# Generalising the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking

- 862 Downloads
- 32 Citations

## Abstract

A common approach used for introducing algebra to young adolescents is an exploration of visual growth patterns and expressing these patterns as functions and algebraic expressions. Past research has indicated that many adolescents experience difficulties with this approach. This paper explores teaching actions and thinking that begins to bridge many of these difficulties at an early age. A teaching experiment was conducted with two classes of students with an average age of eight years and six months. From the results it appears that young students are capable not only of thinking about the relationship between two data sets, but also of expressing this relationship in a very abstract form.

## Keywords

Algebraic thinking Elementary students Visual growth patterns Semiotics Student thinking Teaching actions## Notes

### Acknowledgement

This research is funded by a grant from the Australian Research Council (No. LP0348820)

## References

- Arzarello, F. (1998). The role of natural language in prealgebraic and algebraic thinking. In H. Steinbring, M. Bussi, & A. Sierpinska (Eds.),
*Language and communication in the mathematics classroom*(pp. 249–261). Reston: National Council of Teachers of Mathematics.Google Scholar - Bennett, A. (1988). Visual thinking and number relationships.
*Mathematics Teacher, 81*(4), 267–272.Google Scholar - Blanton, M., & Kaput, J. (2004). Elementary grades students’ capacity for functional thinking. In M. J. Hoynes & A. B. Fuglestad (Eds.),
*Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education*(vol. 2, pp. 135–142). Oslo.Google Scholar - Carpenter, T. P., Franke, M. L., & Levi, L. W. (2003).
*Thinking mathematically: Integrating arithmetic and algebra in elementary school.*Portsmouth: Heinemann.Google Scholar - Cobb, P. (2000). Conducting teaching experiments in collaboration with teachers. In A. Kelly & R. A. Lesh (Eds.),
*Handbook of research design in mathematics and science education*(pp. 307–333). Mahwah: Erlbaum.Google Scholar - Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research.
*Educational Researcher, 32*(1), 9–13.CrossRefGoogle Scholar - Confrey, J., & Lachance, A. (2000). Transformative teaching experiments through conjecture-drive research design. In A. Kelly & R. A. Lesh (Eds.),
*Handbook of research design in mathematics and science education*(pp. 231–265). Mahwah: Erlbaum.Google Scholar - Ernest, P. (2002).
*A semiotic perspective of mathematical activity*. Paper presented at PME 26 2002, Norwich.Google Scholar - Johnassen, D. H., Beissner, K., & Yacci, M. (1993).
*Structural knowledge: Techniques for representing, conveying, and acquiring structural knowledge*. Hillsdale: Erlbaum.Google Scholar - Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DRN based instruction. In S. Campbell & R. Zaskis (Eds.),
*Learning and teaching number theory, journal of mathematical behavior*(pp. 185–212). New Jersey: Albex.Google Scholar - Kaput, J., & Blanton, M. (2001). Algebrafying the elementary mathematics experience. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.),
*The future of the teaching and learning of algebra. Proceedings of the 12th ICMI study conference*(vol. 1, pp. 344–352). Melbourne: ICMI.Google Scholar - MacGregor, M., & Stacey, K. (1996). Origins of students’ interpretation of algebraic notation. In L. Puig & A. Gutierrez (Eds.),
*Proceedings of the 20th International Conference for Psychology of Mathematics Education*(vol. 3, pp. 289–296). Valencia.Google Scholar - Malara, N., & Navarra, G. (2003).
*ArAl Project: Arithmetic pathways towards favouring pre-algebraic thinking*. Bologna: Pitagora Editrice.Google Scholar - Otte, M. (2001).
*Mathematical expistemology from a semiotic point of view*. Paper presented in the Discussion Group for Semiotics in Mathematics Education at PME 25, Utrecht.Google Scholar - Otte, M. (2006). Mathematical epistemology from a Peircean semiotic point of view.
*Educational Studies in Mathematics, 61*, 11–38.CrossRefGoogle Scholar - Peirce, C. S. (1960).
*Collected papers*. Cambridge: Harvard University Press.Google Scholar - Piaget, J. (1970).
*Genetic epistemology*. New York: Columbia University Press.Google Scholar - Presmeg, N. (1997).
*A semiotic framework for linking cultural practice and classroom mathematics*. ERIC Document Reproduction Service No. Ed 425 257.Google Scholar - Radford, L. (2001).
*On the relevance of semiotics in mathematics education*. Paper presented in the Discussion Group for Semiotics in Mathematics Education at PME 25*,*Utrecht.Google Scholar - Redden, T. (1996). “Wouldn’t it be good if we had a symbol to stand for any number”: The relationship between natural language and symbolic notation in pattern description. In L. Puig & A. Gutierrez (Eds.),
*Proceedings of the 20th International Conference, Psychology of Mathematics Education*(vol. 4, pp. 195–202). Valencia.Google Scholar - Saenz-Ludlow, A. (2001).
*Classroom mathematics discourse as an evolving interpreting game*. Paper presented in the Discussion Group for Semiotics in Mathematics Education at PME 25, Utrecht.Google Scholar - Saenz-Ludlow, A. (2006). Classroom interpreting games with an illustration.
*Educational Studies in Mathematics, 61*(2), 183–218.CrossRefGoogle Scholar - Sfard, A. (1991). On the dual nature of mathematical concepts: Reflections on processes and objects as different sides of the same coin.
*Educational Studies in Mathematics, 22*(1), 191–228.CrossRefGoogle Scholar - Stacey, K., & MacGregor, M. (1995). The effect of different approaches to algebra on students’ perceptions of functional relationships.
*Mathematics Education Research Journal, 7*, 69–85.Google Scholar - Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. E. Kelly & R. A. Lesh (Eds.),
*Handbook of research design in mathematics and science education*(pp. 267–306). Mahwah: Erlbaum.Google Scholar - Vygotsky, L. (1934/1986).
*Thought and language*. Cambridge: MIT Press.Google Scholar - Warren, E. (1996).
*Interaction between instructional approaches, students’ reasoning processes, and their understanding of elementary algebra*. Dissertation, Queensland University of Technology.Google Scholar - Warren, E. (2000). Visualisation and the development of early understanding in algebra. In T. Nakahara & M. Koyama (Ed.),
*Proceedings of the 24th conference of the International Group for the Psychology of Mathematics Education*(vol. 4, pp. 273–280). Hiroshima.Google Scholar - Warren, E. (2006). Learning comparative mathematical language in the elementary school: A longitudinal study.
*Educational Studies in Mathematics, 62*(2), 169–189.CrossRefGoogle Scholar