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Historical formation and student understanding of mathematics

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History in Mathematics Education

Abstract

The use of history of mathematies in the teaching and learning of mathematics requires didactical reflection. A crucial area to explore and analyse is the relation between how students achieve under standing in mathematics and the historical construction of mathematical thinking.

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References for §5.1

  • Bringuier, J-C. 1980. Conversations with Jean Piaget, Chicago: Chicago University Press

    Google Scholar 

  • Gillies, D. (ed.) 1992. Revolutions in mathematics, Oxford: Clarendon Press

    Google Scholar 

  • Hoyrup, J. 1995. In measure, number and weight, Albany: State University of New York Press

    Google Scholar 

  • Lizcano, E. 1993. Imaginario colectivo y creación matemática, Barcelona: Editorial Gedisa

    Google Scholar 

  • Mengal, P. 1993. ‘Psychologie et loi de récapitulation’, in: Histoire du concept de récapitulation, P. Mengal (ed.), Paris: Masson, 93–109

    Google Scholar 

  • Piaget, J. and Garcia, R. 1989. Psychogenesis and the history of science, New York: Columbia University Press

    Google Scholar 

  • Vygotsky, L. S. 1994. ‘Tool and symbol in child development’, in: Rent van der Veer and Jann Valsiner, The Vygotsky reader, Oxford UK and Cambridge USA: Blackwell

    Google Scholar 

  • Vygotsky, L. S. 1997. The history of the development of higher mental functions, (Coll. wks of L. S. Vygotsky, Vol. 4), R. W. Rieber (ed.), New York: Plenum Press

    Google Scholar 

References for §5.2

  • Burn, R. P. 1993. ‘Individual development and historical development: a study of calculus’, International journal of mathematics education, science and technology, 24, 429–433

    Google Scholar 

  • Cornu, B. 1991. ‘Limits’, in D. Tall (ed.), Advanced mathematical thinking, Dordrecht: Kluwer, 153–166

    Google Scholar 

  • Dorier, J.-L. 1998. ‘The role of formalism in the teaching of the theory of vector spaces’, Linear algebra and its applications, 275, 1–4, 141–160

    Google Scholar 

  • Hefendehl-Hebeker, L. 1991. ‘Negative numbers: obstacles in their evolution from intuitive to intellectual constructs’, For the learning of mathematics, 11 (1), 26–32

    Google Scholar 

  • Peacock, George 1830. A treatise on algebra, Cambridge

    Google Scholar 

  • Piaget, J. and Garcia, R. 1989. Psychogenesis and the history of science (trans. by Helga Feider), New York: Columbia University Press

    Google Scholar 

  • Radford, L. 1995. ‘Before the other unknowns were invented: didactic inquiries on the methods and problems of medieval Italian algebra’, For the learning of mathematics, 15 (3), 28–38

    Google Scholar 

  • Radford, L. and Grenier, M. 1996a. ‘Entre les choses, les symboles et les idees... une sequence ďenseignement ďintroduction a I’algèbre’, Revue des sciences de ľéducation 22, 253–276

    Google Scholar 

  • Radford, L. and Grenier, M. 1996b. ‘On the dialectical relationships between symbols and algebraic ideas’, in: L. Puig and A. Gutierrez (eds.), Proceedings of the 20th International Conference for the Psychology of Mathematics Education, Valencia: Universidad de Valencia, vol. 4, 179–186

    Google Scholar 

  • Radford, L. 1997. ‘Ľinvention ďune idée mathématique: la deuxieme inconnue en algebra’, Repères, Revue des IREMs 28 (July), 81–96

    Google Scholar 

  • Schneider M. 1988. Des objets mentaux ‘aire’ et ‘volume’ au calcul des primitives, Thèse de doctorat, Louvain-la-Neuve

    Google Scholar 

  • Sfard, Anna 1995. ‘The development of algebra: confronting historical and psychological perspectives’, Journal of mathematical behavior 14, 15–39

    Google Scholar 

  • Sierpinska, Anna 1988. ‘Sur un programme de recherche lit a la notion ďobstacle epistemologique’, in N. Bednarz and C. Gamier (eds.). Construction des savoirs: obstacles et conflicts, Ottawa: Agence ďArc Inc., 130–148

    Google Scholar 

References for §5.3

  • Bartolini Bussi M. 1996. ‘Mathematical discussion and perspective drawing in primary school’. Educational studies in mathematics 31, 11–41

    Google Scholar 

  • Bartolini Bussi M., Boni M., Ferri F. and Garuti R. 1999. ‘Early approach to theoretical thinking: gears in primary school’, Educational studies in mathematics 39, 67–87

    Google Scholar 

  • Bartolini Bussi, M., and M. A. Mariotti, 1999. ‘Semiotic mediation: from history to the mathematics classroom’, For the learning of mathematics 19 (2), 27–35

    Google Scholar 

  • Boero P., Pedemonte B. and Robotti E. 1997. ‘Approaching theoretical knowledge through voices and echoes: a Vygotskian perspective’, Proceedings of the 2lst International Conference on the Psychology of Mathematics Education, Lahti, Finland, vol. 2, 81–88

    Google Scholar 

  • Boero P., Pedemonte B., Robotti E. and Chiappini G. 1998. ‘The ‘voices and echoes game’ and the interiorization of crucial aspects of the oretical knowledge in a Vygotskian perspective: ongoing research’, Proceedings of the 22nd International Conference on the Psychology of Mathematics Education, Stellenbosch, South Africa, vol. 2, 120–127

    Google Scholar 

  • Dorier, J.-L., Robert, A., Robinet, J. A. and Rogalski, M. 1997. ‘Ľalgébre linéaire: I’obstacie du formalisme a travers diverses recherches de 1987 a 1995’, in: J.-L. Dorier (ed.), Ľenseignement de ľalgèbre linéaire en question, Grenoble: La Pensee Sauvage Editions, 105–147

    Google Scholar 

  • Dorier, J.-L. 1995a. ‘A general outline of the genesis of vector space theory’, Historia mathematica 22, 227–261

    Google Scholar 

  • Dorier, J.-L. 1995b. ‘Meta level in the teaching of generalizing concepts in mathematics’, Educational studies in mathematics 29, 175–197

    Google Scholar 

  • Dorier, J.-L. 1997. ‘Une lecture épistémologique de la genése de la théorie des espaces vectoriels’, in: J.-L. Dorier ed., Ľenseignement de ľalgèbre linéaire en question, Grenoble: La Pensée Sauvage Editions, 27–105

    Google Scholar 

  • Dorier, J.-L. 1998a. ‘The role of formalism in the teaching of the theory of vector spaces’ Linear algebra and its applications 275, 141–160

    Google Scholar 

  • Dorier, J.-L. 1998b. ‘État de I’art de la recherche en didactique à propos de I’enseignement de I’algèbre linéaire’, Recherches en didactique des mathématiques 18 (2), 191–230

    Google Scholar 

  • Garuti, R, P Boero & G Chiappini, 1999. ‘Bringing the voice of Plato in the classroom to detect and overcome conceptual mistakes’, Proceedings of the 23rd PME Conference, Haifa, Israel iii, 9–16

    Google Scholar 

  • Hillel, J. & Sierpinska, A. 1994. ‘On one persistent mistake in linear algebra’, Proceedings of the 18th International Conference on the Psychology of Mathematics Education, Lisbon, Portugal, vol. 3, 65–72

    Google Scholar 

  • Mariotti M. A., Bartolini Bussi M., Boero P., Ferri F. and Garuti R. 1997. ‘Approaching geometry theorems in contexts: from history and epistemology to cognition’, Proceedings of the 2lst International Conference on the Psychology of Mathematics Education, Lahti, Finland, vol. 1, 180–195

    Google Scholar 

  • Panza, M. 1996. ‘Concept of function, between quantity and form, in the 18th century’, in: H.N. Jahnke, N. Knoche and M. Otte (eds.), History of mathematics and education: ideas and experiences, Göttingen: Vandenhoeck and Ruprecht, 241–269

    Google Scholar 

  • Schneider M. 1988. Des objets mentaux ‘aire’ et ‘volume’ au calcul des primitives, Thése de doctorat, Louvain-la-Neuve

    Google Scholar 

  • Schneider M. 1992. ‘A propos de I’apprentissage du taux de variation instantané’, Educational studies in mathematics 23, 317–350

    Google Scholar 

  • Sierpinska, A. 1994. Under standing in mathematics, London: Falmer Press

    Google Scholar 

  • Sierpinska, A,, Defence, A., Khatcherian, T., Saldanha, L. 1997. ‘A propos de trois modes de raisonnement en algèbre Iinéaire’, in: J.-L. Dorier (ed.), Ľenseignement de ľalgèbre linéaire en question, Grenoble: La Pensée Sauvage Editions, 249–268

    Google Scholar 

  • Sierpinska, A., Dreyfus, T., Hillel, J. 1999. ‘Evaluation of a teaching design in linear algebra: the case of linear transformations’ Recherches en didactique des mathématiques 19, 7–40

    Google Scholar 

  • Vasco, C. E. 1995. ‘History of mathematics as a tool for teaching mathematics for understanding’, in: D. N. Perkins, J. L. Schwartz, M. M. West, and M. S. Wiske, Software goes to school: teaching for understanding with new technologies, New York: Oxford University Press, 56–69

    Google Scholar 

References for §5.4

  • Bachtin, M. 1968. Dostoevskij, poetica e stilistica, Torino: Einaudi

    Google Scholar 

  • Boero P., Pedemonte B. and Robotti E. 1997. ‘Approaching theoretical knowledge through voices and echoes: a Vygotskian perspective’, Proceedings of the 21st International Conference on the Psychology of Mathematics Education, Lahti, Finland, vol. 2, 81–88

    Google Scholar 

  • Boero P., Pedemonte B., Robotti E. and Chiappini G. 1998. ‘The ‘voices and echoes game’ and the interiorization of crucial aspects of theoretical knowledge in a Vygotskian perspective: ongoing research’, Proceedings of the 22nd International Conference on the Psychology of Mathematics Education, Stellenbosch (South Africa), 2, 120–127

    Google Scholar 

  • Brousseau, G. 1983. ‘Les obstacles épistémologiques et les problémes en mathématiques’, Recherches en Didactique des Mathématiques, 4 (2), 165–198

    Google Scholar 

  • Brousseau, G. 1989. ‘Les obstacles épistémologiques et la didactique des mathématiques’, in N. Bednarz et C. Garnier (eds), Construction des savoirs, obstacles et conflits, Montreal: Agence ďArc, 41–64

    Google Scholar 

  • Brousseau, G 1997. Theory of didactical situations in mathematics, ed & tr N Balacheff et al, Dordrecht: Kluwer

    Google Scholar 

  • Garuti, R. 1997. ‘A classroom discussion and a historical dialogue: a case study’, Proceedings of the 21st International Conference on the Psychology of Mathematics Education, Lahti, Finland, 2, 297–304.

    Google Scholar 

  • Ilyenkov, E. V. 1977. Dialectical logic, Moscow: Progress Publishers

    Google Scholar 

  • Lladò, C. & Boero, P. 1997. ‘Les interactions sociales dans la classe et le role mediateur de I’enseignant’, Actes de la CIEAEM-49, Setubal, 171–179

    Google Scholar 

  • Leonťev, A. N. 1978. Activity, consciousness and personality, Englewod Cliffs: Prentice-Hall

    Google Scholar 

  • Mikhailov, F. T. 1980. The riddle of the self; Moscow: Progress Publishers

    Google Scholar 

  • O’Meara, D. J. 1989. Pythagoras revived, Oxford: Clarendon Press.

    Google Scholar 

  • Radford, L. 1995. ‘La transformación de una teoría matemática: el caso de los nùmeros poligonales’, Mathesis 11 (3), 217–250.

    Google Scholar 

  • Radford, L. and Guérette, G. 1996 ‘Quadratic equations: re-inventing the formula: a teaching sequence based on the historical development of algebra’, in Proc. HEM Braga ii, 301–308.

    Google Scholar 

  • Radford, L. 1997a. ‘On psychology, historical epistemology and the teaching of mathematics: towards a socio-cultural history of mathematics’, For the learning of mathematics 17 (1), 26–33.

    Google Scholar 

  • Radford, L. 1997b. ‘Ľinvention ďune idee mathématique: la deuxième inconnue en algèbre’, Repères (Revue des instituts de Recherche sur I’enseignement des Mathématiques), juillet, 28, 81–96.

    Google Scholar 

  • Radford, L. 1998. ‘On signs and representations: a cultural account’, Scientia paedagogica experimentalis, 35, 277–302.

    Google Scholar 

  • Radford, L. (submitted). ‘On mind and culture: A post-Vygotskian semiotic perspective, with an example from Greek mathematical thought.’

    Google Scholar 

  • Sfard, A. 1997. ‘Framing in mathematical discourse’, Proceedings of the 21st International Conference on the Psychology of Mathematics Education, Lahti, Finland, 4, 144–151.

    Google Scholar 

  • Sierpinska, A. 1994. Understanding in mathematics, London: Falmer Press

    Google Scholar 

  • Tizzani, P. & Boero, P. 1997. ‘La chute des corps de Aristote à Galilee: voix de I’histoire et echos dans la classe’, Actes de la CIEAEM-49, Setubal, 369–376

    Google Scholar 

  • Voloshinov, V. N. 1973. Marxism and the philosophy of language, Cambridge, Mass: Harvard University Press

    Google Scholar 

  • Vygotsky, L. S. 1990. Pensiero e linguaggio, edizione critica a cura di L. Mecacci, Bari

    Google Scholar 

  • Wartofsky, M. 1979. Models, representations and the scientific understanding, Dordrecht: Reidel

    Google Scholar 

  • Wertsch, J. V. 1991. Voices of the mind, Wheatsheaf, Harvester

    Google Scholar 

  • Wittgenstein, L. 1969. On certainty, Oxford: Basil Blackwell

    Google Scholar 

Reference for §5.5

  • Chevallard, Y. 1991. La transposition didactique, Grenoble: La Pensée Sauvage.

    Google Scholar 

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© 2002 Kluwer Academic Publishers

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Radford, L. et al. (2002). Historical formation and student understanding of mathematics. In: Fauvel, J., Van Maanen, J. (eds) History in Mathematics Education. New ICMI Study Series, vol 6. Springer, Dordrecht. https://doi.org/10.1007/0-306-47220-1_5

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  • DOI: https://doi.org/10.1007/0-306-47220-1_5

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-0-7923-6399-6

  • Online ISBN: 978-0-306-47220-6

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