# Developing students’ functional thinking in algebra through different visualisations of a growing pattern’s structure

## Abstract

Spatial visualisation of geometric patterns and their generalisation have become a recognised pathway to developing students’ functional thinking and understanding of variables in algebra. This design-based research project investigated upper primary students’ development of explicit generalisation of functional relationships and their representation descriptively, graphically and symbolically. Ten teachers and their classes were involved in a sequence of tasks involving growing patterns and geometric structures over 1 year. This article focuses on two aspects of the study: visualising the structure of a geometric pattern in different ways and using this to generalise the functional relationship between two quantifiable aspects (variables). It was found that in an initial assessment task (*n* = 222), students’ initial visualisations could be categorised according to different types and some of these were more likely to lead either to recursive or explicit generalisation. In a later task, a small number of students demonstrated the ability to find more than one way to visualise the same geometric structure and thus represent their explicit generalisations as different but equivalent symbolic equations (using pronumerals). Implications for the teaching of functional thinking in middle-school algebra are discussed.

## Keywords

Algebra Functional thinking Pattern generalisation Visualisation Middle years of schooling## Notes

### Acknowledgments

The authors would like to acknowledge with appreciation the teacher participants and School Mathematics Leaders from the *Contemporary Teaching and Learning of Mathematics* project who contributed to the study on which this article is based. The CTLM project was conducted by the Mathematics Teaching and Learning Research Centre, Australian Catholic University, and funded by the Catholic Education Office, Melbourne, for 5 years (2008–2012).

## References

- Australian Curriculum Assessment and Reporting Authority. (2009, January, 2011). The Australian curriculum: mathematics Retrieved October 1, 2011, from http://www.australiancurriculum.edu.au/Mathematics/Curriculum/F-10.
- Baumgartner, E., Bell, P., Hoadley, C., Hsi, S., Joseph, D., Orrill, C., & Tabak, I. (2003). Design-based research: an emerging paradigm for educational inquiry.
*Educational Researcher, 32*(1), 5–8.CrossRefGoogle Scholar - Becker, J. R., & Rivera, F. (2006).
*Sixth graders’ figural and numerical strategies for generalizing patterns in algebra.*Paper presented at the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.Google Scholar - Blanton, M., & Kaput, J. (2004). Elementary grades students’ capacity for functional thinking. In M. Høines & A. Fuglestad (Eds.),
*Proceedings of the 28th annual meeting of International Group for the Psychology of Mathematics Education*(pp. 135-142): IGPME.Google Scholar - Cai, J., & Moyer, J. (2008). Developing algebraic thinking in earlier grades: some insights from international comparative studies. In C. Greenes & R. Rubenstein (Eds.),
*Algebra and algebraic thinking in school mathematics*(pp. 169–180). Reston: National Council of Teachers of Mathematics.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 - Cobb, P. (2000). Conducting teaching experiments in collaboration with teachers. In A. E. Kelly & R. A. Lesh (Eds.),
*Handbook of research design in mathematics and science education*(pp. 307–333). Mahwah: Lawrence Erlbaum Associates.Google Scholar - Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit.
*Educational Studies in Mathematics, 26*(2/3), 135–164.CrossRefGoogle Scholar - English, L. D., & Warren, E. (1998). Introducing the variable through pattern exploration.
*The Mathematics Teacher, 91*(2), 166–170.Google Scholar - Fischbein, E. (1987).
*Intuition in science and mathematics: an educational approach*. Dordrecht: Reidel.Google Scholar - Friel, S. N., & Markworth, K. A. (2009). A framework for analyzing geometric pattern tasks.
*Mathematics Teaching in the Middle School, 15*(1), 24–33.Google Scholar - Gravemeijer, K., & van Eerde, D. (2009). Design research as a means for building a knowledge base for teachers and teaching in mathematics education.
*The Elementary School Journal, 109*(5), 510–524.CrossRefGoogle Scholar - Greenes, C., Cavanagh, M., Dacey, L., Findell, C., & Small, M. (2001).
*Navigating through algebra in prekindergarten—grade 2*. Reston: National Council of Teachers of Mathematics.Google Scholar - Hershkowitz, R., Arcavi, A., & Bruckheimer, M. (2001). Reflections on the status and nature of visual reasoning—the case of the matches.
*International Journal of Mathematical Education in Science and Technology, 32*(2), 255–265.CrossRefGoogle Scholar - Hiebert, J., & Stigler, J. W. (2000). A proposal for improving classroom teaching: lessons from the TIMSS video study.
*The Elementary School Journal, 101*(1), 3–20.CrossRefGoogle Scholar - Hunter, J. (2010). ‘You might say you’re 9 years old but you’re actually
*B*years old because you’re always getting older’: facilitating young students’ understanding of variables. In L. Sparrow, B. Kissane, & C. Hurst (Eds.),*Shaping the future of mathematics education (proceedings of the 33rd annual conference of the mathematics education research group of Australasia)*(Vol. 1, pp. 256–263). Fremantle: MERGA.Google Scholar - Jurdak, M. E., & Mouhayar, R. R. E. (2014). Trends in the development of student level of reasoning in pattern generalization tasks across grade level.
*Educational Studies in Mathematics, 85*, 75–92. doi: 10.1007/s10649-013-9494-2.CrossRefGoogle Scholar - Kaput, J. J. (1999). Teaching and learning a new algebra. In E. Fennema & T. Romberg (Eds.),
*Mathematics classrooms that promote understanding*(pp. 133–155). Mahwah: Erlbaum.Google Scholar - Kaput, J. J. (2008). What is algebra? What is algebraic reasoning? In J. L. Kaput, D. W. Carraher, & M. L. Blanton (Eds.),
*Algebra in the early grades*(pp. 5–17). New York: Taylor & Francis Group.Google Scholar - Kieran, C. (2004). Algebraic thinking in the early grades: what is it?
*The Mathematics Educator, 8*(1), 139–151.Google Scholar - Kruteskii, V. (1976).
*The psychology of mathematical ability in school children*. Chicago: University of Chicago Press.Google Scholar - Kuchemann, D. (2010). Using patterns generically to see structure.
*Pedagogies, 5*(3), 233–250.CrossRefGoogle Scholar - Lannin, J. K. (2005). Generalization and justification: the challenge of introducing algebraic reasoning through patterning activities.
*Mathematical Thinking and Learning, 7*(3), 231–258.CrossRefGoogle Scholar - Lee, L., & Freiman, V. (2004).
*Tracking primary students’ understanding of patterns.*Paper presented at the Annual Meeting - Psychology of Mathematics & Education of North America, Toronto.Google Scholar - MacGregor, M., & Stacey, K. (1995). The effect of different approaches to algebra on students’ perceptions of functional relationships.
*Mathematics Education Research Journal, 7*(1), 69–85.CrossRefGoogle Scholar - Markworth, K. A. (2010).
*Growing and growing: promoting functional thinking with geometric growing patterns.*Unpublished doctoral dissertation, University of North Carolina at Chapel Hill. Available from ERIC (ED519354).Google Scholar - Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.),
*Approaches to algebra: perspectives for research and teaching*(pp. 65–86). Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar - Moss, J., Beatty, R., Barkin, S., & Shillolo, G. (2008). "What is your theory? What is your rule? Fourth graders build an understanding of function through patterns and generalising problems. In C. Greenes & R. Rubenstein (Eds.),
*Algebra and algebraic thinking in school mathematics (70th yearbook of the National Council of Teachers of Mathematics)*(pp. 155–168). Reston: NCTM.Google Scholar - Mullis, I. V. S., Martin, M. O., Gonzalez, E. J., & Chrostowski, S. J. (2004). TIMSS 2003 international mathematics report
*Findings from IEAs Trends in International Mathematics and Science Study at the fourth and eighth grades.*: Lynch School of Education, Boston College: TIMSS and PIRLS International Study Center.Google Scholar - National Council of Teachers of Mathematics. (2000).
*Principles and standards for school mathematics*. Reston: NCTM.Google Scholar - Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics. In A. Gutiérrez & P. Boero (Eds.),
*Handbook of research on the psychology of mathematics education*(pp. 205–235). Rotterdam: Sense Publishers.Google Scholar - Radford, L. (2010). Algebraic thinking from a cultural semiotic perspective.
*Research in Mathematics Education, 12*(1), 1–19.CrossRefGoogle Scholar - Radford, L. (2014). The progressive development of early embodied algebraic thinking.
*Mathematics Education Research Journal, 26*(2), 257–277.CrossRefGoogle Scholar - Radford, L., & Peirce, C. S. (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 conference of the international group for the psychology of mathematics education, North American chapter*(Vol. 1, pp. 2–21). Mérida: Universidad Pedagógica Nacional.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*(5), 507–530.Google Scholar - Rivera, F. (2010). Visual templates in pattern generalization activity.
*Educational Studies in Mathematics, 73*(3), 297–328.CrossRefGoogle Scholar - Siegler, R. S. (2000). The rebirth of children’s learning.
*Child Development, 71*(1), 26–35.CrossRefGoogle Scholar - Smith, E. (2008). Representational thinking as a framework for introducing functions in the elementary curriculum. In J. L. Kaput, D. W. Carraher, & M. L. Blanton (Eds.),
*Algebra in the early grades*(pp. 133–160). New York: Taylor & Francis Group.Google Scholar - Stacey, K. (1989). Finding and using patterns in linear generalising problems.
*Educational Studies in Mathematics, 20*(2), 147–164.CrossRefGoogle Scholar - Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. F. Coxford & A. P. Shulte (Eds.),
*The ideas of algebra, K-12: NCTM 1988 yearbook*(pp. 8–19). Reston: National Council of Teachers of Mathematics.Google Scholar - Warren, E. (2000). Visualisation and the development of early understanding in algebra. In T. Nakahara & M. Koyama (Eds.),
*Proceedings of the 24th conference of the international group for the psychology of mathematics education*(Vol. 4, pp. 273–280). Hiroshima: PME.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*(2), 171–185.CrossRefGoogle Scholar - Warren, E., & Pierce, R. (2004). Learning and teaching algebra. In B. Perry, G. Anthony, & C. Diezmann (Eds.),
*Research in mathematics education in Australasia 2000-2003*(pp. 291–312). Flaxton: Post Pressed.Google Scholar - Wilkie, K. J. (2014). Upper primary school teachers' mathematical knowledge for teaching functional thinking in algebra.
*Journal of Mathematics Teacher Education, 17*(5), 397–428.Google Scholar - Wilkie, K. J., & Clark, D. M. (2015). Pathways to professional growth: investigating upper primary school teachers’ perspectives on learning to teach algebra.
*Australian Journal of Teacher Education, 40*(4), 87–118.Google Scholar - Wright, V. (1997). Assessing mathematical processes in algebra. Unpublished Research dissertation. University of Waikato.Google Scholar