Figure 8.3: Learning Path

It should be obvious that not all possible subsets of the domain are feasible knowledge states. For instance, every student having mastered "long division" would also have mastered "addition of decimal numbers." Thus, there is no knowledge state containing the "long division" item that does not also contain the "addition of decimal numbers" item. The collection of all feasible knowledge states is referred to as the knowledge structure. The very large number of states for any product means that there are many possible ways of acquiring knowledge, i.e., many learning paths (Fig. 8.3). In the ALEKS knowledge structure there are literally billions of such learning paths. A "knowledge space" is a particular kind of knowledge structure.

As in many real-life applications, "noise" and errors of various sorts often creep in, which require the elaboration of a probabilistic theory. The ALEKS System is based on such a probabilistic theory, which makes it capable of recovering from errors. For instance, ALEKS is capable of deciding that a student has mastered an item, even though the student has actually made an error when presented with a problem instantiating this item. This is not mysterious: a sensible examiner in an oral exam, observing an error to a question about addition would nevertheless conclude that the student has mastered addition, for example, if that student had given evidence of skillful manipulation of fractions.