Generalization process by second grade students
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Lista de autores
Torres, María D., Moreno, Antonio y Cañadas, María C.
Resumen
This study is part of a broader study on algebraic reasoning in elementary education. The research objective of the present survey, namely to describe generalization among second grade (7- to 8-year-old) students, was pursued through semi-structured interviews with six children in connection with a contextualized generalization task involving the function y = x + 3. Particular attention was paid to the structures recognized and the type of generalization expressed by these students as they reasoned. In all six, we observed three phases of inductive reasoning: (a) abductive, (b) inductive and (c) generalization. The students correctly recognized the structure at least once during the interview and expressed generalization in three ways.
Fecha
2021
Tipo de fecha
Estado publicación
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Nivel educativo
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Revisado por pares
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Referencias
1. Pólya, G. How to Solve It; University Press: Princeton, NJ, USA, 1945; (Zugazagoitia, J. Cómo Plantear y Reslover Problemas; Traducción al Castellano; Trillas: León, México, 1965). 2. Pólya, G. Matemáticas y Razonamiento Plausible [Mathematics and Plausible Reasoning]; Tecnos: Madrid, Spain, 1966. 3. Radford, L. En torno a tres problemas de la generalización [Around three problems of generalization]. In Investigación en Didáctica de la Matemática. Homenaje a Encarnación Castro; Rico, L., Cañadas, M.C., Gutiénez, J., Molina, M., Segovia, I., Eds.; Editorial Comares: Granada, Spain, 2013; pp. 3–12. 4. Kaput, J.J. What is algebra? What is algebraic reasoning. In Algebra in the Early Grades; Kaput, J.J., Carraher, D.W., Blanton, M.L., Eds.; Routledge: New York, NY, USA, 2008; pp. 5–17. 5. Blanton, M.; Levi, L.; Crites, T.; Dougherty, B. Developing Essential Understanding of Algebraic Thinking for Teaching Mathematics in Grades 3–5; NCTM: Reston, VA, USA, 2011. 6. Doorman, M.; Drijvers, P.; Gravemeijer, K.; Boon, P.; Reed, H. Tool use and the development of the function concept: From repeated calculations to functional thinking. Int. J. Sci. Math. Educ. 2012, 10, 1243–1267. [CrossRef] 7. Morales, R.A.; Cañadas, M.C.; Brizuela, B.M.; Gómez, P. Relaciones funcionales y estrategias de alumnos de primero de Educación Primaria en un contexto funcional [Functional relationships and strategies of first graders in a functional context]. Enseñ. Cienc. 2018, 36, 59–78. [CrossRef] 8. Mason, J.; Stephens, M.; Watson, A. Appreciating. mathematical structure for all. Math. Educ. Res. J. 2009, 21, 10–32. [CrossRef] 9. Cañadas, M.C.; Castro, E. A proposal of categorisation for analysing inductive reasoning. PNA 2007, 1, 67–78. 10. Kaput, J.J.; Blanton , M.L.; Moreno, L. 2 Algebra from a symbolization point of view. In Algebra in the Early Grades; Kaput, J.J., Carraher, D.W., Blanton, M.L., Eds.; Lawrence Erlbaum Associates: New York, NY, USA, 2008; pp. 19–55. 11. Radford, L. The emergence of symbolic algebraic thinking in primary school. In Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds: The Global Evolution of an Emerging Field of Research and Practice; Kieran, C., Ed.; Springer: New York, NY, USA, 2018; pp. 3–25. [CrossRef] 12. Torres, M.D.; Cañadas, M.C.; Moreno, A. Estructuras, generalización y significado de letras en un contexto funcional por estudiantes de 2º de primaria [Structures, generalization and meaning of letters in a functional context for 2nd primary students]. In Investigación en Educación Matemática XXII; Rodríguez-Muñiz, L.J., Muñiz- Rodríguez, L., Aguilar-González, A., Alonso, P., García García, F.J., Bruno, A., Eds.; SEIEM: Gijón, Spain, 2018; pp. 574–583. 13. Vergel, R. Sobre la Emergencia del Pensamiento Algebraico Temprano y su Desarrollo en la Educación Primaria; UD: Bogotá, Colombia, 2016. 14. Ayala-Altamirano, C.; Molina, M. Meanings attributed to letters in functional contexts by Primary School students. Int. J. Sci. Math. Educ. 2020, 18, 1271–1291. [CrossRef] 15. Blanton, M.; Brizuela, B.; Murphy Gardiner, A.; Sawrey, K.; Newman-Owens, A. A learning trajectory in 6-year-olds’ thinking about generalizing functional relationships. J. Res. Math. Educ. 2015, 46, 511–558. [CrossRef] 16. Aliseda, A. La abducción como cambio epistémico: C. S. Peirce y las teorías epistémicas en inteligencia artificial. Analogía 1998, 12, 125–144. 17. Stephens, A.C.; Ellis, A.B.; Blanton, M.; Brizuela, B.M. Algebraic thinking in the elementary and middle grades. In Compendium for Research in Mathematics Education; Cai, J., Ed.; National Council of Teachers of Mathematics: Reston, VI, USA, 2017; pp. 386–420. 18. Peirce, C. Collected Papers of Charles Sanders Peirce; Hartshorne, C., Weiss, P., Eds.; The Belknap Press of Harvard University Press: Cambridge, MA, USA, 1965. 19. Nepomuceno, A. Modelos de razonamiento abductivo [Models of abductive reasoning]. Contrastes 2005, 10, 155–180. 20. Niño, D. Abducción e Inducción en Peirce: Evolución y criterios [Abduction and induction in Peirce: Evolution and criteria]. Designis 2012, 20, 153–161. 21. Rivera, F. Abduction and the emergence of necessary mathematical knowledge. In Springer Handbook of Model-Based Xcience; Magnani, L., Bertolotti, T., Eds.; Springer: Cham, Switzerland, 2017. [CrossRef] 22. Nubia Soler-Álvarez, M.; Manrique, V.H. El proceso de descubrimiento en la clase de matemáticas: Los razonamientos abductivo, inductivo y deductivo [Discovery process in mathematics class: Abductive, inductive and deductive reasoning]. Ensenñ. Cienc. 2014, 32, 191–219. [CrossRef] 23. Aliseda, A. A Unified Framework for Abductive and Inductive Reasoning in Philosophy y AI. In Contributing Paper to the ECAI’96 Workshop on Abductive and Inductive Reasoning; Budapest, Hungary, 12 August 1996; pp. 1–6. Available online: http://citeseer.ist.psu.edu/viewdoc/versions;jsessionid=B03C9D84448240C8BC1DA9A01CC9476B?doi=10.1.1.37.5343 (accessed on 13 February 2021). 24. Aguayo, P. La teoría de la abducción de Peirce: Lógica, metodología e instinto [The Peirce’ theory of abduction: Logic, methodology and instint]. Ideas Valores 2011, 60, 33–53. 25. Rivera, F.; Becker, J.R. Abduction-induction (generalization) processes of elementary majors on figural patterns in algebra. J. Math. Behav. 2007, 26, 140–155. [CrossRef] 26. Vergel, R. Generalización. de patrones y formas de pensamiento algebraico temprano. PNA 2015, 9, 193–215. 27. Barrera, V.J.; Castro, E.; Cañadas, M.C. Cuaderno de trabajo sobre razonamiento inductivo para profesores de primaria en formación [Workbook about inductive reasoning for elementary teachers training]. In Trabajo Presentado en el Grupo de Investigación Pensamiento Numérico y Algebraico del XIII Congreso de la SEIEM; SEIEM: Santander, España, 2009. 28. Pinto, E.; Cañadas, M.C. Generalización y razonamiento inductivo en una estudiante de cuarto de primaria. Un estudio de caso desde el pensamiento funcional [Generalisation and inductive reasoning by a fourth grader. A case study from the functional thinking approach]. In Investigación en Educación Matemática XXII; Rodríguez-Muñiz, L.J., Muñiz-Rodríguez, L., Aguilar González, A., Alonso, P., García García, F.J., Bruno, Y.A., Eds.; SEIEM: Gijón, Spain, 2018; pp. 457–466. 29. Torres, M.D.; Cañadas, M.C.; Moreno, A. Structures identified by second graders in a teaching experiment in a functional approach to early algebra. In Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education; Jankvist, U.T., Van den Heuvel-Panhuizen, M., Veldhuis, M., Eds.; Institute of Education and Erme: Utrecht, The Netherlands, 2019. 30. Castro, E.; Cañadas, M.C.; Molina, M. El Razonamiento Inductivo Como gGenerador de Conocimiento Matemático [Inductive Reasoning as a Generator of Mathematical Knowledge]. Uno 2010, 54, 55–67. Available online: http://hdl.handle.net/10481/26 079 (accessed on 13 February 2021). 31. Kieran, C. The early learning of algebra: A structural perspective. In Research Issues in the Learning and Teaching of Algebra; Wagner, S., Kieran, C., Eds.; NCTM: Reston, VA, USA, 1989; Volume 4, pp. 33–56. 32. Pinto, E.; Cañadas, M.C. Generalization in fifth graders within a functional approach. In Proceedings of the 41st Conference of the International Group for the Psychology of Mathematics Education, Singapore, 17–22 July 2017; Kaur, B., Ho, W.K., Toh, T.L., Choy, Y.B.H., Eds.; PME: Singapore, 2017; Volume 4, pp. 49–56. 33. Pinto, E.; Cañadas, M.C. Estructuras y generalización de estudiantes de tercero y quinto de primaria: Un estudio comparativo [Structures and generalization of third and fifth graders: A comparative study]. In Investigación en Educación Matemática XXI; Muñoz-Escolano, J.M., Arnal-Bailera, A., Beltrán-Pellicer, P., Carrillo, M.L.C.Y.J., Eds.; Sociedad Española de Investigación en Educación Matemática, SEIEM: Zaragoza, Spain, 2017; pp. 407–416. 34. Driscoll, M.J. Fostering Algebraic Thinking: A Guide for Teachers, Grades 6–10; Heinemann: Portsmouth, NH, USA, 1999. 35. Schifter, D.; Monk, S.; Russell, S.J.; Bastable, V. Early algebra: What does understanding the laws of arithmetic mean in the elementary grades. In Algebra in the Early Grades; Kaput, J.J., Carraher, D.W., Blanton, M.L., Eds.; Lawrence Erlbaum Associates: New York, NY, USA, 2008; pp. 413–448. 36. Radford, L. Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to students’ types of generalization. Math. Think. Learn. 2003, 5, 37–70. [CrossRef] 37. Radford, L. Layers of generality and types of generalization in pattern activities. PNA 2010, 4, 37–62. 38. Steffe, L.; Thompson, P.W. Teaching experiment methodology: Underlying principles and essential elements. In Handbook of Research Design in Mathematics and Science Education; Lesh, R., Kelly, Y.A.E., Eds.; LAE.Stacey, K: Mahwah, NJ, USA, 2000; pp. 267–306. 39. Stacey, K. Finding and using patterns in linear generalising problems. Educ. Stud. Math. 1989, 20, 147–164. [CrossRef] 40. Abe, A. Abduction and analogy in chance discovery. In Chance Discovery; Ohsawa, Y., McBurney, M., Eds.; Springer: Berlin, Heidelberg, 2003; pp. 231–248. 41. Cobb, P. Conducting teaching experiments in collaboration with teachers. In Handbook of Research Design in Mathematics and Science Education; Kelly, A.E., Lesh, R.A., Eds.; Lawrence Erlbaum Associates: Mahwah, NJ, USA, 2000; pp. 307–333.
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