# The Emergence of Symbolic Algebraic Thinking in Primary School

## Abstract

This chapter presents the results of a longitudinal investigation on the emergence of symbolic algebraic thinking in young students in the context of sequence generalization. The investigation rests on a characterization of algebraic thinking based on its *analytic* nature and a careful attention to the *semiotic systems* through which students express the mathematical variables involved. Attention to the semiotic systems and their interplay led us to identify non-symbolic and symbolic (alphanumeric) early algebraic generalizations and the students’ evolving intelligibility of the variables and their relationships, and mathematical sequence structure. The results shed some light on the transition from non-symbolic to symbolic algebraic thinking in primary school.

## Keywords

Early algebra Semiotic systems Pattern generalization Algebraic generalizations## Notes

### Acknowledgements

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).

## References

- Ainley, J. (1999). Doing algebra-type stuff: Emergent algebra in the primary school. In O. Zaslavsky (Ed.),
*Proceedings of the 23rd Annual Conference of the International Group for the Psychology of Mathematics Education*(Vol. 4, pp. 9–16). Haifa, Israel: PME.Google Scholar - Bednarz, N., & Janvier, B. (1996). Emergence and development of algebra as a problem-solving tool: continuities and discontinuities with arithmetic. In N. Bednarz, C. Kieran, & L. Lee (Eds.),
*Approaches to algebra, perspectives for research and teaching*(pp. 115–136). Dordrecht: Kluwer.Google Scholar - Bednarz, N., Radford, L., Janvier, B., & Lepage, A. (1992). Arithmetical and algebraic thinking in problem-solving. In W. Geeslin & K. Graham (Eds.),
*Proceedings of the 16*^{th}*Conference of the International Group for the Psychology of Mathematics Education*(Vol. I, pp. 65–72). Durham, NH: PME.Google Scholar - Bednarz, N., Kieran, C., & Lee, L. (Eds.). (1996).
*Approaches to algebra, perspectives for research and teaching.*Dordrecht: Kluwer.Google Scholar - Blanton, M., Brizuela, B., Gardiner, A., Sawrey, K., & Newman-Owens, A. (2017). A progression in first-grade children’s thinking about variable and variable notation in functional relationships.
*Educational Studies in Mathematics*,*Online First.*doi: 10.1007/s10649-016-9745-0. - Cai, J., & Knuth, E. (Eds.). (2011).
*Early algebraization.*New York: Springer.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 - Cole, M. (1996).
*Cultural psychology.*Cambridge, MA: The Belknap Press of Harvard University Press.Google Scholar - Condillac, E. (2001).
*Essay on the origin of human knowledge.*(H. Aarsleff, Trans.) Cambridge, UK: Cambridge University Press. (Original work published 1746).Google Scholar - Damerow, P. (1996).
*Abstraction and representation. Essays on the cultural evolution of thinking.*Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar - Davis, R. B. (1975). Cognitive processes involved in solving simple algebraic equations.
*Journal of Children’s Mathematical Behaviour*,*1*(3), 7–35.Google Scholar - Descartes, R. (1954).
*The geometry*. New York: Dover. Original work published 1637.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 - Gori, D. (1984).
*Libro e trattato della praticha d’alcibra.*[Book and treatise of the practice of algebra]. A cura e introduziene di L. T. Rigatelli. Siena, IT: Quaderni del centro studi della matematica medioevale, N. 9. (Original work published 1544).Google Scholar - Hardcastle, J. (2009). Vygotsky’s enlightenment precursors.
*Educational Review*,*61*(2), 181–195. doi: 10.1080/00131910902846890. - Kaput, J.J. (1998). Transforming algebra from an engine of inequity to an engine of mathematical power by “algebrafying” the K-12 curriculum. In
*The nature and role of algebra in the K-14 curriculum: Proceedings of a national symposium.*Washington, DC: National Academy of Sciences. https://doi.org/10.17226/6286. - Kaput, J.J., Blanton, M., & Moreno, L. (2008a). Algebra from a symbolization point of view. In J.J. Kaput, D.W. Carraher, & M.L. Blanton (Eds.),
*Algebra in the early grades*(pp. 19–55). New York: Routledge.Google Scholar - Kaput, J.J, Carraher, D.W., & Blanton, M.L. (Eds.). (2008b).
*Algebra in the early grades.*New York: Routledge.Google Scholar - Kieran, C. (1989a). A perspective on algebraic thinking. In G. Vergnaud, J. Rogalski, & M. Artigue (Eds.),
*Proceedings of the 13th Conference of the International Group for the Psychology of Mathematics Education*(v. 2, pp. 163–171). Paris: PME.Google Scholar - Kieran, C. (1989b). The early learning of algebra: A structural perspective. In S. Wagner & C. Kieran (Eds.),
*Research issues in the learning and teaching of algebra*(pp. 33–56). Reston, VA: Lawrence Erlbaum Associates and National Council of Teachers of Mathematics.Google Scholar - Locke, J. (1825).
*An essay concerning human understanding.*London: Thomas Davison. (Original work published 1690).Google Scholar - Mason, J., Graham, A., Pimm, D., & Gowar, N. (1985).
*Routes to/ roots of algebra.*Milton Keynes, UK: Open University Press.Google Scholar - Peirce, C. S. (1958).
*Collected papers, vol. I-VIII.*Cambridge, MA: Harvard University Press.Google Scholar - Radford, L. (2006). 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 Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education*(Vol. 1, pp. 2–21). Mérida, Mexico: PME-NA.Google Scholar - Radford, L. (2008a). 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. (2008b). Iconicity and contraction: A semiotic investigation of forms of algebraic generalizations of patterns in different contexts.
*ZDM*,*40*(1), 83–96.Google Scholar - Radford, L. (2010a). Layers of generality and types of generalization in pattern activities.
*PNA – Pensamiento Numérico Avanzado*,*4*(2), 37–62.Google Scholar - Radford, L. (2010b). The eye as a theoretician: Seeing structures in generalizing activities.
*For the Learning of Mathematics*,*30*(2), 2–7.Google Scholar - 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. (2012). On the development of early algebraic thinking.
*PNA*,*6*(4), 117–133.Google Scholar - Radford, L. (2014). The progressive development of early embodied algebraic thinking.
*Mathematics Education Research Journal*,*26*, 257–277.Google Scholar - Radford, L. (2015). Methodological aspects of the theory of objectification.
*Perspectivas Da Educação Matemática*,*9*(3), 129–141.Google Scholar - Radford, L., & Sabena, C. (2015). The question of method in a Vygotskian semiotic approach. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.),
*Approaches to qualitative research in mathematics education*(pp. 157–182). New York: Springer.Google Scholar - Radford, L., Arzarello, F., Edwards, L., & Sabena, C. (2017). The multimodal material mind: Embodiment in mathematics education. In J. Cai (Ed.),
*Compendium for research in mathematics education*(pp. 700–721). Reston, VA: NCTM.Google Scholar - Rideout, B. (2008).
*Pappus reborn. Pappus of Alexandria and the changing face of analysis and synthesis in late antiquity*. Master of Arts in History and Philosophy of Science Thesis. University of Canterbury.Google Scholar - Rivera, F. D. (2010). Second grade students’ preinstructional competence in patterning activity. In M. F. Pinto & T. F. Kawasaki (Eds.),
*Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 4, pp. 81–88). Belo Horizonte, Brazil: PME.Google Scholar - Vergel, R. (2015). Generalización de patrones y formas de pensamiento algebraico temprano [Pattern generalization and forms of early algebraic thinking].
*PNA. Special Issue on Generalization*,*9*(3), 193–215.Google Scholar - Viète, F. (1983).
*The analytic art.*(Translated by T. Richard Witmer). New York: Dover. (Original work published 1591).Google Scholar - Vološinov, V. N. (1973).
*Marxism and the philosophy of language.*Cambridge, MA: Harvard University Press.Google Scholar - Vygotsky, L. S. (1986).
*Thought and language.*(Edited by A. Kozulin). Cambridge: MIT Press.Google Scholar - Wagner, S., & Kieran, C. (Eds.). (1989).
*Research issues in the learning and teaching of algebra.*Reston, VA: NCTM.Google Scholar