This research is part of the London-based Concepts in Secondary Mathematics and Science Project. It is derived from developmental psychology, with particular emphasis on the implications of the results to the teaching of mathematics in secondary schools. A basic premise is that if one first identifies the child's present level of understanding of a particular mathematical topic, it will then be possible to select appropriate experiences to facilitate the movement of the child from this level to a higher one. In this portion of the research, tests were developed to enable teachers to classify children on the continuum of understanding, and data from large-scale testing were used to generate detailed descriptions of these levels for a number of mathematical topics. The eight chapters of the report focus on needs, a review of related literature, and an outline of the methodology used; the development of items, including a summary of the content of each test; the development of a marking scheme; selection of the sample and administration of tests; statistical methods of forming a hierarchy; the hierarchies for each topic area; strategies and errors; and a summary. Eight appendices, most related to statistical concerns, are also included. (MNS)