Recognizing schemes, which are different from strategies, can help teachers understand their students' thinking about fractions. Using Steffe's advanced fraction schemes, the authors describe a progression of development that upper elementary and middle school students might follow in understanding fractions. Each scheme can be viewed as a reorganization of the preceding scheme. For example, when students develop a splitting operation, they can reverse the operations of their partitive fractional scheme to produce a reversible partitive fractional scheme. Similarly, when students have developed a reversible partitive fractional scheme, they may then be ready to extend that understanding into the context of improper fractions, fostering the development of an iterative fractional scheme. Teachers can support students' reorganizations if they understand students' current schemes and provide tasks that appropriately frustrate those ways of operating. The authors hope that, in addition to describing the schemes themselves, they have presented useful ideas about how to make such instructional decisions. (Contains 2 tables and 5 figures.)