Abstract
A semiotic perspective on mathematical activity provides a way of conceptualizing the teaching and learning of mathematics that transcends and encompasses both psychological perspectives focussing exclusively on mental structures and functions, and performance-focussed perspectives concerned only with student' behaviours. Instead it considers the personal appropriation of signs and the underlying meaning structures embodying relationships between signs. It is concerned with patterns of sign use and sign production, including individual creativity in sign use, and the underlying social rules and contexts of sign use. It is based on the concept of a semiotic system, comprising signs, rules of sign production, and an underpinning meaning structure. This theorisation is applied to the learning of number, from counting to calculation. Historical, foundational and developmental (i.e., learning) perspectives are explored and contrasted. It is argued that in each of these domains, the dominant significant activity concerns the production of sequences of signs.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Anderson, M., Saenz-Ludlow, A., Zellweger, S. and Cifarelli, V.V. (eds.): 2003, Educational Perspectives on Mathematics as Semiosis: From Thinking to Interpreting to Knowing, Legas Publishing, New York, Ottawa and Toronto.
Ausubel, D.P.: 1968, Educational Psychology, a Cognitive View, Holt, Rinehart and Winston, New York.
Bachelard, G.: 1951, L'activité rationaliste de la physique contemporaine, Paris.
Bell, A.W., Costello, J. and Küchemann, D.: 1983, Research on Learning and Teaching, Part A. A Review of Research in Mathematical Education, NFER-Nelson, Windsor.
Benacerraf, P. and Putnam, H. (eds.): 1964, Philosophy of Mathematics: Selected Readings, Englewood Cliffs, Prentice-Hall, New Jersey.
Brousseau, G.: 1997, Theory of Didactical Situations in Mathematics, Kluwer, Dordrecht.
Bruner, J.: 1960, The Process of Education, Harvard University Press, Cambridge, Massachusetts.
Bruner, J.: 1964, Towards a Theory of Instruction, Harvard University Press, Cambridge, Massachusetts.
Buxton, L.: 1981, Do You Panic About Maths? Coping with Maths Anxiety, Heinemann Educational Books, London.
Carpenter, T.P. and Moser, J.M.: 1982, ‘The development of addition and subtraction problem-solving skills,’ in T.P. Carpenter, J.M., Moser, and T.A. Romberg (eds.), Addition and Subtraction: A Cognitive Perspective, Erlbaum, Hillsdale, New Jersey.
Chapman, A.P.: 1992, Language Practices in School Mathematics: A Social Semiotic Perspective, Unpublished Ph.D. Thesis, Murdoch University, Australia.
Denvir, B. and Brown, M.: 1986, ‘Understanding of number concepts in low attaining 7–9 year olds (parts I and II),’ Educational Studies in Mathematics 17, 15–36, 143–164.
Dowling, P.: 1988, ‘The contextualising of mathematics: Towards a theoretical map’, in M. Harris (ed.), Schools, Mathematics and Work, Falmer press, London, pp. 93–120.
Ernest, P.: 1991, The Philosophy of Mathematics Education, Falmer Press, London.
Ernest, P.: 1994a, ‘The dialogical nature of mathematics’ in P. Ernest (ed.), Mathematics, Education and Philosophy: An International Perspective, Falmer press, London, pp. 33–48.
Ernest, P.: 1997, ‘Semiotics, mathematics and mathematics education’, The Philosophy of Mathematics Education Journal, 10 (http://www.ex.ac.uk/~PErnest/).
Ernest, P.: 1998a, Social Constructivism as a Philosophy of Mathematics, SUNY Press, Albany, New York.
Ernest, P.: 1998b ‘The relation between personal and public knowledge from an epistemological perspective’, in F. Seeger, J. Voigt and U. Waschescio (eds.), The Culture of the Mathematics Classroom, pp. 245–268, Cambridge University Press, Cambridge.
Ernest, P.: 1999, ‘Forms of knowledge in mathematics and mathematics education: philosophical and rhetorical perspectives’, Educational Studies in Mathematics 38(1–3), 67–83.
Ernest, P.: 2003, ‘The Epistemic subject in mathematical activity’, in M, Anderson, A. Saenz-Ludlow, S Zellweger and V.V. Cifarelli (eds.) Educational Perspectives on Mathematics as Semiosis: From Thinking to Interpreting to Knowing, Legas Publishing, New York, Ottawa and Toronto, pp. 81–106.
Ernest, P., (2005) in ‘Agency and creativity in the semiotics of learning mathematics’ forthcoming. M. Hoffmann, J. Lenhard and F. Seeger, (eds.), Activity and Sign,—Grounding Mathematics Education (Festschrift for Michael Otte), Kluwer, Dordrecht.
Ernest, P. (ed.): 1994b, Constructing Mathematical Knowledge: Epistemology and Mathematics Education, Falmer Press, London.
Evans, J.: 2000, Adults' Mathematical Thinking and Emotions, RoutledgeFalmer Press, London and Philadelphia.
Fischbein, E.: 1994, in Biehler, R. Scholz, R.W., Straesser, R. and Winkelmann (eds.), The Didactics of Mathematics as a Scientific Discipline, Kluwer, Dordrecht.
Foucault, M.: 1981, The History of Sexuality (Part 1), Penguin Books, Harmondsworth.
Foucault, M.: 1982, ‘The subject and power’, in H.L. Dreyfus and P. Rabinow (eds.), Michel Foucault: Beyond Structuralism and Hermeneutics, Harvester Press, Brighton, pp. 208–226.
Fuson, K.C.: 1982, ‘Analysis of the counting-on procedure’, in T.P. Carpenter, J.M. Moser, and T.A. Romberg, (eds.), Addition and Subtraction: A Cognitive Perspective, Erlbaum, Hillsdale, New Jersey.
Gelman, R. and Galistel, C.R.R.: 1978, The Child's Understanding of Number, Harvard University Press, Cambridge, Massachusetts.
Ginsberg, H.: 1977, Children's Arithmetic: How They Learn It and How You Teach It, Pro-Ed, Austin, Texas.
Gödel, K.: 1931, ‘Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte fur Mathematik und Physik 38, 173–198 (Trans. in Heijenoort, 1967), pp. 592–617.
Grattan-Guiness, I. (ed.): 1994, Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, 2 Vols., Routledge, London.
Halliday, M.A.K.: 1978, Language as a Social Semiotic: The Social Interpretation of Language and Meaning, Edward Arnold, London.
Halliday, M.A.K. and Martin, J.R.: 1993, Writing Science: Literacy and Discursive Power, Falmer Press, London.
Heijenoort, J. van, (ed.).: 1967, From Frege to Gödel: A Source Book in Mathematical Logic, Cambridge, Harvard University Press, Massachusetts.
Høyrup, J.: 1994, In Measure, Number, and Weight, SUNY Press, New York.
Hughes, M.: 1986, Children and Number: Difficulties in Learning Mathematics, Blackwell, Oxford.
Ifrah, G.: 1998, The Universal History of Numbers, Harvill Press, London.
Jeffrey, B.: 1981, ‘Happy numbers’, in D. Lingard (ed.). Mathematical Investigations in the Classroom, Derby: Association of Teachers of Mathematics, 1981, pp. 8–11.
Kirshner, D. and Whitson, J.A. (eds.): 1997, Situated Cognition: Social, Semiotic, and Psychological Perspectives, Erlbaum, Lawrence.
Kleene, S.C.: 1967, Introduction to Metamathematics, Amsterdam, North-Holland Piblishing Co.
Lakatos, I.: 1976, Proofs and refutations: The logic of mathematical discovery, in J. Worrall and E. Zahar (eds.), Cambridge University Press, Cambridge.
Lakatos, I.: 1978, Mathematics, Science and Epistemology (Philosophical Papers), Vol. 2, Cambridge University Press, Cambridge.
Lakoff, G. and Nunez, R.: 1997, ‘The metaphorical structure of mathematics: Sketching out cognitive foundations for a mind-based mathematics’, in L. English (ed.), Mathematical Reasoning: Analogies, Metaphors and Images, Mahwah, Erlbaum, New Jersey, 1997, pp. 21–89.
Lakoff, G. and Nunez, R.: 2000, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, Basic Books, New York.
Lambek, J.: 1996, ‘Number words and language origins,’ The Mathematical Intelligencer 18(4), 69–72.
Lave, J. and Wenger, E.: 1991, Situated Learning: Legitimate Peripheral Participation, Cambridge University Press, Cambridge, Massachusetts.
Machover, M.: 1996, Set Theory, Logic and their Limitations, Cambridge University Press, Cambridge.
Maxwell, J.: 1989, ‘Mathephobia’, in P. Ernest, (ed.), Mathematics Teaching: The State of the Art, Falmer Press, London, pp. 221–226.
Morgan, C.: 1998, Writing Mathematically: The Discourse of Investigation, Falmer Press, London.
Nunes, T.: 1992, ‘Ethnomathematics and Everyday Cognition’, in D.A. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, pp. 557–574.
Nunes, T., Schliemann, A. and Carraher, D.: 1988, Street Mathematics and School Mathematics, Cambridge University Press, Cambridge.
Piaget, J.: 1952, The Child's Conception of Number, Norton, New York.
Piaget, J.: 1969, The Psychology of the Child, Basic Books, New York.
Pickering, A.: 1995, ‘Concepts and the mangle of practice: The case of quaternions’, South Atlantic Quarterly, (Special issue: Mathematics, Science and Postclassical Theory, Ed. by B. H. Smith and A. Plotinsky), 94(2), 417–465 (Spring 1995).
Pimm, D.: 1995, Symbols and Meanings in School Mathematics, London: Routledge.
Pinxten, R.: 1987, Towards a Navajo Indian Geometry, Ghent, Belgium: Kultuur, Kennis en Integratie (Communication and Cognition).
Plunkett, S.: 1979, ‘Decomposition and all that rot,’ Mathematics in School 8, 2–7.
Polanyi, M.: 1958, Personal Knowledge, Routledge and Kegan Paul, London (Revised edn. 1964, Harper and Row, New York).
Radford, L.: 1998, On Signs and Representations a Cultural Account, Pre-Print Series, No. 1, School of the Sciences of Education, University Laurentienne, Sudbury, Ontario.
Resnick, L.B. and Ford, W.W.: 1981, The Psychology of Mathematics for Instruction, Lawrence Erlbaum, Hillsdale, New Jersey.
Robitaille, D.F. and Garden, R.A. (eds.): 1989, The IEA Study of Mathematics II: Contexts and Outcomes of School Mathematics, Pergamon, Oxford.
Rotman, B.: 1987, Signifying Nothing: The Semiotics of Zero, Routledge, London.
Rotman, B.: 1993, Ad Infinitum, Stanford University Press, Stanford California.
Saussure, F. de: 1916, Course in General Linguistics, Payot, Paris (Trans. by W. Baskin, Philosophical Library, New York, 1959).
Saxe, G.B.: 1991, Culture and Cognitive Development: Studies in Mathematical Understanding, L. Erlbaum, Hillsdale, New York.
Schmandt-Besserat, D.: 1978, ‘The earliest precursor of writing’, Scientific American 238(6).
Seeger, F. and Steinbring, H. (eds.): 1990, The Dialogue between Theory and Practice in Mathematics Education, IDM, University of Bielefeld, Bielefeld, Germany.
Sierpinska, A.: 1987, ‘Humanities students and epistemological obstacles related to limits’, Educational Studies in Mathematics 18, 371–397.
Skemp, R.R.: 1976, ‘Relational understanding and instrumental understanding’, Mathematics Teaching 77, 20–26.
Smith, D.E. (ed.) 1959, A Source Book in Mathematics (2 Vols.), Dover Press, New York.
Steffe, L.P. and Gale, J. (Eds.): 1995, Constructivism in Education, Erlbaum, Hillsdale, New Jersey.
Thompson, I.: 1998, ‘The influence of structural aspects of the English counting word system on the teaching and learning of place value’, Research in Education 59, 1–8.
Thompson, I. (ed.): 1999, Issues in Teaching Numeracy in Primary Schools Open University Press, Buckingham.
Vygotsky, L.: 1978, Mind in Society, Cambridge, Harvard University Press, Massachusetts.
Wilder, R.L.: 1974, Evolution of Mathematical Concepts, Transworld Books, London.
Wittgenstein, L.: 1953, Philosophical Investigations, Basil Blackwell, Oxford.
Zaslavsky, C.: 1973, Africa Counts, Prindle, Weber and Schmidt, Boston.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ernest, P. A Semiotic Perspective of Mathematical Activity: The Case of Number. Educ Stud Math 61, 67–101 (2006). https://doi.org/10.1007/s10649-006-6423-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10649-006-6423-7