The use and understanding of mathematics is a crucial component of the physical sciences. Much work has been done in physics education research and science education more broadly to determine persistent difficulties with mathematics. This work has led to the development of numerous problem solving strategies aimed at helping learners approach problems more like experts as well as frameworks used by researchers to describe the role of mathematics in physics problem solving. A focus on how students understand the meaning of mathematical formalisms in physical contexts has led to studies of "mathematical sense making" (MSM), though no specific definition of this concept exists. We present a novel framework that operationalizes MSM through the categorization of student moves that contribute to the larger activity of sense making. This framework links prior work on MSM, mathematical problem solving, and conceptual understanding and has utility for both researchers and instructors. In detailed studies of student reasoning, we show that the framework can be applied to describe student sense making across multiple modalities of work (verbal, written, multiple choice) and across contexts (think-aloud interview settings as well as homework, exams, and other artifacts more commonly seen in physics and PER). The framework has descriptive utility for both the nuanced, individual reasoning evident in extended episodes as well as sparser forms of reasoning with much larger sample sizes. In certain instances, the framework also has predictive utility in terms of student sense making and answer making. This predictive utility supports the use of the framework in the analysis and design of curriculum meant to support sense making. In this work, we additionally present two in-depth case studies of extended, collaborative reasoning to show how the framework can be used to describe sense making in terms of the combination and coordination of smaller scale modes of MSM. This study is then expanded to sparser forms of reasoning present on homework and exam questions for N > 100. Finally, the fine-grained approach presented in the case studies is extended to analyze individual curricular items for the reasoning structures they support and the sparser approach is used in a cross-curricular analysis, indicating the varying opportunities for MSM provided by several nationally recognized curricula developed at other institutions. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com.bibliotheek.ehb.be/en-US/products/dissertations/individuals.shtml.]