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Grade 2 Students’ Non-Symbolic Algebraic Thinking

  • Chapter
Early Algebraization

Part of the book series: Advances in Mathematics Education ((AME))

Abstract

The learning of arithmetic, it has recently been argued, need not be a prerequisite for the learning of algebra. From this viewpoint, it is claimed that young students can be introduced to some elementary algebraic concepts in primary school. However, despite the increasing amount of experimental evidence, the idea of introducing algebra in the early years remains clouded by the lack of clear distinctions between what is arithmetic and what is algebraic. The goal of this chapter is twofold. First, at an epistemological level, it seeks to contribute to a better understanding of the relationship between arithmetic and algebraic thinking. Second, at a developmental level, it explores 7–8-years old students’ first encounter with some elementary algebraic concepts and inquires about the limits and possibilities of introducing algebra in primary school.

This chapter is a result of a research program funded by The Social Sciences and Humanities Research Council of Canada/Le Conseil de recherches en sciences humaines du Canada (SSHRC/CRSH).

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References

  • Arzarello, F., & Robutti, O. (2004). Approaching functions through motion experiments. In R. Nemirovsky, M. Borba & C. DiMattia (Eds.), Educational Studies in Mathematics: Vol. 57(3). PME Special Issue of “Approaching functions through motion experiments”. CD-Rom, Chap. 1.

    Google Scholar 

  • Balacheff, N. (1987). Processus de preuve et situations de validation. Educational Studies in Mathematics, 18, 147–176.

    Article  Google Scholar 

  • Becker, J., & Rivera, F. (2006a). Establishing and justifying algebraic generalization at the sixth grade level. In J. Novotná, H. Moraová, M. Krátkná, & N. Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 465–472). Prague, Czech Republic.

    Google Scholar 

  • Becker, J., & Rivera, F. (2006b). Sixth graders’ figural and numerical strategies for generalizing patterns in algebra. In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 95–101). Mérida, México: Universidad Pedagógica Nacional.

    Google Scholar 

  • Bednarz, N., Kieran, C., & Lee, L. (Eds.) (1996). Approaches to Algebra: Perspectives for Research and Teaching. Kluwer: Dordrecht.

    Google Scholar 

  • Blanton, M., & Kaput, J. (2000). Generalizing and progressively formalizing in a third grade mathematics classroom: Conversations about even and odd numbers. In M. Fernandez (Ed.), Proceedings of the Twenty-second Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education PME-NA (pp. 115–119).

    Google Scholar 

  • Brizuela, B., & Schliemann, A. (2004). Ten-year-old students solving linear equations. For the Learning of Mathematics, 24(2), 33–40.

    Google Scholar 

  • Carraher, D. W., & Schliemann, A. (2007). Early algebra and algebraic reasoning. In F. K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 669–705). Greenwich, CT: Information Age Publishing.

    Google Scholar 

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

    Google Scholar 

  • Dougherty, B. (2003). Voyaging from theory to practice in learning: Measure up. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education (PME & PMENA) (Vol. 1, pp. 17–23). Honolulu: University of Hawai’i.

    Google Scholar 

  • Edwards, L. (2009). Gestures and conceptual integration in mathematical talk. Educational Studies in Mathematics, 70(2), 127–141. doi:110.1007/s10649-10008-19124-10646.

    Article  Google Scholar 

  • Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19–25.

    Google Scholar 

  • Filloy, E., Rojano, T., & Puig, L. (2007). Educational Algebra: A Theoretical and Empirical Approach. New York: Springer.

    Google Scholar 

  • Fujii, T., & Stephens, M. (2008). Using number sentences to introduce the idea of variable. In C. Greenes & R. Rubenstein (Eds.), Algebra and Algebraic Thinking in School Mathematics: Seventieth Yearbook (pp. 127–140). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Gómez, J. C. (2004). Apes, Monkeys, Children, and the Growth of Mind. Cambridge: Harvard University Press.

    Google Scholar 

  • Kieran, C. (Ed.) (1989). The Early Learning of Algebra: A Structural Perspective. Virginia: Lawrence Erlbaum Associates and National Council of Teachers of Mathematics.

    Google Scholar 

  • MacGregor, M., & Stacey, K. (1992). A comparison of pattern-based and equation-solving approaches to algebra. In B. Southwell, K. Owens, & B. Penny (Eds.), Proceedings of the 15th Annual Conference, Mathematics Education Research Group of Australasia (MERGA) (pp. 362–370). University of Western Sydney, July 4–8.

    Google Scholar 

  • Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to Algebra (pp. 65–86). Dordrecht: Kluwer.

    Google Scholar 

  • Moss, J., & Beatty, R. (2006). Knowledge building and knowledge forum: Grade 4 students collaborate to solve linear generalizing problems. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (pp. 193–199). Prague, Czech Republic.

    Google Scholar 

  • Nemirovsky, R., & Ferrara, F. (2009). Mathematical imagination and embodied cognition. Educational Studies in Mathematics, 70(2), 159–174. doi:110.1007/s10649-10008-19150-10644.

    Article  Google Scholar 

  • Ontario Ministry of Education (1997). The Ontario Curriculum, Grades 1–8, Mathematics. Toronto: Queen’s Printer for Ontario (Revised version, 2005).

    Google Scholar 

  • Presmeg, N. (1986). Visualization in high school mathematics. For the Learning of Mathematics, 6(3), 42–46.

    Google Scholar 

  • Radford, L. (1997). L’invention d’une idée mathématique: la deuxième inconnue en algèbre. Repères (Revue des instituts de Recherche sur l’enseignement des Mathématiques), 28, 81–96.

    Google Scholar 

  • Radford, L. (2000). Signs and meanings in students’ emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42(3), 237–268.

    Article  Google Scholar 

  • Radford, L. (2001). The historical origins of algebraic thinking. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives in School Algebra (pp. 13–63). Dordrecht: Kluwer.

    Google Scholar 

  • Radford, L. (2003). Gestures, speech and the sprouting of signs. Mathematical Thinking and Learning, 5(1), 37–70.

    Article  Google Scholar 

  • Radford, L. (2006a). Algebraic thinking and the generalization of patterns: A semiotic perspective. In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, North American Chapter (Vol. 1, pp. 2–21). Mérida, Mexico.

    Google Scholar 

  • Radford, L. (2006b). The cultural-epistomological conditions of the emergence of algebraic symbolism. In F. Furinghetti, S. Kaijser, & C. Tzanakis (Eds.), Proceedings of the 2004 Conference of the International Study Group on the Relations between the History and Pedagogy of Mathematics & ESU 4—Revised Edition (pp. 509–524). Uppsala, Sweden.

    Google Scholar 

  • Radford, L. (2008a). Iconicity and contraction: A semiotic investigation of forms of algebraic generalizations of patterns in different contexts. ZDM—The International Journal on Mathematics Education, 40(1), 83–96.

    Article  Google Scholar 

  • Radford, L. (2008b). The ethics of being and knowing: Towards a cultural theory of learning. In L. Radford, G. Schubring, & F. Seeger (Eds.), Semiotics in Mathematics Education: Epistemology, History, Classroom, and Culture (pp. 215–234). Rotterdam: Sense Publishers.

    Google Scholar 

  • Radford, L. (2009a). Signs, gestures, meanings: Algebraic thinking from a cultural semiotic perspective. Plenary Lecture presented at the Sixth Conference of European Research in Mathematics education (CERME 6). Université Claude Bernard, Lyon, France, January 28–February 1, 2009.

    Google Scholar 

  • Radford, L. (2009b). Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings. Educational Studies in Mathematics, 70(2), 111–126.

    Article  Google Scholar 

  • Radford, L., Bardini, C., & Sabena, C. (2007). Perceiving the general: The multisemiotic dimension of students’ algebraic activity. Journal for Research in Mathematics Education, 38, 507–530.

    Google Scholar 

  • Roth, M.-W. (2001). Gestures: Their role in teaching and learning. Review of Educational Research, 71(3), 365–392.

    Article  Google Scholar 

  • Sabena, C., Radford, L., & Bardini, C. (2005). Synchronizing gestures, words and actions in pattern generalizations. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (pp. 129–136). University of Melbourne, Australia.

    Google Scholar 

  • Savage-Rumbaugh, E. S., Rumbaugh, D. M., Smith, S. T., & Lawson, J. (1980). Reference: The linguistic essential. Science, 210(4472), 922–925.

    Article  Google Scholar 

  • Serfati, M. (1999). La dialectique de l’indéterminé, de Viète à Frege et Russell. In M. Serfati (Ed.), La recherche de la vérité (pp. 145–174). Paris: ACL—Les éditions du kangourou.

    Google Scholar 

  • Serfati, M. (2006). La constitution de l’ecriture symbolique mathematique. Symbolique et invention. SMF—Gazette, 108, 101–118.

    Google Scholar 

  • Viète, F. (1983). Nine Studies in Algebra, Geometry and Trigonometry from the Opus Restitutae Mathematicae Analyseos, Seu Algebrâ Novâ (Translated by T. Richard Witmer). New York: Dover.

    Google Scholar 

  • Warren, E. (2006). Teacher actions that assist young students write generalizations in words and in symbols. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 5, pp. 377–384). Prague, Czech Republic.

    Google Scholar 

  • Warren, E., & Cooper, T. (2008). Generalising the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking. Educational Studies in Mathematics, 67, 171–185.

    Article  Google Scholar 

  • Woepcke, F. (1853). Extrait du Fakhrî. Paris: Imprimérie Impériale.

    Google Scholar 

Download references

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Authors and Affiliations

  1. École des science de l’education, Laurentian University, Sudbury, Ontario, Canada

    Luis Radford

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  1. Luis Radford
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Correspondence to Luis Radford .

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Editors and Affiliations

  1. Dept. Mathematical Sciences, University of Delaware, Ewing Hall 501, Newark, 19716, Delaware, USA

    Jinfa Cai

  2. University of Wisconsin-Madison, Teacher Education Building, 225 N. Mills Street, Madison, 53706, Wisconsin, USA

    Eric Knuth

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Radford, L. (2011). Grade 2 Students’ Non-Symbolic Algebraic Thinking. In: Cai, J., Knuth, E. (eds) Early Algebraization. Advances in Mathematics Education. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17735-4_17

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