This is the final report of a research project. The goal of the project was to record a large collection of mathematical performances by students and to develop a system for analyzing these performances. The subjects observed included students in grades 3, 4, 5, 6, 7, and 9 and three adult groups. Four specific results are reported: (1) a large collection of instances of mathematical behavior; (2) a conceptual framework for interpreting and analyzing these instances; (3) a set of specific phenomena of special interest; and (4) some specific suggestions for further work. Also included are chapters entitled The Present Importance of Testing and Education as an Applied Science. (MK)