equivalence transformations, but soon afterwards most remembered only the rules, and some did not even remember that much.58 According to another study, recency of experience seems to account best for students’ ability to carry out certain transformational activities.59 Regardless of the teaching approach used, whether reform-based or traditional (i.e., oriented toward symbol manipulation), students’ ability to carry out successfully the transformational activities of algebra by the end of their high school career appears to be severely limited. This result has been found repeatedly, even in recent studies: “Few students [can] do the kinds of basic symbolic calculation that are common fare on college-admission and placement tests.”60

The Role of Technology

Transformational activities of algebra have benefited substantially less than representational activities from the use of computer technology to help develop meaning and skill. Nevertheless, a few researchers have used graphing technology as a means of providing a foundation for simplifying expressions and solving equations.61 This research is based on the idea that an important aspect of students’ mathematical development is their ability to support the symbolic transformations of algebraic objects by means of visual representations. For instance, the graphs of two functions can be added geometrically to arrive at a third graph whose expression is their algebraic sum. Equations also can be solved by graphing the functional expressions on each side of an equation on a computer or graphing calculator, zooming in on the point of intersection, and finding the approximate value of x for which the two functions are equal.

In one study the students had become so skilled at graphing linear functions by focusing on the y-intercept and slope that they could do it mentally (see Box 8–7). Although most teachers of algebra would be happy if a student could solve equations mentally by visualizing graphs, they would not be satisfied with solutions found by such informal methods. The issue is not, however, simply being able to produce a more accurate solution than one obtained by examining a graph. If it were, computer software and calculators that can do symbol manipulation could be called on to generate solutions that are as accurate as desired. The issue is the role the process plays in learning:

When symbol manipulators become widely available, we will probably take the same view with equation solving that we do with graphing. That is, we will continue to teach students paper-and-pencil means for solving linear equations because the idea is important and the process



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