Many problems seem to require a flash of insight to solve. What form do these sudden insights take, and what impact do they have on how people approach similar problems in the future? In this work, we prompted participants (N = 189) to think aloud as they attempted to solve a sequence of five "matchstick-arithmetic" problems. These problems either all relied on the same kind of non-obvious solution (Same group) or a different kind each time (Different group). Our first observation was that Same participants improved more rapidly than Different participants. We then leveraged techniques from natural language processing to analyze participants' speech, and found that this accelerated improvement for Same participants was accompanied by changes in both how much they spoke and what they said. In particular, they were more likely to spontaneously label the kind of problem they were working on. Taken together, these findings suggest that a hallmark of transferable insights is their accessibility for verbal report, even if the underlying precursors of insight remain difficult to articulate.