1
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Chan JYC, Mazzocco MMM. New measures of number line estimation performance reveal children's ordinal understanding of numbers. J Exp Child Psychol 2024; 245:105965. [PMID: 38823358 DOI: 10.1016/j.jecp.2024.105965] [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: 08/06/2023] [Revised: 04/17/2024] [Accepted: 04/18/2024] [Indexed: 06/03/2024]
Abstract
Children's performance on the number line estimation task, often measured by the percentage of absolute error, predicts their later mathematics achievement. This task may also reveal (a) children's ordinal understanding of the target numbers in relation to each other and the benchmarks (e.g., endpoints, midpoint) and (b) the ordinal skills that are a necessary precursor to children's ability to understand the interval nature of a number line as measured by percentage of absolute error. Using data from 104 U.S. kindergartners, we measured whether children's estimates were correctly sequenced across trials and correctly positioned relative to given benchmarks within trials at two time points. For both time points, we found that each ordinal error measure revealed a distinct pattern of data distribution, providing opportunities to tap into different aspects of children's ordinal understanding. Furthermore, children who made fewer ordinal errors scored higher on the Test of Early Mathematics Ability and showed greater improvement on their interval understanding of numbers as reflected by a larger reduction of percentage of absolute error from Time 1 to Time 2. The findings suggest that our number line measures reveal individual differences in children's ordinal understanding of numbers, and that such understanding may be a precursor to their interval understanding and later mathematics performance.
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Affiliation(s)
- Jenny Yun-Chen Chan
- Institute of Child Development, University of Minnesota-Twin Cities, Minneapolis, MN 55455, USA; Early Childhood Education, The Education University of Hong Kong, Tai Po, New Territories, Hong Kong Special Administrative Region.
| | - Michèle M M Mazzocco
- Institute of Child Development, University of Minnesota-Twin Cities, Minneapolis, MN 55455, USA
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2
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Yu S, Sidney P, Kim D, Thompson CA, Opfer JE. From integers to fractions: The role of analogy in transfer and long-term learning. J Exp Child Psychol 2024; 243:105918. [PMID: 38569300 DOI: 10.1016/j.jecp.2024.105918] [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: 10/01/2023] [Revised: 03/12/2024] [Accepted: 03/13/2024] [Indexed: 04/05/2024]
Abstract
Fractions are the gatekeepers to advanced mathematics but are difficult to learn. One powerful learning mechanism is analogy, which builds fraction understanding on a pre-existing foundation of integer knowledge. Indeed, a short intervention that aligned fractions and integers on number lines improved children's estimates of fractions (Yu et al., 2022). The breadth and durability of such gains, however, are unknown, and analogies to other sources (such as percentages) may be equally powerful. To investigate this issue, we randomly assigned 109 fourth and fifth graders to one of three experimental conditions with different analogical sources (integers, percentages, or fractions) or a control condition. During training, children in the experimental conditions solved pairs of aligned fraction number line problems and proportionally-equivalent problems expressed in integers, percentages, or fractions (e.g., 3/8 on a 0-1 number line aligned with 3 on a 0-8 number line). Children in the control group solved fraction number-line problems sequentially. At pretest and a two-week delayed posttest, children completed a broad fraction knowledge battery, including estimation, comparison, categorization, ordering, and arithmetic. Results showed that aligning integers and fractions on number lines facilitated better estimation of fractional magnitudes, and the training effect transferred to novel fraction problems after two weeks. Similar gains were not observed for analogies using percentages. These findings highlight the importance of building new mathematical knowledge through analogies to familiar, similar sources.
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Affiliation(s)
- Shuyuan Yu
- Department of Cognitive Science, Carleton University, Ottawa, ON, K1S5B6, Canada; Department of Psychology, The Ohio State University, Columbus, OH, 43210, USA.
| | - Pooja Sidney
- Department of Psychology, University of Kentucky, Lexington, KY, 40506, USA.
| | - Dan Kim
- Department of Psychology, The Ohio State University, Columbus, OH, 43210, USA.
| | - Clarissa A Thompson
- Department of Psychological Sciences, Kent State University, Kent, OH, 44242, USA.
| | - John E Opfer
- Department of Psychology, The Ohio State University, Columbus, OH, 43210, USA.
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3
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Ünal ZE, Terzi Z, Yalvaç B, Geary DC. The relation between number line performance and mathematics outcomes: Two meta-analyses. Dev Sci 2024; 27:e13509. [PMID: 38576189 DOI: 10.1111/desc.13509] [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: 10/11/2023] [Revised: 03/07/2024] [Accepted: 03/13/2024] [Indexed: 04/06/2024]
Abstract
Understanding the magnitudes represented by numerals is a core component of early mathematical development and is often assessed by accuracy in situating numerals and fractions on a number line. Performance on these measures is consistently related to performance in other mathematics domains, but the strength of these relations may be overestimated because general cognitive ability has not been fully controlled in prior studies. The first of two meta-analyses (162 studies, 33,101 participants) confirmed a relation between performance on whole number (r = 0.33) and fractions number (r = 0.41) lines and overall mathematics performance. These relations were generally consistent across content domains (e.g., algebra and computation) and other moderators. The second (71 studies, 14,543 participants) used meta-analytic structural equation modeling to confirm these relations while controlling general cognitive ability (defined by IQ and working memory measures) and, in one analysis, general mathematics competence. The relation between number line performance and general mathematics competence remained significant but reduced (β = 0.13). Controlling general cognitive ability, whole number line performance consistently predicted competence with fractions but not performance on numeracy or computations measures. The results suggest an understanding of the magnitudes represented by whole numbers might be particularly important for students' fractions learning. RESEARCH HIGHLIGHTS: Two meta-analyses examined the link between the number line and mathematics performance. The first revealed significant relations across domains (e.g., algebra and computation). The second controlled for general cognitive ability and resulted in reduced but still significant relations. The relation between number line and fractions performance was stronger than relations to other domains.
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Affiliation(s)
- Zehra E Ünal
- Department of Psychological Sciences, University of Missouri, Columbia, USA
| | - Züleyha Terzi
- Department of Mathematics and Science Education, Bogaziçi University, Istanbul, Turkey
| | - Beyzanur Yalvaç
- Department of Mathematics and Science Education, Bogaziçi University, Istanbul, Turkey
| | - David C Geary
- Department of Psychological Sciences, University of Missouri, Columbia, USA
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4
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Flores P, Coelho E, Mourão-Carvalhal MI, Forte P. Motor activities to improve maths performance in pre-school children with typical development. Front Psychol 2024; 15:1332741. [PMID: 38840733 PMCID: PMC11150562 DOI: 10.3389/fpsyg.2024.1332741] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/29/2023] [Accepted: 04/22/2024] [Indexed: 06/07/2024] Open
Abstract
Poor maths skills are associated with negative outcomes throughout life, such as lower academic qualifications, decreased professional success and socio-economic results. Mathematical skills emerge continuously throughout childhood and those that children acquire in pre-school are crucial for activities that support analytical thinking, problem-solving and reasoning and argumentation skills. Many of these activities are related to motor skills, since certain cognitive and motor areas of the brain are activated simultaneously when solving maths problems. Of all motor skills, visuomotor integration skills have been documented as those that are most consistently positively and significantly associated with maths performance in pre-school children. These skills are influenced by visual perception (spatial and attention skills), fine motor coordination and gross motor skills. Early intervention can improve visuomotor integration skills in pre-school children. Of all skills that make up visuomotor integration, spatial skills, in addition to being the first skills to influence numerical knowledge and the recognition of geometric shapes, are also those skills that form part of the majority of programs and activities to be worked on with pre-school children for the development of mathematical concepts. However, most intervention programs or activities to develop spatial skills are carried out in the classroom, usually through activities involving handling small objects. In this sense and given the significant association between visuomotor integration skills and gross motor skills, the main objective of this study was to list a set of activities to develop spatial skills, with a strong involvement of gross motor skills, in a classroom, playground or home context.
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Affiliation(s)
- Pedro Flores
- Department of Sports, Higher Institute of Education and Sciences of the Douro, Penafiel, Portugal
- Department of Sports, University of Trás-os-Montes and Alto Douro, Vila Real, Portugal
- Research Center in Sports, Health and Human Development, Covilhã, Portugal
| | - Eduarda Coelho
- Department of Sports, University of Trás-os-Montes and Alto Douro, Vila Real, Portugal
- Research Center in Sports, Health and Human Development, Covilhã, Portugal
| | - Maria Isabel Mourão-Carvalhal
- Department of Sports, University of Trás-os-Montes and Alto Douro, Vila Real, Portugal
- Research Center in Sports, Health and Human Development, Covilhã, Portugal
| | - Pedro Forte
- Department of Sports, Higher Institute of Education and Sciences of the Douro, Penafiel, Portugal
- Research Center in Sports, Health and Human Development, Covilhã, Portugal
- Research Center for Active Living and Well Being (Livewell), Instituto Politécnico de Bragança, Bragança, Portugal
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Sidney PG, Shirah JF, Blake J, Kruczkowski AL. Adaptive variability in children's conceptual models of division. J Exp Child Psychol 2023; 236:105743. [PMID: 37467598 DOI: 10.1016/j.jecp.2023.105743] [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: 03/29/2023] [Revised: 05/25/2023] [Accepted: 06/27/2023] [Indexed: 07/21/2023]
Abstract
This study examined whether different types of commonly used mathematical tasks affect how children think about whole number division problems. Prior research suggested that children tend to rely on the partitive model to understand whole number division, which is likely problematic when students transition to learning about fraction division. We assessed variability in correct whole number division problem-solving strategies among 63 elementary school children (41.5% female, 58.5% male, 0% nonbinary/gender expansive; 69.2% White, 10.7% multiracial, 6.1% Black, 4.6% Latino, 3.3% other/unidentified, 6.1% preferred not to answer). Each participant was asked to demonstrate four whole number division problems in each of three contexts (within participants): objects, story problems, and number lines. Most children displayed understanding of multiple conceptual models of division, but strategies varied by context. Story problems elicited partitive models, number lines elicited quotative models, and objects elicited both. Finally, elementary school children used strategies adaptively. Number line representations may afford conceptual connections between earlier-learned whole number concepts and analogous later-learned fraction concepts, supporting the integration of children's whole number and fraction knowledge.
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Affiliation(s)
- Pooja G Sidney
- Department of Psychology, University of Kentucky, Lexington, KY 40506, USA.
| | - Julie F Shirah
- Department of Psychology, University of Kentucky, Lexington, KY 40506, USA
| | - Jessica Blake
- Department of Psychology, University of Kentucky, Lexington, KY 40506, USA
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Hilz A, Guill K, Roloff J, Sommerhoff D, Aldrup K. How to Continue? New Approaches to Investigating the Effects of Adaptive Math Learning Programs on Students' Performance, Self-Concept, and Anxiety. J Intell 2023; 11:108. [PMID: 37367510 DOI: 10.3390/jintelligence11060108] [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: 03/17/2023] [Revised: 05/13/2023] [Accepted: 05/25/2023] [Indexed: 06/28/2023] Open
Abstract
Math learning programs were expected to revolutionize students' learning, but their effects so far have mostly been disappointing. Following the debate about why to continue research on math learning programs, we aimed to reformulate this question into how to continue this research. Investigations to date have neither considered a sufficiently wide set of outcome variables nor differentiated between performance measures (e.g., measuring addition and subtraction performance separately) and affective-motivational variables. Moreover, as students can only benefit from a program if they use it, researchers need to take practice behavior into account. Thus, we investigated whether the adaptive arithmetic learning program Math Garden fostered students' addition and subtraction performance, their math self-concept, and a reduction of their math anxiety. We also investigated how practice behavior (practiced tasks/weeks) affected these outcomes. We used a randomized pretest-posttest control group design with 376 fifth-grade students in Germany. Students in the experimental condition practiced with Math Garden for 20.7 weeks and had an increase in math self-concept. The more subtraction tasks the students practiced, the more they improved their subtraction performance. We found no effects on math anxiety. The results are discussed in terms of providing a starting point for new directions in future research.
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Affiliation(s)
- Anna Hilz
- Department of Educational Research and Educational Psychology, IPN-Leibniz Institute for Science and Mathematics Education, Olshausenstr. 62, 24118 Kiel, Germany
| | - Karin Guill
- Department of Educational Research and Educational Psychology, IPN-Leibniz Institute for Science and Mathematics Education, Olshausenstr. 62, 24118 Kiel, Germany
| | - Janina Roloff
- Department of Educational Research and Educational Psychology, IPN-Leibniz Institute for Science and Mathematics Education, Olshausenstr. 62, 24118 Kiel, Germany
| | - Daniel Sommerhoff
- Department of Mathematics Education, IPN-Leibniz Institute for Science and Mathematics Education, 24118 Kiel, Germany
| | - Karen Aldrup
- Department of Educational Research and Educational Psychology, IPN-Leibniz Institute for Science and Mathematics Education, Olshausenstr. 62, 24118 Kiel, Germany
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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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8
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Adams SR, Maki KE. Differential effectiveness, efficiency, and acceptability of drill‐based interventions targeting multiplication facts. PSYCHOLOGY IN THE SCHOOLS 2023. [DOI: 10.1002/pits.22880] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/11/2023]
Affiliation(s)
- Sarah R. Adams
- Family Medicine Franciscan Health Indianapolis Indiana USA
| | - Kathrin E. Maki
- Department of Special Education, School Psychology, and Early Childhood Studies University of Florida Gainesville Florida USA
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9
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Wortha SM, Klein E, Lambert K, Dackermann T, Moeller K. The relevance of basic numerical skills for fraction processing: Evidence from cross-sectional data. PLoS One 2023; 18:e0281241. [PMID: 36719901 PMCID: PMC9888716 DOI: 10.1371/journal.pone.0281241] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2021] [Accepted: 01/18/2023] [Indexed: 02/01/2023] Open
Abstract
Recent research indicated that fraction understanding is an important predictor of later mathematical achievement. In the current study we investigated associations between basic numerical skills and students' fraction processing. We analyzed data of 939 German secondary school students (age range = 11.92 to 18.00 years) and evaluated the determinants of fraction processing considering basic numerical skills as predictors (i.e., number line estimation, basic arithmetic operations, non-symbolic magnitude comparison, etc.). Additionally, we controlled for general cognitive ability, grade level, and sex. We found that multiplication, subtraction, conceptual knowledge, number line estimation, and basic geometry were significantly associated with fraction processing beyond significant associations of general cognitive ability and sex. Moreover, relative weight analysis revealed that addition and approximate arithmetic should also be considered as relevant predictors for fraction processing. The current results provide food for thought that further research should focus on investigating whether recapitulating basic numerical content in secondary school mathematics education can be beneficial for acquiring more complex mathematical concepts such as fractions.
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Affiliation(s)
- Silke M. Wortha
- LEAD Graduate School & Research Network, University of Tuebingen, Tuebingen, Germany
- Department of Neurology, University Medicine Greifswald, Greifswald, Germany
- Centre for Mathematical Cognition, School of Science, Loughborough University, Loughborough, United Kingdom
- * E-mail:
| | - Elise Klein
- University of Paris, LaPsyDÉ, CNRS, Paris, France
- Leibniz-Institut für Wissensmedien, Tuebingen, Germany
| | - Katharina Lambert
- LEAD Graduate School & Research Network, University of Tuebingen, Tuebingen, Germany
- University of Education, Heidelberg, Germany
| | | | - Korbinian Moeller
- LEAD Graduate School & Research Network, University of Tuebingen, Tuebingen, Germany
- Centre for Mathematical Cognition, School of Science, Loughborough University, Loughborough, United Kingdom
- Leibniz-Institut für Wissensmedien, Tuebingen, Germany
- Individual Development and Adaptive Education Center, Frankfurt am Main, Germany
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10
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Nuraydin S, Stricker J, Ugen S, Martin R, Schneider M. The number line estimation task is a valid tool for assessing mathematical achievement: A population-level study with 6484 Luxembourgish ninth-graders. J Exp Child Psychol 2023; 225:105521. [PMID: 35973280 DOI: 10.1016/j.jecp.2022.105521] [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: 02/22/2021] [Revised: 07/15/2022] [Accepted: 07/15/2022] [Indexed: 01/29/2023]
Abstract
The number line estimation task is an often-used measure of numerical magnitude understanding. The task also correlates substantially with broader measures of mathematical achievement. This raises the question of whether the task would be a useful component of mathematical achievement tests and instruments to diagnose dyscalculia or mathematical giftedness and whether a stand-alone version of the task can serve as a short screener for mathematical achievement. Previous studies on the relation between number line estimation accuracy and broader mathematical achievement were limited in that they used relatively small nonrepresentative samples and usually did not account for potentially confounding variables. To close this research gap, we report findings from a population-level study with nearly all Luxembourgish ninth-graders (N = 6484). We used multilevel regressions to test how a standardized mathematical achievement test relates to the accuracy in number line estimation on bounded number lines with whole numbers and fractions. We also investigated how these relations were moderated by classroom characteristics, person characteristics, and trial characteristics. Mathematical achievement and number line estimation accuracy were associated even after controlling for potentially confounding variables. Subpopulations of students showed meaningful differences in estimation accuracy, which can serve as benchmarks in future studies. Compared with the number line estimation task with whole numbers, the number line estimation task with fractions was more strongly related to mathematical achievement in students across the entire mathematical achievement spectrum. These results show that the number line estimation task is a valid and useful tool for diagnosing and monitoring mathematical achievement.
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Affiliation(s)
| | | | - Sonja Ugen
- University of Luxembourg, L-4365 Esch-sur-Alzette, Luxembourg
| | - Romain Martin
- Ministry of Higher Education and Research, L-2327 Luxembourg, Luxembourg
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11
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Maki KE, Moody ME, Cullins SL, Griffin TL. Examination of a Modified Incremental Rehearsal Approach to Explore Causal Mechanisms. JOURNAL OF BEHAVIORAL EDUCATION 2022:1-24. [PMID: 35919840 PMCID: PMC9333052 DOI: 10.1007/s10864-022-09482-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Accepted: 06/14/2022] [Indexed: 06/15/2023]
Abstract
Incremental rehearsal (IR) has consistently been shown to improve students' math fact retention and fluency (Maki et al., Journal of Behavioral Education 30:534-558, 2021). However, less is known about how intervention modifications may support longer-term skill maintenance. The purpose of this study was to compare traditional IR with a modified IR (shuffle IR; ShIR) in which known multiplication facts were shuffled between sequences using a cumulative acquisition design with six fourth- and fifth-grade students. All participants retained and maintained more facts in IR and ShIR compared to a control condition. However, IR or ShIR did not consistently result in greater retention than the other, with three students demonstrating greater retention in the IR condition and three students demonstrating greater retention in the ShIR condition. Most participants demonstrated greater fact maintenance in the ShIR condition than in the IR condition. All participants made fewer intervention session errors in the condition in which they retained more multiplication facts.
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Affiliation(s)
- Kathrin E. Maki
- Department of Special Education, School Psychology, and Early Childhood Studies, University of Florida, Gainesville, FL USA
| | - Mary Elizabeth Moody
- Department of Special Education, School Psychology, and Early Childhood Studies, University of Florida, Gainesville, FL USA
| | - Siera L. Cullins
- Department of Special Education, School Psychology, and Early Childhood Studies, University of Florida, Gainesville, FL USA
| | - Taylor L. Griffin
- Department of Special Education, School Psychology, and Early Childhood Studies, University of Florida, Gainesville, FL USA
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12
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Seitz M, Weinert S. Numeracy skills in young children as predictors of mathematical competence. BRITISH JOURNAL OF DEVELOPMENTAL PSYCHOLOGY 2022; 40:224-241. [PMID: 35262211 DOI: 10.1111/bjdp.12408] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/22/2021] [Revised: 01/21/2022] [Indexed: 11/28/2022]
Abstract
As mathematical competence is linked to educational success, professional achievement, and even a country's economic growth, researchers have been interested in early predictors for quite some time. Although there have been numerous studies on domain-specific numerical abilities predicting later mathematical competence in preschool children, research in toddlers is scarce, especially regarding additional influential aspects, such as domain-general cognitive abilities and the children's social background. Using a large-scale dataset, the present study examined predictive effects of numeracy skills in 17-month-olds for later mathematical achievement. We found small, positive effects, even when controlling for child-related variables (i.e., age and sex) and the children's social background (i.e., maternal education and household language). Additionally, we compared results with a domain-general categorization task and found no distinct effect on mathematical competence. The present results are discussed with regard to the specificities of the dataset, as well as implications for future studies on predictors of mathematical competence.
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Affiliation(s)
- Maximilian Seitz
- Leibniz Institute for Educational Trajectories, Bamberg, Germany
| | - Sabine Weinert
- Department of Developmental Psychology, University of Bamberg, Germany
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13
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Codding RS, Nelson PM, Parker DC, Edmunds R, Klaft J. Examining the impact of a tutoring program implemented with community support on math proficiency and growth. J Sch Psychol 2022; 90:82-93. [PMID: 34969489 DOI: 10.1016/j.jsp.2021.11.002] [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: 03/05/2020] [Revised: 08/05/2021] [Accepted: 11/08/2021] [Indexed: 10/19/2022]
Abstract
The current study evaluated the impact of a math tutoring program delivered in 20 schools to students in 4th through 8th grades by community members over one academic year. Students were randomly assigned to treatment and control groups. Multi-level linear and generalized linear mixed models were used to evaluate group differences in post-test scores and the probability of attaining the spring proficiency benchmark on two increasingly distal measures of math achievement. Intent-to-treat analyses identified higher achievement scores among students assigned to treatment on a measure of fact fluency and a computer adaptive measure of overall math achievement. Students assigned to treatment also had a higher probability of reaching grade-level benchmarks on the computer adaptive test. No statistically significant effects were observed on a state proficiency test. Implications for significant and null findings are discussed within the context of intervention content and delivery.
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Affiliation(s)
- Robin S Codding
- Department of Applied Psychology, Northeastern University, United States of America.
| | - Peter M Nelson
- ServeMinnesota, 120 South 6th Street, Suite 2260, Minneapolis, MN 55402, United States of America.
| | - David C Parker
- ServeMinnesota, 120 South 6th Street, Suite 2260, Minneapolis, MN 55402, United States of America.
| | - Rebecca Edmunds
- Department of Educational Psychology, University of Minnesota, United States of America.
| | - Jenna Klaft
- Department of Educational Psychology, University of Minnesota, United States of America.
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14
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Cirino PT, Tolar TD, Fuchs LS. Domain general and specific contributions to algebra: A sequenced longitudinal path model. CONTEMPORARY EDUCATIONAL PSYCHOLOGY 2022. [DOI: 10.1016/j.cedpsych.2021.102026] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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15
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Siegler RS, Tian J. Why do we have three rational number notations? The importance of percentages. ADVANCES IN CHILD DEVELOPMENT AND BEHAVIOR 2022; 63:1-33. [DOI: 10.1016/bs.acdb.2022.05.001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
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Abstract
A robust left digit effect arises in number line estimation such that adults' estimates for numerals with different hundreds place digits but nearly identical magnitudes are systematically different from one another (e.g., 299 is placed too far to the left of 302). In two experiments, we investigate whether brief feedback interventions designed to increase task effort can reduce or eliminate the left digit effect in a self-paced 0-1,000 number line estimation task. Participants were assigned to complete three blocks of 120 trials each where the middle block contained feedback or no feedback. Feedback was in the form of summary accuracy scores (Experiment 1; N = 153) or competitive (summary) accuracy scores (Experiment 2; N = 145). In both experiments, planned analyses revealed large left digit effects in all blocks regardless of feedback condition. Feedback did not lead to a reduction in the left digit effect in either experiment, but improvements in overall accuracy were observed. We conclude that there are no changes in the left digit effect resulting from either summary accuracy feedback or competitive accuracy feedback. Also reported are exploratory analyses of trial characteristics (e.g., whether 299 is presented before or after 302) and the left digit effect.
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McMullen J, Hannula-Sormunen MM, Lehtinen E, Siegler RS. Predicting adaptive expertise with rational number arithmetic. BRITISH JOURNAL OF EDUCATIONAL PSYCHOLOGY 2021; 92:e12471. [PMID: 34747023 PMCID: PMC9298780 DOI: 10.1111/bjep.12471] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/30/2021] [Accepted: 10/01/2021] [Indexed: 11/29/2022]
Abstract
Background Adaptive expertise is a highly valued outcome of mathematics curricula. One aspect of adaptive expertise with rational numbers is adaptive rational number knowledge, which refers to the ability to integrate knowledge of numerical characteristics and relations in solving novel tasks. Even among students with strong conceptual and procedural knowledge of rational numbers, there are substantial individual differences in adaptive rational number knowledge. Aims We aimed to examine how a wide range of domain‐general and mathematically specific skills and knowledge predicted different aspects of rational number knowledge, including procedural, conceptual, and adaptive rational number knowledge. Sample 173 6th and 7th grade students from a school in the southeastern US (51% female) participated in the study. Methods At three time points across 1.5 years, we measured students’ domain‐general and domain‐specific skills and knowledge. We used multiple hierarchal regression analysis to examine how these predictors related to rational number knowledge at the third time point. Result Prior knowledge of rational numbers, general mathematical calculation knowledge, and spontaneous focusing on multiplicative relations (SFOR) tendency uniquely predicted adaptive rational number knowledge, after taking into account domain‐general and mathematically specific skills and knowledge. Although conceptual knowledge of rational numbers and general mathematical achievement also predicted later conceptual and procedural knowledge of rational numbers, SFOR tendency did not. Conclusion Results suggest expanding investigations of mathematical development to also explore different features of adaptive expertise as well as spontaneous mathematical focusing tendencies.
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Affiliation(s)
- Jake McMullen
- Department of Teacher Education, University of Turku, Finland
| | | | - Erno Lehtinen
- Department of Teacher Education, University of Turku, Finland.,Vytautas Magnus University, Kaunas, Lithuania
| | - Robert S Siegler
- Teacher's College, Columbia University, New York City, New York, USA.,The Siegler Center for Innovative Learning (SCIL), Advanced Technology Center, Beijing Normal University, China
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Lin X, Powell SR. Examining the relation between whole numbers and fractions: A meta-analytic structural equation modeling approach. CONTEMPORARY EDUCATIONAL PSYCHOLOGY 2021. [DOI: 10.1016/j.cedpsych.2021.102017] [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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19
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Rebholz F, Golle J, Moeller K. The association of basic numerical abilities and math achievement: The mediating role of visuospatial and arithmetical abilities. Q J Exp Psychol (Hove) 2021; 75:841-853. [PMID: 34344247 PMCID: PMC8958554 DOI: 10.1177/17470218211040060] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
Abstract
Basic numerical abilities such as number line estimation have been observed repeatedly to be associated with mathematical achievement. Recently, it was argued that the association between basic numerical abilities and mathematical achievement is fully mediated by visuospatial abilities. However, arithmetical abilities have not yet been considered as influencing this association, even though solution strategies in number line estimation as well as mathematical achievement often involve arithmetical procedures. Therefore, we investigated the mediating role of arithmetical and visuospatial abilities on the association between number line estimation and mathematical achievement in a sample of n = 599 German elementary school students. The results indicated that arithmetical abilities as well as visuospatial abilities mediated the association between number line estimation and mathematical achievement. However, neither visuospatial nor arithmetical abilities fully mediated the association between number line estimation and mathematical achievement when considered in isolation. This substantiates the relevance of the intertwined development of visuospatial and arithmetical abilities as well as basic numerical abilities such as number line estimation (i.e., the combination of domain-specific numerical and domain-general abilities) driving mathematical achievement.
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Affiliation(s)
- Franziska Rebholz
- Hector Research Institute of Education Sciences and Psychology, University of Tübingen, Tübingen, Germany.,LEAD Graduate School & Research Network, University of Tübingen, Tübingen, Germany
| | - Jessika Golle
- Hector Research Institute of Education Sciences and Psychology, University of Tübingen, Tübingen, Germany.,LEAD Graduate School & Research Network, University of Tübingen, Tübingen, Germany
| | - Korbinian Moeller
- LEAD Graduate School & Research Network, University of Tübingen, Tübingen, Germany.,Centre for Mathematical Cognition, School of Science, Loughborough University, Loughborough, UK.,Leibniz-Institut für Wissensmedien, Tübingen, Germany.,Department of Psychology, University of Tübingen, Tübingen, Germany
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20
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Barbieri CA, Young LK, Newton KJ, Booth JL. Predicting Middle School Profiles of Algebra Performance Using Fraction Knowledge. Child Dev 2021; 92:1984-2005. [PMID: 33929044 DOI: 10.1111/cdev.13568] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
Fraction knowledge and algebraic skill are closely linked. Algebra is a gatekeeper for advanced courses (Booth & Newton, 2012; Brown & Quinn, 2007). This study uses the person-centered approach of latent profile analysis to examine individual differences in middle schoolers' (N = 350) algebra performance at the end of the year (EOY). The relative importance of a range of fraction skills is considered in predicting the likelihood of displaying a particular profile of EOY algebra knowledge, measured by feature knowledge, equation encoding, equation solving, and word-problem skills. Notably, fraction number line estimation did not predict algebra performance on any of the four measures. Fraction arithmetic at the beginning of the year is most predictive of displaying the highest performing EOY algebra profile.
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21
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Blume F, Dresler T, Gawrilow C, Ehlis AC, Goellner R, Moeller K. Examining the relevance of basic numerical skills for mathematical achievement in secondary school using a within-task assessment approach. Acta Psychol (Amst) 2021; 215:103289. [PMID: 33711503 DOI: 10.1016/j.actpsy.2021.103289] [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: 10/10/2019] [Revised: 11/27/2020] [Accepted: 02/19/2021] [Indexed: 10/21/2022] Open
Abstract
Previous research repeatedly found basic numerical abilities (e.g., magnitude understanding, arithmetic fact knowledge, etc.) to predict young students' current and later arithmetic achievement as assessed by achievement tests - even when controlling for the influence of domain-general abilities (e.g., intelligence, working memory). However, to the best of our knowledge, previous studies hardly addressed this issue in secondary school students. Additionally, they primarily assessed basic numerical abilities in a between-task approach (i.e., using different tasks for different abilities). Finally, their relevance for real-life academic outcomes such as mathematics grades has only rarely been investigated. The present study therefore pursued an approach using one and the same task (i.e., a within-task approach) to reduce confounding effects driven by between-task differences. In particular, we evaluated the relevance of i) number magnitude understanding, ii) arithmetic fact knowledge, and iii) conceptual and procedural knowledge for the mathematics grades of 81 students aged between ten and thirteen (i.e., in Grades 5 and 6) employing the number bisection task. Results indicated that number magnitude understanding, arithmetic fact knowledge, and conceptual and procedural knowledge contributed to explaining mathematics grades even when controlling for domain-general cognitive abilities. Methodological and practical implications of the results are discussed.
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22
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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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24
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Nazari MA, Sabaghypour S, Pezhmanfard M, Azizi K, Vahedi S. The influence of children's mathematical competence on performance in mental number line, time knowledge and time perception. PSYCHOLOGICAL RESEARCH 2020; 85:2023-2035. [PMID: 32623512 DOI: 10.1007/s00426-020-01380-7] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2019] [Accepted: 06/26/2020] [Indexed: 11/28/2022]
Abstract
A growing body of research suggests that space, time and number are represented within a common system. Other studies have shown this relationship is related to the mathematical competency. Here we examined the influence of the mathematical capacities of 8-12 years old children, grouped into high (n = 63) and low (n = 58) on performance in mental number line, time knowledge and time perception. The results revealed that mathematical competency influences mental number line and time knowledge, but with regard to time perception the effects were only observed in time production task. In addition, the results of correlation analysis revealed interaction between time knowledge, time production (but not reproduction) and mental number line. Finally, the findings are discussed within the framework of the recent theories regarding representation of space, time and number.
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Affiliation(s)
- Mohammad Ali Nazari
- Department of Neuroscience, Faculty of Advanced Technologies in Medicine, Iran University of Medical Sciences, Shahid Hemmat Exp. way, Tehran, Iran. .,Division of Cognitive Neuroscience, University of Tabriz, Tabriz, Iran.
| | - Saied Sabaghypour
- Division of Cognitive Neuroscience, University of Tabriz, Tabriz, Iran
| | - Mina Pezhmanfard
- Department of Educational Psychology, University of Tabriz, Tabriz, Iran
| | - Kiana Azizi
- Department of Psychology, University of Tabriz, Tabriz, Iran
| | - Shahram Vahedi
- Department of Educational Psychology, University of Tabriz, Tabriz, Iran
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25
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Pouta M, Lehtinen E, Palonen T. Student Teachers’ and Experienced Teachers’ Professional Vision of Students’ Understanding of the Rational Number Concept. EDUCATIONAL PSYCHOLOGY REVIEW 2020. [DOI: 10.1007/s10648-020-09536-y] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
Abstract
AbstractThe aim of this study was to investigate differences in student teachers’ and experienced teachers’ professional vision in natural settings and to elicit clues of the relation of in-the-moment noticing and instruction quality of students’ understanding of rational number concept. Rational number concept challenges both students and teachers because of natural number bias that learning of rational numbers is vulnerable to. Accurate professional vision and adequate instructions are needed to enhance students’ understanding of rational number concept. Mobile eye-tracking technique enables video recording of natural teaching situations from a teacher’s perspective with more specific information of teacher’s in-the-moment noticing. Combined with cued retrospective reporting, this approach can gather more explicit evidence of teachers’ professional vision and instructions. Results indicated that both student teachers and experienced teachers attended to mathematical and fraction-related aspect similarly but differed in interpreting and instructing students’ fraction understanding. Student teachers made more advanced interpretations but their instructions were less adequate, whereas among experienced teachers, it was just the opposite. Furthermore, student teachers made more attempts to shared attention when using fraction understanding non-supporting instructions, whereas experienced teachers’ attempts to shared attention were related to fraction understanding supporting instructions. Results indicate student teachers’ difficulty to transfer pedagogical content knowledge from noticing to actions and experienced teachers to have more enhanced in-the-moment professional vision and its application to teaching. Practical implications for teacher training as well as methodological decisions of in-the-moment professional vision studies in natural settings are discussed.
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26
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Lambert K, Wortha SM, Moeller K. Time Reading in Middle and Secondary School Students: The Influence of Basic-Numerical Abilities. The Journal of Genetic Psychology 2020; 181:255-277. [PMID: 32406334 DOI: 10.1080/00221325.2020.1760778] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
Abstract
Time reading skills are central for the management of personal and professional life. However, little is known about the differential influence of basic numerical abilities on analog and digital time reading in general and in middle and secondary school students in particular. The present study investigated the influence of basic numerical skills separately for analog and digital time reading in N = 709 students from 5th to 8th grade. The present findings suggest that the development of time reading skills is not completed by the end of primary school. Results indicated that aspects of magnitude manipulation and arithmetic fact knowledge predicted analog time reading significantly over and above the influence of age. Furthermore, results showed that spatial representations of number magnitude, magnitude manipulation, arithmetic fact knowledge, and conceptual knowledge were significant predictors of digital time reading beyond general cognitive ability and sex. To the best of our knowledge, the present study is the first to show differential effects of basic numerical abilities on analog and digital time reading skills in middle and secondary school students. As time readings skills are crucial for everyday life, these results are highly relevant to better understand basic numerical processes underlying time reading.
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Affiliation(s)
- Katharina Lambert
- Hector Research Institute of Education Sciences and Psychology, University of Tuebingen, Tuebingen, Germany.,LEAD Graduate School & Research Network, University of Tuebingen, Tuebingen, Germany
| | - Silke M Wortha
- LEAD Graduate School & Research Network, University of Tuebingen, Tuebingen, Germany.,Leibniz-Institut für Wissensmedien, Tuebingen, Germany
| | - Korbinian Moeller
- LEAD Graduate School & Research Network, University of Tuebingen, Tuebingen, Germany.,Leibniz-Institut für Wissensmedien, Tuebingen, Germany.,Department of Psychology, University of Tuebingen, Tuebingen, Germany
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27
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Are the acuities of magnitude representations of different types and ranges of numbers related? Testing the core assumption of the integrated theory of numerical development. COGNITIVE DEVELOPMENT 2020. [DOI: 10.1016/j.cogdev.2020.100888] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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28
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Tikhomirova T, Kuzmina Y, Lysenkova I, Malykh S. The Relationship Between Non-symbolic and Symbolic Numerosity Representations in Elementary School: The Role of Intelligence. Front Psychol 2019; 10:2724. [PMID: 31866910 PMCID: PMC6906201 DOI: 10.3389/fpsyg.2019.02724] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/18/2019] [Accepted: 11/18/2019] [Indexed: 01/29/2023] Open
Abstract
This study aimed to estimate the extent to which the development of symbolic numerosity representations relies on pre-existing non-symbolic numerosity representations that refer to the Approximate Number System. To achieve this aim, we estimated the longitudinal relationships between accuracy in the Number Line (NL) test and “blue–yellow dots” test across elementary school children. Data from a four-wave longitudinal study involving schoolchildren in grades 1–4 in Russia and Kyrgyzstan (N = 490, mean age 7.65 years in grade 1) were analyzed. We applied structural equation modeling and tested several competing models. The results revealed that at the start of schooling, the accuracy in the NL test predicted subsequent accuracy in the “blue–yellow dots” test, whereas subsequently, non-symbolic representation in grades 2 and 3 predicted subsequent symbolic representation. These results indicate that the effect of non-symbolic representation on symbolic representation emerges after a child masters the basics of symbolic number knowledge, such as counting in the range of twenty and simple arithmetic. We also examined the extent to which the relationships between non-symbolic and symbolic representations might be explained by fluid intelligence, which was measured by Raven’s Standard Progressive Matrices test. The results revealed that the effect of symbolic representation on non-symbolic representation was explained by fluid intelligence, whereas at the end of elementary school, non-symbolic representation predicted subsequent symbolic representation independently of fluid intelligence.
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Affiliation(s)
- Tatiana Tikhomirova
- Department of Psychology, Lomonosov Moscow State University, Moscow, Russia.,Psychological Institute of Russian Academy of Education, Moscow, Russia
| | - Yulia Kuzmina
- Department of Psychology, Lomonosov Moscow State University, Moscow, Russia.,Psychological Institute of Russian Academy of Education, Moscow, Russia
| | - Irina Lysenkova
- Department of Psychology, Kyrgyz-Russian Slavic University, Bishkek, Kyrgyzstan
| | - Sergey Malykh
- Department of Psychology, Lomonosov Moscow State University, Moscow, Russia.,Psychological Institute of Russian Academy of Education, Moscow, Russia
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29
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Lamnina M, Chase CC. Developing a thirst for knowledge: How uncertainty in the classroom influences curiosity, affect, learning, and transfer. CONTEMPORARY EDUCATIONAL PSYCHOLOGY 2019. [DOI: 10.1016/j.cedpsych.2019.101785] [Citation(s) in RCA: 29] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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30
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Liu Y, Wong TTY. The growth rates of dot enumeration ability predict mathematics achievements: A 5-year longitudinal study. BRITISH JOURNAL OF EDUCATIONAL PSYCHOLOGY 2019; 90:604-617. [PMID: 31513291 DOI: 10.1111/bjep.12318] [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: 12/12/2018] [Revised: 07/10/2019] [Indexed: 11/30/2022]
Abstract
BACKGROUND Dot enumeration is the basic mathematics competency in young children and a significant indicator of later mathematics achievement. AIM The present study focused on (1) how children's dot enumeration ability changed as they progressed from late kindergarten years (K3) to the second year of primary school in Hong Kong (P2), and (2) the extent to which such changes are associated with students' mathematics outcomes assessed at the fourth grade, including standardized mathematics achievement, whole number magnitude understanding, and rational number concept. SAMPLE(S) Two hundred and eleven Hong Kong kindergarteners were recruited. METHODS The participants' dot enumeration was assessed from K3 to P2. Their mathematics outcomes were assessed at P4, including standardized mathematics achievement, whole number magnitude understanding, and rational number concept. RESULTS The changes in their dot enumeration speed reflected a linear growth pattern. Further, both the initial level and growth rate of dot enumeration predicted standardized mathematics achievement and whole number magnitude understanding 2 years later while only the latter predicted rational number concept. CONCLUSIONS The results indicate the importance of focusing on children's growth in a specific mathematics skill, in addition to their status at one single time point. Practical implications are discussed in this article.
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Affiliation(s)
- Yingyi Liu
- Department of Psychology, The Education University of Hong Kong, Hong Kong
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31
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Morra S, Bisagno E, Caviola S, Delfante C, Mammarella IC. Working Memory Capacity and the Development of Quantitative Central Conceptual Structures. COGNITION AND INSTRUCTION 2019. [DOI: 10.1080/07370008.2019.1636797] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
Affiliation(s)
- Sergio Morra
- DISFOR (Department of Education), Università di Genova, Genova, Italy
| | - Elisa Bisagno
- DISFOR (Department of Education), Università di Genova, Genova, Italy
| | - Sara Caviola
- School of Psychology, University of Leeds, Leeds, UK
| | - Chiara Delfante
- CEDIA (Data, Information and Telematic Center), Università di Genova, Genova, Italy
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Ludewig U, Lambert K, Dackermann T, Scheiter K, Möller K. Influences of basic numerical abilities on graph reading performance. PSYCHOLOGICAL RESEARCH 2019; 84:1198-1210. [DOI: 10.1007/s00426-019-01144-y] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/15/2018] [Accepted: 01/03/2019] [Indexed: 11/24/2022]
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33
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Tam YP, Wong TTY, Chan WWL. The relation between spatial skills and mathematical abilities: The mediating role of mental number line representation. CONTEMPORARY EDUCATIONAL PSYCHOLOGY 2019. [DOI: 10.1016/j.cedpsych.2018.10.007] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
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34
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Bailey DH, Duncan GJ, Watts T, Clements DH, Sarama J. Risky business: Correlation and causation in longitudinal studies of skill development. ACTA ACUST UNITED AC 2018; 73:81-94. [PMID: 29345488 DOI: 10.1037/amp0000146] [Citation(s) in RCA: 46] [Impact Index Per Article: 7.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/08/2022]
Abstract
Developmental theories often posit that changes in children's early psychological characteristics will affect much later psychological, social, and economic outcomes. However, tests of these theories frequently yield results that are consistent with plausible alternative theories that posit a much smaller causal role for earlier levels of these psychological characteristics. Our article explores this issue with empirical tests of skill-building theories, which predict that early boosts to simpler skills (e.g., numeracy or literacy) or behaviors (e.g., antisocial behavior or executive functions) support the long-term development of more sophisticated skills or behaviors. Substantial longitudinal associations between academic or socioemotional skills measured early and then later in childhood or adolescence are often taken as support of these skill-building processes. Using the example of skill-building in mathematics, we argue that longitudinal correlations, even if adjusted for an extensive set of baseline covariates, constitute an insufficiently risky test of skill-building theories. We first show that experimental manipulation of early math skills generates much smaller effects on later math achievement than the nonexperimental literature has suggested. We then conduct falsification tests that show puzzlingly high cross-domain associations between early math and later literacy achievement. Finally, we show that a skill-building model positing a combination of unmeasured stable factors and skill-building processes can reproduce the pattern of experimental impacts on children's mathematics achievement. Implications for developmental theories, methods, and practice are discussed. (PsycINFO Database Record
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Affiliation(s)
- Drew H Bailey
- School of Education, University of California, Irvine
| | - Greg J Duncan
- School of Education, University of California, Irvine
| | - Tyler Watts
- School of Education, University of California, Irvine
| | | | - Julie Sarama
- Morgridge College of Education, University of Denver
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Wong TTY. The roles of place-value understanding and non-symbolic ratio processing system in symbolic rational number processing. BRITISH JOURNAL OF EDUCATIONAL PSYCHOLOGY 2018; 89:635-652. [PMID: 30307030 DOI: 10.1111/bjep.12249] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/13/2017] [Indexed: 11/29/2022]
Abstract
BACKGROUND While it has been widely demonstrated that children's and adolescents' understanding of rational number plays an important role in their mathematics achievement, we have limited knowledge about the cognitive correlates of this understanding. AIMS The current study aimed at examining whether children's non-symbolic ratio processing and their understanding of place-value structure of whole numbers play a role in their understanding of fractions and decimals and whether their roles are different for fractions versus decimal understanding. SAMPLE A sample of 124 fourth graders was tested. METHODS Participants were tested on their symbolic rational number processing, non-symbolic ratio processing, place-value understanding of whole numbers, mathematics achievement, as well as a series of domain-general and domain-specific cognitive skills related to symbolic rational number processing. RESULTS The findings suggest that, while the understanding of place value of whole numbers significantly predicted the understanding of both fractions and decimals, non-symbolic ratio processing specifically predicted the understanding of fractions, but not decimals. CONCLUSIONS The findings highlight the roles of place-value understanding and non-symbolic ratio processing in the acquisition of symbolic rational numbers.
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Siemann J, Petermann F. Innate or Acquired? - Disentangling Number Sense and Early Number Competencies. Front Psychol 2018; 9:571. [PMID: 29725316 PMCID: PMC5917196 DOI: 10.3389/fpsyg.2018.00571] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/03/2017] [Accepted: 04/04/2018] [Indexed: 01/29/2023] Open
Abstract
The clinical profile termed developmental dyscalculia (DD) is a fundamental disability affecting children already prior to arithmetic schooling, but the formal diagnosis is often only made during school years. The manifold associated deficits depend on age, education, developmental stage, and task requirements. Despite a large body of studies, the underlying mechanisms remain dubious. Conflicting findings have stimulated opposing theories, each presenting enough empirical support to remain a possible alternative. A so far unresolved question concerns the debate whether a putative innate number sense is required for successful arithmetic achievement as opposed to a pure reliance on domain-general cognitive factors. Here, we outline that the controversy arises due to ambiguous conceptualizations of the number sense. It is common practice to use early number competence as a proxy for innate magnitude processing, even though it requires knowledge of the number system. Therefore, such findings reflect the degree to which quantity is successfully transferred into symbols rather than informing about quantity representation per se. To solve this issue, we propose a three-factor account and incorporate it into the partly overlapping suggestions in the literature regarding the etiology of different DD profiles. The proposed view on DD is especially beneficial because it is applicable to more complex theories identifying a conglomerate of deficits as underlying cause of DD.
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Affiliation(s)
- Julia Siemann
- Department of Medical Psychology and Medical Sociology, University Medical Center Schleswig-Holstein, Kiel, Germany
| | - Franz Petermann
- Center for Clinical Psychology and Rehabilitation, University of Bremen, Bremen, Germany
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Schneider M, Merz S, Stricker J, De Smedt B, Torbeyns J, Verschaffel L, Luwel K. Associations of Number Line Estimation With Mathematical Competence: A Meta-analysis. Child Dev 2018; 89:1467-1484. [DOI: 10.1111/cdev.13068] [Citation(s) in RCA: 98] [Impact Index Per Article: 16.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/29/2023]
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Symbolic magnitude processing in elementary school children: A group administered paper-and-pencil measure (SYMP Test). Behav Res Methods 2018; 49:1361-1373. [PMID: 27553484 DOI: 10.3758/s13428-016-0792-3] [Citation(s) in RCA: 29] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/08/2022]
Abstract
The ability to compare symbolic numerical magnitudes correlates with children's concurrent and future mathematics achievement. We developed and evaluated a quick timed paper-and-pencil measure that can easily be used, for example in large-scale research, in which children have to cross out the numerically larger of two Arabic one- and two-digit numbers (SYMP Test). We investigated performance on this test in 1,588 primary school children (Grades 1-6) and examined in each grade its associations with mathematics achievement. The SYMP Test had satisfactory test-retest reliability. The SYMP Test showed significant and stable correlations with mathematics achievement for both one-digit and two-digit comparison, across all grades. This replicates the previously observed association between symbolic numerical magnitude processing and mathematics achievement, but extends it by showing that the association is observed in all grades in primary education and occurs for single- as well as multi-digit processing. Children with mathematical learning difficulties performed significantly lower on one-digit comparison and two-digit comparison in all grades. This all suggests satisfactory construct and criterion-related validity of the SYMP Test, which can be used in research, when performing large-scale (intervention) studies, and by practitioners, as screening measure to identify children at risk for mathematical difficulties or dyscalculia.
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Namkung JM, Fuchs LS, Koziol N. Does Initial Learning about the Meaning of Fractions Present Similar Challenges for Students with and without Adequate Whole-Number Skill? LEARNING AND INDIVIDUAL DIFFERENCES 2017; 61:151-157. [PMID: 29276363 DOI: 10.1016/j.lindif.2017.11.018] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
The purposes of this study were to (a) explore whether early fractions understanding at 4th grade is differentially challenging for students with versus without adequate whole-number competence and (b) identify specific whole-number skill associated with difficulty in fractions understanding. Based on initial whole-number competence, 1,108 4th graders were classified as having (a) adequate whole-number competence (n = 775), (b) less severe whole-number difficulty (n = 201), and (c) severe whole-number difficulty (n = 132). At the end of 4th grade, they were assessed on fractions understanding and further classified as with versus without difficulty in fractions understanding. Multi-level logistic regression indicated that compared to students with adequate whole-number competence, those with less severe whole-number difficulty were almost 5 times as likely to experience difficulty with fractions understanding whereas those with severe whole-number difficulty were about 32 times as likely to experience difficulty with fractions understanding. Students with severe whole-number difficulty were about 7 times as likely to experience difficulty with fractions understanding compared to those with less severe whole-number difficulty. Among students with adequate whole-number competence, the pretest whole-number skill distinguishing those with versus without difficulty in fractions understanding was basic division facts (i.e., 2-digit dividend ÷ 1-digit divisor) and simple multiplication (i.e., 3-digit × 1-digit without regrouping). The role of whole-number competence in developing initial fractions understanding and implications for instruction are discussed.
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Co-development of fraction magnitude knowledge and mathematics achievement from fourth through sixth grade. LEARNING AND INDIVIDUAL DIFFERENCES 2017. [DOI: 10.1016/j.lindif.2017.10.005] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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Krowka SK, Fuchs LS. Cognitive Profiles Associated With Responsiveness to Fraction Intervention. LEARNING DISABILITIES RESEARCH & PRACTICE : A PUBLICATION OF THE DIVISION FOR LEARNING DISABILITIES, COUNCIL FOR EXCEPTIONAL CHILDREN 2017; 32:216-230. [PMID: 29123333 PMCID: PMC5672985 DOI: 10.1111/ldrp.12146] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
This study examined differences in cognitive processing between 4th-grade students who respond adequately, as opposed to inadequately, to intervention on 3 fraction outcomes: number-line estimation, calculation, and word problems. Students were assessed on 7 cognitive processes and on the 3 fraction outcomes. Students were grouped as adequate or inadequate responders, using as the cut-point the control-group mean on pre-to-post improvement on the relevant measure. Between-group differences identified reasoning, concept formation, and listening comprehension related to all 3 fraction outcomes. On the number-line outcome, within-group profile analysis indicated that inadequate responders experienced low reasoning ability relative to their other forms of cognitive processing.
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Liu Y. Fraction magnitude understanding and its unique role in predicting general mathematics achievement at two early stages of fraction instruction. BRITISH JOURNAL OF EDUCATIONAL PSYCHOLOGY 2017; 88:345-362. [PMID: 28884808 DOI: 10.1111/bjep.12182] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/08/2016] [Revised: 07/07/2017] [Indexed: 11/30/2022]
Abstract
BACKGROUND Prior studies on fraction magnitude understanding focused mainly on students with relatively sufficient formal instruction on fractions whose fraction magnitude understanding is relatively mature. AIM This study fills a research gap by investigating fraction magnitude understanding in the early stages of fraction instruction. It extends previous findings to children with limited and primary formal fraction instruction. SAMPLE(S) Thirty-five fourth graders with limited fraction instruction and forty fourth graders with primary fraction instruction were recruited from a Chinese primary school. METHODS Children's fraction magnitude understanding was assessed with a fraction number line estimation task. Approximate number system (ANS) acuity was assessed with a dot discrimination task. Whole number knowledge was assessed with a whole number line estimation task. General reading and mathematics achievements were collected concurrently and 1 year later. RESULTS In children with limited fraction instruction, fraction representation was linear and fraction magnitude understanding was concurrently related to both ANS and whole number knowledge. In children with primary fraction instruction, fraction magnitude understanding appeared to (marginally) significantly predict general mathematics achievement 1 year later. CONCLUSIONS Fraction magnitude understanding emerged early during formal instruction of fractions. ANS and whole number knowledge were related to fraction magnitude understanding when children first began to learn about fractions in school. The predictive value of fraction magnitude understanding is likely constrained by its sophistication level.
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Affiliation(s)
- Yingyi Liu
- Department of Educational Psychology, Faculty of Education, The Chinese University of Hong Kong, Shatin, Hong Kong
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Identifying learning difficulties with fractions: A longitudinal study of student growth from third through sixth grade. CONTEMPORARY EDUCATIONAL PSYCHOLOGY 2017. [DOI: 10.1016/j.cedpsych.2015.11.002] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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Träff U, Skagerlund K, Olsson L, Östergren R. Pathways to arithmetic fact retrieval and percentage calculation in adolescents. BRITISH JOURNAL OF EDUCATIONAL PSYCHOLOGY 2017. [DOI: 10.1111/bjep.12170] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Ulf Träff
- Department of Behavioural Sciences and Learning; Linköping University; Sweden
| | - Kenny Skagerlund
- Department of Behavioural Sciences and Learning; Linköping University; Sweden
| | - Linda Olsson
- Department of Behavioural Sciences and Learning; Linköping University; Sweden
| | - Rickard Östergren
- Department of Behavioural Sciences and Learning; Linköping University; Sweden
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REFERENCES. Monogr Soc Res Child Dev 2017. [DOI: 10.1111/mono.12286] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
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Affiliation(s)
- Robert S. Siegler
- Department of Psychology, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213;
- The Siegler Center for Innovative Learning (SCIL), Beijing Normal University, Beijing 100875, China
| | - David W. Braithwaite
- Department of Psychology, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213;
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Kainulainen M, McMullen J, Lehtinen E. Early Developmental Trajectories Toward Concepts of Rational Numbers. COGNITION AND INSTRUCTION 2016. [DOI: 10.1080/07370008.2016.1251287] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
Affiliation(s)
- Mikko Kainulainen
- Department of Teacher Education and Centre for Learning Research, University of Turku, Finland
| | - Jake McMullen
- Department of Teacher Education and Centre for Learning Research, University of Turku, Finland
| | - Erno Lehtinen
- Department of Teacher Education and Centre for Learning Research, University of Turku, Finland
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Pathways to fraction learning: Numerical abilities mediate the relation between early cognitive competencies and later fraction knowledge. J Exp Child Psychol 2016; 152:242-263. [PMID: 27572521 DOI: 10.1016/j.jecp.2016.08.001] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/11/2016] [Revised: 08/02/2016] [Accepted: 08/03/2016] [Indexed: 11/23/2022]
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Van Hoof J, Verschaffel L, Van Dooren W. Number sense in the transition from natural to rational numbers. BRITISH JOURNAL OF EDUCATIONAL PSYCHOLOGY 2016; 87:43-56. [PMID: 27796031 DOI: 10.1111/bjep.12134] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/28/2014] [Revised: 09/29/2016] [Indexed: 11/30/2022]
Abstract
BACKGROUND Rational numbers are of critical importance both in mathematics and in other fields of science. However, they form a stumbling block for learners. One widely known source of the difficulty learners have with rational numbers is the natural number bias, that is the tendency to (inappropriately) apply natural number properties in rational number tasks. Still, it has been shown that a good understanding of natural numbers is highly predictive for mathematics achievement in general, and for performance on rational number tasks in particular. AIMS In this study, we further investigated the relation between learners' natural and rational number knowledge, specifically in cases where a natural number bias may lead to errors. SAMPLE Participants were 140 sixth graders from six different primary schools. METHOD Participants completed a symbolic and a non-symbolic natural number comparison task, a number line estimation task, and a rational number sense test. RESULTS Learners' natural number knowledge was found to be a good predictor of their rational number knowledge. However, after first controlling for learners' general mathematics achievement, their natural number knowledge only predicted the subaspect of operations with rational numbers. CONCLUSION The results of this study suggest that the relation between learners' natural and rational number knowledge can largely be explained by their relation with learners' general mathematics achievement.
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Affiliation(s)
- Jo Van Hoof
- Centre for Instructional Psychology and Technology, University of Leuven, Belgium
| | - Lieven Verschaffel
- Centre for Instructional Psychology and Technology, University of Leuven, Belgium
| | - Wim Van Dooren
- Centre for Instructional Psychology and Technology, University of Leuven, Belgium
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Mou Y, Li Y, Hoard MK, Nugent LD, Chu FW, Rouder JN, Geary DC. Developmental Foundations of Children's Fraction Magnitude Knowledge. COGNITIVE DEVELOPMENT 2016; 39:141-153. [PMID: 27773965 DOI: 10.1016/j.cogdev.2016.05.002] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
Abstract
The conceptual insight that fractions represent magnitudes is a critical yet daunting step in children's mathematical development, and the knowledge of fraction magnitudes influences children's later mathematics learning including algebra. In this study, longitudinal data were analyzed to identify the mathematical knowledge and domain-general competencies that predicted 8th and 9th graders' (n=122) knowledge of fraction magnitudes and its cross-grade gains. Performance on the fraction magnitude measures predicted 9th grade algebra achievement. Understanding and fluently identifying the numerator-denominator relation in 7th grade emerged as the key predictor of later fraction magnitudes knowledge in both 8th and 9th grades. Competence at using fraction procedures, knowledge of whole number magnitudes, and the central executive contributed to 9th but not 8th graders' fraction magnitude knowledge, and knowledge of whole number magnitude contributed to cross-grade gains. The key results suggest fluent processing of numerator-denominator relations presages students' understanding of fractions as magnitudes and that the integration of whole number and fraction magnitudes occurs gradually.
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