Fine Grained Assesment

Counting Learning Events

We also assumed that a step is the product of one or more learning events, and defined a Learning Event to be a mental event based on a Knowledge Component. Learning events and knowledge components are not directly observable, but steps are.
Mastery really means the probability that a Knowledge Component will be applied when it should be applied. If the student’s competence was frozen instead of constantly changing due to learning and forgetting, then mastery could be estimated by counting the number of times a Knowledge Component was applied and dividing by the number of times it should have been applied. Thus, we need to discuss three Issues: (1) How to detect applications of a Knowledge Component, (2) how to detect times when a Knowledge Component should have been applied, and (3) how to adjust for instruction, learning and forgetting.

Counting only the successful applications of a Knowledge Component is not enough; we need to know how many times the student failed to apply it as well.
One approach is to detect failed attempts at steps. Suppose for simplicity that there is a one-to-one correspondence between a step and a Learning Event
If the student fails to make the step, then the student must lack the knowledge that underlies the Learning Event.