Secondary Students’ Implicit Conceptual Knowledge of Algebraic Symbolism. An Exploratory Study through Problem Posing

Referencias

Álvarez, I. & Gómez-Chacón, I. M. (2015). Understanding the algebraic variable: Comparative Study of Mexican and Spanish students. Eurasia Journal of Mathematics, Science & Technology Education, 11(6), 1507-1529 Arcavi, A. (1994). Symbol sense: informal sense-making in formal mathematics. For the Learning of Mathematics, 1(3), 24-35. Arcavi, A. (2005). Developing and using symbol sense in mathematics. For the learning of mathematics, 25(2), 42-47. Arnau, D. & 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. DOI: https://doi.org/10.5565/rev/ec/v31n3.967 Bills, L. (2001). Shifts in the Meanings of Literal Symbols. In M. Van den Heuvel-Panhuizen (Ed.), Proceedings of the Twenty-fifth PME International Conference (pp. 2-161, 2-168). Utrecht, The Neatherlands: PME. Booth, L.R. (1984). Algebra: Children’s strategies and errors. Windsor, UK: NFER-Nelson. Capraro, M. & Joffrion, H. (2006). Algebraic Equations: Can Middle-School Students Meaningfully Translate from Words to Mathematical Symbols? Reading Psychology, 27(2-3), 147-164. Carpenter, T. & Moser, J. (1982). The development of addition and subtraction problem solving skills. In T. Carpenter, J. Moser & T. Romberg (Eds.), Addition and subtraction: Developmental perspective (pp. 9-24). Hillsdale, N. Jersey: Lawrence Erlbaum Associates. Castro, E. (2001). Multiplicación y división. In E. Castro (Ed.), Didáctica de la matemática en educación primaria. Madrid, España: Síntesis, pp. 203-230. Castro, E. & Castro, E. (1997). Representaciones y modelización. In L. Rico (Ed.), La educación matemática en la enseñanza secundaria (pp. 95-124). Barcelona, España: Horsori. Castro, A., Prat, M. & Gorgorió, N. (2016). Conocimiento conceptual y procedimental en matemáticas: su evolución tras décadas de investigación. Revista de Educación, 374, 43-68 Cedillo, T. E. (2001). Toward an algebra acquisition support system: A study based on using graphic calculators in the classroom. Mathematical Thinking and Learning, 3, 221–259. Cerdán, F. (2010). Las igualdades incorrectas producidas en el proceso de traducción algebraico: un catálogo de errores. PNA, 4(3), 99-110. Chalouh, L. & Herscovics, N. (1988). Teaching algebraic expressions in a meaningful way. In A.F. Coxford & A.P. Shulte (Eds.), The ideas of algebra K-12 (pp. 33-42). Reston, Virginia: NCTM. Chappell, M. F. (2001). Creating connections: Promoting algebraic thinking with concrete models. Mathematics Teaching in the Middle School, 7, 20–25. Crooks, N. & Alibali, M. (2014). Defining and measuring conceptual knowledge in mathematics. Developmental Review, 34(4), 344-377 Fernández-Millán E. y Molina, M. (2016). Indagación en el conocimiento conceptual del simbolismo algebraico de estudiantes de secundaria mediante la invención de problemas. Enseñanza de las Ciencias, 34(1), 53-71. DOI: http://dx.doi.org/10.5565/rev/ensciencias.1455 Ferrucci, B. J., Kaur, B., Carter, J. A. & Yeap, B. (2008). Using a model approach to enhance algebraic thinking in the elementary school mathematics classroom. In C. E. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics: 2008 Yearbook (pp. 195–209). Reston, VA: National Council of Teachers of Mathematics. Filloy, E. & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the learning of Mathematics, 9(2), 19-25. Filloy, E., Rojano T. & Puig, L. (2008). Educational algebra. A theoretical and empirical approach. New York, NY: Springer. Fujii, T. & Stephens, M. (2008). Using number sentences to introduce the idea of variable. In C. E. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics: 2008 Yearbook (pp. 127–140). Reston, VA: National Council of Teachers of Mathematics. Furinghetti, F. & Paola, D. (1994). Parameters, unknowns and variables: a little difference? In J.P. da Ponte & J.F. Matos (Eds.), Proceedings of the XVIII International Conference for the Psychology of Mathematics (vol. 2, pp. 368-375). Lisboa, Portugal: Universidad de Lisboa. Gómez, P. (2007). Desarrollo del conocimiento didáctico en un plan de formación inicial de profesores de matemáticas de secundaria. PhD Thesis. Granada: Universidad de Granada. González-Calero, J. A., Arnau, D. & Puig, L. (2013). Dificultades en la construcción de nombres de cantidades durante la resolución algebraica de problemas verbales por estudiantes de primaria. In A. Berciano, G. Gutiérrez, A. Estepa, & N. Climent (Eds.), Investigación en Educación Matemática XVII (pp. 301-310). Bilbao, Spain: Sociedad Española de Investigación en Educación Matemática. Herscovics, N. & Kieran, C. (1980). Constructing meaning for the concept of equation. Mathematics Teacher, 73(8), pp. 572-580. Hiebert, J. & Lefevre, P. (1986). Conceptual and Procedural Knowledge in Mathematics: An Introductory Analysis. In J. Hiebert (Ed.), Conceptual and Procedural Knowledge: The Case of Mathematics (pp. 1-27). Hillsdale, NJ: Lawrence Erlbaum Associates. Hoch, M. & Dreyfus, T. (2005). Students’ difficulties with applying a familiar formula in an unfamiliar context. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th conference of the international group for the psychology of mathematics education (Vol. 3, pp. 145–152). Melbourne, Australia: University of Melbourne. Kaput, J. (1989). Linking representations in the symbolic systems of algebra. In S. Wagner & C. Kieran (Eds.), Research agenda for mathematics education: Research issues in the learning and teaching of algebra (pp. 167-194). Reston, VA: NCTM. Küchemann, D.E. (1981). Algebra. In K.M. Hart, M.L. Brown, D.E. Kuchemann, D. Kerslake, G. Ruddock & M. McCartney (Eds.), Children’s understanding of mathematics: 11-16 (pp. 102-119). London, Reino Unido: John Murray, Lin, P. J. (2004). Supporting teachers on designing problem-posing tasks as a tool of assessment to understand students' mathematical learning. In M. Hoines y A. Fuglestad (Eds.), Proceedings of the 28th annual meeting of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 257-264). Bergen, Noruega: Bergen University College. Mestre J. P. (2002). Probing adults' conceptual understanding and transfer of learning via problem posing. Journal of Applied Developmental Psychology, 23(1), 9-50. Mitchell, J. M. (2001). Interactions between natural language and mathematical structures: the case of “wordwalking”. Mathematical Thinking and Learning, 3(1), 29-52. Molina, M. (2014). Traducción del simbolismo algebraico al lenguaje verbal: indagando en la comprensión de estudiantes de diferentes niveles educativos. La Gaceta de la RSME, 17(3), 559-579. Molina, M., Rodríguez-Domingo, S., Cañadas, M.C. & Castro, E. (2016). Secondary School Student’s Errors in the Translation of Algebraic Statements. International Journal of Science and Mathematics Education, 13(3), 1-20. Ng, S. F. & 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. Orrantia, J., González, L.B. & Vicente, S. (2005). Un análisis de los problemas aritméticos en los libros de texto de Educación Primaria. Infancia y Aprendizaje, 28, 420-451. http://dx.doi.org/10.1174/021037005774518929 Resnick, L.B., Cauzinille-Marmeche, E. & Mathieu, J. (1987). Understanding algebra. In J.A. Sloboda & D. Rogers (Eds.), Cognitive process in mathematics (pp. 169-203). Oxford, Reino Unido: Clarendon Press. Rittle-Johnson, B. & Schneider, M. (2015). Developing Conceptual and Procedural Knowledge of Mathematics. In R.C. Kadosh & A. Dowker (Eds.), Oxford Handbook of Numerical Cognition (pp. 1118-1134). Oxford, UK: Oxford University Press Rittle-Johnson, B. & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99, 561–574. doi: 10.1037/0022–0663.99.3.561. Rodríguez-Domingo, S. (2015). Traducción entre los sistemas de representación simbólico y verbal: un estudio con alumnado que inicia su formación algebraica en secundaria. PhD thesis. Granada: Universidad de Granada. Ross, A. & Willson, V. (2012). The Effects of Representations, Constructivist Approaches, and Engagement on Middle School Students’ Algebraic Procedure and Conceptual Understanding. School, Science and Mathematics, 112(2), 117-128. Sheikhzade, M. (2008, July). Promoting skills of problem-posing and problem- solving in making a creative social studies classroom. Presented at 4th Global Conference, Oxford. Available at http://www.inter-disciplinary.net/ati/education/cp/ce4/Sheikhzade%20paper.pdf. Stoyanova, E. & Ellerton, N. F. (1996). A framework for research into students' problem posing in school mathematics. In P. C. Clarkson (Ed.), Technology in mathematics education (Proceedings of the 19th annual conference of the Mathematics Education Research Group of Australasia) (pp. 518–525). Melbourne: Mathematics Education Research Group of Australasia. Usiskin Z. (1988). Conceptions of School Algebra and Uses of Variables. In Coxford A.F. & Shulte A. P. (Eds.), The Ideas of Algebra K-12 (pp. 8-19). VA: National Council of Teachers of Mathematics. Vega-Castro, D., Molina, M. & Castro, E. (2012). Sentido estructural de estudiantes de bachillerato en tareas de simplificación de fracciones algebraicas que involucran igualdades notables. Relime, 15(2), 233–258. Wagner, S. (1981). An analytical framework for mathematical variables. In Equipe de Recherche Pedagogique (Eds.), Proceedings of the 5th International PME Conference (pp.165-170). Grenoble, France: PME.