A Comparison of Algebraic Resolutions of Age Problems Using and Without Using Tables
Abstract
This article studies the similarities and differences, when algebraically solving age problems, in the construction of the problem model, and the errors in the equations and its underlying processes, depending on whether the solver uses, or not, tables. The discourse, gestures, and written outputs of eight pairs of secondary school students were analysed by means of qualitative techniques. The results show that as long as students used tables, they managed to construct a problem model, but when they did not, in some cases they conducted a merely syntactic strategy. The errors in the equations were analogous in both solving approaches –with and without using tables– and the fragmented translation from the natural to the algebraic language was an underlying process to the errors in both.
Keywords
Secondary education, Algebraic language, Algebraic solving, TableReferences
Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. Learning and Instruction, 16, 183-198. https://doi.org/10.1016/j.learninstruc.2006.03.001
Arnau, D. y Puig, L. (2013). Actuaciones de alumnos instruidos en la resolución algebraica de problemas en la hoja de cálculo y su relación con la competencia en el método cartesiano. Enseñanza de las Ciencias, 31(3), 49-66. https://doi.org/10.5565/rev/ec/v31n3.967
Arzarello, F., Paola, D., Robutti, O. y Sabena, C. (2009). Gestures as semiotic resources in the mathematics classroom. Educational Studies in Mathematics, 70, 97-109. https://doi.org/10.1007/s10649-008-9163-z
Bednarz, N., Kieran, C. y Lee, L. (Eds.) (1996). Approaches to algebra: Perspectives for research and teaching. Kluwer Academic Publishers.
Bieda, K. N. y Nathan, M. J. (2009). Representational disfluency in algebra: evidence from student gestures and speech. ZDM Mathematics Education, 41, 637-650. https://doi.org/10.1007/s11858-009-0198-0
Bloedy-Vinner, H. (1996). The analgebraic mode of thinking and other errors in word problem solving. En L. Puig y A. Gutiérrez (Eds.), Proceedings of the 20th International Conference for the Psychology of Mathematics Education (pp. 105-112). International Group for the Psychology of Mathematics Education.
Cerdán, F. (2010). Las igualdades incorrectas producidas en el proceso de traducción algebraico: un catálogo de errores. PNA, 4(3), 99-110. https://doi.org/10.30827/pna.v4i3.6164
Clement, J. (1982). Algebra word problem solutions: Thought processes underlying a common misconception. Journal for Research in Mathematics Education, 13(1), 16-30. https://doi.org/10.2307/748434
Ding, M., Chen, W. y Hassler, R. S. (2019). Linear quantity models in US and Chinese elementary mathematics classrooms. Mathematical Thinking and Learning, 21(2), 105-130. https://doi.org/10.1080/10986065.2019.1570834
Ding, M. y Li, X. (2014). Transition from concrete to abstract representations: The distributive property in a Chinese textbook series. Educational Studies in Mathematics, 87, 103-121. https://doi.org/10.1007/s10649-014-9558-y
Duval, R. (2017). Understanding the mathematical way of thinking - The Registers of semiotic representations. Springer International Publishing.
Ernest, P. (2006). A semiotic perspective of mathematical activity: the case of number. Educational Studies in Mathematics, 61, 67-101. https://doi.org/10.1007/s10649-006-6423-7
Filloy, E., Puig, L. y Rojano, T. (2008). El estudio teórico local del desarrollo de competencias algebraicas. Enseñanza de las Ciencias, 25(3), 327-342. https://doi.org/10.5565/rev/ensciencias.3746
Filloy, E., Rojano, T. y Puig, L. (2008). Educational algebra. A theoretical and empirical approach. Springer.
Filloy, E., Rojano, T. y Solares, A. (2010). Problems Dealing with Unknown Quantities and Two Different Levels of Representing Unknowns. Journal for Research in Mathematics Education, 41(1), 52-80. https://doi.org/10.5951/jresematheduc.41.1.0052
Kaput, J. J. (1989). Toward a theory of symbol use in Mathematics. En C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 159-196). Lawrence Erlbaum Associates, Inc.
Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. En F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707-762). Information Age Publishing.
Kintsch, W. (1998). Comprehension. A paradigm for cognition. Cambridge University Press.
McNeil, N. M. y Fyfe, E. R. (2012). «Concreteness fading» promotes transfer of mathematical knowledge. Leaning and Instruction, 22(6), 440-448. https://doi.org/10.1016/j.leaminstruc.2012.05.001
Ministerio de Educación, Cultura y Deporte (2014). Real Decreto 1105/2014, por el que se esta¬blece el currículo básico de la Educación Secundaria Obligatoria y del Bachillerato. Boletín Oficial del Estado, 3, 169-546.
Molina, M., Rodríguez-Domingo, S., Cañadas, M. C. y Castro, E. (2017). Secondary school students’ errors in the translation of algebraic statements. International Journal of Science and Mathematics Education, 15(6), 1137-1156. https://doi.org/10.1007/s10763-016-9739-5
Nathan, M. J., Kintsch, W. y Young, E. (1992). A theory of algebra-word-problem comprehension and its implications for the design of learning environments. Cognition and Instruction, 9(4), 329-389. https://doi.org/10.1207/s1532690xci0904_2
Neuman, Y. y Schwarz, B. (2000). Substituting one mystery for another: the role of self-explanations in solving algebra word-problems. Learning and Instruction, 10, 203-220. https://doi.org/10.1016/S0959-4752(99)00027-4
Ng, S. y Lee, K. (2009). The Model Method: Singapore Children’s Tool for Representing and Solving Algebraic Word Problems. Journal for Research in Mathematics Education, 40(3), 282-313. https://doi.org/10.5951/jresematheduc.40.3.0282
Peräkylä, A. (2005). Analysing talk and text. En N. K. Denzin y Y. S. Lincoln (Eds.), The SAGE handbook of qualitative research (pp. 869-886). SAGE Publications.
Phan, H. P., Ngu, B. H. y Yeung, A. S. (2017). Achieving optimal best: Instructional efficiency and the use of cognitive load theory in mathematical problem solving. Educational Psychology Review, 29(4), 667-692. https://doi.org/10.1007/s10648-016-9373-3
Radford, L. (2000). Signs and meanings in students’ emergent algebraic thinking: a semiotic analysis. Educational Studies in Mathematics, 42, 237-268. https://doi.org/10.1023/A:1017530828058
Radford, L. (2003). Gestures, speech, and the sprouting of signs. Mathematical Thinking and Learning, 5(1), 37-70. https://doi.org/10.1207/S15327833MTL0501_02
Radford, L., Bardini, C. y Sabena, C. (2007). Perceiving the general: The multisemiotic dimension of students’ algebraic activity. Journal for Research in Mathematics Education, 38(5), 507-530. https://doi.org/10.2307/30034963
Rellensmann, J., Schukajlov, S. y 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, 53-78. https://doi.org/10.1007/s10649-016-9736-1
Sanjosé, V., Solaz-Portolés, J. J. y Valenzuela, T. (2009). Transferencia inter-dominios en resolución de problemas: una propuesta instruccional basada en el proceso de traducción algebraica. Enseñanza de las Ciencias, 27(2), 169-184. https://doi.org/10.5565/rev/ensciencias.3729
Schoenfeld, A. H. (1985). Mathematical problem solving. Academic Press.
Soneira, C. (2022). The Use of Representations when Solving Algebra Word Problems and the Sources of Solution Errors. International Journal of Science and Mathematics Education, 20(5), 1037-1056. https://doi.org/10.1007/s10763-021-10181-2
Soneira, C., González-Calero, J. A. y Arnau, D. (2018). Indexical Expressions in Word Problems and their Influence on Multiple Referents of the Unknown. International Journal of Science and Mathematics Education, 16(6), 1147-1167. https://doi.org/10.1007/s10763-017-9824-4
Stacey, K. y MacGregor, M. (2000). Learning the Algebraic Method of Solving Problems. The Journal of Mathematical Behavior, 18(2), 149-16. https://doi.org/10.1016/S0732-3123(99)00026-7
Uesaka, Y. y Manalo, E. (2012). Task-Related Factors that Influence the Spontaneous Use of Diagrams in Math Word Problems. Applied Cognitive Psychology, 26(2), 251-260. https://doi.org/10.1002/acp.1816
Recepción: septiembre 2021 • Aceptación: enero 2023 • Publicación: marzo 2023
Soneira,C. (2023). Comparación de resoluciones algebraicas con y sin tablas de problemas de edades. Enseñanza de las Ciencias, 41(1), 43-61. https://doi.org/10.5565/rev/ensciencias.5546
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