The ideas and techniques involved in learning about fractions were investigated with students in grades 1-12, in the first 3 years of colleges, in community college mathematics courses, and in graduate school. Also included were some high school mathematics teachers, some mathematicians, and some retired persons. Part I provides the rationale and an overview of the project. Part II presents the theoretical background, plans for interviews, and the findings, with excerpts from tape-recorded interviews. An appendix contains transcripts of the interviews. It was concluded that nearly all students have access to good processing capabilities when dealing with concrete materials, and a major task of school programs is to build on these capabilities. The importance of good mental representations is stressed. (MNS)