ARE REGISTERS OF REPRESENTATIONS AND PROBLEM SOLVING PROCESSES ON FUNCTIONS COMPARTMENTALIZED IN STUDENTS' THINKING?

Athanasios Gagatsis; Iliada Elia; Nikos Mousoulides

Artículo Especial

Vol. 9 Núm. 4 (2006): Número Especial

ARE REGISTERS OF REPRESENTATIONS AND PROBLEM SOLVING PROCESSES ON FUNCTIONS COMPARTMENTALIZED IN STUDENTS' THINKING?

Athanasios Gagatsis^▸^▾
Iliada Elia^▸^▾
Nikos Mousoulides^▸^▾

PDF

Enviado: octubre 28, 2024
Publicado: 2006-12-30

Resumen

El objetivo de este artículo es doble. En primer lugar, se hace un resumen superficial de investigaciones sobre la compartimentación de diferentes registros de representación, así como de las aproximaciones de resolución de problemas, relacionadas con el concepto de función. En segundo lugar, se aportan elementos que clarifican las posiblesmaneras que permiten superar el fenómeno de la compartimentación. Investigaciones precedentes muestran que la mayoría de los alumnos de secundaria e, incluso de universidad, tienen dificultades para cambiar, de forma flexible, los sistemas de representación de funciones, de seleccionar y de utilizar aproximaciones apropiadas de resolución de problemas. Los resultados de dos estudios experimentales previos, llevados a cabo por miembros de nuestro equipo de investigación, centrados sobre la utilización de aproximaciones no tradicionales de enseñanza y sobre el empleo de software matemático, proveen pistas preliminares, en cuanto a la manera de cómo puede superarse con éxito el fenómeno de la compartimentación.

Citas

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, 301-317.
Bodin, A., Coutourier, R., & Gras, R. (2000). CHIC : Classification Hiérarchique Implicative et Cohésive-Version sous Windows – CHIC 1.2 , Rennes : Association pour la Recherche en Didactique des Mathématiques.
Cheng, P.C.H. (2000). Unlocking conceptual learning in mathematics and science with effective representational systems. Computers and Education, 33, 109-130.
D’Amore B. (1998). Relational objects and different representative registers: cognitive difficulties and obstacles. L’educazione matematica, 1, 7-28.
Dubinsky, E., & Harel, G. (1992). The nature of the process conception of function. In E. Dubinsky & G. Harel (Eds.), The Concept of Function. Aspects of Epistemology and Pedagogy (pp. 85-106). United States: The Mathematical Association of America.
Duval, R. (1993). Registres de Représentation Sémiotique et Fonctionnement Cognitif de la Pensée. Annales de Didactique et de Sciences Cognitives, 5, 37-65.
Duval, R. (2002). The cognitive analysis of problems of comprehension in the learning of mathematics. Mediterranean Journal for Research in Mathematics Education, 1(2), 1- 16.
Eisenberg, T., & Dreyfus, T. (1991). On the reluctance to visualize in mathematics. In W. Zimmermann & S. Cunningham (Eds.), Visualization in Teaching and Learning Mathematics (pp. 9-24). United States: Mathematical Association of America.
Elia, I., Gagatsis, A., & Gras, R. (2005). Can we “trace” the phenomenon of compartmentalization by using the I.S.A.? An application for the concept of function. In R. Gras, F. Spagnolo & J. David (Eds.), Proceedings of the Third International Conference I.S.A. Implicative Statistic Analysis (pp. 175-185). Palermo, Italy: Universita degli Studi di Palermo.
Evangelidou, A., Spyrou, P., Elia, I., & Gagatsis, A. (2004). University students’ conceptions of function. In M. Johnsen Høines & A. Berit Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, (Vol. 2, pp. 351-358). Bergen, Norway: Bergen University College.
Even, R. (1998). Factors involved in linking representations of functions. The Journal of Mathematical Behavior, 17(1), 105-121.
Gagatsis, A., & Elia, I. (2005a).A review of some recent studies on the role of representation in mathematics education in Cyprus and Greece. Paper presented at CERME 4, 2005, February, Saint Feliux de Guixols, Spain.
Gagatsis, A., & Elia, I. (2005b). Il concetto di funzione e le sue rappresentazioni nell’educazione secondaria. Bollettino dei Docenti di Matematica, 50, 41-54.
Gagatsis, A., & Shiakalli, M. (2004). Ability to translate from one representation of the concept of function to another and mathematical problem solving. Educational Psychology, 24(5), 645-657.
Gagatsis, A., Elia, I., & Mougi, A. (2002). The nature of multiple representations in developing mathematical relations. Scientia Paedagogica Experimentalis, 39(1), 9-24.
Gagatsis, A., Kyriakides, L., & Panaoura, A. (2004). Assessing the cross-cultural applicability of number lines in conducting arithmetic operations using structural equation modeling : A comparative study between Cypriot, Italian and Greek primary pupils. World Studies in Education, 5(1), 85-101.
Gagatsis, A., Shiakalli, M.,& Panaoura, A.(2003). La droite arithmétique comme modèle géométrique de l’ addition et de la soustraction des nombres entiers. Annales de didactique et de sciences cognitives, 8, 95-112.
Gras, R. (1996). L’implication statistique. Collection associée à ‘ Recherches en Didactique des Mathématiques’. Grenoble : La Pensée Sauvage.
Hitt, F. (1998). Difficulties in the articulation of different representations linked to the concept of function. The Journal of Mathematical Behavior, 17(1), 123-134.
Kaldrimidou, M., & Ikonomou, A. (1998). Factors involved in the learning of mathematics: The case of graphic representations of functions. In H. Stenbring, M.G. Bartolini Bussi & A. Sierpinska (Eds.), Language and Communication in the Mathematics Classroom (pp. 271-288). Reston, Va: NCTM.
Kaput, J. J. (1987). Representation systems and mathematics. In C. Janvier (Ed.), Problems of Representation in the Teaching and Learning of Mathematics (pp. 19-26). Hillsdale, NJ: Lawrence Erlbaum.
Knuth, J. E. (2000). Student understanding of the Cartesian Connection: An exploratory study. Journal of Research in Mathematics Education, 31(4), 500-508.
Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of Educational Research, 60, 1–64.
Lerman, I.C. (1981). Classification et Analyse Ordinale des Données. Paris: Dunod.
Lesh, R., Post, T. & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of Representation in the Teaching and Learning of Mathematics (pp. 33-40). Hillsdale, NJ: Lawrence Erlbaum.
Marcou, A., & Gagatsis, A. (2003). Rappresentazioni e apprendimento matematico: applicazioni nel campo delle frazioni. La matematica e la sua didattica, 2,124-138.
Markovits, Z., Eylon, B. and Bruckheimer, M. (1986). Functions today and yesterday. For the Learning of Mathematics, 6(2) 18-28.
Michaelidou, N., Gagatsis, A., & Pitta-Pantazi, D. (2004). The number line as a representation of decimal numbers: A research with sixth grade students. In M. Johnsen Høines & A. Berit Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 305-312). Bergen, Norway: Bergen University College.
Moschkovich, J., Schoenfeld, A. H., & Arcavi, A. (1993). Aspects of understanding: On multiple perspectives and representations of linear relations and connections among them. In T. A. Romberg, E. Fennema, & T. P. Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 69–100). Hillsdale, NJ: Lawrence Erlbaum Associates.
Mousoulides, N., & Gagatsis, A. (2004). Algebraic and geometric approach in function problem solving. In M. Johnsen Høines & A. Berit Fuglestad (Eds.), Proceedings of 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 385-392). Bergen, Norway: PME.
Mousoulides, N., Philippou, G., Hoyles, C (2005). Mathematical discovery in the context of number sequences. Paper presented at the 11th European for Research on Learning and Instruction, 2005, August.
Mousoulides, N., & Gagatsis, A. (2006). The role of new technologies in improving problem-solving in functions. Pre-print, Department of Education, University of Cyprus.
Romberg, T. A., Fennema, E., & Carpenter, T. P. (Eds.). (1993). Integrating research on the graphical representation of functions. Hillsdale, NJ: Erlbaum.
Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification - The case of function. In E. Dubinsky & G. Harel (Eds.), The Concept of Function: Aspects of Epistemology and Pedagogy (pp. 59-84). United States: The Mathematical Association of America.
Sierpinska, A. (1992). On understanding the notion of function. In E. Dubinsky & G. Harel (Eds.), The Concept of Function. Aspects of Epistemology and Pedagogy (pp. 25- 28). United States: The Mathematical Association of America.
Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356-266.
Yerushalmy, M., & Schwartz, J. L. (1993). Seizing the opportunity to make algebra mathematically and pedagogically interesting. In T. A. Romberg, E. Fennema & T. P. Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 41–68). Hillsdale, NJ: Lawrence Erlbaum Associates.

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Palabras clave

Aproximación algebraica
compartimentalización
función
aproximación geométrica
resolución de problemas
registros de representación
transformación de representaciones

Cómo citar

Gagatsis, A., Elia, I., & Mousoulides, N. (2006). ARE REGISTERS OF REPRESENTATIONS AND PROBLEM SOLVING PROCESSES ON FUNCTIONS COMPARTMENTALIZED IN STUDENTS’ THINKING?. Revista Latinoamericana De Investigación En Matemática Educativa, 9(4), 197–224. Recuperado a partir de https://relime.org/index.php/relime/article/view/483

Esta obra está bajo una licencia internacional Creative Commons Atribución-NoComercial 4.0.

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[1] 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, 301-317.

[2] Bodin, A., Coutourier, R., & Gras, R. (2000). CHIC : Classification Hiérarchique Implicative et Cohésive-Version sous Windows – CHIC 1.2 , Rennes : Association pour la Recherche en Didactique des Mathématiques.

[3] Cheng, P.C.H. (2000). Unlocking conceptual learning in mathematics and science with effective representational systems. Computers and Education, 33, 109-130.

[4] D’Amore B. (1998). Relational objects and different representative registers: cognitive difficulties and obstacles. L’educazione matematica, 1, 7-28.

[5] Dubinsky, E., & Harel, G. (1992). The nature of the process conception of function. In E. Dubinsky & G. Harel (Eds.), The Concept of Function. Aspects of Epistemology and Pedagogy (pp. 85-106). United States: The Mathematical Association of America.

[6] Duval, R. (1993). Registres de Représentation Sémiotique et Fonctionnement Cognitif de la Pensée. Annales de Didactique et de Sciences Cognitives, 5, 37-65.

[7] Duval, R. (2002). The cognitive analysis of problems of comprehension in the learning of mathematics. Mediterranean Journal for Research in Mathematics Education, 1(2), 1- 16.

[8] Eisenberg, T., & Dreyfus, T. (1991). On the reluctance to visualize in mathematics. In W. Zimmermann & S. Cunningham (Eds.), Visualization in Teaching and Learning Mathematics (pp. 9-24). United States: Mathematical Association of America.

[9] Elia, I., Gagatsis, A., & Gras, R. (2005). Can we “trace” the phenomenon of compartmentalization by using the I.S.A.? An application for the concept of function. In R. Gras, F. Spagnolo & J. David (Eds.), Proceedings of the Third International Conference I.S.A. Implicative Statistic Analysis (pp. 175-185). Palermo, Italy: Universita degli Studi di Palermo.

[10] Evangelidou, A., Spyrou, P., Elia, I., & Gagatsis, A. (2004). University students’ conceptions of function. In M. Johnsen Høines & A. Berit Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, (Vol. 2, pp. 351-358). Bergen, Norway: Bergen University College.

[11] Even, R. (1998). Factors involved in linking representations of functions. The Journal of Mathematical Behavior, 17(1), 105-121.

[12] Gagatsis, A., & Elia, I. (2005a).A review of some recent studies on the role of representation in mathematics education in Cyprus and Greece. Paper presented at CERME 4, 2005, February, Saint Feliux de Guixols, Spain.

[13] Gagatsis, A., & Elia, I. (2005b). Il concetto di funzione e le sue rappresentazioni nell’educazione secondaria. Bollettino dei Docenti di Matematica, 50, 41-54.

[14] Gagatsis, A., & Shiakalli, M. (2004). Ability to translate from one representation of the concept of function to another and mathematical problem solving. Educational Psychology, 24(5), 645-657.

[15] Gagatsis, A., Elia, I., & Mougi, A. (2002). The nature of multiple representations in developing mathematical relations. Scientia Paedagogica Experimentalis, 39(1), 9-24.

[16] Gagatsis, A., Kyriakides, L., & Panaoura, A. (2004). Assessing the cross-cultural applicability of number lines in conducting arithmetic operations using structural equation modeling : A comparative study between Cypriot, Italian and Greek primary pupils. World Studies in Education, 5(1), 85-101.

[17] Gagatsis, A., Shiakalli, M.,& Panaoura, A.(2003). La droite arithmétique comme modèle géométrique de l’ addition et de la soustraction des nombres entiers. Annales de didactique et de sciences cognitives, 8, 95-112.

[18] Gras, R. (1996). L’implication statistique. Collection associée à ‘ Recherches en Didactique des Mathématiques’. Grenoble : La Pensée Sauvage.

[19] Hitt, F. (1998). Difficulties in the articulation of different representations linked to the concept of function. The Journal of Mathematical Behavior, 17(1), 123-134.

[20] Kaldrimidou, M., & Ikonomou, A. (1998). Factors involved in the learning of mathematics: The case of graphic representations of functions. In H. Stenbring, M.G. Bartolini Bussi & A. Sierpinska (Eds.), Language and Communication in the Mathematics Classroom (pp. 271-288). Reston, Va: NCTM.

[21] Kaput, J. J. (1987). Representation systems and mathematics. In C. Janvier (Ed.), Problems of Representation in the Teaching and Learning of Mathematics (pp. 19-26). Hillsdale, NJ: Lawrence Erlbaum.

[22] Knuth, J. E. (2000). Student understanding of the Cartesian Connection: An exploratory study. Journal of Research in Mathematics Education, 31(4), 500-508.

[23] Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of Educational Research, 60, 1–64.

[24] Lerman, I.C. (1981). Classification et Analyse Ordinale des Données. Paris: Dunod.

[25] Lesh, R., Post, T. & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of Representation in the Teaching and Learning of Mathematics (pp. 33-40). Hillsdale, NJ: Lawrence Erlbaum.

[26] Marcou, A., & Gagatsis, A. (2003). Rappresentazioni e apprendimento matematico: applicazioni nel campo delle frazioni. La matematica e la sua didattica, 2,124-138.

[27] Markovits, Z., Eylon, B. and Bruckheimer, M. (1986). Functions today and yesterday. For the Learning of Mathematics, 6(2) 18-28.

[28] Michaelidou, N., Gagatsis, A., & Pitta-Pantazi, D. (2004). The number line as a representation of decimal numbers: A research with sixth grade students. In M. Johnsen Høines & A. Berit Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 305-312). Bergen, Norway: Bergen University College.

[29] Moschkovich, J., Schoenfeld, A. H., & Arcavi, A. (1993). Aspects of understanding: On multiple perspectives and representations of linear relations and connections among them. In T. A. Romberg, E. Fennema, & T. P. Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 69–100). Hillsdale, NJ: Lawrence Erlbaum Associates.

[30] Mousoulides, N., & Gagatsis, A. (2004). Algebraic and geometric approach in function problem solving. In M. Johnsen Høines & A. Berit Fuglestad (Eds.), Proceedings of 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 385-392). Bergen, Norway: PME.

[31] Mousoulides, N., Philippou, G., Hoyles, C (2005). Mathematical discovery in the context of number sequences. Paper presented at the 11th European for Research on Learning and Instruction, 2005, August.

[32] Mousoulides, N., & Gagatsis, A. (2006). The role of new technologies in improving problem-solving in functions. Pre-print, Department of Education, University of Cyprus.

[33] Romberg, T. A., Fennema, E., & Carpenter, T. P. (Eds.). (1993). Integrating research on the graphical representation of functions. Hillsdale, NJ: Erlbaum.

[34] Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification - The case of function. In E. Dubinsky & G. Harel (Eds.), The Concept of Function: Aspects of Epistemology and Pedagogy (pp. 59-84). United States: The Mathematical Association of America.

[35] Sierpinska, A. (1992). On understanding the notion of function. In E. Dubinsky & G. Harel (Eds.), The Concept of Function. Aspects of Epistemology and Pedagogy (pp. 25- 28). United States: The Mathematical Association of America.

[36] Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356-266.

[37] Yerushalmy, M., & Schwartz, J. L. (1993). Seizing the opportunity to make algebra mathematically and pedagogically interesting. In T. A. Romberg, E. Fennema & T. P. Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 41–68). Hillsdale, NJ: Lawrence Erlbaum Associates.