctness of a proposed arithmetic solution (e.g., 2 × 4 = 9) (Martinez-Lincoln et al., 2015; Niedeggen & Rösler, 1999; Niedeggen et al., 1999; Salillas & Wicha, 2012). In these studies, the operands are presented one at a time, allowing participants to generate expectations of what the solution of a given problem should be (Dickson & Federmeier, 2017; Niedeggen & Rösler, 1999). When judging the correctness of a multiplication problem, around 250 ms after a solution is presented, a correctness effect occurs in the ERP with more positive-going amplitude to correct solutions (e.g., 2 × 4 = 8) than incorrect solutions (e.g., 2 × 4 = 9). This arithmetic correctness effect has been compared to the N400 effect that is observed during language comprehension (Niedeggen et al., 1999). The N400 is a robust negativegoing ERP response that peaks around 400 ms after stimulus onset, and is thought to reflect differences in memory-driven processes associated with the automatic spread of activation within a memory network (see Friederici, 1995; Osterhout & Holcomb, 1996; Petten & Kutas, 1991). For example, words that are more strongly anticipated based on sentence context would receive greater spread of activation in the memory network, eliciting a smaller N400 amplitude compared to less strongly anticipated or incongruous words. Given that the N400 response is an index of semantic memory processes independent of the stimulus format (similar brain responses are observed for pictures, and written, spoken and signed words, see Federmeier & Kutas, 2001 for a review), it is logical to assume that memorized multiplication facts might elicit similar N400 effects as other learned concepts in semantic memory. 2

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Indeed, in children, and in adults when using written number words1, the arithmetic correctness effect appears as a modulation of the N400, reflecting semantic-level processes (Grenier et al., 2018; Moore, Drollette, Scudder, Bharij, & Hillman, 2014; Prieto-Corona et al., 2010; Wicha et al., 2018). Given that monolingual children exhibit an N400 effect when verifying the correctness of multiplication problems (Grenier et al., 2018; Moore et al., 2014; Prieto-Corona et al., 2010; Wicha et al., 2018), bilingual children should similarly show an arithmetic N400 correctness effect, at least in their LA+. ERPs have been used to study bilingual arithmetic in only two studies to date, and both have tested adults. Salillas and Wicha (2012) presented fluent adult bilinguals with simple multiplication problems, as three consecutive written number words in their LA+ and LA−, separately. They showed an amplitude modulation across languages on the N400 correctness effect, with a larger correctness effect (difference between correct and incorrect trials) for the LA+ than LA−. This was interpreted as showing less automatic spread of activation in LA−, leading to less facilitation for the correct responses in that memory network. Martinez-Lincoln et al. (2015) later showed that this difference in the N400 effect across languages could be mitigated through practice using arithmetic in the weaker language (i.e., school teachers teaching math in their LA−). These studies yielded two important findings. First, proficient adult bilinguals (who learned both languages early and spoke both fluently) process arithmetic problems presented as written number words in a qualitatively similar way, eliciting N400 effects across both of their languages. Second, the language that was used during childhood, as well as experience using each language as an adult, can modulate the efficiency of access to multiplication facts. These findings suggest that bilingual adults do not use qualitatively different neurocognitive processes for arithmetic facts in each language (as implied by the Triple Code Model, Dehaene, 1992; Spelke & Tsivkin, 2001). However, it is possible that adults share neurocognitive processes across languages only after a lifetime of experience as bilinguals (e.g., language co-activation may lead to shared processes; Dijkstra & van Heuven, 2002; Hermans et al., 1998; Thierry & Wu, 2007). Perhaps bilingual children who have just recently learned arithmetic facts in only one language might show differences in processing across their languages. No studies to date have investigated the neurocognitive basis of bilingual arithmetic in children. The current study aimed to test whether bilingual children use quantitatively or qualitatively different processes for math fact verification in LA+ and LA− (e.g., larger/faster but similar brain responses versus completely different brain responses). As in previous studies (Bernardo & Calleja, 2005; Saalbach, Eckstein, Andri, Hobi, & Grabner, 2013; Van Rinsveld, Schiltz, Brunner, Landerl, & Ugen, 2016), bilingual children should show a behavioral advantage in one language over the other. If bilingual children use qualitatively equivalent processes when accessing math facts in each language, similarly to bilingual adults, then they should show N400 modulations in both languages. Quantitative differences in the N400 response across languages would support the idea that differences in access to semantic memory contribute to the behavioral advantage in LA+, as suggested by the Encoding Complex Model, as opposed to other cognitive or perceptual causes (e.g., lexical processing differences across languages). However, if we observe a qualitatively different ERP responses when processing arithmetic in LA−, such as a late positivity reflecting translation or more effortful processing (Wicha, Moreno Montes, & Carrasco Ortiz, 2019), this would support a single memory representation with different strategies for accessing these representations across languages, as suggested by the Triple Code Model. The current study tests this by measuring coupled ERP and behavioral

responses while bilingual children verified simple multiplication problems in each of their languages. 2. Methods The multiplication verification task used herein was modelled after studies with monolingual and bilingual adults, with a few important modifications (Dickson et al., 2018; Salillas & Wicha, 2012). Typically, studies of bilingual arithmetic have used written number words as operands (Frenck-Mestre & Vaid, 1993; Martinez-Lincoln et al., 2015; Saalbach et al., 2013; Salillas & Wicha, 2012; Spelke & Tsivkin, 2001). This poses a potential confound, especially with young children, from differences in exposure to written number words across languages. The current study adopted a novel approach in which the multiplication problems were presented as spoken number words followed by an Arabic numeral solution. Using spoken number words approximates typical classroom use and practice for multiplication problems (i.e., children do not typically learn multiplication tables as written number words). Moreover, by using spoken number words, rather than the traditional written words, we eliminate any potential differences in processing that might be driven by reading fluency across languages. In addition, by using Arabic digits for the solution rather than number words, we avoid potential differences driven by lexical features of the solution (e.g., orthographic features unique to one language, like the hyphenation of 2-digit numbers in English, twenty-two). Given that the ERPs are measured from the onset of identical Arabic digit solutions, any differences across language conditions would necessarily be driven by the language of the preceding operands, when retrieval processes presumably occur. A similar version of this cross-modal paradigm was tested successfully in monolingual adults, showing robust correctness effects at the Arabic digit solution following spoken number words in English (Dickson et al., 2018). Additionally, in prior studies, adults were asked to withhold a response until given a cue to respond. To avoid imposing this additional cognitive load of inhibiting a response and maintaining it in working memory, children in this study were allowed to respond as soon as they saw the solution. We determined in a pilot study that motor response in children were naturally delayed long enough to not interfere with the ERPs of interest (see also Van Beek, Ghesquiere, De Smedt, & Lagae, 2014). Finally, a simple language task known to elicit a robust N400 in children was included here for comparison with the arithmetic task. In this way, bilingual arithmetic abilities were directly assessed against general language processing abilities to address whether simple arithmetic representations are accessed differently across languages. 2.1. Participants Thirty-four healthy 3rd to 5th grade bilingual children participated in both the multiplication and word-picture verification (WPV) tasks. To ensure an adequate signal-to-noise ratio in the ERPs, 4 subjects were excluded from the grand average for having less than 10 trials in at least one of the critical conditions for a task (average number of trials: LA+ correct 34, incorrect 29; LA− correct 32, incorrect 29). The multiplication task included 28 children (12 female; mean age 10.7 years, range 8.5–11.9 years; mean grade 5.03, range 3.30–5.90). The language task included 24 children, of which 22 also participated in the math task (12 female; mean age 10.6 years, range 8.5–11.9 years; mean grade 5.0, range 3.3–5.9). Behavioral analyses included only children that had ERP data. Participants were recruited from Spanish-English dual language programs in the San Antonio area. These programs typically enroll a balance of native Spanish speakers hoping to maintain proficiency in both languages (SDL children) and native English speakers hoping to become proficient in a foreign language (EDL children). Based on standardized measures detailed below, all SDL children were equally

1 Other arithmetic effects have been reinterpreted as a P300 (not an N400) for adults who approach the task differently than children, especially when using highly rehearsed forms of number, such as digits and spoken words (Dickson et al., 2018; Dickson & Federmeier, 2017; Jasinski & Coch, 2012).

3

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2.3. Multiplication task

proficient across languages and all EDL children were strongly English dominant. All children were exposed to both languages by 3rd grade (average age of L2 acquisition: 1.8 years; range: 0–5 years of age) and reported learning multiplication in only one of their languages. Of the 28 children who were included in the multiplication task, 10 learned multiplication in English (i.e., LA+ = English) and 18 learned multiplication in Spanish (LA+ = Spanish). Note that any differences in comparing LA+ and LA− are not due to differences in Spanish versus English, per se. Critically, comparisons between languages were always done using a within-language contrast (correct versus incorrect). All participants were right-handed (as assessed by an abridged Edinburgh Inventory: average: 0.82; range: 0.11–1.0; Oldfield, 1971). Participants had normal or corrected-to-normal vision, no history of cognitive or perceptual (visual or auditory) deficits, and were not taking medication that would affect their cognitive abilities. No participants had been diagnosed with language delays or learning disabilities.

2.3.1. Stimuli Stimulus materials consisted of all single digit multiplication problems with operands ranging from 2 to 9. After elimination of tie problems (e.g., 2 × 2, 3 × 3, etc.) and problems containing 0 and 1 (see Campbell & Graham, 1985), a set of 28 core problems remained. Each problem was repeated four times, twice with a correct solution (e.g., 2 × 4 = 8 and 4 × 2 = 8) and twice with an incorrect solution (e.g., 2 × 4 = 12 and 4 × 2 = 10) creating a total of 112 problems presented per language. All incorrect solutions were a multiple of either the first or second operand (never more than 3 multiples larger or smaller) making them all table related. Trials were pseudorandomized separately for each language, so that no three incorrect or correct problems were presented in a row. All operands were presented as spoken number words in either English or Spanish (normalized to a 450 ms duration) followed by a visual Arabic numeral solution. Operands were recorded by a female, fluent Spanish-English bilingual using a natural rate of speech. Sound files were normalized for volume and cropped to the sound onset/offset, avoiding distortions (e.g., clipping).

2.2. Offline behavioral assessments To determine proficiency across both languages, two subtests (Test 14: Picture Vocabulary and Test 15: Oral Comprehension) of The Woodcock-Johnson Tests of Achievement (Woodcock, McGrew, & Mather, 2001a) and one subtest (Test 8: Incomplete Words) of the Woodcock-Johnson Tests of Cognitive Abilities (Woodcock, McGrew, & Mather, 2001b) were used. Equivalent versions of these tests in Spanish from the Batería III Woodcock-Muñoz (Batería III; Muñoz-Sandoval, Woodcock, McGrew, & Mather, 2005) were also administered. These tests required subjects to name pictures, complete oral sentences, and demonstrate phonological awareness in each language. Aged-normalized classifications were obtained from the Woodcock-Johnson Proficiency Battery (WJPB-R) based on a subject’s raw assessment scores in each language. These classifications were then compared across each of the parallel subtests from the WJPB-R and the Batería III. All SDL children had proficiency scores that were within 1 +/- classification between at least 2 out of the 3 tests across languages, reflecting fairly balanced proficiency in both languages. EDL bilinguals were dominant in English, with English always at least 2.5 classifications higher than Spanish (Max: 4 classification difference). Regression analyses showed that WJPB subtests were predictive of each other within a language (all were significant at p < 0.05), but not across languages, confirming that these tests are independent measures of fluency in each language. Subjects’ working memory and math fluency was also assessed using the Numbers Reversed subtest of the WJ III COG and the Wechsler Individual Achievement Test (math fluency-multiplication, WIAT III, Wechsler, 1992), respectively.

2.3.2. Procedure All participants performed the multiplication tasks in separate blocks of English and Spanish trials in a single session, with alternating language order across participants. Within each language block, subjects were given instructions and feedback only in one language (see Grosjean, 1998). To make the task child friendly, participants were instructed to solve multiplication problems sent by aliens during a space mission game. Each task consisted of eight blocks (“levels” in the game for the children) of 14 trials each. Children were encouraged to respond as quickly and as accurately as possible to earn coins toward a prize given at the end of the session. Immediately after completion of each block, they were given feedback about how many coins they had collected and were permitted to take a self-timed break to avoid fatigue. Multiplication problems were presented as 3 consecutive numbers, one number at a time with no symbols between operands. The first two numbers were presented as spoken number words (450 ms duration; 250 ms interstimulus interval - ISI) in the same language. As the spoken operands were presented, a gold coin with an embedded multiplication symbol remained in the center of the screen. The coin served as a fixation point to prevent eye movements during auditory stimulation. Additionally, the “X” on the coin reminded the children to solve the problems as multiplications (see Fig. 1 for an example of presentation and timing). The sounds were played through EEG-compatible insert headphones (ER1; Etymotic Research Inc.). The second operand was followed by a 1000 ms ISI, then an Arabic digit solution appeared on the screen replacing the fixation “coin”. Solutions were presented in

Multiplication Verification Task Blink 1000 ms

Fixation Point 1000 ms

1st Operand 450 ms (250 ms ISI)

2nd Operand 450 ms (1000 ms ISI)

Solution 350 ms

Blank Screen 5000 ms max

12 Time

“two”

“four”

Button Press

Fig. 1. An example multiplication trial in English showing the trial structure from left to right over time in ms. First and second operands were presented as spoken number words. Trial structure was identical for the Spanish multiplication task with the spoken number words presented in Spanish. Children were asked to respond from the onset of the visual solution. 4

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white text on a black background in the center of a 19” LCD Dell monitor. Participants were asked to indicate solution correctness as soon as possible after viewing the solution, by pressing one of two buttons on a game controller (Logitech Gamepad F310) with their index fingers. Finger-response mapping was reversed for half of the participants.

The data were sampled at 256 Hz (2048 Hz with a decimation factor of 1/8) using a 5th order sinc low-pass digital filter to remove high frequencies. ERPLab software was used to process and take measurements of the EEG data (Lopez-Calderon & Luck, 2014). Raw EEG data cut into 1 s epochs (−100 to 900 ms) around the onset of the solution (multiplication task) or the line drawing (WPV task) were assessed for inclusion using artifact rejection algorithms. Thresholds were calibrated for each participant separately through visual inspection of the data, and the algorithms were applied to the entire data set to exclude epochs containing blinks, horizontal eye movement, excessive muscle artifact, and EEG drift or blocking. EEG epochs were then averaged by condition, and 2nd order Butterworth digital filters with a low cutoff at 0.1 Hz and a high cutoff at 30 Hz were applied before analysis.

2.4. Word-picture verification task 2.4.1. Stimuli and procedure Immediately following the multiplication task in one language, children performed a separate word-picture verification task. The children performed this task to generate an N400 effect, an index of semantic-level processing, thereby enabling comparison of this effect across languages. To make the task child-friendly, participants were instructed to play a game in which they helped aliens learn English or Spanish by verifying the semantic fit of word-picture pairs. A prerecorded spoken noun in English or Spanish (word duration varied, English average word length: 645 ms; range of 410–930 ms; Spanish average word length: 643 ms; range of 390–930 ms; ISI 250 ms) was presented through EEG-compatible insert headphones. As the words were presented, a fixation coin was present on the screen to prevent eye movements during auditory stimulation. This was followed by a line drawing of an object presented in white on the center of a black background for 600 ms (images modified from Snodgrass & Vanderwart, 1980). The spoken word was either the most common name for the pictured object (congruent; based on Davis, 2005 for English; Colin J. Davis & Perea, 2005 for Spanish) or a semantic anomaly (incongruent). All words were matched for frequency (using the CELEX for English words and LEXESP for Spanish words) and number of syllables (average: 2; range 1–4) across languages. Additionally, incongruent trials were matched for animacy (i.e., animate vs inanimate objects) and were different across languages to avoid predictability. Participants were asked to identify congruent and incongruent word-picture pairs as quickly and accurately as possible by pressing one of two buttons on a game controller (Logitech Gamepad F310) as in the multiplication task. The task was divided into four blocks (“levels” in the game for children) with 20 trials each. Select trials were removed post-hoc due to coding errors and/or having low response accuracy across all participants (i.e., disagreement with an object being a match for a word), leaving a set of 39 match trials and 41 mismatch trials in the English task and 39 match trials and 39 mismatch trials in the Spanish Task. Trials were pseudorandomized so that no three match or mismatch trials were presented in a row. Two lists of stimuli were created per language where match and mismatched trials were reversed. These lists were alternated across participants.

3. Results 3.1. Multiplication task 3.1.1. Behavior Factorial analysis of variance was used to analyze accuracy and response times, separately, measured from the onset of the solution with two levels of correctness (correct, incorrect) and two levels of language (LA+, LA−). Children performed well above chance on the task in both languages (LA+ accuracy: 75.25%; LA− average accuracy: 73.53%). The 2 × 2 ANOVA revealed significant main effects of correctness on both accuracy (F(1,27) = 12.58, p < 0.01) and response time (F(1,27) = 42.74, p < 0.001), with children responding faster and more accurately to correct (1308.96 ms; 78.12%) than incorrect (1507 ms; 70.66%) solutions (Fig. 2). There was no main effect of language for either measure (RT: F(1,27) = 0.50, p = 0.48; Accuracy: F (1,27) = 1.36, p = 0.25), indicating that children were overall comparably fast and accurate when responding to the LA+ and LA− trials. There was an interaction, however, between correctness and language for both accuracy (F(1,27) = 4.92, p < 0.05) and response time (F (1,27) = 4.28, p < 0.05), with children showing larger effects of correctness in LA+ than LA−. The interaction between language and correctness on accuracy was driven by significantly higher accuracy on correct trials in LA+ compared to LA− (t(27) = 2.15; p < 0.05). This

2.5. EEG recording Participants sat in a dimly lit, sound-attenuating chamber while wearing a custom electrode cap fitted with 26 Ag‐AgCl sintered electrodes (Electro‐Cap International Inc. using BioSemi ActiveTwo electrodes). Electrodes were geodesically arranged on the head (see headplot in Fig. 3). Continuous EEG was recorded using a BioSemi ActiveTwo bioamplifier running ActiveView software. A signal from each electrode was recorded with respect to a common mode sense/ driven right leg (CMS/DRL) active/ground electrode, and was referenced offline to the average of two electrodes placed over the right and left mastoid processes. Noise induced by the amplification and digitization system are reduced by buffer amplifiers contained within the electrodes (BioSemi ActiveTwo, BioSemi B.V., Amsterdam, Netherlands) allowing for measurements of potentials from the surface of the skin with high electrode impedances. Electrode offsets were kept below 50 millivolts. A fixed first order analog antialiasing filter with a half‐power cutoff at 3.6 kHz was applied (see https://www.biosemi.com).

Fig. 2. Behavioral results for the multiplication verification task. On the left, percent accuracy and response time in milliseconds measured from solution onset (n = 28). Dark bars indicate trials ending with correct multiplication solutions and light bars indicate trials ending with incorrect multiplication solutions. Graphs on the right indicate the absolute difference between correct and incorrect solutions beside raw accuracy and response time graphs. LA+ behavior is shown in black and LA− shown in red. Response time measurements include only trials that participants judged correctly. *p < 0.05. 5

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suggests that LA+ has more efficient access to the correct solutions than LA−, confirming prior findings of a language of learning advantage (Marsh & Maki, 1976; Salillas & Wicha, 2012; Spelke & Tsivkin, 2001), and validating our task. In contrast to the accuracy data, the interaction between language and correctness on response time was driven by both slower responses for incorrect and faster responses for correct problems in LA+ compared to LA− (t(27)’s < 2; p’s > 0.1) (see Fig. 2). Additionally, standardized measures of multiplication skill (math fluency-multiplication, WIAT III, Wechsler, 1992) correlated with online accuracy measurements (at p < 0.05), confirming that performance on this multiplication verification task is sensitive to math fluency. Overall, children who had faster response times were faster across all tasks and languages (although this is not theoretically significant here).

than incorrect solutions (1.23 µv, SE = 0.91). There was an interaction between Correctness and Electrode (F(1,27) = 2.90; p < 0.001), with a widely distributed effect largest over midline electrodes. An additional distributional analysis replacing Electrodes with 2 levels of Laterality, 2 levels of Hemisphere, and 4 levels of Anteriority confirmed the presence of a larger effect over medial sites (Correctness × Laterality; F (1,27) = 7.57; p < 0.05); no other interactions reached significance. Critically, there was no main effect of Language (F(1,27) = 1.64; p = 0.21) or interaction between Language and Correctness (F (1,27) = 0.356; p = 0.55) indicating that the size of the N400 effect did not differ when operands were presented in LA+ (−2.98 µv, SE = 0.87) versus LA− (−2.39 µv, SE = 0.87). Fig. 4 shows a strip chart (bottom center) of the correctness effect in each language for each participant, and illustrates no consistent pattern between languages across individuals. Regression analyses with the N400 effect were conducted using the standardized measures and online behavior as regressors, separately. No theoretically significant relationships reached significance to explain the pattern of individual variability (at p < 0.05). Since previous studies have also reported differences driven by a bilingual’s native language (L1), rather than language of learning for arithmetic (LA+), the data was reanalyzed with this grouping in exploratory analyses (see introduction). Only 9 children switched language groups when comparing the data this way. That is, 9 children learned math in their L2. The main effect of Correctness held (F (1,27) = 23.43; p < 0.001), with correct solutions eliciting a more positive response (4.49 µv, SE = 0.73) than incorrect solutions (1.83 µv, SE = 0.80). There was no significant difference based on language on the N400 effect in either latency (t(27) = −1.69; p = 0.10; L1: 356 ms, SE = 8.77; L2: 371.66 ms, SE = 8.23) or amplitude (263–463 ms window: F(1,27) = 0.34; p = 0.56), and no interaction between Correctness and Language (F(1,27) = 0.32; p = 0.57). To confirm that the languages did not differ based on mean amplitude within the N400 time window (300–500 ms) and the later time window (600–900 ms), we also modeled our data using Bayesian analysis. The data were modeled based on Correctness, Language, and the additive effect with and without the interaction between the two. A Watanabe information criteria comparison (Watanabe, 2013) showed that the models without the interactions best fit the data, and posterior summaries of the model parameters provide Bayesian p-values, which reflect the likelihood of disconfirming a null effect (i.e., zero difference across conditions) with p-values considered significant when the 95% highest posterior density for each parameter did not contain 0. For mean amplitudes within the N400 time window, the model revealed a Bayesian p-value of 99.45% for Correctness and an 84.1% Bayesian pvalue for Language. The interaction term showed insignificant parameter values, with a Bayesian p-value of 62.5%. For the later time window, the model revealed a Bayesian p-value of 80.3% for Correctness and an 89.3% Bayesian p-value for Language. The interaction term again showed insignificant parameter values, with a Bayesian p-value of 68.3%. In brief, the Bayesian modeling confirms the ANOVAs showing no main effect or interaction with language.

3.1.2. Qualitative waveform morphology All ERPs were measured from the onset of the Arabic digit solutions in both languages (LA+ and LA−). ERPs were averaged separately for correct and incorrect solutions; only trials that subjects judged accurately were included in analyses. Subjects were excluded from the grand average if they had less than 10 trials in any of the critical conditions to ensure an adequate signal-to-noise ratio in extracting the ERPs. The average number of trials was similar across languages and conditions (LA+ correct: 34 trials; LA+ incorrect: 29 trials; LA− correct: 32 trials; LA− incorrect: 29 trials). Visual inspection of the grand average ERPs revealed typical sensory components, N1-P2, across the head in response to the Arabic digit solutions for both language conditions. Around 250 ms after solution onset, a widespread negative-going deflection is observed with larger amplitude for incorrect than correct solutions, an N400 correctness effect (Figs. 3 and 4). Importantly, this effect was present regardless of the language in which the operands were presented. Based on visual inspection, mean amplitudes were measured between 600 and 900 ms time window to determine if differences occurred across languages after the N400. A 2 (Language: LA+, LA−) × 2 (Correctness: correct, incorrect) ANOVA was performed. There were no main effects of Language (F(1,27) = 3.45; p = 0.07) or Congruency (F(1,27) = 2.44; p = 0.129), and no interaction between Language and Congruency (F(1,27) = 1.22; p = 0.27). The same analysis was performed on mean amplitudes measured between 700 and 900 ms. Similarly, there were no main effects of Language (F (1,27) = 3.54; p = 0.07) or Congruency (F(1,27) = 3.68; p = 0.06), and no interaction between Language and Congruency (F(1,27) = 1.79; p = 0.19). 3.1.3. Quantitative waveform measurements To determine the timing of the N400 effect across languages, averaged responses to correct solutions were subtracted from incorrect solutions to create difference waves. A broad window (250–550 ms) was used to determine when the differences were maximal across channels and across participants. The peak of the effect in LA+ occurred at 397 ms (SE = 8.75 ms) and 402 ms (SE = 11.03) in LA− after solution onset. Given that the timing of the N400 effect across languages did not significantly differ (t(27) = −0.42, p = 0.67)2, a single time window (300–500 ms) was used to measure the average amplitude of the effect for both languages. A 2 (Language: LA+, LA−) × 2 (Correctness: correct, incorrect) × 26 (Electrodes) ANOVA was performed. This analysis revealed a main effect of Correctness (F(1,27) = 20.1; p < 0.001), where correct solutions were more positive in amplitude (4.00 µv, SE = 0.82)

3.1.4. Spanish dual language bilingual subset Given that the specific language of learning differed across children, dividing by English and Spanish across the sample might have introduced confounds. However, we performed an analysis of a subset of children whose LA+ and L1 was Spanish. Of the total sample, 19 children were native speakers of Spanish in a dual language program, or SDLs; 14 reported learning multiplication in Spanish; 5 reported learning multiplication in English. We conducted a post-hoc exploratory analysis with the subset of 14 SDL bilinguals who reported learning math in Spanish (i.e., LA+ = Spanish). Although 14 subjects might be small for power, a similar number of subjects has been used in other studies (Martinez-Lincoln et al., 2015; Niedeggen & Rösler, 1999; Niedeggen et al., 1999; Prieto-Corona et al., 2010), and this subgroup

2 Additionally, a 2 (Correctness: correct, incorrect) × 2 (Language: LA+, LA−) revealed no main effects or interactions (F(1,27) = 0.858; p = 0.36) between Language and Correctness on timing of the raw N400 peaks either.

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Fig. 3. Grand average ERPs (n = 28) to correct (solid lines) and incorrect (dotted lines) solutions when operands were presented in LA+ (black) and LA− (red). Xaxis is time in milliseconds with 0 marking the onset of the solution. Y-axis is voltage in microvolts with negative plotted up. Electrodes plotted here are arranged as indicated on the head plot above. Robust N400 correctness effects are seen for both languages. There were no significant differences across languages after the N400 time window.

was most likely to show any language differences given their extensive exposure to their LA+ both in school and at home. Accuracy and response times were analyzed using the same 2 × 2 ANOVA structure, as above. The ANOVA revealed a significant main effect of correctness on response times (F(1,13) = 32.74, p < 0.001), with children responding faster to correct (1250.69 ms) than incorrect (1423.35 ms) solutions, and no effect of correctness on accuracy (F (1,13) = 2.69; p = 0.12). Similar to the whole group, there was no main effect of language for either measure (RT: F(1,13) = 0.003, p = 0.95; Accuracy: F(1,13) = 1.30, p = 0.27). There was no interaction between correctness and language for either accuracy (F (1,13) = 2.97, p = 0.11) or response time (F(1,13) = 0.22, p = 0.64). Grand average ERPs were created with only this subset of bilinguals

comparing responses to correct and incorrect solutions across languages. The timing of the N400 effect was not significantly different across the two languages (t(13) = -1.06; p = 0.30) with the LA+ effect at 398.48 ms (SE = 10.06) and the LA− effect at 415.72 ms (SE = 15.02), so the same time window (300–500 ms) was used for both languages. An ANOVA on the mean amplitudes revealed a main effect of correctness (F(1,13) = 10.65; p < 0.01), where correct solutions were more positive (3.46 µv, SE = 1.10) than incorrect solutions (1.08 µv, SE = 1.09). Again, there was no interaction between language and correctness (F(1,13) = 0.79; p = 0.38) indicating that the size of the effect did not differ when operands were presented in LA+ (-3.04µv, SE = 1.03) versus LA− (−1.72 µv, SE = 1.04), even for this subset of children who learned math in Spanish. 7

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Arithmetic Correctness Effect by Language LA+ Correct LA+ Incorrect Left

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Fig. 4. Representative electrodes showing grand average ERPs (n = 28) to correct (solid lines) and incorrect (dotted lines) solutions separately for LA+ (black) and LA− (red) The representative electrodes plotted here are indicated by “X”s on the head plot (top center). Below the ERP traces are topographic (isovoltage) maps of the correctness effect (incorrect – correct) for each language between 300 and 500 ms after solution onset. Between the topographic maps is a strip chart reflecting the mean amplitude differences for each individual participant, averaged across all electrodes, for LA+ (black dots) and LA− (red dots).

3.2. Word-picture verification task

p = 0.09) or response time (F(1,27) = 0.004, p = 0.94). The ERPs revealed a classic N400 effect beginning at around 200 ms from the onset of the picture, with larger amplitude for incongruent than congruent pictures (Fig. 5). The N400 congruency effect was present for language conditions, here L1 and L2 for first and second learned languages. The timing of the peak of the effect was not different across languages (t(23) = 0.749; p = 0.46; L1: 384.39 ms, SE = 9.19; L2: 373.52 ms, SE = 11.30), so a single time window (279–479 ms) was used to measure mean amplitude for both L1 and L2. A 2(Language: L1, L2) × 2(Congruency: congruent, incongruent) × 26(Electrode) ANOVA revealed a main effect of Congruency (F(1,23) = 41.48; p < 0.001) where incongruent pictures were more negative (-3.35µv, SE = 0.92) than congruent pictures (2.45 µv, SE = 1.12). There was an interaction between Congruency and Electrode (F(1,23) = 11.38; p < 0.001), with a widely distributed effect that was largest over midline electrodes. An additional distributional analysis replaced Electrodes with 2 levels of Laterality, 2 levels of Hemisphere, and 4 levels of Anteriority and confirmed the presence of a larger effect over medial parietal electrodes (Congruency × Laterality × Anteriority: F(3,21) = 4.58; p < 0.05). Critically, there were no significant differences in the size of the N400

In comparing performance on the language task (WPV) between L1 and L2, a 2 × 2 ANOVA revealed significant main effects of Congruency on both accuracy (F(1,23) = 13.07, p < 0.01) and response time (F (1,27) = 42.74, p < 0.001), with children responding more accurately to incongruent pictures (94.63%) than congruent pictures (91.62%) and faster to congruent (961.04 ms) than incongruent (1031.23 ms) pictures. The slower and more accurate responses for incongruent trials likely reflects a speed/accuracy tradeoff.3 There was no main effect of language for either measure (RT: F(1,23) = 4.03, p = 0.06; accuracy: F (1,23) = 0.55, p = 0.46), indicating that children were similarly fast and accurate in both languages. Additionally, there was no interaction between Congruency and Language for either accuracy (F(1,23) = 2.98, 3

The congruency effect might also be explained by the children making more confident judgments for the grossly anomalous incongruent pairings (“cow” followed by a picture of a bear) than for the congruent pairings that could in principle be depicted many ways (“cow” followed by a specific picture of a cow that might not be exactly what the child imagined). 8

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4.1. Behavioral findings

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As expected, children were faster and more accurate at identifying correct than incorrect solutions, regardless of the language in which the operands were presented. Interestingly, although there were no gross differences in speed and accuracy across languages, the children did show larger behavioral effects (i.e., a larger difference between correct and incorrect solutions) in LA+ than LA− (see Fig. 2). Namely, bilinguals were better at identifying correct solutions in their LA+ than their LA−. These findings are similar to behavioral findings in previous studies showing a response advantage in one language over the other (Bernardo & Calleja, 2005; Bernardo, 2002; Frenck-Mestre & Vaid, 1993; Marsh & Maki, 1976; Salillas & Wicha, 2012; Spelke & Tsivkin, 2001). Based on these behavioral findings alone, it would seem that children are able to distinguish between different solutions more efficiently in their LA+. However, the ERP results paint a different picture.

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Fig. 5. Word Picture Verification task. On the left, grand average ERPs (n = 24) shown at the medial parietal electrode time-locked to picture onset. Congruent (solid lines) and incongruent (dotted lines) items were preceded by spoken nouns in L1 (blue) and L2 (green). It should be noted that L1 and L2 do not necessarily correspond to LA+ and LA− in all individuals. Robust N400 congruency effects are seen for both languages with no differences in the size of the effect across languages. The distribution of the N400 effect, shown on the right as isovoltage maps, reveals the typical broad, but slightly more frontal distribution for pictures.

4.2. ERP morphology – qualitative measures In line with results from monolingual children (Grenier et al., 2018; Prieto-Corona et al., 2010; Wicha et al., 2018), the N400 was overall smaller in amplitude for correct than incorrect solutions (see Figs. 3 and 4). As discussed above, this N400 effect indexes that semantic-level processing of correct solutions was facilitated compared to incorrect solutions based on the preceding operands (e.g., Dickson et al., 2018; Kutas & Federmeier, 2011). Critically, both LA+ and LA− showed this N400 effect, indicating that bilingual children, like adults, use similar neurocognitive processes to verify the arithmetic facts in both languages. If bilingual children had been using different strategies in each language to access the multiplication facts, say retrieval in LA+ versus translation from LA− to LA+, then this should have been reflected in qualitatively different ERP components. Instead, the equivalent N400s reflect access to semantic memory in both languages. Because our operands were spoken, children were forced to retrieve/process the operation in that language. The digit solution acted as a language-neutral probe of the level of activation in the semantic network based on the prior language specific retrieval process. Using digit solutions for both languages meant that any differences observed at the solution would have been driven by the operation (e.g., arithmetic fact retrieval in that language), and not the solution, per se (e.g., differences in decoding the number word for the solution). Although it is possible that presenting identical solutions lead to the similar ERPs observed across languages, this is unlikely. In the adult bilingual studies mentioned above (Martinez-Lincoln et al., 2015; Salillas & Wicha, 2012) the solutions were written words, and therefore different across languages. Still, the brain response was the same in both languages. Additionally, it is, in fact, possible to see different brain responses across conditions when using identical solutions. For example, monolingual adults show an N400 to digit solutions preceded by spoken number words and a P300 to the same solutions preceded by digits (Dickson et al., 2018). Dickson et al. concluded that these different components reflected different levels of processing at the digit solutions based on the format of the preceding operands. As further evidence of equivalent processes in LA+ and LA−, visual inspection of the ERPs to the preceding spoken operands also revealed no measurable differences across languages. Do the equivalent effects across languages reflect separate memory representations in each language? Not necessarily. In principle, the equivalent N400 effects across languages imply that accessing arithmetic facts engages a similar cognitive process in both languages. This may support the structure of the Encoding Complex Model (ECM), which proposes direct access to independent representations of arithmetic facts in LA+ and LA−. Based on this model, differences in processing stem from differences in efficiency of access. However, differences in access do not preclude a single representation. For example, the Abstract-Modular Model proposes a

effect across languages (F(1,23) = 0.07; p = 0.78), similar to the multiplication task. 4. Discussion The goal of this study was to determine if bilingual children process simple multiplication facts differently in their two languages. This question stems from studies measuring response time and accuracy where bilinguals have shown a disparity across languages for verifying simple arithmetic facts. Usually the language in which the facts were learned (LA+) is faster and more accurate than the other language (LA−). Models of how these arithmetic facts are stored and accessed in the bilingual brain have attempted to explain these behavioral effects. One model suggests that arithmetic facts are encoded in a single language-specific code (TCM; Dehaene, 1992; Dehaene & Cohen, 1995, 1997). Although this model does not explicitly address bilinguals, it can be inferred from subsequent research that bilinguals might use qualitatively different neurocognitive processes in each language for arithmetic, as discussed in the introduction. Another model proposes that bilinguals have similar representations in each language that are modulated by experience with differential ease of access to these facts in each arithmetic memory network (ECM; Campbell & Epp, 2004). By inference bilinguals should show quantitative but not qualitative differences in the neurocognitive processes underlying arithmetic in each language. Although the first ERP studies with adult bilinguals provided evidence of similar processing mechanisms in both languages, this study aimed to determine if different mechanisms might be engaged in bilingual children who might be more likely to show a preference for the LA+ for their recently learned multiplication facts. Our behavioral findings replicated prior studies, revealing a small but significant difference in the effect of correctness across languages. Namely, LA+ elicited larger correctness effects in both accuracy and response time compared to the LA−. Our ERP findings revealed that, similar to bilingual adults, these behavioral differences were likely not due to a qualitative difference in processing across languages, given that an N400 effect was observed in both languages. Moreover, the behavioral differences may not be driven by semantic level processes at all, given that there were no significant quantitative differences across languages in this N400 effect. We discuss the implications of the present findings in more detail below. 9

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single format-independent representation for arithmetic facts at the level of meaning (McCloskey & Macaruso, 1994, 1995; McCloskey, 1992). This is contrary to ECM and TCM, where representation refers to the memory of the arithmetic fact, which is separate from their meaning. This model would also account for differences across languages based on efficiency of access, for example, if encoding number words in one language was slower than in the other. However, several studies have provided evidence that processing differences across formats are not simply due to differences in encoding (Campbell & Epp, 2004; Noël, Fias, & Brysbaert, 1997). Another possible explanation is that children automatically translate from the spoken number word to the digit equivalent.4 When they see the solution, they may have already retrieved the digit representation of the multiplication fact. Under this interpretation, the results would index equivalent transcoding from each language to the digit format. Our data cannot speak to whether arithmetic facts are represented as format-specific (digits, LA+, LA−) or format-independent. However, the equivalent N400 effects imply that both languages engage similar cognitive processes, in contrast to the suggestion that LA− might have to be translated to LA+ (e.g., Spelke & Tsivkin, 2001).

measures. The N400 is automatic and does not require a specific behavioral output (Kutas & Federmeier, 2011). In principle, the N400 effect would be present even without a button press response. The children may be processing the multiplication problems similarly to sentences. N400 amplitude to words in a sentence varies with the person’s expectation for an upcoming word based on the preceding context, but not with the person’s subjective rating of how well a presented incongruous word fits in the sentence (Federmeier & Kutas, 1999). This might help explain why the correctness rating at the solution lead to a behavioral effect, but no modulation of the N400 amplitude. We can only conjecture that the behavioral difference across languages may instead arise from differences downstream from semantic access, such as response bias, response selection, or other aspects of task demands that do not modulate the N400. This is something that will need to be addressed in future research. 4.4. Comparison of bilingual adults and children The current ERP findings from bilingual children appear to contradict findings from bilingual adults, where larger ERP correctness effects have been reported in LA+ than LA− (Martinez-Lincoln et al., 2015; Salillas & Wicha, 2012). However, this outcome might be explained by differences in experimental design and/or differences between the groups in their experience with arithmetic, as laid out below. First, the previous ERP studies with bilingual adults showed language differences in the N400 arithmetic correctness effect measured from the onset of written number-word solutions. Therefore, the ERPs might reflect processing differences for written number words rather than differential access to the number facts. That is, reading ability/ frequency differences across languages could have caused less efficient processing in the LA−. Further, arithmetic facts are rarely encountered as written words. Campbell (1994) suggests that written number word stimuli would be processed less efficiently even compared to Arabic digit stimuli. It is possible then that these studies identified differences in semantic access across languages that were driven by weak associations between written number words and simple math facts, rather than differences in accessing language-based memory representations, per se.5 Second, a key difference between children and adults is their amount of experience with simple multiplication facts. In line with training studies in adults, we hypothesized that bilingual children might have been more likely than adult bilinguals to show differences across languages, given that they had recently learned multiplication facts in only one of their languages (e.g.,Saalbach et al., 2013; Spelke & Tsivkin, 2001; Venkatraman, Siong, Chee, & Ansari, 2006). However, it might be the opposite, in that a lifetime of favoring one language over another strengthens access to LA+, leading to differences in processes across languages in adulthood that are not observed in childhood. After learning multiplication facts around 3rd grade, children continue to use multiplication facts when learning more complex mathematic concepts (i.e. algebra, geometry) and in daily life transactions well into adulthood. In monolinguals it has been shown that children process

4.3. ERP amplitude and timing – quantitative measures ECM also suggests that experience should modulate the efficiency of access to arithmetic facts in each format. Indeed, children responded faster (1302.71 ms versus 1315.20 ms) and more accurately (80.2% versus 76.0%) to correct solutions in LA+ than LA−. However, these small effects in behavior were not reflected on the N400, as there were no significant differences in amplitude or timing of the effect across languages. ERPs are an incomplete measure of neural activity, thus it is possible that differences in brain activity occurred across languages that are below measurable threshold. However, given the robustness of the N400 components observed in both languages, any potential difference would be relatively small. This has two important implications. First, the equivalent effect on N400 amplitude across languages suggests that these young bilinguals constructed expectations in semantic memory prior to solution presentation equivalently in LA+ and LA−. As discussed above, this could be through direct retrieval in each language or through transcoding from spoken number words to digits. Similarly, the equivalent effect on the N400 timing suggests that the speed in which these early bilinguals are able to process the meaning of arithmetic solutions is equivalent in both languages. This is perhaps a reflection of this sample’s balanced proficiency in both languages overriding any potential advantage of LA+. Thus, although most studies argue for an advantage in LA+, our data suggest that the language of learning is not the primary determiner of performance in these young bilinguals. A second implication of the equivalent N400 effects across languages is that semantic-level processes do not account for the modest behavioral differences in response time and accuracy. Where then do the behavioral effects stem from if not from the stage of processing indexed by the N400? The behavioral responses likely reflect the culmination of all processing stages from the onset of the multiplication problem until a response is given. The differences in button presses may index differences in lexical access of the number words before the solution appears. However, there were no differences across languages at the second operand either. Similarly, these children did not show differences across languages on the word-picture verification task, reflecting proficiency in both languages. Therefore, the behavioral effects are likely not due to differences in lexical access across languages. Importantly, the N400 is not necessarily linked to performance

5 There were other experimental differences in response demands and stimuli. Responses were delayed for adults but immediate for children, but it is not clear how this would affect differences across languages. Another difference was the nature of the incorrect solutions. Previous bilingual studies that had reported language differences did so on effects of solution relatedness (Martinez-Lincoln et al., 2015; Salillas & Wicha, 2012), a factor not manipulated here due to limitations with experiment length for children. Relatedness effects even in monolingual children are not well-established, so it is unclear how outcomes on this variable would be interpreted in this bilingual sample. Moreover, the reproducibility of the relatedness effect even in monolingual adults remains controversial, with nuances of stimulus design as well as important differences in component identification across studies (Dickson & Federmeier, 2017; cf. Niedeggen & Rösler, 1999).

4 It should be noted that this would be opposite of what models like the Triple Code Model propose where transcoding should occur in the other direction (from digit to verbal code).

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multiplication facts differently than adults even when they are presented as Arabic digits (Grenier et al., 2018; Moore et al., 2014; PrietoCorona et al., 2010; Wicha et al., 2018). Therefore, it might not be too surprising to observe differences between bilingual children and adults. There is evidence that language use can modulate the arithmetic N400 effect in adults. Martinez-Lincoln et al. (2015) showed that bilingual teachers who taught math in their LA− showed no difference in the N400 effect across their languages, in contrast to teachers who taught in their LA+. Since the children in the current study use both languages daily in their dual-language schools, this may drive more efficient access to math facts (or number words) in both languages, even though they report learning arithmetic in only one of their languages. Here, we included independent measures of language and math ability (a practice that is unfortunately still not standard in bilingualism research, where language proficiency is often inferred from self-report). These measures allowed us to analyze subgroups of children with more homogenous language skills. That is, the natural heterogeneity of our dual language sample allowed for a more acute test of language experience on the N400 effect. Even in the subset of SDL bilinguals, whose LA+ was also their home language, there were equivalent N400 arithmetic effects in both languages. These bilingual children also showed equivalent N400s across languages in the semantic word-picture verification task (Fig. 5). Similarly, Moreno and Kutas (2005) found that bilingual adults show equivalent N400 effects across dominant and nondominant languages when reading sentences. This supports the idea that access to basic semantic information across languages in fluent bilinguals does not differ for simple familiar concepts, either in language or math. This comparison provides further support that children access simple multiplication facts from memory in the same way as other meaningful information, and equivalently across languages. Previous research has focused on how different levels of representation are shared versus separate across a bilingual’s languages (e.g., Francis, 1999; Shook & Marian, 2013). Future work should address the importance of the language of learning for other facts, both within and beyond the math domain.

of the bilingual brain in handling cognitive challenges, contradicting biases against embracing bilingualism in and out of the classroom. Acknowledgements This study was funded by awards R21HD079884 from the Eunice Kennedy Shriver National Institute of Child Health and Human Development, and BRAIN EAGER award 1451032 from the National Science Foundation. V.R.C. was funded by RISE R25-GM060655. This work received computational support from the UTSA’s HPC Cluster Shamu, operated by the Office of Information Technology. We thank Dr. Danielle Dickson for all of her careful edits and thoughtful feedback. Thanks also to Jacob Momsen for technical assistance with the sound files, and Maria Estefania and Ana Sophia Niembro, the voices of the number words and nouns, respectively, and to our subject recruitment team, Bianca Obinyan, Ana Godinez, and Paola Montufar. Lastly, we thank all the bilingual children and their families for their time and participation in this study. 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4.5. Conclusions This study aimed to determine if bilingual children use qualitatively different cognitive processes for solving arithmetic problems in each of their languages. We replicated behavioral differences, showing larger effects of correctness on both accuracy and response time in the LA+ than LA−. However, ERP results revealed that bilingual children use similar processing mechanisms, namely semantic level processing, when verifying math facts in each of their languages. This finding is inconsistent with models of arithmetic that suggest different neurocognitive processes should be engaged in each language. Moreover, the finding of equivalent N400s across languages indicates that the behavioral differences are not driven by semantic-level processes. Importantly, this study suggests that access to arithmetic facts across languages seems to resemble general language processing in proficient bilingual children. Given that early math fluency can predict later math outcomes, our findings reassuringly suggest that bilingual children are quite flexible in their ability to retrieve math facts from memory, regardless of the language used. This is not to disregard the real differences across languages in performance measures reported throughout the literature. Our findings imply that the origin of these differences is somewhere beyond semantic access. Further research is needed to elucidate exactly where. Moreover, our sample included mostly children who were proficient in both languages. It is still to be determined how differences in language experience, such as age of acquisition and language proficiency, might modulate the effects that we observed in proficient early bilinguals. Nevertheless, our results do not support a strict view that bilinguals access arithmetic facts in only one language, i.e., the language of learning. Instead, our findings speak strongly to the flexibility 11

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