Visual salience of algebraic transformations

Journal for Research in Mathematics Education

Volumn 35, Issue 4, 2004, Pages 224-257

  Kirshner, David ^a   Awtry, Thomas ^b  

^a LOUISIANA STATE UNIVERSITY   (United States)

^b Louisiana State University at Alexandria   (United States)

Author keywords

Algebra; Cognitive theory; Epistemology; Language and mathematics; Logic and proof; Oral communication; Reform in mathematics education; Spatial visualization

Indexed keywords

EID: 3042818128     PISSN: 00218251     EISSN: None     Source Type: Journal    
DOI: 10.2307/30034809     Document Type: Article

References (75)

1. Anderson, J. R. (1983). Acquisition of proof skill in geometry. In R. S. Michalski, J. G. Carbonell, & T. M. Mitchell (Eds.), Machine learning: An artificial intelligence approach (pp. 191-219). Palo Alto, CA: Tioga Publishing Company.
2. Anderson, J. R., & Lebiere, C. (1998a). Introduction. In J. R. Anderson & C. Lebiere (Eds.), The atomic components of thought (pp. 1-17). Mahwah, NJ: Lawrence Erlbaum Associates.
3. Anderson, J. R., & Lebiere, C. (1998b). Learning. In J. R. Anderson & C. Lebiere (Eds.), The atomic components of thought (pp. 101-142). Mahwah, NJ: Lawrence Erlbaum Associates.
4. Awtry, T. (1993). Visual transformations in symbolic elementary algebra. Unpublished doctoral dissertation, Louisiana State University.
5. Awtry, T., & Kirshner, D. (1994). Visual salience in algebraic transformations. In D. Kirshner (Ed.), Proceedings of the Sixteenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 131-137). Baton Rouge, LA: Louisiana State University.
6. Bechtel, W., & Abrahamsen, A. (2002). Connectionism and the mind: Parallel processing, dynamics, and evolution in networks (2nd ed.). Malden, MA: Blackwell.
7. Bell, E. T. (1936). The meaning of mathematics. In W. D. Reeve (Ed.), The place of mathematics in modern education (pp. 136-181). Eleventh Yearbook of the National Council of Teachers of Mathematics. New York: Bureau of Publications, Teachers College, Columbia University.
8. Bereiter, C. (1991). Implications of connectionism for thinking about rules. Educational Researcher, 20, 10-16.
9. Booth, L. R. (1984). Algebra: Children's strategies and errors. Windsor, Berkshire, UK: NFER-NELSON Publishing Company.
10. Booth, L. R. (1989). A question of structure. In C. Kieran & S. Wagner (Eds.), Research agenda for mathematics education: Research issues in the learning and teaching of algebra (pp. 57-59). Reston, VA: National Council of Teachers of Mathematics; Hillsdale, NJ: Lawrence Erlbaum Associates.
11. Brooks, R. A. (1991). Intelligence without representation. Artificial Intelligence, 47, 139-159.
12. Bundy, A., & Welham, B. (1981). Using meta-level inference for selective application of multiple rewrite rule sets in algebraic manipulation. Artificial Intelligence, 16, 189-212.
13. Cajori, F. (1924). A history of elementary mathematics: With hints on methods of teaching. New York: The Macmillan Company.
14. Carry, L. R., Lewis, C., & Bernard, J. E. (1980). Psychology of equation solving: An information processing study (Tech. Rep. SED 78-22293). Austin: The University of Texas at Austin, Department of Curriculum and Instruction.
15. Clancey, W. J. (1999). Conceptual coordination: How the mind orders experience in time. Mahwah, NJ.: Lawrence Erlbaum Associates.
16. College Entrance Examination Board. (1959). Report of the Commission on Mathematics. New York: Author.
17. Cuoco, A., Goldenberg, E. P, & Mark, J. (1996). Habits of mind: An organizing principle for mathematics curricula. Journal of Mathematical Behavior, 15 (4), 375-402.
18. Davis, R. B. (1979, April). Error analysis in high school mathematics, conceived as information-processing pathology. Paper presented at the annual meeting of the American Educational Research Association, San Francisco. (ERIC Document Reproduction Service No. ED171551).
19. Davis, R. B. (1984). Learning mathematics. Norwood, NJ: Ablex.
20. Davis, R. B., & McKnight, C. C. (1979). Modeling the processes of mathematical thinking. Journal of Children's Mathematical Behavior, 2 (2), 91-113.
21. Dreyfus, H. L. (2002). Intelligence without representation - Merleau-Ponty's critique of mental representation: The relevance of phenomenology to scientific explanation. Phenomenology and the Cognitive Sciences, 1 (4), 367-383.
22. Drouhard, J. (1988, July). Syntactic analysis of algebra and teaching. Paper presented at the Sixth International Congress on Mathematics Education, Budapest, Hungary.
23. Dupuy, J-P. (2000). The mechanization of the mind: On the origins of cognitive science. (M. B. DeBevoise, Trans.). Princeton, NJ: New French Thought.
24. Erlwanger, S. H. (1973). Benny's conception of rules and answers in IPI mathematics. Journal of Mathematical Behavior, 1 (2), 7-26.
25. Ernest, P. (1985). The philosophy of mathematics and mathematics education. International Journal of Mathematics, Education, Science, and Technology, 16 (5), 603-612.
26. Ernest, P. (1987). A model of the cognitive meaning of mathematical expressions. British Journal of Educational Psychology, 57, 343-370.
27. Ernest, P. (1998). Social constructivism as a philosophy of mathematics. Albany, NY: State University of New York Press.
28. Estep, M. (2003). A theory of immediate awareness: Self-organization and adaptation in natural intelligence. Dordrecht, The Netherlands: Kluwer Academic Publishers.
29. Evans, J. St. B. T. (1982). The psychology of deductive reasoning. London: Routledge & Kegan Paul.
30. Fey, J. T. (1992). Computer-intensive algebra: An overview of fundamental mathematical and instructional themes. Unpublished manuscript.
31. Fey, J. T., & Graeber, A. (2003). From the new math to the Agenda for Action. In G. Stanic & J. Kilpatrick (Eds.) History of school mathematics (Vol. 1, pp. 521-558). Reston, VA: National Council of Teachers of Mathematics.
32. Freeman, W. J. (1991). The physiology of perception. Scientific American, 264, 78-85.
33. Gee, J. P. (1992). The social mind: Language, ideology, and social practice. New York: Bergin & Garvey.
34. Greeno, J. G. (1982, April). A cognitive learning analysis in algebra. Paper presented at the annual meeting of the American Educational Research Association, Boston, MA.
35. Haag, V. H. (1961). Studies in mathematics: Vol. 3. Structure of elementary algebra (Rev. ed.). New Haven, CT: School Mathematics Study Group, Yale University.
36. Haberlandt, K. (1997). Cognitive psychology. Boston: Allyn & Bacon.
37. Hamilton, W. R. (1836). A review of Rev. William Whewell's Thoughts on the Study of Mathematics as a part of Liberal Education. Edinburgh Review, 62, 218-252.
38. Hamilton, W. R. (1837). Theory of conjugate functions, or algebraic couples; with a preliminary and elementary essay on algebra as the science of pure time. Transactions of the Royal Irish Academy, XVII, 293-422.
39. Haugland, J. (1985). Semantic engines: An introduction to mind design. In J. Haugland (Ed.), Mind design (pp. 1-34). Cambridge, MA: The MIT Press.
40. Hobson, J. A. (1988). The dreaming brain. New York: Basic Books.
41. Hofstadter, D. R. (1985). Waking up from the Boolean dream, or, subcognition as computation. In D. R. Hofstadter (Ed.), Metamagical themas: Questing for the essence of mind and pattern (pp. 635-665). New York: Basic Books.
42. Kaput, J. J. (1995). A research base supporting long term algebra reform. In D. T. Owens, M. K. Reed, & G. M. Millsaps (Eds.), Proceedings of the Seventeenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 71-94). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
43. Kirshner, D. (1987). The grammar of symbolic elementary algebra. Unpublished doctoral dissertation. University of British Columbia, Vancouver, Canada.
44. Kirshner, D. (1989). The visual syntax of algebra. Journal for Research in Mathematics Education, 20, 274-287.
45. Kirshner, D. (1998, October) Discursive algebra: Explicit talk about implicit processes. Paper and poster presented at the twentieth annual meeting of North American Chapter of the International Group for the Psychology of Mathematics Education, Raleigh, NC.
46. Kirshner, D. (2001). The structural algebra option revisited. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school algebra (pp. 83-98). Dordrecht, The Netherlands: Kluwer Academic Publishers.
47. Kirshner, D. (2002). Untangling teachers' diverse aspirations for student learning: A crossdisciplinary strategy for relating psychological theory to pedagogical practice. Journal for Research in Mathematics Education, 33, 46-58.
48. Kline, M. (1980). Mathematics: The loss of certainty. New York: Oxford University Press.
49. Lloyd, D. (1989). Simple minds: Mental representations from the ground up. Cambridge, MA: MIT Press.
50. Macintyre, A. (1979/1980). The laws of exponentiation. In A. Dold & B. Eckmann (Proceedings Editors of Model theory and arithmetic, Paris). Published in Lecture Notes in Mathematics, 890, 185-197.
51. Matz, M. (1980). Towards a computational theory of algebraic competence. Journal of Mathematical Behavior, 3, 93-166.
52. Merleau-Ponty, M. (1962). Phenomenology of perception. London: Routledge & Kegan Paul.
53. National Advisory Committee on Mathematics Education. (1975). Overview and analysis of school mathematics grades K-12. Washington, DC: Conference Board of the Mathematical Sciences.
54. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics, Reston, VA; Author.
55. National Council of Teachers of Mathematics Algebra Working Group. (1998). A framework for constructing a vision of algebra: A discussion document. In National Research Council, The nature and role of algebra in the K-14 curriculum: Proceedings of a national symposium, May 27 and 28, 1997 (Appendix E, pp. 145-190). Washington, DC: National Academy Press. Available on the Web at http://books.nap.edu/catalog/6286.html.
56. National Research Council. (1998). High school mathematics at work: Essays and examples of for the education of all students. Washington, DC: National Academy Press.
57. Newell, A., & Simon, H. A. (1985) Computer science as empirical inquiry: Symbols and search. In J. Haugland (Ed.), Mind design (pp. 35-66). Cambridge: The MIT Press.
58. Osborne, A., and Kasten, M. (1992). Change and An Agenda for Action: A reconsideration. In R. Morris (Ed.), Studies in mathematics education: Moving into the twenty-first century (Vol. 8, pp. 21-42). Paris: The United Nations Educational, Scientific and Cultural Organization.
59. Payne, S. J., & Squibb, H. R. (1990). Algebra mal-rules and cognitive accounts of error. Cognitive Science, 14, 445-481.
60. Peacock, G. (1833). Report on recent progress and present state of certain branches of analysis. In Report of the Third Meeting of the British Association for the Advancement of Science (pp. 185-352). London: J. Murray.
61. Pimm, D. (1995). Symbols and meanings in school mathematics. New York: Routledge.
62. Radatz, H. (1979). Error analysis in mathematics education. Journal for Research in Mathematics Education, 10, 163-172.
63. Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93, 346-362.
64. Romberg, T. A., Fennema, E., & Carpenter, T. P. (Eds.). (1993). Integrating research on the graphical representation of functions. Hillsdale, NJ: Lawrence Erlbaum Associates.
65. Royden, H. L. (1968). Real analysis. London: Macmillan.
66. Rumelhart, D. E., Hinton, G. E., & Williams, R. J. (1986). Learning internal representations by error propagation. In D. E. Rumelhart, & J. L., McClelland (Eds.), Parallel distributed processing: Explorations in the microstructure of cognition (pp. 318-362). Cambridge, MA: MIT Press.
67. Schoenfeld, A. H. (1987). Cognitive science and mathematics education: An overview. In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 1-32). Hillsdale, NJ: Lawrence Erlbaum Associates.
68. Sfard, A. (1995). The development of algebra: Confronting historical and psychological perspectives. Journal of Mathematical Behavior, 14 (1), 15-39.
69. Sleeman, D. (1982). Assessing aspects of competence in basic algebra. In D. Sleeman & J. S. Brown (Eds.), Intelligent tutoring systems (pp. 185-199). New York: Academic Press.
70. Sleeman, D. (1984). An attempt to understand students' understanding of basic algebra. Cognitive Science, 8, 387-412.
71. Sleeman, D. (1986). Introductory algebra: A case study of student misconceptions. Journal of Mathematical Behavior, 5 (1), 25-52.
72. St. Julien, J. (1994). Cognition and learning: The implications of a situated connectionist perspective for theory and practice in education. Unpublished doctoral dissertation, Louisiana State University.
73. Thompson, P. W. (1989). Artificial intelligence, advanced technology, and learning and teaching algebra. In C. Kieran & S. Wagner (Eds.), Research agenda for mathematics education: Research issues in the learning and teaching of algebra (pp. 135-161). Reston, VA: National Council of Teachers of Mathematics; Hillsdale, NJ: Lawrence Erlbaum Associates.
74. Thompson, P. W., & Thompson, A. G. (1987). Computer presentations of structure in algebra. In J. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the Eleventh Annual Meeting of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 248-254). Montreal: University of Quebec at Montreal.
75. Wagner, S., Rachlin, S. L., & Jensen, R. J. (1984). Algebra learning project: Final report (Contract No. 400-81-0028). Washington, DC: National Institute of Education.