| The differential calculus provides various ways to conceptualize change, any of which can be employed with applied problems. The present study used a cognitive-historical method to investigate how experts formulate a representation of a problem that requires calculus for its exact solution. Think aloud protocols were obtained with three University faculty members--a mathematician, a chemist, and a physicist--as they were solving a mixture problem called the Flask Problem, which can be modeled by a first-order linear ordinary differential equation. The protocols were analyzed into problem solving episodes and interpreted in light of a historical-cognitive framework, supported by an account of the conceptual development of the representational multiplicity of differential calculus. Differing approximation procedures were central in the mathematician's and the chemist's approach to the problem. The mathematician's representational practice was further characterized as Leibnizian, the chemist's as Newtonian. The physicist employed a genuine modeling approach interpreted in light of a function-based calculus. Comparisons of the representational practices were made in three respects: (1) the nature of the representation of change (static vs. dynamic), (2) the nature of the time dimension, and (3) the kind of agency entailed by the representation. A fourth "virtual" participant was also constructed in an attempt to answer the question of how Isaac Newton might have solved the Flask Problem. The construction of the virtual participant was based on Newton's Law of Cooling which can be represented using the same differential equation as models the Flask Problem. The comparison of Newton's actual practice of calculus and "Newtonian calculus" illuminates the latter. In conclusion, the representational multiplicity characteristic of the calculus, and present in similar ways in other domains of mathematics, provides an account of the relation between computation and representation and constitutes an important theme for a cognitive psychology of mathematics. |
