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The Role of Theory in Mathematics Education and Research

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International Handbook of Mathematics Education

Part of the book series: Kluwer International Handbooks of Education ((SIHE,volume 4))

Abstract

Theory is seen as playing both a foreground role, in the sense that the enterprise of research in mathematics education is to develop theory, and a background role, since all observation is theory laden. After considering different theoretical stances towards observation, data, and analysis, three broad approaches to research in mathematics education are identified and the role played by theory in each of these is discussed. The chapter concludes with some speculations concerning possible directions for development, moving from language as core domain of problematicity, to images, icons, and the impact of electronic media on interpretation in the future.

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References

  • Artigue, M. and Perrin-Glorian, M-J.: 1991, ‘Didactic Engineering Research and Development: Some Theoretical Problems Linked to this Duality’, For the Learning of Mathematics 11(1),13–17.

    Google Scholar 

  • Bachelard, D.: 1938 (republished 1983), La Formation de l’Esprit Scientifique, Presse Universitaire de France, Paris.

    Google Scholar 

  • Bakhtin, M.: (1986a), ‘Extracts from “Notes” (1970–1971)’, in G.S. Morson (ed.) Bakhtin: Essays and Dialogues on his Work, The University of Chicago Press, Chicago.

    Google Scholar 

  • Bakhtin, M.: (1986b). ‘Extracts from “The Problem of Speech Genres”’, in G.S. Morson (ed.) Bakhtin: Essays and Dialogues on his Work, The University of Chicago Press, Chicago.

    Google Scholar 

  • Balacheff, N.: 1990, ‘Towards a Problématique for Research on Mathematics Teaching’, Journal for Research in Mathematics Education 21(4), 258–272.

    Article  Google Scholar 

  • Baron-Cook, S.: 1995, Mindblindness, MIT Press, Cambridge, Boston

    Google Scholar 

  • Bauersfeld, H.: 1990, ‘Activity Theory and Radical Constructivism: What do they have in Common?’ in F. Steier (ed.) Ecological Understanding, Annual Conference of the American Society for Cybernetics, Oslo.

    Google Scholar 

  • Bauersfeld, H.: 1991, ‘Structuring the Structures’, in L. Steffe (ed.) Constructivism and Education, Lawrence Erlbaum, Hillsdale, NJ.

    Google Scholar 

  • Bauersfeld, H.: 1992, ‘Integrating Theories for Mathematics Education’, For the Learning of Mathematics 12(2), 19–28.

    Google Scholar 

  • Bennett, J.: 1993, Elementary Systematics: A Tool for Understanding Wholes, Bennett Books, Santa Fe.

    Google Scholar 

  • Bishop, A.: 1992, ‘International Perspectives on Research in Mathematics Education’, in D. Grouws, (ed.) Handbook of Research on Mathematics Teaching and learning, MacMillan, Oxford, 710–723.

    Google Scholar 

  • Boero, P.: 1990, ‘On Long Term Development of Some General Skills in Problem-solving: A Longitudinal Comparative Study’, in G. Booker, P. Cobb, and T.N. Mendicuti (eds.) Proceedings of the Fourteenth PME Conference, International Group for the Psychology of Mathematics Education, Mexico, vol. 2, 169–176.

    Google Scholar 

  • Brousseau, G.: 1981, ‘Problèmes de Didactique des Décimaux’, Recherches en Didactique des Mathématiques 2(1), 37–125.

    Google Scholar 

  • Brousseau, G.: 1984, ‘The Crucial Role of the Didactical Contract in the Analysis and Construction of Situations in Teaching and Learning Mathematics’, in H-G Steiner (ed.) Theory of Mathematics Education, Paper 54, Institut fur Didaktik der Mathematik der Universitat Bielefeld.

    Google Scholar 

  • Brown S., Collins A. and Duguid P.: 1989, ‘Situated Cognition and the Culture of Learning’, Educational Researcher 18(1), 32–41.

    Article  Google Scholar 

  • Bruner, J.: 1986, Actual Minds, Possible Worlds, Harvard University Press, Cambridge.

    Google Scholar 

  • Carpenter, T. and Fennema, E.: 1992, ‘Cognitively Guided Instruction: Building on the Knowledge of Students and Teachers’, in W. Secada (ed.) Curriculum Reform: The Case of Mathematics Education in the US.

    Google Scholar 

  • Carraher, T., Carraher, D. and Schliemann, A.: 1985, ‘Mathematics in the Streets and in Schools’, British Journal of Developmental Psychology 3, 21–29.

    Article  Google Scholar 

  • Carr, W.: 1986, ‘Theories of Theory and Practice’, Journal of Philosophy of Education 20(2), 177–186.

    Article  Google Scholar 

  • Cobb, P.: 1990, ‘A Constructivist Perspective on Information-processing: Theories of Mathematical Activity’, International Journal of Educational Research 14, 67–92.

    Article  Google Scholar 

  • Cobb, P.: 1995, ‘Cultural Tools and Mathematical Learning: A Case Study’, Journal for Research in Mathematics Education 26(4), 362–385.

    Article  Google Scholar 

  • Cohen, L.: 1976, Educational Research in Classrooms and Schools: A Manual of Materials and Methods, Harper & Row, London.

    Google Scholar 

  • Confrey, J.: 1995, ‘A Theory of Intellectual Development’ (in three parts), For The Learning of Mathematics 15(1) pp.2–8,

    Google Scholar 

  • Confrey, J.: 1995, ‘A Theory of Intellectual Development’ (in three parts), For The Learning of Mathematics 15(2), pp.38–47,

    Google Scholar 

  • Confrey, J.: 1995, ‘A Theory of Intellectual Development’ (in three parts), For The Learning of Mathematics 15(3) pp.36–47.

    Google Scholar 

  • Davydov, V.: 1990, Soviet Studies in Mathematics Education: Types of Generalization in Instruction, vol. 2, National Council of Teachers of Mathematics, Reston, VA.

    Google Scholar 

  • Dawood, J. (trans): 1956, The Koran, Penguin Classics, Harmondsworth.

    Google Scholar 

  • Donaldson, M.: 1978, Children’s Minds, Norton, New York.

    Google Scholar 

  • Douady, R, and Mercier, A.: 1992 (eds.), Research in Didactique of Mathematics, La Pensée Sauvage, Grenoble.

    Google Scholar 

  • Elbez, F.: 1983, Teachers’ Thinking: A Study of Practical Knowledge, Nichols, New York.

    Google Scholar 

  • Frankenstein, M.: 1989, Relearning Mathematics: A Different Third R—Radical Mathematics, Free Association Press, London.

    Google Scholar 

  • Freudenthal, H.: 1971, ‘Geometry Between the Devil and the Deep Blue See’, Educational Studies in Mathematics 5,413–435.

    Article  Google Scholar 

  • Freudenthal, H.: 1991, Revisiting Mathematics Education, Kluwer, Dordrecht.

    Google Scholar 

  • Gattegno, C: 1970, What We Owe Children: The Subordination of Teaching to Learning, Routledge & Kegan Paul, London.

    Google Scholar 

  • Goodman, N.: 1978, Ways of World Making, Harvester press, Hassocks.

    Google Scholar 

  • Gravemeijer, K.: 1994, ‘Educational Development and Developmental Research in Mathematics Education’, Journal for Research in Mathematics Education 25(5), 443–471.

    Article  Google Scholar 

  • Gray, E. and Tall, D.: 1994, ‘Duality, Ambiguity, and Flexibility: A ’ProceptuaP View of Simple Arithmetic’, Journal for Research in Mathematics Education 25(2), 116–140.

    Article  Google Scholar 

  • Hamilton, E. and Cairns, H. (eds.): (1961, reprinted 1985) Plato: The Collected Dialogues, Bollingen Series LXXI, Princeton University Press, Princeton.

    Google Scholar 

  • Hanson, N.: 1958, Patterns of Discovery, Cambridge University Press, Cambridge, UK.

    Google Scholar 

  • Hart, K.: 1993, ‘Confidence in Success’, in I. Hirabayashi, N. Nohda, K. Shigematsu and Fou-Lai Liu (eds.) Proceedings of the Seventeenth International PME Conference, vol. 1, Tsukuba, Ibaraki, Japan, 1–17.

    Google Scholar 

  • Holquist, M. (ed.): 1981, The Dialogic Imagination, University of Texas Press, Austin.

    Google Scholar 

  • Howie, G.: 1969, St. Augustine on Education, Henry Regnery Co., Chicago.

    Google Scholar 

  • Howson, G., Keitel, C. and Kilpatrick, J.: 1981, Curriculum Development in Mathematics, Cambridge University Press, Cambridge, UK.

    Book  Google Scholar 

  • Jaynes, J.: 1976, The Origins of Consciousness in the Breakdown of the Bicameral Mind, University of Toronto, Toronto.

    Google Scholar 

  • Kilpatrick, J.: 1981, ‘Research on Mathematical Learning and Thinking in the United States’, Proceedings of PME I, vol. 2, 18–29.

    Google Scholar 

  • Kitcher, P.: 1984, The Nature of Mathematical Knowledge, Oxford University Press, Oxford.

    Google Scholar 

  • Koerner, K.: 1992, ‘The Sapir-Whorf Hypothesis: Leibniz to Lucy’, Journal of Linguistic Anthropology 2(2).

    Google Scholar 

  • Kuhn, T.: 1970, The Structure of Scientific Revolutions, University of Chicago, Chicago.

    Google Scholar 

  • Laborde, C: 1989a, ‘Audacity and Reason: French Research in Mathematics Education’, For the Learning of Mathematics 9(3), 31–36.

    Google Scholar 

  • Laborde, C: 1989b, ‘Hardiesse et Raison des Recherches Françaises en Didactique des Mathématiques’, in G. Vergnaud, J. Rogalski and M. Artigue (eds.) Actes de la 13e Conférence Internationale PME, vol. 1,46–61.

    Google Scholar 

  • Lacan, J.: 1977, ‘The Mirror Stage’, in A. Sheridan (trans.), Écrits: A Selection, Tavistock, London.

    Google Scholar 

  • Lakatos, I.: 1976, Proofs and Refutations, Cambridge University Press, Cambridge.

    Book  Google Scholar 

  • Lave, J.: 1988, Cognition in Practice: Mind, Mathematics, and Culture in Everyday Life, Cambridge University Press, Boston.

    Book  Google Scholar 

  • Lincoln, Y. and Guba, E.: 1985, Naturalistic Enquiry, Sage, Beverly Hills.

    Google Scholar 

  • Lyotard, J.-F.: 1984, The Postmodern Condition: A Report on Knowledge, Theory and History of Literature V10, Manchester University Press, Manchester.

    Google Scholar 

  • Lyotard, J.-F.: 1991, Phenomenology, (trans. B. Beakley), State University of New York Press, Albany.

    Google Scholar 

  • Lyotard, J.-F.: 1992. ‘From the Postmodern Condition: A Report on Knowledge’, in D. Ingram, and J. Ingram (eds.) Critical Theory: The Essential Readings, Paragon House, New York.

    Google Scholar 

  • Lyotard, J.-F.: 1994, ‘Bureaucracy: Resistance, Witnessing, Writing’, L’Esprit Creative 34(1), 101–108.

    Google Scholar 

  • Marton, F.: 1981, ‘Phenomenography: Describing Conceptions of the World Around Us’, Instructional Science 10, 177–200.

    Article  Google Scholar 

  • Marton, F.: 1988, ‘Phenomenography: A Research Approach to Investigating Different Understandings of Reality’, in J. Sherman and N. Webb (eds.) Qualitative Research in Education: Focus and Methods, Falmer press, London, 141–161.

    Google Scholar 

  • Mason, J. and Davis, J.: 1989, ‘Notes on a Radical Constructivist Epistemethodology Applied to Didactic Situations’, Journal of Structural Learning 10, 157–176.

    Google Scholar 

  • Mason, J.: 1979, ‘Which Medium, Which Message?’ Visual Education, Feb. 1979, 29–33.

    Google Scholar 

  • Mason, J.: 1994, ‘Researching from the Inside in Mathematics Education: Locating an I-You Relationship’, in J.P. da Ponte and J.F. Matos (eds.) Proceedings of the Eighteenth Conference for the Psychology of Mathematics Education, vol. 1, PME, Lisbon, 176–190.

    Google Scholar 

  • Mason, J.: 1996, Personal Enquiry: Moving from Concern Towards Research, Open University, Milton Keynes.

    Google Scholar 

  • Mason, J.: in press, ‘Researching from the Inside in Mathematics Education’, in A. Sierpinska and J. Kilpatrick (eds.) ICMI Study: What is Research in Mathematics Education and What are its Results? Kluwer, Dordrecht.

    Google Scholar 

  • Maturana, H. and Varela, F.: 1988, The Tree of Knowledge: The Biological Roots of Human Understanding, Shambala, Boston.

    Google Scholar 

  • Mellin-Olsen, S.: 1987, The Politics of Mathematics Education, Reidel, Dordrecht.

    Google Scholar 

  • Mühäusler, P. and Harré, R.: 1990, Pronouns and People: The Linguistic Construction of Personal Identity, Basil Blackwell, Oxford.

    Google Scholar 

  • Nagel, E.: 1961, The Structure of Science, Routledge & Kegan Paul, London.

    Google Scholar 

  • Nunes, T.: 1992, ‘Ethnomathematics and Everyday Cognition’, in D. Grouws (ed.) Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, 557–574.

    Google Scholar 

  • Orton, R.: 1988, ‘Two Theories of’Theory’ in Mathematics Education: Using Kuhn and Lakatos to Examine Four Fundamental Issues’, For the Learning of Mathematics 8(2), 36–43.

    Google Scholar 

  • Open University: 1980, PME233 Mathematics Across the Curriculum, Open University, Milton Keynes.

    Google Scholar 

  • Perry, W.: 1968, Forms of Intellectual and Ethical Development in the College Years: A Scheme, Holt, Rinehart & Winston, New York.

    Google Scholar 

  • Romberg, T.: 1983, ‘Toward ‘Normal Science’ in Some Mathematics Education Research’, Zentralblatt fur Didaktik der Mathematik 15(8), 89–92.

    Google Scholar 

  • Royce, J.: 1978, ‘How Can We Best Advance the Construction of a Theory in Psychology?’ Canadian Psychological Review 19(4), 259–276.

    Article  Google Scholar 

  • Salomon, G.: 1979, Interaction of Media, Cognition and Learning, Jossey-Bass, London.

    Google Scholar 

  • Schaeffer, J.D.: 1990, Senus Communis: Vico, Rhetoric, and the Limits of Relativism, Duke University Press, London.

    Google Scholar 

  • Schoenfeld, A., Smith, J. and Arcavi, A.: 1993, ‘The Microgenetic Analysis of One Student’s Evolving Understanding of Complex Subject Matter Domain’, in R. Glaser (ed.) Advances in Instructional Psychology, vol. 4, Lawrence Erlbaum, Hillsdale, NJ.

    Google Scholar 

  • Schwab, J.: 1962, ‘The Teaching of Science as Enquiry’, in J. Schwab and P. Brandwein (eds.) The Teaching of Science, Harvard University Press, Cambridge.

    Google Scholar 

  • Sfard, A.: 1991, ‘On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin’, Educational Studies in Mathematics 22, 1–36.

    Article  Google Scholar 

  • Simon, M.: 1995, ‘Reconstructing Mathematics Pedagogy from a Constructivist Point of View’, Journal for Research in Mathematics Education 26(2), 114–145.

    Article  Google Scholar 

  • Skemp, R.: 1983, ‘The International Group for the Psychology of Mathematics Education: Past, Present and Future’, Zentralblatt fur Didaktik der Mathematik 15, 112–117.

    Google Scholar 

  • Stallings, W.: 1995, ‘Confessions of a Quantitative Educational Researcher Trying to Teach Qualitative Research’, Educational Researcher 24(3), 31–32.

    Google Scholar 

  • Steffe, L. and Cobb, P.: 1983, Construction of Arithmetical Meanings and Strategies, Springer-Verlag, New York.

    Google Scholar 

  • Steinbring, H.: 1989, ‘Routine and Meaning in the Mathematics Classroom’, For the Learning of Mathematics 9(1), 24–33.

    Google Scholar 

  • Steiner, H-G.: 1985, ‘Theory of Mathematics Education (TME): An Introduction’, For the Learning of Mathematics 5(2), 11–17.

    Google Scholar 

  • Tahta, D.: 1991, ‘Three Apples Fell From Heaven in..., in ... Reflections on a Day...’, IF 2, Centre for Mathematics Education, Open University, Milton Keynes, 25–30.

    Google Scholar 

  • Thorn, R.: 1973, ‘Modern Mathematics: Does It Exist?’ in G. Howson (ed) Developments in Mathematical Education, Cambridge University Press, Cambridge.

    Google Scholar 

  • Todorov, T.: 1984, Mikhail Bakhtin: The Dialogical Principle (trans. W. Godzich), University of Minnesota Press, Minneapolis.

    Google Scholar 

  • Usiskin, Z.: 1982, Van Hiele Levels and Achievement in Secondary School Geometry, University of Chicago, Chicago.

    Google Scholar 

  • Varela, F., Thompson, E. and Rosch, E.: 1991, The Embodied Mind: Cognitive Science and Human Experience, MIT press, Cambridge.

    Google Scholar 

  • Vergnaud, G.: 1981, ‘Quelques Orientations Théoriques et Méthodlogiques des Recherches Françaises en Didactique des Mathématiques’, in Actes du Vième Colloque de PME, Grenoble, Edition IMAG, vol. 2, 7–17.

    Google Scholar 

  • Vergnaud, G.: 1990, ‘La Théorie des Champs Conceptuels’, Recherches en Didactique des Mathématiques 10(2–3), 133–170.

    Google Scholar 

  • Vico, G-B.: 1744, Principi di Scienza Nuova, Republished as Bergin & Fisch (trans.) 1961, Anchor Books, Garden City.

    Google Scholar 

  • von Glasersfeld, E.: 1984, ‘An Introduction to Radical Constructivism’, in P. Watzlawick (ed.), The Invented Reality, Norton, London, 17–40.

    Google Scholar 

  • von Glasersfeld, E.: 1995, Radical Constructivism, Falmer Press, London.

    Book  Google Scholar 

  • Vygotsky L.: 1978, Mind in Society: The Development of the Higher Psychological Processes, Harvard University Press, London.

    Google Scholar 

  • Watzlawick P.: 1984, The Invented Reality: How do we Know what we Believe we Know? — Contributions to Constructivism, Norton, London.

    Google Scholar 

  • Waywood, A.: 1993, ‘Looking for an Appropriate Object of Educational Research: A Reflection on Three Presentations’, Mathematics Education Research Journal 2(5), 83–89.

    Article  Google Scholar 

  • Wertsch, J.: 1985, Culture, Communication and Cognition: Vygotskian Perspectives, Cambridge University Press, Cambridge, UK.

    Google Scholar 

  • Winnicot, D.: 1971, Playing and Reality, Tavistock, London

    Google Scholar 

  • Woodrow, D.: 1995, ‘The Quest for Multiple Beliefs in Learning Theories: Its Frustration by Single Faiths’, Chreods 9, 10–14.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Open University, UK

    John Mason & Andrew Waywood

  2. United Kingdom and Australian Catholic University, Australia

    John Mason & Andrew Waywood

Authors
  1. John Mason
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  2. Andrew Waywood
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Editor information

Editors and Affiliations

  1. Monash University, Clayton, Victoria, Australia

    Alan J. Bishop

  2. Universiti Brunei, Darussalam, Brunei

    Ken Clements

  3. Freie Universität Berlin, Germany

    Christine Keitel

  4. The University of Georgia, USA

    Jeremy Kilpatrick

  5. Université Joseph Fourier, Grenoble, France

    Colette Laborde

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© 1996 Kluwer Academic Publishers

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Mason, J., Waywood, A. (1996). The Role of Theory in Mathematics Education and Research. In: Bishop, A.J., Clements, K., Keitel, C., Kilpatrick, J., Laborde, C. (eds) International Handbook of Mathematics Education. Kluwer International Handbooks of Education, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1465-0_29

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  • DOI: https://doi.org/10.1007/978-94-009-1465-0_29

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  • Online ISBN: 978-94-009-1465-0

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