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Krajcsi A, Reynvoet B. Miscategorized subset-knowers: Five- and six-knowers can compare only the numbers they know. Dev Sci 2024; 27:e13430. [PMID: 37392074 DOI: 10.1111/desc.13430] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/15/2022] [Revised: 05/21/2023] [Accepted: 06/15/2023] [Indexed: 07/02/2023]
Abstract
Initial acquisition of the first symbolic numbers is measured with the Give a Number (GaN) task. According to the classic method, it is assumed that children who know only 1, 2, 3, or 4 in the GaN task, (termed separately one-, two-, three-, and four-knowers, or collectively subset-knowers) have only a limited conceptual understanding of numbers. On the other hand, it is assumed that children who know larger numbers understand the fundamental properties of numbers (termed cardinality-principle-knowers), even if they do not know all the numbers as measured with the GaN task, that are in their counting list (e.g., five- or six-knowers). We argue that this practice may not be well-established. To validate this categorization method, here, the performances of groups with different GaN performances were measured separately in a symbolic comparison task. It was found that similar to one to four-knowers, five-, six-, and so forth, knowers can compare only the numbers that they know in the GaN task. We conclude that five-, six-, and so forth, knowers are subset-knowers because their conceptual understanding of numbers is fundamentally limited. We argue that knowledge of the cardinality principle should be identified with stricter criteria compared to the current practice in the literature. RESEARCH HIGHLIGHTS: Children who know numbers larger than 4 in the Give a Number (GaN) task are usually assumed to have a fundamental conceptual understanding of numbers. We tested children who know numbers larger than 4 but who do not know all the numbers in their counting list to see whether they compare numbers more similar to children who know only small numbers in the GaN task or to children who have more firm number knowledge. Five-, six-, and so forth, knowers can compare only the numbers they know in the GaN task, similar to the performance of the one, two, three, and four-knowers. We argue that these children have a limited conceptual understanding of numbers and that previous works may have miscategorized them.
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Affiliation(s)
- Attila Krajcsi
- Department of Cognitive Psychology, Institute of Psychology, ELTE Eötvös Loránd University, Budapest, Hungary
| | - Bert Reynvoet
- Brain and Cognition, KU Leuven, Leuven, Belgium
- Faculty of Psychology and Educational Sciences, KU Leuven Kulak, Leuven, Belgium
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2
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How many seconds was that? Teaching children about time does not refine their ability to track durations. Cognition 2023; 235:105410. [PMID: 36848703 DOI: 10.1016/j.cognition.2023.105410] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/08/2022] [Revised: 01/09/2023] [Accepted: 02/13/2023] [Indexed: 02/27/2023]
Abstract
Over development, children acquire symbols to represent abstract concepts such as time and number. Despite the importance of quantity symbols, it is unknown how acquiring these symbols impacts one's ability to perceive quantities (i.e., nonsymbolic representations). While it has been proposed that learning symbols shapes nonsymbolic quantitative abilities (i.e., the refinement hypothesis), this hypothesis has been understudied, especially in the domain of time. Moreover, the majority of research in support of this hypothesis has been correlational in nature, and thus, experimental manipulations are critical for determining whether this relation is causal. In the present study, kindergarteners and first graders (N = 154) who have yet to learn about temporal symbols in school completed a temporal estimation task during which they were either (1) trained on temporal symbols and effective timing strategies ("2 s" and counting on the beat), (2) trained on temporal symbols only ("2 s"), or (3) participated in a control training. Children's nonsymbolic and symbolic timing abilities were assessed before and after training. Results revealed a correlation between children's nonsymbolic and symbolic timing abilities at pre-test (when controlling for age), indicating this relation exists prior to formal classroom instruction on temporal symbols. Notably, we found no support for the refinement hypothesis, as learning temporal symbols did not impact children's nonsymbolic timing abilities. Implications and future directions are discussed.
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3
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Krajcsi A, Kojouharova P. Stimulus frequency alone can account for the size effect in number comparison. Acta Psychol (Amst) 2023; 232:103817. [PMID: 36571893 DOI: 10.1016/j.actpsy.2022.103817] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/25/2022] [Revised: 11/24/2022] [Accepted: 12/16/2022] [Indexed: 12/25/2022] Open
Abstract
In a number comparison task, the size effect (i.e, smaller values are easier to compare than larger values) is usually attributed to a psychophysics-based representation. However, alternative models assume that the size effect is a frequency effect: Smaller numbers are easier to process because they are observed more frequently. Previous studies have demonstrated that the frequency of the digits fundamentally influences the comparison size effect: In new number symbols, the frequency entirely determines the size effect. In contrast, in Arabic notation, the size effect aggregates the frequency in the actual session and the previous regular size effect. Here, we investigate whether the previously acquired regular size effect can depend on the frequency of the stimuli as well or on a psychophysics-based representation that is not yet active in new symbols. Participants in the study compared numbers that were denoted with new symbols, with the frequency of the symbols being changed throughout the session. We found that the frequency of the stimuli in both halves of the session was aggregated in the size effect. In addition, no psychophysics-based size effect was found throughout the session. These results confirm that the size effect can be created and shaped purely by the frequency of the symbols, while a psychophysics-based representation is not necessary to account for these size effect-related phenomena.
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Affiliation(s)
- Attila Krajcsi
- Cognitive Psychology Department, Institute of Psychology, ELTE Eötvös Loránd University, Budapest, Hungary.
| | - Petia Kojouharova
- Research Centre for Natural Sciences, Institute of Cognitive Neuroscience and Psychology, Budapest, Hungary
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4
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The approximate number system cannot be the leading factor in the acquisition of the first symbolic numbers. COGNITIVE DEVELOPMENT 2023. [DOI: 10.1016/j.cogdev.2022.101285] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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5
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A refined description of initial symbolic number acquisition. COGNITIVE DEVELOPMENT 2023. [DOI: 10.1016/j.cogdev.2022.101288] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
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6
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Fang S, Zhou X. Form perception speed is critical for the relationship between non-verbal number sense and arithmetic fluency. INTELLIGENCE 2022. [DOI: 10.1016/j.intell.2022.101704] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/05/2022]
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7
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Yang M, Liang J. Early number word learning: Associations with domain-general and domain-specific quantitative abilities. Front Psychol 2022; 13:1024426. [DOI: 10.3389/fpsyg.2022.1024426] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/21/2022] [Accepted: 10/07/2022] [Indexed: 11/13/2022] Open
Abstract
Cardinal number knowledge-understanding “two” refers to sets of two entities-is a critical piece of knowledge that predicts later mathematics achievement. Recent studies have shown that domain-general and domain-specific skills can influence children’s cardinal number learning. However, there has not yet been research investigating the influence of domain-specific quantifier knowledge on children’s cardinal number learning. The present study aimed to investigate the influence of domain-general and domain-specific skills on Mandarin Chinese-speaking children’s cardinal number learning after controlling for a number of family background factors. Particular interest was paid to the question whether domain-specific quantifier knowledge was associated with cardinal number development. Specifically, we investigated 2–5-year-old Mandarin Chinese-speaking children’s understanding of cardinal number words as well as their general language, intelligence, approximate number system (ANS) acuity, and knowledge of quantifiers. Children’s age, gender, parental education, and family income were also assessed and used as covariates. We found that domain-general abilities, including general language and intelligence, did not account for significant additional variance of cardinal number knowledge after controlling for the aforementioned covariates. We also found that domain-specific quantifier knowledge did not account for significant additional variance of cardinal number knowledge, whereas domain-specific ANS acuity accounted for significant additional variance of cardinal number knowledge, after controlling for the aforementioned covariates. In sum, the results suggest that domain-specific numerical skills seem to be more important for children’s development of cardinal number words than the more proximal domain-general abilities such as language abilities and intelligence. The results also highlight the significance of ANS acuity on children’s cardinal number word development.
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Cheung P, Toomey M, Jiang YH, Stoop TB, Shusterman A. Acquisition of the counting principles during the subset-knower stages: Insights from children's errors. Dev Sci 2022; 25:e13219. [PMID: 34935245 DOI: 10.1111/desc.13219] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/17/2021] [Revised: 11/06/2021] [Accepted: 11/29/2021] [Indexed: 11/28/2022]
Abstract
Studies on children's understanding of counting examine when and how children acquire the cardinal principle: the idea that the last word in a counted set reflects the cardinal value of the set. Using Wynn's (1990) Give-N Task, researchers classify children who can count to generate large sets as having acquired the cardinal principle (cardinal-principle-knowers) and those who cannot as lacking knowledge of it (subset-knowers). However, recent studies have provided a more nuanced view of number word acquisition. Here, we explore this view by examining the developmental progression of the counting principles with an aim to elucidate the gradual elements that lead to children successfully generating sets and being classified as CP-knowers on the Give-N Task. Specifically, we test the claim that subset-knowers lack cardinal principle knowledge by separating children's understanding of the cardinal principle from their ability to apply and implement counting procedures. We also ask when knowledge of Gelman & Gallistel's (1978) other how-to-count principles emerge in development. We analyzed how often children violated the three how-to-count principles in a secondary analysis of Give-N data (N = 86). We found that children already have knowledge of the cardinal principle prior to becoming CP-knowers, and that understanding of the stable-order and word-object correspondence principles likely emerged earlier. These results suggest that gradual development may best characterize children's acquisition of the counting principles and that learning to coordinate all three principles represents an additional step beyond learning them individually.
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Affiliation(s)
- Pierina Cheung
- National Institute of Education, Nanyang Technological University, Singapore
| | - Mary Toomey
- Department of Psychology, Wesleyan University, Middletown, Connecticut, USA
| | - Yahao Harry Jiang
- Department of Psychology, Wesleyan University, Middletown, Connecticut, USA
| | - Tawni B Stoop
- Department of Psychology, Penn State University, State College, Pennsylvania, USA
| | - Anna Shusterman
- Department of Psychology, Wesleyan University, Middletown, Connecticut, USA
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9
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Symbolic number comparison and number priming do not rely on the same mechanism. Psychon Bull Rev 2022; 29:1969-1977. [PMID: 35503169 PMCID: PMC9568444 DOI: 10.3758/s13423-022-02108-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 04/13/2022] [Indexed: 11/23/2022]
Abstract
In elementary symbolic number processing, the comparison distance effect (in a comparison task, the task is more difficult with smaller numerical distance between the values) and the priming distance effect (in a number processing task, actual number is easier to process with a numerically close previous number) are two essential phenomena. While a dominant model, the approximate number system model, assumes that the two effects rely on the same mechanism, some other models, such as the discrete semantic system model, assume that the two effects are rooted in different generators. In a correlational study, here we investigate the relation of the two effects. Critically, the reliability of the effects is considered; therefore, a possible null result cannot be attributed to the attenuation of low reliability. The results showed no strong correlation between the two effects, even though appropriate reliabilities were provided. These results confirm the models of elementary number processing that assume distinct mechanisms behind number comparison and number priming.
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10
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He X, Zhou X, Zhao J, Zhang Y. Visual Perception Supports Adults in Numerosity Processing and Arithmetical Performance. Front Psychol 2021; 12:722261. [PMID: 34744887 PMCID: PMC8570262 DOI: 10.3389/fpsyg.2021.722261] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/08/2021] [Accepted: 09/24/2021] [Indexed: 11/20/2022] Open
Abstract
Previous studies have found a correlation between numerosity processing and arithmetical performance. Visual perception has already been indicated as the shared cognitive mechanism between these two; however, these studies mostly focused on children. It is not clear whether the association between numerosity processing and arithmetical performance still existed following the development of individual arithmetical performance. Consequently, the underlying role of visual perception in numerosity processing and arithmetical performance has not been sufficiently studied in adults. For this study, researchers selected a total of 205 adult participants with an average age of 22years. The adults were administered arithmetic tests, numerosity comparison, and visual figure matching. Mental rotation, choice reaction time, and nonverbal intelligence were used as cognitive covariates. Results showed that numerosity comparison of adults correlated with their arithmetical performance, even after controlling for age and gender differences as well as general cognitive processing. However, after controlled for visual figure matching, the well-established association between numerosity comparison and arithmetic performance disappeared. These results supported the visual perception hypothesis, that visual perception measured by visual figure matching can account for the correlation between numerosity comparison and arithmetic performance. This indicated that even for adult populations, visual perceptual ability was the underlying component of numerosity processing and arithmetic performance.
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Affiliation(s)
- Xinyao He
- School of Psychology, Liaoning Normal University, Liaoning, China
| | - Xinlin Zhou
- State Key Laboratory of Cognitive Neuroscience and Learning, Siegler Center for Innovative Learning, Advanced Innovation Center for Future Education, Beijing Normal University, Beijing, China
| | - Jin Zhao
- Dalian Institute of Science and Technology, Liaoning, China
| | - Yiyun Zhang
- School of Psychology, Liaoning Normal University, Liaoning, China
- State Key Laboratory of Cognitive Neuroscience and Learning, Siegler Center for Innovative Learning, Advanced Innovation Center for Future Education, Beijing Normal University, Beijing, China
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11
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Sella F, Slusser E, Odic D, Krajcsi A. The emergence of children’s natural number concepts: Current theoretical challenges. CHILD DEVELOPMENT PERSPECTIVES 2021. [DOI: 10.1111/cdep.12428] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Francesco Sella
- Centre for Mathematical Cognition Loughborough University Loughborough UK
| | - Emily Slusser
- Department of Child and Adolescent Development San Jose State University San Jose California USA
| | - Darko Odic
- Department of Psychology The University of British Columbia Vancouver BC Canada
| | - Attila Krajcsi
- Department of Cognitive Psychology Eötvös Loránd University Budapest Hungary
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12
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Longitudinal relations between the approximate number system and symbolic number skills in preschool children. J Exp Child Psychol 2021; 212:105254. [PMID: 34352660 DOI: 10.1016/j.jecp.2021.105254] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/24/2021] [Revised: 07/05/2021] [Accepted: 07/07/2021] [Indexed: 01/29/2023]
Abstract
This study examined the longitudinal relation between the approximate number system (ANS) and two symbolic number skills, namely word problem-solving skill and number line skill, in a sample of 138 Chinese 4- to 6-year-old children. The ANS and symbolic number skills were measured first in the second year of preschool (Time 1 [T1], mean age = 4.98 years; SD = 0.33) and then in the third year of preschool (Time 2 [T2]). Cross-lagged analyses indicated that word problem-solving skill at T1 predicted ANS acuity at T2 but not vice versa. In addition, there were bidirectional relations between children's word problem-solving skill and number line estimation skill. The observed longitudinal relations were robust to the control of child's sex, age, maternal education, receptive vocabulary, spatial visualization, and working memory except for the relation between T1 word problem-solving skill and T2 number line estimation skill, which was explained by child's age.
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13
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Long-term relevance and interrelation of symbolic and non-symbolic abilities in mathematical-numerical development: Evidence from large-scale assessment data. COGNITIVE DEVELOPMENT 2021. [DOI: 10.1016/j.cogdev.2021.101008] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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14
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Follow-up questions influence the measured number knowledge in the Give-a-number task. COGNITIVE DEVELOPMENT 2021. [DOI: 10.1016/j.cogdev.2020.100968] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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15
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Wilkey ED, Ansari D. Challenging the neurobiological link between number sense and symbolic numerical abilities. Ann N Y Acad Sci 2019; 1464:76-98. [PMID: 31549430 DOI: 10.1111/nyas.14225] [Citation(s) in RCA: 27] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/23/2019] [Revised: 07/25/2019] [Accepted: 08/06/2019] [Indexed: 01/29/2023]
Abstract
A significant body of research links individual differences in symbolic numerical abilities, such as arithmetic, to number sense, the neurobiological system used to approximate and manipulate quantities without language or symbols. However, recent findings from cognitive neuroscience challenge this influential theory. Our current review presents an overview of evidence for the number sense account of symbolic numerical abilities and then reviews recent studies that challenge this account, organized around the following four assertions. (1) There is no number sense as traditionally conceived. (2) Neural substrates of number sense are more widely distributed than common consensus asserts, complicating the neurobiological evidence linking number sense to numerical abilities. (3) The most common measures of number sense are confounded by other cognitive demands, which drive key correlations. (4) Number sense and symbolic number systems (Arabic digits, number words, and so on) rely on distinct neural mechanisms and follow independent developmental trajectories. The review follows each assertion with comments on future directions that may bring resolution to these issues.
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Affiliation(s)
- Eric D Wilkey
- Brain and Mind Institute, Western University, London, Ontario, Canada
| | - Daniel Ansari
- Brain and Mind Institute, Western University, London, Ontario, Canada
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16
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Carey S, Barner D. Ontogenetic Origins of Human Integer Representations. Trends Cogn Sci 2019; 23:823-835. [PMID: 31439418 DOI: 10.1016/j.tics.2019.07.004] [Citation(s) in RCA: 62] [Impact Index Per Article: 12.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/01/2019] [Revised: 07/15/2019] [Accepted: 07/16/2019] [Indexed: 11/30/2022]
Abstract
Do children learn number words by associating them with perceptual magnitudes? Recent studies argue that approximate numerical magnitudes play a foundational role in the development of integer concepts. Against this, we argue that approximate number representations fail both empirically and in principle to provide the content required of integer concepts. Instead, we suggest that children's understanding of integer concepts proceeds in two phases. In the first phase, children learn small exact number word meanings by associating words with small sets. In the second phase, children learn the meanings of larger number words by mastering the logic of exact counting algorithms, which implement the successor function and Hume's principle (that one-to-one correspondence guarantees exact equality). In neither phase do approximate number representations play a foundational role.
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Affiliation(s)
- Susan Carey
- Department of Psychology, Harvard University, Cambridge, MA 02138, USA.
| | - David Barner
- Department of Psychology, University of California, San Diego, La Jolla, CA 92093, USA; University of California, San Diego, La Jolla, CA 92093, USA
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17
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Zhang Y, Liu T, Chen C, Zhou X. Visual form perception supports approximate number system acuity and arithmetic fluency. LEARNING AND INDIVIDUAL DIFFERENCES 2019. [DOI: 10.1016/j.lindif.2019.02.008] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/29/2023]
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18
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The influence of visual-spatial skills on the association between processing of nonsymbolic numerical magnitude and number word sequence skills. J Exp Child Psychol 2018; 178:184-197. [PMID: 30388483 DOI: 10.1016/j.jecp.2018.09.018] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/23/2017] [Revised: 09/21/2018] [Accepted: 09/29/2018] [Indexed: 11/24/2022]
Abstract
Nonsymbolic numerical magnitude processing skills are assumed to be fundamental to mathematical learning. Recent findings suggest that visual-spatial skills account for associations between children's performance in visually presented nonsymbolic numerical magnitude comparison tasks and their performance in visually presented arithmetic tasks. The aim of the current study was to examine whether associations between children's performance in visually presented tasks assessing nonsymbolic numerical magnitude processing skills and their performance in tasks assessing early mathematical skills, which do not involve visual stimulation, may also be mediated by visual-spatial skills. This line of reasoning is based on the assumption that children make use of mental visualization processes when working on tasks assessing early mathematical skills, such as knowledge of the sequence of number words, even when these tasks do not involve visual stimulation. We assessed 4- to 6-year-old children's performance in a nonsymbolic numerical magnitude comparison task, in tasks concerning knowledge of the sequence of number words, and in a developmental test to assess visual-spatial skills. Children's nonsymbolic numerical magnitude processing skills were found to be associated with their number word sequence skills. This association was fully mediated by interindividual differences in visual-spatial skills. The effect size of this mediation effect was small. We assume that the ability to construct mental visualizations constitutes the key factor underlying this mediation effect.
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Guillaume M, Van Rinsveld A. Comparing Numerical Comparison Tasks: A Meta-Analysis of the Variability of the Weber Fraction Relative to the Generation Algorithm. Front Psychol 2018; 9:1694. [PMID: 30271363 PMCID: PMC6142874 DOI: 10.3389/fpsyg.2018.01694] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/15/2018] [Accepted: 08/22/2018] [Indexed: 01/29/2023] Open
Abstract
Since more than 15 years, researchers have been expressing their interest in evaluating the Approximate Number System (ANS) and its potential influence on cognitive skills involving number processing, such as arithmetic. Although many studies reported significant and predictive relations between ANS and arithmetic abilities, there has recently been an increasing amount of published data that failed to replicate such relationship. Inconsistencies lead many researchers to question the validity of the assessment of the ANS itself. In the current meta-analysis of over 68 experimental studies published between 2004 and 2017, we show that the mean value of the Weber fraction (w), the minimal amount of change in magnitude to detect a difference, is very heterogeneous across the literature. Within young adults, w might range from < 10 to more than 60, which is critical for its validity for research and diagnostic purposes. We illustrate here the concern that different methods controlling for non-numerical dimensions lead to substantially variable performance. Nevertheless, studies that referred to the exact same method (e.g., Panamath) showed high consistency among them, which is reassuring. We are thus encouraging researchers only to compare what is comparable and to avoid considering the Weber fraction as an abstract parameter independent from the context. Eventually, we observed that all reported correlation coefficients between the value of w and general accuracy were very high. Such result calls into question the relevance of computing and reporting at all the Weber fraction. We are thus in disfavor of the systematic use of the Weber fraction, to discourage any temptation to compare given data to some values of w reported from different tasks and generation algorithms.
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Affiliation(s)
- Mathieu Guillaume
- Cognitive Science and Assessment Institute (COSA), University of Luxembourg, Luxembourg, Luxembourg
| | - Amandine Van Rinsveld
- Centre for Research in Cognitive Neuroscience (CRCN), Université Libre de Bruxelles, Brussels, Belgium
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20
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Park J. A neural basis for the visual sense of number and its development: A steady-state visual evoked potential study in children and adults. Dev Cogn Neurosci 2018; 30:333-343. [PMID: 28342780 PMCID: PMC6969086 DOI: 10.1016/j.dcn.2017.02.011] [Citation(s) in RCA: 22] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/22/2016] [Revised: 12/22/2016] [Accepted: 02/28/2017] [Indexed: 01/29/2023] Open
Abstract
While recent studies in adults have demonstrated the existence of a neural mechanism for a visual sense of number, little is known about its development and whether such a mechanism exists at young ages. In the current study, I introduce a novel steady-state visual evoked potential (SSVEP) technique to objectively quantify early visual cortical sensitivity to numerical and non-numerical magnitudes of a dot array. I then examine this neural sensitivity to numerical magnitude in children between three and ten years of age and in college students. Children overall exhibit strong SSVEP sensitivity to numerical magnitude in the right occipital sites with negligible SSVEP sensitivity to non-numerical magnitudes, the pattern similar to what is observed in adults. However, a closer examination of age differences reveals that this selective neural sensitivity to numerical magnitude, which is close to absent in three-year-olds, increases steadily as a function of age, while there is virtually no neural sensitivity to other non-numerical magnitudes across these ages. These results demonstrate the emergence of a neural mechanism underlying direct perception of numerosity across early and middle childhood and provide a potential neural mechanistic explanation for the development of humans' primitive, non-verbal ability to comprehend number.
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Affiliation(s)
- Joonkoo Park
- Department of Psychological and Brain Sciences, Commonwealth Honors College, University of Massachusetts, 135 Hicks Way, Amherst, MA 01003, United States.
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21
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Costa HM, Nicholson B, Donlan C, Van Herwegen J. Low performance on mathematical tasks in preschoolers: the importance of domain-general and domain-specific abilities. JOURNAL OF INTELLECTUAL DISABILITY RESEARCH : JIDR 2018; 62:292-302. [PMID: 29349826 DOI: 10.1111/jir.12465] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/19/2017] [Revised: 11/10/2017] [Accepted: 11/29/2017] [Indexed: 06/07/2023]
Abstract
BACKGROUND Different domain-specific and domain-general cognitive precursors play a key role in the development of mathematical abilities. The contribution of these domains to mathematical ability changes during development. Primary school-aged children who show mathematical difficulties form a heterogeneous group, but it is not clear whether this also holds for preschool low achievers (LAs) and how domain-specific and domain-general abilities contribute to mathematical difficulties at a young age. The aim of this study was to explore the cognitive characteristics of a sample of preschool LAs and identify sub-types of LAs. METHODS 81 children were identified as LAs from 283 preschoolers aged 3 to 5 years old and were assessed on a number of domain-general and domain-specific tasks. RESULTS Cluster analysis revealed four subgroups of LAs in mathematics: (1) a weak processing sub-type; (2) a general mathematical LAs sub-type; (3) a mixed abilities sub-type; and (4) a visuo-spatial deficit sub-type. Whilst two of the groups showed specific domain-general difficulties, none showed only domain-specific difficulties. CONCLUSIONS Current findings suggest that preschool LAs constitute a heterogeneous group and stress the importance of domain-general factors for the development of mathematical abilities during the preschool years.
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Affiliation(s)
- H M Costa
- Department of Psychology, Kingston University London, UK
- Department of Psychology, Anglia Ruskin University, UK
| | - B Nicholson
- Department of Psychology, Kingston University London, UK
| | - C Donlan
- Department of Developmental Science, University College London, UK
| | - J Van Herwegen
- Department of Psychology, Kingston University London, UK
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The contributions of numerical acuity and non-numerical stimulus features to the development of the number sense and symbolic math achievement. Cognition 2017; 168:222-233. [DOI: 10.1016/j.cognition.2017.07.004] [Citation(s) in RCA: 42] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/30/2016] [Revised: 07/06/2017] [Accepted: 07/07/2017] [Indexed: 01/29/2023]
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23
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What counts in preschool number knowledge? A Bayes factor analytic approach toward theoretical model development. J Exp Child Psychol 2017; 166:116-133. [PMID: 28888192 DOI: 10.1016/j.jecp.2017.07.016] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/29/2016] [Revised: 05/29/2017] [Accepted: 07/26/2017] [Indexed: 11/22/2022]
Abstract
Preschool children vary tremendously in their numerical knowledge, and these individual differences strongly predict later mathematics achievement. To better understand the sources of these individual differences, we measured a variety of cognitive and linguistic abilities motivated by previous literature to be important and then analyzed which combination of these variables best explained individual differences in actual number knowledge. Through various data-driven Bayesian model comparison and selection strategies on competing multiple regression models, our analyses identified five variables of unique importance to explaining individual differences in preschool children's symbolic number knowledge: knowledge of the count list, nonverbal approximate numerical ability, working memory, executive conflict processing, and knowledge of letters and words. Furthermore, our analyses revealed that knowledge of the count list, likely a proxy for explicit practice or experience with numbers, and nonverbal approximate numerical ability were much more important to explaining individual differences in number knowledge than general cognitive and language abilities. These findings suggest that children use a diverse set of number-specific, general cognitive, and language abilities to learn about symbolic numbers, but the contribution of number-specific abilities may overshadow that of more general cognitive abilities in the learning process.
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Libertus ME, Forsman L, Adén U, Hellgren K. Deficits in Approximate Number System Acuity and Mathematical Abilities in 6.5-Year-Old Children Born Extremely Preterm. Front Psychol 2017; 8:1175. [PMID: 28744252 PMCID: PMC5504250 DOI: 10.3389/fpsyg.2017.01175] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/09/2016] [Accepted: 06/27/2017] [Indexed: 01/29/2023] Open
Abstract
Preterm children are at increased risk for poor academic achievement, especially in math. In the present study, we examined whether preterm children differ from term-born children in their intuitive sense of number that relies on an unlearned, approximate number system (ANS) and whether there is a link between preterm children’s ANS acuity and their math abilities. To this end, 6.5-year-old extremely preterm (i.e., <27 weeks gestation, n = 82) and term-born children (n = 89) completed a non-symbolic number comparison (ANS acuity) task and a standardized math test. We found that extremely preterm children had significantly lower ANS acuity than term-born children and that these differences could not be fully explained by differences in verbal IQ, perceptual reasoning skills, working memory, or attention. Differences in ANS acuity persisted even when demands on visuo-spatial skills and attention were reduced in the ANS task. Finally, we found that ANS acuity and math ability are linked in extremely preterm children, similar to previous results from term-born children. These results suggest that deficits in the ANS may be at least partly responsible for the deficits in math abilities often observed in extremely preterm children.
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Affiliation(s)
- Melissa E Libertus
- Department of Psychology, Learning Research and Development Center, University of PittsburghPittsburgh, PA, United States
| | - Lea Forsman
- Department of Women's and Children's Health, Karolinska InstitutetStockholm, Sweden
| | - Ulrika Adén
- Department of Women's and Children's Health, Karolinska InstitutetStockholm, Sweden
| | - Kerstin Hellgren
- Department of Clinical Neuroscience, Karolinska InstitutetStockholm, Sweden
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Schwenk C, Sasanguie D, Kuhn JT, Kempe S, Doebler P, Holling H. (Non-)symbolic magnitude processing in children with mathematical difficulties: A meta-analysis. RESEARCH IN DEVELOPMENTAL DISABILITIES 2017; 64:152-167. [PMID: 28432933 DOI: 10.1016/j.ridd.2017.03.003] [Citation(s) in RCA: 32] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/01/2016] [Revised: 02/21/2017] [Accepted: 03/06/2017] [Indexed: 05/23/2023]
Abstract
Symbolic and non-symbolic magnitude representations, measured by digit or dot comparison tasks, are assumed to underlie the development of arithmetic skills. The comparison distance effect (CDE) has been suggested as a hallmark of the preciseness of mental magnitude representations. It implies that two magnitudes are harder to discriminate when the numerical distance between them is small, and may therefore differ in children with mathematical difficulties (MD), i.e. low mathematical achievement or dyscalculia. However, empirical findings on the CDE in children with MD are heterogeneous, and only few studies assess both symbolic and non-symbolic skills. This meta-analysis therefore integrates 44 symbolic and 48 non-symbolic response time (RT) outcomes reported in nineteen studies (N=1630 subjects, aged 6-14 years). Independent of age, children with MD show significantly longer mean RTs than typically achieving controls, particularly on symbolic (Hedges' g=0.75; 95% CI [0.51; 0.99]), but to a significantly lower extent also on non-symbolic (g=0.24; 95% CI [0.13; 0.36]) tasks. However, no group differences were found for the CDE. Extending recent work, these meta-analytical findings on children with MD corroborate the diagnostic importance of magnitude comparison speed in symbolic tasks. By contrast, the validity of CDE measures in assessing MD is questioned.
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Affiliation(s)
- Christin Schwenk
- Institute of Psychology, University of Münster, Fliednerstraße 21, 48149 Münster, Germany.
| | - Delphine Sasanguie
- Brain and Cognition, KU Leuven, Tiensestraat 102 - Box 3711, 3000 Leuven, Belgium; Faculty of Psychology and Educational Sciences@Kulak, KU Leuven Kulak, Etienne Sabbelaan 53, 8500 Kortrijk, Belgium.
| | - Jörg-Tobias Kuhn
- Institute of Psychology, University of Münster, Fliednerstraße 21, 48149 Münster, Germany.
| | - Sophia Kempe
- Institute of Psychology, University of Münster, Fliednerstraße 21, 48149 Münster, Germany.
| | - Philipp Doebler
- TU Dortmund University, Faculty of Statistics, Technische Universität Dortmund, 44221 Dortmund, Germany.
| | - Heinz Holling
- Institute of Psychology, University of Münster, Fliednerstraße 21, 48149 Münster, Germany.
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Collins E, Park J, Behrmann M. Numerosity representation is encoded in human subcortex. Proc Natl Acad Sci U S A 2017; 114:E2806-E2815. [PMID: 28320968 PMCID: PMC5389276 DOI: 10.1073/pnas.1613982114] [Citation(s) in RCA: 30] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/29/2023] Open
Abstract
Certain numerical abilities appear to be relatively ubiquitous in the animal kingdom, including the ability to recognize and differentiate relative quantities. This skill is present in human adults and children, as well as in nonhuman primates and, perhaps surprisingly, is also demonstrated by lower species such as mosquitofish and spiders, despite the absence of cortical computation available to primates. This ubiquity of numerical competence suggests that representations that connect to numerical tasks are likely subserved by evolutionarily conserved regions of the nervous system. Here, we test the hypothesis that the evaluation of relative numerical quantities is subserved by lower-order brain structures in humans. Using a monocular/dichoptic paradigm, across four experiments, we show that the discrimination of displays, consisting of both large (5-80) and small (1-4) numbers of dots, is facilitated in the monocular, subcortical portions of the visual system. This is only the case, however, when observers evaluate larger ratios of 3:1 or 4:1, but not smaller ratios, closer to 1:1. This profile of competence matches closely the skill with which newborn infants and other species can discriminate numerical quantity. These findings suggest conservation of ontogenetically and phylogenetically lower-order systems in adults' numerical abilities. The involvement of subcortical structures in representing numerical quantities provokes a reconsideration of current theories of the neural basis of numerical cognition, inasmuch as it bolsters the cross-species continuity of the biological system for numerical abilities.
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Affiliation(s)
- Elliot Collins
- Department of Psychology, Carnegie Mellon University, Pittsburgh PA 15213-3890
- Center for the Neural Basis of Cognition, Carnegie Mellon University, Pittsburgh PA 15213-3890
- School of Medicine, University of Pittsburgh, Pittsburgh, PA 15261
| | - Joonkoo Park
- Department of Psychological and Brain Sciences, University of Massachusetts, Amherst, MA 01003
| | - Marlene Behrmann
- Department of Psychology, Carnegie Mellon University, Pittsburgh PA 15213-3890;
- Center for the Neural Basis of Cognition, Carnegie Mellon University, Pittsburgh PA 15213-3890
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27
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Chew CS, Forte JD, Reeve RA. Cognitive factors affecting children's nonsymbolic and symbolic magnitude judgment abilities: A latent profile analysis. J Exp Child Psychol 2016; 152:173-191. [PMID: 27560661 DOI: 10.1016/j.jecp.2016.07.001] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/04/2016] [Revised: 07/01/2016] [Accepted: 07/01/2016] [Indexed: 10/21/2022]
Abstract
Early math abilities are claimed to be linked to magnitude representation ability. Some claim that nonsymbolic magnitude abilities scaffold the acquisition of symbolic (Arabic number) magnitude abilities and influence math ability. Others claim that symbolic magnitude abilities, and ipso facto math abilities, are independent of nonsymbolic abilities and instead depend on the ability to process number symbols (e.g., 2, 7). Currently, the issue of whether symbolic abilities are or are not related to nonsymbolic abilities, and the cognitive factors associated with nonsymbolic-symbolic relationships, remains unresolved. We suggest that different nonsymbolic-symbolic relationships reside within the general magnitude ability distribution and that different cognitive abilities are likely associated with these different relationships. We further suggest that the different nonsymbolic-symbolic relationships and cognitive abilities in combination differentially predict math abilities. To test these claims, we used latent profile analysis to identify nonsymbolic-symbolic judgment patterns of 124, 5- to 7-year-olds. We also assessed four cognitive factors (visuospatial working memory [VSWM], naming numbers, nonverbal IQ, and basic reaction time [RT]) and two math abilities (number transcoding and single-digit addition abilities). Four nonsymbolic-symbolic ability profiles were identified. Naming numbers, VSWM, and basic RT abilities were differentially associated with the different ability profiles and in combination differentially predicted math abilities. Findings show that different patterns of nonsymbolic-symbolic magnitude abilities can be identified and suggest that an adequate account of math development should specify the inter-relationship between cognitive factors and nonsymbolic-symbolic ability patterns.
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Affiliation(s)
- Cindy S Chew
- Melbourne School of Psychological Sciences, University of Melbourne, Parkville, Victoria 3010, Australia.
| | - Jason D Forte
- Melbourne School of Psychological Sciences, University of Melbourne, Parkville, Victoria 3010, Australia
| | - Robert A Reeve
- Melbourne School of Psychological Sciences, University of Melbourne, Parkville, Victoria 3010, Australia.
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28
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Merkley R, Ansari D. Why numerical symbols count in the development of mathematical skills: evidence from brain and behavior. Curr Opin Behav Sci 2016. [DOI: 10.1016/j.cobeha.2016.04.006] [Citation(s) in RCA: 64] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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29
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Acquisition of the Cardinal Principle Coincides with Improvement in Approximate Number System Acuity in Preschoolers. PLoS One 2016; 11:e0153072. [PMID: 27078257 PMCID: PMC4831828 DOI: 10.1371/journal.pone.0153072] [Citation(s) in RCA: 49] [Impact Index Per Article: 6.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/26/2014] [Accepted: 03/23/2016] [Indexed: 01/29/2023] Open
Abstract
Human mathematical abilities comprise both learned, symbolic representations of number and unlearned, non-symbolic evolutionarily primitive cognitive systems for representing quantities. However, the mechanisms by which our symbolic (verbal) number system becomes integrated with the non-symbolic (non-verbal) representations of approximate magnitude (supported by the Approximate Number System, or ANS) are not well understood. To explore this connection, forty-six children participated in a 6-month longitudinal study assessing verbal number knowledge and non-verbal numerical acuity. Cross-sectional analyses revealed a strong relationship between verbal number knowledge and ANS acuity. Longitudinal analyses suggested that increases in ANS acuity were most strongly related to the acquisition of the cardinal principle, but not to other milestones of verbal number acquisition. These findings suggest that experience with culture and language is intimately linked to changes in the properties of a core cognitive system.
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30
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How approximate and exact number skills are related to each other across development: A review☆. DEVELOPMENTAL REVIEW 2016. [DOI: 10.1016/j.dr.2014.11.001] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/29/2023]
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31
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Benavides-Varela S, Butterworth B, Burgio F, Arcara G, Lucangeli D, Semenza C. Numerical Activities and Information Learned at Home Link to the Exact Numeracy Skills in 5-6 Years-Old Children. Front Psychol 2016; 7:94. [PMID: 26903902 PMCID: PMC4750023 DOI: 10.3389/fpsyg.2016.00094] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/10/2015] [Accepted: 01/18/2016] [Indexed: 01/29/2023] Open
Abstract
It is currently accepted that certain activities within the family environment contribute to develop early numerical skills before schooling. However, it is unknown whether this early experience influences both the exact and the approximate representation of numbers, and if so, which is more important for numerical tasks. In the present study the mathematical performance of 110 children (mean age 5 years 11 months) was evaluated using a battery that included tests of approximate and exact numerical abilities, as well as everyday numerical problems. Moreover, children were assessed on their knowledge of number information learned at home. The parents of the participants provided information regarding daily activities of the children and socio-demographic characteristics of the family. The results showed that the amount of numerical information learned at home was a significant predictor of participants' performance on everyday numerical problems and exact number representations, even after taking account of age, memory span and socio-economic and educational status of the family. We also found that particular activities, such as board games, correlate with the children's counting skills, which are foundational for arithmetic. Crucially, tests relying on approximate representations were not predicted by the numerical knowledge acquired at home. The present research supports claims about the importance and nature of home experiences in the child's acquisition of mathematics.
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Affiliation(s)
- Silvia Benavides-Varela
- Department of Developmental Psychology and Socialization, University of Padova Padova, Italy
| | - Brian Butterworth
- Institute of Cognitive Neuroscience and Psychology Department, University College London London, UK
| | - Francesca Burgio
- Neuropsychology Unit, Istituto di Ricovero e Cura a Carattere Scientifico San Camillo Hospital FoundationLido-Venice, Italy; Neuroscience Department, University of PadovaPadova, Italy
| | - Giorgio Arcara
- Neuroscience Department, University of Padova Padova, Italy
| | - Daniela Lucangeli
- Department of Developmental Psychology and Socialization, University of Padova Padova, Italy
| | - Carlo Semenza
- Neuropsychology Unit, Istituto di Ricovero e Cura a Carattere Scientifico San Camillo Hospital FoundationLido-Venice, Italy; Neuroscience Department, University of PadovaPadova, Italy
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32
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Hyde D, Berteletti I, Mou Y. Approximate numerical abilities and mathematics. PROGRESS IN BRAIN RESEARCH 2016; 227:335-51. [DOI: 10.1016/bs.pbr.2016.04.011] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/29/2023]
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33
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Identifying the cognitive predictors of early counting and calculation skills: Evidence from a longitudinal study. J Exp Child Psychol 2015. [PMID: 26218332 DOI: 10.1016/j.jecp.2015.06.011] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
Abstract
The extent to which phonological, visual-spatial short-term memory (STM), and nonsymbolic quantitative skills support the development of counting and calculation skills was examined in this 14-month longitudinal study of 125 children. Initial assessments were made when the children were 4 years 8 months old. Phonological awareness, visual-spatial STM, and nonsymbolic approximate discrimination predicted growth in early calculation skills.These results suggest that both the approximate number system and domain-general phonological and visual-spatial skills support early calculation. In contrast, only performance on a small nonsymbolic quantity discrimination task (where the presented quantities were always within the subitizing range) predicted growth in cardinal counting skills. These results suggest that the development of counting and the development of calculation are supported by different cognitive abilities.
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Abstract
Many studies rely on estimation of Weber ratios (W) in order to quantify the acuity an individual's approximate number system. This paper discusses several problems encountered in estimating W using the standard methods, most notably low power and inefficiency. Through simulation, this work shows that W can best be estimated in a Bayesian framework that uses an inverse (1/W) prior. This beneficially balances a bias/variance trade-off and, when used with MAP estimation is extremely simple to implement. Use of this scheme substantially improves statistical power in examining correlates of W.
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35
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Ebersbach M, Erz P. Symbolic versus non-symbolic magnitude estimations among children and adults. J Exp Child Psychol 2014; 128:52-68. [DOI: 10.1016/j.jecp.2014.06.005] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/05/2013] [Revised: 06/11/2014] [Accepted: 06/23/2014] [Indexed: 10/25/2022]
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36
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Negen J, Sarnecka BW. Is there really a link between exact-number knowledge and approximate number system acuity in young children? BRITISH JOURNAL OF DEVELOPMENTAL PSYCHOLOGY 2014; 33:92-105. [PMID: 25403910 DOI: 10.1111/bjdp.12071] [Citation(s) in RCA: 35] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/23/2014] [Revised: 08/24/2014] [Indexed: 11/30/2022]
Abstract
Although everyone perceives approximate numerosities, some people make more accurate estimates than others. The accuracy of this estimation is called approximate number system (ANS) acuity. Recently, several studies have reported that individual differences in young children's ANS acuity are correlated with their knowledge of exact numbers such as the word 'six' (Mussolin et al., 2012, Trends Neurosci. Educ., 1, 21; Shusterman et al., 2011, Connecting early number word knowledge and approximate number system acuity; Wagner & Johnson, 2011, Cognition, 119, 10; see also Abreu-Mendoza et al., 2013, Front. Psychol., 4, 1). This study argues that this correlation should not be trusted. It seems to be an artefact of the procedure used to assess ANS acuity in children. The correlation arises because (1) some experimental designs inadvertently allow children to answer correctly based on the size (rather than the number) of dots in the display and/or (2) young children with little exact-number knowledge may not understand the phrase 'more dots' to mean numerically more. When the task is modified to make sure that children respond on the basis of numerosity, the correlation between ANS acuity and exact-number knowledge in normally developing children disappears.
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Affiliation(s)
- James Negen
- University of California, Irvine, California, USA
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37
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Fazio LK, Bailey DH, Thompson CA, Siegler RS. Relations of different types of numerical magnitude representations to each other and to mathematics achievement. J Exp Child Psychol 2014; 123:53-72. [PMID: 24699178 DOI: 10.1016/j.jecp.2014.01.013] [Citation(s) in RCA: 246] [Impact Index Per Article: 24.6] [Reference Citation Analysis] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/17/2013] [Revised: 01/20/2014] [Accepted: 01/22/2014] [Indexed: 01/29/2023]
Affiliation(s)
- Lisa K Fazio
- Department of Psychology, Carnegie Mellon University, Pittsburgh, PA 15213, USA.
| | - Drew H Bailey
- Department of Psychology, Carnegie Mellon University, Pittsburgh, PA 15213, USA
| | | | - Robert S Siegler
- Department of Psychology, Carnegie Mellon University, Pittsburgh, PA 15213, USA; Siegler Center for Innovative Learning, Beijing Normal University, Beijing 100875, China
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38
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Association between individual differences in non-symbolic number acuity and math performance: a meta-analysis. Acta Psychol (Amst) 2014; 148:163-72. [PMID: 24583622 DOI: 10.1016/j.actpsy.2014.01.016] [Citation(s) in RCA: 248] [Impact Index Per Article: 24.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/05/2013] [Revised: 01/15/2014] [Accepted: 01/28/2014] [Indexed: 01/29/2023] Open
Abstract
Many recent studies have examined the association between number acuity, which is the ability to rapidly and non-symbolically estimate the quantity of items appearing in a scene, and symbolic math performance. However, various contradictory results have been reported. To comprehensively evaluate the association between number acuity and symbolic math performance, we conduct a meta-analysis to synthesize the results observed in previous studies. First, a meta-analysis of cross-sectional studies (36 samples, N = 4705) revealed a significant positive correlation between these skills (r = 0.20, 95% CI = [0.14, 0.26]); the association remained after considering other potential moderators (e.g., whether general cognitive abilities were controlled). Moreover, a meta-analysis of longitudinal studies revealed 1) that number acuity may prospectively predict later math performance (r = 0.24, 95% CI = [0.11, 0.37]; 6 samples) and 2) that number acuity is retrospectively correlated to early math performance as well (r = 0.17, 95% CI = [0.07, 0.26]; 5 samples). In summary, these pieces of evidence demonstrate a moderate but statistically significant association between number acuity and math performance. Based on the estimated effect sizes, power analyses were conducted, which suggested that many previous studies were underpowered due to small sample sizes. This may account for the disparity between findings in the literature, at least in part. Finally, the theoretical and practical implications of our meta-analytic findings are presented, and future research questions are discussed.
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39
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Mussolin C, Nys J, Content A, Leybaert J. Symbolic number abilities predict later approximate number system acuity in preschool children. PLoS One 2014; 9:e91839. [PMID: 24637785 PMCID: PMC3956743 DOI: 10.1371/journal.pone.0091839] [Citation(s) in RCA: 60] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/27/2013] [Accepted: 02/15/2014] [Indexed: 01/29/2023] Open
Abstract
An ongoing debate in research on numerical cognition concerns the extent to which the approximate number system and symbolic number knowledge influence each other during development. The current study aims at establishing the direction of the developmental association between these two kinds of abilities at an early age. Fifty-seven children of 3-4 years performed two assessments at 7 months interval. In each assessment, children's precision in discriminating numerosities as well as their capacity to manipulate number words and Arabic digits was measured. By comparing relationships between pairs of measures across the two time points, we were able to assess the predictive direction of the link. Our data indicate that both cardinality proficiency and symbolic number knowledge predict later accuracy in numerosity comparison whereas the reverse links are not significant. The present findings are the first to provide longitudinal evidence that the early acquisition of symbolic numbers is an important precursor in the developmental refinement of the approximate number representation system.
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Affiliation(s)
- Christophe Mussolin
- Center for Research in Cognition and Neurosciences, Laboratory Cognition Language Development, Université Libre de Bruxelles, Brussels, Belgium
- * E-mail:
| | - Julie Nys
- Center for Research in Cognition and Neurosciences, Laboratory Cognition Language Development, Université Libre de Bruxelles, Brussels, Belgium
| | - Alain Content
- Center for Research in Cognition and Neurosciences, Laboratory Cognition Language Development, Université Libre de Bruxelles, Brussels, Belgium
| | - Jacqueline Leybaert
- Center for Research in Cognition and Neurosciences, Laboratory Cognition Language Development, Université Libre de Bruxelles, Brussels, Belgium
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Hoffmann D, Mussolin C, Martin R, Schiltz C. The impact of mathematical proficiency on the number-space association. PLoS One 2014; 9:e85048. [PMID: 24416338 PMCID: PMC3885673 DOI: 10.1371/journal.pone.0085048] [Citation(s) in RCA: 51] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/31/2013] [Accepted: 11/25/2013] [Indexed: 01/29/2023] Open
Abstract
A specific instance of the association between numerical and spatial representations is the SNARC (Spatial Numerical Association of Response Codes) effect. The SNARC effect describes the finding that during binary classification of numbers participants are faster to respond to small/large numbers with the left/right hand respectively. Even though it has been frequently replicated, important inter-individual variability has also been reported. Mathematical proficiency is an obvious candidate source for inter-individual variability in numerical judgments, but studies investigating its influence on the SNARC effect remain scarce. The present experiment included a total of 95 University students, divided into three groups differing significantly in their mathematical proficiency levels. Using group analyses, it appeared that the three groups differed significantly in the strength of their number-space associations in a parity judgment task. This result was further confirmed on an individual level, with higher levels in arithmetic leading to relatively weaker SNARC effects. To explain this negative relationship we propose accounts based on differences in access to qualitatively different numerical representations and also consider more domain general factors, with a focus on inhibition capacities.
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41
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Hoffmann D, Hornung C, Martin R, Schiltz C. Developing number–space associations: SNARC effects using a color discrimination task in 5-year-olds. J Exp Child Psychol 2013; 116:775-91. [DOI: 10.1016/j.jecp.2013.07.013] [Citation(s) in RCA: 67] [Impact Index Per Article: 6.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/04/2013] [Revised: 07/22/2013] [Accepted: 07/24/2013] [Indexed: 01/29/2023]
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42
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Abstract
Human infants in the first year of life possess an intuitive sense of number. This preverbal number sense may serve as a developmental building block for the uniquely human capacity for mathematics. In support of this idea, several studies have demonstrated that nonverbal number sense is correlated with mathematical abilities in children and adults. However, there has been no direct evidence that infant numerical abilities are related to mathematical abilities later in childhood. Here, we provide evidence that preverbal number sense in infancy predicts mathematical abilities in preschool-aged children. Numerical preference scores at 6 months of age correlated with both standardized math test scores and nonsymbolic number comparison scores at 3.5 years of age, suggesting that preverbal number sense facilitates the acquisition of numerical symbols and mathematical abilities. This relationship held even after controlling for general intelligence, indicating that preverbal number sense imparts a unique contribution to mathematical ability. These results validate the many prior studies purporting to show number sense in infancy and support the hypothesis that mathematics is built upon an intuitive sense of number that predates language.
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43
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Magnitude representations in Williams syndrome: differential acuity in time, space and number processing. PLoS One 2013; 8:e72621. [PMID: 24013906 PMCID: PMC3755976 DOI: 10.1371/journal.pone.0072621] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/19/2013] [Accepted: 07/11/2013] [Indexed: 01/29/2023] Open
Abstract
For some authors, the human sensitivity to numerosities would be grounded in our ability to process non-numerical magnitudes. In the present study, the developmental relationships between non numerical and numerical magnitude processing are examined in people with Williams syndrome (WS), a genetic disorder known to associate visuo-spatial and math learning disabilities. Twenty patients with WS and 40 typically developing children matched on verbal or non-verbal abilities were administered three comparison tasks in which they had to compare numerosities, lengths or durations. Participants with WS showed lower acuity (manifested by a higher Weber fraction) than their verbal matched peers when processing numerical and spatial but not temporal magnitudes, indicating that they do not present a domain-general dysfunction of all magnitude processing. Conversely, they do not differ from non-verbal matched participants in any of the three tasks. Finally, correlational analyses revealed that non-numerical and numerical acuity indexes were both related to the first mathematical acquisitions but not with later arithmetical skills.
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Sasanguie D, Defever E, Maertens B, Reynvoet B. The approximate number system is not predictive for symbolic number processing in kindergarteners. Q J Exp Psychol (Hove) 2013; 67:271-80. [PMID: 23767979 DOI: 10.1080/17470218.2013.803581] [Citation(s) in RCA: 79] [Impact Index Per Article: 7.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
Abstract
The relation between the approximate number system (ANS) and symbolic number processing skills remains unclear. Some theories assume that children acquire the numerical meaning of symbols by mapping them onto the preexisting ANS. Others suggest that in addition to the ANS, children also develop a separate, exact representational system for symbolic number processing. In the current study, we contribute to this debate by investigating whether the nonsymbolic number processing of kindergarteners is predictive for symbolic number processing. Results revealed no association between the accuracy of the kindergarteners on a nonsymbolic number comparison task and their performance on the symbolic comparison task six months later, suggesting that there are two distinct representational systems for the ANS and numerical symbols.
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