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.

Is thirty-two three tens and two ones? The embedded structure of cardinal numbers

The acquisition and representation of natural numbers have been a central topic in cognitive science. However, a key question in this topic about how humans acquire the capacity to understand that numbers make 'infinite use of finite means' (or that numbers are generative) has been left unanswered. While previous theories rely on the idea of the successor principle, we propose an alternative hypothesis that children's understanding of the syntactic rules for building complex numerals-or numerical syntax-is a crucial foundation for the acquisition of number concepts. In two independent studies, we assessed children's understanding of numerical syntax by probing their knowledge about the embedded structure of cardinal numbers using a novel task called Give-a-number Base-10 (Give-N10). In Give-N10, children were asked to give a large number of items (e.g., 32 items) from a pool that is organized in sets of ten items. Children's knowledge about the embedded structure of numbers (e.g., knowing that thirty-two items are composed of three tens and two ones) was assessed from their ability to use those sets. Study 1 tested English-speaking 4- to 10-year-olds and revealed that children's understanding of the embedded structure of numbers emerges relatively late in development (several months into kindergarten), beyond when they are capable of making a semantic induction over a local sequence of numbers. Moreover, performance in Give-N10 was predicted by other task measures that assessed children's knowledge about the syntactic rules that govern numerals (such as counting fluency), demonstrating the validity of the measure. In Study 2, this association was tested again in monolingual Korean kindergarteners (5-6 years), as we aimed to test the same effect in a language with a highly regular numeral system. It replicated the association between Give-N10 performance and counting fluency, and it also demonstrated that Korean-speaking children understand the embedded structure of cardinal numbers earlier in the acquisition path than English-speaking peers, suggesting that regularity in numerical syntax facilitates the acquisition of generative properties of numbers. Based on these observations and our theoretical analysis of the literature, we propose that the syntax for building complex numerals, not the successor principle, represents a structural platform for numerical thinking in young children.

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.

Do children use language structure to discover the recursive rules of counting?

We test the hypothesis that children acquire knowledge of the successor function - a foundational principle stating that every natural number n has a successor n + 1 - by learning the productive linguistic rules that govern verbal counting. Previous studies report that speakers of languages with less complex count list morphology have greater counting and mathematical knowledge at earlier ages in comparison to speakers of more complex languages (e.g., Miller & Stigler, 1987). Here, we tested whether differences in count list transparency affected children's acquisition of the successor function in three languages with relatively transparent count lists (Cantonese, Slovenian, and English) and two languages with relatively opaque count lists (Hindi and Gujarati). We measured 3.5- to 6.5-year-old children's mastery of their count list's recursive structure with two tasks assessing productive counting, which we then related to a measure of successor function knowledge. While the more opaque languages were associated with lower counting proficiency and successor function task performance in comparison to the more transparent languages, a unique within-language analytic approach revealed a robust relationship between measures of productive counting and successor knowledge in almost every language. We conclude that learning productive rules of counting is a critical step in acquiring knowledge of recursive successor function across languages, and that the timeline for this learning varies as a function of count list transparency.

Units-first or tens-first: Does language matter when processing visually presented two-digit numbers?

The linguistic structure of number words can influence performance in basic numerical tasks such as mental calculation, magnitude comparison, and transcoding. Especially the presence of ten-unit inversion in number words seems to affect number processing. Thus, at the beginning of formal math education, young children speaking inverted languages tend to make relatively more errors in transcoding. However, it remains unknown whether and how inversion affects transcoding in older children and adults. Here we addressed this question by assessing two-digit number transcoding in adults and fourth graders speaking French and German, that is, using non-inverted and inverted number words, respectively. We developed a novel transcoding paradigm during which participants listened to two-digit numbers and identified the heard number among four Arabic numbers. Critically, the order of appearance of units and tens in Arabic numbers was manipulated mimicking the "units-first" and "tens-first" order of German and French. In a third "simultaneous" condition, tens and units appeared at the same time in an ecological manner. Although language did not affect overall transcoding speed in adults, we observed that German-speaking fourth graders were globally slower than their French-speaking peers, including in the "simultaneous" condition. Moreover, French-speaking children were faster in transcoding when the order of digit appearance was congruent with their number-word system (i.e., "tens-first" condition) while German-speaking children appeared to be similarly fast in the "units-first" and "tens-first" conditions. These findings indicate that inverted languages still impose a cognitive cost on number transcoding in fourth graders, which seems to disappear by adulthood. They underline the importance of language in numerical cognition and suggest that language should be taken into account during mathematics education.

English and Chinese Children's Performance on Numerical Tasks

East Asian pupils have consistently outperformed Western pupils in international comparisons of mathematical performance at both primary and secondary school level. It has sometimes been suggested that a contributory factor is the transparent counting systems of East Asian languages, which may facilitate number representation. The present study compared 35 7-year-old second-year primary school children in Oxford, England and 40 children of similar age in Hong Kong, China on a standardized arithmetic test; on a two-digit number comparison test, including easy, misleading and reversible comparisons; and on a number line task, involving placing numbers in the appropriate position on four number lines: 1-10, 1-20, 1-100, and 1-1000. The Chinese children performed significantly better than the English children on the standardized arithmetic test. They were faster but not significantly more accurate on the Number Comparison and Number Line tasks. There were no interactions between language group and comparison type on the number comparison task, though the performance of both groups was faster on easy pairs than those where there was conflict between the relative magnitudes of the tens and the units. Similarly, there were no interactions between group and number line range, though the performance of both groups was influenced by the range of the number line. The study supports the view that counting systems affect aspects of numerical abilities, but cannot be the full explanation for international differences in mathematics performance.

Bogdanova O, Kovas Y. Cognition, emotion, and arithmetic in primary school: A cross-cultural investigation

The study investigated cross-cultural differences in variability and average performance in arithmetic, mathematical reasoning, symbolic and non-symbolic magnitude processing, intelligence, spatial ability, and mathematical anxiety in 890 6- to 9-year-old children from the United Kingdom, Russia, and China. Cross-cultural differences explained 28% of the variance in arithmetic and 17.3% of the variance in mathematical reasoning, with Chinese children outperforming the other two groups. No cross-cultural differences were observed for spatial ability and mathematical anxiety. In all samples, symbolic magnitude processing and mathematical reasoning were independently related to early arithmetic. Other factors, such as non-symbolic magnitude processing, mental rotation, intelligence, and mathematical anxiety, produced differential patterns across the populations. The results are discussed in relation to potential influences of parental practice, school readiness, and linguistic factors on individual differences in early mathematics. Statement of contribution What is already known on this subject? Cross-cultural differences in mathematical ability are present in preschool children. Similar mechanisms of mathematical development operate in preschool children from the United Kingdom, Russia, and China. Tasks that require understanding of numbers are best predictors of arithmetic in preschool children. What does this study add? Cross-cultural differences in mathematical ability become greater with age/years of formal education. Similar mechanisms of mathematical development operate in early primary school children from the United Kingdom, Russia, and China. Symbolic number magnitude and mathematical reasoning are the main predictors of arithmetic in all three populations.

Producing Spatial Words Is Not Enough: Understanding the Relation Between Language and Spatial Cognition

Prior research has investigated the relation between children's language and spatial cognition by assessing the quantity of children's spatial word production, with limited attention to the context in which children use such words. This study tested whether 4-year-olds children's (N = 41, primarily white middle class) adaptive use of task-relevant language across contexts predicted their spatial skills. Children were presented with a spatial scene description task, four spatial tasks, and vocabulary assessments. Children's adaptive use of task-relevant language was more predictive of their spatial skills than demographic and language factors (e.g., quantity of spatial words produced). These findings identify new links between language and spatial cognition and highlight the importance of understanding the quality, not just quantity, of children's language use.

Comparisons of Children's Cognitive Representation of Number: China, France, Japan, Korea, Sweden, and the United States

Cross-national comparisons of mathematics achievement have shown differences in favour of Asian students. This study examined the idea that the superior mathematics performance of students from Japan, Korea, and China may be due, in part, to differences in cognitive representation of number that is affected by numerical language characteristics differentiating Asian and non-Asian language groups. First-graders from the People's Republic of China, Japan, Korea, France, Sweden, and the United States were asked to construct various numbers using Base 10 blocks. Chinese, Japanese, and Korean children showed a preference for using a construction of tens and ones to show numbers; place value appeared to be clearly represented in those constructions. French, Swedish, and US children, in contrast, showed a preference for using a collection of units, suggesting that they represent number as a grouping of counted objects. More Asianlanguage speakers than non-Asian-language speakers were also able to make two correct constructions for each number, which suggests greater flexibility of mental number manipulation. Thus, the unique characteristics of the Asian number language system may facilitate the teaching and learning of mathematics, especially computation, for speakers of those languages.

Preschool Origins of Cross-National Differences in Mathematical Competence: The Role of Number-Naming Systems

Differences in mathematical competence between U S and Chinese children first emerge during the preschool years, favor Chinese children, and are limited to specific aspects of mathematical competence The base-10 structure of number names is less obvious in English than in Chinese, differences between these languages are reflected in children's difficulties learning to count Language differences do not affect other aspects of early mathematics, including counting small sets and solving simple numerical problems Because later mathematics increasingly involves manipulation of symbols, this early deficit in apprehending the base-10 structure of number names may provide a basis for previously reported differences in mathematical competence favoring Chinese schoolchildren

Speaking two languages with different number naming systems: What implications for magnitude judgments in bilinguals at different stages of language acquisition?

Differences between languages in terms of number naming systems may lead to performance differences in number processing. The current study focused on differences concerning the order of decades and units in two-digit number words (i.e., unit-decade order in German but decade-unit order in French) and how they affect number magnitude judgments. Participants performed basic numerical tasks, namely two-digit number magnitude judgments, and we used the compatibility effect (Nuerk et al. in Cognition 82(1):B25-B33, 2001) as a hallmark of language influence on numbers. In the first part we aimed to understand the influence of language on compatibility effects in adults coming from German or French monolingual and German-French bilingual groups (Experiment 1). The second part examined how this language influence develops at different stages of language acquisition in individuals with increasing bilingual proficiency (Experiment 2). Language systematically influenced magnitude judgments such that: (a) The spoken language(s) modulated magnitude judgments presented as Arabic digits, and (b) bilinguals' progressive language mastery impacted magnitude judgments presented as number words. Taken together, the current results suggest that the order of decades and units in verbal numbers may qualitatively influence magnitude judgments in bilinguals and monolinguals, providing new insights into how number processing can be influenced by language(s).

Language supports young children's use of spatial relations to remember locations

Two experiments investigated the role of language in children's spatial recall performance. In particular, we assessed whether selecting an intrinsic reference frame could be improved through verbal encoding. Selecting an intrinsic reference frame requires remembering locations relative to nearby objects independent of one's body (egocentric) or distal environmental (allocentric) cues, and does not reliably occur in children under 5 years of age (Nardini, Burgess, Breckenridge, & Atkinson, 2006). The current studies tested the relation between spatial language and 4-year-olds' selection of an intrinsic reference frame in spatial recall. Experiment 1 showed that providing 4-year-olds with location-descriptive cues during (Exp. 1a) or before (Exp. 1b) the recall task improved performance both overall and specifically on trials relying most on an intrinsic reference frame. Additionally, children's recall performance was predicted by their verbal descriptions of the task space (Exp. 1a control condition). Non-verbally highlighting relations among objects during the recall task (Exp. 2) supported children's performance relative to the control condition, but significantly less than the location-descriptive cues. These results suggest that the ability to verbally represent relations is a potential mechanism that could account for developmental changes in the selection of an intrinsic reference frame during spatial recall.

Cultural context of cognitive development

The cognitive problems that children formulate and solve in their daily lives necessarily take form in a cultural context. We review and illustrate two dominant approaches to study relations between cultural context and cognitive development, and we point to the limitations and affordances of each. Using a dichotomous approach, scholars employ a methodology that sharply differentiates cognition from cultural context, treating elements of cultural context as independent variables and elements of cognition as dependent variables. The approach often leads to propositions about transcultural features of context that influence the cognitive development of individuals. In contrast, using an intrinsic relations approach, researchers create units of analysis that capture relations between cognition and cultural context, investigating their mutual grounding in daily activities. We also review a small but important body of research that extends these approaches to diachronic analysis. This research seeks to understand shifting relations between cultural context and cognitive development over historical time. WIREs Cogn Sci 2014, 5:447-461. doi: 10.1002/wcs.1300 For further resources related to this article, please visit the WIREs website.

CONFLICT OF INTEREST

The authors have declared no conflicts of interest for this article.

On the limits of language influences on numerical cognition - no inversion effects in three-digit number magnitude processing in adults

The inversion of number words influences numerical cognition even in seemingly non-verbal tasks, such as Arabic number comparison. However, it is an open question whether inversion of decades and units also influences number processing beyond the two-digit number range. The current study addresses this question by investigating compatibility effects in both German- (a language with inverted) and English-speaking (a language with non-inverted number words) university students (mean age 22 years) in a three-digit number comparison task. We observed reliable hundred-decade as well as hundred-unit compatibility effects for three-digit number comparison. This indicates that, comparable two-digit numbers, three-digit numbers are processed in a parallel decomposed fashion. However, in contrast to previous results on two-digit numbers as well as on children’s processing of three-digit numbers, no reliable modulation of these compatibility effects through language was observed in adults. The present data indicate that inversion-related differences in multi-digit number processing are limited. They seem to be restricted to the number range involving those digits being inverted (i.e., tens and units in two-digit numbers) but do not generalize to neighboring digits. Possible reasons for this lack of generalization are discussed.

Intransparent German number words complicate transcoding - a translingual comparison with Japanese

Superior early numerical competencies of children in several Asian countries have (amongst others) been attributed to the higher transparency of their number word systems. Here, we directly investigated this claim by evaluating whether Japanese children’s transcoding performance when writing numbers to dictation (e.g., “twenty five” → 25) was less error prone than that of German-speaking children – both in general as well as when considering language-specific attributes of the German number word system such as the inversion property, in particular. In line with this hypothesis we observed that German-speaking children committed more transcoding errors in general than their Japanese peers. Moreover, their error pattern reflected the specific inversion intransparency of the German number-word system. Inversion errors in transcoding represented the most prominent error category in German-speaking children, but were almost absent in Japanese-speaking children. We conclude that the less transparent German number-word system complicates the acquisition of the correspondence between symbolic Arabic numbers and their respective verbal number words.

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.

Linguistic influence on mathematical development is specific rather than pervasive: revisiting the Chinese Number Advantage in Chinese and English children

The relative linguistic transparency of the Asian counting system has been used to explain Asian students' relative superiority in cross-cultural comparisons of mathematics achievement. To test the validity and extent of linguistic transparency in accounting for mathematical abilities, this study tested Chinese and British primary school children. Children in Hong Kong can learn mathematics using languages with both regular (Chinese) and irregular (English) counting systems, depending on their schools' medium of instruction. This makes it possible to compare groups with varying levels of exposure to the regular and irregular number systems within the same educational system, curriculum, and cultural environment. The study included three groups of first/second graders and third/fourth graders with varying degrees of experience to the Chinese language and counting systems: no experience (UK; n = 49); spoke Chinese at home and learnt to count in English at school (HK-E; n = 43); spoke Chinese at home and learnt to count in Chinese at school (HK-C; n = 47). They were compared on counting, numerical abilities and place value representation. The present study also measured nonverbal reasoning, attitude toward mathematics, involvement of parents, and extra-curricular mathematics lessons to explore alternative explanations of children's numeric ability. Results indicated that students in HK-C were better at counting backward and on the numeric skills test than those in HK-E, who were in turn better than the UK students. However, there was no statistical difference in counting forward, place value understanding, and a measure of arithmetic. Our findings add to existent literature suggesting that linguistic transparency does not have an all-pervasive influence on cross-national differences in arithmetic performance.

Reexamining the language account of cross-national differences in base-10 number representations

East Asian students consistently outperform students from other nations in mathematics. One explanation for this advantage is a language account; East Asian languages, unlike most Western languages, provide cues about the base-10 structure of multi-digit numbers, facilitating the development of base-10 number representations. To test this view, the current study examined how kindergartners represented two-digit numbers using single unit-blocks and ten-blocks. The participants (N=272) were from four language groups (Korean, Mandarin, English, and Russian) that vary in the extent of "transparency" of the base-10 structure. In contrast to previous findings with older children, kindergartners showed no cross-language variability in the frequency of producing base-10 representations. Furthermore, they showed a pattern of within-language variability that was not consistent with the language account and was likely attributable to experiential factors. These findings suggest that language might not play as critical a role in the development of base-10 representations as suggested in earlier research.

Promovendo a compreensão da composição aditiva em crianças surdas

O estudo tem como objetivo analisar a compreensão da composição aditiva dos números em crianças surdas. Este artigo descreve dois estudos que utilizaram a Tarefa de Compra de Nunes e Schliemann (1990) para investigar a compreensão da composição aditiva em crianças no contexto de contar dinheiro. No primeiro estudo, comparamos a compreensão da composição aditiva em crianças surdas à de crianças ouvintes da mesma idade. No segundo estudo, realizamos uma breve intervenção para avaliar a possibilidade de melhorar sua compreensão de composição aditiva. Concluímos que intervenções ainda que breves, mas teoricamente significativas e claramente direcionadas, podem ser usadas para melhorar o desempenho das crianças surdas nas tarefas que avaliam a composição aditiva.

The effectiveness of Korean number naming on insight into numbers in Dutch students with mild intellectual disabilities

BACKGROUND

Children from Asian countries score higher on early years' arithmetic tests than children from Europe or the United States of America. An explanation for these differences may be the way numbers are named. A clear ten-structure like in the Korean language method leads to a better insight into numbers and arithmetic skills. This assumption forms the basis of the current study.

METHOD

Examined is whether an intervention with number naming in the Korean way influences number awareness of students with mild intellectual disabilities (N=70; mean age: 9;0 years).

RESULTS

The results indicate a positive effect of this alternative method of number naming on the insight into numbers up to 20. However, the effect did not generalize to insight into numbers 21-100.

CONCLUSIONS

The Korean method of number naming seems to be a promising way to teach students with mild intellectual disabilities insight into numbers.

The effectiveness of Korean number naming on insight into numbers in Dutch students with mild intellectual disabilities

Children from Asian countries score higher on early years' arithmetic tests than children from Europe or the United States of America. An explanation for these differences may be the way numbers are named. A clear ten-structure like in the Korean language method leads to a better insight into numbers and arithmetic skills. This assumption forms the basis of the current study. Examined is whether an intervention with number naming in the Korean way influences number awareness of students with mild intellectual disabilities (N=70; mean age: 9.0 years). The results indicate a positive effect of this alternative method of number naming on the insight into numbers up to 20. However, the effect did not generalize to insight into numbers 21-100. The Korean method of number naming seems to be a promising way to teach students with mild intellectual disabilities insight into numbers.

Language effects in addition: How you say it counts

This study aimed to investigate the role of language in calculation. Two populations were compared, one with a base-10 language, and another (Basque) in which number words are constructed by combining multiples of 20 and units or teens (e.g., “35” is said “twenty and fifteen”). Experiment 1 asked participants to verbally solve additions presented as Arabic digits. Basque participants solved the additions that consisted of a multiple of 20 and a teen (e.g., 20 + 15) faster than controls with identical answers (e.g., 25 + 10). No differences were found in the base-10 language group. Experiment 2 replicated this result even if participants had to type the answer on a numerical keypad, instead of saying it. Hence, the structure of number words in each of the languages influenced the way additions were solved, even if language was not necessary for conducting the task. Finally, in Experiment 3, both language groups performed a numerical comparison task in which no effects of the structure of number words were obtained. Results of the three experiments are discussed in light of current models of numerical cognition.

Counting-On, Trading and Partitioning: Effects of Training and Prior Knowledge on Performance on Base-10 Tasks

Factors affecting performance on base-10 tasks were investigated in a series of four studies with a total of 453 children aged 5-7 years. Training in counting-on was found to enhance child performance on base-10 tasks (Studies 2, 3, and 4), while prior knowledge of counting-on (Study 1), trading (Studies 1 and 3), and partitioning (Studies 1 and 4) were associated with enhanced base-10 performance. It emerged that procedural knowledge of counting-on, trading, and partitioning can lead to improvements in procedural knowledge of the base-10 system. The findings lend support to the model of iterative development of conceptual and procedural knowledge advanced by Rittle-Johnson et al. (2001).

Culture and children's cosmology

In this investigation, we examined children's knowledge of cosmology in relation to the shape of the earth and the day-night cycle. Using explicit questioning involving a choice of alternative answers and 3D models, we carried out a comparison of children aged 4-9 years living in Australia and England Though Australia and England have a close cultural affinity, there are differences in children's early exposure to cosmological concepts. Australian children who have early instruction in this domain were nearly always significantly in advance of their English counterparts. In general, they most often produced responses compatible with a conception of a round earth on which people can live all over without falling off. We consider coherence and fragmentation in children's knowledge in terms of the timing of culturally transmitted information, and in relation to questioning methods used in previous research that may have underestimated children's competence.

U.S. and Korean children's comprehension of fraction names: a reexamination of cross-national differences

Two experiments tested the claim that the transparency of Korean fraction names promotes fraction concepts (Miura, Okamoto, Vlahovic-Stetic, Kim, & Han, 1999). In Experiment 1, U.S. and Korean first and second graders made similar errors on a fraction-identification task, by treating fractions as whole numbers. Contrary to previous findings, Korean children performed at chance when a whole-number representation was included. Nonetheless, Korean children outperformed their U.S. peers overall. In Experiment 2, U.S. children's performance improved when fraction names were used that explicitly referred to part-whole relations like Korean fraction names. U.S. children's scores actually exceeded those of Korean children. Thus, although the differences in fraction names may influence children's performance, this may not account for the reported cross-national differences.

Developmental cognitive neuropsychology of number processing and calculation: varieties of developmental dyscalculia

This article provides a brief overview about the current state of cognitive developmental neuropsychology of developmental dyscalculia (DD) as well as results from a Zurich study that investigates different subtypes of DD according to various aspects of numerical abilities that are impaired or preserved. The differential effects of impairments of one particular numerical area on the development of other numerical abilities are highlighted in the case of a 17 year old boy with severe DD and Developmental Gerstmann Syndrome. A comprehensive model of developmental dynamics of number processing and calculation abilities will be proposed in the last section with respect to the development of intelligence theory.

Reasoning about conventional time as a function of conventional time systems

This study investigated how reasoning about conventional time information varied as a function of conventional time systems by using the Chinese month and Jieqi systems. Twenty Chinese students were asked to answer month-related questions and another 20 were asked to answer Jieqi-related questions. Reaction time and accuracy were the dependent measures. A cross-boundary effect was observed in processing months, and distance and direction effects were obtained when participants judged the interval of Jieqi. These results suggested that arithmetic operations were used in Chinese reasoning about months and verbal-articulatory processes were used for the Jieqi. The effects of mode of language representation on cognition and the strategies for cross-linguistic study are discussed.

Linguistic Relativity: The Case of Place Value in Multi-Digit Numbers

Differences between languages have been implicated recently in explanations of the cross-cultural disparities observed in children's mathematical performance on place value tasks (e.g., Miura, Okamaoto, Kim, Chang, Steere & Fayol, 1994). Children's understanding of place value was investigated here with 93 English-speaking children and 50 Japanese-speaking children (aged 6 and 7 years). Cubes denoting units and tens were made available to children for producing representations of multi-digit numerals. It was found that subtle shifts in task instructions produced a marked influence on children's performance. In particular, differences between English and Japanese participants disappeared when the use of tens cubes was demonstrated in practice trials. More generally, the findings indicate that the influence of language on the cognitive representation of number is less direct than has previously been suggested. Copyright 1998 Academic Press.

Children and arithmetic

The development of children's understanding of mathematical relations and of their grasp of the number system is described. It is discussed that children easily recognise one-way part-part relations but that the number system at first causes them difficulty. Children's relational understanding allows them to solve addition and subtraction problems fairly well when these deal with simple increases and decreases in quantity. It also helps them to make proportional judgements when these involve part-part relations. However, problems that involve relations between parts and wholes are at first extremely difficult. The review also deals with the effects of context and shows the considerable aptitudes that are handed on to children in informal settings.

Architectures for numerical cognition

Current theories of numerical cognition differ in assumptions about the componential architecture of number processing and about the extent of notation-specific processes. To investigate these issues, 64 adult subjects were tested on simple addition and multiplication problems presented in Arabic digit or English number-word format. Overall, response times and error rates were much higher with the word format, but more importantly, presentation format interacted with arithmetic operation and problem size. Operation errors (2 + 4 = 8), operand-naming errors (2 + 8 = 8), and operand-intrusion errors (9 x 6 = 36) were each characterized by a different format x operation interaction, and analysis of inter-trial error priming showed selective interference from preceding trials as a function of number format. These types of format-specific retrieval interference and operation-specific effects of format are problematic for models that hypothesize notation-independent memory processes for arithmetic. Furthermore, analyses of operand-naming errors, operand-intrusion errors, and other operand-priming effects, revealed strong interactions of number reading and number-fact retrieval processes; processes that are typically posited to be functionally independent. The results suggest a complex encoding architecture that incorporates notation-dependent activation of addition and multiplication facts, as well as interpenetration of number reading and number-fact retrieval processes.