8.2.4 Inner and Outer Fringes of a Knowledge State
Figure 8.4: Outer Fringe of a Knowledge State
Figure 8.5: Inner Fringe of a Knowledge State
An item that has not yet been mastered by a student may not be immediately
learnable by that student. Learning one or more prerequisite items
may be necessary. Consider a student in a particular knowledge state
K. The set of all items that may be learned
immediately by a student in that state K is called
the outer fringe of the state K.
The outer fringe of a state K is defined as
the set of all items, any one of which may be the
next one learned. An item is in the outer fringe of the state
K if the addition of that item to the state
K forms a new, feasible knowledge state
(Fig. 8.4). Typically, the outer
fringe of a knowledge state will contain between one and several items.
Similarly, an item is in the inner fringe of a state
K if there is some other knowledge state to which
that item may be added to form state K
(Fig. 8.5). The inner
fringe of a state K is thus defined as
the set of all items, any one of which may have
been the last one learned.
These two concepts of inner and outer fringes are used in powerful ways
in the Learning Mode of the ALEKS system. For example, the system
always offers a student problems to solve that are based on items in
the outer fringe of his or her state. If ALEKS judges that a student
is experiencing difficulties in learning some new item, ALEKS
typically reviews the mastery of items in the inner fringe of the
student's state that are also related to the new item to be learned.