Learning Curves for Problems with Multiple Knowledge Components

Learning curves have proven to be a useful tool for understanding how a student learns a given skill as they progress through a curriculum. A learning curve for a given Knowledge Component (KC) is a plot of some measure of competence as a function of the number of opportunities the student has had to apply that KC. Consider the case where each problem-solving step is recorded by, for instance, by an intelligent tutoring system. In this case, one normally assigns a unique KC to each problem-solving step and the construction of the associated learning curves is straightforward. On the other hand, many online homework systems only evaluate the student’s final answer to a problem. In that case, the student has generally applied a number of KCs to find the answer and their performance on the problem is some composite of their mastery of all of the requisite KCs. In this paper, we propose a simple method for generating learning curves for multipleKC problems that is independent of any particular theory of learning. In the case where there is only one KC per problem, the method reduces to the ordinary learning curves. We demonstrate this method using a set of artificially generated student data. Author