Michèle Artigue Université Paris 7 & IREM

1. Learning mathematics in a CAS environment : the genesis of a reflection about instrumentation and the dialectics between technical and conceptual work Michèle Artigue Université Paris 7 & IREM

2. Summary • Introduction • The progressive development of the theoretical frame • The unexpected complexity of instrumental genesis • The institutional status of « instrumented techniques » • Revisiting the relationships between technical and conceptual work: the epistemic value of instrumented work and techniques

3. Introduction An evident contrast between: • the impregnation by technology of most social and professional practices • the limited penetration of these into the educational world Behind this constrast, an opposition of values that CAS technology makes specially clear

4. From the professional and social sides: • A pragmatic relationship to technology, • Recognising that technology is not of an evident access, • Accepting the fact that technology strongly shapes our practices and perspectives

5. From the educational side: • A vision of technology which remains mainly a pedagogical one, technology being asked to serve values defined independently from it • An attitude that leads to see technology as something transparent with respect to knowledge and values

6. Some dangerous associations Computer technologies desqualify teaching pratices orientated towards skill acquisition, and support constructivist approaches Technology, by taking in charge the technical side, leaves the mind free for conceptual reasoning and provides new means for that The use of technology favours conceptual reasoning Reality is much more complex…

7. In fact, our first observations of students working with DERIVE clearly showed: 1. The existence of two opposite tendencies : • One favouring reflexive and strategic work • One tending to save reflection or reduce the global coherence of action 2. The fact that technical work changes but does not disappear at all 3. The essential role played by the didactic situations at stake and by the teacher’s management of these

8. This was the starting point of a collective reflection on: the relationships between conceptual and technical work which made us more and more sensitive to instrumental issues and to the key role these play in learning and teaching processes in CAS environments, both at individual and institutional level ERES (Montpellier) DIDIREM (Paris) EQUIPE TICE (Rennes)

9. The development of a theoretical frame • Taking some distance from the dominant constructivist perspectives And: • Engaging in an approach that would force us • to integrate the institutional dimension, • not to underestimate the role of techniques and instrumental mediations to mathematical knowledge

10. 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 tasks and techniques

11. 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 But the sensitivity to these changes depends on the research frames one adopts

12. Complementing the didactic anthropological approach by an ergonomic one (P. Rabardel) From the artefact To the instrument Instrumental genesis Instrumentalisation Instrumentation Schemes Constraints New potential

13. One paticular example : framing schemes f(x)=x(x+7)+9/x

14. The unexpected complexity of instrumental genesis (L. Trouche) The diversity of students’ profiles Theorist Rationalist Resources Meta-knowledge Validation Tinkerer Scholastic Experimentalist

15. The unexpected complexity of instrumental genesis (L. Trouche) • The development of specific schemes and their evolution when passing from graphic calculators to symbolic calculators • Their interaction with global schemes such as the scheme of « approximative detour »: anticipation and control – substitute – stratageme (guessing)

16. The instrumental genesis of variation (B. Defouad) First interview : understanding the variations of f(x)=x(x+7)+9/x

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

18. Then, coming back to the graphic application

19. Further verifications using tables and zooms

20. The third interview

22. 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 ?

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

24. 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 remain private objects which are not officially worked out

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

26. Becoming aware of such constraints and difficulties: the second experimentation Some essential changes: • drastic selection • official work of institutionalisation and routinisation • management of the didactic contract taking into account its necessary evolution With evident positive effects

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

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

29. 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

30. Understanding exact – approximate computation modes

31. 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 linking in a dialectic way technical and conceptual work

33. What is expected from technology? Must help students to adapt to the technological world Must allow students to master current math practices Must help to renew pedagogical practices, must provide new teaching tools, for visualising, communicating… Must save teaching and learning time Must help to understand mathematical concepts, must increase students’ math power Must make teaching and learning easier and better