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Artículos

Vol. 11 No. 1 (2008): Marzo

A SEARCH FOR A CONSTRUCTIVIST APPROACH FOR UNDERSTANDING THE UNCOUNTABLE SET P(N)

Submitted
May 22, 2024
Published
2008-01-24

Abstract

This study considers the question of whether individuals build mental structures for the set P(N) that give meaning to the phrase, "all subsets of N." The contributions of our research concerning this question are two-fold. First, we identified constructivist perspectives that have been, or could be used to describe thinking about infinite sets, specifically, the set of natural numbers N. Second, to determine whether individuals' thinking about the set P(N) can be interpreted in terms of one or more of the perspectives we considered, we analyzed the thinking of eight mathematicians. Beyond negative conceptions, that is, what P(N) is not, the results of our analysis cast doubt on whether individual understanding of the set P(N) extends beyond the formal definition. We discuss the possible implications of our findings, and indicate further research arising from this study.

References

  1. Asiala, M., Brown, A., DeVries, D., 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 II (pp. 1-32). Providence: American Mathematical Society.
  2. Aspinwall, L., Shaw, K.L., & Presmeg, N.C. (1997). Uncontrollable mental imagery: Graphical connections between a function and its derivative. Educational Studies in Mathematics, 33(3),301-317.
  3. Brown, A., McDonald, M., & Weller, K. (in press). Step by step: Infinite iterative processes and actual infinity. In F. Hitt, D. Holton, & P. Thompson (Eds.), Research in collegiate mathematics education VII. Providence: American Mathematical Society.
  4. Burton, L. (2001). Research mathematicians as learners and what mathematics education can learn from them. British Education Research Journal, 27(5), 589-599.
  5. Cantor, G. (1941). The theory of transfinite numbers (Phillip E. B. Jourdain, Trans.). La Salle, IL Open Court Publishing.
  6. Chartrand, G., Polimeni, A., & Zhang, P. (2007). Mathematical Proofs: A transition to advance mathematics (2nd ed.). Boston: Addison-Wesley.
  7. Chomsky, N. (2006). Language and mind (3rd ed.). New York: Cambridge University Press.
  8. Cobb, P., Boufi, A., McClain, K., & Whitenack, J. (1997). Reflective discourse and collective reflection. Journal for Research in Mathematics Education, 28(3), 258-277.
  9. Cobb, P., Yackel, E., & McClain, K. (Eds.). (2000). Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design. Mahwah, NJ: Lawrence Erlbaum Associates.
  10. Cooley, L., Trigueros, M., & Baker, B. (in press). Schema thematization: A framework and an example. Journal for Research in Mathematics Education.
  11. Davis, R. B. & Maher, C. A. (1997), How students think: The role of representations. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors and images (pp. 93-115). Mahwah, NJ: Lawrence Erlbaum Associates.
  12. Dreyfus, T. (1991). On the status of visual reasoning in mathematics and mathematics education. In F. Furinghetti (Ed.), Proceedings of the 15th Annual Conference of the International Group for the Psychology of Mathematics Education (Volume 1, pp. 33-48). Assisi, Italy.
  13. Dreyfus, T. (1995). Imagery for diagrams. In R. Sutherland and J. Mason (Eds.), Exploiting mental imagery with computers in mathematics education (pp. 3-19). Berlin: Springer.
  14. Dubinsky, E. (1999). Mathematical reasoning: Analogies, metaphors, and images. Notices of the AMS, 46(5), 555-559.
  15. Dubinsky, E. (2000). Meaning and formalism in mathematics. International Journal of Computers for Mathematical Learning, 5(3), 211-240.
  16. Dubinsky, E. & McDonald, M. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research. In Derek Holton et al. (Eds.), The teaching and learning of mathematics at university level: An ICMI study (pp. 273-280). Netherlands: Kluwer.
  17. Dubinsky, E., Weller, K., McDonald, M., & Brown, A. (2005a). Some historical issues and paradoxes regarding the concept of infinity: An APOS analysis: Part 1. Educational Studies in Mathematics, 58(3), 335-359.
  18. Dubinsky, E., Weller, K., McDonald, M., & Brown, A. (2005b). Some historical issues and paradoxes regarding the concept of infinity: Part II. Educational Studies in Mathematics, 60(2), 253-266.
  19. Ernest, P. (1997). Social constructivism as a philosophy of mathematics. Albany, New York: State University of New York Press.
  20. Ernest, P. (1998). The relation between personal and public knowledge from an epistemological perspective. In F. Seeger, J. Voight, and U. Waschescio, (Eds.), The culture of the mathematics classroom (pp. 245-268). Cambridge: Cambridge University Press.
  21. Fischbein, E. (1987). Intuition in science and mathematics. Dordrecht: Kluwer.
  22. Fischbein, E. (2001). Tacit models of infinity. Educational Studies in Mathematics, 48(2/3), 309- 329.
  23. Fischbein E., Tirosh D., & Hess P. (1979). The intuition of mfinity. Educational Studies in Mathematics, 10(1), 491-512
  24. Gold, B. (2001). Where mathematics comes from: How the embodied mind brings mathematics into being. Read this! The MAA Online book review column Retrieved September 2005 at http://www.maa.org/reviews/wheremath html.
  25. Goldenberg E. P. (1988). Mathematics, metaphors and human factors: Mathematical, technical and pedagogical challenges in the educational use of graphical representation of functions. Journal of Mathematical Behavior, 7(2), 135-173.
  26. Hadamard, J. (1945). The Psychology of invention in the mathematical field. New York: Princeton University Press.
  27. Hersh, R. (1997). What is mathematics really? New York: Oxford University Press.
  28. Jahnke, H. (2001). Cantor's cardinal and ordinal infinities: An epistemological and didactic view. Educational Studies in Mathematics, 48(2/3), 175-197.
  29. Kidron, I. (2003). Polynomial approximations of functions: Historical perspectives and new tools. International Journal of Computers for Mathematical Learning, 8(3), 299-331.
  30. Lakoff G. & Núñez, R. (2000). Where mathematics comes from. New York: Basic Books.
  31. Madden, J. (2001). Where mathematics comes from: How the embodied mind brings mathematics into being. Notices of the AMS, 48(10), 1182-1188.
  32. MacLane, S. (1981). Mathematical models: A sketch for the philosophy of mathematics. The American Mathematical Monthly, 88(7), 462-472.
  33. MacLane, S. (1986). Mathematics, form, and function. New York: Springer-Verlag.
  34. Moore, A.W.: 1999, The Infinite (2nd ed.). London: Routledge & Paul.
  35. Moreno, L. E. & Waldegg, G. (1991). The conceptual evolution of actual infinity. Educational Studies in Mathematics, 22(3), 211-231.
  36. Piaget, J. & Garcia, R. (1989). Psychogenesis and the history of science. New York: Columbia University Press.
  37. Piaget, J. & Inhelder, B. (1971). Mental imagery in the child (P.A. Chilton, Trans., original work published in 1966). London: Routledge & Kegan Paul.
  38. Presmeg, N.C. (1985). The role of visually mediated processes in high school mathematics: A Classroom investigation. Unpublished doctoral dissertation, University of Cambridge.
  39. Presmeg, N.C. (1986). Visualisation in high school mathematics. For the Learning of Mathematics, 6(3), 42-48.
  40. Presmeg, N. C. (1998). A semiotic analysis of students' own cultural mathematics. Research Forum Report. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education (Volume 1, pp. 136-151). Stellenbosch, South Africa.
  41. Rodd, M. M. (2000). On mathematical warrants: Proof does not always warrant, and a warrant may be other than proof. Mathematical Thinking and Learning, 2(3), 221-244.
  42. Rotman, B. (1988). Toward a semiotics of mathematics. Semiotica, 72(1/2), 1-35.
  43. Rotman, B. (1993). Taking God out of mathematics and putting the body back in: An essay in corporeal semiotics. Stanford, CA: Stanford University Press.
  44. Rotman, B. (2000). Mathematics as sign: Writing, imagining, counting. Stanford, CA: Stanford University Press.
  45. Schiralli M. & and Sinclair, N. (2003). A constructive response to "Where Mathematics Comes From". Educational Studies in Mathematics, 52(1), 79-91.
  46. Sfard, A. (1994). Reification as the birth of metaphor. For the Learning of Mathematics, 14(1), 44- 55.
  47. Sfard, A. (1997). A commentary: On metaphorical roots of conceptual growth. In L. English (Ed.). Mathematical reasoning: Analogies, metaphors, and images (pp. 339-371). Mahwah, NJ: Lawrence Erlbaum Associates.
  48. Sfard, A. (1998). Symbolizing mathematical reality into being or how mathematical discourse and mathematical objects create each other. In P. Cobb, K. E. Yackel, & K. McClain (Eds), Symbolizing and Communicating: Perspectives on mathematical discourse, tools, and instructional design (pp. 37-98). Mahwah, NJ: Lawrence Erlbaum Associates.
  49. Sierpinska, A. & Viwegier, M. (1989). How and when attitudes towards mathematics and infinity become constituted into obstacles in students. Proceedings of the 13th Annual Meeting for the Psychology of Mathematics Education (Volume 3, pp. 166-173). Paris
  50. Stenger, C., Vidakovic, D., & Weller, K. (2005, February). Students' conceptions of infinite iteration: A follow-up study. Paper presented at the SIGMAA on RUME conference, Phoenix, AZ.
  51. Tall, D. (1980). The notion of infinite measuring number and its relevance in the intuition of infinity. Educational Studies in Mathematics, 11(3), 271-284.
  52. Tall, D. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity, and proof. In D.A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 495-511). New York: MacMillan.
  53. Tall, D. (2001). Conceptual and formal infinities. Educational Studies in Mathematics, 48(2/3), 199-238.
  54. Tall, D. O, Gray, E M., Bin Ali, M., Crowley, L., DeMarois, P., McGowen, M., Pitta, D., Pinto, M., Thomas, M., & Yusof, Y. (2001). Symbols and the bifurcation between procedural and conceptual thinking. Canadian Journal of Science, Mathematics and Technology Education, 1(1), 81-104.
  55. Tall, D. (2005, July). A theory of mathematical growth through embodiment, symbolism, and proof. Paper presented at the International Colloquium on Mathematical Learning from early Childhood to Adulthood (Organized by the Centre de Recherche sur 'lEnsiegnement des Mathematiques), Nivelles, Belgium.
  56. Tall, D. O. & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in mathematics, 12(2), 151- 169.
  57. Tirosh, D. (1991). The role of students' intuitions of infinity in teaching the Cantorian theory. In D. Tall (Ed.), Advanced Mathematical Thinking (pp. 199-214). Dordrecht: Kluwer.
  58. Tirosh, D. (1999). Finite and infinite sets: Definitions and intuitions. International Journal of Mathematical Education in Science & Technology, 30(3), 341-349,
  59. Tsamir, P. (1999). The transition from comparison of finite to the comparison of infinite sets: Teaching prospective teachers. Educational Studies in Mathematics, 38(1/3), 209-234.
  60. Vidakovic, D. (1997). Learning the concept of inverse function in a group versus individual environment. In E. Dubinsky, D. Mathews, & B. Reynolds (Eds.), Readings in cooperative learning (MAA Notes No. 44, pp.175-195). Washington, D.C.: Mathematical Association of America.
  61. Vidakovic, D. & Martin, B. (2004). Small-group searches for mathematical proofs and individual reconstructions of mathematical concepts. Journal of Mathematical Behavior, 23(4), 465-492
  62. Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. O. Tall (Ed.), Advanced Mathematical Thinking (pp. 65-81). Dordrecht: Kluwer.
  63. Vygotsky, L.S. (1978). Mind in Society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.
  64. Vygotsky, L. S. (1981). The development of higher mental functions. In J. V. Wertsch (Ed.), The concept of activity in Soviet psychology (pp. 144-188). Armonk, N.Y.: Sharpe.
  65. Weller, K., Clark, J., Dubinsky, E., Loch, S., McDonald, M., & Merkovsky, R. (2003). Student performance and attitudes in courses based on APOS Theory and the ACE Teaching Cycle. In A. Selden, E. Dubinsky, G. Harel, & F. Hitt (Eds.), Research in collegiate mathematics education V (pp. 97-131). Providence: American Mathematical Society.
  66. Wheatly, G. (1991). Enhancing mathematics learning through imagery. Arithmetic Teacher, 39(1). 34-36.
  67. Zazkis, R., Dubinsky, E., & Dautermann, J. (1996). Coordinating Visual and Analytic Strategies: A study of students' understanding of the group D₁. Journal for Research in Mathematics Education, 27(4), 435-457.
  68. Zimmerman, W. & Cunningham, S. (Eds.) (1991). Visualization in teaching and learning mathematics (MAA Notes No. 19). Washington, D.C.: Mathematical Association of America.

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