This document discusses the importance of acceleration vs. sophistication in mathematics programs and curricula for gifted students. The discussion proceeds through three steps. First, an attempt is made to model how various kinds of gifted students interact with three dimensions of the curriculum. These dimensions are acceleration, sophistication (depth), and enrichment (breadth). Second, a description is given of the way in which one program has tried to respond to this model. This program is the mathematics program for gifted students at the University High School, the laboratory school of the University of Illinois. Finally, an attempt is made to draw a few conclusions and practical guidelines for dealing with gifted students. (HM)