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Signs and meanings in students' emergent algebraic thinking: A semiotic analysis

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Abstract

The purpose of this article, which is part of a longitudinal classroom research about students' algebraic symbolizations, is twofold: (1) to investigate the way students use signs and endow them with meaning in their very first encounter with the algebraic generalization of patterns and(2) to provide accounts about the students' emergent algebraic thinking. The research draws from Vygotsky's historical-cultural school of psychology, on the one hand, and from Bakhtin and Voloshinov's theory of discourse on the other, and is grounded in a semiotic-cultural theoretical framework in which algebraic thinking is considered as a sign-mediated cognitive praxis. Within this theoretical framework, the students' algebraic activity is investigated in the interaction of the individual's subjectivity and the social means of semiotic objectification. An ethnographic qualitative methodology, supported by historic, epistemological research, ensured the design and interpretation of a set of teaching activities. The paper focuses on the discussion held by a small group of students of which an interpretative, situated discourse analysis is provided. The results shed some light on the students' production of (oral and written) signs and their meanings as they engage in the construction of expressions of mathematical generality and on the social nature of their emergent algebraic thinking.

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REFERENCES

  • Artigue, M.: 1988, 'Ingénierie Didactique', Recherches en Didactique des Mathématiques 9(3), 281-308.

    Google Scholar 

  • Arzarello, F.: 1991, 'Pre-algebraic problem solving', in J.P. Ponte, J.F. Matos, J.M. Matos and D. Fernandes (eds), Mathematical problem solving and new information technologies (NATO ASI Series F, Vol. 89, 155-166), Springer-Verlag, Berlin.

    Google Scholar 

  • Arzarello, F., Bazzini, L. and Chiappini, C.: 1993, 'Cognitive processes in algebraic thinking: Towards a theoretical framework', in Ichiei Hirabayashi, Nobuhiko Nohda, Keiichi Sheigematsu and Fou-Lai Lin (eds), Proceedings of the 17th International Conference for the Psychology of Mathematics Education, Vol. 1, University of Tsukuba, Tsukuba, Ibaraki, Japan, pp. 138-145.

    Google Scholar 

  • Arzarello, F., Bazzini, L. and Chiappini, C.: 1994a, 'The process of naming in algebraic problem solving', in João Pedro da Ponte and João Filipe Matos (eds), Proceedings of the 18th International Conference for the Psychology of Mathematics Education, Vol. 2, University of Lisbon, Portugal, pp. 40-47.

    Google Scholar 

  • Arzarello, F., Bazzini, L. and Chiappini, C.: 1994b, L'algebra come strumento di pensiero, Analisi teorica e considerazioni didattiche, Progetto Strategico del C.N.R., Pavia: Dipartimento di Matemática, Università di Pavia, Quaderno nº 6.

    Google Scholar 

  • Astington, J.W.: 1995, 'Theory of Mind Development and Social Understanding', Cognition and Emotion 9(2/3), 151-165.

    Google Scholar 

  • Bakhtin, M.M.: 1981, The Dialogical Imagination, University of Texas Press, Austin.

    Google Scholar 

  • Bakhtin, M.M.: 1986, Speech Genres and Other Late Essays, University of Texas Press, Austin.

    Google Scholar 

  • Bakhtin, M.M.: 1993, Toward a philosophy of the act, University of Texas Press, Austin.

    Google Scholar 

  • Bartolini Bussi, M.G.: 1995, 'Analysis of Classroom Interaction Discourse from a Vygotskian Perspective', in L. Meira and D. Carraher (eds), Proceedings of the 19th International Conference for the Psychology of Mathematics Education, Vol. 1, Universidade Federal de Pernambuco, Brazil, pp. 95-98.

    Google Scholar 

  • Bartolini Bussi, M.G.: 1998, 'Verbal interaction in the mathematics classroom: A Vygotskian Analysis', in H. Steinbring, M. Bartolini Bussi and A. Sierpinska (eds), Language and Communication in the Mathematics Classroom, National Council of Teachers of Mathematics, Reston, Virginia, pp. 65-84.

    Google Scholar 

  • Bateson, G.: 1973, Steps to an Ecology of Mind, Paladin, Frogmore.

    Google Scholar 

  • Bauersfeld, H.: 1995, 'Languages Games. In the Mathematical Classroom: Their Function and Their Effects', in P. Cobb and H. Bauersfeld (eds), The Emergence of Mathematical Meaning, Interaction in Classroom Cultures, Erlbaum Associates, Hillsdale, New Jersey.

    Google Scholar 

  • Boero, P., Pedemonte, B. and Robotti, E.: 1997, 'Approaching theoretical knowledge through voices and echoes: a Vygotskian perspective, Proceedings of the XXI International Conference for the Psychology of Mathematics Education, Vol. 2, Lahti (Finland), pp. 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 XXII International Conference for the Psychology ofMathematics Education, Vol. 2, Stellenbosch (South Africa), pp. 120-127.

  • Cajori, F.: 1993, A History of Mathematical Notations, Dover, New York.

    Google Scholar 

  • Chaiklin, S. and Lave, J.: 1993, Understanding Practice. Perspectives on activity and context, Cambridge University Press, Cambridge.

    Google Scholar 

  • Clement, J.: 1982, 'Algebra Word problem solutions: Thought processes underlying a common misconception', Journal for Research in Mathematics Education 13(1), 16-30.

    Article  Google Scholar 

  • Clement, J., Lochhead, J. and Monk, G.: 1981, 'Translation Difficulties in Learning Mathematics', American Mathematical Monthly 88, 286-289.

    Article  Google Scholar 

  • Cobb, P., Boufi, A., McClain, K. and Whitenack, J.: 1997, 'Reflective Discourse and Collective Reflection', Journal for Research in Mathematics Education 28(3), 258-277.

    Article  Google Scholar 

  • Côté, J., Salmela, J.H., Baria, A. and Russell, S.: 1993, 'Organizing and interpreting unstructured qualitative data', The Sport Psychologist 7, 127-137.

    Google Scholar 

  • Coulthard, M.: 1977, An Introduction to Discourse Analysis, Longman, London.

    Google Scholar 

  • Davis, R.B.: 1975, 'Cognitive processes involved in solving simple algebraic equations', Journal of Children's Mathematical Behaviour 1(3), 7-35.

    Google Scholar 

  • Eco, U.: 1988, Le signe, Éditions Labor, Bruxelles.

    Google Scholar 

  • Fairclough, N.: 1995, Critical Discourse Analysis, Longman, London/New York.

    Google Scholar 

  • Frege, G.: 1971, Écrits logiques et philosophiques, Éditions du Seuil, Paris.

    Google Scholar 

  • Gahan, C. and Hannibal, M.: 1998, Doing Qualitative Research Using QSR Nud_ist, London, Thousand Oaks, Sage, New Delhi.

    Google Scholar 

  • Geertz, C.: 1973, The Interpretation of Cultures, Basic Books, New York.

    Google Scholar 

  • Goetz, J.P. and LeCompte, M.D.: 1984, Ethnography and Qualitative Design in Educational Research, Academic Press, San Diego, CA.

    Google Scholar 

  • Hall, J.K.: 1995, '(Re)creating our Worlds with Words: A Sociohistorical Perspective of Face-to-Face Interaction', Applied Linguistics 16(2), 206-232.

    Article  Google Scholar 

  • Kieran, C.: 1989, 'The early learning of algebra: A structural perspective', in S. Wagner and C.Kieran (eds), Research Issues in the Learning and Teaching of Algebra, Lawrence Erlbaum Associates and National Council of Teachers of Mathematics, Virginia, pp. 33-56.

    Google Scholar 

  • Kirshner, D.: 2001, 'The Structural Algebra Option Revisited', in R. Sutherland, T. Rojano, A. Bell and R. Lins (eds), Perspectives on School Algebra, Kluwer, Dordrecht/Boston/London, pp. 83-98.

    Google Scholar 

  • Lee, L.: 1996. 'An initiation into algebraic culture through generalization activities', in N. Bednarz, C. Kieran and L. Lee (eds), Approaches to Algebra, Kluwer, Dordrecht/Boston/London, pp. 87-106.

    Google Scholar 

  • Leontiev, A.A.: 1981, 'Sign and Activity', Translated by J.V.Wertsch (ed.), The concept of activity in Soviet Psychology, New York, M.E. Sharpe, pp. 241-255.

    Google Scholar 

  • Leontiev, A.N.: 1984, Activité, Conscience, personnalité, Éditions du Progrès, Moscou.

    Google Scholar 

  • Lotman, Y.: 1990, Universe of the Mind. A Semiotic Theory of Culture, I.B. Taurus, London/New York.

    Google Scholar 

  • MacGregor, M. and Stacey, K.: 1992, 'A Comparison of Pattern-Based and Equation-Solving Approaches to Algebra', in B. Southwell, K. Owens and B. Penny (eds), Proceedings of the 15th Annual Conference, Mathematics Education Research Group of Australasia (MERGA), University of Western Sydney, July 4-8, pp. 362-370.

  • MacGregor, M. and Stacey, K.: 1993, 'Seeing a pattern and writing a rule', in I. Hirabayashi, N. Nohda, K. Shigematsu and F. Lin (eds), Proceedings of the 17th International Conference for the Psychology of Mathematics Education, Vol. 1, University of Tsukuba, Japan, pp. 181-188.

    Google Scholar 

  • MacGregor, M. and Stacey, K.: 1997, 'Students' understanding of algebraic notation: 11-15', Educational Studies in Mathematics 33, 1-19.

    Article  Google Scholar 

  • Mason, J.: 1987, 'What do symbols represent?', in C. Janvier (ed.), Problems of Representation in the Teaching and Learning of Mathematics, Lawrene Erlbaum, Hillsdale, New Jersey/London, pp. 73-81.

    Google Scholar 

  • Mason, J.: 1996, 'Expressing generality and roots of algebra', in N. Bednarz, C. Kieran and L. Lee (eds), Approaches to Algebra, Kluwer, Dordrecht/Boston/London, pp. 65-86.

    Google Scholar 

  • Manitius, K.: 1888, Des Hypsikles Schrift Anaphorikos nach Ñberlieferung und Inhalt kritisch behandelt, Lehmannsche Buchdruckerei, Dresden.

    Google Scholar 

  • Meira, L.: 1995, 'The Microevolution of Mathematical Representations in Children's Activity', Cognition and Instruction 13(2), 269-313.

    Article  Google Scholar 

  • Meira, L.: 1996, 'Students' early algebraic activity: Sense making and the production of meanings in mathematics', in L. Puig and A. Gutiérrez (eds), Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education, Vol. 3, pp. 377-384.

  • Mikhailov, F.T.: 1980, The Riddle of the Self, Progress Publishers, Moscow.

    Google Scholar 

  • Miller, C.R.: 1984, 'Genre as social action', Quarterly Journal of Speech 70, 151-167.

    Article  Google Scholar 

  • Ministry of Education and Training: 1997, 'The Ontario Curriculum, Grades 1-8', Queen's Printer for Ontario, Toronto.

  • Moerman, M.: 1988, Talking Culture, Ethnography and Conversational Analysis, University of Pennsylvania Press, Philadelphia.

    Google Scholar 

  • Nemirovsky, R.: 1994, 'On ways of symbolizing: The case of Laura and the Velocity sign', Journal of Mathematical Behavior 13, 389-422.

    Article  Google Scholar 

  • NÚñez, R.: 2000, 'Mathematical Idea Analysis: What embodied cognitive science can say about the human nature of mathematics', in T. Nakahara and M. Koyama (eds), Proceedings of the 24th Conference of the International Group for the psychology of Mathematics Education (PME-24), Vol. 1, Hiroshima University, Japan, pp. 3-22.

    Google Scholar 

  • NÚñez, R., Edwards, L. and Matos, J.F.: 1999, 'Embodied cognition as grounding for situatedness and context in mathematics education', Educational Studies in Mathematics 39(1-3), 45-65.

    Article  Google Scholar 

  • Nyckees, V.: 1998, La sémantique, Belin, Paris.

    Google Scholar 

  • Parmentier, R.: 1994, Signs in Society-Studies in Semiotic Anthropology, Indiana University Press, Bloomington/Indianapolis.

    Google Scholar 

  • Peirce, C.S.: 1955, Philosophical Writings, Selected by J. Buchler (ed.), Dover, New York.

    Google Scholar 

  • Presmeg, N.C.: 1998, 'Metaphoric and Metonymic signification in mathematics', Journal of Mathematical Behaviour 17(1), 25-32.

    Article  Google Scholar 

  • Radford, L.: 1995, 'La transformación de una teoría matemática: el caso de los NÚmeros Poligonales', Mathesis 11(3), 217-250.

    Google Scholar 

  • Radford, L.: 1998a, 'On Signs and Representations. A Cultural Account', Scientia Paedagogica Experimentalis, 35(1), 277-302.

    Google Scholar 

  • Radford, L.: 1998b, On Culture and Mind, a post-Vygotskian Semiotic Perspective, with an Example from Greek Mathematical Thought, Paper presented at the 23rd Annual Meeting of the Semiotic Society of America, Victoria College, University of Toronto, October 15-18, 1998.

  • Radford, L.L.: 1999, 'The Rhetoric of Generalization', Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, Haifa, Technion-Israel Institute of Technology, pp. 89-96.

    Google Scholar 

  • Radford, L.: 2000, 'Sujeto, objeto, cultura y la formación del conocimiento', Educación Matemática 12(1), 51-69.

    Google Scholar 

  • Radford, L.: 2001, 'The Historical Origins of Algebraic Thinking', in R. Sutherland, T. Rajano, A. Bell and R. Lins (eds), Perspectives on School Algebra, Kluwer, Dordrecht/Boston/london, pp. 13-36.

    Google Scholar 

  • Radford, L. and Grenier, M.: 1996, 'Entre les idées, les choses et les symboles. Une séquence d'enseignement d'introduction à l'algèbre', Revue des sciences de l'éducation 22, 253-276.

    Google Scholar 

  • Rico, L., Castro, E. and Romero, I.: 1996, 'The role of representation systems in the learning of numerical structures', in Luis Puig and Angel Gutiérrez (eds), Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education, Vol. 1, University of Valencia, Spain, pp. 87-102.

    Google Scholar 

  • Sasman, M., Olivier, A. and Linchevski, L.: 1999, 'Factors influencing student's generalization thinking processes', in O. Zaslavsky (ed.), Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, Haifa, Israel Institute of Technology, pp. 161-168.

    Google Scholar 

  • Seeger, F.: 1991, 'Interaction and knowledge in mathematics education', Recherches en Didactique des Mathématiques 11(2/3), 125-166.

    Google Scholar 

  • Sfard, A.: 1994, 'Reification as the birth of metaphor', For the Learning of Mathematics 14(1), 44-55.

    Google Scholar 

  • Steinbring, H.: 1998, 'From 'Stoffdidaktik' to Social Interactionism: An evolution of approaches to the study of language and communication in German mathematics education research', in H. Steinbring, M. Bartolini Bussi and A. Sierpinska (eds), Language and Communication in the Mathematics Classroom, National Council of Teachers of Mathematics, Reston, Virginia, pp. 102-119.

    Google Scholar 

  • Todorov, T.: 1984, Mikhail Bakhtin: The Dialogical Principle, University of Minnesota Press, Minneapolis, London.

    Google Scholar 

  • Vile, A. and Lerman, S.: 1996, 'Semiotics as a descriptive framework in mathematics education', in Luis Puig and Angel Gutiérrez (eds), Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education,Vol. 4, University of Valencia, Spain, pp. 395-402.

    Google Scholar 

  • Voloshinov, V.N.: 1973, Marxism and the Philosophy of Language, Harvard University Press, Cambridge, Massachusetts/London, England.

    Google Scholar 

  • Vygotsky, L.S.: 1962, Thought and Language, MIT Press, Cambridge.

    Google Scholar 

  • Vygotsky, L.S.: 1981, 'The instrumental method in psychology', in J.V. Wertsch (ed.), The concept of activity in Soviet psychology, Sharpe, Armonk, N.Y., pp. 135-143.

    Google Scholar 

  • Vygotsky, L.S.: 1997, in R. Rieber (ed.), Collected Works., Vol. 4, Plenum Press, New York / London.

    Google Scholar 

  • Vygotsky, L.S. and Luria, A.: 1994, 'Tool and symbol in child development', in R. van der Veer and J. Valsiner (eds), The Vygotsky Reader, Blackwell Publishers, Oxford, pp. 99-174.

    Google Scholar 

  • Waschescio, U.: 1998, 'The missing link: Social and cultural aspects in social constructivist theories', in F. Seeger, J. Voigt and U.Waschescio (eds), The Culture of the Mathematics Classroom, Cambridge University Press, Cambridge, pp. 221-241.

    Google Scholar 

  • Wertsch, J.V.: 1991, Voices of the Mind-A Sociocultural Approach to Mediate Action, Harvard University Press, Cambridge, MA.

    Google Scholar 

  • Zinchenko, V.P.: 1985, 'Vygotsky's ideas about units for the analysis of mind', in J.V. Wertsch (ed.), Culture, communication and cognition: Vygotskian perspectives, Cambridge University Press, pp. 94-118.

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  1. École des sciences de l'éducation, Université Laurentienne, Sudbury, Ontario, Canada, P3E 2C6

    Luis Radford

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Radford, L. Signs and meanings in students' emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics 42, 237–268 (2000). https://doi.org/10.1023/A:1017530828058

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