PERSONALISED E-LEARNING PROCESS
The Case of Geometry in IWT
Giovannina Albano and Giuseppe Maresca
Dipartimento di Ingegneria dell’Informazione e Matematica Applicata, Università di Salerno
Via Ponte don Melillo, I-84084 Fisciano – SA, Italy
Keywords: e-Learning, Mathematics Education, Personalisation.
Abstract: This paper is concerned with the personalisation of teaching/learning paths in mathematics education. Such
personalisation would exploit the features offered by the e-learning platform IWT, which allows to manage
both the knowledge domain and the student’s profile. We will present the implementation of a personalised
course of “Geometry” in IWT, describing the given representation of the knowledge domain and on the
other side the design of various learning objects both meaningful from educational viewpoint and according
to different learning style. The genesis of the course is based on the integration of research in mathematics
education and e-learning. The course has been experimented at the Faculty of Engineering of the University
of Salerno (Italy). The analysis of the outcomes is in progress taking into account both pedagogical and
technical issues.
1 BACKGROUND
This paper starts from one of the main hypotheses of
e-learning (Nichols, 2003), which considers the
facilitation of education processes, providing the
learners with many personalised learning
opportunities. According to this perspective, we
want to present the case of the Geometry learning at
University level.
In the next sections we will focus on some issues
which are regarded as critical by research in
education and in particular in mathematics education
and could be dealt with in a more appropriate way
with the help of an e-learning platform.
1.1 Personalisation/ Individualisation
of the Learning Process
The individualisation of teaching is one of the most
critical issues in instructional practice. It is well
known that some instructional strategies are more or
less effective for particular individuals depending
upon their specific abilities. According to Cronbach
& Snow (1997) the best learning achievements occur
when the instruction is exactly matched to the
aptitudes of the learner. At first, we can say that
individualisation regards how much the instruction
fits students’ characteristics, creating learning
situations suitable to different students. In particular
we refer to the individualisation at the teaching level
which, according to Baldacci (1999), means the
adjustment of the teaching to the individual students’
characteristics, by means of specific and concrete
teaching practices. Another major goal is the
personalisation of the teaching, which refers to the
set of activities directed to stimulate each specific
person in order to achieve the maximum of his/her
intellectual capability. It is clear that neither
individualisation nor personalisation are possible at
undergraduate level, especially with large classes of
freshman students, if teaching is still based on
standard lectures.
From the viewpoint of individualisation the
teaching procedures included in the platform should
get the students to attain the basic skills, by means of
a choice of different learning paths, whereas from
that of personalisation teaching activities should be
planned in order to allow each student to get his/her
own way to excellence, through specific
opportunities to develop his/her own cognitive
potential. In order to develop each student’s specific
skills of, it is necessary to let him/her free to move,
to choose, to plan and to manage some suitable
cognitive situations.
138
Albano G. and Maresca G. (2010).
PERSONALISED E-LEARNING PROCESS - The Case of Geometry in IWT.
In Proceedings of the 2nd International Conference on Computer Supported Education, pages 138-143
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1.2 Mathematics e-Learning
In the case of mathematics, e-learning offers new,
almost unexplored opportunities, especially as
concerns personalisation, cooperative and
constructive methods, language and representations
(Albano&Ferrari, 2008).
At the International Conference of
Mathematicians (2006), a discussion panel has been
devoted to the integration of e-learning with the
teaching and the learning of mathematics. The object
is to understand how to direct the technological
potential in order to improve quality and quantity of
mathematics learning. Clearly enough needs are
different according to the kind and the level of
instruction considered, and general answers are not
available.
Bass (2006) recognizes five topics in
mathematics education which can be helped by
technology:
1) Drawing mathematically accurate and
pedagogically valid graphs. Graphs can be used for
different purposes: exploring, investigating what
happens if some elements vary, prove/show ideas,
explanations, solutions.
2) Keeping trace of the classroom work and
errors. This from the one hand provides indications
(mainly to the teacher) to re-direct subsequent work,
from the other hand allows (mainly the student) to
"see" his/her own improvements increasing his/her
sense of self-efficacy (Zan, 2000).
3) Coordinating lectures and textbooks.
Technology gives the opportunity to design tasks
and additional activities for the student.
4) Easy access for the teacher. Technology gives
the opportunity to adopt a flexible timetable for
meeting students. Also the exchanges of messages
between teacher and student contribute to trace each
student's history and his/her advancements.
5) The repetitive nature of individual, out-of-
schedule sessions. Most often some understanding
problems cyclically recur and the teacher is
compelled to replicate his/her explanations each
time. FAQ's and for a allow teachers to make
accessible to all students topical discussions
potentially useful.
2 THE CASE OF THE
GEOMETRY COURSE
In this section we will see in details the work
underlain to allow IWT to support students with
personalised learning paths. IWT (Intelligent Web
Teacher), realised at Italian Pole of Excellence on
Learning&Knowledge, is a distance learning
platform , whose innovative features are openness,
flexibility and extensibility, in particular given the
presence of three models (Didactic, Student,
Knowledge) allowing the student to reach the
defined didactical objectives delivering a
personalised course with respect to his/her specific
needs, previous own knowledge, preferred learning
styles, didactical model more suitable to the
knowledge at stake and to the mental model (then
engagement) of the learner.
The case we will examine regards the scientific
domain of the Geometry addressed the first year
University level. The work has been consisted on
one side in representing the knowledge domain and
on the other side in designing and implementing
various learning objects both meaningful from
educational viewpoint and according to different
learning style.
2.1 The Representation of the
Knowledge Domain
One of the fundamental steps of the model is the
construction of a sufficiently rich structure on the
raw set of data, notions and exercises. Indeed, while
a traditional textbook’s structure is essentially linear,
a more ramified type of backbone appears necessary
in this context. In fact, the richer the structure, the
greater the information that can be recovered from
data and feedback – the simplest example of this
being a time series as a part of the real line, where
both algebraic and order structures play a
meaningful role and convey information.
According to this very general idea, the IWT
Knowledge Model allows the experts to define and
structure disciplinary domains, by constructing
domain dictionaries, composed by a list of terms
representing the relevant concepts of the disciplinary
domain that we are modelling, and constructing
some ontologies on such dictionaries that are
modelled using graphs structure. The first step is to
choose a suitable level of granularity in splitting the
various types of knowledge into atomic parts. An
irreducibility criterion appears to be reasonable in
the notion joining sense. More precisely, we may
define a semigroup structure on notions, where the
internal operation is given by joining notions;
irreducibility now means that a given notion is a
“prime” i.e. it cannot be expressed (in a non-trivial
way) as the product of two or more notions. Of
course we may not have a unique factorisation so
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that, although irreducible elements appear to be well
defined, an arbitrary element may be factorised in
more than one way into “primes” – thus possibly
requiring random choices during the factorisation
process, or additional nodes and links to account for.
According to this, a suitable decomposition of the
Geometry domain has been done. A dictionary
consisting of about 150 terms, i.e. elementary
concepts, has been created. A graph, whose nodes
are the elements of the dictionary, has been
designed.
The arcs connecting the nodes are mainly related
to two order relations called “Is Required By” (pre-
requisite) and “Suggested Order”, and a
decomposition relation called “Has Part”. The
following figure zooms in on the created ontology,
to better show the relations:
Figure 1: A zoom on the Geometry ontology.
As you can see, the concept “Matrici” (i.e.
Matrices) has been split into five sub-concepts,
which are connected by the relation “Has Part” with
“Matrici”. Among these nodes, some order relation
is mandatory, e.g. you need to know what is a
determinant (node “Determinante”) in order to learn
what is the rank of a matrix. So the relation “Is
Required By” connects the node “Determinante” to
the node “Rango” (i.e. rank). On the other hand,
there is no pre-requisite relation between the
concepts of rank and echelon matrix (node
“MatriceAScalini”). Anyway the author of the
ontology (an expert of the knowledge domain) may
suggest a preference, according to his/her
educational experience or to the addressed
educational context. This is why in the figure you
can see that the node “Rango” is linked to the node
“MatriceAScalini” by the relation “Suggested
Order”. If this latter relation is present, the platform
will take into account, otherwise a random choice is
done.
2.2 The Learning Objects
In this section we will describe the various types of
learning objects (LO) created for each concept of the
knowledge domain, and the educational ratio of their
creation. The IWT Knowledge Model allows to
annotate each LO with a metadata, which requires to
specify a concept (or more than one) inside a domain
which the content of the learning object itself is
referring to. In this way, it is possible to link the
learning object to the concepts of the ontologies,
indeed, by associating a learning object with one or
more concepts, we can assume that the content of
such learning object “explains” the correlated
concepts.
2.2.1 Hypermedia
In the school practice it is evident the change in the
students’ style of studying/working, which is too
often based on patterns of mnemonic learning and
on a very focused study, neglecting variation and
connections. So the study appears strongly split,
pieces of knowledge are memorised being absolutely
disjointed from the context where they born and live
and often the involved concepts themselves become
“words with no sense” repeated as they appear on
the textbook.
As stressed by the National Council of Teachers
in Mathematics (2000), when the students are able to
see connections among various mathematical
contents, they arrive to have a global and integrated
vision of the mathematics. It is important, as the
students learn new concepts, to make evident the
connections with the knowledge they already have.
The connections they develop give them a grater
mathematical power.
Linked to what said above, we also cite the need
of putting each content in a suitable framework,
which means to give the right level of detail (e.g.
some technical steps are fundamental because they
allow to understand, some others no or not always).
Moreover we consider the reification, that is how to
compress pieces of knowledge or procedures,
uncompressing them only if necessary).
According to the above framework, some
learning objects which are a generalization of the
hyperrmedia have been constructed. They are
composed of a main HTML text with keywords. The
links bring to other learning objects, which differ as
both typology of resources (e.g. plain texts,
animated slides, exercises or algorithms, figures,
simulations, video, etc.) and educational parameters
(didactical approach, semantic density, difficulty,
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level of interactivity, etc.).
The links in the main text have been designed in
order to allow the students to make connections
among different topics of the mathematical
knowledge, and in particular of the geometry one; to
see the same concept from different viewpoint (e.g.
geometrical meaning of an algebraic concept such as
the determinant of a matrix); to deepen historical or
motivational references; to explicit technical details
(e.g. in a proof); to use various semiotic
representations and their coordination (Duval, 2006),
that is to make a treatment in a fixed semiotic
system (e.g. algorithmic procedures) or a conversion
from a semiotic representation to another one (e.g.
among verbal formulation, symbolic one and
figures), the latter being the key of the
comprehension in mathematics; to recall definitions
or theorems which are pre-requisites of the topic at
stake.
2.2.2 Structured Video
According to Rav (1999), the whole mathematical
know-how is plunged in the proofs, which contains
all the mathematical methodologies, concepts,
strategies for problem solving, connections among
theories and so on. Based on the Rav thought, some
reflections have guided the creation of suitable
learning objects regarding proofs. Our starting
remark is that in general a proof is not a whole
inseparable text, but it is possible to single out a
structure composed by several autonomous blocks,
which have a proper meaning and a specific role
within the proving path (e.g. sub-goals). Each of
such blocks can be considered as a module which it
is possible to refer to in a concise manner or in wide
manner depending on the advisability. The
composition of more modules leads to the
construction of new knowledge, that is it allows to
prove the thesis of the theorem at stake. It is
worthwhile to note that various theorems may share
same modules within their proofs. Moreover some
proofs have a non linear path, that is some pieces are
non depending one on the other, so the ordering of
the corresponding modules is not univocally
determined and thus the proving flow also.
Thus it becomes crucial for effective learning
that the students are able to identify such modules
and to understand their role in the context, because
this allows them from one hand to look at the text
with more different levels (a whole text, a list of
modules, list of expanded modules), and on the other
hand it makes evident proving strategies and solving
techniques.
In the above framework, some learning objects
consisting in structured video, realised with a
multimedia blackboard, have been designed and
implemented. The videos reproduce something like a
face-to-face lecture, focused on the written steps and
their audio comments. Various colours have been
used to address attention balancing. Pieces of
previous knowledge (even in a different digital
format) can be stored in other pages of the
blackboard and then suitably recalled.
According to what said above, in order to make
evident the modules constituting the proof, the
videos have been further managed. They have been
split into more pieces corresponding to an
educational splitting of the proof, that are the
modules. Each piece has a title, which is a synthetic
phrase describing the characterisation of the module
(e.g. the sub-goal the module bring to). The list of
these titles constitutes a lateral index, moving along
it the student can access directly the related part of
the video. It is obvious that, where more
decompositions are possible, just one choice has
presented and it will be given the students as
homework to create other possible lateral indices.
Moreover, we note that various granularity can be
chosen in splitting the videos. Some realised videos
have a very fine granularity, whilst other ones have
macro-decomposition. This is because we want to
leave up to students as homework to go on by using
subsequent refinements.
A similar reasoning has been done with respect
to solving techniques. So videos, illustrating step by
step how to solve some exercises or how to apply an
algorithm, have been created. The videos are
supplied with a lateral index corresponding to
elementary steps the procedure can be split into.
2.2.3 Static and Dynamic Exercises
In order to cover the knowledge domain with
problem solving competences, attention has been
paid to offer learning objects on basic solving
techniques. Thus two type of exercises have been
implemented: a static one and a dynamic one.
The first one consists in a solving model in plain
text for various exercises, supplied with many
comments and theoretical recalls in order to contrast
the mnemonic acquisition of some procedures
usually applied automatically from the students,
without a previous analysis of the exercise at stake.
This means that students often does not think of the
correctness of the application of a certain solving
procedure, which often leads to incorrect outcomes
as the chosen procedure is not applicable. Moreover,
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often they do not take into account some specific
conditions of the exercise at stake and the automatic
application of standard procedure leads to waste
time (even if they reach the correct result), which is
an important variable to be faced to in a written
time-restricted examination.
Dynamic exercises have also been designed and
implemented. To this aim, Mathematica and
WebMathematica have been used to create suitable
algorithms generating on the fly infinity (and always
different) exercises. All the algorithms are based on
the divide and conquer strategy, splitting each
exercise into one or more elementary steps; that is
the student is guided to the solution facing easier
sub-problems. We define “elementary” step as a
sub-problem which is seen as first time (very fine
granularity) or a sub-problem which corresponding
to an exercise already developed step by step. For
instance, the exercise “Echelon Form of a matrix” is
split into elementary steps corresponding to the steps
described by the Gauss algorithm. At the same time
“Echelon Form of a matrix” is an elementary step if
it is used to prove the linear independence of
vectors. At each elementary step a hint is given and
an interaction is required, so that students have to
give an answer to the current sub-problem. An
automatic evaluation of the correctness of the given
answer is done, using Mathematica. The algorithms
have been suitably thought in order to recognize and
distinguish errors of a (most probably) theoretical
character (e.g. logical inconsistencies) and
computational errors. Correspondingly, a different
warning message is generated, suggesting the most
likely nature of the error and suitable means of
correcting it. This feature proved particularly useful
in saving time during the error correction phase,
since students did not have to uselessly repeat the
whole theoretical background in case of mere
computational errors and, conversely, receiving a
timely warning when they needed to get a better
understanding of the underlying theory.
Moreover if the student is wrong in his answer,
the system at the first time force to re-insert the
answer in order to stimulate the students to try again,
then if the student made mistake again, the system
gives the chance of viewing the correct result if the
student wants.
2.2.4 Animated Slides
Animated slides are particular meaningful when
some figures comes into play. The construction of a
figure is often the first and the key task to correctly
solve a problem. To this aim, the conversion
between verbal description and figural
representation is crucial. Ferrari (2004) note that a
large share of students’ failures can be ascribed to
linguistic issues. The animation and the
synchronisation between the textual description and
the corresponding graphical representation allow to
guide the student in such conversion. Also in this
case, the animation has been designed according to
some suitable elementary steps. The main topics
treated in this way concern the analytic geometry.
Here the conversion among verbal, graphic and
symbolic representations has been treated by
suitable animations, which allow to see step by step
for instance the construction of the equations of the
line or the plane in two and three dimension through
a continuous migration from the graphical situation
to the verbal description and to the algebraic
formula. This way the learner experiences the
genesis of the known equation of the line and at the
same time gains experience in the coordination of
different semiotic systems. The latter is a
worthwhile learning activity, as such coordination is
not spontaneous and it is the key of comprehension
in mathematics (Duval, 2006).
2.2.5 Lessons
According to the viewpoint of having various
learning objects with different granularity, we have
created some modules, called Lesson, which consists
of a collection of elementary learning objects among
the types seen above.
2.2.6 Junction Elements
A further element to enrich the connection structure
available in IWT, besides the arcs of the ontology
and the links of the hypermedia, consists in the so
called “Junction Elements”. They allow to add a new
learning object acting as connectors between
adjoining learning objects which are apparently
disjointed, giving the learning path a non
homogeneous look. For instance, this is the case of a
plain text based on historical approach to a given
concept followed by a dynamic exercise. Then a
junction element allow to bridge the gap between
them. IWT offers three types of junction elements
according the following goals: to fix the objective, in
order to bridge the gap between theoretical notions
and their applications; to settle the learning process,
fostering curiosity, connections and so on; to
stimulate the fruition of further learning objects
which make evident interesting aspects related to the
concept at stake.
Some few experimental junction elements of the
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cited previous types have been designed,
respectively in form of: a student report of his/her
experience in using his/her acquired knowledge to
solve a given problem; a simulated forum, in order
to stimulate curiosity; a collection of questions, with
or without answers, in order to address the students
to interact with other specific learning objects.
2.3 The Personalised Geometry Course
Starting from what said above, we will see how IWT
is able to create a personalised Geometry course. At
first the teacher will select the Geometry ontology,
the target concepts for his/her course and,
eventually, some milestones (e.g. intermediate tests).
When student accesses to the course the first time,
IWT is able to automatically generate for each
student the best possible learning path according to
the information available in the Student Model, to
the course specifications and to the learning objects
available in the repository (Albano et al., 2007). At
first the ontology is used to create the list of the
concepts needed to reach the target concept of the
course. Then the information of the Student Model is
used to update this list according to the cognitive
state and to choose the more suitable learning
objects according to the learner preferences. The
choice is made possible taking the learning objects
whose metadata better matches with the learner
preferences data. Moreover the platform is able to
dynamically update the learning path according to
the outcomes of the intermediate tests.
The student has also chance to personalize
himself the course. In fact, for each didactical
resource of the course he/she has the possibility to
access alternative resources, so to explore and
choose what he/she considers the more suitable to
better understand the topic at stake. Moreover,
he/she can create his/her own resources, adding
annotations (textual or multimedia), and also decide
to let them public or not. In such a way students
interact with the learning material in a tri-
dimensional relationship: they do not restrict
themselves to receive and elaborate some objects
(such as in the case of the book), but produce new
learning objects starting from the ones placed at
their disposal by the platform (Maragliano, 2000).
3 FUTURE TRENDS
In this paper we have presented a personalised
Geometry course based on the integration of
research in mathematics education and e-learning. It
has been experimented at the University of Salerno.
The data already available on IWT show a highly
level of interactions of the students with the course
material. Some first feedbacks report their
enthusiasm for the wide range of different resources
available, and their preference for videos, interactive
exercises and hypermedia. Specific tasks have been
designed in order to test the pedagogical efficacy of
the different kind of the created resources, which
have been assigning to the students along this term.
The analysis of the outcomes is in progress taking
into account: average trend of the tasks and done
mistakes; academic achievements comparison with
standard course; answers to a submitted
questionnaire to explore their feeling regarding the
personalised course; IWT reports about interaction
with the learning objects.
REFERENCES
Albano, G., Gaeta, M., Ritrovato, P. (2007) - IWT: an
innovative solution for AGS e-Learning model.
International Journal of Knowledge and Learning,
3(2/3), 209-224.
Albano, G., Ferrari, P.L. (2008). Integrating technology
and research in mathematics education: the case of e-
learning. In Garcia Peñalvo (ed.): Advances in E-
Learning: Experiences and Methodologies (132-148).
Baldacci M. (1999). L’individualizzazione. Basi
psicopedagogiche e didattiche. Bologna: Pitagora.
Bass, H. (2006). The instructional potential of digital
technologies. In Proc. of International Conference of
Mathematicians, Madrid, 3, 1747-1752.
Cronbach, L., Snow, R. (1977). Aptitudes and
Instructional Methods: A Handbook for Research on
Interactions. New York: Irvington.
Duval, R. (2006). The cognitive analysis of problems of
comprehension in the learning of mathematics.
Educational Studies in Mathematics, 61(1), 103-131.
Ferrari, P.L. (2004). Mathematical Language and
Advanced Mathematics Learning. In Johnsen Høines,
M. & Berit F., A. (Eds.), Proc. of the 28th Conf. of the
International Group for the Psychology of
Mathematics Education, Bergen (N), (2, 383-390).
Maragliano R. (2000). Nuovo manuale di didattica
multimediale. Editori Laterza
Nichols, M. (2003). A theory for eLearning. Educational
Technology & Society, 6(2), 1-10.
Rav, Y. (1999). Why Do We Prove Theorems?.
Philosophia Mathematica, 7(3), 5-41.
The National Council of Teachers in Mathematics. (2000).
Principles and standards of school mathematics.
Reston (VA).
Zan R., (2000). A metacognitive intervention in
mathematics at university level. International Journal
of Mathematical Education in Science and
Technology, 31 (1), 143-150.
PERSONALISED E-LEARNING PROCESS - The Case of Geometry in IWT
143