PITTSBURGH UNIV PA LEARNING RESEARCH AND DEVELOPMENT CENTER
At some time in our lives, we have all been forced to learn the procedural skills which supposedly comprise mathematical literacy e.g., place value addition through the process of rote memorization, perhaps, enhanced by the use of models e.g., the abacus. These models were intended to provide an intuitive basis for a given procedure. This formalizes the concept of an analogy between procedures based on a Sacerdoti-like representation called planning nets. A planning net represents the synthesis of a given procedure from a set of constraints that define the properties of the arithmetic operation being implemented and the representation of the objects numbers being manipulated. An analogy between procedures is represented as a maximal partial isomorphism between the planing nets of the two procedures. The planning net representation turns out to provide an elegant framework for defining the teleologic semantics of a procedure as well as for investigating how to construct a natural sequence of models or microworlds for a student to use in inventing his own procedure. Since both utilize the same framework, we have an extraordinarily powerful way to explain or teach the underlying teleology by showing how to relate it to a sequence of intuitively understood models.