A number of educational researchers have developed pedagogical approaches that involve the teacher in discovering and helping to correct misconceptions that students bring to their study of their subject matter. During the last decade, several computer systems have been developed to support teaching and learning using this kind of approach. A central conceptual construct used by these systems is the "facet" of understanding: an atomic diagnosable unit of belief. A formidable challenge to applying such pedagogical approaches to new topic areas is the task of discovering and organizing the facets for the new subject area. This paper presents a taxonomy of misconceptions and a methodology for going about the task of preparing a database of facets. Important issues include the generality and diagnosability of facets, granularity of facets, and their placement on a scale of problematicity. Examples are drawn from the subjects of physics and computer science and in the context of two computer systems: the Diagnoser and INFACT. (Contains 2 figures.)