Skip to main content

First Encounter with Variables by First and Third Grade Spanish Students

  • Chapter
  • First Online:
Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds

Part of the book series: ICME-13 Monographs ((ICME13Mo))

Abstract

This chapter presents findings from a teaching experiment on the initial understandings that primary Spanish students demonstrated when they were first introduced to the use of letters to stand for an indeterminate varying quantity in a functional relationship. We provide a detailed account of our task design and class activity to show how understanding of variable notation for functional relationships was cultivated. We discuss the degree to which results from previous studies generalize to the Spanish context. Our results, similar to those of previous studies, support the introduction of variables in elementary grades.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions

Notes

  1. 1.

    Names of students used in this chapter are pseudonyms.

  2. 2.

    We remind the reader that five in Spanish is “cinco.”

References

  • Arcavi, A. (2005). Developing and using symbol sense in mathematics. For the learning of mathematics, 25(2), 42–47.

    Google Scholar 

  • 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 (pp. 5–23). Berlin: Springer.

    Google Scholar 

  • Blanton, M., Stephens, A., Knuth, E., Gardiner, A. M., Isler, I., & Kim, J.-S. (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.

    Google Scholar 

  • Brizuela, B. M., Blanton, M., Gardiner, A. M., Newman-Owens, A., & Sawrey, K. (2015a). A first grade student’s exploration of variable and variable notation [Una alumna de primer grado explora las variables y su notación]. Estudios de Psicología, 36(1), 138–165.

    Google Scholar 

  • Brizuela, B. M., Blanton, M., Sawrey, K., Newman-Owens, A., & Gardiner, A. M. (2015b). Children’s use of variables and variable notation to represent their algebraic ideas. Mathematical Thinking and Learning, 17(1), 34–63.

    Google Scholar 

  • Cañadas, M. C., & Castro, E. (2007). A proposal of categorisation for analysing inductive reasoning. PNA, 1(2), 69–81.

    Google Scholar 

  • Carraher, D. W., Schliemann, A. D., Brizuela B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37(2), 87–115.

    Google Scholar 

  • Carraher, D. W., Schliemann, A. D. & Schwartz, J. (2008). Early algebra is not the same as algebra early. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 235–272). Mahwah, NJ: Erlbaum.

    Google Scholar 

  • Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational psychologist, 31(3–4), 175–190.

    Google Scholar 

  • Ender, A. (2014). Implicit and explicit cognitive processing in incidental vocabulary acquisition. Applied Linguistics, 37(4), 536–560.

    Google Scholar 

  • Fernández-Millán E., & Molina, M. (2016). Indagación en el conocimiento conceptual del simbolismo algebraico de estudiantes de secundaria mediante la invención de problemas [Inquiry into secondary students’ conceptual knowledge of algebraic symbolism through problem posing]. Enseñanza de las Ciencias, 34(1), 53–71. DOI:http://dx.doi.org/10.5565/rev/ensciencias.1455.

  • Filloy, E., Rojano T., & Puig, L. (2008). Educational algebra. A theoretical and empirical approach. New York: Springer.

    Google Scholar 

  • Furinghetti, F., & Paola, D. (1994). Parameters, unknowns and variables: a little difference? In J. P. Ponte & J. F. Matos (Eds.), Proceedings of the 20 th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 368–375) Lisbon, Portugal: PME.

    Google Scholar 

  • Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2005). Middle-school students’ understanding of core algebraic concepts: Equality and variable. Zentralblatt fur Didaktik der Mathematik—International Reviews on Mathematical Education, 37, 1–9.

    Google Scholar 

  • Küchemann, D. E. (1981). Algebra. In K. M. Hart (Eds.), Children’s understanding of mathematics: 11–16 (pp. 102–119). London, UK: John Murray.

    Google Scholar 

  • Lobato, J., Clarke, D., & Ellis, A. (2005). Initiating and eliciting in teaching: A reformulation of telling. Journal for Research in Mathematics Education, 36(2), 101–136.

    Google Scholar 

  • MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation: 11–15. Educational Studies in Mathematics, 33, 1–19.

    Google Scholar 

  • Molina, M., Castro, E., Molina, J. L., & Castro, E. (2011). Un acercamiento a la investigación de diseño a través de los experimentos de enseñanza [An approach to design research through teaching experiments]. Enseñanza de las Ciencias, 29(1), 75–88.

    Google Scholar 

  • 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. DOI:10.1007/s10763-016-9739-5.

  • Radford, L. (2011). Grade 2 students’ non-symbolic algebraic thinking. In J. Cai & E. Knuth (Eds.), Early algebraization (pp. 303–322). Berlin: Springer.

    Google Scholar 

  • Radford, L. (2014). The progressive development of early embodied algebraic thinking. Mathematics Education Research Journal, 26(2), 257–277.

    Google Scholar 

  • Sfard, A. (2000). Symbolizing mathematical reality into being: How mathematical discourse and mathematical objects create each other. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating: Perspectives on mathematical discourse, tools, and instructional design (pp. 37–98). Mahwah, NJ: Erlbaum.

    Google Scholar 

  • 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. 133–160). New York: Routledge.

    Google Scholar 

  • Spanish Ministry of Education and Science. (2014). Real Decreto 126/2014 de 28 de febrero, por el que se establece el currículo básico de la Educación Primaria [Royal decree 126/2014 of February 28th, establishing Primary Education basic curriculum] (vol. BOE nº 52, pp. 19349–19420). Madrid: Ministry of Education and Science.

    Google Scholar 

  • Tirosh, D., Even, R., & Robinson, N. (1998). Simplifying algebraic expressions: Teacher awareness and teaching approaches. Educational Studies in Mathematics, 35(1), 51–64.

    Google Scholar 

  • Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. Coxford & A. Shulte (Eds.), The ideas of algebra, K-12 (Yearbook of the National Council of Teachers of Mathematics, pp. 8–19). Reston, VA: NCTM.

    Google Scholar 

Download references

Acknowledgements

This study was developed within the Spanish projects of Research and Development with reference code EDU2013-41632-P and EDU2016-75771-P, financed by the Spanish Ministry of Economy and Competitiveness.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Marta Molina .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Molina, M., Ambrose, R., del Rio, A. (2018). First Encounter with Variables by First and Third Grade Spanish Students. In: Kieran, C. (eds) Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-68351-5_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-68351-5_11

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-68350-8

  • Online ISBN: 978-3-319-68351-5

  • eBook Packages: EducationEducation (R0)

Publish with us

Policies and ethics