ABSTRACT
When the concept of analytic geometry evolved in the mathematics of 17th-century Europe, the meaning of the term was quite different from our modern notion. The main conceptual difference was that curves were thought of as having a primary existence apart from any analysis of their numeric or algebraic properties. Equations did not create curves; curves gave rise to equations. When Descartes published his Geometry in 1638 (trans. 1952), he derived for the first time the algebraic equations of many curves, but never once did he create a curve by plotting points from an equation. Geometrical methods for drawing each curve were always given first, and then, by analyzing the geometrical actions involved in a physical curve-drawing apparatus, he would arrive at an equation that related pairs of coordinates (Dennis, 1995; Dennis & Confrey, 1995). Descartes used equations to create a taxonomy of curves (Lenoir, 1979). This tradition of seeing curves as the result of geometrical actions continued in the work of Roberval, Pascal, Newton, and Leibniz. As analytic geometry evolved toward calculus, a mathematics developed that involved going back and forth between curves and equations. Operating within an epistemology of multiple representations entailed a constant checking back and forth between curve-generating geometrical actions and algebraic language (Confrey & Smith, 1991). Mechanical devices for drawing curves played a fundamental, coequal role in creating new symbolic languages (e.g., calculus) and establishing their viability. The tangents, areas, and arc lengths associated with many curves were known before any algebraic equations were written. Critical experiments using curves allowed for the coordination of algebraic representations with independently established results from geometry (Dennis, 1995). What we present here is a description of one student’s investigation of two curve-drawing devices. The usual approach to analytic geometry in which a student studies the graphs of equations has been reversed, in that the student primarily confronted curves created without any preexisting coordinate system. This student first physically established certain properties of and Interrelationships between curves and only afterward came to represent these beliefs using the language of symbolic algebra. This student’s actions are interpreted within Confrey’s (1993, 1994) framework, which views mathematics as dialogue between “grounded activity” and “systematic inquiry.” In this study, we provided curve-drawing devices and posed problems that allowed the student an opportunity to voice both sides of this dialogue.
