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Applying the Formal Concept Analysis to Introduce Guidance in an Inquiry-Based Learning Environment


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The European research project weSPOT aims at supporting science learning in secondary and higher education. The underlying pedagogical approach, inquiry-based learning, is often criticized for the lack in teaching learning content and for overburden novice learners. To fill this gap, we developed the Formal Concept Analysis (FCA) tool which is used by teachers to define a knowledge domain, i.e. the objects, attributes and their relations to each other. Learning resources can be assigned to subsets of objects and attributes. By navigating through the concept lattice students get an overview of the topic. They learn by consuming learning resources, either in a self-regulated way, by interacting with the nodes of the lattice, or when following a recommender system which suggests learning resources based on the domain model and defined pedagogical rules. The paper describes the FCA tool and how it is used by teachers and students, and the recommendation strategy that supports students when browsing the knowledge domain.
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Applying the Formal Concept Analysis to Introduce Guidance
in an Inquiry-based Learning Environment
Michael A. Bedek
Knowledge Technologies Institute
Graz University of Technology
Graz, Austria
Simone Kopeinik
Bernd Prünster
Dietrich Albert
Abstract—The European research project weSPOT aims at
supporting science learning in secondary and higher education.
The underlying pedagogical approach, inquiry-based learning,
is often criticized for the lack in teaching learning content and
for overburden novice learners. To fill this gap, we developed
the Formal Concept Analysis (FCA) tool which is used by
teachers to define a know ledge domain, i.e. the objects,
attributes and their relations to each other. Learning resources
can be assigned to subsets of objects and attributes. By
navigating through the concept lattice students get an overview
of the topic. They learn by consuming learning resources,
either in a self-regulated way, by interacting with the nodes of
the lattice, or when following a recommender system which
suggests learning resources based on the domain model and
defined pedagogical rules. The paper describes the FCA tool
and how it is used by teachers and students, and the
recommendation strategy that supports students when
browsing the knowledge domain.
Keywords—Technology-enhanced learning; Open learner
modeling; Recommender system; Formal Concept Analysis
In a European project called weSPOT, educational
researchers and software developers from nine countries are
developing a theoretical framework and a corresponding
multimedia toolkit for science learning and teaching in
combination with today’s curricula and teaching practices
[1]. weSPOT stands for Working Environment with Social
and Personal Open Tools for inquiry-based learning and
aims to support the meaningful contextualization of
scientific concepts by relating them to personal curiosity,
experiences, and reasoning of students. As the title suggests,
the underlying pedagogical paradigm is inquiry-based
learning (IBL) which combines science knowledge with
practicing skills that are often described as skills of the XXI
century, such as critical thinking, creativity, digital literacy,
and problem-solving [2]. With the availability of ubiquitous
web-based, mobile tools and technologies, learning can take
place anywhere and anytime, with and without monitoring
or supervision of teachers. However, in the background
teachers are challenged to provide appropriate learning
resources and to keep up with students’ learning progress
without reverting to exams or tests [3].
IBL is embedded in constructivist learning paradigms in
which students are encouraged to construct their own factual
and conceptual knowledge about a particular domain [4].
Despite a wide range of advantages that IBL offers, there
are also some disadvantages. As for all constructivist
learning approaches, it requires extra effort from students,
since it demands them to be active learners when taking
control of their learning and when creating their own
knowledge. Minimal guidance imposes a great cognitive
load on learners. Thus, providing guidance specifically
designed to support the cognitive processes necessary for
the particular knowledge domain leads to a more effective
and efficient learning process [5]. Additionally, problem
solving, which is a central aspect of IBL, places a huge
burden on working memory [6]. This is in particular
relevant for novice learners, who lack proper schemas to
integrate the new information with their prior knowledge
The Formal Concept Analysis (FCA) tool that is
presented in this paper is part of the weSPOT platform. It
combines self-directed learning by interacting with concept
structures and open learner modeling (see [8] and [9] for an
overview). In addition to that, the recommendation of
learning resources guides a learner through a predefined
knowledge domain, systematically adapting to the learner’s
current knowledge level. In practice, the teacher defines the
domain model, enriches it with learning resources, and
monitors students’ learning. In collaborative learning
experiences, students are allowed and challenged to add
learning resources and to share them with their peers. The
combination of guided, self-directed and open learner
modeling aims to neutralize the above outlined
disadvantages of IBL.
The FCA describes concepts and concept hierarchies in
mathematical terms, based on the application of order and
lattice theory [10]. The starting point for identifying formal
concepts is the definition of the formal context K. The
formal context K is a triple (O, A, I) consisting of the set O
of objects, the set A of attributes and I as binary relation
which connects objects and attributes, i.e. oIa means that
object o has an attribute a. A formal context can be best
represented as a cross table, with objects in the rows,
attributes in the columns and assigned relations as selected
cells (see Fig. 1). For all subsets of objects X O and all
subsets of attributes Y A the following derivation
operators need to be defined:
X X´ := {a A | oIa for all o X} which is the set of
common attributes of the objects in X, and
Y Y´ := {o O| oIa for all a Y} which is the set of
objects that have all attributes in Y.
A formal concept is a pair (X, Y) with the subsets X O
and Y A which fulfill X’ = Y and Y´ = X. The set of objects
X is called the extension of the formal concept; it is the set
of objects that belong to the formal concept. The set Y is
called the formal concept’s intension, i.e. the set of
attributes, which apply to all objects of the formal concept.
The formal concepts can be ordered by a sub-supra concept
relation and represented as a labelled line diagram (see Fig.
2). This hierarchical representation of the formal concepts is
called the concept lattice B(K) (see [11] for details). The
concept lattice can be “read” as follows: The extension X of
a particular formal concept comprises all object-labels
which can be reached through the descending paths from
that concept node. As an example, the node with the
attribute label “Grass snake” has the extension {Grass
snake, Tree frog}. The intension Y is represented by all
attributes that can be reached by an ascending path from that
node. In our example, the intension consists of the attributes
{hatched from egg, is able to swim}. Sub-concepts are more
specific than supra-concepts, i.e. their intension is larger,
and thus, they are located below the supra-concepts in the
concept lattice. In comparison, supra-concepts have smaller
intensions (since they are less specific). Therefore, they are
located above their sub-concepts in the concept lattice.
There are several tools and software applications to
extract formal concepts from a formal context, and to
visualize concept lattices. Some of them support interaction
or more advanced FCA methods, such as attribute
exploration [12]. One example, we would like to mention is
the Concept Explorer1 which served as a starting point for
our visualization of concept lattices. A more comprehensive
overview on FCA-related tools may be found online 2.
Recently, the FCA as a theory and related tools became
more popular for learner´s assessment and for supporting
(technology-enhancing) learning. As an example, [13]
formulated how FCA can be applied to extract skills from a
formal context that consists of students and their answering
patterns from exams. Also, it has been argued that FCA is
suitable for modelling misconceptions of learners [14].
Recently, [15] supported computer science students in
overcoming conceptual difficulties when writing Java
Codes. The software FcaStone 3 has been used to report
concept lattice visualizations of students and their
misconceptions when submitting Java exercises. The work
presented in [15] served as conceptual starting point when
applying and refining pedagogical underpinnings described
within the next section.
The FCA tool follows two main goals: First, to provide
users with a simple way to introduce domain knowledge
into a technology enhanced learning (TEL) environment
(Editor View), and second, to support users while exploring
the incorporated knowledge structure (Lattice View).
Therefore, the tool distinguishes between two user roles, the
teacher and the learner. Teachers have an extended set of
rights and views that allow looking at learner models of
their students (as described in section V) and to add domain
models using the Editor View (see Fig 1).
The Editor View provides a graphical user interface
(GUI) for the creation of knowledge domains in form of a
formal context. With the matrix-like GUI the teacher creates
a domain by entering objects (e.g. Bumble-bee), attributes
(e.g. is toxic) and relations between them (via check boxes).
Alternatively, the teacher may load and draw on a domain
model that has been created and shared previously. Learning
resources (URLs or files) can then be added to elements of
both sets (i.e. objects and attributes). The tool is
implemented in two components: a web-based application in
form of an Elgg4 plugin, and a java web-service component.
When the teacher saves a domain, data is sent to the back-
end where the formal concepts are identified and an
4 Elgg is an open source social networking engine that serves as weSPOT's
learning environment (see
according lattice is calculated using the library Colibri-
Java5. This resulting concept lattice visualizes the domain
model and acts as a learner model representation.
Figure 1. The Editor view for creating and editing a learning domain.
A. Domain Learning
When the teacher assigns a domain model to an inquiry,
learner models are instantiated. Students can explore their
learner models engaging in interactive graph visualizations
(see Fig. 2), that have been implemented based on the
jQuery library arborjs 6. Nodes of the graph describe
concepts consisting of objects and attributes, links between
nodes indicate sub- and supra-concept relations. By
selecting a node, the corresponding concept´s extension and
intension are illustrated in a highlighted manner. The
concept lattice makes the structure of the knowledge domain
and the interrelations of its concepts explicit. Similar as for
concept maps, this kind of graphic organizer aims to
facilitate meaningful learning by activating prior knowledge
and illustrating its relationship with new concepts [16].
The extendable panel on the right side of the plugin
illustrates relevant concept information textually. It also
provides links to learning resources that can be displayed in
a pop up window. The learner model is updated
continuously, with assumptions regarding a learner’s
knowledge that are based on learning resources a learner has
engaged with. Therefore, the following learner actions are
taken into account: i) selecting a learning resource from the
lattice, ii) selecting a learning resource within the Elgg
environment and iii) adding a learning resource to the
To support the learner when navigating through the
learning domain, these actions influence the colors of graph
nodes as explained in the next section.
Figure 2. Learner model displayed in Lattice View
B. Open Learner Modeling
In a nutshell, open learner models make the learner model
(i.e. the “systems” assumption about a learner´s current
knowledge and lack of knowledge) explicit to the learner
him- or herself and in some cases also to other stakeholders
such as teachers. Visualizations of open learner models (for
an overview see [9]) support reflection on the side of the
learner and assist teachers in better understanding strengths
and weaknesses of their students. Visualizing the learner
model in form of a concept hierarchy has shown to improve
student's domain understanding even more than other
approaches, such as concept tag clouds or tree maps [17].
The FCA tool's Lattice View applies the often used
traffic-light analogy [18] to show the learner the extent to
which she already consumed learning resources that are
assigned to formal concepts´ objects and attributes. Since a
concept consists of objects and attributes the nodes are split
in two halves: the upper half representing attributes and the
lower half objects. A green filling indicates that all learning
resources assigned to the attributes or objects of the formal
concept have been interacted with. The orange color applies
when more than 50% of the attributes´ or objects´ learning
resources have been consumed, red otherwise.
By clicking on the node, a side panel appears which
shows all learning resources in the columns, the attributes
and objects of the node in the rows and their connections as
crossed cells. Again, consumed learning resources are
highlighted in green, supporting the learner to easily identify
unattended learning material.
A further level of support is given by the implementation
of resource recommendations. The application of
recommender systems has become very popular in e-
commerce platforms but is also gaining more and more
importance in TEL-research where different approaches
have been successfully applied within the last few years
[19]. In TEL, recommendation engines are usually based on
static and dynamic learner information that is derived from
learning activities. Our approach builds upon the FCA based
learner model as described in section V. Using this theory-
driven model, teachers and students design and organize
their own learning domain and with this, define the
candidate set of learning resources that are recommended.
Using the domain model as a basis, the algorithm considers
critical aspects [20], such as the learning environment and
its aims and the targeted learning domain.
Aim of the algorithm is to guide a learner through the
domain as efficiently as possible, thereby suggesting
resources that match a learner’s knowledge gap. A structural
overview is outlined in Fig. 3. Based on the FCA domain
model a learning resource lr is selected from a set of
resources LR={lr1, lr2, … lrn} that a learner has not taken
into account in her previous learning history (see DP1). The
set LR is further split into LRA (all resources related to
attributes) and LRO (all resources related to objects). At the
beginning the model distinguishes between two branches
(DP2): the “initialization”, if the learner has not made use of
any learning resource lri in the past, i.e. |
|=0 and the
“run”, if a learner has already visited some of the available
resources, i.e. |
In case of the initialization we aim to return a learning
resource that is neither too specific nor too general, or in
other words, a learning resource which is assigned to a
medium level of objects O={o1, o2, … on} and attributes
A={a1, a2, … an}. In both branches we first determine
whether the formal context K has more attributes or more
objects (DP3). If there are fewer objects than attributes we
select a learning resource lr LRO otherwise we consider
the candidate as lr LRA. This should reduce the amount of
learning resources the student has to engage with to get an
overview of the domain. If the learner has already visited
some of the learning resources (run) the algorithm selects
those resources that involve the least amount of
attributes/objects that are unknown to the learner.
This paper presents a tool which enables to easily
structure and visualize learning domains based on the FCA.
The FCA is an algebraic theory, which provides additional
insights into given domains by calculating sub- and supra-
concepts. The formal concepts and their relations are
illustrated via the Lattice View. Extracted concept lattices
further serve as a basis for learner models. In the weSPOT
project the tool's Editor View is used by teachers to define a
knowledge domain and assign it to an inquiry. However,
other possible scenarios are for students to create their
learning domains themselves or to apply the tool in an
assessment procedure where students are asked to correctly
align given attributes with objects.
The weSPOT platform is still under development;
however, several evaluation studies have been conducted.
These evaluation studies were of a formative character, i.e.
the main aim was to gather critical comments or suggestions
for improvement from teachers and students [21]. Thus, the
main variables of interest were usability, perceived
Figure 3. Simplified schema of the recommendation strategy based on a user's learning resource engagement
usefulness as well as likeability. So far, these formative
evaluation results are quite encouraging since they have
shown that in particular older students (16-17 years) like to
engage with the lattice view. Also, the tool has been
perceived as useful since it makes information and their
interrelations more explicit: Some of these students reported
that the lattice view and their interactions with the nodes
increased their awareness of previously hidden interrelations
between the knowledge domains´ elements.
Upcoming evaluation studies will have a summative
character. For example, it will be compared how different
combinations of the weSPOT toolkit have an impact on
learning outcomes such as (increase in) domain-specific
knowledge and general inquiry skills, but also on intrinsic
motivation and cognitive load since IBL in general might
overburden learners (in particular young and novice ones)
At the moment, development work focuses on including
additional learning analytics for updating the learner model.
Currently, the learner model gets updated whenever a
student “views” a learning resource or adds one to the lattice
by him- or herself. Obviously these actions do not
necessarily indicate the students' understanding of a learning
resource´s inherent content. Thus, in a next step additional
learning analytics (such as time spent on a resource) and
also solution patterns on test items will be incorporated in
the updating process of the learner model. This will allow a
fair evaluation of the open learner modelling approach and
the validity of the recommender, as these aspects are
dependent on a valid learner model.
The research leading to these results has received
funding from the European Community's Seventh
Framework Program (FP7/2007-2013) under grant
agreement no 318499 (weSPOT project).
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... Within this study, the annotation process consisted of two steps: firstly, the selection of semantic features (attributes) from a provided dropdown menu (1): the attributes were drawn from the inquiry's domain model which has been provided by the teacher. Further information on the domain model and related tools can be found in Bedek et al. [4]. Secondly, the assignment of tags: after the student closed the dropdown menu, tag recommendations (3) appeared just below the tags input text field (2). ...
... The attributes were drawn from the inquiry's domain model which was provided by the teacher. For further information on the domain model please see [4]. ...
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