Tareas con diversas soluciones: estructura conceptual en profesores de matemáticas
DOI:
https://doi.org/10.24320/redie.2017.19.1.971Palabras clave:
Procesos de Aprendizaje, resolución de problemas, educación de profesoresResumen
En el artículo se analizan las diferentes soluciones que un grupo de 15 profesores de matemáticas propuso para resolver un problema rutinario de aritmética, con el objetivo de identificar qué elementos específicos puede aportar el uso de tareas con múltiples soluciones para la formación y actualización docente. La posición teórica que sustenta este trabajo tiene como elemento fundamental la identificación y discusión de rutas de solución para fortalecer el conocimiento matemático y didáctico de los profesores, y favorecer el desarrollo de una postura crítica respecto a su práctica profesional.Descargas
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Referencias
Bingolbali, E. (2011). Multiple solutions to problems in mathematics teaching: Do teachers really value them? Australian Journal of Teacher Education, 36(1), 18-31.
Cai, J. y Nie, B. (2007). Problem solving in chinese mathematics education: research and practice. ZDM Mathematics Education, 39, 459-473.
Duval, R. (2006). A cognitive analysis of problems of comprehension in learning of mathematics. Educational Studies in Mathematics, 61, 103-131.
Fennema, E. y Romberg, T. A. (1999). Mathematics classroom that promote understanding. Mahwah, NJ: Lawrence Erlbaum.
Gu, L, Huang, R. y Marton, F. (2004). Teaching with variation: a Chinese way of promoting effective mathematics learning. En L. Fan, N.-Y. Wong., J. Cai y S. Li (Eds.), How chinese learn mathematics: Perspectives from insiders (pp. 309-347). NJ: World Scientific.
Guberman, R. y Leikin, R. (2013). Interesting and difficult mathematical problems: changing teachers’ views by employing multiple-solution tasks. Journal of Mathematics Teachers Education, 16, 33-56.
Henningsen, M. y Stein, M. K. (1997). Mathematical tasks and students cognition: Classroom-based factor that support and inhibit high level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524-549.
Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., Olivier, A. y Human, P. (1997). Making sense: teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.
Kieran, C. (2013). The false dichotomy in mathematics education between conceptual understanding and procedural skills: An example from algebra. En K. Leatham (Ed.), Vital directions in mathematics education research (pp. 153-171). Nueva York: Springer.
Lamon, S. J. (2006). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. Mahwah, NJ: Lawrence Erlbaum Associates.
Leikin, R. (2007). Habits of mind associated with advanced mathematical thinking and solution spaces of mathematical tasks. The Fifth Conference of the European Society for Research in Mathematics Education (CERME-5) (pp. 2330-2339).
Leikin, R. (2010). Learning through teaching through the lens of multiple solution tasks. En R. Leikin y R. Zaskis (Eds.), Learning through teaching mathematics: development of teachers’ knowledge and expertise in practice (pp. 69-85). Dordrecht, Holanda: Springer.
Leikin, R. (2011). Multiple-solution tasks: from a teacher education course to teacher practice. ZDM Mathematics Education, 43, 993-1006.
Leikin, R. (2014). Challenging mathematics with multiple solution tasks and mathematical investigations in geometry. En Y. Li, E. A. Silver y S. Li (Eds.), Transforming mathematics instruction: Multiple approaches and practices (pp. 59-80). Cham, Alemania: Springer.
Leikin, R. y Lev, M. (2007). Multiple solution tasks as a magnifying glass for observation of mathematical creativity. En J. H. Woo, H. C. Lew, K. S. Park y D. Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 161-168). Seúl: PME.
Leikin, R. y Levav-Waynberg, A. (2007). Exploring mathematics teacher knowledge to explain the gap between theory-based recommendations and school practice in the use of connecting tasks. Educational Studies in Mathematics, 66, 349-371.
Leikin, R. y Levav-Waynberg, A. (2008). Solution spaces of multiple-solution connecting tasks as a mirror of the development of mathematics teachers’ knowledge. Canadian Journal of Science, Mathematics and Technology Education, 8(3), 233-251.
Ma, L. (2010). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Nueva York: Routledge.
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Autor.
Polya, G. (1945). How to solve it: A new aspect of mathematical method. NJ: Princeton University Press.
Romberg, T. A. (1994). Classroom instruction that fosters mathematical thinking and problem solving: Connections between theory and practice. En A. H. Schoenfeld (Ed.), Mathematical thinking and problem solving (pp. 287-304). Hillsdale, NJ: Lawrence Erlbaum.
Rubio, C. J. (2006). Problemas para la 20a. Olimpiada Mexicana de Matemática en San Luis Potosí. Mérida: Universidad Autónoma de Yucatán.
Santos-Trigo, M. (1996). An exploration of strategies used by students to solve problems with multiple ways of solution. Journal of Mathematical Behavior, 15, 263-284.
Santos-Trigo, M. (1997). La transferencia del conocimiento y la formulación o rediseño de problemas en el aprendizaje de las matemáticas. Revista Mexicana de Investigación Educativa, 2(3), 11-30.
Santos-Trigo, M. (2007). La resolución de problemas matemáticos: fundamentos cognitivos. México: Trillas.
Santos-Trigo, M. (2009). Innovación e investigación en educación matemática. Innovación Educativa, 9(46), 5-13.
Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press.
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. En D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334-371). Nueva York: Macmillan.
Shimada, S. y Becker, J. (1997). The open ended approach: A new proposal for teaching mathematics. Reston, VA: NCTM.
Shulman, L. S. (2005). Conocimiento y enseñanza: fundamentos de la nueva reforma. Profesorado. Revista de currículum y formación del profesorado, 9(2), 1-30.
Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM Mathematics Education, 29(3), 75-80.
Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C. y Font Strawhum, B. T. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. Journal of Mathematical Behavior, 24, 287-301.
Silver, E. A. y Kenney, P. A. (1995). Sources of assessment information for instructional guidance in mathematics. En T. A. Romberg (Ed.), Reform in school mathematics and authentic assessment (pp. 38-68). Albany: State University of New York.
Simon, M. A. (1994). Learning mathematics and learning to teach: Learning cycles in mathematics teacher education. Educational Studies in Mathematics, 26, 71-94.
Simon, M. y Schifter, D. (1991). Towards a constructivist perspective: An intervention study of mathematics teacher development. Educational Studies in Mathematics, 22, 309-331.
Simon, M. A. y Tzur, R. (2004). Explicating the role of mathematical tasks in conceptual learning: an elaboration of the hypothetical learning trajectory. Mathematical Thinking and Learning, 6(2), 91-104.
Steen, L. A. (1988). The science of patterns. Science, 240, 611-616.
Stein, M. K. y Smith M. S. (1998). Mathematical tasks as a framework for reflection: From research to practice. Mathematics Teaching in the Middle School, 3, 268-275.
Stigler, J. y Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. Nueva York: The Free Press.
Tsamir, P., Tirosh, D., Tabach, D. y Levenson, E. (2010). Multiple solution methods and multiple outcomes– is it a task for kindergarten children? Educational Studies in Mathematics, 73, 217-231.
Yackel, E. y Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477.
Cai, J. y Nie, B. (2007). Problem solving in chinese mathematics education: research and practice. ZDM Mathematics Education, 39, 459-473.
Duval, R. (2006). A cognitive analysis of problems of comprehension in learning of mathematics. Educational Studies in Mathematics, 61, 103-131.
Fennema, E. y Romberg, T. A. (1999). Mathematics classroom that promote understanding. Mahwah, NJ: Lawrence Erlbaum.
Gu, L, Huang, R. y Marton, F. (2004). Teaching with variation: a Chinese way of promoting effective mathematics learning. En L. Fan, N.-Y. Wong., J. Cai y S. Li (Eds.), How chinese learn mathematics: Perspectives from insiders (pp. 309-347). NJ: World Scientific.
Guberman, R. y Leikin, R. (2013). Interesting and difficult mathematical problems: changing teachers’ views by employing multiple-solution tasks. Journal of Mathematics Teachers Education, 16, 33-56.
Henningsen, M. y Stein, M. K. (1997). Mathematical tasks and students cognition: Classroom-based factor that support and inhibit high level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524-549.
Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., Olivier, A. y Human, P. (1997). Making sense: teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.
Kieran, C. (2013). The false dichotomy in mathematics education between conceptual understanding and procedural skills: An example from algebra. En K. Leatham (Ed.), Vital directions in mathematics education research (pp. 153-171). Nueva York: Springer.
Lamon, S. J. (2006). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. Mahwah, NJ: Lawrence Erlbaum Associates.
Leikin, R. (2007). Habits of mind associated with advanced mathematical thinking and solution spaces of mathematical tasks. The Fifth Conference of the European Society for Research in Mathematics Education (CERME-5) (pp. 2330-2339).
Leikin, R. (2010). Learning through teaching through the lens of multiple solution tasks. En R. Leikin y R. Zaskis (Eds.), Learning through teaching mathematics: development of teachers’ knowledge and expertise in practice (pp. 69-85). Dordrecht, Holanda: Springer.
Leikin, R. (2011). Multiple-solution tasks: from a teacher education course to teacher practice. ZDM Mathematics Education, 43, 993-1006.
Leikin, R. (2014). Challenging mathematics with multiple solution tasks and mathematical investigations in geometry. En Y. Li, E. A. Silver y S. Li (Eds.), Transforming mathematics instruction: Multiple approaches and practices (pp. 59-80). Cham, Alemania: Springer.
Leikin, R. y Lev, M. (2007). Multiple solution tasks as a magnifying glass for observation of mathematical creativity. En J. H. Woo, H. C. Lew, K. S. Park y D. Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 161-168). Seúl: PME.
Leikin, R. y Levav-Waynberg, A. (2007). Exploring mathematics teacher knowledge to explain the gap between theory-based recommendations and school practice in the use of connecting tasks. Educational Studies in Mathematics, 66, 349-371.
Leikin, R. y Levav-Waynberg, A. (2008). Solution spaces of multiple-solution connecting tasks as a mirror of the development of mathematics teachers’ knowledge. Canadian Journal of Science, Mathematics and Technology Education, 8(3), 233-251.
Ma, L. (2010). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Nueva York: Routledge.
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Autor.
Polya, G. (1945). How to solve it: A new aspect of mathematical method. NJ: Princeton University Press.
Romberg, T. A. (1994). Classroom instruction that fosters mathematical thinking and problem solving: Connections between theory and practice. En A. H. Schoenfeld (Ed.), Mathematical thinking and problem solving (pp. 287-304). Hillsdale, NJ: Lawrence Erlbaum.
Rubio, C. J. (2006). Problemas para la 20a. Olimpiada Mexicana de Matemática en San Luis Potosí. Mérida: Universidad Autónoma de Yucatán.
Santos-Trigo, M. (1996). An exploration of strategies used by students to solve problems with multiple ways of solution. Journal of Mathematical Behavior, 15, 263-284.
Santos-Trigo, M. (1997). La transferencia del conocimiento y la formulación o rediseño de problemas en el aprendizaje de las matemáticas. Revista Mexicana de Investigación Educativa, 2(3), 11-30.
Santos-Trigo, M. (2007). La resolución de problemas matemáticos: fundamentos cognitivos. México: Trillas.
Santos-Trigo, M. (2009). Innovación e investigación en educación matemática. Innovación Educativa, 9(46), 5-13.
Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press.
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. En D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334-371). Nueva York: Macmillan.
Shimada, S. y Becker, J. (1997). The open ended approach: A new proposal for teaching mathematics. Reston, VA: NCTM.
Shulman, L. S. (2005). Conocimiento y enseñanza: fundamentos de la nueva reforma. Profesorado. Revista de currículum y formación del profesorado, 9(2), 1-30.
Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM Mathematics Education, 29(3), 75-80.
Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C. y Font Strawhum, B. T. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. Journal of Mathematical Behavior, 24, 287-301.
Silver, E. A. y Kenney, P. A. (1995). Sources of assessment information for instructional guidance in mathematics. En T. A. Romberg (Ed.), Reform in school mathematics and authentic assessment (pp. 38-68). Albany: State University of New York.
Simon, M. A. (1994). Learning mathematics and learning to teach: Learning cycles in mathematics teacher education. Educational Studies in Mathematics, 26, 71-94.
Simon, M. y Schifter, D. (1991). Towards a constructivist perspective: An intervention study of mathematics teacher development. Educational Studies in Mathematics, 22, 309-331.
Simon, M. A. y Tzur, R. (2004). Explicating the role of mathematical tasks in conceptual learning: an elaboration of the hypothetical learning trajectory. Mathematical Thinking and Learning, 6(2), 91-104.
Steen, L. A. (1988). The science of patterns. Science, 240, 611-616.
Stein, M. K. y Smith M. S. (1998). Mathematical tasks as a framework for reflection: From research to practice. Mathematics Teaching in the Middle School, 3, 268-275.
Stigler, J. y Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. Nueva York: The Free Press.
Tsamir, P., Tirosh, D., Tabach, D. y Levenson, E. (2010). Multiple solution methods and multiple outcomes– is it a task for kindergarten children? Educational Studies in Mathematics, 73, 217-231.
Yackel, E. y Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477.
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2017-01-10
