The semantics and acquisition of number words: integrating linguistic and developmental perspectives

This article brings together two independent lines of research on numerally quantified expressions, e.g. two girls. One stems from work in linguistic theory and asks what truth conditional contributions such expressions make to the utterances in which they are used--in other words, what do numerals mean? The other comes from the study of language development and asks when and how children learn the meaning of such expressions. My goal is to show that when integrated, these two perspectives can both constrain and enrich each other in ways hitherto not considered. Specifically, work in linguistic theory suggests that in addition to their 'exactly n' interpretation, numerally quantified NPs such as two hoops can also receive an 'at least n' and an 'at most n' interpretation, e.g. you need to put two hoops on the pole to win (i.e. at least two hoops) and you can miss two shots and still win (i.e. at most two shots). I demonstrate here through the results of three sets of experiments that by the age of 5 children have implicit knowledge of the fact that expressions like two N can be interpreted as 'at least two N' and 'at most two N' while they do not yet know the meaning of corresponding expressions such as at least/most two N which convey these senses explicitly. I show that these results have important implications for theories of the semantics of numerals and that they raise new questions for developmental accounts of the number vocabulary.     

The SNARC effect does not imply a mental number line

In this study, we directly contrast two approaches that have been proposed to explain the SNARC effect. The traditional direct mapping account suggests that a direct association exists between the position of a number on the mental number line and the location of the response. On the other hand, accounts are considered that propose an intermediate step in which numbers are categorized as either small or large between the number magnitude and the response representations. In a magnitude comparison task, we departed from the usual bimanual left/right response dimension and instead introduced the unimanual close/far dimension. A spatial-numerical association was observed: small numbers were associated with a close response, while large numbers were associated with a far response, regardless of the movement direction (left/right). We discuss why these results cannot be explained by assuming a direct mapping from the representation of numbers on a mental number line to response locations and discuss how the results can be explained by the alternative accounts.     

Find the picture of eight turtles: a link between children's counting and their knowledge of number word semantics

An essential part of understanding number words (e.g., eight) is understanding that all number words refer to the dimension of experience we call numerosity. Knowledge of this general principle may be separable from knowledge of individual number word meanings. That is, children may learn the meanings of at least a few individual number words before realizing that all number words refer to numerosity. Alternatively, knowledge of this general principle may form relatively early and proceed to guide and constrain the acquisition of individual number word meanings. The current article describes two experiments in which 116 children (2½- to 4-year-olds) were given a Word Extension task as well as a standard Give-N task. Results show that only children who understood the cardinality principle of counting successfully extended number words from one set to another based on numerosity—with evidence that a developing understanding of this concept emerges as children approach the cardinality principle induction. These findings support the view that children do not use a broad understanding of number words to initially connect number words to numerosity but rather make this connection around the time that they figure out the cardinality principle of counting.     

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 Spencer made number: First uses of the number words

This article describes the development of number concepts between infancy and early childhood. It is based on a diary study that tracked number word use in a child from 12 to 38 months of age. Number words appeared early in the child’s vocabulary, but accurate reference to specific numerosities evolved gradually over the entire 27-month period. Reliable performance on standardized number and counting tasks lagged behind accurate number use in informal settings by many months. Although there was evidence that this child labeled mentally represented sets, these instances comprised a very small proportion of the total sample and were usually references to anticipated sets rather than remembered sets.     

The logical syntax of number words: Theory, acquisition and processing

Recent work on the acquisition of number words has emphasized the importance of integrating linguistic and developmental perspectives [Musolino, J. (2004). The semantics and acquisition of number words: Integrating linguistic and developmental perspectives. Cognition93, 1-41; Papafragou, A., Musolino, J. (2003). Scalar implicatures: Scalar implicatures: Experiments at the semantics-pragmatics interface. Cognition, 86, 253-282; Hurewitz, F., Papafragou, A., Gleitman, L., Gelman, R. (2006). Asymmetries in the acquisition of numbers and quantifiers. Language Learning and Development, 2, 76-97; Huang, Y. T., Snedeker, J., Spelke, L. (submitted for publication). What exactly do numbers mean?]. Specifically, these studies have shown that data from experimental investigations of child language can be used to illuminate core theoretical issues in the semantic and pragmatic analysis of number terms. In this article, I extend this approach to the logico-syntactic properties of number words, focusing on the way numerals interact with each other (e.g. Three boys are holding two balloons) as well as with other quantified expressions (e.g. Three boys are holding each balloon). On the basis of their intuitions, linguists have claimed that such sentences give rise to at least four different interpretations, reflecting the complexity of the linguistic structure and syntactic operations involved. Using psycholinguistic experimentation with preschoolers (n = 32) and adult speakers of English (n = 32), I show that (a) for adults, the intuitions of linguists can be verified experimentally, (b) by the age of 5, children have knowledge of the core aspects of the logical syntax of number words, (c) in spite of this knowledge, children nevertheless differ from adults in systematic ways, (d) the differences observed between children and adults can be accounted for on the basis of an independently motivated, linguistically-based processing model [Geurts, B. (2003). Quantifying kids. Language Acquisition, 11(4), 197-218]. In doing so, this work ties together research on the acquisition of the number vocabulary with a growing body of work on the development of quantification and sentence processing abilities in young children [Geurts, 2003; Lidz, J., Musolino, J. (2002). Children’s command of quantification. Cognition, 84, 113-154; Musolino, J., Lidz, J. (2003). The scope of isomorphism: Turning adults into children. Language Acquisition, 11(4), 277-291; Trueswell, J., Sekerina, I., Hilland, N., Logrip, M. (1999). The kindergarten-path effect: Studying on-line sentence processing in young children. Cognition, 73, 89-134; Noveck, I. (2001). When children are more logical than adults: Experimental investigations of scalar implicature. Cognition, 78, 165-188; Noveck, I., Guelminger, R., Georgieff, N., & Labruyere, N. (2007). What autism can tell us about every . . . not sentences. Journal of Semantics,24(1), 73-90. On a more general level, this work confirms the importance of integrating formal and developmental perspectives [Musolino, 2004], this time by highlighting the explanatory power of linguistically-based models of language acquisition and by showing that the complex structure postulated by linguists has important implications for developmental accounts of the number vocabulary.     

Number words in young children's conceptual and procedural knowledge of addition, subtraction and inversion

Three studies addressed children's arithmetic. First, 50 3- to 5-year-olds judged physical demonstrations of addition, subtraction and inversion, with and without number words. Second, 20 3- to 4-year-olds made equivalence judgments of additions and subtractions. Third, 60 4- to 6-year-olds solved addition, subtraction and inversion problems that varied according to the inclusion of concrete referents and number words. The results indicate that number words play a different role in conceptual and procedural development. Children have strong addition and subtraction concepts before they can translate the physical effects of these operations into number words. However, using number words does not detract from their calculation procedures. Moreover, consistent with iterative relations between conceptual and procedural development, the results suggest that inversion acquisition depends on children's calculation procedures and that inversion understanding influences these procedures.     

Numerical morphology supports early number word learning: Evidence from a comparison of young Mandarin and English learners

Previous studies showed that children learning a language with an obligatory singular/plural distinction (Russian and English) learn the meaning of the number word for one earlier than children learning Japanese, a language without obligatory number morphology (Barner, Libenson, Cheung, & Takasaki, 2009; Sarnecka, Kamenskaya, Yamana, Ogura, & Yudovina, 2007). This can be explained by differences in number morphology, but it can also be explained by many other differences between the languages and the environments of the children who were compared. The present study tests the hypothesis that the morphological singular/plural distinction supports the early acquisition of the meaning of the number word for one by comparing young English learners to age and SES matched young Mandarin Chinese learners. Mandarin does not have obligatory number morphology but is more similar to English than Japanese in many crucial respects. Corpus analyses show that, compared to English learners, Mandarin learners hear number words more frequently, are more likely to hear number words followed by a noun, and are more likely to hear number words in contexts where they denote a cardinal value. Two tasks show that, despite these advantages, Mandarin learners learn the meaning of the number word for one three to six months later than do English learners. These results provide the strongest evidence to date that prior knowledge of the numerical meaning of the distinction between singular and plural supports the acquisition of the meaning of the number word for one.     

From grammatical number to exact numbers: early meanings of 'one', 'two', and 'three' in English, Russian, and Japanese

This study examined whether singular/plural marking in a language helps children learn the meanings of the words 'one,' 'two,' and 'three.' First, CHILDES data in English, Russian (which marks singular/plural), and Japanese (which does not) were compared for frequency, variability, and contexts of number-word use. Then young children in the USA, Russia, and Japan were tested on Counting and Give-N tasks. More English and Russian learners knew the meaning of each number word than Japanese learners, regardless of whether singular/plural cues appeared in the task itself (e.g., "Give two apples" vs. "Give two"). These results suggest that the learning of "one," "two" and "three" is supported by the conceptual framework of grammatical number, rather than that of integers.     

The connection between cognitive development and specific fears and worries in normal children and children with below-average intellectual abilities: a preliminary study.

The present study explored the relationship between cognitive development and anxiety phenomena in 4-12-year-old children. Fears and worries of normal children (n=176) were compared to those of children with below-average intellectual abilities (children with BAIA; n=105). We evaluated to what extent level of cognitive development as indexed by a Piagetian conservation task was associated with the presence of fears and worries. While normal children and children with BAIA did not differ with regard to the content of their fears and worries, normal children more frequently reported such anxiety phenomena during the semi-structured Anxiety Interview than did children with BAIA. Furthermore, in normal children, evidence was found to suggest that level of cognitive development contributes to the experience of fears and worries. That is, anxiety phenomena were more prevalent among those children who passed a Piagetian conservation task. However, when anxiety phenomena were assessed by means of the Koala Fear Questionnaire (KFQ), a different picture emerged. KFQ data suggested that fears were less frequent in normal children and those children with BAIA who had a higher level of cognitive functioning. Apparently, the Anxiety Interview and the KFQ tap quite different aspects of anxiety. The KFQ seems to measure primitive fears that are likely to be prevalent among children with limited cognitive capacity, whereas the Anxiety Interview assesses more sophisticated anxiety phenomena that probably depend on high levels of cognitive functioning.     

Shyness as a risk factor for second language acquisition of immigrant preschoolers

The aim of this study was to examine whether shyness is a risk factor for second language acquisition in immigrant preschoolers. Results from studies on first language acquisition indicate that shy children show less favorable language development; however, it remains unclear how shyness affects second language acquisition. As second language skills are often acquired in interactions outside the family where shyness is more evident, we postulate that shyness has a strong negative effect on second language acquisition. This hypothesis was examined using standardized tests and parental ratings in a sample of 330 immigrant preschoolers cross-sectionally and with 130 immigrant preschoolers longitudinally. The analyses revealed lower second language competence as well as slower language development in shy immigrant children compared to their non-shy counterparts. The present study highlights that not only contextual but also personality factors need to be considered for a comprehensive understanding of second language acquisition in immigrant children.     

How counting represents number: what children must learn and when they learn it

This study compared 2- to 4-year-olds who understand how counting works (cardinal-principle-knowers) to those who do not (subset-knowers), in order to better characterize the knowledge itself. New results are that (1) Many children answer the question "how many" with the last word used in counting, despite not understanding how counting works; (2) Only children who have mastered the cardinal principle, or are just short of doing so, understand that adding objects to a set means moving forward in the numeral list whereas subtracting objects mean going backward; and finally (3) Only cardinal-principle-knowers understand that adding exactly 1 object to a set means moving forward exactly 1 word in the list, whereas subset-knowers do not understand the unit of change.     

A validation of eye movements as a measure of elementary school children's developing number sense

The number line estimation task captures central aspects of children's developing number sense, that is, their intuitions for numbers and their interrelations. Previous research used children's answer patterns and verbal reports as evidence of how they solve this task. In the present study we investigated to what extent eye movements recorded during task solution reflect children's use of the number line. By means of a cross-sectional design with 66 children from Grades 1, 2, and 3, we show that eye-tracking data (a) reflect grade-related increase in estimation competence, (b) are correlated with the accuracy of manual answers, (c) relate, in Grade 2, to children's addition competence, (d) are systematically distributed over the number line, and (e) replicate previous findings concerning children's use of counting strategies and orientation-point strategies. These findings demonstrate the validity and utility of eye-tracking data for investigating children's developing number sense and estimation competence.     

Toddlers' language-mediated visual search: They need not have the words for it

Eye movements made by listeners during language-mediated visual search reveal a strong link between visual processing and conceptual processing. For example, upon hearing the word for a missing referent with a characteristic colour (e.g., “strawberry”), listeners tend to fixate a colour-matched distractor (e.g., a red plane) more than a colour-mismatched distractor (e.g., a yellow plane). We ask whether these shifts in visual attention are mediated by the retrieval of lexically stored colour labels. Do children who do not yet possess verbal labels for the colour attribute that spoken and viewed objects have in common exhibit language-mediated eye movements like those made by older children and adults? That is, do toddlers look at a red plane when hearing “strawberry”? We observed that 24-month-olds lacking colour term knowledge nonetheless recognized the perceptual–conceptual commonality between named and seen objects. This indicates that language-mediated visual search need not depend on stored labels for concepts.     

Taking the task seriously: reflections on measures of color acquisition

The experiments described in the lead articles by Kowalski and Zimiles and by O'Hanlon and Roberson examine factors that lead to color term acquisition. These experiments touch on the debate regarding the relative contributions of language and concepts in word learning. In this reflection, we examine how conclusions concerning the debate depend deeply on the particular task presented to children, and we propose an alternative approach to studying color term acquisition.     

When the mental number line involves a delay: the writing of numbers by children of different arithmetical abilities

Our study focused on number transcoding in children. It investigated how 9-year-olds with and without arithmetical disabilities wrote Arabic digits after they had heard them as number words. Planning time before writing each digit was registered. Analyses revealed that the two groups differed not only in arithmetical abilities but also in verbal and reading abilities. Children with arithmetical disabilities were overall slower in planning Arabic digits than were control children with normal arithmetical abilities. In addition, they showed a number size effect for numbers smaller than 10, suggesting a semantically mediated route in number processing. Control children did not need more planning time for large numbers (e.g., 8) than for small numbers (e.g., 3), suggesting a direct nonsemantic route. For both two- and three-digit numbers, both groups of children showed a number size effect, although the effect was smaller each time for control children. The presence of the stronger number size effect for children with arithmetical disabilities was seen as a delay in the development of quick and direct transcoding. The relation between transcoding problems and arithmetical disabilities is discussed. A defect in the linking of numerical symbols to analog numerical representations is proposed as an explanation for the transcoding problems found in some children.     

Developing access to number magnitude: a study of the SNARC effect in 7- to 9-year-olds

The SNARC (spatial-numerical association of response codes) effect refers to the finding that small numbers facilitate left responses, whereas larger numbers facilitate right responses. The development of this spatial association was studied in 7-, 8-, and 9-year-olds, as well as in adults, using a task where number magnitude was essential to perform the task and another task where number magnitude was irrelevant. When number magnitude was essential, a SNARC effect was found in all age groups. But when number magnitude was irrelevant, a SNARC effect was found only in 9-year-olds and adults. These results are taken to suggest that (a) 7-year-olds represent number magnitudes in a way similar to that of adults and that (b) when perceiving Arabic numerals, children have developed automatic access to magnitude information by around 9 years of age.     

Moderating effects of maternal emotional availability on language and cognitive development in toddlers of mothers exposed to a natural disaster in pregnancy: The QF2011 Queensland Flood Study

BackgroundPrenatal maternal stress exposure has been linked to sub-optimal developmental outcomes in toddlers, while maternal emotional availability is associated with better cognitive and language abilities. It is less clear whether early care-giving relationships can moderate the impact of prenatal stress on child development. The current study investigates the impact of stress during pregnancy resulting from the Queensland Floods in 2011 on toddlers’ cognitive and language development, and examines how maternal emotional availability is associated with these outcomes.MethodsData were available from 131 families. Measures of prenatal stress (objective hardship, cognitive appraisal, and three measures of maternal subjective stress) were collected within one year of the 2011 Queensland floods. Maternal emotional availability was rated from video-taped mother-child play sessions at 16 months: sensitivity (e.g., affective connection, responsiveness to signals) and structuring (e.g., scaffolding, guidance, limit-setting). The toddlers’ cognitive and language development was assessed at 30 months. Interactions were tested to determine whether maternal emotional availability moderated the relationship between prenatal maternal stress and toddler cognitive and language functioning.ResultsPrenatal stress was not correlated with toddlers’ cognitive and language development at 30 months. Overall, the higher the maternal structuring and sensitivity, the better the toddlers’ cognitive outcomes. However, significant interactions showed that the effects of maternal structuring on toddler language abilities depended on the degree of prenatal maternal subjective stress: when maternal subjective stress was above fairly low levels, the greater the maternal structuring, the higher the child vocabulary level.ConclusionThe current study highlights the importance of maternal emotional availability, especially structuring, for cognitive and language development in young children. Findings suggest that toddlers exposed to higher levels of prenatal maternal stress in utero may benefit from high maternal structuring for their language development.     

Basic numerical skills in children with mathematics learning disabilities: a comparison of symbolic vs non-symbolic number magnitude processing

Forty-five children with mathematics learning disabilities, with and without comorbid reading disabilities, were compared to 45 normally achieving peers in tasks assessing basic numerical skills. Children with mathematics disabilities were only impaired when comparing Arabic digits (i.e., symbolic number magnitude) but not when comparing collections (i.e., non-symbolic number magnitude). Moreover, they automatically processed number magnitude when comparing the physical size of Arabic digits in an Stroop paradigm adapted for processing speed differences. Finally, no evidence was found for differential patterns of performance between MD and MD/RD children in these tasks. These findings suggest that children with mathematics learning disabilities have difficulty in accessing number magnitude from symbols rather than in processing numerosity per se.