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Abstract

This article is a critical synthesis of the work which has been done in the field of bilingualism and calculation. The theoretical frameworks dealing with the question of the possible influence of the format or of the language on the retrieval of arithmetical facts are described. The results of the research are discussed in detail by examining the possible alternative interpretations. Encoding and production processes, as well as the possible influence of short-term memory (which may vary according to the subject's language), are the main considerations.
... Various studies have tried to address the issue of language-specificity in the storage of arithmetic facts by comparing the performance of monolinguals and bilinguals (in between-group designs), aimed to establish whether bilinguals possess one or two distinct language-related memory stores for arithmetic facts (see Noël & Fias 1998, for a review). However, this approach has produced somewhat inconsistent results, likely due to the adoption of different inclusion criteria for bilingual participants and methodological procedures (cf. ...
... McClain and Shih Huang (1982) confirmed and extended the results obtained by Marsh and Maki, by presenting the problems in auditory (verbal) format. The lack of significance of the interaction between language (L1 vs. L2) and number of additions may be taken as tentative evidence in support of models that posit the existence of a single memory store containing arithmetic facts (Noël & Fias 1998). On the other hand, the main effect reported by Marsh and Maki concerning the fact that the participants were faster when they had to respond in L1 may also be interpreted as evidence consistent with the Encoding Complex view (Campbell 1994). ...
... Even more important, it has been shown that the associative confusion effect does not fulfill the so called "resistance to suppression" criterion for automaticity (Zbrodoff & Logan 1986), and it can be abolished by loading working memory under dual-task conditions (Rusconi et al. 2004). Consistent with this analysis, Noël and Fias (1998) have suggested that differences between languages in arithmetic performance can be attributed to working memory rather than to access to arithmetic facts. In addition, as we pointed out earlier, in the context of arithmetic problem solving, people seem not to rely exclusively on direct retrieval processes when performing addition (e.g., they may opt for back-up strategies to reach an optimal performance; LeFevre et al. 1996;Shrager & Siegler 1998). ...
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In this chapter we aim to give a taste of the current research in bilingualism and numerical cognition. Far from being an exhaustive review, our chapter is intended to frame the relationship between bilingualism and number processing into a novel perspective, by reporting some of the most recent empirical findings. We will first introduce the topic by focusing more generally on the link between language and number processing. A great deal of our knowledge of numbers is traded, thought and manipulated by means of language. But how essential is verbal language to numerical knowledge itself? Recent studies have tentatively addressed the question by looking at numerical abilities in human populations having a very poor lexicon for numbers. Their results will be briefly outlined before addressing the more specific issue of bilingualism. We will then turn to the field of cognitive arithmetic, which has attracted most efforts and produced most of the studies lying at the convergence between bilingualism and mathematical cognition. As a general principle, basic experimental paradigms and effects that were originally described with monolingual participants were employed to explore arithmetic competence in the first and second languages of bilingual participants. We will end by showing how the introduction of novel experimental paradigms in the last few years promises to challenge the conclusions drawn by previous studies.
... In experiment 2, we will further elaborate the findings by looking at an output modality (typing instead of naming) that requires the Dutch-speaking subjects to respond like the French-speaking (i.e. by starting with the value of the ten). It may be noted that by looking at the performance of people with a different native language, we circumvent one of the main problems in research dealing with the relationship between input format and mathematical cognition ( Noël and Fias, 1998). Because most of the research is based on designs with repeated measures, it is often difficult to determine whether the input effects are due to differences in the semantic numerical system or to differences in the encoding and production stage of the different types of materials. ...
... For instance, it has repeatedly been shown that bilinguals need more time to calculate in their second language than in their first language. However, this effect may be caused by a difficulty in translating a numeral of the second language to an abstract representation or to the corresponding numeral of the first language, as well as to differences in the semantic system(s) underlying the mathematical operation (Noël and Fias, 1998). This is particularly important for the evaluation of a model like Dehaene's, which claims that basic number fact retrieval relies on auditory verbal representations of the first language (Dehaene, 1992, p. 33). ...
Article
Recent theoretical developments have redefined a Whorfian effect as a processing difference due to the language of the individual, and no longer as a marker for or against linguistic determinism. Within this framework, Whorfian effects can be used to investigate whether a particular part of the cognitive system is penetrable by language processes or forms an encapsulated module, provided the experimenter ensures that the target language difference is not caused by peripheral input or output processes. In this article, we examine the possibility of a Whorfian effect in numerical cognition by making use of the fact that in the Dutch number naming system the order of tens and units is reversed (i.e. 24 is read `four-and-twenty'). In a first experiment, we asked native French- and Dutch-speaking students to name the solution of addition problems with a two-digit and a single-digit operand (e.g. 20+4=?, 24+1=?). The order of the operands was manipulated (20+4 vs. 4+20) as well as the presentation modality (Arabic vs. verbal). Three language differences emerged from this study. Experiment 2, however, showed that these differences were all due to input or output processes rather than differences in the addition operation (i.e. the differences between Dutch and French disappeared when subjects were asked to type the answer rather than pronounce it). On the basis of these findings, we question the idea that mathematical operations are based on verbal processes.
... Intrusions are one source of evidence for a linguistic representation of number facts. Nonetheless, the assumption that there are linguistic codes for arithmetic facts is a controversial one (e.g., Noël & Fias, 1999;Noël, Robert, & Brysbaert, 1998). Later, we review the available evidence on this issue. ...
... Language and Simple Arithmetic A central feature of the encoding-complex model is the assumption that number-fact memory involves language, and we have argued that the arithmetic performance of Chinese-English bilinguals implies languagebased number-fact representations. The data appear to be incompatible with the amodal, abstract code proposed by McCloskey (1992), but establishing the positive case for language-specific codes is more difficult (see Noël & Fias, 1999, for a recent review). Here, to further substantiate our claims, we review research that has investigated whether number facts are stored in a linguistic code. ...
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We present a model of the cognitive architecture of basic numerical skills in adult Chinese-English bilinguals. The model is based on data reported by Campbell, Kanz, and Xue (1999) and combines Dehaene and Cohen's triple-code theory with Campbell and Clark's encoding-complex approach to modeling number processing. Participants were required to name, add or multiply Arabic or Mandarin numerals and to respond in English or Chinese. They also performed magnitude comparisons on pairs of Arabic or Mandarin numerals. The proposed model of their performance on this set of tasks assumes 1) that number processing is modular with respect to representational code (e.g., visual, visuo-spatial, verbal) rather than with respect to numerical function, 2) task-specific communication between representational codes is interactive rather than additive, and 3) memory for arithmetic facts is at least partially language-based and our Chinese-English bilinguals possessed both Chinese and English-language number-fact representations. We provide new analyses of the arithmetic data and a review of research on the role of language in simple arithmetic to substantiate our claims about linguistic codes for number-fact memory.
... There is an ongoing debate about the representational code of arithmetic fact knowledge, While some authors claimed that arithmetic memory may be based on abstract or language independent representations81828384 the dominant view in numerical cognition is that arithmetic facts (e.g., 2+3 = 5) are based on language-specific codes [8,10,85,86], The present study is informative on this aspect as well. Not only did we find activation in language areas along the perisylvian sulcus for the case of easier, more retrieval related problems. ...
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Numerical cognition is a case of multi-modular and distributed cerebral processing. So far neither the anatomo-functional connections between the cortex areas involved nor their integration into established frameworks such as the differentiation between dorsal and ventral processing streams have been specified. The current study addressed this issue combining a re-analysis of previously published fMRI data with probabilistic fiber tracking data from an independent sample. We aimed at differentiating neural correlates and connectivity for relatively easy and more difficult addition problems in healthy adults and their association with either rather verbally mediated fact retrieval or magnitude manipulations, respectively. The present data suggest that magnitude- and fact retrieval-related processing seem to be subserved by two largely separate networks, both of them comprising dorsal and ventral connections. Importantly, these networks not only differ in localization of activation but also in the connections between the cortical areas involved. However, it has to be noted that even though seemingly distinct anatomically, these networks operate as a functionally integrated circuit for mental calculation as revealed by a parametric analysis of brain activation.
... p < .05]. Campbell et al. (1999) suggested that this Size × Language interaction might indicate that both small and large problems are available by direct retrieval in Chinese, whereas smaller problems are more likely than larger problems to be directly available in English (but see Noël and Fias, 1999). As in the naming task, arithmetic language errors were more common on switch trials (15.0% ...
Article
Meuter and Allport (1999) demonstrated greater RT (response time) costs for bilinguals to switch to their first language (L1) from their second language (L2) relative to switching to L2 from L1. Here, analyses of digit naming and simple arithmetic (from 2+2 to 9+9 and from 2×2 to 9×9) by Chinese–English bilinguals demonstrated that these asymmetrical language switching costs can vary with stimulus format (Arabic or Mandarin numerals), and that the asymmetry is observed both with direct retrieval (e.g. naming the digit “8”) and indirect retrieval from the lexicon (e.g. answering “2+6”). Inhibitory processes in language selection entail format- and task-specific skills.
... Derakshan and Eysenck (1998) suggest that anxious thoughts and feelings ''pre-empt some of the resources of working memory, and thus impair performance when the task demands on working memory are great'' (1998: 711). Mental calculation is one of the tasks where demands on working memory are very high (Ellis 1992;Noël & Fias 1998). Multilinguals will avoid using a language in which they feel more anxious as this would hamper cognitive processing and disrupt the outcome. ...
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The present study investigates self-reported language choice for mental calculations among 1,454 adult multilinguals from a variety of linguistic, so-cial and ethnic backgrounds. As mental calculation is a complex cognitive operation involving both language-dependent and language independent processes, we sought to establish a baseline of first language (L1) or for-eign language(s) (LX) use for mental calculation and identify the factors that influence multilinguals' choice of language for mental calculation. A series of multiple regression analyses on calculation in the L1, L2, L3 and L4 showed that the following variables (in decreasing order) are the best predictors of language choice: frequency of general use, self-perceived pro-ficiency in writing, socialization in the LX, context of acquisition, commu-nicative and/or foreign language anxiety, perceived usefulness, and age of onset of acquisition. These variables explained over 40 percent of the vari-ance in the foreign languages.
... & Job, 2007). One view is that memory for arithmetic facts (e.g., 2 + 3 0 5, 6 × 5 0 30, etc.) is often based on languagespecific codes (Dehaene, Spelke, Pinal, Stanescu, & Tsivkin, 1999;Rusconi et al., 2007;Venkatraman, Siong, Chee, & Ansari, 2006), whereas others have argued that arithmetic memory is based on abstract or languageindependent representations (Brysbaert, Fias, & Noël, 1998;McCloskey, 1992;Noël & Fias, 1999;Whalen, McCloskey, Lindemann, & Bouton, 2002). ...
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We examined interoperation transfer of practice in adult Chinese-English bilinguals' memory for simple multiplication (6 × 8 = 48) and addition (6 + 8 = 14) facts. The purpose was to determine whether they possessed distinct number-fact representations in both Chinese (L1) and English (L2). Participants repeatedly practiced multiplication problems (e.g., 4 × 5 = ?), answering a subset in L1 and another subset in L2. Then separate groups answered corresponding addition problems (4 + 5 = ?) and control addition problems in either L1 (N = 24) or L2 (N = 24). The results demonstrated language-specific negative transfer of multiplication practice to corresponding addition problems. Specifically, large simple addition problems (sum > 10) presented a significant response time cost (i.e., retrieval-induced forgetting) after their multiplication counterparts were practiced in the same language, relative to practice in the other language. The results indicate that our Chinese-English bilinguals had multiplication and addition facts represented in distinct language-specific memory stores.
... The study of how bilinguals process number facts allows for more fine-tuned tests of propositions regarding the effects of format on number cognition. The results of most recent studies on number cognition among bilinguals seem to be more consistent with the encoding complex model (but see Noël & Fias, 1998, for a contrary opinion). For example, Marsh and Maki (1976) gave addition problems in digit format to bilinguals and asked them to reply in either their first or second language. ...
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In two experiments, Filipino-English bilinguals were asked to verify simple addition equations that were presented either in digit, verbal-Filipino, or verbal-English formats and that included different types of sum probes. The main results show (1) faster and more accurate processing of digit and English items than of Filipino items, (2) stronger associative interference by type of probe with the digit and English items compared with the Filipino items, and (3) priming of responses from English to digit codes, and from Filipino to digit codes, but not vice versa. The results were explained by using an elaborated version of Campbell's (1994) encoding complex model with additional assumptions to address the experience of bilinguals. The additional assumptions relate to the preference among the bilingual's two verbal formats, the different strengths of activation pathways within each format, and the asymmetric activation across formats.
Article
This chapter presents evidence from a group of experiments comparing the perceptual strategies employed by Spanish-English bilinguals to those used by their monolingual counterparts. The data show that language history affects the way readers assign structure to ambiguous syntactic constituents, but in different ways off-line and on-line. Off-line, bilinguals depart from the monolingual model in their attachment of relative clauses in their non-dominant language, and exhibit language-independent strategy use—with input in either language, bilinguals have behavior patterns similar to those of monolinguals of their dominant language. On-line, however, bilinguals appear to have no reliable attachment preferences, unlike monolinguals, who uniformly prefer attachment to the low site. The bilinguals' departure from the monolingual model is not taken to be indicative of a qualitative difference between monolingual and bilingual sentence processing. Instead, this result is attributed to a lack of sensitivity in the experimental procedure, which taps early processing with monolinguals, but misses the opportunity with bilinguals, whose reading times are overall slower.
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Native speakers of six languages (Chinese, English, Finnish, Greek, Spanish and Swedish) were tested for digit span with and without articulatory suppression. The results showed that under control conditions Chinese speakers obtained a larger digit span than speakers of the remaining languages, who did not differ among themselves. However, under articulatory suppression, these differences were eliminated and suppressed digit span was equivalent across the languages. These findings provide empirical support for the view that attributes cross-linguistic differences in digit span to variation in the articulatory duration of digit names and the rate of subvocal rehearsal between languages.
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In different languages the names of numbers take different times to articulate. This chapter considers the role of language and representation in arithmetic. It reviews studies which demonstrate that digit word-length limits the short-term memory for digit sequences (such as telephone numbers or digit span as used in many intelligence tests). Three experiments are reported which show that a language's number name word-lengths have a determinative influence upon the ease of mental calculation and counting in that language – some languages are more conducive to mental arithmetic than others. More general aspects of the effects of language word-length are also considered.
Article
Four experiments examined the memory processes used to maintain location in a counting sequence. In Experiment 1, subjects who rapidly counted forward omitted many repeated-digit numbers (e.g., 77), as found previously with backward counting. Subjects in Experiment 2 counted backward with normal auditory feedback or with headphones through which white noise was channeled. In both cases, repeated-digit errors predominated, suggesting that the contents of short-term memory, rather than auditory sensory memory, are checked during counting. In Experiment 3, subjects silently wrote counting responses, and the omission errors resembled those in vocal counting. Repetition errors were also found and attributed to phonological recoding failures. Articulatory suppression in Experiment 4 greatly increased the number of repetition errors in the written counting task. A model of the counting process was proposed according to which subjects keep track of their location in the counting sequence by monitoring phonologically coded short-term memory representations of the numbers.
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A cross-sectional approach covering an age range of 10 years was used to compare the developmental changu in picture naming md number naming of 77 high scbool md 74 elementary students who came from Germany to Sweden. For the two different groups of students, length of residcoce in Swedcn was the main independent variable and reaction time on simple naming tasks of pictures and numbers in German and Swedish was the dependent variable. The results provide evidence that elementuy school students achieved a balanced form of bilingualism 2 years earlier than high school students on the picture naming task. Naming two-digit numbers was shown to be a relatively difficult task for elementary school children, as indiated by markedly prolonged response times. Despite this fact, after about 4 years of residence in Sweden, both groups of students reached language ballaced on this task in that they showed identical response times in both languages. The results on the picture naming task were taken as support for the optimal age hypothesis.
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Two experiments compared rates of solving simple and complex addition and multiplication problems in groups of speakers of French or English in Experiment 1 (n = 35) and Spanish or English in Experiment 2 (n = 84). Subjects were divided into groups of English unilinguals, weak bilinguals, and strong bilinguals according to their performance on a naming task. In both experiments, simple problems consisted of two single-digit numbers. At least three single-digit numbers were used for complex problems in Experiment 1 and double-digit numbers in Experiment 2. Mean solution times, particularly for complex problems, were lowest for the monolingual group, followed in turn by the weak bilingual and strong bilingual groups, but these differences were not statistically reliable in either experiment. In Experiment 2, however, componential analyses of solution times indicated that strong bilingual subjects were slower at executing the carry operation when solving complex problems, relative to the two remaining groups. Results were interpreted in terms of the relationship between bilingualism and the representation and processing of numerical information.