This study focuses on the nature of preservice elementary school teachers' understandings of several concepts in elementary number theory that are evoked by a computer-based microworld called "Number Worlds". In particular, the focus is on the concepts of factor, multiple and prime number. The notion of "thickness" is examined with respect to understanding mathematical concepts and alludes to various attributes of "Number Worlds" that may be responsible for provoking different layers and depths of understanding, termed "thick understanding". (Contains 10 figures.)