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Structuring and Merging

Distributed Content

Luca Stefanutti, Dietrich Albert, Cord Hockemeyer

Department of Psychology, University of Graz

Universit¨

atsplatz 2/III, 8010 Graz, AT

luca.stefanutti, dietrich.albert, cord.hockemeyer@uni-graz.at

A ﬂexible approach for structuring and merging distributed learning object is

presented. At the basis of this approach there is a formal representation of a learning

object, called attribute structure. Attribute structures are labeled directed graphs

representing structured information on the learning objects. When two or more

learning objects are merged, the corresponding attribute structures are uniﬁed, and

the uniﬁed structure is attached to the resulting learning object.

Keywords: distributed learning objects, knowledge structures, structuring content

1. INTRODUCTION

In order to decide which object comes next in presenting a collection of learning objects to a

learner, one might establish some order. Given a set Oof learning objects, a surmise relation

on Ois any partial order ‘4’ on the learning objects in O. The interpretation of ‘4’ is that, given

any two learning objects oand o′in O,o4o′holds if a learner who masters o′also masters o.

The concept of a surmise relation was introduced by [5] as one of the fundamental concepts of

a theoretical framework called Knowledge Space Theory. According to this theory the knowledge

state of a learner is the subset Kof all learning objects in Othat (s)he masters. A subset K⊆Q

is said to be a knowledge state of the surmise relation ‘4’ if o∈Kand o′4oimplies o′∈K

for all learning objects o, o′∈O. Then the collection Kof all knowledge states of ‘4’ is called the

knowledge space derived from ‘4’. Knowledge spaces are used for representing and assessing

the knowledge state of a learner (see, e.g.,[5] and [6] in this connection).

The construction of a surmise relation may follow different approaches. After a brief presentation

of an existing approach based on vectors of components of a learning object, we extend

this approach to a more ﬂexible representation called attribute structure [2]. The mathematical

properties of attribute structures make it possible to compare distributed learning objects in terms

of how much informative and how much demanding they are.

2. THE COMPONENT APPROACH

According to the component approach [1, 7], every content object oin Ois equipped with an

ordered n-tuple A=ha1, a2, . . . , aniof attributes where the length nof the attribute n-tuple A

is ﬁxed for all objects. Each attribute aiin Acomes from a corresponding attribute set Cicalled

the i-th component of the content object. In this sense, given a collection C={C1, C2, . . . , Cn}

of disjoint attribute sets (or components), each object o∈Ois equipped with an element of the

Cartesian product P=C1×C2× · · · × Cn. Usually each component Ciis equipped with a partial

order ‘6i’ so that hCi,6iiis in fact a partially ordered set of attributes. The partial order ‘6i’ is

interpreted in the following way: for a, b ∈Ci, if a6ibthen a learning object characterized by

attribute ais less demanding than a learning object characterized by attribute b. To give a simple

example, it might be postulated that ‘computations involving integer numbers’ (attribute a) are less

demanding than ‘computations involving rational numbers’ (attribute b). A natural order 6on the

elements in P, the so-called coordinatewise order [4], is derived from the npartial orders ‘6i’ by

hx1, x2, . . . , xni6hy1, y2, . . . , yni ⇐⇒ ∀i:xi6iyi

4th International LeGE-WG Workshop:

Towards a European Learning Grid Infrastructure

Progressing with a European Learning Grid 1

Structuring and Merging Distributed Content

Stefanutti, L., Albert, D., & Hockemeyer, C. (2005). Structuring and Merging Distributed Content. In

P. Ritrovato, C. Allison, S. A. Cerri, T. Dimitrakos, M. Gaeta & S. Salerno (Eds.), Towards the

Learning Grid: Advances in Human Learning Services (Vol. 127 Frontiers in Artificial Intelligence

and Applications, pp. 113-118). IOS Press.

Structuring and Merging Distributed Content

If f:O→ P is a mapping assigning an attribute n-tuple to each learning object, then a surmise

relation ‘4’ on the learning objects is established by

o4o′⇐⇒ f(o)6f(o′)

for all o, o′∈O. The mapping fcan easily be established even when the learning objects are

distributed (see, e.g., [8]).

3. ATTRIBUTE STRUCTURES

An attribute structure is used to represent structured information on a learning object or an asset

and in this sense it represents an extension of the attribute n-tupel discussed in Section 2. From

a mathematical standpoint attribute structures correspond to the feature structures introduced by

[3] in computational linguistics. Let Cbe a set of components and Aa collection of attributes, with

A∩C=∅. An attribute structure is a labeled directed graph A=hQ, ¯q, α, ηiwhere:

•Qis a set of nodes of the graph;

•¯q∈Qis the root node of the graph;

•α:Q→Ais a partial function assigning attributes to some of the nodes;

•η:Q×C→Qis a partial function specifying the edges of the graph.

As an example, let

C′={picture,topic,subtopic,text,language}

be a set of components, and

A′={PICTURE1,TEXT1,ENGLISH,MATH,MATRIX INVERSION}

be a collection of attributes. Suppose moreover that a simple learning object is described by

the asset structure A1=hQ1,¯q1, α1, γ1i, where Q1is the set of nodes, ¯q1is the root node, and

α1and γ1are deﬁned as follows: α1(0) is not deﬁned, α1(1) = PICTURE1,α1(2) = TEXT1,

α1(3) = MATH,α1(4) = ENGLISH, and α1(5) = MATRIX INVERSION;η1(0,picture) = 1,

η1(0,text) = 2,η1(0,topic) = 3,η1(1,topic) = 3,η1(2,topic) = 3,η1(2,language) = 4, and

η1(3,subtopic) = 5. The digraph representing this attribute structure is displayed in Figure 1.

The structure A1describes a very simple learning object containing a picure along with some text

explanation. Both text and picture have MATH as topic and MATRIX INVERSION as subtopic.

The root node of the structure is node 0and it can be easily checked from the ﬁgure that each

0

1

2

picture

text

3topic 5

topic

topic

sub-topic

4language

ENGLISH

PICTURE 1

TEXT1

MATH MATRIX_INVERSION

LO1

FIGURE 1: The attribute structure A1describes a simple learning object on ‘matrix algebra’

node can be reached from this node following some path in the graph. The root node is the

entry node in the asset structure of the learning object, and the edges departing from this node

specify the main components of the learning object itself. Thus, in our example, the learning

object represented by A1is deﬁned by three different components: picture,text and topic. The

values of these three components are the attributes given by α1(η1(0,picture)) = PICTURE1,

α1(η1(0,text)) = DESCRIPTION,α1(η1(0,topic)) = MATH.

Observe, for instance that node 5can be reached from node 0following the path

hpicture,topic,subtopici. The fact that, in this example, each node is reachable from the root

node through some path is not a coincidence. It is explicitly required that every node in an attribute

structure be reachable from the root node.

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Structuring and Merging Distributed Content

4. COMPARING AND COMBINING ATTRIBUTE STRUCTURES

Attribute structures can be compared one another. Informally, an attribute structure Asubsumes

another attribute structure A′(denoted by A ⊑ A′) if A′contains at least the same information as

A. In this sense an attribute structure can be thought as a class of learning objects (the class of

all learning objects represented by that structure), and ‘⊑’ can be regarded as a partial order on

such classes. Formally, an attribute structure A=hQ, ¯q, α, ηisubsumes another attribute structure

A′=hQ′,¯q′, α′, η′iif there exists a mapping h:Q→Q′fulﬁlling the three conditions

(1) h(¯q) = ¯q′;

(2) for all q∈Qand all c∈C, if η(q, c)is deﬁned then h(η(q, c)) = η′(h(q), c);

(3) for all q∈Q, if α(q)is deﬁned then α(q) = α′(h(q)).

In Section 2 the attribute sets were assumed to be partially ordered according to pedagogical

criteria and/or cognitive demands. Similarly we assume now that a partial order ‘6’ is deﬁned

on the set Aof attributes so that, given two attributes a, b ∈A, if a6bthen a learning object

deﬁned by attribute ais less demanding than a learning object deﬁned by attribute b. Then the

subsumption relation ‘⊑’ is made consistent with ‘6’ if condition (3) is replaced by

(4) for all q∈Q, if α(q)is deﬁned then α(q)6α′(h(q)).

According to this new deﬁnition, if A ⊑ B then Ais either less informative than Bor less

demanding than Bor both.

As an example consider the three attribute structures depicted in Figure 2. Assuming that

MATRIX PRODUCT 6MATRIX INVERSION, both mappings gand hfulﬁll conditions (1), (2)

and (4), thus both attribute structures labeled by LO2 and LO3 subsume the attribute structure

labeled by LO1. However there is neither mapping from LO2 to LO3 fulﬁlling the subsumption

conditions, nor the opposite, thus these last two structures are incomparable to each other. The

picture text

topic

topic topic

sub-topic

language

ENGLISH

PICTURE 1 TEXT1

MATH

MATRIX_INVERSION

LO1

text

topic

sub-topic

language

ENGLISH

TEXT1

MATH

MATRIX_INVERSION

LO3

picture

topic

topic

sub-topic

PICTURE1

MATH

MATRIX_PRODUCT

LO2

g h

FIGURE 2: Both LO2 and LO3 subsume LO1. However LO2 and LO3 are incomparable.

derivation of a surmise relation for the learning objects parallels that established in section 2. If

s:o7→ s(o)is a mapping assigning an attribute structure to each learning object, then a surmise

relation ‘4’ on the learning objects is derived by

o4o′⇐⇒ s(o)⊑s(o′)

for all o, o′∈O.

Two binary operations are deﬁned on attribute structures: uniﬁcation and generalization.

Mathematically, the uniﬁcation of two attribute structures Aand B(denoted by A⊔B), when exists,

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Structuring and Merging Distributed Content

picture

topic

topic

sub-topic

PICTURE1

MATH

MATRIX_PRODUCT

text

topic

sub-topic

language

ENGLISH

TEXT1

MATH

MATRIX_INVERSION

topic

=

sub-topic

MATH

topic

MATRIX_PRODUCT

FIGURE 3: Generalization of two asset structures

is the least upper bound of {A,B} with respect to the subsumption relation. Dually,generalization

(denoted by A ⊓ B) is the greatest lower bound. In particular, for any two attribute structures A

and Bit holds that

A ⊑ A ⊔ B,B ⊑ A ⊔ B

A ⊓ B ⊑ A,A ⊓ B ⊑ B

When two different learning objects are merged together, or when different assets are assembled

into a single learning object, the corresponding attribute structures are uniﬁed, and the

resulting attribute structure is assigned to the resulting learning object. On the other hand, the

generalization operation is used to ﬁnd the common structure of two or more learning objects

or, stated another way, to classify learning objects. An example of the generalization operation

applied to two attribute structures is shown in Figure 3. Here, the resulting structure shows that

two learning objects have in common topic and subtopic. Generalized attribute structures can also

be used e.g. for searching a distributed environment for all learning objects whose structure is

consistent with a certain ‘template’ (for instance to ﬁnd out all learning objects that are ‘problems’

involving, as cognitive operation, ‘recognition’ rather than ‘recall’).

REFERENCES

[1] Albert D., Held T. (1999) Component-based knowledge spaces in problem solving and

inductive reasoning. In D. Albert and J. Lukas (Eds.), Knowledge Spaces. Theories, Empirical

Research, Applications. Mahwah, NJ: Lawrence Erlbaum Associates.

[2] Albert D., Stefanutti L. (2003) Ordering and Combining Distributed Learning Objects through

Skill Maps and Asset Structures. Proceedings of the International Conference on Computers

in Education (ICCE 2003). Hong Kong, 2-5 December.

[3] Carpenter B. (1992). The logic of typed feature structures. Cambridge Tracts in Theoretical

Computer Science. Cambridge University Press, Cambridge.

[4] Davey B.A., Priestley H.A. (2002). Introduction to lattices and order. Second edition.

Cambridge University Press.

[5] Doignon J.-P., Falmagne J.-C. (1985). Spaces for the assessment of knowledge. International

Journal of Man-Machine Studies, 23, 175–196.

[6] Doignon J.-P., Falmagne J.-C. (1999). Knowledge Spaces. Berlin: Springer-Verlag.

[7] Schrepp M., Held T., Albert D. (1999). Component-based construction of surmise relations

for chess problems. In D. Albert and J. Lukas (Eds.), Knowledge Spaces. Theories, Empirical

Research, Applications. Mahwah, NJ: Lawrence Erlbaum Associates.

[8] Stefanutti L., Albert D., Hockemeyer C. (2003). Derivation of knowledge structures for

distributed learning objects. Proceedings of the 3rd International LeGE-WG Workshop, 3rd

December.

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