All of us who teach mathematics make decisions every day based on our descriptions, implicit or explicit, of students' understanding of the mathematics we teach.  The way we present particular material, our responses to students' questions, the exams we write, and the grades we give are all based on inferences we make about what they are thinking from the things they write and say.  These inferences and the decisions based on them are shaped in crucial ways by the informal models and images of mathematical understanding we hold.  In this seminar we will study a number of papers in which diverse models of students' understanding of mathematics are presented, and we will analyze various forms of evidence of college students' mathematical thinking, such as videotape and students' written work.  The purpose here is to understand these authors' models and to apply them to examples close to our own teaching.  In the process, our own models of students' understanding should become more explicit and be more clearly articulated as they are brought into contact with the more formal models found in these papers.