The purpose of this study is to describe how university honors calculus students negotiate meaning and language for conceptually important ideas through mathematical discourse. Mathematical discourse has been recognized as an important topic by mathematics education researchers of various theoretical perspectives. This study is written from a perspective that merges symbolic interactionism (Blumer, 1969) with personal agency (Walter & Gerson, 2007) to assert that human choice reflects, but is not determined by, meanings that are primarily developed through social interaction. The process of negotiation of meaning is identified, described, and analyzed in the discourse of four students and their professor as they draw conclusions about the volume of water in a reservoir based on graphs of inflow and outflow. Video data, participant work, and transcript were analyzed using grounded theory and other qualitative techniques to develop three narrative accounts. The first narrative highlights the participants' use of personal pronouns and personal experience to negotiate meaning for the conventional mathematical terms "inflection" and "concavity." The second narrative describes how the participants' choices in discourse reflect an effort to represent both their mathematical and experiential understandings correctly as they negotiate language to describe critical "zero points." The third narrative describes the participants' process of mapping analogical language and meaning from the context of motion to the context of water in a reservoir. Analysis of these three narratives from the perspective of conventional and ordinary mathematical language suggests that the contextualized study of mathematics may provide students access to mathematical discourse if the relevant mappings between mathematical language and language from other appropriate contexts are made explicit. Analysis from the perspective of social speech (Piaget 1997/1896) suggests that specific uses of personal pronouns, personal experience, and revoicing (O'Connor & Michaels, 1996) may serve to invite students to become participants in mathematical discourse. An agency-based definition of mathematical discourse is suggested for application in research and practice.