One of the main aims of LEA’s BOX is to provide a competencecentred
and noninvasive methodology for the assessment of the learning
progress of individual learners as well as groups of learners.
The notion of learning progress implies the change of a learner´s
current state of knowledge, abilities, skills and competences
over time. A valid assessment of such changes over time, or
in other words, a valid and noninvasive assessment of learning
by means of Learning Analytics, requires a precise and welldescribed
representation of the learning domain. LEA´s BOX applies
two psychopedagogically sound frameworks to describe the
learning domain in a formalized and precise way: The Formal
Concept Analysis and the (Competencebased) Knowledge Space
Theory.
Another framework with a similar mathematical background,
definitions, and objectives is the Competencebased Knowledge
Space Theory which provides a theoretical framework for knowledge
and competence modeling (Albert & Lukas, 1999; Falmagne
& Doignon, 2011; Falmagne, Albert, Doble, Eppstein, &
Hu, 2013). It is a powerful approach for structuring and representing
domain and learner knowledge. In its original formalisation,
a knowledge domain is characterized by a set of problems or
test items. The knowledge state of an individual is identified
with the subset of problems this person is able to solve.
Due to mutual dependencies between the problems, not all potential
knowledge states will occur. These dependencies are captured
by the socalled prerequisite relation or its generalisation,
the prerequisite function. The collection of all possible
states is called a knowledge structure.

Competencebased
extensions of the original framework (Albert & Lukas,
1999; Heller, Ünlü, & Albert, 2013; Heller,
Steiner, Hockemeyer, & Albert, 2006) consider the
latent cognitive constructs underlying observable behaviour
and assume a competence structure on a set of abstract
skills underlying the problems and learning objects
of the domain. By associating skills to the problems
and learning objects of a domain, knowledge and learning
structures on the problems and, respectively, learning
objects are induced. The skills, which are not directly
observable, can be uncovered on the basis of a person’s
observable performance. Skills are thereby commonly
defined adopting learning and teaching goals as they
can be identified from the curriculum (Korossy, 1997)
and by combining action/procedural and conceptual/declarative
components (Marte, Steiner, Heller, & Albert, 2008).
These skills can be related to existing educational
taxonomies (e.g. Anderson & Krathwohl, 2001); the
skill modelling approach of CbKST is therefore in line
with approaches aiming at the standardised and comparable
representation of competence as an outcome of educational
programs or school types and at providing a supporting
frame for competenceoriented and learnercentred instruction
(e.g. BMUKK, 2012; European Communities, 2007, 2008;
European Commission, 2012).

The structures
CbKST formulates on skills (or problems) in terms of prerequisite
relations or functions can be graphically depicted by
Hasse diagrams (e.g. Pemmaraju & Skiena, 1990) and,
respectively, And/Or graphs, which are directed graphs
with the nodes representing the problems of a domain and
the arcs representing prerequisite relationships among
those problems. These structures are traditionally been
used at the backend of learning technologies, as a basis
for adaptation mechanisms. In the iClass project an approach
of opening the structures on domain skills and their association
with learning objects and assessment problems to end users
has been taken. A range of visual tools has been developed
to empower learners and teachers in planning and performing
their learning and teaching, and to help them in reflecting
on the learning and teaching process (Nussbaumer, Steiner,
& Albert, 2008; Steiner, Nussbaumer, & Albert,
2009). In particular, one of these tools – in line
with ideas of open learner models  visualises assessment
results on skills and reports them back to learners (and
teachers) to enable reflection on acquired skills and
identification of existing competence gaps. 

CbKST provides the basis for adaptive assessment procedures
of a learner’s current competence and knowledge state
as well as for the realisation of intelligent educational
adaptation and has been successfully applied as a cognitive
basis for realising in terms of personalising learning experiences
in different learning systems (Albert, Hockemeyer, & Wesiak,
2002; Conlan, O’Keeffe, Hampson, & Heller, 2006;
Falmagne, Cosyn, Doignon, & Thiéry, 2006). The
socalled microadaptivity approach (Augustin, Hockemeyer,
KickmeierRust, & Albert, 2011; KickmeierRust & Albert,
2010) has been developed and applied in the context of gamebased
learning (KickmeierRust, Mattheiss, Steiner, & Albert,
2011) and integrates CbKST with theory of human problem solving
(Newell & Simon, 1972) in order to model learners’
behaviour and skills in problems solving during learning and
assessment situations. The approach enables noninvasive assessment
of learners’ available and lacking skills by monitoring
and interpreting their (inter)actions in the learning environment
during problem solving and the gathered assumptions on a learner’s
skills serve the provision of adaptive hints, prompts or feedback
tailored to the learner’s available and lacking skills
(e.g. KickmeierRust, Steiner, & Albert, 2011). Microadaptivity
can therefore be understood as an approach to formative assessment
and tailored educational interventions.
Formal Concept Analysis (FCA), established by Wille (1982),
aims to describe concepts and concept hierarchies in mathematical
terms. The starting point of the FCA is the specification
of a “formal context” (also called learning domain).
The formal context K is defined as a triple (G, M, I) with
G as a set of objects which belong to the learning domain,
M as a set of attributes which describe the learning domain,
and finally, I as a binary relation between G and M. The relation
I connects objects and attributes, i.e. (g, m) ∈ I means the
object g has the attribute m. The formal context K can be
best read when depicted as a cross table, with the objects
in the rows, the attributes in the columns and relations between
them by assigning “X” in the according cells.
A formal concept
is a pair (A, B) with A as a subset of objects and B
as a subset of attributes. A is called the extension
of the formal concept; it is the set of objects which
belong to the formal concept. B is called the intension,
it is the set of attributes which apply to all objects
of the extension. The ordered set of all formal concepts
is called the concept lattice B(K) (see Wille, 2005)
Every node of the Concept Lattice represents a single
formal concept. The extension of a particular formal
concept can be read off from the lattice by gathering
all objects which can be reached by descending paths
from that node. The intension is represented by all
attributes which can be reached by an ascending path
from that node. For example, the node with the label
“Leech” represents a formal concept with
{Leech, Goldfish) as extension and {m1, m2} as intension.

Learning Domain Biotope (Formal
Context based on Ganter and Wille, 1996)
Notes
regarding attributes: m1…lives solely in the water,
m2…is able to change location,
m3… has limbs, m4…breastfeeds descendants,
m5…applies photosynthesis 


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