PROBLEM-BASED LEARNING
A Graph Theory Experience
Mar´ıa Ang´elica Pinninghoff J., Ricardo Contreras A.
Department of Computer Science, University of Concepci´on, Chile
Pedro Salcedo L.
Department of Research and Educational Informatics, University of Concepci´on, Chile
Keywords:
Problem-based learning, Graph theory, Engineering education.
Abstract:
Problem-based learning is a widely used technique in supporting different educational efforts, in which the
core idea is to understand a real problem and, starting from this point, to seek the elements that are necessary
to solve it. This work presents an experience that shows the results of applying this technique in a typical com-
puter science topic in a classical engineering curriculum. Preliminary results obtained by using this technique
are discussed, and some directions for future work are proposed.
1 INTRODUCTION
Problem-Based Learning (PBL) can be defined as an
inquiry process that resolves questions, curiosities,
doubts, and uncertainties about complex phenomena
in life. A problem is any doubt, difficulty, or uncer-
tainty that invites or needs some kind of resolution.
According to (Barell, 2007), student inquiry is very
much an integral part of PBL and problem resolution.
In other words, PBL requires to establish a classroom
environment that welcomes questioning and differ-
ent points of view, and that thrives on collaboration
among all participants.
PBL was first applied in the 60s, to health sci-
ences teaching; in the Faculty of Health Sciences of
McMaster University, Canada (Walsh, 2005), and in
the School of Medicine of Case Western Reserve Uni-
versity, USA, trying to reach two fundamental goals:
to develop problem solving skills in students and to
place learning closer to real problems. This success-
ful experience was rapidly spread from medical and
professional schools to different disciplines.
Currently, this strategy is implemented as a learn-
ing environment in which the problem drives the
learning, i.e., a problem is presented to the students
before they rally acquire any knowledge. The idea
is that the students have to discover that they need
to learn some new knowledge, before they can solve
the problem. If a problem is presented to a student
before learning, for them it becomes a more motivat-
ing activity. This happens because in the seek of new
knowledge, students really know why they are learn-
ing. Additionally, learning triggered by the need to
solve a problem, allows to store that knowledge in
the form of patterns that later, when necessary, can
be easily retrieved.
Table 1 shows different learning structures in
PBL and traditional learning methodologies (Garcia-
Famoso, 2005):
One important point to remark is that PBL is not
problem solving. Problem solving is the process used
to solve a problem, so the students involved in a PBL
process should be skilled in solving problems or, at
least, in critical thinking. The case-based approach
strategy, on the other hand, usually integrates exist-
ing knowledge; and therefore is not, according to the
previous assessment, problem-based learning.
Keeping in mind the idea of a new learning
paradigm, it is important to be aware that students
should be guided towards solutions, and not be driven
towards them. This means that tutors must let each
group make a reasonable attempt at the set problems
before intervening. This can lead to the fact that the
same explanation should be given to different groups
at different times.
79
Angélica Pinninghoff J. M., Contreras A. R. and Salcedo L. P. (2010).
PROBLEM-BASED LEARNING - A Graph Theory Experience.
In Proceedings of the 2nd International Conference on Computer Suppor ted Education, pages 79-83
DOI: 10.5220/0002768700790083
Copyright
c
SciTePress
Table 1: PBL and traditional learning structures.
Traditional Methodologies Problem-Based Learning
Teacher centered Student centered
Linear and rational Coherent and relevant
Teacher as transmitter Instructor as facilitator or collaborator
Students as passive receivers Students as constructors. Active
Structured environment Flexible environment
Individual and competitive Co-operative
Assessment is a teacher responsibility Assessment is a shared responsibility
The aim of this work is to show the results of intro-
ducing PBL in an engineering course. The inclusion
of PBL in the course Graphs and Algorithms was ac-
complished having in mind that the key objective is
to familiarize students with this model. Furthermore,
the core idea is to show that this strategy can help in
keeping students’ interest when exposed to particular
subjects. As expected, this approach presents some
pros and cons, that are mentioned in the conclusions.
This article is structured as follows; the first sec-
tion is made up of the present introduction; the second
section describes some previous experiences in com-
puter science related to the specific topic to be dealt
with; the third section is devoted to our particular ex-
perience, while the fourth section shows the prelimi-
nary results we obtained with our approach, and the
final section shows the conclusions of the work.
2 PBL IN COMPUTER SCIENCE
The traditional approach, for teaching specific top-
ics in computer science, has been carried out in a
sequence that considers the introduction of concep-
tual material in lectures. This material is comple-
mented with a set of examples and a set of exercises.
Sometimes, these activities are supported with practi-
cal work in laboratory sessions.
While pioneer work in PBL is associated to med-
ical learning; computer science, as a discipline, is
playing an important role in the application of this
paradigm. We can mention the work in the School
of Computer Science and Information Technology in
Melbourne, Australia, which describes the experience
of dealing with a specific topic: computing theory.
We share with the authors the perception that by us-
ing PBL, students are more involved in the teaching
learning process (Hamilton et al., 2003).
Positive results in computer science are also re-
ported in (Chinn and Martin, 2005), where authors
describe the experience carried out in the Institute
of Technology, University of Washington, Tacoma.
They combine the use of PBL with collaborative
learning, and describe the design and implementation
of such learning environment for courses in the be-
ginning sequence of a computer science program. In
computer science, thetop-down approach of introduc-
ing different topics in lectures, reinforcing it with ex-
amples, and applying it as a kind of magic recipe, fre-
quently leads to a lack of motivation, due in part to
the nature of the topics, and in part (and most impor-
tant) because its applications are not clear in advance
(Hamilton et al., 2003).
A recent work (dos Santos et al., 2009) propose an
innovativepedagogicalmethodology based on PBL to
improve the learning effectiveness in software engi-
neering, through the implementation of software fac-
tories, where students can work together to solve real
problems.
PBL has been applied in graduate programs, as re-
ported in (Linge and Parsons, 2006), that addresses
the challenge of developing techniques for the ef-
fective teaching of computer network design, within
the context of a master’s program module and using
a two-threaded approach, that comprised a problem-
based learning thread and a conventional lecture
thread.
3 TEACHING GRAPH THEORY
WITH PBL
We have selected a specific course in computer sci-
ence: Graphs and Algorithms. It is necessary to no-
tice that the first step was to switch from the classic
concept of chapter to the concept of unit. Units help
ensure that all the various activities add up to mean-
ingful learnings. Chapters, on the other hand, don’t
help to students’ inquiry, because of the long period
devoted to a chapter coverage, compared to the lim-
ited, and typically short time associated to a specific
unit.
PBL can be carried out in a collaborative learning
environment, which is the case under discussion, or
CSEDU 2010 - 2nd International Conference on Computer Supported Education
80
it can be carried out as an individual project; in the
latter it doesn’t require cooperation among students.
We have chosen the collaborative environment as op-
posed to a competitive one, where students are in a
race against each other in order to get best marks, be-
cause this is the model they will find in a real working
place.
3.1 Basic Considerations
We have selected Graph Theory based on the premise
that this subject is widely considered conceptually
difficult, but accepting that it’s strongly connected
to interesting and real problems. The traditional ap-
proach to this subject finishes a specific topic by
showing a well known problem that students under-
stand but that simply appears. We think that, from
this perspective, students need to start a backtracking
process to grasp how different concepts, previously
seen, are assembled to obtain a solution. The PBL
approach, on the other hand, encourages a search pro-
cess having in mind the problem to be solved.
As a first step in the introduction of PBL, we de-
cided to address the effort devoted to introduce the
new paradigm in a specific unit: spanning trees. In
simple words, a spanning tree is a set of nodes (e.g.,
points representing locations) having one condition;
all of them are connected, i.e., given two arbitrary
chosen nodes, there is a path between them. If we add
the condition that the cost of connecting all the nodes
is as low as possible, we are dealing with a mini-
mum spanning tree. In doing so, we decided to imple-
ment a two thread view for the specific unit (spanning
trees). It means that the PBL strategy was adopted
when working with the specific problem concerning
minimum spanning trees, as described in the next sub-
section, and as a second thread we supported students
with conventional lectures, aimed to solve doubts and
to provide key concepts.
3.2 The Particular Assignment
The specifications of the first assignment used in 2009
is shown below.
Main objective from the teacher perspective: To
develop a graph-based model to describe an optimiza-
tion problem, finding a spanning tree that solves the
problem.
Learning objectives from the teachers perspective:
i) to understand the benefits of modeling problems by
using graphs; students have to construct a graph with
labeled arcs that represent an instance of cost; ii) to
know how specific algorithms (i.e., Prim and Kruskal)
help to get spanning trees; iii) to develop the capa-
bility of modeling problems using graphs; iv) to im-
plement a mechanism that allows to solve the posed
problem.
The problem presented to the students is the fol-
lowing: This class is going to develop a physical con-
nection using optic fiber, in order to have all the stu-
dents connected to the university network. Due to
economic constraints, students divided into groups,
each of which is independent in terms of connectivity.
Students belonging to a particular group can be con-
nected among themselves and, at least one member
of each group is connected to the university network,
trying to find a minimum cost solution. It is neces-
sary to determine how much optic fiber is necessary
to implement the solution.
Students had to submit two written reports, a soft-
ware operating on real data and a final presentation.
Students were classified in groups of 3 or 4 members,
randomly chosen. For the first report delivery, each
one of these groups modeled the problem concern-
ing only students involved in the group. This helped
to understand the problem and to take into account
the criteria of the group in order to decide the selec-
tion of a particular model. For the second report de-
livery, each group shares the collected data with all
the other groups. In doing so, each group is able to
implement a particular solution, based on their own
models, by considering the complete population to be
connected. The implemented solution is a software
product which is described in the second report. Fi-
nally, each one of the groups developed a presentation
for analyzing the experience from their own points of
view.
4 PRELIMINARY RESULTS
In this first experience with PBL, the results havebeen
interesting. Students showed a high level of inter-
action, and class attendance (that is not mandatory)
presented very good ratings, compared to the tradi-
tional scheme. In the different steps during this ex-
perience, students became actively involved, giving
opinions and asking for different concepts related to
the topic.
It is a policy in the University, to ask the students
for an anonymous evaluation about teaching. This is
accomplished via web, so students do not need to be
physically at the university; in fact, most of them an-
swer the questions during vacations. Among the dif-
ferent aspects considered in this anonymous require-
ment, the ability to teach is evaluated. The main point
PROBLEM-BASED LEARNING - A Graph Theory Experience
81
here is that the lecturer for this class is the same one
they’ve had for a long period, starting in 2000. The
best previous mark the lecturer was given was 5.48
(in a 1 to 7 interval), whereas in 2009, with students
emphasizing the PBL experience, this mark raised up
to 5.8. This is illustrated in Figure 1.
Figure 1: Teacher evaluation evolution.
Figure 2: Evolution of marks in Graphs and Algorithms.
In respect to marks obtained by the students, a
statistic for Graphs and Algorithms shows how marks
fit in the different considered ranges. To simplify
this, Figure 2 shows marks corresponding to the upper
range (that goes from 4.6 to 7) starting with data from
2004. In other words, for the first time in six years,
more than 70% of the students got marks in the upper
range of the evaluation interval, whereas the previous
top value recorded was 65% of the students.
Even though it is very early in the process to infer
that PBL improves understanding, interest and, as a
consequence, better marks, there is a strong evidence,
based on the students’ opinions, that it can be a help-
ful tool in the teaching process.
These increased marks, and a better perception
about the teacher’s capability for teaching may be a
singular point. The novelty of this approach may lead
to students to accept changes in the teaching-learning
process more easily, and time will let us judge more
accurately this particular issue. But we firmly believe
that the positive outcomes PBL offers to the teaching-
learning process should not be ignored. And this is the
reason why we are encouraging this teaching strategy.
5 CONCLUSIONS
The idea of introducing a new teaching paradigm that
challenges traditional teaching has been positively
considered by the actors involved in the process. Stu-
dents became gradually interested in the process, and
from a passive attitude the level of interaction evolved
to a very participative process.
Students have found that PBL approach is more
stimulating than standard lectures. The need to deal
with a problem that can be easily understood, al-
though solving the problem may be a hard work, it
motivates students to work in a collaborative perspec-
tive. To place the problem as a central issue that needs
to be solved in teamwork, is a new approach for most
of the students, where they understand and experience
the effects of working in a non competitive approach.
The positive students’ evaluation highlights the
fact that they had the opportunity of working in dif-
ferent scenarios that had a very important practical
dimension and realism.
It is important to notice that students were really
involved in this process, we can illustrate this through
a simple fact: in order to have a closer contact with
the problem, they used Google Earth to obtain precise
distances to work with. The importance of having an
almost real problem acts as a motivating element. In
future planning this fact needs to be taken into ac-
count. An important difference deals with evaluation.
In this experience, the evaluation process considers
a project in which students are actively involved, in-
stead of passive assessments, as it is usually done in
classic teaching.
The teacher’s opinion is that the additional effort,
devoted to material preparation and time for answer-
ing questions about the problem is highly rewarding,
as students get, in their own style and time, a way to
solve a particular situation.
No doubt, this experience needs to be spread out
by including more topics and more courses, and there
is still a lot of work to be done. Nevertheless, it seems
to be a good starting point.
Future directions to be implemented in 2010, con-
sider to increase the number of topics in Graph and
Algorithms, in order to have a PBL strategy for a com-
plete course. These topics should include graph color-
ing and planar graphs, among others. Besides that, it
is time to include the PBL strategy in another course,
and Software Engineering looks like a good choice.
CSEDU 2010 - 2nd International Conference on Computer Supported Education
82
ACKNOWLEDGEMENTS
This work has been partially supported by grant
DIRDOC 07-014, University of Concepci´on, Chile.
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