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Memory in mathematical understanding

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Abstract

Whatever cognitive processes are involved in understanding mathematics, it is clear that one of them is learning. No one is born with an understanding of measure theory, abstract algebra or general topology; the very name ‘mathematics’ means “that which is to be learned” (Boyer, 1968). One of the outcomes of learning is remembered knowledge. Indeed it is our contention that memory plays an essential role in the understanding of mathematics, However, what it is that is remembered by students who ‘understand mathematics’ in contrast to those who do not is by no means a trivial question. In fact, we would like to suggest that there is a gap in this connection between recent developments in memory research and the theory and practice of mathematics education. A second purpose of the present article is to survey briefly the orgins of this gap.

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References

  • Allport, D. A.: 1980, ‘Patterns and actions: cognitive mechanisms are content-specific’, in G.Claxton (ed.):Cognitive Psychology: New Directions, Routledge and Kegan Paul, London, pp. 26–64.

    Google Scholar 

  • Anderson, J. R. and Bower, G. H.: 1973,Human Associative Memory, Wiley, Halstead Press.

  • Ausubel, D. P.: 1968,Educational Psychology: A Cognitive View, Holt, Rinehart and Winston, pp. 113–114.

  • Ausubel, D. P.: 1971, ‘Some psychological and educational limitations of learning by discovery’, in D. B. Aichele and R. E. Reyes (eds.):Reading in Secondary School Mathematics, Prindle, Weber and Schmidt, pp. 193–209.

  • Baddeley, A. D.: 1976,The Psychology of Memory, Basic Books.

  • Bartlett, F. C.: 1932,Remembering: A Study in Experimental and Social Psychology, Cambridge University Press.

  • Bell, A. W.: 1976,The Learning of General Mathematical Strategies, Ph.D. Thesis, Centre for Mathematical Education, University of Nottingham, p. 9.18.

  • Boyer, C. B.: 1968,A History of Mathematics, Wiley, p. 53.

  • Branca, N. A.: 1980, ‘Communication of mathematical structure and its relationship to achievement’,Journal for Research in Mathematics Education 11, 37–49.

    Google Scholar 

  • Bruner, J. S.: 1961, ‘The act of discovery’,Harvard Educational Review 31, 21–32.

    Google Scholar 

  • Bruner, J. S.: 1962,The Process of Education, Harvard University Press.

  • Bruner, J. S.: 1973,Beyond the Information Given: Studies in the Psychology of Knowing, Norton.

  • Burton, L.: 1984, ‘Mathematical thinking: the struggle for meaning’,Journal for Research in Mathematics Education 15, 1, 35–49.

    Google Scholar 

  • Butler, H. C. and Wren, F.: 1941,The Teaching of Secondary Mathematics, 1 st ed., McGraw-Hill.

  • Byers, V.: 1980, ‘What does it mean to understand mathematics?’,International Journal of Mathematics Education Science and Technology 11, 1, 1–10.

    Google Scholar 

  • Byers, V. and Erlwanger, S. H.: 1984, ‘Content and form in mathematics’,Educational Studies in Mathematics 15, 259–275.

    Google Scholar 

  • Byers, V. and Herscovics, N.: 1977, ‘Understanding school mathematics’,Mathematics Teaching 81.

  • Carry, L. B., Lewis, C. and Bernard, J. F.: 1980,Psychology of Equation Solving: An Information Processing Study, Final Technical Report, Dept. of Curriculum & Instruction, University of Texas At Austin.

  • Claxton, G.: 1980, ‘Remembering and understanding’, in G. Claxton (ed.):Cognitive Psychology: New Directions, Routledge and Kegan Paul, pp. 197–235.

  • Claxton, G.: 1980, ‘Cognitive psychology: a suitable case for what sort of treatment?’, in G. Claxton (ed.):Cognitive Psychology: New Directions, Routledge and Kegan Paul, pp. 1–25.

  • Collins, A. M. and Quillian, M. R.: 1972, ‘How to make a language user’, in E. Tulvig and W. Donaldson (eds.):Organization of Memory, Academic Press, pp. 309–351.

  • Davis, E. J.: 1978, ‘A model for understanding understanding in mathematics’,Arithmetic Teacher, September.

  • Davis, R. R.: 1983, ‘Complex mathematical cognition’, in H. P.Ginsburg (ed.):The Development of Mathematical Thinking, Academic Press, New York, pp. 254–290.

    Google Scholar 

  • Erlwanger, S. H.: 1973, ‘Benny's conception of rules and answers in IPI mathematics’,Journal of Childrens' Mathematical Behaviour 1, 2.

    Google Scholar 

  • Flavell, J. H.: 1977,Cognitive Development, Prentice-Hall.

  • Freemont, H.: 1971, ‘New mathematics and old dilemmas’, in D. B. Aichele and R. E. Reyes (eds.):Readings in Secondary School Mathematics, Prindle, Weber and Schmidt, p. 4.

  • Gagné, R. M.: 1968, ‘Learning hierarchies’,Educational Psychologist 6, 1–9.

    Google Scholar 

  • Gagné, R. M.: 1970,Conditions of Learning, 2nd Ed., Holt Rinehart and Winston.

  • Gagné, R. M.: 1983, ‘Some issues in the psychology of mathematics instruction’,Journal for Research in Mathematics Education 14, 7–18.

    Google Scholar 

  • Gagné, R. M. and White, R. T.: 1978, ‘Memory structures and learning outcomes’,Review of Educational Research,48, 2, 187–222.

    Google Scholar 

  • Geeslin, W. E. and Shavelson, R. J.: 1975, ‘Comparison of content structure and cognitive structure in high school students' learning of probability’,Journal for Research in Mathematics Education 6, 109–120.

    Google Scholar 

  • Ginsburg, H.: 1977,Children's Arithmetic: The Learning Process, D. van Nostrand, pp. 128–129.

  • Greeno, J. G.: 1973, ‘The structure of memory and the process of solving problems’, in R. L. Solso, (ed.):Contemporary Issues in Cognitive Psychology: The Loyola Symposium, Winston, pp. 103–134.

  • Greer, B.: 1981, ‘Cognitive psychology and mathematical thinking’,For the Learning of Mathematics 1, 3, 22–24.

    Google Scholar 

  • Hebb, D. O.: 1972,Textbook of Psychology, 3rd ed., Saunders, p. 99.

  • Hilgard, E. R.: 1957,Introduction to Psychology, 2nd ed., Harcourt Brace, pp. 285–302.

  • Jenks, S. M., Peck, D. M. and Chatterley, L. J.: 1980, ‘Why blame the kids? We teach mistakes!’,Arithmetic Teacher 28, 2, 38–41.

    Google Scholar 

  • Krutetskii, V. A.: 1976,The Psychology of Mathematical Abilities in School Children, in J. Kilpatrick and I. Wirszup (eds.): University of Chicago Press.

  • Langford, F. G.: 1972,Some Computational Strategies of Seventh Grade Pupils, University of Virginia, Charlottesville.

    Google Scholar 

  • Lehman, H.: 1977, ‘On understanding mathematics’,Educational Theory 27, 2.

    Google Scholar 

  • Mandler, G.: 1972, ‘Organization and recognition’, in E. Tulvig and W. Donaldson (eds.):Organization of Memory, Academic Press, pp. 146–167.

  • Mayer, R. E.: 1977,Thinking and Problem Solving: An Introduction to Human Cognition and Learning, Scott, Foresman & Co.

  • Mayer, R. E. and Greeno, J. G.: 1972, ‘Structural differences between learning outcomes produced by different instructional methods’,Journal of Educational Psychology 63, 165–173.

    Google Scholar 

  • Michener, E. R.: 1978, ‘Understanding understanding mathematics’,Cognitive Science 2, 361–383.

    Google Scholar 

  • Miller, G. A.: 1956, ‘The magical number seven, plus or minus two: some limits on our capacity for processing information’,Psychological Review 63, 81–96.

    Google Scholar 

  • Morgan, C. T. and Deese, J.: 1957,How to Study, McGraw-Hill.

  • Morris, J.: 1981, ‘Math anxiety: teaching to avoid it’,Mathematics Teacher 74, 6, 414.

    Google Scholar 

  • Neisser, U.: 1978, ‘What are the important questions?’, in M. M. Guneberg, P. E. Morris, and R. N. Sykes (eds.):Practical Aspects of Memory, Academic Press, pp. 3–24.

  • Osgood, C. E.: 1953,Method and Theory in Experimental Psychology, Oxford, p. 570.

  • Paivio, A.: 1971,Imagery and Verbal Processes, Holt, Reinhart & Winston.

  • Postman, L.: 1972, ‘A pragmatic view of organization theory’, in E. Tulvig and W. Donaldson (eds.):Organization of Memory, Academic Press, pp. 3–48.

  • Piaget, J. and Inhelder, B.: 1973,Memory and Intelligence, Basic Books.

  • Radatz, H.: 1979, ‘Error analysis in mathematics education’,Journal for Research in Mathematics Education 10, 163–171.

    Google Scholar 

  • Resnick, L. B.: 1983, ‘A developmental theory of number understanding’, in H. P.Ginsburg (ed.):The Development of Mathematical Thinking, Academic Press, New York, pp. 109–151.

    Google Scholar 

  • Riley, M. S., Green, J. G. and Heller, J. I.: 1983, ‘Development of children's problem-solving ability in arithmetic’, in H. P.Ginsburg (ed.):The Development of Mathematical Thinking, Academic Press, New York, pp. 153–196.

    Google Scholar 

  • Sadowski, B. R. and McIlveen, D. H.: 1982, ‘Diagnosis and remediation of sentence-solving error patterns’,Arithmetic Teacher 31, 5, 42–45.

    Google Scholar 

  • Scopes, P. G.: 1973,Mathematics in Secondary Schools—A Teaching Approach, Cambridge University Press, p. 86.

  • Shavelson, R. J.: 1972, ‘Some aspects of the correspondence between content structure and cognitive structure in physics instruction’,Journal of Educational Psychology 63, 225–234.

    Google Scholar 

  • Shavelson, R. J.: 1974, ‘Methods for examining representations of a subject-matter structure in a student's memory’,Journal of Research in Science Teaching 11, 231–249.

    Google Scholar 

  • Shulman, L. S.: 1970, ‘Psychology of mathematics education’, inMathematics Education, Yearbook 69, National Society for the Study of Education, Part I, University of Chicago Press.

  • Skemp, R. R.: 1976, ‘Relational understanding and instrumental understanding’,Mathematics Teaching 77.

  • Skemp, R. R.: 1979, ‘Goals of learning and qualities of understanding’,Mathematics Teaching 88.

  • Skemp, R. R.: 1979,Intelligence, Learning and Action, Wiley, pp. 259–260.

  • Suydam, M. N. and Dessart, D. J.: 1980, ‘Skill learning’, in R. J.Shumway (ed.):Research in Mathematics Education, National Council of Teachers of Mathematics, Reston, pp. 207–243.

    Google Scholar 

  • Tulvig, E.: 1972, ‘Episodic and semantic memory’, in E. Tulvig and W. Donaldson (eds.):Organization of Memory, Academic Press, pp. 381–403.

  • Vinner, S.: 1981, Personal communication.

  • Wertheimer, M.: 1959,Productive Thinking, enl. ed., Harper, p. 11.

  • Wittrock, M. C.: 1974, ‘A generative model of mathematics learning’,Journal of Research in Mathematics Education 5, 181–196.

    Google Scholar 

  • Woodworth, R. S. and Schlossberg, H.: 1954,Experimental Psychology, 2nd ed., Holt, p. 730.

Download references

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Byers, V., Erlwanger, S. Memory in mathematical understanding. Educ Stud Math 16, 259–281 (1985). https://doi.org/10.1007/BF00776733

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