Fundamentos cognitivos para la iniciación en el aprendizaje de las matemáticas
Tipo de documento
Lista de autores
NRC, National Research Council of the National Academies
Resumen
En este artículo, sobre fundamentos cognitivos para la iniciación en el aprendizaje de las matemáticas, se realiza una revisión de investigaciones sobre el aprendizaje de las matemáticas en educación infantil. Esta revisión está estructurada según los siguientes apartados: Evidencias sobre la comprensión temprana del número, desarrollo del pensamiento espacial y la geometría, desarrollo de la medición, y regulación de la conducta y la atención.
Fecha
2014
Tipo de fecha
Estado publicación
Términos clave
Enfoque
Nivel educativo
Idioma
Revisado por pares
Formato del archivo
Volumen
3
Número
1
Rango páginas (artículo)
21-48
ISSN
22548351
Referencias
Aguiar, A., and Baillargeon, R. (1998). Eight-and-a-half-month-old infants’ reasoning about containment events. Child Development, 69, 636-653. Ansari, D., Donlan, C., Thomas, M.S.C., Ewing, S.A., Peen, T., and Karmiloff-Smith, A. (2003). What makes counting count: Verbal and visuo-spatial contributions to typical and atypical number development. Journal of Experimental Child Psychology, 85, 50-62. Antell, S.E., and Keating, L.E. (1983). Perception of numerical invariance by neonates. Child Development, 54, 695-701. Baillargeon, R. (1991). Reasoning about the height and location of a hidden object in 4.5 and 6.5 month-old children. Cognition, 38, 13-42. Baillargeon, R. (1995). Physical reasoning in infancy. In M.S. Gazzaniga (Ed.), The Cognitive Neurosciences (pp. 181-204). Cambridge, MA: Bradford Press. Barth, H., LaMont, K., Lipton, J. and Spelke, E.S. (2005). Abstract number and arithmetic in preschool children. Proceedings of the National Academy of Sciences, 102, 14116-14121. Barth, H., LaMont, K., Lipton, J., Dehaene, S., Kanwisher, N., and Spelke, E. (2006). Nonsymbolic arithmetic in adults and young children. Cognition, 98, 199-222. Blair, C., and Razza, R.P. (2007). Relating effortful control, executive function, and false belief understanding to emerging math and literacy ability in kindergarten. Child Development, 78, 647-663. Blair, C., Knipe, H., Cummings, E., Baker, D.P., Gamson, D., Eslinger, P., and Thorne, S.L. (2007). A developmental neuroscience approach to the study of school readiness. In R.C. Pianta, M.J. Cox, and K.L. Snow (Eds.), School Readiness and the Transition to Kindergarten in the Era of Accountability (pp. 149-174). Baltimore, MD: Paul H. Brookes. Bomba, P.C., and Siqueland, E.R. (1983). The nature and structure of infant form categories. Journal of Experimental Child Psychology, 35, 294-328. Boulton-Lewis, G.M. (1987). Recent cognitive theories applied to sequential length measuring knowledge in young children. British Journal of Educational Psychology, 57, 330-342. Boysen, S.T. and Berntson, G.G. (1989). Numerical competence in a chimpanzee (Pan troglodytes). Journal of Comparative Psychology, 103, 23-31. Brainerd, C.J. (1973). Mathematical and behavioral foundations of number. Journal of General Psychology, 11, 369-381. Brannon, E.M. (2002). The development of ordinal numerical knowledge in infancy. Cognition, 83, 223-240. Brannon, E.M., and Terrace, H.S. (1998). Ordering of the numerosidades 1 to 9 by monkeys. Science, 282, 746-749. Brannon, E.M., and Terrace, H.S. (2000). Representation of the numerosidades 1-9 by rhesus macaques (Macaca mulatta). Journal of Experimental Psychology: Animal Behavior Processes, 26, 31-49. Brannon, E.M., Wushoff, C.J., Gallistel, C.R., and Gibbon, J. (2001). Numerical subtraction in the pigeon: Evidence for a linear subjective number scale. Psychological Science, 12, 238-243. Brannon, E.M., Abbott, S., and Lutz, D. (2004). Number bias for the discrimination of large visual sets in infancy. Cognition, 93, B59-B68. Bruner, J., Goodnow, J., and Austin, A. (1956). A Study of Thinking. New York: Wiley. Bryant, P.E. (1974). Perception and Understanding in Young Children. London: Methuen. Burger, W.F., and Shaughnessy, J.M. (1986). Characterizing the van Hiele levels of development in geometry. Journal for Research in Mathematics Education, 17, 31-48. Bushnell, E.W., McKenzie, B.E., Lawrence, D., and Connell, S. (1995). The spatial coding strategies of 1-year-old infants in a locomotor search task. Child Development, 66, 937-958. Butterworth, B. (2005). The development of arithmetical abilities. Journal of Child Psychology and Psychiatry, 46(1), 3-18. Cannon, J., Levine, S.C., and Huttenlocher, J. (2007, March). Sex Differences in the Relation of Early Puzzle Play and Mental Rotation Skill. Paper presented at the biennial meeting of the Society for Research on Child Development, Boston, MA. Cantlon, J.F., and Brannon, E.M. (2006). Shared system for ordering small and large numbers in monkeys and humans. Psychological Science, 17, 401-406. Cantlon, J., Fink, R., Safford, K., and Brannon, E. (2007). Heterogeneity impairs numerical matching but not numerical ordering in preschool children. Developmental Science, 10, 431-441. Carey, S. (2004). Bootstrapping and the origins of concepts. Daedalus, 59-68. Casasola, M., and Cohen, L.B. (2002). Infant categorization of containment, support and tight-fit spatial relationships. Developmental Science, 5, 247-264. Clark, E. (1972). On the child’s acquisition of antonyms in two semantic fields. Journal of Verbal Learning and Verbal Behavior, 11, 750-758. Clearfield, M.W. (2004). Infants’ enumeration of dynamic displays. Cognitive Development, 19(3), 309-324. Clearfield, M.W., and Mix, K.S. (1999). Number versus contour length in infants’ discrimination of small visual sets. Psychological Science, 10, 408-411. Clearfield, M.W., and Mix, K.S. (2001). Infants’ use of area and contour length to discriminate small sets. Journal of Cognition and Development, 2, 243-260. Clements, D.H., and Battista, M.T. (1992). Geometry and spatial reasoning. In D.A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 420-464). New York: Macmillan. Clements, D.H., and Sarama, J. (2007). Early childhood mathematics learning. In F.K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 461-555). New York: Information Age. Clements, D.H., Swaminathan, S., Hannibal, M.A.Z., and Sarama, J. (1999). Young children’s concept of shape. Journal for Research in Mathematics Education, 30, 192-212. Cohen, L.B., and Marks, K.S. (2002). How infants process addition and subtraction events. Developmental Science, 5, 186-201. Cooper, R.G., Jr. (1984). Early number development: Discovering number space with addition and subtraction. In C. Sophian (Ed.), Origins of Cognitive Skills (pp. 157-192). Hillsdale, NJ: Erlbaum. Copeland, R.W. (1979). How Children Learn Mathematics: Teaching Implications of Piaget’s Research (3rd ed.). New York: Macmillan. Cordes, S., and Brannon, E.M. (2008). The difficulties of representing continuous extent in infancy: Using numbers is just easier. Child Development, 79, 476-489. Cordes, S., and Brannon, E.M. (in press). The relative salience of discrete and continuous quantity in infants. Developmental Science. Curry, M., and Outhred, L. (2005). Conceptual understanding of spatial measurement. In P. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce, and A. Roche (Eds.), Building Connections: Theory, Research and Practice (Proceedings of the 27th Annual Conference of the Mathematics Education Research Group of Australasia, Melbourne, pp. 265-272). Sydney: MERGA. Dehaene, S. (1997). The Number Sense: How the Mind Creates Mathematics. New York: Oxford University Press. Dehaene, S., Dehaene-Lambertz, G., and Cohen, L. (1998). Abstract representations of numbers in the animal and human brain. Trends in Neurosciences, 21, 355-361. DeLoache, J.S., Strauss, M., and Maynard, J. (1979). Picture perception in infancy. Infant Behavior and Development, 2, 77-89. Demany, L., McKenzie, B., and Vurpillot, E. (1977). Rhythm perception in early infancy. Nature, 266, 218-219. Diamond, A. (2008, June). Cognitive control (executive functions) in young children: Relevance of what we know to what can be done to help children. In Emotion Regulation, Children’s Brains and Learning, plenary session presented at Head Start’s Ninth National Research Conference, Washington, DC. Diamond, A., Barnett, W.S., Thomas, J., and Munro, S. (2007). Preschool program improves cognitive control. Science, 318, 1387-1388. Duffy, S., Huttenlocher, J., and Levine, S.C. (2005a). It’s all relative: How young children encode extent. Journal of Cognition and Development, 6, 51-63. Duffy, S., Huttenlocher, J., and Levine, S.C. (2005b). How infants encode spatial extent. Infancy, 8, 81-90. Espy, K.A., McDiarmid, M.M., Cwik, M.F., Stalets, M.M., Hamby, A., and Senn, T.E. (2004). The contribution of executive functions to mathematic skills in preschool children. Developmental Neuropsychology, 26, 465-486. Feigenson, L., and Carey, S. (2003). Tracking individuals via object-files: Evidence from infants’ manual search. Developmental Science, 6, 568-584. Feigenson, L., and Carey, S. (2005). On the limits of infants’ quantification of small object arrays. Cognition, 97, 295-313. Feigenson, L., and Halberda, J. (2004). Infants chunk object arrays into sets of individuals. Cognition, 91, 173-190. Feigenson, L., Carey, S., and Hauser, M. (2002). The representations underlying infants’ choice of more: Object files vs. analog magnitudes. Psychological Science, 13, 150-156. Feigenson, L., Carey, S., and Spelke, L. (2002). Infants’ discrimination of number vs. continuous extent. Cognitive Psychology, 44, 33-66. Feigenson, L., Dehaene, S., and Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8, 307-314. Fennema, E.H., and Sherman, J.A. (1977). Sex-related differences in mathematics achievement, spatial visualization, and affective factors. American Education Research Journal, 14, 51-71. Fennema, E.H., and Sherman, J.A. (1978). Sex-related differences in mathematics achievement and related factors. Journal for Research in Mathematics Education, 9, 189-203. Fuson, K.C., and Murray, C. (1978). The haptic-visual perception, construction, and drawing of geometric shapes by children aged two to five: A Piagetian extension. In R. Lesh and D. Mierkiewicz (Eds.), Concerning the Development of Spatial and Geometric Concepts (pp. 49-83). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education. Galperin, P., and Georgiev, L. (1969). The formation of elementary mathematical notions. In J. Kilpatrick and I. Wirzup (Eds.), The Learning of Mathematical Concepts. Soviet Studies in the Psychology of Learning and Teaching Mathematics (Vol. 1). Palo Alto, CA: SMSG. Gao, F., Levine, S.C., and Huttenlocher, J. (2000). What do infants know about continuous quantity? Journal of Experimental Child Psychology, 77, 20-29. Gelman, R. (1972). Logical capacity of very young children: Number invariance rules. Child Development, 43, 75-90. Gelman, R. (1980). What young children know about numbers. Educational Psychologist, 15, 54-68. Gentner, D. (2003). Why we’re so smart. In D. Gentner and S. Goldin-Madow (Eds.), Language in Mind: Advances in the Study of Language and Thought (pp. 195-235). Cambridge: MA: MIT Press. Ginsburg, H.P., Cannon, J. Eisenband, and Pappas, S. (2006). Mathematical thinking and learning. In K. McCartney and D. Phillips (Eds.), Handbook of Early Child Development (pp. 208-229). Oxford, England: Blackwell. Guay, R.B., and McDaniel, E. (1977). The relationship between mathematics achievement and spatial abilities among elementary school children. Journal for Research in Mathematics Education, 8, 211-215. Hannibal, M.A.Z., and Clements, D.H. (2008). Young children’s understanding of basic geometric shapes. Manuscript submitted for publication, Indiana University of Pennsylvania, Indiana, PA. Hart, K. (1984). Which comes first—length, area, or volume? Arithmetic Teacher, 31, 16-18, 26-27. Hedges, L.V., and Chung, V. (in preparation). Does spatial ability predict STEM college major and employment?: An examination of two longitudinal studies. University of Chicago. Heibeck, T.H., and Markman, E.M. (1987). Word learning in children: An examination of fast mapping. Child Development, 67, 850-866. Hermer, L., and Spelke, E.S. (1994). A geometric process for spatial reorientation in young children. Nature, 37 0, 57-59. Hermer, L., and Spelke, E.S. (1996). Modularity and development: The case of spatial reorientation. Cognition, 61, 195-232. Hespos, S.J., and Spelke, E.S. (2004). Conceptual precursors to language. Nature, 43 0, 453-456. Hiebert, J. (1981). Cognitive development and learning linear measurement. Journal for Research in Mathematics Education, 12, 197-210. Hiebert, J. (1984). Why do some children have trouble learning measurement concepts? Arithmetic Teacher, 31, 19-24. Holloway, I., and Ansari, D. (2008). Domain-specific and domain-general changes in children’s development of number comparison. Developmental Science, 11, 644-649. Huttenlocher, J., Jordan, N.C., and Levine, S.C. (1994). A mental model for early arithmetic. Journal of Experimental Psychology: General, 123, 284-296. Huttenlocher, J., Newcombe, N., and Sandberg, E.H. (1994). The coding of spatial location in young children. Cognitive Psychology, 27, 115-148. Huttenlocher, J., Newcombe, N. and Vasilyeva, M. (1999). Spatial scaling in young children. Psychological Science, 10, 393-398. Huttenlocher, J., Duffy, S., and Levine, S.C. (2002). Infants and toddler discriminate amount: Are they measuring? Psychological Science, 13, 244-249. Jordan, N.C., Kaplan, D., Nabors Oláh, L. and Locuniak, M.N. (2006). Number sense growth in kindergarten: A longitudinal investigation of children at risk for mathematics difficulties. Child Development, 77, 153-175. Kail, R., Carter, P. and Pellegrino, J. (1979). The locus of sex differences in spatial skill. Perception and Psychophysics, 26, 182-186. Kato, Y. (1986). Development of spatial recognition in preschool children: On Piaget and Inhelder’s hypothesis of topological space. Perceptual and Motor Skills, 63, 443-450. Klein, A., Starkey, P., and Wakesley, A. (1999). Enhancing Pre-kindergarten Children’s Readiness for School Mathematics. Paper presented at the American Educational Research Association, Montreal, Canada. Klein, J.S., and Bisanz, J. (2000). Preschoolers doing arithmetic: The concepts are willing but the working memory is weak. Canadian Journal of Experimental Psychology, 54, 105-114. Kobayashi, T., Hiraki, K., Mugitani, R., and Hasegawa, T. (2004). Baby arithmetic: One object plus one tone. Cognition, 91, B23-B34. Kwon, M.-K., Levine, S.C., Suriyakham, L.W., and Ehrlich, S.B. (2009). Infants’ Quantitative Sensitivity: Number, Continuous Extent or Both. Presented at the Society for Research on Child Development Biennial Meeting, Denver, CO. Lean, G., and Clements, M.A. (1981). Spatial ability, visual imagery, and mathematical performance. Educational Studies in Mathematics, 12, 267-299. Learmonth, A., Newcombe, N., and Huttenlocher, J. (2001). Toddlers’ use of metric information and landmarks to reorient. Journal of Experimental Child Psychology, 80(3), 225-244. Leong, D.J. (n.d.). Tools of the Mind: Pre-K, Preschool. Available in: http://www.mscd.edu/extendedcampus/toolsofthemind/assets/pdf/Preschool%20Brochure%20(acrobat).pdf [accessed June 2008]. Levine, S.C., Jordan, N.C., and Huttenlocher, J. (1992). Development of calculation abilities in young children. Journal of Experimental Child Psychology, 53, 72-103. Levine, S.C., Huttenlocher, J., Taylor, A., and Langrock, A. (1999). Early sex differences in spatial ability. Developmental Psychology, 35, 940-949. Levine, S.C., Vasilyeva, M., Lourenco, S., Newcombe, N., and Huttenlocher, J. (2005). Socioeconomic status modifies the sex difference in spatial skill. Psychological Science, 16, 841-845. Levine, S.C., Huttenlocher, J., Pruden, S., Ratliff, K, and Saunders, J. (2008, June). Learning to Think Spatially: Role of Early Spatial Language and Activities. Paper presented at The Ins and Outs of Spatial Language: From Theory to Practice, Chicago, IL. Liben, L.S., and Downs, R.M. (1989). Understanding maps as symbols: The development of map concepts in children. Advances in Child Development and Behavior, 22, 145-201. Liben, L.S., and Yekel, C.A. (1996). Preschoolers’ understanding of plan and oblique maps: The role of geometric and representational correspondence. Child Development, 67, 2780-2796. Linn, M.C., and Peterson, A.C. (1985). Emergence and characterization of sex difference in spatial ability: A meta-analysis. Child Development, 56, 1479-1498. Lipton, J., and Spelke, E.S. (2003). Origins of number sense: Large number discrimination in human infants. Psychological Science, 14, 396-401. Lipton, J., and Spelke, E.S. (2004). Discrimination of large and small numerosidades by human infants. Infancy, 5, 271-290. Loewenstein, J., and Genter, D. (2005). Relational language and the development of relational mapping. Cognitive Psychology, 50, 315-353. Lourenco, S.F., Huttenlocher, J., and Fabian, L. (under review). Early sex difference in weighting geometric cues. Lovell, K. (1959). A follow-up study of some aspects of the work of Piaget and Inhelder on the child’s conceptions of space. British Journal of Educational Psychology, 29, 104-117. Mandler, J.M. (1992). How to build a baby II: Conceptual primitives. Psychological Review, 99, 587-604. Marmor, G.S. (1975). Development of kinetic images: When does the child first represent movement in mental images? Cognitive Psychology, 7, 548-559. McClelland, M.M., Cameron, C.E., Connor, C.M., Farris, C.L., Jewkes, A.M., and Morrison, F.J. (2007). Links between behavioral regulation and preschoolers’ literacy, vocabulary, and math skills. Developmental Psychology, 43, 947-959. McDonough, L., Choi, S., and Mandler, J.M. (2003). Understanding spatial relations: Flexible infants, lexical adults. Cognitive Psychology, 46, 229-259. Meck, W.H., and Church, R.M. (1983). A mode control model of counting and timing processes. Journal of Experimental Psychology: Animal Behavior Processes, 9, 320-334. Miller, K.F. (1984). Child as the measurer of all things: Measurement procedures and the development of quantitative concepts. In C. Sophian (Ed.), Origins of Cognitive Skills (pp. 193-228). Hillsdale, NJ: Erlbaum. Miller, K.F. (1989). Measurement as a tool for thought: The role of measurement procedures in children’s understanding of quantitative invariance. Developmental Psychology, 25, 589-600. Mix, K.S. (1999a). Preschoolers’ recognition of numerical equivalence: Sequential sets. Journal of Experimental Child Psychology, 74, 309-322. Mix, K.S. (1999b). Similarity and numerical equivalence: Appearances count. Cognitive Development, 14, 269-297. Mix, K.S. (2002). The construction of number concepts. Cognitive Development, 17, 1345‑1363. Mix, K.S. (2008). Surface similarity and label knowledge impact early numerical comparisons. British Journal of Developmental Psychology, 26, 13-32. Mix, K.S., Huttenlocher, J., and Levine, S.C. (1996). Do preschool children recognize auditoryvisual numerical correspondences? Child Development, 67, 1592-1608. Mix, K.S., Levine, S.C., and Huttenlocher, J. (1997). Numerical abstraction in infants: Another look. Developmental Psychology, 33, 423-428. Mix, K.S., Huttenlocher, J, and Levine, S.C. (2002). Quantitative Development in Infancy and Early Childhood. New York: Oxford University Press. Mix, K.S., Sandhofer, C.M., and Baroody, A. (2005). Number words and number concepts: The interplay of verbal and nonverbal processes in early quantitative development. In R.V. Kail (Ed.), Advances in Child Development and Behavior (vol. 3, pp. 305-346). New York: Elsevier. Moore, D.S., and Johnson, S.P. (2008). Mental rotation in human infants: A sex difference. Psychological Science, 19(11), 1063-1066. Moore, D., Benenson, J., Reznick, J.S., Peterson, M., and Kagan, J. (1987). Effect of auditory numerical information on infants’ looking behavior: Contradictory evidence. Developmental Psychology, 23, 665-670. National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics: An Overview. Reston, VA: Author. Newcombe, N., and Huttenlocher, J. (2000). Making Space: The Development of Spatial Representation and Reasoning. Cambridge, MA: MIT Press. Newcombe, N., and Huttenlocher, J. (2006). Development of spatial cognition. In D. Kuhn and R.S. Siegler (Eds.), Handbook of Child Psychology (6th ed., pp. 734-776). New York: Wiley. Newcombe, N., Huttenlocher, J., and Learmonth, A. (1999). Infants’ coding of location in continuous space. Infant Behavior and Development, 22, 483-510. Nunes, T., and Bryant, P. (1996). Children Doing Mathematics. Cambridge, MA: Blackwell. Nunes, T., Light, P., and Mason, J. (1993). Tools for thought: The measurement of length and area. Learning and Instruction, 3, 39-54. O’Hanlon, C.G., and Roberson, D. (2006). Learning in context: Linguistic and attentional constraints on children’s color term learning. Journal of Experimental Child Psychology, 94, 25-300. Ozer, D. (1987). Personality, intelligence, and spatial visualization: Correlates of mental rotation test performance. Journal of Personality and Social Psychology, 53, 129-134. Passolunghi, M.C., Vercelloni, B., and Schadee, H. (2007). The precursors of mathematics learning: Working memory, phonological ability, and numerical competence. Cognitive Development, 22, 165-184. Pettito, A.L. (1990). Development of numberline and measurement concepts. Cognition and Instruction, 7, 55-78. Piaget, J. (1941/1965). The Child’s Conception of Number. New York: Norton. Piaget, J., and Inhelder, B. (1967). The Child’s Conception of Space. (F.J. Langdon and J.L. Lunzer, Trans.). New York: Norton. (Original work published in 1948). Piaget, J., Inhelder, B., and Szeminska, A. (1960). The Child’s Conception of Geometry. London: Routledge and Kegan Paul. Quinn, P.C. (1994). The categorization of above and below spatial relations by young infants. Child Development, 65, 58-69. Quinn, P.C. (2004). Spatial representation by young infants: Categorization of spatial relations or sensitivity to a crossing primitive? Memory and Cognition, 32, 852-861. Quinn, P.C., and Liben, L.S. (2008). A sex difference in mental rotation in young infants. Psychological Science, 19(11), 1067-1070. Quinn, P.C., Norris, C.M., Pasko, R.N., Schmader, T.M. and Mash, C. (1999). Formation of a categorical representation for the spatial relation between by 6- to 7 month old infants. Visual Cognition, 6, 569-585. Rips, L.J., Bloomfield, A., and Asmuth, J. (2008). From numerical concepts to concepts of number. Behavioral and Brain Sciences, 6, 623-642. Sandberg, E.H., and Huttenlocher, J. (2001). Advanced spatial skills and advance planning: Components of 6-year-olds navigational map use. Journal of Cognition and Development, 2, 51-70. Sarnecka, B.W., and Carey, S. (2008). How counting represents numbers: What children must learn and when they learn it. Cognition, 108, 662-674. Satlow, E., and Newcombe, N. (1998). When is a triangle not a triangle? Young children’s developing concepts of geometric shape. Cognitive Development, 13, 547-559. Saxe, G.B., Guberman, S.R., and Gearhart, M. (1987). Social processes in early number development. Monographs of the Society for Research in Child Development, 52(2), iii-viii. Seo, K.-H., and Ginsburg, H.P. (2004). What is developmentally appropriate in early childhood mathematics education? In D.H. Clements, J. Sarama and A.-M. DiBiase (Eds.), Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education (pp. 91-104). Mahwah, NJ: Erlbaum. Shallcross, W.L., Göksun, T., Golinkoff, R. M., Hirsh-Pasek, K., Lloyd, M., Newcombe, N., and Roseberry, S. (2008, March). Building Talk: Parental Utterances During Construction Play. Poster presented at the 16th International Conference on Infant Studies, Vancouver, Canada. Shea, D.L., Lubinski, D., and Benbow, C.P. (2001). Importance of assessing spatial ability in intellectually talented young adolescents. Journal of Educational Psychology, 93, 604-614. Shipley, E.F., and Shepperson, B. (1990). Countable entities: developmental changes. Cognition, 32, 109-136. Skolnick, J., Langbort, C., and Day, L. (1982). How to Encourage Girls in Mathematics and Science: Strategies for Parents and Educators. Englewood Cliffs, NJ: Prentice-Hall. Slater, A., and Morison, V. (1985). Shape constancy and slant perception at birth. Perception, 14, 337-344. Smothergill, D.W., Hughes, F.P., Timmons, S.A., and Hutko, P. (1975). Spatial visualizing in children. Developmental Psychology, 11, 4-13. Sophian, C. (2002). Learning about what fits: Preschool children’s reasoning about effects of object size. Journal of Research in Mathematics Education, 33, 290-302. Sophian, C., and Kailihiwa, C. (1998). Units of counting: Developmental changes. Cognitive Development, 13, 561-585. Sophian, C., Garyantes, D., and Chang, C. (1997). When three is less than two: Early development in children’s understanding of fractional quantities. Developmental Psychology, 33, 731-744. Spelke, E.S. (1990). Principles of object perception. Cognitive Science, 14, 29-56. Spelke, E.S., and Kinzler, K.D. (2007). Core knowledge. Developmental Science, 10, 89-96. Starkey, P., and Cooper, R. (1980). Perception of numbers by human infants. Science, 210, 1033-1034. Starkey, P., Spelke, E.S., and Gelman, R. (1983). Detection of intermodal numerical correspondences by human infants. Science, 222, 179-181. Starkey, P., Spelke, E.S., and Gelman, R. (1990). Numerical abstraction by human infants. Cognition, 36, 97-128. Stea, D., Kerkman, D.D., Phinon, M.F., Middlebrook, N.N., and Rice, J.L. (2004). Preschoolers use maps to find a hidden object outdoors. Journal of Environmental Psychology, 24, 341-345. Stewart, R., Leeson, N., and Wright, R.J. (1997). Links between early arithmetical knowledge and early space and measurement knowledge: An exploratory study. In F. Biddulph and K. Carr (Eds.), Proceedings of the Twentieth Annual Conference of the Mathematics Education Research Group of Australasia (vol. 2, pp. 477-484). Hamilton, New Zealand: MERGA. Strauss, M.S., and Curtis, L.E. (1981). Infant perception of numerosidad. Child Development, 52, 1146-1152. Strauss, M.S., and Curtis, L.E. (1984). Development of numerical concepts in infancy. In C. Sophian (Ed.), Origins of Cognitive Skills (pp. 131-155). Hillsdale, NJ: Erlbaum. Szechter, L.E., and Liben, L. (2004). Parental guidance in preschoolers’ understanding of spatial-graphic representation. Child Development, 75, 869-885. Usiskin, Z. (1987). Resolving the continuing dilemmas in school geometry. In M.M. Lindquist and A.P. Shulte (Eds.), Learning and Teaching Geometry, K-12 (pp. 1-31), Reston, VA: National Council of Teachers of Mathematics. Uttal, D.H. (1996). Angles and distances. Children’s and adults’ reconstruction and scaling of spatial configurations. Child Development, 67, 2763-2779. van Hiele, P.M. (1986). Structure and Insight: A Theory of Mathematics Education. Orlando, FL: Academic Press. Van Loosbroek, E., and Smitsman, A.W. (1990). Visual perception of numerosidad in infancy. Developmental Psychology, 26, 916-922. Wallace, J.R., and Veek, A.L. (1995). Children’s Use of Maps for Direction and Distance Estimation. Paper presented at the Biennial Meeting of the Society for Research on Child Development, Indianapolis, IN. Wang, R.F., and Spelke, E.S. (2002). Human spatial representation: Insights from animals. Trends in Cognitive Sciences, 6, 376-382. Wheatley, G.H. (1990). Spatial sense and mathematics learning. Arithmetic Teacher, 37, 10-11. Wood, J.N., and Spelke, E.S. (2005). Infants’ enumeration of actions: Numerical discrimination and its signature limits. Developmental Science, 8, 173-181. Wynn, K. (1990). Children’s understanding of counting. Cognition, 36, 155-193. Wynn, K. (1992a). Addition and subtraction by human infants. Nature, 358, 749-750. Wynn, K. (1992b). Children’s acquisition of the palabras numéricas and the counting system. Cognitive Psychology, 24, 220-251. Wynn, K. (1996). Infants’ individuation and enumeration of actions. Psychological Science, 7, 164-169. Wynn, K., Bloom, P., and Chiang, W-C. (2002). Enumeration of collective entities by 5-monthold infants. Cognition, 83, B55-B62. Xu, F. (2003). Numerosidad discrimination in infants: Evidence for two systems of representations. Cognition, 89, B15-B25. Xu, F., and Arriaga, R.I. (2007). Number discrimination in 10-month-old infants. British Journal of Developmental Psychology, 25, 103-108. Xu, F., and Spelke, E.S. (2000). Large number discrimination in 6-month-old infants. Cognition, 74, B1-B11. Xu, F., Spelke, E., and Goddard, S. (2005). Number sense in human infants. Developmental Science, 8, 88-101.