elementary school number curriculum for students to gain experience with these more abstract forms of equivalence. It would be helpful, for example, if the curriculum included perimeter problems in which students were asked to calculate the perimeter of a 7-by-4 rectangle in three ways that yield equivalent expressions: 2(7+4), (2×7)+(2×4), and 7+7+4+4. Such situations are ideal for initiating discussions of the equivalence of arithmetic expressions and of the properties underlying that equivalence. Because such occasions are currently quite rare in the part of the curriculum dealing with number, however, notions of equivalence generally have to be further developed when arithmetic is extended to algebra.

Students’ notions of equality and equivalence, as well as their deepening understanding of the relationship between operations and their inverses, are developed through the transformational activities of algebra, especially those related to simplifying expressions and solving equations. A great deal of research has been carried out on this sphere of algebraic activity.

Performing the same operation on both sides of the equation is an important formal equation-solving procedure. This method, however, is often not the first one taught to students. Trial-and-error substitution of values for the unknown and other informal techniques such as the cover-up method and working backwards (undoing) are used to introduce equation solving (see Box 8–6).

In one comparison of the cover-up method with the formal procedure of performing the same operation on both sides of the equation in six seventh-grade classes, the students who learned to solve equations by means of the cover-up method performed better than those who learned both methods in close proximity.^{56} The students who learned to solve equations using only the formal method performed worse than those who learned both methods. These findings suggest that students learning formal methods of equation solving may benefit from well-timed prior instruction in the informal technique of “cover up.”

Another study found that students who were entering their first algebra course showed one of two preferences when solving simple linear equations in which there was only one operation: Some used trial-and-error substitution; the others used undoing.^{57} For two-step equations involving two operations such as 2*x*—5=11, the latter group of students spontaneously extended their right-to-left undoing technique: Take 11, add 5 to it, then divide by 2.