Graph theory and algorithms: various approaches to utilization of virtual learning environment Eva Milková Faculty of Informatics and Management University of Hradec Kralove Hradec Kralove, Czech Republic
[email protected] Abstract—The paper focuses on experiences gained during the last years with the WebCT virtual learning environment together with multimedia applications supported education of three subjects. WebCT has been used at the Faculty of Informatics and Management since the beginning of this century. The author of the paper has been prepared with her students various multimedia applications dealing with objects appropriate to subject matter for more than 15 years. In the paper we discuss a benefit of multimedia applications and utilization of the WebCT courses used as a support of subjects dealing with graph theory and combinatorial optimization, i.e. Discrete Mathematics and Discrete Methods and Optimization. The paper also mentions the WebCT course prepared for the subject Algorithms and Data Structures aimed at the development of students’ algorithmic thinking essential for the mentioned subjects. Keywords-algorithmic thinking; graph theory; combinatorial optimization; virtual learning environment; multimedia applications, on-line testing, team work
I.
INTRODUCTION
Computer technology has progressed quickly and personal computers and the Internet have become closely linked to human life. Technology has also substantially influenced education. Multimedia applications together with individual approaches within the didactics process give teachers an excellent chance to increase the quality of education. With help of a virtual learning environment teachers can make students’ study more effective and time-efficient. Carefully prepared multimedia study material placed into a virtual learning environment enables to demonstrate and visualize the subject matter more clearly and comprehensible, to develop students’ logical thinking, increase their imagination and help them to solve various problems. At our university the professional virtual WebCT learning environment has been used. In the paper we emphasize those WebCT tools whose inclusion into the courses prepared for the following three subjects is important. Along with suitable multimedia applications we discuss the subject Algorithms and Data Structures first. Then we discuss two subjects dealing with graph theory and combinatorial optimization, namely the subjects Discrete Mathematics and Discrete Methods and Optimization.
A. Algorithms and Data Structures The subject Algorithms and Data Structures is a compulsory subject taught in the first term. The aim of the subject is to develop logical thinking of students, teach them to create simpler algorithms and to understand more difficult ones. The subject is placed before the other subjects dealing with algorithmic and programming skills. There are many different theoretical researches, which deal with the question of how to start developing algorithmic thinking in students. For example, works written by Seymour Papert deal with children’s programming languages (see e.g. [1]) and similar works based on his approach have been developed by our colleagues from the Slovak Republic (see e.g. [2], [3]) as well. Several authors have introduced basic algorithms using spread sheets (as e.g. [4], [5]). Our approach is based on a brick-box and we have been successfully using it for more than ten years in the subject Algorithms and Data Structures. Remark: One can imagine a brick-box, a nice and useful game for children. There are only several base elements available from which children are able to create incredible buildings. Our approach to the creation of algorithms is based exactly on this image. Thus, when we lead our students’ first steps in the creation of algorithms we explain to them that it is like building interesting objects out of just a few basic elements. In the subject Algorithms and Data Structures it means that we start our teaching with basic algorithmic structures (basic elements from the brick-box) and typical algorithmic structures (a few parts made out of these elements) and then we let students get into the secrets of making whole algorithms (building whole constructions). Students create algorithms in the Czech pseudo-language (based on Pascal programming language – see [6]), they write them on papers. The subject matter is explained on the lectures and also in the textbook [7], where more than 150 problem assignments, questions and exercises are presented. The accuracy of a solution can be verified with the help of the ALGORITHMS program (see thereinafter). The program is enclosed, together with solutions of all the textbook's given tasks, on the CD attached to the textbook. During lessons students apply the acquired knowledge to a variety of tasks. After some time when students prepare their
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solutions on paper, each task is illustrated by two or three students at the blackboard and their solutions are compared and discussed by all students. On the one hand, this means that students are led to try to find more solutions to the given task and to be able to understand the efficiency of algorithms as well. On the other hand when incorrect solutions occur among the presented solutions the teacher has an opportunity to discuss with students where the problem is. Mistakes in incorrect algorithms are emphasized on suitable entrance dates. B. Discrete Mathematics The subject Discrete Mathematics (DIMA) is a compulsory subject taught in the fourth term. Its aim is to develop and deepen students’ capacity for logical thinking. Students gain a basic level of competence in graph theory and combinatorial optimization. In the second half of the last century graph theory experienced enormous growth particularly due to its close relations with computer science. Now the mentioned disciplines belong to the fastest growing areas of mathematics and computer science due to their wide ranging applications. On the one hand, usually, there are more methods which can be used for solving the same graph-problem, while on the other hand, using effective modifications of one algorithm, we can devise methods of solving various other tasks. In the DIMA lectures and lessons we deal with nondirected graphs and devote enough time to the theoretical background of each discussed concept and problem. Concerning graph algorithms, we try to make students familiar with certain algorithms in contexts to be able to get deeper insight into each problem and entirely understand it. We always try to examine the given topic as thoroughly as possible and find a “bridge” to another topic. Well-prepared students should be able to describe various practical situations with the aid of graphs, solve the given problem expressed by the graph, and translate the solution back into the initial situation. More precisely, our approach can be described in the following four steps: When starting explanation of new subject matter, we describe a particular problem with a real life example and/or puzzle to enable students to get an idea about its use. If possible, we examine each concept and problem from more than one point of view and discuss various approaches to the given problem solution. We let students practice and give their own examples describing the topic. Using the constructed knowledge and suitable modification of the problem solution, we proceed to new subject matter. Students can find the explained subject matter partly in [8] and partly in electronic study text. They have also the GrAlg program (see thereinafter) available. The main purpose of the program is visual representation of basic graph-concepts and
graph-algorithms using a coloring process on graphs created within the program. C. Discrete Methods and Optimization The subject Discrete Methods and Optimization (DMO) is a compulsory subject taught in the seventh term. Its aim is not only to develop students’ knowledge gained in the subject Discrete Mathematics and to focus on directed graphs, but also to enhance students’ skills in self study of a given new part from the area of graph theory and combinatorial optimization and their ability to explain it to the others. In the first half of the term lectures and lessons are organized in the similar way as lectures and lessons of the subject Discrete Mathematics. However, the second half of the term is focused on enhancing students’ skills in self-study. Students are given some time for studying new matter and preparing presentations. They work in teams of four students. Two of them prepare a presentation containing appropriate theory illustrated on an example. The other members of the team serve as reviewers. Their task is to check the presentation and prepare examples that prove correctness or incorrectness of the explained matter. Defense and discussion with other students take place in remaining lectures and lessons. An indivisible part of the exam is a presentation on an optional topic from the area of DIMA or DMO describing a practical task. This part of the examination runs as a colloquium and each student shows his/her work to the colleagues taking part on the exam. The GrAlg program (see thereinafter) is at student’s disposal again. Using it students can easily create needed graphs and place them into their presentations. II. MULTIMEDIA APPLICATIONS CREATED ON A SCRIPT GIVEN BY THE TEACHER WITH REGARD TO STUDENTS NEEDS Multimedia applications play an important role among the electronic study materials assigned to the appropriate subject. Along with large software products dealing with a wide spectrum of objects developed by a team of professionals there are various smaller presentations and programs dealing with objects appropriate to course subject matter created on a script given by the teacher with regard to students needs. The author of the paper has been interested in creation of such study material for many years. With her students she prepares large programs and presentations for more than one year, usually within their thesis. Students create smaller ones during the term. Multimedia presentations are usually used to describe a topic and illustrate it using visualization of objects and processes. They mostly enable testing the coursework using several prepared exercises. Multimedia programs offer not only the same possibilities as presentations, but their biggest advantage is the possibility to create an infinite number of needed examples. With the help of multimedia applications students can revise the topic when it is needed; they can use them as a useful
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complement of the printed study text. Some programs and presentations offer complete revision of the large subject matter, another serve as detailed visualization of the given topic explained within the lecture. Short animations can serve teacher as an understandable motivation to the given topic. Let us briefly introduce multimedia applications created for the discussed subjects and emphasize their main benefit for students’ self-study. A. Algorithms and Data Structures In the subject Algorithms and Data Structures students create or complete various algorithms. To assist students a program called ALGORITHMS was created. They can use it to get feedback for the solutions they wrote in hand in the Czech pseudo-language when studying at home and preparing for lessons. The ALGORTIHMS program was developed in Borland Delphi environment within the thesis [9]. It is user friendly and provides entire graphical comfort for users. The design of the program is shown on Fig. 1.
(see e.g. [6], [10], [11]). We also remark this illustration; however, we explain the main idea of each simple sorting algorithm as the way children rank in a line. In the appropriate lecture there are always several students willing to by rank to illustrate the sorting processes. Students can later remind this “show” using the animations sorting six snowmen. The short presentations were created by students through their optional work and are available together with the ALGDS web application, which enables to test the individual level of knowledge in an entertaining way, not only on CD to the textbook [7] but also at http://lide.uhk.cz/fim/ucitel/milkoev1/ in the section ALGDS/ Lectures. For those students who were not able to participate in the classroom lesson, and also for students who were present but want to go through a review, the video-tutorials have been created by a postgraduate student (for more information see [12]). Prepared video-tutorials are aimed at the demonstation of various solutions to the same problem and they are included into the WebCT (see the following section III). B. Discrete Mathematics The essential program developed for the subject Discrete Mathematics is the program GrAlg (Graph Algorithms) created in the Delphi environment by our student within his thesis [13]. The main purpose of the application is the easy creation and modification of graphs and the possibility to emphasize with colours basic graph-concepts and graph algorithms on graphs created within the program. The program enables the creation of a new graph, editing it, saving graph in the program, in its matrix representation and also saving graph in bmp format. It also makes it possible to display some graph properties of the given graph represented by figure, to add color to vertices and edges, and to change positions of vertices and edges by “drop and draw a vertex (an edge respectively)”.
Figure 1. Program ALGORITHMS - main window with an algorithm, two right windows displaying the actual values of used variables and two windows below - left window displaying output values and right window displaying errors
The biggest advantage of the GrAlg program is the possibility to run programs visualizing all of the subjects explained algorithms on nondirected graphs in a way from which the whole process and used data structures can clearly be seen (see Fig. 2).
Using the program students can place their solution of the given task in the program and the program shows them step by step how their algorithm work, if it is correct or not. The program also shows actual values of used variables in each step of the algorithm’s process. In this way they can go at their own speed through each algorithm explained on the lectures and in the textbook [7] and better understand the explained matter. Next to the ALGORITHMS program as an essential material for student’s feedback, there are also several very useful smaller digital objects supporting students’ imagination. As an example let us mention animations, created in Macromedia Flash environment, visualizing three simple sorting techniques: Selection Sort, Insertion Sort and Bubble Sort. It is known that the main idea of Insertion Sort is usually described as the way many people sort a hand of playing cards
Figure 2. Program GrAlg – vizalizatoon of the BFS algorithm
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The program allows the user to open more than one window so that two (or more) objects or algorithms can be compared at once (see Fig. 3). Using the GrAlg program students can revise subjectmatter and more deeply understand it. They can use not only graphs prepared by the teacher but also graphs created by themselves and explore the properties of these graphs and run in the program offered algorithms on these graphs. The possibility to open more than one window enables them to follow mutual relations among used concepts and algorithms. Remark: The GrAlg program is not only a substantial help to students in their self-study but it also helps teachers explain all needed concepts and the process of particular algorithms on lectures and seminars. More preciously, the program enables the teacher to complete his/her explanation within lectures in such a way that the topic is more comprehensible; the possibility to use colors allows the teacher to emphasize needed objects and relations; the option to open more than one window enables to explain the problem from more points of view and show mutual relations among used concepts and algorithms. Moreover, the possibility to save each created graph in bmp format allows teachers easy insertion of needed graphs into the study material and thus saves their time when preparing text material and presentations.
C. Discrete Methods and Optimization As we have already mentioned, the subject Discrete Methods and Optimization is aimed to increase students’ knowledge within graph theory and combinatorial optimization and also to develop their ability to study an appropriate topic (an easier one) on their own and be able to explain it to other students. The GrAlg program is at student’s disposal again. The main benefits of the program used as a complement to the DMO subject is the possibility to revise subject matter explained in the DIMA subject. Moreover, the option “Save Graph in bmp format” enables them to easily create needed graphs for their own tasks (see above the section I part C). III.
VIRTUAL LEARNING ENVIRONMENT
Engineering education at the Faculty of Informatics and Management of the University of Hradec Králové is supported in the WebCT virtual learning environment. At the faculty there are part-time students in combined study-programs in addition to full-time students. The studyprograms operate by linking distance studies with consultations at sessions with teachers every two weeks resulting to approximately three face-to-face sessions for each subject within one term. The final credit test and exams take place face to face at the faculty. Concerning full-time students, nowadays more and more students juggle their studies with work and other activities. Students have less time for their studies and are spending less time on campus to attend lectures and to interact with their peers and teachers. However, student engagement is crucial to the success of education. Students learn more when they are intensely involved in their education; are asked to think about what they are learning and apply it in different settings. The virtual learning environment is an opportunity for students to combine their occupation with studies. Relevant study material is accessible whenever and wherever. In this way also students who live abroad for several months as part of a student exchange program are able to follow the subject.
Figure 3. Program GrAlg - three opened window to solve a puzzle
Although all of the subjects' explained algorithms are comprehensibly visualized in the GrAlg program on graphs represented by figures, the program does not visualize algorithms on graphs represented by matrices. However, to deeply understand a graph-algorithm it is important to develop students’ ability to solve a task using the appropriate algorithm on a graph represented in a way suitable for computers, e.g. represented by matrix. Thus besides the GrAlg program several presentations demonstrating step by step solutions of various algorithms on graphs represented by matrices and presentations describing practical tasks assist students in their self study. These presentations were created in Macromedia Flash environment by students within their optional works or in Power Point by students studying the subject Discrete Methods and Optimization (see above the section I part C).
One of the biggest advantages of virtual learning environment for teacher is the possibility to work with appropriate groups of students (i.e. with the students of the same study form studying the same subject in the same academic year). Let us remember that the content of study and study requirements are the same for all students, and that on the one hand full-time students juggle their studies with work and other activities and on the other hand part-time students have face-to-face lessons. Thus our approach to all students is the same and, although we prepare WebCT course for a subject for both forms of study, the courses use the same tools and the same study materials. (Remark: WebCT enables the use of study material saved in the WebCT environment for other courses. This capability is also very practical when we need to use study materials prepared for a course and again repeatedly in the subjects that follow for revision of acquired knowledge.)
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There are many tools each virtual learning environment offers. However, virtual courses have to be clearly organized and should not contain redundant information. Let us review in alphabetical order the tools that we use in the ALGDS, DIMA and DMO courses created in the WebCT environment to support subjects of discussion and let us describe their functions in our courses. Announcements: This tool serves the teacher as a reminder of important dates (written credit tests, examination etc.) and offers necessary information to students. Assignments: We use this tool in the DMO course to support teamwork necessary in the DMO subject. The WebCT environment is well-suited to this. Students can communicate with the teacher and each other, and discussions are accessible to members of the team only. Calendar: The tool is used by students to remark information about important dates. Course Content: The tool serves as the main course page. Here, students find a detailed plan of lectures, large multimedia programs assigned to the appropriate subject (ALGORITHMS, GrAlg), links to all electronic study texts and small multimedia applications visualizing the explained matter saved in the Module tool, problem statements of tasks solved in lessons, samples of credit and exam tests, and information concerning recommended literature and credit and exam conditions. Discussion: Using this tool, students have the opportunity to discuss their questions and problems concerning the appropriate subject with other students. The tool is displayed in our courses mainly as a means of communication among students. Thanks to the lectures, lessons and face to face sessions, the Discussion tool is only seldom used by teachers. Evaluation: The lecturer and assistants insert evaluations of credit tests into the WebCT environment and every student can find an evaluation of his/her work using the My Evaluation tool. Modules: We use this tool to assign all electronic study materials that correspond to a lecture to the appropriate module. The number of modules is the same as the number of face-to-face lectures (sessions) provided during a term. My Files: This tool enables students to place their own various files connected with a subject in a course. Web Links: Here, students interested in an area explained within a subject can find links to additional information outside the immediate framework of the subject.
IV.
RESULTS AND DISCUSSION
For all three subjects - Algorithms and Data Structure, Discrete Mathematics and Discrete Methods and Optimization - we display all the tools mentioned in the previous section, except Assignments, in the appropriate courses. A. Algorithms and Data Structures Thanks to the textbook [7] containing solved problem assignments and CD of the ALGORITHMS program and solutions of all the given tasks in the textbook, the ALGDS course serves as an optional complement to the subject Algorithms and Data Structures. However, all students attend the course early at the start of the term to download the detailed plan of lectures (lectures closely correspond to the content in the textbook; the plan includes links to pages whose content will be discussed in the appropriate week) and information concerning recommended literature as well as credit and exam conditions. Beginners are encouraged to attend all lectures (and, of course, all lessons). They are recommended to have the textbook [7] at their disposal at each lecture and lesson. On the other hand, more advanced students in the area of programming can decide, using the detailed plan of lectures, which lectures would be beneficial to attend. They carefully study the samples of credit and exam tests placed in the Course Content tool. They solve the tasks using the ALGORITHMS program, placed also in the Course Content tool, to be aware what pseudo-language is demanded in the subject. They practice their knowledge using the ALGDS web application. The link to the application is given in the Web Link tool. The number of advanced students is very low among fulltime students. On the contrary, part-time students belong rather to advanced students rather than beginners. Summary: The ALGDS course has been used more often by advanced students than by beginners. B. Discrete Mathematics The DIMA course serves as an optional complement to the subject Discrete Mathematics. However, all information (detailed plan of lectures, recommended literature, and credit and exam conditions) and study material, including the electronic version of publication [8] and the GrAlg program, are available in this virtual environment. Electronic texts containing the subject matter explained in the lecture are placed in the appropriate module (in Modules tool) together with multimedia presentations describing the content in addition to graphs used during the lecture. Problem statements of all exercises solved in lessons as well as samples of credit and exam tests are provided in the Course Content tool. Summary: All students regularly attend the course.
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C. Discrete Methods and Optimization The DMO course is organized in a similar way to the DIMA course. Nevertheless, the course is an obligatory complement of the subject Discrete Methods and Optimization due to the teamwork that is an inevitable part of the DMO subject taught in the seventh term. All electronic texts used in the DIMA course are placed in the DIMA-repetition module. In this way students have the opportunity to freshen their knowledge gained in the fourth term. Other study material is placed in modules correspondent to the lectures. Summary: All students regularly attend the course. D. Recapitulation Last term 79 full-time students attended the subject Discrete Mathematics. During the term a tracing of the utilization of the above mentioned tools in the DIMA course was made (see Tab. 1- the number of visits and the spent time.
In the paper we introduced our experience in the utilization of the WebCT environment. We also focused on multimedia applications created by students and discussed how they can add to the enhancing of students’ self study. Let us emphasize again that visual presentations improve understanding of explained subject matter, and their location within the virtual learning environment enables the students to acquire, complete, test and deepen their knowledge and increase their imagination. Students admire quality multimedia applications prepared by their colleagues who, on the other hand, are proud that their works serve as a useful study material. ACKNOWLEDGMENT This research was supported by the ESF project REFIMAT, No. CZ.1.07/2.2.00/15.0016. REFERENCES [1]
TABLE I.
PROPORTION OF THE TOOLS USAGE
tool Announcements Calendar Course Content Discussion My Evaluation Modules My Files Web Links
session 262 128 2613 321 716 2209 1 29
[2]
time 2:39:29 1:39:06 376:41:00 5:04:51 24:42:03 638:43:18 0:00:20 1:46:12
We can say that the gained proportion of the tools usage shown in the Table 1 has confirmed our expectation. The Course Content and Modules tools have been used by each student about twice a week. Obviously, the Course Content is the most often used tool also in the ALGDS and DMO courses. Moreover, in the DMO course the Modules tool together with the Assignments tool have been used very often as well. The Discussion tool has been often used by part-time students. V.
[3] [4]
[5] [6] [7]
[8] [9]
[10] [11]
CONCLUSION
Since 2000 the professional virtual WebCT environment has been used as a complement to subjects studied at our faculty both in full-time and part-time study programs. “The faculty has devoted its attention to the problems and issues associated with eLearning since the beginning of 1997 and belongs to pioneers in utilization of ICT in the process of education in the Czech Republic.” [14] WebCT, like all the other similar virtual learning environments (see e.g. [15]), offers tools enabling planning and controlling studies as well as preparing and presenting study materials. WebCT also offers communication between teachers and their students and among students themselves. It enables test setting, receiving solutions, informing students of evaluations of their work, monitoring students’ activities, checking the students’ levels of knowledge, etc. WebCT supports team work as well.
[12] [13]
[14]
[15]
S. Papert, Mindstorms: Children, computers, and powerful ideas. 2nd ed. Da Capo Press, 1993. A. Blaho, I. Kalaš, “Playing, developing and computing with images in Comenius Logo for Windows,” in Proceedings of EuroLogo'95, Birmingham 1995. A. Blaho, I. Kalaš, Imagine Logo Primary workbook. Logotron Limited: Cambridge, 2004. Š. Hubálovský, “Modelling of real kinematics situation as a method of the system approach to the algorithm development thinking,” International journal of applied mathematics and informatics, 4/4, pp. 83-91, 2010. G. Lovaszova, J. Hvorecký, “Using spreadsheet calculations to demonstrate concepts of programming”, IJCEELL, 15(3-6), pp.162-84, 2005. N. Wirth, Algoritmy a datové struktúry údajov. Alfa, Bratislava, 1989. (translation of “Algorithms and data structures”, Prentice-Hall, 1987.) E. Milková, J. Haviger, F. Rubáček, P. Voborník, Algoritmy – základní konstrukce v příkladech a jejich vizualizace (in English: Algorithms – basic contructions in examples and their vizualization). Hradec Králové, Gaudeamus, 2010. E. Milková, Problém minimální kostry grafu (in English: The Mimimum Spanning Tree Problem). Hradec Králové, Gaudeamus, 2001. P. Voborník, Programovací jazyk pro podporu výuky algoritmů (in English: “Programming language supporting education of algorithms”). Thesis, University of Hradec Králové, Czech Republic, June 2006. H. T. Cormen, H. E. Leiserson, R.L. Rivest, C. Stein, Introduction to algorithms. MIT Press, Massachusetts Institute of Technology, 2009. P. Wróblewski, Algoritmy: Datové struktury a programovací techniky. Computer Press, Brno, 2004. (translation of “Algorytmy, struktury danych i techniki programowania”, Helion S.A. 2003, in English: Algorithms: Data structures and programming techniques) E. Milková, “Multimedia tools for the development of algorithmic thinking”, Recent Patents on Computer Science, 4, pp. 98-107, 2011. J. Šitina, Grafové algoritmy a jejich vizualizace (in English: “Graph algorithms and their vizualiaztion”). Thesis, University of Hradec Králové, Czech Republic, June 2010. P. Poulová, M. Černá, I. Šimonová, “e-Learning reflecting individual learning styles,” in i-Society 2010:international conference:proceedings, Infonomics society, London 2010, pp. 92-97. B. Zoltán, M. Turčáni, “Creation of education activities models with elearning support 2010,” in Information&Communication Technology in Natural Science Education - 2010 : problems of Education in the 21st Century, vol. 21/4, pp. 30-38, 2010.
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