What Do Students Need to Learn about Division of Fractions?

What should students know and be able to do after learning division of fractions? Many mathematics educators believe that it is not enough for students' knowledge of the division of fractions to be limited to the invert-and-multiply algorithm. In fact, many in the field have argued that students' learning of mathematics needs to go beyond rote memorization of procedures (Brownell 1938; NCTM 2000). It is important for teachers to carefully consider what students need to learn beyond the algorithmic procedure; developing students' conceptual understanding of division of fractions is not a trivial task.