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Long experience watching students do arithmetic with a four-function calculator has convinced most mathematics teachers that students need to know a lot about arithmetic to use a calculator well. The ability to carry out algorithmic routines accurately and efficiently with long numbers has become unimportant of itself, but students instead need "number sense" to guide sensible calculator use. McIntosh, Reys, and Reys (1992) have carefully analyzed the components of number sense and present them in a framework for use by teachers and researchers. This number sense includes understanding such concepts as place value for estimation and being able to identify the operations to use for solving problems.
The emphasis on different aspects of algebraic knowledge will similarly change when computer algebra systems (CASs) are readily available. Fey (1990) and Arcarvi (1994) have set out what a parallel "symbol sense" may entail and have identified some important components that we have used as a basis for this article. The fundamental thinking involved in symbol sense is important regardless of the level of technology that is used, but it assumes special relevance when we consider teaching algebra with CASs. In this article, we focus specifically on that part of symbol sense that is most affected by the availability of having CASs and we present a framework that can be used by teachers and researchers to guide in planning both curriculum and assessment.
ALGEBRAIC INSIGHT: A SUBSET OF SYMBOL SENSE
Figure 1 shows a model for mathematical problem solving with and without the use of CAS. Consider, for example, making a choice between two payment plans for a cell telephone.
Plan 1: The base cost is $15 plus $0.85 per minute during peak times and $0.35 per minute during off-peak hours.
Plan 2: The base cost is $30, which includes ten free minutes of peak time and ten free minutes of off-peak. Additional peak minutes are $0.40, and off-peak minutes are $0.22.
Students then solve the mathematical problem by finding the range of values ofp and q for which Ci is less than C2. The thinking required in this initial phase is unaffected by the availability of CASs. The right-side vertical arrow in figure 1 indicates the mathematical solution of this problem. Students can obtain this solution...