In this review, we attempt to integrate two crucial aspects of numerical development: learning the magnitudes of individual numbers and learning arithmetic. Numerical magnitude development involves gaining increasingly precise knowledge of increasing ranges and types of numbers: from nonsymbolic to small symbolic numbers, from smaller to larger whole numbers, and from whole to rational numbers. One reason why this development is important is that precision of numerical magnitude knowledge is correlated with, predictive of, and causally related to both whole and rational number arithmetic. Rational number arithmetic, however, also poses challenges beyond understanding the magnitudes of the individual numbers. Some of these challenges are inherent; they are present for all learners. Other challenges are culturally contingent; they vary from country to country and classroom to classroom. Generating theories and data that help children surmount the challenges of rational number arithmetic is a promising and important goal for future numerical development research.


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Literature Cited

  1. Agrillo C, Piffer L, Bisazza A, Butterworth B. 2012. Evidence for two numerical systems that are similar in humans and guppies. PLOS ONE 7:2e31923 [Google Scholar]
  2. Alibali MW, Goldin-Meadow S. 1993. Gesture-speech mismatch and mechanisms of learning: what the hands reveal about a child's state of mind. Cogn. Psychol. 25:4468–523 [Google Scholar]
  3. Andres M, Michaux N, Pesenti M. 2012. Common substrate for mental arithmetic and finger representation in the parietal cortex. NeuroImage 62:31520–28 [Google Scholar]
  4. Ashcraft MH. 1995. Cognitive psychology and simple arithmetic: a review and summary of new directions. Math. Cogn. 1:13–34 [Google Scholar]
  5. Ashcraft MH, Moore AM. 2012. Cognitive processes of numerical estimation in children. J. Exp. Child Psychol. 111:2246–67 [Google Scholar]
  6. Bailey DH, Hoard MK, Nugent L, Geary DC. 2012. Competence with fractions predicts gains in mathematics achievement. J. Exp. Child Psychol. 113:3447–55 [Google Scholar]
  7. Bailey DH, Siegler RS, Geary DC. 2014. Early predictors of middle school fraction knowledge. Dev. Sci. 17:5775–85 [Google Scholar]
  8. Bailey DH, Zhou X, Zhang Y, Cui J, Fuchs LS. et al. 2015. Development of fraction concepts and procedures in U.S. and Chinese children. J. Exp. Child Psychol. 129:68–83 [Google Scholar]
  9. Baroody AJ, Tiilikainen SH. 2003. Two perspectives on addition development. The Development of Arithmetic Concepts and Skills: The Construction of Adaptive Expertise AJ Baroody, A Dowker 75–125 Mahwah, NJ: Erlbaum [Google Scholar]
  10. Berch DB, Foley EJ, Hill RJ, McDonough-Ryan PM. 1999. Extracting parity and magnitude from Arabic numerals: developmental changes in number processing and mental representation. J. Exp. Child Psychol. 74:286–308 [Google Scholar]
  11. Berteletti I, Booth JR. 2015. Perceiving fingers in single-digit arithmetic problems. Front. Psychol. 6:226 [Google Scholar]
  12. Berteletti I, Lucangeli D, Piazza M, Dehaene S, Zorzi M. 2010. Numerical estimation in preschoolers. Dev. Psychol. 41:545–51 [Google Scholar]
  13. Berteletti I, Man G, Booth JR. 2015. How number line estimation skills relate to neural activations in single digit subtraction problems. NeuroImage 107:198–206 [Google Scholar]
  14. Booth JL, Siegler RS. 2006. Developmental and individual differences in pure numerical estimation. Dev. Psychol. 42:1189–201 [Google Scholar]
  15. Booth JL, Siegler RS. 2008. Numerical magnitude representations influence arithmetic learning. Child Dev 79:41016–31 [Google Scholar]
  16. Brown JS, Van Lehn K. 1982. Toward a generative theory of “bugs.”. Addition and Subtraction: A Cognitive Perspective TP Carpenter, JM Moser, TA Romberg 117–36 Hillsdale, NJ: Erlbaum [Google Scholar]
  17. Brownell WA. 1947. The place of meaning in the teaching of arithmetic. Elem. Sch. J. 47:5256–65 [Google Scholar]
  18. Bugden S, Price GR, McLean DA, Ansari D. 2012. The role of the left intraparietal sulcus in the relationship between symbolic number processing and children's arithmetic competence. Dev. Cogn. Neurosci. 2:4448–57 [Google Scholar]
  19. Bulthé J, de Smedt B, Op de Beeck HP. 2014. Format-dependent representations of symbolic and non-symbolic numbers in the human cortex as revealed by multi-voxel pattern analyses. NeuroImage 87:311–22 [Google Scholar]
  20. Byrnes JP, Wasik BA. 1991. Role of conceptual knowledge in mathematical procedural learning. Dev. Psychol. 27:5777–86 [Google Scholar]
  21. Campbell JID, Fugelsang J. 2001. Strategy choice for arithmetic verification: effects of numerical surface form. Cognition 80:3B21–30 [Google Scholar]
  22. Campbell JID, Xue Q. 2001. Cognitive arithmetic across cultures. J. Exp. Psychol.: Gen. 130:2299–315 [Google Scholar]
  23. Cantrell L, Smith LB. 2013. Open questions and a proposal: a critical review of the evidence on infant numerical abilities. Cognition 128:3331–52 [Google Scholar]
  24. Carpenter TP, Corbitt M, Kepner H, Lindquist M, Reys R. 1980. Results of the second NAEP mathematics assessment: secondary school. Math. Teach. 73:329–38 [Google Scholar]
  25. Carpenter TP, Lindquist MM, Matthews W, Silver EA. 1983. Results of the third NAEP mathematics assessment: secondary school. Math. Teach. 76:9652–59 [Google Scholar]
  26. Case R, Okamoto Y. 1996. The role of central conceptual structures in the development of children's thought. Monogr. Soc. Res. Child Dev. 61:1–21–295 [Google Scholar]
  27. Castronovo J, Göbel SM. 2012. Impact of high mathematics education on the number sense. PLOS ONE 7:4e33832 [Google Scholar]
  28. CCSSI (Common Core State Stand. Initiat.) 2010. Common Core State Standards for Mathematics. Washington, DC: Natl. Gov. Assoc. Cent. Best Pract. Counc. Chief State School Off.
  29. Chen Q, Li J. 2014. Association between individual differences in nonsymbolic number acuity and math performance: a meta-analysis. Acta Psychol. 148:163–72 [Google Scholar]
  30. College Board. 2015. Advanced Placement Physics 1 equations, effective 2015 New York: College Board https://secure-media.collegeboard.org/digitalServices/pdf/ap/ap-physics-1-equations-table.pdf
  31. Cordes S, Brannon EM. 2008. Quantitative competencies in infancy. Dev. Sci. 11:6803–8 [Google Scholar]
  32. Cramer KA, Post TR, del Mas RC. 2002. Initial fraction learning by fourth- and fifth-grade students: a comparison of the effects of using commercial curricula with the effects of using the rational number project curriculum. J. Res. Math. Educ. 33:2111–44 [Google Scholar]
  33. Davidson D. 2012. Making it in America. The Atlantic Jan./Feb. 65–83
  34. Davis J, Choppin J, McDuffie AR, Drake C. 2013. Common Core State Standards for Mathematics: middle school mathematics teachers' perceptions. Rep.,: Univ. Rochester Warner Cent. Prof. Dev. Educ. Reform, Rochester, NY
  35. de Smedt B, Verschaffel L, Ghesquière P. 2009. The predictive value of numerical magnitude comparison for individual differences in mathematics achievement. J. Exp. Child Psychol. 103:4469–79 [Google Scholar]
  36. Dehaene S. 2011. The Number Sense: How the Mind Creates Mathematics New York: Oxford Univ. Press
  37. Dehaene S, Dupoux E, Mehler J. 1990. Is numerical comparison digital? Analogical and symbolic effects in two-digit number comparison. J. Exp. Psychol.: Hum. Percept. Perform. 16:3626–41 [Google Scholar]
  38. Depaepe F, Torbeyns J, Vermeersch N, Janssens D, Janssen R. et al. 2015. Teachers’ content and pedagogical content knowledge on rational numbers: a comparison of prospective elementary and lower secondary school teachers. Teach. Teach. Educ. 47:82–92 [Google Scholar]
  39. DeWolf M, Grounds MA, Bassok M, Holyoak KJ. 2014. Magnitude comparison with different types of rational numbers. J. Exp. Psychol.: Hum. Percept. Perform. 40:171–82 [Google Scholar]
  40. Dotan D, Dehaene S. 2013. How do we convert a number into a finger trajectory?. Cognition 129:3512–29 [Google Scholar]
  41. Dowker A. 1997. Young children's addition estimates. Math. Cogn. 3:2140–53 [Google Scholar]
  42. Duncan GJ, Dowsett CJ, Claessens A, Magnuson K, Huston AC. et al. 2007. School readiness and later achievement. Dev. Psychol. 43:1428–46 [Google Scholar]
  43. Fazio LK, Bailey DH, Thompson CA, Siegler RS. 2014. Relations of different types of numerical magnitude representations to each other and to mathematics achievement. J. Exp. Child Psychol. 123:53–72 [Google Scholar]
  44. Feigenson L, Dehaene S, Spelke E. 2004. Core systems of number. Trends Cogn. Sci. 8:307–14 [Google Scholar]
  45. Fischbein E, Deri M, Sainati Nello M, Sciolis Marino M. 1985. The role of implicit models in solving verbal problems in multiplication and division. J. Res. Math. Educ. 16:13–17 [Google Scholar]
  46. Fuchs LS, Geary DC, Compton DL, Fuchs D, Hamlett CL, Bryant JD. 2010. The contributions of numerosity and domain-general abilities to school readiness. Child Dev. 81:51520–33 [Google Scholar]
  47. Fuchs LS, Schumacher RF, Long J, Namkung J, Hamlett CL. et al. 2013. Improving at-risk learners’ understanding of fractions. J. Educ. Psychol. 105:3683–700 [Google Scholar]
  48. Fuchs LS, Schumacher RF, Long J, Namkung J, Malone A. et al. 2016. Effects of intervention to improve at-risk fourth graders’ understanding, calculations, and word problems with fractions. Elem. Sch. J. 1164625–51
  49. Fuchs LS, Schumacher RF, Sterba SK, Long J, Namkung J. et al. 2014. Does working memory moderate the effects of fraction intervention? An aptitude-treatment interaction. J. Educ. Psychol. 106:2499–514 [Google Scholar]
  50. Gabriel F, Coché F, Szucs D, Carette V, Rey B, Content A. 2012. Developing children's understanding of fractions: an intervention study. Mind Brain Educ. 6:3137–46 [Google Scholar]
  51. Geary DC, Berch DB, Mann-Koepke K. 2015. Evolutionary Origins and Early Development of Number Processing 1 Mathematical Cognition and Learning San Diego, CA: Elsevier Acad.
  52. Geary DC, Berch DB, Ochsendorf R, Mann-Koepke K. 2017. Acquisition of Complex Arithmetic Skills and Higher-Order Mathematical Concepts 3 Mathematical Cognition and Learning San Diego, CA: Elsevier Acad In press
  53. Geary DC, Hoard MK, Byrd-Craven J, Nugent L, Numtee C. 2007. Cognitive mechanisms underlying achievement deficits in children with mathematical learning disability. Child Dev. 78:41343–59 [Google Scholar]
  54. Givvin KB, Stigler JW, Thompson BJ. 2011. What community college developmental mathematics students understand about mathematics, part II: the interviews. MathAMATYC Educ. 2:34–18 [Google Scholar]
  55. Goncu A, Gauvain M. 2012. Sociocultural approaches to educational psychology: theory, research, and application. APA Educational Psychology Handbook, 1 Theories, Constructs, and Critical Issues KR Harris, S Graham, T Urdan, CB McCormick, GM Sinatra, J Sweller 125–54 Washington, DC: Am. Psychol. Assoc. [Google Scholar]
  56. Gunderson EA, Ramirez G, Beilock SL, Levine SC. 2012. The relation between spatial skill and early number knowledge: the role of the linear number line. Dev. Psychol. 48:51229–41 [Google Scholar]
  57. Halberda J, Feigenson L. 2008. Developmental change in the acuity of the “number sense”: the approximate number system in 3-, 4-, 5-, and 6-year-olds and adults. Dev. Psychol. 44:51457–65 [Google Scholar]
  58. Halberda J, Ly R, Wilmer JB, Naiman DQ, Germine L. 2012. Number sense across the lifespan as revealed by a massive Internet-based sample. PNAS 109:2811116–20 [Google Scholar]
  59. Halberda J, Mazzocco MMM, Feigenson L. 2008. Individual differences in non-verbal number acuity correlate with maths achievement. Nature 455:7213665–68 [Google Scholar]
  60. Hanson SA, Hogan TP. 2000. Computational estimation skill of college students. J. Res. Math. Educ. 31:4483–99 [Google Scholar]
  61. Hecht SA, Close L, Santisi M. 2003. Sources of individual differences in fraction skills. J. Exp. Child Psychol. 86:4277–302 [Google Scholar]
  62. Hecht SA, Vagi KJ. 2010. Sources of group and individual differences in emerging fraction skills. J. Educ. Psychol. 102:4843–59 [Google Scholar]
  63. Hiebert J, Wearne D. 1985. A model of students’ decimal computation. Cogn. Instr. 2:3–4175–205 [Google Scholar]
  64. Hiebert J, Wearne D. 1986. Procedures over concepts: the acquisition of decimal number knowledge. Conceptual and Procedural Knowledge: The Case of Mathematics J Hiebert 199–223 Hillsdale, NJ: Erlbaum
  65. Hoffer TB, Venkataraman L, Hedberg EC, Shagle S. 2007. Final Report on the National Survey of Algebra Teachers (for the National Mathematics Advisory Panel Subcommittee) Washington, DC: US Dep. Educ.
  66. Hoffmann D, Hornung C, Martin R, Schiltz C. 2013. Developing number-space associations: SNARC effects using a color discrimination task in 5-year-olds. J. Exp. Child Psychol. 116:4775–91 [Google Scholar]
  67. Holloway ID, Ansari D. 2009. Mapping numerical magnitudes onto symbols: the numerical distance effect and individual differences in children's mathematics achievement. J. Exp. Child Psychol. 103:117–29 [Google Scholar]
  68. Hubbard EM, Piazza M, Pinel P, Dehaene S. 2005. Interactions between number and space in parietal cortex. Nat. Rev. Neurosci. 6:6435–48 [Google Scholar]
  69. Hyde DC, Khanum S, Spelke ES. 2014. Brief non-symbolic, approximate number practice enhances subsequent exact symbolic arithmetic in children. Cognition 131:192–107 [Google Scholar]
  70. Ischebeck A, Schocke M, Delazer M. 2009. The processing and representation of fractions within the brain: an fMRI investigation. NeuroImage 47:403–13 [Google Scholar]
  71. Iuculano T, Butterworth B. 2011. Understanding the real value of fractions and decimals. Q. J. Exp. Psychol. 64:112088–98 [Google Scholar]
  72. Izard V, Dehaene-Lambertz G, Dehaene S. 2008. Distinct cerebral pathways for object identity and number in human infants. PLOS Biol 6:2e11 [Google Scholar]
  73. Jacob SN, Vallentin D, Nieder A. 2012. Relating magnitudes: the brain's code for proportions. Trends Cogn. Sci. 16:3157–66 [Google Scholar]
  74. Jordan NC, Glutting J, Dyson N, Hassinger-Das B, Irwin C. 2012. Building kindergartners’ number sense: a randomized controlled study. J. Educ. Psychol. 104:3647–60 [Google Scholar]
  75. Jordan NC, Hansen N, Fuchs LS, Siegler RS, Gersten R, Micklos D. 2013. Developmental predictors of fraction concepts and procedures. J. Exp. Child Psychol. 116:145–58 [Google Scholar]
  76. Jordan NC, Kaplan D, Oláh LN, Locuniak MN. 2006. Number sense growth in kindergarten: a longitudinal investigation of children at risk for mathematics difficulties. Child Dev. 77:1153–75 [Google Scholar]
  77. Kant I. 2003 (1781). The Critique of Pure Reason Mineola, NY Dover:
  78. Kieren TE. 1976. On the mathematical, cognitive, and instructional foundations of rational numbers. Number and Measurement: Papers from a Research Workshop RA Lesh, DA Bradbard 101–50 Columbus, OH: ERIC/SMEAC [Google Scholar]
  79. Kucian K, Grond U, Rotzer S, Henzi B, Schönmann C. et al. 2011. Mental number line training in children with developmental dyscalculia. NeuroImage 57:782–95 [Google Scholar]
  80. Laski EV, Siegler RS. 2014. Learning from number board games: You learn what you encode. Dev. Psychol. 50:3853–64 [Google Scholar]
  81. Le Corre M, Carey S. 2007. One, two, three, four, nothing more: an investigation of the conceptual sources of the verbal counting principles. Cognition 105:395–438 [Google Scholar]
  82. LeFevre JA, Smith-Chant BL, Hiscock K, Dale KE, Morris J. 2003. Young adults’ strategic choices in simple arithmetic: implications for the development of mathematical representations. The Development of Arithmetic Concepts and Skills: Constructing Adaptive Expertise AJ Baroody, A Dowker 203–28 Mahwah, NJ: Erlbaum [Google Scholar]
  83. Libertus ME, Feigenson L, Halberda J. 2011. Preschool acuity of the approximate number system correlates with school math ability. Dev. Sci. 14:61292–300 [Google Scholar]
  84. Linsen S, Verschaffel L, Reynvoet B, De Smedt B. 2015. The association between numerical magnitude processing and mental versus algorithmic multi-digit subtraction in children. Learn. Instr. 35:42–50 [Google Scholar]
  85. Lortie-Forgues H, Siegler RS. In press. Conceptual knowledge of decimal arithmetic. J. Educ. Psychol.
  86. Lortie-Forgues H, Tian J, Siegler RS. 2015. Why is learning fraction and decimal arithmetic so difficult. Dev. Rev. 38:201–21 [Google Scholar]
  87. Luo F, Lo J-J, Leu Y-C. 2011. Fundamental fraction knowledge of preservice elementary teachers: a cross-national study in the United States and Taiwan. Sch. Sci. Math. 111:4164–77 [Google Scholar]
  88. Lyons IM, Price GR, Vaessen A, Blomert L, Ansari D. 2014. Numerical predictors of arithmetic success in grades. 1–6 Dev. Sci. 17:5714–26 [Google Scholar]
  89. Ma L. 1999. Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States Mahwah, NJ: Erlbaum
  90. Mack NK. 1995. Confounding whole-number and fraction concepts when building on informal knowledge. J. Res. Math. Educ. 26:5422–41 [Google Scholar]
  91. McCloskey M. 2007. Quantitative literacy and developmental dyscalculias. Why Is Math So Hard for Some Children? The Nature and Origins of Mathematical Learning Difficulties and Disabilities DB Berch, MMM Mazzocco 415–29 Baltimore, MD: Paul H. Brookes Publ. [Google Scholar]
  92. McCrink K, Wynn K. 2004. Large-number addition and subtraction by 9-month-old infants. Psychol. Sci. 15:11776–81 [Google Scholar]
  93. McCrink K, Wynn K. 2007. Ratio abstraction by 6-month-old infants. Psychol. Sci. 18:8740–45 [Google Scholar]
  94. McNeil NM. 2014. A change-resistance account of children's difficulties understanding mathematical equivalence. Child Dev. Perspect. 8:142–47 [Google Scholar]
  95. Meert G, Grégoire J, Noël M-P. 2009. Rational numbers: componential versus holistic representation of fractions in a magnitude comparison task. Q. J. Exp. Psychol. 62:81598–616 [Google Scholar]
  96. Moss J, Case R. 1999. Developing children's understanding of the rational numbers: a new model and an experimental curriculum. J. Res. Math. Educ. 30:2122–47 [Google Scholar]
  97. Moyer RS, Landauer TK. 1967. Time required for judgements of numerical inequality. Nature 215:51091519–20 [Google Scholar]
  98. Natl. Math. Advis. Panel 2008. Foundations for Success: The Final Report of the National Mathematics Advisory Panel Washington, DC: US Dep. Educ.
  99. Ni YJ, Zhou Y-DD. 2005. Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educ. Psychol. 40:127–52 [Google Scholar]
  100. Nieder A, Dehaene S. 2009. Representation of number in the brain. Annu. Rev. Neurosci. 32:1185–208 [Google Scholar]
  101. Opfer JE, Siegler RS. 2007. Representational change and children's numerical estimation. Cogn. Psychol. 55:3169–95 [Google Scholar]
  102. Östergren R, Träff U. 2013. Early number knowledge and cognitive ability affect early arithmetic ability. J. Exp. Child Psychol. 115:3405–21 [Google Scholar]
  103. Park J, Park DC, Polk TA. 2013. Parietal functional connectivity in numerical cognition. Cereb. Cortex 23:92127–35 [Google Scholar]
  104. Parnas M, Lin AC, Huetteroth W, Miesenböck G. 2013. Odor discrimination in Drosophila: from neural population codes to behavior.. Neuron 79:5932–44 [Google Scholar]
  105. Piaget J. 1952. The Child's Concept of Number New York: Norton
  106. Piazza M. 2011. Neurocognitive start-up tools for symbolic number representations. Space, Time, and Number in the Brain: Searching for the Foundations of Mathematical Thought S Dehaene, E Brannon 267–85 London: Elsevier [Google Scholar]
  107. Piazza M, Pinel P, Le Bihan D, Dehaene S. 2007. A magnitude code common to numerosities and number symbols in human intraparietal cortex. Neuron 53:2293–305 [Google Scholar]
  108. Piffer L, Petrazzini MEM, Agrillo C. 2013. Large number discrimination in newborn fish. PLOS ONE 8:4e62466 [Google Scholar]
  109. Putt IJ. 1995. Preservice teachers’ ordering of decimal numbers: When more is smaller and less is larger!. Focus Learn. Probl. Math. 17:31–15 [Google Scholar]
  110. Ramani GB, Siegler RS. 2008. Promoting broad and stable improvements in low-income children's numerical knowledge through playing number board games. Child Dev. 79:2375–94 [Google Scholar]
  111. Reeve RA, Paul JM, Butterworth B. 2015. Longitudinal changes in young children's 0–100 to 0–1000 number-line error signatures. Front. Psychol. 6:647 [Google Scholar]
  112. Resnick I, Jordan NC, Hansen N, Rajan V, Carrique J. et al. 2016. Developmental growth trajectories in understanding of fraction magnitude from fourth through sixth grade. Dev. Psychol. 52:5746–57 [Google Scholar]
  113. Resnick LB, Nesher P, Leonard F, Magone M, Omanson S, Peled I. 1989. Conceptual bases of arithmetic errors: the case of decimal fractions. J. Res. Math. Educ. 20:18–27 [Google Scholar]
  114. Reyna VF, Brainerd CJ. 1991. Fuzzy-trace theory and children's acquisition of mathematical and scientific concepts. Learn. Individ. Differ. 31:127–59 [Google Scholar]
  115. Ritchie SJ, Bates TC. 2013. Enduring links from childhood mathematics and reading achievement to adult socioeconomic status. Psychol. Sci. 24:71301–8 [Google Scholar]
  116. Robinson KM. 2016. The understanding of additive and multiplicative arithmetic concepts. Math. Cogn. Learn. In press [Google Scholar]
  117. Rugani R, Vallortigara G, Priftis K, Regolin L. 2015. Number-space mapping in the newborn chick resembles humans’ mental number line. Science 347:6221534–36 [Google Scholar]
  118. Schneider M, Siegler RS. 2010. Representations of the magnitudes of fractions. J. Exp. Psychol.: Hum. Percept. Perform. 36:51227–38 [Google Scholar]
  119. Sformo T. 2008. Practical Problems in Mathematics: For Automotive Technicians Clifton Park, NY: Cengage Learn.
  120. Shrager J, Siegler RS. 1998. SCADS: a model of children's strategy choices and strategy discoveries. Psychol. Sci. 9:5405–10 [Google Scholar]
  121. Siegler RS. 1988. Strategy choice procedures and the development of multiplication skill. J. Exp. Psychol.: Gen. 117:3258–75 [Google Scholar]
  122. Siegler RS. 1989. Hazards of mental chronometry: an example from children's subtraction. J. Educ. Psychol. 81:4497–506 [Google Scholar]
  123. Siegler RS. 1996. Emerging Minds: The Process of Change in Children's Thinking New York: Oxford Univ. Press
  124. Siegler RS. 2006. Microgenetic analyses of learning. Handbook of Child Psychology, 2 Cognition, Perception, and Language W Damon, RM Lerner, D Kuhn, RS Siegler 464–510 Hoboken, NJ: Wiley, 6th ed.. [Google Scholar]
  125. Siegler RS. 2016. Magnitude knowledge: the common core of numerical development. Dev. Sci. 19:3341–61 [Google Scholar]
  126. Siegler RS, Araya R. 2005. A computational model of conscious and unconscious strategy discovery. Advances in Child Development and Behavior 33 RV Kail 1–42 Oxford, UK: Elsevier [Google Scholar]
  127. Siegler RS, Booth JL. 2004. Development of numerical estimation in young children. Child Dev. 75:2428–44 [Google Scholar]
  128. Siegler RS, Crowley K. 1994. Constraints on learning in nonprivileged domains. Cogn. Psychol. 27:2194–226 [Google Scholar]
  129. Siegler RS, Jenkins EA. 1989. How Children Discover New Strategies Hillsdale, NJ: Erlbaum
  130. Siegler RS, Lortie-Forgues H. 2015. Conceptual knowledge of fraction arithmetic. J. Educ. Psychol. 107:3909–18 [Google Scholar]
  131. Siegler RS, Mu Y. 2008. Chinese children excel on novel mathematics problems even before elementary school. Psychol. Sci. 19:8759–63 [Google Scholar]
  132. Siegler RS, Pyke AA. 2013. Developmental and individual differences in understanding of fractions. Dev. Psychol. 49:101994–2004 [Google Scholar]
  133. Siegler RS, Ramani GB. 2009. Playing linear number board games—but not circular ones—improves low-income preschoolers’ numerical understanding. J. Educ. Psychol. 101:3545–60 [Google Scholar]
  134. Siegler RS, Shrager J. 1984. Strategy choices in addition and subtraction: How do children know what to do?. The Origins of Cognitive Skills C Sophian 229–93 Hillsdale, NJ: Erlbaum [Google Scholar]
  135. Siegler RS, Thompson CA, Schneider M. 2011. An integrated theory of whole number and fractions development. Cogn. Psychol. 62:4273–96 [Google Scholar]
  136. Skagerlund K, Träff U. 2016. Processing of space, time, and number contributes to mathematical abilities above and beyond domain-general cognitive abilities. J. Exp. Child Psychol. 143:85–101 [Google Scholar]
  137. Stigler J, Givvin K, Thompson A. 2010. What community college developmental mathematics students understand about mathematics. MathAMATYC Educ. 1:34–16 [Google Scholar]
  138. Sullivan J, Barner D. 2014. The development of structural analogy in number-line estimation. J. Exp. Child Psychol. 128:171–89 [Google Scholar]
  139. Thompson CA, Opfer JE. 2008. Costs and benefits of representational change: effects of context on age and sex differences in magnitude estimation. J. Exp. Child Psychol. 101:120–51 [Google Scholar]
  140. Thompson CA, Opfer JE. 2010. How 15 hundred is like 15 cherries: effect of progressive alignment on representational changes in numerical cognition. Child Dev. 81:61768–86 [Google Scholar]
  141. Torbeyns J, Schneider M, Xin Z, Siegler RS. 2015. Bridging the gap: Fraction understanding is central to mathematics achievement in students from three different continents. Learn. Instr. 37:5–13 [Google Scholar]
  142. Venkatraman V, Ansari D, Chee M. 2005. Neural correlates of symbolic and non-symbolic arithmetic. Neuropsychologia 43:5744–53 [Google Scholar]
  143. Verschaffel L, Greer B, De Corte E. 2007. Whole number concepts and operations. Second Handbook of Research on Mathematics Teaching and Learning F Lester 557–628 Charlotte, NC: Inf. Age [Google Scholar]
  144. Watts TW, Duncan GJ, Siegler RS, Davis-Kean PE. 2014. What's past is prologue: relations between early mathematics knowledge and high school achievement. Educ. Res. 43:7352–60 [Google Scholar]
  145. White SLJ, Szűcs D, Soltész F. 2012. Symbolic number: the integration of magnitude and spatial representations in children aged 6 to 8 years. Front. Psychol. 2:392 [Google Scholar]
  146. Wilson AJ, Revkin SK, Cohen D, Dehaene S. 2006. An open trial assessment of “The Number Race,” an adaptive computer game for remediation of dyscalculia. Behav. Brain Funct. 2:11–16 [Google Scholar]
  147. Wynn K. 1992. Children's acquisition of the number words and the counting system. Cogn. Psychol. 24:2220–51 [Google Scholar]
  148. Xu X, Chen C, Pan M, Li N. 2013. Development of numerical estimation in Chinese preschool children. J. Exp. Child Psychol. 116:2351–66 [Google Scholar]

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