Saïd Assar
email: said.assar@it-sudparis.eu

Redouane El
email: redouane.elamrani@reims-ms.fr

Richard T Watson
email: rwatson@terry.uga.edu

Nelson Mandela

ICT and education: A critical role in human and social development Editorial Introduction to the Special Issue

observed that, "Education is the most powerful weapon which you can use to change the world." He left unanswered two questions: why do we want to change the world and what is it that needs to be changed. The articles in this special issues address the questions of how ICTs can enable education to change or improve the lives of people, important educational goals in all parts of the world, but particularly so in developing regions and underserved communities.

In order to achieve development, measurable improvements in people's lives have to be considered. Theses changes can be assessed in terms of social, economical and political development. In many regions of the world, there is an urgent need to raise living standards by providing a sustainable access to modern infrastructures in terms of water, electricity, transport, to raise income generation and job creation, to solve conflict, establish peace and limit access to weapons, to address the needs of vulnerable and oppressed groups and to encourage political systems based on democracy and the guarantee of human rights. However, what might be the most important thing to change is access to education. In many countries, access to education has become a strong focus of their governments. The need for economic and social development is a strong driver for governments and other interested institutions to invest in educational reform.

Education involves transmission of knowledge, accurate thinking, skills, habits and shared values. It is one of the main keys to social and economic development and improvements in human welfare as it is instrumental to transforming humans and their ways of life and death, and to improve their natural and artificial environment. As global economic competition grows sharper, education becomes an important source of competitive advantage, closely linked to economic growth. Providing education to everyone is a continuing challenge for developing regions and countries. It is universally recognized as a fundamental building block for human development and poverty reduction. To be effective, it requires appropriately trained human resources, specific material infrastructures, and adequate organizational guidelines and institutions. It requires also a long term strategic vision together with a strong and persistent political will.

The underserved regions of the world face many constraints in delivering education to the right people at the right time. Budgets are always tight, educational material is unavailable, human resources for teaching are scare and the best teachers rarely want to work in remote rural areas. All these factors have encouraged an interest in the use of information and communications technology (ICT) to deliver education and training. The significance of ICT is a recurrent theme in the global economy. ICT can play a significant role in generating economic growth. [START_REF] Gholami | Is ICT Key to Development?[END_REF] found that investment in ICT, education, and economic literacy can play a significant role in equalizing opportunities for marginalized groups and communities. Their adoption and large-scale diffusion are recognized as a strong enabler for development but do not lead deterministically to economic growth.

However, the issue of the direction of causality in the context of developing countries is still ambiguous, not fully understood, and needs to be formally addressed. Many studies show the importance of the local context and the importance of adapting IT-based practices when implementing them [START_REF] Sahay | Scaling of health information systems in India: challenges and approaches[END_REF]. Implementing ICT In education is one of the major challenges facing developing countries as it is impossible to imagine future learning environments that are not supported, in one way or another, by ICT. The importance of the introduction of ICT into education systems is linked to the pressure on societies and economies to respond dynamically to globalization. ICTs are recognized to have the potential to reduce barriers to education for many as they can provide the underserved and geographically and culturally disadvantaged with access to educational resources. Moreover, the adoption of ICT by education has often been seen as a way to contribute to educational change (enable reform or better management of education systems), better prepare students for the information age, improve learning outcomes and the skills of learners and prepare them for the information society. These initiatives are helping also to bridge the so-called digital divide: the gap between those who have access to and control of technology and those who do not. It is a new expression of the traditional technological dualism between rich and poor countries (Margarita et al., 2009). Some authors consider the failure to use ICT is itself a result of the digital and knowledge divides that exist. This situation raises the following question: could developing countries and underserved regions accelerate national development efforts by simply installing more computers, educational software, and Internet access in their educational system? The findings bring to light contradictory results and the difficulties of such projects. Some research has shown that ICTs are used differently in underserved regions in terms of their goals for the introduction of ICT and in the pathways they have chosen to achieve those goals [START_REF] Cairncross | ICTs for education and building human capital[END_REF][START_REF] Reynolds | ICT in Education: The Future Research and Policy Agenda, Keynote presentation at conference on[END_REF]. Schools can use ICT to merely access existing information and knowledge, rather than as part of a new 'transformative' pedagogy of teaching and learning. The effective use of ICT is harder than was initially expected. Many teachers in developing regions are often minimal users or non-users of ICT in their classrooms.

Empirical evidence has uncovered many limitations on the value returned by investments in ICT in education [START_REF] Cuban | Oversold and underused: Computers in the classroom[END_REF]. There are many reasons explaining this situation: a lack of capacity in third world countries in general to build, maintain, and utilize ICT to address educational needs, lack of ICT policy, poor ICT project management and corruption. Perceived difficulty in the integration of ICT in education has led to ineffective implementation, adoption of different systems and standards, duplication of effort, and waste of technology resources. Consequently, some of the earlier enthusiasm for educational ICT has waned. The cost of adopting ICT including acquiring hardware and software, setting up telecommunication networks, and the maintenance and repair of facilities is often prohibitive for developing countries. Given the low per capita income and declining standards of living, many policy makers focus policy formulation and policies on meeting the basic needs of the poor. The use of OER (Open Educational Resources) [START_REF] Johnstone | Open educational resources serve the world[END_REF] and OSS (Open Source Software) could be an alternative option that allows users to freely create and distribute software and educational resources without the prohibitive costs or licensing limitations.

There is practical importance to understanding the reasons for this non-use in relation to teacher and students' age, the subjects being taught, geographical, social, and cultural context. Several researchers argue that the development of effective ICT programs in education depends on a complex network of other social and economic factors that restrains developing countries from utilizing ICTs [START_REF] Kendall | Formulating Information and Communication Technology (ICT) Policy Through Discourse: How Internet Discussions Shape Policies on ICTs for Developing Countries[END_REF]. Van Reijswoud (2009) argues that ICT projects in developing countries become successful when adapted to local conditions and introduce the appropriate technology to increase the success of such projects. In other words, the educational effectiveness of ICTs in undeserved countries will depend on several variables, including the appropriate design of software and hardware, the training and attitude of instructors, and the realization that different students have different requirements. It also requires a willingness to experiment with effective use of ICT in education through pedagogical techniques that are different from traditional classroom teaching.

While there is a substantial literature on the possible determinants of ICT in changing and modernizing educational systems and ways of learning in developed countries, scientific evidence of the concrete contributions of ICTs to the learning domain is less evident in developing countries. A recent report by the World Bank's Information for Development Programme [START_REF] Trucano | Knowledge Maps: ICT in Education. What do we know about the effective uses of information and communication technologies in education in developing countries?[END_REF] specified that there is limited data to support the perceived benefits from ICT and very little systematic researches about how ICT is actually used in the classroom and even less about its impact on educational outcomes, social behaviour, and the performance indicators to monitor the use and impact of ICTs in education.

Given the increasing strategic importance of ICT in education in developing countries, there is a pressing need for additional work and further investigation and research to be done in this area. For ICTs become effective and integral tools in education, their impact needs to be quantitatively and convincingly demonstrated.

Special issue on ICT and education in underserved regions

This special issue investigates ICT and education in undeserved regions and developing countries through multiple aspects: what is the actual status of ICT usage for education? How can ICT help in providing high quality training programs adapted to the specific needs of students and trainees? What are the problems and barriers related to ICT usage for education? How do the specific social, economic, cultural, and geographical contexts in developing regions impact the usage of ICT in education? We sought significant case studies, theoretical, exploratory, or position papers, which could help in understanding these problems and in shaping good practices and recommendations for educational practitioners and researchers interested in ICT usage in the classroom.

The preliminary ideas behind this special issue were initially experimented with in a workshop entitled "Information Systems research and education in developing countries" which was organized at ICIS 2008 in Paris [START_REF] Assar | Information Systems research and education in developing countries[END_REF]. The objective of this workshop was to gather researchers to discuss and exchange experiences and ideas concerning ICT and information systems education in developing countries. Submissions were allowed in either English or French, and the presentations were in English. The workshop attracted 42 submissions from 31 countries, of which 18 were accepted for presentation and published in the workshop's proceedings. 36 participants attended the workshop. The majority of the papers were empirical and practical and tackled a variety of topics involving IS and IT educational projects and uses in developing countries.

This workshop was organized with the help of UNESCO Israel who provided a USD 10,000 budget, which was used to pay ICIS conference fees and hotel expenses for workshop participants from low and upper-middle income countries. Participants were selected based on the quality of their research as well as the merits (candidates have academic career in Information Systems; some publications on national and international IS journals). We are particularly thankful to Professor Dov Te'eni (Tel-Aviv University, Israel) and Professor Frantz Rowe (Nantes University, France) who initiated the workshop project, worked on getting funding, and helped in organizing the workshop in the context of ICIS 2008 conference.

As part of the workshop organization, we were encouraged by ITD's Editor-in-Chief Sajda Qureshi to plan this special issue. Although it targeted authors of papers presented at the workshop, the issue was open to any submission and the Call For Papers was widely diffused. Twelve manuscripts were submitted, of which three were from workshop authors.

Organization of the special issue

This special issue has four research papers and one view from practice. The papers cover different topics related to ICT and education. We now briefly summarize each of these papers and discuss its relevance to the questions raised in the first section of this paper.

The first contribution, by Khitam Shraim and Zuheir Khlaif, is entitled "An e-learning approach to secondary education in Palestine: Opportunities and challenges". In a region where the ongoing Palestinian-Israeli conflict has deeply affected population movements and has separated student and teachers from their educational institutions, ICT based distance learning has become a necessity to improve access to quality education for all Palestinians students. While substantial investment and considerable efforts have been made to develop e-learning approaches since 2002, its introduction into primary and secondary education is still in the planning process. Through a mixed quantitative and qualitative study, this paper explores the potential of e-learning methods to enhance the educational process and to provide continuous learning for secondary students in Palestine. An interactive Web-based application prototype called the Alaws Educational Network (AEN) provides a variety of methods for a student-centered learning process, including virtual classrooms, discussion forums, and e-training courses. Students and teachers were asked to evaluate different aspects of the AEN in terms of usefulness, self-efficacy, willingness, and challenges as indications of their ability and readiness to embrace e-learning. The results show that both students and teachers have positive attitudes towards the usefulness of e-learning methods but that they might not yet be ready to adopt them. Open comments of respondents pointed to issues that need to be addressed related to existing digital divide and technical limitations of the network, lack of elearning skills, lack of autonomy, poor time management and workload pressure. They also pointed to more complex issues concerning institutional support and language barriers. This paper shows that although final users are willing to embrace ICT supported distance teaching and learning, other problems remain: top managers are not fully aware of its potential, best practices need to be disseminated, the necessity of developing IT literacy programs, digital divide between schools in urban and those in rural areas. Authors consider that a holistic approach to the problem needs to be developed so that the combination of social, technical and organizational factors contributes to the success of the project.

The second paper makes a contribution to adaptability of education to people, and explores the impact of language and cultural barriers on the success of introducing ICT in the classroom. It is authored by Greta Björk Gudmundsdottir and is entitled "When does ICT support education in South Africa? The importance of teachers' capabilities and the relevance of language". In a region that once has been a highly segregated nation, and despite emphasis on the equal rights of all ethnic groups as mentioned in the new South African progressive constitution, South Africa remains a country of great disparities. The paper seeks to understand how cultural aspects intertwine and possibly influence computer use in South African schools. The main focus is on a broad understanding of the concept of digital divide, and how cultural complexity, with special emphasis on language, can affect the digital divide in schools that already have material access to ICT. One of the distinctive aspects of cultural complexity in South Africa is the language dimension. The country has 11 official languages, and the languages used in South Africa are related to status, class, and political context. While learners are taught in their mother tongue in the first three grades of school (grade one to three), learners with African home languages participate in English or possibly Afrikaans classes for the rest of their school attendance. Strong connection has been suggested in some research works between the home language as the language of instruction and learners' positive learning processes. So for the purpose of her research, the author develop a broad definition of the digital divide concept which, beyond having or not access to computers and the internet, includes linguistic and cultural divide as the dominance of the English language and culture restrain access to and use of knowledge. The study is a comparative case study and is based on fieldwork in seventh-grade classes in four primary schools in Cape Town. The learners answered a questionnaire regarding their ICT use and skills, and interviews were conducted with learners, teachers, and principals. In the findings, two underlying explanations have been discussed when seeking to answer the question of what is influencing the digital divide in schools that already have material access to computers: (i) the importance of raising the ICT capability of the teachers and their opportunities to obtain training, and (ii) the role of language in conjunction with learners' ICT use. It is argued that the challenges of languages in South African schools can exacerbate or maintain the digital divide among learners who are already disadvantaged due to a range of social inequalities. The author considers that ICT needs to be put in a local context, which includes use in a familiar language so that learners can fully master the use of ICT in today's global knowledge society. Moreover, she suggests that greater opportunities for teacher training are needed in order to enhance culturally sensitive and appropriate ICT integration based on local needs and capacity.

The third paper is by Godfred Bonnah Nkansah and Tim Unwin is entitled «The contribution of ICTs to the delivery of Special Educational Needs in Ghana: practices and potential". Those with special educational needs are underserved in many communities, irrespective of the stage of development of the economy. Using Ghana as the site for their research, Nkansah and Unwin investigate how ICT can support those with special educational needs (SEN). To address this issue, the authors used qualitative interviews, focus groups and questionnaire. They illustrate the power of ICT to benefit those with major disabilities and note that ICT can reduce their marginalization, but this is an issue rarely considered in developing economies where there is so much competition for resources, particularly in the education sector. Moreover, the other issue is about the cost and the inadequate human and physical infrastructure at the school level to accommodate and manage such technologies effectively. The contribution of this article is to raise awareness of the potential of ICT to educate one of the most underserved communities in the world. It is a call for governments across Africa, and indeed the world, to examine how assistive technologies can help those with SEN to be supported to achieve the dignity associated with education: "The time has come for a globally concerted effort to place accessibility at the heart of ICT for education initiatives".

The fourth paper in this issue, written by Negash Solomon is entitled "Learning Assessment of a Videoconference-based Training: Lessons from medical training between US and Ethiopia". The main research question guiding this study was whether videoconferencing can serve as an effective medium to facilitate training programs conducted between high and low-income countries? To address this issue, the authors focused on a two-way videoconferencing training program conducted between the US and Ethiopia. Specifically, four videoconference locations in the US were connected to a similar center in Ethiopia in order to provide high level HIV/AIDS medical management training to Ethiopian medical professionals. In the context of developing countries, this paper raises an interesting and relevant issue: given that knowledge experts are typically concentrated in highincome countries, how can we increase the knowledge of medical professionals in underserved communities through a videoconference-based training? In the health sector, ICT can be a cornerstone for efficient and effective services. Over the last decade, the need to develop and organize new ways of providing health services has been accompanied by major advances in ICT, enabling better support for health services and systems and simultaneously improving global awareness of health issues. As a consequence, the World Bank and UNESCO have insisted on member nations developing long term e-health strategies. In the context of underserved countries, this approach can potentially solve problems such as the shortage of skills of health professionals and the training of medical workers. The results of this study indicate that the videoconference-based training had a positive impact on Ethiopian physicians' attitudes and resulted in significant learning gains. It shows that videoconference-based training can bridge the knowledge gap in low-income countries by linking them with high-income countries and may become the catalyst for developing advanced telemedicine services. Nevertheless, while this potential is apparent, it is not yet clear whether it can be realized in an environment in which there are human resource constraints at both the sending and receiving sites. This study also identified constraining and promoting factors of the videoconference-based training. The paper concludes with three lessons learned in the following three areas: cultural context, IT infrastructure, and sustainable community. Each lesson has implications for research and practice.

A view from practice paper by Paolo Brunello provides a critical review of the success factors of ICT projects in developing regions, and is entitled "ICT for Education projects: a look from behind the scene". Based on the author's direct experience in multiple ICT for education projects in Burundi, he argues that factors other than lack of maintenance and local ownership of technology can have strong impact on the success or failure of such projects. A first identified factor is bureaucracy compliance, as the donor from developed countries often expects a high level of procedures' traceability. The result is a bureaucratic burden which is due to critical infrastructural limitations in the developing countries. This can lead to a bureaucratic paradox where administrative rules are trespassed in different ways so that to carry out the planned activities. A second identified factor is related to the predefined roles played by actors on the ground. "Developer" and "developee" are constantly negotiating their respective roles and mutually reconfirming them, and this sets the stage for a predefined, often unadapted, way of transmitting knowledge and running development projects. A third identified factor is transaction costs and relates to running and maintaining technological infrastructures implemented during the development project. Running costs expenditures (i.e specific printer cartridges) are generally intended to be provided by the beneficiaries, and this rapidly can become a major reason for ICT project to fail on the long run. A fourth identified factor is the bias towards setting up physical infrastructures over setting educational use of these infrastructures. This also can be a strong reason for ICT for education projects to fail as training and usage are not emphasized enough. To conclude, the author calls for a radical change in the overarching mindset dominating both aid and education, moving away from the "medical metaphor" with donor countries playing the doctors and recipient countries the patients. He suggests a constructionist approach where the role of donor countries is one of a mentor and a supervisor of technological transfer, a facilitator for peer learning and a partner for knowledge sharing.

Acknowledgments

We acknowledge the authors of all the submitted articles and all who helped in the review process of the special issue. We particularly would like to acknowledge the work and efforts of the following reviewers for this special issue: Patience Akpan-Obong (Arizona State University, USA); Ghada Alaa (American University at Cairo, Egypt); Mokhtar Amami (Royal Military College of Canada); Reza Barkhi (Virginia Tech, USA); Madeleine Besson (Telecom Business School, France); Cécile Clergeau (University of Angers, France); Sam Dzever (Telecom Business School, France); Zarinah M. Kasirun (University of Malaysia); Beatriz B. B. Lanza (Companhia de Informática do Paraná -Celepar, Brazil); Allyson Macdonald (University of Iceland); Ravishankar Mayasandra (Loughborough University, UK); Becky Matter (University of Washington, USA); Philip Musa (University of Alabama at Birmingham, USA); Ståle Angen Rye (Agder University College, Norway); Matti Rossi (Aalto University School of Economics, Finland); Elmarie Venter (University of Pretoria, South-Africa); Rebecca Walton (University of Washington, USA); Sidne Ward (University of Missouri, USA); Yingqin Zheng (London School of Economics, UK). We also give a special acknowledgement to Sajda Qureshi, the editor-inchief of this journal, who gave encouragement, support and advice throughout the development of the special issue.

 Van Reijswoud, V. (2009)


Ghislaine Gueudet-Chartier

Thematic Group 7 EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III

1.Introduction

When students encounter linear algebra for the first time, they find it very abstract, and disconnected from what they have learned before (Dorier 2000). Geometry is often mentioned by teachers as a way to help the students in their learning of linear algebra, because it seems to provide an opportunity to introduce concepts, that will be then generalised, within a more concrete context. The actual facts are more intricate; there is no natural generalisation process leading from geometry to linear algebra [START_REF] Gueudet-Chartier | Rôle du géométrique dans l'enseignement et l'apprentissage de l'algèbre linéaire[END_REF].

However, when the linear algebra concepts are formed, geometry can intervene in problem solving processes.

In his analysis of the genesis of linear algebra, Dorier (2000) distinguishes its analytical origin, linked with the solving of linear systems, and its geometrical origin, related with synthetic geometry. Both sides: analytic, and synthetic, are still present within modern linear algebra. It suggests the possibility of existence of a similar distinction in the processes of solving linear algebra problems.

The existence of analytical processes is obvious. Is it really possible to develop solving processes linked with synthetic geometry? And do the students use such processes? I provide here answers to these questions. I first justify (section 2.1) the need for a "concrete" model in linear algebra. I explain then (section 2.2) the precise meaning that I confer here to the expression "geometric thinking", and I display possibilities of geometric thinking in linear algebra.

In section 3, I study the example of a particular problem, and the solving processes actually developed by the students.

Linear algebra and geometric thinking

The need for a model

According to Fischbein (1987) the reasoning activity in mathematics always rely upon conceptions that seem certain to the person producing that reasoning. Even when dealing with very abstract structures, the mathematician (or the student) acts in accordance with a credible reality. For Fischbein, "credible" is synonymous with "behaviourally meaningful".

That credible reality will often be provided by the use of a model. Fischbein defines a model as follows : "A system B represents a model of system A if, on the basis of a certain isomorphism, a description or solution produced in terms of A may be reflected consistently in terms of B and vice-versa"(Fischbein, 1987, p.121).

A model can be extramathematical, for example when drawings are used. But it can also stem from a mathematical theory, if the concepts involved have the appearance of a credible reality in the eyes of the student, or mathematician, who uses them.

Harel's researches about linear algebra lead to results very similar to Fischbein's theory.

Harel posits three principles for the learning of linear algebra. The Concreteness Principle is specially relevant here: "For students to abstract a mathematical structure from a given model of that structure, the elements of that model must be conceptual entities in the student's eyes: that is to say, the student has mental procedures that can take these objects as inputs." (Harel, 2000, p.180) That principle is inspired by Piaget's and Greeno's work. Harel shows in particular that geometry can provide a model helping to construct linear algebra concepts, if the geometrical concepts are for the students mental entities on which mental operations can be performed.

Harel studies mostly the construction of the linear algebra concepts by the students. But models will also intervene in the problem solving processes of students already familiar with linear algebra concepts. Their reasoning must indeed be grounded on a credible reality, as Fischbein states it.

The notion of model permits to confer a precise meaning to the expression "geometric thinking" in linear algebra; I present it in the next subsection.

"Geometric thinking" in dimension n

I only study here the case of finite dimensional vector spaces, with an inner product.

Because these spaces are isomorphic to IR n , a first possibility of concrete model is provided by the use of coordinates. A problem proposed within IR n can be first formulated and solved in IR 2 and/or IR 3 , and then generalised to dimension n. That analytical process can be completely disconnected from any kind of geometry, for example to prove properties of matrices, or to compute determinants. It can also be associated with a geometric interpretation of the problem in the plane, or in the 3space; but also in that case, I will call it "analytic thinking".

Other possibilities exist, directly linked with synthetic geometry. A first step towards linear algebra was made by Leibniz when he tried to develop a geometrical calculus, opposed to analytic geometry. Using analogies between synthetic geometry and functions spaces has been determining in the genesis of linear algebra (see Dorier 2000). How can synthetic geometry intervene in modern linear algebra? I will term here "geometric thinking", in the solving of linear algebra problems, the use of a geometric model, that can thus be associated with drawings, and involves no coordinates. The model is limited to dimension 1, 2 or 3; it can stem from traditional Euclidean geometry, vector geometry, or even from linear algebra.

So "geometric thinking" can intervene when solving a problem in dimension n>3, if a model restricted to dimension 2 or 3 can help the solving process.

An analysis of university textbooks indicates two main procedures (because that analysis was intended to ground a questionnaire for students at the University of Rennes, I retained for it the books officially recommended to these students during their two first university years: Grifone 1990, Guinin 1993, Liret&Martinet 1997[START_REF] Liret | Outils d'analyse des contenus mathématiques à enseigner au lycée et à l'université[END_REF].

-Decreasing the dimension: in many problems, central elements of a solution take place in a 2 or 3-dimensional well chosen space. For example, to prove that the union of two subspaces is not a subspace if one of the subspaces is not included in the other, it is sufficient to consider two vectors, one in each subspace, that do not belong to the second subspace. Their sum does not belong to the union, thus it is not a vector space. The important point happens in a plane; it can be associated with a drawing, or a mental picture.

-Increasing the dimension: some linear algebra problems in dimension n can be proved by induction, or at least require the understanding of the process involved in moving up from n to (n+1). The associated solving process can in such cases be grounded on the understanding of the process leading from dimension 1 to dimension 2, or from 2 to 3. A typical example is provided by the Gram-Schmidt orthonormalisation process.

Both cases correspond to "expert processes" (Robert 1998), commonly developed by the mathematicians.

But do students actually use such models in their solving processes? We will see in next section, on a particular example, that the answer is positive.

Geometric thinking in students' problem solving processes

During the second semester of the academic year 2000-2001, I observed a six-weeks long linear algebra course 1 for second year students. It focused mainly on quadratic forms, and vector spaces with an inner product (all the students had learned elementary linear algebra during their first year).

After these observations, I met the teacher, and then eight students for individual interviews. I have chosen to encounter four students who were following that course for the first time, and four students who had the same course the preceding year with another teacher. I also retained students with various levels (according to their results to an examination they had two weeks before). The interviews were recorded and transcripted. A short description of the whole students' questionnaire is given in the appendix.

It included in particular the following exercise:

"Find the length of a diagonal of a cube with edges of length 1 in IR n .'' During all the interviews, I have chosen to intervene as little as possible in order to avoid influencing the students solving processes. For that particular exercise, the only hint I proposed was to indicate where does the diagonal of a cube in dimension 3 lie, in case they drew such a cube, and needed such a hint.

I present here an analysis of the students' solving processes.

1 That teaching consisted weekly in two lectures (1h15 each) and two tutorials (2h each). I observed all the lectures, and some of the tutorials.

Presentation of the exercise

A first difficulty when solving this exercise is linked with the geometrical vocabulary: these students never met "cubes'' and "diagonals'' in IR n . It could embarrass some of them, and prevent them from solving the exercise, even if it was possible to answer without a clear insight of the n-cube's significance.

Two main ways of solving that exercise were accessible to these students.

The first one is the analytic solution. One of the diagonals of the cube is a vector OA, were O has coordinates (0,0...0) and A has coordinates (1,1 ...1). Thus the length of the diagonal is OA = n .

Studying the problem for n=2 or n=3 can be naturally helpful in that case; it allows to discover the coordinates of A for these values of n, and that result can be immediately generalised.

The second solution is more interesting for our purpose, because it can be interpreted as "geometric thinking", instead of the "analytic thinking" of the first case. It uses an induction process. For n=2, the length of the diagonal is 2 . For n=3, it is √3, it can therefore be induced that the length of the (n-1)-cube's diagonal is (n-1) The diagonal of the n-cube is the sum of a diagonal of the (n-1)-cube and of a unit vector, orthogonal to it. According to Pythagorean Theorem, the length of the diagonal is thus n (this is the vector version; the diagonals can also be interpreted as the sides of a right-angled triangle). That solution requires the study of the case n=2, but also of the process that leads from n=2 to n=3. That process provides indeed the key of the induction; the associated drawing can display the diagonal of a side, and a unit vector orthogonal to it. Such a reasoning involves geometric thinking. The induction can also stay more or less analytic, as we will see it in the next paragraph.

Students' solving processes

Considering the students' answers and solving processes leads to distinguish three groups among the eight students that I interviewed.

Group 1: The obstacle of geometrical vocabulary (Students A, B, C, D).

These students did not overcome the obstacle of vocabulary. They were not able to confer a meaning to the term "cube'' in IR n .

Student A: "For n=3, it's a cube... For n? I do not see what a cube can be! For n=3, the diagonal of a face is √2; it gives √3... But for dimension n...''

These students solved the cases n=2 and n=3, but were unable to identify a generalisable process. They remained trapped in the geometric context, and were unable to develop a geometric thinking in higher dimensions.

Group 2: Using coordinates (Student E).

There is only one student in that "group''. His reasoning process is very different from all the others. He started with dimension 2, and immediately made the associated drawing, figuring only two vectors and two points O and A, and not the complete square.

He made the same kind of drawing in dimension 3, deduces that the length of the diagonal is √3 is that case, and then that the coordinates of A will always be (1, 1, 1..), thus the length of the diagonal would be n .

That student thinks in terms of linear algebra: he draws vectors, considers a basis; that allows him to use coordinates, and then the generalisation appears as a natural process. Although drawings are central in his solving process, his reasoning is more analytical than geometric.

Group 3: Induction process (Students F, G, H).

These three students started by calculating √2 for n=2 (they first drew a square); then they drew a cube, and used Pythagorean theorem, quoting it or not, to compute √3 for n=3. They all claimed then that the general result was n , using a kind of induction process, but none of them produced a rigorous proof.

Students G and H base their reasoning on geometrical statements, while student F grounded is answer only on the aspect of the result ("it is √2 for n=2, √3 for n=3, so it must be √n").

I give here the example of student H solving process. After calculating √2 for n=2, and using it to deduce √3 for n=3, he said:

A O 2 e 2 e 1
Student H: "I would say then there are 1, and √3, and it is orthogonal, so it gives √4 = 2, but I'm not sure, I dot not see it clearly... But I think it works, because √3 is the diagonal, and the last edge is orthogonal to it. So it is actually 2 for n=4. And it is always the same, so you obtain √√√(((1 + 1) 2 + .... + 1) 2 + 1) = n .'' He first felt embarrassed because he was missing a picture figuring exactly the ncube. But his drawing of the 3-cube displayed in particular a right-angled triangle formed by the diagonal of a face, an edge of the cube, and the corresponding diagonal of the cube:

Because of that drawing, he focused on the orthogonality of the diagonal of a face and an edge of the cube. Even if he did not explicitly interpret the diagonal of the 3cube as the diagonal of a "face" of the 4-cube, he used it that way; that allowed him to compute the result for n=4. He did not try to provide any further geometrical interpretation, but immediately generalized his result for any value of n.

In his course, the teacher spent a lot of time presenting IR 2 and IR 3 , and gave many examples in IR 2 and IR 3 when presenting more general results. When I asked him about the possible use of IR 2 and IR 3 for the study of IR n , his assumption was: "For quadratic forms, everything already happens in dimension 3. The important point is to be able to move from dimension 2 to dimension 3."

In the particular case I presented here, the observations confirm that statement. The four students who did not identify the process that led from n=2 to n=3 failed. The four others found the result, and two of them really developed a reasoning involving "geometric thinking".

The observations related in this section result from a clinical study, about a precise task, conducted with a small number of students. However, the behaviours described indicate more general phenomena.

4.Conclusion

Fischbein, and Harel, state that "concrete" models are needed to learn linear algebra. The study of IR 2 and IR 3 with coordinates provides such a model. But other possibilities exist. Some students actually use, when possible, "geometric thinking" to solve linear algebra problems; it seems to be really helpful for them. Proposing several tasks, to groups of students of several levels [START_REF] Gueudet-Chartier | Rôle du géométrique dans l'enseignement et l'apprentissage de l'algèbre linéaire[END_REF], indicates that it is only done by a minority of students. Why? Let us reconsider the two procedures presented in section 2.2.

-Decreasing the dimension: It is a familiar process for mathematicians to think in a well chosen 2-dimensional space (and it is then possible to help the reasoning with a drawing). The actual n-dimensional object can not be represented; but it is always possible to cut it along a plane, and represent the obtained section.

-Increasing the dimension: Moving up from dimension (n-1) to dimension n is a complex process. It is in particular necessary to interpret the space of dimension (n-1) as an hyperplane of the n-space. Mathematicians are quite used to such processes; they consider the first space as an hyperplane, and then add a supplementary line to it, to obtain the whole n-space. It is a difficult process for students, that requires to be familiar enough with subspaces.

These possibilities of using geometric models have never been explicitly mentioned in the courses I observed, although the teacher himself used them. Students were encouraged to use induction with coordinates; but "geometric thinking" was not taught to them. The teacher seemed to expect that the students develop it spontaneously. Some students succeed in it, but not all of them. There is no evidence that "geometric thinking" can be taught. However, it would be interesting to design a linear algebra course including a detailed presentation of geometric models, and of their use in the solving of n-dimensional problems; it would allow us to observe the consequences of such a design on the students reasoning processes. Part 3 : In the part 3, I asked questions related with the course and the tutorials, and linked with the use of drawings.

For example, the teacher was used to represent the elements of a vector space sometimes as vectors, and sometimes as points. He never explained the reasons for his choice of one or another representation.

He also used drawings (with arrows) to illustrate properties in vector spaces of polynomials.

So I investigated the students' interpretations of these pictures, and their own use of such drawings.

Bibliography

  DORIER, J.L. (ed.) (1997): L'enseignement de l'algèbre linéaire en question, Grenoble : La Pensée sauvage éditions. DORIER, J-L. (ed.) (2000): On the teaching of linear algebra, Dordrecht : Kluwer Academic Publisher FISCHBEIN, E. (1987): Intuition in science and Mathematics, An Educational Approach, Dordrecht: D.Reidel Publishing Company. GRIFONE, J. (1990) : Algèbre linéaire, Toulouse : Cepaduès éditions.

G. Gueudet-Chartier

Appendix : short description of the whole student's questionnaire Part 1: The following two exercises were proposed in an examination the students sitted two weeks before. During the interview, we discussed their answers, and specially the drawing they produced (or not).

Exercise 1.1: Let q be the quadratic form defined in IR 2 as q(x) = (2x 1 + 3x 2 ) 2 . Write the corresponding matrix, compute its rank, and draw its isotropic cone.

Exercise 1.2: Let q be the quadratic form defined in IR 3 as q(x) =3x 1 2 + 2x 2 2 -x 3 2 . Give its rank, and draw its isotropic cone . Is q a positive form?

Part 2: In the part 2, I proposed three exercises that the students did not encounter in the tutorials.

Exercise 2.1: Four drawings (in a plane with a system of axes) were proposed. The task was to decide whether the represented sets can be the kernel, or the isotropic cone of a well chosen quadratic form.

Exercise 2.2: Let q be a quadratic form on IR 2 [t], and {P, Q, R} an orthonormal basisof IR 2 [t] for q. Compute, if possible, q(P+Q).

Exercise 2.3: Find the length of a diagonal of a cube with edges of length 1 in IR n


Bernard Jacquemin
email: bernard.jacquemin@uha.fr

A Derivational Rephrasing Experiment for Question Answering

In Knowledge Management, variations in information expressions have proven a real challenge. In particular, classical semantic relations (e.g. synonymy) do not connect words with different parts-of-speech. The method proposed tries to address this issue. It consists in building a derivational resource from a morphological derivation tool together with derivational guidelines from a dictionary in order to store only correct derivatives. This resource, combined with a syntactic parser, a semantic disambiguator and some derivational patterns, helps to reformulate an original sentence while keeping the initial meaning in a convincing manner This approach has been evaluated in three different ways: the precision of the derivatives produced from a lemma; its ability to provide well-formed reformulations from an original sentence, preserving the initial meaning; its impact on the results coping with a real issue, ie a question answering task. The evaluation of this approach through a question answering system shows the pros and cons of this system, while foreshadowing some interesting future developments.

Introduction

With the exponential increase of available textual documents, it has become impossible for anyone to read all of them, or manage all the information they contain. Automatic methods are thus necessary to deal with these masses of text and to provide quick and easy access to a piece of information lost in data. Among Knowledge Management disciplines that try to solve this issue, the question answering task [which consists in supplying the text phrase that contains the answer to a question] is particularly fussy: on the one hand the answer supplied has to be as concise and precise as in Information Extraction, and on the other, the system must adapt to the varying queries and address changing information types in order to find an answer, as does Information Retrieval. The major obstacle with which the question answering task is confronted consists in identifying text meaning: a difficult job for a computer. The same piece of information is indeed phrased in different ways in a question and in the questioned text base. These differences prevent the system from matching data and consequently from extracting the right answer [Grau et Magnini, 2005[START_REF] Strzalkowski | [END_REF]].

Several approaches have been proposed to tackle this problem. Some of them attempted, and sometimes succeeded, in building semantic representations for query and textual utterances that were next matched [Grois et Wilkins, 2005[START_REF] Harabagiu | [END_REF]. But the query expansion method, although simpler to carry out, is a very common choice in the discipline, because it covers a large amount of different phrasing of the same piece of information [START_REF] Grau | Coping with Alternate Formulations of Questions and Answers[END_REF], Dang et al., 2006]. The process consists generally in constituting, for each significant word in the query, a disjunctive list of terms with the same meaning as the original word. In order to find equivalent terms, classical semantic relations make it possible to draw up lists of synonyms, hyperonyms, etc. But these semantic relations do not give the opportunity to extend the rewording beyond the limits of the part-of-speech of the original word, and even more so to explore new syntactic schemata.

In order to free themselves from this part-ofspeech constraint, many researchers have followed the morphological derivation trail, considering that members of the same derivative family have roughly the same meaning [START_REF] Church | One term or two?[END_REF], Jacquemin, 1996, Hull, 1996]. Nevertheless, the results reached by morphological derivation in a query expansion task are often inconclusive. Far from improving the quality and precision of answers, derivation systems tend to provoke numerous incorrect answers. At present, the current derivation systems are not able to generate the whole derivation family of a given word, without generating simultaneously several words incorrectly associated with the authentic derivatives, but morphologically, and above all semantically, distinct from them. On the contrary, some parameters and constraints can be defined for these generation tools in such a way that priority is given to precision, minimizing noise and eliminating scoria from candidates-derivatives. But if this method significantly reduces the error rate, it also entails a dramatic reduction in recall that dispels the query expansion interest for textual information man-agement [START_REF] Gaussier | Recherche d'information en français et traitement automatique des langues[END_REF], Bilotti et al., 2004].

Despite this assessment, a method that uses both a morphological derivation tool and a general French dictionary is proposed in order to build a rich and accurate derivational resource by filtering candidates-derivatives with derivational instructions. To take advantage of the just-generated derivatives in query expansion, parse the utterance is parsed and a Word Sense Derivation system applied [START_REF] Jacquemin | Enriching a text by semantic disambiguation for information extraction[END_REF]. The tool used to generate candidates-derivatives and the dictionary with the filtering instructions are presented herein, as are the means of constructing the derivational resource and a short evaluation of its quality; thereafter, the approach to the formulation of derivational rephrasings as close as possible to the original utterance meaning are outlined. Finally, the derivative rephrasing approach in the absolute and in terms of its impact on our question answering system's performance are evaluated.

From generation to filtering of derivatives

The method proposed consists first in generating as many words as possible that are likely to belong to the same derivational family as a given term in such a way as to get as many actual derivatives as possible among the candidates, whilst not taking into account the number of incorrect creations. Then all the inaccurate candidates-derivatives are excised from the list by filtering all the propositions that do not match the derivational instructions from the dictionary. For many years, research has been undertaken in the automatic derivational morphology field [Lovins, 1968, Porter, 1980].

Consequently, several tools can, for a given term, provide a list of candidates-derivatives likely to belong to its derivational family. Some of these systems are based upon derivation learning [START_REF] Snover | Unsupervised Learning of Morphology Using a Novel Directed Search Algorithm: Taking the First Step[END_REF], whereas others apply general and ad hoc rules to generate derivatives [Namer, 2003]. This research employs a probabilistic system that searches the term's stem and attaches successively to that stem all the suffixes it knows in order to return a derivational list for this term [Gaussier, 1999]. This tool is based on stemming and suffixation learning from an inflectional lexicon. It meets the requirements of the method described above: the stemming learning parameters can be set up more or less strictly, and the weakest constraints make it possible to generate so many candidatesderivatives that the whole of the derivational family is created, or almost -valuing recall over precision since the noise filtering happens after the derivatives are generated.

Following this method, each significant entry (nouns, verbs, adjectives, adverbs) was addressed by means of the French dictionary used for the filtering process (the Dubois dictionary, see below) with the generation tool. For each entry a list of candidatesderivatives was obtained, covering at best the derivational field of this entry, and more besides. It should be noted that entries shorter than 3 syllables have been ignored by the method, because the tool cannot find a stem for shorter words, and it is absolutely necessary for the suffixation process. Another restriction is applied on generated forms: each proposition is compared to a lexicon extracted from a large corpus (5 years of Le Monde newspaper, 100 millions words) in order to eliminate chimaeras and nonexistent words from the derivational resource.

Furthermore, the Dubois dictionary utilised is a general electronic French dictionary that contains derivational information for each entry [START_REF] Dubois | [END_REF]]. The dictionary is made up of 2 computer files respectively dedicated to the description of verbs (12,309 verbal entries) and of other words (102,917 non-verbal entries) in French1 . As shown in the table 1, the Dubois dictionary contains very rich and varied information, particularly in the verb component, which is considerably more detailed (conjugation, syntactic schema. . . ).

A specificity of the Dubois lies in providing all the information types for each meaning, which is more rigorous than most dictionaries tend to be. Information types concern semantics (domain, class, sense), syntax (operator, syntactic construction. . . ) and morphology (conjugation, derivatives, name). Construction and conjugation fields only appear in verbs, and each type of information is consistent in the two parts of the Dubois dictionary.

1---------- -Q---RB---- Name 6L 6L Level 2 5
Table 1: Entry formaliser in the Dubois of verbs.

The guidelines in the dictionary are provided as alphanumeric codes and are therefore easier for an automatic system to read than a human being. For example, the 1aZ code from the conjugation field in the table 1 indicates that the corresponding sense (here the two senses for the entry formaliser have the same conjugation code, as usual) belongs to the regular version (a) of the first (1) conjugation pattern aimer (to love), and that the auxiliary (Z) for composed active tenses is avoir (to have)2 . In spite of this formalised aspect, some information fields cannot be used directly by a computer. In particular, derivational instructions are not explicit enough to give the opportunity to generate the right derivatives from the entry: instructions generally indicate which suffix to use, but not how to find the stem, nor whether the stem and the affixes undergo morphological changes because of their mutual influence. For example, the derivation fields in the table 1 provide a Q code for the second sense of formaliser (to formalise), that indicates the existence of a verbal adjective with the suffix -é in both positive (formalisé, formalised) and negative (informalisé, unformalised) forms. But the instruction is no more explicit regarding the negative prefix that could be founded on the privative a-(with possible euphonious consonants, depending on the stem) or on in-(with a possible consonant variation, depending on the stem). Because of this lack of precision, we had to use the derivation tool described above. Nonetheless, the instructions provided give enough information to take out incorrect candidates-derivatives.

Candidates-derivatives

Dubois' instructions It is quite easy to filter out wrong candidatesderivatives, by comparing the affixal characteristics of each candidate for a term with all the instructions in the derivative field of the corresponding entry: the suffix identifies the candidates that are well conformed. In the left hand column, the table 2 shows some of the candidates-derivatives generated by the derivation tool for the entry couper (to cut). The bold font indicates the candidates that matched a derivational instruction (in the right hand column) in the dictionary. These candidates are thus considered as real derivatives for the current entry, and the candidates whose suffix does not match the derivational instructions are deemed erroneous, and deleted from the derivational list.

coup (knock) - coupure (cut) nominal derivative in -ure coupable (guilty) - coupage (cutting) nominal derivative in -age coupant (sharp) verbal adjective in -ant coupeur (cutter) nominal derivative in -eur coupé (cut) verbal adjective in -é coupon (remnant) - . . . . . .
When the 115,226 Dubois' entries were submitted to the derivation tool, about 2 million candidatesderivatives were returned. Among those candidates, 502,429 were identified as real derivatives by our methodology, i.e. about 5 derivatives per entry on

From expansion to rephrasing

This derivation-filtering method is at the origin of a very rich and precise derivational resource, which is particularly useful for query expansion. However, even if the semantic link between the members of a derivational family is effective, it is not stable between all the meanings of every member of the family. For example, in the entry formaliser (formalise, see table 1), the derivational field differs between the two senses involved, and among the derivational family for the entry, some derivatives are related to one sense and not to the other: the B code gives an instruction to generate formalisation (formalisation), that corresponds to sense 2 (formalise) of the entry and not to sense 1 (take offence). Thus, even if derivation is a morphological process, some semantic constraints have to be taken into account when it is used in Knowledge Management. Consequently, the use of all the derivatives proposed by the derivational resource for utterance expansion is likely to throw up some inappropriate meanings, and then some noise. However, derivational instructions are displayed only for the corresponding senses in the Dubois dictionary. In view of this peculiarity, it seemed necessary to take into account the original sense of every term so that only derivatives matching the derivational instructions for this sense were produced.

The issue is to identify the sense of the term in the utterance needing expansion, and then to select the derivatives suggested by the derivational field matching the sense. In the perspective of Information Extraction and synonymic expansion, we designed a Word Sense Disambiguation system based on syntactic analysis and applying disambiguation rules extracted from the Dubois dictionary [Jacquemin, 2004a, Jacquemin, 2004b] . Lexical, syntactic and semantic information provided by the Dubois made it possible to create rules for every sense of each entry. Each type of information is converted into dependencies, terms and features schemas relative to the corresponding sense. For example, the entry prendre (to take) has for its sense "to escape" an example field containing the sentence il prend la fuite (he takes flight). This sentence produces a disambiguation rule that selects the sense "to escape" for the word prendre when its direct object is the word fuite (flight). These rules can match (or not) dependencies between words extracted by the XIP parser [Roux, 1999, Aït-Mokhtar et al., 2002] from the utterances to disambiguate. So nearly 45% of the significant words in submitted texts can be disambiguated, by associating the polysemantic words to one of their meanings in the dictionary. For these disambiguated terms, only the derivatives that match the derivational instructions for the selected sense may be used for expanding the text. For monosemantic terms and terms for which no disambiguation rule worked, the derivational expansion cannot be specified. Thus all the derivatives for a term in our derivational resource are used for expansion.

Moreover, when the WSD method is applied to a sentence for text expansion, the syntactic analysis performed by XIP produces a dependencies structure. This structure offers great advantages. All the syntactic dependencies constitute in one way or another a formal representation of the parsed sentence, since on the one hand the dependencies describe evenly the links between the words of the sentence, and, on the other, the lexical units in the sentence are identified as the arguments of the dependencies and their linguistic characteristics are expressed as features based on the arguments. The formal representation is propitious for standardisation of the word contents (lemmatisation, normalisation) and of the structure. Lexical and syntactic information is expressed in an optimised way to store data within a database, where it is indexed and easy to retrieve. In this form, it is easier to match information from a query with information from text containing the answer: it is associated if their respective structures coincide. Syntactic structure also makes it possible to remedy a weakness in the derivational expansion method. In spite of the real meaning closeness generally ob-served between derivatives from the same derivational family, and in spite of the semantic subgroups established in the derivational family in order to ascribe the derivatives selection to the ones with the same meaning as the original term in context, the sense challenge inherent to the derivation phenomenon has not yet been overcome: members of the same derivational family show meaning variations in relation to the nature of the suffix used, but above all because of the rewording from a lexical category into another [START_REF] Hathout | [END_REF]. The syntactic structure of the utterance itself cannot deal with a simple expansion by a disjunctive list of derivatives, even if their sense is similar. For example, the sentence il a coupé le courant (he cut off the power) can be expanded by a derivative coupure (power cut) coming from couper (to cut off). But the corresponding utterance *il a coupure le courant (he power cut the power) is unsatisfactory. In order to build a correct expanded utterance, successively replacing the original terms by a list of their derivatives is not enough: it is necessary to rephrase the sentence. This action must be taken on the syntactic structure of the original sentence: the structure must be modified in such a way that a derivative can be substituted for the original term in the sentence without rendering it ungrammatical. The syntactic dependencies structure produced during the word sense disambiguation process provides the opportunity to simulate the rephrasing through the dependencies structure in order to avoid the generation issue. Ideally, an automatic system is needed that can easily and correctly rephrase a sentence such as il a coupé le courant (he cut off the power) as la coupure de courant (the power cut). However, text generation is still a research issue confronted with tricky problems in morphology, syntax, semantics and even pragmatics. However, if the dependencies structure coming from the morpho-syntactic analysis of the original sentence constitutes a standardized representation, the same is true of the reformulation. Therefore, it is possible to rephrase an utterance virtually without generating a real sentence: one only has to build the dependen-cies structure where the dependencies are the same as the ones that would have been produced by an XIP analysis of the rephrased sentence had it been generated. Thus the issue is to build a correct new dependencies structure from the original one. Designing syntactic derivation patterns that would make it possible to induce the derived dependencies structure from the original one was also considered as part of this research. Figure 1 shows the simulated rephrasing process: the original sentence is processed by the morpho-syntactic analyzer XIP and syntactic dependencies are extracted. Word Sense Disambiguation rules are applied to select the contextual meaning of the terms (not shown on figure 1) in order to establish the correct derivatives. A derivation pattern depending on the original syntactic structure, on the category of the original term and on one of the derivatives is applied in order to create a new syntactic structure where the derivative is an argument instead of the original term. The new structure corresponds to the XIP analysis of the rephrased utterance (simulated in brackets) that should not be effectively generated.

It was determined that a derivational XIP grammar should be created in order to simulate correct derivational rephrasing in most cases. For this purpose, the derivational rephrasing process was considered on a relatively large scale and in a real life environment. Certain changing parameters had to be studied: the lexical category of the original word and of the derivative, the suffix in the original word and in the derivative, and for verbs, the syntactic schema. By successively varying the value of these parameters, all the possible combinations of authentic original texts and corresponding sentences rephrased by the research team were duly tested. For each combination, 3 instances were randomly selected from among the Dubois dictionary entries (for example, 3 direct transitive verbal entries with the -iser suffix that comprise instructions for a nominal derivative with the -ation suffix). By successively questioning Google with each entry as a request, the first 20 different phrastic contexts where the entry appeared were chosen. Every selected sentence was then submitted to morpho-syntactic analysis by XIP in order to extract syntactic dependencies. The original sentence was also rephrased by using the derivative corresponding to the parameters combination, and the new sentence submitted to XIP. Taking into account the recurrence of an initial syntactic schema (at least 5 occurrences for every entry in the same parameters combination) and the regularity of the corresponding dependencies structure in the rephrasings (at least 2/3 of the instances of the recurrent initial syntactic schema are rephrased into the same dependency structure), 54 derivation patterns were drawn such as the one shown in figure 1 including 34 patterns for a derivation from a verb.

The derivation patterns were tested by rephrasing the sentences from a corpus to as great an extent as possible. The corpus was drawn from a general encyclopedic dictionary, the Encyclopédie Hachette Multimédia [START_REF] Alcouffe | Concevoir une encyclopédie multimédia[END_REF]. This corpus contains 50,000 words from articles with the tag Roman Antiquity. The corpus was morpho-syntactically analyzed and submitted to Word Sense Disambiguation in order to select derivatives that could be used for rephrasing. From this result, 807 derivative patterns were applied to reformulate sentences. In order to evaluate the quality of the new dependencies structures, we generated sentences where the selected derivative took the original word's place, modifying the syntactic structure to keep the sentence well-formed, and submitted the new sentences to XIP analysis. For 656 reformulations (81.29%), the dependency produced by the derivative pattern matched the XIP analysis of the sentence as originally written. The nonmatching cases were due mainly to errors in the partof-speech tagging of the original word (102 occurrences, 12.64%) or to syntax analysis in either the original or the rephrased sentences (37 occurrences, 4.58%). Only 12 errors (1.49%) may be legitimately attributed to the derivative patterns, when the original sentence has a particular syntactic schema.

4 Rephrasing evaluation in a question answering task

Derivational rephrasing in a QA system

This research has thus produced a rich and precise derivational lexicon that will associate to a word's specific sense only those derivatives with a similar meaning. A method that can rephrase utterances through morphological derivation of a term was also developed, which takes into account both the original term meaning when proposing derivatives and the syntactic structure of the rephrased utterance. This method simulates the rephrasing into a dependencies structure in order to avoid the text generation issue. The next step is thus to integrate the method in a question answering system and to supply more textual formulations in order to match elements from the question and from the answer. Since a major issue in question answering is how to match texts with an identical meaning but a different wording, a derivational rephrasing module should help the existing synonymic rephrasing module in the question answering system employed in this research. The question answering system developed [Jacquemin, 2005] employs an original methodology to find textual answers to a question by matching dependencies structures. Such structures are extracted by morpho-syntactic analysis of both question and text; then Word Sense Disambiguation LREC 2010 (Sous presse)

Figure 2: The architecture of our question answering system is performed in order to select correct synonyms according to the initial meaning. It is then possible to simulate a synonymic rephrasing by enriching the dependencies structure. A feature of this approach is the special deep pre-processing undergone by the text base instead of the question. The method uses only minimal analysis to extract dependencies from the question. This approach is connected with the fact that XIP is better at analysing normal text than questions, and above all it is related to the necessity to have as much syntactic context as possible in order to improve the Word Sense Disambiguation results [Weaver, 1949, Reifler, 1955]. The classical approach in query expansion was improved by rephrasing performed on the texts, first through synonymy and subsequently through derivation. The search for an answer is performed by comparing the question minimal structure with the text enriched structures, and matching the inner dependencies (see figure 2). Since the derivational method employed in this research also uses XIP morpho-syntactic analysis and the Word Sense Disambiguation system to collect information from an utterance and to simulate rephrasing with the same meaning, it seemed natural to integrate it into the question answering system. Fig-

ure 3 shows the mechanism of the question answering system. A minimal morpho-syntactic analysis is performed on the question in order to extract the dependencies structure (Question's structure) that has to be matched with the text enriched structures. Furthermore, the text base to question has been pre-processed: morphological, syntactic and semantic analysis as well as rephrasing are performed before the request phase. The morpho-syntactic analysis produces the base dependencies corresponding to the sentence structure. When the Word Sense Disambiguation rules have been applied to the terms in the syntactic structure, both synonyms and derivatives that match the original senses are selected to perform rephrasing: synonyms are inserted into the existing dependencies (in red), disjunctively to the corresponding original terms that belong to the same lexical category; and for derivatives, the corresponding derivation patterns are applied in order to create new dependencies structures simulating rephrasing (derivational dependencies). Answering the question consists in returning sentences from the text that contain the same data as does the question. In figure 3, the question is answered by matching its structure with dependencies from a text structure. All the matching dependencies in the text come from derivational and synonymic rephrasing.

The current version of the question answering system developed in this research programme cannot be entered in competitions like TREC (Text REtrival Conference, [Harman, 1992[START_REF] Voorhees | [END_REF]) or CLEF (Cross-Language Evaluation Forum, [START_REF] Peters | Advances in Cross-Language Information Retrieval[END_REF], Peters et al., 2005]). On the one hand the system cannot select precisely the answer to the question since no such module has yet been developed to perform this selection: the answer to the question appears in a full sentence. On the other hand, the system is currently based on one reference dictionary, the Dubois, that exists only for French: the rephrasing methods, and consequently the answering process, can only be applied to French questions and texts. Furthermore, lack of time and of human resources prevented the organisation and evaluation of this system on a larger scale. However, considerable efforts were made to measure impartially the efficiency of the question answering system and the impact of derivational rephrasing on the results. The TREC 8 campaign [Voorhees, 1999] evaluated question answering systems for English and the evaluation design has characteristics very close to an experiment that this research was indeed able to implement. In this evaluation, 200 questions are submitted to systems, which have to return up to 5 answers, sorted by relevance. All the questions have at least one correct answer in the text base, and a correct answer should appear in a 50 words window. A score is assigned to each question, depending on the inverse rank of the first correct answer: the score is 1/1 if the first answer is correct, 1/2 if the second answer is correct and the first one is wrong, 1/3 if the third answer is correct and the first two are wrong and so on until the fifth answer. The global score for a system is the mean of every question's scores. The text base questioned is drawn from 500 articles with an Antiquité (Antiquity) tag extracted from the Encyclopédie Hachette Multimédia. After reading all the articles and without the texts in front of them, 8 people from outside the project proposed 25 questions each (i.e. a total of 200 questions as in TREC 8) about information content in the texts. All the questions are in correct French. The answers are full sentences, which seemed more relevant than a 50 words window. In order to highlight the real influence of the derivational rephrasing in the answering process, the system was made to question the texts at several levels of pre-processing (table 3): for the baseline, only the significant terms (nouns, verbs, adjectives, adverbs) were stored in an index; for the base rephrasing, the base structure is extracted by an XIP analysis and a first synonymic rephrasing is performed with the few synonyms coming from the Dubois dictionary; the derivational rephrasing corresponds to the base rephrasing with the derivational rephrasing method described above; the highest level of rephrasing includes all the enrichments, i.e. a derivatives structure that contains the derivational rephrasing, the synonymic rephrasing with synonyms that come from several dictionaries (Dubois, EuroWordNet, Bailly, Memodata) and a pronominal co-reference procedure.

Rephrasing levels

Results and discussion

Further to this evaluation, despite the quality of the derivational resource created and in spite of the capacity of this particular method to simulate grammatical derivational rephrasings of texts very close to their original meaning, it can be observed that this enrichment does not greatly improve the results achieved. The derivational rephrasing provides only one more answer. However, no answer would be found for this question without the derivation process. Moreover, the proposed answer is correct and first-ranked for the question (see figure 3). It is also remarkable that the derivation process did not damage the results [START_REF] Clarke | Question answering by passage selection (multitext experiments for trec-9)[END_REF], Monz, 2003]. By examining the system performance in greater detail, as much in the successful answers as in the weaknesses, certain error explanations and several ways to improve the system were identified.

Firstly, at least 11 cases were noted as being without answer where an idea was expressed with a verbal construction in the question and with a nominal or adjectival expression in the text. At this point, all the rephrasing processes are applied to texts and none to the questions. The exceptional wealth of information contained in the Dubois of verbs can be confirmed; the Dubois of words is not as complete, and the derivation field is often poorer than in the verbal part: in the case of a verbal entry, all meanings are drawn together, providing instructions for the whole derivational family, whereas the nouns, adjectives or adverbs sometimes have omissions, and do not provide instructions by means of which the corresponding verb, adjective, adverb or noun may be found. Consequently the derivational rephrasing is incomplete, and no match can be made with a missing derivative that appears in the question. Thus the gaps in the Dubois words derivation fields must be filled in by symmetrising the derivation instructions from the Dubois verbs. Semantic fields like Domain (see table 1), that are consistent in the two parts of the dictionary, should share the instructions between the senses. Secondly, in 8 cases neither the derivational rephrasing, nor the synonymic rephrasing could simulate the question formulation, and thus provide an answer, because in the questions the two types are combined: the question addresses the same notion as the text, but the part of speech and the word are occasionally different. Thus derivational and synonymic rephrasing should also be combined, by derivational rephrasing after synonymic rephrasing or by synonymic rephrasing after derivational rephrasing (or both).

The implementation of these propositions might bring a significant improvement to the system. It should provide correct answers to the 19 marked questions that did not get any answer from the current system. If this proves to be the case, the results would be improved by nearly 10%. Finally, a small test was undertaken using the derivational resource created to perform a classical derivational expansion on 5 articles from the corpus. Five questions from the evaluation were posed, whose answer was in one of these articles. These questions were correctly answered at the derivational rephrasing level of the evaluation. In this test, a dramatic reduction in successful performance ensued in that two questions did not generate any correct answer, and only one had the correct answer in the first place. The mean score for these questions is 0.367 (to 0.767 with the derivational rephrasing). This scaled-down test was too small to be strictly interpreted, but it shows a distinct tendency of the approach taken by this research to preserve the quality of the results, contrary to the classical derivational expansion, which usually impacts negatively upon accuracy [Hull, 1996].

LREC 2010 (Sous presse)

Conclusion

In textual Knowledge Management disciplines, and more specifically in question answering, the different means of expressing the same information in a sentence can be a major source for identifying the meaning of the contents. Classical semantic relations such as synonymy or hyperonymy often provide new wordings in most of the current approaches, but the partof-speech variations are still an issue that needs to be worked on, especially through morphological derivation.

The combined use of a derivation tool with few constraints for a very large recall, and a general French dictionary that provides derivational guidelines made it possible to create a derivational French lexicon that is particularly rich and precise. Moreover, the specific lexico-semantic characteristics of the Dubois dictionary -mainly the systematic association of the derivational guidelines with the corresponding meaning -and the Word Sense Disambiguation process developed by this research combine to provide access to derivatives with a close meaning for a term in a selected sense. By using the XIP morpho-syntactic analyser to apply Word Sense Disambiguation rules, a syntactic dependencies structure that constitutes a formal representation of the utterance was extracted for each disambiguated utterance. In this structure it is possible to simulate derivational rephrasing of the original sentence: applying some derivation patterns leads to designing new dependencies involving the proposed derivative; these represent a rephrased utterance without generating it.

The derivational rephrasing process was integrated with the question answering system in order to measure its quality and impact on performance. The evaluation design followed for the French language copies the question answering track used in the TREC 8 competition. In spite of the modest results increase due to the derivational rephrasing method employed, observation confirms that it never damages performance in the way derivational expansion usually does. Moreover, following careful analysis of the results of the questions as well as those of the questioned texts, some promising ideas emerged to enable system improvements, notably by enriching the dictionary's derivational information field of non-verbal entries (symmetrisation from the verbal entries), and by performing a derivational rephrasing on the dependencies after the synonymic enrichment application, or by performing a synonymic rewording on the dependencies after the derivation patterns application, or both. Plans are currently being advanced to investigate the opportunity to integrate this procedure into the QALC question answering system [de Chalendar et al., 2002], based on deep processing of the questions and working on French language inter alia.

Figure 1 :

 1 Figure 1: Rephrasing into a dependency structure with the help of a derivation pattern

«

  De quel chef Domitien est-il le successeur? » (Of which chief is Domitian the successor?) Question's structure: PrepPh(successeur,de,chef) ATTRIBUTE(Domitien,successeur) Text's structure: SUBJECT(succéder OR remplacer,Domitien) ATTRIBUTE(empereur OR chef,Titus) Base dependencies DirObj(succéder OR remplacer,empereur OR chef) ATTRIBUTE(Domitien,successeur) PrepPh(successeur,de,empereur OR chef) Derivational dependencies « . . . Domitien succéda à l'empereur Titus. . . » (Domitian succeeded to the emperor Titus)

Figure 3 :

 3 Figure 3: Questioning a dependencies structure with synonymic and derivational rephrasing

Table 2 :

 2 Filtering candidates-derivatives produced by the derivation tool.

Table 3 :

 3 Evaluation results

		Score	No answer
	Baseline	0.295	139
	Base rephrasing	0.462	105
	Derivational rephrasing	0.467	104
	All the enrichments	0.504	97

In order to make the text clearer, we designate the whole dictionary by the name Dubois, and the two parts respectively by Dubois of verbs and Dubois of words.

In French, two auxiliaries may be used in composed active tenses: avoir (to have) and être (to be), but only one is correct for a given verb and this information is needed for nonnative speakers and computers.average. An evaluation of these derivatives was then undertaken. Randomly taking 10,000 derivatives from the derivational resource just created, only 24 wrong creations were identified, i.e. precision was at 99.76%. The wrong derivatives were generally created on the basis of a long original term for which the derivation tool found two different plausible stems. For each suffixation, two derivatives were generated every time, one for each stem. For example, the noun compartiment (compartment) produced two stems, which in turn were used to generate two candidates: compartimentable (compartmentalisable) and *comparable with the same suffixation. Since in our method candidate control is based on the suffix, false stemming cannot be corrected, or even detected automatically. Nonetheless, the very low error rate should entail an insignificant quantity of noise in a Knowledge Management application.However, the derivational field in the dictionary gives instructions for 542,296 derivations, which leaves out

a further 39,867 derivations. The omission of these derivations is accounted for by the derivatives created by prefixation, which is not assumed by the derivation tool. Consequently, neither negative forms, nor other prefixations can be generated in the current state of this approach, unless we use another derivation tool. It can, however, be noted that when derivatives are created by prefixation, the derivation process causes a larger lexico-semantic variation in relation with the original term than does suffixation, particularly in the case of a negative prefix.

LREC 2010 (Sous presse) Bernard Jacquemin, A derivational rephrasing experiment for question answering