Generalization
For the construction of knowledge in general and mathematical knowledge in particular, generalization is essential (Castro, Cañadas and Molina, 2010). Mason et al. (1985)
among other authors, highlighted this as the root of algebra. Through different approaches, most authors have pointed out that generalizing is recognizing a regularity that goes beyond the particular cases given and their representation, whether specifying the present similarity or broadening reasoning towards patterns, procedures, structures and relations between them (Kaput, 1999). Stephens, Ellis, Blanton and Brizuela (2017) differentiated generalization as process and product. The product is derived from three processes: (a) identify the regularity of a set of elements, (b) reason beyond the cases at hand, and (c) broaden the results beyond the particular cases. The product is the result of those processes.
In our study, we assumed generalization as the quintessential cross-cutting aspect in most notions of algebra. It can refer to the generalization of arithmetic relations, of patterns, of functional relations, of structures, of methods to solve problems, or rather to solving generalization problems or the representations used in generalization (with algebraic language being one of them).
Generalization allows more flexible mathematical thinking among students, helping them (a) dismiss irrelevant information, (b) adapt, adjust and reorganize prior experiences, (c) pay attention to certain ideas, skills and properties involving different situations, and (d) solve problems and understand different mathematical situations (Carpenter & Levi 2000; Carraher & Schliemann, 2015; Warren, 2005).
Carpenter, T. P., & Levi, L. (2000). Developing conceptions of algebraic reasoning in the primary grades (Res. Rep. 00-2). Madison, WI: National Center for Improving Student Learning and Achievement in Mathematics and Science.
Carraher, D. W., & Schliemann, A. D. (2015). Powerful ideas in elementary school mathematics. In L. D. English & D. Kirshner (Eds.), Handbook of international research in mathematics education (pp. 191- 218). New York, NY: Taylor &Francis.
Castro, E., Cañadas, M. C., & Molina, M. (2010). El razonamiento inductivo como generador de conocimiento matemático. Uno, 54, 55-67.
Kaput, J. J. (1999). Teaching and learning a new algebra. En E. Fennema y T. A. Romberg (Eds.), Mathematics classrooms that promote understanding (pp.133-155). Lawrence Erlbaum Associates.
Mason, J., Graham, A., Pimm, D., & Gowar, N. (1985). Routes to roots of algebra. London, United Kingdom: The Open University.
Stephens, A., Ellis, A., Blanton, M. L., & Brizuela, B. M. (2017). Algebraic thinking in the elementary and middle grades. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 386-420). Reston, VA: NCTM.
Warren, E. (2005). Young children’s ability to generalise the pattern rule for growing patterns. In H. Chick & J. Vincent (Eds.), Proceedings of the 29th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 305- 312). Melbourne, Australia: Program Committee.
