More often than not, I notice that students tend to have difficulty finding meaning in doing algebra when they are first introduced to its concept at the secondary level. It is rather common for them to struggle in making sense of what algebra is about.
The Meaning of Algebra
The attempts made at introducing lower secondary students to algebra seem to be abrupt and overwhelming, which is very much akin to a “culture shock” (Lee, 1996, p. 87). This manner of introducing algebra places an overemphasis on the use of algebraic symbols in expressing and manipulating generalities (Mason, 1996) without much or any attempt to make sense of these symbols, let alone understand the purpose of these symbols in the broader context of algebra.
While I acknowledge that the skills of expressing and manipulating algebraic symbols are important, have we however become too obsessed with the functions of algebra that we neglect its meaning and purpose in the first place? Have we placed too much emphasis on the “how” and inadvertently neglected the “why” behind algebra?
A Culture of Algebraic Compliance
Even as far back as the late 1980s and early 1990s, it was found that a disproportionately high amount of focus has been placed on the “how” rather than the “why” behind algebra (Mason, 1996).
In other words, a heavy emphasis was bestowed on behaviours as opposed to the thinking processes in algebra. This has led to, what I would term as, a “culture of algebraic compliance”. Such is a culture that encourages systematic compliance to the algebraic procedures in order to achieve an ultimate goal of solving a problem.
While such a culture has its merits, it unfortunately takes away the sense of meaning attached to algebra. This is akin to being able to execute the drills of a sport, such as badminton, very well but not knowing why the drills need to be executed in a particular way and for what purpose they serve in a game.
So what exactly can we do about this culture of algebraic compliance?
Understanding Algebra through Pattern Generalization
I contend that the issue lies with the way algebra is perceived and introduced.
Algebra, at its core, is generalized arithmetic (Usiskin, 1988; Sfard, 1995). By generalized arithmetic, I mean perceiving the variables as “pattern generalizers” (Usiskin, 1988, p. 9). In other words, the variables are used to represent relationships in a pattern.
For example, the linear numerical increasing pattern of 2, 5, 8, 11, etc. can be represented by a functional rule, which governs the relationship between the term position and the term itself. Through means of inductive reasoning, a conjecture can be formed that relates the term position and the term itself. This conjecture can be extended to all cases in order to obtain a generalization of , where denotes the nth term and n, the term position.
Benefits of Pattern Generalization
Using such a pattern generalization approach has two benefits. First, it helps students recognize that variables are generalized arithmetic. Second, the variables serve a purpose in representing a relationship between the term position and the term itself. This is a very powerful move as it brings into picture the purpose of variables in algebra, thus enabling students to make sense of them.
The attachment of meaning to algebra cannot be substituted by merely performing algebraic procedures. What is needed, therefore, is a deliberate exposition of pattern generalization activities in order to foster such algebraic thinking. This will be crucial if we want to move from a “culture of algebraic compliance” to a “culture of algebraic sense making”.
In conclusion, an introduction to algebra is somewhat similar to assimilating into a culture (Lee, 1996). However, the question beckons: What type of algebraic culture do we want to create? A culture that is entirely focused on the “how” or a culture that strikes a balance between the “why” and the “how” of algebra?
These are important questions that we need to consider seriously. I would definitely advocate using pattern generalization to introduce algebra at the secondary level as there are strong historical, philosophical and psychological groundings and has been proven to be pedagogically advantageous (Lee, 1996).
Lee, L. (1996). An initiation into algebraic culture through generalising activities. In N. Bednarz, C. Kieran & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 87–106). Dordrecht, The Netherlands: Kluwer Academic Publishers.
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, The Netherlands: Kluwer Academic Publishers.
Usiskin, Z. (1988). Conceptions of school algebra and uses of variable. In A. F. Coxford & A. P. Shulte (Eds.), The ideas of algebra, K-12 (1988 Yearbook of the National Council of Teachers of Mathematics, pp. 8–19). Reston, VA: NCTM.