Keywords:

Cellular phones allow us to communicate instantly and comfortably across vast distances.

Zsuzsanna Kondor

Being mobile has become crucial for the organization of our everyday work and life. New communications technologies provide the opportunity to organize activities more and more flexibly and effectively. However, this increasing ease of organizational activities has its special price: the traditional division between the different fields of activities -work and spare time, official engagement and privacy -and concomitant scheduling has started to break down. Mobile phones have often been considered devices of disturbance. At the same time, with the collapse of the traditional framework of everyday work and life, cellular phones are becoming indispensable devices to manage exactly these activities.

This apparent contradiction is easily dissolvable by making its roots and workings visible. According to the position I will side with in this paper, the means used to express and disseminate ideas are in close inter-relation with our cognitive capacities. The rapid adaptation of cellular phones indicates that some newly needed cognitive capacities are actually in our possession, ready to be exploited. I will here rely on Merlin Donald's cognitive-evolutionary work on the one hand, and the conclusions of the socalled Toronto School on the other. I would like to propose that aversions to new instruments are rooted in the habits and institutions created by devices of communications we had become accustomed to earlier. First, I will point to some main characteristics of mobile usage, from the specific perspective of the history of communications technologies. I will then analyze the changes indicated above from a cognitive evolutionary perspective. By way of conclusion, I will consider attempts at modelling cognition utilizing the conceptual means of communications technology. and friends. Mobile telephony provides the possibility of "perpetual presence"; we can be reached for whatever purpose and we can keep in touch with others regardless of their local conditions and engagement. This kind of freedom provides a feeling of security: one is always able to ask for help. On the other hand, there arise the difficulties caused by perpetual multitasking, in particular the switch between various physical, psychological, and computational conditions. 1 The quick adaptation to and the spread of the cellular phone do not mirror these difficulties. Deliberating on the reasons for them, we encounter institutions that have developed during the age of literacy. Describing these institutions, I will first compare the communicational patterns of orality with that of literacy, and then go on to sketch some special characteristics of verbal expression.

In the age of primary orality the only possibility to maintain and preserve knowledge was communicating; that is, sustaining living language -a storage language, as Havelock puts it -which was "devised orally for the purpose of survival". 2 Such restriction required a special technology to weave ideas together, and to transmit awareness of the new facts of life. According to this special technology the mode of expression was additive, redundant, and the expressions and words used were very closely embedded in concrete situations. Intercourse was empathetic, participatory and agonistic. 3 Literacy, due to the invention of alphabetical writing as a new kind of storage system, with its new kinds of limits,4 created new habits in cognition compared to those of orality. Alphabetical writing reinforced the dominance of verbal expression and created a special one-sidedness in thinking. Verbal face-to-face communication is situational and multime-80 1 Cf. Joshua S. Rubinstein, David E. Meyer and Jeffrey E. Evans, "Executive Control of Cognitive Processes in Task Switching", Journal of Experimental Psychology -Human Perception and Performance, vol. 27, no. 4 (2001). The conclusion of the study is that multitasking may actually take more time in the end, i.e. the outcome regarding efficiency is questionable. Let us note however that, considering the potentials of cognitive evolution and the plasticity of the human brain, the idea of an adaptation to new circumstances is not at all far-fetched. dial, it does not have to be complete in its verbal aspects. Ever since verbally expressed knowledge became recordable in writing, a new way of composing ideas became necessary to compensate for the absence of the living situation. This necessity induced thinking to retire from everyday life: there emerged general subjects, abstract concepts detached from the human life-world, linearly structured arguments, and the linear notion of time. 5 The new means of expressing ideas made it possible to record general and comprehensive statements. The written text became a reliable and effective medium of preserving knowledge. But let us note that during the centuries of literacy, for a long time to come, the same did not hold for the dimension of pictorial representation.

Considering the representational frameworks of verbal versus pictorial expression, we can see that verbal representation demands a certain re-structuring of any phenomenon. In the case of pictorial representations, we can capture the whole as well as the details of an image to gain a holistic view of the given phenomenon. By contrast, words can represent a given phenomenon only re-structured: general features, circumstances, details, modalities, etc. have to be described as being ordered according to certain priorities. In live intercourse, the particular situation furnishes some components or parts of the information, but in the case of verbal expression, there is a certain degree of generality. Moreover, as William M. Ivins, Jr. put it: "All that words can deal with … are similarities. The simple reason for all this is that words, with the exception of proper names, relation words, and syntactical devices, are mere conventional symbols for similarities." Differences, Ivins suggests, are just as perceptible as similarities; however, words are not able to cope with them. Ivins offers considerations about Western thinking as the outcome of a special technological inability: "I have a notion that much of the philosophical theory of the past can eventually be traced back to the fact that, whereas it was possible after a fashion to describe or define objects by the use of arbitrary and exactly repeatable word symbols addressed, mediately or immediately, to the ear, it was not possible to describe or define them by exactly repeatable images addressed to the eye" (ibid., p. 62). Cf. also Merlin Donald's emphasis on differences between pictorial and verbal representation. This suggests that pic-This structure of verbal representation creates a certain distance from the phenomena by adapting to the possibilities and limitations created by the use of words. Being able to represent something via words is tantamount to being able to think according to structured priorities. The use of words, i.e. verbal representation in a given situation, requires a special point of view, a special perspective, to use the terminology of Michael Tomasello. 7 Accordingly, in everyday practice intentions and perspectives will dominate verbal expression, as opposed to the holistic grasp of the world which is mirrored by pictorial representation.

This tendency became stronger due to the invention of external storage systems whose most effective version was phonetic writing. However, replacing the live situation with mute words required considerable cognitive and intellectual effort and caused difficulties. Epistemology as formed during the centuries of fully developed literacy is determined by the dualism of the knowing subject opposed to the object to be known. Moreover, the subject is considered to be an individual mind which can gain knowledge about the external world by its own cognitive efforts and/or private experiences. 8 Written texts and the concomitant activities created the basis for the conviction that it is systematic individual effort which can yield true value. Still, language, the main tool to express ideas, became an object of criticism because it often proved insufficient for the purpose of rational argumentation, exact description and regulation. Language was increasingly considered to be the means of rational thinking which is to be improved. 82 torial representation might be more direct and less mediated than verbal representation. Donald differentiates between three kinds of visual symbolic innovation: pictorial, ideographic, and phonological. The main difference between them is that "writing generally does not depend upon perceptual isomorphisms and metaphors, whereas the pictorial modes does. The pictorial mode is primarily dependent upon visual norms and does not necessarily engage linguistic structures" (Merlin Donald, Origins of the Modern Mind: Three Stages in the Evolution of Culture and Cognition, Cambridge, MA: Harvard University Press, 1991, p. 278), and words "must have been inherently general-purpose constructs from the start" (ibid., p. 252).

The 20th century was a "language-textured" age, [START_REF] Cf | Think of the dominance of the general over the particular due to the dominance of verbal expression. Think, also, of the dilemma of the precision vs. richness of an expression, reflecting the role of tacit knowledge in language as well as in scientific rationality[END_REF] which is not really surprising. Since grasping reality and language are hardly separable as long as the main representational framework is provided by language, metaphysics and the philosophy of language overlapped. This overlap became very conspicuous by the 20th century. Surprisingly, or perhaps not so surprisingly, that century, in its later decades, came to labelled as the age of "secondary orality". During this period several inventions occurred in communications technology: photography, film, gramophone, radio were introduced into everyday life. As the details and elements of dayto-day activities became easily communicable, philosophical reflection, which hitherto had been bound up with texts, turned to phenomena not previously transmittable. With particular, everyday, details of life appearing on the horizon of philosophical meditations, unexpected anomalies occurred within the texture of traditional philosophy.

The development described above can be considered as a change in the framework of cognitive and social activities. Accordingly, at first, individuals communal and communicative, then atomistic and rational, and recently, re-integrated into communities, were the order of day. In what follows, I will now bring into the picture the cognitive evolutionary theory of Merlin Donald.

Morals from Cognitive Evolution

Donald summed up the changes in cognitive skills during the last two millennia as a development leading to "a new human representational apparatus". As he writes: "The entire theoretic complex of visual symbolic devices, memory management skills, and related metalinguistic skills was completely novel." 10 The steps towards this new representational apparatus presuppose the existence of certain "external devices" which "allow an iterative, interactive thought process to operate repetitively on its own products; and, more importantly, the thought process itself can be largely externalized and institutionalized. Since external memory devices can provide a linkage to all preceding levels of cognitive evolu-83 tion, they also serve in the construction of an integrative field, where the products of various types of thinking can be juxtaposed and combined." 11 The development of external memory 12 was a long-term evolution of representational skills which were in close relation to social intelligence. 13 Donald emphasizes that "it is clear that language was the final step, and that presymbolic forms of social intelligence must have been its foundation". 14 Donald distinguishes three main transitions in human cognitive evolution: a shift from episodic to mimetic, then to mythical, and finally to theoretical culture. Each of these changes means the emergence of a new kind of representation as well as an increasing load on biological memory. 15 Episodic memory supplemented with mimetic representation created community with special habits and organization, conferring a sense of identity on the members of the community. Verbal representation built upon the mimetic one was the first such stage when mythical constructions playing the role of orienting world-views and structures emerged. The transition from mythical to theoretical culture presupposes the existence of an effective external storage system. "[T]he first two evolutionary transitions would have greatly increased the load on biological memory. However, the final step in this tremendous cognitive expansion might 84 11 Ibid. 12 I would like to emphasize that verbal and pictorial representation from a cognitiveevolutionary point of view seem to bear different functions. Although both of them have special importance in the maintenance of social cohesion, it was pictorial representation that first provided the opportunity to bridge the gap between, on the one hand, internally processed knowledge and experience, and, on the other, a kind of external storage system. Donald supposes that pictorial representations emerged in the framework of oralmythical culture and he considers pictorial representation as the first step towards an external storage system. As he puts it: "the first pictorial images themselves were also external representations. They existed outside of the individual, rather than in visual memory. Therefore a technological bridge was under construction that would eventually connect the biological individual with an external memory architecture" (Donald, op. cit., p. 284). 13 Regarding social intelligence Donald relies on the investigations of Robin I. M. Dunbar. See Donald, op. cit., pp. 10 and 137.

14 Donald, op. cit., p. 137.

15 "Human memory had, from its inception, expanded the range of primate memory. The earliest form of hominid culture, mimetic culture, depended on an expansion in the self-representational systems of the brain and created the initial base for semantic memory storage, which consisted initially of representational action scenarios reflected in mime, gesture, craft, and skill. With the evolution of speech and narrative ability, there were even greater increases in the load on biological memory, adding not only the storage networks for phonological rules and the lexicon in its entirety but also a very large store of narrative conceptual knowledge", ibid., p. 319.

have reduced the load on some aspects of biological memory, by gradually shifting many storage tasks onto the newly developed E[xternal]S[ymbolic] S[torage System]. At the very least, the existence of the ESS must have forced a great change in priorities and memory organization." 16 Donald's reconstruction of cognitive evolution and the argumentation for it clearly reveal that the evolution of cognitive structure at the modular level might have continued well beyond the point at which physical evolution had stopped. Cultures restructure the mind, not only in terms of its specific contents, which are obviously culture-bound, but also in terms of its fundamental neurological organization. Whether the organization is vested in a parallel set of specific brain adaptations or not (and obviously at times it is not), the brain sets fewer constraints than formerly thought on the process of cognitive evolution. Culture can literally reconfigure the use patterns of the brain; and it is probably a safe inference from our current knowledge of cerebral plasticity that those patterns of use determine much about how the exceptionally plastic human central nervous system is ultimately organized, in terms of cognitive structure.17 Donald realizes the difficulties regarding the cognitive function of external memory. Reckoning with the fact that "our view of humanity is evolving alongside the machinery of our minds",18 he tries to make its function clear with the help of metaphors. He uses the metaphor of computer R[andom]A[ccess]M[emory] which is supplemented by an external memory for the purpose of expanding the computer's capabilities. Donald stresses that though the capability of RAM can be increased, it has physical limits just as in the case of biological memory. The other metaphor he uses is also from the realm of computer science. As he puts it: "External memory is best defined in functional terms: it is the exact external analog of internal, or biological memory, namely, a storage and retrieval system that allows humans to accumulate experience and knowledge. We do not possess any ready theoretical frameworks in psychology from which to view external memory. Fortunately, there is an excellent point of comparison in the field of computing science: networks." 19 Networks can be seen as the society of computers. Connectedness provides the possibility that computers do not have to retain the same "cognitive"/computational abilities. They can use "the resources of the system … shared, and the system functions as a unit larger than any of its individual components." 20 Of course the emergence a functioning network presupposes definite structural changes as well. Being equipped with an interface is necessary for the use of external memory and external capabilities. However, "once plugged in [humans'] skills are determined by both the network and their own biological inheritance". 21 The cognitive evolutionary point of view sheds light on the modifications of cognitive structure. It provides solid background for the culturalhistorical observations of the scholar. The development of social organization and regulation and, last but not least, the history of philosophy can be easily considered to be attempts at solving difficulties raised by the limitations of the dominant representational framework of the time. 22

Modelling Cognition

Attempts to describe human understanding have deep roots in Western metaphysics and epistemology. Recent investigations in different fields like psychology, artificial intelligence, neurobiology and cognitive science reveal that the traditional metaphysical model of the human mind is mistaken. Let us note that contemporary scientific research aiming at modelling cognition merits philosophical consideration not only in its achievements, but also in its failures. The consequences and force of the traditional notion of dualism -res extensa and res cogitans -and the limitations of the literate and rational mind can be highlighted by reflecting upon insufficiencies emerging in the framework of cognitive science.

One main branch of cognitive science is classical cognitivism, to use the term by Andy Clark. 23 Classical cognitivism considers the mind to be a calculating and symbol-manipulating machinery, i.e. it approaches the mind in the spirit of Cartesian dualism and focusses on abilities which are amenable through language. The other main branch of cognitive science is connectionism which is built on the model of the neural network of the brain. This is clearly a useful approach, but the question of choice regarding the necessary input remains rather artificial, since connectionist models can best work in the framework of restricted, welldesigned, micro-worlds.

The tendency to focus on only restricted areas of the external world is an obvious consequence of the dualistic approach. A similar, earlier consequence was the Cartesian ego, isolated, and supposed to follow its systematically constructed set of rules. The dualism of the knowing subject and the object to be known suggests that the connection between the two poles is one-sided. The subject wants to gain information about the world as it is in order to gain true knowledge and to find general rules of nature. Moreover, this subject is considered to be individual, general in its functioning, and independent of its circumstances. Accordingly, modelling cognition easily tends to be restricted to a special problem, and of course cognition is easily identifiable with symbol-manipulation driven by certain algorithms, i.e. a kind of calculation. Andy Clark points out some other difficulties arising from the traditional approach. As he puts it:

In fact, the methodology of rational reconstruction can mislead in several crucial ways. First, the immediate replacement of real physical quantities with symbolic items can obscure opportunistic strategies that involve acting upon or otherwise exploiting the real world as an aid to problem solving. (Recall the 007 principle.) Second, conceptualizing the problem in terms of an input-output mapping likewise invites a view of cognition as passive computation. That is, it depicts the output phase as the rehearsal of problem solution. 24 Considering cognition as symbol-manipulation is only apparently obvious. Language is the main forum of formulating and disseminating ideas. Due to the institutions developed during the high tide of literacy, and, in particular, due to the philosophical delusions arising from the former, language is considered to be insufficient for the purpose of rational, i.e. clear-cut and exact thinking: additional means of logic seem to be necessary in order to improve the workings of language. From the perspective of writing -which is actually a manipulation of arbitrary signs while serving the technical means of rationality25 -and that of logic, symbolmanipulation seems to be the only effective way by which to describe cognition. However, the results of developmental psychology and evolutionary surveys suggest quite a different concept of cognition.

Andy Clark calls attention to the fact that the gradually evolved structure of different biological and cognitive functions is hardly understandable from a teleological perspective. He further refers to certain achievements of developmental psychology, and, finally, to cases which reveal how the external world provides an orientational potential. With the help of this array of references Clark shows that cognition is not isolated from its environment and is indeed biologically, socially, and culturally embedded. Clark's famous concept of the extended mind is built upon the expedient scaffolding which can be found in the functioning of the simplest as well as the most complicated biological organisms. "We build 'designer environments' in which human reason is able to far outstrip the computational ambit of the unaugmented biological brain. Advanced reason is thus above all the realm of the scaffolded brain: the brain in its bodily context, interacting with a complex world of physical and social structures." 26 Although Clark refers to Donald's theory, 27 he does not exploit its argumentative potential. The reconstruction of the evolution of representational abilities is, I believe, convincing enough not to question the importance of external structures and institutions. [START_REF]The regions of the brain that are most characteristically human -especially the great expanses of the frontal and anterior temporal lobes -are likely to be the most malleable neurological structures in nature, taking on many forms[END_REF] Modifications in the way humans handle and grasp their world as it is revealed from the perspec-88 tive of communications technologies suggest accepting external institutions as important organizers of cognition.

Conclusion

As Clark points out: "Mind cannot usefully be extended willy-nilly into the world. … But the overall picture is of a rather special kind of user/ artifact relationship -one in which the artifact is reliably present, frequently used, personally 'tailored', and deeply trusted." 29 Just like in the case of mobile phones. Indeed, cellular phones are conducive to the above conditions. Judging by the widespread use of mobile telephony, one can hardly question that it is becoming an important part of our everyday life. The use of mobile phones caused perceptible modifications in the structure of everyday activities as it became an essential element of the institutional framework of communication. These modifications and the quick adaptation reveal ancient capacities which were concealed by the institutions of literacy. Currently we create external structures which require exactly those patterns of behaviour which were buried by the institutions that emerged as answers to the demands of literacy. Shifting the emphasis let me quote Kristóf Nyíri: a "mobile phone is … a machine which corresponds to deep, primordial human communicational urges". 30 Finally, I would like to focus briefly on the work of István Hajnal, the Hungarian scholar whose basic contribution was a theory of objectification. Hajnal points out that humans tend to create a structure of "scaffolding", both material and institutional. This is the process of objectification; it causes experiences, emotions and skills to become visible, manifest. To make them manifest means to detach them from the natural ground they originate from. Hence they can serve as a support for, as well as a limitation of, our activities. He claims that from the perspective of the social sciences, the process of objectification renders the general structure of a community or society and the particularities of its every-In effect, the physical structure of mind has become less and less fixed as neocortical evolution has progressed. This leaves room not only for the kinds of radical reconfiguration introduced by literacy but also (presumably) for larger differences between the brains of individual human beings than are possible in primates. It also leaves room for further cognitive restructuring, possibly in fundamental ways" (Donald, op. cit., pp. 380 f.). 29 Clark, Being There, p. 217. 30 Kristóf Nyíri, "Introduction: From the Information Society to Knowledge Communities", in Kristóf Nyíri (ed.), Mobile Communication: Essays on Cognition and Community, Vienna: Passagen Verlag, 2003, p. 12. day life manageable in a single coherent theoretical framework.

Hajnal's concept of literacy defines a special culture which becomes possible due to the technical-practical invention of alphabetical writing, and leads to ways of thinking and cognition never before possible. The invention of a new recording system of alphabetical writing has created the demand for novel ways of thinking and, accordingly, new cognitive skills. The mere technical means of writing was able to affect human intelligence so deeply simply because it was a special kind of objectification, a kind of material support for the thought process.

Clark's emphasis is quite similar even though he looks at the matter from a very different point of view. He refers to an important distinction in Donald's theory. Donald differentiates mythic from theoretic culture and, as Clark stresses, the necessary conditions of the latter were not simply symbolic invention, but "[it was] the process of externally encoded cognitive change and discovery". 31 Clark speaks about "objectification" with regard to language. As he puts it: "The cultural tool of public language gives us not just labels but whole, structured, recursive systems for the encoding, objectification, and communication of thoughts and ideas." 32 If one considers Hajnal's comprehensive theory (especially his concept of literacy), Donald's reconstruction of cognitive evolution, and the Toronto School's findings as to the impact of various technological means of communication on everyday activity, as well as on intellectual disposition and cognitive capacity, Clark's theory is not really ground-breaking. The three former theories attempt to clarify the close inter-relatedness of the individual with its environment, while Clark takes this idea further by focusing on the fact that the individual cannot be detached from its (technical-technological, institutional, and cultural) environment. The concept of the extended mind is in harmony with some main tendencies occurring in twentieth-century philosophy (recall Heidegger, recall Wittgenstein) and it has scientific support as well. This is clearly a significant step towards a holistic grasp of cognition, as opposed to a dualistic one. Epistemology seems well on its way to leave behind the bogus idea of the immanence of the individual mind. 90 31 Clark, Natural-Born Cyborgs, p. 79, quoting Donald, Origins of the Modern Mind, p. 343. 32 Ibid., p. 72.

  New Haven, CT: Yale University Press, 1986, p. 59.

	3 See Walter J. Ong, Orality and Literacy: The Technologizing of the Word, London:
	Methuen, 1982, pp. 36-46.

2 E. A. Havelock, The Muse Learns to Write: Reflections on Orality and Literacy from Antiquity to the Present,

The holistic perception of situations which is given in live intercourse, here has to be constructed with only the help of written words. The situation, gestures, and mimics have to be substituted by words.

See Michael Tomasello, The Cultural Origins of Human Cognition, Cambridge, MA: Harvard University Press, 1999.

Cf. Ernest Gellner: "there is atomistic individualism, which sees the individual building his cognitive world (and indeed any other) by orderly, step-by-step, individual effort, possibly maintaining cooperative relationships with others similarly engaged, but without this fundamentally affecting the nature of the enterprise, which in the end is solitary" (E. Gellner, Language and Solitude: Wittgenstein, Malinowski and the Habsburg Dilemma, Cambridge: Cambridge University Press, 1998, p. 181).

Ibid. p. 320.

Ibid. p. 14.

Ibid. p. 382.

Ibid. p. 309.

Andrew J. Clark, Being There: Putting Brain, Body, and World Together Again, Cambridge, MA: MIT Press, 1997, p. 81. Regarding the 007 principle see ibid., p. 46.

Cf. István Hajnal's views on literacy: "Things, that had earlier happened instinctively in the human being's inner and outer life, started to turn conscious with the appearance of literacy. This sphere of life becomes objectified and abstracted; the human being projects this sphere in front of himself, and examines it consciously and from the outside. There arises the possibility for methodical purposefulness, for the conscious handling of concepts and for combinational and complicated working." (From the posthumous collec-


Zsuzsanna Kondor
email: kondor@webmail.phil-inst.hu

INTEGRATING TRADITIONS COMMUNICATION REVISITED

Technology-enhanced Learning" gains special importance if we look closely at the concept of technology on the one hand, and learning activity placed into the framework of communications technology on the other. The importance of communications technology has been emphasized by many Hungarian scholars in the first third of the 20 th century and due to historian István Hajnal technology has become deeply integrated into the framework of human activities. I'd like to delineate the main sources of Hungarian philosophy of communication.

As a researcher who is engaged in philosophy, I shall outline some main philosophical considerations related to TEL. In conclusion, I will show the convergence of recent philosophy and the Hungarian traditions regarding communications. This convergence could serve as a basis for collaboration as well as a conceptual framework for further research.

Hungarian Traditions in Communication Philosophy

There were at least four Hungarians who were decisive forerunners of the philosophy of communication: Melchior Palágyi, József Balogh, István Hajnal, and Béla Balázs. They contributed to the understanding of human intelligence mainly by revealing the importance of communication and communications technology.

The philosopher and p hysicist Melchior Palágyi's most important findings regarding communication were first, that language is not just means of communication but also the vehicle of thinking. Secondly, he thought that alphabetical writing leads to abstract conceptual distinctions 1 . Thirdly, he argued that the phenomenon of silent thinking gives the impression of "pure" reason.

The classical scholar József Balogh's main research interest, (silent reading and writing) shed further light on Palágyi's latter remark. That is, this practice which was bound up with different intellectual activities became silent after the invention of the printing press. When only manuscripts were available, reading was practiced aloud and others participated. It was not a solitary activity. The practice of reading and writing was accompanied by the conviction that the activity of audire, legere and intelligere constituted a complete unity -to listen, to read and to understand were not 1 "I can not imagine that a primitive human being, who never learnt to read and write, could in his soul discover two kinds of thoughts, that is concepts and judgments… Indeed, of someone who could write, but had no idea of alphabetic writing, and used only a picture of language like e.g. Egyptians, of this human being one could hardly assume that it was inclined to find … in its soul concepts on the one hand and judgments on the other…" (Melchior Palágyi, Az ismerettan alapvetés [Foundations of Epistemology], Athenaeum, 1904 quoted by K. Nyíri, "From Palágyi to Wittgenstein", in: P. Fleissner and J. C. Nyíri (eds.), Philosophy of Culture and the Politics of Electronic Networking, 1999, p. 4) separable activities. After the invention of the printing press, it became much easier to read. However, it subsequently provided an opportunity to dissolve the unity of audire, legere and intelligere. This was the first step towards the over-mechanization of the word. Balogh regarded all instruments of communication -such as typewriters, dictating and speaking machines, telegraphs and telephones -to be tools of mechanization. These instruments made reading and writing more comfortable, but this convenience resulted in superficiality and speediness, through which the "art of writing degenerated into mass production".2 István Hajnal's concept of literacy led to a similar conclusion, though its theoretical background is totally different. Literacy, according to Hajnal, defined a special culture which became possible due to the invention of the new recording system of alphabetical writing. This system created new demands on thought processes, and new cognitive skills to deal with the demands. As he put it: "With the appearance of literacy, we know that what had been happening instinctively to this point in a human being's inner and outer life, now starts to become conscious. This sphere of life becomes objectified and abstracted; the human being projects this sphere in front of himself and examines it consciously and from the outside. There arises the possibility for methodical purposefulness, for the conscious handling of concepts and for combinational and complicated work."3 Literacy was able to impact on different spheres of human consciousness because it was a kind of objectification. As Hajnal formulated,"[m]ovements and sounds do disappear, still, humans can use them and their matter-relatedness to produce something that is objective, something that functions as an extrinsic intermediator for inner life."4 There are different ways to objectify ideas. Hajnal grasps technology through the process of objectification and internalization. New technologies should be rooted in everyday life (i.e. it should fit earlier experiences and knowledge of materials and different kinds of power of nature). They should harmonize with the needs of human interrelationships, social customs and institutions. The success of the printing press depended on previous technical inventions, the blossoming writing culture and the "reading audience" of the Middle Ages. 5 Writing is actually an artificial instrument for thinking, through which ideas can free themselves from the context of real life that gave them life. In the long run "[l]etters produced letters, writing produced writing. Purely speculative thinking was highly refined, even in the smallest of tasks, and more perfectly so since the more one-sidedly it functioned, the more it divided the professions into mechanical details, the more it exempted them from sensing the heavy material of life." 6 As we can see Hajnal's criticism is the obvious consequence of the process of objectification. Béla Balázs, the film aesthete, called attention the importance of non-verbal communication as opposed to the dominance of verbal expression. As he wrote, "In film … speaking is a play o f facial gestures and immediately visual facial expression. They, who see speaking, will learn things very different from they who hear the words". And "ideas did not always first appear in concepts and words, so that painters would only subsequently provide illustrations for them with their pictures." However, words heavily form our thoughts. "Psychological and logical analyses have proven that our words are not subsequent representations of our thoughts, but forms which will from the beginning determine the latter." 7

Main Research Perspectives

The relatedness of thinking and language, the notion of literacy, and a certain criticism of the dominance of written language are very similar concerns to the findings of McLuhan's Toronto Circle. The impact of Hajnal is obvious as Walter J. Ong referred to him explicitly. [START_REF] Ong | referred t o one of Hajnal's paper, L'Enseignement de l'écriture aux universités médiévales[END_REF] The basic idea common to both is that the mode of expression is decisive regarding its content. Before I delineate some main concerns of philosophy related to TEL, I will sketch the periodization elaborated by Walter J. Ong, a prominent member of the Toronto Circle. He distinguishes three main epochs of communications technology.

In the age of primary orality, according to him, the only possibility to maintain and preserve knowledge was communication, i.e. sustaining living language. This restriction required a special technology. Expression itself was additive, redundant, and the expressions and words used were very heavily embedded in concrete situations. Intercourse was empathetic, participatory and agonistic. Literacy emerged as verbally expressed knowledge became recordable due to alphabetical writing. [START_REF] Havelock | About the special character of alphabetical writing[END_REF] With this invention, a new way of interweaving ideas became necessary as a substitute for the living situation, i.e. to make it complete. This necessity induced thinking to retire from everyday life: general subjects, abstract concepts detached from the human life-world, linearly structured arguments, and the time-scale emerged. Secondary orality set in with telephone, radio, television and various kinds of sound tape, electronic technology. "This new orality " -as Ong put it -"has striking resemblances to the old in its participatory mystique, its fostering of communal sense, its concentration on the present moment, and even in its formulas.

But it is essentially a more deliberate and self-conscious orality based permanently on the use of writing and print, which are essential for the manufacture and operation of the equipment and for its use as well."10 In the last decade, of course we can see some new tendencies in connection with tools we use to communicate, which suggests the consideration of a certain revival of literacy.11 In any case, for each main epoch a special comprehension of the surrounding world, characteristic intellectual attitude and demands regarding the horizon of knowledge are distinguishable.

These modifications are easily comprehensible in the framework of three different perspectives: historical, cognitive and representational point of view.

From a historical point of view the main endeavor of metaphysics shows an impressive trajectory: from the Pre-Socratic notion of arkhé, which is tailored according to the physical world, to the Heideggerian "thrown-ness" into the world which suggests to leave behind the analyzing, speculative and from the everyday activity alienated tradition of Western philosophy. To look not at the hardcore and sometimes quite esoteric part of philosophy, but at the concept of language, which was a crucial topic of philosophy in the 20 th century, we find a trajectory similar to that of metaphysics.

At the very start of Western Philosophy, Heraclitus said that Logos controls how things function, and it is common in everyone. Logos is in one way a kind of ruling principle and in anothe r way, some sort of articulation of this functioning, i.e. a certain revealing force.

After the invention and spread of alphabetical writing, Logos gained new characteristics. The main point of Aristotelian logic is to provide a proper instrument (i.e. a set of rules which can help to decide whether an argument is true or false), and to arrange knowledge gained by everyday experience. Modern logic eliminated the importance of the reference to the facts of everyday life and the algorithm gained crucial importance. According to Leibniz, signs are instead of things. However this logic aimed to improve language, it eliminated the anchor of language in everyday practice, i.e. there were no need for reference.

The emergence of the communal character of language was a turning point and it happened in the age of secondary orality. Wittgenstein considered language a changing phenomenon which changes in accordance to its use. "[T]he speaking of language is part of an activity, or a form of life." The term "language-game" emphasizes this fact. The so-called "private language argument" stresses the communal character of language, i.e. the fact that speaking a language means obeying rules -agreement in definitions and implicit presuppositions which gain sense only in a community. "If language is to be a means of communication there must be agreement not only in definitions but also … in judgements." 12 Heidegger in his famous work Being and Time calls attention to the fact that language can be seen as something which is ready-at-hand [zuhandene] in the everyday activity we articulate our comprehension "Being-with others". In that case language is "a tool to be used in social intercourse", i.e. an instrument of communication on the one hand, but language is also a constituent element of beingin-the-world as it is an important moment of revealing, on the other hand. As we can see, language became anchored again in the realm of everyday intercourse, albeit with some distinctive consciousness due to centuries of literacy.

If we take a look at the concept of the individual, crucial alterations are apparent. We can distinguish three main epochs: the age of involvement, then a long period of solitude and recently, the time of the reintegrated individual. "Once upon a time"as Erne st Gellner put it -"mankind lived, by and large, in closed intimate communities, governed by practices simultaneously geared both to maintaining internal order and adjusting to nature …. The criteria adapted for judging the acceptability of practices … were, so to speak, self-validating, traditional." 13 Later on the individual emerged whose attitude towards their micro and macro world changed. With literacy, solitary reason set in. Atomistic individualism, labeled thus by Gellner, and started by Descartes, considered human knowledge a step by step development stemming from the intellectual effort of the individual. This conviction led to the division of intellectual integrity -that is, the concern of solitary reason and the experience of everyday practice (which shows humans to be socially committed beings) created anomalies which w ere explicitly realized only much later. Epistemology, philosophy of history and ethics were hardly comprehensible in the same coherent framework. As new communications technology provides the opportunity to communicate continuously, new forms of life and new norms arose which started to dissolve both epistemological individualism and the ivory tower of the solitary reason. The individual is on the way to being integrated in their worlds again.

Cognitive perspective convinces us of the importance of social intelligence as opposed to the immanence of the individual mind on the one hand, and calls attention to the importance of the surrounding world in which humans are immersed on the other hand. In his famous book Origins of the Modern Mind 14 , Merlin Donald outlined the history of cognitive evolution. According to his theory, the development of representational skills was closely related to social intelligence. He distinguished three main transitions in human cognitive evolution: the shift from episodic to mimetic, then from mimetic to mythical, and finally from mythical to theoretical culture. Each of these changes means the emergence of a new kind of representation as well as an increased load on biological memory. Episodic memory completed with mimetic representation could now create community with special habits and organization which would provide a certain kind of identity. Attaching verbal representation to the mimetic one was the first stage, when mythical constructions as orienting world-views and orders could come to life. The transition from mythical to theoretical culture presupposes the existence of an effective external storing system. "[ T]he first two evolutionary transitions would have greatly increased the load on biological memory. However, the final step in this tremendous cognitive expansion might have reduced the load on some aspects of biological memory, by gradually shifting many storage tasks onto the newly deve loped E[xternal]S[ymbolic]S[toring system]. At the very least, the existence of the ESS must have forced a great change in priorities and memory organization." 15 Andy Clark similarly stressed the importance of the environment, although in a wider sense than the realm of ESS. "We build 'designer environments' in which human reason is able to far outstrip the computational ambit of the unaugmented biological brain. Advanced reason is thus above all the realm of the scaffolded brain: the brain in its bodily context, interacting with a complex world of physical and social structures." 16 The third research perspective is served by representation. In the age of multimedia, the ability to clarify the difference between verbal and pictorial representation gains special importance. Verbal representation has some special requirements and limits which presuppose some cognitive capabilities. The structure of verbal representation unintentionally creates a distance from the phenomena, by just obeying the possibilities and limitations provided by the use of words. Verbal expression is linear, detailed and ordered according to structured priorities. The use of words in a given situation means a special point of view, a special perspective. 17 "All that words can deal with, however, are similarities. The simple reason for all this is that words, with the exception of proper names, relation words, and syntactical devices, are mere conventional symbols for similarities. Although differences are just as perceptible as similarities," words are not able to cope with them. "But that these differences are not expressible in words does not mean that they are ineffable, for they are clearly communicable in non-verbal ways." 18 Pictorial representation of course "can deal with" both similarities and differences. It provides a holistic view, i.e. we can capture the whole as well as the details of an image at a glance. An image, as opposed to verbal expression, is able to mediate "the immense wealth of living shapes" without decreasing exactne ss. 19 The research of William M. Ivins Jr. clearly proves that our intellectual horizon is deeply determined by the means we can formulate and communicate our experiences, thoughts and problems. As he put it: "I have a notion that much of the philosophical theory of the past can eventually be traced back to the fact, that whereas it was possible after a fashion to describe or define objects by the use of arbitrary and 15 Donald op. cit. p. 320 16 19 "Higher degrees of formalization make statements of science more precise, its inferences more impersonal and correspondingly more 'reversible'; but every step towards this ideal is achieved by a progressive sacrifice of content. The immense wealth of living shapes governed by the descriptive sciences is narrowed down to bare pointer-readings for the purpose of the exact sciences, and experience vanishes altogether from our direct sight as we pass on to pure mathematics. -There is a corresponding variation in the tacit coefficient of speech. In order to describe experience more fully language must be less precise. But greater imprecision brings more effectively into play the powers of inarticulate judgement required to resolve the ensuing indeterminacy of speech. So it is our personal participation that governs the richness of concrete experience to which our speech can refer. Only by the aid of this tacit coefficient could we ever say anything at all about experience -a conclusion I have reached -said Polányi -already by showing that the process of denotation is itself unformalizable." (M. Polányi, Personal Knowledge -Towards a Post-Critical Philosophy, London: Routledge & Kegan Paul, 1962, pp. 86f. ) exactly repeatable word symbols addressed, mediately or immediately, to the ear, it was not possible to describe or define them by exactly repeatable images addressed to the eye." 20 This rough draft of the main characteristics of these two important forms of expression poses the question, what kind of modifications can be expected due to the easy use of both kinds of representations. Could it affect our conceptual framework directly? Could we comprehend a wider horizon for our micro and macro world due to these easily accessible and different forms of expression?

Conclusion

As we can see, communications technology gains special importance regarding cognitive skills, social institutions, norms and ideals. The environment of learning is also changing, even some institutional changes are imminent. The idea of "Life Long Learning" stems from the increasing mobility and the speed of technological change. As the different spheres of individual life (public and private), the scheduling which was traditionally static and organization of everyday activities were differentiated according to the workplace and home) are being altered radically, the traditional role of the school must be changing as well. Considering computer program-design, these technologically determined social conditions are definitive. Taking into consideration the Hajnalian concept of technology, cognitive abilities and the notion of scaffolding in the sense of Clark, it becomes clear that extant forms of communication have crucial importance. Accordingly, everyday use, (i.e. the way pupils/students communicate) is a good basis to create new and helpful tools for learning.

The popularity of different chat forums calls attention to some of these changes. 21 As our cognitive settings are considerably determined by our social makeup, this new way of communication can modify our close circle of friends. According to Robin Dunbar's "social brain hypothesis" there is certain limit to the number of individuals we can have in a certain sense intensive relationship. 22 The intensity is highly dependant on the intensity of connectedness. So it is not possible to extend the circle of relationships which are close to some extent, but the visible awareness of the other's presence could restructure the membership of closer circles. The other important finding is the phenomenon of media convergence. In a multimedial framework, chat makes the differences visible among live voice, written text, image and animation on the one hand, and helps to use them simultaneously (i.e. to create a certain merged literacy) on the other hand.

Last but not at least, I think, it is obvious that collaboration is the best way of integration. If we consider the Hungarian sources of communication philosophy, recent findings of cognitive science and psychology, and tendencies in 20 th century philosophy, the convergence we can experience suggests a reconfiguration of the network of basic concepts, like traditional dichotomies of individual vs. social, natural vs. artificial, and knowledge vs. skill.

  with a declining level of emotional intensity and closeness. It is as though each of us sits in the centre of a series of expanding circles at 5, 15, 35, 80 and 150 individuals." R. I. M. Dunbar: "Are There Cognitive Constraints on an E-World?", in: Kristóf Nyíri (ed.), Mobile Communication. Essays on Cognition and Community, Vienna: Passagen Verlag, 2003 pp. 58

  Andy Clark, Being There. Putting Brain, Body, and World Together Again, Cambridge: MIT Press, 2001, p. 191 17 See Tomasello, Michael, The Cultural Origins of Human Cognition, Cambridge: Harvard Univ. Press, 1999 18 William M. Ivins Jr.,Prints and Visual Communication, Cambridge, Massachusetts: Harvard University Press 1953, p. 139

J. Balogh, Voces Paginarium. Beträge zur Geschichte des lauten Lesens und Schreibens, Budapest, 1921, p. 29

I. Hajnal, "Európai kultúrtörténet -írástörténet" (1932), in: F. Glatz (ed.), Technika, muvelodés, Budapest 1993, p. 18

I. Hajnal, "Történelem és szociológia", (1939), in: Glatz (ed.) op. cit. p. 203

"Book printing press, as it is well-known, the produce of the exact melting of many details of masterly transcribing work, of school dictata, of paper manufacturing, of several types of inks/paints, of the metal industry and the finest engraving craft." (Hajnal, "A európai város kialakulása", 1941, in: Glatz (ed.) op. cit. p. 228) "The invention of printing press was only an inevitable consequence of the deep-rooted and largescale writing culture of the late Middle Ages. As many people understood written texts, there was a large reading audience and also a wide-scale book-production in the last centuries of the Middle Ages, which brought forth by pure practical expediency the invention of printing." (Hajnal, "Európai kultúrtörténet -írástörténet", 1932 in: Glatz (ed.) op. cit. p. 24)

Hajnal, "Évforduló" (1948), in: Glatz (ed.) op. cit. p. 449

Walter J. Ong, Oralit and Literacy: The Technologizing of the Word, London: Methuen, 1982, p. 136

See Maurizio F erraris, Where Are You? Mobile Ontology (http://www.socialscience.tmobile.hu/2005/index.htm) The paper is going to be published in the volume of Kristóf Nyíri (ed.), Mobile Understanding The Epistemology of Ubiquitous Communication, Vienna: Passagen Verlag, 2006.

 12 Ludwig Wittgenstein, Philosophical Investigtions § 242

E. Gellner, Language and Solitude. Wittgenstein, Malinowski and the Habsburg Dilemma, Cambridge: University Press 1998, p. 189

Merlin Donald, Origins of the Modern Mind. Three Stages in the Evolution of Culture and Cognition Cambridge: Harvard Univ. Press, (1991) 1993

W. M. Jr. Ivins, Prints and Visual Communication, Cambridge, Massachusetts: Harvard University Press 1953 p. 62. Cf. Ivins's further remark: "Plato's Ideas and Aristotle's forms, essences, and definitions, are specimens of this transference of reality from the object to the exactly repeatable and therefore seemingly permanent verbal formula. An essence, in fact, is not part of the object but part of the definition. Also, I believe, the well-known notions of substance and attributable qualities can be derived from this operational dependence upon exactly repeatable verbal descriptions and definitions -for the very linear order in which words have to be used results in a syntactical time order analysis of qualities that actually are simultaneous and so intermingled and interrelated that no quality can be removed from one of the bundles of qualities we call objects without changing both it and all the other qualities." (Ivins, op. cit. p. 63)

The last workshop of our team, which is engaged in the Philosophy of TEL, was to clarify the role and importance of chat. The next few sentences focus on some of its results.

"Although humans can obviously cope with very large urban environments and even nation-states, the number of people within those large population units with whom one can say that one has a direct personal relationship is very much smaller. Censuses of the population units of hunter-gatherers, the size of scientific sub-disciplines, the number of people to whom one sends Christmas cards and the number of people of whom one can ask a favour all turn out to be about 150 in number... ... number of individuals that can be held in a relationship of a given degree of intensity. There is some longstanding evidence, for example, that the number of individuals we can have a particularly close bond with is limited to around 12-15, and that within this there may be an inner circle of about 5 individuals with whom this relationship is especially strong. There is, in addition, evidence to suggest that there may in fact be a series of layers, with boundaries at around 35 and 80-100, each associated


Jean-François Nicaud
email: jean-francois.nicaud@imag.fr

Hamid Chaachoua
email: hamid.chaachoua@imag.fr

Marilena Bittar
email: marilena@nin.ufms.br

Denis Bouhineau
email: denis.bouhineau@imag.fr

Student's modelling with a lattice of conceptions in the domain of linear equations and inequations

Keywords: student's modelling, algebra, rewriting rules, conceptions

We present a student's modelling process in algebra which consists of two phases. The first phase is a local diagnosis where a student's transformation of an expression A into an expression B is diagnosed with a sequence of rewriting rules. A library of correct and incorrect rules has been built for that purpose. The second phase uses a lattice of conceptions built for modelling students more globally. Conceptions are attributed to students according to a mechanism using the local diagnoses as input.

This modelling process has been applied off-line to data gathered in France and Brazil with 13-16 years old students who used the Aplusix learning environment. The results are described and discussed. The process will be included later inside Aplusix in order to model students on-line and to provide the students' conceptions to the teachers.

Introduction

Since 2003, we are engaged in a research work 1 devoted to automatic student's modelling in algebra. Our general model is based on correct and incorrect rules allowing to interpret the local behaviour of the student, and on conceptions allowing to model the student in a more global way. Our methodology contains five components: [START_REF] Artigue | Epistémologie et didactique[END_REF] Gathering a large set of data concerning students' calculations; (2) Construction of a library of correct and incorrect rules;

(3) Automatic diagnosis of the student's calculations in term of sequences of rules; (4) Construction of a library of conceptions; [START_REF] Buchanan | Rule-Based Expert Systems, The Mycin experiments of the Stanford Heuristic Programming Project[END_REF] Automatic diagnosis of the student's calculations in term of conceptions. This paper mainly describes the parts devoted to conceptions.

According to Artigue [START_REF] Artigue | Epistémologie et didactique[END_REF], a conception is related to a concept and is characterized by three components: (1) a set of situations which give meaning to the concept; (2) a set of significations (mental images, representations, symbolic expressions); (3) tools (rules, theorems-in-act, algorithms). This paper is mainly devoted to theorems-in-act associated to the concept of movement in (in)equations. An example of theorems-in-act is the following: When a sub-expression is moved from one side to the other side in an (in)equation, its sign is always changed. Note that this is correct for additive movements (e.g., x-3 = 5 → x = 5+3) and incorrect for multiplicative movements (e.g., -3x = 5 → x = 5/3). This theorem-in-act is not a rewriting rule: it applies to all the movement rules (and generally, several theorems-inact apply to a given rule). In the rest of the paper, we will use the term conception instead of the term theorem-in-act for fluidity reasons. The reader needs just to remember that conception means, in the paper, a theorem-in-act part of a conception. Several research works have been devoted to conceptions and some conceptions have been built by hand [START_REF] Artigue | Epistémologie et didactique[END_REF][START_REF] Balacheff | Students conceptions: an introduction to a formal characterization Cahier du laboratoire Leibniz # 65[END_REF][START_REF] Brousseau | Theory of didactical situations in mathematics[END_REF]. However, automatic diagnosis of conceptions is a difficult problem which has been little investigated. This is the part of our research which is mainly described in this paper. Examples of previous works in this field are Pépite [START_REF] Delozanne | A diagnostic based on a qualitative model of competence in elementary algebra[END_REF] and Baghera [START_REF] Webber | A multi-agent and emergent approach to learner modelling[END_REF]. We will compare our work with Pépite in the last section.

For gathering the data to be analysed, we have used the Aplusix learning environment [START_REF] Nicaud | Mixing microworld and CAS features in building computer systems that help students to learn algebra[END_REF][START_REF]The Aplusix Website[END_REF] which allows students to freely make calculation steps, as they do in the paper environment, and which records all the students' actions in log files. Except for 6 classes, Aplusix has been used in an ecological context, i.e., by students supervised by their mathematics teachers during the normal schedule of the class. Data have been gathered in France and Brazil from about 3000 students of grades 8 to 11.

The long term goal of this work is twofold: First, we aim at building a map of conceptions covering the domain of elementary algebra (algebra for grades 8 to 11) and to diagnose thousands of students from many countries, to get a repartition of the obtained conceptions; this is our academic goal. Second, we plan to encompass the map of conceptions and the diagnosis mechanism in the Aplusix system to improve its interest by providing to the teachers the conceptions of their students, and by selecting the best exercises to give to the students according to the calculated conceptions (in particular to correct the erroneous parts of these conceptions); this is our applied goal which is similar to Cognitive Tutors [START_REF] Koedinger | Intelligent tutoring goes to school in the big city[END_REF] in a larger domain. This paper presents the ongoing work on the movement concept in (in)equations. A detailed description of the different parts of the work is available in a research report [START_REF]Modélisation cognitive d'élèves en algèbre et construction de stratégies d'enseignement dans un contexte technologique[END_REF].

Modelling student's calculations with rewriting rules

Most of the calculations in formal algebra concern the application of rewriting rules, according to the "replacement of equals" principle. Given two expressions A and B, a rewriting rule R: A → B can be applied to an expression E if there is unification between A and a subexpression U of E. The application of the rule, when possible, consists of replacing U in E by B. Rewriting rules usually come from algebraic identities provided by axioms and theorems. For example, the identity A(B+C)=AB+AC produces two rewriting rules, one is the expansion rule A(B+C) → AB+AC, the other is the factorisation rule AB+AC → A(B+C). Most of the rewriting rules of elementary algebra can be classified in reduction and simplification rules (numerical calculations, like term collection, e.g., 3x+5x → 8x, etc.), factorisation rules (which provides additional factors e.g., AB+AC → A(B+C)), expansion rules (inverses of factorisation rules), and rules on relations (see below). When a student solves an exercise in formal algebra, he/she produces calculation steps. Our model interprets these steps as the application of correct or incorrect rules. For example, if the student transforms 2x(3x 2 -4) into 6x 3 -4, we can interpret this calculation step by the application of the incorrect rule A(B+C) → AB+C followed by the application of correct reduction rules.

The main strategy for solving linear equations and inequations consists of expanding both sides, if necessary, then isolating the variable, using rules which carry out identical operations on both sides or using movement rule that move an additive or multiplicative expression from one side to the other. Examples of these rules are: A=B → A+C=B+C, addition to both sides; A+C=B → A=B-C, additive movement; AC=B → A=B/C (C≠0), multiplicative movement. There are 12 movement rules for equations (4 additives and 8 multiplicative) and also 12 for ≠ inequations. For the other inequations (< ≤ > ≥), multiplicative movement rules are duplicated according to the semantic sign of C, because when this sign is negative, we have an inversion of the inequality (e.g., AC ≤ B → A ≥ B/C (C<0)). Therefore, there are 20 movement rules for each sort of inequations (4 additives and 16 multiplicative). In the school practice, operations on both sides of (in)equations are progressively replaced by movement rules which are the fundamental rules for solving linear (in)equations. Other rules are expansion and reduction rules, which do not apply to the entire (in)equation but to a sub-expression.

The very large quantity of movement rules (104 = 2*12+4*20) questions the relevance of the rule model. Is a student supposed to have all these detailed rules in mind when he/she solves an (in)equation? Certainly not in this form. Hence, we propose another model which emphasizes the main features and we consider at the same time incorrect movements. A movement has an argument which is the element moving from one side to the other side. In an additive movement, the argument is located in a sum or is the whole side. When the movement is carried out correctly, the argument remains additive, on the other side, with an opposite syntactic sign. In a multiplicative movement, the argument can be multiplicative within a numerator, or multiplicative within a denominator. When the movement is carried out correctly, the argument is still multiplicative in the other side, and its place changes (denominator ↔ numerator). In the case of an inequation, the orientation of the inequation changes if the semantic sign of the argument is negative. We can describe movements with only one general rule, entitled "Movement", associated to the vector presented in table 1. For example, the incorrect transformation 2x-4 < 5 → 2x > 5-4 is represented by a Movement with the argument "-4" and the vector (<, LefToRight, NumeratorToNumerator, InitAdditive, FinAdditive, SignUnchanged, OrientationChanged).

A local diagnosis algorithm

We have implemented an algorithm to diagnose student's local algebraic transformations in terms of rewriting rules. The algorithm uses a library of 260 correct and incorrect rules. This library has been obtained from a cognitive analysis (we observed students' actions in various situations using the replay system of Aplusix) and from an epistemic analysis (as the one carried out for the movement rule), see details in [START_REF] Sander | From usage analysis to automatic diagnosis: The case of the learning of algebra[END_REF][START_REF]Modélisation cognitive d'élèves en algèbre et construction de stratégies d'enseignement dans un contexte technologique[END_REF]. The incorrect rules of the library currently covers operations on (in)equations, reductions (except operations on fractions and square roots) and expansions. Fractions, square roots and factorisations will be studied later.

The local diagnosis algorithm has three phases. The first phase isolates the subexpressions where the transformation occurs. As a result, an expression A has been transformed into an expression B by the student. The second phase is a heuristic search algorithm which develops a tree from the starting expression A. At each step, the node of the search space which is the closest to B, according to a distance between expressions, is chosen and developed. The development consists of applying the applicable rules of the library. When the development produces the expression B, the goal is reached and the path from A to B in the tree is a sequence of rules that explains the transformation of A into B. The second phase stops when a chosen number of nodes have been developed or when no more rules are applicable. So this phase can fail. It can fail because of: a missing incorrect rule, an early stop of the process, a student's behaviour that has not to be understood. The third phase consists of the evaluation of the different diagnoses and of the choice of the best one. For example, the transformation of 2x-6 = 7x-8 into -5x = -14 is diagnosed with 4 rules: (1) Incorrect additive move of 6 leading to 2x = 7x-8-6; (2) Correct additive move of 7x leading to 2x-7x = -8-6;

(3) Correct additive reduction, leading to -5x = -8-6; (4) Correct additive reduction, leading to -5x = -14.

The current performance of the local diagnosis, in terms of success/failure, for classes of grades 8 and 9, in France (540 students) and Brazil (2500 students), is the following: Between 90% and 100% of success for correct transformations, depending of the class, and between 74% and 93% of success for incorrect transformations. Note that a failure is sometimes the best diagnosis (researchers don't always explain a student's transformation). The correctness of the diagnoses has been studied by three researchers for two French classes, grades 8 and 9, for incorrect expansions, incorrect reductions and incorrect transformations on in(equation): between 82% and 97% diagnoses have been considered to be correct, depending of the class and the category of rule (expansions, etc.).

A lattice of conceptions

We have designed a set of conceptions for the concept of movement which is organised as a lattice. Each conception which is not a micro-conception has two subconceptions and is the union of these two subconceptions. The conceptions at the first level are called "global conceptions" with respect to an aspect of the concept of movement. We have considered three aspects of the concept of movement: the sign aspect (whether the sign of the argument is changed or not), the inequality orientation for inequations (whether the orientation of the inequality is changed or not) and the operator evolution (what happens to the operator linking the argument to the (in)equation in the movement). Figure 1 shows a part of this lattice. We have defined five global conceptions for the sign aspect:

-CorrectSign: Correct treatment of the sign of the argument; -AbsoluteValue: Change of the sign of the argument if and only if this sign is "-"; -SemiAbsoluteValue: Change of the sign of the argument if and only if: this sign is "-" and the argument is multiplicative; or the argument is additive; -SaveSign: Never change the sign of the argument; -ChangeSign: Always changes the sign of the argument. Of course, CorrectSign is the only correct conception in that list. The other conceptions produce correct or incorrect calculations depending on the context. The inequality orientation and the operator evolution aspects lead to similar decompositions in global conceptions.

Let us detail the ChangeSign global conception of the sign aspect. This global conception consists of "always changing the sign" of the argument of a movement. It can be decomposed in partial conceptions: ChangeSign-eq for "always changing the sign in an equation" and ChangeSign-ineq for "always changing the sign in an inequation". Again, ChangeSign-eq can be decomposed in more partial conceptions ChangeSign-eq-add for "always changing the sign in an equation for an additive argument" and ChangeSign-eq-mult for "always changing the sign in an equation for a multiplicative argument". In this framework, some students may have a ChangeSign-eq and not a ChangeSign-ineq conception. Those who have both ChangeSign-eq and ChangeSign-ineq conceptions have the ChangeSign conception.

For ChangeSign-eq-add (Y-eq-add in figure 1), we have two micro-conceptions named 1b and 2b. The first one is changing the sign when it is "+" in an equation and when the argument is additive; the second one is changing the sign when it is "-" in an equation and when the argument is additive.

In order to have a lattice, we add some nodes considered as abstract conceptions: three nodes containing the aspects (SignAspect, InequalityOrientation and OperatorEvolution) and having the global conceptions as direct descendants, a "Top" node having these three nodes as direct descendants, and a "Bottom" node being direct descendant of every microconception. The lattice has a total of 146 nodes. . The abstract nodes are not represented. X has an Xbis conception which is another decomposition of the same conception (X is first decomposed in eq/ineq then add/mult; Xbis is first decomposed in add/mult then eq/ineq). X and Xbis are active at the same time. Xbis is necessary to allow to have X-add and X-mult in the lattice.

Computation of the student's conceptions

The computation of the student's conceptions uses first the local diagnoses described in section 3, and an intermediary construction: the Local Behaviour Vector (LBV) which links movement rules and micro-conceptions. The movement rules of the local diagnoses which match LBVs are counted for each LBV. When a condition is verified for a LBV, a corresponding micro-conception is activated. Then a propagation mechanism is launched to activate other conceptions.

For building the Local Behaviour Vectors, we use some elements of the vector of the unique movement rule and some other useful elements. This is the decision of the observer, as emphasised by [START_REF] Balacheff | Students conceptions: an introduction to a formal characterization Cahier du laboratoire Leibniz # 65[END_REF]: "Modelling behaviour requires a first level of interpretation, that of the organization of reality". The LBVs we have chosen for the sign aspect is described in table 2. The first three variables define the context, the last one corresponds to the action. When the first three variables are chosen, the two possible values of the last one determine a couple of opposite LBVs. An example of a couple of opposite LBVs is the following: the context is (Equation, InitAdditive, SignArgMinus); one LBV has the action SignChanged, the other has the action SignUnchanged.

When we have a list of rules attributed to a student by the local diagnoses of a set of transformations, we match each movement rule to each LBV and count the number of occurrences. A student who has a rational behaviour, with respect to the model, would mark only one LBV of a couple of opposite LVB. However, we cannot expect to have this level of rationality. As a consequence, we have chosen the following mechanism to activate LBVs: Let LBV1 and LBV2 be two opposite LBVs. Let n1 and n2 be the respective numbers of occurrences of LBV1 and LBV2 for a given student:

IF n1+n2 = 0 THEN there is no LBV activation ELSE IF n1 / (n1+n2) ≥ 2/3 THEN LBV1 is activated with coefficient n1 / (n1+n2) ELSE IF n2 / (n1+n2) ≥ 2/3 THEN LBV2 is activated with coefficient n2 / (n1+n2)
ELSE there is no LBV activation An activated LBV becomes a micro-conception which can be expressed by: "In the context of this LBV, the student generally performs the action of the LBV". Generally is expressed by the coefficient which is a sort of Certainty Factors [START_REF] Buchanan | Rule-Based Expert Systems, The Mycin experiments of the Stanford Heuristic Programming Project[END_REF].

The micro-conceptions are the lowest real conceptions in the lattice (having Bottom as direct descendant). After the activation of the micro-conceptions, each upper conception is determined by calculating recursively its coefficient as the geometrical average of the coefficients of its two direct descendants (if the coefficients of the two direct descendants are a and b, the result is sqrt(ab) ).

A micro-conception is correct or incorrect. When it is incorrect, any behaviour that matches the micro-conception is incorrect. A conception which is not a micro-conception is correct if and only if its two direct descendants are correct. When a conception, which is not a micro-conception, is incorrect, some behaviours that match the conception are incorrect but, generally, others are correct. For example, the ChangeSign conception in which "the student always changes the sign" contains correct and incorrect behaviours.

Experimental study

Since 2003, we have conducted several experiments with classes in France and Brazil of grades 8, 9 and 10 and we have recorded thousands of hours of students' activities. We have recently applied the modelling process to a part of the data. It produced a description of each student in terms of a list of conceptions attributed to this student, and a summary table containing the number of occurrences of each conception.

An experiment with a group of 342 students of grade 9 was conducted in Campo Grande (Brazil) in 2004 with 20 minutes of use of Aplusix for familiarisation and 1 hour of use in the test mode where no feedback was given to the students. The modelling process has been applied to the data corresponding to the test phase. The analysis of the distribution of the conceptions with respect to the context shows what follows:

○ Type of exercise: 56% of conceptions concern equations (97% correct and 3% incorrect); 44% of conceptions concern inequations (71% correct and 29% incorrect). ○ Initial position of the argument: 62% of conceptions concern an additive position (95% correct and 5% incorrect); 38% of conceptions concern a multiplicative position (68% correct and 32% incorrect).

Most of the conceptions concern equations with an additive position of the argument. The high rate of correct conceptions cannot be viewed as certitude of a good result because the level of generality of the conceptions has to be taken into account, the more general ones being the "global conceptions" defined in section 4. Actually, we had only 2% of correct global conceptions (e.g., CorrectSign). At the other levels (the level is the depth in the lattice, the global conception having level 1), we have 32% correct conceptions for level 2 (e.g., CorrectSign-eq), and 66% correct conceptions concerning very specific contexts of levels 3 and 4. These results are coherent with hand analysis made for a few students and with the general opinion of the teachers of the classes. The distribution of the number of conceptions per student is a Gaussian distribution, see table 3. The distribution of the conceptions with respect to the aspects of the movement concept is shown in table 4. There are many correct conceptions for Sign aspect and Operator evolution, but just a few of them are at level 1. There is an important amount of incorrect conceptions at level 1 for Inequality orientation. This is coherent with the fact that these students have had many exercises about equations and not many about inequations in the preceding school year. We have applied the modelling process to 221 French grade 10 students, obtaining similar results. The main result for both populations is the following: incorrect conceptions concern two contexts: (1) inequations, (2) multiplicative initial position of the argument. Last, we have modelled 30 French grade 10 students who used Aplusix during the whole school year 2003-2004. The analysis data collected at the end of the school year shows:

○ A total of 171 conceptions with 18% for level 1, 15.2% for level 2 and 30% for level 3; ○ 70% students have the CorrectSign correct global conception; ○ 40% have the SaveOrientation incorrect global conception (never change the sign of an inequality). We note that these students, who had a longer training, have more general conceptions, and that the sign aspect is rather well acquired but the orientation aspect of the inequality is not.

Discussion and future work

This work is a significant step towards the achievement of the goals we have presented in the introduction. The obtained results are coherent with opinions of teachers and with analyses "by hand" of a few students' data, but a deeper study of the coherence is necessary and will be carried out. However, we need to analyse in depth the data that are not captured by the process. For example when we find that a student has 3 conceptions, we have an interesting result, but in order to achieve a completion goal, we would like to have an opinion about the behaviours of this student that do not participate to these 3 conceptions. Some may be sleeps, other random behaviour, other rational behaviour not captured by the model.

Let us summarize our work and make a comparison with Pépite [START_REF] Delozanne | A diagnostic based on a qualitative model of competence in elementary algebra[END_REF]: ○ We collect very large sets of data in ecological situation; Pépite collect limited sets of data in specific situations; ○ Our diagnoses are based on implemented detailed knowledge which means that our system has a deep algebraic understanding of the phenomena; this is not the case of Pépite; ○ We are building a wide map of conceptions understandable by teachers. So does Pépite which domain is larger than ours, including other registers than the formal one. ○ We are preparing an operational process in which the students will learn with Aplusix in ecological situations and in which Aplusix will calculate on-line the students' conce-ptions and tell them to the teacher; this is not the case of Pépite. As Aplusix has proved to help students learn algebra, and as publishers are very interested in selling Aplusix, we hope that many students of many countries will benefit of this work in a few years. This way of modelling students is not a dynamic way in the sense of building automatically pieces of knowledge to attribute to students like [START_REF] Baffes | Refinement-Based Student Modeling and Automated Bug Library Construction[END_REF]. There are several reasons for that. First, we think that algebra of grades 8 to 10 is a too complex domain for that goal, when a deep modelling is expected; this complexity led us to consider two levels: the rule level and the conception level. Second, we are fundamentally interested by the map of conceptions and by a capacity to provide understandable descriptions of conceptions to the teachers.

Last, we began to think of a new goal which consists of making a benchmark for student's modelling in algebra. This means to give access to the data we collect and to ask the interested research teams to use these data to model students with their own methods and tools. Results of the researches would be published on a Website and compared. Two levels would be possible: a rule level and a conception level. Tools made by teams could be accessible on the Website.

Figure 1 .

 1 Figure 1. Part of the lattice for two global conceptions of the sign aspect: CorrectSign (X in the figure) and ChangeSign (Y in the figure). The abstract nodes are not represented. X has an Xbis conception which is another decomposition of the same conception (X is first decomposed in eq/ineq then add/mult; Xbis is first decomposed in add/mult then eq/ineq). X and Xbis are active at the same time. Xbis is necessary to allow to have X-add and X-mult in the lattice.

Table 1 .

 1 The seven dimensions of the general Movement rule.

	Dimension	Possible values
	Symbol of relation	= ≠ < ≤ > ≥
	Horizontal orientation	LeftToRight, RightToLeft
	Vertical orientation	NumeratorToNumerator, DenominatorToNumerator,
		NumeratorToDenominator, DenominatorToDenominator
	Initial position of the argument	InitAdditive, InitMultiplicative
	Final position of the argument	FinAdditive, FinMultiplicative
	Change of syntactic sign of the argument	SynSignChanged, SynSignUnchanged
	Change of the inequality orientation	OrientationChanged, OrientationUnchanged

Table 2 .

 2 The seven dimensions of the Local Behaviour Vector for the sign aspect.

	Dimensions (or variables)	Possible values
	Type of the exercise	Equation, Inequation
	Initial position of the argument	InitAdditive, InitMultiplicative
	Sign of the argument	SignArgPlus, SignArgMinus
	Change of syntactic sign of the argument	SynSignChanged, SynSignUnchanged

Table 3 .

 3 Distribution of the numbers of conceptions. Note that the ideal student has 3 conceptions: the correct global conception of each of the 3 aspects of the movement concept.

	Number of conceptions	0	1	2	3	4	5	6	7	8	Total
	Number of students	11	37	58	80	38	47	48	22	1	342
	Percentage	3.2	10.8	16.9	23.4	11.1	13.7	14	6.4 0.3	100

Table 4 .

 4 Distribution of the conceptions with respect to the three aspects of the movement concept.

	Level of the	Sign aspect	Inequality orientation	Operator evolution
	conception	Correct	Incorrect	Correct	Incorrect	Correct	Incorrect
	1	20	0	0	37	0	0
	2	158	0	85	4	158	0
	3	129	11	0	0	186	0
	4	0	0	0	0	335	9

This work is funded by the programme 'ACI, Ecole et Sciences Cognitives' of the French Ministry of Research.