toires and a better understanding of functions. The students who used the graphing calculator for only a short period of time did no better on the posttest than the students in the control group. They merely replaced their algebraic and guess-and-test procedures with graphing methods. Unlike the students who spent more time using the graphing calculator, they were not able to enrich their conceptual understanding of functions.

The widespread availability of computer and graphing-calculator technologies has dramatically affected the kinds of representational activities that have been developed and studied since the 1980s. Today’s graphing programs, curve fitters, spreadsheets, and spreadsheet-like generators of tables of values and so on have been found to provide more effective environments than pencil and paper for introducing students to variables, algebraic expressions, and equations in a problem-solving context. Research has documented that the visual and numerical supports provided for symbolic expressions by digital representations of graphs and tables help students create meaning for expressions and equations in ways difficult to manage in learning environments not supported by computers or calculators. More research is needed into the ways that computers and graphing calculators are being used and can be used effectively in the early grades.

The Transformational Activities of Algebra

What the Number-Proficient Child Brings

In the previous section, we discussed some of the perspectives brought to the study of algebra by students emerging from traditional elementary school arithmetic. These perspectives included the following:

  • An orientation to execute operations rather than to use them to represent relationships; which leads to

  • Use of the equal sign to announce a result rather than signify an equality;

  • Use of inverse or undoing operations to solve a problem and the corresponding absence of a notion of describing a situation with the stated operations of a problem; and

  • A perception of letters as representing unknowns but not variables.

In this section, we discuss additional features of arithmetic thinking that must be addressed when students encounter the transformational activities of algebra.

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