Numerical landmarks are useful--except when they're not.

Placing landmarks on number lines, such as marking each tenth on a 0-1 line with a hatch mark and the corresponding decimal, has been recommended as a useful tool for improving children's number sense. Four experiments indicated that some landmarks do have beneficial effects, others have harmful effects, and yet others have no effects on representations of common fractions (N/M). The effects of the landmarks were seen not only on the number line task where they appeared but also on a subsequent magnitude comparison task and on correlations with mathematics achievement tests. Landmarks appeared to exert their effects through the encodings and strategies that they promoted. Theoretical and educational implications are discussed.

[1]  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.

[2]  Andreas Nieder,et al.  Neuronal population coding of continuous and discrete quantity in the primate posterior parietal cortex , 2007, Proceedings of the National Academy of Sciences.

[3]  Xinlin Zhou,et al.  Chinese kindergartners’ automatic processing of numerical magnitude in Stroop-like tasks , 2007, Memory & cognition.

[4]  Michael Schneider,et al.  Representations of the magnitudes of fractions. , 2010, Journal of experimental psychology. Human perception and performance.

[5]  Lynn S. Fuchs,et al.  Improving At-Risk Learners' Understanding of Fractions , 2013 .

[6]  Portia C. Elliott,et al.  The Learning of Mathematics , 2007 .

[7]  S. Dehaene,et al.  The mental representation of parity and number magnitude. , 1993 .

[8]  J. Huttenlocher,et al.  The Coding of Spatial Location in Young Children , 1994, Cognitive Psychology.

[9]  Pierre Pica,et al.  Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures , 2008, Science.

[10]  Lieven Verschaffel,et al.  A validation of eye movements as a measure of elementary school children's developing number sense , 2008 .

[11]  Marco Zorzi,et al.  The mental representation of numerical fractions: real or integer? , 2007, Journal of experimental psychology. Human perception and performance.

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

[13]  E. Spelke,et al.  Log or linear? Distinct intuitions of the number scale in Western and Amazonian cultures , 2008 .

[14]  J. Rieser,et al.  Visual experience, visual field size, and the development of nonvisual sensitivity to the spatial structure of outdoor neighborhoods explored by walking. , 1992, Journal of experimental psychology. General.

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

[16]  Justin Halberda,et al.  Individual differences in non-verbal number acuity correlate with maths achievement , 2008, Nature.

[17]  Marie-Pascale Noël,et al.  Rational numbers: Componential versus holistic representation of fractions in a magnitude comparison task , 2009, Quarterly journal of experimental psychology.

[18]  M. Ashcraft,et al.  Cognitive processes of numerical estimation in children. , 2012, Journal of experimental child psychology.

[19]  D. Ansari.  Effects of development and enculturation on number representation in the brain , 2008, Nature Reviews Neuroscience.

[20]  S. Dehaene,et al.  Interactions between number and space in parietal cortex , 2005, Nature Reviews Neuroscience.

[21]  Klaus Willmes,et al.  On the development of the mental number line: more, less, or never holistic with increasing age? , 2004, Developmental psychology.

[22]  H. Barth,et al.  Developmental change in numerical estimation. , 2013, Journal of experimental psychology. General.

[23]  Dana Ganor-Stern,et al.  The Representation of Negative Numbers: Exploring the Effects of Mode of Processing and Notation , 2009, Quarterly journal of experimental psychology.

[24]  James W. Stigler,et al.  What Community College Developmental Mathematics Students Understand about Mathematics. , 2010 .

[25]  M. Mazzocco,et al.  Parts and 'holes': gaps in rational number sense among children with vs. without mathematical learning disabilities. , 2008, Developmental science.

[26]  J. G. Bremner,et al.  The development of relational landmark use in six- to twelve-month-old infants in a spatial orientation task. , 2000, Child development.

[27]  Douglas A. Grouws,et al.  Handbook of research on mathematics teaching and learning , 1992 .

[28]  Lisa K. Fazio,et al.  Strategy use and strategy choice in fraction magnitude comparison. , 2016, Journal of experimental psychology. Learning, memory, and cognition.

[29]  Aryn A Pyke,et al.  Developmental and individual differences in understanding of fractions. , 2013, Developmental psychology.

[30]  Steven A. Hecht,et al.  Toward an information processing account of individual differences in fraction skills , 1998 .

[31]  Hilary C Barth,et al.  The development of numerical estimation: evidence against a representational shift. , 2011, Developmental science.

[32]  Robert S. Siegler,et al.  Fractions: the new frontier for theories of numerical development , 2013, Trends in Cognitive Sciences.

[33]  Melissa W. Clearfield,et al.  The role of crawling and walking experience in infant spatial memory. , 2004, Journal of experimental child psychology.

[34]  Jack L. Vevea,et al.  Why do categories affect stimulus judgment? , 2000, Journal of experimental psychology. General.

[35]  Russell Gersten,et al.  Developmental predictors of fraction concepts and procedures. , 2013, Journal of experimental child psychology.

[36]  T L Hubbard,et al.  Spatial memory averaging, the landmark attraction effect, and representational gravity , 2000, Psychological research.

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

[38]  K. Priftis,et al.  Brain damage: Neglect disrupts the mental number line , 2002, Nature.

[39]  Mathematics Skills of 17-Year-Olds in the United States: 1978 to 2004. , 2010 .

[40]  Kathleen A. Cramer,et al.  Using manipulative models to build number sense for addition of fractions: NCTM 2002 yearbook , 2002 .

[41]  B. Rittle-Johnson,et al.  Developing Conceptual Understanding and Procedural Skill in Mathematics: An Iterative Process. , 2001 .

[42]  Michael Schneider,et al.  Mental number line, number line estimation, and mathematical achievement : Their interrelations in grades 5 and 6 , 2009 .

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

[44]  D. Geary.  Cognitive predictors of achievement growth in mathematics: a 5-year longitudinal study. , 2011, Developmental psychology.

[45]  Stella F. Lourenco,et al.  General Magnitude Representation in Human Infants , 2010, Psychological science.

[46]  D. Cohen,et al.  Numerical bias in bounded and unbounded number line tasks , 2011, Psychonomic bulletin & review.

[47]  Sue Betka,et al.  National Center for Education Evaluation and Regional Assistance , 2015 .

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

[49]  A. Lew.  Looking beyond the boundaries: time to put landmarks back on the cognitive map? , 2011, Psychological bulletin.

[50]  Robert S Siegler,et al.  The powers of noise-fitting: reply to Barth and Paladino. , 2011, Developmental science.

[51]  E. Spelke,et al.  Number-Space Mapping in Human Infants , 2010, Psychological science.

[52]  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.

[53]  W. Gevers,et al.  The SNARC effect does not imply a mental number line , 2008, Cognition.

[54]  E. Miller,et al.  Analog Numerical Representations in Rhesus Monkeys: Evidence for Parallel Processing , 2004, Journal of Cognitive Neuroscience.

[55]  Steven A Hecht,et al.  Sources of Group and Individual Differences in Emerging Fraction Skills. , 2010, Journal of educational psychology.

[56]  Clarissa A. Thompson,et al.  An integrated theory of whole number and fractions development , 2011, Cognitive Psychology.