Abstract
Algebraic symbols are investigated from both a linguistic and a semiotic perspective. In the first part of the chapter, a theoretical framework is presented based on Frege’s notions of sense and denotation and language aspects of the algebraic symbol system that affect how individuals read collections of symbols are discussed. The chapter then focuses on the interpretative nature of assigning meaning to symbols, including discussion of a hierarchical framework based on an interpretation of Deacon’s work. Meaning-making is also related to the activity of symbolising. The notion of a language focus is related to teacher preparation, and the chapter concludes with a brief examination of possible future directions for research related to algebraic symbols and language. Examples from research are given throughout the chapter to illustrate instructional aspects of the topics discussed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Artigue, M., Abboud, M., Drouhard, J-Ph., & Lagrange, J-B. (1994). Integrating DERIVE® in secondary level mathematical teaching: Theoretical potentialities and the real life of teachers and students. In J. P. da Ponte & J. Matos (Eds.), Proceedings of the 18thannual conference of the International Group for the Psychology of Mathematics Education (Vol. 1, p. 30). Lisbon, Portugal: Program Committee.
Arzarello, F., Bazzini, L., & Chiappini, G. (1994). Intentional semantics as a tool to analyze algebraic thinking. Rendiconti del Seminario Matematico dell’Università e del Politecnico di Torino 52(2), 105–125.
Baker, B., Hemenway, C, & Trigueros, M. (2001). On transformations of basic functions. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 41–47). Melbourne, Australia: The University of Melbourne.
Bednarz, N. (2001). A problem-solving approach to algebra: Accounting for the reasonings and notations developed by students. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 69–78). Melbourne, Australia: The University of Melbourne.
Boero, P. (1993). About the transformation function of the algebraic code. In R. Sutherland (Ed.), Algebraic processes and the role of symbolism (working conference of the ESRC seminar group, pp. 48–55). London: University of London, Institute of Education.
Brown, T. (1997). Mathematics education and language: Interpreting hermeneutics and post-structuralism. Dordrecht, The Netherlands: Kluwer Academic.
Buchler, J. (Ed.) (1955). The philosophical writings of Peirce. New York: Dover Books.
Chomsky, N. (1965). Aspects of the theory of syntax. Cambridge, MA: MIT Press.
Cobb, P. (2000). From representations to symbolizing: Introductory comments on semiotics and mathematical leraning. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design (pp. 17–36). Mahwah, NJ: Lawrence Erlbaum.
Davis, G. E., & McGowen, M. A. (2001). Embodied objects and the signs of mathematics. A discussion paper prepared for the PME discussion group “Symbolic Cognition in Advanced Mathematics” at the 25th annual conference of the International Group for the Psychology of Mathematics Education Freudenthal Institute, University of Utrecht, The Netherlands (pp. 1–25). Available at http://www.harpercollege.edu/~mmcgowan/publications
Deacon, T. W. (1997). The symbolic species: The co-evolution of language and the brain. New York: W. W. Norton & Company.
Devlin, K. (2000). The math gene: How mathematical thinking evolved and why numbers are like gossip. New York: Basic Books.
Donald, M. (1991). Origins of the modern mind: Three stages in the evolution of culture and cognition. Cambridge, MA: Harvard University Press.
Dörfler, W. (2000). Means for meaning. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design (pp. 99–132). Mahwah, NJ: Lawrence Erlbaum.
Drouhard, J-Ph. (1989). Une étude pro-didactique: l’analyse syntaxique des expressions de l’algèbre élémentaire [Synatactic analysis of elementary algebra writings]. In A. Bessot (Ed.), Actes du Séminaire de Didactique des Mathématiques et de l’Informatique (pp. 5–13). Grenoble: Université Joseph Fourier.
Drouhard, J-Ph. (1992). Les écritures symboliques de l’algèbre élémentaire [Elementary algebra symbolic writings]. Unpublished Doctoral Dissertation. Université Paris 7.
Drouhard. J-Ph., Léonard F., Maurel M., Pécal M., & Sackur C. (1994). ‘Blind calculators’, ‘denotation’ of algebra symbolic expressions, and ‘write false’ interviews. In D. Kirshner (Ed), Proceedings of the Sixteenth Annual Meeting, North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 152–158). Baton Rouge, LA: Louisiana State University.
Drouhard, J-Ph. (1995). Algébre, calcul symbolique et didactique [Algebra, symbolic calculation and didactics]. In R. Noirfalise & M.-J. Perrin-Glorian (Eds.), Actes de la VIIIèmeÉcole d’Été de Didactique des Mathématiques (pp. 325–344). Clermont-Ferrand, France: IREM de Clermont-Ferrand.
Drouhard, J-Ph. (2001). Research in language aspects of algebra: a turning point? In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 238–242). Melbourne, Australia: The University of Melbourne.
Ducrot, O., & Todorov T. (1972). Dictionnaire encyclopédique des sciences du langage [Encyclopaedic dictionary of language sciences]. Paris: Le Seuil.
Durkin, K. (1991). Language in mathematical education: An introduction. In K. Durkin & B. Shire (Eds.), Language in mathematical education: Research and practice (pp. 3–16). Milton Keynes, UK: Open University Press.
Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. In F. Hitt & M. Santos (Eds.), Proceedings of the Twenty First Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 3–26). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
Duval, R. (2000). Basic issues for research in mathematics education. In T. Nakahara & M. Koyama (Eds.), Proceedings of 24thannual conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 55–69). Hiroshima, Japan: Program Committee.
Esty, W. W. (1999). The Language of Mathematics, 16thEdition. Bozeman, MT: Montana State University Technical Report.
Esty, W. W., & Teppo, A. R. (1994). A general-education course emphasizing mathematical language and reasoning. Focus on Learning Problems in Mathematics, 16(1), 13–35.
Fearnley-Sander, D. (2001). Algebra worlds. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 243–251). Melbourne, Australia: The University of Melbourne.
Freudenthal, H. (1991). Revisiting Mathematics Education: China Lectures. Dordrecht, The Netherlands: Kluwer Academic.
Freudenthal, H. (1980). Weeding and sowing. Dordrecht, The Netherlands: D. Reidel Publishing Company.
Filloy, E., & Rojano, T. (1984). From an arithmetical to an algebraic thought. In J. M. Moser (Ed.), Proceedings of the Sixth Annual Meeting of PME-NA (pp. 51–56). Madison: University of Wisconsin.
Goldin, G. A., & Kaput, J. J. (1996). A joint perspective on the idea of representation in learning and doing mathematics. In L. P. Steffe, P. Nesher, P. Cobb, G. Goldin, & B. Greer (Eds.), Theories of Mathematical Learning (pp. 397–430). Mahwah, NJ: Lawrence Erlbaum.
Goody, J. (1987). The interface between the written and the oral. (Studies in literacy, family, culture and the state). Cambridge, MA: Cambridge University Press.
Goody, J. (1994). Entre l’Oralité et l’écriture [The interface between the written and the oral]. Paris: PUF.
Gray, E., & Tall, D. (1994). Duality, ambiguity, and flexibility: A “proceptual” view of simple arithmetic. Journal for Research in Mathematics Education, 25(2), 116–140.
Gravemeijer, K., Cobb, P., Bowers, J., & Whitenack, J. (2001). Symbolizing, modeling, and instructional design. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design (pp. 225–274). Mahwah, NJ: Lawrence Erlbaum.
Guillerault, M., & Laborde C. (1982). Ambiguities in the description of a geometrical figure. New York: Plenum Press.
Harel, G. & Kaput, J. (1991). The role of conceptual entities and their symbols in building mathematical concepts. In D. Tall (Ed), Advanced mathematical thinking (pp. 82–94). Dordrecht, The Netherlands: Kluwer Academic.
Havelock, E. A. (1986). The Muse learns to write. Reflections on orality and literacy from antiquity to the present. New Haven & London: Yale University Press.
Hewitt, D. (2001). On learning to adopt formal algebraic notation. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 305–312). Melbourne, Australia: The University of Melbourne.
Kieran, C, & Sfard, A. (1991). Seeing through symbols: The case of equivalent expressions. Focus on Learning Problems in Mathematics, 21(1), 1–17.
Kinzel, M. T. (2001). Analyzing college calculus students’ interpretation and use of algebraic notation. In R. Speiser, C. A. Maher, & C. N. Walter, (Eds.), Proceedings of the 21stAnnual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 109–116). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
Kirshner, D. (1987a). Linguistic analysis of symbolic elementary algebra, Unpublished Doctoral Dissertation, University of British Columbia.
Kirshner, D. (1987b). The myth about binary representation in algebra. In J. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11thannual conference of of the the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 308–315). Montreal, Canada: Program Committee.
Kirshner, D. (1989). The visual syntax of algebra. Journal for Research in Mathematics Education, 20(3), 276–287.
Laborde, C. (1982). Langue naturelle et écriture symbolique deux codes en interaction dans l’enseignement mathématique [Natural language and symbolic writings: Two interacting codes in the teaching of mathematics]. Thèse d’Etat, Université Joseph Fourier, Grenoble.
Laborde, C. (1990). Language and Mathematics. In J. Kilpatrick & P. Nesher (Eds.), Mathematics and cognition (pp. 4–19). Cambridge: Cambridge University Press.
Lagrange, J-B. (2001). Tasks for students using computer algebra: a study of the design of educational computer applications. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 376–383.). Melbourne, Australia: The University of Melbourne.
Linchevski, L., & Livneh, D. (1999). Structure sense: The relationship between algebraic and numerical contexts. Educational Studies in Mathematics, 40, 173–196.
Malara, N. A., & Navarra, G. (2001). “Brioshi” and other mediation tools employed in a teaching of arithmetic from a relational point of view with the aim of approaching algebra as a language. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 412–419). Melbourne, Australia: The University of Melbourne.
McArthur, T. (Ed.) (1992). The Oxford companion to the English language. Oxford: Oxford University Press.
MacGregor, M., & Price, E. (1999). An exploration of aspects of language proficiency and algebraic learning. Journal for Research in Mathematics Education, 30(4), 449–467.
McGuinness, B. (Ed.) (1984). Collected papers on mathematics, logic, and philosophy. Oxford: Blackwell.
Menzel, B. (2001). Language conceptions of algebra are idiosyncratic. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 446–453). Melbourne, Australia: The University of Melbourne.
Miles, D. D. (1999). Algebra vocabulary as an improvement factor in algebra achievement. Focus on Learning Problems in Mathematics, 21(2), 37–49.
Miller, D. L. (1993). Making the connection with language. Arithmetic Teacher, (February, 311–316.
Nicaud, J.-F., Bouhineau, D., & Gélis, J.-M. (2001). Syntax and semantics in algebra. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 475–486). Melbourne, Australia: The University of Melbourne.
Novotná, J., & Kubínová, M. (2001). The influence of symbolic algebraic descriptions in word problem assignments on grasping process and on solving strategies. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 495–500). Melbourne, Australia: The University of Melbourne.
Panizza, M (2002). Generalización y control en álgebra [Generalisation and control in algebra]. In C. Crespo (Ed.), Proceedings of the 15th Conference RELME (pp. 213–218). México: Grupo Editorial Iberoamericana.
Pirie, S. E. B., & Martin, L. (1997). The equation, the whole equation, and nothing but the equation: One approach to the teaching of linear equations. Educational Studies in Mathematics, 34, 159–181.
Quinlan, C. (2001). Importance of view regarding algebraic symbols. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 507–514). Melbourne, Australia: The University of Melbourne.
Radford, L. (2002). On heroes and the collapse of narratives: A contribution to the study of symbolic thinking. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of 26thannual conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 81–88). Norwich, UK: Program Committee.
Rasmussen, C. L., Zandieh, M., King, K., & Teppo, A. (In press). Advancing mathematical activity: A practice-oriented view of advanced mathematical thinking. Mathematical Thinking and Learning.
Rojano, T, & Sutherland, R. (2001). Arithmetic world-Algebra world. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 515–522). Melbourne, Australia: The University of Melbourne.
Sackur, C. (1995). Blind calculators in algebra: Write false interviews. In E. Cohors-Fresenborg (Ed.), Proceedings of the first European Research Conference on Mathematics Education (ERCME’ 95) (pp. 82–85). Osnabrück (Germany): University of Osnabrück.
Sfard, A. (2000). Symbolizing mathematical reality into being-or how mathematical discourse and mathematical objects create each other. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design (pp. 37–98). Mahwah, NJ: Lawrence Erlbaum.
Sfard, A., & Linchevski, L. (1994). The gains and pitfalls of reification-the case of algebra. Educational Studies in Mathematics, 26, 191–228.
Sfard, A., & Thompson, P. (1994). Problems of reification: Representations and mathematical objects. In D. Kirshner (Ed), Proceedings of the Sixteenth Annual Meeting, North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 3–34). Baton Rouge, LA: Louisiana State University.
Sutherland, R. (2001). Algebra as an emergent language of expression. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 570–576). Melbourne, Australia: The University of Melbourne.
Teppo, A. R., & Esty, W. W. (2001). Mathematical contexts and the perception of meaning in algebraic symbols. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 577–581). Melbourne, Australia: The University of Melbourne.
Trigueros, M., & Ursini, S. (2001). Approaching the study of algebra through the concept of variable. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra (Proceedings of the 12th ICMI Study Conference, pp. 598–605). Melbourne, Australia: The University of Melbourne.
Vergnaud, G. (1996). The theory of conceptual fields. In L. P. Steffe, P. Nesher, P. Cobb, G. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 219–239). Mahwah, NJ: Lawrence Erlbaum.
Villarreal, M. (2000). Mathematical thinking and intellectual technologies: The visual and the algebraic. For the Learning of Mathematics, 20(2), 2–7.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer Science + Business Media, Inc.
About this chapter
Cite this chapter
Drouhard, JP., Teppo, A.R. (2004). Symbols and Language. In: Stacey, K., Chick, H., Kendal, M. (eds) The Future of the Teaching and Learning of Algebra The 12th ICMI Study. New ICMI Study Series, vol 8. Springer, Dordrecht. https://doi.org/10.1007/1-4020-8131-6_9
Download citation
DOI: https://doi.org/10.1007/1-4020-8131-6_9
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-8130-9
Online ISBN: 978-1-4020-8131-6
eBook Packages: Springer Book Archive