Jean-baptiste Lagrange Laboratoire de Didactique André Revuz Université Paris-Diderot

1. Digital representations of mathematical objects in the teaching-learning process: a cross European research project Jean-baptiste Lagrange Laboratoire de Didactique André Revuz Université Paris-Diderot http://www.lar.univ-paris-diderot.fr

2. The ReMath project Representing Mathematics with digital medias Communication, cooperation and collaboration …for connecting ideas about representations networking theoretical frameworks

3. Plan • ReMath: questions and working plan • The ReMath’s approach • Cross case studies as a methodology • The Casyopée cross-case • Presentation • Small group work • Report and Discussion • Conclusion on ReMath 35mn 20mn 25mn 10mn

4. Representing Mathematics with Digital Media • STREP Number IST4-26751 (FP 6) • 42 months (Dec. 2005 - May 2009) • Six teams • Instituto Technologie Didattiche, ITDGenova • Università degli Studi, UNISISiena • National Kapodistrian University, ETL Athens • Institute of Education, IOE London • Université Joseph Fourier, Mehta Grenoble • Université Paris Diderot,DidiremParis

5. FP6 • European research activities are structured around consecutive programmes, or so-called Framework Programmes. 1984: First Framework Programme (1984-1987)1987:“European Single Act” -science becomes a Community responsibility1987: SecondFramework Programme (1987-1991) 1990: Third Framework Programme (1990-1994) 1993:Treaty on European Union; role of RTD in the EU enlarged 1994: Fourth Framework Programme (1994-1998) 1998: FifthFramework Programme (1998-2002) 2002: SixthFramework Programme (2002-2006) 2007: Seventh Framework Programme (2007-2013)

6. The Sixth Framework Programme (FP6) 2002-2006. • the Priority – Information Society Technologies • IST 2005-06 Work Program. • strategic objective 2.4.10 “Technology-enhanced learning (TEL)” • To explore interactions between the learning of the individual and that of the organisation … • To contribute to new understandings of the learning processes by exploring links between human learning, cognition and technologies.

7. Key Objectives • To bridge the gap between technology and pedagogy • A representations-based approach to cognition in learning mathematics • we can only access and operate on Mathematical objects by means of representations. • the potential impact of ICT tools on mathematical learning seen through the filter of representations. • Support to teachers and learnersoffering tools that address • not only individual cognition • but also the entire learning situation • Integration of efforts in the European context • how different theoretical frameworks deal with the question of representations.

8. Methodology A cyclical process of a) desiging and developing six state-of-the-art DYNAMIC DIGITAL ARTEFACTS for representing mathematics, b) developing scenarios for the use of these artefacts for educational added value c) carrying out empirical research involving cross-experimentation in realistic educational contexts

9. The project’s structure • WP1 : Theoretical integration • WP2 : Software developpement • WP3 : Scenarios (pedagogical plans) • WP4 : (cross) experimentations • WP5 : Multilingual repository and communication platform (Math.Di.L.S.) http://remath.cti.gr

11. DDAs: Digital Didactical Artefacts

12. Diversity in ReMath: DDAs • Domains and objects: • algebra, functions, 3D geometry, cinematic, geography… • Representations • connections between them, • means of action • possibilities of evolution • Distance with • usual systems of representation, • usual software used in education, • with the curriculum

13. Initial Diversity : Frameworks ETL

14. Theoretical Integration Progressive elaboration of a shared theoretical basis about representations Extension of the connections between frameworks Specific common research tools Distinction between metaphoric and functionaluse of theories The language of “concerns” The idea of “didactical functionalities” (1)Tool features, (2) Educational goals, (3) Modalities of employment

15. A special methodology: the cross case studies ETL Didirem UNISI alien familiar alien familiar alien Casyopée Cruislet

16. Epistemological Profile Casyopée • Objects represented mathematical function of one variable(families of) • dependencies in a physical system (2D geometry) • algebraic functions • Curriculum compatible but innovative • Connections and activities • Crossing two entries • Three different levels where functions can be represented • Two types of representations, with specific activities

17. Three different levels where covariation and dependency can be experienced and/or represented. Casyopée • Physical systems (dynamic geometry) • Magnitudes and measures • MathematicalFunctions

18. Representations and Types of activities Enactive-iconic Representations (Tall) Experience of movements inside physical systems Work on graphical or tabular representatives issued of physical systems ‘Explorations’ on graphs and tables of mathematical functions Algebraic Representations Semiotic Registers of representations (Duval 1999) Treatments - Conversions Three categories of activities (Kieran 2004) generational transformational global / meta-level Casyopée

19. Casyopée PME 33

20. a, b, c, 3 parameters >0 A(-a,0); B(0,b); C(c,0) Find a rectangle MNPQ of maximal area with M on [OA] ; Q on [OC] ; N on [AB] and P on [BC] Casyopée An optimisation problem

21. CONNECTIONS BETWEEN ACTIVITIES Casyopée

22. Casyopée CONNECTIONS BETWEEN ACTIVITIES

23. CONNECTIONS BETWEEN ACTIVITIES Casyopée

24. Casyopée cross-case study • Context in the two experiments • A DDA innovative but highly compatible with the curriculum. • Close epistemological and didactic references • Previous collaboration teachers/researchers • Grade levels: Grade 11 (France), Grade 12 (Italy) • Institutional pressure: High (France)/Moderate (Italy). • Teachers Familiar with DDA:Yes (France)/No (Italy).

25. Casyopée cross-case study DIDIREM and UNISI Theoretical frameworks Theory of Semiotic Mediations Activity Theory Instrumental Approach UNISI Background on Functions Semiotic Register Theory of Didactical Situations Anthropo. Theory of Didiactics DIDIREM

26. DIDIREM Anthropological theory of didactics (ATD) Ecological perspective Sensibility to institutional constraints and norms Attention to (instrumented) techniques Theory of didactic situations (TDS) Attention to students’ a‑didactic interaction with the milieu of the situation. Careful choice of tasks and control of the didactic variables

27. UNISI Theory of semiotic mediations (TSM) Gives much attention to the collective progression of mathematics knowledge Through the progressive evolution of systems of signs: students’ personal signs first linked to their activity with the artifact shared during collective activities purposefully designed, develop with the help of the teachers into semiotic chains towards mathematical signs.

28. Casyopée cross-case study The UNISI scenario The DIDIREM scenario

29. Questions: 1. What important similarities and differences between the two scenarios? Hypotheses about factors explaining these. 2. What research outcomes can be expected from a cross-study with regard to: (a) representations (b) theoretical integration (c) role of the context? Casyopée cross-case study

30. Similarities and differences Casyopée cross-case study • Two scenarios with slightly different educational goals but favouring the same type of tasks: • functions approached in terms of co-variation; • functions approached as modeling tools for problems arising in geometrical context. • Two scenarios giving high importance to the interaction between the different semiotic registers offered by Casyopée. The intertwined influence of differences in grade levels and close epistemological views.

31. Casyopée cross-case study Similarities and differences • Two scenarios paying evident attention to the process of instrumental genesis but managing it in different ways • in the UNISI scenario, an organization of the instrumentalization process mainly concentrated in the first session; • in the DIDIREM scenario, a progressive organization of the instrumentalization process along the whole scenario. The influence of a shared instrumental concern combined with the differences induced by its inscription into two different theoretical frames.

32. Casyopée cross-case study Similarities and differences Two scenarios giving high importance to the students’ autonomous work. A characteristic transcending the theoretical diversity. But a different balance between individual/group work and collective work, and a very different management of collective phases. The evident influence of differences in theoretical frameworks.

33. Summarizing the cross case study Maracci M., Cazes C., Vandebrouck F., Mariotti M-A. (2009) Casyopée in the classroom: two different theory-driven pedagogical approaches, Proceedings of CERME 6 Casyopée cross-case study • The Unisi team has mainly structured its pedagogical plan according to the Theory of Semiotic Mediation • The teacher plays a crucial role throughout the whole pedagogical plan, especially for • fostering the evolution of students’ personal meanings towards the targeted mathematical meanings • facilitating the students’ consciousness-raising of those mathematical meanings • The Didirem team : several theoretical frames. • Attention to students’ instrumental genesis • Compatibility with institutional demand • Process of learning designed through a careful choice of mathematical tasks, with an adidactical potential • But the teacher's actions and role escapes the PP’s design

34. Context

36. DIDIREM difficulties with Cruislet Cruislet cross-case analysis Instrumental sensitivity Controlled design The DIDIREM culture Anticipating the potential and limit of adidactic adaptations AAnticipating possible cognitive outcomes Epistemological concern CruisletCharacteristics Curricular distance Implementation of mathematical objects Technological distance

37. Context

38. Conclusion: Beyond ReMath Different conception of the theoretical work connections based on concrete common practice on “boundary objects” understanding • the necessity of theoretical constructs • their influence on tool and scenario design Research practices as objects for study • understanding the consistency of ‘alien’ choices • awareness of the crucial role of context in didactical research, and the need for better conceptualization An inspiration for (young) researchers?