The algorithm for fraction multiplication is relatively easy to memorize and implement with accuracy. The simplicity of the algorithm masks the conceptual complexity involved in making sense of what fraction multiplication means in quantitative situations. One important interpretation of fraction multiplication involves fraction composition, (i.e., finding a fraction of a fraction). Using this interpretation, "a/b x c/d" might be interpreted to mean "a/b" of "c/d." Previous teaching experiments have suggested that developing a general approach to fraction composition, one that allows students to find "a/b "of "c/d" for any two rational fractions, requires the development of distributive partitioning operations. In this dissertation, I describe a different general approach to fraction composition that relies not on distributive partitioning, but rather on partitioning an equivalent fraction. This approach to fraction composition emerged as two students, Wes (age 12) and Iris (age 13), participated in an online teaching experiment over the course of seven months in which they used length models to reason quantitatively about contextual situations that from the researcher's perspective involved multiplication and division of fractions. The design of the larger study and the analysis of the results were informed by radical constructivism, a theoretical framework that derives from Piaget's genetic epistemology. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com.bibliotheek.ehb.be/en-US/products/dissertations/individuals.shtml.]