Teaching experiments have generated several hypotheses concerning the construction of fraction schemes and operations and relationships among them. In particular, researchers have hypothesized that children's construction of splitting operations is crucial to their construction of more advanced fractions concepts (Steffe, 2002). The authors propose that splitting constitutes a psychological structure similar to that of a mathematical group (Piaget, 1970b): a structure that introduces mutual reversibility of students' partitioning and iterating operations that the authors refer to as the "splitting loope". Data consisted of 66 sixth-grade students' written performance on 20 tasks designed to provoke responses that would indicate particular fractions schemes and operations. Findings are consistent with hypotheses from related teaching experiments. In particular, they demonstrate--consistent with the notion of the splitting loope--that equipartitioning and the partitive unit fraction scheme mediate the construction of splitting from partitioning and iterating operations. (Contains 7 footnotes, 11 tables, and 8 figures.)