1. Pragmatikus és episztemikus érték a CAS segítségével megvalósított oktatásban Pragmatic and epistemological values in CAS enhanced mathematics education Sárvári Csaba Pécsi Tudományegyetem Pollack Mihály Műszaki Főiskolai Kar Matematikai Tanszék 2004. május 21-22.

3. Material component Psychological component {Schemes organizing the activities} Instrument Mental component Technical component An instrument is the result of construction by a subject, in a community of practice, from a given tool, through a process, the instrumental genesis. (Luc Trouche, 2003)

4. Scheme: invariant organization of behavior for a given class of situations, the psychological locus of the dialectical relationship between gestures and operative invariants, i.e. between activity and thought. Technical component: sequence of actions that one performs on a machine in order to obtain a certain goal. Mental component: mathematical objects involved; mental image of the problem solving process and the machine actions.

5. Instrumental genesis A tool ? Its constraints, its possibilities A subject Her/his knowledge, her/his work method Instrumentation Instrumentalization An Instrument Tool + instrumented action schems

6. Instrumental orchestration (Luc Trouche): External guidance of student’s instrumental genesis A plan of action, partaking in a didactic exploitation system which an institution organizes with the view of guiding students’ instrumented action. Components of instrumented orchestration: a set of individuals a set of objectives a didactic configuration a set of exploitation of this configuration

7. Instrumental orchestration on our courses E-learning web-portal

8. Paul Drivers (2002): The value of the instrumentation theory is that it provides a specific look at the interaction between student and technological tool, and in particular it shows how seemingly technical obstacles can be related to conceptual difficulties. Anthropological and socio-cultural approach (Michele Artigue): Mathematical objects are not absolute objects, but are entities which arise from practices of given institutions. Sensitive to the role played by instruments in mathematical activity Wish to rehabilitate the technical work.

9. Technique: a complex assembly of reasoning and of routine subtasks. Techniques don’t have only a pragmatic value which permits them to produce results, they have also an epistemological value. Epistemological value of technique By understanding of the objects Source of new questions

10. Complementary sources in what the mathematical work instrumented by CAS can offer to conceptualisation, to the progression of mathematical knowledge (Artigue): 1. Mastering of instrumented techniques Partially compensation for the reduction in epistemic value resulting from the immediateness of results! o Algebraic equivalence and simplifications o Exact-approximate o Relation between semantics and syntax o Status of numbers

11. 2. New potential offered by instrumented techniques o Access to generalisation through symbolic computation integrating parameters o Assistance to computation and symbolic proofs o CAS allow the students to prove in a symbolic way the regularities graphically observed (Lagrange, Trouche ( 1996), Drivers (2000,Freudenthal Institute)

12. New epistemic value, new equilibrium, between conceptual and technical work, between epistemic and pragmatic value in our praxis: New pragmatic value of procedures involves new epistemic value (Newton)Newton Possibility operative using of representations such as recursive representations, earlier didn’t have pragmatic value (recursive)recursive Possibility to negotiate problems without CAS- in our praxis- unapproachable (simplex) Multilateral concept building combining different methods and representations (Varianten)Varianten Possibility of model building in teaching process (Kepler)Kepler Chance to build more connected the knowledge- representation net (L’Hospital)L’Hospital

13. Köszönöm a figyelmet! Thank you for your attention!