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.

ALEKS® is a registered trademark of ALEKS Corporation.