Neurocognitive Architectures and the Nonsymbolic Foundations of Fractions Understanding

Development of Mathematical Cognition: Neural Substrates and Genetic Influences: Volume 2

Volumn 2, Issue , 2015, Pages 141-164

  Lewis, Mark Rose ^a   Matthews, Percival G ^a   Hubbard, Edward M ^a  

^a UNIVERSITY OF WISCONSIN MADISON   (United States)

Author keywords

Educational neuroscience; Fractions; Fractions instruction; Neurocognitive architecture; Nonsymbolic ratios; Perceptually based training; Ratio processing system

Indexed keywords

EID: 85152858153     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1016/B978-0-12-801871-2.00006-X     Document Type: Chapter

References (102)

1. D.H. Bailey, M.K. Hoard, L. Nugent and D.C. Geary (2012) Competence with fractions predicts gains in mathematics achievement. Journal of Experimental Child Psychology 113(3), 447-455. 10.1016/j.jecp.2012.06.004
2. M. Bonato, S. Fabbri, C. Umiltà and M. Zorzi (2007) The mental representation of numerical fractions: Real or integer? Journal of Experimental Psychology. Human Perception and Performance 33(6), 1410-1419. 10.1037/0096-1523.33.6.1410
3. J.L. Booth and K.J. Newton (2012) Fractions: Could they really be the gatekeeper’s doorman? Contemporary Educational Psychology 37(4), 247-253. 10.1016/j.cedpsych.2012.07.001
4. T.W. Boyer, S.C. Levine and J. Huttenlocher (2008) Development of proportional reasoning: Where young children go wrong. Developmental Psychology 44(5), 1478-1490. 10.1037/a0013110
5. E.M. Brannon and J.D. Roitman (2003) Nonverbal representations of time and number in animals and human infants. W.H. Meck (Eds) Functional and neural mechanisms of interval timing Boca Raton, FL: CRC Press 143-182.
6. D. Briars and R.S. Siegler (1984) A featural analysis of preschoolers' counting knowledge. Developmental Psychology 20(4), 607-618. 10.1037/0012-1649.20.4.607
7. B. Butterworth (2005) Developmental dyscalculia. J. Campbell (Eds) Handbook of mathematical cognition New York: Psychology Press 455-467.
8. B. Butterworth, S. Varma and D. Laurillard (2011) Dyscalculia: From brain to education. Science 332(6033), 1049-1053. 10.1126/science.1201536
9. J.F. Cantlon, E.M. Brannon, E.J. Carter and K.A. Pelphrey (2006) Functional imaging of numerical processing in adults and 4-y-old children. Public Library of Science, Biology 4(5), e125. 10.1371/journal.pbio.0040125
10. T.P. Carpenter, M.K. Corbitt, H. Kepner, M.M. Lindquist and R. Reys (1981) Results from the Second Mathematics Assessment of the National Assessment of Educational Progress. Reston, VA: National Council of Teachers of Mathematics
11. F.W. Chu, K. vanMarle and D.C. Geary (2013) Quantitative deficits of preschool children at risk for mathematical learning disability. Frontiers in Psychology 4 10.3389/fpsyg.2013.00195 Article 195
12. R. Cohen Kadosh, A. Henik, O. Rubinsten, H. Mohr, H. Dori, V. van de Ven, et al. (2005) Are numbers special?: The comparison systems of the human brain investigated by fMRI. Neuropsychologia 43(9), 1238-1248. 10.1016/j.neuropsychologia.2004.12.017
13. R. Cohen Kadosh and V. Walsh (2009) Numerical representation in the parietal lobes: Abstract or not abstract? Behavioral and Brain Sciences 32(3-4), 313-328. 10.1017/S0140525X09990938
14. K. Cramer and T. Wyberg (2009) Efficacy of different concrete models for teaching the part-whole construct for fractions. Mathematical Thinking and Learning 11(4), 226-257. 10.1080/10986060903246479
15. V.V. Davydov and Z. Tsvetkovich (1991) On the objective origin of the concept of fractions. Focus on Learning Problems in Mathematics 12(1), 13-64.
16. B. De Smedt, M.P. Noël, C. Gilmore and D. Ansari (2013) How do symbolic and non-symbolic numerical magnitude processing relate to individual differences in children's mathematical skills? A review of evidence from brain and behavior. Trends in Neuroscience and Education 2(2), 48-55. 10.1016/j.tine.2013.06.001
17. S. Dehaene (1997/2011) The number sense: How the mind creates mathematics. New York: Oxford University Press 978-0199753871
18. S. Dehaene and L. Cohen (2007) Cultural recycling of cortical maps. Neuron 56(2), 384-398. 10.1016/j.neuron.2007.10.004
19. S. Dehaene, G. Dehaene-Lambertz and L. Cohen (1998) Abstract representations of numbers in the animal and human brain. Trends in Neurosciences 21(8), 355-361. 10.1016/S0166-2236(98)01263-6
20. S. Duffy, J. Huttenlocher and S. Levine (2005) It is all relative: How young children encode extent. Journal of Cognition and Development 6(1), 51-63. 10.1207/s15327647jcd0601_4
21. L. Feigenson, S. Dehaene and E. Spelke (2004) Core systems of number. Trends in Cognitive Sciences 8(7), 307-314. 10.1016/j.tics.2004.05.002
22. W. Fias, V. Menon and D. Szucs (2013) Multiple components of developmental dyscalculia. Trends in Neuroscience and Education 2(2), 43-47. 10.1016/j.tine.2013.06.006
23. C.R. Gallistel and R. Gelman (1992) Preverbal and verbal counting and computation. Numerical Cognition 44(1–2), 43-74. 10.1016/0010-0277(92)90050-R
24. D.C. Geary (1993) Mathematical disabilities: Cognitive, neuropsychological, and genetic components. Psychological Bulletin 114(2), 345-362. 10.1037/0033-2909.114.2.345
25. D.C. Geary (2007) Development of mathematical understanding. D. Kuhn, R.S. Siegler, W. Damon, R.M. Lerner (Eds) 6th ed. Handbook of child psychology: Vol. 2 Hoboken, NJ: John Wiley & Sons, Inc. 777-810. Retrieved from http://dx.doi.org/10.1002/9780470147658.chpsy0218
26. D.C. Geary, D.B. Berch and K. Mann Koepke (Eds) (2015) Evolutionary origins and early development of number processing San Diego, CA: Academic Press
27. R. Gelman and E.M. Williams (1998) Enabling constraints for cognitive development and learning: Domain specificity and epigenesis. D. Kuhn, R. Siegler (Eds) Cognition, perception, and language Vol. 2 New York: John Wiley & Sons Inc 575-630.
28. J.N. Giedd, A. Raznahan, A. Alexander-Bloch, E. Schmitt, N. Gogtay and J.L. Rapoport (2015) Child psychiatry branch of the National Institute of Mental Health longitudinal structural magnetic resonance imaging study of human brain development. Neuropsychopharmacology 40(1), 43-49. 10.1038/npp.2014.236
29. C. Gilmore, N. Attridge, S. Clayton, L. Cragg, S. Johnson, N. Marlow, et al. (2013) Individual differences in inhibitory control, not non-verbal number acuity, correlate with mathematics achievement. PloS One 8(6), 10.1371/journal.pone.0067374 e67374
30. U. Goswami (1989) Relational complexity and the development of analogical reasoning. Cognitive Development 4(3), 251-268. 10.1016/0885-2014(89)90008-7
31. U. Goswami (1995) Transitive relational mappings in three- and four-year-olds: The analogy of Goldilocks and the three bears. Child Development 66(3), 877-892. 10.1111/j.1467-8624.1995.tb00911.x
32. K. Grill-Spector, R. Henson and A. Martin (2006) Repetition and the brain: Neural models of stimulus-specific effects. Trends in Cognitive Sciences 10(1), 14-23. 10.1016/j.tics.2005.11.006
33. K. Grill-Spector and R. Malach (2001) fMR-adaptation: A tool for studying the functional properties of human cortical neurons. Acta Psychologica 107(1–3), 293-321. 10.1016/S0001-6918(01)00019-1
34. J. Halberda, M.M.M. Mazzocco and L. Feigenson (2008) Individual differences in non-verbal number acuity correlate with maths achievement. Nature 455(7213), 665-668. 10.1038/nature07246
35. S.A. Hecht (1998) Toward an information-processing account of individual differences in fraction skills. Journal of Educational Psychology 90(3), 545-559. 10.1037/0022-0663.90.3.545
36. S.A. Hecht, L. Close and M. Santisi (2003) Sources of individual differences in fraction skills. Journal of Experimental Child Psychology 86(4), 277-302. 10.1016/j.jecp.2003.08.003
37. S.A. Hecht and K.J. Vagi (2010) Sources of group and individual differences in emerging fraction skills. Journal of Educational Psychology 102(4), 843-859. 10.1037/a0019824
38. J.G. Hollands and B.P. Dyre (2000) Bias in proportion judgments: The cyclical power model. Psychological Review 107(3), 500-524. 10.1037/0033-295X.107.3.500
39. A. Ischebeck, M. Schocke and M. Delazer (2009) The processing and representation of fractions within the brain: An fMRI investigation. NeuroImage 47(1), 403-413. 10.1016/j.neuroimage.2009.03.041
40. V. Izard, G. Dehaene-Lambertz and S. Dehaene (2008) Distinct cerebral pathways for object identity and number in human infants. Public Library of Science, Biology 6(2), e11. 10.1371/journal.pbio.0060011
41. S.N. Jacob and A. Nieder (2009) Notation-independent representation of fractions in the human parietal cortex. The Journal of Neuroscience 29(14), 4652-4657. 10.1523/JNEUROSCI.0651-09.2009
42. S.N. Jacob and A. Nieder (2009) Tuning to non-symbolic proportions in the human frontoparietal cortex. European Journal of Neuroscience 30(7), 1432-1442. 10.1111/j.1460-9568.2009.06932.x
43. S.N. Jacob, D. Vallentin and A. Nieder (2012) Relating magnitudes: The brain’s code for proportions. Trends in Cognitive Sciences 16(3), 157-166. 10.1016/j.tics.2012.02.002
44. Y. Jeong, S.C. Levine and J. Huttenlocher (2007) The development of proportional reasoning: Effect of continuous versus discrete quantities. Journal of Cognition and Development 8(2), 237-256. 10.1080/15248370701202471
45. A.Y. Kallai and J. Tzelgov (2009) A generalized fraction: An entity smaller than one on the mental number line. Journal of Experimental Psychology. Human Perception and Performance 35(6), 1845-1864. 10.1037/a0016892
46. A.Y. Kallai and J. Tzelgov (2012) When meaningful components interrupt the processing of the whole: The case of fractions. Acta Psychologica 139(2), 358-369. 10.1016/j.actpsy.2011.11.009
47. B. Krekelberg, G.M. Boynton and R.J.A. van Wezel (2006) Adaptation: From single cells to BOLD signals. Trends in Neurosciences 29(5), 250-256. 10.1016/j.tins.2006.02.008
48. K. Kucian, U. Grond, S. Rotzer, B. Henzi, C. Schönmann, F. Plangger, et al. (2011) Mental number line training in children with developmental dyscalculia. NeuroImage 57(3), 782-795. 10.1016/j.neuroimage.2011.01.070
49. M. Le Corre and S. Carey (2007) One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles. Cognition 105(2), 395-438. 10.1016/j.cognition.2006.10.005
50. M.E. Libertus, L. Feigenson and J. Halberda (2011) Preschool acuity of the approximate number system correlates with school math ability. Developmental Science 14(6), 1292-1300. 10.1111/j.1467-7687.2011.01080.x
51. I.M. Lipkus, G. Samsa and B.K. Rimer (2001) General Performance on a Numeracy Scale among Highly Educated Samples. Medical Decision Making 21(1), 37-44. 10.1177/0272989X0102100105
52. I.M. Lyons, G.R. Price, A. Vaessen, L. Blomert and D. Ansari (2014) Numerical predictors of arithmetic success in grades 1-6. Developmental Science 17(5), 714-726. 10.1111/desc.12152
53. N.K. Mack (1990) Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education 16–32
54. N.K. Mack (1995) Confounding whole-number and fraction concepts when building on informal knowledge. Journal for Research in Mathematics Education 422–441
55. M.M.M. Mazzocco, L. Feigenson and J. Halberda (2011) Impaired acuity of the approximate number system underlies mathematical learning disability (dyscalculia). Child Development 82(4), 1224-1237. 10.1111/j.1467-8624.2011.01608.x
56. M.M.M. Mazzocco, L. Feigenson and J. Halberda (2011) Preschoolers’ precision of the approximate number system predicts later school mathematics performance. PloS One 6(9), 10.1371/journal.pone.0023749 e23749
57. K. McCrink, E.S. Spelke, S. Dehaene and P. Pica (2013) Non-symbolic halving in an Amazonian indigene group. Developmental Science 16(3), 451-462. 10.1111/desc.12037
58. K. McCrink and K. Wynn (2007) Ratio abstraction by 6-month-old infants. Psychological Science 18(8), 740-745. 10.1111/j.1467-9280.2007.01969.x
59. G. Meert, J. Grégoire, X. Seron and M.-P. Noël (2011) The mental representation of the magnitude of symbolic and nonsymbolic ratios in adults. The Quarterly Journal of Experimental Psychology 65(4), 702-724. 10.1080/17470218.2011.632485
60. G. Meert, J. Grégoire, X. Seron and M.-P. Noël (2013) The processing of symbolic and nonsymbolic ratios in school-age children. PloS One 8(11), 10.1371/journal.pone.0082002 e82002
61. K.S. Mix, S.C. Levine and J. Huttenlocher (1999) Early fraction calculation ability. Developmental Psychology 35(1), 164-174. 10.1037/0012-1649.35.1.164
62. J. Moss and R. Case (1999) Developing children’s understanding of the rational numbers: A new model and an experimental curriculum. Journal for Research in Mathematics Education 30(2), 122-147. 10.2307/749607
63. C. Mussolin, S. Mejias and M.-P. Noël (2010) Symbolic and nonsymbolic number comparison in children with and without dyscalculia. Cognition 115(1), 10-25. 10.1016/j.cognition.2009.10.006
64. National Governors Association Center for Best Practices, & Council of Chief State School Officers (2010) Common core state standards. Washington, DC: National Governors Association Center for Best Practices, Council of Chief State School Officers
65. National Mathematics Advisory Panel (2008) Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education
66. K.J. Newton (2008) An extensive analysis of preservice elementary teachers’ knowledge of fractions. American Educational Research Journal 45(4), 1080-1110. 10.3102/0002831208320851
67. Y. Ni and Y.-D. Zhou (2005) Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist 40(1), 27-52. 10.1207/s15326985ep4001_3
68. A. Nieder (2005) Counting on neurons: The neurobiology of numerical competence. Nature Reviews Neuroscience 6(3), 177-190.
69. A. Nieder and S. Dehaene (2009) Representation of number in the brain. Annual Review of Neuroscience 32 185-208.
70. A. Obersteiner, W. Van Dooren, J. Van Hoof and L. Verschaffel (2013) The natural number bias and magnitude representation in fraction comparison by expert mathematicians. Learning and Instruction 28 64-72. 10.1016/j.learninstruc.2013.05.003
71. M. Piazza (2010) Neurocognitive start-up tools for symbolic number representations. Trends in Cognitive Sciences 14(12), 542-551. 10.1016/j.tics.2010.09.008
72. M. Piazza, A. Facoetti, A.N. Trussardi, I. Berteletti, S. Conte, D. Lucangeli, et al. (2010) Developmental trajectory of number acuity reveals a severe impairment in developmental dyscalculia. Cognition 116(1), 33-41. 10.1016/j.cognition.2010.03.012
73. M. Piazza, P. Pinel, D. Le Bihan and S. Dehaene (2007) A magnitude code common to numerosities and number symbols in human intraparietal cortex. Neuron 53(2), 293-305. 10.1016/j.neuron.2006.11.022
74. P. Pinel, M. Piazza, D. Le Bihan and S. Dehaene (2004) Distributed and overlapping cerebral representations of number, size, and luminance during comparative judgments. Neuron 41(6), 983-993. 10.1016/S0896-6273(04)00107-2
75. A. Pitkethly and R. Hunting (1996) A review of recent research in the area of initial fraction concepts. Educational Studies in Mathematics 30(1), 5-38.
76. T.R. Post, G. Harel, M. Behr and R. Lesh (1991) Intermediate teachers’ knowledge of rational number concepts. E. Fennema, T. Carpenter, S. Lamon (Eds) Integrating research on teaching and learning mathematics New York: State University of NY Press 177-198.
77. G.R. Price, I. Holloway, P. Räsänen, M. Vesterinen and D. Ansari (2007) Impaired parietal magnitude processing in developmental dyscalculia. Current Biology 17(24), R1042-R1043. 10.1016/j.cub.2007.10.013
78. S. Qin, S. Cho, T. Chen, T. Rosenberg-Lee, D.C. Geary and V. Menon (2014) Hippocampal-neocortical functional reorganization underlies children's cognitive development. Nature Neuroscience 17(9), 1263-1269. 10.1038/nn.3788
79. V.F. Reyna and C.J. Brainerd (2008) Numeracy, ratio bias, and denominator neglect in judgments of risk and probability. Learning and Individual Differences 18(1), 89-107. 10.1016/j.lindif.2007.03.011
80. V.J. Schmithorst and R.D. Brown (2004) Empirical validation of the triple-code model of numerical processing for complex math operations using functional MRI and group Independent Component Analysis of the mental addition and subtraction of fractions. NeuroImage 22(3), 1414-1420. 10.1016/j.neuroimage.2004.03.021
81. R.S. Siegler, T. Carpenter, F. Fennell, D. Geary, J. Lewis, Y. Okamoto, et al. (2010) Developing effective fractions instruction for kindergarten through 8th grade: A practice guide (NCEE #2010-4039). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education
82. R.S. Siegler, G.J. Duncan, P.E. Davis-Kean, K. Duckworth, A. Claessens, M. Engel, et al. (2012) Early predictors of high school mathematics achievement. Psychological Science 23(7), 691-697. 10.1177/0956797612440101
83. R.S. Siegler, L.K. Fazio, D.H. Bailey and X. Zhou (2013) Fractions: The new frontier for theories of numerical development. Trends in Cognitive Sciences 17(1), 13-19. 10.1016/j.tics.2012.11.004
84. R.S. Siegler and A.A. Pyke (2013) Developmental and individual differences in understanding of fractions. Developmental Psychology 49(10), 1994-2004. 10.1037/a0031200
85. R.S. Siegler, C.A. Thompson and M. Schneider (2011) An integrated theory of whole number and fractions development. Cognitive Psychology 62(4), 273-296. 10.1016/j.cogpsych.2011.03.001
86. K.E. Singer-Freeman and U. Goswami (2001) Does half a pizza equal half a box of chocolates?: Proportional matching in an analogy task. Cognitive Development 16(3), 811-829. 10.1016/S0885-2014(01)00066-1
87. C. Sophian (2000) Perceptions of proportionality in young children: Matching spatial ratios. Cognition 75(2), 10.1016/S00100277(00)00062-7
88. A.G. Spinillo and P. Bryant (1991) Children’s proportional judgments: The importance of “half.”. Child Development 62(3), 427-440. 10.1111/j.1467-8624.1991.tb01542.x
89. A.G. Spinillo and P.E. Bryant (1999) Proportional Reasoning in Young Children: Part-Part Comparisons about Continuous and Discontinuous Quantity. Mathematical Cognition 5(2), 181-197. 10.1080/135467999387298
90. S. Stafylidou and S. Vosniadou (2004) The development of students’ understanding of the numerical value of fractions. Learning and Instruction 14(5), 503-518. 10.1016/j.learninstruc.2004.06.015
91. S.S. Stevens and E.H. Galanter (1957) Ratio scales and category scales for a dozen perceptual continua. Journal of Experimental Psychology 54(6), 377-411. 10.1037/h0043680
92. J.W. Stigler, K.B. Givvin and B.J. Thompson (2010) What community college developmental mathematics students understand about mathematics. MathAMATYC Educator 1(3),
93. D. Szűcs, A. Devine, F. Soltesz, A. Nobes and F. Gabriel (2013) Developmental dyscalculia is related to visuo-spatial memory and inhibition impairment. Cortex 49(10), 2674-2688. 10.1016/j.cortex.2013.06.007
94. D. Szűcs and U. Goswami (2013) Developmental dyscalculia: Fresh perspectives. Trends in Neuroscience and Education 2(2), 33-37. 10.1016/j.tine.2013.06.004
95. A.A. Taubman (2007) Threshold Resistance: The Extraordinary Career of a Luxury Retailing Pioneer. New York: Harper Business 978-0061235375
96. D. Vallentin and A. Nieder (2008) Behavioral and prefrontal representation of spatial proportions in the monkey. Current Biology 18(18), 1420-1425. 10.1016/j.cub.2008.08.042
97. D. Vallentin and A. Nieder (2010) Representations of visual proportions in the primate posterior parietal and prefrontal cortices. European Journal of Neuroscience 32(8), 1380-1387. 10.1111/j.1460-9568.2010.07427.x
98. X. Vamvakoussi and S. Vosniadou (2010) How many decimals are there between two fractions? Aspects of secondary school students’ understanding of rational numbers and their notation. Cognition and Instruction 28(2), 181-209. 10.1080/07370001003676603
99. D.C. Van Essen (2005) A Population-Average, Landmark- and Surface-based (PALS) atlas of human cerebral cortex. NeuroImage 28(3), 635-662. 10.1016/j.neuroimage.2005.06.058
100. D.C. Van Essen, H.A. Drury, J. Dickson, J. Harwell, D. Hanlon and C.H. Anderson (2001) An integrated software suite for surface-based analyses of cerebral cortex. Journal of the American Medical Informatics Association 8(5), 443-459. 10.1136/jamia.2001.0080443
101. G. Woodruff and D. Premack (1981) Primative mathematical concepts in the chimpanzee: Proportionality and numerosity. Nature 293(5833), 568-570. 10.1038/293568a0
102. H. Wu (2008) Fractions, decimals, and rational numbers. www.math.berkeley.edu/~wu/NMPfractions4.pdf Retrieved from