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• #### Index 441-454

Several of the teaching approaches discussed in the following sections have profitably used computer technologies, especially graphics, as a means of making algebraic symbolism more meaningful. These studies provide evidence of the positive role that computer-supported approaches can play in the learning of algebra, as well as suggesting that technology can be a means for making algebra accessible to all students, including those who, for whatever reason, lack skill in pencil-and-paper computation.20 Thus, these examples suggest that some version of “algebra for all” may be viable.

#### The Representational Activities of Algebra

##### What the Number-Proficient Student Brings

Traditional representational activities of algebra center on the formation of algebraic expressions and equations. Creating these expressions and equations involves understanding the mathematical operations and relations and representing them through the use of letters and—for equations—the equal sign. It also requires thinking that proceeds in rather different ways from the thinking that develops in traditional arithmetic.

In the transition from arithmetic to algebra, students need to make many adjustments, even those students who are quite proficient in arithmetic. At present, for example, elementary school arithmetic tends to be heavily answer oriented and does not focus on the representation of relations.21 Students beginning algebra, for whom a sum such as 8+5 is a signal to compute, will typically want to evaluate it and then, for example, write 13 for the box in the equation 8+5=□+9 instead of the correct value of 4. When an equal sign is present, they treat it as a separator between the problem and the solution, taking it as a signal to write the result of performing the operations indicated to the left of the sign.22 Or, when doing a sequence of computations, students often treat the equal sign as a left-to-right directional signal. For example, consider the following problem:

Daniel went to visit his grandmother, who gave him \$1.50. Then he bought a book for \$3.20. If he has \$2.30 left, how much money did he have before visiting his grandmother?

In solving this problem, sixth graders will often write 2.30+3.20=5.50–1.50 =4.00, tacking the second computation onto the result of the first.23 Since 2.30+3.20 equals 5.50, not 5.50–1.50, the string of equations they have written violates the definition of equality. To modify their interpretation of the equality sign in algebra, students must come to respect the true meaning

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