1
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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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Rees A, Carter E, Bott L. Priming scalar and ad hoc enrichment in children. Cognition 2023; 239:105572. [PMID: 37494789 DOI: 10.1016/j.cognition.2023.105572] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2022] [Revised: 03/21/2023] [Accepted: 07/20/2023] [Indexed: 07/28/2023]
Abstract
Sentences can be enriched by considering what the speaker does not say but could have done. Children, however, struggle to derive one type of such enrichments, scalar implicatures. A popular explanation for this, the lexical alternatives account, is that they do not have lexical knowledge of the appropriate alternatives to generate the implicature. Namely, children are unaware of the scalar relationship between some and all. We conducted a priming study with N = 72 children, aged 5;1 years, and an adult sample, N = 51, to test this hypothesis. Participants were exposed to prime trials of strong, alternative, or weak sentences involving scalar or ad hoc expressions, and then saw a target trial that could be interpreted in either way. Consistent with previous studies, children were reluctant to derive scalar implicatures. However, there were two novel findings. (1) Children responded with twice the rate of ad hoc implicature responses than adults, suggesting that the implicature was the developmentally prior interpretation for ad hoc expressions. (2) Children showed robust priming effects, suggesting that children are aware of the scalar relationship between some and all, even if they choose not to derive the implicature. This suggests that the root cause of the scalar implicature deficit is not due to the absence of lexical knowledge of the relationship between some and all.
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3
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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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4
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Schneider RM, Brockbank E, Feiman R, Barner D. Counting and the ontogenetic origins of exact equality. Cognition 2021; 218:104952. [PMID: 34801862 DOI: 10.1016/j.cognition.2021.104952] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/20/2021] [Revised: 10/28/2021] [Accepted: 11/03/2021] [Indexed: 01/29/2023]
Abstract
Humans are unique in their capacity to both represent number exactly and to express these representations symbolically. This correlation has prompted debate regarding whether symbolic number systems are necessary to represent large exact number. Previous work addressing this question in innumerate adults and semi-numerate children has been limited by conflicting results and differing methodologies, and has not yielded a clear answer. We address this debate by adapting methods used with innumerate populations (a "set-matching" task) for 3- to 5-year-old US children at varying stages of symbolic number acquisition. In five studies we find that children's ability to match sets exactly is related not simply to knowing the meanings of a few number words, but also to understanding how counting is used to generate sets (i.e., the cardinal principle). However, while children were more likely to match sets after acquiring the cardinal principle, they nevertheless demonstrated failures, compatible with the hypothesis that the ability to reason about exact equality emerges sometime later. These findings provide important data on the origin of exact number concepts, and point to knowledge of a counting system, rather than number language in general, as a key ingredient in the ability to reason about large exact number.
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Affiliation(s)
- Rose M Schneider
- Department of Psychology, University of California, San Diego, CA, United States of America; Department of Cognitive, Linguistic, and Psychological Sciences, Brown University, Providence, RI, United States of America.
| | - Erik Brockbank
- Department of Psychology, University of California, San Diego, CA, United States of America; Department of Cognitive, Linguistic, and Psychological Sciences, Brown University, Providence, RI, United States of America
| | - Roman Feiman
- Department of Psychology, University of California, San Diego, CA, United States of America; Department of Cognitive, Linguistic, and Psychological Sciences, Brown University, Providence, RI, United States of America
| | - David Barner
- Department of Psychology, University of California, San Diego, CA, United States of America; Department of Cognitive, Linguistic, and Psychological Sciences, Brown University, Providence, RI, United States of America
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5
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Williams K, Zax A, Patalano AL, Barth H. Left Digit Effects in Numerical Estimation across Development. JOURNAL OF COGNITION AND DEVELOPMENT 2021. [DOI: 10.1080/15248372.2021.1984243] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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6
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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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7
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Horn P, Fritzsche T, Ehlert A, Adani F. Tapping into the interplay of lexical and number knowledge using fast mapping: A longitudinal eye-tracking study with two-year-olds. Infant Behav Dev 2021; 64:101573. [PMID: 34058633 DOI: 10.1016/j.infbeh.2021.101573] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2020] [Revised: 05/07/2021] [Accepted: 05/07/2021] [Indexed: 11/18/2022]
Abstract
Language skills and mathematical competencies are argued to influence each other during development. While a relation between the development of vocabulary size and mathematical skills is already documented in the literature, this study further examines how children's ability to map a novel word to an unknown object as well as their ability to retain this word from memory may be related to their knowledge of number words. Twenty-five children were tested longitudinally (at 30 and at 36 months of age) using an eye-tracking-based fast mapping task, the Give-a-Number task, and standardized measures of vocabulary. The results reveal that children's ability to create and retain a mental representation of a novel word was related to number knowledge at 30 months, but not at 36 months while vocabulary size correlated with number knowledge only at 36 months. These results show that even specific mapping processes are initially related to the acquisition of number words and they speak for a parallelism between the development of lexical and number-concept knowledge despite their semantic and syntactic differences.
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Affiliation(s)
- Peter Horn
- Department of Primary Education, University of Potsdam, Germany.
| | - Tom Fritzsche
- Department of Linguistics, University of Potsdam, Germany
| | - Antje Ehlert
- Department of Inclusive Education, University of Potsdam, Germany
| | - Flavia Adani
- Department of Education and Psychology, Free University of Berlin, Germany
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8
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Gordon R, Scalise NR, Ramani GB. Give yourself a hand: The role of gesture and working memory in preschoolers' numerical knowledge. J Exp Child Psychol 2021; 208:105145. [PMID: 33848695 DOI: 10.1016/j.jecp.2021.105145] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/12/2020] [Revised: 02/02/2021] [Accepted: 03/02/2021] [Indexed: 10/21/2022]
Abstract
Hand gestures can be beneficial in math contexts to reduce the user's cognitive load by supporting domain-general abilities such as working memory. Although prior work has shown a strong relation between young children's early math performance and their general cognitive abilities, it is important to consider how children's working memory ability may relate to their use of spontaneous gesture as well as their math-specific abilities. The current study examined how preschool-aged children's gesture use and working memory relate to their performance on an age-appropriate math task. Head Start preschoolers (N = 81) were videotaped while completing a modified version of the Give-N task to measure their cardinality understanding. Children also completed a forward word span task and a computerized Corsi Block task to assess their working memory. The results showed that children's spontaneous gesture use and working memory were related to their performance on the cardinality task. However, children's gestures were not significantly related to working memory after controlling for age. Findings suggest that young children from low-income backgrounds use gestures during math contexts in similar ways to preschoolers from higher-income backgrounds.
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Affiliation(s)
- Raychel Gordon
- Department of Human Development and Quantitative Methodology, University of Maryland, College Park, College Park, MD 20742, USA.
| | - Nicole R Scalise
- Department of Human Development and Quantitative Methodology, University of Maryland, College Park, College Park, MD 20742, USA; Department of Cognitive Sciences, University of California, Irvine, Irvine, CA 92617, USA
| | - Geetha B Ramani
- Department of Human Development and Quantitative Methodology, University of Maryland, College Park, College Park, MD 20742, USA
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9
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Barner D. Numerical Symbols as Explanations of Human Perceptual Experience. MINNESOTA SYMPOSIA ON CHILD PSYCHOLOGY 2021. [DOI: 10.1002/9781119684527.ch7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
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10
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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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11
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Georges C, Cornu V, Schiltz C. The importance of visuospatial abilities for verbal number skills in preschool: Adding spatial language to the equation. J Exp Child Psychol 2020; 201:104971. [PMID: 32916593 DOI: 10.1016/j.jecp.2020.104971] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2019] [Revised: 07/24/2020] [Accepted: 07/29/2020] [Indexed: 10/23/2022]
Abstract
Children's verbal number skills set the foundation for mathematical development. Therefore, it is central to understand their cognitive origins. Evidence suggests that preschool children rely on visuospatial abilities when solving counting and number naming tasks despite their predominantly verbal nature. We aimed to replicate these findings when controlling for verbal abilities and sociodemographic factors. Moreover, we further characterized the relation between visuospatial abilities and verbal number skills by examining the role of spatial language. Because spatial language encompasses the verbalization of spatial thinking, it is a key candidate supporting the interplay between visuospatial and verbal processes. Regression analysis indicated that both visuospatial and verbal abilities, as assessed by spatial perception and phonological awareness, respectively, uniquely predicted verbal number skills when controlling for their respective influences, age, gender, and socioeconomic status. This confirms the spatial grounding of verbal number skills. Interestingly, adding spatial language to the model abolished the predictive effects of visuospatial and verbal abilities, whose influences were completely mediated by spatial language. Verbal number skills thus concurrently depend on specifically those visuospatial and verbal processes jointly indexed through spatial language. The knowledge of spatial terms might promote verbal number skills by advancing the understanding of the spatial relations between numerical magnitudes on the mental number line. Promoting spatial language in preschool thus might be a successful avenue for stimulating mathematical development prior to formal schooling. Moreover, measures of spatial language could become an additional promising tool to screen preschool children for potential upcoming difficulties with mathematical learning.
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Affiliation(s)
- Carrie Georges
- Institute of Cognitive Science and Assessment, Department of Behavioral and Cognitive Sciences, Faculty of Humanities, Education and Social Sciences, University of Luxembourg, L-4366 Esch-sur-Alzette, Luxembourg.
| | - Véronique Cornu
- Centre pour le développement des apprentissages Grande-Duchesse Maria Teresa (CDA), Ministère de l'Éducation nationale, L-1445 Strassen, Luxembourg
| | - Christine Schiltz
- Institute of Cognitive Science and Assessment, Department of Behavioral and Cognitive Sciences, Faculty of Humanities, Education and Social Sciences, University of Luxembourg, L-4366 Esch-sur-Alzette, Luxembourg
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12
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Haman M, Lipowska K. Moving attention along the mental number line in preschool age: Study of the operational momentum in 3- to 5-year-old children's non-symbolic arithmetic. Dev Sci 2020; 24:e13007. [PMID: 32567767 DOI: 10.1111/desc.13007] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/16/2018] [Revised: 05/03/2020] [Accepted: 06/02/2020] [Indexed: 01/29/2023]
Abstract
People tend to underestimate subtraction and overestimate addition outcomes and to associate subtraction with the left side and addition with the right side. These two phenomena are collectively labeled 'operational momentum' (OM) and thought to have their origins in the same mechanism of 'moving attention along the mental number line'. OM in arithmetic has never been tested in children at the preschool age, which is critical for numerical development. In this study, 3-5 years old were tested with non-symbolic addition and subtraction tasks. Their level of understanding of counting principles (CP) was assessed using the give-a-number task. When the second operand's cardinality was 5 or 6 (Experiment 1), the child's reaction time was shorter in addition/subtraction tasks after cuing attention appropriately to the right/left. Adding/subtracting one element (Experiment 2) revealed a more complex developmental pattern. Before acquiring CP, the children showed generalized overestimation bias. Underestimation in addition and overestimation in subtraction emerged only after mastering CP. No clear spatial-directional OM pattern was found, however, the response time to rightward/leftward cues in addition/subtraction again depended on stage of mastering CP. Although the results support the hypothesis about engagement of spatial attention in early numerical processing, they point to at least partial independence of the spatial-directional and magnitude OM. This undermines the canonical version of the number line-based hypothesis. Mapping numerical magnitudes to space may be a complex process that undergoes reorganization during the period of acquisition of symbolic representations of numbers. Some hypotheses concerning the role of spatial-numerical associations in numerical development are proposed.
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Affiliation(s)
- Maciej Haman
- Faculty of Psychology, University of Warsaw, Warszawa, Poland
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13
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Song Y, Broekhuizen ML, Dubas JS. Happy Little Benefactor: Prosocial Behaviors Promote Happiness in Young Children From Two Cultures. Front Psychol 2020; 11:1398. [PMID: 32714246 PMCID: PMC7346734 DOI: 10.3389/fpsyg.2020.01398] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2019] [Accepted: 05/25/2020] [Indexed: 11/24/2022] Open
Abstract
Evidence that young children display more happiness when sharing than receiving treats supports that humans, by nature, are prosocial. However, whether this "warm glow" is also found for other prosocial behaviors (instrumental helping and empathic helping) and/or in different cultures is still unclear. Dutch (studies 1 and 2) and Chinese (study 3) young children participated in a sharing task, followed by instrumental helping and empathic helping tasks in which they were praised (thanked) if they helped. Consistent results were found across three studies, showing that (1) participants displayed more happiness after giving than receiving treats; (2) toddlers displayed more happiness after instrumental helping than initially interacting with the experimenter; and (3) toddlers' happiness remained the same after positive social feedback (i.e., being thanked). Taken together, these results indicate that independent of culture, both sharing and instrumental helping are emotionally rewarding, supporting an evolutionary origin of these behaviors.
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Affiliation(s)
- Yue Song
- Developmental Psychology, Utrecht University, Utrecht, Netherlands
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14
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When one-two-three beats two-one-three: Tracking the acquisition of the verbal number sequence. Psychon Bull Rev 2020; 27:122-129. [PMID: 31900801 DOI: 10.3758/s13423-019-01704-8] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/29/2023]
Abstract
Learning how to count is a crucial step in cognitive development, which progressively allows for more elaborate numerical processing. The existing body of research consistently reports how children associate the verbal code with exact quantity. However, the early acquisition of this code, when the verbal numbers are encoded in long-term memory as a sequence of words, has rarely been examined. Using an incidental assessment method based on serial recall of number words presented in ordered versus non-ordered sequences (e.g., one-two-three vs. two-one-three), we tracked the progressive acquisition of the verbal number sequence in children aged 3-6 years. Results revealed evidence for verbal number sequence knowledge in the youngest children even before counting is fully mastered. Verbal numerical knowledge thus starts to be organized as a sequence in long-term memory already at the age of 3 years, and this numerical sequence knowledge is assessed in a sensitive manner by incidental rather than explicit measures of number knowledge.
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Koopman SE, Arre AM, Piantadosi ST, Cantlon JF. One-to-one correspondence without language. ROYAL SOCIETY OPEN SCIENCE 2019; 6:190495. [PMID: 31824689 PMCID: PMC6837223 DOI: 10.1098/rsos.190495] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/15/2019] [Accepted: 10/01/2019] [Indexed: 05/10/2023]
Abstract
A logical rule important in counting and representing exact number is one-to-one correspondence, the understanding that two sets are equal if each item in one set corresponds to exactly one item in the second set. The role of this rule in children's development of counting remains unclear, possibly due to individual differences in the development of language. We report that non-human primates, which do not have language, have at least a partial understanding of this principle. Baboons were given a quantity discrimination task where two caches were baited with different quantities of food. When the quantities were baited in a manner that highlighted the one-to-one relation between those quantities, baboons performed significantly better than when one-to-one correspondence cues were not provided. The implication is that one-to-one correspondence, which requires intuitions about equality and is a possible building block of counting, has a pre-linguistic origin.
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Affiliation(s)
- Sarah E. Koopman
- Brain and Cognitive Sciences, University of Rochester, 500 Wilson Boulevard, Rochester, NY, USA
| | | | - Steven T. Piantadosi
- Brain and Cognitive Sciences, University of Rochester, 500 Wilson Boulevard, Rochester, NY, USA
- Psychology, University of California, Berkeley, CA, USA
| | - Jessica F. Cantlon
- Brain and Cognitive Sciences, University of Rochester, 500 Wilson Boulevard, Rochester, NY, USA
- Psychology, Carnegie Mellon University, Pittsburgh, PA, USA
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16
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Allen K, Higgins S, Adams J. The Relationship between Visuospatial Working Memory and Mathematical Performance in School-Aged Children: a Systematic Review. EDUCATIONAL PSYCHOLOGY REVIEW 2019. [DOI: 10.1007/s10648-019-09470-8] [Citation(s) in RCA: 34] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
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17
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Contrast and entailment: Abstract logical relations constrain how 2- and 3-year-old children interpret unknown numbers. Cognition 2018; 183:192-207. [PMID: 30496910 DOI: 10.1016/j.cognition.2018.11.005] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/30/2017] [Revised: 10/31/2018] [Accepted: 11/14/2018] [Indexed: 11/20/2022]
Abstract
Do children understand how different numbers are related before they associate them with specific cardinalities? We explored how children rely on two abstract relations - contrast and entailment - to reason about the meanings of 'unknown' number words. Previous studies argue that, because children give variable amounts when asked to give an unknown number, all unknown numbers begin with an existential meaning akin to some. In Experiment 1, we tested an alternative hypothesis, that because numbers belong to a scale of contrasting alternatives, children assign them a meaning distinct from some. In the "Don't Give-a-Number task", children were shown three kinds of fruit (apples, bananas, strawberries), and asked to not give either some or a number of one kind (e.g. Give everything, but not [some/five] bananas). While children tended to give zero bananas when asked to not give some, they gave positive amounts when asked to not give numbers. This suggests that contrast - plus knowledge of a number's membership in a count list - enables children to differentiate the meanings of unknown number words from the meaning of some. Experiment 2 tested whether children's interpretation of unknown numbers is further constrained by understanding numerical entailment relations - that if someone, e.g. has three, they thereby also have two, but if they do not have three, they also do not have four. On critical trials, children saw two characters with different quantities of fish, two apart (e.g. 2 vs. 4), and were asked about the number in-between - who either has or doesn't have, e.g. three. Children picked the larger quantity for the affirmative, and the smaller for the negative prompts even when all the numbers were unknown, suggesting that they understood that, whatever three means, a larger quantity is more likely to contain that many, and a smaller quantity is more likely not to. We conclude by discussing how contrast and entailment could help children scaffold the exact meanings of unknown number words.
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18
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Nava E, Rinaldi L, Bulf H, Macchi Cassia V. Visual and proprioceptive feedback differently modulate the spatial representation of number and time in children. J Exp Child Psychol 2017; 161:161-177. [DOI: 10.1016/j.jecp.2017.04.012] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/14/2016] [Revised: 04/07/2017] [Accepted: 04/08/2017] [Indexed: 01/29/2023]
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19
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Sella F, Berteletti I, Lucangeli D, Zorzi M. Preschool children use space, rather than counting, to infer the numerical magnitude of digits: Evidence for a spatial mapping principle. Cognition 2017; 158:56-67. [DOI: 10.1016/j.cognition.2016.10.010] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/27/2015] [Revised: 10/06/2016] [Accepted: 10/17/2016] [Indexed: 11/29/2022]
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21
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Jara-Ettinger J, Piantadosi S, Spelke ES, Levy R, Gibson E. Mastery of the logic of natural numbers is not the result of mastery of counting: evidence from late counters. Dev Sci 2016; 20. [PMID: 27546566 DOI: 10.1111/desc.12459] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/17/2015] [Accepted: 04/26/2016] [Indexed: 01/29/2023]
Abstract
To master the natural number system, children must understand both the concepts that number words capture and the counting procedure by which they are applied. These two types of knowledge develop in childhood, but their connection is poorly understood. Here we explore the relationship between the mastery of counting and the mastery of exact numerical equality (one central aspect of natural number) in the Tsimane', a farming-foraging group whose children master counting at a delayed age and with higher variability than do children in industrialized societies. By taking advantage of this variation, we can better understand how counting and exact equality relate to each other, while controlling for age and education. We find that the Tsimane' come to understand exact equality at later and variable ages. This understanding correlates with their mastery of number words and counting, controlling for age and education. However, some children who have mastered counting lack an understanding of exact equality, and some children who have not mastered counting have achieved this understanding. These results suggest that understanding of counting and of natural number concepts are at least partially distinct achievements, and that both draw on inputs and resources whose distribution and availability differ across cultures.
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Affiliation(s)
- Julian Jara-Ettinger
- Department of Brain & Cognitive Sciences, Massachusetts Institute of Technology, USA
| | - Steve Piantadosi
- Department of Brain & Cognitive Sciences, University of Rochester, USA
| | | | - Roger Levy
- Department of Brain & Cognitive Sciences, Massachusetts Institute of Technology, USA
| | - Edward Gibson
- Department of Brain & Cognitive Sciences, Massachusetts Institute of Technology, USA
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22
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Moore AM, vanMarle K, Geary DC. Kindergartners' fluent processing of symbolic numerical magnitude is predicted by their cardinal knowledge and implicit understanding of arithmetic 2years earlier. J Exp Child Psychol 2016; 150:31-47. [PMID: 27236038 DOI: 10.1016/j.jecp.2016.05.003] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/06/2015] [Revised: 05/04/2016] [Accepted: 05/05/2016] [Indexed: 11/19/2022]
Abstract
Fluency in first graders' processing of the magnitudes associated with Arabic numerals, collections of objects, and mixtures of objects and numerals predicts current and future mathematics achievement. The quantitative competencies that support the development of fluent processing of magnitude, however, are not fully understood. At the beginning and end of preschool (M=3years 9months at first assessment, range=3years 3months to 4years 3months), 112 children (51 boys) completed tasks measuring numeral recognition and comparison, acuity of the approximate number system, and knowledge of counting principles, cardinality, and implicit arithmetic and also completed a magnitude processing task (number sets test) in kindergarten. Use of Bayesian and linear regression techniques revealed that two measures of preschoolers' cardinal knowledge and their competence at implicit arithmetic predicted later fluency of magnitude processing, controlling domain-general factors, preliteracy skills, and parental education. The results help to narrow the search for the early foundation of children's emerging competence with symbolic mathematics and provide direction for early interventions.
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Affiliation(s)
- Alex M Moore
- Department of Psychological Sciences, University of Missouri, Columbia, MO 65211, USA
| | - Kristy vanMarle
- Department of Psychological Sciences, University of Missouri, Columbia, MO 65211, USA
| | - David C Geary
- Department of Psychological Sciences, University of Missouri, Columbia, MO 65211, USA.
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Wagner K, Kimura K, Cheung P, Barner D. Why is number word learning hard? Evidence from bilingual learners. Cogn Psychol 2015; 83:1-21. [PMID: 26413888 DOI: 10.1016/j.cogpsych.2015.08.006] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/03/2015] [Revised: 08/11/2015] [Accepted: 08/24/2015] [Indexed: 01/29/2023]
Abstract
Young children typically take between 18 months and 2 years to learn the meanings of number words. In the present study, we investigated this developmental trajectory in bilingual preschoolers to examine the relative contributions of two factors in number word learning: (1) the construction of numerical concepts, and (2) the mapping of language specific words onto these concepts. We found that children learn the meanings of small number words (i.e., one, two, and three) independently in each language, indicating that observed delays in learning these words are attributable to difficulties in mapping words to concepts. In contrast, children generally learned to accurately count larger sets (i.e., five or greater) simultaneously in their two languages, suggesting that the difficulty in learning to count is not tied to a specific language. We also replicated previous studies that found that children learn the counting procedure before they learn its logic - i.e., that for any natural number, n, the successor of n in the count list denotes the cardinality n+1. Consistent with past studies, we found that children's knowledge of successors is first acquired incrementally. In bilinguals, we found that this knowledge exhibits item-specific transfer between languages, suggesting that the logic of the positive integers may not be stored in a language-specific format. We conclude that delays in learning the meanings of small number words are mainly due to language-specific processes of mapping words to concepts, whereas the logic and procedures of counting appear to be learned in a format that is independent of a particular language and thus transfers rapidly from one language to the other in development.
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Affiliation(s)
- Katie Wagner
- Department of Psychology, University of California, San Diego, United States.
| | - Katherine Kimura
- Department of Psychology, University of California, San Diego, United States; Department of Psychology, University of California, Berkeley, United States
| | - Pierina Cheung
- Department of Psychology, University of California, San Diego, United States; Department of Psychology, University of Waterloo, Canada
| | - David Barner
- Department of Psychology, University of California, San Diego, United States; Department of Linguistics, University of California, San Diego, United States
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24
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Stapel JC, Hunnius S, Bekkering H, Lindemann O. The development of numerosity estimation: Evidence for a linear number representation early in life. JOURNAL OF COGNITIVE PSYCHOLOGY 2015. [DOI: 10.1080/20445911.2014.995668] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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25
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Alvarado M. The utility of written numerals for preschool children when solving additive problems / La utilidad de los numerales escritos en la resolución de problemas aditivos en niños preescolares. STUDIES IN PSYCHOLOGY 2015. [DOI: 10.1080/02109395.2014.1000026] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022] Open
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26
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Piantadosi ST, Jara-Ettinger J, Gibson E. Children's learning of number words in an indigenous farming-foraging group. Dev Sci 2014; 17:553-63. [PMID: 24766463 DOI: 10.1111/desc.12078] [Citation(s) in RCA: 33] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2012] [Accepted: 02/26/2013] [Indexed: 01/29/2023]
Abstract
We show that children in the Tsimane', a farming-foraging group in the Bolivian rain-forest, learn number words along a similar developmental trajectory to children from industrialized countries. Tsimane' children successively acquire the first three or four number words before fully learning how counting works. However, their learning is substantially delayed relative to children from the United States, Russia, and Japan. The presence of a similar developmental trajectory likely indicates that the incremental stages of numerical knowledge - but not their timing - reflect a fundamental property of number concept acquisition which is relatively independent of language, culture, age, and early education.
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Izard V, Streri A, Spelke ES. Toward exact number: young children use one-to-one correspondence to measure set identity but not numerical equality. Cogn Psychol 2014; 72:27-53. [PMID: 24680885 DOI: 10.1016/j.cogpsych.2014.01.004] [Citation(s) in RCA: 37] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/17/2012] [Revised: 01/23/2014] [Accepted: 01/30/2014] [Indexed: 01/29/2023]
Abstract
Exact integer concepts are fundamental to a wide array of human activities, but their origins are obscure. Some have proposed that children are endowed with a system of natural number concepts, whereas others have argued that children construct these concepts by mastering verbal counting or other numeric symbols. This debate remains unresolved, because it is difficult to test children's mastery of the logic of integer concepts without using symbols to enumerate large sets, and the symbols themselves could be a source of difficulty for children. Here, we introduce a new method, focusing on large quantities and avoiding the use of words or other symbols for numbers, to study children's understanding of an essential property underlying integer concepts: the relation of exact numerical equality. Children aged 32-36 months, who possessed no symbols for exact numbers beyond 4, were given one-to-one correspondence cues to help them track a set of puppets, and their enumeration of the set was assessed by a non-verbal manual search task. Children used one-to-one correspondence relations to reconstruct exact quantities in sets of 5 or 6 objects, as long as the elements forming the sets remained the same individuals. In contrast, they failed to track exact quantities when one element was added, removed, or substituted for another. These results suggest an alternative to both nativist and symbol-based constructivist theories of the development of natural number concepts: Before learning symbols for exact numbers, children have a partial understanding of the properties of exact numbers.
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Affiliation(s)
- Véronique Izard
- Laboratoire Psychologie de la Perception, Université Paris Descartes, Sorbonne Paris Cité, 45 rue des Saints-Pères, 75006 Paris, France; CNRS UMR 8158, 45 rue des Saints-Pères, 75006 Paris, France; Department of Psychology, Harvard University, 33 Kirkland St., Cambridge, MA 02138, USA.
| | - Arlette Streri
- Laboratoire Psychologie de la Perception, Université Paris Descartes, Sorbonne Paris Cité, 45 rue des Saints-Pères, 75006 Paris, France; CNRS UMR 8158, 45 rue des Saints-Pères, 75006 Paris, France
| | - Elizabeth S Spelke
- Department of Psychology, Harvard University, 33 Kirkland St., Cambridge, MA 02138, USA
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Le Corre M. Children acquire the later-greater principle after the cardinal principle. BRITISH JOURNAL OF DEVELOPMENTAL PSYCHOLOGY 2013; 32:163-77. [PMID: 24372336 DOI: 10.1111/bjdp.12029] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/02/2013] [Revised: 11/20/2013] [Indexed: 12/01/2022]
Abstract
Many have proposed that the acquisition of the cardinal principle (CP) is a result of the discovery of the numerical significance of the order of the number words in the count list. However, this need not be the case. Indeed, the CP does not state anything about the numerical significance of the order of the number words. It only states that the last word of a correct count denotes the numerosity of the counted set. Here, we test whether the acquisition of the CP involves the discovery of the later-greater principle - that is, that the order of the number words corresponds to the relative size of the numerosities they denote. Specifically, we tested knowledge of verbal numerical comparisons (e.g., Is 'ten' more than 'six'?) in children who had recently learned the CP. We find that these children can compare number words between 'six' and 'ten' only if they have mapped them onto non-verbal representations of numerosity. We suggest that this means that the acquisition of the CP does not involve the discovery of the correspondence between the order of the number words and the relative size of the numerosities they denote.
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Affiliation(s)
- Mathieu Le Corre
- Centro de Investigación Transdisciplinar en Psicología, Universidad Autonoma del Estado de Morelos, Cuernavaca, México
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29
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Mix KS, Prather RW, Smith LB, Stockton JD. Young children's interpretation of multidigit number names: from emerging competence to mastery. Child Dev 2013; 85:1306-1319. [PMID: 24354885 DOI: 10.1111/cdev.12197] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
This study assessed whether a sample of two hundred seven 3- to 7-year-olds could interpret multidigit numerals using simple identification and comparison tasks. Contrary to the view that young children do not understand place value, even 3-year-olds demonstrated some competence on these tasks. Ceiling was reached by first grade. When training was provided, there were significant gains, suggesting that children can improve their partial understandings with input. Findings add to what is known about the processes of symbolic development and the incidental learning that occurs prior to schooling, as well as specifying more precisely what place value misconceptions remain as children enter the educational system.
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30
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Connecting numbers to discrete quantification: a step in the child's construction of integer concepts. Cognition 2013; 129:31-41. [PMID: 23831562 DOI: 10.1016/j.cognition.2013.05.011] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/15/2012] [Revised: 03/01/2013] [Accepted: 05/18/2013] [Indexed: 11/20/2022]
Abstract
The present study asks when young children understand that number words quantify over sets of discrete individuals. For this study, 2- to 4-year-old children were asked to extend the number word five or six either to a cup containing discrete objects (e.g., blocks) or to a cup containing a continuous substance (e.g., water). In Experiment 1, only children who knew the exact meanings of the words one, two and three extended higher number words (five or six) to sets of discrete objects. In Experiment 2, children who only knew the exact meaning of one extended higher number words to discrete objects under the right conditions (i.e., when the problem was first presented with the number words one and two). These results show that children have some understanding that number words pertain to discrete quantification from very early on, but that this knowledge becomes more robust as children learn the exact, cardinal meanings of individual number words.
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31
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Sarnecka BW, Wright CE. The idea of an exact number: children's understanding of cardinality and equinumerosity. Cogn Sci 2013; 37:1493-506. [PMID: 23672476 DOI: 10.1111/cogs.12043] [Citation(s) in RCA: 56] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/03/2010] [Revised: 10/16/2012] [Accepted: 10/17/2012] [Indexed: 01/29/2023]
Abstract
Understanding what numbers are means knowing several things. It means knowing how counting relates to numbers (called the cardinal principle or cardinality); it means knowing that each number is generated by adding one to the previous number (called the successor function or succession), and it means knowing that all and only sets whose members can be placed in one-to-one correspondence have the same number of items (called exact equality or equinumerosity). A previous study (Sarnecka & Carey, 2008) linked children's understanding of cardinality to their understanding of succession for the numbers five and six. This study investigates the link between cardinality and equinumerosity for these numbers, finding that children either understand both cardinality and equinumerosity or they understand neither. This suggests that cardinality and equinumerosity (along with succession) are interrelated facets of the concepts five and six, the acquisition of which is an important conceptual achievement of early childhood.
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Syrett K. The Role of Cardinality in the Interpretation of Measurement Expressions. LANGUAGE ACQUISITION 2013; 20:228-240. [PMID: 23888105 PMCID: PMC3718074 DOI: 10.1080/10489223.2013.796949] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
The purpose of this brief article is to investigate four-year-olds' interpretation of attributive measure phrases (MPs), such as 3-pound, and the role of cardinality in mediating children's responses. In two experiments, I demonstrate that children at this age are starting to recognize that such MPs refer to a property of an individual, such as weight per unit (rather than the weight of an entire collection). Accordingly, they distinguish between attributive and pseudopartitive MPs. However, when the opportunity presents itself to treat the number word as referring to the cardinality of a set, some children succumb to this pressure, deviating from adult-like responses. I argue that the fundamental aspect of number word meaning that children take the first few years of life to master - that number words denote exact cardinality of a set of discrete objects - is precisely the aspect they must overcome when interpreting these MPs. However, the evidence shows that four-year-olds are well on their way to doing so.
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34
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Sarnecka BW, Negen J. A number of options: rationalist, constructivist, and Bayesian insights into the development of exact-number concepts. ADVANCES IN CHILD DEVELOPMENT AND BEHAVIOR 2012. [PMID: 23205414 DOI: 10.1016/b978-0-12-397919-3.00009-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/29/2023]
Abstract
The question of how human beings acquire exact-number concepts has interested cognitive developmentalists since the time of Piaget. The answer will owe something to both the rationalist and constructivist traditions. On the one hand, some aspects of numerical cognition (e.g. approximate number estimation and the ability to track small sets of one to four individuals) are innate or early-developing and are shared widely among species. On the other hand, only humans create representations of exact, large numbers such as 42, as distinct from both 41 and 43. These representations seem to be constructed slowly, over a period of months or years during early childhood. The task for researchers is to distinguish the innate representational resources from those that are constructed, and to characterize the construction process. Bayesian approaches can be useful to this project in at least three ways: (1) As a way to analyze data, which may have distinct advantages over more traditional methods (e.g. making it possible to find support for a nuli hypothesis); (2) as a way of modeling children's performance on specific tasks: Peculiarities of the task are captured as a prior; the child's knowledge is captured in the way the prior is updated; and behavior is captured as a posterior distribution; and (3) as a way of modeling learning itself, by providing a formal account of how learners might choose among alternative hypotheses.
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Affiliation(s)
- Barbara W Sarnecka
- Department of Cognitive Sciences, University of California, Irvine, CA 92697-5100, USA.
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35
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Abstract
Number-knower levels are a series of stages of number concept development in early childhood. A child's number-knower level is typically assessed using the give-N task. Although the task procedure has been highly refined, the standard ways of analyzing give-N data remain somewhat crude. Lee and Sarnecka (Cogn Sci 34:51-67, 2010, in press) have developed a Bayesian model of children's performance on the give-N task that allows knower level to be inferred in a more principled way. However, this model requires considerable expertise and computational effort to implement and apply to data. Here, we present an approximation to the model's inference that can be computed with Microsoft Excel. We demonstrate the accuracy of the approximation and provide instructions for its use. This makes the powerful inferential capabilities of the Bayesian model accessible to developmental researchers interested in estimating knower levels from give-N data.
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36
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Piantadosi ST, Tenenbaum JB, Goodman ND. Bootstrapping in a language of thought: a formal model of numerical concept learning. Cognition 2012; 123:199-217. [PMID: 22284806 DOI: 10.1016/j.cognition.2011.11.005] [Citation(s) in RCA: 69] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/31/2010] [Revised: 11/02/2011] [Accepted: 11/11/2011] [Indexed: 01/29/2023]
Abstract
In acquiring number words, children exhibit a qualitative leap in which they transition from understanding a few number words, to possessing a rich system of interrelated numerical concepts. We present a computational framework for understanding this inductive leap as the consequence of statistical inference over a sufficiently powerful representational system. We provide an implemented model that is powerful enough to learn number word meanings and other related conceptual systems from naturalistic data. The model shows that bootstrapping can be made computationally and philosophically well-founded as a theory of number learning. Our approach demonstrates how learners may combine core cognitive operations to build sophisticated representations during the course of development, and how this process explains observed developmental patterns in number word learning.
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Affiliation(s)
- Steven T Piantadosi
- Department of Brain and Cognitive Sciences, University of Rochester, United States.
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37
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Davidson K, Eng K, Barner D. Does learning to count involve a semantic induction? Cognition 2012; 123:162-73. [PMID: 22245033 DOI: 10.1016/j.cognition.2011.12.013] [Citation(s) in RCA: 76] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/02/2010] [Revised: 12/21/2011] [Accepted: 12/23/2011] [Indexed: 11/25/2022]
Abstract
We tested the hypothesis that, when children learn to correctly count sets, they make a semantic induction about the meanings of their number words. We tested the logical understanding of number words in 84 children that were classified as "cardinal-principle knowers" by the criteria set forth by Wynn (1992). Results show that these children often do not know (1) which of two numbers in their count list denotes a greater quantity, and (2) that the difference between successive numbers in their count list is 1. Among counters, these abilities are predicted by the highest number to which they can count and their ability to estimate set sizes. Also, children's knowledge of the principles appears to be initially item-specific rather than general to all number words, and is most robust for very small numbers (e.g., 5) compared to larger numbers (e.g., 25), even among children who can count much higher (e.g., above 30). In light of these findings, we conclude that there is little evidence to support the hypothesis that becoming a cardinal-principle knower involves a semantic induction over all items in a child's count list.
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Affiliation(s)
- Kathryn Davidson
- Department of Linguistics, University of California, San Diego, CA 92093, United States.
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38
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Tardif T, Gelman SA, Fu X, Zhu L. Acquisition of generic noun phrases in Chinese: learning about lions without an '-s'. JOURNAL OF CHILD LANGUAGE 2012; 39:130-161. [PMID: 21849102 PMCID: PMC3592979 DOI: 10.1017/s0305000910000735] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
English-speaking children understand and produce generic expressions in the preschool years, but there are cross-linguistic differences in how generics are expressed. Three studies examined interpretation of generic noun phrases in three- to seven-year-old child (N=192) and adult speakers (N=163) of Mandarin Chinese. Contrary to suggestions by Bloom (1981), Chinese-speaking adults honor a clear distinction between generics (expressed as bare NPs) and other quantified expressions ('all'/suo3you3 and 'some'/you3de). Furthermore, Mandarin-speaking children begin to distinguish generics from 'all' or 'some' as early as five years, as shown in both confirmation (Study 2) and property-generation (Study 3) tasks. Nonetheless, the developmental trajectory for Chinese appears prolonged relative to English and this seems to reflect difficulty with 'all' and 'some' rather than difficulty with generics. Altogether these results suggest that generics are primary, and that the consistency of markings affects the rate at which non-generic NPs are distinguished from generics.
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Affiliation(s)
- Twila Tardif
- Department of Psychology, University of Michigan, Ann Arbor MI 48109-1043, USA.
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39
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40
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Slusser EB, Sarnecka BW. Find the picture of eight turtles: a link between children's counting and their knowledge of number word semantics. J Exp Child Psychol 2011; 110:38-51. [PMID: 21524422 PMCID: PMC3105118 DOI: 10.1016/j.jecp.2011.03.006] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/17/2010] [Revised: 02/21/2011] [Accepted: 03/11/2011] [Indexed: 10/18/2022]
Abstract
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.
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Affiliation(s)
- Emily B Slusser
- Department of Psychology, Wesleyan University, Middletown, CT 06457, USA.
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41
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Nicoladis E, Pika S, Marentette P. Are number gestures easier than number words for preschoolers? COGNITIVE DEVELOPMENT 2010. [DOI: 10.1016/j.cogdev.2010.04.001] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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42
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Abstract
We develop and evaluate a model of behavior on the Give-N task, a commonly-used measure of young children's number knowledge. Our model uses the knower-level theory of how children represent numbers. To produce behavior on the Give-N task, the model assumes children start out with a base-rate that make some answers more likely a priori than others, but is updated on each experimental trial in a way that depends on the interaction between the experimenter's request and the child's knower-level. We formalize this process as a generative graphical model, so that the parameters-including the base-rate distribution and each child's knower-level-can be inferred from data using Bayesian methods. Using this approach, we evaluate the model on previously published data from 82 children spanning the whole developmental range. The model provides an excellent fit to these data, and the inferences about the base-rate and knower-levels are interpretable and insightful. We discuss how our modeling approach can be extended to other developmental tasks, and can be used to help evaluate alternative theories of number representation against the knower-level theory.
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Affiliation(s)
- Michael D Lee
- Department of Cognitive Sciences, University of California, Irvine
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43
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Sarnecka BW, Lee MD. Levels of number knowledge during early childhood. J Exp Child Psychol 2009; 103:325-37. [PMID: 19345956 PMCID: PMC3127737 DOI: 10.1016/j.jecp.2009.02.007] [Citation(s) in RCA: 75] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/06/2008] [Revised: 02/06/2009] [Accepted: 02/27/2009] [Indexed: 11/20/2022]
Abstract
Researchers have long disagreed about whether number concepts are essentially continuous (unchanging) or discontinuous over development. Among those who take the discontinuity position, there is disagreement about how development proceeds. The current study addressed these questions with new quantitative analyses of children's incorrect responses on the Give-N task. Using data from 280 children, ages 2 to 4 years, this study showed that most wrong answers were simply guesses, not counting or estimation errors. Their mean was unrelated to the target number, and they were lower-bounded by the numbers children actually knew. In addition, children learned the number-word meanings one at a time and in order; they treated the number words as mutually exclusive; and once they figured out the cardinal principle of counting, they generalized this principle to the rest of their count list. Findings support the 'discontinuity' account of number development in general and the 'knower-levels' account in particular.
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Affiliation(s)
- Barbara W Sarnecka
- Department of Cognitive Sciences, University of California, Irvine, CA 92697, USA.
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44
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Barth H, Starr A, Sullivan J. Children's mappings of large number words to numerosities. COGNITIVE DEVELOPMENT 2009. [DOI: 10.1016/j.cogdev.2009.04.001] [Citation(s) in RCA: 45] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/29/2023]
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45
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Abstract
Mathematicians frequently evoke their "intuition" when they are able to quickly and automatically solve a problem, with little introspection into their insight. Cognitive neuroscience research shows that mathematical intuition is a valid concept that can be studied in the laboratory in reduced paradigms, and that relates to the availability of "core knowledge" associated with evolutionarily ancient and specialized cerebral subsystems. As an illustration, I discuss the case of elementary arithmetic. Intuitions of numbers and their elementary transformations by addition and subtraction are present in all human cultures. They relate to a brain system, located in the intraparietal sulcus of both hemispheres, which extracts numerosity of sets and, in educated adults, maps back and forth between numerical symbols and the corresponding quantities. This system is available to animal species and to preverbal human infants. Its neuronal organization is increasingly being uncovered, leading to a precise mathematical theory of how we perform tasks of number comparison or number naming. The next challenge will be to understand how education changes our core intuitions of number.
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Affiliation(s)
- Stanislas Dehaene
- INSERM, Cognitive Neuro-imaging Unit, IFR 49, Gif sur Yvette, France.
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46
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Musolino J. The logical syntax of number words: Theory, acquisition and processing. Cognition 2009; 111:24-45. [PMID: 19217616 DOI: 10.1016/j.cognition.2008.12.008] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/22/2005] [Revised: 12/17/2008] [Accepted: 12/19/2008] [Indexed: 11/20/2022]
Affiliation(s)
- Julien Musolino
- Department of Psychology and Center for Cognitive Science, Rutgers University, 152 Frelinghuysen Road, Piscataway, NJ 08854-8020, USA.
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47
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Abstract
Cardinality principle refers to the fact that the last number tag used in counting determines the cardinality of a set. Macedonian kindergarten children were tested with the give-a-number task for understanding of this principle. It was found that Macedonian children, unlike their western counterparts, pass through an additional stage, 5-knowers, before they master the cardinality principle. Also, the age at which they pass through the individual stages is somewhat higher than the age of children coming from western samples. Possible explanations are offered and discussed.
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48
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Abstract
AbstractRips et al.'s critique is misplaced when it faults the induction model for not explaining the acquisition of meta-numerical knowledge: This is something the model was never meant to explain. More importantly, the critique underestimates what children know, and what they have achieved, when they learn the cardinal meanings of the number words “one” through “nine.”
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49
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Canobi KH, Bethune NE. Number words in young children’s conceptual and procedural knowledge of addition, subtraction and inversion. Cognition 2008; 108:675-86. [DOI: 10.1016/j.cognition.2008.05.011] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/14/2007] [Revised: 05/07/2008] [Accepted: 05/19/2008] [Indexed: 10/21/2022]
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50
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Izard V, Pica P, Spelke E, Dehaene S. Exact Equality and Successor Function: Two Key Concepts on the Path towards understanding Exact Numbers. PHILOSOPHICAL PSYCHOLOGY 2008; 21:491. [PMID: 20165569 DOI: 10.1080/09515080802285354] [Citation(s) in RCA: 40] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 01/29/2023]
Abstract
Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of exact numbers: the fact that all numbers can be generated by a successor function, and the fact that equality between numbers can be defined in an exact fashion. We discuss some recent findings assessing how speakers of Mundurucu (an Amazonian language), and young western children (3-4 years old) understand these fundamental properties of numbers.
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Affiliation(s)
- Véronique Izard
- Department of Psychology, Harvard University, Cambridge MA02138, USA
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