A taxonomy for mathematical knowledge-expertise was developed during a year-long study of students' understanding of proof across the undergraduate mathematics major at a medium-size comprehensive university. The taxonomy takes the form of a matrix with elements adapted from science assessment and expertise theory. The matrix was developed in order to more accurately describe the performance of 12 students and one faculty expert on a "proof-aloud" task. The taxonomy matrix characterizes three stages of mathematical expertise across six cognitive and two affective components of mathematical knowledge. This paper will describe how the taxonomy was developed, what it implies about teaching practice, and its subsequent application to analyzing mathematical tasks, student work and instructor feedback.