# Students’ use of variables and multiple representations in generalizing functional relationships prior to secondary school

- 1.2k Downloads
- 4 Citations

## Abstract

Algebra has been explicit in many school curriculum programs from the early years but there are competing views on what content and approaches are appropriate for different levels of schooling. This study investigated 12–13-year-old Australian students’ algebraic thinking in a hybrid environment of functional and equation-based approaches to learning algebra. This article reports on a survey of 102 students examining their generalization ability and knowledge of multiple representations and variables prior to formal study of algebraic expressions and equations at secondary school. Nearly half of the students demonstrated the ability to generalize explicitly with one fifth able to construct a symbolic functional rule. Nearly half were able to represent a real-life scenario of a linear relationship algebraically. There was little evidence yet of connecting a growing pattern or real-life scenario to an appropriate graphical representation. Level of pattern generalization ability was found to be associated with flexible thinking for exploring functional relationships in reverse and with explaining the inappropriateness of proportional reasoning for linear functions with a constant. Implications for the teaching and learning of algebra are presented.

## Keywords

Algebra teaching and learning Functional thinking Multiple representations Variables Middle years of schooling## References

- Australian Curriculum Assessment and Reporting Authority. (2009/2011, January). The Australian curriculum: Mathematics. Retrieved October 1, 2011, from http://www.australiancurriculum.edu.au/Mathematics/Curriculum/F-10
- Bardini, C., Pierce, R., & Vincent, J. L. (2013).
*First year university students’ understanding of functions: Over a decade after the introduction of CAS in Australian high schools, what is new?*Paper presented at the Lighthouse Delta 2013: The 9th Delta Conference on teaching and learning of undergraduate mathematics and statistics, Kiama, New South Wales.Google Scholar - Becker, J. R., & Rivera, F. (2004). An investigation of beginning algebra students’ ability to generalize linear patterns. In M. J. Høines & A. B. Fuglestad (Eds.),
*Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 1, pp. 286–293). Bergen: PME.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). Bergen, Norway: IGPME.Google Scholar - Brizuela, B. M., Blanton, M., Sawrey, K., Newman-Owens, A., & Murphy Gardiner, A. (2015). Children’s use of variables and variable notation to represent their algebraic ideas.
*Mathematical Thinking and Learning, 17*(1), 34–63. doi: 10.1080/10986065.2015.981939 - 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: The 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 - Chazan, D., & Yerushalmy, M. (2003). On appreciating the cognitive complexity of school algebra: research on algebra learning and directions of curricular change. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.),
*A research companion to principles and standards for school mathematics*(pp. 123–135). Reston: National Council of Teachers of Mathematics.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.Google Scholar - Creswell, J. W. (2007).
*Qualitative inquiry and research design: choosing among five approaches*(2nd ed.). Thousand Oaks: Sage.Google Scholar - Davis, J. D. (2007). Real-world contexts, multiple representations, student-invented terminology, and y-intercept.
*Mathematical Thinking and Learning, 9*(4), 387–418.CrossRefGoogle Scholar - Goldin, G., & Shteingold, N. (2001). Systems of representations and the development of mathematical concepts. In A. A. Cuoco & F. Curcio (Eds.),
*The roles of representation in school mathematics*(pp. 1–23). Reston: National Council of Teachers of Mathematics.Google Scholar - Gonzales, P., Pahlke, E., Guzman, J. C., Partelow, L., Kastberg, D., Jocelyn, L., & Williams, T. (2004).
*Pursuing excellence*: Eighth-grade mathematics and science achievement in the United States and other countries from the Trends in International Mathematics and Science Study (TIMSS) 2003 (NCES 2005–007). Washington, DC: US Department of Education. National Center for Education Statistics.Google Scholar - Greenes, C., Cavanagh, M., Dacey, L., Findell, C., & Small, M. (2001).
*Navigating through algebra in prekindergarten–grade 2*. Reston: The National Council of Teachers of Mathematics.Google Scholar - Kaput, J. (1993). The urgent need for proleptic research in the representation of quantitative relationships. In T. A. Romberg, T. P. Carpenter, & E. Fennema (Eds.),
*Integrating research on the graphical representation of functions*(pp. 279–312). Hillsdale: Lawrence Erlbaum Associates.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 - Kaput, J., Blanton, M., & Moreno, A. L. (2008). Algebra from a symbolization point of view. In J. Kaput, D. W. Carraher, & M. Blanton (Eds.),
*Algebra in the early grades*(pp. 19–56). Mahwah: Lawrence Erlbaum Associates.Google Scholar - Kieran, C. (2004). Algebraic thinking in the early grades: What is it?
*The Mathematics Educator, 8*(1), 139–151.Google Scholar - Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. K. Lester Jr. (Ed.),
*Second handbook of research on mathematics teaching and learning*(Vol. 2, pp. 707–762). Charlotte: National Council of Teachers of Mathematics, Information Age Publishing.Google Scholar - Knuth, E. J. (2000). Student understanding of the cartesian connection: An exploratory study.
*Journal for Research in Mathematics Education, 31*(4), 500–507.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., Townsend, B., & Barker, D. (2006). The reflective cycle of student error analysis.
*For the Learning of Mathematics, 26*(3), 33–38.Google 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, Canada.Google Scholar - Lesh, R. (1981). Applied mathematical problem solving.
*Educational Studies in Mathematics, 12*(2), 235–264.CrossRefGoogle 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, 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 - Mason, J. (2008). Making use of children’s powers to produce algebraic thinking. In J. Kaput, D. W. Carraher, & M. Blanton (Eds.),
*Algebra in the early grades*(pp. 57–94). Mahwah: Lawrence Erlbaum Associates. Google Scholar - McPhan, G., Morony, W., Pegg, J., Cooksey, R., & Lynch, T. (2008).
*Maths? Why not? Final report prepared for the Department of Education, Employment and Workplace Relations (DEEWR)*. Retrieved from http://www.aamt.edu.au/index.php/Activities-and-projects/Previous-projects/Maths-Why-Not - Miles, M. B., & Huberman, A. M. (1994).
*Qualitative data analysis*(2nd ed.). Thousand Oaks: Sage.Google Scholar - Moschkovich, J., Schoenfeld, A. H., & Arcavi, A. (1993). Aspects of understanding: On multiple perspectives and representations of linear functions and connections among them. In T. A. Romberg, T. P. Carpenter, & E. Fennema (Eds.),
*Integrating research on the graphical representation of functions*(pp. 69–100). Hillsdale: Lawrence Erlbaum Associates.Google Scholar - National Council of Teachers of Mathematics. (2000).
*Principles and standards for school mathematics*. Reston: NCTM.Google Scholar - O’Toole, J., & Beckett, D. (2010).
*Educational research: Creative thinking & doing*. Melbourne: Oxford 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 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. (2010a). Algebraic thinking from a cultural semiotic perspective.
*Research in Mathematics Education, 12*(1), 1–19.CrossRefGoogle Scholar - Radford, L. (2010b). Layers of generality and types of generalization in pattern activities.
*PNA, 4*(2), 37–62.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 - Rakes, C. R., Valentine, J. C., McGatha, M. B., & Ronau, R. N. (2010). Methods of instructional improvement in algebra: a systematic review and meta-analysis.
*Review of Educational Research, 80*(3), 372–400. doi: 10.3102/0034654310374880 - Rivera, F. (2010). Visual templates in pattern generalization activity.
*Educational Studies in Mathematics, 73*(3), 297–328.CrossRefGoogle Scholar - Romberg, T. A., Carpenter, T. P., & Fennema, E. (1993). Toward a common research perspective. In T. A. Romberg, T. P. Carpenter, & E. Fennema (Eds.),
*Integrating research on the graphical representation of functions*(pp. 1–9). Hillsdale: Lawrence Erlbaum Associates.Google 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 generalizing problems.
*Educational Studies in Mathematics, 20*(2), 147–164.CrossRefGoogle Scholar - Stacey, K., & Chick, H. (2004). What is the problem with algebra? In K. Stacey, H. Chick, & M. Kendal (Eds.),
*The future of the teaching and learning of algebra*(pp. 1–20). Boston: Kluwer Academic Publishers.Google Scholar - Sutherland, R. (2002).
*A comparative study of algebra curricula*: London: Qualifications and Curriculum Authority.Google Scholar - Swafford, J. O., & Langrall, C. W. (2000). Grade 6 students’ preinstructional use of equations to describe and represent problem situations.
*Journal for Research in Mathematics Education, 31*(1), 89–112.CrossRefGoogle Scholar - Thomson, S., De Bortoli, L., & Buckley, S. (2013).
*PISA 2012: How Australia measures up*. Melbourne: Australian Council for Educational Research.Google Scholar - Turner, J. C., & Meyer, D. K. (2009). Understanding motivation in mathematics: What is happening in classrooms? In K. Wentzel & D. Miele (Eds.),
*Handbook of motivation at school*(pp. 527–552). New York: Routledge.Google 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 - Victorian Curriculum and Assessment Authority. (2007/2011, February 22). Victorian essential learning standards: Mathematics. Retrieved May 2, 2012, from http://vels.vcaa.vic.edu.au/vels/maths.html
- Walkington, C., Petrosino, A., & Sherman, M. (2013). Supporting algebraic reasoning through personalized story scenarios: How situational understanding mediates performance.
*Mathematical Thinking and Learning, 15*(2), 89–120. doi: 10.1080/10986065.2013.770717 - Warren, E., & Cooper, T. (2008). Generalizing the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking.
*Educational Studies in Mathematics, 67*(2), 171–185.CrossRefGoogle 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., & Clarke, D. M. (2015). Developing students’ functional thinking in algebra through different visualizations of a growing pattern’s structure.
*Mathematics Education Research Journal*. Advance online publication.Google Scholar