A Shortest Learning Path Selection Algorithm in E-learning
Chengling Zhao and Liyong Wan
Department of Information Technology, Central China Normal University, Wu-Han, China
[email protected]
Abstract
Generally speaking, in the e-learning systems, a
course is modeled as a graph, where each node
represents a knowledge node (KU) and two nodes are
connected to form a semantic network. The desired
knowledge is provided by the student as a direct
request or from search results, mapping the owned
knowledge onto the target knowledge. How to select a
learning path which costs the least time and effort is
critical. In this paper, we described the relationships
between different nodes in the graph structure of
knowledge units and propose an algorithm to select the
shortest learning paths to learn the target knowledge.
1. Introduction
With the development of the Internet, e-learning
system has become more and more popular because it
can make learner study at any time and any location
conveniently. In the e-learning systems, a course is
modeled as a graph, where each node represents a
knowledge unit (KU) and two nodes are connected
only if the knowledge gained by learning the former
node is needed as a prerequisite to learn the latter node
[1]. A KU includes the course theory, principle,
concept, definition, worked example and exercise, etc.
the relationship between two KUs can be divided into
three types: precedence relationship, succession
relationship and parallel relationship. The desired
knowledge is provided by the student as a direct
request or from search results, mapping the owned
knowledge onto the target knowledge. In this case the
student may have a number of learning paths to reach
the target knowledge. How to select a learning path
which costs the least time and effort is critical.
2. Learning path and arrowhead weight
2.1. Learning path
All KUs belonging to the same course are
connected together into a graph structure using
oriented arrowheads; the orientation represents the
three types of relationships between two units. The
target is to create and propose to the student all the
possible paths starting from his/her knowledge and
directed to the desired topic of interest according to the
connections between different nodes. A graph structure
of KUs is illustrated in figure 1.
Figure 1. A graph structure of KUs.
From figure 1 we can see that there are 10 KUs in
the graph. 6 and 9 form a precedence relationship; 1
and 2 form a succession relationship; 7 and 8 form a
parallel relationship. We suppose the target KU is the
node 6 and the student owns the knowledge node 2.
Figure 2 shows all paths from the node 2 to the node 6.
Figure 2. All the paths from the node 2 to the
node 6.
2.2. Arrowhead weight
When we list all learning paths from the KU the
learner owns to the KU the learner want to reach, we
must consider an issue: which learning path is the best
for the learners. We define the best learning path is the
learning process through which will cost the least time
and effort. Then we should consider weight of an
arrowhead represents the difficulty to access a topic
coming from a previous one. Arrowhead weights are
managed by teachers, usually by establishing their
initial values, which are further refined with the system
help, e.g. through a statistical analysis of previous
students sessions [1]. Each oriented arrowhead
Proceedings of the Sixth International Conference on Advanced Learning Technologies (ICALT'06)
0-7695-2632-2/06 $20.00 © 2006
IEEE
connecting two nodes is weighted in order to measure
the difficulty to face the next node. Here we suppose
the average weight between two nodes is 1.0. The
higher value of weight means that the learner needs
more time and effort to learn the next KU. Figure 3
shows an example of graph consists of arrowhead
weights.
Figure 3. An example of graph consists of
arrowhead weights.
The weights are beside the arrowheads. If the
student have learned node A, but B, C, D and E have
not yet learned. Now his target KU is E, obviously
there have been 9 learning paths: A-B-E, A-B-C-E,
A-B-C-D-E, A-C-E, A-C-B-E, A-C-D-E, A-D-C-E and
A-D-C-B-E.
If
we
select
A-B-E,
weights= ¦ weight =0.7+1.1=1.8; if we select
a) If vij=i , the shortest path is vi to vj;
b) If vij z i , we suppose vij=k, the shortest path is vi
–vk- vj ,then if vik=i, it means that there is no medial
node between vi and vk; if vkj=k , it means that there is
no medial node between vk and vj , else we can find a
medial node k.
c) Repeat a) and b) until we find out all of the
medial nodes.
d) Connect vi and vj with these medial nodes, we
can get the concrete shortest learning paths (one path
or more than one).
Now we use the algorithm to select the shortest
learning path in figure 3. we suppose the IDs of A,B,C,
and D are:1,2,3,4 and 5.
So D= M 0.7 1.2 1.0 M
V= § 1 1 1 1 1 ·
§
¨
¨M
¨M
¨
¨M
¨M
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·
¸
M 0.3 M 1.1 ¸
0.4 M 0.2 0.9 ¸
¸
M 0.6 M 0.8 ¸
M M M M ¸¹
3. The algorithm
We suppose a graph structure consists of n nodes,
and then the algorithm is below:
(1) Construct a initial adjacency matrix D=pij , pij is
the arrowhead weight from node i to node j. if there is
no arrowhead exist between the two nodes or the
orientation of the arrowhead is reverse or i=j, pij=M.
[2]
(2) Construct a medial node matrix V=vij , the value
of vij is the ID of node i.
(3) Start iterative operation˄the times of iterative
operation equal to the number of nodes. we first
compare pij (i, j=2,3…, n)with pi1+p1j, there are three
cases:
a) If pij> pi1+p1j, replace pi1+p1jwith pij in the next
adjacency matrix, and vij=1;
b) If pij<pi1+p1j, there is no change in the next
adjacency matrix and node matrix;
c) If pij=pi1+p1j, there are two results, one is as a)
and the other is as b)
So we get the next adjacency matrix D1 and node
matrix V1, then begin the next iterative operation until
we get Dn and Vn .
(4) Search for the shortest path from vi to vj,
¨
¨M
¨M
¨
¨M
¨M
©
IEEE
¸
0.7 0.3 0.5 1.1 ¸
0.4 0.7 0.2 0.9 ¸
¸
1.0 0.6 0.8 0.8 ¸
M M M M ¸¹
The other case is D5=
§M
¨
¨M
¨M
¨
¨M
¨M
©
0.7 1.0 1.0 1.8 ·
¸
0.7 0.3 0.5 1.1 ¸
0.4 0.7 0.2 0.9 ¸
¸
1.0 0.6 0.8 0.8 ¸
M M M M ¸¹
¨2
¨3
¨
¨4
¨5
©
3 2 3 2¸
3 2 3 3¸
¸
3 4 3 4¸
5 5 5 5 ¸¹
V 5= § 1
¨
¨2
¨3
¨
¨4
¨5
©
1
3
3
3
2
2
2
4
1
3
3
3
2·
¸
2¸
3¸
¸
4¸
5 5 5 5 ¸¹
We can see that p15=1.8 (A-E), then we search for the
medial nodes can get the shortest path is 1-2-5 (A-B-E)
and 1-4-5(A-D-E).
4. Conclusions and future work
In e-learning courses, how to select a learning path
which costs the least time and effort is critical. In this
paper, we describe the relationships between different
nodes in the graph structure of knowledge units and
propose an algorithm to select the shortest learning
paths to learn the target knowledge. In the future, we
will use this algorithm to construct an adaptive
learning system, which can provide the best and easiest
learning experience to the learners.
5. References
[1]Vincenza Carchiolo, Alessandro Longheu, Michele
Malgeri. Adaptive Formative Paths in a Web-based Learning
Environment. Educational Technology & Society. 4(2002),
pp.64-75.
[2] Wang Ying. Algorithm Research of the Optimal Path in
Intelligent Traffic System. Computer Development &
Applications.5(2005), pp. 45-47.
Proceedings of the Sixth International Conference on Advanced Learning Technologies (ICALT'06)
0-7695-2632-2/06 $20.00 © 2006
2 2 2 2 ¸¸
3 3 3 3¸
¸
4 4 4 4¸
5 5 5 5 ¸¹
Through 5 times of iterative operation, the adjacency
matrix and medial node matrix are as follows:
One case is D5= § M 0.7 1.0 1.0 1.8 · V5= §¨ 1 1 2 1 4 ·¸
A-C-B-E, weights= ¦ weight =1.2+0.4+1.1=2.7. The
smallest weights mean the shortest path. In the next
section we will propose an algorithm to select the
shortest learning path.
¨2
¨
¨3
¨
¨4
¨5
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