Generalization in fifth graders within a functional approach

  1. Pinto, Eder 1
  2. Cañadas, María C. 1
  1. 1 Universidad de Granada
    info

    Universidad de Granada

    Granada, España

    ROR https://ror.org/04njjy449

Revue:
PNA: Revista de investigación en didáctica de la matemática

ISSN: 1887-3987

Année de publication: 2018

Volumen: 12

Número: 3

Pages: 173-184

Type: Article

DOI: 10.30827/PNA.V12I3.6643 DIALNET GOOGLE SCHOLAR lock_openDIGIBUG editor

D'autres publications dans: PNA: Revista de investigación en didáctica de la matemática

Objectifs de Développement Durable

Résumé

This article discusses evidence of 24 fifth graders’ (10-11 year olds’) ability to generalize when solving a problem which involves a linear function. Analyzed in the context of the functional approach of early algebra, the findings show that 3 students generalized both when solving specific instances and when asked to provide the general formula; while 15 students generalized only when asked to define the general formula. The results are described in terms of the functional relationship identified, the types of representation used to express them and the type of questions in which students generalized their answers. Most of the pupils who generalized did so based on the correspondence between pairs of values in the function at issue.Generalización de estudiantes de quinto de primaria desde un enfoque funcional En este artículo presentamos evidencias de generalización de 24 estudiantes de quinto de primaria (10-11 años) al resolver un problema que involucra una función lineal. Desde el enfoque funcional del early algebra, los hallazgos muestran que 3 estudiantes generalizaron al trabajar con casos particulares y cuando se les pide expresar la regla general; mientras que 16 estudiantes solo lo hicieron cuando les pedimos expresar la regla general. Describimos los resultados en términos de la relación funcional identificada, los tipos de representaciones que emplearon para expresar dichas relaciones y el tipo de pregunta en la cual los estudiantes generalizaron. La mayoría de los estudiantes que generalizaron establecieron una relación de correspondencia entre los pares de valores de la función. 

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Information sur le financement

This study forms part of National R&D Projects EDU2013-41632-P and EDU2016-75771-P funded by the Spanish Ministry of Economy and Competitiveness; the first author was supported by a PhD scholarship granted by Chilean Government through the CONICYT, folio 72160307-2015.

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Références bibliographiques

  • Blanton, M. L. (2008). Algebra and the elementary classroom: Transforming thinking, transforming practice. Portsmouth, NH: Heinemann.
  • Blanton, M. L., & Kaput, J. J. (2005). Helping elementary teachers build mathematical generality into curriculum and instruction. ZDM, 37(1), 34-42.
  • Blanton, M. L., & Kaput, J. J. (2011). Functional thinking as a route into algebra in the elementary grades. In J. Cai, & E. Knuth (Eds.), Early algebraization. A Global dialogue from multiple perspectives (pp. 5-23). Berlin, Germany: Springer.
  • Blanton, M. L., Brizuela, B. M., Gardiner, A., Sawrey, K., & Newman-Owens, A. (2015). A learning trajectory in 6-year-olds’ thinking about generalizing functional relationships. Journal for Research in Mathematics Education, 46(5), 511-558.
  • Blanton, M. L., Levi, L., Crites, T., & Dougherty, B. (Eds.) (2011). Developing essen-tial understanding of algebraic thinking for teaching mathematics in grades 3-5. Reston, VA: National Council of Teachers of Mathematics.
  • Cañadas, M. C. & Molina, M. (2016). Una aproximación al marco conceptual y prin-cipales antecedentes del pensamiento funcional en las primeras edades [Approach to the conceptual framework and background of functional thinking in early years]. In E. Castro, E. Castro, J. L. Lupiáñez, J. F. Ruiz, & M. Torralbo (Eds.), Investigación en Educación Matemática. Homenaje a Luis Rico (pp. 209-218). Granada, Spain: Comares.
  • Carpenter, T., & Levi, L. (2000). Developing conceptions of algebraic reasoning in the primary grades. Madison, WI: National Center for Improving student Learning and Achievement in Mathematics and Science.
  • Carraher, D. W., & Schliemann, A. D. (2002). The transfer dilemma. The Journal of the Learning Sciences, 11(1), 1-24.
  • Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebraic reasoning. En F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 669-705). Reston, VA: NCTM.
  • Carraher, D. W., & Schliemann, A. D. (2016). Powerful ideas in elementary school mathematics. In L. English, & D. Kirshner (Eds.), Handbook of international re-search in mathematics education. Third edition (pp. 191-218). New York, NY: Routledge.
  • Carraher, D. W., Martinez, M. V., & Schliemann, A. D. (2008). Early algebra and mathematical generalization. ZDM, 40(1), 3-22.
  • Castro, E., Cañadas, M. C., & Molina, M. (in press). Pensamiento funcional mostrado por estudiantes de Educación Infantil. Edma 0-6. Educación Matemática en la Infancia. Dienes, Z. (1961). On abstraction and generalization. Harvard Educational Review, 31(3), 281-301.
  • Ellis, A. B. (2007). A taxonomy for categorizing generalizations. The Journal of the Learning Sciences, 16(2), 221-262.
  • Harel, G., & Tall, D. (1991). The general, the abstract, and the generic in advanced mathematics. For the Learning of Mathematics, 11(1), 38-42.
  • Kaput, J. J. (1999). Teaching and learning a new algebra. In E. Fennema & T. A. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 133-155). Mahwah, NJ: Lawrence Erlbaum Associates.
  • Kaput, J. J. (2008). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 5-17). New York, NY: Lawrence Erlbaum.
  • Kieran, C. (2004). Algebraic thinking in the early grades: What is it? The Mathemat-ics Educator, 8(1), 139-151. Krutetskii, V. A. (1976). The psychology of mathematical abilities in schoolchildren. Chicago, IL: University of Chicago Press.
  • Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra (pp. 65-86). Dordrecht, The Neth-erlands: Kluwer Academic Publishers.
  • Mason, J., & Pimm, D. (1984). Seeing the general in the particular. Educational Stud-ies in Mathematics, 15(3), 277-289.
  • Mason, J., Burton, L., & Stacey, K. (1988). Pensar matemáticamente [Thinking math-ematically]. Barcelona, Spain: Labor.
  • Merino, E., Cañadas, M. C., & Molina, M. (2013). Uso de representaciones y patrones por alumnos de quinto de educación primaria en una tarea de generalización [Representations and patterns used by fifth grade students in a generalization task]. Edma 0-6: Educación Matemática en la Infancia, 2(1), 24-40.
  • Mitchelmore, M., & White, P. (2007). Abstraction in mathematics learning. Mathe-matics Education Research Journal, 19(2), 1-9.
  • Pinto, E. & Cañadas, M. C. (2017). Generalization in fifth graders within a functional approach. In B. Kaur, W. Kin Ho, T. Lam Toh, & B. Heng Choy (Eds.), Proceed-ings of the 41st Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 49-56). Singapore: PME.
  • Radford, L. (2010). Layers of generality and types of generalization in pattern activi-ties. PNA, 4(2), 37-62. Romberg, T., Fennema, E., & Carpenter, T. (1993). Integrating research on the graphical representation of functions. New York, NY: Routledge.
  • Schifter, D., Monk, S., Russell, S., & Bastable, V. (2008). Early algebra: What does understanding the laws of arithmetic mean in the elementary grades? In J. J. Kaput, D. W.
  • Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 389-412). New York, NY: Lawrence Erlbaum. Smith, E. (2008). Representational thinking as a framework for introducing functions in the elementary curriculum. In J. J.
  • Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 133-163). New York, NY: Lawrence Erl-baum.
  • Stacey, K. (1989). Finding and using patterns in linear generalising problems. Educa-tional Studies in Mathematics, 20(2), 147-164.
  • Warren, E., Miller, J., & Cooper, T. J. (2013). Exploring young students’ functional thinking. PNA, 7(2), 75-84.
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