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From Concrete Representations to Abstract Symbols
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accurate, relevant, and important in doing real mathematics in the sense discussed by Davis and Maher. The students expressed the way they thought about ideas that were new to them by referring to their own personal representations. In so doing, they recognized the structural equivalence among three problems that, on the surface, did not appear to be the same. Moreover, this understanding was durable over time.
REFERENCES
Davis, R. B. & Maher, C. A. (1997). How students think: The role of representations. In
English, L. D. (Ed.). Mathematical Reasoning: Analogies, Metaphors and Images, pp. 93-115. Mahwah, NJ: Lawrence Erlbaum Associates.
Kiczek, R. D., Maher, C. A., & Speiser, R. (2001). Tracing the Origins and Extensions of
Michael’s Representation. In Cuoco, A. A. and Curcio, F. R. (Eds.). The Roles of Representation in School Mathematics, 2001 Yearbook (pp. 201-214). Reston, VA: National Council of Teachers of Mathematics.
Maher, C. A. & Kiczek, R. D. (2000). Long-term building of mathematical ideas related to proof
making. Contribution to: Boero, P., Harel, G., Maher, C., & Miyzai, M. (organizers). Proof and proving in mathematics education. ICME9 TSG 12.
Maher, C. A. (2002). How students structure their own investigations and educate us: What we
have learned from a fourteen year study. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the Twenty-sixth Annual Meeting of the International Group for the Psychology of Mathematics Education (PME26) (Vol. 1, pp. 31-46). Norwich, England: School of Education and Professional Development, University of East Anglia.
Maher, C. A. & Martino, A. M. (1996). The development of the idea of Mathematical proof: A
5-year case study. Journal for Research in Mathematics Education, 27, 194-214.
Muter, E. M. & Maher, C. A. (1999). Recognizing isomorphism and building proof: Revisiting
earlier ideas. In Proceedings of the Twentieth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA 20) (Vol. 2, pp. 461-467). Raleigh, NC: North Carolina State University.
Powell, A. B. (2003) ”So Let’s Prove It!”: Emergent and Elaborated Mathematical Ideas and
Reasoning in the Discourse and Inscriptions of Learners Engaged in a Combinatorial Task.Unpublished doctoral dissertation, Rutgers University, New Brunswick, NJ.
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| | Authors: Uptegrove, Elizabeth. and Maher, Carolyn. |
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accurate, relevant, and important in doing real mathematics in the sense discussed by Davis and
Maher. The students expressed the way they thought about ideas that were new to them by
referring to their own personal representations. In so doing, they recognized the structural
equivalence among three problems that, on the surface, did not appear to be the same. Moreover,
this understanding was durable over time.
REFERENCES
Davis, R. B. & Maher, C. A. (1997). How students think: The role of representations. In
English, L. D. (Ed.). Mathematical Reasoning: Analogies, Metaphors and Images, pp. 93-
115. Mahwah, NJ: Lawrence Erlbaum Associates.
Kiczek, R. D., Maher, C. A., & Speiser, R. (2001). Tracing the Origins and Extensions of
Michael’s Representation. In Cuoco, A. A. and Curcio, F. R. (Eds.). The Roles of
Representation in School Mathematics, 2001 Yearbook (pp. 201-214). Reston, VA: National
Council of Teachers of Mathematics.
Maher, C. A. & Kiczek, R. D. (2000). Long-term building of mathematical ideas related to proof
making. Contribution to: Boero, P., Harel, G., Maher, C., & Miyzai, M. (organizers). Proof
and proving in mathematics education. ICME9 TSG 12.
Maher, C. A. (2002). How students structure their own investigations and educate us: What we
have learned from a fourteen year study. In A. D. Cockburn & E. Nardi (Eds.), Proceedings
of the Twenty-sixth Annual Meeting of the International Group for the Psychology of
Mathematics Education (PME26) (Vol. 1, pp. 31-46). Norwich, England: School of
Education and Professional Development, University of East Anglia.
Maher, C. A. & Martino, A. M. (1996). The development of the idea of Mathematical proof: A
5-year case study. Journal for Research in Mathematics Education, 27, 194-214.
Muter, E. M. & Maher, C. A. (1999). Recognizing isomorphism and building proof: Revisiting
earlier ideas. In Proceedings of the Twentieth Annual Meeting of the North American
Chapter of the International Group for the Psychology of Mathematics Education (PME-NA
20) (Vol. 2, pp. 461-467). Raleigh, NC: North Carolina State University.
Powell, A. B. (2003) ”So Let’s Prove It!”: Emergent and Elaborated Mathematical Ideas and
Reasoning in the Discourse and Inscriptions of Learners Engaged in a Combinatorial Task.
Unpublished doctoral dissertation, Rutgers University, New Brunswick, NJ.
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