Meanings given to algebraic symbolism in problem posing

Referencias

Arcavi, A. (1994). Symbol sense: Informal sense–making in formal mathematics. For the Learning of Mathematics, 1(3), 24-35. Arcavi, A. (2006). El desarrollo y el uso del sentido de los símbolos [The development and use of the sense of symbols]. In I. Vale, T. Pimentel, A. Barbosa, L. Fonseca, L. Santos, & P. Canavarro (Eds.), Números e álgebra na aprendizagem da Matemática e na formaçâo de profesores (pp. 29-47). Caminha, Portugal: Sociedade Portugesa de Ciências de Eduacaçâo. Bonotto, C., & Dal Santo, L. (2015). On the relationship between problem posing, Problem solving, and creativity in the primary school. In F. M. Singer, N. F. Ellerton, & J. Cai (Eds), Mathematical problem posing. From research to effective practice (pp. 103-123). New York, NY: Springer. Booth, L. R. (1982). Ordering your operations. Mathematics in School, 11(3), 5-6. Bossé, M. J., Adu-Gyamfi, K., & Cheetham, M. R. (2011a). Assessing the difficulty of mathematical translations: Synthesizing the literature and novel findings. International Electronic Journal of Mathematics Education, 6(3), 113-133. Bossé, M. J., Adu-Gyamfi, K., & Cheetham, M. R. (2011b). Translations among mathematical representations: Teacher beliefs and practices. International Journal of Mathematics Teaching and Learning, June. Retrieved from http://www.cimt.plymouth.ac.uk/journal/bosse4.pdf Brown, S., & Walter, M. (2005). The art of problem posing. The 3rd addition. New York, NY: Routledge. Cai, J. (1998). An investigation of U.S. and Chinese students’ mathematical problem posing and problem solving. Mathematics Education Research Journal, 10, 37-50. Cai, J., & Hwang, S. (2002). Generalized and generative thinking in US and Chinese students’ mathematical problem solving and problem posing. Journal of Mathematical Behavior, 21(4), 401-421. Cai, J., Hwang, S., Jiang, C., & Silber, S. (2015). Problem-posing research in mathematics education: Some answered and unanswered questions. In F. M. Singer, N. F. Ellerton, & J. Cai (Eds), Mathematical problem posing. From research to effective practice (pp. 3-39). New York, NY: Springer. Cañadas, M. C., & Figueiras, L. (2011). Uso de representaciones y generalización de la regla del producto [Use of representations and generalization in the product principle]. Infancia y Aprendizaje, 34(4), 409-425. Castro, E. (2011). La invención de problemas y sus ámbitos de investigación [Problem posing and their research areas]. In J. L. Lupiáñez, M. C. Cañadas, M. Molina, M. Palarea, & A. Maz (Eds.), Investigaciones en Pensamiento Numérico y Algebraico e Historia de la Matemática y Educación Matemática – 2011 (pp. 1-16). Granada, Spain: Dpto. Didáctica de la Matemática, Universidad de Granada. Retrieved from http://funes.uniandes.edu.co/2015/ Castro, E. (2012). Dificultades en el aprendizaje del álgebra escolar [Difficulties in school algebra learning]. In A. Estepa, A. Contreras, J. Deulofeu, M. C. Penalva, F. J. García, & L. Ordóñez (Eds.), Investigación en Educación Matemática XVI (pp. 75-94). Baeza, Spain: SEIEM. Castro, E. (Ed.) (2001). Didáctica de la matemática en la Educación primaria [Mathematics Education on Elementary Education]. Madrid, Spain: Síntesis. Castro, E., Rico, L., & Gil, F. (1992). Enfoques de investigación en problemas verbales aritméticos aditivos [Approaches to research in additive verbal problems]. Enseñanza de las Ciencias, 10(3), 243-253. Cerdán, F. (2010). Las igualdades incorrectas producidas en el proceso de traducción algebraico: un catálogo de errores [The incorrect equalities developed in the algebraic translation process: A catalog of errors]. PNA 4(3), 99-110. Dede, Y. (2005). Interpretation of the first-degree equations: A study on freshmen students in education faculty. Cumhuriyet University Social Sciences Journal, 29(2), 197-205. Fernández-Millán, E., & Molina, M. (2016). Indagación en el conocimiento conceptual del simbolismo algebraico de estudiantes de secundaria mediante la invención de problemas [Investigating secondary students’ conceptual knowledge through problema posing]. Enseñanza de las Ciencias, 34(1), 53-71. Goldin, G. A. (1998). Representational systems, learning, and problem solving in mathematics. Journal of Mathematical Behavior, 17(2), 137-165. Goldin, G. A., & Mcclintock, C. E. (Eds.) (1980). Task variables in mathematical problem solving. Pensilvania, PA: The Franklin Institute Press. Greer, B. (1992). Multiplication and division as models of situations. In D. A. Grouws (Ed.), Handbook for research on mathematics teaching and learning (pp. 276-295). New York, NY: Macmillan. Heller, J. I., & Greeno, J. G. (1979). Information processing analysis of mathematical problem solving. In R. W. Tyler & S. H. White (Eds.), Testing, teaching, and learning: Report of a conference on research on testing. Washington, National Institute of Education. Isik, C., & Kar, T. (2012). The analysis of the problems the pre-service teachers experience in posing problems about equations. Australian Journal of Teacher Education, 37(9), Article 6. Available at: http://ro.ecu.edu.au/ajte/vol37/iss9/6 Kaput, J. (1987). Representation and mathematics. En C. Janvier (Ed.), Problems of representation in the learning of mathematics (pp. 19-26). Hillsdale, NJ: Lawrence Earlbaum Associates. Kieran, C. (2006). Research on the learning and teaching of algebra. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 11-50). Rotterdam, The Netherlands: Sense. Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning. Nueva York, NJ: NCTM. Kilpatrick, J. (1978). Variables and methodologies in research on problem solving. In L. L. Hatfield & D. A. Bradbard (Eds.), Mathematical problem solving: Papers from a research workshop (pp. 7-20). Columbus, OH: ERICISMEAC. Kirshner, D. (1989). The visual syntax of algebra. Journal for Research in Mathematics Education, 20, 274-289. Koichu, B. & Kontorovich I. (2012). Dissecting success stories on mathematical problem posing: A case of the billiard task. Educational Studies in Mathematics, 83(1), 71-86. Leikin, R. (2015). Problem posing for and through investigations in a dynamic geometry environment. In F. M. Singer, N. Ellerton, & J. Cai (Eds.), Problem posing: From research to effective practice (pp. 373-391). Dordrecht, The Netherlands: Springer. MacGregor, M. & Stacey, K. (1993). Cognitive models underlying students' formulation of simple linear equations. Journal for Research in Mathematics Education, 24(3), 217-232. Marshall, S. P. (1995). Schemas in problem solving. New York, NY: Cambridge University Press. Ministerio de Educación y Ciencia (2015). Real Decreto 1105/2014, de 26 de diciembre, por el que se establece el currículo básico de la Educación Secundaria Obligatoria y del Bachillerato [Royal Decree that establishes the basic curriculum of secondary education]. BOE, 3, 169-546. Molina, M. (2009). Una propuesta de cambio curricular: integración del pensamiento algebraico en educación primaria [A proposal of curricular change: Integration of algebraic thinking in elementary education]. PNA, 3(3), 135-156. Molina, M., Rodríguez-Domingo, S., Cañadas, M. C., & 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 National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author. Ngah, N., Ismail, Z., Tasir, Z., Said, M., & Haruzuan, M. N. (2016). Students’ ability in free, semi-structured and structured problem posing situations. Advanced Science Letters, 22(12), 4205-4208. OECD (2016). PISA 2015 Assessment and analytical framework: Science, reading, mathematic and financial literacy, PISA, OECD Publishing, Paris. http://dx.doi.org/10.1787/9789264255425-en Orrantia, J., González, L. B., & Vicente, S. (2005). Un análisis de los problemas aritméticos en los libros de texto de Educación Primaria [An analysis of the arithmetic problems in textbooks of Elementary Education]. Infancia y Aprendizaje, 28, 420-451. Ponte, J. P., & Henriques, A. (2013). Problem posing based on investigation activities by university students. Educational Studies in Mathematics, 83(1), 145-156 Puig, L. (1996). Elementos de resolución de problemas [Elements on problem solving]. Granada, Spain: Comares. 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. 1102-1118). Oxford, United Kingdom: Oxford University Press. Resnick, L. B., Cauzinille-Marmèche, E., Mathieu, J. (1987). Understanding algebra. In J. A. Sloboda & D. Rogers (Eds), Cognitive processes in mathematics (pp. 169-203). Oxford, United Kingdom: Clarendon Press. Rodríguez-Domingo, S., & Molina, M. (2013). De lo verbal a lo simbólico: un paso clave en el uso del álgebra como herramienta para la resolución de problemas y la modelización matemática [From verbal to symbolic: A key step in the use of algebra as a tool for problem solving and mathematical modeling]. In L. Rico, M. C. Cañadas, J. Gutiérrez, M. Molina, & I. Segovia (Eds.), Investigación en Didáctica de la Matemática. Homenaje a Encarnación Castro (pp. 111-118). Granada, Spain: Comares. Rodríguez-Domingo, S., Molina, M., Cañadas, M. C., & Castro, E. (2015). Errores en la traducción de enunciados algebraicos entre los sistemas de representación simbólico y verbal [Errors in the translation of algebraic statements between symbolic and verbal representation systems]. PNA, 9(4), 273-293. Ruano, R. M., Socas, M., & Palarea, M. M. (2008). Análisis y clasificación de errores cometidos por alumnos de secundaria en los procesos de sustitución formal, generalización y modelización en álgebra [Secondary students’ error analysis and classification in formal substitution, generalization and modelling process in algebra]. PNA, 2(2), 61-74. Schmidt, S., & Weiser, W. (1995). Semantic structures of one-step word problems involving multiplication or division. Educational Studies in Mathematics, 28(1), 55-7. Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics 14, 19-28. Silver, E. A., & Cai, J. (1996). An analysis of arithmetic problem posing by middle school students. Journal for Research in Mathematics Education, 27, 521-539. Silver, E. A., Mamona-Downs, J., Leung, S., & Kenney, P. A. (1996). Posing mathematical problems: An exploratory study. Journal for Research in Mathematics Education, 27(3), 293-309. 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. Star, J. (2005). Re-«conceptualizing» procedural knowledge in mathematics. Journal for Research in Mathematics Education, 36(5), 404-411. Stephens, A. (2003). Another look at words problems, The Mathematics Teacher, 96(1), 63-66. Stoyanova, E., & Ellerton, N. F. (1996). A framework for research into students’ problem posing. In P. Clarkson (Ed.), Technology in mathematics education (pp. 518-525). Melbourne, Australia: Mathematics Education Research Group of Australasia. Van Harpen, X. Y. & Presmeg, N. C. (2013). An investigation of relationships between students’ mathematical problem-posing abilities and their mathematical content knowledge. Educational Studies in Mathematics, 83, 117-132. Wheeler, D. (1989). Contexts for research on the teaching and learning of algebra. In S. Wagner & C. Kieran (Eds.), Research issues in the learning and teaching of algebra (pp. 278-287). Reston, VA: Lawrence Erlbaum Associates.