The purpose of this paper is to share the strategies that children invented to solve problems involving division of fractions in the absence of algorithmic instruction. The data come from two sources, the first being a teaching experiment and the second being the regular classroom practice of this teacher/researcher. It was found that when the conditions of the teaching experiment were replicated, the three methods that students constructed to solve the identical problems involving division of fractions were the same as those seen in the original group. The three methods involved reasoning concerning natural numbers, measurement and fraction knowledge, all of which relate to counting. (Contains 5 footnotes.) [For complete proceedings, see ED500859.]