Learning Mathematics pp 125-170 | Cite as

# Images of Rate and Operational Understanding of the Fundamental Theorem of Calculus

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## Abstract

Conceptual analyses of Newton’s use of the Fundamental Theorem of Calculus and of one 7th-grader’s understanding of distance traveled while accelerating suggest that concepts of rate of change and infinitesimal change are central to understanding the Fundamental Theorem. Analyses of a teaching experiment with 19 senior and graduate mathematics students suggest that students’ difficulties with the Theorem stem from impoverished concepts of rate of change and from poorly-developed and poorly coordinated images of functional covariation and multiplicatively-constructed quantities.

## Keywords

Average Rate Teaching Experiment Fundamental Theorem Circular Cross Section Functional Covariation
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© Springer Science+Business Media Dordrecht 1994