The fifteenth Journal for Research in Mathematics Education monograph had its origins in a conference titled An Interdisciplinary Conference on Assessment in K--12 Mathematics: Collaborations Between Mathematics Education and Psychometrics, which was held in 2011 in Atlanta, Georgia. The basis for the conference was the renaissance in the field of psychometrics in which an increasing variety of psychometric models are becoming available through advances in computer hardware and software. This is opening new avenues for studying the mathematical knowledge of teachers and students. The overarching purpose of the monograph is to guide further interdisciplinary collaborations between mathematics education researchers and psychometricians by examining theoretical and conceptual issues that have arisen in recent efforts to apply contemporary psychometric models to mathematics education research. Specifically, its chapters are intended to (a) illustrate for mathematics education researchers the two main categories of psychometric models-models that locate individuals along continua and models that place individuals into discrete groups, as well as hybrids of these approaches; (b) provide examples that apply these different categories of psychometric models to mathematics education research; (c) illustrate how researchers have selected different psychometric models on the basis of the researchers' goals and (d) demonstrate issues related to item development. With these goals in mind, the monograph will enhance an awareness among mathematics education researchers that it is increasingly possible to select from a variety of model options when pursuing particular research goals and that choosing among models involves trade-offs.