Advertisement

Educational Studies in Mathematics

, Volume 38, Issue 1–3, pp 135–161 | Cite as

Beyond mere knowledge of mathematics: The importance of knowing-to act in the moment

  • John Mason
  • Mary Spence
Article

Abstract

Knowing-to is active knowledge which is present in the moment when it is required. To try to produce knowing-to, formal education focuses on forms of knowing which are easier to teach and to test: knowing-that (factual), knowing-how (technique and skills), and knowing-why (having a story in order to structure actions and from which to reconstruct actions). Together these constitute knowing-about the subject. Expertise is demonstrated by being able to respond to assessments: to write essays and to solve routine problems. The central problem of education is that knowing-about does not in itself guarantee knowing-to, as teachers have attested throughout the ages. For example, Edward Fitzgerald (Harrison, 1937) captures it beautifully in one stanza of his purported translation of the Rubaiyat of Omar Khayyam:

Myself when young did eagerly frequent,

Doctor and Saint and heard great argument,

About it and about: but ever more Came out by the same door as in I went (p. 341).

Instead of trying to reach definitions, we illustrate distinctions amongst kinds of knowing as used by various authors in the past. Then we turn to our own experience, for it is in one's own experience that one can locate and enliven sources of metaphoric resonances and metonymic triggers which constitute understanding. Drawing on our experience we distinguish knowing-to from other forms of knowing, and explore implications of that distinction for teaching and learning mathematics. We propose that knowing-to act in the moment depends on the structure of attention in the moment, depends on what one is aware of. Educating this awareness is most effectively done by labelling experiences in which powers have been exhibited, and developing a rich network of connections and triggers so that actions ‘come to mind’. No-one can act if they are unaware of a possibility to act; no-one can act unless they have an act to perform.

Keywords

Formal Education Central Problem Learning Mathematic Labelling Experience Active Knowledge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bauersfeld, H.: 1993, 'Theoretical perspectives on interaction in the mathematics classroom', in R. Biehler, R. Scholz, R. Sträßer and B. Winkelmann (eds.), The Didactics of Mathematics as a Scientific Discipline, Kluwer, Dordrecht, The Netherlands.Google Scholar
  2. Bereiter, C. and Scardammalia, M.: 1989, Knowing Learning and Instruction: Essays in Honor of Robert Glaser, Lawrence Erlbaum, Hillsdale.Google Scholar
  3. Biggs, J.: 1994, 'Modes of learning, forms of knowing, and ways of schooling', in A. Demetriou, M. Shayer and A. Efklides (eds.), Neo-Piagetian Theories of Cognitive Development, Routledge, London, pp. 31–51.Google Scholar
  4. Brousseau, G.: 1984, 'The crucial role of the didactical contract in the analysis and construction of situations in teaching and learning mathematics', in H. Steiner (ed.), Theory of Mathematics Education, Paper 54, Institut fur Didaktik der Mathematik der Universität Bielefeld, pp. 110–119.Google Scholar
  5. Brousseau, G.: 1997, Theory of Didactical Situations in Mathematics: Didactiques des Mathématiques, 1970–1990, (translated by N. Balacheff, M. Cooper, R. Sutherland, V. Warfield), Kluwer, Dordrecht, The Netherlands.Google Scholar
  6. Brown, J., Collins, A. and Duguid, P.: 1989, 'Situated cognition and the culture of learning', Educational Researcher 18(1), 32–41.Google Scholar
  7. Brown, M.: 1981, 'Is it an add, or a multiply, Miss?' (part 3), Mathematics in Schools 10(1), 26–28.Google Scholar
  8. Bruner, J.: 1966, Towards a Theory of Instruction, Harvard University Press, Cambridge.Google Scholar
  9. Burton, L.: 1995, 'Moving towards a feminist epistemology of mathematics', Educational Studies in Mathematics 28(3), 275–291.CrossRefGoogle Scholar
  10. Burton, L. (ed.): (in press), Learning Mathematics: From Hierarchies to Networks, Falmer Press, London.Google Scholar
  11. Chevellard, Y.: 1985, La Transposition Didactique, La Pensée Sauvage, Grenoble.Google Scholar
  12. Clancey, W.: 1993, 'Situated Action: a neuropsychological interpretation response to Vera & Simon', Cognitive Science 17, 87–116.Google Scholar
  13. Collins, A., Brown, J. and Newman, S.: 1989, 'Cognitive apprenticeship: teaching the crafts of reading, writing, and mathematics', in L. Resnick (ed.), Knowing, Learning, and Instruction: essays in honor of Robert Glaser, Lawrence Erlbaum, Hillsdale, pp. 453–494.Google Scholar
  14. Davidov, D.: 1990, 'Types of Generalisation in Instruction', Soviet Studies in Mathematics Education Vol. 2, NCTM, Reston.Google Scholar
  15. Detterman D. and Sternberg, R. (eds.): 1993, Transfer on Trial: Intelligence, Cognition, and Instruction, Ablex, Norwood N.J.Google Scholar
  16. De Jong, T. and Ferguson-Hessler, M.: 1996, 'Types and qualities of knowledge', Educational Psychologist 31(2), 105–113.Google Scholar
  17. Dewey, J.: 1902, The Child and the Curriculum, U of Chicago Press, Chicago.Google Scholar
  18. Dubinsky E. and Levin P.: 1986, 'Reflective abstraction and mathematics education: the genetic decomposition of induction and compactness', Journal of Mathematical Behaviour 5, 55–92.Google Scholar
  19. Fennema, E. and Franke, M.: 1992, 'Teachers' knowledge and its impact', in D. Grouws (ed.), Handbook of Research on Mathematics Teaching, MacMillan, New York, pp. 147–164.Google Scholar
  20. Frayn, M.: 1998, Copenhagen, Methuen Drama, Methuen, London.Google Scholar
  21. Gates, P.: 1993, '“I just didn't think of it”: learning to teach mathematics', Proceedings of the British Society for Research in Learning Mathematics Meeting May 22, pp. 29–34.Google Scholar
  22. Gattegno, C.: 1987, The Science of Education Part I: Theoretical Considerations, Educational Solutions, New York.Google Scholar
  23. Greeno, J. Smith, D. and Moore, J.: 1993, 'Transfer of situated learning', in D. Detterman and R. Sternberg (eds.), Transfer on Trial: Intelligence, Cognition, and Instruction, Abbex, Norwood, NJ, pp. 99–167.Google Scholar
  24. Harrison, G. (ed.): 1937, A Book of English Poetry, Penguin, Harmondsworth.Google Scholar
  25. Hewitt, D.: 1994, The Principle of Economy in the Learning and Teaching of Mathematics, unpublished PhD dissertation, Open University, Milton Keynes.Google Scholar
  26. Hiebert, J. and Carpenter, T.: 1992, 'Learning and teaching with understanding', in D. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning, MacMillan, New York, pp. 65–93.Google Scholar
  27. Hofer, B. and Pintrich, P.: 1997, 'The development of epistemological theories: beliefs about knowledge and knowing and their relation to learning', Review Of Educational research 67(1), 88–140.Google Scholar
  28. Honderich, T.: 1995, The Oxford Companion to Philosophy, Oxford University Press, Oxford.Google Scholar
  29. Hoyles, C.: 1998, 'A culture of proving in school maths?', in D. Tinsley and D. Johnson (eds.), Information and Communications Technologies in School Mathematics, Chapman & Hall, London, pp. 169–182.Google Scholar
  30. Kang, W. and Kilpatrick, J.: 1992, 'Didactic transposition in mathematics textbooks', For The Learning of Mathematics 12(1), 2–7.Google Scholar
  31. Kieren, T.: 1994, 'Bonuses of understanding mathematical understanding', in D. Robitaille, D. Wheeler and C. Kieran (eds.), Selected Lectures from the 7th International Congress on Mathematical Education, Les Presses de l'Université Laval, Quebec, pp. 211–228.Google Scholar
  32. Krutetskii, V.: 1976, The Psychology of Mathematical Abilities in Schoolchildren, University of Chicago Press, Chicago.Google Scholar
  33. Lacan, J.: 1985, 'Sign, symbol and imagery', in Blonsky (ed.), On Signs, Blackwell, Oxford.Google Scholar
  34. Lave, J.: 1988, Cognition in Practice: Mind, Mathematics and Culture in Everyday Life, Cambridge University Press, Cambridge.Google Scholar
  35. Locke, J.: 1894, (Dover reprint 1959): An Essay Concerning Human Understanding, 2 Vols, Dover, London.Google Scholar
  36. Marton, F. and Fazey, J.: (preprint), Understanding as Space of Experienced Variation.Google Scholar
  37. Mason, J., Burton, L. and Stacey, K.: 1982, Thinking Mathematically, Addison Wesley, London.Google Scholar
  38. Mason, J. and Davis, J.: 1988, 'Cognitive and metacognitive shifts', PME XII, (ed. Borbás) Vol. 2, pp. 487–494.Google Scholar
  39. Mason, J. and Davis, J.: 1989, 'The inner teacher, the didactic tension, and shifts of attention', in G. Vergnaud, M. Rogalski and M. Artigue (eds.), Proceedings of PME XIII, Paris, Vol. 2, pp. 274–281.Google Scholar
  40. Mason, J. and Spence, M.: 1998, 'Towards a psychology of knowing-to', in C. Kane, M. Goos and E. Warren (eds.), Proceedings of MERGA 21, Coolangatta, New South Wales, Australia.Google Scholar
  41. Mason, J.: 1989, 'Mathematical abstraction seen as a delicate shift of attention', For the Learning of Mathematics 9(2). 2–8.Google Scholar
  42. Mason, J.: 1993, 'Working on awareness', in J. Searl (ed.), Proceedings of the Edinburgh Mathematics Teaching Conference, University of Edinburgh, Edinburgh.Google Scholar
  43. Mason, J.: 1996, Personal Enquiry: Moving from Concern towards Research, Open University, Milton Keynes.Google Scholar
  44. Mason, J.: 1998, 'Structure of attention in teaching mathematics, Plenary address to Canadian Mathematics Education Study Group 15th meeting, May', in Y. Pothier (ed.), Proceedings of cmESG 15, in preparation.Google Scholar
  45. Maturana, H. and Varela, F.: 1972, 'Autopoesis and cognition: the realization of the living', D. Reidel, Dordrecht, The Netherlands.Google Scholar
  46. Maturana, H. and Varela, F.: 1988, The Tree of Knowledge: the Biological Roots of Human Understanding, Shambala, Boston.Google Scholar
  47. Maturana, H.: 1988, 'Reality: the search for objectivity or the quest for a compelling argument', Irish Journal of Psychology 9(1), 25–82.Google Scholar
  48. McGowen, M.: 1998, 'Cognitive units, concept images, and cognitive collages', Unpublished PhD thesis, Warwick University, Coventry.Google Scholar
  49. Michener, E.: 1978, 'Understanding understanding mathematics', Cognitive Science 2, 361–383.CrossRefGoogle Scholar
  50. Miller, L., Malone, J. and Kandl, T.: 1992, 'A study of secondary teachers' perceptions of the meaning of understanding', AERA paper, San Francisco.Google Scholar
  51. Northfield, J. and Baird, J.: 1992, Learning from the PEEL Experience, Monash University Printing Service, Melbourne.Google Scholar
  52. Noss, R. and Hoyles, C.: 1996, Windows on Mathematical Meanings, Mathematics Education Library, Kluwer, Dordrecht, The Netherlands.Google Scholar
  53. Perry, W.: 1968, Forms of Intellectual and Ethical Development in the College Years: a scheme, Holt, Rhinehart & Winston, New York.Google Scholar
  54. Piaget, J.: 1950, 'Introduction á l'epistemologie genetique', presses Univ. de France, Paris.Google Scholar
  55. Piaget, J.: 1977, The Development of Thought: Equilibration of Cognitive Structures, (translated by A. Rosin), Harvard University Press, Cambridge.Google Scholar
  56. Pimm, D.: 1987, Speaking Mathematically, Hodder & Stoughton, London.Google Scholar
  57. Pirie, S. and Kieren, T.: 1994, 1Growth in mathematical understanding: how can we characterise it and how can we represent it?', Educational Studies in Mathematics 26(2–3), 165–190.CrossRefGoogle Scholar
  58. Polya, G.: 1945, How To Solve It, Princeton University Press, Cambridge.Google Scholar
  59. Polya, G.: 1962, Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving, Combined edition, Wiley, New York.Google Scholar
  60. Reid, B.: 1980, The Haida Legend of The Raven and The First Humans, Museum Note No. 8, U. B. C. Museum of Anthropology, Vancouver.Google Scholar
  61. Renkl, A., Mandl, H. and Gruber, H.: 1996, 'Inert knowledge: analyses and remedies', Educational Psychologist 31(2), 115–121.Google Scholar
  62. Russell, B.: 1914, Our Knowledge of the External World as a Field for Scientific Mathod in Philosophy, George Allen & Unwin, London.Google Scholar
  63. Ryle, G.: 1949, The Concept of Mind, Hutchinson, London.Google Scholar
  64. Schoenfeld, A.: 1988, 'When good teaching leads to bad results: The disasters of 'well taught' mathematics classes', Educational Psychologist 23, 145–166.Google Scholar
  65. Schön, D.: 1983, The Reflective Practitioner: How Professionals Think in Action, Temple Smith, London.Google Scholar
  66. Schön, D.: 1987, Educating the Reflective Practitioner, Jossey-Bass, London.Google Scholar
  67. Shah, I.: 1968, The Way of the Sufi, Jonathon Cape, London.Google Scholar
  68. Shulman, L.: 1987, 'Knowledge and teaching: Foundations of the new reform', Harvard Educational Review 57(1), 1–22.Google Scholar
  69. Sierpinska, A.: 1994, Understanding in Mathematics, Falmer Press, London.Google Scholar
  70. Skemp, R.: 1979, Intelligence, Learning and Action, Wiley, Chichester.Google Scholar
  71. Spence, M.: 1996, 'Psychologising algebra: case studies of knowing in the moment', unpublished PhD thesis, Madison Wisconsin.Google Scholar
  72. Tahta, D.: 1972, A Boolean Anthology: Selected Writings of Mary Boole on Mathematics Education, Association of Teachers of Mathematics, Derby.Google Scholar
  73. van Hiele, P.: 1986, Structure and Insight: A Theory of Mathematics Education, Academic Press, Orlando.Google Scholar
  74. Vergnaud, G.: 1981, 'Quelques orientations théoriques et méthodologiques des recherches Françaises en didactique des mathématiques', Actes duVième Colloque de PME, Vol. 2, pp. 7–17, Edition IMAG, Grenoble.Google Scholar
  75. Whitehead, A.: 1932, The Aims of Education and Other Essays, Williams & Norgate, London.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • John Mason
    • 1
  • Mary Spence
    • 1
  1. 1.Centre for Mathematics EducationOpen UniversityMilton KeynesUK

Personalised recommendations