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Navigating scientific challenges, persevering through difficulties, and coping with failure are considered hallmarks of a successful scientist. However, relatively few studies investigate how undergraduate science, technology, engineering, and mathematics (STEM) students develop these skills and dispositions or how instructors can facilitate this development in undergraduate STEM learning contexts. This is a critical gap, because the unique cultures and practices found in STEM classrooms are likely to influence how students approach challenges and deal with failures, both during their STEM education and in the years that follow. To guide research aimed at understanding how STEM students develop a challenge-engaging disposition and the ability to adaptively cope with failure, we generate a model representing hypotheses of how students might approach challenges and respond to failures in undergraduate STEM learning contexts. We draw from theory and studies investigating mindset, goal orientations, attributions, fear of failure, and coping to inform our model. We offer this model as a tool for the community to test, revise, elaborate, or refute. Finally, we urge researchers and educators to consider the development, implementation, and rigorous testing of interventions aimed at helping students develop a persevering and challenge-engaging disposition within STEM contexts.