Language and number: a bilingual training study

Thinking about numbers in different tongues: An overview of the influences of multilingualism on numerical and mathematical competencies

In an increasingly multilingual and multicultural world, understanding the interactions between language and mathematics is critical, especially when individuals must acquire and exercise their mathematical competencies in multiple languages. Indeed, research shows that, overall, L2 language learners are at an academic disadvantage compared to their L1 peers. The current article briefly overviews how multilingualism influences basic and advanced mathematical skills and interacts with mathematical learning difficulties. We first outline the traditional cognitive models of number learning and language processing. We then discuss the particularities of multilingualism and how it impacts numerical skills such as counting and building lexical-semantic associations, transcoding and arithmetic, mathematical word problems and mathematical performance tests, and dyscalculia diagnosis. We end this review by outlining challenges, recommendations, and solutions for multilingual educational settings. The article is intended as a guide for numerical cognition researchers who work with diverse populations and for mathematics educators and educational policy-makers facing the challenges of a multilingual classroom.

Multiple number-naming associations: How the inversion property affects adults' two-digit number processing

Some number-naming systems are less transparent than others. For example, in Dutch, 49 is named "negenenveertig," which translates to "nine and forty," i.e., the unit is named first, followed by the decade. This is known as the "inversion property," where the morpho-syntactic representation of the number name is incongruent with its written Arabic form. Number word inversion can hamper children's developing mathematical skills. But little is known about its effects on adults' numeracy, the underlying mechanism, and how a person's bilingual background influences its effects. In the present study, Dutch-English bilingual adults performed an audiovisual matching task, where they heard a number word and simultaneously saw two-digit Arabic symbols and had to determine whether these matched in quantity. We experimentally manipulated the morpho-syntactic structure of the number words to alter their phonological (dis)similarities and numerical congruency with the target Arabic two-digit number. Results showed that morpho-syntactic (in)congruency differentially influenced quantity match and non-match decisions. Although participants were faster when hearing traditional non-transparent Dutch number names, they made more accurate decisions when hearing artificial, but morpho-syntactically transparent number words. This pattern was partly influenced by the participants' bilingual background, i.e., their L2 proficiency in English, which involves more transparent number names. Our findings suggest that, within inversion number-naming systems, multiple associations are formed between two-digit Arabic symbols and number names, which can influence adults' numerical cognition.

Making sense of domain specificity

The notion of domain specificity plays a central role in some of the most important debates in cognitive science. Yet, despite the widespread reliance on domain specificity in recent theorizing in cognitive science, this notion remains elusive. Critics have claimed that the notion of domain specificity can't bear the theoretical weight that has been put on it and that it should be abandoned. Even its most steadfast proponents have highlighted puzzles and tensions that arise once one tries to go beyond an initial intuitive sketch of what domain specificity involves. In this paper, we address these concerns head on by developing an account of what it means for a cognitive mechanism to be domain specific that overcomes the obstacles that have made domain specificity seem so problematic. We then apply this understanding of domain specificity to one of the key debates that it has figured prominently in-the rationalism-empiricism debate concerning the origins of cognitive traits-and introduce several related theoretical notions that work alongside domain specificity in helping to clarify what makes a view more (or less) rationalist. This example illustrates how the notion of domain specificity can, and should, continue to play a central role in ongoing debates in cognitive science.

Language-switching and retrieval-based learning: an unfavorable combination

Language-switching costs arise when learners encode information in one language and subsequently recall that information in a different language. The assumed cognitive mechanism behind these costs is the principle of encoding specificity that implies language-dependent representations of information. The aim of our study was to test this mechanism and to gain insights into the impact of language-switching on subsequent learning. To this end, we used retrieval-based learning as a carrier-paradigm. In a 2×3-design, 117 participants learned mathematical concepts with a practice-test or a restudy opportunity (within-subjects factor). In addition, the sample was divided into three groups regarding language-switching (between-subjects factor): one group without switching, one switched for the final tests, and one switched between initial learning and subsequent learning. Results show the expected main effects: participants performed better for the items learned via retrieval-based learning (testing-effect) and worse in conditions with language-switching (language-switching-costs). Most importantly, we were able to find an interaction between learning condition and language-switching: retrieval-based learning suffers particularly from language-switching. Additionally, our results indicate that language switching before subsequent learning seems to be particularly detrimental. These results provide both validation for encoding specificity as mechanism underlying language-switching costs and new information on the impact of the time of language-switching that can be considered in educational designs such as "Content and Language Integrated Learning."

Dissociating Language and Thought in Human Reasoning

What is the relationship between language and complex thought? In the context of deductive reasoning there are two main views. Under the first, which we label here the language-centric view, language is central to the syntax-like combinatorial operations of complex reasoning. Under the second, which we label here the language-independent view, these operations are dissociable from the mechanisms of natural language. We applied continuous theta burst stimulation (cTBS), a form of noninvasive neuromodulation, to healthy adult participants to transiently inhibit a subregion of Broca's area (left BA44) associated in prior work with parsing the syntactic relations of natural language. We similarly inhibited a subregion of dorsomedial frontal cortex (left medial BA8) which has been associated with core features of logical reasoning. There was a significant interaction between task and stimulation site. Post hoc tests revealed that performance on a linguistic reasoning task, but not deductive reasoning task, was significantly impaired after inhibition of left BA44, and performance on a deductive reasoning task, but not linguistic reasoning task, was decreased after inhibition of left medial BA8 (however not significantly). Subsequent linear contrasts supported this pattern. These novel results suggest that deductive reasoning may be dissociable from linguistic processes in the adult human brain, consistent with the language-independent view.

Number transcoding in bilinguals—A transversal developmental study

Number transcoding is the cognitive task of converting between different numerical codes (i.e. visual “42”, verbal “forty-two”). Visual symbolic to verbal transcoding and vice versa strongly relies on language proficiency. We evaluated transcoding of German-French bilinguals from Luxembourg in 5^th, 8^th, 11^th graders and adults. In the Luxembourgish educational system, children acquire mathematics in German (LM1) until the 7^th grade, and then the language of learning mathematic switches to French (LM2). French `70s `80s `90s are less transparent than `30s `40s `50s numbers, since they have a base-20 structure, which is not the case in German. Transcoding was evaluated with a reading aloud and a verbal-visual number matching task. Results of both tasks show a cognitive cost for transcoding numbers having a base-20 structure (i.e. `70s, `80s and `90s), such that response times were slower in all age groups. Furthermore, considering only base-10 numbers (i.e. `30s `40s `50s), it appeared that transcoding in LM2 (French) also entailed a cost. While participants across age groups tended to read numbers slower in LM2, this effect was limited to the youngest age group in the matching task. In addition, participants made more errors when reading LM2 numbers. In conclusion, we observed an age-independent language effect with numbers having a base-20 structure in French, reflecting their reduced transparency with respect to the decimal system. Moreover, we find an effect of language of math acquisition such that transcoding is less well mastered in LM2. This effect tended to persist until adulthood in the reading aloud task, while in the matching task performance both languages become similar in older adolescents and young adults. This study supports the link between numbers and language, especially highlighting the impact of language on reading numbers aloud from childhood to adulthood.

The Effects of Language Instruction on Math Development

How does language shape mathematical development? In this article, we consider this question by reviewing findings from cross-sectional and longitudinal research. In this literature, we find that differences in the structures of languages and individual variation in language ability are associated with mathematical performance in both obvious and unexpected ways. We then consider the causal nature of these relations, with a focus on experimental studies that have tested the effects of language instruction on mathematical outcomes. Findings from this work show that certain forms of language instruction meaningfully improve performance in several mathematical domains, providing strong evidence of a linguistic pathway in mathematical development. However, much additional research is needed to understand how language instruction may be integrated optimally into math education. We conclude with recommendations for research.

Reevaluating the Language of Learning Advantage in Bilingual Arithmetic: An ERP Study on Spoken Multiplication Verification

Many studies of bilingual arithmetic report better performance when verifying arithmetic facts in the language of learning (LA+) over the other language (LA−). This could be due to language-specific memory representations, processes established during learning, or to language and task factors not related to math. The current study builds on a small number of event-related potential (ERP) studies to test this question while controlling language proficiency and eliminating potential task confounds. Adults proficient in two languages verified single-digit multiplications presented as spoken number words in LA+ and LA−, separately. ERPs and correctness judgments were measured from solution onset. Equivalent P300 effects, with larger positive amplitude for correct than incorrect solutions, were observed in both languages (Experiment 1A), even when stimuli presentation rate was shortened to increase difficulty (Experiment 1B). This effect paralleled the arithmetic correctness effect for trials presented as all digits (e.g., 2 4 8 versus 2 4 10), reflecting efficient categorization of the solutions, and was distinct from an N400 generated in a word–picture matching task, reflecting meaning processing (Experiment 2). The findings reveal that the language effects on arithmetic are likely driven by language and task factors rather than differences in memory representation in each language.

How numerical surface forms affect strategy execution in subtraction? Evidence from behavioral and ERP measures

In this study, to investigate the influence of numerical surface forms on strategy execution in subtraction and addition, 25 college students were required to perform a mental arithmetic task. Behavioral and electroencephalography responses were recorded to explore the stages of strategy execution. Arithmetic problems were presented within 11-20 by Arabic, Chinese and English numerals, respectively. The results showed that, compared with Arabic and Chinese numerals, participants had longer RTs under the condition of English numerals. English numerals induced stronger N100, P200 and P300 amplitudes, and longer latency in P200 and P300 amplitudes than in other conditions. These findings indicated that numerical surface forms affected the execution of mental arithmetic strategies, with Chinese and English numerals being processed deeper in the brain. Especially, the English digital word form occupied more psychological resources, and it was activated more in the frontal area and the parietal area, which would obviously hinder the processing of arithmetic information.

Testing the role of symbols in preschool numeracy: An experimental computer-based intervention study

Numeracy is of critical importance for scholastic success and modern-day living, but the precise mechanisms that drive its development are poorly understood. Here we used novel experimental training methods to begin to investigate the role of symbols in the development of numeracy in preschool-aged children. We assigned pre-school children in the U.S. and Italy (N = 215; Mean age = 49.15 months) to play one of five versions of a computer-based numerical comparison game for two weeks. The different versions of the game were equated on basic features of gameplay and demands but systematically varied in numerical content. Critically, some versions included non-symbolic numerical comparisons only, while others combined non-symbolic numerical comparison with symbolic aids of various types. Before and after training we assessed four components of early numeracy: counting proficiency, non-symbolic numerical comparison, one-to-one correspondence, and arithmetic set transformation. We found that overall children showed improvement in most of these components after completing these short trainings. However, children trained on numerical comparisons with symbolic aids made larger gains on assessments of one-to-one correspondence and arithmetic transformation compared to children whose training involved non-symbolic numerical comparison only. Further exploratory analyses suggested that, although there were no major differences between children trained with verbal symbols (e.g., verbal counting) and non-verbal visuo-spatial symbols (i.e., abacus counting), the gains in one-to-one correspondence may have been driven by abacus training, while the gains in non-verbal arithmetic transformations may have been driven by verbal training. These results provide initial evidence that the introduction of symbols may contribute to the emergence of numeracy by enhancing the capacity for thinking about exact equality and the numerical effects of set transformations. More broadly, this study provides an empirical basis to motivate further focused study of the processes by which children’s mastery of symbols influences children’s developing mastery of numeracy.

Relations among spatial skills, number line estimation, and exact and approximate calculation in young children

Decades of research have established that spatial skills correlate with numerical skills. However, because both spatial and numerical skills are multidimensional, we sought to determine how specific spatial skills relate to specific numeracy skills. We used a cohort-sequential design, assessing a large diverse sample of students (N = 612, initially in pre-kindergarten [pre-K]-3rd grade, 4-9 years of age) at four time points spanning 2 years. We examined how initial levels of five spatial skills (visuospatial working memory [VSWM], mental transformation, mental rotation, proportional reasoning, and analog magnitude system [AMS] acuity) related to initial levels and growth rates in exact and approximate calculation skills, and we further investigated number line estimation as a potential mediator. We found unique patterns of relations between spatial skills and numeracy. Initial levels of mental rotation, proportional reasoning, and AMS acuity related to initial levels of exact calculation skill; initial levels of AMS acuity related to initial levels of approximate calculation; and initial levels of proportional reasoning related to initial levels of number line estimation. VSWM and mental transformation did not relate to numeracy skills after controlling for other spatial skills. Initial levels of number line estimation related to both exact and approximate calculation after controlling for spatial skills. Notably, neither spatial skills nor number line estimation predicted growth in exact or approximate calculation skills. These results indicate that there is specificity in the time-invariant relations between spatial skills and numeracy, and they suggest that researchers and educators should treat spatial skills and numeracy as multidimensional constructs with complex and unique interrelations.

Learning a new geometric concept: The role of working memory and of domain-specific abilities

It has been suggested that not only domain‐specific factors but also working memory (WM) may play a crucial role in mathematical learning included Geometry, but the issue has not been deeply explored. In the present study, we examined the role of domain‐specific factors and of verbal versus visuospatial WM on geometric learning of a new geometrical figure (trapezoid), never presented previously by the teachers participating to the study, after a lecture also involving manipulatives. Results on 105 children in their Year 4 indicated that not only some domain‐specific components (geometric declarative knowledge and calculation) but also visuospatial working memory had a significant specific impact on the ability of solving geometric problems requiring to calculate the perimeter and the area of the new figure. On the contrary, verbal WM and geometrical mental imagery did not offer a specific contribution. These findings could have important educational implications, stressing the importance of taking into account the main different aspects supporting the acquisition of geometry.

Tapping into the interplay of lexical and number knowledge using fast mapping: A longitudinal eye-tracking study with two-year-olds

Language skills and mathematical competencies are argued to influence each other during development. While a relation between the development of vocabulary size and mathematical skills is already documented in the literature, this study further examines how children's ability to map a novel word to an unknown object as well as their ability to retain this word from memory may be related to their knowledge of number words. Twenty-five children were tested longitudinally (at 30 and at 36 months of age) using an eye-tracking-based fast mapping task, the Give-a-Number task, and standardized measures of vocabulary. The results reveal that children's ability to create and retain a mental representation of a novel word was related to number knowledge at 30 months, but not at 36 months while vocabulary size correlated with number knowledge only at 36 months. These results show that even specific mapping processes are initially related to the acquisition of number words and they speak for a parallelism between the development of lexical and number-concept knowledge despite their semantic and syntactic differences.

How the Language of Instruction Influences Mathematical Thinking Development in the First Years of Bilingual Schoolers

The present research study focuses on how the language of instruction has an impact on the mathematical thinking development as a consequence of using a language of instruction different from the students' mother tongue. In CLIL (Content and Language Integrated Learning) academic content and a foreign language are leant at the same time, a methodology that is widely used in the schools in the present times. It is, therefore, our main aim to study if the language of instruction in second language immersion programs influences the development of the first formal mathematical concepts. More specifically, if the learning of mathematical concepts in the early ages develops in a similar way if it is taught in the students' mother tongue and is not influenced by the language used for teaching. Or else, if it can influence the development of the first skills only in the students' general performance or in certain areas. The results of both the analysis of variance and multiple regression confirm how influencing the language of instruction is when mathematical thinking is developed teaching formal contents in a non-coincidence language. The second language is affecting the resolution of daily life problems, being more competent those students in 1st grades whose language of instruction matched with their mother tongue.

Using Occam's razor and Bayesian modelling to compare discrete and continuous representations in numerosity judgements

Previous research has established that numeric estimates are based not just on perceptual data but also past experience, and so may be influenced by the form of this stored information. It remains unclear, however, how such experience is represented: numerical data can be processed by either a continuous analogue number system or a discrete symbolic number system, with each predicting different generalisation effects. The present paper therefore contrasts discrete and continuous prior formats within the domain of numerical estimation using both direct comparisons of computational models of this process using these representations, as well as empirical contrasts exploiting different predicted reactions of these formats to uncertainty via Occam's razor. Both computational and empirical results indicate that numeric estimates commonly rely on a continuous prior format, mirroring the analogue approximate number system, or 'number sense'. This implies a general preference for the use of continuous numerical representations even where both stimuli and responses are discrete, with learners seemingly relying on innate number systems rather than the symbolic forms acquired in later life. There is however remaining uncertainty in these results regarding individual differences in the use of these systems, which we address in recommendations for future work.

Language-specific numerical estimation in bilingual children

We tested 5- to 7-year-old bilingual learners of French and English (N = 91) to investigate how language-specific knowledge of verbal numerals affects numerical estimation. Participants made verbal estimates for rapidly presented random dot arrays in each of their two languages. Estimation accuracy differed across children's two languages, an effect that remained when controlling for children's familiarity with number words across their two languages. In addition, children's estimates were equivalently well ordered in their two languages, suggesting that differences in accuracy were due to how children represented the relative distance between number words in each language. Overall, these results suggest that bilingual children have different mappings between their verbal and nonverbal counting systems across their two languages and that those differences in mappings are likely driven by an asymmetry in their knowledge of the structure of the count list across their languages. Implications for bilingual math education are discussed.

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

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.

Language and Math: What If We Have Two Separate Naming Systems?

The role of language in numerical processing has traditionally been restricted to counting and exact arithmetic. Nevertheless, the impact that each of a bilinguals’ languages may have in core numerical representations has not been questioned until recently. What if the language in which math has been first acquired (LLmath) had a bigger impact in our math processing? Based on previous studies on language switching we hypothesize that balanced bilinguals would behave like unbalanced bilinguals when switching between the two codes for math. In order to address this question, we measured the brain activity with magneto encephalography (MEG) and source estimation analyses of 12 balanced Basque-Spanish speakers performing a task in which participants were unconscious of the switches between the two codes. The results show an asymmetric switch cost between the two codes for math, and that the brain areas responsible for these switches are similar to those thought to belong to a general task switching mechanism. This implies that the dominances for math and language could run separately from the general language dominance.

Neuroplasticity, bilingualism, and mental mathematics: A behavior-MEG study

Bilingual experience alters brain structure and enhances certain cognitive functions. Bilingualism can also affect mathematical processing. Reduced accuracy is commonly reported when arithmetic problems are presented in bilinguals' second (L2) vs. first (L1) language. We used MEG brain imaging during mental addition to characterize spatiotemporal dynamics during mental addition in bilingual adults. Numbers were presented auditorally and sequentially in bilinguals' L1 and L2, and brain and behavioral data were collected simultaneously. Behaviorally, bilinguals showed lower accuracy for two-digit addition in L2 compared to L1. Brain data showed stronger response magnitude in L2 versus L1 prior to calculation, especially when two-digit numbers were involved. Brain and behavioral data were significantly correlated. Taken together, our results suggest that differences between languages emerge prior to mathematical calculation, with implications for the role of language in mathematics.

The word order of languages predicts native speakers' working memory

The relationship between language and thought is controversial. One hypothesis is that language fosters habits of processing information that are retained even in non-linguistic domains. In left-branching (LB) languages, modifiers usually precede the head, and real-time sentence comprehension may more heavily rely on retaining initial information in working memory. Here we presented a battery of working memory and short-term memory tasks to adult native speakers of four LB and four right-branching (RB) languages from Africa, Asia and Europe. In working memory tasks, LB speakers were better than RB speakers at recalling initial stimuli, but worse at recalling final stimuli. Our results show that the practice of parsing sentences in specific directions due to the syntax and word order of our native language not only predicts the way we remember words, but also other non-linguistic stimuli.

A Mechanistic Study of the Association Between Symbolic Approximate Arithmetic Performance and Basic Number Magnitude Processing Based on Task Difficulty

Two types of number magnitude processing – semantic and spatial – are significantly correlated with children’s arithmetic performance. However, it remains unclear whether these abilities are independent predictors of symbolic approximate arithmetic performance. The current study addressed this question by assessing 86 kindergartners (mean age of 5 years and 7 months) on semantic number processing (number comparison task), spatial number processing (number line estimation task), and symbolic approximate arithmetic performance with different levels of difficulty. The results showed that performance on both tasks of number magnitude processing was significantly correlated with symbolic approximate arithmetic performance, but the strength of these correlations was moderated by the difficulty level of the arithmetic task. The simple symbolic approximate arithmetic task was equally related to both tasks. In contrast, for more difficult symbolic approximate arithmetic tasks, the contribution of number comparison ability was smaller than that of the number line estimation ability. These results indicate that the strength of contribution of the different types of numerical processing depends on the difficulty of the symbolic approximate arithmetic task.

Cortical circuits for mathematical knowledge: evidence for a major subdivision within the brain's semantic networks

Is mathematical language similar to natural language? Are language areas used by mathematicians when they do mathematics? And does the brain comprise a generic semantic system that stores mathematical knowledge alongside knowledge of history, geography or famous people? Here, we refute those views by reviewing three functional MRI studies of the representation and manipulation of high-level mathematical knowledge in professional mathematicians. The results reveal that brain activity during professional mathematical reflection spares perisylvian language-related brain regions as well as temporal lobe areas classically involved in general semantic knowledge. Instead, mathematical reflection recycles bilateral intraparietal and ventral temporal regions involved in elementary number sense. Even simple fact retrieval, such as remembering that 'the sine function is periodical' or that 'London buses are red', activates dissociated areas for math versus non-math knowledge. Together with other fMRI and recent intracranial studies, our results indicated a major separation between two brain networks for mathematical and non-mathematical semantics, which goes a long way to explain a variety of facts in neuroimaging, neuropsychology and developmental disorders.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Bilingual education for young children: review of the effects and consequences

Bilingual education has been an educational option in many countries for over 50 years but it remains controversial, especially in terms of its appropriateness for all children. The present review examines research evaluating the outcomes of bilingual education for language and literacy levels, academic achievement, and suitability for children with special challenges. The focus is on early education and the emphasis is on American contexts. Special attention is paid to factors such as socioeconomic status that are often confounded with the outcomes of bilingual education. The conclusion is that there is no evidence for harmful effects of bilingual education and much evidence for net benefits in many domains.

Mental arithmetic in the bilingual brain: Language matters

How do bilinguals solve arithmetic problems in each of their languages? We investigated this question by exploring the neural substrates of mental arithmetic in bilinguals. Critically, our population was composed of a homogeneous group of adults who were fluent in both of their instruction languages (i.e., German as first instruction language and French as second instruction language). Twenty bilinguals were scanned with fMRI (3T) while performing mental arithmetic. Both simple and complex problems were presented to disentangle memory retrieval occuring in very simple problems from arithmetic computation occuring in more complex problems. In simple additions, the left temporal regions were more activated in German than in French, whereas no brain regions showed additional activity in the reverse constrast. Complex additions revealed the reverse pattern, since the activations of regions for French surpassed the same computations in German and the extra regions were located predominantly in occipital regions. Our results thus highlight that highly proficient bilinguals rely on differential activation patterns to solve simple and complex additions in each of their languages, suggesting different solving procedures. The present study confirms the critical role of language in arithmetic problem solving and provides novel insights into how highly proficient bilinguals solve arithmetic problems.

Numbers and functional lateralization: A visual half-field and dichotic listening study in proficient bilinguals

Potential links between language and numbers and the laterality of symbolic number representations in the brain are still debated. Furthermore, reports on bilingual individuals indicate that the language-number interrelationships might be quite complex. Therefore, we carried out a visual half-field (VHF) and dichotic listening (DL) study with action words and different forms of symbolic numbers used as stimuli to test the laterality of word and number processing in single-, dual-language and mixed -task and language- contexts. Experiment 1 (VHF) showed a significant right visual field/left hemispheric advantage in response accuracy for action word, as compared to any form of symbolic number processing. Experiment 2 (DL) revealed a substantially reversed effect - a significant right ear/left hemisphere advantage for arithmetic operations as compared to action word processing, and in response times in single- and dual-language contexts for number vs. action words. All these effects were language independent. Notably, for within-task response accuracy compared across modalities significant differences were found in all studied contexts. Thus, our results go counter to findings showing that action-relevant concepts and words, as well as number words are represented/processed primarily in the left hemisphere. Instead, we found that in the auditory context, following substantial engagement of working memory (here: by arithmetic operations), there is a subsequent functional reorganization of processing single stimuli, whether verbs or numbers. This reorganization - their weakened laterality - at least for response accuracy is not exclusive to processing of numbers, but the number of items to be processed. For response times, except for unpredictable tasks in mixed contexts, the "number problem" is more apparent. These outcomes are highly relevant to difficulties that simultaneous translators encounter when dealing with lengthy auditory material in which single items such as number words (and possibly other types of key words) need to be emphasized. Our results may also shed a new light on the "mathematical savant problem".

Attaching meaning to the number words: contributions of the object tracking and approximate number systems

Children's understanding of the quantities represented by number words (i.e., cardinality) is a surprisingly protracted but foundational step in their learning of formal mathematics. The development of cardinal knowledge is related to one or two core, inherent systems - the approximate number system (ANS) and the object tracking system (OTS) - but whether these systems act alone, in concert, or antagonistically is debated. Longitudinal assessments of 198 preschool children on OTS, ANS, and cardinality tasks enabled testing of two single-mechanism (ANS-only and OTS-only) and two dual-mechanism models, controlling for intelligence, executive functions, preliteracy skills, and demographic factors. Measures of both OTS and ANS predicted cardinal knowledge in concert early in the school year, inconsistent with single-mechanism models. The ANS but not the OTS predicted cardinal knowledge later in the school year as well the acquisition of the cardinal principle, a critical shift in cardinal understanding. The results support a Merge model, whereby both systems initially contribute to children's early mapping of number words to cardinal value, but the role of the OTS diminishes over time while that of the ANS continues to support cardinal knowledge as children come to understand the counting principles.

The relationship between non-verbal systems of number and counting development: a neural signatures approach

Two non-verbal cognitive systems, an approximate number system (ANS) for extracting the numerosity of a set and a parallel individuation (PI) system for distinguishing between individual items, are hypothesized to be foundational to symbolic number and mathematics abilities. However, the exact role of each remains unclear and highly debated. Here we used an individual differences approach to test for a relationship between the spontaneously evoked brain signatures (using event-related potentials) of PI and the ANS and initial development of symbolic number concepts in preschool children as displayed by counting. We observed that individual differences in the neural signatures of the PI system, but not the ANS, explained a unique portion of variance in counting proficiency after extensively controlling for general cognitive factors. These results suggest that differences in early attentional processing of objects between children are related to higher-level symbolic number concept development.

Dissociation between exact and approximate addition in developmental dyslexia

Previous research has suggested that number sense and language are involved in number representation and calculation, in which number sense supports approximate arithmetic, and language permits exact enumeration and calculation. Meanwhile, individuals with dyslexia have a core deficit in phonological processing. Based on these findings, we thus hypothesized that children with dyslexia may exhibit exact calculation impairment while doing mental arithmetic. The reaction time and accuracy while doing exact and approximate addition with symbolic Arabic digits and non-symbolic visual arrays of dots were compared between typically developing children and children with dyslexia. Reaction time analyses did not reveal any differences across two groups of children, the accuracies, interestingly, revealed a distinction of approximation and exact addition across two groups of children. Specifically, two groups of children had no differences in approximation. Children with dyslexia, however, had significantly lower accuracy in exact addition in both symbolic and non-symbolic tasks than that of typically developing children. Moreover, linguistic performances were selectively associated with exact calculation across individuals. These results suggested that children with dyslexia have a mental arithmetic deficit specifically in the realm of exact calculation, while their approximation ability is relatively intact.

Cross-linguistic patterns in the acquisition of quantifiers

Learners of most languages are faced with the task of acquiring words to talk about number and quantity. Much is known about the order of acquisition of number words as well as the cognitive and perceptual systems and cultural practices that shape it. Substantially less is known about the acquisition of quantifiers. Here, we consider the extent to which systems and practices that support number word acquisition can be applied to quantifier acquisition and conclude that the two domains are largely distinct in this respect. Consequently, we hypothesize that the acquisition of quantifiers is constrained by a set of factors related to each quantifier's specific meaning. We investigate competence with the expressions for "all," "none," "some," "some…not," and "most" in 31 languages, representing 11 language types, by testing 768 5-y-old children and 536 adults. We found a cross-linguistically similar order of acquisition of quantifiers, explicable in terms of four factors relating to their meaning and use. In addition, exploratory analyses reveal that language- and learner-specific factors, such as negative concord and gender, are significant predictors of variation.

Early Literacy and Numeracy Skills in Bilingual Minority Children: Toward a Relative Independence of Linguistic and Numerical Processing

Many studies have suggested that the concept of "number" is relatively independent from linguistic skills, although an increasing number of studies suggest that language abilities may play a pivotal role in the development of arithmetic skills. The condition of bilingualism can offer a unique perspective into the role of linguistic competence in numerical development. The present study was aimed at evaluating the relationship between language skills and early numeracy through a multilevel investigation in monolingual and bilingual minority children attending preschool. The sample included 156 preschool children. Of these, 77 were bilingual minority children (mean age = 58.27 ± 5.90), and 79 were monolinguals (mean age = 58.45 ± 6.03). The study focused on three levels of analysis: group differences in language and number skills, concurrent linguistic predictors of early numeracy and, finally, profile analysis of linguistic skills in children with impaired vs. adequate numeracy skills. The results showed that, apart from the expected differences in linguistic measures, bilinguals differed from monolinguals in numerical skills with a verbal component, such as semantic knowledge of digits, but they did not differ in a pure non-verbal component such as quantity comparison. The multigroup structural equation model indicated that letter knowledge was a significant predictor of the verbal component of numeracy for both groups. Phonological awareness was a significant predictor of numeracy skills only in the monolingual group. Profile analysis showed that children with a selective weakness in the non-verbal component of numeracy had fully adequate verbal skills. Results from the present study suggest that only some specific components of language competence predict numerical processing, although linguistic proficiency may not be a prerequisite for developing adequate early numeracy skills.

Verbal-Spatial and Visuospatial Coding of the SNARC Effect: A Comparative Study in Cultures With Different Reading Directions

Using the magnitude comparison task, the current study tested verbal-spatial and visuospatial coding accounts by comparing the spatial-numerical association of response codes effect in 30 Chinese-speaking and 60 bilingual Uighur Chinese participants. The experimental tasks were presented using Chinese words for 30 Chinese-speaking and 30 bilingual Uighur Chinese participants, while only Uighur words were presented to the remaining 30 bilingual Uighur Chinese participants. Overall, the results of the current study showed that verbal-spatial coding could be applied to the experimental task when presented using Chinese words, but visuospatial coding could be applied to the experimental task when presented using Uighur words. The findings demonstrated that spatial coding was flexible and depended upon the language context in which the experimental task was presented.

Representations of numerical and non-numerical magnitude both contribute to mathematical competence in children

A growing body of evidence suggests that non-symbolic representations of number, which humans share with nonhuman animals, are functionally related to uniquely human mathematical thought. Other research suggesting that numerical and non-numerical magnitudes not only share analog format but also form part of a general magnitude system raises questions about whether the non-symbolic basis of mathematical thinking is unique to numerical magnitude. Here we examined this issue in 5- and 6-year-old children using comparison tasks of non-symbolic number arrays and cumulative area as well as standardized tests of math competence. One set of findings revealed that scores on both magnitude comparison tasks were modulated by ratio, consistent with shared analog format. Moreover, scores on these tasks were moderately correlated, suggesting overlap in the precision of numerical and non-numerical magnitudes, as expected under a general magnitude system. Another set of findings revealed that the precision of both types of magnitude contributed shared and unique variance to the same math measures (e.g. calculation and geometry), after accounting for age and verbal competence. These findings argue against an exclusive role for non-symbolic number in supporting early mathematical understanding. Moreover, they suggest that mathematical understanding may be rooted in a general system of magnitude representation that is not specific to numerical magnitude but that also encompasses non-numerical magnitude.

Origins of the brain networks for advanced mathematics in expert mathematicians

The origins of human abilities for mathematics are debated: Some theories suggest that they are founded upon evolutionarily ancient brain circuits for number and space and others that they are grounded in language competence. To evaluate what brain systems underlie higher mathematics, we scanned professional mathematicians and mathematically naive subjects of equal academic standing as they evaluated the truth of advanced mathematical and nonmathematical statements. In professional mathematicians only, mathematical statements, whether in algebra, analysis, topology or geometry, activated a reproducible set of bilateral frontal, Intraparietal, and ventrolateral temporal regions. Crucially, these activations spared areas related to language and to general-knowledge semantics. Rather, mathematical judgments were related to an amplification of brain activity at sites that are activated by numbers and formulas in nonmathematicians, with a corresponding reduction in nearby face responses. The evidence suggests that high-level mathematical expertise and basic number sense share common roots in a nonlinguistic brain circuit.

Spatial Ability Explains the Male Advantage in Approximate Arithmetic

Previous research has shown that females consistently outperform males in exact arithmetic, perhaps due to the former's advantage in language processing. Much less is known about gender difference in approximate arithmetic. Given that approximate arithmetic is closely associated with visuospatial processing, which shows a male advantage we hypothesized that males would perform better than females in approximate arithmetic. In two experiments (496 children in Experiment 1 and 554 college students in Experiment 2), we found that males showed better performance in approximate arithmetic, which was accounted for by gender differences in spatial ability.

Why is number word learning hard? Evidence from bilingual learners

Young children typically take between 18 months and 2 years to learn the meanings of number words. In the present study, we investigated this developmental trajectory in bilingual preschoolers to examine the relative contributions of two factors in number word learning: (1) the construction of numerical concepts, and (2) the mapping of language specific words onto these concepts. We found that children learn the meanings of small number words (i.e., one, two, and three) independently in each language, indicating that observed delays in learning these words are attributable to difficulties in mapping words to concepts. In contrast, children generally learned to accurately count larger sets (i.e., five or greater) simultaneously in their two languages, suggesting that the difficulty in learning to count is not tied to a specific language. We also replicated previous studies that found that children learn the counting procedure before they learn its logic - i.e., that for any natural number, n, the successor of n in the count list denotes the cardinality n+1. Consistent with past studies, we found that children's knowledge of successors is first acquired incrementally. In bilinguals, we found that this knowledge exhibits item-specific transfer between languages, suggesting that the logic of the positive integers may not be stored in a language-specific format. We conclude that delays in learning the meanings of small number words are mainly due to language-specific processes of mapping words to concepts, whereas the logic and procedures of counting appear to be learned in a format that is independent of a particular language and thus transfers rapidly from one language to the other in development.

24-Month-Old Children With Larger Oral Vocabularies Display Greater Academic and Behavioral Functioning at Kindergarten Entry

Data were analyzed from a population-based, longitudinal sample of 8,650 U.S. children to (a) identify factors associated with or predictive of oral vocabulary size at 24 months of age and (b) evaluate whether oral vocabulary size is uniquely predictive of academic and behavioral functioning at kindergarten entry. Children from higher socioeconomic status households, females, and those experiencing higher quality parenting had larger oral vocabularies. Children born with very low birth weight or from households where the mother had health problems had smaller oral vocabularies. Even after extensive covariate adjustment, 24-month-old children with larger oral vocabularies displayed greater reading and mathematics achievement, increased behavioral self-regulation, and fewer externalizing and internalizing problem behaviors at kindergarten entry.

Language-specific memory for everyday arithmetic facts in Chinese-English bilinguals

The role of language in memory for arithmetic facts remains controversial. Here, we examined transfer of memory training for evidence that bilinguals may acquire language-specific memory stores for everyday arithmetic facts. Chinese-English bilingual adults (n = 32) were trained on different subsets of simple addition and multiplication problems. Each operation was trained in one language or the other. The subsequent test phase included all problems with addition and multiplication alternating across trials in two blocks, one in each language. Averaging over training language, the response time (RT) gains for trained problems relative to untrained problems were greater in the trained language than in the untrained language. Subsequent analysis showed that English training produced larger RT gains for trained problems relative to untrained problems in English at test relative to the untrained Chinese language. In contrast, there was no evidence with Chinese training that problem-specific RT gains differed between Chinese and the untrained English language. We propose that training in Chinese promoted a translation strategy for English arithmetic (particularly multiplication) that produced strong cross-language generalization of practice, whereas training in English strengthened relatively weak, English-language arithmetic memories and produced little generalization to Chinese (i.e., English training did not induce an English translation strategy for Chinese language trials). The results support the existence of language-specific strengthening of memory for everyday arithmetic facts.

Number word structure in first and second language influences arithmetic skills

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

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

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

Does spatial locative comprehension predict landmark-based navigation?

In the present study we investigated the role of spatial locative comprehension in learning and retrieving pathways when landmarks were available and when they were absent in a sample of typically developing 6- to 11-year-old children. Our results show that the more proficient children are in understanding spatial locatives the more they are able to learn pathways, retrieve them after a delay and represent them on a map when landmarks are present in the environment. These findings suggest that spatial language is crucial when individuals rely on sequences of landmarks to drive their navigation towards a given goal but that it is not involved when navigational representations based on the geometrical shape of the environment or the coding of body movements are sufficient for memorizing and recalling short pathways.