This paper reports on a project which has developed the general latent class model as a framework for representation of item responses. This framework can be used to represent data in applications such as mastery tests and other kinds of achievement tests, where there is reason to believe that current foundations are deficient. Methods of estimation for the latent class model have been improved and hypothesis tests addressing issues important in developing specific models for test data have been devised. These hypothesis tests include a test for monotone homogeneity of items, tests of invariance of item parameters between groups and over time, a test for the significance of inclusion of a new state in a model, and other tests. A nonparametric approach to maximum likelihood estimation of item response functions for monotonely homogeneous sets of items has been devised. The use of this general approach has been illustrated by developing models which successfully represent signed-number addition test data gathered by Tatsuoka and Birenbaum (1979). A number of technical issues relating to these models are discussed. (Author/JAZ)