How does the idea of cognitive technologies relate to mathematics education? A few historical notes prepare the stage. We may recall Ernst Mach's (189311960) statement, in his seminal work on the science of
mechanics earlier this century, that the purpose of mathematics should be to save mental effort. Thus arithmetic procedures allow one to bypass counting procedures. and algebra substitutes "relations for values, symbolizes and definitively fixes all numerical operations that follow the same rule" (p. 583). When numerical operations are symbolized by mechanical operations with symbols, he notes, "our brain energy is spared for more important tasks" (p. 584), such as discovery or planning. Although overly neural in his explanation, his point about freeing up mental capacity by making some of the functions of problem solving automatic is a central theme in cognitive science today.
Whitehead (1948) made a similar point: "By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power" (p. 39). He noted that a Greek mathematician would be astonished to learn that today a large proportion of the population can perform the division operation on even extremely large numbers (Menninger, 1969). He would be more astonished still to learn that with calculators, knowledge of long division algorithms is now altogether unnecessary. Further arguments about the transformational roles of symbolic notational systems in mathematical thinking are offered by Cajori (1929a, 1929b), Grabiner (1974), and, particularly, Kaput (in press).
Although long on insight, Mach and Whitehead lacked a cognitive psychology that explicated the processes through which new technologies could facilitate and reorganize mathematical thinking. What aspects of
mathematical thinking can new cognitive technologies free up, catalyze, or uncover? The remainder of this chapter is devoted to exploring this central question.
A historical approach is critical because it enables us to see how looking only at the contemporary situation limits our thinking about what it means to think mathematically and to be mathematically educated (cf. Resnick & Resnick, 1977, on comparable historical redefinitions of "literacy" in American education). These questions become all the more significant when we realize that our cognitive and educational research conclusions to date on what student of a particular age or Piagetian developmental level can do in mathematics are restricted to the static medium of mathematical thinking with paper and pencil.' The dynamic
and interactive media provided by computer software make gaining an intuitive understanding (traditionally the province of the professional mathematician) of the interrelationships among graphic, equational, and pictorial representations more accessible to the software user. Doors to mathematical thinking are opened, and more people may wander in.
Thus, the basic findings of mathematical education will need to be rewritten, so that they do not contain our imagination of what students might do, thereby hindering the development of new cognitive technologies
for mathematics education.
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