Teaching the (Elusive) Algebra


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“If there is a heaven for school subjects, algebra will never go there. It is the one subject in the curriculum that has kept children from finishing high school, from developing their special interests and from enjoying much of their home study work. It has caused more family rows, more tears, more headaches, and more sleepless nights than any other school subject.” (NCTM yearbook, 2008, p. 3)

Four Conceptions About Algebra

There are many conceptions about algebra in the literature. According to Usiskin (1988), there are four fundamental conceptions of algebra.

First Conception: Algebra is considered as a generalized arithmetic. In this conception, a variable is considered as a pattern generalizer. The key instructions for students in this conception are “translate and generalize”. For example, the use of the four fundamental operations in arithmetic when used can be generalized. For example, understanding the arithmetic expression “5 – 2” as “5 take away 2″ is not a generalization but ‘5 – 2” should be understood as the difference, so that “x – y” would make sense. Another example is  the arithmetic expression -5 x 3 = -15, could be generalized as −a × b=−ab.

Second conception: This suggests that algebra is a study of procedures for solving certain kinds of problems. In this conception, one has to find a generalization for a particular question and solve it for the unknown. For example, if we consider the problem “When 3 is added to 5 times a certain number, the sum is 40. Find the number.” (Usiskin, 1988, p. 12). The problem translated into algebraic language will be an equation of the form “ 5 x + 3 = 40 ” with a solution of x = 7.4. Therefore, in this conception, variables are either unknowns or constants. The key instruction here is “simplify and solve”.

Third conception: Algebra is considered to be the study of relationships among quantities. Here, variables really tend to vary. For example, a formula for the area of a rectangle is A = LW . This is a relationship among three quantities. There is no feeling of an unknown here. Instead all A, L, and W can take many values. In such an example, no solution process is involved.

Fourth conception: This accepts algebra as the study of structures. Under this notion, the variable is little more than an arbitrary symbol. The variable will become an arbitrary object in a structure related by certain properties. This is the view of variable found in abstract algebra. Here, the variable neither acts as an unknown nor is it an argument.

References:

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: 1988 yearbook (pp. 8-19). National Council of Teachers of Mathematics

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