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.     

Numerical ordering ability mediates the relation between number-sense and arithmetic competence

What predicts human mathematical competence? While detailed models of number representation in the brain have been developed, it remains to be seen exactly how basic number representations link to higher math abilities. We propose that representation of ordinal associations between numerical symbols is one important factor that underpins this link. We show that individual variability in symbolic number-ordering ability strongly predicts performance on complex mental-arithmetic tasks even when controlling for several competing factors, including approximate number acuity. Crucially, symbolic number-ordering ability fully mediates the previously reported relation between approximate number acuity and more complex mathematical skills, suggesting that symbolic number-ordering may be a stepping stone from approximate number representation to mathematical competence. These results are important for understanding how evolution has interacted with culture to generate complex representations of abstract numerical relationships. Moreover, the finding that symbolic number-ordering ability links approximate number acuity and complex math skills carries implications for designing math-education curricula and identifying reliable markers of math performance during schooling.     

The relation between syllable number and visual complexity in the acquisition of word meanings

Four experiments were conducted to explore the correlation between syllable number and visual complexity in the acquisition of novel words. In the first experiment, adult English speakers invented nonsense words as names for random polygons differing in visual complexity. Visually simple polygons received names containing fewer syllables than visually complex polygons did. In addition, analyses of English word-object pairings indicated that a significant correlation between syllable number and visual complexity exists in the English lexicon. In Experiments 2 and 3, adult English speakers matched monosyllabic novel words more often than trisyllabic novel words with visually simple objects, whereas trisyllabic matches were more common for visually complex objects. Experiment 4 replicated these findings with children, indicating that the assumption of a correlation between word and visual complexity exists during the period of intense vocabulary growth. Although the actual correlation between syllable number and visual complexity is small, other posited constraints on word meaning are also limited in strength. However, an increasing number of small, language-specific word-meaning correlations are being uncovered. Given the documented ability of speakers to detect and use these subtle correlations, we argue that a more fruitful approach to word-meaning acquisition would forgo the search for a few broad, powerful word-meaning constraints, and we attempt to uncover individually weak, but perhaps jointly powerful word-meaning correspondences.     

How to learn the natural numbers: inductive inference and the acquisition of number concepts

Theories of number concepts often suppose that the natural numbers are acquired as children learn to count and as they draw an induction based on their interpretation of the first few count words. In a bold critique of this general approach, Rips, Asmuth, Bloomfield [Rips, L., Asmuth, J. & Bloomfield, A. (2006). Giving the boot to the bootstrap: How not to learn the natural numbers. Cognition, 101, B51-B60.] argue that such an inductive inference is consistent with a representational system that clearly does not express the natural numbers and that possession of the natural numbers requires further principles that make the inductive inference superfluous. We argue that their critique is unsuccessful. Provided that children have access to a suitable initial system of representation, the sort of inductive inference that Rips et al. call into question can in fact facilitate the acquisition of larger integer concepts without the addition of any further principles.     

The relation between subitizable symbolic and non‐symbolic number processing over the kindergarten school year

A long‐standing debate in the field of numerical cognition concerns the degree to which symbolic and non‐symbolic processing are related over the course of development. Of particular interest is the possibility that this link depends on the range of quantities in question. Behavioral and neuroimaging research with adults suggests that symbolic and non‐symbolic quantities may be processed more similarly within, relative to outside of, the subitizing range. However, it remains unclear whether this unique link exists in young children at the outset of formal education. Further, no study has yet taken numerical size into account when investigating the longitudinal influence of these skills. To address these questions, we investigated the relation between symbolic and non‐symbolic processing inside versus outside the subitizing range, both cross‐sectionally and longitudinally, in 540 kindergarteners. Cross‐sectionally, we found a consistently stronger relation between symbolic and non‐symbolic number processing within versus outside the subitizing range at both the beginning and end of kindergarten. We also show evidence for a bidirectional relation over the course of kindergarten between formats within the subitizing range, and a unidirectional relation (symbolic → non‐symbolic) for quantities outside of the subitizing range. These findings extend current theories on symbolic and non‐symbolic magnitude development by suggesting that non‐symbolic processing may in fact play a role in the development of symbolic number abilities, but that this influence may be limited to quantities within the subitizing range.     

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 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.     

Language counts: Early language mediates the relationship between parent education and children's math ability

Children's early math skills have been hailed as a powerful predictor of academic success. Disparities in socioeconomic context, however, also have dramatic consequences on children's learning. It is therefore critical to investigate both of these distinct contributors in order to better understand the early foundations of children's academic outcomes. This study tests an integrated model of children's developing math ability so as to (1) identify the specific skills and abilities most clearly linked to early math achievement and (2) measure the influence of children's socioeconomic context on each of these skills. We first evaluated the early vocabulary, number word knowledge (knower level), and Approximate Number System (ANS) acuity of a diverse group of preschoolers. Then, approximately 1 year later as they entered Kindergarten, we administered a test of early math achievement. We find that children's early language (general vocabulary and number word knowledge) fully mediates the relationship between parent education and math ability. Additionally, number word knowledge mediates the relationship between ANS acuity and early math. We argue that increased focus on number word knowledge, as well as general vocabulary, may help to minimize disparities in math ability as children enter kindergarten. We also highlight the role of parent education on children's learning and note that this may be an important locus for intervention.     

The relation between the sense of agency and the experience of flow

This article investigates the relation between people’s feelings of agency and their feelings of flow. In the dominant model describing how people are able to assess their own agency—the comparator model of agency—when the person’s intentions match perfectly to what happens, the discrepancy between intention and outcome is zero, and the person is thought to interpret this lack of discrepancy as being in control. The lack of perceived push back from the external world seems remarkably similar to the state that has been described as a state of flow. However, when we used a computer game paradigm to investigate the relation between people’s feelings of agency and their feelings of flow, we found a dissociation between these two states. Although these two states may, in some ways, seem to be similar, our data indicate that they are governed by different principles and phenomenology.     

The relation between spatial skill and early number knowledge: The role of the linear number line

[Correction Notice: An Erratum for this article was reported in Vol 48(5) of Developmental Psychology (see record 2012-11771-001). The grey boxes around the faces in Figure 2 are missing. The correct version is presented in the erratum.] Spatial skill is highly related to success in math and science (e.g., Casey, Nuttall, Pezaris, & Benbow, 1995). However, little work has investigated the cognitive pathways by which the relation between spatial skill and math achievement emerges. We hypothesized that spatial skill plays a crucial role in the development of numerical reasoning by helping children to create a spatially meaningful, powerful numerical representation-the linear number line. In turn, a strong linear number representation improves other aspects of numerical knowledge such as arithmetic estimation. We tested this hypothesis using 2 longitudinal data sets. First, we found that children's spatial skill (i.e., mental transformation ability) at the beginning of 1st and 2nd grades predicted improvement in linear number line knowledge over the course of the school year. Second, we found that children's spatial skill at age 5 years predicted their performance on an approximate symbolic calculation task at age 8 and that this relation was mediated by children's linear number line knowledge at age 6. The results are consistent with the hypothesis that spatial skill can improve children's development of numerical knowledge by helping them to acquire a linear spatial representation of numbers. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

The relation between the intensity of excitation and the number of neurones traversed

In previous formulations the assumption has been made that the intensity of excitation or inhibition varies with distance. In this paper the simple case of a series of identical neurones is treated. The development shows five possible relations. In future theoretical formulations it is no longer necessary to postulate arbitrary relations in case any of the derived forms can be used.     

An exploration of the interplay between psychological sense of community,social identification and salience

Past research indicates that there is a strong relationship between the constructs of psychological sense of community (PSOC) and social identification. The current study draws on data (N = 219) examining participants' membership in a number of different communities to present an examination of the relationship between these constructs. In particular, the study examines the relative strength of the separate aspects of social identification (based on Cameron's 2004, Three Factor Model of Social Identification) as predictors of overall PSOC, accounting for situational salience. Results indicate that Ingroup Ties is consistently the strongest predictor of PSOC and that the strength of Ingroup Affect and Centrality alter according to the group or community context. The theoretical implications of these results are discussed in terms of the interplay and overlap of these important community processes. Copyright © 2005 John Wiley & Sons, Ltd.     

The prospective relationship between child personality and perceived parenting: Mediation by parental sense of competence

This study examined the prospective relationship between childhood Big Five personality characteristics and perceived parenting in adolescence. In addition, we investigated whether this relationship was mediated by parental sense of competence, and whether associations were different for mothers and fathers. For 274 children, teachers reported on children’s Big Five personality characteristics at Time 1, mothers and fathers reported on their sense of competence at Time 2, and the children (who had now become adolescents) rated their parents’ warmth, overreactivity and psychological control at Time 3. Mediation analysis revealed both direct and indirect effects. No differences in associations were found for perceived parenting of mothers and fathers. This study demonstrates that child personality in late childhood is significantly related to perceived parental warmth, overreactivity and psychological control in adolescence. In addition, parental sense of competence mediates the relationship between child conscientiousness and perceived parental warmth, overreactivity and psychological control.     

On the language specificity of basic number processing: transcoding in a language with inversion and its relation to working memory capacity

Transcoding Arabic numbers from and into verbal number words is one of the most basic number processing tasks commonly used to index the verbal representation of numbers. The inversion property, which is an important feature of some number word systems (e.g., German einundzwanzig [one and twenty]), might represent a major difficulty in transcoding and a challenge to current transcoding models. The mastery of inversion, and of transcoding in general, might be related to nonnumerical factors such as working memory resources given that different elements and their sequence need to be memorized and manipulated. In this study, transcoding skills and different working memory components in Austrian (German-speaking) 7-year-olds were assessed. We observed that inversion poses a major problem in transcoding for German-speaking children. In addition, different components of working memory skills were differentially correlated with particular transcoding error types. We discuss how current transcoding models could account for these results and how they might need to be adapted to accommodate inversion properties and their relation to different working memory components.     

Search for a curvilinear relationship between the sense of coherence and the intensity of PTSD in MVA survivors

Abstract

The relationship between the sense of coherence (SOC) and the intensity of posttraumatic stress disorder (PTSD) was examined in order to determine its nature and to resolve the inconsistencies between (1) a growing body of empirical research that indicates a linear relationship between these variables, and (2) the schema-based theories of PTSD that suggest a curvilinear relationship between cognitions and the intensity of PTSD. In this cross-sectional study an attempt was also made to identify some psychological factors that moderate this relationship. Participants were a sample of 1132 motor vehicle accident (MVA) survivors. The results showed that gender and temperamental predisposition to PTSD constituted moderator variables of the relationship between SOC and the intensity of PTSD. This supported both the empirical evidence on the linear and negative relationship between SOC and the intensity of PTSD and the theories that postulated the curvilinear relationship.     

The role of numerical and non-numerical cues in nonsymbolic number processing: Evidence from the line bisection task

In line bisection tasks, adults and children bisect towards the numerically larger of two nonsymbolic numerosities [de Hevia, M. D., & Spelke, E. S. (2009 de Hevia, M. D., & Spelke, E. S. (2009). Spontaneous mapping of number and space in adults and young children. Cognition, 110, 198–207. doi:10.1016/j.cognition.2008.11.003[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). Spontaneous mapping of number and space in adults and young children. Cognition, 110, 198–207. doi:10.1016/j.cognition.2008.11.003]. However, it is not clear whether this effect is driven by number itself or rather by visual cues such as subtended area [Gebuis, T., & Gevers, W. (2011 Gebuis, T., & Gevers, W. (2011). Numerosities and space: Indeed a cognitive illusion! A reply to de Hevia and Spelke (2009). Cognition, 121, 248–252. doi:10.1016/j.cognition.2010.09.008[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). Numbers and space: Indeed a cognitive illusion! A reply to de Hevia and Spelke (2009 de Hevia, M. D., & Spelke, E. S. (2009). Spontaneous mapping of number and space in adults and young children. Cognition, 110, 198–207. doi:10.1016/j.cognition.2008.11.003[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). Cognition, 121, 248–252. doi:10.1016/j.cognition.2010.09.008]. Furthermore, this effect has only been demonstrated with flanking displays of two and nine items. Here, we report three studies that examined whether this “spatial bias” effect occurs across a range of absolute and ratio numerosity differences; in particular, we examined whether the bias would occur when both flankers were outside the subitizing range. Additionally, we manipulated the subtended area of the stimulus and the aggregate surface area to assess the influence of visual cues. We found that the spatial bias effect occurred for a range of flanking numerosities and for ratios of 3:5 and 5:6 when subtended area was not controlled (Experiment 1). However, when subtended area and aggregate surface area were held constant, the biasing effect was reversed such that participants bisected towards the flanker with fewer items (Experiment 2). Moreover, when flankers were identical, participants bisected towards the flanker with larger subtended area or larger aggregate surface area (Experiments 2 and 3). On the basis of these studies, we conclude that the spatial bias effect for nonsymbolic numerosities is primarily driven by visual cues.     

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.