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.