by Julie M. Clark,
Francisco Cordero,
Jim Cottrill,
Bronislaw
Czarnocha, David J. DeVries,
Denny St. John,
Georgia Tolias,
and Draga
Vidakovic.
This paper is part of a series of studies by the Research in Undergraduate Mathematics Education Community (RUMEC), concerning the nature and development of college students' mathematical knowledge. This project began as an attempt to explore calculus students' understanding of the chain rule and its applications. Based on the initial description of how the chain rule concept may be learned (genetic decomposition) an attempt to interpret the data using the Action-Process-Object theoretical framework is made. The insufficiency of this alone led to an extension of the Action-Process-Object-Schema epistemological framework (APOS) which includes a theory of schema development based on ideas of Piaget and Garcia. The Piagetian Triad is suggested as a mechanism for describing schema development in general, and the chain rule is used as an example. The Triad of the Intra, Inter and Trans stages of schema development provides the structure for interpreting the students' understanding of the chain rule and classifying their responses to interview questions about the chain rule. The results of this data analysis allowed for a proposed revised epistemological analysis of the chain rule. Finally, several suggestions and questions for future study are presented.