1. The integration of computer technologies in mathematics education: what is offered by an instrumental approach? Michèle Artigue Université Paris 7 & IREM

2. The begining of the story : a command from the Ministry of Education asking me  To enter the CAS group created by the Ministry two years before for identifying the potential of CAS for mathematics teaching and learning, and for reflecting on the necessary curricular changes required by their integration into mathematics teaching  To help this group draw the lessons of the innovative and empirical work it had developed in this area, to articulate the kind of knowledge it had built, and to make it widely accessible.

3. The first phase: entering the group, listening and learning…  A group of 15 experts in the educational use of computer environments familiar with CAS,  Strongly convinced that CAS could make their teacher’s life as well as their students’ mathematical life better,  Having developed a lot of reflections on CAS and a lot of classroom situations using these,  But meeting evident difficulties at identifying what could be considered as the results of these.

4. Six months later, a first research project: addressing the issue of CAS potential by  Investigating what was identified by experts as potential of CAS, with what kind of evidence through questionnaires and survey literature.  Investigating how these potential actualized in experts’ teaching practices through the observation of selected classroom sessions.

5. What results ?  The homogeneity of experts’ discourse about the potential of CAS for maths teaching and learning, and the limited evidence underpinning it,  The evident gap between discourses and practices.

6. An homogeneous discourse about the potential of CAS  Supporting experimental approaches, the management of more complex problems, of more realistic problems.  Providing immediate feed back to students’ actions.  Scaffolding for students meeting difficulty with algebraic techniques.  Allowing the devolution to students of mathematical work on syntactic aspects of algebra.  Supporting conceptual work and learning by freeing students from the technical algebraic burden.

7. But, contrasting with this…  At the exception of two cases, answers to questionnaires evidencing not a real integration of CAS, but only an episodic use in the best cases.  Evident discrepancies between predictions, teachers’ visions and the reality of students’ functionning in classroom observations.

8. Observations of students in classroom sessions showed: 1. The existence of two opposite tendencies : One favouring reflective and strategic work, One tending to save reflection or reduce the global coherence of action. 2. The cognitive cost of interpreting feed-back / the diversity of possible actions. 3. The fact that technical work changed but did not disappear at all.

9. What lessons?  The sign of an evident under-estimation of crucial phenomena such as the computer transposition of mathematical knowledge, of instrumental issues,  The sign of an inadequate vision of the relationships between conceptual and technical work in such environments

10. The second research project around the TI92: an opportunity for investigating  The relationships between conceptual and technical work,  Instrumental issues and the key role these play in learning and teaching processes, both at individual and institutional level. ERES (Montpellier) DIDIREM (Paris) EQUIPE TICE (Rennes)

11. The need for an adequate theoretical frame  Taking some distance from the dominant perspectives in CAS research at that time And:  Engaging in an approach that would force us: not to underestimate the role of techniques and instrumental mediations to mathematical knowledge, to integrate the institutional dimension of learning processes into the reflection.

12. Some key points in the anthropological approach (Chevallard) Mathematical objects arise from institutional practices : « praxeologies » Praxeologies can be seen as complexes of tasks- techniques-technology- theory Techniques have both a pragmatic and epistemic value The advance of knowledge goes with the routinisation of some tasks and techniques

13. Some resulting conjectures: The technological evolution breaks the traditional balance between conceptual and technical work:  by reducing the cost of the technical work, and thus the routinisation needs,  by changing the pragmatic and epistemic values of techniques,  by introducing new conceptual needs through the computer transposition of mathematics knowledge What are exactly these changes in the case of CAS technology and how to efficiently cope with these ?

14. Complementing the didactic anthropological approach by an ergonomic one (Rabardel) To the instrument Instrumental genesis From the artefact InstrumentalisationInstrumentation ConstraintsNew potential

15. One particular example : framing schemes f(x)=x(x+7)+9/x

16. Why choosing to rely on cognitive ergonomy?  A long tradition of scientific cooperation with the Rabardel’s team  The publication of a synthetic book about instrumental approaches at the time the research was begining  But nevertheless a clear awareness of essential differences as regards instrumented learning between professional and educational contexts: the crucial problem of legitimacy and values

17. New research questions  What about instrumental genesis in the specific case of CAS technology ?  Can such a genesis be efficiently managed in the current secondary mathematical culture ? Under what conditions ?  Focusing on one particular emblematic theme: the theme of functional variation at grade eleven in scientific sections (Defouad).

18. The methodology A qualitative study based on (the selection and following up all the year long of 8 students, through regular questionnaires, classroom observations, interviews, and systematic collect of written productions (beyond the data collected for the global project).

19. The interviews  Informations about the use out of school, in the most recent assessment, about personal structuration…  Conjecturing the variations of a particular function and trying to prove the articulated conjectures – free use of the TI92 – a function out of the range of the students’ familiar objects

20. What results ? The unexpected complexity of instrumental genesis First interview : understanding the variations of f(x)=x(x+7)+9/x

21. The second step: symbolic computations CAS gives you everything you need but…

22. Then, coming back to the graphic application

23. Further verifications using tables and zooms

24. The third interview

26. The instrumental genesis of variation  A slow progression from the graphic calculator culture towards the CAS culture  The resulting change in the status of the different applications (Home, Graphic, Table)  An evident dependence of this progression on the evolution of mathematical knowledge  Specific phenomena : zapping, over- verification strategies, explosion-reduction phases How to explain such results ?

27. The ordinary life of techniques in their relationship with conceptualisation Solving new problems Exploratory phase: Craft work Selection, improvment, institutionalisation of some techniques Routinisation and investment in more complex situations Development of a « theoretical » discourse Personal techniques Offical techniques

28. What changes with instrumented techniques? During the first experimentation:  no official selection,  legitimation but not institutionalisation,  a « theoretical discourse » reserved to paper and pencil techniques Instrumented techniques remained private objects which were not officially worked out

29. Why? Some specific difficulties… The diversity of commands and possible techniques The mixture of computer and mathematics knowledge engaged in explanation and jusitication, including new math. knowledge The problematic accessibility of technical knowledge The distance with ordinary norms and values of mathematics teaching

30. Becoming aware of such constraints and difficulties: the second experimentation Some essential changes:  drastic selection of commands officially used  Institutionalisation of a selected set of instrumented techniques and development of an associated ‘technological’ discourse  official work of routinisation of instrumented techniques  management of the didactic contract as regard instrumented techniques and their relationships with paper/pencil techniques taking into account its necessary evolution All of this with evident positive effects

31. Revisiting the dialectics technical/ conceptual: the epistemic value of instrumented work and techniques Standard environment CAS environment Immediate results Step by step solving Multiplicity of accessible results Surprising results New mathematical needs

32. Understanding discretisation processes and their graphic effects : f(x)=sin(x)/x

33. Understanding CAS algebraic transformations and simplifications and learning to efficiently use these An opportunity for deepening knowledge about algebraic equivalence, relationships between « sense » and « denotation », and for addressing syntactic issues

34. Understanding exact – approximate computation modes

35. Two different kinds of situations  Those arising from the use of the technology itself, and especially from the new mathematical needs induced by the computer transposition of mathematical knowledge  Those which take benefit from the pragmatic potential of CAS for introducing generalisation issues, modelling activities, and solving more complex problems Balancing the pragmatic / epistemic valences of instrumental use for efficiently linking technical and conceptual work

36. Summarizing  A progressive and dialectic development of research questions and theoretical frames  Identifying phenomena, finding reasonable explanations to these, integrating these phenomena in coherent systems  A better understanding of the cost and conditions for effective integration of CAS  New conceptual tools in order to address issues linked to learning and teaching processes in computer environments more generally