Representation in the solution of mathematical problems: an analysis of metacognitive strategies of secondary education students
DOI:
https://doi.org/10.15359/ru.34-1.15Keywords:
problem solving, representation records, mathematics teaching, secondary education, representation register, metacognitionAbstract
This article seeks to show the importance of solving verbal mathematics problems, and the role of metacognitive regulation in this process. To do so, awareness will be raised in secondary school teachers regarding the use of problem-solving as an exploratory resource in the metacognitive strategies applied by students, which teachers may use as input to modify methodologies in the classroom, both individually and collectively. A theoretical framework focused on the use of problem solving is provided as a didactic resource, paying special attention to the representations students use during problem solving to facilitate their understanding of a problem. Quasi-experimental research was carried out on a non-random sample of 99 students from the first and third years of secondary school, who solved problems with guidance focused on handling numerical and geometrical concepts. The results show clear differences in the metacognitive strategies used by students during the solution of these types of problems; teachers must therefore adapt their guidance strategies based on the mathematical content addressed.
References
Arteaga, B. & Macías, J. (2016). La representación en la resolución de problemas matemáticos como diagnóstico de estrategias metacognitivas. En F. España (ed.) XVI Congreso de Enseñanza y Aprendizaje de las Matemáticas (pp. 118-126). Cádiz, España: Sociedad Andaluza de Educación Matemática THALES. Recuperado de https://thales.cica.es/xviceam/actas/pdf/actas.pdf
Artigue, M. & Blomhøj, M. (2013). Conceptualizing inquiry-based education in mathematics. ZDM Mathematics Education, 45(6), 797-810. doi: https://doi.org/10.1007/s11858-013-0506-6
Blanco, L. & Cárdenas, J. A. (2013). La resolución de problemas como contenido en el currículo de primaria y secundaria. Campo Abierto, 32(1), 137-156. Recuperado de https://goo.gl/3pnaQF
Coleoni, E. & Buteler, L. (2008). Recursos metacognitivos durante la resolución de un problema de Física. Investigações em Ensino de Ciências, 13(3), 371-383. Recuperado de https://www.if.ufrgs.br/cref/ojs/index.php/ienci/article/view/447/265
Contreras, J. & Del Pino, C. (2007). Resolución de problemas en contextos matemáticos. Unión. Revista iberoamericana de Educación Matemática, 12, 27-36. Recuperado de http://www.fisem.org/www/union/revistas/2007/12/Union_012_005.pdf
Crespo, S. (2003). Learning to pose mathematical problems: exploring changes in preservice teachers’ practices. Educational Studies in Mathematics, 52(3), 243-270. doi: https://doi.org/10.1023/A:1024364304664
Desoete, A. (2007). La evaluación y mejora del proceso de enseñanza-aprendizaje de las matemáticas a través de la metacognición. Revista Electrónica de Investigación Psicoeducativa, 5(13), 705-730. doi: https://doi.org/10.25115/ejrep.v5i13.1243
Domenech, M. (2004). El papel de la inteligencia y de la metacognición en la resolución de problemas (Tesis doctoral). Universidad Rovira I Virgili. Tarragona (España). https://www.tdx.cat/handle/10803/8958
Duval, R. (1993). Registres de représentation sémiotique et fonctionnement cognitif de la pensée. Annales de Didactique et de Science Cognitives, 5, 37-65. Traducción: Registros de representación semiótica y funcionamiento cognitivo del pensamiento. E. Hitt, (Ed.) Investigaciones en Matemática Educativa II (pp. 173-201). México: Grupo Editorial Iberoamérica.
Duval, R. (2004). Como hacer que los alumnos entren en las representaciones geométricas. Cuatro entradas y... una quinta. En AA.VV. Números, formas y volúmenes en el entorno del niño (pp. 159-187). Madrid: Ministerio de Educación y Ciencia.
Duval, R. (2016). Un análisis cognitivo de problemas de comprensión en el aprendizaje de las matemáticas. Comprensión y aprendizaje en matemáticas: perspectivas semióticas seleccionadas (pp. 61-94). Colombia: Universidad Distrital Francisco José de Caldas.
García, P.; San José, V. & Solaz-Portolés, J. J. (2015). Efectos de las características del problema, captación de su estructura y uso de analogías sobre el éxito de los estudiantes de secundaria en la resolución de problemas. Teoría de la Educación. Revista Interuniversitaria, 27(2), 221-244. doi: http://dx.doi.org/10.14201/teoredu2015272221244
Gaulin, C. (2001). Tendencias actuales de la resolución de problemas. Sigma, 19, 51-63. Recuperado de http://www.hezkuntza.ejgv.euskadi.eus/r43-573/es/contenidos/informacion/dia6_sigma/es_sigma/adjuntos/sigma_19/7_Tendencias_Actuales.pdf
Elosua, M. R. (1993). Estrategias para enseñar y aprender a pensar. Madrid: Ediciones Narcea.
Ellerton, N. F. (2013). Engaging pre-service middle-school teacher-education students in mathematical problem posing: development of an active learning framework. Educational Studies in Mathematics, 83(1), 87-101. doi: https://doi.org/10.1007/s10649-012-9449-z
English, L. D. & Gainsburg, J. (2016). Problem solving in a 21st-century mathematics curriculum. In L. D. English & D. Kirshner (Eds.), Handbook of international research in mathematics education (3rd ed., pp. 313–335). New York: Taylor & Francis.
Flavell, J. H. (1999). Cognitive development: Children’s knowledge about the mind. Annual Review of Psychology, 50, 21-45. doi: https://doi.org/10.1146/annurev.psych.50.1.21
Kapa, E. (2007). Transfer from structured to open-ended problem solving in a computerized metacognitive environment. Learning and Instruction, 17, 688-707. doi: https://doi.org/10.1016/j.learninstruc.2007.09.019
Kramarski, B., Mevarech, Z. R. & Arami, M. (2002). The effects of metacognitive instruction on solving mathematical authentic tasks. Educational studies in mathematics, 49(2), 225-250. doi: https://doi.org/10.1023/A:1016282811724
Lampert, M. (1990). When the Problem Is Not the Question and the Solution Is Not the Answer: Mathematical Knowing and Teaching. American Educational Research Journal, 27(1), 29-63. doi: https://doi.org/10.3102/00028312027001029
Leikin, R. & Grossman, D. (2013). Teachers Modify Geometry Problems: From proof to investigation. Educational Studies in Mathematics, 82(3), 515-531. doi: https://doi.org/10.1007/s10649-012-9460-4
Lesh, R. & Zawojewski, J. S. (2007). Problem solving and modeling. In F. Lester (Ed.), The Second Handbook of Research on Mathematics Teaching and Learning (pp. 763-804). Charlotte, NC: Information Age Publishing.
Lester, F. K. (1983). Trends and issues in mathematical problem-solving research. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 229-261). Orlando, FL: Academic Press.
Lester, F. K. & Kehle, P. E. (2003). From problem solving to modeling: The evolution of thinking about research on complex mathematical activity. En R. Lesh y H. M. Doerr (Eds.), Beyond constructivism. Models and modeling perspectives on mathematical problem solving, learning, and teaching (pp. 501-517). Mahwah, NJ: Lawrence Erlbaum Associates.
Loh, M. Y. & Lee, N. H. (2019). The Impact of Various Methods in Evaluating Metacognitive Strategies in Mathematical Problem Solving. In P. Liljedahl & M. Santos-Trigo (Ed.). Mathematical Problem Solving (pp. 155-176). Cham: Springer.
National Council of Teachers of Mathematics NCTM (2000). Principles and Standards for School Mathematics. Reston: NCTM.
Otani, H. & Widner, R. L. (2005). Metacognition: New Issues and Approaches Guest Editors' Introduction. The Journal of General Psychology, 132(4), 329-334. doi: https://doi.org/10.3200/GENP.132.4.329-334
Özsoy, G. & Ataman, A. (2009). The effect of metacognitive strategy training on mathematical problem solving achievement. International Electronic Journal of Elementary Education, 1(2), 67-82. doi: https://files.eric.ed.gov/fulltext/ED508334.pdf
Pelczer, I. & Gamboa, F. (2008). Problem posing strategies of mathematically gifted students. En R. Leikin (Ed.). Proccedings of the 5th International Conference on Creativity in Mathematics and the Education of Gifted Students (pp. 193-199). Tel Aviv: Center for Educational Technology.
Peñalva, L. P. (2010). Las matemáticas en el desarrollo de la metacognición. Política y cultura, 33, 135-151. Recuperado de https://polcul.xoc.uam.mx/index.php/polcul/article/view/1112/1087
Pifarré, M. & Sanuy, J. (2001). La enseñanza de estrategias de resolución de problemas matemáticos en la ESO: un ejemplo concreto. Revista Enseñanza de las ciencias, revista de investigación y experiencias didácticas, 19(2), 297-308. Recuperado de https://www.raco.cat/index.php/Ensenanza/article/view/21745/21579
Puente, A. (1993). Modelos mentales y habilidades en la solución de problemas aritméticos verbales. Revista de Psicología General y Aplicada, 46(2), 149-160.
Rellensmann, J.; Schukajlow, S. & Leopold, C. (2017). Make a drawing. Effects of strategic knowledge, drawing accuracy, and type of drawing on students’ mathematical modelling performance. Educational Studies in Mathematics, 95(1), 53-78. doi: https://doi.org/10.1007/s10649-016-9736-1
Rigo, M.; Paez, D. A. & Gómez, B. (2010). Prácticas metacognitivas que el profesor de nivel básico promueve en sus clases ordinarias de matemáticas. Un marco interpretativo. Enseñanza de las Ciencias 28(3), 405-416. Recuperado de https://www.raco.cat/index.php/Ensenanza/article/view/210808/353417
Rodríguez, E. (2005). Metacognición, resolución de problemas y enseñanza de las matemáticas. Una propuesta integradora desde un enfoque antropológico (Tesis doctoral). Universidad Complutense de Madrid. Recuperado de https://eprints.ucm.es/7256/1/T28687.pdf
Rowland, T.; Huckstep, P. & Thwaites, A. (2003). Observing Subject Knowledge in Primary Mathematics Teaching. Proceedings of the British Society for Research into Learning Mathematics, 23(1), 37-42. Recuperado de http://www.skima.maths-ed.org.uk/BSRLMNotts16Nov03.pdf
Santos, D. A. & Lozada, G. A. (2013). ¿Es posible hacer evidentes los procesos de metacognición en la resolución de problemas, fase 2? Revista Científica (especial), 42-45. doi: https://doi.org/10.14483/23448350.5482
Schoenfeld, A. H. (1987). What's all the fuss about metacognition? En A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 189-215). Hillsdale, NJ: Lawrence Erlbaum Associates.
Silva, C. (2004). Educación en matemática y procesos metacognitivos en el aprendizaje. Revista del Centro de Investigación Universidad La Salle, 7, 81-91.
Singer, F. M.; Ellerton, N. & Cai, J. (2013). Problem-posing research in mathematics education: New questions and directions. Educational Studies in Mathematics, 83(1), 1-7. doi: https://doi.org/10.1007/s10649-013-9478-2
Stacey, K. (2005). The place of problem solving in contemporary mathematics curriculum documents. The Journal of Mathematical Behavior, 24(3-4), 341-350. doi: https://doi.org/10.1016/j.jmathb.2005.09.004
Stanic, G. & Kilpatrick, J. (1989). Historical perspectives on problem solving in the mathematics curriculum. En R. Charles y E. Silver (Eds.), The teaching and assessing of mathematical problem solving (pp. 1-22). Reston, VA: NCTM.
Sternberg, R. J. (1988). Intelligence. En R. J. Sternberg y E. E. Smith, The psychology of Human Thought. Cambridge. Cambridge University Press.
Swanson, H. L. (1990). Influence of metacognitive knowledge and aptitude on problem solving. Journal of Educational Psychology, 82(2), 306-314. doi: http://dx.doi.org/10.1037/0022-0663.82.2.306
Vila, A. & Callejo, M. L. (2004). Matemáticas para aprender a pensar. El papel de las creencias en la resolución de problemas. Madrid: Narcea.
Downloads
Published
Issue
Section
License
Authors who publish with this journal agree to the following terms:
1. Authors guarantee the journal the right to be the first publication of the work as licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
2. Authors can set separate additional agreements for non-exclusive distribution of the version of the work published in the journal (eg, place it in an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
3. The authors have declared to hold all permissions to use the resources they provided in the paper (images, tables, among others) and assume full responsibility for damages to third parties.
4. The opinions expressed in the paper are the exclusive responsibility of the authors and do not necessarily represent the opinion of the editors or the Universidad Nacional.
Uniciencia Journal and all its productions are under Creative Commons Atribución-NoComercial-SinDerivadas 4.0 Unported.
There is neither fee for access nor Article Processing Charge (APC)
