We present results of an investigation of university students' development of mathematical models of motion in a physical science course for preservice teachers and graduate students in science and mathematics education. Although some students were familiar with the standard concepts of position, velocity, and acceleration from physics classes, most students had difficulty using these concepts to characterize actual or hypothetical motions. Furthermore, some students developed their own nonstandard method of describing accelerated motion in terms of changes in the "average" velocity, from the start of the motion up to a given time. This is in contrast to the physics community's use of the acceleration construct, defined in terms of changes in the "instantaneous" velocity, to describe such motion. Although the change in average velocity is not typically identified as an important construct in traditional physics texts, some students found it intuitively appealing, and were able to use it successfully to describe and predict motion. We conclude that by focusing on standard constructs, and ignoring possible intuitive ways that students might view motion, standard kinematics instruction may miss an opportunity to maximize student understanding. (Contains 1 table and 3 figures.)