View More View Less
  • 1 Pszichológiai Intézet, Iskolapszichológia Tanszék, ELTE-PPK, Budapest
Restricted access

Absztrakt

A matematikai megismerés, a számfeldolgozás folyamatának feltérképezése az elmúlt évtizedekben gyors ütemben elindult, mégis számos kérdés továbbra is nyitott. A fejlődési szempont és különösen a matematikai képességek tipikus fejlődésének vizsgálata sajnos viszonylag háttérbe szorult, pedig ez mind gyakorlati, mind elméleti szempontból releváns téma a kognitív pszichológia számára. Napjaink egyre inkább elfogadott nézete szerint a tipikus fejlődésmenet megismerése nélkülözhetetlen az atipikus fejlődés, így a fejlődési diszkalkulia megértéséhez, továbbá a felnőtt számfeldolgozás olyan modelljeinek megalkotásához, amelyek fejlődési szempontból is plauzibilisek.

Összefoglaló tanulmányunkban ezért a tipikus fejlődésre fókuszálunk, amelyben kitüntetett jelentőségű a veleszületett számérzék, illetve a biológiailag elsődleges matematikai képességek kibontásának, kiterjesztésének időszaka, az óvodáskor. A számnevek elsajátításával lehetővé válik a mennyiségek pontos reprezentációja, a számlálás, és alapvető számtani műveletek elvégzése nagyobb számkörben. A számfogalom kialakulásával, a számok terén szerzett tapasztalatok bővülésével a formális matematikatanulás első éveiben további mérföldköveit találjuk a számolási képességek fejlődésének. A számok reprezentációját végző mentális számegyenes egyre pontosabb, a számtani műveletek elvégzése során a kisiskolások egyre inkább számtani emlékezetükre támaszkodnak és egyre hatékonyabban alkalmazzák a rendelkezésükre álló számolási stratégiákat.

  • 1. Ashcraft, M. H. (1995): Cognitive psychology and simple arithmetic: A review and summary of new directions. Mathematical Cognition, 1, 334.

    • Search Google Scholar
    • Export Citation
  • 2. Baroody, A. J. (1999): The roles of estimation and the commutativity principle in the development of third-graders’ mental multiplication. Journal of Experimental Child Psychology, 75, 157193.

    • Search Google Scholar
    • Export Citation
  • 3. Briars, D. J., Siegler, R. S. (1984): A featural analysis of preschoolers’ counting knowledge. Developmental Psychology, 20, 607618.

    • Search Google Scholar
    • Export Citation
  • 4. Brown, J. S., Burton, R. B. (1978): Diagnostic models for procedural bugs in basic mathematical skills. 5. Cognitive Science, 2 (2), 155192.

    • Search Google Scholar
    • Export Citation
  • 5. Bryant, P., Christie, C., Rendu, A. (1999): Children's understanding of the relation between addition and subtraction: Inversion, identity, and decomposition. Journal of Experimental Child Psychology, 74, 194212.

    • Search Google Scholar
    • Export Citation
  • 6. Butterworth, B. (1999): The Mathematical Brain. London, Macmillan.

  • 7. Butterworth, B. (2005): The development of arithmetical abilities. Journal of Child Psychology and Psychiatry, 46 (1), 318.

  • 8. Butterworth, B., Girelli, L., Zorzi, M., Jonckheere, A. R. (2001): Organisation of addition facts in memory. Quarterly Journal of Experimental Psychology, 54A, 10051029.

    • Search Google Scholar
    • Export Citation
  • 9. Butterworth, B., Marchesini, N., Girelli, L. (2003): Organisation of multiplication facts in memory: Developmental trends. In: A. J. Baroody, A. Dowker (eds), The Development of Arithmetical Concepts and Skills: Constructing adaptive expertise. Mahwah, NJ, LEA, 189202.

    • Search Google Scholar
    • Export Citation
  • 10. Camos, V. (2003): Counting strategies from 5 years to adulthood: Adaptation to structural features. European Journal of Psychology of Education, 18 (3), 251265.

    • Search Google Scholar
    • Export Citation
  • 11. Campbell, J. I. D. (1994): Architectures for numerical cognition. Cognition, 53, 144.

  • 12. Campbell, J. I. D., Graham, D. J. (1985): Mental multiplication skill: Structure, process and acquisition. Canadian Journal of Psychology, 39, 338366.

    • Search Google Scholar
    • Export Citation
  • 13. Campbell, J. I. D., Gunter, R. (2002): Calculation, culture, and the repeated operand effect. Cognition, 86, 7196.

  • 14. Carpenter, T. P., Moser, J. M. (1982): The development of addition and subtraction problem solving skills. In: T. P. Carpenter, J. M. Moser, T. A. Romberg (eds), Addition and Subtraction: A cognitive perspective. Hillsdale, NJ, LEA, 924.

    • Search Google Scholar
    • Export Citation
  • 15. Chinn, C. A. (2006): The microgenetic method: Current work and extensions to classroom research. In: J. L. Green, G. Camilli, P. Elmore (eds), Handbook of Complementary Methods in Education Research. Washington, DC, American Educational Research Association, 439456.

    • Search Google Scholar
    • Export Citation
  • 16. Cooney, J. B., Swanson, H. L., Ladd, S. F. (1988): Acquisition of mental multiplication skill: Evidence for the transition between counting and retrieval strategies. Cognition and Instruction, 5, 323345.

    • Search Google Scholar
    • Export Citation
  • 17. Csépe Valéria (2005): Kognitív fejlődés-neuropszichológia. Budapest, Gondolat Kiadó.

  • 18. De Brauwer, J., Verguts, T., Fias, W. (2006): The representation of multiplication facts: Developmental changes in the problem size, five, and tie effects. Journal of Experimental Child Psychology, 94 (1), 4356.

    • Search Google Scholar
    • Export Citation
  • 19. Dehaene, S. (1992): Varieties of numerical abilities. Cognition, 44, 142.

  • 20. Dehaene, S. (2003): A számérzék: miként alkotja meg az elme a matematikát? Budapest, Osiris Kiadó.

  • 21. Dehaene, S., Cohen, L. (1997): Cerebral pathways for calculation: Double dissociation between rote verbal and quantitative knowledge of arithmetic. Cortex, 33, 219250.

    • Search Google Scholar
    • Export Citation
  • 22. Dehaene, S., Spelke, E., Pinel, P., Stanescu, R., Tsivkin, S. (1999): Sources of mathematical thinking: Behavioral and brain-imaging evidence. Science, 284, 970974.

    • Search Google Scholar
    • Export Citation
  • 23. De Rammelaere, S., Stuyven, E., Vandierendonck, A. (2001): Verifying simple arithmetic sums and products: Are the phonological loop and the central executive involved? Memory and Cognition, 29 (2), 267273.

    • Search Google Scholar
    • Export Citation
  • 24. Dowker, A. (2003): Young children's estimates for addition: The zone of partial knowledge and understanding. In: A. J. Baroody, A. Dowker (eds), The Development of Arithmetic Concepts and Skills: Constructing adaptive expertise. Mahwah, NJ, LEA, 243265.

    • Search Google Scholar
    • Export Citation
  • 25. Duncan, E. M., McFarland, C. E. (1980): Isolating the effect of symbolic distance and semantic congruity in comparative judgments: An additive-factors analysis. Memory & Cognition, 2, 95110.

    • Search Google Scholar
    • Export Citation
  • 26. Ebersbach, M., Luwel, K., Frick, A., Onghena, P., Verschaffel, L. (2008): The relationship between the shape of the mental number line and familiarity with numbers in 5- to 9-year old children: Evidence for a segmented linear model. Journal of Experimental Child Psychology, 99, 117.

    • Search Google Scholar
    • Export Citation
  • 27. Ellis, S. (1997): Strategy choice in sociocultural context. Developmental Review, 17, 490524.

  • 28. Fayol, M., Barrouillet, P., Marinthe, C. (1998): Predicting arithmetical achievement from neuro-psychological performance: A longitudinal study. Cognition, 68, 6370.

    • Search Google Scholar
    • Export Citation
  • 29. Fuson, K. C. (1988): Children's Counting and Concept s of Number. New York, Springer-Verlag.

  • 30. Fuson, K. C. (1992): Research on whole number addition and subtraction. In: D. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning. New York, Macmillan, 243275.

    • Search Google Scholar
    • Export Citation
  • 31. Fuson, K. C., Burghardt, B. H. (2003): Multi-digit addition and subtraction methods invented in small groups and teacher support of problem solving and reflection. In: A. J. Baroody, A. Dowker (eds), The Development of Arithmetic Concepts and Skills: Constructing adaptive expertise. Hillsdale, NJ, Erlbaum, 267304.

    • Search Google Scholar
    • Export Citation
  • 32. Fuson, K. C., Kwon, Y. (1992): Learning addition and subtraction: Effects of number words and other cultural tools. In: J. Bideaud, C. Meljac, J-P. Fischer (eds), Pathways to Number. Hillsdale, NJ, Erlbaum, 283306.

    • Search Google Scholar
    • Export Citation
  • 33. Fuson, K. C., Wearne, D., Hiebert, J., Human, P., Murray, H., Olivier, A., Carpenter, T. P., Fennema, E. (1997): Children's conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28, 130162.

    • Search Google Scholar
    • Export Citation
  • 34. Geary, D. C. (1994): Children's Mathematical Development: Research and practical applications. Washington, DC, American Psychological Association.

    • Search Google Scholar
    • Export Citation
  • 35. Geary, D. C. (1995): Reflections of evolution and culture in children's cognition: Implications for mathematical development and instruction. American Psychologist, 50, 2437.

    • Search Google Scholar
    • Export Citation
  • 36. Geary, D. C. (2000): From infancy to adulthood: The development of numerical abilities. European Child & Adolescent Psychiatry, 9 (2), 1116.

    • Search Google Scholar
    • Export Citation
  • 37. Geary, D. C. (2004): Mathematics and learning disabilities. Journal of Learning Disabilities, 37, 415.

  • 38. Geary, D. C., Brown, S. C. (1991): Cognitive addition: Strategy choice and speed-of-processing differences in gifted, normal, and mathematically disabled children. Developmental Psychology, 27, 398406.

    • Search Google Scholar
    • Export Citation
  • 39. Geary, D. C., Hamson, C. O., Hoard, M. K. (2000): Numerical and arithmetical cognition: A longitudinal study of process and concept deficits in children with learning disability. Journal of Experimental Child Psychology, 77, 236263.

    • Search Google Scholar
    • Export Citation
  • 40. Geary, D. C., Widaman, K. F. (1992): Numerical cognition: On the convergence of componential and psychometric models. Intelligence, 16, 4780.

    • Search Google Scholar
    • Export Citation
  • 41. Gelman, R., Gallistel, C. R. (1978): The Child's Understanding of Number. Cambridge, MA, Harvard University Press.

  • 42. Gelman, R., Meck, E. (1983): Preschoolers’ counting: Principles before skill. Cognition, 13, 343359.

  • 43. Gibbon, J., Church, R. M. (1981): Time left: Linear versus logarithmic subjective time. Journal of the Experimental Analysis of Behavior, 7, 87107.

    • Search Google Scholar
    • Export Citation
  • 44. Gordon, P. (2004): Numericalcognitionwithoutwords: EvidencefromAmazonia. Science, 306, 496499.

  • 45. Gyarmathy Éva (1998): A tanulási zavarok szindróma a szakirodalomban I. Új Pedagógiai Szemle, 10, 5968.

  • 46. Huntley-Fenner, G. (2001): Children's understanding of number is similar to adults’ and rats’: Numerical estimation by 5–7-year-olds. Cognition, 78, 2740.

    • Search Google Scholar
    • Export Citation
  • 47. Imbo, I., Vandierendonck, A. (2007): The development of strategy use in elementary school children: Working memory and individual differences. Journal of Experimental Child Psychology, 96, 284309.

    • Search Google Scholar
    • Export Citation
  • 48. Józsa Krisztián (2003): A számolási készség fejlesztése. In: Dubiczné Mile Katalin és Farkas Istvánné (szerk.), Az általános iskola alapozó szakaszának megújítása. Székesfehérvár, Fejér Megyei Pedagógiai Szakmai és Szakszolgáltató Intézet, 2744.

    • Search Google Scholar
    • Export Citation
  • 49. Kanayet, F., Opfer, J. E. (2009): Why children's number-line estimates follow Fechner's law. In: N. Taatgen, H. van Rijn (eds), Proceedings of the XXXI Annual Cognitive Science Society. Mahwah, NJ, Erlbaum.

    • Search Google Scholar
    • Export Citation
  • 50. Kaufman, E. L., Lord, M. W., Reese, T. W., Volkmann, J. (1949): The discrimination of visual number. American Journal of Psychology, 62, 498525.

    • Search Google Scholar
    • Export Citation
  • 51. Kaufmann, L., Nuerk, H-C. (2005): Numerical development: Current issues and future perspectives. Psychology Science,42, 142170.

  • 52. Laski, E. V., Siegler, R. S. (2007): Is 27 a big number? Correlational and causal connections among numerical categorization, number line estimation, and numerical magnitude comparison. Child Development, 76, 17231743.

    • Search Google Scholar
    • Export Citation
  • 53. Lee, K. M., Kang, S. Y. (2002): Arithmetic operation and working memory: Differential suppression in dual tasks. Cognition, 83, 6368.

    • Search Google Scholar
    • Export Citation
  • 54. Lemaire, P., Fayol, M., Abdi, H. (1991): Associative confusion effect in cognitive arithmetic: Evidence for partially autonomous processes. European Bulletin of Cognitive Psychology, 11, 587604.

    • Search Google Scholar
    • Export Citation
  • 55. Lemaire, P., Siegler, R. S. (1995): Four aspects of strategic change: contributions to children's learning of multiplication. Journal of Experimental Psychology: General, 124 (1), 8397.

    • Search Google Scholar
    • Export Citation
  • 56. Márkus Attila (2007): Számok, számolás, számolászavarok. Budapest, Pro Die Kiadó.

  • 57. McCloskey, M. (1992): Cognitive mechanisms in numerical processing: Evidence from acquired dyscalculia. Cognition, 44, 107157.

  • 58. Mix, K. S. (1999): Preschoolers’ recognition of numerical equivalence: sequential sets. Journal of Experimental Child Psychology, 74, 309332.

    • Search Google Scholar
    • Export Citation
  • 59. Moeller, K., Klein, E., Fischer, M. H., Nuerk, H-C., Willmes, K. (2011): Representation of multiplication facts – Evidence for partial verbal coding. Behavioral and Brain Function, 7, 19.

    • Search Google Scholar
    • Export Citation
  • 60. Moyer, R. S., Landauer, T. K. (1967): Time required for judgments of numerical inequalities. Nature, 215, 15191520.

  • 61. Opfer, J. E., Siegler, R. S. (2007): Representational change and children's numerical estimation. Cognitive Psychology, 55, 169195.

    • Search Google Scholar
    • Export Citation
  • 62. Osborne, R. T., Lindsey, J. M. (1967): A longitudinal investigation of change in the factorial composition of intelligence with age in young school children. Journal of Genetic Psychology, 110, 4958.

    • Search Google Scholar
    • Export Citation
  • 63. Pesenti, M., Thioux, M., Seron, X., De Volder, A. (2000): Neuroanatomical substrate of Arabic number processing, numerical comparison and simple addition: A PET study. Journal of Cognitive Neuroscience, 12 (3), 61479.

    • Search Google Scholar
    • Export Citation
  • 64. Piaget, J. (1952): The Child's Conception of Number. London, Routledge & Kegan Paul.

  • 65. Pica, P., Lemer, C., Izard, V., Dehaene, S. (2004): Exact and approximate arithmetic in an Amazonian indigene group. Science, 306, 499503.

    • Search Google Scholar
    • Export Citation
  • 66. Potter, M. C., Levy, E. I. (1968): Spatial enumeration without counting. Child Development, 39, 265272.

  • 67. Rubinstein, O., Henik, A., Berger, A., Shahar-Shalev, S. (2002): The development of internal representations of magnitude and their association with arabic numerals. Journal of Experimental Child Psychology, 81, 7492.

    • Search Google Scholar
    • Export Citation
  • 68. Sekular, R., Mierkiewicz, D. (1977): Children's judgment of numerical inequality. Child Development, 48, 630633.

  • 69. Shrager, J., Siegler, R. S. (1998): SCADS: A model of children's strategy choices and strategy discoveries. Psychological Science, 9, 405410.

    • Search Google Scholar
    • Export Citation
  • 70. Siegler, R. S. (1988a): Strategy choice procedures and the development of multiplication skill. Journal of Experimental Psychology: General, 117, 258275.

    • Search Google Scholar
    • Export Citation
  • 71. Siegler, R. S. (1988b): Individual differences in strategy choices: good students, not-so-good students, and perfectionists. Child Development, 59 (4), 833851.

    • Search Google Scholar
    • Export Citation
  • 72. Siegler, R. S. (1996): Emerging Minds: The process of change in children's thinking. Oxford University Press.

  • 73. Siegler, R. S. (1999): Strategic development. Trends in Cognitive Sciences, 3 (11), 430435.

  • 74. Siegler, R. S., Booth, J. L. (2004): Development of numerical estimation in young children. Child Development, 75, 428444.

  • 75. Siegler, R. S., Jenkins, E. (1989): How Children Discover New Strategies. Hillsdale, NJ, Lawrence Erlbaum Associates.

  • 76. Siegler, R. S., Opfer, J. E. (2003): The development of numerical estimation: Evidence for multiple representations of numerical quantity. Psychological Science, 14, 237243.

    • Search Google Scholar
    • Export Citation
  • 77. Siegler, R. S., Shipley, C. (1995): Variation, selection, and cognitive change. In: T. Simon, G. Halford (eds), Developing Cognitive Competence: New approaches to process modeling. Hillsdale, NJ, Erlbaum, 3176.

    • Search Google Scholar
    • Export Citation
  • 78. Soltész Fruzsina (2010): Typical and atypical development of magnitude processing. Ph.D. Thesis, ELTE, Budapest.

  • 79. Song, M. J., Ginsburg, H. P. (1987): The development of informal and formal mathematical thinking in Korean and U. S. children. Child Development, 58, 12861296.

    • Search Google Scholar
    • Export Citation
  • 80. Spelke, E. S., Tsivkin, S. (2001): Language and number: A bilingual training study. Cognition, 78, 4588.

  • 81. Stazyk, E. H., Ashcraft, M. H., Hamann, M. S. (1982): A network approach to simple mental multiplication. Journal of Experimental Psychology: Learning, Memory, and Cognition, 8, 320335.

    • Search Google Scholar
    • Export Citation
  • 82. Temple, E., Posner, M. I. (1998): Brain mechanisms of quantity are similar in 5-year-olds and adults. Proceedings of the National Academy of Sciences USA, 95, 78367841.

    • Search Google Scholar
    • Export Citation
  • 83. Tóth Dénes , Csépe Valéria (2008): Azolvasás fejlődésekognitív pszichológiaiszempontból. Pszichológia, 28 (1), 3552.

  • 84. Vandenberg, S. G. (1962): The hereditary abilities study: Hereditary components in a psychological test battery. American Journal of Human Genetics, 14, 220237.

    • Search Google Scholar
    • Export Citation
  • 85. Vandenberg, S. G. (1966): Contributions of twin research to psychology. Psychological Bulletin, 66, 327352.

  • 86. Wynn, K. (1990): Children's understanding of counting. Cognition, 36, 155193.

  • 87. Xu, F., Spelke, E. S., Goddard, S. (2005): Number sense in human infants. Developmental Science, 8, 88101.

  • 88. Zhou, X., Booth, J. R., Lu, J., Zhao, H., Butterworth, B., Chen, C., Dong, Q. (2011): Age-independent and age-dependent neural substrate for single-digit multiplication and addition arithmetic problems. Developmental Neuropsychology, 36 (3), 338352.

    • Search Google Scholar
    • Export Citation
  • 89. Zhou, X., Chen, C., Zang, Y., Dong, Q., Chen, C., Qiao, S. (2007): Dissociated brain organizations for single-digit addition and multiplication. NeuroImage, 35, 871880.

    • Search Google Scholar
    • Export Citation