the only conceptual demand that is somewhat independent of the context is manipulating the algebraic expression to yield simpler algebraic expressions. That activity is very important, however, since it allows the student to see at a glance why the result for the above problem is always *x*+3, whatever the value of *x*. The evolving sequence of simplified algebraic expressions can permit a perception of “*x*+3-ness” in a way that is not so readily available from simply reading the problem. Thus, the algebraic representation can induce an awareness of structure that is much more difficult, if not impossible, to achieve using everyday language.

One hundred eighteen algebra students who had already taken algebra for a year were given Problem A. Only nine set up the expression (5*x*+12–*x*)/4 and then reduced it algebraically to *x*+3. Four of them went on to “demonstrate further” by substituting a couple of numerical values for *x*. Thirty-four others set up the equation (5*x*+12–*x*)/4=*x*+3 and then proceeded to simplify the left side, yet they did not base their conclusions on their algebraic work. Instead, they worked numerical examples and drew conclusions from them.

For the great majority of students, therefore, this task posed enormous problems both in representing a general statement and in using that statement to justify numerical arguments. According to the researchers, these students seemed completely lost when asked to use algebra. “Formulating the algebraic generalization was not a major problem for the [few] students who chose to do so; using it and appreciating it as a general statement was where these students failed.”^{66} Therefore, for the students who responded to the request to use algebra, their difficulties were related not to the simplification of the expression but to the third of the conceptual demands outlined above: being aware that the algebraic result constitutes a proof or justification of the arithmetical result that one obtains empirically by trying several numbers. This research also suggests that even when students are successfully taught symbolic manipulation, they may fail to see the power of algebra as a tool for representing the general structure of a situation. Without some skill with symbolic manipulations, however, students are unlikely to use algebra to justify generalizations.

Even when students are successfully taught symbolic manipulation, they may fail to see the power of algebra as a tool for representing the general structure of a situation.

Tasks involving geometric and numerical patterns are a frequent means of introducing students to the use of algebra for predicting. Problem B in Box 8–8 is typical. To help students find a pattern in the arrangement of dots