Carsten Ullrich
email: ullrich_c@sjtu.edu.cn

Kerstin Borau
email: kerstinborau@onlinesjtu.com

Heng Luo
email: hengluo@sjtu.edu.cn

Xiaohong Tan
email: xhtan@sjtu.edu.cn

Liping Shen
email: lpshen@sjtu.edu.cn

Ruimin Shen
email: rmshen@sjtu.edu.cn

Why Web 2.0 is Good for Learning and for Research: Principles and Prototypes

Keywords: K, 3, 1 [Computing Milieux]: Computers and Education-Computer Uses in Education; H, 4, 3 [Information Systems]: Information Systems ApplicationsCommunications Applications

The term "Web 2.0" is used to describe applications that distinguish themselves from previous generations of software by a number of principles. Existing work shows that Web 2.0 applications can be successfully exploited for technologyenhance learning. However, in-depth analyses of the relationship between Web 2.0 technology on the one hand and teaching and learning on the other hand are still rare. In this article, we will analyze the technological principles of the Web 2.0 and describe their pedagogical implications on learning. We will furthermore show that Web 2.0 is not only well suited for learning but also for research on learning: the wealth of services that is available and their openness regarding API and data allow to assemble prototypes of technology-supported learning applications in amazingly small amount of time. These prototypes can be used to evaluate research hypotheses quickly. We will present two example prototypes and discuss the lessons we learned from building and using these prototypes.

1. WEB 2.0 AND LEARNING

Introduction

Today, the term Web 2.0 is used to describe applications that distinguish themselves from previous generations of software by a number of principles. These new, Web 2.0, applications take full advantage of the network nature of the Web: they encourage participation, are inherently social and open. Whereas Web 2.0 is not characterized by a new step of technology as is the Semantic Web [START_REF] Berners-Lee | The semantic web[END_REF], in the last years the Web changed from a medium to a platform, from a read-web to a read-write-web, thereby fulfilling Berners-Lee's original vision of the Web [START_REF] Berners-Lee | Weaving the Web[END_REF].

These principles are in line with modern educational theories such as constructivism and connectionism and thus make Web 2.0 applications very attractive for teachers and learners. Wikis, blogs, and social bookmarking are now commonly used in learning [START_REF] Alexander | Web 2.0: A new wave of innovation for teaching and learning?[END_REF]. Recent work at the crossroads of technology-enhanced learning (TEL) and Web research focuses on exploiting Web 2.0 features such as tags for user modeling [START_REF] Carmagnola | User modeling in the social web[END_REF], personalization of mashups [START_REF] Thang | Personalised mashups: Opportunities and challenges for user modelling[END_REF], and ontologies and authoring [START_REF] Al-Khalifa | Folksannotation: A semantic metadata tool for annotating learning resources using folksonomies and domain ontologies[END_REF].

While this research shows convincing examples of using Web 2.0 services for learning, in-depth analyses of the relationship between Web 2.0 technology on the one hand and teaching and learning on the other hand are still rare. However, the significance to understand the implications of technology is underlined by Noss [START_REF] Noss | The Challenge of New Technologies: Doing Old Things in a New Way, or Doing New Things?[END_REF] who asks "[w]hat kinds of pedagogy are appropriate to using [a new] technology and, more fundamentally, how does this technology change the epistemologies of what may or may not be taught in schools". Being able to evaluate the implications of an technology, i. e., "critical media literacy" [START_REF] Collins | Teaching and learning with multimedia[END_REF], requires understanding the very basics of this technology.

With respect to education, it is indeed the case that technical features often impose a specific pedagogical approach. For instance, typical learning management systems such as Blackboard1 present courses as mostly static Web pages, enriched with multiple choice questions. These traditional learning management systems implement what Schulmeister [START_REF] Schulmeister | Grundlagen hypermedialer Lernsysteme[END_REF] calls "administered learning", which is based on the knowledge-transfer paradigm of behaviorist learning. In con-trast, Intelligent Tutoring Systems (ITS) are associated with cognitive theories of learning: an ITS usually implements user-adaptive teaching based on a cognitive theory of the mind [START_REF] Lelouche | Intelligent tutoring systems from birth to now[END_REF], such as the Pact-tutors building on Anderson's ACT-* theory [START_REF] Anderson | The Architecture of Cognition[END_REF][START_REF] Anderson | Cognitive tutors: Lessons learned[END_REF].

Obviously, such a classification is very broad and does not capture the field of TEL in all details (e.g. ActiveMath [START_REF] Melis | Semantic-aware components and services of ActiveMath[END_REF] is a Web-based learning environment based on constructivist paradigms), but the relevant point is that one often observes that technical characteristics lead to specific pedagogical characteristics. Thus, it is important to analyze the technological basics underlying Web 2.0.

Overview on the Article

The contribution of this paper is as follows:

• First, we will analyze the technological principles of the Web 2.0 and elaborate on their pedagogical implications on learning. The focus on technological principles distinguished our work from prior research in this area, such as [START_REF] Downes | E-learning 2.0[END_REF][START_REF] Alexander | Web 2.0: A new wave of innovation for teaching and learning?[END_REF] that explores the potential of Web 2.0 for education in general or [START_REF] Anderson | What is Web 2.0? Ideas, technologies and implications for education[END_REF] that describes the Web 2.0 principles for an educational audience .

• Secondly, we intend to show that Web 2.0 is not only well suited for learning but also for research on learning: the wealth of services that is available and their openness regarding programming interfaces and data allow to assemble prototypes of technology-supported learning applications in amazingly small amount of time. These prototypes can be used to evaluate research hypotheses quickly.

The in-depth analysis is subject of Section 2. In Section 3, we will elaborate on what we mean by rapid prototyping of learning applications. We will present two examples that we realized: micro-blogging for language learning 3.1 and social bookmarking for authoring learning object annotations 3.2. These two projects allowed us to learn practical lessons about using Web 2.0 services for prototyping, which we will discuss in Section 4. We will start by briefly introducing one of today's prevalent learning theory, constructivism, which will provide a context for the subsequent analysis.

Parenthesis: Constructivism

Constructivism is based on the premise that knowledge cannot be transmitted but has to be constructed by the individual. Therefore, learning is an active process of integrating information with pre-existing knowledge.

Cognitively oriented constructivist theories such as discovery learning [START_REF] Bruner | On knowing: Essays for the left hand[END_REF] and microworlds [START_REF] Papert | Mindstorms: Children, Computers, and Powerful Ideas[END_REF] emphasize exploration and discovery. Socially oriented constructivist theories, such as social constructivism [START_REF] Vygotsky | Mind in society[END_REF] and cognitive apprenticeships [START_REF] Brown | Situated cognition and the culture of learning[END_REF] stress the collaborative efforts of groups of learners as sources of learning.

In constructivism, the control over the learning process shifts from the teacher to student, with the learner playing an active role in the learning process. Learning takes place in context and in collaboration and provides opportunities to solve realistic and meaningful problems. In contrast, the teachers focus mainly on preparatory activities and provide support in case assistance is needed. Consequently, the teacher is an initiator of and an adviser in the learning process.

The last years have seen an increasing research in and appliance of constructivist approaches. Pure constructivist approaches, however, are not unchallenged: instructions and drill still need to play a part in the classroom [START_REF] Mantyka | The Math Plague: How to Survive School Mathematics[END_REF]. Hence, the moderate constructivist theory has developed as a pragmatic approach which integrates instructions into a theory that has a clear constructivist tendency.

ANALYZING WEB 2.0 PRINCIPLES FROM A LEARNING PERSPECTIVE

The term "Web 2.0" runs danger of becoming a buzzword as empty as "e-learning": some years ago, every learning software that used the Internet in some way was coined as "e-learning software", regardless of whether it was innovative or helpful for learning. Today, Web 2.0 faces the same fate: every Web-site with a fancy interface is sold as Web 2.0.

However, careful analysis shows that Web 2.0 can be used to describe a new set of software applications that distinguish itself from previous applications by a number of principles. While in the beginning of the Internet, software was modeled after the practices prevalent at that time, slowly a new kind of applications emerged. The defining principles of these applications were first captured by O'Reilly [START_REF] Reilly | What is Web 2.0: Design Patterns and Business Models for the next generation of software[END_REF]. Anderson [START_REF] Anderson | What is Web 2.0? Ideas, technologies and implications for education[END_REF] used a slightly different categorization of what he calls the "big ideas" of Web 2.0 and gives examples of current usages of Web 2.0 services for learning and digital libraries.

None of the principles is a new technology or development for itself. As the inventor of the Web, Tim Berners-Lee, points out, Web 2.0 works on the same standards as Web 1.0 [START_REF] Anderson | What is Web 2.0? Ideas, technologies and implications for education[END_REF], and his original vision of the Web was the one of a "read-write-Web", where everyone could add and edit Web pages [START_REF] Berners-Lee | Weaving the Web[END_REF]. However, taken together the "big ideas of Web 2.0" have reached a critical mass that transforms the way of publishing and information exchange so distinctively that the term "Web 2.0" is warranted.

In the following, we will analyze Web 2.0 principles (based on [START_REF] Reilly | What is Web 2.0: Design Patterns and Business Models for the next generation of software[END_REF][START_REF] Anderson | What is Web 2.0? Ideas, technologies and implications for education[END_REF]) and analyze each principle with respect to the implication on technology-enhanced learning. This will allow us to make general statements about the implications of Web 2.0 on pedagogy based on the inherent technical features of Web 2.0.

Individual Creativity

Web 2.0 enables and facilitates the active participation of each user. Web 2.0 applications and services allow publishing and storing of textual information, by individuals (blogs) and collectively (wikis), of audio recordings (podcasts), of video material (vidcasts), of pictures, etc. Authoring of this user generated content is greatly facilitated by providing easy to use desktop-like interfaces. While some time ago, Web applications were easily distinguishable from their desktop counterparts due to their design and pointclick-wait interaction, today's Web 2.0 applications are often recognizable as being Web application only at second glance. Due to techniques such as Ajax and Flash, responses from the user interface now behave similar to desktop applications (as long as a fast Internet connection is available). Additionally, Web 2.0 services typically put much effort in usability and aim at simplifying the interactions as much as possible by concentrating on the task or service the application provides. Customers often have several similar services to choose from, and as consequence a service is designed to please in order to attract and keep the customers.

This stimulation of active participation distinguishes Web 2.0 based learning from traditional "Web 1.0" learning, which is exemplified in traditional learning management systems, where users read Web pages and solve exercises but cannot contribute and social interactions are restricted to forums. Together with the social dimension captured by the "harnessing of the power of the crowds", these two principles are the most visible ones and as such the most analyzed and stressed by pedagogical research: Downes stresses the constructivist nature of these principles and contrasts the delivered learning of learning management systems with the learner-centered activities triggered by Web 2.0 applications: "the Web was shifting from being a medium, in which information was transmitted and consumed, into being a platform, in which content was created, shared, remixed, repurposed, and passed along" [START_REF] Downes | E-learning 2.0[END_REF].

However, this raises the question how the learner is supported in his usage of these tools and resources. Studies show that students rarely develop explicit learning strategies on their own. According to [START_REF] Tergan | Hypertext und Hypermedia: Konzeption, Lernmöglichkeiten, Lernprobleme und Perspektiven[END_REF], disorientation and cognitive overload are the principal obstacles of self-regulated learning in technology-enhanced learning. This and similar studies [START_REF]Learning for Tomorrows World -First Results from PISA 2003[END_REF] provide evidence that students must learn to self-regulate their learning process since most of them do not posses this skill.

While in traditional Web-based technology-enhanced learning existing research shows how to provide pedagogically supported access to resources and learning supporting tools within a server-based learning environment [START_REF] Ullrich | Course Generation as a Hierarchical Task Network Planning Problem[END_REF], only little research has investigated on using such techniques for Web 2.0 based learning (e. g., [START_REF] Thang | Personalised mashups: Opportunities and challenges for user modelling[END_REF] for personalization of mashups in the domain of tourism).

Harnessing the Power of the Crowd

Web 2.0 services are characterized by the fact that their value increases the more people are using it. A traditional static Web site does not "improve" when visited by large amounts of surfers since it presents its content the same, static way. In contrast, Web 2.0 sites use information provided by the visitors explicitly (user contributions build up the site or part of it) or implicitly (user activities on the site are used for adapting its content or presentation).

The explicit and implicit harnessing of the power of the crowd are best exemplified by Wikipedia and Amazon. In the online encyclopedia Wikipedia, the user explicitly contribute to the encyclopedia by adding and editing articles. In the online shop Amazon, collaborative filtering based on the shopping behavior is used for making suggestions to the customers. Each page describing an article contains suggestions of related products ("Customers who bought this item also bought . . . ").

What are the implications of the principle of harnessing collective intelligence for education? First of all, in Web 2.0 services potentially large amounts of other users will exists and active contribution is encouraged (see the previous principle). Each user is therefore immediately a member of a community with a low barrier to participate.

Constructivism sees learning as activity that takes place in a social context [START_REF] Vygotsky | Mind in society[END_REF]. Thus generally speaking, these innate properties of Web 2.0 services are beneficial for learning. An obvious example is learning of a foreign language. In Sec-tion 3.1 we describe how we used micro-blogging, i. e., SMSlike news exchange between users, for learning English as a second language. The large community of pre-existing users of this service allowed the learners to observe communication of native speakers and to practice by communicating with other users who were no member of the class.

Even in domains other than language learning, the value of communicating is important. For instance, in mathematics learning being able to verbally explain results and problem solving steps is increasingly emphasized. Competencybased learning as, among others, put forward by Niss [START_REF] Niss | Mathematical competencies and the learning of mathematics: the danish KOM project[END_REF] regards communication of results as similar important as formal proving skills.

On the other hand, having the learners engaged in an unrestricted community can be distracting. During our microblogging usage, learners suddenly started to post German and Japanese messages, which in itself is no bad thing but distracted from the goal of practicing English. Additionally, unmoderated contributions can be problematic if offending content is posted. However most Web 2.0 services have builtin quality control mechanisms.

Collective contribution of content can be problematic in case assumptions are made regarding the data provided by the users. The second case study described in this article uses social bookmarking services for easy authoring of learning objects. There, bookmarks are stored on a Web 2.0 service2 annotated with tags that provide additional information about the bookmarked resources. In the case study, the tags were predefined and carried specific semantics that were exploited by a learning management system to suggest new learning resources. However, such a fixed semantic can be enforced only in closed communities. Otherwise, problems might arise if by chance other users employ the same tags but with a different semantic. While most Web 2.0 services offer a possibility to define closed communities, 3 this of course undermines the benefits of an open community.

Some research was performed on harnessing the collective intelligence for e-learning applications. For instance, [START_REF] Al-Khalifa | Folksannotation: A semantic metadata tool for annotating learning resources using folksonomies and domain ontologies[END_REF][START_REF] Bateman | Collaborative tagging approaches for ontological metadata in adaptive e-learning systems[END_REF][START_REF] Wu | Exploring social annotations for the semantic web[END_REF] investigate the usage of tags for ontology generation and authoring support.

Diverse Data on an Epic Scale

In Web 2.0, data is often as important as function. Take del.icio.us as an example: its functionality is voluntarily limited to the basic function of bookmarking with tags, however, the value of del.icio.us emerges from the massive amount of annotated resources. Thus Web 2.0 services employ different measures for increasing user contributions and participations, for instance by building trust (e. g., offering users to leave with an export of their data), by explicit licenses (often open licenses such as Creative Commons), and paradoxically, by making content accessible through RSS syndication and APIs. Behind the user-provided data of Web 2.0 lies the Semantic Web [START_REF] Berners-Lee | The semantic web[END_REF] with its vision to make the data currently hidden in databases available for usage by machines.

As a result, Web 2.0 enables access to data at an unprecedented scale, such as pictures (e. g., Flickr) 4 ), bookmarks, mapping data (e. g., Google Maps), but also indexed data, such as the Google search index. Some of this data was available in the Web 1.0, however not as well-annotated and centralized.

This content can be exploited for learning in various ways. First, learners (and teachers) can use existing resources during their learning process as information sources (e. g., Wikipedia articles as starting points to learn about concepts). Furthermore, active knowledge construction is supported: learners can use the data as building blocks for creating new content (mixing or mash-up of content).

Most of this content is not designed for instruction, but is real world data, uttered by real people in real contexts. As such, it is better suited for constructivist approaches than for Instructional Design with its emphasis on very specifically designed content elements that try to elicit specific instructional outcomes [START_REF] Merrill | Instructional transaction theory: Instructional design based on knowledge objects[END_REF]. This is not to say that constructivism does not value carefully authored content. However, the content takes a less prominent role.

The abundance of available data has also advantages for the e-learning researcher. For instance, it is now easier than ever to build tools that exploit the data to enrich the learning experiences. A vocabulary trainer could automatically enrich the words to be learned by retrieving pictures from Flickr or videos from YouTube.

The main problems consist of license problems, plagiarism and disappearing data. Often, remixed content is based on copy-protected material since especially young users are not aware of potential restrictions. Closely related is the problem of plagiarism. While reusing previous work is not a bad thing in itself, pretending to be the original author of a work certainly is, and copy-and-pasting related work without citation is more and more wide-spread [START_REF] Smith | Plagiarism and google generation under spotlight[END_REF]. The problem of no longer available data is as old as the Web, but is potentially order of magnitudes larger: whereas previously referenced data consisted of a single document (the link target), now an API change (or the closing of a service) can make the complete data of a Web 2.0 service unavailable. However, such actions will almost certainly face massive opposition from the users, and the fear of loss of users will (hopefully) prevent such actions.

Architecture of Assembly

Similar to traditional Web services, the Web 2.0 makes data and functionality accessible. Users can access Web 2.0 services by browsing the Web sites but also through APIs. Typically, APIs allow to add, change, and retrieve data. Content is disseminated by RSS/Atom feeds that allows users to pull the data without ever visiting the site itself. Most content created in the Web 2.0 is micro-content: small, self-contained units, such as blog entries, images and other multimedia content well suited for remixing [START_REF] Dash | Introducing the microcontent client[END_REF]. This micro-content can be combined with other data and services, e. g., tags of Flickr photos can be used to show the location in Google Maps. In difference to traditional Web-services, the Web 2.0 approach is characterized by pragmatic solutions and lightweight formats.

Additionally, existing Web 2.0 services often disseminate their functionality by plug-in modular components, so called widgets [START_REF] Caceres | Widgets 1.0. W3C working draft[END_REF]. This allows integrating the service on a given Web-site by adding only several lines of code. For instance, the microblogging service used in our second use case can be added in the blogs of the learners very easily.

The potential for education resulting from this mix-up / mash-up culture is twofold. First, individual creativity can take place at a level higher than content: just like new content is created by combining other content, new functionality is created by mash-ups. Secondly, the syndication of functionality by widgets allows extending existing learning environments. This way, additional expression channels can be added at a very low cost. Personal Learning Environments (PLE) takes this concept one step further. A PLE [START_REF] Van Harmelen | Personal learning environments[END_REF] is not a pre-build collection of tools and content but a framework that allows a learner to assemble his own suite of applications and content sources. It is build on Web 2.0 technology and uses Web 2.0 services, such as blogs, wikis and social bookmarking. Various content sources, applications, such as e-portfolios [START_REF] Barrett | Electronic portfolios: Digital stories of lifelong and lifewide learning[END_REF], and information from social networks are integrated by the learner with the PLE.

Being able to construct your own learning environment is surely a constructivist notion. But again the problem arises that especially weak learners may not be able to assemble and use such tools efficiently.

Additionally, the "architecture of assembly" and its extensive usage of third-party services might result in technical problems. APIs of a service can change and unlike traditional software there is no prior version to revert to. Additionally, rate limits can raise problems. Often, Web 2.0 services impose rate limits, e. g., by specifying a maximum number of request a client can send per hour. This can interfere with educational applications, for instance if interactions or content created by learners is to be downloaded for off-line analysis (see Section 3.1).

Also, one needs to be aware that using third-party content within an applications can lead to security problems, such as Cross Site Scripting (XSS) attacks [START_REF] Lawton | Web 2.0 creates security challenges[END_REF].

Independent Access to Data

Ideally, Web 2.0 services reach for a wider range of clients than the PC browser. They allow access from and dissemination of data to devices such as mobile phones, PDAs, game consoles, etc. By offering multiple sources of input, this principle increases potential participation of the user. Additionally, the location-awareness that often comes along with mobile devices makes new applications possible, e. g., location-aware dating, sight-seeing, etc.

In addition to device independence, the data on a Web page itself becomes independent of the intended usage of the server and, as a consequence, resources located at an URL become usable in a number of ways. Rendering the data for presentation in a browser, the standard processing of today, is only one of the multiple potential applications. For instance, micro-formats can be used to annotate part of the data. These are predefined mark-up languages that allow to a annotate data on Web pages and thereby assigning a semantic [START_REF] Khare | Microformats: The next (small) thing on the semantic web[END_REF][START_REF] Birbeck | RDFa primer. W3C working draft[END_REF]. This semantic is exploited not by the server but by the client using plug-ins. For instance, contact information on a Web page that is annotated with the hcard micro-format can be directly exported to a mail program or address book. It is no longer the server that defines the usage of the Web data, but the client. The browser starts to act a as information broker.

From a pedagogical viewpoint, having additional means for active participation is advantageous. Being able to participate from everywhere using mobile device will lead to less artificial learning situations, which do not take place in front of the computer but in the context of the real life, a characteristic stressed by constructivist learning. The microblogging service described later in this article allows sending blogging-messages from the mobile phone. This way, the learners could participate wherever they were. The additional benefit of location-awareness can give rise to new learning scenarios [START_REF] Sharples | Towards a Theory of Mobile Learning[END_REF].

For the researcher who is using Web 2.0 services, the additional features offered by multiple devices come for free. There is no extra development and programming effort necessary.

Leveraging the Long Tail

The Long Tail captures the observation that often demand (for a product, information or service) is characterized by a power law, i. e., a decreasing curve with a small number of highly frequently occurring events and a very large number of events that occur with a low frequency [START_REF] Anderson | The Long Tail: Why the Future of Business Is Selling Less of More[END_REF][START_REF] Brynjolfsson | Goodbye Pareto Principle, Hello Long Tail: The Effect of Search Costs on the Concentration of Product Sales[END_REF]. In the physical world, the limited shelf-space cuts the growth of the tail. However in the digital world, with its virtually unlimited space, no barrier hinders the growth and thus enables reaching critical mass of demand even for niche-products. Web 2.0 sites often especially cater for the Long Tail, for instance by facilitating community building.

For learning, the interest of Web 2.0 services to enable the Long Tail has the effect that on the one hand creation and construction of content but also making content public has become easier than ever. This allows producing, publishing, receiving and giving feedback. Again, this is not restricted to language learning: creating, discussing and subsequently revising the creations are essential parts of learning as seen in constructivism.

In addition, social networking can be used to find peers with similar interest both for school and college students but also for vocational learners. Matchmaking between peers with common interests is also a hot topic in Web 2.0 as exemplified by services that employ user information to find users with similar interest, e. g., based on their browsing behavior. 5 Similar to previous patterns, this pattern involves connecting between learners and "strangers", which might result in problems. Lately, the social network Myspace was criticized for not taking sufficient measures to protect minors in their network. While such events are not representative, they show that teaching personnel should keep an eye on what the learners are doing, for instance by letting them report (in class or by blog) about their learning progress.

Perpetual Beta

"Perpetual beta" and the following principle "lightweight models" capture organizational issues related to the software development that only indirectly influence the user. However, they are listed here since they capture parts of what defines the Web 2.0 and they have an effect on the e-learning researcher.

In contrast to traditional software, Web 2.0 applications are no longer released in version-based software packages, one version at a time, but are constantly refined and improved. Changes to services happen gradually, there is no Google 1.2. This is facilitated by the ability of Web applications to track the user's interaction with the service and 5 http://www.stumbleupon.com/ thereby gathering data about interaction patterns that is nearly impossible to collect for desktop applications.

While constant improvement of a service is not a bad thing, new or changed features may lead to confusion of learners who are using the service and may lead to distract from the task at hand. Changes in functionality and user interfaces require adapting previously written manuals. Sometimes parts of services are stopped completely, as it happened recently with the fee-based Google Video. This principle does not endorse a pedagogical theory. However, it has an effect on teachers/researchers that use a specific service. One advantage of the perpetual beta is that the developers are usually open to suggestions from adaptors. They often set-up developer discussion boards and use these to receive additional feedback. In our micro-blogging use case, we made the experience that the main developers of the service were strongly open to suggestions and adapted existing APIs and introduced new ones in response.

Lightweight Models

Software development in Web 2.0 is characterized by its focus on high-level functionality. Today, freely available software has reached a level of maturity that allows to build applications by focusing on high-level functionality. The LAMP stack (Linux, Apache, MySQL, and PHP) and frameworks such as Ruby on Rails and Django6 enable the creation of Web 2.0 services in a very limited amount of time. The effect of the availability of such high-level functionality is explored in Section 3, where we describe how we used existing services for rapid prototyping of technology-enhanced learning applications.

Summary

The term "E-Learning 2.0" coined by Downes [START_REF] Downes | E-learning 2.0[END_REF] illustrates the observation that the characteristics of the Web as it manifests itself today, the "Web 2.0", bring forth implications on teaching and learning. Indeed, the above analysis shows that in so far as one can speak of an "innate" pedagogy of a technological artifact, the Web 2.0 is characterized by social learning and active participation, as advocated by constructivism.

In the following section, we will illustrate the potential of the Web 2.0 extends to research itself. It has never been easier to assemble prototypes that help to evaluate the potential of new e-learning applications.

WEB 2.0 AS A RESEARCH TOOL

Just like the Web 2.0 has transformed the Web and the way people use the Web, we believe that Web 2.0 has the potential to transform research in technology-enhanced learning. We base this claim on two observations: first, on the wealth and variety of Web 2.0 services that are available today, and second, on the low complexity of exploiting the functionality of the services by APIs, which allows to implement functionality on top of these services.

• The wealth of functionality (mostly) freely available today is unparalleled. To cite but a few examples relevant for learning: e-portfolios 7 , navigation analysis 8 , conversion of spoken word to text and vice versa9 , collaborative office functionality10 , video editing11 , etc.

• The Web technology has matured to such a degree that complex applications can be assembled in a very short time. Mash-up engines like Yahoo Pipes and Microsoft's Popfly12 allow a drag&drop construction of applications that integrate functionality from existing services without any programming knowledge. More complex exploitation is achieved by access through open APIs.

We argue that until now the potential that arises from these facts has not been sufficiently stressed: instead of (re)implementing services, the researcher in the area of technology enhanced learning should reuse existing services. The wealth and accessibility of functionality allows to assemble complex prototypes in a small matter of time. These prototypes are usually stable enough for evaluation and validation of research hypotheses. In order to illustrate these claims, in the following, we will describe how we used two existing Web 2.0 service for rapid prototyping

Micro-Blogging for Language Learning

The research described in this section was performed at the distant university of Shanghai Jiao Tong university whose students to a large percentage consist of vocational learners. The research question investigated was how to increase active participation of the students in oral communication courses for English as a second language. In the traditional Chinese education system, knowledge transmission from the teacher to the learner is valued higher than active participation of the learner in the classroom [START_REF] Zhang | A cultural look at information and communication technologies in eastern education[END_REF]. Therefore, even adult learners do not feel comfortable practicing in front of fellow learners. Additionally, being vocational learners, the students did often not find the time to practice outside the classroom hours.

Our research therefore focused on the question how to provide practice possibilities that the students could use in their limited time and without fear of loosing their face. We decided to investigate micro-blogging, a type of blogging that enables users to post short messages that are distributed within their community. Users can post messages from their mobile devices, a Web page, and from Instant Messengers. The same channels are used for receiving messages from other users.

Micro-blogging was used in the lecture in the following way: the lecturer created an account at an existing microblogging service (Twitter13 ). As a homework, the students of the lecture were prompted to create their own account and to become "friends" with the lecturer as well as with the other students. Since each Twitter user receives the messages of his or her friends, each student who followed the instruction would receive the messages of his/her fellow students. The students were then told to post a least seven micro-blogging messages a week and to read the incoming messages of their fellow students.

During the lecture, our expectations that micro-blogging offers a number of possibilities, services and advantages which differ from the standard classroom interaction were confirmed. In the classroom, when they students have to talk to their foreign teacher in a foreign language, they feel very shy and timid and often find themselves at loss of words. This is partly due to the nervousness caused by the situation; and it is partly due to their limited vocabulary. Using microblogging, the students can take their time before responding or commenting and are thus relieved of some of their pressure; and they can also consult a dictionary to find a suitable but unknown word or expression.

From a linguistic point of view, micro-blogging is a communicative approach to the teaching and learning of a foreign language. The students use the foreign language not to fulfill a task, but to communicate about their daily chores, activities, current events, etc-very much the same thing they would do in their native tongue. And in contrast to classroom interaction, Twitter offers the students do so at any time they like.

The aspect of free choice of time is especially suitable for our target group. This group consists of students enrolled at a college offering a combination of evening classes and distance learning courses. The majority of the students has job; and since they are already taking classes in the evening, their busy schedule allows them little time to study and practice. However, most of the students have computer access during their working hours, and so they can visit the micro-blogging website and read the messages sent by the classmates and respond, or simply write a message by themselves whenever they have some minutes of spare time.

The interactional aspect of the micro-blogging site offers a further advantage: probably no student would consider it a "break from work" to practice drill sentences or review a grammar lesson, but visiting the micro-blogging website and communicating with classmates is perceived as "taking a break from work" and relaxing for a few minutes, although the students are actually practicing their language communication skills. In fact, some students noted that microblogging had an additional advantageous side effect: the atmosphere between the classmates improved, and meeting on the micro-blogging Web site gave them the impression of meeting on a virtual schoolyard.

Since Twitter is an open community where messages can be read by all users, not only friends, the students were contacted and contacted other users from outside the class; among them native speakers of English. As a consequence, they do not only use micro-blogging among each other but also with native speakers, which will further improve their communication skills concerning "real-use" of English.

The micro-blogging service we used offers the possibility to send direct messages. Then only two parties can view the message. This is particularly helpful when a student has a question which he does not want to ask "in public" (fear of face loss), or to clarify a point, etc. The students can use this service to communicate directly with each other, or to communicate directly with their teacher. On the other hand, the teacher can correct mistakes by sending a direct message to the student who made the mistake without involving other students.

In order to facilitate the teacher's access of the microblogging service, we implemented an "offline" application. The application automatically downloaded all the students messages and implements a limited form of automatic grading (the more messages sent, the higher the score). It also allows an easier analysis of the amount and content of the interactions. The application accessed the Twitter service through its open APIs: for each learner, the list of recent messages was requested and the content of each message was downloaded.

How are the Web 2.0 principles previously identified reflected in the micro-blogging project? When planning the project, we expected the following: the stimulation of individual creativity would motivate the students to participate actively at times suitable for them, using several devices, i. e., Web interface, instant messenger and SMS. The power of the crowd and the long tail of interest of Twitter users would allow the students to find native speakers with similar interests. At the same time, being able to see their fellow students messages, would improve the sense of community. From a technical site, the architecture of assembly and the lightweight models would allow to construct the prototype in a limited amount of time.

Most of our expectations were confirmed and will be reported in detail in Section 4. Regarding the technical efforts required to use the Web 2.0 as a research tool, they were indeed low. By building on an existing and open service (Principle "architecture of assembly") we were able to offer the students a environment much more attractive than would have been possible by implementing it from scratch. At the same time, due to the open API, we could access all the data required for investigating our research hypotheses. Thanks to existing Web frameworks ("lightweight models") such as Ruby on Rails, we were able to build the download tool in about a person month, a time that includes getting familiar with the Rails framework.

Social Bookmarking for Learning Object Annotation

Authoring learning resources is a time consuming and difficult task. In this project, we focused on the question how lecturers with no or little knowledge of the technical details of learning resources and metadata standards can be supported while collecting content for their courses. Firstly, to be accepted by the users, this activity needs to integrate in their usual work-flow, without imposing additional load. Secondly, student research assistants should be able contribute to the collected resources, too. A third constraint was imposed by the organizational factors: any solution needed to be integrated in the learning management system used by the participating organization (the distant university of Shanghai Jiao Tong University, NEC).

Our solution is based on social bookmarking. Not surprisingly, we found that in their daily routine, lecturers search the Web for resources they can use in their lectures and mark them with bookmarks. While standard bookmarks are stored locally and hence only available for a single user and not reusable by other users, social bookmarking allows storing bookmarks on a server, accessible by other users and additionally annotated with user-defined tags. We therefore encouraged the lecturers to use social bookmarking service to store their bookmarks. Additionally, we instructed the lecturers to use a mixture of user-defined and predefined tags (in contrast to purely user-defined approaches as described in [START_REF] Alexander | Web 2.0: A new wave of innovation for teaching and learning?[END_REF]). For each lecture, predefined tags were specified in a domain ontology that modeled the concepts and relationships of the subject domain (a standard technique today, going back to [START_REF] Vassileva | Dynamic Courseware Generation: at the Cross Point of CAL, ITS and Authoring[END_REF]). Part of the ontology for the lecture on data structure is shown in Figure 1. Using the terms defined in the ontologies, the lecturers were able to specify the concepts a Web resource described.

In addition to the information about the domain concepts, the lecturers were able to specify the instructional type of a resource [START_REF] Ullrich | The learning-resource-type is dead, long live the learning-resource-type! Learning Objects and Learning Designs[END_REF] and its difficulty level. In order to minimize the chance that tags of other users of the social bookmarking service would interfere with our predefined tags, we prefixed the tags with our own character sequence. All in all, we used the following tags (with a different prefix):

• sjtu:type:TYPE, where TYPE is one of definition, example, exercise, elaboration and introduction.

• sjtu:kp:CONCEPTID, where CONCEPTID is the identifier of a concept as defined in the domain ontology.

• sjtu:difficulty:VALUE, where VALUE is one of very_ easy, easy, medium, difficult, very_difficult.

The learning management system used by NEC employs the above domain ontology for the organization and presentation of learning resources. For each concept, the learning management system presents the available resources on a dynamically generated Web page. The resources can be especially authored for the learning management system, but also include resources that were bookmarked in the social bookmarking service. The learning management system retrieves these resources by using the API provided by social bookmarking service. This simple mechanism allows the learning management system to include resources found in the Web.

This work is comparable to previous research on open corpus hypermedia [START_REF] Henze | Adaptation in open corpus hypermedia[END_REF]. The basic principle is the same: a domain description is linked with resources that describe the concepts of the domain. This domain knowledge is then exploited by the learning management system, which distinguishes our work from usages of social bookmarking as described in [START_REF] Alexander | Web 2.0: A new wave of innovation for teaching and learning?[END_REF], where groups of lectures use a joint collection of bookmarks.

One lecturer who used the prototyped appreciated the ease of adding and collecting additional resources so much that she used it in an unexpected way by asking her students to add resources. Her course, "computer networks", was a small course attended by 20 students. Despite the enthusiasm of the lecturer, students made little use of the tool, only a total of eight resources were added. The lecturer attributed the limited participation to the fact that the course content followed a comprehensive textbook and there was no real need for additional resources. Our hypothesis is that in a course without good textbooks, the degree of participation will increase. The lecturer will test this hypothesis in a new lecture.

The important point of this project for this paper is the effort spent in realizing the prototype: the implementation was almost trivial. For the basic functionality, we used the service del.icio.us that allows access via API. On the learning management system side, we only had to add code that performed the lookup of the additional resources (including a cache in order to minimize load) and dynamically added the links to the resources. By building on the functionality available in the Web 2.0, we were able to assemble a prototype in very little time. Due to the popularity of the tool, it is now reimplemented within the NEC learning management system, with the specific functionality identified as relevant by the lecturers (e. g., extended by a grading system).

The next session summarizes the lessons we learned from using Web 2.0 services for rapid prototyping of technologyenhanced learning applications.

LESSONS LEARNED

The two examples of micro-blogging and social bookmarking allowed us to gain practical experience regarding the usage of Web 2.0 for learning. We will first discuss the two examples from the viewpoint of the analysis of Web 2.0 principles applications for learning done in the first part of this paper. Then, we elaborate on the suitability for rapid prototyping of technology-enhanced learning applications (as described the second part of this paper).

Learning in the Web 2.0

Our experiments have shown Web 2.0 services indeed stimulate active participation (principle "individual creativity"). Participation in the micro-blogging service was high and continues until four months after the lecture has finished. Usability of the services was judged as being rather good. About 75% of the microblogging users reported that registration and usage of the service was easy. This supports our claim that using existing Web 2.0 services can hold advantages since the burden of designing an easy to use interface is taken from the researcher.

An analysis of a questionnaire (83 participants) showed that three quarter of the students liked to use the service; about 50% used it at least once a day or more often. Only 5% disagreed to the statement that the usage of the microblogging service improved the overall atmosphere of the course, and again only 5% did not find that the microblogging improved the sense of community. 94% had the impression that they were able to improve their English.

The social dimension ("power of the crowd" and "long tail") was important, both within the course and outside. Within the lecture, we found that the students encouraged each other to participate: in some weeks they held "competitions" who contributes the most. Some of the micro-blogging messages show that students had conversations among each other, thus communication was not only one-way. As a re-sult, despite being a distance course with optional attendance of class, a sense of community was created. Students also established contact to external people, e. g., native speakers (50% of the students according to the questionnaire).

However, we could also observe that unconstrained active participation results in distraction: e. g., in the microblogging service, students posted messages in other languages than English. Additionally, some users seemed not to be aware (or not to care) that their messages were readable by all other users and posted pejorative messages ("I'm too fat", "My boss is an idiot"). It is therefore important to note that although Web 2.0 applications fulfill their goal of stimulating participation, their usage in a classroom does not relieve the teacher of his role as a moderator.

The problem that the openness of a service can potentially interfere with the intended functionality of the learning application that uses the service was also relevant for the social bookmarking application. There, in theory, foreign users might add resources using the same tags, but with a different semantic. In this concrete example, we avoided the problem by using specially marked tags. This is not a bullet-proof solution, but minimizes potential problems.

We also noted that mobile access to the Web 2.0 is still to come (Principle "independent access"). Most students used the standard Web access to the services. In the microbloggin service, only about 10% used the mobile phone regularly to send/receive messages. Paradoxically, one problem with receiving the messages via SMS was the success of the service: on some days, more than 50 messages were sent by the students, quickly cluttering any SMS inbox.

To our knowledge, the students did not use any of the advanced assembly features ("architecture of assembly"), such as the integration of the services in their homepage or blog. We believe that this is caused by the students still being unfamiliar with such possibilities, and that we did not provide instructions on how to do so. However, we did observe that students used standard personalization features, such as uploading background pictures and personal avatar pictures.

Web 2.0 for Prototypes

The exploitation of existing Web 2.0 services for our research allowed to realize running prototypes quickly and relieved us from the burden of technical details such as managing user load and of the difficult task of designing user interfaces. Nevertheless, such an approach is not without drawbacks and one needs to be aware of potential shortcomings.

First of all, in the "architecture of assembly" one becomes dependent of an API controlled by a third party. This is more severe than using a programming library, since API changes directly affect the application: the perpetual beta stage of Web 2.0 applications has the consequence that one cannot download a specific version and stick to it. Similarly, conditions of use can change. During our Twitter usage, the service introduced rate limits that regulated the amount of requests per hour a client could send. This conflicted with our goal to download all of the students' messages.

However, for most part Web 2.0 service owners aim at keeping close and good contact to their users, including developers. The "perpetual beta" of Web 2.0 applications aims at increasing the value a user gets from using the service. Additionally, the architecture of participation brings forward an interest of developers in new usages of their service. Thus, in our experience, they are very interested in feedback and open to suggestions. For instance, most sites will make exceptions to rate limits upon request.

Having to rely on external APIs also involves the problem that sometimes not all required functionality is implemented. However, as long as the wished-for functionality is there in principle (as a rule of thumb: as long as it is achievable by the Web interface), it is possible to artificially "extend" the functionalities. In the micro-blogging prototype, the API allowed only to retrieve the last 20 messages of each user, which sometimes was insufficient. Despite the API limitation, the Web interface allowed to browser all messages of a user. Thus, we implemented a crawler that retrieved the necessary messages by parsing the Web page.

An additional problem raised by using services not under direct control concerns user identification. More often than not, each service requires the creation of a new login; only rarely can existing accounts on one service be reused for a different service. The non-existing single-sign-on is a well-known problem in the Web, however decentralized approaches such as OpenID14 might answer these problems. But currently, students still need to create their own accounts and these accounts needs to be matched to the student accounts used within the educational organization.

One drawback of the "perpetual beta" is that user interfaces will change. Usually the change improves the service, but still, the changes need to be reflected in user manuals written for the students. During our usage of the microblogging service, the user interface and underlying metaphors were changed twice. Since the changes happened after the students had become familiar with the service, there was not immediate need for adapting the user manual. The situation would have been different if the change had taken place during the start of the lecture.

Depending on the degree of integrating of the third party Web 2.0 services and one's own application, security becomes an issue. A client application such as an learning management system cannot directly control the service provider and depends on his reliability. This matter is complicated when mashups or widgets are used since these include additional sources of vulnerability [START_REF] Lawton | Web 2.0 creates security challenges[END_REF].

In our experiments, we did not use the "diverse data on an epic scale" for learning support, but are using it for research: we have now a corpus of 5400 Twitter updates to be analyzed for linguistic patterns, such as typical language learning problems. For instance, we were able to identify typical mistakes by Chinese learners and are now preparing special Twitter lessons to overcome these problems.

CONCLUSIONS

This paper made the following contributions: we showed that insofar as one can speak of an innate pedagogy of technology, the Web 2.0 pedagogy is best associated with constructivism and social learning. This claim is based on an analysis of the technological principles of Web 2.0. Thus, our work extends prior research that explores the potential of Web 2.0 for education in general by [START_REF] Downes | E-learning 2.0[END_REF][START_REF] Alexander | Web 2.0: A new wave of innovation for teaching and learning?[END_REF] and explains Web 2.0 principles for an educational audience [START_REF] Anderson | What is Web 2.0? Ideas, technologies and implications for education[END_REF].

Furthermore, we showed that Web 2.0 yields potential for research on technology-supported learning. By exploiting

Figure 1 :

 1 Figure 1: Part of the domain ontology used for the lecture on data structures

http://www.blackboard.com

http://del.icio.us

See, e. g., http://www.corank.com/ as an communitybased alternative to del.icio.us.

[START_REF] Anderson | The Architecture of Cognition[END_REF] http://www.flickr.com

http://www.rubyonrails.org/, http://www. djangoproject.com/

http://www.google.com/notebook

http://www.google.com/analytics

http://www.jott.com/, http://www.spokentext.net/

http://docs.google.com/

http://www.jumpcut.com/

http://pipes.yahoo.com/pipes/, http://www.popfly. com/

http://twitter.com.

http://openid.net/ existing Web 2.0 services and thanks to the openness of these services, it is possible to assemble in limited amount of time complex prototypes that allow to assess the validity of research hypotheses. We discussed two example applications that illustrate this claim and presented the lessons we learned from these examples.To summarize, Web 2.0 offers an intriguing and unparalleled wealth of functionality at a very high level. The exploitation of this functionality offers a high potential for the future of technology-enhanced learning. However, it is important not to forget that even if technology can be inspiring, the main focus in e-learning should still lie on the needs of the learner.


Hashim Habiballa
email: hashim.habiballa@osu.cz

Tibor Kmet'

Mathematical Logic and Deduction in Computer Science Education

Keywords:

Mathematical logic is a discipline used in sciences and humanities with different point of view. Although in tertiary level computer science education it has a solid place, it does not hold also for secondary level education. We present a heterogeneous study both theoretical based and empirically based which points out the key role of logic in computer science, computer science education and knowledge representation. We focus on the key contrast of semantics and syntax, the resolution principle as a leading inference technique (giving also interesting non-clausal generalization of the rule). Further we discuss the possibilities of inclusion the non-classical (many-valued) logics in education together with the original generalization of the non-clausal resolution rule into fuzzy logic. The last part describes partial results of the research concerning the secondary education in the Czech Republic especially in the mathematical logic field. The generalization of the presented ideas entails the article.

Introduction and Motivation

Logic is a well-established branch, in comparison with the other theoretical computer science disciplines, with deep tradition and its roots could be found in ancient history. Although the questions that logic encountered in the past were different, we could formulate some common issues namely the effort to simulate the human reasoning by exact (symbolic) way. Although a mathematician rather uses logic as a tool for expressing and proving properties of mathematical objects, for a computer scientist logic is a full member of theoretical computer science. Its main task is to provide formal (symbolic) framework for knowledge representation and deduction [START_REF] Lukasová | Formal Logic in Artificial Intelligence[END_REF]. As it is common in computer science, we also try to find effective algorithms solving this task. The symbolic approach means the ability to express the knowledge by "symbols" and it could be perceived as a concurrent approach to the connectionistic one. We would like to provide the reader with some issues concerning logic and especially automated deduction in computer science education (CSE). The problem of automated deduction is the key role of logic in computer science. We will focus not only to the tertiary level of education, but also to the upper secondary level of education. Formal deduction forms a key issue concerning mathematical and computer science aspects of logic teaching. The main focus on the secondary level education is given to the logical semantics and the formal aspect is usually overlooked [START_REF] Lukasová | Formal Logic in Artificial Intelligence[END_REF]. We are trying at present to establish more mathematical topics into computer science education (CSE). Our main current interest is closely related with teaching formal methods of deduction on secondary level and tertiary level education. The experiment for such a teaching has been prepared and it will contain topics described in [START_REF] Habiballa | Logic (I, II, III). Matematika-Fyzika-Informatika[END_REF]. The methodology is inspired by our previous works [START_REF] Habiballa | Didactic experiment in teaching theoretical computer science[END_REF][START_REF] Habiballa | Theoretical branches in teaching computer science[END_REF]. It is based especially on two formal methodstableau method [START_REF] Fitting | First-order Logic and Automated Theorem Proving[END_REF] and the wide-used resolution principle [START_REF] Bachmair | Resolution theorem proving[END_REF].

Logic and Knowledge Representation in CSE

The classical (two-valued) logic is a formalism known for many years. We are able to communicate by the natural language and further we process this knowledge and deduce conclusions. The natural language is unfortunately too complex to be effectively processed by a machine. That's why logic provides a tool of expressing the knowledge in a relatively narrow class of symbols. On the other hand logic brings methods for the automated (algorithmical) solution of deduction.

On the secondary level education the classical logic is typically taught in the frame of mathematics. The lessons contain the propositional logic with semantically oriented explanation and basics of predicate logic. The main intent is mathematically focused with the accent to usage of the logic as a tool for establishing mathematical theories. The formal aspect of logic is suppressed, but such an approach is not suitable for CSE. Logic in computer science has a solid place and it provides an important theoretical background for applied disciplines like database technology, algorithmization etc. Its leading role in knowledge representation has even more significant impact. We should be aware of the fact that these key applications of logic use its formal (syntactical) aspects too. Both of these sides are reasons to consider the formal logic in secondary level education. We recommend the reader to study very interesting research performed in Israel concerning theoretical computer science education in secondary (high) school [START_REF] Gal-Ezer | Curriculum and course syllabi for a high-school program in computer science[END_REF][START_REF] Armoni | On the achievements of high school students studying computational models[END_REF]. The research is based on the experimental education consisting of several themes that we are used to encounter only on university level education in computer science. We would like to state especially the CM (computational model) unit providing students with theoretical model of computation like finite automata, push-down automata or Turing's machines and further the second paradigm module providing basics of logic programming. When we hear some voices requiring even the algorithmization teaching to be reduced on secondary level or theoretical computer science to be reduced on tertiary level, it seems to be revolutionary project held by Israeli CSE researchers with positive results [START_REF] Armoni | On the achievements of high school students studying computational models[END_REF]. We would like to show that also logic (and especially its formal aspects) should have its place in secondary level curricula. But the recommendations mentioned could be even used for tertiary level education especially in introductory courses.

Syntax vs. Semantics -Key Problem

The key problem concerning right teaching of classical logic concerns to the essential need to explain the students the difference between syntax and semantics. The standard lesson focuses on the explanation of the logical connectives interpretation. Such an approach could satisfy the needs of mathematical education but could be hardly satisfying for CSE. We will present this issue by a simple example of deduction. Note that this type of logical tasks is perceived as suitable since abstract exercises are not motivating in contrast to these ones describing "real-life" situations. We will use only propositional logic in this example since its expressive power is sufficient for such a simple deduction and the simplicity has the didactic advantage. EXAMPLE 1 (knowledge representation part). We would like to prove the knowledge about a court trial. We have three suspects -A, B, C and we know the following observations:

1. There are no other persons related with this case. 2. A never works alone. 3. C is not guilty.

We will use atomic propositions for the fact that A is guilty (A), B is guilty (B) and C is guilty (C). The first observation could be expressed by disjunction of the every atomic proposition. The second one could be symbolically represented by implication with A as antecedent and disjunction of B and C as consequent. The third one is simply a negation of C.

1. A ∨ B ∨ C, 2. A → (B ∨ C), 3. ¬C.
The symbolic representation of the knowledge is only a half solution of the deduction problem. Before we follow the next step, we will formulate the specific notions to be taught in the logic lessons. At first the notions of satisfiability, validity and inconsistency should be explained and consequently the key notion of logical consequence should be well presented. These notions are mostly reduced to the notion of the tautology, which is not so important as the problem of consequence (the deduction bases on consequence). We follow with the semantical (unsuitable) solution.

EXAMPLE 2 (semantical solution part). If we work with semantics (interpretation) we should simply use the table method to prove (find) conclusion (logical consequence) of Table 1 Semantic solution of deduction

A B C A∨ B ∨ C A → (B ∨ C) ¬C A ¬A B ¬B 0 0 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 0 1 1 1 * 0 / 1 * 1 * 0 / 0 1 1 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 1 1 0 1 0 0 1 1 1 0 1 1 1 * 1 * 0 / 1 * 0 / 1 1 1 1 1 0 1 0 1 0
premises. We find the models of the premises and compare the behavior of potential conclusions (A, ¬A, B, ¬B). The suspect C is not guilty as it is stated in the third observation (Table 1).

From the interpretation table we can observe that there are two models of the premises (marked with *). We mark potential conclusions with the symbol *, where it is also true and we mark it with the symbol / if it is false. It is clear that right logical consequence should be true in all models of premises. The table semantically proves that the proposition B is guilty is consequence, but we cannot infer any conclusion about A.

Above presented example shows a semantical way to prove the consequence. It is very important to concentrate the attention of students on the complexity of such an approach. In this case we have to create all possible interpretations of premises, which depends on the number of atomic propositions. Although semantically oriented table method is an algorithm deciding the problem of logical consequence, its time complexity is exponential w.r.t. the number of atomic propositions. It means if we will solve the similar court trial with 10 suspects we have to create about 1000 interpretations, for 20 suspects we will create over one million interpretations and so on. It is clear that even for a computer such a method is very hard and practically unusable.

Therefore it is very important to introduce formal logical methods in teaching. Formal (syntactical) methods in contrast with semantical ones do not require any interpretation of formulae. Instead of interpretation they work with the sound rules handling formulae only like symbols without any sense. These methods should keep the properties of soundness and completeness (these notions also to be explained). Several formal methods are known from the tertiary level education, but we recommend using two of them. The first one is the tableau method that is based on decomposition of an inconsistent formula into tree with closed branches. The method is relatively simple and straightforward. The second method -resolution principle -is interesting from another point of view. Many of practically successful applications of automated deduction are based on this principle, e.g., logic programming (PROLOG) or symbolic based expert systems. Formal methods can be also divided into direct and indirect methods. The first ones lead from premises directly into logical consequence. The second ones use the fact, that the set of premises and the negated logical consequence is inconsistent and that's why we will conclude a contradiction (false formula). Indirect methods are more favorable for automated deduction since the objective of the process is the same for every logical consequence. It provides a possibility to find effective heuristics and algorithms reducing the complexity of the proving mechanism. The resolution principle in its clausal form works with clauses (disjunctive formulae with literals). It can be expressed in the following form.

Resolution Principle

C 1 ∨ x C 2 ∨ ¬x C 1 ∨ C 2 , ( 1
)
where C 1 , C 2 are parts of clauses and x is a resolving atom.

EXAMPLE 3 (formal solution part). When trying to use the resolution principle for proving, we have to transform all premises into clauses. It could be simply done by the rewrite rules.

1. A ∨ B ∨ C 2. A → (B ∨ C) 3. ¬C Premises 1. and 3. are clauses, but premise 2. is transformed into disjunction. 1. A ∨ B ∨ C (axiom), 2. ¬A ∨ B ∨ C (axiom), 3
. ¬C (axiom), then we can construct a direct proof with some common simplification rewrite rules (e.g., if a clause contains occurrence of the same literal we may remove all redundant occurrences).

4. (resolution on A in 1. a 2.): We formally proved that B is guilty. Of course we can also show indirect proof (and we recommend it in teaching). The main advantage of formal methods should be explained to students -these methods have the complexity depending on the structure of a formula (which is more sensible for practical usage).

(B ∨ C) ∨ (B ∨ C) ⇒ B ∨ C.

Automated Deduction Implementations in Teaching

We give the reader an interesting link to the implementation of the above-presented ideas. Libuse Pavliskova has developed very useful application and didactic material in the frame of her teacher studies [START_REF] Pavlisková | Deduction principles in propositional logic[END_REF], which can be used for secondary and tertiary education. It presents semantical and formal methods for deduction together with its automatization through computer software. An important advantage is also the presence of motivating solved examples like the court trial (Fig. 1).

Traditional Clausal Form Resolution in Education

The resolution principle (rule) is a well known and probably the most used deductive (inference) approach to automated deduction in practice (logic programming, expert sys- tems, clausal form logic etc.). The rule in its clausal form works with clauses (disjunctive formulae with literals).

Standard view of resolution principle has several educational problems:

1. The requirement of clausal form transformation (it requires conjunctive normal form transformation in propositional logic and furthermore in non-propositional case it needs skolemization). 2. No direct semantical explication of soundness (it requires usage of some semantically oriented proof method). 3. Transformation brings significant structure loss (one information-keeping structure -a formula -is decomposed into several clauses).

Non-clausal resolution (mixture of syntax and semantics)

Resolution principle could be also generalized for the usage with general propositional logic formulae (not only clauses).

General resolution -propositional version

F [G] F [G] F [G/⊥] ∨ F [G/T ] , ( 2
)
where the propositional logic formulae F and F are the premises of inference and G is an occurrence of a subformula of both F and F . The expression Such a rule has several educational advantages in contrast to clausal form resolution:

F [G/⊥] ∨ F [G/T ]
1. Soundness of the general resolution is straight-forward and could be explicated to students in direct form. 2. The structure of original formulae is preserved. 3. The general resolution rule does not require any transformation of formulae. It means no other explication of these non-trivial procedures is needed. 4. There are possibilities to extend the rule for many valued logics (many-valued formalism based on standard residuated lattices or MV-algebras is sufficiently simple to be presented to students).

Many-Valued (Fuzzy) Logics in Education

We have addressed another interesting topic related with the teaching mathematical logic and deduction. The secondary level education rarely touches the presence of non-classical logics both in theoretical level and real-life. We often hear the notion "fuzzy logic" in relation with wide-used electrical appliances like washing machines. Of course this notion in the lay perception will not lead directly to the logical deduction. But this example could lead to the intuitive teaching of many-valued (fuzzy) logics. The two-valued (classical) logic is based on two truth values -true and false. Such an approach is not fully convincing for modelling of human thinking. People often use concepts that cannot be expressed only by full true or false. For example the concept of contented person could not be described by two logical values. We are contented in some degree. The most used approach for modelling of vagueness -fuzzy mathematics -use this intuitive tool as a many-valued interpretation. Fuzzy mathematics works with a fuzzy set that is based on membership degree to be a complex structure instead of the poor set {0,1} (non-member, member of a set). The fuzzy logic uses this principle of interpretation to be also complex structures (e.g., where interpretation of a formula is classical two valued case -false (0), true (1); in fuzzy logic there is a possibility to interpret a formula by a member of complex structure mostly by the interval of real or rational numbers [0,1]). Then we can express the contented person by some rational number (degree) (e.g., perfectly contented person in 1, almost perfectly contented person in 0.9, etc.). It allows us to model real-life a situation in soft way and that is why fuzzy logic is closely related with soft computing disciplines (evolutionary algorithms, neural networks, fuzzy modeling etc.).

Even we can come to believe that fuzzy logic is very simple formalism, we should be aware that the correct usage of fuzzy logics need to work proved structures as truth degrees. We usually work with standard algebras -structures with not only truth-value sets but also with fixed operators (conjunction, disjunction, implication) as functions. There are several such algebras with different properties (e.g., Gödel algebra, Goguen algebra, Łukasewicz algebra). Fuzzy logics are generalizations of classical logic and therefore it is not surprising that not all laws and properties are preserved like in two-valued case. When using these algebras as truth value sets we call resulting logics by appropriate name (e.g., Gödel propositional logic). When trying to perform formal deduction we often use Łukasewicz logic since it has several "nice" properties like holding law of double negation.

We will assume Łukasewicz algebra to be:

L Ł = [0, 1], ∧, ∨, ⊗, →, 0, 1 ,
where [0, 1] is the interval of reals between 0 and 1, which are the smallest and greatest elements respectively. Basic and additional operations are defined as follows:

a ⊗ b = 0 ∨ (a + b -1), a → b = 1 ∧ (1 -a + b), a ⊕ b = 1 ∧ (a + b), ¬a = 1 -a.
Then we bound the new (fuzzy) logical connectives with the operators of the algebra. The interpretation of Lukasiewicz conjunction is bound with the operator ⊗, interpretation of Lukasiewicz disjunction is bound with the operator ⊕, interpretation of implication is bound with the operator → and the negation is bound with ¬. The following sample calculations in Łukasewicz algebra show also a little bit unreal behavior of its certain operators (Łukasewicz conjunction and disjunction). Nevertheless other operators have reasonable interpretation (implication and negation). As stated above the main reason why choose Łukasewicz algebra is that several essential laws comparable to two-valued logic are preserved.

EXAMPLE 5 (Łukasewicz algebra). a = 0.5, b = 0.5 ⇒ a ⊗ b = 0 ∨ (0.5 + 0.5 -1) = 0 (in the logical meaning it states two "half" true propositions in conjunction lead to no truth at all). a = 0.5, b = 0.6 ⇒ a → b = 1 ∧ (1 -0.5 + 0.6) = 1 (if consequent has the interpretation greater or equal than antecedent then the implication is fully true). a = 0.5, b = 0.3 ⇒ a → b = 1 ∧ (1 -0.5 + 0.3) = 0.8 (if consequent has the interpretation less than antecedent then implication is less than 1). [START_REF] Novák | Mathematical Principles of Fuzzy Logic[END_REF] assigns grade to every axiom, in which the formula is valid. It will be written as a/A, where A is a formula and a is a syntactic evaluation. We will need to introduce several notions from fuzzy logic, in order to present the fuzzy graded calculus.

Graded fuzzy propositional calculus

Inference rule

An n-ary inference rule r in the graded logical system is a scheme

R: a 1 /A 1 , . . . , a n /A n r evl (a 1 , . . . , a n )/r syn (A 1 , . . . , A n ) , ( 3
)
using which the evaluated formulas a 1 /A 1 , . . . , a n /A n are assigned the evaluated formula r evl (a 1 , . . . , a n )/r syn (A 1 , . . . , A n ). The syntactic operation r syn is a partial nary operation on F J and the evaluation operation r evl is an n-ary lower semicontinous operation on L (i.e., it preserves arbitrary suprema in all variables).

Evaluated proof, refutational proof and refutation degree

An evaluated formal proof of a formula A from the fuzzy set X\mathrel ⊂∼ F J is a finite sequence of evaluated formulas

w := a 0 /A 0 , a 1 /A 1 , . . . , a n /A n , ( 4
)
such that A n := A and for each i n, either there exists an m-ary inference rule r such that a i /A i := r evl (a i1 , . . . , a im )/r syn (A i1 , . . . , A im ), i 1 , . . . , i m < n or

a i /A i := X(A i )/A i .
We will denote the value of the evaluated proof by V al(w) = a n , which is the value of the last member in (4).

EXAMPLE 6 (many-valued logics). For educational purposes we can start with the simple but illustrative "sorites paradox". The well-known paradox shows the weakness of two-valued logic. We can model (using predicate logic) the situation stated in the following two sentences: 1. The heap without a stone is small. 2. If a heap with n stones is small then this heap with one added stone is also small.

When using the classical logic we can model this situation by two formulae (with the predicate):

1. small heap(0), 2. small heap(n) → small heap(n + 1). It is clear that by these two formulae we will prove that every heap (with any number of stones) is small, which is nonsense (paradox following from two-valued truth).

Using Lukasiewicz predicate logic (generalization of classical predicate logic) we are able to model formulae with truth degree from interval [0,1]:

1. small heap(0) -true in the degree 1, 2. small heap(n) → small heap(n + 1) -true in the degree 0.999. Formula 2. (implication) is in contrast with classical logic holding not fully but only in the degree 0.999. This fact allows us to avoid the paradox. Due to the degree 0.999 every application of the implication will lead to the inference of the fact that heap with one added stone is small in the degree decreased by 0.001. If we would like to prove the formula small_heap(1000) (heap with 1000 stones is small) we follow the following sequence (I means interpretation of the implication according to the operator in previously defined algebra):

I(small heap( 0)) = 1, then we will use that in equation, 0.999 = I(1 ∧ 1 -1 + small heap( 1)) ⇒ I(small heap( 1)) = 0.999 0.999 = I(1 ∧ 1 -0.999 + small heap( 2)) = 0.001 + I(small heap( 2)) ⇒ I(small heap( 2)) = 0.998 . . . I(small heap( 1000)) = 0.

It means that we proved a heap with 1000 stones is not small in any degree higher that 0.

Fuzzy logic has also another interesting applications (used in industry, decisionmaking, etc.) based in linguistic expressions. The idea of these expressions lies in the formalization of natural language expressions that we often use in real-life like "something is very small", "something is rather big". We can model such expressions by fuzzy sets and use them in linguistic variables included in "IF -THEN" type complex formulae. These formulae can form large knowledge bases that can be used to control various processes, to decide hard problems according to the formulations of IF -THEN rules given by experts. For example we can model the process of car driving:

IF traffic lights changed to yellow AND car speed is small AND distance from crossing is small THEN press down brake by medium power.

IF traffic lights changed to yellow AND car speed is medium AND distance from crossing is very small THEN press down accelerator by big power.

There are a lot of such real applications for example Linguistic Fuzzy Logic Controller designed on University of Ostrava. Such popular examples of fuzzy logic usage may be used in education also on secondary level.

The authors have proposed also generalization of the general resolution rule for Lukasiewicz Predicate Logic with evaluated syntax and such an approach is also applicable to mathematical and computer science education [START_REF] Habiballa | Non-clausal resolution in fuzzy predicate logic with evaluated syntax (background and implementation)[END_REF]. The rule could be with simplification stated as follows (of course we should also define the most general unifier, other notions for refutational proofs in fuzzy logic).

General Resolution for Fuzzy Propositional Logic

r GR : a/F [G 1 , . . . , G k ], b/F [G 1 , . . . , G n ] a ⊗ b/F [G/⊥]∇F [G/T ] ,
where F and F are premises and G resolved subformula and ⊗ is the Lukasiewicz multiplication.

EXAMPLE 7 (evaluated proof in fuzzy propositional logic using resolution). 1. a ↔ b ∧ ¬d/1 (axiom with evaluation degree 1). 2. a/0.7 (axiom with evaluation degree 0.7).

Automated Non-Clausal Resolution Implementations in Teaching

The authors has already developed also a tool for automated theorem proving based on general resolution (Fig. 2) (Fuzzy Predicate Logic Generalized Deductive system) -enriched application for classical logic [6]. The application could be used for presentation of the rule and also several inference strategies developed especially for this formalism. The future extensions consist of presentation of the tree of resolution proof visualization, improvement of object-oriented implementation, free distribution of the package and also some "real-life" applications.

Current State of Logic Education in Czech Republic

We have already presented the theoretic principles of effective teaching of logic together with interesting generalisation of the resolution principle gaining from syntactic-semantic properties. At the end we would like to shortly summarize the results of a research concerning the current state of high school education. It was performed through a web questionnaire sent to all secondary schools in the Czech Republic. The research concerning the actual content of teaching computer science was established on University of Ostrava in 2003. We have used the research instrument of on-line questionnaire. Link to this questionnaire was sent to 863 secondary level schools in the Czech Republic (according to the database of Institute for Information in Education of the Czech Republic); total amount of available responses was 132 from the entire Czech Republic. Although the research has wider focus we present only the results concerning the logic education. Fig. 3 shows the structure of institutions that responded to the questionnaire.

The general secondary schools are the institutions primarily preparing for tertiary level education (mainly university studies). Other schools in the sample include mainly technical schools with various study programmes (electrical, chemical, telecommunication etc.) preparing for direct fulfillment in practice. Trade schools educate the secondary level professionals in economic field and other vocational schools include rather the humanities (pedagogical, social-juridical, nursing).

We will analyze only the results for general secondary schools (total number of 55) since these schools form the most consistent and the most frequent type of secondary schools in the research. We should observe that relatively high amount of schools teach at least the basics of propositional logic, which typically concern to semantics (symbols are bound with its interpretation). Then we can distinct the second sort of topics taught at lower level. Student are taught the key formula properties, but still in semantic level. Further we follow with topics that have syntactic basis. All these sorts have relatively high extent in schools, but the topics addressed in the first part of the article are taught at very low level (formal deduction and methods).

We should observe not surprising results, which show that the teaching of key topics of logic from computer scientist's point of view is at very low level in secondary school system in the Czech Republic. The principal notion of deduction is taught only at 2 schools from 55, even deduction forms the leading role in the usage of logic in practical "real-life" application (logic programming, expert systems based on logical knowledge representation). So wide used resolution principle, that is the most expanded method implemented in deductive inference systems, is taught only in one school. All these topics (as we tried to show in brief previous examples) can be explicated in relatively intuitive manner (free of formalization known from tertiary level education). There is also important fact that 77% of teaching logic has place in the frame of mathematical education (47 of 58, counting occurrences in all grades in one school).

Conclusions

We have presented a brief survey into the possibilities and advantages of computer science based education mathematical logic. These ideas are part of the research performed on University of Ostrava trying to improve secondary education in the Czech Republic.

We have already published several papers in the national didactic journals and proceedings, which described in detail topics, have to be included into the secondary education. Israeli CSE researchers already successfully implement some of the similar ideas in the Second Paradigm module of the high-school curricula that include theoretical and practical teaching of deductive systems through logic programming [START_REF] Gal-Ezer | Curriculum and course syllabi for a high-school program in computer science[END_REF]. Our approach should emphasize theoretical learning of the inference method principles, including non-clausal resolution and also give a short introduction into the theory and application of the many-valued logics.

We follow these principles:

-teaching of mathematical logic should be performed not only in the frame of mathematical education, but also in the frame of computer science education; such education should have interdisciplinary aspect to be effective (it means mathematics and computer science have to cooperate); -mathematical education should inhere the basics -language, semantics, connectives, interpretation, tautologies; -computer science education should give advancement -syntax vs. semantics, knowledge representation, problem tasks from "real-life", deduction, formal deduction and methods, resolution, many-valued logics; -we should use existing ICT tools to make the learning interesting for students (logic programming, visual inference engines with prepared examples).

H. Habiballa 1976 graduated at the Department of Computer Science, University of Ostrava, Czech Republic in computer science and technology -information systems (MSc -1999); further he holds advanced degree in computer science education (EdD -2003) at the Constantine the Philosopher University, Nitra, Slovak Republic and he defended doctoral thesis (PhD -2004) in computer science education at the same institution. Further he defended rigorous thesis and second dissertation in information systems -logic (PhD -2007). He works as an assistant professor at the Department of Computer Science, University of Ostrava and also participates as a research assistant in the Institute for Research and Applications of Fuzzy Modeling. His interests include research both in theoretical computer science and computer science education (several national and international grant projects). He is author of about 100 publications (articles in reviewed journals, proceedings of conferences, chapters in monographs, teaching texts etc.).

T. Kmet' full professor of mathematics, Constantine the Philosopher University in Nitra; interests in mathematical modelling and didactics of computer science.

Matematinė logika ir dedukcija informatikos kurse

Hashim HABIBALLA, Tibor KMET'

Matematinė logika ivairiais būdais naudojama gamtos ir humanitariniuose moksluose. Informatikos mokymas labai svarbus aukštajame moksle, tačiau vidurinėje pakopoje taip nėra. Straipsnyje pateikiamas ivairiarūšis teorinis ir empirinis tyrimas, kuris atkreipia dėmes i i logikos svarb a informatikoje, informatikos mokyme ir žini u atvaizdavime. Straipsnyje aptariami semantikos ir sintaksės klausimai, problem u sprendimo principas kaip išvad u konstravimo būdas (pateikiant idomius taisykli u apibendrinimo būdus). Taip pat aptariama galimybė itraukti neklasikin e (daugiareikšm e) logik a i mokym a. Paskutinėje dalyje pateikiami daliniai tyrimai, atlikti Čekijos Respublikos viduriniame lavinime, būtent, matematinės logikos srityje. Straipsnyje itin pabrėžiamas pateikt u idėj u apibendrinimas.

  5. (resolution on C in 4. a 3. -formula 3. completely resolved): B.

Fig. 1 .

 1 Fig. 1. Application "Logical deduction".

  is the resolvent of the premises on G. Every occurrence of G is replaced by false symbol in the first formula and by true symbol in the second one. It is called F the positive, F the negative premise, G the resolved subformula. EXAMPLE 4 (general resolution with equivalence). 1. a ↔ b ∧ ¬d (axiom). 2. a (axiom). 3. b ∨ d → e (axiom). 4. ⊥ ∨ [T ↔ b ∧ ¬d] (resolved from (2), (1) on a) ⇒ b ∧ ¬d. 5. ⊥ ∧ ¬d ∨ T ∨ d → e (resolved from (4), (3) on b) ⇒ e.

  3. b ∨ d → e/0.75 (axiom with evaluation degree 0.75). 4. ⊥ ∨ [T ↔ b ∧ ¬d]/0.7 ⊗ 1 (resolved from (2), (1) on a) ⇒ b ∧ ¬d/0.7. 5. ⊥ ∧ ¬d ∨ T ∨ d → e/0.7 ⊗ 0.75 (resolved from (4), (3) on b) ⇒ e/0.45.

Fig. 2 .

 2 Fig. 2. Fuzzy Predicate Logic GEneralized Deductive System.

Fig. 3 .

 3 Fig. 3. Secondary schools structure in the research (GYM -general secondary schools, SPS -technical schools, OA -trade schools, SOS -vocational schools, Other -training schools).

Fig. 4 .

 4 Fig. 4. Logic in general secondary schools (number of schools).

Table 2 Basic principles Basic Principles Teaching Logic on Secondary Schools

 2

	Issue	Number (Percentage) of Schools
	Proposition and truth value	42 (76%)
	Propositional logic symbols	40 (73%)
	Equivalence of formulae	30 (56%)

Table 3

 3 Advanced semantics

	Advanced Semantic Notions
	Issue	Number (Percentage) of Schools
	Boolean functions	26 (47%)
	Satisfiability, tautology, contradiction	23 (42%)

Table 4 Syntax Syntactic Notions Issue Number (Percentage) of Schools

 4

	Propositional Formula Construction	22 (40%)
	Normal Forms	17 (31%)

Table 5 Formal deduction Formal Deduction and Methods Issue Number (Percentage) of Schools

 5

	Tableau method	6 (11%)
	Syntactic trees	4 (7%)
	Decidability and satisfiability	3 (5%)
	Deduction	2 (4%)
	Formula complexity	2 (4%)
	Resolution principle	1 (4%)
	Non-classical logics	1 (2%)

Informatics inEducation, 2008, Vol. 7, No. 1, 75-90