A hierarchy for learning to solve different types of addition with fractions problems was hypothesized on the basis of both content analysis and psychological considerations. Problem types were defined according to the relationship of the two denominators to each other (e.g., equal, prime, etc.) Students in grades 4 through 8 were each given 45…

This paper reviews methods which have been used to identify and validate learning hierarchies. Specifically, three methods which reflect the notion of a relatively inviolate sequence are discussed and compared. Two of the psychometric methods compare pairs of skills, the ordering-theoretic method (Bart and Krus) and the test of inclusion (White…