In order to build toward an algorithmic theory of education we construct simple, idealized mathematical models of students, learning, educational software and educational software usage data. The models are created by taking concepts from the psychology literature and commercial educational software, stripping them down to bare mathematical essentials, and then rigorously analyzing these models. We consider the spacing effect from the psychology literature and model the notion of spaced repetition as simple constraints on mathematical sequences. Though the constraints are simply stated - that each occurrence of any element in the sequence fall within a given interval of possible distances beyond the previous occurrence - the mathematical problems that arise from these constraints are subtle. We present novel mathematical techniques suited to these problems. We also consider educational software usage data, and consider the task of measuring the amount of educational content a student must have mastered at any given time given that they produced some specific usage data. We find that once properly defined, the task is again subtle and requires carefully constructed algorithms, which in turn require careful mathematical analysis. Finally we consider the notion that it is easier for students to learn new concepts that are related to already-familiar concepts, and we present a novel network optimization problem inspired by this notion.