Sixteen middle school students ranked the lengths of various paths in two audiotaped problem-solving interviews. Every student invoked at least one of four intuitions that originated from their everyday experiences: compression, detour, complexity, and straightness. After their intuitions proved inadequate in the pretest, they were taught an applicable algorithm. However, they used their intuitions again during the posttest before applying the instructed algorithm. The reuse of the inadequate intuitions demonstrates their robustness and their continued higher cueing priority despite the presence of the successful algorithm. When students applied multiple intuitions that conflicted, they often vacillated. Eventually, most students chose one. Nevertheless, they continued using the rejected intuition to compare other paths. As a result, their problem solving suggests that their intuitions are sparsely connected fragments. On the other hand, when intuitions support a common conclusion, students may integrate them to create a larger knowledge structure. Copyright © 1996 National Council of Teachers of Mathematics.