As Relações De Primeira-Ordem Em Tarefas De Proporção:
Uma Outra Explicação Quanto Às Dificuldades Das Crianças
Palavras-chave:
Proporção, Relações de primeira-ordem, Dificuldades das crianças, Relações parte-parte/parte-todoResumo
Problemas de proporção são considerados muito difíceis para crianças, sendo esta dificuldade explicada em termos da incapacidade em estabelecer relações de segunda-ordem. Este artigo evidencia que a causa das dificuldades reside muitas vezes, nas relações iniciais de primeiraordem e que quando estas são fáceis as crianças são capazes de estruturar as relações de segunda-ordem. Ambos os tipos de relações são detalhadamente considerados na análise dos resultados de várias investigações, bem como a natureza das relações de primeira-ordem (parte-parte e parte-todo) e das dimensões (complementares e não-complementares) envolvidas nos problemas. As discussões apontam a existência de diferentes níveis de compreensão por parte das crianças acerca do conceito de proporção. Implicações educacionais, metodológicas e quanto ao desenvolvimentocognitivo são apresentadas.
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Referências
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Proceedings of the 2nd. International Conference for the Psychology of Mathematics
Education, Osnabruck.
Brown, G. & Müller, D.J. (1976). Studies investigating the development of the concept
of proportion. Em S. Modgil & G. Modgil (Eds.), Piagetian research: Compilation
and commentary, Vol. 3 (pp. 66-67). Windsor: NFER-Nelson Publishing Company.
Bruner, J.S. & Kenney, H. (1966). On relational concepts. Em J.S. Bruner; R.R. Olver
& P.M. Greenfield (Eds.), Studies in cognitive growth (pp. 168-182). New York:
John Willey & Sons.
Bryant, P.E. (1974). Perception and understanding in young children: An experimental
approach. London: Methuen.
Carraher, T.N. (1986). From drawings to buildings; working with mathematical scales.
International Journal of Behavioral Development, 9, 527-544.
Carraher, T.N.; Carraher, D.W. & Schliemann, A.D. (1986). Proporcionalidade na
educação científica e matemática, III: Desenvolvimento cognitivo e aprendizagem.
Revista Brasileira de Estudos Pedagógicos, 67(157), 586-602.
Carraher, T.N.; Carraher, D.W.; Schliemann, A.D. & Ruiz, E.L. (1986). Proporcionalidade
na educação científica e matemática, I: Quantidades medidas por
razões. Revista Brasileira de Estudos Pedagógicos, 67(155), 93-107.
Chapman, R. (1975). The development ofchildren's understanding of proportions. Child
Development, 46, 141-148.
Hart, K.M. (1981). Ratio and proportions. Em K.M. Hart (Ed.) Children's understanding
of mathematics (pp. 11-16). London: John Murray.
Hart, K.M. (1984). Ratios: Childrer's strategies and errors. Windsor: NFER-Nelson
Publishing Company.
Inhelder, B. & Piaget, J. (1958). The growth of logical thinking from childhood to
adolescence. London: Routledge and Kegan Paul.
Inhelder, B. & Piaget, J. (1964). The early growth of logicin the child. New York: Harper.
Karplus, R. & Peterson, R.W. (1970). Intellectual development beyond elementary
school II: Ratio, a survey. School-Science and Mathematics, 70 (9), 813-820.
Karplus, R. & Karplus, E.F. (1972). Intellectual development beyond elementary school
III: Ratio, a longitudinal survey. School-Science and Mathematics, 72(3), 813-820.
Karplus, R.; Pulos, S. & Stage, E.K. (1983). Proportional reasoning in early adolescents.
Em R. Lesh e M. Landau (Eds.), Acquisition of mathematics concepts and
processes (pp. 45-90). New York: Academic Press.
Lesh, R., Post, T. & Behr, M. (1988). Proportional reasoning. Em J. Hiebert & M. Behr
(Eds.), Reasearch agenda for mathematics education. Number, concepts and
operations in the middle grades. Vol.2 (pp. 93-118). Lawrence Erlbaum Associates,
National Council of Teachers of Mathematics.
Müller, D.J. (1978). ChildrerTs concepts of proportion: An investigation into the claims
of Bryant & Piaget. British Journal of Educational Psychology, 48, 29-35
Múller, D.J. (1979). Perceptual reasoning and proportion. Mathematics Teaching, 87,
20-22.
Noelting, R. (1976). Stages and mechanisms in the development of the concept of
proportion in children and adolescents. Em M.A. Poulsen e colaboradores. (Eds.),
Piagetian theory and the helping professions: 5th Annual Conference. Los
Angeles, California.
Noelting, R. (1980a). The development of proportional reasoning and the ratio concept.
Part I - Diff erentiation of stages. Educational Studies in Mathematics, 11,217-253.
Noelting, R. (1980b). The development of proportional reasonig and the ratio concept
Part II - Problem-structure at sucessive stages: Problem-solving strategies and
the mechanism of adaptative restructuring. Educational Studies in Mathematics,
11, 331-363.
Piaget, J. (1952). The child's conception of number. London: Routledge and Kegan Paul.
Piaget, J. & Inhelder, B. (1975). The origin of the idea of chance in children. New York:
W.Norton.
Piaget, J., Grize, J.B., Szeminska, A. & Bang, V. (1968). Epistemologie et psichologie
de la fonction. Paris: Presses Universitaires de France.
Siegler, R.S. (1976). Three aspects of cognitive development. Cognitive Psychology, 8,
481-520.
Siegler, R.S. (1978). The origin of scientific reasoning. Em R.S. Siegler (Ed.) Children's
thinking: What develops? (pp. 109-149). Hillsdale, N.J.: Lawrence Erlbaum Associates.
Siegler, R.S. & Vago, S. (1978). The development of proportionality concept: Judging
relative fullness. Journal of Experimental Child Psychology, 25, 371-395,
Spinillo, A.G. (1987). The development of proportional reasoning in young children.
Trabalho apresentado e publicado nos Resumos da 15a. Annual Conference of
the British Psychological Society - Developmental Section, York, U.K.
Spinillo, A.G. (1990). The development of the concept of proportion in young children.
Tese de Doutorado não publicada. University of Oxford, Oxford, U.K.
Spinillo, A.G. (1992). A importância do referencial de 'metade' e o desenvolvimento do
conceito de proporção. Psicologia: Teoria e Pesquisa, 8(3), 305-317.
Spinillo, A.G. & Bryant, P.E. (1988). The importance of 'half in children's proportional judgements. 3rd. European Conference on Developmental Psychology, (Resumos).
Budapest, Hungary.
Spinillo, A.G. & Bryant, P.E. (1989). The initial understanding of ratio and proportion.
13th. International Conference for the Psychology of Mathematics Education,
(Resumos) Paris, France.
Spinillo, A.G. & Bryant, P.E. (1990). Ratio and proportion: Judging discrete and
continuous quantities. 4th European Conference on Developmental Psychology
(Resumos). Stirling, U.K.
Spinillo, A.G. & Bryant, P.E. (1991). Children's proportional judgements: The importance
of 'half. Child Development, 62, 427-440.
Tourniaire, F. (1986). Proportions in elementary school. Educational Studies in Mathematics,
77,401-412.
Tourniaire, F. & Pulos, S. (1985). Proportional reasoning: a review of the literature.
Educational Studies in Mathematics, 16,181 -204.
Wilkening, F. & Anderson, N.H. (1990). Representation and diagnosis of knowledge
structures in developmental psychology. Em N.H. Anderson (Ed.), Contributions
to information integration theory (pp. 1-30). Hillsdale, N.J.: Lawrence Erlbaum
Associates.
Proceedings of the 2nd. International Conference for the Psychology of Mathematics
Education, Osnabruck.
Brown, G. & Müller, D.J. (1976). Studies investigating the development of the concept
of proportion. Em S. Modgil & G. Modgil (Eds.), Piagetian research: Compilation
and commentary, Vol. 3 (pp. 66-67). Windsor: NFER-Nelson Publishing Company.
Bruner, J.S. & Kenney, H. (1966). On relational concepts. Em J.S. Bruner; R.R. Olver
& P.M. Greenfield (Eds.), Studies in cognitive growth (pp. 168-182). New York:
John Willey & Sons.
Bryant, P.E. (1974). Perception and understanding in young children: An experimental
approach. London: Methuen.
Carraher, T.N. (1986). From drawings to buildings; working with mathematical scales.
International Journal of Behavioral Development, 9, 527-544.
Carraher, T.N.; Carraher, D.W. & Schliemann, A.D. (1986). Proporcionalidade na
educação científica e matemática, III: Desenvolvimento cognitivo e aprendizagem.
Revista Brasileira de Estudos Pedagógicos, 67(157), 586-602.
Carraher, T.N.; Carraher, D.W.; Schliemann, A.D. & Ruiz, E.L. (1986). Proporcionalidade
na educação científica e matemática, I: Quantidades medidas por
razões. Revista Brasileira de Estudos Pedagógicos, 67(155), 93-107.
Chapman, R. (1975). The development ofchildren's understanding of proportions. Child
Development, 46, 141-148.
Hart, K.M. (1981). Ratio and proportions. Em K.M. Hart (Ed.) Children's understanding
of mathematics (pp. 11-16). London: John Murray.
Hart, K.M. (1984). Ratios: Childrer's strategies and errors. Windsor: NFER-Nelson
Publishing Company.
Inhelder, B. & Piaget, J. (1958). The growth of logical thinking from childhood to
adolescence. London: Routledge and Kegan Paul.
Inhelder, B. & Piaget, J. (1964). The early growth of logicin the child. New York: Harper.
Karplus, R. & Peterson, R.W. (1970). Intellectual development beyond elementary
school II: Ratio, a survey. School-Science and Mathematics, 70 (9), 813-820.
Karplus, R. & Karplus, E.F. (1972). Intellectual development beyond elementary school
III: Ratio, a longitudinal survey. School-Science and Mathematics, 72(3), 813-820.
Karplus, R.; Pulos, S. & Stage, E.K. (1983). Proportional reasoning in early adolescents.
Em R. Lesh e M. Landau (Eds.), Acquisition of mathematics concepts and
processes (pp. 45-90). New York: Academic Press.
Lesh, R., Post, T. & Behr, M. (1988). Proportional reasoning. Em J. Hiebert & M. Behr
(Eds.), Reasearch agenda for mathematics education. Number, concepts and
operations in the middle grades. Vol.2 (pp. 93-118). Lawrence Erlbaum Associates,
National Council of Teachers of Mathematics.
Müller, D.J. (1978). ChildrerTs concepts of proportion: An investigation into the claims
of Bryant & Piaget. British Journal of Educational Psychology, 48, 29-35
Múller, D.J. (1979). Perceptual reasoning and proportion. Mathematics Teaching, 87,
20-22.
Noelting, R. (1976). Stages and mechanisms in the development of the concept of
proportion in children and adolescents. Em M.A. Poulsen e colaboradores. (Eds.),
Piagetian theory and the helping professions: 5th Annual Conference. Los
Angeles, California.
Noelting, R. (1980a). The development of proportional reasoning and the ratio concept.
Part I - Diff erentiation of stages. Educational Studies in Mathematics, 11,217-253.
Noelting, R. (1980b). The development of proportional reasonig and the ratio concept
Part II - Problem-structure at sucessive stages: Problem-solving strategies and
the mechanism of adaptative restructuring. Educational Studies in Mathematics,
11, 331-363.
Piaget, J. (1952). The child's conception of number. London: Routledge and Kegan Paul.
Piaget, J. & Inhelder, B. (1975). The origin of the idea of chance in children. New York:
W.Norton.
Piaget, J., Grize, J.B., Szeminska, A. & Bang, V. (1968). Epistemologie et psichologie
de la fonction. Paris: Presses Universitaires de France.
Siegler, R.S. (1976). Three aspects of cognitive development. Cognitive Psychology, 8,
481-520.
Siegler, R.S. (1978). The origin of scientific reasoning. Em R.S. Siegler (Ed.) Children's
thinking: What develops? (pp. 109-149). Hillsdale, N.J.: Lawrence Erlbaum Associates.
Siegler, R.S. & Vago, S. (1978). The development of proportionality concept: Judging
relative fullness. Journal of Experimental Child Psychology, 25, 371-395,
Spinillo, A.G. (1987). The development of proportional reasoning in young children.
Trabalho apresentado e publicado nos Resumos da 15a. Annual Conference of
the British Psychological Society - Developmental Section, York, U.K.
Spinillo, A.G. (1990). The development of the concept of proportion in young children.
Tese de Doutorado não publicada. University of Oxford, Oxford, U.K.
Spinillo, A.G. (1992). A importância do referencial de 'metade' e o desenvolvimento do
conceito de proporção. Psicologia: Teoria e Pesquisa, 8(3), 305-317.
Spinillo, A.G. & Bryant, P.E. (1988). The importance of 'half in children's proportional judgements. 3rd. European Conference on Developmental Psychology, (Resumos).
Budapest, Hungary.
Spinillo, A.G. & Bryant, P.E. (1989). The initial understanding of ratio and proportion.
13th. International Conference for the Psychology of Mathematics Education,
(Resumos) Paris, France.
Spinillo, A.G. & Bryant, P.E. (1990). Ratio and proportion: Judging discrete and
continuous quantities. 4th European Conference on Developmental Psychology
(Resumos). Stirling, U.K.
Spinillo, A.G. & Bryant, P.E. (1991). Children's proportional judgements: The importance
of 'half. Child Development, 62, 427-440.
Tourniaire, F. (1986). Proportions in elementary school. Educational Studies in Mathematics,
77,401-412.
Tourniaire, F. & Pulos, S. (1985). Proportional reasoning: a review of the literature.
Educational Studies in Mathematics, 16,181 -204.
Wilkening, F. & Anderson, N.H. (1990). Representation and diagnosis of knowledge
structures in developmental psychology. Em N.H. Anderson (Ed.), Contributions
to information integration theory (pp. 1-30). Hillsdale, N.J.: Lawrence Erlbaum
Associates.
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Publicado
2013-04-25
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Galvão Spinillo, A. (2013). As Relações De Primeira-Ordem Em Tarefas De Proporção:: Uma Outra Explicação Quanto Às Dificuldades Das Crianças. Psicologia: Teoria E Pesquisa, 9(2), 349–364. Recuperado de https://periodicos.unb.br/index.php/revistaptp/article/view/17223
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