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Theorising the dialectic between technique and concept in mathematical activity: Reflections on recent French studies of the educational use of computer algebra systems Kenneth Ruthven University of Cambridge School of Education

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Key orientations Overarching aim - to find a systematic way of thinking about the learning of mathematics in technological environments. Primary motivation - to devise powerful theoretical tools with which to gain greater control over practical situations. Important influence - reservation about dichotomisation of technical proficiency and conceptual understanding.

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Virtual mathematics Our sense of interacting with mathematical objects is shaped by, and expressed through, the tasks and techniques which give them form and substance, and the technical and theoretical discourses surrounding these. New technologies have the potential to provoke a rebalancing of our senses, or to reorganise a particular sense.

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Diamathematical variation While in society at large there are multiple, dynamic diamathematics, the normalising function and institutionalised character of schooling tend to maintain a stable, standardised school diamathematic. This gives rise to a tendency for new technologies to be envisioned as instruments of pedagogical transformation rather than mathematical redefinition.

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Idiomathematical invention While idiomathematical invention plays an important part in thinking and learning, concerns for effective communication and favourable evaluation encourage adoption of the standardised school diamathematic. This leaves teachers and students ill prepared to handle a congestion of idiosyncratic and improvised techniques in the absence of established norms for the use of new tools.

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Culture, cognition and machine To understand the dual processes whereby machine operations are instrumentalised and student actions instrumented, we must link the anthropological concept of technique and the psychological concept of scheme. These might be taken as two faces of an invariant organisation of actions in relation to a particular class of situations, with scheme referring to a cognitive structure, and technique to a cultural system.

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The cognitive concept On the cognitive face, even the meanest perceptual or gestural act depends on a corresponding scheme; and those forms of action and expression which we typically deem mathematical depend on schemes within schemes within schemes. In this sense, no technique can be anything other than conceptual; and no development of technique can proceed without the development of concepts.

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The cultural concept There remains a tendency to accept a reductive characterisation of human technique as lower- order mechanical skill and to press for higher-order conceptual understanding. This resonates with an emphasis on broad unifying ideas within the scholarly culture of mathematics; and the high valuation placed on the compact and elegant codification of these ideas at the apex of the cultural system.

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Integrating concepts Unifying ideas are careful orchestrations of successive layers of more fundamental ideas. Students develop schemes well adapted to the tasks, techniques and discourses they meet. If tasks are strongly compartmentalised, techniques highly prescribed, and discourses severely restricted, then the mathematical competences that students develop will be correspondingly fragmented and inflexible.

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The communicative interface The proposed theoretical framework seeks to create a system through which the social and individual faces of thinking, learning and teaching can be analysed in conjunction. This requires deeper analysis of the communicative interface where cognitive scheme meets cultural technique.

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Routinisation and accounting An important feature of routinisation is the way in which it tends to curtail -even suspend- the social custom of accounting for ones actions. Accounting underpins the inter-psychological processes of appropriation which play a central part in classroom teaching and learning. It is through accounting that schemes (cognitive structures) and techniques (cultural systems) are publicly manifested and socially co-ordinated.

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Classroom accounting An account should be just as full as is necessary for it to be intelligible to its intended audience. Often, the component inscriptional actions of the technique are sufficient to provide an account. This depends on the audience being able to bring to bear schemes which enable them to recognise - and usually elaborate- the component actions. Where task and/or technique are not standard for the audience, the inscriptional record has to be filled out by some degree of commentary.

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CAS simplification While automatic simplification by CAS may be highly efficient in practical terms, in epistemic terms it deprives users -notably learners- of information -and commentary- on intermediate states and procedures which would help in conceptualising the simplification relationship between initial and final expressions. On the other hand, classroom use of a CAS exposes students to a less controlled and more varied range of expressions than customary.

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An example from Artigue

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The teacherly CAS It was students whose prior grasp of algebra was weakest for whom the gaps in these CAS- generated accounts were most disruptive. One suspects that students epistemic stance towards the CAS, as well as their level of algebraic expertise, affected what they learnt. The interactions above can be treated as a form of dialogue in which the CAS rephrases the students contributions and then gives the answer to the simplification problem.

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An example from Lagrange

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The c[l]ueless CAS The CAS is unable to operate pragmatically; it can only operate syntactically, in terms of elections made by the user, or of preset defaults which may serendipitously coincide with the pragmatic sense, or conflict with it. To the extent that simplification in particular - and mathematical reasoning in general- is event-driven, goal-oriented, situated action, the instrumentalisation of a CAS depends heavily on the pragmatic framing of commands and reframing of results by the user.

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Technical and conceptual work Effective use of computational technologies requires their instrumentalisation to create mathematical tools. Far from eliminating technical work, or setting it apart from conceptual work, instrumentation involves these aspects of mathematical thinking being reorganised jointly.

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