The phenomenological, epistemological, and semiotic components of generalization
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Radford, Luis
Resumen
In the first part of this article, I argue that generalization involves three related components: phenomenological, epistemological, and semiotic. I also argue that the concept of generalization conveyed by theories of knowing (e.g., rationalist and empiricist) depends on the manner in which these theories understand the above three components and their interrelations. I elaborate my argument in reference to a cultural-historical dialectical concept of generalization. In the second part of the article, I provide an overview of the articles contained in this special issue and discuss their contributions to educational research.
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2015
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Referencias
Baxandall, M. (1972). Painting and experience in fifteenth century Italy - A primer in the social history of pictorial style. Oxford, United Kingdom: Clarendon Press. Becker von, O. (1936). Die lehre vom geraden und ungeraden im neunten buch der euklidischen elemente [The theory of odd and even in the ninth book of Euclid's Elements]. Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, 3(B), 533-553. D’Eredità, G. (2012). Chess and mathematical thinking. Cognitive, epistemological and historical issues. PhD Dissertation. Palermo, Italy: University of Palermo. D’Ereditá, G., & Ferro, F. (2015). Generalization in chess thinking. PNA, 9(3), 241-255. Ferro, M. (2012). Chess thinking and configural concepts. Acta Didactica Universitatis Comenianae, 15-30. Ferro, M. (2013). A multimodal semiotic approach to investigate on synergies between geometry and chess. PhD Dissertation. Palermo, Italy: University of Palermo. Foucault, M. (1966). Les mots et les choses [The order of things]. Paris, France: Éditions Gallimard. Heath, L. T. (1956). Euclid, the thirteen books of the elements (Vol. 2). New York, NY: Dover. Kant, I. (1770/1894). Inaugural dissertation. New York, NY: Columbia College. (Original work published 1770) Kant, I. (1787/2003). Critique of pure reason. (N. K. Smith, Trans.) New York, NY: St. Marin’s Press. (Original work published 1787) Lefèvre, W. (1981). Rechenstein und sprache [Computing stones and language]. In P. Damerow & W. Lefèvre (Eds.), Rechenstein, experiment, sprache. Historische fallstudien zur entstehung der exakten wissenschaften [Computing stones, experiment, language. Historical case studies on the emergence of the exact sciences] (pp. 115-169). Stuttgart, Germany: Klett-Cotta. Leont'ev, A. N. (1978). Activity, consciousness, and personality. Englewood Cliffs, NJ: Prentice-Hall. Netz, R. (1999). The shaping of deduction in Greek mathematics. Cambridge, United Kingdom: Cambridge University Press. Otte, M. (2003). Does mathematics have objects? In what sense? Synthese, 134(1-2), 181-216. Otte, M. F., Mendonça, T. M., Gonzaga, L., & de Barros, L. (2015). Generalizing is neces-sary or even unavoidable. PNA, 9(3), 141-162. Panofsky, E. (1991). Perspective as symbolic form. New York, NY: Zone Books. Radford, L. (2003). On culture and mind. A post-Vygotskian semiotic perspective, with an example from greek mathematical thought. In M. Anderson, A. Sáenz-Ludlow, S. Zellweger, & V. V. Cifarelli (Eds.), Educational perspectives on mathematics as semiosis: From thinking to interpreting to knowing (pp. 49-79). Ottawa, Canada: Legas Publishing. Radford, L. (2008). The ethics of being and knowing: Towards a cultural theory of learning. In L. Radford, G. Schubring, & F. Seeger (Eds.), Semiotics in mathematics education: Epistemology, history, classroom, and culture (pp. 215-234). Rotterdam, The Netherlands: Sense Publishers. Radford, L. (2012). On the development of early algebraic thinking. PNA, 6(4), 117-133. Radford, L. (2013a). En torno a tres problemas de la generalización [Concerning three problems of generalization]. In L. Rico, M. C. Cañadas, J. Gutiérrez, M. Molina, & I. Segovia (Eds.), Investigación en didáctica de la matemática. Homenaje a Encarnación Castro (pp. 3-12). Granada, Spain: Editorial Comares. Radford, L. (2013b). Three key concepts of the theory of objectification: Knowledge, knowing, and learning. Journal of Research in Mathematics Education, 2(1), 7-44. Radford, L. (2014a). On the role of representations and artefacts in knowing and learning. Educational Studies in Mathematics, 85, 3, 405-422. doi:10.1007/s10649-013-9527-x Radford, L. (2014b). De la teoría de la objetivación [On the theory of objetivation] Revista Latinoamericana de Etnomatemática, 7(2), 132-150. Radford, L. (2014c). The progressive development of early embodied algebraic thinking. Mathematics Education Research Journal, 26(2), 257-277. Radford, L., Bardini, C., & Sabena, C. (2006). Perceptual semiosis and the microgenesis of algebraic generalizations. In M. Bosch, G. de Abreu, M. Artaud, M. Artigue, M. Bartolini-Bussi, B. Barzel, et al. (Eds.), Proceedings of the fourth congress of the European Society for Research in Mathematics Education (CERME 4) (pp. 684-695). Sant Feliu de Guíxols, Spain: FUNDEMI IQS – Universitat Ramon Llull. Rivera, F. (2015). The distributed nature of pattern generalization. PNA, 9(3), 163-190. Santi, G., & Baccaglini-Frank, A. (2015). Forms of generalization in students experiencing mathematical learning difficulties. PNA, 9(3), 215-240. Vergel, R. (2015). Generalización de patrones y formas de pensamiento algebraico temprano. PNA, 9(3), 191-213. Vygotsky, L. S. (1987). Collected works Vol. 1. New York, NY: Plenum Press. Wartofsky, M. (1968). Conceptual foundations of scientific thought. New York, NY: The Macmillan Company. Wartofsky, M. (1977). Consciousness, praxis, and reality: Marxism vs. Phenomenology. In D. Ihde & R. Zaner (Eds.), Interdisciplinary phenomenology (pp. 133-151). Dordrecht, The Netherlands: Springer. Wartofsky, M. (1979). Models, representation and the scientific understanding. Dordrecht, The Netherlands: D. Reidel.