Toward the end of a study of equation solving by means of a graphical representation, a seventh grader was asked to solve the equation 7 Interviewer: Can you solve 7 Jer: Well, you could, see, it would be like start at 4 and 8, this one would go up by 7, hold on, 8, 8 and 7, hold on, no, 4 and 7, 4 and 7 is 11. They’d be equal, like, 2 or 3 or something like that. Interviewer: How are you getting that 2 or 3? Jer: I’m just like graphing it in my head. SOURCE: Kieran and Sfard, 1999, p. 15. Used by permission of the author. |

*is generalizable,* but we will also teach how to use symbol manipulators to solve these and more-complicated equations [emphasis added].^{62}

Thus, most teachers—for the time being, at least—remain insistent that students learn to do by hand the various algebraic transformations of expressions and equations. In 1989 one mathematics educator noted that “the unanswered question standing in the way of reducing the manipulative skills agenda of secondary school algebra is whether students can learn to plan and interpret manipulations of symbolic forms without being themselves proficient in the execution of those transformations.”^{63} Very little research has been conducted since then to help resolve the question; however, the research that has been done is quite telling. A recent study investigated the impact on algebra achievement of a three-year integrated mathematics curriculum in which technology was used to perform symbolic manipulations as well as to link various representations of problem situations.^{64} In this study, which involved over 300 high school students in 12 schools, some support was found for the notion that learning how to interpret results of algebraic calculations is not highly dependent on the ability to perform the calculations themselves.