THE GENESIS OF SIGNS BY GESTURES. THE CASE OF GUSTAVO Ferdinando Arzarello , Francesca Ferrara

1. THE GENESIS OF SIGNS BY GESTURES. THE CASE OF GUSTAVO Ferdinando Arzarello , Francesca Ferrara, Ornella Robutti, Domingo Paola

2. This work is within a research project, aimed at studyng the relationships among gestures, signs, instruments and concepts in math learning. It involves my group in Turin (L. Bazzini, O. Robutti, F. Ferrara, C. Sabena, D. Paola & others) and other researchers in Modena (M. Bartolini Bussi) and Siena (M.A. Mariotti). There are also links with people in other countries, e.g.: L. Edwards, R. Nemirovsky, L. Radford.

3. OVERVIEW The theoretical framework The experiment Comments and conclusion

4. Researchers from psychology claim that gestures play an active role in thinking, intending, communicating not as mutually exclusive functions (McNeill, 1992; Goldin-Meadow, 2003). Recent results from neurosciences (the mirror neurons: Rizzolatti & Arbib, 1998) give new evidence on the link between action, gesture and speech.

5. GENERAL PROBLEM What happens in math learning? Are there specific features, which shape differently the role of gestures?

6. CLAIM Gestures and actions with artefacts support the genesis of new signs and concepts.

7. FOCUS OF THE PAPER: An example of this genesis.

8. General Frame: a semiotic analysis (1) Mathematics As abstract entities, mathematics objects require signs to be made somehow perceivable. 1 + 2 = 3

9. General Frame: a semiotic analysis (2) Gestures are signs Vygotsky: “A gesture is specifically the initial visual sign in which the future writing of the child is contained as the future oak is contained in the seed. The gesture is a writing in the air and the written sign is very frequently simply a fixed gesture.”

10. Symbols Artefacts Gestures Speech General Frame: a semiotic analysis (3) Gestures Speech ACTIONS

11. 2. The experiment

12. 1 2 The experiment Students of the 8th grade, working group activity. Task: Find the solid figure that fits the 3D gap obtained if two congruent regular squared-based pyramids are placed (on the same plane) with two sides of the bases touching each other. The pupils are asked to imagine the geometric configuration and get the solution ‘in their mind’, without using any kind of concrete support (paper and pencil not allowed) and to tell what is the solid. The teacher observes the group work without interfering.

13. We will focus our analysis around two pupils: Sara and Gustavo. Gustavo Sara

14. Three phases: • EXPLORING • CONJECTURING • SOLVING

15. 1) EXPLORING

16. REPRESENTATIONAL GESTURES (S. Kita, L. Edwards) in the GESTURE SPACE TO EXPLORE

17. CLIP 1

18. 2) CONJECTURING

19. Phase A: a new gesture space In order to allow Gustavo’s mates to see the shape of the unknown solid, new signs are introduced, with a more “concrete” nature than the previous gestures: imaginary segments that are handled as if they were real; in fact they can support the imaging of pupils in a more stable way.

20. CLIP 2

21. 2) Conjecturing: Phase B Along the discussion, a conflict appears into the whole group: the opposite behaviour of the two skew sides above and below. Gustavo elaborates a dynamic explanation, expressed through speech + gestures. To illustrate the figure he involves his mates in a fresh shared gesture space.

22. CLIP 3

23. Gustavo: It is made of two triangles with the bases below, and two triangles with the basesabove. … Gustavo: Yeah, it is a solid, made of two triangles placed with the basesbelow, which are those starting in this way and going up, and two triangles with the basesabove that are those going in this way A mixed language is used: math, space & dynamics indexes

24. 3) SOLVING

25. But gesture and speech are not enough! To overcome the obstacle, something different is to be used: something really existing, that can be effectively seen, touched, that is acted on. At this point, a last phase of production begins: a tool enters the scene:…

26. Gustavo: Guys, we’ve found the solution to all our problems! [he has taken a piece of plasticine by the hands of a group mate]

27. Through the plasticine, the students are able to check their conjectures and to reach the solution: the new solid is identified! Angela: It’s a thing made in this way, it’s strange Lucy: Try to put it Sara: But it is a pyramid with triangular bases!

28. 3. Comments and conclusions

29. SPACE OF ACTION, PRODUCTION & COMMUNICATION clip1 clip2 clip3 Indexical fnctn Symbolic fnctn A triangle with a thickness A tent Two triangles with the bases below, and two triangles with the bases above A pyramid with a triangular basis GESTURE SPACE METAPHORS GEOMETRY DYNAMIC GENESIS 2D Vs 3D

30. We can interpret this last moment using the theory of instrumentation of Rabardel (1995): Our students take an instrument they have and know, with its schemes of use, and, through their perceptuo-motor information about it, they add new schemes of use in order to solve the problem.

31. The relevant point of the activity is the evolution of processes that generate (virtual / concrete) signs, to make palpable what is imagined. Gestures are used as a tool for this and in the end are subsituted by a new tool, the plasticine. Symbols Speech Gestures Gestures and artefacts as tools, which generate new signs Artefacts Conclusions

32. Thinking is not a process that takes place “behind” or “underneath” bodily activity, but it is the bodily activities themselves. Nemirovsky (2003)

33. Thank you!