Life is full of word problems, yet computers do not do word problems. The mathematical thinking skills students must learn now include deciding when and how to use a computer or calculator as well as the ability to determine what the output from the machine signifies. Punching the key marked π on a calculator is a far cry from the qualitative abstraction skills developed in a student when he attempts to understand what π means (see, for example, §A.3).
My experiences as both high school and college teacher and in the computer labs at several universities suggest to me that directed, supplementary, computer and calculator exercises can improve mathematical fluency among students at every level. When a student can see polynomial after polynomial graphed clearly and correctly, she moves towards developing insight into what a polynomial is, when it has ``roots,'' and what sort of ``zeroes'' there are [Simpson & Tall, 1998].
At UNC I have taught nine sections of the quantitative reasoning course called Math 120 - Mathematics & Liberal Arts. Students in my sections are required to obtain a graphing calculator. Of the 30 instructional days of the course, an average of 15 have included classwork which required the use of the graphing calculator (as a graphic tool, adding machine, or for programming). Students in these classes have seemed to meet the challenge of articulating mathematical ideas through programming with much less intensely negative emotions than when faced with the struggle to articulate without the calculator. Research on computer programming by students in the context of functions, suggests that even very basic programming requires students to engage in a higher level of clarity and involvement with mathematical concepts [Breidenbach et al., 1992].
Though many students in Math 120 grumbled at first about the calculators, most admitted at the end of the course that their grumbling had been more from discomfort with the technology than anything else. Many were prouder of their newfound calculator programming skills than of their budding abilities to face, evaluate and solve word problems about home mortgages. The omnipresence of the calculator allowed students to direct their frustration with the concepts they were learning at the machine rather than at themselves: short-circuiting the self-doubt that so often is distilled into debilitating, 100 proof, ``math anxiety.'' As Berger has noted,
What makes computers interesting for belief researchers is the fact that they also have an implicit function. The computer serves as an objective for human projections, as a multifarious metaphor for human thinking, for the human brain, and even for a human being itself. Thus the computer is both an instrumental and projective medium, a tool and a metaphor. [Berger, 1999]So, the programming of a small computer like a graphic calculator allows students physical manifestation of themselves as learners. The metaphysical becomes tactile and confusion is communicated quite clearly on the screen with the single word ERROR. Students become consciously self-reflective, mathematically self-aware, in order to resolve whatever mis-step(s) resulted in the error message. There is the potential to unravel the tangle of misunderstanding while projecting any deficiencies onto the machine, thus students may be spared from painful blows to self-esteem [Turkle, 1995].