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Highlighting the role of applying mathematics within the life and social sciences through research produced by undergraduate participants of Arizona State University's (ASU) Mathematical and Theoretical Biology Institute (MTBI), we offer a brief account of what this community has learned from a research model that deliberately relinquishes the scientific agenda to its undergraduate participants. Over the past 21 years MTBI's summer research program has focused on training a diverse population of student researchers in the art of identifying and investigating questions primarily at the interface of the mathematical, life, and social sciences. At the heart of this paper are insights gained from the research leadership of twenty-one cohorts of undergraduate students. We highlight three selected projects that capture MTBI's philosophy of student-driven research, a model that has motivated hundreds of students to enroll in quantitative graduate programs across the nation. The first models collaborative active learning and its role in building robust communities of learners; the second studies the effects of oil spills on the spatial dynamics of loggerhead sea turtles; and the third develops a new, student-created version of pair-approximation modeling which has led to new research and a wide variety of applications. [ABSTRACT FROM AUTHOR]