Modelos mentais e metáforas na resolução de problemas matemáticos verbais

Medeiros, Cleide Farias de

doi:10.1590/S1516-73132001000200006

Resumos

Neste presente trabalho é feita, inicialmente, uma breve retrospectiva sobre os focos mais comuns dos estudos no campo da resolução de problemas matemáticos. É defendido que há algo mais a ser considerado nessa tarefa. Adotando a concepção de que o ser humano produz "modelos" ou "metáforas" na resolução de problemas matemáticos verbais inseridos em contextos reais é, primeiro, apontada a necessidade de entendimento do significado de modelo mental como sendo uma representação mental de um "todo" em que uma "teia" cognitiva que envolve significados acerca das suas "partes" encontram-se estrutural e coerentemente interligados. Em seguida é explorada a complexidade cognitiva envolvida na resolução de uma situação problemática idealizada na esfera de um modelo mental (ou metáfora) combinatorial. Algumas formas de soluções são exibidas apresentando as "armadilhas" do próprio problema, os aspectos culturais e a subjetividade do resolvedor como três elementos que parecem interferir na produção de um modelo mental bem-sucedido. São apresentadas sugestões para outros estudos.

Analogia; Idealização; Campo perceptual; Campo observacional; Combinação; Arranjo

In this paper a brief survey is initially undertaken about the usual focuses of studies in the field of mathematical problem solving. It is claimed that more has to be considered in this task. By conceiving that human being produces 'models' or 'metaphors' in the resolution of mathematical verbal problems imbedded in real contexts, it is, firstly, pointed out the need for understanding the meaning of 'mental model' as a cognitive structure where meanings related to the 'parts' of a 'whole' are structurally interconnected and, secondly, the cognitive complexity involved in the resolution of a problematic situation which can be idealised within a combinatorial 'mental model' or combinatorial 'metaphor', is explored. Some ways of solutions are exhibited showing the problem 'pitfalls' as well as some aspects of the problem solver's culture and subjectivity as three elements which seem to interfere in the production of a successful 'mental model'. Suggestions for other studies are presented.

analogy; idealisation; perceptual field; observacional field; combination; arrangement

ARTIGOS

Modelos mentais e metáforas na resolução de problemas matemáticos verbais

Cleide Farias de Medeiros

Professora Adjunto IV, PhD, Departamento de Educação, Universidade Federal Rural de Pernambuco,UFRPE, Recife, Pe, Brasil (e-mail: cfmed@hotlink.com.br)

RESUMO

Neste presente trabalho é feita, inicialmente, uma breve retrospectiva sobre os focos mais comuns dos estudos no campo da resolução de problemas matemáticos. É defendido que há algo mais a ser considerado nessa tarefa. Adotando a concepção de que o ser humano produz "modelos" ou "metáforas" na resolução de problemas matemáticos verbais inseridos em contextos reais é, primeiro, apontada a necessidade de entendimento do significado de modelo mental como sendo uma representação mental de um "todo" em que uma "teia" cognitiva que envolve significados acerca das suas "partes" encontram-se estrutural e coerentemente interligados. Em seguida é explorada a complexidade cognitiva envolvida na resolução de uma situação problemática idealizada na esfera de um modelo mental (ou metáfora) combinatorial. Algumas formas de soluções são exibidas apresentando as "armadilhas" do próprio problema, os aspectos culturais e a subjetividade do resolvedor como três elementos que parecem interferir na produção de um modelo mental bem-sucedido. São apresentadas sugestões para outros estudos.

Unitermos: Analogia, Idealização, Campo perceptual, Campo observacional, Combinação, Arranjo.

ABSTRACT

In this paper a brief survey is initially undertaken about the usual focuses of studies in the field of mathematical problem solving. It is claimed that more has to be considered in this task. By conceiving that human being produces 'models' or 'metaphors' in the resolution of mathematical verbal problems imbedded in real contexts, it is, firstly, pointed out the need for understanding the meaning of 'mental model' as a cognitive structure where meanings related to the 'parts' of a 'whole' are structurally interconnected and, secondly, the cognitive complexity involved in the resolution of a problematic situation which can be idealised within a combinatorial 'mental model' or combinatorial 'metaphor', is explored. Some ways of solutions are exhibited showing the problem 'pitfalls' as well as some aspects of the problem solver's culture and subjectivity as three elements which seem to interfere in the production of a successful 'mental model'. Suggestions for other studies are presented.

Keywords: analogy, idealisation, perceptual field, observacional field, combination, arrangement.

Texto completo disponível apenas em PDF.

Full text available only in PDF format.

Referências bibliográficas

AUSUBEl, D.; NOVAK, J.; HANESIAN, H. Trad. Eva Nick. Psicologia Educacional. Rio de Janeiro: Editora Interamericana, 1980.

BENKO, A.; LOAIZIA, R.; LONG, R.; et al. Math word problem remediation with elementary students. Illinois: Saint Xavier University, IRI/Skylight, 1999. (Master's Action Research Project)

BLACK, M. More about metaphor. In: ORTONY, A. (ed.) Metaphor and Thought. Cambridge: Cambridge University Press, 1979.

BOYD, R. et al. (ed.) The Philopsophy of Science. Cambridge: The MIT Press, 1991. BORGES, A. T. Um estudo de modelos mentais. Investigações em Ensino das Ciências, v. 2, dez. 1997.

CARPENDALE, J. I.; Language and operations in children's class inclusion reasoning: the operational semantic theory of reasoning. Developmental Review, v. 16, n. 4, p. 391-415, Dec.1996.

CHAPMAN, O. Inservice teacher development in Mathematical problem solving. Journal of Mathematics Teacher Education, v. 2, n. 2, p. 121-42, 1999.

CLEMENT, J. A method experts use to evaluate the validity of models used as problem representations in Science and Mathematics. In: ANNUAL MEETING OF THE AMERICAN EDUCATIONL RESEARCH ASSOCIATION, april 1985, Chicago.

COFFMAN, J. What to Solve? Oxford: Oxford University Press, 1990.

CCOPER, D. E. Metaphor. England: Basil Blackwell Publisher Ltd, 1989.

ELIADE, M. Metaphor and Reference. In: RICOEUR, P. (ed). The rule of metaphor: multidisciplinary studies of the creation of meaning in language. London: Routledge & Kegan Paul, 1986.

FICHBEIN, E. The autonomy of mental models for the learning of mathematics. International Journal of Mathematics Education, v. 10, n. 1, p. 23-30, Feb. 1990.

FRANCO, C.; COLLINVAUX, D. Grasping Mental Models. In: ENCONTRO NACIONAL DE PESQUISA EM EDUCAÇÃO, 2, 1999, Valinhos. Atas... Valinhos, 1999.

GAROFALO, J.; LESTER, F. Metacognition, Cognitive Monitoring and Mathematical Performance. Journal for Research in Mathematics Education, v. 16, n. 3, p. 163-176, 1985.

HARRISON, G.; TREAGUST, F. Secondary students' mental models of atoms and molecules: implications for teaching Chemistry. Science Education, v. 80, n 5, p.509-34, Sep. 1996.

HART, K.(ed). Children's Understanding of Mathematics: 11-16. London: John Murray, 1987.

HONG, E. Mental models in word problem solving: an analysis of Korean Elementary Students. In: ANNUAL MEETING OF THE AMERICAN EDUCATIONAL RESEARCH ASSOCIATION, 1993, Atlanta. 37 p.

HUGGINS, L. The influence of specific thinking skills training on Mathematics problem solving performance. University of South Carolina, 1988.

HUSSERL, E. The crisis of european sciences and transcendental phenomenology. Illinois: Northwestern University Press, 1970.

JEFFERSON, D. Experiment using schematic mapping skills to teach problem solving skills in vocational education. PhD Thesis. DAI, v. 50 A, 673, 1990.

KOYAMA, M. Research on the complementarity of intuition and logical thinking in the process of understanding mathematics: an examination of the two-axes process model by analyzing an elementary school mathematics class. Hiroshima Journal of Mathematics Education, v. l5, p.21-33, Mar. 1997.

LACEY, A. A dictionary of Philosophy. London: Routledge & Kegan Paul, 1986. LAKATOS, I. Proofs and Refutations. Cambridge: Cambridge University Press, 1976.

LANGFORD, G. Philosophy and Education. London: Mcmillan, 1971.

LEBLANC, J. You can teach problem solving. Arithmetic Teacher, p. 16-20, nov.1977.

LEECH, J. Semantics: the study of meaning. England: Penguin Books, 1981.

LONGMAN DICTIONARY OF CONTEMPORARY ENGLISH, New Edition, England: Longman, 1987.

LOW, G.D. On teaching metaphor. Applied Linguistics, v. 9, n. 2, p. 125-147,1988.

MAC CORMAC, E. R. A Cognitive Theory of Metaphor. London: The MIT Press, 1988.

MASON, J. Modelling: what do we really want pupils to learn? In: PIMM, D. (ed.) Mathematics, Teachers and Children. Great Britain: The Open University, 1988.

MEDEIROS, C.F. An Investigation into Errors Made in Attempts to Solve Mathematical Problems. PhD Thesis, Center for Studies in Science and Mathematics Education, Leeds, England, 1992.

ORTON, A. Learning Mathematics. London: Cassell Educational Limited, 1987.

PIMM, D. Mathematical Metaphor. For the Learning of Mathematics, v. 8, n. 1, p. 30-34, 1988.

POLYA, G. How to solve it. New Jersey: Princeton University Press,1973.

________. Mathematical Discovery. New York: John Willey & Sons, 1968.

REBER, A. The Penguin Dictionary of Psychology. England: Viking, 1985.

RICOUER, P. The Rule of Metaphor: multidisciplinary studies of the creation of meaning in language. London: Routledge & Kegan Paul, 1986.

SCHEERER, M. Problem Solving. Scientific American, v. 208, p. 118-128, 1987.

SHAPPIRO, W., Paper presented at the Annual Meeting of the American Educational Research Association, Montreal, Quebec, Canada, April 19-23, 1999.

SCHOENFELD, A. Beyond the purely cognitive: belief systems, social cognitions and metacognitions as driving forces in intellectual performance. Cognitive Science, v. 7, p. 329-366, 1979.

STRASSER, S. Phenomenology and the Human Sciences. Pittsburg: Duquesne University Press, 1963.

SUYDAM, M. Update on research on problem solving: implications for classroom teaching. Arithmetic teacher, v. 30, p. 56-60, fev. 1982.

SYDNEY, C. Fundamental Mathematics: the principles and their applications to basic topics. London: Draft Edition, Rotex Publications, 1990.

THOMSON, R. The Psychology of Thinking. Middlesex: Penguin Books, 1969.

VERCHAFFEL, L.; CORTE, E.; VIERSTRAET, H. Upper elementary school pupils' difficulties in modeling and solving non-standard additive word problems involving ordinal numbers. Journal for Research in Mathematics Education, v. 30, n. 3, p. 265-85, May 1999.

WERTHEIMER, M. Productive Thinking. New York: Harper & Brothers Publishers, 1959.

Artigo Recebido em: 30/08/00

Artigo Aceito para Publicação em: 08/06/01

AUSUBEl, D.; NOVAK, J.; HANESIAN, H. Trad. Eva Nick. Psicologia Educacional. Rio de Janeiro: Editora Interamericana, 1980.
BENKO, A.; LOAIZIA, R.; LONG, R.; et al Math word problem remediation with elementary students Illinois: Saint Xavier University, IRI/Skylight, 1999. (Master's Action Research Project)
BLACK, M. More about metaphor. In: ORTONY, A. (ed.) Metaphor and Thought. Cambridge: Cambridge University Press, 1979.
BOYD, R. et al. (ed.) The Philopsophy of Science. Cambridge: The MIT Press, 1991.
BORGES, A. T. Um estudo de modelos mentais. Investigações em Ensino das Ciências, v. 2, dez. 1997.
CARPENDALE, J. I.; Language and operations in children's class inclusion reasoning: the operational semantic theory of reasoning. Developmental Review, v. 16, n. 4, p. 391-415, Dec.1996.
CHAPMAN, O. Inservice teacher development in Mathematical problem solving. Journal of Mathematics Teacher Education, v. 2, n. 2, p. 121-42, 1999.
CLEMENT, J. A method experts use to evaluate the validity of models used as problem representations in Science and Mathematics. In: ANNUAL MEETING OF THE AMERICAN EDUCATIONL RESEARCH ASSOCIATION, april 1985, Chicago.
COFFMAN, J. What to Solve? Oxford: Oxford University Press, 1990.
CCOPER, D. E. Metaphor. England: Basil Blackwell Publisher Ltd, 1989.
ELIADE, M. Metaphor and Reference. In: RICOEUR, P. (ed). The rule of metaphor: multidisciplinary studies of the creation of meaning in language. London: Routledge & Kegan Paul, 1986.
FICHBEIN, E. The autonomy of mental models for the learning of mathematics. International Journal of Mathematics Education, v. 10, n. 1, p. 23-30, Feb. 1990.
FRANCO, C.; COLLINVAUX, D. Grasping Mental Models. In: ENCONTRO NACIONAL DE PESQUISA EM EDUCAÇÃO, 2, 1999, Valinhos. Atas.. Valinhos, 1999.
GAROFALO, J.; LESTER, F. Metacognition, Cognitive Monitoring and Mathematical Performance. Journal for Research in Mathematics Education, v. 16, n. 3, p. 163-176, 1985.
HARRISON, G.; TREAGUST, F. Secondary students' mental models of atoms and molecules: implications for teaching Chemistry. Science Education, v. 80, n 5, p.509-34, Sep. 1996.
HART, K.(ed). Children's Understanding of Mathematics: 11-16. London: John Murray, 1987.
HONG, E. Mental models in word problem solving: an analysis of Korean Elementary Students. In: ANNUAL MEETING OF THE AMERICAN EDUCATIONAL RESEARCH ASSOCIATION, 1993, Atlanta. 37 p.
HUGGINS, L. The influence of specific thinking skills training on Mathematics problem solving performance. University of South Carolina, 1988.
HUSSERL, E. The crisis of european sciences and transcendental phenomenology. Illinois: Northwestern University Press, 1970.
JEFFERSON, D. Experiment using schematic mapping skills to teach problem solving skills in vocational education. PhD Thesis. DAI, v. 50 A, 673, 1990.
KOYAMA, M. Research on the complementarity of intuition and logical thinking in the process of understanding mathematics: an examination of the two-axes process model by analyzing an elementary school mathematics class. Hiroshima Journal of Mathematics Education, v. l5, p.21-33, Mar. 1997.
LACEY, A. A dictionary of Philosophy. London: Routledge & Kegan Paul, 1986.
LAKATOS, I. Proofs and Refutations. Cambridge: Cambridge University Press, 1976.
LANGFORD, G. Philosophy and Education. London: Mcmillan, 1971.
LEBLANC, J. You can teach problem solving. Arithmetic Teacher, p. 16-20, nov.1977.
LEECH, J. Semantics: the study of meaning. England: Penguin Books, 1981.
LONGMAN DICTIONARY OF CONTEMPORARY ENGLISH, New Edition, England: Longman, 1987.
LOW, G.D. On teaching metaphor. Applied Linguistics, v. 9, n. 2, p. 125-147,1988.
MAC CORMAC, E. R. A Cognitive Theory of Metaphor. London: The MIT Press, 1988.
MASON, J. Modelling: what do we really want pupils to learn? In: PIMM, D. (ed.) Mathematics, Teachers and Children. Great Britain: The Open University, 1988.
MEDEIROS, C.F. An Investigation into Errors Made in Attempts to Solve Mathematical Problems PhD Thesis, Center for Studies in Science and Mathematics Education, Leeds, England, 1992.
ORTON, A. Learning Mathematics. London: Cassell Educational Limited, 1987.
PIMM, D. Mathematical Metaphor. For the Learning of Mathematics, v. 8, n. 1, p. 30-34, 1988.
POLYA, G. How to solve it. New Jersey: Princeton University Press,1973.
________. Mathematical Discovery. New York: John Willey & Sons, 1968.
REBER, A. The Penguin Dictionary of Psychology. England: Viking, 1985.
RICOUER, P. The Rule of Metaphor: multidisciplinary studies of the creation of meaning in language. London: Routledge & Kegan Paul, 1986.
SCHEERER, M. Problem Solving. Scientific American, v. 208, p. 118-128, 1987.
SHAPPIRO, W., Paper presented at the Annual Meeting of the American Educational Research Association, Montreal, Quebec, Canada, April 19-23, 1999.
SCHOENFELD, A. Beyond the purely cognitive: belief systems, social cognitions and metacognitions as driving forces in intellectual performance. Cognitive Science, v. 7, p. 329-366, 1979.
STRASSER, S. Phenomenology and the Human Sciences. Pittsburg: Duquesne University Press, 1963.
SUYDAM, M. Update on research on problem solving: implications for classroom teaching. Arithmetic teacher, v. 30, p. 56-60, fev. 1982.
SYDNEY, C. Fundamental Mathematics: the principles and their applications to basic topics. London: Draft Edition, Rotex Publications, 1990.
THOMSON, R. The Psychology of Thinking. Middlesex: Penguin Books, 1969.
VERCHAFFEL, L.; CORTE, E.; VIERSTRAET, H. Upper elementary school pupils' difficulties in modeling and solving non-standard additive word problems involving ordinal numbers. Journal for Research in Mathematics Education, v. 30, n. 3, p. 265-85, May 1999.
WERTHEIMER, M. Productive Thinking. New York: Harper & Brothers Publishers, 1959.

Datas de Publicação

Publicação nesta coleção
05 Jan 2012
Data do Fascículo
2001

Histórico

Aceito
08 Jun 2001
Recebido
30 Ago 2000

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

[1] AUSUBEl, D.; NOVAK, J.; HANESIAN, H. Trad. Eva Nick. Psicologia Educacional. Rio de Janeiro: Editora Interamericana, 1980.

[2] BENKO, A.; LOAIZIA, R.; LONG, R.; et al Math word problem remediation with elementary students Illinois: Saint Xavier University, IRI/Skylight, 1999. (Master's Action Research Project)

[3] BLACK, M. More about metaphor. In: ORTONY, A. (ed.) Metaphor and Thought. Cambridge: Cambridge University Press, 1979.

[4] BOYD, R. et al. (ed.) The Philopsophy of Science. Cambridge: The MIT Press, 1991.

[5] BORGES, A. T. Um estudo de modelos mentais. Investigações em Ensino das Ciências, v. 2, dez. 1997.

[6] CARPENDALE, J. I.; Language and operations in children's class inclusion reasoning: the operational semantic theory of reasoning. Developmental Review, v. 16, n. 4, p. 391-415, Dec.1996.

[7] CHAPMAN, O. Inservice teacher development in Mathematical problem solving. Journal of Mathematics Teacher Education, v. 2, n. 2, p. 121-42, 1999.

[8] CLEMENT, J. A method experts use to evaluate the validity of models used as problem representations in Science and Mathematics. In: ANNUAL MEETING OF THE AMERICAN EDUCATIONL RESEARCH ASSOCIATION, april 1985, Chicago.

[9] COFFMAN, J. What to Solve? Oxford: Oxford University Press, 1990.

[10] CCOPER, D. E. Metaphor. England: Basil Blackwell Publisher Ltd, 1989.

[11] ELIADE, M. Metaphor and Reference. In: RICOEUR, P. (ed). The rule of metaphor: multidisciplinary studies of the creation of meaning in language. London: Routledge & Kegan Paul, 1986.

[12] FICHBEIN, E. The autonomy of mental models for the learning of mathematics. International Journal of Mathematics Education, v. 10, n. 1, p. 23-30, Feb. 1990.

[13] FRANCO, C.; COLLINVAUX, D. Grasping Mental Models. In: ENCONTRO NACIONAL DE PESQUISA EM EDUCAÇÃO, 2, 1999, Valinhos. Atas.. Valinhos, 1999.

[14] GAROFALO, J.; LESTER, F. Metacognition, Cognitive Monitoring and Mathematical Performance. Journal for Research in Mathematics Education, v. 16, n. 3, p. 163-176, 1985.

[15] HARRISON, G.; TREAGUST, F. Secondary students' mental models of atoms and molecules: implications for teaching Chemistry. Science Education, v. 80, n 5, p.509-34, Sep. 1996.

[16] HART, K.(ed). Children's Understanding of Mathematics: 11-16. London: John Murray, 1987.

[17] HONG, E. Mental models in word problem solving: an analysis of Korean Elementary Students. In: ANNUAL MEETING OF THE AMERICAN EDUCATIONAL RESEARCH ASSOCIATION, 1993, Atlanta. 37 p.

[18] HUGGINS, L. The influence of specific thinking skills training on Mathematics problem solving performance. University of South Carolina, 1988.

[19] HUSSERL, E. The crisis of european sciences and transcendental phenomenology. Illinois: Northwestern University Press, 1970.

[20] JEFFERSON, D. Experiment using schematic mapping skills to teach problem solving skills in vocational education. PhD Thesis. DAI, v. 50 A, 673, 1990.

[21] KOYAMA, M. Research on the complementarity of intuition and logical thinking in the process of understanding mathematics: an examination of the two-axes process model by analyzing an elementary school mathematics class. Hiroshima Journal of Mathematics Education, v. l5, p.21-33, Mar. 1997.

[22] LACEY, A. A dictionary of Philosophy. London: Routledge & Kegan Paul, 1986.

[23] LAKATOS, I. Proofs and Refutations. Cambridge: Cambridge University Press, 1976.

[24] LANGFORD, G. Philosophy and Education. London: Mcmillan, 1971.

[25] LEBLANC, J. You can teach problem solving. Arithmetic Teacher, p. 16-20, nov.1977.

[26] LEECH, J. Semantics: the study of meaning. England: Penguin Books, 1981.

[27] LONGMAN DICTIONARY OF CONTEMPORARY ENGLISH, New Edition, England: Longman, 1987.

[28] LOW, G.D. On teaching metaphor. Applied Linguistics, v. 9, n. 2, p. 125-147,1988.

[29] MAC CORMAC, E. R. A Cognitive Theory of Metaphor. London: The MIT Press, 1988.

[30] MASON, J. Modelling: what do we really want pupils to learn? In: PIMM, D. (ed.) Mathematics, Teachers and Children. Great Britain: The Open University, 1988.

[31] MEDEIROS, C.F. An Investigation into Errors Made in Attempts to Solve Mathematical Problems PhD Thesis, Center for Studies in Science and Mathematics Education, Leeds, England, 1992.

[32] ORTON, A. Learning Mathematics. London: Cassell Educational Limited, 1987.

[33] PIMM, D. Mathematical Metaphor. For the Learning of Mathematics, v. 8, n. 1, p. 30-34, 1988.

[34] POLYA, G. How to solve it. New Jersey: Princeton University Press,1973.

[35] ________. Mathematical Discovery. New York: John Willey & Sons, 1968.

[36] REBER, A. The Penguin Dictionary of Psychology. England: Viking, 1985.

[37] RICOUER, P. The Rule of Metaphor: multidisciplinary studies of the creation of meaning in language. London: Routledge & Kegan Paul, 1986.

[38] SCHEERER, M. Problem Solving. Scientific American, v. 208, p. 118-128, 1987.

[39] SHAPPIRO, W., Paper presented at the Annual Meeting of the American Educational Research Association, Montreal, Quebec, Canada, April 19-23, 1999.

[40] SCHOENFELD, A. Beyond the purely cognitive: belief systems, social cognitions and metacognitions as driving forces in intellectual performance. Cognitive Science, v. 7, p. 329-366, 1979.

[41] STRASSER, S. Phenomenology and the Human Sciences. Pittsburg: Duquesne University Press, 1963.

[42] SUYDAM, M. Update on research on problem solving: implications for classroom teaching. Arithmetic teacher, v. 30, p. 56-60, fev. 1982.

[43] SYDNEY, C. Fundamental Mathematics: the principles and their applications to basic topics. London: Draft Edition, Rotex Publications, 1990.

[44] THOMSON, R. The Psychology of Thinking. Middlesex: Penguin Books, 1969.

[45] VERCHAFFEL, L.; CORTE, E.; VIERSTRAET, H. Upper elementary school pupils' difficulties in modeling and solving non-standard additive word problems involving ordinal numbers. Journal for Research in Mathematics Education, v. 30, n. 3, p. 265-85, May 1999.

[46] WERTHEIMER, M. Productive Thinking. New York: Harper & Brothers Publishers, 1959.