Sør-Trøndelag University College, March 2013

1. Learning mathematics in rich environments: solving problems by building efficient systems of instrumentsMathematical modelling as an approach to teaching and learning mathematics in (lower secondary) school Sør-Trøndelag University College, March 2013 Luc Trouche French Institute of Education, ENS de Lyon

2. Early instruments for navigation. Plate XX from N. Bion ‘s The Construction and Principal Uses of Mathematical Instruments. Translated from the French. To Which Are Added The Construction and Uses of Such Instruments as Are Omitted by M. Bion; Particularly of Those Invented or Improved by the English. By Edmund Stone. . . (London, 1723). http://libweb5.princeton.edu/visual_materials/maps/websites/pacific/introduction/introduction.html

3. Plan Mathematics and tools, a very ancient common story Elements of an instrumental approach of didactics A first example Orchestration, a teaching challenge Examples to work with Discussion and perspectives towards new collaborative tools

4. Mathematics and tools, a very ancient common story Two sides of an old Babylonian tablet (2000 BC), 10cm x 10cm, highly structured (5 levels), bearing about one hundred of mathematical problems…

5. Mathematics and tools… The Mohr–Mascheroni theorem states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone (Mohr, 1672, Maschieroni1797)

6. Mathematics and tools… The four colors theorem states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. It was proven in 1976 by K. Appel and W. Haken. It was the first major theorem to be proved using a computer.

7. Mathematics and tools… A permanent coexistence of several artefacts influencing the way of doing and thinking mathematics… In this case: a new way of computing (“indian computation”) and an old way (abacus)… A transition during, from the south to the north of France, several centuries…

8. Mathematics and tools… A permanent coexistence of several artefacts influencing the way of doing and thinking mathematics… In this case: two sides of the same medal…

9. Mathematics and tools… A permanent coexistence of several artefacts influencing the way of doing and thinking mathematics… The digital metamorphosis: a set of tools in a single envelope, portable, tactile…

10. Mathematics and tools… Finally, what artefacts are involved in the practice of mathematics? Material or symbolic: language, semiotic registers (integer numbers, plane geometrical figures…); algorithms; compass and rulers; calculators; various software… At three levels: primary artefacts, mode of use, internal representation of the artefact itself Mathematical artefacts, or artefacts use for mathematical purpose… Used by an individual as an isolated artefact, or in combination with other artefacts Implicit or explicit use Individual or collective artefact…

11. Mathematics and tools… A set of artefacts A set of artefacts always changed by the adding of new artefacts, leading to an internal reorganization For today: the toolkit will include Geogebra and a Pad, specific artefact dedicated to collaborative work (and reflective practice)

12. An instrumental approach of didactics Nec manus nuda, nec intellectus sibi permissus, multum valet; instrumentis et auxiliis res perficitur; quibus opus est, non minus ad intellectum, quam ad manum. Neither the naked hand nor the understanding left to itself can effect much. It is by instruments and helps that the work is done, which are as much wanted for the understanding as for the hand. Francis Bacon, London, 1561-1626

13. An instrumental approach of didactics • A tradition : • The idea of technè (Plato) • Working, tools and learning, existence and conscience (Descartes, Diderot, Marx) • Heir of this tradition, Vygotski (quoting Bacon) situates each piece of learning in a world of culture where the instruments (material as well as psychological) play an essential role. • Same idea in the Activity Theory (Engeström 1999) [who refers to the word tätigkeit, implying the principle of historicity]

14. An instrumental approach of didactics • A tradition : • The idea of technè (Plato) • Working, tools and learning, existence and conscience (Descartes, Diderot, Marx) • Heir of this tradition, Vygotski (quoting Bacon) situates each piece of learning in a world of culture where the instruments (material as well as psychological) play an essential role. • Same idea in the Activity Theory (Engeström 1999) [who refers to the word tätigkeit, implying the principle of historicity]

15. An instrumental approach of didactics Artefacts are only propositions exploited or not by users (Rabardel 1995/2002) Two processes closely interrelated, instrumentation and instrumentalisation : “Students’ activity is shaped by the tools, while at the same time they shape the tools to express their arguments” (Noss & Hoyles 1996) An instrument as a result of an individual and social construction, oriented by tasks, then context dependent, in a given community Task to perform, context of work Instrumental genesis A subject An artefact Instrumentation Instrumentalisation An instrument (to do something) = an artefact (or a part of) and an instrumented scheme

16. An instrumental approach of didactics Task to perform, context of work Instrumentation is a process through which the constraints and potentialities of an artifact shape the subject’s activity. It develops through the emergence and evolution of schemes while performing tasks Instrumental genesis A subject An artefact Instrumentation

17. An instrumental approach of didactics

18. An instrumental approach of didactics Task to perform, context of work A process of personalisation and transformation of the artefact Externalization, vs. internalization. “Vygotsky (…) not only examined the role of artefacts as mediators of cognition, but was also interested in how children created artefacts of their own to facilitate their performance” (Engeström 1999) Neither a diversion, nor a poaching… But an essential contribution of users to the conception of artefacts Instrumental genesis A subject An artefact Instrumentalisation

19. An instrumental approach of didactics Task to perform, context of work A process of personalisation and transformation of the artefact Externalization, vs. internalization. “Vygotsky (…) not only examined the role of artefacts as mediators of cognition, but was also interested in how children created artefacts of their own to facilitate their performance” (Engeström 1999) Neither a diversion, nor a poaching… But an essential contribution of users to the conception of artefacts Instrumental genesis A subject An artefact Instrumentalisation

20. An instrumental approach of didactics A set of artefacts intervening in each mathematical task Being able to articulate them, an essential objective of mathematics learning A challenge for conceptualisation (coordinating several semiotic registers, a need to distinguish a concepts and its representations – see the case of function) A powerful way for solving problems Instrumental geneses A subject Several artefacts Instrumentation Instrumentalisation A set, or a system of instruments ?

21. An instrumental approach of didactics Coordinating several semiotic registers, a need to distinguish a concepts and its representations

22. First exercise Three circles have the same radius, and pass through the same point O. What about the three other intersection points I, J and K?

23. First exercise Three circles have the same radius, and pass through the same point O. What about the three other intersection points I, J and K?

24. First exercise Three circles have the same radius, and pass through the same point O. What about the three other intersection points I, J and K?

25. First exercise Three circles have the same radius, and pass through the same point O. What about the three other intersection points I, J and K?

26. First exercise Three circles have the same radius, and pass through the same point O. What about the three other intersection points I, J and K?

27. Orchestration, a teaching challenge A great diversity of environments, a very rapid evolution A necessity to think how to monitor students instrumental geneses, according to the mathematical situations that student face, and to the technological environments where these mathematical situations take place.

28. Orchestration, a teaching challenge A “milieu” for mathematics learning An orchestration (= a scenario) An environment (= a set of artifacts) A crucial need to think the space and time of the students’ mathematics work. A crucial need to organize the artefacts (available, or to be introduced), in relation with the problem, the phases of its solving, the didactical variables, the learning objectives. A mathematical situation A AB = AC = 5 What is the aera of the triangle ABC? B C

29. Second exercise… A “milieu” for mathematics learning An orchestration An environment (= a set of artifacts) A problem to solve, in a reflective way (what artefacts could be used, what combination of artefacts…) Then, some elements of a possible orchestration to design, for implementation of this situation in a mathematics classroom (grade 10 students) Different scenarios, according to different pedagogical objectives… A mathematical situation A AB = AC = 5 What is the area of the triangle ABC? B C

30. Second exercise… Objective : the concept of function Environment: rulers and compass, and network of calculators Measures of the different data (BC, height), computation of the corresponding aera, and gathering by the teacher of the couples (BC, area) on the shared screen

31. Second exercise… Looking for a formula, co-elaboration of a solution modelling the given problem Is there a maximum, where and why? area Measure of BC

32. Second exercise… Second environment Objetivo: the concept of function Environment including Geogebra Students working by pairs

33. Second exercise…

34. Second exercise…

35. Second exercise… Extension of the problem AB = 5, AC = 4

36. Second exercise…

37. Discussion and perspectives Orchestration in a double perspective: Articulating the different instruments beeing developed by all the students in a given classroom Articulating the different instruments being developed by a given student in his/her mind (instrument for analysing the variation of a function, instrument for analysing a geometrical figure, etc.) Complex processes, needing to careful prepare un teaching session… Dynamic + collaborative artefacts: to be carefully implemented…

38. References Engeström, Y. & al. (1999). Perspectives on Activity Theory. Cambridge: Cambridge University Press Gueudet, G., & Trouche, L. (2011). Mathematics teacher education advanced methods: an example in dynamic geometry. ZDM, The International Journal on Mathematics Education, 43(3), 399-411. Guin, D., & Trouche, L. (1999). The Complex Process of Converting Tools into Mathematical Instruments. The Case of Calculators. The International Journal of Computers for Mathematical Learning, 3(3), 195-227. Maschietto, M., & Trouche, L. (2010). Mathematics learning and tools from theoretical, historical and practical points of view: the productive notion of mathematics laboratories. ZDM, The International Journal on Mathematics Education,42(1), 33-47. Noss, R., & Hoyles, C. (1996). Windows on Mathematical Meanings: ‪Learning Cultures and Computers. New York: Springer Rabardel P. (1995, 2002). People and technology, a cognitive approach to contemporary instruments (retreived from http://ergoserv.psy.univ-paris8.fr/) Trouche, L., Drijvers, P., Gueudet, G., & Sacristan, A. I. (2013). Technology-Driven Developments and Policy Implications for Mathematics Education. In A.J. Bishop, M.A. Clements, C. Keitel, J. Kilpatrick, & F.K.S. Leung (Eds.), Third International Handbook of Mathematics Education (pp. 753-790). New York: Springer. Trouche, L., & Drijvers, P. (2010). Handheld technology for mathematics education, flashback to the future. ZDM, The International Journal on Mathematics Education, 42(7), 667-681. Trouche, L. (2004). Managing the complexity of human/machine interactions in computerized learning environments: guiding students’ command process through instrumental orchestrations. TheInternational Journal of Computers for Mathematical Learning, 9, 281-307.