The development of students' graphical understanding of the derivative

• Published in: The Journal of mathematical behavior 1997, Vol.16 (4), p.399-431
• Main Authors: Asiala, Mark, Cottrill, Jim, Dubinsky, Ed, Schwingendorf, Keith E.
• Format: Article
• Language: English
• ISSN: 0732-3123; 1873-8028
• DOI: 10.1016/S0732-3123(97)90015-8

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. The present study explores calculus students' graphical understanding of a function and its derivative. An initial theoretical analysis of the cognitive constructions necessary for this understanding is given. An instructional treatment designed to help foster the formation of these mental constructions is described, and results of interviews, conducted after the implementation of the instructional treatment, are discussed. The understanding demonstrated by these students is analyzed according to the Action-Process-Object-Schema (APOS) theoretical framework. Based on the data collected as part of this study, a revised epistemological analysis for the graphical understanding of the derivative is proposed. Moreover, a comparative analysis is made of performance of students using the instructional treatment we designed with students taking a traditional calculus course. Although this analysis is flawed in many ways, it does suggest that the students whose course was based on the theoretical analysis of learning that we give here may have had more success in developing a graphical understanding of a function and its derivative, than students from traditional courses.