Costs and benefits of representational change: effects of context on age and sex differences in symbolic magnitude estimation.

Studies have reported high correlations in accuracy across estimation contexts, robust transfer of estimation training to novel numerical contexts, and adults drawing mistaken analogies between numerical and fractional values. We hypothesized that these disparate findings may reflect the benefits and costs of learning linear representations of numerical magnitude. Specifically, children learn that their default logarithmic representations are inappropriate for many numerical tasks, leading them to adopt more appropriate linear representations despite linear representations being inappropriate for estimating fractional magnitude. In Experiment 1, this hypothesis accurately predicted a developmental shift from logarithmic to linear estimates of numerical magnitude and a negative correlation between accuracy of numerical and fractional magnitude estimates (r=-.80). In Experiment 2, training that improved numerical estimates also led to poorer fractional magnitude estimates. Finally, both before and after training that eliminated age differences in estimation accuracy, complementary sex differences were observed across the two estimation contexts.

[1]  Robert S. Siegler,et al.  Development of Numerical Estimation: A Review , 2004 .

[2]  D. Rimoin.  Emery and Rimoin's Principles and Practice of Medical Genetics and Genomics , 1997 .

[3]  S. Dehaene,et al.  THREE PARIETAL CIRCUITS FOR NUMBER PROCESSING , 2003, Cognitive neuropsychology.

[4]  R. Cohen,et al.  Development of spatial representations: role of task demands and familiarity with the environment. , 1979, Child development.

[5]  Philippe Pinel,et al.  Distributed and Overlapping Cerebral Representations of Number, Size, and Luminance during Comparative Judgments , 2004, Neuron.

[6]  Klaus Willmes,et al.  On the functional role of human parietal cortex in number processing: How gender mediates the impact of a ‘virtual lesion’ induced by rTMS , 2006, Neuropsychologia.

[7]  Margariete Montague Wheeler,et al.  The Development of Concepts and Strategies Used in Computational Estimation. , 1989 .

[8]  Gender‐Based Differential Item Performance in Mathematics Achievement Items , 1987 .

[9]  William P. Banks,et al.  The apparent magnitude of number scaled by random production , 1974 .

[10]  J. Huttenlocher,et al.  Early sex differences in spatial skill. , 1999, Developmental psychology.

[11]  John R. Anderson,et al.  The dynamics of scaling: a memory-based anchor model of category rating and absolute identification. , 2005, Psychological review.

[12]  K. McGilly,et al.  Classroom lessons: Integrating cognitive theory and classroom practice. , 1994 .

[13]  ROBERT S. MOYER,et al.  Time required for Judgements of Numerical Inequality , 1967, Nature.

[14]  Robert S. Siegler,et al.  Rightstart: Providing the central conceptual prerequisites for first formal learning of arithmetic to students at risk for school failure. , 1994 .

[15]  Geetha B. Ramani,et al.  Promoting broad and stable improvements in low-income children's numerical knowledge through playing number board games. , 2008, Child development.

[16]  R. Solomon,et al.  A control group design for experimental studies of developmental processes. , 1968, Psychological bulletin.

[17]  E. Fennema,et al.  Gender differences in mathematics performance: a meta-analysis. , 1990, Psychological bulletin.

[18]  M. Potter,et al.  Time to understand pictures and words , 1975, Nature.

[19]  Jo-Anne LeFevre,et al.  The Development of Procedural and Conceptual Knowledge in Computational Estimation. , 1993 .

[20]  D. Grimes,et al.  Patients' understanding of medical risks: implications for genetic counseling. , 1999, Obstetrics and gynecology.

[21]  E. Spelke,et al.  Large number discrimination in 6-month-old infants , 2000, Cognition.

[22]  Herbert A. Simon,et al.  The Sciences of the Artificial , 1970 .

[23]  R. Siegler.  Microgenetic Studies of Self-Explanation , 2002 .

[24]  M. Potter,et al.  Pictures in sentences: understanding without words. , 1986, Journal of experimental psychology. General.

[25]  J. Hiebert.  Conceptual and procedural knowledge : the case of mathematics , 1987 .

[26]  R. Siegler,et al.  The Development of Numerical Estimation , 2003, Psychological science.

[27]  D. Berch,et al.  Extracting parity and magnitude from Arabic numerals: developmental changes in number processing and mental representation. , 1999, Journal of experimental child psychology.

[28]  Julie L. Booth,et al.  Numerical magnitude representations influence arithmetic learning. , 2008, Child development.

[29]  A. Su,et al.  The National Council of Teachers of Mathematics , 1932, The Mathematical Gazette.

[30]  G. Roland,et al.  Council of Teachers of Mathematics , 1999 .

[31]  Susan M. Barnett,et al.  When and where do we apply what we learn? A taxonomy for far transfer. , 2002, Psychological bulletin.

[32]  Julie L. Booth,et al.  Development of numerical estimation in young children. , 2004, Child development.

[33]  M. Ashcraft.  Cognitive arithmetic: A review of data and theory , 1992, Cognition.

[34]  David J. Freedman,et al.  Representation of the Quantity of Visual Items in the Primate Prefrontal Cortex , 2002, Science.

[35]  Daniel Ansari,et al.  Neural correlates of symbolic number processing in children and adults , 2005, Neuroreport.

[36]  Mary K. Hoard,et al.  Cognitive mechanisms underlying achievement deficits in children with mathematical learning disability. , 2007, Child development.

[37]  David C. Geary,et al.  Development of Number Line Representations in Children With Mathematical Learning Disability , 2008, Developmental neuropsychology.

[38]  N. Granott,et al.  Microdevelopment: Transition Processes in Development and Learning , 2009 .

[39]  N. Cole,et al.  Gender and fair assessment , 1997 .

[40]  E. Spelke,et al.  Language and Conceptual Development series Core systems of number , 2004 .

[41]  Michelene T. H. Chi,et al.  Self-Explanations: How Students Study and Use Examples in Learning To Solve Problems. Technical Report No. 9. , 1987 .

[42]  Rochel Gelman,et al.  Measurement Estimation: Learning to Map the Route From Number to Quantity and Back , 1998 .

[43]  Thomas P. Carpenter,et al.  Results from the Second Mathematics Assessment of the National Assessment of Educational Progress , 1981 .

[44]  B. Kolb,et al.  Sex differences in spatial learning and prefrontal and parietal cortical dendritic morphology in the meadow vole, Microtus pennsylvanicus , 1998, Brain Research.

[45]  N. Makris,et al.  Normal sexual dimorphism of the adult human brain assessed by in vivo magnetic resonance imaging. , 2001, Cerebral cortex.

[46]  C. Gallistel,et al.  Preverbal and verbal counting and computation , 1992, Cognition.

[47]  Robert S. Siegler,et al.  Representational change and children’s numerical estimation , 2007, Cognitive Psychology.

[48]  E. Spelke,et al.  Sources of mathematical thinking: behavioral and brain-imaging evidence. , 1999, Science.

[49]  Clarissa A. Thompson,et al.  The trouble with transfer: insights from microgenetic changes in the representation of numerical magnitude. , 2008, Child development.

[50]  Clarissa A. Thompson,et al.  Early development of spatial-numeric associations: evidence from spatial and quantitative performance of preschoolers. , 2010, Developmental science.

[51]  S. Dehaene,et al.  Abstract representations of numbers in the animal and human brain , 1998, Trends in Neurosciences.

[52]  Geetha B. Ramani,et al.  Playing linear numerical board games promotes low-income children's numerical development. , 2008, Developmental science.

[53]  Terry L. Childers,et al.  1 When Two and Two is Not Equal to Four : Errors in Processing Multiple Percentage Changes HAIPENG ( , 2007 .

[54]  R. Siegler,et al.  Strategy choice procedures and the development of multiplication skill. , 1988, Journal of experimental psychology. General.

[55]  John E. Opfer,et al.  Representational change and magnitude estimation: Why young children can make more accurate salary comparisons than adults , 2008, Cognition.

[56]  Julie L. Booth,et al.  Developmental and individual differences in pure numerical estimation. , 2006, Developmental psychology.

[57]  H. Ten Have,et al.  Genetic Counseling , 2019, Dictionary of Global Bioethics.

[58]  J. Burkell,et al.  What are the chances? Evaluating risk and benefit information in consumer health materials. , 2004, Journal of the Medical Library Association : JMLA.

[59]  D Kahneman,et al.  On the reality of cognitive illusions. , 1996, Psychological review.

[60]  S. Dehaene,et al.  The priming method: imaging unconscious repetition priming reveals an abstract representation of number in the parietal lobes. , 2001, Cerebral cortex.

[61]  Ann Dowker,et al.  Estimation Strategies of Four Groups , 1996 .

[62]  G. Murphy,et al.  The Big Book of Concepts , 2002 .

[63]  David C. Geary,et al.  Sexual selection and sex differences in mathematical abilities , 1996, Behavioral and Brain Sciences.

[64]  J. Hagen,et al.  Estimates and Estimate-Based Inferences in Young Children. , 1971 .

[65]  Z Kourtzi,et al.  Representation of Perceived Object Shape by the Human Lateral Occipital Complex , 2001, Science.

[66]  C. Brainerd.  Young Children's Mental Arithmetic Errors: A Working-Memory Analysis. , 1983 .

[67]  Elida V. Laski,et al.  Is 27 a big number? Correlational and causal connections among numerical categorization, number line estimation, and numerical magnitude comparison. , 2007, Child development.

[68]  D. Geary,et al.  PSYCHOLOGICAL SCIENCE IN THE PUBLIC INTEREST The Science of Sex Differences in Science and Mathematics , 2022 .

[69]  Jonathan Evans,et al.  Frequency versus probability formats in statistical word problems , 2000, Cognition.

[70]  Jamie I. D. Campbell.  Handbook of mathematical cognition , 2004 .

[71]  Judith F. Kroll,et al.  Conceptual representation of pictures and words: Reply to Clark. , 1987 .

[72]  E. Spelke.  Sex differences in intrinsic aptitude for mathematics and science?: a critical review. , 2005, The American psychologist.

[73]  Richard De Lisi,et al.  Biology, society, and behavior : the development of sex differences in cognition , 2002 .

[74]  V Menon,et al.  Cerebral Cortex doi:10.1093/cercor/bhi055 Developmental Changes in Mental Arithmetic: Evidence for Increased Functional Specialization in the Left Inferior Parietal Cortex , 2005 .