Deliverable <n° 20.3.1>: < Intermediate Report on Learning Contexts> Editor: <Chronis Kynigos>, <ETL/NKUA> Contributors: <Rosa Maria Bottino, ITD, J. Lagrange, DIDIREM, Candia Morgan, KnowledgeLab UNILON, A. Mariotti, Sienna, M. Kuhn, UDUE, J.-F. Nicaud, MeTAH and Leibnitz, IMAG> Date of Delivery: <31.12.04>

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Contents Rationale of this Document...................................................................... 3 TELMA Teams ........................................................................................ 4 1. Introduction ...................................................................................... 4 2. <Title> ............................................................................................. 7 <n-1>. Future work ............................................................................. 54 <n>. References.................................................................................. 54

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Rationale of this Document <text>

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TELMA Teams <teams affiliation and acronym used in the text, to be checked> (ITD-CNR) Consiglio Nazionale Ricerche – Istituto Tecnologie Didattiche – Genova - Italy (UDUE) Universitat Duisburg, Essen - Collide Research Group – Duisburg - Germany (UNILON) University of London - Institute of Education – London - UK (DIDIREM) University Paris 7 Denis Diderot - DIDIREM – Paris - France (NKUA-ETL) National Kapodistrian University of Athens - Educational Technology Lab – Athens - Greece (MeTAH) MeTAH and Leibniz - IMAG, Grenoble, France (Siena) Departiment of Mathematics, University of Siena. – Italy

1. Introduction: a structure for describing contexts of work

in an integrated way

In the old days, the research paradigm within the education field was almost exclusively experimental. It seemed as though there was only implicit recognition of the complexity of human learning and the processes for supporting this learning. In the cases where it was recognized that there might be factors influencing some educational process outside the ones under scrutiny, the researcher’s stance was to try to ‘neutralize’ them rather than to include them in the analysis of the data (as for example in control and experimental group situations). In the wake of this initial trend, most of the research was of a ‘diagnostic’ character, aiming to identify learning difficulties and student misconceptions by means of research ‘instruments’ (questionnaires, tests). From as far back as the end of the 70’s (for an inspiring critique of experimental design see Donaldson, 1978), a questioning of this approach led to alternative research paradigms including factors influencing the learning process, such as the situated cognition movement (Gene and Lave, …). Since then, there has been a vast move towards the recognition of the complexity of human thinking and learning, perceiving humans as social beings using cultural

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artifacts for communication, one of which is language and another written expression and technology (Bartolini Bussi, ). The research paradigm has since then evolved containing a large variety of approaches from both psychology and sociology origins including action research and participant observation studies. Analysis of in-vivo realistic learning situations is highly based on discourse analysis enriched by interview and actor profile data. An example of this evolution is the research carried out in the 90’s by Paul Cobb and his group (Cobb, … ) coining the term ‘emergent perspectives’ to signify how analyzing the data through one lens left the group unhappy about the extent to which they could explain teaching and learning phenomena in mathematics classrooms. This resulted in the group going back to the same data and looking at it from different perspectives (in this case, bringing into account the classroom social norms influencing teacher and student behaviors). Within the framework of Kaleidoscope, the exercise of synthesizing the different contexts of the TELMA groups and the way they are treated in the corresponding research publications is meant to provide a first attempt at developing integrated ways of perceiving the various contexts. In research involving the development and use of technology, however, there are specific contextual issues other than the ones usually treated in education research which have to do with the educational process in general. A large number (if not the majority) of technology-based research in education involves design of some educational process, intervention in the everyday life and habits within learning situations and the infusion of some kind of innovation (i.e. change) with respect to practices, tools, learning domain, social milieu e.t.c. Design studies aiming to somehow intervene in normal educational life, inevitably cause some kind of perturbation (to employ a term used by Laborde, 200…). This perturbation is not only at the level of the actual educational process in the classroom. It is also at the socio-systemic level (Jaworski, 2004) of how this intervention is materialized between organizations and in what type of organizational context this collaboration takes place. Finally, some of the TELMA teams are involved in design and development of technological tools. This brings about another contextual complexity which should not be overlooked. The one involving the types and methods of design and development work and how these influence the characteristics of the tools and the nature of intervention in the school, as well as the research questions themselves. Apart from the educational process aspects of context, the other types of aspect (socio-systemic, technology development) are seldom explicitly described in educational research publications. However, as argued by diSessa in a forthcoming special issue in the Interactive Learning Environments Journal, these issue do matter and greatly influence the types of technology emerging from the teams and the kinds of use they are put to in the educational context. Teams of TELMA have been aware that teaching and learning involve complex processes, and that bringing in ICT adds even more complexity

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and of the subsequent need for investigating as many as possible aspects of the context. From a review of literature in the years 1994-1998, Lagrange & al (2003) identified five dimensions (semio-epistemological, cognitive, instrumental, institutional, situational). The first two dimensions were widely considered in the literature, whereas the three latter, related to the educational context, were emerging. They mentioned that, at this time, important dimensions like teacher and technological design, also important for taking the context into account were rarely considered. In the next section four clusters of contextual issues are described briefly since they provide the structure for the subsequent synthesis.

Cluster one: Context of Educational Environments taken into account in data analyses The aspects which were apparent in the TELMA teams’ work were:

• use of language • social aspects of learning – socio-mathematical norms • teacher mediation • communities of practice • process of mathematical reasoning • nature of tasks (structured, game-like, projects etc)

Cluster two: Intervention into a Socio-Systemic Milieu • classroom research • lab situation • school research: How are the socio-systemic factors addressed?

• administration, teachers in daily action • daily program (time, curriculum, method) • society – parents • roles and relationship with researchers • relationships between organizations • existence or absence of institutional mechanism (e.g. part of an institutionalized

pairing of University – school etc) • middle/large scale

Cluster three: teacher education and support • Institutionalized channel or ad-hoc project based • Relations and stakes between researchers and teachers • Frequency, longevity, take-up and how do the teachers use the course

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Cluster four: Technology Design and Development • Institutions • Collaboration with company or other development institution a) one – off collaboration b)

discrete sequence of projects ad-hoc c) longitudinal sustainable collaboration • In-built development – how is it payed for and sustained vis-à-vis persons and know-how

2. Context of Educational Environments taken into account in data analyses

Social Aspects of Learning – Socio-Mathematical Norms In mathematics education, there has been a general trend in the past 10 years or so in perceiving the process of mathematical learning as an essentially social (versus individual) phenomenon. The socio-cultural perspective in mathematics education looks at meaning making as a process of interaction between people and participation in communities and cultures. The TELMA teams essentially analyze mathematical learning through the use of technological tools, most often developed by the teams themselves. The complex agenda of understanding the interactions with the tools themselves and amongst students and teachers has been operationalised in different ways by the teams.

ETL-NKUA The group has been carrying out a series of classroom based research studies. The classroom activities have been perceived as innovative for the actors involve since they consisted of small group project work based on the use of exploratory software. The innovation was not only to do with the use of the tools but mainly with the didactical milieu of such activity and was organized at a time slot which was tangential to the rest of the school program. In the look for the kinds of learning emerging from such activity, the group progressively realized that it was not possible to gain interpretative power unless the interaction and social aspects of the classroom milieu were taken seriously into account. They based their analyses largely on the work by Paul Cobb’s group on emergent interpretative perspectives and socio-mathematical norms. In some cases these norms were the object of study (Kynigos and Theodossopoulou, 2001, Kynigos and Argyris, 2004) while in others, the context within which the interpretations of learning processes of specific conceptual fields took place (Giannoutsou and Kynigos, 2004, Psycharis and Kynigos, 2004).

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ITD-CNR The ITD/CNR group addresses the social nature of learning within the framework of elaborating meanigful practices through which technology can be used effectively. From a theoretical point of view the team has focalised on the social nature of cognition and meaning (Resnick, 1987) and has adopted Salomon’s olistic view of learning environment (Salomon, 1996). According to this view, a learning environment is something that encompasses the whole context within which a technology is being put into use, and cannot be identified merely with the technology itself, as often red in the literature. In building a meaningful educational environment, a software system can have an important role as a tool mediating teaching and learning processes, but it is only one of the components of the whole environment. Not less important are the pedagogic activities in which the use of the tool is integrated, the way in which these activities evolves, the social interactions that take place, and the way in which the work is organised and embedded in the general structure of the school and of the educational institution. As the matter of fact, the mediation offered by a given software to cognition is not sufficient to explain the learning aspects related with motivation, with goals formation and with the attribution of a meaning to the whole activity which goes beyond the meaning of the single actions involved in the performance of a task. The analysis of these aspects requires looking at learning not only as an individual construction developed during the interaction with the computer but also as a social construction developed within the whole learning environment. From a theoretical point of view, the acknowledgment of the crucial role of contextual and social aspects in teaching and learning processes, has increased the interest to theories that highlight the importance of studying the relations among individuals, mediating tools, and the social group. In particular, ITD team refers to Activity Theory and, in particular, to the work of Cole and Engeström (1993), as a framework for analysing the learning environment where ICT tools are integrated. Referring to Activity Theory, ITD team interpret the learning environment as constituted by the enactment, within a cultural context, of a teaching and learning activity oriented to an educational object, involving students, teachers, and tools. The adoption of this theoretical perspective has influenced the way in which the design and the use of the technology-mediated tools for mathematics learning has been carried out by the team. A concept elaborated within the ITD team is that of Situated Multi-Environment Learning Systems (see, Bottino, 2004). Such systems can be defined as characterised by a strict integration of tools for supporting visualisation, re-elaboration of knowledge, communication and collaboration among users. The underlying idea is to support socially situated interaction and investigation, having in mind the classroom and

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not only the individual. This imply considering the design and the implementation of tools able to support not only the relationship of the student with the knowledge to be learnt but also all the relationships that are established between participants during a teaching and learning activity. For example, one of the tools elaborated by ITD team is ARI-LAB-2 (Bottino & Chiappini, 2003) a situated multi-environment system aimed to support the development of arithmetic problem solving abilities with compulsory school pupils. ARI-LAB-2 allows the user to interact with a structured and interconnected set of environments that includes microworlds for visually representing data and solution processes, an environment for building problems solutions, data-bases of solved problems, a communication environment, and a teacher environment. During the problem solving activity, the user, interacting with the communication environment, can establish a connection with other users, and with the teacher, and share messages and problem solutions with them. A variety of interaction modes are supported. The user can choose the partner/s to communicate with at a given time, decide whether or not to read a message or a solution when received, look at them later, display the entire dialogue held with partners, etc. The opportunities given by the communication environment allow to insert the problem solving activity in a social interaction practice which can change students' attitude towards the problem, the validation context in which the resolution process is set, and the way in which assistance can be given to students. For example, the teacher can build examples of problems solutions that can be sent to students for reference. The exchange of solutions, not only with the teacher but also with other students, allow the student to compare them with his/her own thus favouring the enactment of learning by analogy strategies. Moreover, different pedagogical strategies can be orchestrated by the teacher to promote the progressive acquisition of competencies by students. For example, a student can be given responsability for controlling the solution produced by a partner; pair of students can be given symmetrical roles and tasks to develop a joint problem solving activity (e.g. buyer and seller), games involving communication between partners can be proposed, etc. The strict interralation between tools functionalities and meaningful dicactical activities can change the perspective through which looking at learning as a social practice.

SIENNA This team developed and employed the theory of instruments of semiotic mediation with respect to two particular "fields of experience" (Boero et al., 1995), strictly related to the use of a didactic software: the "geometric constructions in the Cabri environment" (Mariotti, 1996; 1998; 2001); and the field of “proving equivalencies of algebraic expressions in the

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L’Algebrista environment”. The study is based on the theoretic framework of Vygotskij, with particular attention to the social construction of knowledge. The main findings on which the proposed research is founded concern the process of semiotic mediation related to some particular cultural artifacts, i.e. software environments or ICT. Did@TIC-MeTAH Since 2000, the objectives of the Did@TIC team have been directed towards the realization of an environment, called Aplusix, that will help students to learn algebra skills and that will be a product widely distributed all over the world. The main features of this environment are: To present traditional tasks to students: (1) formal exercises like expand or factor polynomial expressions, solve equations, inequations or systems of equations, on the one hand; (2) word problems, on the other hand. To let students make their own calculations, thanks to an advanced editor of algebraic expressions, To provide the fundamental feedbacks to students in a training mode: The correctness/incorrectness of their calculation steps The correct/incorrect end of their task. To have a test mode and a self-correction mode. This orientation was original and still is. Computer systems for learning algebra skills are mainly: (1) menu-driven systems that ask students to choose operators in menus and that perform the calculations (Beeson, 1990); (2) systems devoted to a very small domain of algebra, generally first degree equations (Koedinger et al. 1997); (3) systems with prerecorded answers. How can an orientation consisting in implementing traditional activities be original? Probably because the realization is complex, requiring a good two-dimension editor of algebraic expressions, a module for the calculation of the equivalence between expressions and a module for the evaluation of the correct end of an exercise. There were several reasons for this orientation: An epistemic reason: The learning of algebra requires the acquisition of algebraic skills (ref. xx), Didactical and psychological reasons: (1) Skills are mainly acquired through practice, i.e., learning by doing (ref. xx); (2) The awareness of their errors are essential to the students’ learning process (ref. xx), An experimental reason: Menu-driven systems are not very attractive to teachers and students, even when they have proved to favor learning (Beeson, 2002), A good knowledge of the team in the design and the implementation of algorithms for TEL systems in algebra. To be more suitable to teachers, the system includes a wide set of exercises (so teachers have a lot of ready made exercises), and an editor of exercises (so teachers can input their own exercises).

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At the present time, almost all the goals are reached: the program has proved its capacity to help the students; the students and the teachers who have used Aplusix are willing to continue to use it. The remaining goal is the wide distribution: The commercialization is now beginning in several countries. The social aspect in the sense of exchange between several actors has not been developed first by the team, because the first goal was to obtain a meaningful interaction between the student and the computer program at an epistemic level. Algebra is a domain that requires the acquisition of important skills and the team thinks that an individual training is necessary for that, that the students must face their errors and try to correct some of these errors by themselves. However, without having an explicit social goal at the beginning, the result had a social aspect that has been revealed by teachers who said that, when the students were working in the computer room with Aplusix, (1) their role moved from a verification agent who mainly answers to the question “Is it correct?” to an explanation agent who often answers to questions like “Why is it incorrect?”; (2) they devoted more time to help weak students (Nicaud et al., 2002). Recently, a module allowing the teacher to observe the activities of the students, and to have statistics on these activities, has been added to Aplusix. This increases the insertion of the system in the class: on his/her computer, the teacher can see the final form of exercises solved by any student. Later, other features will be added like messages or solved exercises exchanged between the teacher and students or between students.

UDUE UDUEs software framework Cool Modes (“COllaborative Open Learning, MOdelling and DEsigning System”) supports the learning activities of the students which allows co-constructive activities. In shared workspaces, the co-learners can synchronously and jointly explore micro worlds or solve problems using elements of the mathematical toolboxes. Such, cooperation and collaboration is enabled and facilitated through ICT and anchored in the learning objects.

Nature of Tasks (Structured, Game-Like, Projects Etc) An important aspect influencing the ways in which research interpretations are made is the nature of the activities and tasks given to students. The TELMA team activities range from well structured and strictly defined tasks aimed at identifying students reasoning on specific curriculum based concepts to loosely defined exploratory activities aimig tooelicit the generation of meaning in a constructionist or experimental or even playful way.

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DIDIREM As regard the nature of tasks also, the research carried out by DIDIREM is multi-faceted, depending on the technological contexts at stake. Nevertheless, from a theoretical point of view, these different facets share some common points: firstly their common use of the theory of didactical situations (Brousseau, 1997) both for analysing the a-didactic potential of given tasks and for designing tasks; secondly a specific sensitiveness to the possible ecology of the situations at stake in the educational system. In the Lingot project for instance, two different kinds of tasks are designed: the first ones serve the diagnostic aimed at by the software Pépite. In that case, what is essential is to design tasks able to approach the multidimensionality of algebraic competence, and to give access to the supposed student's cognitive coherence. Through a reduced amount of tasks, the tool and object dimensions of algebra have to be expressed, values for the different cognitive criteria introduced in the model have to become accessible, potential coherences have to be identified and tested. The learning tasks, which have been designed in the second phase of the Lingot project, serve another aim. What is then at stake is to think about tasks presenting some kind of genericity as regard algebraic learning (in the terms of the theory of didactical situations, these would be in some sense "fundamental situations"). These are not singular tasks but families of tasks associated with fundamental aspects of algebraic knowledge. These families are described through a general pattern, and parameters, which correspond to didactical variables, associated with the generic pattern. The Cime prototype (Coulange & Grugeon, 2003) illustrates this case. Students are asked to find numerical values allowing the matching between a problem expressed in natural language and a system of equations, one or both these two modes of description of the same situation being only partially given. Among the didactic variables are for instance the number and place of the missing data, the level of congruence between the expression in natural language and the symbolic expression. What is also at stake is the design of the interaction between the student and the problem, with different possible modes of interaction which can be chosen according to what has been revealed by the Pépite diagnostic, and then by the student's behaviour when solving problems of the same family. In the spreadsheet research, what is the main ambition in the task design is to build a progression organizing the joint development of algebraic knowledge and spreadsheet knowledge. This design is supported by a theoretical work inspired by the instrumental approach (Guin & Trouche, 2002). This theoretical work has allowed the researcher to build a hypothetical instrumental genesis, taking into account both its personal and institutional dimensions, and the progression designed aims at supporting a learning trajectory in accordance with the hypothesized genesis. Another facet of this research, nevertheless, is of a different nature, and this time what is at stake is a systematic analysis of the

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spreadsheet resources listed on the web site of the Ministry of Education, in France. The task analysis is once more framed by categories issued from the instrumental approach, and a specific attention is provided to the kind of information given to the teacher, and to the distance with standard tasks and practices. In the survey of French classroom use of CAS, the anthropological and instrumental approaches helped our team to be aware of a difficult viability of classroom tasks that teachers prepared. (Artigue 1997, Lagrange 1996). Because calculation is easier with CAS, innovative teachers tended to offer students more conceptual tasks and situations. These tasks brought students a conceptual load that teachers often underestimated and students could not really tackle them without a great deal of instrumentation. Another problem was that these tasks often appeared to students as very different from the everyday tasks and thus they did not connect them to their mathematical knowledge. This survey was followed by the TI-92 experiment observing and designing lessons for secondary classrooms using this CAS calculator. Our team worked to design tasks in two directions. The first consisted in problems where symbolic calculation “enhanced” the ordinary “by hand” resolution with a reasonable increase of the cognitive load and need for instrumentation. The other direction was to design tasks specific of the use of the calculator also challenging students’ mathematical understanding. For instance, tasks about algebraic equivalence of expressions helped students to know the calculator functionalities for handling expressions as well as to understand different forms of algebraic expressions (Lagrange 2003, Artigue to appear). In this experiment as well as in the Casyopée project, task design and analysis is done in relationship with the curriculum, considering its possible evolution and the potentialities and limitations of technological tools. The design of Casyopée involves an analysis of limitations of tasks that the curriculum offers to solve with existing software (spreadsheet, dynamic geometry, CAS...) A method consists in analyzing a family of tasks and to specify didactical variables in relationship with technology’s potentialities. From this analysis, generic instrumented tasks are designed (Lenne & al. 2003). In the O2U project, we broadly distinguish technical exercise and problems. Technical exercises allow interactivity: UeL propose exercises using technological tools, which could not exist within a traditional tutorial. The presence of such exercises is probably one of the reasons for the students’ motivation. But very often these exercises propose only drill on technical tasks. Due to their technical, even boring aspect, the teachers might not propose some of these exercises in a traditional tutorial. However, the students seem to work on them seriously during a computer session. The exercises prompt a good rereading of the course, which the students might not have done so deeply at home by themselves. For the learning of

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new knowledge, the personal, sometimes repetitive work of each student, at his (her) own pace, certainly favors progress thanks to the quantity of handled exercises. Let us assume that a critical quantity of exercises or examples of applications is necessary to learn a theorem, acquire a technical method, or accumulate a stock of examples, this quantity is certainly more easily attained in such an educational design. However, the effects on learning of these exercises are positive but also known as limited (Robert 2003). Real mathematical problems require self-correction: some of the exercises are real mathematical problems. Thus they require a real problem-solving activity, and a written solution: they must be solved in a traditional way. The student decides the correctness of his answer by comparing it with the solution proposed by the computer. That new task proved difficult, but very interesting, and led to discussions between the students working by pairs. In UeL, particular care is put on the elaboration of the corrections by delivering them not at once but step by step and if possible by urging the student towards an activity.

Did@TIC-MeTAH The tasks studied by the team are the classic structured tasks given to students in their traditional curriculum and word problems. The emphasis is on supporting solution processes through the use of the Aplusix system providing several kind of feedback, in particular on correct (in the algebraic sense) and incorrect actions. However, the Aplusix system allows the students to explore any kind of algebraic task on formal expression, an “algebraic task” being defined as a task for which equivalent expressions provides the same solved forms.

ITD-CNR ITD considers the activity as the unit of analysis for the teaching and learning process. The term activity has to be interpreted within the teoretical framework of the Activity Theory. According to this theory, an activity is a cooperative activity, that is a goal-directed social interaction. The use of activity as unit of analysis allows a reformulation of the relationship between the student and the cultural and social environment in which teaching and learning processes take place. The main objectives of the research worked out by ITD are twofold: studying how the use of advanced technology can affect the reformulation of the relationship between the student and the cultural and social environment; studying how technology can support the teacher in designing meaningful and efficient activities for the teaching and learning process. ITD uses the notion of field of experience (Boero et Al., 1995) as reference to frame the relationship between students and cultural environment.

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A field of experience is a sector of human culture that the teacher and the students can recognize and consider as unitary and homogeneous. From the didactical viewpoint, the field of experience notion allows ITD team to anchor the general reference to concrete situation to the need to design learning activities in homogeneous, unitary cultural fields, ones that are meaningful for both the student and the teacher. The main goal of a pedagogy based on fields of experience is to master systematically the field in which work is being done and to make explicit the mathematical knowledge built within the activities performed in that field. In order to promote the development of a didactical practice based on fields of experience, ITD has designed and experimented several microworlds that have been integrated within the different versions of the ARI-LAB system. Such microworlds model the resources and constraints of real-world and arithmetic fields of experience by means of computational objects. The students can interact with these objects producing effects and receiving feedback that can be interpreted as mathematical phenomena (within the context of the field of experience). Exploiting the action potentialities of these microworld and those of the Communication environment (one of the tools integrated in ARI-LAB), two differet types of tasks had been proposed to the students in classroom experiments. The first type of tasks required that the the individual student solved traditional text-based problems through the use and the mediation of the microworlds. Then the solutions worked out by the students became an object of reflexion for the class by means of an activity of comparison guided by the teacher. Comparison was accomplished both by means of collective discussions with the whole class or within smaller groups, or through computer mediated communications. ITD work has pointed out that this activity of comparison, in a context in which the students share the same tools for developing the solution process, is helpful to promote forms of learning based on imitation and on the reconstruction of strategies performed by others. The second type of tasks experimented during class work had a more cooperative nature. For example, pairs of students had been engaged in the construction of a problem solution exploiting the communication possibilities offered by ARI-LAB. In the development of these tasks the students took on specific roles to meet the requirements of the situation presented to them (i.e. seller and buyer, competitive roles in playing a game, etc.). Students could play these roles because they shared the same tools, and because the communication possibilities embedded in ARI-LAB allowed the development of the social interactions that were necessary to solve them.

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SIENA Τhe team focuses on the construction task taking into account issues concerning the conceptual characteristics of pupil’s thinking while introduced to proof as well as the social interchange in the classroom where different solutions can be reported and compared. “The Construction Task From our considerations of the notion of geometrical construction in relation to the Cabri environment, we formulated the hypotheses that it may serve as a key to accessing the meaning of proof. In other words, we choose the ‘construction task’ as the core of all activities in our teaching experiment aimed to introduce students to proof. Let us analyze the characteristics of the construction task, as it is presented to the students. When students are asked to construct a geometrical figure, two distinct requests are made: 1. a procedure aimed to obtain a specific drawing/figure (for instance a Cabri-figure); 2. a justification of such a procedure, explaining the reasons for its correctness. The two requests correspond to two distinct parts in the expected written answers. In the Cabri environment, the construction activity is integrated with the dragging function, that is, the construction of a figure can be associated with control by dragging. In this case, the necessity of justifying the solution comes from the need of validating one’s own construction, in order to explain why it works and/or foresee that it will function. Of course, dragging the figure may be sufficient to convince one of the correctness of the solution, but at this point the second component of the teaching/learning activities comes into play. Construction problems also become part of a social interchange, where different solutions are reported and compared.” (Μariotti, 2001)

ETL-NKUA This group has objectified the development of educational activity plans (scenarios). Each scenario serves one or more the following purposes:

• a document to be used by teachers/students, • a piece of academic – oriented pedagogical design, • a research instrument, • a document providing developers with information for building functional specifications of

microworlds. In the case where it serves more than one purpose, the group’s usual method for developing scenarios is to initially construct one document and then progressively modify it into more discrete versions. The main thread between all the scenarios developed is the explicit reference to as many contextual aspects as possible, so that the innovative character of the

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scenario is made clear. The scenario constitutes the basis of the method and strategy analysis according to which the team proposes to apply the planned activities in the classroom and of the range of recommended roles for the participants (students, teachers, teacher educators, school administration), but also of the structure of collaboration among different groups (the classroom as a whole, small groups of pupils in the same classroom or in different ones). There is thus reference to the social orchestration, the kinds of use of the microworld expected, the kinds of knowledge expected to emerge referring to concepts, processes of learning and technical knowledge, didactic strategies etc. Some scenarios have been associated with pieces of curriculum more than others (e.g. the ‘parallelogram microworld’, Kynigos 2001), some are quite explicitly game oriented (e.g. the ‘juggler’ microworld, Learning Games project in progress). They all perceive learning as a meaning generation, constructionist process, i.e. involving the development, editing and experimentation with models either of mathematical entities (geometrical figures) or physical objects (a bridge, a map).

UDUE The nature and domain of the stochastic tasks are compliant with the curriculum and the school routine. Structured tasks were used during the didactic sequences to organize students group work. A longer material for self regulated learning in stochastics is in preparation. In other domains students work is organized in projects dealing with non standard problems (lunar heights, orientation in a maze). In both cases tasks are closely related to real world problems.

Process of Mathematical Reasoning The process of mathematical reasoning is of course central to all the groups in one way or another. It is interesting however, to consider the extent to which the interaction of mathematical reasoning with various aspects of the context of learning situations is made explicit. How is mathematical reasoning integrated in learning situations?

DIDIREM The possible links between advanced technology and processes of mathematical reasoning are approached in different ways in the DIDIREM team, depending on the technological context. In the research carried out about professional software such as CAS and spreadsheet, the emphasis is put on the fact that in a technological environment, the development of mathematical knowledge intertwines with the development of technological knowledge, through the so-called process of instrumental genesis. DIDIREM research is thus especially sensitive to the resulting effects on mathematical reasoning processes and

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on their development. For instance, research carried out at high school level has shown that, in algebra, the amazing diversity of symbolic expressions offered by CAS when compared with paper and pencil environments favours the emergence of a reflection about the equivalence of algebraic expressions, and leads to the development of specific reasoning processes about this equivalence. Such mathematical issues are no longer for the student only a matter of didactic contract as is often the case in ordinary classrooms, and their "devolution" to the students becomes easier. Research carried out with younger students (grade 7) entering the world of algebra in the environment of spreadsheets also evidences that the processes they develop for manipulating algebraic objects and solving algebraic problems are strongly influenced by the technology at stake and its characteristics from an instrumental point of view (Haspekian, 2003). For instance, working with spreadsheet favours the development of reasoning processes that have an intermediate status between arithmetic and algebraic reasoning processes: formulas used for representing the relationships between data in the spreadsheet are very close to the algebraic representation of the problem, but they can be obtained by manipulating only cells and numbers; moreover the global resolution process is very close the arithmetic "trial and refinement" process, except that calculations are structured and automated. This leads to see spreadsheet as a technology adding "an algebraic organisation to an arithmetic resolution". Both research carried out about CAS and spreadsheets by DIDIREM nevertheless shows a positive influence of such professional environments on mathematical reasoning processes provided that a specific attention is given to instrumental genesis, which is not so easily achieved today, as the academic institutions generally consider these as something transparent. A motivation of the Casyopée project is that CAS could really help students in their algebraic reasoning, understanding that a same expression can be written in several different but equivalent forms (factored, expanded), choosing the relevant form for a given task, by interpreting forms of expressions in term of properties of functions (sign, variations, zeros…) and developing a “strategic” view in problems. A concern is that professional CAS are designed without considering the mathematical knowledge of the user and its cultural background (mathematical notations, habits…). Regarding their use in secondary schools we can say more precisely that the “instrumental gap” between these applications and the ‘ordinary’ classroom mathematics is very wide. Another drawback of standard CAS is that they put no emphasis on proof. Algebraic transformations are performed, but a student is not helped to build a proof using these transformations. Proof is important mathematical reasoning. We do not see it as a formal activity but rather as a process of

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interweaved exploration in several registers (graphic, numerical, algebraic…), using algebraic theorems, and writing. The Casyopée environment that we are developing should offer CAS functionalities that a secondary student might be able to link with his usual mathematical practice. It should support approaches interweaving exploration and algebraic strategic reasoning, and help the student to organize his work. The Lingot project offers another facet of the way the links between technology and reasoning processes are approached (Delozanne, 2003). In this project, the starting point is a multidimensional model of algebraic competence, resulting from didactical research (Grugeon, 1995). In this model, algebraic competence is approached both in its object and tool dimensions (Douady, 1986). Through the object dimension, what is meant is the relationship the student has developed with the algebraic objects (expressions and formulas, equations, functions...), forming and transforming these, passing from one semiotic representation to another one. Through the tool dimension, what is approached are the functionalities the student give to algebra (expressing generality and patterns, modeling functional situations, solving problems leading to equations and inequations, proving arithmetic properties...). This time technology is used as a tool which allows to get more easily and systematically than it could be the case in a paper-pencil environment a multidimensional image of the relationship that a given student has with algebra, to understand what makes his or her cognitive coherence, and then to think about his or her possible cognitive development in that area. In O2U project, observing students provided useful indication on how students reasoning can be affected by working in a web based learning environment. Abusive use of hints: some students were always asking for hints from the computer even before making any attempt to find a personal solution. Some would need no help to find a solution; and all would have benefited from a personal research preceding the seek for help. This is still a major problem. Fake mathematical activity: the computer gives often an immediate feedback. It is generally useful and students appreciate it much. But it also allows guessing the right answer without slightest idea of mathematics involved. Students can for instance answer at random and proceed by attempt / error for some exercises. The computer modifies the contract: giving a feedback on the student’s answer, it plays a role usually devoted to the teacher. This indicates that, in a traditional teaching at the university in France, student self-validation is not a usual practice. Students expect validation from the teacher. But a teacher would never term “correct” a mere numerical answer, with no justification. The computer does so. Artigue (1988) reports similar observation. This problem can be avoided by modifying the kind of feedback given by the computer, or by choosing to ask for detailed proofs in the associated

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setting. Whatever the solution retained is, the new didactic contract, including the computer, must be explicitly established. So, we want to emphasize a fundamental conflict that the designer of such software must face. He (or she) wants to optimize the computer’s possibilities, especially in terms of interactivity: helps for the student, feedback to his answers…. But the possibilities of abusive use of hints, fake mathematical activity … grows with the interactivity of the software. The designer and the user must thus find a compromise permitting both a real problem-solving activity and a rich environment for the exercises. Did@TIC-MeTAH The team has implemented in the Aplusix system the algebraic reasoning by equivalence of the rewriting rule theory. Solving a formal exercise consists of developing a set of transformations from the given expression, until a solved form is obtained (e.g., a product of prime polynomials for the factoring problem type). A correct transformation is performed by applying a correct rewriting rule R to a sub-expression of the current expression. See Nicaud et al., 2004 for a detailed description of the algebraic reasoning. The Aplusix systems reifies the reasoning by equivalence by placing each step in a rectangle and by drawing an equivalence sign between these rectangles when the calculation is correct (i.e., when the expressions are equivalent). The experiments conducted by the team have proved the didactical interest of this choice. The learning of the mathematical reasoning is placed by the team within learning milieux (Brousseau, 1985) based on the use of Aplusix and are interested in this specific type of student activity and clearly defined domain in the quest for more elaborate and succinct descriptions and representations of the notion of ‘knowledge’, both with respect to students and with respect to designing intelligent computational environments. “We placed this usage in a constructivist approach of learning where students are learning by adaptation to a milieu providing contradictions, difficulties, disturbing situations… (Brousseau, 1997). In this framework, knowledge construction is the result of the interaction of the student with one particular environment. This one should be organized by the teacher with adequate problems, adequate sort of actions available to students in the environment, and adequate sort of feedback provided by the environment. We viewed APLUSIX as a milieu for learning in which the actions concerned mainly the editing of the algebraic expressions, and providing three categories of feedback: feedback about the equivalence of expressions; feedback on the state of the current step provided by indicators; feedback provided by textual messages.”

ETL-NKUA The group aims to gain insight into the nature of mathematical meanings constructed by pupils during their explorations and the ways in which

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meaning generation interacts with the use of the available tools. Meaning making is seen as inextricably linked to interaction with the tools at hand, to the small group social milieu, to that of the classroom and the socio-systemic context and history of the innovative activity taking place. Within this framework, the group have found the ideas of situated abstractions (Nos and Hoyles, 1996), constructionism, (Harel and Papert, 1991) and conceptual fields (Vergnaud, 1987) as particularly useful to think about students reasoning. Looking at students’ construction of meaning made the group keep an open mind about the concepts students employ, the relationships between them and the ways they can be organized and structured in the interpretation process. This is in contrast to taking the structure and concepts of a given curriculum as the starting point for student understandings, pinpointing the group’s tendency to think of the activities as innovative with respect to traditional classroom practice. The mathematical domain has included concepts from geometry, such as figural properties, Thales’ theorem, curvature and angles, concepts from arithmetic-algebra, such as proportional reasoning and units of measurement and concepts from spatial reasoning such as orientation and positioning within cartography tasks. (Kynigos & Keisoglu, Giannoutsou & Kynigos, Psycharis and Kynigos, 2004)

SIENA Τhe analysis of the process of mathematical reasoning is based on the idea of “Field of experience” (Boero et al., 1995) as a system constituted by three types of context: external context; student internal con-text; teacher internal context. “According to Boero et al. (1995, p. 153), the term ‘field of experience’ is used to intend the system of three evolutive components (external context; student internal con-text; teacher internal context), referred to a sector of human culture which the teacher and students can recognise and consider as unitary and homogeneous. The development of the field of experience is realised through the social activities of the class; in particular, verbal interaction is realised in collective activities aimed at a social construction of knowledge: i.e. ‘Mathematical Discussions’, that is polyphony of articulated voices on a mathematical object, that is one of the objects - motives of the teaching – learning activity (Bartolini Bussi, 1996, p. 16).” (Mariotti, 2000) “According to Mariotti, the nature of the Cabri environment, may foster a shift from the practical to the theoretical meaning of geometrical constructions, nevertheless the environment itself is not enough, and the intervention of the teacher becomes determinant. However, some elements of the Cabri environment are presented as key elements for the

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development of a dialectic between the practical and the theoretical level, in fact they can be interpreted as external signs, standing for elements of a geometrical theory: “- the primitive commands and macros, realising the geometrical relationship characterising geometrical figures, are the external signs of the basic elements which constitute the theory; - the dragging function which starts as a perceptual control tool to check the correctness of the construction, then becomes the external sign of the theoretical control.” (Mariotti, 2002) When geometrical activities are concerned, these elements of the software, are viewed as the external signs on which “the evolution of pupils' internal context is based”; such evolution concerns the development of both, the aimed geometrical theory and at the meaning of theory itself. In the teaching experiment, presented by Mariotti, the meaning of geometrical construction emerges from activities of construction, within Cabri, and from related mathematical discussions; through the practice of mathematical discussion, the way pupils make sense of the construction activities within Cabri, is elaborated and developed under the guidance of the teacher.” (Cerulli 2004) ITD-CNR One of the main research interest of ITD is the study of how the use of advanced technologies can affect the process of mathematical reasoning in the students. This study has beeen worked out in the domain of arithmetic problem solving and in the approach to algebra, observing students while solving tasks with the mediation of the ARI-LAB system. The term mediation is used in the sense of Vygotsky who defined semiotic mediation as the use of signs to produce effects on other subjects or on oneself (Vygotsky, 1978). ITD research work pointed out that the main effects of this mediation are the emergence of goals for tackling the problem at hand and the development of suitable action schemes for solving the problem. The mediation offered by the microworlds of ARI-LAB to the process of mathematical reasoning of the students is determined by two complementary factors that help to make sense of the mathematical concepts involved in the solution strategy: mediation provided by reference to a field of experience, and the possibility to express action and solution schemes through conceptual metaphors that are accessible to students and grounded in physical, lived reality (Nunez et Al., 1999). Some studies accomplished by the ITD team shows how classroom practice with microworlds leads to the acquisition of mathematical tools and thinking strategies that are specific to the field of experience and that allow the pupil to think and act coherently with reference to the external world. Reference to the knowledge, the solution patterns and the linguistic expressions of the field of experience is what enables the student to exploit the microworld’s operative possibilities in order to build solution

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strategies for the problems set. The cognitive mechanisms underpinning interaction with the ARI-LAB microworlds are ordinary ones, such as those used for basic spatial relations: groupings, motion, distribution of things in space, basic manipulation of objects, iterated actions, and so on. Drawing on the theory of Embodied Cognition (see Nunez et Al., 1999) ITD research has pointed out that these mechanisms are grounded on image-schemes that are perceptual-conceptual primitives which allow the organisation of experience involving spatial relations. The solution strategies adopted by the students can be seen as visual-conceptual extensions of these image schemas. What allows these conceptual extensions to be produced is the meaning provided by making reference to the field of experience, reified in the physically and bodily grounded operative possibilities of the microworld. This enables the students to control their behaviour during the solution process and defines the nature of the microworld’s mediation in the student’s actions. The appropriation of pertinent solution strategies for problem solution does not derive solely from the result of interaction between pupil and microworlds. It also depends on the pupil’s mastery of the cultural aspects of the field of experience. Where this mastery is lacking, it must be constructed within the social practice enacted in the class. The goal of this practice should be the appropriation of the rules characterising the field of experience and the student’s acceptance of the specific obligations and responsibilities concerning the knowledge involved in problem solving within that field of experience. An important part of the research worked out by the ITD team is addressed to study how the use of ARI-LAB can help in building a learning environment that achieves the aforementioned goals. As far as the appropriation of the rules characterising a field of experience, ITD has documented that the use of the ARI-LAB system can represent an important mediating tool for managing the dialectic between the dual roles that the rules play in a solution strategy: rule as an individual-community mediator defyning what is and what is not acceptable in its application, and rule as an object of learning. The transformation of rules from being individual-community mediators to objects of learning takes place in a network of activities where shifts of focus and breakdowns occur within the system mediation. In some teaching experiments ITD analyzed and traced the actual focus shifts and breakdowns to understand how the ARI-LAB system can contribute to the negotiation and appropriation of the socially shared rules underlying the solution of arithmetic problems (Bottino & Chiappini, 2002). As far as the student’s acceptance of specific obligations and responsibilities in problem solution, ITD team has documented how the ARI-LAB system mediates the construction of a didactical contract and its evolution so as to help students to gradually take on obligations involved in the construction of a solution strategy. For example in a teaching experiment developed for four years with a primary school class, ITD has highlight that the use of ARI-LAB allowed the students to progressively

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build more formal solutions of arithmetic problems. This was done through the following requirements and tasks that evolved during time: to solve the problems presented by working in microworlds and exploiting their posibility of action, copying the meaningful graphic representations thus obtained into the Solution Sheet, then writing appropriate notes to explain them. to convert the solution produced into a discourse expressed in written verbal language. to convert the verbal written solution into arithmetical relations and explain, in verbal language, the meaning of those relations, making reference to the concrete situation at hand (Bottino & Chiappini, 2002). Modelling and experimenting are understood as crucial parts of mathematical reasoning. Following Stachowiak’s general modelling theory (1973), modelling means the process of constructing a model in which the ‘subject’ composes a model in regard to the ‘original’ to serve a specific purpose. So, the process of modelling means setting ‘original’, ‘subject’ and ‘model’ in a specific relation. Modelling is therefore more than reproduction: the whole process is a reflected transformation in which students organise actively their own learning in a creative way. The ‘subject’ decides which attributes and connections out of the context are accepted, emphasised or neglected and how the results are applied to the real world. This process is described in detail in the mathematical modelling cycle (Berry & Houston, 1995). In our context it means putting students in the role of researchers. They solve concrete probability or astronomy problems by modelling experiments, simulate and analyse their outcomes to answer ‘real’ questions. In case of a positive validation they present their solution e.g. in form of a prognosis or a formula to the audience otherwise they can repeat the cycle of scientific experimentation.

Use of Language Actors (students, teachers) talk, while they are engaged in mathematical activity of one form or another, is an important source of information in the research of all the TELMA teams. The teams however, have different ways of perceiving the role of language in mathematical learning and use different theoretical backgrounds to interpret the learning processes within contexts involving mathematical discourse. The IOE group draw from the fields of systemic functional linguistics (Halliday, 1985) to analyze all manifestations of language in the mathematical learning process, from the language used in textbooks to the one developed experientially by the students. The Piza team have made mathematical discussion an object of their research, focusing on the potential for didactical use of student communication strategies. The ETL group have analyzed students’ talk in small group project work focusing on the social aspects of

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communicational intent and how that interacts with the process of learning mathematical meanings.

UNILON The idea of context has an important place within the social semiotic theoretical framework and methodology used by the UNILON team. An important principle for this perspective is the recognition that meaning making occurs in social contexts and that the use of language and other semiotic systems must be understood as functional within those contexts (Halliday, 1978). The nature of context is understood to be broader than the particular episode of activity being studied or the immediate context of the particular classroom, incorporating consideration of the culture outside the classroom. Studying semiotic activity must thus take into account both the immediate situation in which meanings are being exchanged (the context of situation) and the broader culture within which the participants are embedded (the context of culture). These notions originated in the work of the anthropologist Malinowski and are discussed by Halliday & Hasan (1989). The context of situation encompasses the goals of the current activity, the other participants, the tools available and other aspects of the immediate environment. Each situation cannot be considered in isolation but as an example of a situation type or semiotic structure formed out of the socio-semiotic variables: field, tenor and mode. The field of discourse may be thought of not simply as the subject matter but includes the institutional setting of the activity in which a speaker and other participants are engaged. Tenor encompasses the relationships between the participants, and mode refers to the channel of communication (e.g., writing, speech, gesture, graphics). The context of culture includes broader goals, values, history and organising concepts that the participants hold in common. This formulation of context of culture suggests a uniformity of culture both between and within the participants that is not realistic and Morgan (in press) argues for an alternative formulation in terms of drawing on multiple discourses. Importantly, however, the thinking and meaning-making of individuals is not simply set within a social context but actually arises through social involvement in exchanging meanings. This dialectical and dynamic conception of the relationship between the individual and the social is compatible, Hodge and Kress (1988) argue, with the theories of Volosinov and Vygotsky. In relation to the use of technology in mathematics education, there are two strands of interest. First, the study of classroom practices in technologically enhanced environments involves analysis of learners’ and teachers’ interactions with each other and with technological tools. The texts produced in these interactions, including, for example, audio and video recordings of students making dynamic geometry constructions and

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their interactions with their teacher (Mezue), will be analysed using detailed attention to the (choices of) words used by the participants in the classroom, emphasizing the role of language both as a resource, structuring the individual’s participation in the social practice(s), and methodologically as a means by which the researchers construct meanings for the practices they observe. As well as studying texts produced by the participants in a practice, it is important to study the texts available for them to use in so far as these both form part of the immediate context of situation and reflect aspects of the context of culture. (From this perspective, ‘text’ is understood to refer to a socially coherent unit of communication; thus a software package may be considered to be a text.) For example, a software tool may, by the actions it allows to users and the feedback it provides, embody a particular set of values related to mathematics and mathematical activity. Understanding the meanings that learners and teachers may make while using such a tool involves understanding (inter alia) the ways in which these values may relate to the goals of activity in the context of situation and to the values of the mathematical and other discourses available to users as part of the context of culture. The development of ways of describing and analysing the multi-modal and novel semiotic systems involved in technologically enhanced learning environments is less well developed than the analysis of linguistic systems, though Kress & van Leeuwen (2001) propose an approach to this task that is based in social semiotic theory.

SIENA The Pisa team pays attention on the use of language in the context of “Mathematical Discussions” focusing on the cognitive dialectics between personal senses and general meaning. “Mathematical Discussion’ activities. Among the classroom activities, ‘Mathematical discussions’ (Bartolini Bussi, 1991, 1999) take up an essential part in the educational process, with specific aims, which are both cognitive (construction of knowledge) and metacognitive (construction of attitudes towards learning mathematics). Our proposal refers to a specific type of discussion, mathematical discussion, which is not a simple comparison of different points of view, not a simple contrast between arguments. The main characteristic (Bartolini Bussi, 1999) of this kind of discussion is the cognitive dialectics between personal senses (Leont’ev, 1976/1959, p. 244) and general meaning, which is constructed and promoted by the teacher. In this case, the cognitive dialectics takes place between the sense of justification and general meaning of mathematical proof. Different senses of justification correspond to possible different goals of the discussion, whereas moving from one goal

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to another corresponds to the evolution of the sense of justification, which is the main motive of discussion. The role of the teacher is fundamental, in order to direct the goal of the discussion and to guide the evolution of personal senses towards the geometrical meaning of a construction problem, and more generally to the theoretical perspective. This corresponds to two different types of motivation which determine the discussion activity and interrelate and support each other. On the one hand, justifications must be provided within a system of shared principles. On the other hand, the methods of validation must be shared and the rules of mathematical argumentation made explicit. In the following two sections, after a brief general summary of the results coming from the experimental data, we shall discuss the process of semiotic mediation with respect to the evolution of the pupils’ internal context. Some examples will be analysed, drawn from the transcripts of collective discussions and from the written reports provided by the pupils.” Mariotti, 2001

ITD-CNR In an educational perspective the relationship between tools and language is strictly linked with the appropriation of new cultural meanings by the students. For ITD team a tool is always the result of a cultural evolution. It is produced for specific aims and consequently incorporates specific resources. That does not necessarily imply that the tool is the source of the meaning or that, at the didactic level, the meaning can emerge from the interaction between the student and the tool. The meaning resides in the aims for which the tool is used; in the plans which are developed for using it. These plans, expressed in a socially shared language, constitute the meaning of the activity. The socially shared language is gradually built in the course of the activity with the tool, and reflect its development,. When a tool is used within a didactical practice, one always has to deal with a double interpretation of the term meaning: there is a meaning connected to the practical and concrete use of the resources incorporated in the tool in relation to the task at hand, and there is also a meaning related to a rationalization of such use and to its organisation within a theoretical construction. These two types of meaning constrution have to be seen as the result of a social construction within a cooperative activity in which communication play a foundamental role. In this framework studying the relationship between tool and language in the construction of mathematical meaning within an activity means analysing not only the mediation of the tool to the student’s action but

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also the mediation it gives to the communication among partecipants involved in that activity. ITD team research has analysed the role of the tools in the mediation of the following two types of communication: The communication that takes place when the teacher assists students while solving a task. Tharp and Gallimore (1999) pointed out that assistance is best provided through instructional conversation, a dialogue between teacher and learner in which the teacher listens carefully in order to grasp the students’ communicative intent, and tailors the dialogue to foster their emerging understanding. In this dialogue the teacher can help the student according to different modalities: modelling, questioning, instructing, etc. The action opportunities offered by tools and the effects produced in the interaction with them shape the language used in the dialogue and the different form of assistance to student’s performance that characterize it (Bottino & Chiappini, 1998). The communication with the whole class, guided by the teacher, aimed to reflect on the experience developed and to construct meanings that go beyond those involved in task solution. The use of tools in a didactical activity allows partecipants to make a metaphorical use in language of what the interface has exhibited dynamically and concretely in the interaction. The metaphorical use of computational objects, and of the effects visualized in the inteface allows partecipants to speak of abstract mathematical objects, processes and concepts being able to make reference to what the inteface exibits (Chiappini et al., 2003)

UDUE In UDUE usage of ICT is understood to incorporate new digital media as tools for intellectual expression and production. Digital media are used in normal school situations. Students work is not restricted to the manipulation of software tools, but accompanied by supportive “external” talk. Dewey’s notion of media (Dewey, 1934) focuses on their function as means of intellectual and artistic expression, far from the function of representing content! This concept of expressive media subsumes notations as used in mathematics, music and language as well as modern “digital media”. A media-theoretical re-interpretation and adaptation of Dewey’s concept of media has recently been presented by Vogel (2001) and adopted to ICT concerning criteria 5 by Hoppe (Hoppe, et alt. 2002):

“M is a medium if (by definition) M is used intentionally; M is used for expressive (and/or communicative) purposes; M is a genuine constituent of the acts or products generated; The specific use of M is established in the framework of a performative practice (culture), or constituted by social rules;

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M affords representational flexibility through a compositional or combinatorial structuring of expressions. (modified) We claim that interactive digital media constitute new forms of intellectual and artistic expression in the very sense of Dewey’s principles (including our modification).” Therefore designing tools means for the users, and in our case the students, the opportunity to express their thinking and their understanding during the learning process and also while constructing their results. Elements of our expressive media derive from the visual languages (Pinkwart et al, 2001) (e.g. for stochastic experimenting, for computing lunar heights, …) enriched by handwritten annotations in the constructed mathematical models.

Teacher Mediation In mathematics education, there is a trend in the past ten years to focus research on the teacher, either analyzing teachers’ beliefs and practices in the classroom or hw these might be affected in the process of professional development, often within courses organized in the locale of the research teams (see plenary lecture at ICME 2004). The TELMA teams have recognized teachers’ roles in the technology based mathematical environments in different ways. ETL and Pizza have worked with teachers in the design of tools and activities, while ETL has carried out research on teachers’ practices (roles and beliefs-in-practice) in technology based classroom projects. Teachers’ use and appropriation of ICT tools is a main focus in the DIDIREM studies and a consequence is that the team worked with teachers for the design of tools as well as for the study of their implementation.

DIDIREM The group have included their teachers in the research design and analysis of the data. This allows the team to work closely with the teachers with respect to the design and implementation of the courses during the normal mathematics lessons in the classroom. The group also point to the opportunity give to the teachers to gain insight into the ways their students think about the mathematics taught. An important trend in Didirem’s work related to the teacher is the study of the changes brought into the teachers’ work by the introduction of technological tools. New opportunities, but also new constraints, challenge teachers’ consistent and complex systems of practices. In the years 1994-96, a survey of the use of the CAS DERIVE in French secondary classrooms brought a better awareness of this challenge (Lagrange 1996). Recent research about learning environments at university level focuses on specific teachers’ practices much impacted by technology like the helps

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supplied during students’ work. The UeL (University on line) project was initiated by the Ministry of Education in France and realized by a team of 11 universities. Its conception aims to connect didactical reflection and new teaching situation at tertiary level. Now the resource is substantial and the questions are to introduce it in the learning and teaching practice and to track information about its use. More generally the group “observatory of the use”(02U) observes ordinary classes of students working with a web based learning environment. During presential sessions with a web based learning environment, the teacher turns into a counselor and tutor, providing individual advises. In France, during traditional sessions, most teachers make frequent interventions for the whole class. Teachers state that students don’t endeavor to look for solutions, but even for homework, he (she) gives indications or correction of the exercise, thinking that it may be too difficult for the students or simply because of the time he (she) can spare. So, in traditional sessions, students do not really feel in charge of the resolution of the problem. The teacher’s attitude constitutes here an obstacle to the devolution process (Brousseau 1997). On the opposite, during a computer session, students know that there will be few teachers’ interventions. Students do not work on the same exercise at the same time, so teacher’s interventions for the whole class could not be accurate. So the student has to have an active behavior. The resource provides an associated environment (hints, tools for the resolution...) thus students are not left alone in front of the exercises, there exist several real possibilities to act. A new result on the teacher’s attitude stems from our observations: he (she) dedicates a very long time to the solution of the problems proposed. In classical tutorial sessions it is a well-known fact. But we observed that, even in computer sessions, the main part of teacher’s interventions (face to face) are explanations on the correct answer provided by the computer, or questions to ensure that the student has really understood the correct answer. It seems that in computer sessions the teacher has an individual feedback role, while in classical sessions he (she) addresses to the whole class, in order to enhance the students’ activity or to provide the solution for one exercise. Another trend when designing a new environment is to consider the conditions for its appropriation by the teacher. The Lingot project aims to offer the teacher an instrument for students’ diagnosis and uses feedback from teachers’ use of successive versions. Taking into account difficulties that the integration of professional CAS brought to teachers, the Casyopée project considers teachers’ needs as an important base for developing an environment including CAS facilities. Both projects use an iterative design method, confronting successive versions to actual use by teachers.

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These trends implied to work with teachers in actual classroom contexts and to look at the professional development that would be necessary to support new instrumented practices. Did@TIC-MeTAH The Aplusix system has been designed with facilities to be incorporated in the school situations. The “Test Map” of the system is a data basis of exercises. Teachers can use it by asking their students to work on a particular family of exercises. This ready to use feature is particularly appreciated in countries like Brazil where classes are large and teachers have a lot of teaching. Furthermore, two modules are particularly devoted to the teachers: (1) the editor of exercises allowing them to build their own exercises; (2) the observation and statistics module allowing them to observe the students’ behavior. The team also develops researches aiming at modeling students with conceptions which are sets of correct and incorrect knowledge able to solve exercises like a student does. One of the goals of the work is to indicate the students’ conceptions to the teachers in order to help him/her to conduct his/her class. “Several works in mathematics education underline the teacher's role in the design of learning situations. The teacher has to decide the nature of the questions and the moment where he/she has to ask them to the students, the nature of the answers that can be given or not, the problems that can be proposed to the students, etc. For the learning situations, we make the hypothesis that these decisions will be more applicable if the teacher can have a good model of the knowledge state of the learner.”

ETL-NKUA This group has carried out extensive research on two issues involving teachers. Firstly on the nature of teaching practices in the classroom (i.e. roles and beliefs-in-practice) during the infusion of pedadogical innovation with the use of exploratory software for small group project work. Secondly, the group have analyzed how teachers’ beliefs are challenged as the engage in experiential mathematics activities with exploratory software within innovative professional development courses. In the first research, which was extended for a period of four years, the complicated relationship between teachers beliefs and practices has captures the interest of the ETL team. In a correspondent research in order to describe teachers practices they studied teachers verbal and nonverbal communicative behavior in the classroom characterizing their interventions, the roles they constructed and the activities they encouraged. They use particular episodes to highlight a sample of the types of teacher interventions and then used the quantitative picture to discuss the full set of types and the balance of interventions.

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“The study focused on the beliefs and practices these teachers had established while engaged in an innovative mathematical investigations course based on the use of exploratory software. The data itself revealed the complexity of issues, which seemed to play a pertinent role in the forming of these beliefs and practices. Each of the three perspectives with which we analyzed the data (i.e. looking at aspects of teacher interventions in the classroom, at the emerging social roles and at the possible influences of the school and the educational system) provided us with different insights. From the types of comments made by the teachers in the classroom we saw that, although there seemed to be some coherence in the kinds of activities in which they intended their students to be engaged, they referred to different aspects of the learning situations in different ways. Some of these aspects may in fact have diverted teachers’ and students’ attention away for the mathematical ideas in their investigations to issues of work management and collaboration. Furthermore, although the type of intended innovation and the use of exploratory software played a major role in the kind of mathematical activity going on in the classroom, coming up with tangible results in the given time slots was high enough in the teachers’ priorities to influence the types of interventions they made towards the end of sessions.” Kynigos & Argyris, 2004 The second piece of research was based on a study on the ways in which teachers used mathematical exploratory software within an in-service professional development course sets the framework to analyze the kind of perturbations these teachers experienced while integrating getting to grips with the technology and designing courses for their future teacher – students and material for school students. The analysis focuses on the interactions between the teachers and the technology and the emergence of differing roles these teachers attached to the technology in teaching, learning and mathematics.

SIENA The teachers’ role is addressed very seriously by this team, as is apparent by Mariotti’s work on perceving technological tools as instruments of semiotic mediation. The key element of this theory is thus the idea of instrument of semiotic mediation, which refers to a special use of instruments in class practices: the instrument is introduced in the practices on purpose by the teachers, and it is exploited to accomplish communication strategies that aim at developing meanings related to the mathematical contents consistent with the motive of the teaching/learning activity. “In this framework, how theoretical meanings are originated from phenomenological experience depends strictly on how the teacher exploits hybrid signs as pivot, structuring a complex relationship between the considered microworld or artefact, and the mathematical knowledge

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corresponding to her educational objective. Meanings are developed under the guidance, thus under the control, of the teacher who is institutionally in charge of ensuring their consistency with mathematical knowledge”. (Mariotti, 2001) This theory is used by the Pisa team both at the level of developing ICT tools, and employing them in the educational experiment cycle. The framework has been used in order to individuate and exploit the didactical functionalities of Cabri. Furthermore, it has been used in the planning, put in practice and diagnostic phases of an educational experiment cycle aiming at introducing pupils to geometry theory. In particular the framework has been used to individuate special communication strategies to be employed by the teacher. At the same time the framework has been used to develop and experiment with piece of software called ‘L’Algebrista’. In particular it has been used to define the didactical functionalities of the tool, in terms of the educational goal of introducing pupils to algebra as a theory. The tool has been designed in order to favour the employment of particular teacher’s communication strategies that thus influenced the definition of the tool’s didactical functionalities, which in this framework depend strictly on the role played by the teacher.

ITD-CNR An important part of the work of ITD team are the in-depth class experiments performed as an integral part of the process of designing, implementing, and evaluating innovative ICT based systems for mathematics teaching and learning at school level. In the performance of such experiments the teachers of the classes involved were present throughout and participated actively both in planning the educational activities and in following the students’ work. Each experiment was accomplished in a real class situation and during normal class hours not in an “ad hoc” laboratory setting. The experiments carried out by the team focused on the building up and development of mathematical concepts and abilities related to the curriculum through the performance of activities where the use of technological tools was integrated. Such experiments were of medium/long-term (in some cases they last some years). According with Activity Theory framework, in the enactment of the technology-mediated educational activities and in their analysis, attention was focused on the roles played by teachers and students (Bottino & Chiappini, 2002). Such roles defined the system of reciprocal obbligations that mediated the strategy by which community members (teachers and students) interrelated for the social construction of the learning objective. Brousseau (Brousseau, 1986) has shown how the system of reciprocal obligations binding participants in mathematics teaching/learning is regulated by a kind of contract the “didactical contract”.

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The didactical contract defines the set of obligations which, either explicitly or implicitly, determine the area of responsibility to be managed by each participant (student or teacher) within the activity, with respect to the knowledge in question, and for which each will respond to the community. Construction of a suitable didactical contract for the learning of a given knowledge takes place through a dynamic process. ITD work, during the planning and the carrying out of classroom experiments was focused on studying how the employed technology could mediate the contract construction and its evolution. Some features of the ARI-LAB system, such as the possibility to access a communication environment or the possibility to keep track of the work performed, supported the teacher in orchestrating situations where roles different from traditional ones were assumed. For instance, a student, in a carefully planned activity, could assess the work done by other students or could assist other less proficient classmates providing examples and action schemes conducive to problems solutions. This kind of activities, where roles were different by those usually played in classroom, helped students to gradually take on obligations involved in the learning process. Moreover, in ITD work, attention is played to the design and implementation of tools to support the teacher in planning and managing of whole didactical activity. For example, the ARI-LAB-2 system (Bottino & Chiapini, 2003), has been designed to offer the teacher an environment, the Teacher’s Environment, where s/he can write texts of problems, prepare problem solutions, and manage the local network. A number of configurability and personalisation opportunities are offered as well. For example, the teacher can send, through the local network, texts of problem and solutions to the students of her/his classes. It is possible to send them to all the students of a class or only to the ones chosen by the teacher in the class list. Moreover, the teacher can choose the microworlds to be made available for the resolution of a specific problem; he/she can also select the validation tools to be made available during the solution of a specific problem. The teacher can as well impart different problems to different students, and send messages and solutions both to groups of students, to the whole class, or to each individual student. From the teacher environment, through the local network, the teacher can also control the work performed by a student by looking at the problems s/he has solved and insert notes and comments. All these opportunities supports the teacher in carrying out activities that can be more sensitive of the class context (different students’ needs, different roles that students can play, necessity to equilibrate advice to the whole class and to the individual, etc.). The design of tools that can facilitate the teacher in the planning and managing of the class activity is an important aspect to be considered in technology enhanced education, that is ofetn underestimated.

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ITD work is currently oriented in this direction. In particular, the ARI@ITALES system has been developed with the aim of offering a kit of tools (partly based on ARI-LAB-2 environments) to be used by teachers for building components of online courses on curricular mathematics topics. Such tools allow teachers to build and organise interactive and constructive activities and the student to develop them through an usual web browser.

UDUE As our partners und UDUE has found out in SEED using ICT in a way to enrich and enable students’ (group) work leaves less space for a teacher in the role of a lecturer or instructor. This challenge leads to a transformation of his role “in that of a mediator, of a guide, of a motivator, in short, in the position of providing a service to his students. This new role is not less central than the teachers’ traditional role: quite on the contrary as it implies a much wider spectrum of competences, new challenges and a more relevant support and reference role. Furthermore, whenever the learner is attributed a central role in the learning process, the teacher is brought to face the complexity of supporting pupils’ different learning styles and paces” (R. Magli et alt., 2004 Small scale method and Research Instruments Description deliverable of the IST project SEED).

3. Intervention into a Socio-Systemic Milieu As mentioned earlier, most of the research work in which the TELMA teams have been engaged is within the framework of design research. In some sense, a didactical intervention is designed and implemented and the investigation focuses on various aspects of educational practice resulting from this intervention in normal everyday practice. Since most of the work has addressed educational processes within education systems, an important and influential parameter in what goes on is both the researchers and the actors’ socio-systemic contexts. These may address the organizational pragmatics of the University or the lab, the relationship between the researchers’ organization and the educational sites. For instance, when the researchers approach a site, what is their perceived role by the administration of that site and the actors to be involved? What is at stake for the organizations and the individuals? What kind of ‘perturbation’ does the implementation of the research imply and what kinds of permanent changes are the educational institutions prepared to make? These perturbations can involve not only practical issues, like everyday schedules and technology use management, but also much deeper issues like teacher-student roles, social orchestration in the classroom, epistemologies and beliefs about mathematics and the

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educational process. How much personal contact do the researchers have with the actors? These kinds of issues are extremely difficult to tease out and by nature very idiosyncratic for each research setting. However, they do influence greatly the ways on which the technology is used and the kinds of teaching and learning that take place. In this section, some of these issues are elaborated by describing the specifics of the TELMA group’s research sites. The section is organized into subsections which have been selected from a first reading of the groups’ published materials.

DIDIREM Research studies by Didirem all include a kind of participative design. Early research about the use of CAS in French classrooms was done in close connection with a project of the French ministry, elucidating teacher’s expectations and providing a framework to understand classroom phenomenon. A didactical engineering for the use of the TI-92 calculator elaborated from ordinary classroom experimentation with teachers collaborating, was an outcome of a latter project. A year ago, the team started a new study in connection with local authorities’ project to experiment on line resources use by underprivileged area students, aiming at understanding how the educational system could cope with new fast developing practices offered by the Internet. The O2U project also aims to connect didactical reflection and new teaching situations at tertiary level. Did@TIC-MeTAH The research area of the team is the use of the computer to improve the learning of algebra in secondary education. This is decomposed in three projects: (1) Design and implementation of the APLUSIX software; (2) Study of the use of APLUSIX in classes; (3) Student’s modelling. The research is based in classroom activity and the researchers’ focus is on two issues. First on students’ learning processes of manipulating algebraic expressions and second on the ways in which tool functionalities are used with respect to their design to support specific learning activity. The research is of course influenced by the group’s interest and expertise in intelligent systems and the modelling of student learning. The teacher is also in focus and the tool has also been used for diagnosing student skills and misconceptions within this algebraic domain. Experiments with APLUSIX have been conducted in schools to provide indications of user’s interface and robustness of the software. Several test have been made according to the scheme: (1) pre-test without help (on paper or with APLUSIX), (2) training during 1 to 3 hours with APLUSIX with a very little help of the teacher, (3) post-test without help (on paper or with APLUSIX). “A one-year-long usage in a class: integration in a curriculum of APLUSIX as a milieu for learning. A long experiment of APLUSIX has been conducted in 2002-2003 at grade 10, with 33 students of a high school of

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Annemasse in France. At the beginning of the year, before any teaching of algebraic notions, we made a pre-test, using APLUSIX, on different types of algebraic problems already viewed at grade 9. The analysis of the test pointed out some student’s difficulties about algebraic notions learned in grade 8. As a consequence, we organized lessons on the notions of factorisation and equation solving, and activities (2-3 hours on each notion) with APLUSIX during the classroom time. During the rest of the year, the teacher was invited to use APLUSIX every time he thought it was relevant. We observed that he used APLUSIX every time he worked on algebra, especially for inequalities and systems of equations, alternatively with the paper-pencil environment.” (Nicaud et al., 2004) “A regular use in a class of grade 9. In December 2001, 18 students of a class used the system several times a week. It was a class of grade 9 of the middle school of Montfermeil in France. The class was a special class with many students having deep difficulties with mathematics. The students started learning expansions, simplifications and factorisations of simple expressions, and resolution of simple equations with APLUSIX. Some students worked alone, others worked in groups of two. Most of them needed just a few minutes to become familiar with the software, even those who did not have an important experience of the computer. Some of them acquired a good mastery of the drag&drop functionality. The teacher noticed an improvement of the students’ interest for algebra. All the students enjoyed going to the computer room. Some of them, who generally did not listen, began to ask questions. They solved more exercises than usually and more difficult exercises. The teacher noticed also that his relation with the students shifted from the position of judge (who says what is wrong) to the position of interpreter (who explains errors). A test with paper-pencil pre-test and post-test. A test was organised in January 2002, with a small group of 8 volunteers, coming from different classes of the middle school of Montfermeil in France. They worked apart from the normal classroom. The 9th of January, the students had a 30 mn paper-pencil pre-test and a 1h30 session with the system. The 16th, they had a 1h30 session with the system and a 30 mn paper-pencil post-test. The exercises concerned linear equations and inequalities. The verification of the calculations was activated during the sessions and deactivated during the tests. There was no lesson on algebra between the sessions. From the pre-test to the post-test, the average of the group increased from 4.2 out of 10 to 7.9 while the standard deviation decreased from 3.4 to 2.8. Besides these progresses, we noticed, for some students, an evolution in the presentation of the reasoning, see figure 14.

Pre-test Post-test

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Figure 14. The evolution of the presentation of the reasoning for a particular student. During the sessions, the students had the possibility to ask questions to the teacher. Most of the questions were Is it finished? (at this moment, the indicators and the checking of the termination of the exercise were not available) or Why is it not working? The last question generally came from difficulties with fractions and negative integers. The algebraic techniques they used at the end of the second session were what the teacher had expected.” “Teacher’s point of view about the use of APLUSIX. Regularly, during all the year, we discussed with the teacher about the use of APLUSIX in his class. We present, below, the main points evoked during these discussions”. (Nicaud et al., 2004)

ITD-CNR ITD reserch work is characterised by an integrated approach where tools are designed and studied as embedded in educational contexts which are themselves objects of the research. The design and implementation of technology-based tools is informed and evolve as a result of experiments focused on the integration of the tools in the work of actual classes. For example, the ARI-LAB system had been tested in different school situations and with different kinds of students (different school levels, normal students, deaf students, students who are considered as low achievers in mathematics). All the experiments were developed on the long term. With one class, the system was used for almost the whole cycle of primary school (from the second to the fifth grade), with significant integration into the mathematics curriculum. During experiments one computer was available for each student. The experiments were carried out by ITD researchers in cooperation with the class teachers. As the matter of fact, during all experiments, the teachers of the classes involved were present throughout and participated actively both in planning the teaching itineraries and in following the students’ work on the computer. Each experiment was accomplished in a real class situation and during normal class hours not in an “ad hoc” laboratory setting. Evaluation of the experiments had been based on analysis of the written observation protocols taken by researchers during the work with the

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different classes, analysis of the problem solutions devised by the students and saved in the system, analysis of the written dialogues held via the communication environment, and of the exchanged solutions. The recently developed system ARI@ITALES was used by ITD team in two class experiments to test an innovative approach to e-learning. Two on-line courses had been designed by means of the ARI@ITALES tools. Such courses had been tested in two different class situations. One course was devoted to the development of problem solving abilities with primary school pupils. The other was oriented to the introduction of rational numbers properties with secondary school students. The two courses integrate the constructive approach based on the exploration of mathematical properties and on problem solving with the approach based on learning objects (on-line structured courses). The content of the courses had been developed with the class teachers and tested during normal class hours. The final aim of this work was to test if a methodology of the kind proposed could be a useful support for teachers in planning and managing meaningful contexts of use for technology enhanced educational activities in mathematics. The planning and managing of the educational activity is to be considered as an ongoing process where the conceptions and the experience of the individual teacher should be related with those of other teachers, with the results in mathematics education research, with the analysis of experimental data, etc., keeping attention to the constaints and peculiar characteristics of the situation in which each teacher works. One of the research objectives currently under examination by ITD team will be oriented to the analysis of how to support the raising and the establishment of learning communities of teachers where the building of learning environments can result from cooperation, comparison, and sharing of pedagogical experiences among teachers.

ETL-NKUA The group have carried out three types of intervention design. The first involved the level of discrete school organizations. The intervention was at school level, engaging the administration in gradually infusing a slot for alternative kinds of teaching in the daily program. This slot was called the ‘investigation’ hour and typically involved the normal teacher supporting small group project work with exploratory software largely based on the generic authoring system ‘e-slate’. The research group provided systematic teacher education courses to these teachers and participated in teaching on occasion, especially through a series of research projects. The school perceived the group as consultants for implementing an innovative program, the importance of which was of course augmented by the fact that it was based on the use of technology. The group perceived collaboration with a school as a long term investment, beyond the implementation of one piece of research and have included the

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organization in R&D project consortia or alternatively invited specific teachers to collaborate in these projects, either implementing a piece of software in the classroom, participating in the development of a scenario or providing teacher education courses to other schools. The second kind of intervention in which the group participated was the design and implementation of a middle-scale project involving the infusion of technology use in around 10% of the secondary schools in the country (project ‘Odysseia’). This project was managed by the organization called R.A.C.T.I. which is affiliated to the ministry of education and the group played a consultancy role in its design and implementation. It was also involved in a project to train teacher educators, a project to implement robotic technologies in two school sites and a project to design microworlds for investigative activity. This participation of the group was normal since there were more than 100 projects carried out within the framework of ‘Odysseia’. The teacher trainer courses were addressed to teachers selected by the ministry of education, who followed a one year full time course and where then seconded to carry out school based education to five schools neighboring their own. The third type of intervention was the generation and support of a community of practice, the synthesis and activities of which were part of the research design. With respect to the former, the community was based on the institutional distribution of its members and the complementary nature of their expertise (InDiCE community). Their activities involled the conception, design, documentation, implementation and evaluation of innovative educational activities based on prototypical e-slate microworlds. The members were teachers, teacher educators, developers and educational administrators. The domains were not primarily selected to conform to a given curriculum and they consisted of history, language, programming, science, geography and mathematics. The researchers were interested in what could be perceived as involving mathematical reasoning within all of these activities and microworlds (project SEED, 2001-2004).

SIENA Τhe team adopts a “teaching experiment” approach. “A TEACHING EXPERIMENT The research project described in this paper began some years ago. It was conducted in the framework of a long-term teaching experiment, located in the ‘research for innovation’ paradigm, in which action in the classroom is both a means and a result of the evolution of research analysis (Bartolini Bussi, 1994, 1996). One of the main objectives was to investigate the feasibility of a teaching approach centered on the use of the microworld Cabri-Géomètre and aimed at developing theoretical thinking in geometry. Our hypothesis was that the teaching/learning process associated with this development can be expected to be gradual,

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thus the experiment involved following students through two years of study, corresponding to the 9th and 10th grades of schooling. The research study was carried out through a strict collaboration between researchers and school teachers and the teaching experiment set up on the basis of a sequence of activities designed by the whole group. These activities were realized in the classroom by the teacher as a regular development of mathematics class, i.e. the experiment was included in the regular curriculum. The content of the geometry curriculum was not upset, but the general approach changed dramatically. The approach developed involved the integration of the software Cabri-Géomètre into classroom activity, not only as a didactic support, but as an essential part of the teaching/learning process. Three regular classes at the upper secondary school level, and from different schools, participated in the project. At the end of the first two years, a group of teachers decided to continue following the project, which of course has also developed in other directions. The analysis is carried out on the complex of information collected: notes from direct observation, pupils’ protocols and transcripts of collective discussions.”

UDUE classroom research

Most of UDUEs software is developed for education. Especially the math tools are dedicated to school practice. The adequate field for testing, redesign and evaluation is the classroom. Engaging the teacher as a co-designer reduces the trouble students may have with the testing situation. We embed our technology: an interactive white board is used in the usual way offering additional functions (archive, dynamic simulation, …), digital input devices (graphic tablets) support the students during their work with the software tools, computers are embedded in special tables no interfering students communication. This concept succeeds in a smooth transmission from standard to digital media, using the computer and the software as tools, as expressive media.

• lab situation • school research: How are the socio-systemic factors addressed?

a) Level of intervention is low. UDUE uses existing free space or designs for domains compliant with curricula. The emphasize of cooperative and collaborative work (facilitated by the computer) is also covered by German curricula, which set a high value on such work methods.

• administration, teachers in daily action • daily program (time, curriculum, method) • society – parents • roles and relationship with researchers

In UDE we observed that complementary expertise, that of teachers, developers and researchers, forms the basis to design and explore artefacts in form of specified software tools. Our way of action research put the teacher in the crucial role: he brings the ideas, “instructs” the developer and plans the actions in school, supported by the researcher during the design process and esp. in evaluation questions. teache

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• relationships between organizations During the last two years, UDUE has developed a strategy which could be called “docking strategy”. Already existing local communities of teachers or groups of mediators which got in touch with UDUE have been asked for further cooperation. With this strategy, e.g. the contact with ten secondary middle schools has been build up. To mention is one important restriction concerning these contacts. Most of them are communities of interest for their own goals. Since they are restricted by time and money it can be observed that they are not interested in building up further, virtual, communities for a wider field of activity. Thus it is possible to bring such communities in contact with each other for some exchange but this action requires a lot of energy, permanence and time and sustainability is not guaranteed.

• existence or absence of institutional mechanism (e.g. part of an institutionalized pairing of University – school etc)

• middle/large scale

4. Teacher Education and Support Although teachers are often taken into account explicitly in the research of the TELMA group teams, there is very scarce information about contextual issues involving the relationship between the teachers and the group, as is normal in academic publications. Especially with respect to the first two of the items below, we need additional information. Institutionalized channel or ad-hoc project based Relations and stakes between researchers and teachers Frequency, longevity, take-up and how do the teachers use the course The presence of the computer and of particular software certainly represents a perturbation element in the context of the classroom. The teacher has to elaborate a new relationship to mathematical knowledge, augmented by the whole set of relations which link it to the use of technology in general and a specific software in particular. At the same time the teacher has to adapt his/her role of mediator taking into account the new elements offered by the software. All of these issues involve not only teachers’ time and energy but also some kind of perception of the teacher profession as a developing one and of the engagement in professional development activity as a normal part of the teachers’ job

researchdevelope

complementary

expertise

initializing kernel

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(for en extended discussion on the idea of inquiry as a state of mind for the teaching profession look at Jaworski, PME 2004). There are large differences of teachers’ professional contexts amongst educational systems and also within educational systems amongst schools and individual teachers. Furthermore, the ways in which the intervening researchers are perceived (their official ‘capacity’, as well as their actual contribution to the teachers’ work) highly influences the ways in which the technology will be used. There is a lot of mathematics education literature on the relationships between researchers and teachers and the idea of teachers carrying out research for themselves (look at PME over the years apart from research reports, other thematic activities around the notion of teacher – researcher, e.g. Breen, ) . Frequency, longevity, take-up and how do the teachers use the course.

DIDIREM ETL-NKUA We mentioned above that Didirem’s work related to the teacher implied to look at the professional development that would be necessary to support new instrumented practices. For instance, the CAS and TI-92 projects produced pilot situations for teacher training in the use of technology. Software design and associated didactical reflection can also be as basis for sessions of teacher development beyond technology. For instance, Pépite has been used in pre-service training sessions and by experimented teachers in professional development programs about algebra and about student knowledge evaluation. Did@TIC-MeTAH The Aplusix system has been realized by the Did@TIC team to be widely distributed and used. The distribution uses classical publishers for the school version. There is also a home version which is distributed as a shareware. Until recently, the experiments of the Aplusix system were made in several frameworks: an official framework with 4 teachers funded by INRP (National Institute for Pedgogical Research); these teachers use the system all along the school year; punctual experiments in several schools (in France and Brazil) driven by researchers of the project; local experiments conducted by students of researchers in mathematics education who do not belong to the project (in Italy and Canada); autonomous experiments of teachers who downloaded the system. “Secondary school teachers are involved in the study of the use of APLUSIX, sometimes by performing experiments prepared in the laboratory, sometimes by using APLUSIX the way they feel.” (Research team description)

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The use of Applusix was implemented in schools for two years during the normal algebra curriculum courses. Each student worked on their own with a computer. The focus of the pedagogical design is on the student and their interaction with the software. «From the end of 2001 to the end of 2003, we have regularly organised tests of the system with teachers of middle schools and high schools, mainly for grades 9, 10 and 11. The first goal was to test the software in order to see its utility and its usability. The second goal was to collect data for a project aiming at modelling students in algebra with mal-rules (Payne & Squibb, 1990) and conception (Balacheff & Gaudin, 2002). This work is going on and will not be presented in this paper. This goal led us to organise experiments from September 2002 to December 2003 that participate to the evaluation of the usability. Apart form a few special cases, all the tests of APLUSIX were made by the teachers of the classes during the regular school time. Most of the time, there was one student per computer in computer rooms having from 12 to 30 computers. After these tests, the usability of the software is obvious, at least for the basic functions (input, delete, make new steps) and classes of 30 students were able to have a correct activity from the beginning with very little help of the teacher. However, most of the students did not naturally use the cut, copy, paste and drag&drop functions. The use of these functions needed to be suggested. The difficulties faced by the student were mainly mathematical difficulties.» “A one-year-long usage in a class: integration in a curriculum of APLUSIX as a milieu for learning. A long experiment of APLUSIX has been conducted in 2002-2003 at grade 10, with 33 students of a high school of Annemasse in France. At the beginning of the year, before any teaching of algebraic notions, we made a pre-test, using APLUSIX, on different types of algebraic problems already viewed at grade 9. The analysis of the test pointed out some student’s difficulties about algebraic notions learned in grade 8. As a consequence, we organized lessons on the notions of factorisation and equation solving, and activities (2-3 hours on each notion) with APLUSIX during the classroom time. During the rest of the year, the teacher was invited to use APLUSIX every time he thought it was relevant. We observed that he used APLUSIX every time he worked on algebra, especially for inequalities and systems of equations, alternatively with the paper-pencil environment.” (Nicaud et al., 2004) A regular use in a class of grade 9. In December 2001, 18 students of a class used the system several times a week. It was a class of grade 9 of the middle school of Montfermeil in France. The class was a special class with many students having deep difficulties with mathematics. The students started learning expansions, simplifications and factorisations of simple expressions, and resolution of simple equations with APLUSIX. Some students worked alone, others worked in groups of two. Most of them

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needed just a few minutes to become familiar with the software, even those who did not have an important experience of the computer. Some of them acquired a good mastery of the drag&drop functionality. The teacher noticed an improvement of the students’ interest for algebra. All the students enjoyed going to the computer room. Some of them, who generally did not listen, began to ask questions. They solved more exercises than usually and more difficult exercises. The teacher noticed also that his relation with the students shifted from the position of judge (who says what is wrong) to the position of interpreter (who explains errors). A test with paper-pencil pre-test and post-test.” In the case of the above two groups, in some cases the appreciation of the teachers’ support is related to the diagnosing facilities provided to the teacher. Example: Didirem team. “PepiProf supports two teachers’ tasks: completing and studying the coding of student’s answers and scrutinizing the student’s profiles. The interface supporting the coding task is adapted and used by every category of users: �It gives teachers a framework to interpret the students’ answers. �They understand the diagnosis items when they see them in the context of students’ answers. It is particularly appreciated by pre-service teachers and by teachers’ trainers because it gives an entry to understand students difficulties in algebra.” Delozanne et al. In other cases, the appreciation of the teachers’ role relates to the multiplicity of classroom interaction modes. The ITD group is an example.

ITD-CNR ITD team has been involved in many training and teacher education projects which have been mainly centred on technology enhanced learning in mathematics. Aim of the work is the development of models and methods for teachers' education and training, based on information and communication technologies (ICT) (Bottino & Chiappini, 1998). Problems concerned with teachers' education are strictly linked with those related with pedagogical innovation in school and with the transfer of research results. These subjects are topical since, as literature pointed out, it is very difficult to pass from research projects, which are necesserely restricted to limited and controlled contexts, to school practice on a larger scale. This is even

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more true as far as innovative projects based on the use of advanced technology are concerned. ITD work in the area of teacher education has been supported by CNR strategic and special projects, projects with schools, cooperation with school regional institutions, and with the Ministry of Education. The work has been often related with the software development projects developed by ITD. For example, as far as the ARI-LAB, the full accomplishment of the project foreseen the design and development of innovative teachers' education activities, based on classroom experimentations and collaborative work. These activities are accomplished also through distance learning actions. The work performed by ITD considers the formative process as a complex activity determined by the persons who participated in it, by its content, by the mediating tools used, and by the context in which it is developed (Bottino & Arscone, 2000). A formative and training model has been developed by the team. Such model considers the acquisition of new competencies as related with their actual use in school practice. In this model the role of the trainer changes together with that of the teachers. The trainer has to guide teachers towards the autonomy in the use of the acquired knowledge and tools and towards their sharing with other teachers. This model has brought to a set of complementary activities perfromed by the group: • Widespread of products, results and methodologies concerned with the innovative use of ICT which can be meaningful for the enhancement of teaching and learning processes. Particular reference was made to the ARI-LAB project and to the innovative perspectives in mathematics education it entails at compulsory school level. • Development of ICT mediated training activities for teachers based on classroom experimentation and collaborative work. • Realization of informative structures for teachers and trainers which allow to widespread experiences, information, didactical materials useful for class work and professional training. • Study and implementation of methods and models for the development of communication and collaborative work among teachers with particular reference to distance education activities. In the research work carried out by ITD on teacher education and support, ICT plays a double role: they are the object of the formative process and they are the instrument which mediates the possibility to accomplish new forms of education and training. ICT makes it possible to set up information and communication structures which can be defined as a “hypercontext”, i.e. a mechanism allowing various forms of interaction between research, training and teaching practice. This hypercontext makes it possible to set up training methods of a new type compared to those which come into being in traditional training experiences.

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Last but not least, a significant part of ITD team work is played by studies on the national policies and initiatives in promoting the use of ICT in school education, and of their real impact on schools and teachers development (Bottino, 2003; Bottino & Furinghetti, 1998, 1999).

ETL-NKUA The group have carried out research on teacher beliefs in practice and emerging roles in classroom of the school based research described above. This was done in the context of the support provided to these teachers by the research group and after agreement with the teachers to participate by allowing the researchers to be present during classroom practice. In fact, the teachers were willing to allow very detailed data to be collected. Everything they did and said was recorded and analyzed and they took part in semi-structured intensive interviews. The second kind of research has been on processes of professional development courses based on the use of exploratory software providing insights into the ways teachers’ activities during the course influenced and interacted with their knowings and their beliefs about mathematics, technology and pedagogy. The teachers taking part were acting in their capacity as trainee in service teachers participating in an innovation led by their ministry (i.e. they were not there as individuals selecting to get a postgraduate qualification). One of the characteristics of the innovation was precisely their role as school based teacher educators. This role had not been given to teachers before at least officially. The third kind of research involved the teachers in our In.Di.C.E. community of practice, where they had the role to ‘think out of the box’ and come up with activity plans and microworlds constituting innovative educational approaches.

UDUE • Institutionalized channel or ad-hoc project based

In SEED teacher education and support was part of the project. Besides open teacher workshops a teacher community was formed with regular meetings on voluntary basis but without obligations or commitments on both sides. Teachers were trained to use existing software tools and get further support preparing and executing lesson series. In case of own ideas they form an “inner circle” with researchers and developers to design and test new tools. Now support and training of teachers takes place ad hoc in the context of a special, project based activity.

• Relations and stakes between researchers and teachers complementary expertise of teachers, developers and researchers, forms the basis to design and explore tools in classroom situations. the teacher in the centre of the activity: he initiates the topic, designs together with the developer and enriches his school practice. The researcher supports

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him during the design process and observes the process and esp. the field test for his evaluation and research work.

• Frequency, longevity, take-up and how do the teachers use the course Spreading of tools and ideas takes its time. Today usage of our tools is regular, but low level. We have continuous contact to a small group of teachers, who have integrated the usage of the tools in their way of teaching, having already changed their mindsets.

5. Technology Design and Development Almost all of the TELMA teams have been or are engaged in the development of tools for mathematical learning. In mathematics education research at least, the contextual issues emerging from keeping this kind of know-how in an institution/team carrying out educational research have been greatly underrated if not totally overseen. However, developers come from a field of expertise, computer science, which has its own research and development agendas, its own epistemology, time-frames, meanings for epistemic rigor and methods. Most teams include actors from education, social sciences, cognitive sciences, computer science. Genuine communication amongst these groups is necessary for anything fruitful to emerge in the way of products or education processes. How the groups go about it in highly dependent on the context within which each group is set and the methods for supporting this kind of communication, which takes time, often far beyond single projects. One thing that happens is the mergence of hybrid expertise and hybrid actors, i.e. people who come to know enough about alien expertise to be able to deeply incorporate the others’ agendas in their own research or development process. In this process, there are of course problems and obstacles (for an extensive discussion, see Kynigos, 2002 about generating cultures for microworld development and Kynigos, 2002, about generating and Institutionally Distributed Complementary Expertise Community (In.Di.C.E.)). In a study aiming to analytically compare four groups engaged in integrated development and educational research, diSessa (in press, Interactive Learning Environments) points out three contextual issues which seemed to have prominence in the methods and effectiveness of collaboration amongst the different kinds of actors. The issue of divergent views, of social hierarchy and of community-specific practices. ‘Divergent views: Different people, especially if they hold different kinds of expertise, may have different values, priorities and ‘‘lenses’’ to view ‘‘how things should be done,’’ and, particularly, what constitutes high quality and the likelihood of achieving that quality in a particular case. Social hierarchy: Whether institutionalized or not, differential power and authority may develop in collaborations. Technologists tend to have high

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status (or, in a self-fulfilling manner, assume they have high status) compared to educators, especially teachers. Degrees, salaries, assumptions about articulate argument, and so on, can systematically favor some participants over others. Community-specific practices: Participants in collaborations usually retain a main affiliation with their ‘‘home’’ community. Patterns of practice, rewards and sanctions, and so on, all differ, which can cause subtle or explicit misalignments. For example, technologists gain approval in their community for designing and making systems; long periods of waiting for feedback from educational field trials, may cause at least tension, if not disruption.’ (diSessa, 2004, in press) diSessa, then moves on to identify four distinct models of collaboration amongst technologists and educators, the integration teams model where somall integrated groups engage in co-development for the start, the two-legged model where two big teams residing in different institutions collaborate loosely between them, the member sustained model where everybody does their own thing and they collectively gather their work on a visible online repository-forum (see Educational Object Economy) and the ‘Layered Distributed Development of Educational Resources – LaDDER) model where all actors work in one of a number of layers each characterized by a certain degree of expertise in technology and education. It is very hard at this phase to have information like that from the TELMA team publications of course, so hopefully as this deliverable progresses, synthetic paragraphs may become available. The TELMA teams are both developers and users of educational mathematical tools and users of professional ICT tools like Spreadsheets and C.A.S. Some of the TELMA teams researches focus on the use of ICT in educational practices while others they consider the whole lifecycle of the tools, from the design to the actual use in classroom situations. In the case of professional ICT, TELMA teams have been focusing only on the educational use of the software, but not in their development. The Did@TIC team developed and experimented the educational ICT Aplusix, while the I.T.D. team developed and experimented ARI-LAB-2. The Pisa team on the one hand developed and experimented L’Algebrista, and on the other hand experimented the educational ICT Cabri Géomètre (which was not developed by such team). The ETL team is involved in what diSessa called the two legged model of work collaborating with a group of technologists at the Computer Technology Institute. The ETL group co-designed the E-slate authoring system and developed a number of microworlds with it, engaging teachers and students in the development. The DIDIREM team focuses on the educational uses of professional ICT such as Computer Algebra Systems (CAS) and developed a diagnostic system called Pépite/Lingot.. The Knowledge Lab group have

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carried out educational research mainly using Cabri. A common point in the technology development methods is that most of the teams are developing tools (in a sequence of projects) in close relation to its actual use in the classroom. This poses another set of specific contextual issues like the schools tolerance for using software in-the-making, which is often buggy and which changes in interface and functionalities. Also, schools may feel an unwanted dependency on the academic teams since their agenda for using technology involves tools that may not be always available and also may be hard to find and sustain.

DIDIREM The Didirem team tries to consider together curriculum, classroom practices and software design by particularising to situations involving experimental situations for the learning of functions. More precisely, they are looking for ‘sustainable’ evolutions of curriculum and practices helped by the design and experimentation of a new environment dedicated to these situations. Casyopee as well as Lingot project involve collaboration between different teams from different disciplines. Development relying on a participative and iterative perspective, where potential users strongly intertwine. Τhe design is connected with curricular purposes. A first version has been experimented to classrooms before delivery. The Didirem team collaborates with other communities working in the field of educational use of technology. Tight collaboration exists with psychologists in ergonomics, about the themes of instrumentation and of the teacher activity (Robert & Rogalski 2003). The team also collaborates with computer scientists involved in the Artificial Intelligence and Education community. With regards to diSessa (ibid)’s four models, the projects mentioned above mostly work in integrated teams, whereas collaboration in a local federation of research groups belonging to several universities in Paris should help to move towards what diSessa calls the LaDDER model. The objectives of this federation are: to share a common multidisciplinary reflection about educational use of technology, to support team work on projects or key questions, to develop collaboration in doctoral studies. The above-mentioned projects are a topic for collaboration in this federation and the key questions are: modeling the learner pedagogical models’ design dynamic adaptation of a computer environment to the learner description and indexation of pedagogical objects design of tools for authors integration of active components conception, validation and analysis of instrumented didactical situations

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Did@TIC-MeTAH The Did@TIC team works on TEL for algebra. It is a part of the Leibniz laboratory in the IMAG federation of laboratories of Grenoble which is devoted to applied mathematics and computer science. The Did@TIC team is also a part of MeTAH translaboratory team of Grenoble devoted to TEL. The team includes researchers in computer science and in mathematics education. APLUSIX is a pluridisciplinary project concerning the learning of algebra containing functionalities usually found in microworlds and in CAS. This project has several objectives. One is building TEL systems to be used, the others are building models and prototypes to test ideas, improving our knowledge of sub-domains of algebra, studying human learning in algebra with the help of new technologies. However, the team had, from the beginning, that building TEL systems to be used objective. Currently, the APLUSIX system has been experimented during two years and begins to be distributed as a commercial product. The design methodology used to develop the APLUSIX system consists of searching ideas of meaningful interactions between a student and a system and implementing them, when it is possible, either by using existing tools or by developing new ones. The ideas come mainly from an epistemological analysis of the considered domain as well as from pedagogical issues (Nicaud et al., 2004). The current version of APLUSIX required an important amount of development. This development has been done by researchers in computer science. An engineer has been recently attributed to the team for two years. In the future, the team will search a framework for having 3 or 4 engineers to have the development realized by engineers instead of researchers.

ITD-CNR The approach to design and developmend of ICT-based educational tools performed by ITD team is based on two main processes: product development and process development (Bottino et Al., 2003). As far as product development, ITD team has adopted a multi-tools approach. One of the objectives has been offering the possibility to configure tools in different ways for different users. Tools developed include: graphical microworlds based on direct manipulation interfaces, communication environments, environments for building problem solutions, data-bases, problem text editors, environments for building simulations, environments to support the teacher in the management of the pedagogical activity. Such tools have been implemented to be used both on local networks and on the internet through a normal web browser. The product development has been carried out with computer science technicians that have developed the source code on the basis of detailed the specifications produced by ITD researchers. The work has been an

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iterative one. As the matter of fact, software components have been evaluated from the beginning of their development, according to RAD (Rapid Application Development) approach, in order to arrive quite quickly to the first prototypes. As far as process development, prototypes, during their development, have been tested with end users, teachers and students, in class experiments. The tools have been evaluated both technically and pedagogically. Actors involved in the evaluation have been: ITD researchers, mathematics teachers that work part-time with ITD team, and computer science developers. The evaluation has been performed by means of an in-depth analysis of the developments. Evaluation has involved tools use in real class situations. On the basis of these evaluations new implementations of some functions have been performed. Teachers have analysed the system from a pedagogical point of view trying to outline its possible applications, opportunities it offers and possible problems. They have actively worked with ITD researchers to design educational activities in which the use of the tools have been integrated in class work (Bottino & Al., 2004).

UDUE • Institutions

UDE has been developing collaborative learning environments on its own for almost a decade now. Usually available state-of-the-art technology is adopted and integrated in our collaborative platforms. Because computer science and mathematics were closely related organisationally at UDE and also from the scientific tradition, at all times there has been a strong influence in both directions. A lot of members of the research group have roots in mathematics or even mathematics teaching, so applications of our collaborative learning scenarios are naturally drawn to mathematical topics also. Mathematics teachers are usually among the teachers in our partners schools most interested in conducting lessons using our group's tool.

• Collaboration with company or other development institution a) one – off collaboration b) discrete sequence of projects ad-hoc c) longitudinal sustainable collaboration

UDE usually develops tools on its own, but we had some collaborations with other institutions addressing the integration of different platforms' features and potential. This includes a technical integration for interoperability of microworlds with the eSlate platform during the SEED project and recently the integration of our collaborative platform with a cognitive tutor system to generate and apply expert knowledge in specific domains to the support of learning processes (McLaren et al. 2004). One of the domains of interest there is algebra, which is a topic well elaborated on on our partner's side with the AlgebraTutor (Koedinger et al. 1997).

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In general according to diSessa's classification, UDE takes up the approach of an integration team, utilizing the divergent and multiple views of its members, which are mathematicians, mathematics teachers, software engineers, and developers. The development of math-specific tools is driven either by our internal experts or by applying partipatory design principles with taechers from our partner schools. Indeed community-specific practices can matter in our team, since the affiliation with the “original discipline” influences the position with respect to the development: the incremental refinement of our tools is valuable especially for the practitioner, i.e. the teacher using the tool, but not so much for the technologists, because the small increments are not suited to gain approval in their community (this is especially true for a developer, for a software engineer the process of incremental development itself may be a target of additional research).

ETL-NKUA This group co-designed the e-slate authoring system claiming responsibility for its authoring features, its scriptability and the design of a series of components related to mathematical tools. Furthermore, they engaged in secondary development (authoring) of a large number of microworlds (tools for expression, construction, experimentation and investigation in collaborative settings). They perceived the activity of developing such microworlds as part of the education process both with respect to students and with respect to the professional development of teachers. This development went on in five types of context: researchers and research students developing microworlds to be used as instruments for research in educational contexts students (primary, secondary and university undergraduate and post graduate students) developing artifacts – models as part of their engagement in classroom small group project work teachers as part of their activities during professional development courses, training to become teacher educators students developing computer games in a special school club a mix of actors members of our In.Di.C.E. community The group’s perspective has always been to perceive the use of technology for mathematical meaning making as incorporating the idea of development. Used in this sense, development is about constructionism (Harel and Papert, 1991), i.e. learning through bricolage and construction of models, editing, interpreting and discussion feedback, reflecting and manipulating. The essence of being engaged in the two-legged model of collaboration is that both the technical aspects of the development of e-slate and the educational aspects of research into experiential-constructional learning in different settings were achieved by the expert work of two teams which had to be rather large and focused. A smaller group of hybrid actors played the role of liaison between the teams. Of

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course this kind of work has difficulties and these are discussed both in Kynigos in press and Kynigos, 2002.

6. Future work <text>

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