1. 1 The role of gesture, signs and instruments in mathematical conceptualisation Ferdinando Arzarello, Francesca Ferrara, Ornella Robutti (*) Domingo Paola (°), Cristina Sabena (*) (*) Dipartimento di Matematica, Università di Torino (°) Liceo Issel, Finale Ligure ISGS – Interacting Bodies Lyon, 15-18 June 2005

2. 2 General Frame As abstract entities, mathematical objects require signs to be made somehow perceivable to the students. 1 + 2 = 3

3. 3 Gestures are signs “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.” (Vygotsky) General Frame

4. 4 Gestures & Words General Frame Icon Index Symbol (Peirce)

5. 5 General Frame Gestures & Words Artefacts

6. 6 General Frame Artefacts Gestures & Words Students often need to see, to touch, and to manipulate. As a consequence, the environment plays a crucial role in learning math.

7. 7 Students of the 8 th 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 activity 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). The teacher observes the group work without interfering.

8. 8 We will focus our analysis around two pupils: Sara and Gustavo

9. 9 Three phases: 1) EXPLORING 2) CONJECTURING 3) SOLVING

10. 10 1) EXPLORING

11. 11 1)EXPLORING pupils re-construct the geometry of the given configuration

12. 12 REPRESENTATIONAL GESTURES

13. 13 Gustavo’s gestures are first iconic; then become indexes of figures; finally acquire a symbolic status, referring to the world of Mathematics (where entities like triangles, pyramids, … live). GUSTAVO: It’s a triangle, but with a thickness. It’s a solid of pyramid. The transition is supported by and accomplished through verbal cues:

14. 14 2) CONJECTURING

15. 15 2) CONJECTURING Gustavo “draws” on the desk to explain his hypothesis GUSTAVO: It is in this way, in this way, and then in this way.

16. 16 New signs emerge New signs emerge: imaginary segments handled as if they were real 2) CONJECTURING Gustavo “deletes” while explaining his hypothesis

17. 17 GUSTAVO: It is made of two triangles with the bases below, and two triangles with the bases above. SARA: But, it’s a solid. It [the problem] tells a solid, one! GUSTAVO: Yeah, it is a solid, made of two triangles placed with the bases below, which are those starting in this way and going up, and two triangles with the bases above that are those going in this way.

18. 18 3) SOLVING

19. 19 Hands are not enough. Something different is used: something really existing, that can be effectively seen, touched, manipulated. 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]

20. 20 3) SOLVING Through the plasticine, the students are able to check their conjectures and to reach the solution. 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.

21. 21 General Frame Artefacts Problem Solving Gestures & Words