Given the recent release and widespread state adoption of the Common Core State Standards for Mathematics (CCSSM), teachers across the country now have common goals related to what their students should understand about particular mathematical topics. The authors of CCSSM explain, "Asking a student to understand something means asking a teacher to assess whether the student has understood it" (CCSSI 2010, p. 4). Teachers know that although students may be able to perform a computation, state a definition, or provide an example, they may not actually understand the mathematics. So, what does it mean for students to understand mathematics, and how can teachers gauge this understanding? Researchers and educators have proposed several frameworks to use when thinking about what it means to "understand" mathematics. In this article, the authors propose adapting a model originally used for curriculum development (Usiskin 2007) as a multidimensional approach to assessing students' mathematical understanding across four dimensions: (1) Skills; (2) Properties; (3) Uses; and (4) Representations. This multidimensional approach to understanding, described by the acronym SPUR (Usiskin 2007), can give teachers useful information about the depth of their students' understanding of a mathematical topic. Here, the authors briefly examine the results from an international research project on elementary school students' mathematical attainment in relation to SPUR. Then they demonstrate how teachers might use SPUR as a guide for assessment in their classrooms, specifically in relation to CCSSM. (Contains 4 figures.)