Curriculum spaces and mathematical models for curriculum design.

The area of mathematical psychology concerning knowledge spaces is complemented by the theoretical notion of a "curriculum space," which is developed as a mathematical model for curriculum design in order to illuminate the interplay between learners and the curricula in which they enroll. The analysis is carried out in terms of concepts from graph theory and order theory, yielding a precise language in which to communicate aspects of curricular influences on student-achievement. Emphasis is placed on the structure of tasks that are addressed in a curriculum, the structure of prerequisite-course assignments within the curriculum, and the compatibility between these structures. Among the main results, conditions are given under which the tasks in a curriculum induce a partially ordered set, curricula that admit "compatible" prerequisite-structures are completely characterized, and a greedy algorithm is presented for categorizing courses into "curriculum maps" that respect the compatibility condition. • Knowledge space theory is complemented by the notion of a curriculum space. • Bipartite graphs model learners and tasks in a curriculum. • Greedy algorithm for compatible curriculum maps. • Prerequisite structure given by a partially ordered set of tasks in a curriculum. • Students who remain underprepared after completing prerequisite courses. [ABSTRACT FROM AUTHOR]