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Teaching and Learning Logic in a Virtual Learning Environment ANTONIA HUERTAS, Department of Computer Science, Open University of Catalonia, R. Poblenou, 156, 08018 Barcelona, Spain. E-mail: [email protected] Abstract Teaching and learning in a virtual learning environment (VLE) poses some difficulties, but also challenges and opportunities to rethink the whole learning process, particularly in abstract subjects like logic or high level mathematics. On the other hand, resources and ways to work, now available in VLEs, might soon extend to all kinds of environments. In this paper, we will present experiences at the Open University of Catalonia (UOC), a particular VLE, concerning the whole process of teaching logic and mathematics. In addition, we will discuss some challenges and we will present some innovation projects allowed by the present and near future technologies. Keywords: e-learning, teaching logic, virtual learning environment, on-line high education.

1 Introduction In the Information Society, technologies are having a strong influence on the university environment since they imply the appearance of both new educational necessities and new methodological possibilities. As a result of this influence, new learning models arise in which the role that teachers and students develop differs partly from the one established in the traditional model. Hence, a re-definition of the teaching-learning process is taking place. This re-definition affects both the universities as institutions, and the knowledge areas they cover. A university degree is not enough to provide us with all those concepts that we will need in our extensive and changing professional career. Consequently, the traditional paradigm associated with university education is changing due to new lifelong learning necessities and, on the other hand, to new possibilities that information and communication technologies (ICT) offer to us. By educational technologies, we will refer to resources derived from the application of information and communication technologies to the educational environment, i.e. on-line platforms for collaborative learning, digital libraries, materials in electronic format (audio-visual, interactive, etc.), wireless LANs, Internet and intranet communities, mobile networks, social software, etc. All these technological resources are being used more and more in education, a concept which is usually known as e-learning. We refer to on-line education as the one using internet as the only space for communication through students and teachers, and between students, beeing a subset of the more general distance education concept. On-line education and e-learning models are currently practiced widely all over the world due to the rapid growth of distance and global education and the development of internet and ICT. As Seufert, Lechner and Stanoevska [20] point out, e-learning models can provide high quality educational offerings at the same time that they allow for convenient and flexible learning environments without space, distance or time restrictions. E-learning, which represents a new paradigm Vol. 15 No. 4, ß The Author, 2007. Published by Oxford University Press. All rights reserved. For Permissions, please email: [email protected] doi:10.1093/jigpal/jzm023

322 Teaching and Learning Logic in a Virtual Learning Environment in the context of university education, impregnates all environments of an entirely on-line university like the Open University of Catalonia (UOC). It is necessary to point out, however, that these educational technologies are being used more and more by those traditional universities which are based on the existence of a face-to-face campus where students and teachers meet. Nowadays, the traditional educational paradigm (one centred on the figure of a masterful teacher and some lasting long-time knowledge) continues to coexist with the new educational paradigm (one centred on students and their active learning process). In this new paradigm, the teacher’s role is moving from one related to a knowledge transmission agent to another related to a specialist agent who designs the course, guides and supervises the student’s learning process. This paradigm is commonly known as teaching to learn or, as it is mentioned in [18], the post-modern university paradigm. Borders between both paradigms are not very clear since both coexist and they are interrelated. On the one hand, it happens that educational technologies are used in combination with traditional methodologies and curricula. On the other hand, it is frequent to find on-line courses that combine a student-centred methodology with materials that come from the traditional model (lecture notes and blackboard dissertations, for example). Regarding the area of teaching and learning logic and related subjects like mathematics, reforms are coming — not only in the on-line distance arena but also in the traditional face-to-face education, and many teachers have been encouraged to try new strategies such as on-line personalized support, collaborative learning, integration of logical and mathematical software in their courses, and new, engaging curriculum that requires promoting conceptual understanding and critical reasoning by the students instead of procedural knowledge. With respect to the state of the art in logical e-learning, there are many aspects to consider, among others [8]: (a) the type, range and amount of technology being used, (b) the range and level of courses and programs provided, (c) the pedagogical issues and (d) the institutional support provided both for students and teachers. In the first part of this paper, we will discuss the benefits and the challenges that e-learning and in particular on-line courses can provide both for students and instructors and discuss some methodological and technical aspects. In the second part of the paper, we will comment some experiences on logical and mathematical e-learning that have been developed for the last ten years at the Open University of Catalonia, a completely on-line university with more than 32,000 students, and we will present some ongoing related projects.

2 Benefits of e-learning E-learning programs and technologies can offer relevant benefits to students of on-line courses. Some of these benefits can be especially useful for those students who are working adults with severe time constraints, students who live far from where the physical campus is located (that is, students with geographical constraints), or students with some physical disabilities. The following are some of the main benefits for them [25]:  Relief from the constraints imposed by having to attend a class on campus at a scheduled time  Possibility of self-pacing some of the course content and activities.

Teaching and Learning Logic in a Virtual Learning Environment 323  Improvement in access to education for those students with physical disabilities.  Contribution to the development of technical skills. Regarding the list of the benefits mentioned above, it seems clear that the student’s technical skills and competencies may significantly be improved by having to: (a) communicate with teachers and other students via e-mail, posted notes or chat, (b) develop collaborative projects via a web-based platform, and (c) interact with educational software. Of course, these social and technical experiences can be very valuable for the student’s future career in a globalized world. In particular, interaction with logical software, both in face-to-face or in on-line courses, can provide additional benefits to students, since this interaction can help them to gain a better understanding of some logical concepts, procedures and applications. Web-based distance education programs also offer several benefits to instructors and institutions:  The opportunity to revisit and improve curricula, teaching methodologies (for instance, lab tutorials without typical restrictions on lab availability could be prepared), and materials (for instance, the development of multimedia or interactive documents is enhanced).  The possibility to model and solve real life problems that are either too complicated or too time consuming to be analyzed within a limited time or without the assistance of computers.  The opportunity of sharing material and methodologies among universities in order to ensure: (a) compatibility of curricula, and (b) assessment of theoretical, practical and transversal competencies and capabilities [17].

3 Challenges of e-learning Typically, any type of distance (not face-to-face) education presents higher dropout rates than traditional programs. The nature of distance education can create a sense of isolation in learners, and students can feel disconnected from the instructor, the rest of the class, and even the institution. It is necessary, then, that instructors provide just-in-time guidance to student’s activities and also that they provide regular (almost daily) feed-back on these activities (not only scores, but also detailed revisions or, at least, models of the correct solutions). On the other hand, communication among students should also be facilitated and promoted by instructors (by dynamizing the web spaces devoted to that function) and the institution (by providing an easily usable and efficient web platform that plays the role of an on-line campus). Another usual problem is students having difficulties with the use of technology [22]. Both instructors and their institutions should ensure that students are prepared and have the necessary training before starting an e-learning course or program. Finally, an important additional problem is the poor integration of mathematical and logical notation with most of the existing e-learning web platforms and browsers. On the instructor side, one important challenge can be the time required to design the course and, once it has started, to effectively provide guidance and support to students [1]. Part of this time constraint may be associated with a lack of technical skills from that of the faculty. These facts can contribute to faculty resistance to using e-learning technologies and mathematical software. Teaching with technology presents continual demands for faculty training and professional development. Finally, from the institution side,

324 Teaching and Learning Logic in a Virtual Learning Environment e-learning is a cost-intensive method to deliver education. The start-up and ongoing costs of e-learning courses and programs are significant, due in part to the cost of: (a) communication technology (networks and servers) and specialized staff that must support it, (b) licensed software (web platforms, operating systems, databases managers, logical software, utilities, etc.), and (c) formative courses and programs for faculty, both at methodological and technical levels (including annual upgrading courses on new versions of software). On the other hand, one important issue is to address the aspects concerning the efficacy of test and assessment processes. It is critical to define evaluating strategies that allow ensuring the authorship of the on-line activities of the evaluation. At the same time, face-to-face exams have to reproduce those conditions of the study that allow the students to show their competencies and capabilities not only in the academic area but also in the professional one [5].

4 Experiences at the Open University of Catalonia (UOC) The Open University of Catalonia (www.uoc.edu) is an exclusively on-line, web-based University. Since its creation in 1994, the UOC has carved out a broad and diverse university community and has brought together over 37,000 people in more than 45 countries. Nowadays, the UOC offers more than 20 degrees (from computer sciences and business studies to psychology and law degrees), several postgraduate and master programs, and a specialized doctorate on the Information and Knowledge Society. Its mission is to facilitate continuous and distance learning in the Information Society, where individual knowledge must be reviewed and updated throughout our entire life. In order to achieve this, the university offers intensive use of information and communications technologies, thereby enabling to overcome the barriers imposed by time and space and to offer an educational model based on personalized attention for every individual student. At the UOC, students, academic staff and administrative staff interact and co-operate on a virtual campus (a highdeveloped intranet which supports all academic necessities, including registering and technical assistance), constituting a university community which uses the Internet to create, structure, share and disseminate knowledge. The UOC educational model places the student as an active agent at the centre of her learning process [15]. Throughout this process, the student is assisted by a team of teachers (both faculty and consulting teachers) in each subject. Furthermore, a tutor guides and advises the student from the moment she starts her degree until the moment she finishes it. The student is also helped by her partners in the virtual classroom spaces, which are located inside the UOC virtual campus. In these virtual spaces, the student can find a complete course syllabus, the course learning materials and other resources (such as the digital library or academic software). The virtual campus also includes an e-mail service, a chat service and some other spaces (newsgroups and distribution lists) conveniently designed to promote interaction among students and among students and teachers. Teaching and learning in the UOC on-line university environment has forced the institution to overcome difficulties where no traditional models could help, and to invent new ways by using technology. In the case of teaching and learning logic and mathematics, additional difficulties arise from the fact that students must obtain abstract reasoning abilities:  The lack of motivation for abstract subjects within the context of technical studies, since students do not find them useful for their training. As Meyer [11] points

Teaching and Learning Logic in a Virtual Learning Environment 325 out, on-line students need motivation as a very important variable to learn in that environment.  The dedication needed is often higher than that one assigned in the system of credits, especially for students with an initial weaker mathematical background.  Many of the learning activities in order to obtain skills demand interactivity as a pedagogical strategy.  The writing of formulas with a conventional keyboard and in e-mail is a problem additional to the communication via Internet. In this section, we will comment some experiences on logical and mathematical e-learning developed at the UOC, in particular at the department of computing. The computer science studies programme started in 1996 with a few students and now, with more than 3,000 students, this Department is one of the largest in Spain. Each term (semester) of the last course more than 1,000 students followed the subject of Logic, more than 500 Algebra, more than 400 Calculus and Statistics and more than 500 Discrete Mathematics. Throughout these ten years there has been a continuous process of reforming the area of mathematical foundations teaching and learning. In order to explain that process we can define three different phases or moments depending on the way and grade of use and integration of ICT technologies, the methodological reforms that have taken place, or the curriculum changes that have been decided. These three phases are the following: Phase 1 (traditional experience) When computer science degrees started in 1996, curricula, resources and learning methodologies were created as a mirror of the face-to-face universities, based on the existence of a physical campus, however the teacher’s role at the UOC was not the traditional one, but the one of a specialist agent who designs the course, guides and supervises the student’s learning process. Then, new experiences to adequate methodology, resources and math curriculum to that different educational paradigm were needed. At this phase, the most important innovation consisted in the virtual on-line classroom and the way students and professors communicated. Briefly, we can find the following:     

Traditional curricula (the same of the traditional universities). Traditional methodologies for distance education. Mainly traditional resources with some new digital or video material available. New virtual e-learning environment and new professor’s and student’s roles. Traditional (paper and pencil) examinations.

Phase 2 (using more ICT resources) Parallel with the development of ICT technologies and the new role of students and teachers some experiences of innovation started in order to improve the quality of the learning process in logical and mathematical courses. Progressively, teachers draw-up didactical materials using more ICT resources. Those materials were, mainly, a complement of the didactical modules of the course, which generally were more theoretical than applied and still very similar to the traditional text material. Two examples of the new material are:  e-Math project (www.uoc.edu/in3/e-math). The main objective of that project was the use and diffusion of the ICT learning technologies at that moment (internet and specialized software) in the curricula of some mathematical courses.

326 Teaching and Learning Logic in a Virtual Learning Environment  Continuous assessment tests: Continuous assessment at the UOC consists in virtual tests that the students do at home and send to the professor. In the logical and mathematical area, these tests try to stimulate dual treatment of exercises, combining the traditional resolution with a resolution using mathematical, statistical or logical software. At that moment, a progressive use of commercial mathematical software (Minitab, Mathcad, Scientific Notebook, Wiris, etc.) began in the mathematical and statistics courses. Using logical software to support natural deduction tests or model checking also started. Phase 3 (redesign courses in a new paradigm) In this present phase the use of ICT increases in a more integrated way with other resources and methodology, but the main feature is the redesign of the curriculum and the methodological and assessment models. In the process of designing new on-line courses, the first stage consisted in carrying out a preliminary analysis on the current situation of those courses at our university using the approach provided by the new e-learning paradigm. The main points of this phase are:  New curricula is designed according to the e-learning paradigm.  A new methodology adequate for distance education with an intensive use of ICT is implemented. One important fact is that by using logical and mathematical software there is time saved that can be used for better practice and learning concepts.  New material is designed to use software in an integrated way with other resources. Interactive materials are designed and implemented. The correct verbalization of formulas is provided in on-line courses.  Final examinations are partly face-to-face and partly virtual.  Technological solution to the writing of formulas in e-mail is provided.

5 Opportunities. Some e-learning projects at the UOC Current research recognizes the importance of all forms of learning being focused on the students and they being active participants in their learning, where they can make choices on how learning is delivered [8]. Following these tendencies, universities are increasingly using a constructivist approach to their teaching both in face-to-face and in distance education [9]. Lecturers are taking advantage of the opportunities offered by the technology to implement a new pedagogical approach where the students are at the centre of the model and they are acting as guides. Interaction is seen as important, with lecturers using the technology to increase interaction between students and themselves, and among students. Learning logic and mathematics using technology without face-to-face instructional support is not easy at all. Besides that, in the case of the UOC and other on-line universities the majority of students are working adults. In most cases, these students terminated their studies several years ago and, therefore, they do not have a solid foundation which, additionally, causes them some anxiety. Instructors should make use of different available e-learning methods and strategies such as dynamic presentations, lab tutorials, simulations, concepts discussion, interaction and collaboration among students to support activity, exploration and creation, which could assist students in constructing their own mathematical

Teaching and Learning Logic in a Virtual Learning Environment 327 and logical knowledge. Additionally, feedback from instructors and constant consultation by e-mail or posted notes are fundamental components of an efficient web based learning process. As Sakshaug [19] points out, ‘‘educators need to establish relationships between learners and themselves and among learners in distance education settings. They need to create interactive environments in which the learner is engaged and feels part of the learning experience. They also need to have training in the use of distance technology and how it is most effectively used’’. In answer to the challenges and opportunities, some e-learning projects are being developed at the UOC. Let us see a summarized presentation of them:

5.1 Teaching and learning logic and mathematics in a VLE There is a UOC research group working on designing an e-learning model for logic and mathematics for technical students. In that context, several math courses were analyzed. From this analysis, the following key points were isolated [7]:  Generally speaking, students’ math background was deficient. Furthermore, students presented very different math backgrounds.  An important discrepancy existed between the real time needed to achieve the academic goals and the number of nominal credits assigned to each subject.  The following facts were detected: an excess of algorithmic (repetitive) calculation methods, an excess of formalism through the material, and some lack of students’ motivation for the subject.  Many students presented serious difficulties in their conceptual modelling and abstraction skills.  The evaluation criteria should be reviewed and updated. Trying to provide answers to the former questions, the two mentioned teachinglearning paradigms, a traditional face-to-face one and a new e-learning paradigm were tackled [7]. From this confrontation, it was clear that we were using partly traditional and partly e-learning paradigms. First of all, a mathematical e-learning model for computer sciences students had to be derived using the approach provided by the new paradigm. The general guidelines of the model were established:  Instrumental: mathematics and logic should be considered as a tool that can help computer science engineers to solve real problems.  Interdisciplinary: courses should be contextualized in the degrees to which they are associated.  Constructivism: the main methodological approach to use is a top-down one, that is, going from practical cases to theoretical concepts [2].  Conceptual modelling: concepts and procedures are more important than solving algorithms [16].  Personalization of the learning process: special attention must be paid to the determination of the mathematical background of each student.  Workload adjustment: an important effort should be made during the course planning phase so that course nominal credits will be adjusted to real credits.  Continuous evaluation process: it should integrate software use and automatic corrections.

328 Teaching and Learning Logic in a Virtual Learning Environment TABLE 1. Comparing satisfaction indexes of the old and new Algebra course Institutional survey (Algebra)

March05–July05

Sept.05–Febr.06

Contents of the subject are useful Positive evaluation of the subject Course credits correspond to real time Resources are up to date Satisfaction of resources

37% 48% 46% 51% 54%

56% 70% 61% 87% 71%

It is known that e-learning success depends on both its initial adoption and subsequent continued usage (continuance) and, as [4] states, user’s continuance intention is determined by satisfaction. After implementing a new Algebra course following that model (September 2005–February 2006) we have obtained positive results when comparing the satisfaction indexes and results of students in this term and the previous one (March 2005–July 2005). In Table 1 we can see satisfaction indexes of the Algebra course according to the institutional survey results. On the other hand, the rate of students passing the course of Algebra has been 10% above the previous semester. The final examination has been partly traditional (physical presence and paper-pencil examination) and partly virtual (every student had to model and solve personalized tests by intensively using mathematical software). What would be necessary for our pedagogical model is that each student could do the final examination with a computer, in the same way they work at home. The institution is now working on a new final assessment system.

5.2 JEM e-Content Network JEM is a EU-funded thematic network for Joining Educational Mathematics (www. jem-thematic.net). It started in August 2006 and will run for 3 years. The goal of the JEM thematic network is to pool together the required expertise and to contribute to the coordination of content enrichment activities in the area of mathematics and logic, to the maintenance of agreed standards and to the delivery of powerful synoptic high-quality user information and support pages, invoked in e-learning platforms operated by the partners. These activities will have tremendous long-term benefits for the quality of e-learning in mathematics in Europe and the wider world. The real challenge of e-Learning is to produce content that brings a general improvement to the way students learn and instructors teach. The design and the production of high-quality digital content has turned out to be harder than expected, the main reason being that it requires expertise in several subject matters as well as expertise in several technologies. In the case of mathematics, the required areas of expertise include that of professional mathematicians, software engineers, publishers, and perhaps learning theorists [3]. JEM is the entry point to activities in the area of semantic mark-up for mathematics which is directly related to enhancing digital content, in particular e-learning of mathematics. Much of this activity is or has been sponsored by the EU (OpenMath, OpenMath-TN, WebALT,

Teaching and Learning Logic in a Virtual Learning Environment 329 LeActiveMath, Genesis), some is sponsored by national agencies (MatTaFI, Serving Mathematics, EMILeA-stat, Moses, Mumie, MEMBERS, math-kit), some by multi-national consortia (W3C’s MathML Interest Group, OpenMath Society), and some by private enterprise (NAG Ltd., Liguori Editore, Maths For More, ISN).

5.3 Web repository of digital sharable logical material It is part of a Spanish-funded project. It starts in January 2007 and will run for 3 years. The main goal of this project consists in pedagogical investigation, with a particular interest in e-leaning. The digital library Summa Logicae (logicae.usal.es) will be maintained and extended by creating new educational texts. In the same way a new educational software for those logics will be created. A good deal of effort will be devoted to specific e-learning materials, which are going to be multiplatform, fulfilling of course current standards of high interactivity. All of these results will be collected in http://logicae.usal.es

5.4 Personalization of the learning process in a virtual learning environment (VLE) based on itineraries and navigational behaviour To back-up this project, there is now a Spanish-funded project that will last for three years (2007-2009). The students navigate through the web-based virtual campus interacting with learning resources which are structured following the SCORM e-learning standard. The main goal is to design a usage mining tool for analyzing such user navigational behaviour and for extracting relevant information that can be used to determine the optimal scheduling for each course depending on user profile. Next, to extend the sequencing capabilities of the LOM, SCORM and IMS LD standards will be necessary to include the concept of recommended itinerary, by combining teacher expertise with learned experience acquired by system usage analysis [13].

6 Final remarks E-learning is a rapidly growing activity, and more and more students get their education through web-based courses. It seems there is a competition going on, among different institutions and even European regions, in offering web-based education. Until now, though, we have seen very few scientific papers and reports on the students’ opinions of such education. Sikora and Carroll [21] found postsecondary students taking courses both on campus and at distance to be equally satisfied with distance courses and classroom courses. Additional studies have documented the favourable perceptions of students involved in distance education [23]. With regard to student performance in distance education, Misko [12] reported that vocational education distance students generally recorded solid performances on their course assessments. Willis and Joyner [24] found no differences between on-campus students who took an information technology class on-line and offcampus students who also took the class online. In their European survey report, McCullough and Aimard [10] found that 92% of the surveyed population (students and academic staff) considered that e-learning had really supported their teaching/study, mainly because of the advantages it offers in terms of flexibility, better time management and autonomy/responsibility. Surveyed students and teachers considered

330 Teaching and Learning Logic in a Virtual Learning Environment required self-discipline as the main obstacle to teaching/studying in an e-learning environment. On the other hand, involvement intensity of the participants and presence and quality of technical support were considered the key factors of success in e-learning education. Finally, 95% of the surveyed population associated e-learning with innovation, in the sense that it creates new kinds of relationships between the different agents involved in the learning/teaching process and gives wider access to content and knowledge. In conclusion, we can say that the experiences and challenges of teaching and learning logic and mathematics in virtual learning environments are bringing about great opportunities in the whole higher education community in that it is necessary to rethink the whole learning and teaching processes of these subjects, the used tools for teaching and the roles of student and teacher. Among others, innovation in methodology could be the major consequence of the e-learning paradigm. Technology is the tool and methodology is the way. They themselves have to adapt completely in order to respond to the demand of the new students (many of them digital natives). Among other subjects, logic and mathematics in high school need to adapt both curricula and methodology to that new e-learning paradigm.

Acknowledgements This work has partially been supported by the Spanish CICYT project BFF2003 - 08998 C03 - 02 (2004-2006).

References [1] Birnbaum, B. (2001). Foundations and Practices in the Use of Distance Education. Lewiston, NY: Edwin Mellen Press. [2] Bringslid, O. (2002). Mathematical e-learning using interactive mathematics on the Web. European Journal of Engineering Education, 27(3), 249–255. [3] Caprotti, O., Carlson, L., Seppa¨la ¨, M., & Strotmann, A. (2005). Web Advanced Learning Technologies for Assessment in Mathematics. In Proceedings of the 3rd International Conference on Multimedia and Information and Communication Technologies in Education. [4] Chiu, C.-M., Hsu, M.-H., Sun, S.-Y., Lin, T.-C., & Sun, P.-C. (2005). Usability, quality, value and e-learning continuance decisions. Computers & Education, 45, 399–416. [5] Co´rcoles, C., Huertas, M.A., Juan, A.A., Serrat, C., & Steegmann, C. (2006). Math on-line education: state of the art, experiences and challenges. In Proceedings of the International Congress of Mathematicians, Madrid, Spain, August 22–30, pp. 578–579. [6] Daradoumis, T., Xhafa, F., & Juan, A. (2006). A Framework for Assessing Self, Peer and Group Performance in e-Learning. In Self, Peer, And Group Assessment In E-Learning. Idea Group Press, pp. 279–294. [7] Huertas, M.A., Juan, A, Steegmann, C. (2006): TIC y Matemticas en la UOC. In Revista de la Educacin: Educacin y Cultura en la Sociedad de la Informacin, Vol. 7. [8] Lai, K., Pratt, K., & Grant, A. (2003). State of the Art and Trends in Distance, Flexible, and Open Learning: A Review of the Literature. Available at http:// www.otago.ac.nz/courses/distance_study/pdf/distance_lit_review.pdf

Teaching and Learning Logic in a Virtual Learning Environment 331 [9] Lee, Y. (2005). Integrating Constructivism approaches in e-learning to enhance mathematical self-study. In Proceedings of the Eighth International Conference on Reform, Revolution and Paradigm Shifts in Mathematics Education, Johor Bahru, Malaysia, Nov 25–Dec 1, pp. 238–243. [10] McCullough, C., & Aimard, V. (2006). E-Learning in Europe: How do trainers, teachers and learners rate e-learning? Available at: http://elearningeuropa.info [11] Meyer, K. (2002). Quality in distance education. Focus on On-line learning, JosseyBass, Hoboken. [12] Misko, J. (2000). The effects of Different Modes of Delivery: Student Outcomes and Evaluations. Leabrook, Australia: National Centre for Vocational Education Research. [13] Mor, E., & Minguillo´n, J. (2004). E-learning personalization based on itineraries and long-term navigational behaviour. Proceedings of the thirteenth W3C, pp. 264–265. [14] Murfin, B. (2001): A case study of Math and Science Teacher Education in a Collaborative Virtual Learning Environment. Journal of Computers in Mathematics and Science Teaching, Vol. 20, N. 4, pp. 405–425. [15] Pallof, R., & Pratt, K. (2003). The virtual student. San Francisco: Jossey-Bass Wiley. [16] Pierce, R. & Stacey, K. (2001). Reflections on the Changing Pedagogical use of Computer Algebra Systems: Assistance for Doing or Learning Mathematics? Journal of Computers in Mathematics and Science Teaching, 20(2), 143–161. [17] Quality Assurance Agency for Higher Education (2006). Code of Practice for the assurance of academic quality and standards in higher education. United Kingdom. Available at http://www.qaa.ac.uk/academicinfrastructure/codeOfPractice/. [18] Raschke, C. (2003). The Digital Revolution and the coming of the Postmodern University. RoutledgeFalmer. London. [19] Sakshaug, L. (2000). Research on distance education: Implications for learning mathematics. Focus on Learning Problems in Mathematics, 22, 111–124. [20] Seufert, S., Lechner, U., & Stanoevska, K. (2002). A Reference Model for Online Learning Communities. International Journal on E-Learning, 1(1), 43–54. [21] Sikora, A., & and Carroll, C. (2002). A Profile of Participation in Distance Education: 1999-2000. Washington, DC: National Centre for Education Statistics, U.S. Department of Education. [22] Simonson, M., Smaldino, S., Albright, M., & Zvacek, S. (2003). Teaching and Learning at a Distance. Upper Saddle River, NJ: Merrill Prentice Hall. [23] Wagner, R., Werner, J., & Schramm, R. (2002). An Evaluation of Student Satisfaction in Distance Learning Courses. In Proceedings of the 18th Annual Conference on Distance Teaching and Learning, USA. Madison: University of Wisconsin System. [24] Willis, C., & Joyner, R. (2000). Perceptions of an On-campus/On-line and a Off-campus/On-line Information Processing Course. In Proceedings of the 17th Annual Meeting of the Atlantic Coast Business and Marketing Education Conference, Raleigh, North Carolina, February 18–19. [25] Zirkle, C. (2003). Distance education in career and technical education: A review of the research literature. Journal of Vocational Education Research, 28(2), 151–171. Received 30 December 2006