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Vol. 6 N.º 3 (2003): Noviembre

MODELOS E TEORIAS DA COMPREENSÃO MATEMATICA: COMPARANDO O MODELO DE PIRIE E DE KIEREN DO CRESCIMENTO DA COMPREENSÃO MATEMATICA E DA TEORIA DE APOE

Submetido
dezembro 23, 2024
Publicado
novembro 30, 2003

Resumo

A busca para uma descrição cognitive significativa da compreensão seguiu para a metade passada um do século. Dentro das três décadas passadas, os perspectives novos e integrative cresceram fora das distinções de Richard Skemp entre a compreensão instrumental e relacional. O crescimento destes perspectives, acima até de 1987, foi documentado por Tom Schroeder em sua síntese de PME do trabalho na compreensão resultando dos contrastes de instrumental/relational de Richard Skemp. Desde 1987, o trabalho na compreensão progrediu e este papel examina as estruturas teóricas novas, mais recentes de compreender quais se levantaram destas raizes. Este papel focaliza em duas estruturas teóricas, modelo de Pirie e de Kieren do crescimento da compreensão matemática e a teoria de APOS de Dubinsky, e discute outras estruturas teóricas contemporary tais como o trabalho por Cornu e Sierpinska em obstáculos cognitive ou epistemological, nas investigações na definição do conceito e na imagem do conceito por Vinner e em explorações altas, kaput de respresentações múltiplas, e de distinções de Sfard entre conceptions operacionais e estruturais. Além de explicating as definições da compreensão propostas por estas duas estruturas, a discussão dirige-se a seus elementos e construções assim como seus enlaces às caracterizações históricas e recentes da compreensão. O papel discute então porque o modelo de Pirie e de Kieren e a teoria de APOS satisfem aos 2000) critérios de Schoenfeld (para a classificação porque uma teoria e os conclem finalmente com discussões de uma variedade das interconexões entre estas duas teorias as.well.as os elementos que os fazem distintos de se tal como suas origens, organizações, relacionamentos a outras estruturas, e implicações das duas teorias para  a avaliação e práticas pedagogical.

Referências

  1. Asiala, M., Brown, A., DeVries, D. J., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A framework for research and curriculum development in Undergraduate mathematics education. In J. Kaput, A., H. Schoenfeld & E. Dubinsky (Eds.) Research in Collegiate Mathematics Education (pp.1-32). Providence, RI: American Mathematical Society.
  2. Asiala, M., Brown, A., Kleiman, J.& Mathews, D. (1998). The development of students' understanding of permutations and symmetries. International Journal of Computers for Mathematical Learning, 3, 13-43.
  3. Ayers, T., Davis, G., Dubinsky, E., & Lewin, P. (1988). Computer experiences in learning composition of functions. Journal for Research in Mathematics Education, 19(3), 246–259.
  4. Bachelard, G. (1938). La formation de l'esprit scientifique. Editions J. Paris: Vrin.
  5. Ball, D. (1991). Research on teaching mathematics: Making subject matter knowledge part of the equation. In J. Brophy (Ed.), Advances in research on teaching: Vol. 2 Teachers' subject matter knowledge and classroom instruction (pp.1– 48). Greenwich, CT: JAI Press.
  6. Brown, A., DeVries, D. J., Dubinsky, E. & Thomas, K. (1997). Learning binary operations, groups, and subgroups. Journal of Mathematical Behavior, 16(3), 187–239.
  7. Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18(1), 32 – 42.
  8. Brownell, W.A. (1945). Psychological considerations in the learning and teaching of arithmetic. In W. D. Reeve (Ed.), The Teaching of Arithmetic. Tenth yearbook of the National Council of Teachers of Mathematics (pp.1–31). New York: Teachers College, Columbia University.
  9. Brownell, W. A. & Sims, V. M. (1946). The nature of understanding. In J. F. Weaver & J. Kilpatrick (Eds.) (1972), The place of meaning in mathematics instruction: Selected theoretical papers of William A. Brownell (Studies in Mathematics, Vol. 21, pp. 161–179). Stanford University: School Mathematics Study Group. (Originally published in The measurement of understanding, Forty-fifth Yearbook of the National Society
  10. for the Study of Education, Part I, 27--43.)
  11. Burton, L. (1984). Mathematical thinking: The struggle for meaning. Journal for Research in Mathematics Education, 15(1), 35 – 49.
  12. Byers, V. & Erlwanger, S. (1985). Memory in mathematical understanding. Educational Studies in Mathematics, 16, 259 – 281.
  13. Byers, V. & Herscovics, N. (1977). Understanding school mathematics. Mathematics Teaching, 81, 24 – 27.
  14. Clark, J., Cordero, F., Cottrill, J., Czarnocha, B., DeVries, D., St. John, D., Tolias, G. & Vidakovic, D. (1997). Constructing a schema: The case of the chain rule. Journal for Mathematical Behavior, 16(4), 345 – 364.
  15. Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced Mathematical Thinking (pp.153 – 166). Dordrecht: Kluwer.
  16. Cottrill, J. (1999). Students' understanding of the concept of chain rule in first year calculus and the realation to their understanding of composition of functions. Unpublished doctoral dissertation, Purdue University.
  17. Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K. & Vidakovic, D. (1996). Understanding the limit concept: Beginning with a coordinated process scheme. Journal of Mathematical Behavior, 15, 167 – 192.
  18. Davis, E. J. (1978). A model for understanding in mathematics. Arithmetic Teacher, September, 13 – 17.
  19. Davis, R. B. (1984). Learning mathematics: The cognitive science approach to mathematics education. Norwood, NJ: Ablex.
  20. Davis, R. B. & Vinner, S. (1986). The notion of limit: Some seemingly unavoidable misconception stages. Journal of Mathematical Behavior, 5, 281 – 303.
  21. Driver, R. & Easley, J. (1978). Pupils and paradigms. Studies in Science Education, 5, 61 – 84.
  22. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp.95 – 123). Dordrecht: Kluwer.
  23. Dubinsky, E. (1994). A theory and practice of learning college mathematics. In A. Schoenfeld (Ed.), Mathematical thinking and problem solving (pp. 221 – 247). Hillsdale, NJ: Erlbaum.
  24. Dubinsky, E., Dautermann, J., Leron, U., & Zazkis, R. (1994). On learning fundamental concepts of group theory. Educational Studies in Mathematics, 27, 267 – 305.
  25. Dubinsky, E. & McDonald, M. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research. In D. Holton et al. (Eds.), The teaching and learning of mathematics at university level: An ICMI study (pp. 273 – 280). Netherlands: Kluwer Academic Publishers.
  26. Fehr, H. (1955). A philosophy of arithmetic instruction. Arithmetic Teacher, 2, 27 – 32.
  27. Ferrini-Mundy, J. & Gaudard, M. (1992). Secondary school calculus: Preparation or pitfall in the study of college calculus? Journal for Research in Mathematics Education, 23(1), 56 – 71.
  28. Flavell, J. H. (1977). Cognitive development. Englewood Cliffs, New Jersey: Prentice Hall.
  29. Ginsburg, H. P., Lopez, L. S., Mukhopadhyay, S., Yamamota, T., Willis, M. & Kelley, M. S. (1992). Assessing understandings of arithmetic. In R. Lesh & S. Lamon (Eds.), {it Assessment of authentic performance in school mathematics (pp. 265 – 289). Washington, DC: AAAS Press.
  30. Greeno, J. G. (1977). Process of understanding in problem solving. In N.J. Castellan, Jr., D. B. Pisoni & G. R. Potts (Eds.), Cognitive theory (Vol. 2, pp. 43 – 83). Hillsdale, NJ: Erlbaum.
  31. Greeno, J. G. (1991). Number sense as situated knowing in a conceptual domain. Journal for Research in Mathematics Education, 22(3), 170 – 218.
  32. Grinevitch, O. (in preparation). Student understanding of abstract algebra: A theoretical examination. Unpublished doctoral dissertation, Bowling Green State University.
  33. Harel, G. & Tall, D. (1991). The general, the abstract, and the generic in advanced mathematics. For the Learning of Mathematics, 11(1), 38 – 42.
  34. Hatano, G. (1988). Social and motivational bases for mathematical understanding. In G. B. Saxe & M. Gearhart (Eds.), New Directions for Child Development (Vol. 41, pp. 55 – 70).
  35. Herscovics, N. & Bergeron, J. (1988). An extended model of understanding. Proceedings of PME-NA 10 (pp. 15 – 22). Dekalb, Ill: Northern Illinois University.
  36. Hiebert, J. (Ed.) (1986). Conceptual and procedural knowledge: The case of mathematics. Hillsdale, NJ: Erlbaum.
  37. Hiebert, J. & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65 – 97). New York, NY: Macmillan.
  38. Hiebert, J. & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1 – 27). Hillsdale, NJ: Erlbaum.
  39. Hiebert, J. & Wearne, D. (1986). Procedures over concepts: The acquisition of decimal number knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp.199 – 223). Hillsdale, NJ: Erlbaum.
  40. Janvier, C. (Ed.) (1987). Problems of representation in the teaching and learning of mathematics. Hillsdale, NJ: Erlbaum.
  41. Kaput, J. (1985). Representation and problem solving: Methodological issues related to modeling. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 381 – 398). Hillsdale, NJ: Erlbaum.
  42. Kaput, J. (1987a). Representation and mathematics. In C. Janvier (Ed.), Problems of representation in mathematics learning and problem solving (pp. 19 – 26). Hillsdale, NJ: Erlbaum.
  43. Kaput, J. (1987b). Towards a theory of symbol use in mathematics. In C. Janvier (Ed.), Problems of representation in mathematics learning and problem solving (pp. 159 – 195). Hillsdale, NJ: Erlbaum.
  44. Kaput, J. (1989a). Linking representations in the symbol systems of algebra. In C. Kieran & S. Wagner (Eds.), A research agenda for the learning and teaching of algebra (pp. 167 –194). Reston, VA: National Council of Teachers of Mathematics; Hillsdale, NJ: Erlbaum.
  45. Kaput, J. (1989b). Supporting concrete visual thinking in multiplicative reasoning: Difficulties and opportunities. Focus on Learning Problems in Mathematics, 11(1), 35– 47.
  46. Kaput, J. (1992). Technology and mathematics education. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 515 – 556). New York, NY: Macmillan
  47. Kieren, T. E. (1990). Understanding for teaching for understanding. The Alberta Journal of Educational Research, 36(3), 191–201.
  48. Kieren, T. E. (1997). Theories for the classroom: Connections between research and practice. For the Learning of Mathematics, 17(2), 31 – 33.
  49. Lehman, H. (1977). On understanding mathematics. Educational Theory, 27(2).
  50. Leinhardt, G. (1988). Getting to know: Tracing student's mathematical knowledge from intuition to competence. Educational Psychologist, 23(2), 119 – 144.
  51. Mack, N. (1990). Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education, 21(1), 16 – 32.
  52. Mansfield, H. M. & Happs, J. C. (1992). Using grade eight students' existing knowledge to teach about parallel lines. School Science and Mathematics, 92(8), 450 – 454.
  53. McDonald, M. A., Mathews, D. M. & Strobel, K. H. (2000). Understanding sequences: A tale of two objects. In E. Dubinsky, A. H. Schoenfeld & J. Kaput (Eds.), Research in Collegiate Mathematics Education (pp. 77 – 102). Providence, RI: American Mathematical Society.
  54. McLellan, J.A. & Dewey, J. (1895). The psychology of number and its applications to methods of teaching arithmetic. New York: D. Appleton.
  55. Michener, E.~R. (1978). Understanding understanding mathematics. Cognitive Science, 2, 361 – 383.
  56. Nesher, P. (1986). Are mathematical understanding and algorithmic performance related? For the Learning of Mathematics, 6(3), 2 – 9.
  57. Nickerson, R. S. (1985). Understanding understanding. American Journal of Education, 93(2), 201 – 239.
  58. Ohlsson, S. (1988). Mathematical meaning and applicational meaning in the semantics of fractions and related concepts, In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 53 – 91). Reston, VA: National Council of Teachers of Mathematics.
  59. Ohlsson, S., Ernest, A. M., & Rees, E. (1992). The cognitive complexity of learning and doing arithmetic. Journal for Research in Mathematics Education, 23(5), 441 – 467.
  60. Piaget, J. (1970). Genetic epistemology. New York, NY: W.W. Norton.
  61. Piaget, J. (1975). Piaget's theory. In P.B. Neubauer (Ed.), The process of child development (pp. 164 – 212). New York, NY: John Aronson.
  62. Pirie, S. E. B. (1988). Understanding: Instrumental, relational, intuitive, constructed, formalised dots ? How can we know? For the Learning of Mathematics, 8(3), 2 – 6.
  63. Pirie, S. E. B. & Kieren, T. E. (1989). A recursive theory of mathematical understanding. For the Learning of Mathematics, 9(3), 7 – 11.
  64. Pirie, S. E. B. & Kieren, T. E. (1990). A recursive theory for the mathematical understanding: some elements and implications. Paper presented at the Annual Meeting of the American Educational Research Association (Boston, MA, April 1990).
  65. Pirie, S. E. B. & Kieren, T. E. (1991a). A dynamic theory of mathematical understanding: Some features and implications. (ERIC Document Reproduction Service No. ED 347 067)
  66. Pirie, S. E. B. & Kieren, T. E. (1991b). Folding back: Dynamics in the growth of mathematical understanding. Paper presented at the Fifteenth Meeting of the Psychology of Mathematics Education Conference (Assissi, Italy, July 1991).
  67. Pirie, S. E. B. & Kieren, T. E. (1992a). Creating constructivist environments and constructing creative mathematics. Educational Studies in Mathematics, 23, 505 – 528.
  68. Pirie, S. E. B. & Kieren, T. E. (1992b). Watching Sandy's understanding grow. Journal of Mathematical Behavior, 11, 243 – 257.
  69. Pirie, S. E. B. & Kieren, T. E. (1994a). Beyond metaphor: Formalising in mathematical understanding with constructivist environments. For the Learning of Mathematics, 14(1), 39 – 43.
  70. Pirie, S. E. B. & Kieren, T. E. (1994b). Growth in mathematical understanding: How can we characterise it and how can we represent it? Educational Studies in Mathematics, 26, 165 – 190.
  71. Pirie, S. E. B. & Martin, L. (1997). The equation, the whole equation and nothing but the equation! One approach to the teaching of linear equations. Educational Studies in Mathematics, 34, 159 – 181.
  72. Pirie, S. E. B. & Schwarzenberger, R. L. E. (1988). Mathematical discussion and mathematical understanding. Educational Studies in Mathematics, 19, 459 – 470.
  73. Polya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press.
  74. Polya, G. (1962). Mathematical discovery (Vol. 2). New York, NY: Wiley.
  75. Resnick, L. B. & Omanson, S. (1987). Learning to understand arithmetic. In R. Glaser (Ed.), Advances in instructional psychology (Vol. 3, pp.41 – 95). Hillsdale, NJ: Erlbaum.
  76. Resnick, L. B., Nesher, P., Leonard, F., Magone, M., Omanson, S., & Peled, I. (1989). Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics Education, 20(1), 8 – 27.
  77. Saxe, G. B. (1988). Studying working intelligence. In B. Rogoff & J. Lave (Eds.), Everyday cognition (pp. 9 – 40). Cambridge, MA: Harvard University Press.
  78. Schoenfeld, A. H. (1989). Exploring the process problem space: Notes on the description and analysis of mathematical processes. In C. Maher, G. Goldin, & R. B. Davis (Eds.), Proceedings of psychology of mathematics education North America XI, (Vol. 2, pp. 95 – 120). New Brunswick, NJ: Rutgers, Centre for Mathematics, Science and Computer Education.
  79. Schoenfeld, A. H. (1998). Toward a theory of teaching-in-context. Issues in Education, 4(1), 1 – 94.
  80. Schoenfeld, A. H. (2000). Purposes and methods of research in mathematics education. Notices of the American Mathematical Society, 47(6), 641 – 649.
  81. Schroeder, T. L. (1987). Student's understanding of mathematics: A review and synthesis of some recent research. In J. Bergeron, N. Herscovics, & C. Kieren (Eds.), Psychology of Mathematics Education XI, (Vol. 3, pp. 332 – 338). Montreal: PME.
  82. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1 –36.
  83. Sfard, A. (1992). Operational origins of mathematical objects and the quandry of reification: The case of function. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistomology and pedagogy (pp.59 – 84). Washington, DC: MAA.
  84. Sfard, A. (1994). Reification as the birth of metaphor. For the Learning of Mathematics, 14(1), 44 – 55.
  85. Sfard, A. & Linchevski, L. (1994). The gains and the pitfalls of reification: The case of algebra. Educational Studies in Mathematics, 26, 191 – 228.
  86. Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20 – 26.
  87. Skemp, R. R. (1979). Goals of learning and qualities of understanding. Mathematics Teaching, 88, 44 – 49.
  88. Skemp, R. R. (1982). Symbolic understanding. Mathematics Teaching, 99, 59 – 61.
  89. Sierpinska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18, 371 – 397.
  90. Sierpinska, A. (1990a). Remarks on understanding in mathematics. Paper presented at the 1990 meeting of the Canadian Mathematics Education Study Group (Vancouver, CAN, 1990).
  91. Sierpinska, A. (1990b). Some remarks on understanding in mathematics. For the Learning of Mathematics, 10(3), 24--41.
  92. Silver, E. A. (1986). Using conceptual and procedural knowledge: A focus on relationships. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 181 – 189). Hillsdale, NJ: Erlbaum.
  93. Simon, M.~A. (1993). Prospective elementary teachers' knowledge of division. Journal for Research in Mathematics Education, 24(3), 233 – 254.
  94. Steffe, L. P. & Kieren, T. E. (1994). Radical constructivism and mathematics education. Journal for Research in Mathematics Education, 25(6), 711 – 733.
  95. Tall, D. (1978). The dynamics of understanding mathematics. Mathematics Teaching, 84, 50 – 52.
  96. Tall, D. (1989). Concept images, generic organizers, computers, and curriculum change. For the Learning of Mathematics, 9(3), 37 – 42.
  97. Tall, D. (1991). The psychology of advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 3 – 21). Dordrecht: Kluwer.
  98. Tall, D. & Vinner, S. (1981). Concept image and concept definition in mathematics with special reference to limits and continuity. Educational Studies in Mathematics, 12, 151 – 169.
  99. Towers, J., Martin, L. & Pirie, S. E. B. (2000). Growing mathematical understanding: Layered observations. In M. L. Fernandez (Ed.), Proceedings of the Twenty-Second Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 225 – 230). Columbus, OH: ERIC Clearinghouse.
  100. Van Engen, H. (1949). An analysis of meaning in arithmetic. Elementary School Journal, 49, 321 – 329.
  101. VanLehn, K. (1986). Arithmetic procedures are induced from examples. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 133 – 180). Hillsdale, NJ: Erlbaum.
  102. Vinner, S. (1983). Concept definition, concept image and the notion of function. International Journal of Mathematical Education in Science and Technology, 14, 293-305.
  103. Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.), Advanced mathematical thinking (pp. 65 – 81). Dordrecht: Kluwer.
  104. Von Glasersfeld, E. (1987). Learning as a constructive activity. In C. Janvier (Ed.), Problems of representation in the learning and teaching of mathematics (pp. 3 – 18). Hillsdale, NJ: Erlbaum.
  105. Wearne, D. & Hiebert, J. (1988). A cognitive approach to meaningful mathematics instruction: Testing a local theory using decimal numbers. Journal for Research in Mathematics Education, 19(5), 371 – 384.
  106. Wertheimer, M. (1959). Productive Thinking. New York: Harper & Row.

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