Logically Optimal Curriculum Sequences for Adaptive
Hypermedia Systems
Roland Hu¨bscher
107 Dunstan Hall, Department of Computer Science and Software Engineering
Auburn University, Auburn, AL 36849-5347, U.S.A.
roland@eng.auburn.edu
Abstract. Curriculum sequencing is an important technique used in many adaptive hypermedia sys-
tems. When following one of the possible page sequences, visiting some pages may become redundant,
because its content has been covered already by another page. Using disjunctive and conjunctive pre-
requisites instead of partial orders to describe the many possible sequences, logical redundancy between
pages can be computed on the fly without burdening the teaching model with that task [1]. Although
the general case of finding all redundant pages is NP-Complete [2] and thus, intractable unless P = NP,
a large subset can be located efficiently in realtime. The advantage of separating out logical redundancy,
the advantage of using conjunctive and disjunctive prerequisites, and the algorithms to find redundant
pages are discussed. An interesting characteristic of the presented approach is that it can be used
together with a wide variety of user and teaching models.
1 Introduction
The goal of curriculum sequencing is to provide a student with an optimal path through the material to be
learned and tasks to be executed, e.g., practicing a skill or answering questions [3]. Since every student has
different prior knowledge, preferences, and often different learning goals, providing individualized curriculum
sequences for each student using adaptive hypermedia is a promising approach.
But what is an optimal path? In an educational adaptive hypermedia system, an optimal path maximizes
a combination of the student’s understanding of the material and the efficiency of learning the material.
However, it makes more sense to attempt to suggest a few good paths instead of an optimal one, because
the latter does not exist. It should then be left to the learner to select whichever one of the, possibly many,
paths that are expected to lead to effective learning for this student. Otherwise, the hypermedia looses its
big advantage of being freely explorable, providing at least somewhat of an authentic learning experience to
the student [4]. However, providing a hypermedia system with a link structure that allows the user to choose
between good paths could result in redundant visits to certain pages and thus, in a suboptimal curriculum
sequence. This situation cannot be avoided if a wide variety of learners need to be supported. Different
examples, different kinds of representations, different modalities, etc. can be used for explaining the same
concepts in slightly different ways which means that not everybody ought to have to look at everything
everybody else does.
For instance, assume that in one sequence concept A needs to be studied before concept X. In another
sequence, we want the student, again for pedagogical reasons, to read about concept B before X. Looking
at the bigger picture, we realize that either A or B ought to be studied before X. Thus, once the learner
understands A, he or she can ignore B, as it has become redundant with respect to the goal of understanding
X. However, as we will see later, the general case is not as simple as this trivial example may imply.
A guiding principle for our approach is that the space the user may explore is maximized, i.e., we want to
constrain the user’s choice as little as possible. However, navigation support should scaffold [5] the user to take
the best paths with respect to the user’s interest and current understanding of the domain, the domain itself,
and the underlying pedagogical framework. In short, we are interested in effectively supporting scaffolded
exploration.
In this paper, we study how we can find logically optimal paths in hypermedia systems that adaptively
provide many good curriculum sequences. Curriculum sequencing is used as a technical term, although most
of the time this sequencing, or adaptive ordering [6], is applied to smaller parts, like courses. Logically

optimal paths do not contain pages that are redundant based on logical, and not on pedagogical, grounds.
Sometimes, one explicitly wants to be redundant to some degree, e.g., an issue should be covered by at least
two examples. Such situations can also be described by the presented formalism. We show that generally all
redundancy cannot be detected in realtime, however, a large subset can be found efficiently by the presented
algorithms.
2 Sequencing with Prerequisites
The material to be learned and tasks to be executed, from now on simply called “units,” need to be ordered for
each learner dependent on the teaching method employed and the model the system has of the student. The
resulting structure describing all the sequences can be represented as a directed acyclic and-or graph where
the vertices are the units and the edges are prerequisites between the units. This graph is called a prerequisite
graph. Note that we use the term “prerequisite” not in a strictly pedagogical sense, i.e., we do not claim
that organizing a course simply using prerequisites is appropriate. We simply call the temporal constraints
between the units prerequisites. However, we claim that most pedagogically interesting organizations of the
units can formally be reduced to a prerequisite graph.
One might argue that a simple partial-order representation of the units is just as expressive. Although
this is true in principle, the size of the partial-order graph would be exponentially larger than the prerequisite
graph as will become clear below.
A prerequisite is either a conjunctive or a disjunctive. A conjunctive prerequisite u1 ^ u2 ^ : : : ^ un ) u
asserts that all of the units u1; u2; : : : ; un need to be “visited” by the student before unit u may be visited.
Instead of “visited” we could also say “learned,” “seen,” “understood,” etc., depending on the specific design
of the adaptive hypermedia system and the teaching method adopted. This is independent of our concept of
logically optimal paths. Similarly, a disjunctive prerequisite u1 _ u2 _ : : : _ un ) u asserts that at least one
of the units u1; u2; : : : ; un needs to be visited by the student before unit u.
We are currently designing a language that allows course designers to describe teaching methods that
can be translated into the prerequisite formalism introduced here. The following few simple examples will
provide an idea of where this language is going.
Assume that x and y are some knowledge units, for instance, a concept or a topic consisting of several
knowledge units. Sometimes, for instance in Problem-Based Learning [7, 8], it is preferable to have the
students find out that they have a need to learn certain prerequisites. So, one would like to talk first about
y and only then about x. The rule
if x is required to understand y then y ) x
says that prerequisites (in the pedagogical, not formal, sense) should be visited after what they are prereq-
uisite for. This example makes it explicit that we use the prerequisites in the formal sense as a way to order
the units. However, how prerequisites in the pedagogical sense are used depends on the teaching method
used.
The following rule results in a top-down organization of part-whole hierarchies since it says that one
should learn about the whole before learning about its parts.
if y is part of x then x ) y
And finally, the last rule says that related issues should be visited in parallel to the main topic: If x
is strongly related to y, then x is visited while visiting y. This rule only makes sense if x and y are more
complex hypermedia structures than a single page. Then we can refer to the start and the beginning units
of those structures and use our prerequisites again. Virtual units introduced below can be used as special
start and end units if necessary.
if x is strongly related to y then start(y) ) start(x)
if x is strongly related to y then end(x) ) end(y)
In order to make the simple formalism of conjunctive and disjunctive prerequisites a bit more expressive,
we introduce the notion of virtual units. A virtual unit does not exist for the student, however it is an element