Justificación de las propiedades distributiva y conmutativa del producto en primaria
Resumen
Se presenta una propuesta de innovación para la justificación de las propiedades distributiva y conmutativa del producto en Educación Primaria. Atendiendo a las nuevas directrices curriculares que fomentan el desarrollo del pensamiento algebraico, se plantean dos sesiones focalizadas en la comprensión conceptual de ambas propiedades en la construcción de las tablas de multiplicar en segundo ciclo. Para evaluar el impacto de la innovación, analizamos las justificaciones de 28 estudiantes (de 8 a 12 años) al entrevistarles sobre la generalización de las propiedades mediante experimentos cruciales y ejemplos genéricos. Los resultados muestran una mayor frecuencia de justificaciones aritméticas y una mayor complejidad en la propiedad distributiva, y evidencian el potencial en estas edades para elaborar justificaciones generales.
Palabras clave
Aritmética generalizada, Justificación, Propiedad conmutativa, Propiedad distributivaCitas
Anderson, L. W. & Krathwohl, D. R. (2001). A Taxonomy for Learning, Teaching and Assessing: A Revision of Bloom’s Taxonomy of Educational Objectives: Complete Edition. Longman.
Ayala-Altamirano, C. & Molina, M. (2021). El proceso de generalización y la generalización en acto. Un estudio de casos. PNA. Revista de Investigación en Didáctica de la Matemática, 15(3), 211-241. https://doi.org/10.30827/pna.v15i3.18109
Boaler, J. (1993). The Role of Contexts in the Mathematics Classroom: Do they Make Mathematics More «Real»? For the learning of mathematics, 13(2), 12-17.
Bosch, M., Gutierrez, A., & Llinares, S. (2024). A survey of Spanish research in mathematics education. ZDM - Mathematics Education. https://doi.org/10.1007/s11858-024-01638-z
Blanton, M., Levi, L., Crites, T., Dougherty, B., & Zbiek, R. M. (2011). Developing Essential Understanding of Algebraic Thinking for Teaching Mathematics in Grades 3-5. Series en Essential Understandings. National Council of Teachers of Mathematics. 1906 Association Drive, Reston, VA 20191-1502.
Blanton, M., Stephens, A., Knuth, E., Gardiner, A., Isler, I., & Kim, J. (2015). The development of children’s algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46(1), 39-87. https://doi.org/10.5951/jresematheduc.46.1.0039
Brown, S. I. (2013). Insights into mathematical thought: Excursions with distributivity. National Council of Teachers of Mathematics.
Burkhardt, H. & Swan, M. (2013). Task design for systemic improvement: Principles and frameworks. En C. Margolinas (Ed.), Task Design in Mathematics Education (The 22nd ICME study conference) (pp. 433-432). ICME.
Cañadas, M. C., Blanton, M., & Brizuela, B. M. (2019). Special issue on early algebraic thinking. Journal for the Study of Education and Development, 42(3), 469-478. https://doi.org/10.1080/02103702.2019.1638569
Campbell, T. G., King, S., & Zelkowski, J. (2020). Comparing middle grade students’ oral and written arguments. Research in Mathematics Education, 23(1), 21-38. https://doi.org/10.1080/14794802.2020.1722960
Carpenter, T. P. & Levi, L. (2000). Developing conceptions of algebraic reasoning in the primary grades. (Research Report No. 00-2). National Center for Improving Student Learning and Achievement in Mathematics and Science
Crooks, N. M. & Alibali, M. W. (2014). Defining and measuring conceptual knowledge in mathematics. Developmental Review, 34(4), 344-377. https://doi.org/10.1016/j.dr.2014.10.001.
Ellis, A. B. & Özgür, Z. (2024). Trends, insights, and developments in research on the teaching and learning of algebra. ZDM - Mathematics Education, 56(2), 199-210. https://doi.org/10.1007/s11858-023-01545-9
Hallgren, K. A. (2012). Cálculo de la fiabilidad interevaluador para datos observacionales: Resumen y tutorial. Tutoriales en Métodos Cuantitativos para Psicología, 8, 23-34. https://doi.org/10.20982/tqmp.08.1.p023
Kieran, C. (2022). The multi-dimensionality of early algebraic thinking: background, overarching dimensions, and new directions. ZDM - Mathematics Education, 54(6), 1131-1150. https://doi.org/10.1007/s11858-022-01435-6
Knuth, E., Choppin, J., & Bieda, K. (2009). Proof in middle school: Moving beyond examples. Mathematics Teaching in the Middle School, 15(4), 206-211.
Krathwohl, D. R. (2002). A Revision of Bloom´s Taxonomy: An Overview. Theory into practice, 41(4), 212-218.
Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and learning, 7(3), 231-258. https://doi.org/10.1207/s15327833mtl0703_3
Larsson, K. (2015). Sixth grade students’ explanations and justifications of distributivity. En K. Krainer & N. Vondrová (Eds.), Proceedings of the 9th Conference of the European society for research in mathematics education (pp. 295-301). ERME.
Marrades, R. & Gutiérrez, Á. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational Studies in Mathematics, 44(1), 87-125. https://doi.org/10.1023/A:1012785106627
Mayer, R. E. (2002). Rote Versus Meaningful Learning. Theory into practice, 41(4), 226-232.
Ministerio de Educación y Formación Profesional (MEFP) (2022). Real Decreto 217/2022, de 29 de marzo, por el que se establece la ordenación y las enseñanzas mínimas de la Educación Secundaria Obligatoria.
Pang, J. & Kim, J. (2018). Characteristics of Korean students’ early algebraic thinking: A generalized arithmetic perspective. En C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds, ICME-13 Monographs (pp. 141-165). Springer. http://dx.doi.org/10.1007/978-3-319-68351-5_6
Ramírez Uclés, R., Brizuela, B. M., & Blanton, M. (2022). Kindergarten and First-Grade Students’ Understandings and Representations of Arithmetic Properties. Early Childhood Education Journal, 50(2), 345-356. https://doi.org/10.1007/s10643-020-01123-8
Ramírez Uclés, R., Brizuela, B. M., & Blanton, M. (2024). Kindergarten and First-Grade Students’ Understandings of Arithmetic Properties Across Different Kinds of Problems. Canadian Journal of Science, Mathematics and Technology Education. https://doi.org/10.1007/s42330-024-00331-3
Rico, L. (2013). El método del Análisis Didáctico. Unión: Revista Iberoamericana de Educación Matemática, 1(33). 11-27.
Robinson, K. M., Price, J. A., & Demyen, B. (2018). Understanding arithmetic concepts: Does operation matter? Journal of Experimental Child Psychology, 166, 421-436. https://doi.org/10.1016/j.jecp.2017.09.003
Rocha, H. (2019). Mathematical proof: from mathematics to school mathematics. Philosophical Transactions of the Royal Society A, 377(2140), 20180045. http://dx.doi.org/10.1098/rsta.2018.0045
Sánchez, J. M. (2005). La innovación educativa institucional y su repercusión en los centros docentes de Castilla-La Mancha. REICE. Revista Iberoamericana sobre Calidad, Eficacia y Cambio en Educación, 3(1), 638-664.
Seah, R. & Horne, M. (2020). The construction and validation of a geometric reasoning test item to support the development of learning progression. Mathematics Education Research Journal, 32(4), 607-628. https://doi.org/10.1007/s13394-019-00273-2
Skemp, R. R. (2006). Relational understanding and instrumental understanding. Mathematics teaching in the middle school, 12(2), 88-95. https://doi.org/10.5951/MTMS.12.2.0088
Stein, M. K., Smith, M, Henningsen, M., & Silver, E. (2000). Implementing Standards-Based Mathematics Instruction. Teachers College Press.
Strachota, S., Brizuela, B., Gibbins, A., Blanton, M., Gardiner, A. M., & Sawrey, K. (2023). First Graders’ Definitions, Generalizations, and Justifications of Even and Odd Numbers. Canadian Journal of Science, Mathematics and Technology Education, 23(3), 459-478. https://doi.org/10.1007/s42330-023-00297-8
Stylianides, G. J., Stylianides, A. J., & Moutsios-Rentzos, A. (2024). Proof and proving in school and university mathematics education research: a systematic review. ZDM - Mathematics Education, 56(1), 47-59. https://doi.org/10.1007/s11858-023-01518-y
Tsai, Y. L. & Chang, C. K. (2009). Using combinatorial approach to improve students’ learning of the distributive law and multiplicative identities. International Journal of Science and Mathematics Education, 7(3), 501-531. https://doi.org/10.1007/s10763-008-9135-x
Vermeulen, E. M., Spronk, J., & van der Wijst, N. (1993). A new approach to firm evaluation. Annals of Operations Research, 45(1), 387-403. https://doi.org/10.1007/BF02282060
Vermeulen, N., Olivier, A., & Human, P. (1996). Students’ awareness of the distributive property. En L. Puig & A. Gutiérrez (Eds.), PME20: proceedings of the 20th Confference of the International Group of the Psychology for Mathematics Education, 4, 379-386.
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Agencia Estatal de Investigación
Números de la subvención PID2020-113601GB-I00
Derechos de autor 2025 Rafael Ramírez Uclés, Sandra Fuentes, Mathías A. López, Bárbara M. Brizuela
Esta obra está bajo una licencia internacional Creative Commons Atribución 4.0.
