Educational Studies in Mathematics

, Volume 92, Issue 1, pp 1–19 | Cite as

An early algebra approach to pattern generalisation: Actualising the virtual through words, gestures and toilet paper

  • Francesca Ferrara
  • Nathalie Sinclair


This paper focuses on pattern generalisation as a way to introduce young students to early algebra. We build on research on patterning activities that feature, in their work with algebraic thinking, both looking for sameness recursively in a pattern (especially figural patterns, but also numerical ones) and conjecturing about function-based relationships that relate variables. We propose a new approach to pattern generalisation that seeks to help children (grades 2 and 3) work both recursively and functionally, and to see how these two modes are connected through the notion of variable. We argue that a crucial change must occur in order for young learners to develop a flexible algebraic discourse. We draw on Sfard’s (2008) communication approach and on Châtelet’s (2000) notion of the virtual in order to pursue this argument. We also root our analyses within a new materialist perspective that seeks to describe phenomena in terms of material entanglement, which include, in our classroom research context, not just the children and the teacher, but also words, gestures, physical objects and arrangements, as well as numbers, operations and variables.


Discourse Generalisation Gesture Materialism Patterns Variable Virtual 


  1. 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). New York, NY: Springer.CrossRefGoogle Scholar
  2. Carraher, D. W., Martinez, M., & Schliemann, A. D. (2008). Early algebra and mathematical generalization. ZDM The International Journal on Mathematics Education, 40(1), 3–22.CrossRefGoogle Scholar
  3. Carraher, D. W., Schliemann, A. D., & Schwarz, J. L. (2008). Early algebra ≠ algebra early. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 235–272). New York, NY: Lawrence Erlbaum Associates.Google Scholar
  4. Châtelet, G. (2000/1993). Les enjeux du mobile [Figuring space: Philosophy, Mathematics and Physics]. (R. Shore & M. Zagha, Trans.). Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar
  5. de Freitas, E., & Sinclair, N. (2014). Mathematics and the body. Material entanglements in the classroom. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  6. Deleuze, G. (1994). Difference and repetition. (P. Patton, Trans.). New York: Columbia University Press.Google Scholar
  7. Kim, D. J., Ferrini-Mundi, J., & Sfard, A. (2012). How does language impact the learning of mathematics? Comparison of English and Korean speaking university students’ discourses on infinity. International Journal of Educational Research, 51–52, 86–108.CrossRefGoogle Scholar
  8. Moss, J. & Beatty, R. (2010). Knowledge Building and Mathematics: Shifting the Responsibility for Knowledge Advancement and Engagement. Canadian Journal of Learning and Technology, 36(1). Retrieved from
  9. Radford, L. (2008). 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.CrossRefGoogle Scholar
  10. Radford, L. (2010). Elementary forms of algebraic thinking in young students. 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. 73–80). Belo Horizonte: PME.Google Scholar
  11. Radford, L. (2012). On the development of early algebraic thinking. PNA, 6(4), 117–133.Google Scholar
  12. Radford, L. (2014). Towards an embodied, cultural, and material conception of mathematics cognition. ZDM – The International Journal on Mathematics Education, 46, 349–361.CrossRefGoogle Scholar
  13. Rivera, F. D. (2011). Toward a visually-oriented school mathematics curriculum. Research, theory, practice, and issues. New York: Springer.CrossRefGoogle Scholar
  14. Roth, W.-M. (2011). Geometry as objective science in elementary classrooms: Mathematics in the flesh. New York: Routledge.Google Scholar
  15. Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge, England: Cambridge University Press.CrossRefGoogle Scholar
  16. Sfard, A. (2009). What’s all the fuss about gestures? A commentary. Educational Studies in Mathematics, 70(2), 191–200.CrossRefGoogle Scholar
  17. Sinclair, N., de Freitas, E., & Ferrara, F. (2013). Virtual encounters: The murky and furtive world of mathematical inventiveness. ZDM The International Journal on Mathematics Education, 45(2), 239–252.CrossRefGoogle Scholar
  18. Slavit, D. (1997). An alternate route to the reification of function. Educational Studies in Mathematics, 33(3), 259–281.CrossRefGoogle Scholar
  19. 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 Erlbaum Associates.Google Scholar
  20. Stylianides, A. J., & Stylianides, G. J. (2013). Seeking research-grounded solutions to problems of practice: Classroom-based interventions in mathematics education. ZDM The International Journal on Mathematics Education, 45(3), 333–341.CrossRefGoogle Scholar
  21. Tatha, D. (1980). About geometry. For the Learning of Mathematics, 1(1), 2–9.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Dipartimento di Matematica “Giuseppe Peano”Università di TorinoTorinoItaly
  2. 2.Simon Fraser UniversityBurnabyCanada

Personalised recommendations