The Education Technology Center (ETC) Fractions Group works to investigate well-documented difficulties that children have in understanding fractions. A central assumption underlying this group's inquiry is that students' difficulties in manipulating fractions arise from their lack of understanding about the nature of fractions. In order to learn operations with fractions, students must first understand the order properties of fractions and their nature as "numbers between numbers." The notion of betweenness, so evident in the perception of continuous quantity, is neither evident nor even salient in the symbol systems and notation schemes used to describe continuous quantity. Yet, understanding betweenness appears to be a prerequisite for effective manipulation of fractions. From a larger view, betweenness is an important theme pervading much of mathematical thinking. Thus, this project's focus on teaching students to understand betweenness is chosen not only as a means of addressing student difficulties with fractions, but also as a way of teaching a fundamental concept in mathematics. This document includes: (1) discussion of the development of teaching materials; (2) lesson plans; (3) pilot study results; (4) anecdotal notes; and (5) guidelines for observers used in the pilot study. A list of 43 references is appended. (CW)