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Liu S, Su Y, Suo D, Zhao J. Heuristic strategy of intuitive statistical inferences in 7- to 10-year-old children. J Exp Child Psychol 2024; 242:105907. [PMID: 38513328 DOI: 10.1016/j.jecp.2024.105907] [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: 09/18/2023] [Revised: 01/24/2024] [Accepted: 02/26/2024] [Indexed: 03/23/2024]
Abstract
Intuitive statistical inferences refer to making inferences about uncertain events based on limited probabilistic information, which is crucial for both human and non-human species' survival and reproduction. Previous research found that 7- and 8-year-old children failed in intuitive statistical inference tasks after heuristic strategies had been controlled. However, few studies systematically explored children's heuristic strategies of intuitive statistical inferences and their potential numerical underpinnings. In the current research, Experiment 1 (N = 81) examined 7- to 10-year-olds' use of different types of heuristic strategies; results revealed that children relied more on focusing on the absolute number strategy. Experiment 2 (N = 99) and Experiment 3 (N = 94) added continuous-format stimuli to examine whether 7- and 8-year-olds could make genuine intuitive statistical inferences instead of heuristics. Results revealed that both 7- and 8-year-olds and 9- and 10-year-olds performed better in intuitive statistical inference tasks with continuous-format stimuli, even after focusing on the absolute number strategy had been controlled. The results across the three experiments preliminarily hinted that the ratio processing system might rely on the approximate number system. Future research could clarify what specific numerical processing mechanism may be used and how it might support children's statistical intuitions.
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Affiliation(s)
- Siyi Liu
- School of Psychological and Cognitive Sciences, Beijing Key Laboratory of Behavior and Mental Health, Peking University, Beijing 100871, China
| | - Yanjie Su
- School of Psychological and Cognitive Sciences, Beijing Key Laboratory of Behavior and Mental Health, Peking University, Beijing 100871, China.
| | - Dachuan Suo
- Faculty of Psychology, Beijing Normal University, Beijing 100875, China
| | - Jiaxuan Zhao
- Graduate School of Education, University of Pennsylvania, Philadelphia, PA 19104, USA
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Abreu-Mendoza RA, Powell AB, Renninger KA, Rivera LM, Vulic J, Weimar S, Rosenberg-Lee M. Middle-schoolers' misconceptions in discretized nonsymbolic proportional reasoning explain fraction biases better than their continuous reasoning: Evidence from correlation and cluster analyses. Cogn Psychol 2023; 143:101575. [PMID: 37229979 DOI: 10.1016/j.cogpsych.2023.101575] [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: 12/16/2022] [Revised: 04/19/2023] [Accepted: 05/07/2023] [Indexed: 05/27/2023]
Abstract
Early emerging nonsymbolic proportional skills have been posited as a foundational ability for later fraction learning. A positive relation between nonsymbolic and symbolic proportional reasoning has been reported, as well as successful nonsymbolic training and intervention programs enhancing fraction magnitude skills. However, little is known about the mechanisms underlying this relationship. Of particular interest are nonsymbolic representations, which can be in continuous formats that may emphasize proportional relations and in discretized formats that may prompt erroneous whole-number strategies and hamper access to fraction magnitudes. We assessed the proportional comparison skills of 159 middle-school students (mean age = 12.54 years, 43% females, 55% males, 2% other or prefer not to say) across three types of representations: (a) continuous, unsegmented bars, (b) discretized, segmented bars that allowed counting strategies, and (c) symbolic fractions. Using both correlational and cluster approaches, we also examined their relations to symbolic fraction comparison ability. Within each stimulus type, we varied proportional distance, and in the discretized and symbolic stimuli, we also manipulated whole-number congruency. We found that fraction distance across all formats modulated middle-schoolers' performance; however, whole-number information affected discretized and symbolic comparison performance. Further, continuous and discretized nonsymbolic performance was related to fraction comparison ability; however, discretized skills explained variance above and beyond the contributions of continuous skills. Finally, our cluster analyses revealed three nonsymbolic comparison profiles: students who chose the bars with the largest number of segments (whole-number bias), chance-level performers, and high performers. Crucially, students with a whole-number bias profile showed this bias in their fraction skills and failed to show any symbolic distance modulation. Together, our results indicate that the relation between nonsymbolic and symbolic proportional skills may be determined by the (mis)conceptions based on discretized representations, rather than understandings of proportional magnitudes, suggesting that interventions focusing on competence with discretized representations may show dividends for fraction understanding.
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Affiliation(s)
| | - Arthur B Powell
- Department of Urban Education, Rutgers University-Newark, Newark, NJ, United States
| | - K Ann Renninger
- Department of Educational Studies, Swarthmore College, Swarthmore, PA, United States
| | - Luis M Rivera
- Department of Psychology, Rutgers University-Newark, Newark, NJ, United States
| | - John Vulic
- Sydney School of Education and Social Work, The University of Sydney, Australia
| | - Steve Weimar
- 21st Century Partnership for STEM Education, PA, United States
| | - Miriam Rosenberg-Lee
- Department of Psychology, Rutgers University-Newark, Newark, NJ, United States; Behavioral Neuroscience Program Graduate Program, Rutgers University-Newark, Newark, NJ, United States.
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Thompson CA, Mielicki MK, Rivera F, Fitzsimmons CJ, Scheibe DA, Sidney PG, Schiller LK, Taber JM, Waters EA. Leveraging Math Cognition to Combat Health Innumeracy. PERSPECTIVES ON PSYCHOLOGICAL SCIENCE 2023; 18:152-177. [PMID: 35943825 DOI: 10.1177/17456916221083277] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/31/2023]
Abstract
Rational numbers (i.e., fractions, percentages, decimals, and whole-number frequencies) are notoriously difficult mathematical constructs. Yet correctly interpreting rational numbers is imperative for understanding health statistics, such as gauging the likelihood of side effects from a medication. Several pernicious biases affect health decision-making involving rational numbers. In our novel developmental framework, the natural-number bias-a tendency to misapply knowledge about natural numbers to all numbers-is the mechanism underlying other biases that shape health decision-making. Natural-number bias occurs when people automatically process natural-number magnitudes and disregard ratio magnitudes. Math-cognition researchers have identified individual differences and environmental factors underlying natural-number bias and devised ways to teach people how to avoid these biases. Although effective interventions from other areas of research can help adults evaluate numerical health information, they circumvent the core issue: people's penchant to automatically process natural-number magnitudes and disregard ratio magnitudes. We describe the origins of natural-number bias and how researchers may harness the bias to improve rational-number understanding and ameliorate innumeracy in real-world contexts, including health. We recommend modifications to formal math education to help children learn the connections among natural and rational numbers. We also call on researchers to consider individual differences people bring to health decision-making contexts and how measures from math cognition might identify those who would benefit most from support when interpreting health statistics. Investigating innumeracy with an interdisciplinary lens could advance understanding of innumeracy in theoretically meaningful and practical ways.
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Affiliation(s)
| | | | - Ferdinand Rivera
- Department of Mathematics and Statistics, San Jose State University
| | | | | | | | - Lauren K Schiller
- Department of Human Development, Teachers College, Columbia University
| | | | - Erika A Waters
- Department of Surgery, Washington University School of Medicine in St. Louis
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Hurst MA, Levine SC. Children's understanding of most is dependent on context. Cognition 2022; 225:105149. [PMID: 35533418 DOI: 10.1016/j.cognition.2022.105149] [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: 08/13/2021] [Revised: 04/24/2022] [Accepted: 04/25/2022] [Indexed: 11/26/2022]
Abstract
Children struggle with the quantifier "most". Often, this difficulty is attributed to an inability to interpret most proportionally, with children instead relying on absolute quantity comparisons. However, recent research in proportional reasoning more generally has provided new insight into children's apparent difficulties, revealing that their overreliance on absolute amount is unique to contexts in which the absolute amount can be counted and interferes with proportional information. Across two experiments, we test whether 4- to 6-year-old children's interpretation of most is similarly dependent on the discreteness of the stimuli when comparing two different quantities (e.g., who ate most of their chocolate?) and when verifying whether a single amount can be described with the term most (e.g., is most of the butterfly colored in?). We find that children's interpretation of most does depend on the stimulus format. When choosing between absolutely more vs. proportionally more as depicting most, children showed stronger absolute-based errors with discrete stimuli than continuous stimuli, and by 6-years-old were able to reason proportionally with continuous stimuli, despite still demonstrating strong absolute interference with discrete stimuli. In contrast, children's yes/no judgements of single amounts, where conflicting absolute information is not a factor, showed a weaker understanding of most for continuous stimuli than for discrete stimuli. Together, these results suggest that children's difficulty with most is more nuanced than previously understood: it depends on the format and availability of proportional vs. absolute amounts and develops substantially from 4- to 6-years-old.
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Peer Tutoring and Math Digital Tools: A Promising Combination in Middle School. MATHEMATICS 2022. [DOI: 10.3390/math10132360] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/05/2023]
Abstract
Peer tutoring in combination with math digital tools was employed with middle school students learning mathematics. A total of 112 students in 9th grade (14 to 15 years old) participated in the study. A pretest–posttest with control group design was used. Students worked with systems of linear equations during the experience. The effects of the intervention of peer tutoring in combination with math digital tools on students’ mathematics achievement were examined using quantitative methods. The way students in the experimental group learned and their motivation towards using digital tools compared with students in the control group were analysed qualitatively. Statistically significant improvements and a large effect size were reported for students’ mathematics achievement in the experimental group. No statistically significant differences were reported between the pretest and the posttest for the control group. The qualitative analyses revealed that students in the experimental group achieved a higher level of autonomous learning, showed a greater association of mathematical concepts, helped their peers more, did more exercises and problems than students in the control group, and enjoyed the experience.
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Gunderson EA, Hildebrand L. Relations among spatial skills, number line estimation, and exact and approximate calculation in young children. J Exp Child Psychol 2021; 212:105251. [PMID: 34333360 DOI: 10.1016/j.jecp.2021.105251] [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: 09/11/2020] [Revised: 06/25/2021] [Accepted: 07/02/2021] [Indexed: 10/20/2022]
Abstract
Decades of research have established that spatial skills correlate with numerical skills. However, because both spatial and numerical skills are multidimensional, we sought to determine how specific spatial skills relate to specific numeracy skills. We used a cohort-sequential design, assessing a large diverse sample of students (N = 612, initially in pre-kindergarten [pre-K]-3rd grade, 4-9 years of age) at four time points spanning 2 years. We examined how initial levels of five spatial skills (visuospatial working memory [VSWM], mental transformation, mental rotation, proportional reasoning, and analog magnitude system [AMS] acuity) related to initial levels and growth rates in exact and approximate calculation skills, and we further investigated number line estimation as a potential mediator. We found unique patterns of relations between spatial skills and numeracy. Initial levels of mental rotation, proportional reasoning, and AMS acuity related to initial levels of exact calculation skill; initial levels of AMS acuity related to initial levels of approximate calculation; and initial levels of proportional reasoning related to initial levels of number line estimation. VSWM and mental transformation did not relate to numeracy skills after controlling for other spatial skills. Initial levels of number line estimation related to both exact and approximate calculation after controlling for spatial skills. Notably, neither spatial skills nor number line estimation predicted growth in exact or approximate calculation skills. These results indicate that there is specificity in the time-invariant relations between spatial skills and numeracy, and they suggest that researchers and educators should treat spatial skills and numeracy as multidimensional constructs with complex and unique interrelations.
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Affiliation(s)
| | - Lindsey Hildebrand
- Department of Psychology and Neuroscience, Boston College, Chestnut Hill, MA 02467, USA
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Abreu-Mendoza RA, Coulanges L, Ali K, Powell AB, Rosenberg-Lee M. From Non-symbolic to Symbolic Proportions and Back: A Cuisenaire Rod Proportional Reasoning Intervention Enhances Continuous Proportional Reasoning Skills. Front Psychol 2021; 12:633077. [PMID: 34093313 PMCID: PMC8177818 DOI: 10.3389/fpsyg.2021.633077] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/24/2020] [Accepted: 04/23/2021] [Indexed: 12/03/2022] Open
Abstract
The persistent educational challenges that fractions pose call for developing novel instructional methods to better prepare students for fraction learning. Here, we examined the effects of a 24-session, Cuisenaire rod intervention on a building block for symbolic fraction knowledge, continuous and discrete non-symbolic proportional reasoning, in children who have yet to receive fraction instruction. Participants were 34 second-graders who attended the intervention (intervention group) and 15 children who did not participate in any sessions (control group). As attendance at the intervention sessions was irregular (median = 15.6 sessions, range = 1-24), we specifically examined the effect of the number of sessions completed on their non-symbolic proportional reasoning. Our results showed that children who attended a larger number of sessions increased their ability to compare non-symbolic continuous proportions. However, contrary to our expectations, they also decreased their ability to compare misleading discretized proportions. In contrast, children in the Control group did not show any change in their performance. These results provide further evidence on the malleability of non-symbolic continuous proportional reasoning and highlight the rigidity of counting knowledge interference on discrete proportional reasoning.
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Affiliation(s)
| | - Linsah Coulanges
- Department of Psychology, Rutgers University, Newark, NJ, United States
| | - Kendell Ali
- Graduate School of Education, Rutgers, The State University of New Jersey, New Brunswick, NJ, United States
| | - Arthur B. Powell
- Department of Urban Education, Rutgers University, Newark, NJ, United States
| | - Miriam Rosenberg-Lee
- Department of Psychology, Rutgers University, Newark, NJ, United States
- Center for Molecular and Behavioral Neuroscience, Rutgers, The State University of New Jersey, Newark, NJ, United States
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