Engagement events—whether interviews, installations, or participatory encounters—can entail a range of happenings which, in one way or another, “overspill” the empirical, analytic, or political framing of those engagement events. This article looks at how we might attend to these overspills—for instance, forms of “misbehavior” on the part of lay participants—not only to provide accounts of them but also to explore ways of deploying them creatively. In particular, Stengers’ figure of the “idiot” is proposed as a device for deploying those overspills (...) to interrogate “what we are busy doing” as social science researchers in engagement events. This interrogation is furthered by considering the proactive idiocy of “Speculative Design’s” version of the public engagement with science which seems directly to engender “overspilling.” Providing examples of speculative design prototypes and practices, the article develops an ideal typical contrast between social scientific and designerly perspectives on public engagement. It is suggested that speculative design can serve as a resource for supplementing “science and technology studies” conceptualizations of, and practices toward, public, engagement, and science. (shrink)