Theory-based knowledge modeling in a subdomain of elementary algebra

The knowledge structures theory has been developing highly efficient computer-based procedures for the assessment of knowledge. However, a crucial precondition of a practical application of these assessment procedures in some knowledge domain is the establishment of a psychologically valid knowledge structure capturing the solvability dependencies among items of the domain. Several approaches for establishing a knowledge structure have been developed - each with its benefits and drawbacks. In an earlier paper (Held and Korossy, 1998) we contrasted an a-priori structure generated by an expert with an a-posteriori structure generated by a data-analytic method (the Item Tree Analysis) and discussed the possible contribution of data-analytic methods (such as the Item Tree Analysis) to an integrated methodology of establishing well-founded solvability structures on item sets. The current investigation takes up this line of research. Based on a large set of 21 carefully constructed problems, a theory-based knowledge modeling of a subdomain of elementary algebra has been developed and empirically examined. Moreover, the sample of observed response patterns has been taken as input for a data-analytic procedure (the Item Tree Analysis). The comparison of the theory-based knowledge structure with the data-analytic ally generated structure provides convincing arguments for the theory-based construction of knowledge structures. The present paper describes in detail the theoretical modeling of surmised solution dependencies and the results of the empirical examination of the model. A subsequent paper will present the aspects of applying the Item Tree Analysis to the empirical data.