O n embodiment, artifacts, and signs: a semiotic-cultural perspective on mathematical thinking

1. On embodiment, artifacts, and signs:a semiotic-cultural perspective on mathematical thinking Luis Radford*, Caroline Bardini*, Cristina Sabena§, Pounthioun Diallo*, Athanase Simbagoye* * Université Laurentienne, Canada.§ Università di Torino, Italy A research program funded by the Social Sciences and Humanities Research Council of Canada

2. Relocating the Body in the Cognitive Map • Two fundamental claims: • Concepts are crucially shaped by our bodies and our senses. • the body is a locus for the production of meaning and the first opening of intentionality towards the world.

3. The Rationalist Tradition • if we are ever to have pure knowledge of anything, we must get rid of the body and contemplate things by themselves with the soul (Plato, Phaedo, 66b-67b)

4. The sensual and the conceptual • Kant presented a conception of mind wherein the very content of knowledge is determined by sensible as well as intellectual conditions

5. J. Piaget How exactly are we to understand the role of the sensual and the conceptual? • Sensorimotor stage • the ontogenetic evolving logic-mathematical structures pick sensual knowledge up and transform it into abstract thinking.

6. Theories of Embodiment • Sensorimotor activity is not merely a stage of development that fades away or disappears in more advanced stages, but rather is thoroughly present in thinking and conceptualizing.

7. The body is a locus for the production of meaning, but … • the world that the body encounters is a cultural world populated by other bodies, objects, signs, and meanings, a world already endowed with ethical, aesthetical, scientific and other values. • These values provide the world with specific configurations that, instead of being neutral, qualify the body with the historicity of events and concepts deposited in language, artifacts, and institutions. (M. Foucault)

8. A dialogue to overcome criticism • In the end, the embodied perspective is no more than “the return in a more sophisticated register of the old organicism” or else a token of “the post-modern cult of pleasure” and love for the concrete. (Terry Eagleton)

9. Thinking as a form of social praxis • thinking is an active mode of social action in which what we know and the way we come to know it is framed by cultural forms of rationality out of which specific kinds of questions and problems are posed.

10. Pierre and Marthe’s graph walk Thinking as a form of social praxis Grade 9 regular Math lesson

11. In reflecting about this problem … • the students became involved in a socialpraxis that goes back to the 15th century • the problem on which the students had to reflect was framed by a cultural kind of rationality that legitimizes the problem and endows it with meaning

12. Understanding Cultural and subjective meanings … the transformation or interpretation of a sign into a previous sign for which the individual has attained a more or less stable cultural meaning

14. Embodiment and the Language of Space Karla: They’re further! Cindy : No, I think that they are closer here … and there they move away, and they move away here and there they arrive together

15. Thinking as located in body, artifacts, and signs • “the human brain is thoroughly dependent upon cultural resources for its very operation; and those resources are, consequently, not adjuncts to, but constituents of, mental activity” Clifford Geertz

16. Towards a cultural conception of Embodiment • The graph, the calculator, the CBR, are cultural resources which bear an embodied intelligence (Pea) • They carry in themselves, in a compressed way, socio-historical experiences of cognitive activity and artistic and scientific standards of inquiry (Lektorsky).

17. Mesopotamian Math Pre-cuneiform Signs 1642 Pascal’s Arithmetic Machine Abacus and written numeric symbols (16th C.)

18. What allows these students to think mathematically is… • our general human unique biological constitution, • the historical mark that this biological constitution has left in our language (e.g. in the linguistic spatial metaphors that we use), but also • the historically embodied intelligence and symbolic dimension deposited in the social practices and in the whole arsenal of cultural artifacts that are integral part of human thinking.

19. The Tree of Knowledge Maturana and Varela Phylogenesis (cultural development) Ontogenesis (life-span development)

20. Our proposal … • To stretch out our concept of embodiment far beyond the realm of the body itself in order to account for the unique dimension brought forth by our historical and cultural past. • Thinking is also carried out in the signs, the tools and the historical cultural significations of the settings in which we live and grow.