A new diagnostic modeling system for automatically synthesizing a deep structure model of a student's misconceptions or bugs in his/her basic mathematics skills provides a mechanism for explaining why a student is making a mistake as opposed to simply identifying the mistake. This report consists of four sections. The first provides examples of the problems that must be handled by a diagnostic model. It then introduces procedural networks as a general framework for representing the knowledge underlying a skill. The second section discusses some of the pedagogical issues that have emerged from the use of diagnostic models within an instructional system. This discussion is framed in the context of a computer-based tutoring/gaming system developed to teach students and student teachers how to diagnose bugs strategically as well as how to provide a better understanding of the underlying structure of arithmetic skills. The third section describes our uses of an executable network as a tool for automatically diagnosing student behavior, for automatically generating "diagnostic" tests, and for judging the diagnostic quality of a given exam. Included in this section is a discussion of the success of this system in diagnosing 1300 school students from a data base of 20,000 test items. The last section discusses some future research directions. (Author/MN)