This report contains four papers describing research based on the view of mathematical knowledge as a hierarchy of "rules." The first paper: "The Role of Rules in Behavior" was abstracted in ED 040 036 (October 1970). The second paper: "A Theory of Mathematical Knowledge" defends the thesis that rules are the basic building blocks of mathematical knowledge. These rules operate at different levels, for example: addition and subraction at one level, the idea of inverse operations at a higher level. Mathematical creativity then consists of combining rules to produce new results. The third paper: "Deterministic Theorizing in Structural Learning" describes experiments based on this hypothesis, in which a subject's performance on certain tasks is predicted with virtual certainty from his performance on the component rules. The role of the memory is also discussed, and experiments are described supporting the view of the mind as an information processor with a fixed capacity. The fourth paper: "A Research Basis for Teacher Education" argues that teachers need to know more of the aims of the mathematics they teach, more of the way mathematical knowledge is structured, and more of the way students learn. The author's research on structural learning (as described in the second and third papers) is then summarized, and possible developments are outlined. (MM)