# Evolution of a teaching approach for beginning algebra

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## Abstract

The article reports aspects of the evolution of a teaching approach over repeated trials for beginning symbolic algebra. The teaching approach emphasized the structural similarity between arithmetic and algebraic expressions and aimed at supporting students in making a transition from arithmetic to beginning algebra. The study was conducted with grade 6 students over 2 years. Thirty-one students were followed for a year, and data were analysed as they participated in the three trials conducted that year. Analysis of students’ written and interview responses as the approach evolved revealed the potential of the approach in creating meaning for symbolic transformations in the context of both arithmetic and algebra as well as making connections between arithmetic and symbolic algebra. Students by the end of the trials learnt to use their understanding of both procedures and a sense of structure of expressions to evaluate/simplify expressions and reason about equality/equivalence of expressions both in the arithmetic and the algebraic contexts.

## Keywords

Arithmetic Algebra Structure Teaching approach Term Equality Expressions## References

- Ainley, J., Bills, L., & Wilson, K. (2005). Designing spreadsheet based tasks for purposeful algebra.
*International Journal of Computers for Mathematical Learning, 10*(3), 191–215.CrossRefGoogle Scholar - Banerjee, R. (2000).
*Identification of learning difficulties in elementary algebraic concepts. Unpublished masters dissertation, Department of Education*. Delhi: University of Delhi.Google Scholar - Blanton, M., & Kaput, J. J. (2001). Algebrafying the elementary mathematics experience Part II: Transforming practice on a district-wide scale. In H. Chick, K. Stacey, & J. Vincent (Eds.),
*Proceedings of the 12th ICMI Study Conference: The Future of the Teaching and Learning of Algebra*(Vol. 1, pp. 87–95). Melbourne: The University of Melbourne.Google Scholar - Booth, L. R. (1984).
*Algebra: Childrens’ strategies and errors*. Windsor: NFER-Nelson.Google Scholar - Carpenter, T. and Franke, M. (2001). Developing algebraic reasoning in the elementary school: Generalization and proof. In H. Chick, K. Stacey and J. Vincent (Eds.),
*Proceedings of the 12th ICMI Study Conference: The Future of the Teaching and Learning of Algebra*(Vol. 1, pp. 155–162). Melbourne, Australia: The University of MelbourneGoogle Scholar - Cerulli, M. and Mariotti, M. A. (2001). Arithmetic and algebra, continuity or cognitive break? The case of Francesca. In Marja van-den Heuvel Pannhueizen (Ed.),
*Proceedings of the 25*th*Conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 225–232). Utrecht, The Netherlands: PMEGoogle Scholar - Chaachoua, H., Nicaud, J., Bronner, A. and Bouhineau, D. (2004). APLUSIX, a learning environment for algebra, actual use and benefits. ICME 10: 10th International Congress on Mathematical Education Copenhagen, Denmark.Google Scholar
- Chaiklin, S. and Lesgold, S. (1984). Prealgebra students’ knowledge of algebraic tasks with arithmetic expressions.
*Paper presented at the annual meeting of the American Educational Research Association*.Google Scholar - Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schuble, L. (2003). Design experiments in educational research.
*Educational Researcher, 32*(1), 9–13.CrossRefGoogle Scholar - Filloy, E., Rojano, T., & Rubio, G. (2001). Propositions concerning the resolution of arithmetical-algebra problems. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.),
*Perspectives in school algebra*(pp. 155–176). Dordrecht: Kluwer.Google Scholar - Fischbein, E. and Barash, A. (1993). Algorithmic models and their misuse in solving algebraic problems. In I. Hirabayashi, N. Nohda, K. Shigematsu and F. Lin (Eds.),
*Proceedings of the 17th International Group of the Psychology of Mathematics Education*(Vol.1, pp. 162–172). Tsukuba, Ibaraki, Japan: PME.Google Scholar - Fujii, T. and Stephens, M. (2001). Fostering an understanding of algebraic generalization through numerical expressions: the role of quasi-variables. In H. Chick, K. Stacey and J. Vincent (Eds.),
*Proceedings of the 12th ICMI Study Conference: The Future of the Teaching and Learning of Algebra*(Vol. 1, pp. 258–264). Melbourne, Australia: The University of Melbourne.Google Scholar - Kieran, C. (1992). Learning and teaching of school algebra. In D. A. Grows (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 390–419). New York: Macmillan.Google Scholar - Kieran, C. (2004). The core of algebra. In K. Stacey, H. Chick, & M. Kendal (Eds.),
*The future of teaching and learning of algebra: The 12th ICMI Study*(pp. 21–34). Dordrecht: Kluwer.CrossRefGoogle Scholar - Kirshner, D. (2001). The structural algebra option revisited. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.),
*Perspectives on school algebra*(pp. 83–98). Dordrecht: Kluwer.Google Scholar - Kuchemann, D. E. (1981). Algebra. In K. M. Hart, M. L. Brown, D. E. Kuchemann, D. Kerslake, G. Ruddock, & M. McCartney (Eds.),
*Children’s understanding of mathematics: 11–16*(pp. 102–119). London: John Murray.Google Scholar - Liebenberg, R. Linchevski, L. Sasman, M. C. and Olivier, A. (1999). Focusing on the structural aspects of numerical expressions. In J. Kuiper (Ed.),
*Proceedings of the 7th Annual Conference of the Southern African Association for Research in Mathematics and Science Education*(SAARMSE) (pp. 249–256). Harare, Zimbabwe.Google Scholar - Liebenberg, R. Sasman, M. and Olivier, A. (1999). From numerical equivalence to algebraic equivalence. In
*Proceedings of the Fifth Annual Conference of the Association for Mathematics Education in South Africa*(Vol. 2, pp. 173–183), Port Elizabeth: Port Elizabeth Technikon.Google Scholar - Linchevski, L., & Herscovics, N. (1996). Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations.
*Educational Studies in Mathematics, 30*(1), 39–65.CrossRefGoogle Scholar - Linchevski, L., & Livneh, D. (1999). Structure sense: The relationship between algebraic and numerical contexts.
*Educational Studies in Mathematics, 40*(2), 173–196.CrossRefGoogle Scholar - Linchevski, L., & Livneh, D. (2002). The competition between numbers and structure: Why expressions with identical algebraic structures trigger different interpretations.
*Focus on Learning Problems in Mathematics, 24*(2), 20–37.Google Scholar - Livneh, D. and Linchevski, L. (2007). Algebrification of arithmetic: Developing algebraic structure sense in the context of arithmetic. In J. Woo, H. Lew, K. Park and D. Seo (Eds.),
*Proceedings of the 31st Conference of the Psychology of Mathematics Education*(Vol. 3, pp. 217–225). Seoul, Korea: PME.Google Scholar - MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation:11–15.
*Educational Studies in Mathematics, 33*(1), 1–19.CrossRefGoogle Scholar - Malara, N and Iaderosa, R. (1999). The interweaving of arithmetic and algebra: Some questions about syntactic and structural aspects and their teaching and learning. In Inge Schwank (Ed.),
*Proceedings of the First Conference of the European Society for Research in Mathematics Education*(Vol. 2, pp. 159–171). Forschungsinstitut fuer Mathematikdidaktik, Osnabrueck.Google Scholar - Mason, J., Graham, A., Pimm, D., & Gowar, N. (1985).
*Routes to/roots of algebra*. Milton Keynes: The Open University Press.Google Scholar - Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin.
*Educational Studies in Mathematics, 22*(1), 1–36.CrossRefGoogle Scholar - Tabach, M., & Friedlander, A. (2008). Understanding equivalence of symbolic expressions in a speardsheet-based environment.
*International Journal of Computers for Mathematical Learning, 13*, 27–46.CrossRefGoogle Scholar - Tall, D., Thomas, M., Davis, G., Gray, E. M., & Simpson, A. (2000). What is the object of encapsulation of a process?
*The Journal of Mathematical Behavior, 18*(2), 223–241.CrossRefGoogle Scholar - Thompson, P. W. and Thompson, A. G. (1987). Computer presentations of structure in algebra: In J. Bergeron, N. Herscovics and C. Kieran (Eds.),
*Proceedings of the Eleventh Annual Meeting of the International Group of the Psychology of Mathematics Education*(Vol. 1. pp. 248–254). Montreal, Canada: PME.Google Scholar