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Journal

2017 | 1(2017) | 7–26

Article title

Konstruktywizm Jeana Piageta i koncepcja zmysłu liczby a edukacja matematyczna

Title variants

EN
Jean Piaget’s constructivism and the number sense theory vs mathematics education

Languages of publication

PL

Abstracts

PL
Twierdzenie Jeana Piageta, że rozwój poznawczy dziecka jest podstawą nabywania i rozwijania pojęcia liczby oraz umiejętności operowania liczbami, wywarło ogromny wpływ na sposób nauczania matematyki. Od czasu pojawienia się wyników badań wskazujących na istnienie biologicznych podstaw operowania liczbami koncepcja Piageta zaczęła być szeroko krytykowana przez przedstawicieli neuropsychologii poznawczej. Dowody na to, że istnieje wrodzone, niezależne od systemu językowego i edukacji podłoże powstawania i rozwoju reprezentacji liczbowej, wpłynęły na sposób tworzenia programów kształcenia i zaleceń metodycznych w Europie Zachodniej i Stanach Zjednoczonych. Pomimo że dyskusja na temat możliwości nabywania i rozwoju pojęcia liczby przez dzieci toczy się od wielu lat, nadal trudno mówić o jednoznacznych rozstrzygnięciach w tej kwestii. Celem tego artykułu jest krytyczna analiza obu stanowisk teoretycznych oraz formułowanych w ich ramach zaleceń dotyczących praktyki edukacyjnej.
EN
Piaget’s claim that the cognitive development of a child is the basis for acquiring and developing the concept of numbers as well as the ability to work with numbers, had a huge impact on the methods of teaching mathematics in Poland and the world. Since the appearance of research results indicating the existence of a biological basis for the ability to work with numbers, the Piaget’s theory has been widely criticized by representatives of cognitive neuropsychology. The evidence that there is an innate foundation for the formation and development of numerical representation, independent of language and education, has influenced the manner of constructing educational curricula and methodological recommendations in Western Europe and the United States. Although the discussion on the possibility of children acquiring and developing number sense has been ongoing for many years, it is still difficult to point to a conclusive resolution of this issue. The aim of this paper is to present a critical analysis of the two theoretical concepts and recommendations for educational practice, formulated within their frameworks.

Journal

Year

Issue

Pages

7–26

Physical description

Dates

published
2017-03-31

Contributors

  • Instytut Psychologii, Uniwersytet Jagielloński

References

  • Aster, M. G. von i Shalev, R. S. (2007). Number development and developmental dyscalculia. Developmental Medicine & Child Neurology, 49(11), 868–873.
  • Berch, D. B. (2005). Making sense of number sense: implications for children with mathematical disabilities. Journal of Learning Disabilities, 38(4), 333–339.
  • Bijeljac-Babic, R., Bertoncini, J. i Mehler, J. (1991). How do four-day-old infants categorize multisyllabic utterances. Developmental Psychology, 29(4), 711–721.
  • Blakemore, S-J. i Frith, U. (2008). Jak uczy się mózg. Kraków: Wydawnictwo Uniwersytetu Jagiellońskiego.
  • Boltzmann, L. E. (1890). Über die Bedeutung von Theorien. W: L. E. Boltzmann (red.), (1979), Populäre Schriften (s. 54–58). Braunschweig: Vieweg.
  • Bonny, J. W. i Lourenco, S. F. (2013). The approximate number system and its relation to early math achievement: evidence from the preschool years. Journal of Experimental Child Psychology, 114(3), 375–388.
  • Bonny, J. W. i Lourenco, S. F. (2015). Individual differences in children’s approximations of area correlate with competence in basic geometry. Learning and Individual Differences, 44, 16–24.
  • Brożek, B. i Hohol, M. (2014). Umysł matematyczny. Kraków: Copernicus Center Press.
  • Bugden, S. i Ansari, D. (2011). Individual differences in children’s mathematical competencies are related to the intentional but not automatic processing of Arabic numerals. Cognition, 118(1), 32–44.
  • Butterworth, B. (1999). The mathematical brain. London: Macmillan.
  • Carey, S. i Spelke, E. (1996). Science and core knowledge. Philosophy of Science, 63(4), 515–533.
  • Castronovo, J. i Göbel, S. (2012). Impact of high mathematics education on the number sense. PLoS One, 7(4), e33832.
  • Cohen, L. B. i Marks, K. S. (2002). How infants process addition and subtraction events. Developmental Science, 5(2), 186–212.
  • Conklin, M. i Sheffield, S. (2012). It makes sense! Using the hundreds chart to build number sense. Sausalito: Math Solutions.
  • Dackermann, T., Huber, S., Bahnmueller, J., Nuerk, H.-C. i Moeller, K. (2015). An integration of competing accounts on children’s number line estimation. Frontiers in Psychology, 6. doi: 10.3389/fpsyg.2015.00884
  • Dantzig, T. (1954). Number: the language of science. New York: MacMillan.
  • Dehaene, S. (1992). Varieties of numerical abilities. Cognition, 44(1–2), 1–42.
  • Dehaene, S. (2011). The number sense. How the mind creates mathematics. New York: Oxford Univeristy Press.
  • Dehaene, S., Dehaene-Lambartz, G. i Cohen, L. (1998). Abstract representations of numbers in the animal and human brain. Trends in Neuroscience, 21(8), 355–361.
  • Dehaene, S., Molko, N., Cohen, L. i Wilson, A. J. (2004). Arithmetic and the brain. Current Opinion in Neurobiology, 14(2), 218–224.
  • Dehaene, S., Piazza, M., Pinel, P. i Cohen, L. (2003). Three parietal circuits for number processing. W: J. I. D. Campbell (red.), Handbook of mathematical cognition (s. 433–453). New York: Psychology Press.
  • Diamond, A. i Goldman-Rakic, P . S. (1989). Comparison of human infants and rhesus monkeys on Piaget’s A-not-B task: evidence for dependence on dorsolateral prefrontal cortex. Experimental Brain Research, 74(1), 24–40.
  • Dietrich, J., Huber, S. i Nuerk, H.-C. (2015). Methodological aspects to be considered when measuring the approximate number system (ANS)
  • – a research review. Frontiers in Psychology, 6. doi: 10.3389/fpsyg.2015.00295
  • Ditz, H, i Nieder, A. (2015). Neurons selective to the number of visual items in the corvid songbird endbrain. Proceedings of the National Academy of Sciences, 112(25), 7827–7832.
  • Feigenson, L., Dehaene, S. i Spelke, E. (2004). Core system of number. Trends in Cognitive Science, 8(7), 307–314.
  • Feigenson, L., Libertus, M. E. i Halberda, J. (2013). Links between the intuitive sense of number and formal mathematics ability. Child Development Perspectives, 7(2), 74–79.
  • Fuhs, M. W. i McNeil, N. M. (2013). ANS acuity and mathematics ability in preschoolers from low-income homes: contributions of inhibitory control. Developmental Science, 16(1), 136–148. doi: 10.1111/desc.12013
  • Geary, D. C. (1995). Reflections of evolution and culture in children’s cognition: implications for mathematical development and instruction. American Psychologist, 50(1), 24–37.
  • Gelman, R. i Baillargeon, R. (1983). A review of some Piagetian concepts. W: P. H. Mussen (red.), Handbook of child psychology (s. 167–230). New York: John Wiley & Sons.
  • Gilmore, C. K., McCarthy, S. E. i Spelke, E. S. (2010). Non-symbolic arithmetic abilities and achievement in the first year of formal schooling in mathematics. Cognition, 115(3), 394–406.
  • Gilmore, C. K., Attridge, N., Clayton, S., Cragg, L., Johnson, S., Marlow, N., Simms, V. i Inglis, M. (2013). Individual differences in inhibitory control, not non-verbal number acuity, correlate with mathematics achievement. Plos One, 8(6), e67374.
  • Gold, R. (1984). Performance on Donaldson and McGarrigle’s “cars and garages” task as evidence about the reasons for failure on Piaget’s number-conservation task. The Journal of Genetic Psychology, 147(2), 151–165.
  • Gracia-Bafaully, M. i Noël, M. P. (2008). Does finger training increase young children’s numerical performance? Cortex, 44(4), 368–375.
  • Greeno, J. G. (1991). Number sense as situated knowing in a conceptual domain. Journal for Research in Mathematics Education, 22(3), 170–218.
  • Griffin, S. (2004). Building number sense with Number Worlds: a mathematics program for young children. Early Childhood Research Quarterly, 19(1), 173–180.
  • Gruszczyk-Kolczyńska, E. i Skura, M. (2005). Skarbiec matematyczny. Warszawa: Nowa Era.
  • Gruszczyk-Kolczyńska, E. (2012). Dzieci ze specyficznymi trudnościami w uczeniu się matematyki. Warszawa: Wydawnictwa Szkolne i Pedagogiczne.
  • Gruszczyk-Kolczyńska, E. (red.). (2014). Edukacja matematyczna w klasie I. Książka dla nauczycieli i rodziców. Kraków: Wydawnictwo Centrum Edukacyjne Bliżej Przedszkola.
  • Gunderson, E. A., Ramirez, G., Beilock, S. L. i Levine, S. C. (2012). The relation between spatial skill and early number knowledge: the role of the linear number line. Developmental Psychology, 48(5), 1229–1241.
  • Halberda, J. i 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. Developmental Psychology, 44(5), 1457–1465.
  • Halberda, J., Ly, R., Wilmer, J., Naiman, D. i Germine, L. (2012). Number sense across the lifespan as revealed by a massive internet-based sample. Proceedings of the National Academy of Sciences, 109(28), 11116–11120.
  • Halberda, J., Mazzocco, M. M. i Feigenson, L. (2008). Individual differences in non-verbal number acuity correlate with maths achievement. Nature, 445(2), 665–669.
  • Hassinger-Das, B., Jordan, N. C., Glutting, J., Irwin, C. i Dyson, N. (2014). Domain-general mediators of the relation between kindergarten number sense and first-grade mathematics achievement. Journal of Experimental Child Psychology, 118, 78–92.
  • Holloway, I. D. i Ansari, D. (2009). Mapping numerical magnitudes onto symbols: the numerical distance effect and individual differences in children’s mathematics achievement. Journal of Experimental Child Psychology, 103(1), 17–29.
  • Honoré, N. i Noël, M.-P. (2016). Improving preschoolers’ arithmetic through number magnitude training: the impact of non-symbolic and symbolic training. Plos One, 11(11). doi: 10.1371/journal.pone.0166685
  • Hornung, C., Schiltz, C., Brunner, M. i Martin, R. (2014). Predicting first-grade mathematics achievement: the contributions of domain-general cognitive abilities, nonverbal number sense, and early number competence. Frontiers in Psychology, 5. doi: 10.3389/fpsyg.2014.00272
  • Howell, S. C. i Kemp, C. R. (2010). Assessing preschool number sense: skills demonstrated by children prior to school entry. Educational Psychology, 30(4), 411–429.
  • Hyde, D. C. i Spelke, E. S. (2010). Neural signatures of number processing in human infants: evidence for two core systems underlying numerical cognition. Developmental Science, 14(2). doi: 10.1111/j.1467-7687.2010.00987.x
  • Inglis, M., Attridge, N., Batchelor, S. i Gilmore, C. (2011). Non-verbal number acuity correlates with symbolic mathematics achievement: but only in children. Psychonomic Bulletin & Review, 18(6), 1222–1229.
  • Inhelder, B. i Piaget, J. (1970). Od logiki dziecka do logiki młodzieży. Warszawa: PWN.
  • Jordan, N. C. (2007). The need for number sense. Early Intervention at Every Age, 65(2), 63–65.
  • Jordan, N. C., Glutting, J. i Ramineni, C. (2010). The importance of number sense to mathematics achievement in first and third grades. Learning and Individual Differences, 20(2), 82–88.
  • Kalinowska, A. (2013). Wczesnoszkolna edukacja matematyczna – ograniczenia i ich przełamywanie. Olsztyn: Wydawnictwo Uniwersytetu Warmińsko-Mazurskiego.
  • Kaufmann, L., Wood, G., Rubinsten, O. i Henik, A. (2011). Meta-analysis of developmental fMRI studies investigating typical and atypical trajectories of number processing and calculation. Developmental Neuropsychology, 36(6), 763–787.
  • Kaufmann, L., Kucian, K. i Aster, M. von (2015). Development of the numerical brain. W: R. Cohen Kadosh i A. Dowker (red.), The Oxford handbook of numerical cognition (s. 485–501). Oxford: Oxford University Press.
  • Koechlin, E., Dehaene, S. i Mehler, J. (1997). Numerical transformations in five-month-old human infants. Mathematical Cognition, 3(2), 89–104.
  • Klus-Stańska, D. i Kalinowska, A. (2004). Rozwijanie myślenia matematycznego młodszych uczniów. Warszawa: Wydawnictwo Akademickie „Żak”.
  • Klus-Stańska, D. i Nowicka, M. (2014). Sensy i bezsensy edukacji wczesnoszkolnej. Gdańsk: Harmonia Universalis.
  • Libertus, M. E., Feigenson, L. i Halberda, J. (2013). Is approximate number precision a stable predictor of math ability? Learning and Individual Differences, 25(1), 126–133.
  • Libertus, M., Odic, D. i Halberda, J. (2012). Intuitive sense of number correlates with math scores on college-entrance examination. Acta Psychologica, 141(3), 373–379.
  • Lindskog, M., Winman, A. i Juslin, P. (2014). The association between higher education and approximate number system acuity. Frontiers in Psychology, 5. doi:
  • 10.3389/fpsyg.2014.00462
  • Lipton, J. S. i Spelke, E. S. (2004). Discrimination of large and small numerosities by human infants. Infancy, 5(3), 271–290.
  • Lourenço, O. i Machado, A. (1996). In defense of Piaget’s theory: a reply to 10 common criticisms. Psychological Review, 103(1), 143–164.
  • Mackiewicz, R. (2012). Liczby w decyzjach ekonomicznych: instynkt numeryczny i wrażliwość cenowa. W: A. Falkowski i T. Zaleśkiewicz (red.), Psychologia poznawcza w praktyce. Ekonomia, biznes, polityka (s. 137–185). Warszawa: PWN.
  • McGuire, P., Kinzie, M. i Berch, D. (2011). Developing number sense in pre-K with five-frames. Early Childhood Education Journal, 40(4), 213–222.
  • Mehler, J. i Bever, T. G. (1967). Cognitive capacity of very young children. Science, 158(3797), 141–142.
  • Mehler, J. i Bever, T. G. (1968). Reply by J. Mehler and T. G. Bever. Science, 162(3857), 979–981.
  • Menon, V. (2015). Arithmetic in the child and adult brain. W: R. Cohen Kadosh i A. Dowker (red.), The Oxford handbook of numerical cognition (s. 502–
  • –530). Oxford: Oxford University Press.
  • McGarrigle, J. i Donaldson, M. (1974). Conservation accidents. Cognition, 4(3), 341–350.
  • Mix, K. S., Levine, S. C. i Huttenlocher, J. (1997). Numerical abstraction in infants: another look. Developmental Psychology, 33(3), 423–428.
  • Moore, D. S., Benenson, J., Reznick, J. S., Peterson, M. i Kagan, J. (1987). Effect of auditory numerical information on infants’ looking behavior: contradictory evidence. Developmental Psychology, 23(5), 665–670.
  • Nieder, A. (2005). Counting on neurons: the neurobiology of numerical competence. Nature Reviews Neuroscience, 6, 177–190.
  • Nieder, A. i Merten, K. (2007). A labeled-line code for small and large numerosities in the monkey prefrontal cortex. Journal of Neuroscience, 27(22), 5986–5993.
  • Nieder, A. i Miller, E. K. (2003). Coding of cognitive magnitude. Compressed scaling of numerical information in the primate prefrontal cortex. Neuron, 37(1), 149–157.
  • Nieder, A. i Miller, E. K. (2004). A parieto-frontal network for visual numerical information in the monkey. Proceedings of National Academy of Science, 101, 7457–7462.
  • Nosworthy, N., Bugden, S., Archibald, L., Evans, B. i Ansari, D. (2013). A two-minute paper-and--pencil test of symbolic and nonsymbolic numerical magnitude processing explains variability in primary school children’s arithmetic competence. Plos One, 8(7), e67918.
  • Nunes, T. i Bryant, P. E. (1996). Children doing mathematics. Oxford: Blackwell.
  • Oszwa, U. (2009). Psychologiczna analiza procesów operowania liczbami u dzieci z trudnościami w matematyce. Lublin: Wydawnictwo UMCS.
  • Parrish, S. (2010). Number talks: helping children build mental math and computation strategies, grades K-5. Sausalito: Math Solutions.
  • Peters, A. (2013). Realizing utopia as a scholarly endeavor. The European Journal of International Law, 24(2), 533–552.
  • Piaget, J. (1966). Narodziny inteligencji dziecka. Warszawa: PWN.
  • Piaget, J. (1967). Six psychological studies. New York: Random House.
  • Piaget, J. (1968). Quantification, conservation, and nativism. Science, 162(3857), 976–979.
  • Piaget, J. i Inhelder, B. (1993). Psychologia dziecka. Wrocław: Siedmioróg.
  • Piazza, M., Pica, P., Izard, V., Spelke, E. S. i Dehaene, S. (2013). Education enhance the acuity of the nonverbal approximate number system, Psychological Science, 24(6), 1037–143.
  • Piazza, M., Pinel, P., Le Bihan, D. i Dehaene, S. (2007). A magnitude code common to numerosities and number symbols in human intraparietal cortex. Neuron, 53(2), 293–305.
  • Pica, P., Lemer, C., Izard, V. i Dehaene, S. (2004). Exact and approximate arithmetic in an Amazonian indigene group. Science, 306(5695), 499–503.
  • Pina, V., Castillo, A., Cohen Kadosh, R. i Fuentes, L. J. (2015). Intentional and automatic numerical processing as predictors of mathematical abilities in primary school children. Frontiers in Psychology, 6. doi: 10.3389/fpsyg.2015.00375
  • Pixner, S., Moeller, K., Nuerk, H.-C., Hermanova, V. i Kaufmann, L. (2011). Whorf reloaded: language effects on non-verbal number processing in first grade – a trilingual study. Journal of Experimental Child Psychology, 108(2), 371–382.
  • Price, G. R., Palmer, D., Battista, C. i Ansari, D. (2012). Nonsymbolic numerical magnitude comparison: reliability and validity of different task variants and outcome measures, and their relationship to arithmetic achievement in adults. Acta Psychologica, 140(1), 50–57.
  • Ramani, G. B. i Siegler, R. S. (2011). Reducing the gap in numerical knowledge between low- and middle-income preschoolers. Journal of Applied Developmental Psychology, 32(3), 146–159.
  • Reinholz, A., Rychwalska, A., Stefańska, J. i Trojan, M. (2003). Wpływ ilości i układu elementów małych zbiorów na jakość subityzowania. Psychologia–Etologia–Genetyka, 8, 91–112.
  • Reys, B. J. (1994). Promoting number sense in middle grades. Teaching Mathematics in the Middle School, 1, 114–120.
  • Rumbaugh, D. M., Savage-Rumbaugh, S. i Hegel, M. T. (1987). Summation in the chimpanzee (Pan troglodytes). Journal of Experimental Psychology. Animal Behavior Processes, 13(2), 107–115.
  • Sasanguie, D., Göbel, S., Moll, K., Smets, K. i Reynvoet, B. (2013). Approximate number sense, symbolic number processing, or number-space mappings: what underlies mathematics achievement? Journal of Experimental Child Psychology, 114, 418–431.
  • Semadeni, Z. (2016). Podejście konstruktywistyczne do matematycznej edukacji wczesnoszkolnej. Warszawa: Ośrodek Rozwoju Edukacji.
  • Seron, X. i Pesenti, M. (2001). The number sense theory needs more empirical evidence. Mind and Language, 16(1), 76–88.
  • Simon, T. J. (1997). Reconceptualizing the origins of number knowledge: a “non numerical” account. Cognitive Development, 12, 349–372.
  • Simon, T. J., Hespos, S. J. i Rochat, P. (1995). Do infants understand simple arithmetic? A replication of Wynn (1992). Cognitive Development, 10, 253–269.
  • Slusser, E., Ditta, A. i Sarnecka, B. (2013). Connecting numbers to discrete quantification: a step in the child’s construction of integer concepts. Cognition, 129(1), 31–41.
  • Soltesz, F., Szűcs, S. i Szűcs, L. (2010). Relationships between magnitude representation, counting and memory in 4-to 7-year-old children: a developmental study. Behavioral and Brain Functions, 13(6), 1–14.
  • Starkey, P. i Cooper, R. G. (1980). Perception of numbers by human infants. Science, 210(4473), 1033–1035.
  • Starkey, P., Spelke, E. S. i Gelman, R. (1983). Detection of intermodal numerical correspondences by human infants. Science, 222(4620), 179–181.
  • Strauss, M. i Curtis, L. (1981). Infant perception of numerosity. Child Development, 52(4), 1146–1152.
  • Szczygieł, M., Cipora, K. i Hohol, M. (2015). Liczenie na palcach w ontogenezie i jego znaczenie dla rozwoju kompetencji matematycznych. Psychologia Rozwojowa, 20(3), 23–33.
  • Szemińska, A. (1981). Rozwój pojęć matematycznych u dziecka, W: Z. Semadeni (red.). Nauczanie początkowe matematyki. Podręcznik dla nauczyciela (t. 1, s. 112–250). Warszawa: Wydawnictwa Szkolne i Pedagogiczne.
  • Szűcs, D., Devine, A., Soltesz, F., Nobes, A. i Gabriel, F. (2014). Cognitive components of a mathematical processing network in 9-year-old children. Developmental Science, 17(4), 506–524.
  • Wadsworth, B. J. (1998). Teoria Piageta. Poznawczy i emocjonalny rozwój dziecka. Warszawa: Wydawnictwa Szkolne i Pedagogiczne.
  • Wakeley, A., Rivera, S. i Langer, J. (2000). Can young infants add and subtract? Child Development, 71(6), 1525–1534.
  • Walden, T., Kim, G., McCoy, C. i Karrass, J. (2007). Do you believe in magic? Infants’ social looking during violations of expectations. Developmental Science, 10(5), 654–663.
  • Wilson, A. J., Revkin, S. K., Cohen, D., Cohen, L. i Dehaene, S. (2006). An open trial assessment of “The Number Race”, an adaptive computer game for remediation of dyscalculia. Behavioral and Brain Function, 20(2). doi: 10.1186/1744-9081-2-20
  • Wilson, A, Dehaene, S., Dubois, O. i Fayol, M. (2009). Effects of an adaptive game intervention on accessing number sense in low-socioeconomic-status kindergarten children. Mind, Brain and Education, 4(3), 224–234.
  • Wynn, K. (1992). Addition and subtraction by human infants. Nature, 358(6389), 749–750.
  • Xu, F. i Carey, S. (1996). Infants’ metaphysics: the case of numerical identity. Cognitive Psychology, 30(2), 111–153.
  • Xu, F. i Spelke, E. S. (2000). Large number discrimination in 6-month-old infants. Cognition, 74, B1–B11.

Notes

http://www.edukacja.ibe.edu.pl/images/numery/2017/1-1-szczygiel-konstruktywizm-jeana-piageta.pdf

Document Type

Publication order reference

Identifiers

YADDA identifier

bwmeta1.element.desklight-00496fd3-3653-454d-93e6-4882ebfce2b4
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