Aristotle's wheel paradox is a fascinating example of a classical puzzle that can pique a student's interest. The existing explanations in the literature are limited to discussions of the wheel trajectory; in this paper, we study the paradox from the viewpoint of motion decomposition, an approach we consider more intuitive. The motion of the point at the wheel edge can be resolved into a uniform rectilinear motion and a uniform circular motion. By analyzing the velocity of the point on the edge of the small wheel, we conclude that when the large wheel is rolling at a constant speed without slipping, the small wheel does actually both roll and slide.