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Schiller LK, Abreu-Mendoza RA, Thompson CA, Rosenberg-Lee M. Children's estimates of equivalent rational number magnitudes are not equal: Evidence from fractions, decimals, percentages, and whole numbers. J Exp Child Psychol 2024; 247:106030. [PMID: 39167859 DOI: 10.1016/j.jecp.2024.106030] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/02/2024] [Revised: 06/25/2024] [Accepted: 06/25/2024] [Indexed: 08/23/2024]
Abstract
Integration of rational number knowledge with prior whole number knowledge has been theorized as critical for mathematical success. Fractions, decimals, and percentages are generally assumed to differ in difficulty based on the degree to which their structure is perceptually similar to whole numbers. Specifically, percentages are viewed as most similar to whole numbers with their fixed unstated denominator of 100. Decimals are often assumed to be easier than fractions because their place-value structure is an extension of the base-ten system for whole numbers, unlike fractions, which have a bipartite structure (i.e., a/b). However, there has been no comprehensive investigation of how fraction, decimal, and percentage knowledge compares with whole number knowledge. To assess understanding of the four notations, we measured within-participants number line estimation of equivalent fractions and decimals with shorter string lengths (e.g., 8/10 and 0.8) and longer string lengths (e.g., 80/100 and 0.80), percentages (e.g., 80%), and proportionally equivalent whole numbers on a 0-100 scale (e.g., 80.0). Middle school students (N = 65; 33 female) generally underestimated all formats relative to their actual values (whole numbers: 3% below; percentages: 2%; decimals: 17%; fractions: 5%). Shorter string-length decimals and fractions were estimated as smaller than equivalent longer string-length equivalents. Overall, percentages were estimated similarly to corresponding whole numbers, fractions had modest string-length effects, and decimals were the most underestimated, especially for single-digit decimals. These results highlight the strengths and weaknesses of children's understanding of each notation's magnitudes and challenge the assumption that decimals are easier than fractions.
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Affiliation(s)
- Lauren K Schiller
- Department of Psychology, Rutgers University, Newark, Newark, NJ 07102, USA; Teachers College, Columbia University, New York, NY 10027, USA; Department of Psychological Sciences, Kent State University, Kent, OH 44242, USA.
| | | | - Clarissa A Thompson
- Department of Psychological Sciences, Kent State University, Kent, OH 44242, USA
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Reinert RM, Gashaj V, Hartmann M, Moeller K. A potential dissociation between perception and production version for bounded but not unbounded number line estimation. Trends Neurosci Educ 2023; 31:100202. [PMID: 37308259 DOI: 10.1016/j.tine.2023.100202] [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: 11/16/2022] [Revised: 04/05/2023] [Accepted: 04/12/2023] [Indexed: 06/14/2023]
Abstract
BACKGROUND What, exactly, do number line estimation (NLE) tasks measure? Different versions of the task were observed to have different effects on performance. METHOD We investigated associations between the production (indicating the location) and perception version (indicating the number) of the bounded and unbounded NLE task and their relationship to arithmetic. RESULTS A stronger correlation was observed between the production and perception version of the unbounded than the bounded NLE task, indicating that both versions of the unbounded-but not the bounded-NLE task measure the same construct. Moreover, overall low but significant associations between NLE performance and arithmetic were only observed for the production version of the bounded NLE task. CONCLUSION These results substantiate that the production version of bounded NLE seems to rely on proportion judgment strategies, whereas both unbounded versions and the perception version of the bounded NLE task may rely more on magnitude estimation.
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Affiliation(s)
| | - Venera Gashaj
- University of Tübingen, Tübingen, Germany; Centre for Mathematical Cognition, School of Science, Loughborough University, United Kingdom.
| | - Matthias Hartmann
- Department of Psychology, University of Bern, Bern, Switzerland; Faculty of Psychology, UniDistance Suisse, Brig, Switzerland
| | - Korbinian Moeller
- Centre for Mathematical Cognition, School of Science, Loughborough University, United Kingdom; Leibniz-Institut für Wissensmedien, Tübingen, Germany; LEAD Graduate School and Research Network, University of Tübingen, Germany; Individual Development and Adaptive Education Center for Children at Risk, Frankfurt am Main, Germany
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Haman M, Patro K. More linear than log? Non-symbolic number-line estimation in 3- to 5-year-old children. Front Psychol 2022; 13:1003696. [PMID: 36389566 PMCID: PMC9659870 DOI: 10.3389/fpsyg.2022.1003696] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/26/2022] [Accepted: 10/03/2022] [Indexed: 11/04/2022] Open
Abstract
The number-line estimation task has become one of the most important methods in numerical cognition research. Originally applied as a direct measure of spatial number representation, it became also informative regarding various other aspects of number processing and associated strategies. However, most of this work and associated conclusions concerns processing numbers in a symbolic format, by school children and older subjects. Symbolic number system is formally taught and trained at school, and its basic mathematical properties (e.g., equidistance, ordinality) can easily be transferred into a spatial format of an oriented number line. This triggers the question on basic characteristics of number line estimation before children get fully familiar with the symbolic number system, i.e., when they mostly rely on approximate system for non-symbolic quantities. In our three studies, we examine therefore how preschool children (3–5-years old) estimate position of non-symbolic quantities on a line, and how this estimation is related to the developing symbolic number knowledge and cultural (left-to-right) directionality. The children were tested with the Give-a-number task, then they performed a computerized number-line task. In Experiment 1, lines bounded with sets of 1 and 20 elements going left-to-right or right-to-left were used. Even in the least numerically competent group, the linear model better fit the estimates than the logarithmic or cyclic power models. The line direction was irrelevant. In Experiment 2, a 1–9 left-to-right oriented line was used. Advantage of linear model was found at group level, and variance of estimates correlated with tested numerosities. In Experiment 3, a position-to-number procedure again revealed the advantage of the linear model, although the strategy of selecting an option more similar to the closer end of the line was prevalent. The precision of estimation increased with the mastery of counting principles in all three experiments. These results contradict the hypothesis of the log-to-linear shift in development of basic numerical representation, rather supporting the linear model with scalar variance. However, the important question remains whether the number-line task captures the nature of the basic numerical representation, or rather the strategies of mapping that representation to an external space.
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Affiliation(s)
- Maciej Haman
- Faculty of Psychology, University of Warsaw, Warsaw, Poland
- *Correspondence: Maciej Haman,
| | - Katarzyna Patro
- Faculty of Psychology, University of Warsaw, Warsaw, Poland
- Department of Psychology, University of Tuebingen, Tübingen, Germany
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Thompson CA, Fitzsimmons CJ, Mielicki M, Scheibe DA, Sidney PG, Schiller L, Siegler RS, Opfer JE. Comments regarding Numerical Estimation Strategies Are Correlated with Math Ability in School-Age Children. COGNITIVE DEVELOPMENT 2022. [DOI: 10.1016/j.cogdev.2022.101188] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022]
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5
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Numerical estimation strategies are correlated with math ability in school-aged children. COGNITIVE DEVELOPMENT 2021. [DOI: 10.1016/j.cogdev.2021.101089] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/29/2023]
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Silva FJ, Silva PN, Silva KM. Judging Numbers: Global and Local Contextual Effects in Individual and Group Data. PSYCHOLOGICAL RECORD 2021. [DOI: 10.1007/s40732-021-00467-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/29/2023]
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Williams K, Paul J, Zax A, Barth H, Patalano AL. Number line estimation and standardized test performance: The left digit effect does not predict SAT math score. Brain Behav 2020; 10:e01877. [PMID: 33073518 PMCID: PMC7749540 DOI: 10.1002/brb3.1877] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 05/25/2020] [Revised: 08/20/2020] [Accepted: 09/19/2020] [Indexed: 11/09/2022] Open
Abstract
INTRODUCTION Recent work reveals a new source of error in number line estimation (NLE), the left digit effect (Lai, Zax, et al., 2018), whereby numerals with different leftmost digits but similar magnitudes (e.g., 399, 401) are placed farther apart on a number line (e.g., 0 to 1,000) than is warranted. The goals of the present study were to: (1) replicate the left digit effect, and (2) assess whether it is related to mathematical achievement. METHOD Participants were all individuals (adult college students) who completed the NLE task in the laboratory between 2014 and 2019 for whom SAT scores were available (n = 227). RESULTS We replicated the left digit effect but found its size was not correlated with SAT math score, although it was negatively correlated with SAT verbal score for one NLE task version. CONCLUSIONS These findings provide further evidence that individual digits strongly influence estimation performance and suggest that this effect may have different cognitive contributors, and predict different complex skills, than overall NLE accuracy.
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Affiliation(s)
| | - Joanna Paul
- Department of PsychologyWesleyan UniversityMiddletownCTUSA
| | - Alexandra Zax
- Department of PsychologyWesleyan UniversityMiddletownCTUSA
| | - Hilary Barth
- Department of PsychologyWesleyan UniversityMiddletownCTUSA
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8
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Liang Y, Zhang L, Wang C, Liu Y. Performance patterns and strategy use in number line estimation among preschool children with different spontaneous focusing on numerosity tendencies. INFANT AND CHILD DEVELOPMENT 2020. [DOI: 10.1002/icd.2203] [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]
Affiliation(s)
- Yuan Liang
- School of Psychology Shaanxi Normal University Xi'an China
| | - Lijin Zhang
- School of Psychology Shaanxi Normal University Xi'an China
- Shaanxi Key Laboratory of Behavior and Cognitive Neuroscience Xi'an China
- Shaanxi Key Research Center of Child Mental and Behavioral Health Xi'an China
| | - Chunling Wang
- School of Psychology Shaanxi Normal University Xi'an China
| | - Yujuan Liu
- Research Center for Moral Education, Psychology and Special Education National Institute of Education Sciences Beijing China
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9
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Spontaneous partitioning and proportion estimation in children’s numerical judgments. J Exp Child Psychol 2019; 185:71-94. [DOI: 10.1016/j.jecp.2019.04.004] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/30/2018] [Revised: 04/02/2019] [Accepted: 04/03/2019] [Indexed: 01/29/2023]
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10
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Aulet LS, Lourenco SF. The Developing Mental Number Line: Does Its Directionality Relate to 5- to 7-Year-Old Children's Mathematical Abilities? Front Psychol 2018; 9:1142. [PMID: 30034355 PMCID: PMC6043688 DOI: 10.3389/fpsyg.2018.01142] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/29/2018] [Accepted: 06/14/2018] [Indexed: 01/29/2023] Open
Abstract
Spatial representations of number, such as a left-to-right oriented mental number line, are well documented in Western culture. Yet, the functional significance of such a representation remains unclear. To test the prominent hypothesis that a mental number line may support mathematical development, we examined the relation between spatial-numerical associations (SNAs) and math proficiency in 5- to 7-year-old children. We found evidence of SNAs with two tasks: a non-symbolic magnitude comparison task, and a symbolic "Where was the number?" (WTN) task. Further, we found a significant correlation between these two tasks, demonstrating convergent validity of the directional mental number line across numerical format. Although there were no significant correlations between children's SNAs on the WTN task and math ability, children's SNAs on the magnitude comparison task were negatively correlated with their performance on a measure of cross-modal arithmetic, suggesting that children with a stronger left-to-right oriented mental number line were less competent at cross-modal arithmetic, an effect that held when controlling for age and a set of general cognitive abilities. Despite some evidence for a negative relation between SNAs and math ability in adulthood, we argue that the effect here may reflect task demands specific to the magnitude comparison task, not necessarily an impediment of the mental number line to math performance. We conclude with a discussion of the different properties that characterize a mental number line and how these different properties may relate to mathematical ability.
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Affiliation(s)
- Lauren S. Aulet
- Department of Psychology, Emory University, Atlanta, GA, United States
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11
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Slusser E, Barth H. Intuitive proportion judgment in number-line estimation: Converging evidence from multiple tasks. J Exp Child Psychol 2017; 162:181-198. [PMID: 28605698 DOI: 10.1016/j.jecp.2017.04.010] [Citation(s) in RCA: 37] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/20/2016] [Revised: 01/30/2017] [Accepted: 04/16/2017] [Indexed: 11/27/2022]
Abstract
How children's understanding of numerical magnitudes changes over the course of development remains a key question in the study of numerical cognition. In an ongoing debate about the source of developmental change, some argue that children maintain and access different mental representations of number, with evidence coming largely from common number-line estimation tasks. In contrast, others argue that a theoretical framework based on psychophysical models of proportion estimation accounts for typical performance on these tasks. The current study explored children's (n=71) and adults' (n=27) performance on two number-line tasks: the "number to position" (or NP) task and the inverse "position to number" (or PN) task. Estimates on both tasks are consistent with the predictions of the proportion estimation account and do not support the hypothesis that a fundamental shift in mental representations underlies developmental change in numerical estimation and, in turn, mathematical ability. Converging evidence across the tasks also calls into question the utility of bounded number-line tasks as an evaluation of mental representations of number.
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Affiliation(s)
- Emily Slusser
- Department of Child and Adolescent Development, San Jose State University, San Jose, CA 95192, USA.
| | - Hilary Barth
- Department of Psychology, Wesleyan University, Middletown, CT 06457, USA
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Gallagher-Mitchell T, Simms V, Litchfield D. Learning from where 'eye' remotely look or point: Impact on number line estimation error in adults. Q J Exp Psychol (Hove) 2017; 71:1526-1534. [PMID: 28540753 DOI: 10.1080/17470218.2017.1335335] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
Abstract
In this article, we present an investigation into the use of visual cues during number line estimation and their influence on cognitive processes for reducing number line estimation error. Participants completed a 0-1000 number line estimation task before and after a brief intervention in which they observed static-visual or dynamic-visual cues (control, anchor, gaze cursor, mouse cursor) and also made estimation marks to test effective number-target estimation. Results indicated that a significant pre-test to post-test reduction in estimation error was present for dynamic-visual cues of modelled eye-gaze and mouse cursor. However, there was no significant performance difference between pre- and post-test for the control or static anchor conditions. Findings are discussed in relation to the extent to which anchor points alone are meaningful in promoting successful segmentation of the number line and whether dynamic cues promote the utility of these locations in reducing error through attentional guidance.
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Affiliation(s)
- Thomas Gallagher-Mitchell
- 1 Department of Psychology, Edge Hill University, Ormskirk, UK.,2 Department of Psychology, Liverpool Hope University, Liverpool, UK
| | - Victoria Simms
- 3 School of Psychology, Ulster University, Coleraine, UK
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The log–linear response function of the bounded number-line task is unrelated to the psychological representation of quantity. Psychon Bull Rev 2017; 25:447-454. [DOI: 10.3758/s13423-017-1290-z] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/29/2023]
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The Development of Symbolic and Non-Symbolic Number Line Estimations: Three Developmental Accounts Contrasted Within Cross-Sectional and Longitudinal Data. Psychol Belg 2016; 56:382-405. [PMID: 30479447 PMCID: PMC5853816 DOI: 10.5334/pb.276] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/29/2023] Open
Abstract
Three theoretical accounts have been put forward for the development of children's response patterns on number line estimation tasks: the log-to-linear representational shift, the two-linear-to-linear transformation and the proportion judgment account. These three accounts have not been contrasted, however, within one study, using one single criterion to determine which model provides the best fit. The present study contrasted these three accounts by examining first, second and sixth graders with a symbolic and non-symbolic number line estimation task (Experiment 1). In addition, first and second graders were tested again one year later (Experiment 2). In case of symbolic estimations, the proportion judgment account described the data best. Most young children's non-symbolic estimation patterns were best described by a logarithmic model (within the log-to-lin account), whereas those of most older children were best described by the simple power model (within the proportion judgment account).
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Abstract
The three target articles presented in this special issue converged on an emerging theme: the importance of spatial proportional reasoning. They suggest that the ability to map between symbolic fractions (like 1/5) and nonsymbolic, spatial representations of their sizes or magnitudes may be especially important for building robust fractions knowledge. In this commentary, we first reflect upon where these findings stand in a larger theoretical context, largely borrowed from mathematics education research. Next, we emphasize parallels between this work and emerging work suggesting that nonsymbolic proportional reasoning may provide an intuitive foundation for understanding fraction magnitudes. Finally, we end by exploring some open questions that suggest specific future directions in this burgeoning area.
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Gomez A, Piazza M, Jobert A, Dehaene-Lambertz G, Huron C. Numerical abilities of school-age children with Developmental Coordination Disorder (DCD): A behavioral and eye-tracking study. Hum Mov Sci 2016; 55:315-326. [PMID: 27592037 DOI: 10.1016/j.humov.2016.08.008] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/07/2015] [Revised: 08/21/2016] [Accepted: 08/22/2016] [Indexed: 01/29/2023]
Abstract
Developmental Coordination Disorder (DCD) is a disorder of motor coordination which interferes with academic achievement. Difficulties in mathematics have been reported. Performance in the number line task is very sensitive to atypical development of numerical cognition. We used a position-to-number task in which twenty 7-to-10years old children with DCD and 20 age-matched typically developing (TD) children had to estimate the number that corresponded to a hatch mark placed on a 0-100 number line. Eye movements were recorded. Children with DCD were less accurate and slower to respond than their peers. However, they were able to map numbers onto space linearly and used anchoring strategies as control. We suggest that the shift to a linear trend reflects the ability of DCD children to use efficient strategies to solve the task despite a possibly more imprecise underlying numerical acuity.
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Affiliation(s)
- Alice Gomez
- Cognitive Neuroimaging Unit, CEA DRF/I2BM, INSERM, Université Paris-Sud, Université Paris-Saclay, NeuroSpin center, 91191 Gif/Yvette, France
| | - Manuela Piazza
- Cognitive Neuroimaging Unit, CEA DRF/I2BM, INSERM, Université Paris-Sud, Université Paris-Saclay, NeuroSpin center, 91191 Gif/Yvette, France
| | - Antoinette Jobert
- Cognitive Neuroimaging Unit, CEA DRF/I2BM, INSERM, Université Paris-Sud, Université Paris-Saclay, NeuroSpin center, 91191 Gif/Yvette, France
| | - Ghislaine Dehaene-Lambertz
- Cognitive Neuroimaging Unit, CEA DRF/I2BM, INSERM, Université Paris-Sud, Université Paris-Saclay, NeuroSpin center, 91191 Gif/Yvette, France
| | - Caroline Huron
- Cognitive Neuroimaging Unit, CEA DRF/I2BM, INSERM, Université Paris-Sud, Université Paris-Saclay, NeuroSpin center, 91191 Gif/Yvette, France.
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Free versus anchored numerical estimation: A unified approach. Cognition 2016; 149:11-7. [DOI: 10.1016/j.cognition.2015.11.015] [Citation(s) in RCA: 41] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/27/2015] [Revised: 11/09/2015] [Accepted: 11/25/2015] [Indexed: 11/23/2022]
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Reid EE, Baroody AJ, Purpura DJ. Assessing Young Children's Number Magnitude Representation: A Comparison Between Novel and Conventional Tasks. JOURNAL OF COGNITION AND DEVELOPMENT 2015. [DOI: 10.1080/15248372.2014.920844] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/29/2023]
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Hurst M, Leigh Monahan K, Heller E, Cordes S. 123s and ABCs: developmental shifts in logarithmic-to-linear responding reflect fluency with sequence values. Dev Sci 2014; 17:892-904. [DOI: 10.1111/desc.12165] [Citation(s) in RCA: 34] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/14/2013] [Accepted: 10/11/2013] [Indexed: 01/29/2023]
Affiliation(s)
| | | | | | - Sara Cordes
- Department of Psychology; Boston College; USA
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Xu X, Chen C, Pan M, Li N. Development of numerical estimation in Chinese preschool children. J Exp Child Psychol 2013; 116:351-66. [PMID: 23933179 DOI: 10.1016/j.jecp.2013.06.009] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/13/2012] [Revised: 06/19/2013] [Accepted: 06/19/2013] [Indexed: 01/29/2023]
Abstract
Although much is known about the development of mental representations of numbers, it is not clear how early children begin to represent numbers on a linear scale. The current study aimed to examine the development of numerical estimation of Chinese preschoolers. In total, 160 children of three age groups (51 3- and 4-year-olds, 50 5-year-olds, and 59 6-year-olds) were administered the numerical estimation task on three types of number lines (Arabic numbers, dots, and objects). All three age groups took the test on the 0-10 number lines, and the oldest group also took it on the 0-100 and 0-1000 Arabic number lines. Results showed that (a) linear representation of numbers increased with age, (b) representation of numbers was consistent across the three types of tasks, (c) Chinese participants generally showed earlier onset of various landmarks of attaining linear representations (e.g., linearity of various number ranges, accuracy, intercepts) than did their Western counterparts, as reported in previous studies, and (d) the estimates of older Chinese preschoolers on the 0-100 and 0-1000 symbolic number lines fitted the two-linear and linear models better than alternative models such as the one-cycle, two-cycle, and logarithmic models. These results extend the small but accumulating literature on the earlier development of number cognition among Chinese preschoolers compared with their Western counterparts, suggesting the importance of cultural factors in the development of early number cognition.
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Affiliation(s)
- Xiaohui Xu
- School of Preschool Education, Capital Normal University, Beijing 100048, China.
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22
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Landy D, Silbert N, Goldin A. Estimating large numbers. Cogn Sci 2013; 37:775-99. [PMID: 23819823 DOI: 10.1111/cogs.12028] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2011] [Revised: 07/03/2012] [Accepted: 08/17/2012] [Indexed: 01/29/2023]
Abstract
Despite their importance in public discourse, numbers in the range of 1 million to 1 trillion are notoriously difficult to understand. We examine magnitude estimation by adult Americans when placing large numbers on a number line and when qualitatively evaluating descriptions of imaginary geopolitical scenarios. Prior theoretical conceptions predict a log-to-linear shift: People will either place numbers linearly or will place numbers according to a compressive logarithmic or power-shaped function (Barth & Paladino, ; Siegler & Opfer, ). While about half of people did estimate numbers linearly over this range, nearly all the remaining participants placed 1 million approximately halfway between 1 thousand and 1 billion, but placed numbers linearly across each half, as though they believed that the number words "thousand, million, billion, trillion" constitute a uniformly spaced count list. Participants in this group also tended to be optimistic in evaluations of largely ineffective political strategies, relative to linear number-line placers. The results indicate that the surface structure of number words can heavily influence processes for dealing with numbers in this range, and it can amplify the possibility that analogous surface regularities are partially responsible for parallel phenomena in children. In addition, these results have direct implications for lawmakers and scientists hoping to communicate effectively with the public.
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Affiliation(s)
- David Landy
- Department of Psychology, University of Richmond, Richmond, VA23173, USA.
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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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