How Does Space Interact with Numbers

Mathematics requires visuo-spatial processing. This is obvious for geometry which is ostensibly spatial and requires creating, maintaining, and performing transformations on 2–D or 3–D visuospatial models. However, it is also the case for other mathematical dimensions. Understanding the place-value system of Arabic numbers for instance requires the processing of the position of each digit in the number. Doing multi-digit calculation in columns similarly involves the processing of spatial information; and the use of visual images has been shown to be positively correlated with higher mathematical word-problem-solving performance (Van Garderen 2006). Globally, numerous researchers have found close connection between spatial thinking and mathematics (Mix and Cheng 2012) with good math abilities in people with higher spatial skills. In the first part of this chapter, we will examine evidence showing that number magnitude is spatially coded. Then, we will show that the spatial representation of numbers improves with age. Third, we will report the effects of weak visuo-spatial abilities on numerical development in children. Finally, we will consider how this space-number interaction has been considered in training aiming at fostering numerical development.

[1]  Elizabeth S. Spelke,et al.  Non-symbolic arithmetic abilities and mathematics achievement in the first year of formal schooling , 2010, Cognition.

[2]  M. Casey,et al.  Spatial-Mechanical Reasoning Skills versus Mathematics Self-Confidence as Mediators of Gender Differences on Mathematics Subtests Using Cross-National Gender-Based Items , 2001 .

[3]  D. Uttal,et al.  The malleability of spatial skills: a meta-analysis of training studies. , 2013, Psychological bulletin.

[4]  Wim Fias,et al.  The impact of verbal working memory on number-space associations. , 2014, Journal of experimental psychology. Learning, memory, and cognition.

[5]  B. Rourke.  Nonverbal Learning Disabilities: The Syndrome and the Model , 1989 .

[6]  John Bynner,et al.  Numeracy and employment , 1997 .

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

[8]  Martin H. Fischer,et al.  Mental movements without magnitude? A study of spatial biases in symbolic arithmetic , 2008, Cognition.

[9]  Andrea Facoetti,et al.  Developmental trajectory of number acuity reveals a severe impairment in developmental dyscalculia , 2010, Cognition.

[10]  Brian Butterworth,et al.  Developmental dyscalculia and basic numerical capacities: a study of 8–9-year-old students , 2004, Cognition.

[11]  Laura G. Barron,et al.  Relative Validity of Distinct Spatial Abilities: An Example with Implications for Diversity , 2013 .

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

[13]  W. Fias,et al.  Number sense in children with visuospatial disabilities: Orientation of the mental number line , 2005 .

[14]  C. Gallistel,et al.  Non-verbal numerical cognition: from reals to integers , 2000, Trends in Cognitive Sciences.

[15]  C. Cornoldi,et al.  Arithmetic Difficulties in Children With Visuospatial Learning Disability (VLD) , 2003, Child neuropsychology : a journal on normal and abnormal development in childhood and adolescence.

[16]  C. Cornoldi,et al.  An analysis of the criteria used to diagnose children with Nonverbal Learning Disability (NLD) , 2014, Child neuropsychology : a journal on normal and abnormal development in childhood and adolescence.

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

[18]  Ulrike Cress,et al.  Sensori-motor spatial training of number magnitude representation , 2011, Psychonomic bulletin & review.

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

[20]  P. Nichelli,et al.  Right hemisphere developmental learning disability: A case study , 1995 .

[21]  Amy Devine,et al.  Developmental dyscalculia is related to visuo-spatial memory and inhibition impairment☆ , 2013, Cortex.

[22]  K. Wynn,et al.  Large-Number Addition and Subtraction by 9-Month-Old Infants , 2004, Psychological science.

[23]  Stanislas Dehaene,et al.  The neural basis of the Weber–Fechner law: a logarithmic mental number line , 2003, Trends in Cognitive Sciences.

[24]  Koleen McCrink,et al.  Operational momentum in large-number addition and subtraction by 9-month-olds. , 2009, Journal of experimental child psychology.

[25]  A. Zivotofsky,et al.  Bisections in two languages: When number processing, spatial representation, and habitual reading direction interact , 2010, Neuropsychologia.

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

[27]  W. Fias,et al.  Semantic Influences on Feature-Based Attention Due to Overlap of Neural Circuits , 2002, Cortex.

[28]  M. Casey,et al.  The Influence of Spatial Ability on Gender Differences in Mathematics College Entrance Test Scores across Diverse Samples. , 1995 .

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

[30]  S. Rotzer,et al.  Dysfunctional neural network of spatial working memory contributes to developmental dyscalculia , 2009, Neuropsychologia.

[31]  Delinda van Garderen,et al.  Spatial Visualization, Visual Imagery, and Mathematical Problem Solving of Students With Varying Abilities , 2006 .

[32]  F. Restle.  Speed of Adding and Comparing Numbers. , 1970 .

[33]  Michael D. Dodd,et al.  Perceiving numbers causes spatial shifts of attention , 2003, Nature Neuroscience.

[34]  Stanislas Dehaene,et al.  Précis of The Number Sense , 2001 .

[35]  Wim Gevers,et al.  Look, no hands: A perceptual task shows that number magnitude induces shifts of attention , 2008, Psychonomic bulletin & review.

[36]  Freya Acar,et al.  A Working Memory Account of the Interaction between Numbers and Spatial Attention , 2014, Quarterly journal of experimental psychology.

[37]  Peter Brugger,et al.  Stimulus-response compatibility in representational space , 1998, Neuropsychologia.

[38]  Bert De Smedt,et al.  Defective number module or impaired access? Numerical magnitude processing in first graders with mathematical difficulties. , 2011, Journal of experimental child psychology.

[39]  M. Noël,et al.  Basic numerical skills in children with mathematics learning disabilities: A comparison of symbolic vs non-symbolic number magnitude processing , 2007, Cognition.

[40]  J. Lammertyn,et al.  The hunt for SNARC , 2005 .

[41]  B. Reynvoet,et al.  Automatic number priming effects in adults with and without mathematical learning disabilities , 2014, Front. Psychol..

[42]  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.

[43]  D. Lubinski,et al.  Spatial ability for STEM domains: Aligning over 50 years of cumulative psychological knowledge solidifies its importance. , 2009 .

[44]  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.

[45]  Stella F. Lourenco,et al.  Spatial attention and the mental number line: Evidence for characteristic biases and compression , 2007, Neuropsychologia.

[46]  Marco Zorzi,et al.  Numerical estimation in preschoolers. , 2010, Developmental psychology.

[47]  David C. Geary,et al.  External validation of the strategy choice model for addition , 1989 .

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

[49]  Stanislas Dehaene,et al.  Moving along the Number Line: Operational Momentum in Nonsymbolic Arithmetic , 2006 .

[50]  Mark H. Ashcraft,et al.  Cognitive arithmetic: Evidence for retrieval and decision processes in mental addition. , 1978 .

[51]  Mauro Pesenti,et al.  Interference of lateralized distractors on arithmetic problem solving: a functional role for attention shifts in mental calculation , 2016, Psychological research.

[52]  Tracy Packiam Alloway,et al.  Working memory, reading, and mathematical skills in children with developmental coordination disorder. , 2007, Journal of experimental child psychology.

[53]  Geetha B. Ramani,et al.  Playing linear number board games—but not circular ones—improves low-income preschoolers’ numerical understanding. , 2009 .

[54]  Juha Silvanto,et al.  The mental number line modulates visual cortical excitability , 2009, Neuroscience Letters.

[55]  J. Opfer,et al.  How Numbers Bias Preschoolers’ Spatial Search , 2011 .

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

[57]  S. Shaki,et al.  Numbers and space: Associations and dissociations , 2009, Psychonomic bulletin & review.

[58]  Wolf Schwarz,et al.  Moving the eyes along the mental number line: Comparing SNARC effects with saccadic and manual responses , 2004, Perception & psychophysics.

[59]  Martin H. Fischer,et al.  Reading space into numbers – a cross-linguistic comparison of the SNARC effect , 2008, Cognition.

[60]  Stéphane Rusinek,et al.  L’outil « estimateur », la ligne numérique mentale et les habiletés arithmétiques , 2010 .

[61]  S. Dehaene,et al.  Mathematical difficulties in developmental coordination disorder: Symbolic and nonsymbolic number processing. , 2015, Research in developmental disabilities.

[62]  Daniela Lucangeli,et al.  Spatial Working Memory and Arithmetic Deficits in Children With Nonverbal Learning Difficulties , 2010, Journal of learning disabilities.

[63]  S. Dehaene.  Varieties of numerical abilities , 1992, Cognition.

[64]  Kelly S. Mix,et al.  The relation between space and math: developmental and educational implications. , 2012, Advances in child development and behavior.

[65]  Stella F. Lourenco,et al.  Bisecting the mental number line in near and far space , 2010, Brain and Cognition.

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

[67]  Giuseppe Vallar,et al.  Visualizing numbers in the mind's eye: The role of visuo-spatial processes in numerical abilities , 2008, Neuroscience & Biobehavioral Reviews.

[68]  Wim Fias,et al.  Shared spatial representations for numbers and space: the reversal of the SNARC and the Simon effects. , 2006, Journal of experimental psychology. Human perception and performance.

[69]  Marie-Pascale Noël,et al.  Images of numbers, or “when 98 is upper left and 6 sky blue” , 1992, Cognition.

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

[71]  Stanislas Dehaene,et al.  Dynamic representations underlying symbolic and nonsymbolic calculation: Evidence from the operational momentum effect , 2009, Attention, perception & psychophysics.

[72]  Nancy C. Jordan,et al.  Dyscalculia from a developmental and differential perspective , 2013, Front. Psychol..

[73]  Wim Fias,et al.  Verbal-spatial and visuospatial coding of number-space interactions. , 2010, Journal of experimental psychology. General.

[74]  C. Semenza,et al.  The role of working memory in the association between number magnitude and space. , 2008, Acta psychologica.

[75]  Maciej Haman,et al.  The spatial-numerical congruity effect in preschoolers. , 2012, Journal of experimental child psychology.

[76]  S. Dehaene,et al.  Is numerical comparison digital? Analogical and symbolic effects in two-digit number comparison. , 1990, Journal of experimental psychology. Human perception and performance.

[77]  Wim Fias,et al.  Spatial representation of number , 2014 .

[78]  T. Alloway,et al.  Working Memory and Learning in Children With Developmental Coordination Disorder and Specific Language Impairment , 2008, Journal of learning disabilities.

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

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

[81]  Yang Seok Cho,et al.  Polarity correspondence: A general principle for performance of speeded binary classification tasks. , 2006, Psychological bulletin.

[82]  M. H. Fischer,et al.  Number processing induces spatial performance biases , 2001, Neurology.

[83]  K. Moeller,et al.  A general number-to-space mapping deficit in developmental dyscalculia. , 2015, Research in developmental disabilities.

[84]  Ghislain Deslongchamps,et al.  When do spatial abilities support student comprehension of STEM visualizations? , 2013, Cognitive Processing.

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

[86]  Martin H. Fischer,et al.  Spatial representations in number processing--evidence from a pointing task , 2003 .

[87]  Hermann Bulf,et al.  Small on the left, large on the right: numbers orient visual attention onto space in preverbal infants. , 2016, Developmental science.

[88]  Marco Zorzi,et al.  Explicit versus Implicit Processing of Representational Space in Neglect: Dissociations in Accessing the Mental Number Line , 2006, Journal of Cognitive Neuroscience.

[89]  U. Cress,et al.  How to rapidly construct a spatial-numerical representation in preliterate children (at least temporarily). , 2016, Developmental science.

[90]  G. Vallar,et al.  Tapping effects on numerical bisection , 2010, Experimental Brain Research.

[91]  P. Gordon.  Numerical Cognition Without Words: Evidence from Amazonia , 2004, Science.

[92]  F. Rivera-batiz,et al.  Quantitative Literacy and the Likelihood of Employment among Young Adults in the United States , 1992 .

[93]  Yves Rossetti,et al.  Interference between number processing and line bisection: a methodology , 2005, Neuropsychologia.

[94]  S. Dehaene,et al.  Exact and Approximate Arithmetic in an Amazonian Indigene Group , 2004, Science.

[95]  B. Rourke,et al.  Disabilities of Arithmetic and Mathematical Reasoning , 1997, Journal of learning disabilities.

[96]  Virginie Crollen,et al.  Spatial and numerical processing in children with high and low visuospatial abilities. , 2015, Journal of experimental child psychology.

[97]  Pieter Reitsma,et al.  Developing access to number magnitude: a study of the SNARC effect in 7- to 9-year-olds. , 2008, Journal of experimental child psychology.

[98]  J. Moss,et al.  Effects of mental rotation training on children’s spatial and mathematics performance: A randomized controlled study , 2015, Trends in Neuroscience and Education.

[99]  T. Vecchi,et al.  Blind individuals show pseudoneglect in bisecting numerical intervals , 2011, Attention, perception & psychophysics.

[100]  Patrik Vuilleumier,et al.  The Number Space and Neglect , 2004, Cortex.

[101]  Guilherme Wood,et al.  On the Cognitive Link between Space and Number: A Meta-Analysis of the SNARC Effect , 2008 .

[102]  Avishai Henik,et al.  Automatic activation of internal magnitudes: a study of developmental dyscalculia. , 2005, Neuropsychology.

[103]  W. Fias.  The Importance of Magnitude Information in Numerical Processing: Evidence from the SNARC Effect , 1996 .

[104]  M. McCourt,et al.  Pseudoneglect: a review and meta-analysis of performance factors in line bisection tasks , 2000, Neuropsychologia.

[105]  Wim Fias,et al.  Oculomotor bias induced by number perception. , 2004, Experimental psychology.

[106]  Stanislas Dehaene,et al.  Two mental calculation systems: A case study of severe acalculia with preserved approximation , 1991, Neuropsychologia.

[107]  A. Henik,et al.  Attention, automaticity, and developmental dyscalculia. , 2009, Neuropsychology.

[108]  Michael von Aster,et al.  Mental number line training in children with developmental dyscalculia , 2011, NeuroImage.

[109]  W. Fias,et al.  The development of the SNARC effect: evidence for early verbal coding. , 2012, Journal of experimental child psychology.

[110]  Paul B. Buckley,et al.  Comparisons of digits and dot patterns. , 1974, Journal of experimental psychology.

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

[112]  C. Cornoldi,et al.  Intuitive geometry and visuospatial working memory in children showing symptoms of nonverbal learning disabilities , 2013, Child neuropsychology : a journal on normal and abnormal development in childhood and adolescence.

[113]  L. Verschaffel,et al.  The predictive value of numerical magnitude comparison for individual differences in mathematics achievement. , 2009, Journal of experimental child psychology.

[114]  Daniel Ansari,et al.  Mapping numerical magnitudes onto symbols: the numerical distance effect and individual differences in children's mathematics achievement. , 2009, Journal of experimental child psychology.

[115]  Mauro Pesenti,et al.  Attentional Bias Induced by Solving Simple and Complex Addition and Subtraction Problems , 2014, Quarterly journal of experimental psychology.

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

[117]  Guilherme Wood,et al.  Crossed Hands and the Snarc Effect: Afailure to Replicate Dehaene, Bossini and Giraux (1993) , 2006, Cortex.

[118]  Wim Fias,et al.  How number is associated with space? The role of working memory. , 2011 .

[119]  W Fias,et al.  Irrelevant digits affect feature-based attention depending on the overlap of neural circuits. , 2001, Brain research. Cognitive brain research.

[120]  Jean-Philippe van Dijck,et al.  Numbers are associated with different types of spatial information depending on the task , 2009, Cognition.

[121]  Jean-Philippe van Dijck,et al.  A working memory account for spatial–numerical associations , 2011, Cognition.

[122]  S. Sorby.  Developing spatial cognitive skills among middle school students , 2009, Cognitive Processing.

[123]  Wim Fias,et al.  Numbers and space: a computational model of the SNARC effect. , 2006, Journal of experimental psychology. Human perception and performance.

[124]  Sheryl A. Sorby,et al.  The Development and Assessment of a Course for Enhancing the 3‐D Spatial Visualization Skills of First Year Engineering Students , 2000 .

[125]  Kelly S. Mix,et al.  Spatial Training Improves Children's Mathematics Ability , 2014 .

[126]  Samar Zebian,et al.  Linkages between Number Concepts, Spatial Thinking, and Directionality of Writing: The SNARC Effect and the REVERSE SNARC Effect in English and Arabic Monoliterates, Biliterates, and Illiterate Arabic Speakers , 2005 .

[127]  Virginie Crollen,et al.  Spatial and numerical processing in children with non-verbal learning disabilities. , 2015, Research in developmental disabilities.

[128]  D. Uttal,et al.  Spatial Thinking and STEM Education. When, Why, and How? , 2012 .

[129]  Sian L. Beilock,et al.  The relation between spatial skill and early number knowledge: the role of the linear number line. , 2012, Developmental psychology.

[130]  Lori Andersen.  Visual–Spatial Ability: Important in STEM, Ignored in Gifted Education , 2014 .

[131]  Marco Zorzi,et al.  Representation of numerical and non-numerical order in children , 2012, Cognition.