The mathematical models of this paper were developed as an outgrowth of working with the Comprehensive Achievement Monitoring project (Project CAM) which was conceived as a model and application of sampling procedures such as those used in industrial quality control techniques to educational measurement. This paper explores mathematical modeling techniques to gain insight into the role of achievement testing in the instructional process. All assumptions are listed, three models are presented, and computational examples provided. Each model centers on one of the following: 1) instruction given in a finite amount of time; 2) instructional activities in an individually-paced curriculum; and 3) tests used to group students for instruction. (Author/CK)