Sometimes, in the teaching and learning of mathematics, open-ended problems posed by teachers or students can lead to a fuller understanding of mathematical concepts--a depth of understanding that no one could have anticipated. Interesting solutions and ideas emerged unexpectedly when the authors asked prospective and in-service teachers an "old" algebra question in new ways. Their initial goal was to model the National Council of Teachers of Mathematics' (NCTM's) Process Standards in their classrooms (NCTM 2000). The result was a deeper understanding of solutions that emerge from the algorithm for solving equations involving radicals. Teachers gained new insights from students' ideas and strategies. Most important, students were engaged in an exploration of radical equations with true and extraneous solutions without simply relying on an algorithm that they had been taught. These investigations ultimately led them to a deeper understanding of the mathematical concepts. (Contains 4 figures.)