Teaching and Learning Process in Mathematical Education: a Vygostkian approach

The purpose of this paper is to show the application of the theory of semiotic mediation, based on Vygostkij's theory, retracing the theoretical framework that supports the research in the field of mathematical education: the historical-epistemological, the instrumental and the cultural-historical approaches. The paper aims also to highlight the nature of the artifacts as mediation tools handled by teachers and children, retracing Wartofsky and his distinction between primary, secondary and tertiary artifacts. After a literature review of national and international contributions, an analysis is proposed for the main steps carried out during the implementation of the didactical cycle in kindergarten, centered on semiotic mediation with the giant abacus tool, particularly retracing the path of the Research Group in Mathematical Education of Modena and Reggio Emilia.

1. Bagni G.T. (2006). Linguaggio, storia e didattica della matematica. Bologna: Pitagora Editore.
2. Bakhtin M.M. (1986). Speech genres and other late essays. Austun: Univerity Texas Press.
3. Bartolini Bussi M.G., & Mariotti M.A. (1999). Semiotic mediation: from history to mathematics classroom. For the Learning of Mathematics, 19 (2), 27—35.
4. Bartolini Bussi M.G., & Quattrocchi P. (1992). Macchine matematiche e altri oggetti. Modena: Comune di Modena.
5. Bartolini-Bussi M.G. (2010). Historical Artefacts, Semiotic Mediation and Teaching Proof. In G. Hann, H. N. Janke, & H. Pulte, Explanation and Proof in Mathematics: Philosophical and Educational Perspectives (p. 151—168). London: Springer.
6. Bartolini-Bussi M.G. (2011). Artefatti e segni nell'inseg- namento-apprendimento della matematica: i primi anni. In B. D'Amore, & S. Sbaragli, Un Quarto di Secolo al Servizio della Didattica della Matematica (p. 55—60). Bologna: Pitagora Editrice.
7. Bartolini-Bussi M.G., & Boni M. (2011). Numeri. Scuola Materna per l'educazione dell'Infanzia, II—XXIII, n° 11, Anno XCVIII. La Scuola.
8. Bartolini-Bussi M.G., & Mariotti M.A. (2008). Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective. In L. English, Handbook of International Research in Mathematics Education. Routledge.
9. Bartolini-Bussi M.G., & Mariotti M.A. (2009). Mediazione semiotica nella didattica della matematica: arte- fatti e segni nella tradizione di Vygostky. L'Insegnamento della Matematica e delle Scienze Integrate, Vol. 32 A-B 270— 294.
10. Bartolini-Bussi M.G., & Maschietto M. (2006). Mac- chine matematiche: dalla storia alla scuola. Milano: Springer- Verlag.
11. Maschietto M., & Bartolini-Bussi M.G. (2011). Mathematical machines: from history to mathematics classroom. In O. Zaslavsky, & P. Sullivan, Constructing Knowledge for Teaching Secondary Mathematics Tasks to enhance prospec- tive and practicing teacher learning (p. vol. 6, 227—245). New York: Springer Verlag.
12. Duval R. (1993). Registres de representation semio- tique et fonctionnement cognitif de la pensee. Annales de didactique et de sciences cognitives, IREM de Strasbourg, 5, 37—65.
13. Duval R. (2005). Linguaggio, simboli, immagini, sche- mi… In quale modo intervengono nella comprensione in matematica e altrove? Bollettino dei Docenti di Matematica, nr. 50. Bellinzona (Svizzera): UIM, CDC. 19—40.
14. Fiorani H. (in press). Hands-On Maths in Kindergar- ten. In Didactic Strategies and Technologies for Education: Incorporating Advancements. (P.M. Pumilia-Gnarini, E. Favaron, E. Pacetti, J. Bishop, L. Guerra Eds.). USA: IGI- Global.
15. Fiorani H., Impedovo M.A., & Iaquinta R. (2011). Dal Positivismo al Post-Costruttivismo. Teoria, Strumenti, Didattica. Lecce: Libellula Edizioni.
16. Ligorio M.B. (2003). Come si insegna, come si ap- prende,. Roma: Carocci.
17. McLuhan M. (1962). The Gutenberg galaxy. London.
18. Norman D.A. (1993). Things that make us smart. Lon- don: Addison-Wesley.
19. Ong W.J. (1967). The presence of the word. New Haven: CT: Yale University Press.
20. Rabardel P. (1995a). Les hommes et les technologies. Approche Cognitive des instruments contemporains. Paris: Armand Colin.
21. Rabardel P. (1995b). Qu'est-ce qu'un instrument? Appropriation, conceptualisation, mises en situation. Les dossiers de l'Ingenierie educative, 19, 61—65.
22. Sfard A. (1991). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1—36.
23. Sfard A. (2008). Thinking as communicating: Human development, the growth of discourses and mathematizing. Cambridge: University Press.
24. Vygostky L.S. (1978). Mind in Society: Development of Higher Psychological Processes. Cambridge: Harvard Univer- sity Press.
25. Vygotsky L.S. (1981). The genesis of higher mental functions. In J. V. Wertsch, The concept of activity in Soviet psychology (p. 144—188). NY: Sharpe: Armonk.
26. Vygotsky L.S. (1987) Thinking and speech. In R.W. Rieber and A.S. Carton (Eds.), The collected works of L.S. Vygotsky, Volume 1: Problems of general psychology, (Trans. N. Minick). New York: Plenum.
27. Vygotsky L.S. (1997). The collected works of LS. Vygotsky, Vol. 4: The history of the development of high- er mental functions (M.J. Hall, Trans; R.W. Reiber, Ed.) New York:Plenum Press.
28. Vygotsky L. (1986). Thought and language. Cambridge: The MIT Press.
29. Wartofsky M. (1979). Perception, representation and the forms of action: towards an historical epistemology. In M. Wartofsky, Models Representation and the scientific understanding (p. 188-209.). Dordrecht: Reidel.
30. Wells G. (September, 1994). Learning and teaching "scientific concepts". Vygostky's ideas revisited. Paper pre- sented at the Vygostky and Human Sciences Conference. Moscow.