This study aims to begin to uncover the detailed ways that pre-service elementary teachers examine and understand mathematics teaching, the grain size at which they do so, and what constitutes evidence of teaching mathematics for understanding for pre-service mathematics teachers. Thus, the central research questions for this study are: (a) What are pre-service teachers' conceptions of teaching mathematics for understanding?; and (b) What counts as evidence for pre-service teachers of their practices in the planning, enactment, and reflection on teaching? This study takes place in the context of the Performance Assessment for California Teachers (PACT), also known as the Teaching Event. The PACT is a standards-based performance assessment designed to measure pre-service teacher learning. The assessment draws on a variety of data sources, such as lesson plans, student work, videos of teaching, and teacher reflections to measure pre-service teachers' ability to plan, implement, assess, and reflect on an integrated series of lessons that are intended to facilitate quality instruction for all learners. Furthermore, this assessment is broken down by content area, with each defining the particular knowledge and practices that are valued by the discipline. An important feature of the assessment for all disciplines is its focus on application of subject-specific pedagogical knowledge. For mathematics in particular, candidates are prompted to design, enact, and reflect on a series of lessons directed toward building conceptual understanding, computational and procedural fluency, and mathematical reasoning skills. Because the vision of mathematics teaching and learning promoted by the PACT is consistent with that advocated by research on teaching mathematics for understanding, the authors use this set of artifacts to study the specific ways that prospective teachers have come to understand this approach to mathematics instruction. To begin to investigate these questions, the authors conducted a case study of four pre-service teachers' PACT materials, including their written documents and video of teaching. These case studies provide insight into how pre-service teachers make sense of mathematics instruction and how they attempt to implement teaching practices to promote understanding of mathematics. The authors propose that the results of this study can be used to construct a framework to examine pre-service teachers' understanding of mathematics instruction more broadly, as well as to inform the design of teacher education programs that promote learning to teach mathematics for understanding. (Contains 1 table and 2 notes.)