# Reflection and Recursion

## Abstract

Each age defines education in terms of the meanings it gives to teaching and learning, and those meanings arise in part from the metaphors used to characterise teachers and learners. In the ancient world, one of the defining technologies (Bolter, 1984) was the potter’s wheel. The student’s mind became clay in the hands of the teacher. In the time of Descartes and Leibniz, the defining technology was the mechanical clock. The human being became a sort of clockwork mechanism whose mind either was an immaterial substance separate from the body (Descartes) or was itself a preprogrammed mechanism (Leibniz). The mind has also, at various times, been modelled as a wax tablet, a steam engine and a telephone switchboard.

## Keywords

Mathematics Education Mathematics Teacher Cognitive Science Recursive Function Educational Study## Preview

Unable to display preview. Download preview PDF.

## References

- Anderson, L.M., Brubaker, N.L., Alleman-Brooks, J., and Duffy, G.G. (1984).
*Making seatwork work*(Research Series No. 142 ). East Lansing: Michigan State University, Institute for Research on Teaching.Google Scholar - Baird, J.R., and White, R.T. ( 1984, April). `Improving learning through enhanced metacognition: A classroom study’. Paper presented at the meeting of the American Educational Research Association, New Orleans.Google Scholar
- Bartlett, F.C. (1967).
*Remembering: A study in experimental and social psychology*. Cambridge: Cambridge University Press.Google Scholar - Bauersfeld, H. (1979). `Research related to the mathematical learning process’. In B. Christiansen and H.G. Steiner (Eds.),
*New trends in mathematics teaching (Vol*. 4, pp. 199–213 ). Paris: Unesco.Google Scholar - Bauersfeld, H. (1983). ‘Subjektive Erfahrungsbereiche als Grundlage einer Interaktionstheorie des Mathematiklernens und -lehrens’ [Domains of subjective experiences as the basis for an interactive theory of mathematics learning and teaching]. In
*Untersuchungen zum Mathematikunterricht: Vol*. 6. Lernen und Lehren von Mathematik (pp. 1–56 ). Koln: Aulis-Verlag Deubner.Google Scholar - Begle, E.G. (1979).
*Critical variables in mathematics education: Findings from a survey of the empirical literature*. Washington, DC: Mathematical Association of America and National Council of Teachers of Mathematics.Google Scholar - Bolter, J.D. (1984).
*Turing’s man: Western culture in the computer age*Google Scholar - Borkowski, J.G., and Krause, A. (1983)
*. `Racial differences in intelligence: The importance of the executive system.’*Intelligence Vol.*7, pp. 379–395*Google Scholar - Brown, A.L. (1978)
*. `Knowing when, where, and how to remember: A problem of metacognition’. In R. Glaser (Ed.)*Advances in instructional psychology (Vol.*1, pp. 77–165). Hillsdale, NJ: Erlbaum*Google Scholar - Brown, A.L. (in press).
*`Metacognition, executive control, self-regulation and other even more mysterious mechanisms’. In F.E. Weinert and R.H. Kluwe (Eds.)*Learning by thinking.*West Germany: Kuhlhammer*Google Scholar - Brown, A.L., and Palincsar, A.S. (1982)
*. `Inducing strategic learning from texts by means of informed, self-control training’*Topics in Learning and Learning Disabilities Vol.*2 No. 1, pp. 1–17*Google Scholar - Brown, S.I. (1983).
*[Review of*Mindstorms: Children computers and power ful ideas.] Problem Solving Vol.*5 No. 7/8, pp. 3–6*Google Scholar - Campione, J.C., and Brown, A.L. (1978)
*. `Toward a theory of intelligence: Contributions from research with retarded children’*Intelligence Vol.*2, pp. 279–304*Google Scholar - Carroll, J.B. (1976)
*. `Psychometric tests as cognitive tasks: A new `structure of intellect’. In L.B. Resnick (Ed.)*The nature of intelligence (pp.*27–56). Hillsdale, NJ: Erlbaum*Google Scholar - Case, R. (1974). `Structures and strictures: Some functional limitations on theGoogle Scholar
- course of cognitive growth’.
*Cognitive Psychology*,*Vol*6, pp. 544–574. Chomsky, N. (1980).*Rules and representations*New ork: Columbia University Press.Google Scholar - Cooper, D., and Clancy, M. (1982).
*Oh! Pascal*New York: Norton.Google Scholar - Cutland, N. (1980). Computability: An introduction to recursive function theory.
*Cambridge: Cambridge University Press*.Google Scholar - Davis, R.B. (1983).
*`Complex mathematical cognition’. In H.P. Ginsberg (Ed.)*The development of mathematical thinking (pp.*253–290). New York: Academic Press*Google Scholar - Davis, R.B. and McKnight, C.C. (1979)
*. `Modeling the processes of mathematical thinking’*Journal of Children’s Mathematical Behavior*Vol. 2 No. 2, pp. 91–113*Google Scholar - Davis, R.M. (1977).
*`Evolution of computers and computing’*Science*No. 195, pp. 1096–1102*Google Scholar - Dewey, J. (1933). How we think: A restatement of the relation of reflective thinking to the educative process.
*Boston: Heath*.Google Scholar - Dictionary of scientific and technical terms
*(2nd ed.). (1978). New York: McGraw-Hill*Google Scholar *Easley, J. (1984). `Is there educative power in students’ alternative frameworks - or else, what’s a poor teacher to do?’*Problem Solving, Vol. 6 No. 2, pp 1–4.Google Scholar- Eco, U. (1983). The name of the rose (W.
*Weaver, Trans.). New York: Warner Books. (Original work published 1980)*Google Scholar - Editorial Board of Educational Studies in Mathematics.
*(Eds.) (1969)*Proceedings of the First International Congress on Mathematical Education Lyon 24–30 August 1969.*Dordrecht, Holland: Reidel*Google Scholar - Fischer, G., Burton, R.E., and Brown, J.S. (1978). Aspects of a theory of simplification, debugging, and coaching
*(BBN Report No*.*3912)*.Google Scholar - Flavell, J.H. (1976).
*`Metacognitive aspects of problem solving’. In L.B. Resnick (Ed.)*The nature of intelligence*(pp. 231–235). Hillsdale, NJ: Erlbaum*Google Scholar - Flavell, J.H. (1979).
*`Metacognition and cognitive monitoring: A new area of cognitive-developmental inquiry’*American Psychologist*Vol. 34, pp. 906–911*Google Scholar - Fodor, J.A. (1983). The modularity of mind: An essay on faculty psychology.
*Cambridge, MA: MIT Press*.Google Scholar - Freudenthal, H. (1978). Weeding and sowing: Preface to a science of mathematical education.
*Dordrecht, Holland: Reidel*.Google Scholar - Freudenthal, H. (1981a)
*. `Major problems of mathematics education’*Educational Studies in Mathematics*12, pp. 133–150*Google Scholar - Freudenthal, H. (1981b).
*`Should a mathematics teacher know something about the history of mathematics?’*For the Learning of Mathematics*2(1), pp. 30–33*Google Scholar - Freudenthal, H. (1983). Didactical phenomenology of mathematical structures.
*Dordrecht, Holland: Reidel*.Google Scholar - Fuys, D., Geddes, D., and Tischler, R. (Eds.). (1984). English translation of selected writings of Dina van Hiele-Geldof and Pierre M. van Hiele.
*New York: Brooklyn College, School of Education*.Google Scholar - Gardner, H. (1983). Frames of Mind: The theory of multiple intelligences.
*New York: Basic Books*.Google Scholar - Garofalo, J., and Lester, F.K., Jr. (1984). Metacognition, cognitive monitoring and mathematical performance.
*Manuscript submitted for publication*.Google Scholar - Genkins, E.F. (1975)
*. `The concept of bilateral symmetry in young children’. In M.F. Rosskopf (Ed.)*Children’s mathematical concepts: Six Piagetian studies in mathematics education*(pp. 5–43). New York: Teachers College Press*Google Scholar - Gitomer, D.H., and Glaser, R. (in press).
*`If you don’t know it, work on it: Knowledge, self-regulation and instruction’. In R.E. Snow and J. Farr (Eds.)*Aptitude learning and instruction: Vol. 3. Conative and affective process analyses.*Hillsdale, NJ: Erlbaum*Google Scholar - Hawkins, D. (1973).
*`Nature, man and mathematics’. In. A.G. Howson (Ed.)*Developments in mathematics education: Proceedings of the Second International Congress on Mathematical Education*(pp. 115–135). Cambridge: Cambridge University Press*Google Scholar - Hayes-Roth, B., and Hayes-Roth, F. (1979).
*`A cognitive model of planning’*Cognitive Science*Vol. 3 pp. 275–310*Google Scholar - Hilton, P. (1984).
*`Current trends in mathematics and future trends in mathematics education’*For the Learning of Mathematics*Vol. 4 No. 1, pp. 2–8*Google Scholar - Hofstadter, D.R. (1979). Godel, Escher, Bach: An eternal golden braid.
*New York: Basic Books*.Google Scholar - Hofstadter, D.R., and Dennett, D.C
*. (1981)*The mind’s I: Fantasies and reflections on self and soul.*New York: Bantam*Google Scholar - Jahnke, H.N.
*(1983). `Technology and education: The example of the computer’ [Review of*Mindstorms: Children computers and powerful ideas]. Educational Studies in Mathematics*Vol. 14, pp. 87–100*Google Scholar - Jaynes, J. (1976).
*The origin of consciousness in the breakdown of the bicameral mind*. Boston: Houghton Mifflin.Google Scholar - Johnson-Laird, P.N. (1983a). `A computational analysis of consciousness’.
*Cognition and Brain Theory*, Vol. 6, pp. 499–508.Google Scholar - Johnson-Laird, P.N. (1983b).
*Mental models: Towards a cognitive science of language*,*inference*,*and consciousness*. Cambridge, MA: Harvard University Press.Google Scholar - Lawler, R.W. (1981). `The progressive construction of mind’.
*Cognitive Science*, Vol. 5, pp. 1–30.CrossRefGoogle Scholar - Levinas, E. (1973).
*The theory of intuition in Husserl’s phenomenology*(A. Orianne, Trans.). Evanston, IL: Northwestern University Press. ( Original work published 1963 )Google Scholar - Locke, J. (1965).
*An essay concerning human understanding*(Vol. 1). New York: Dutton. (Original work published 1706 )Google Scholar - Loper, A.B. (1982). `Metacognitive training to correct academic deficiency’.
*Topics in Learning and Learning Disabilities*, Vol. 2 No. 1, pp. 61–68.Google Scholar - Mandler, G. (1975). `Consciousness: Repectable, useful, and probably necessary’. In R.L. Solso (Ed.),
*Information processing and cognition: The Loyola Symposium*(pp. 229–254 ). Hillsdale, NJ: Erlbaum.Google Scholar - Maurer, S.B. (1983). The effects of a new college mathematics curriculum on high school mathematics. In A. Ralston and G.S. Young (Eds.),
*The future of college mathematics: Proceedings of a conference/workshop on the first two years of college mathematics*(pp. 153–173 ). New York: Springer-Verlag.CrossRefGoogle Scholar - McCarthy, J. (1983). `Recursion’. In A. Ralston and E.D. Reilly, Jr. (Eds.),
*Encyclopedia of computer science and engineering*(2nd ed., pp. 12731275 ). New York: Van Nostrand Reinhold.Google Scholar - McCloskey, M. (1983, April). `Intuitive physics’.
*Scientific American*, pp. 122–130.Google Scholar - Merleau-Ponty, M. (1973).
*Consciousness and the acquisition of language*(H.J. Silverman, Trans.). Evanston, IL: Northwestern University Press. ( Original work published 1964 )Google Scholar - Minsky, M. (1980). `K-lines: A theory of memory’.
*Cognitive Science*, Vol. 4, pp. 117–133.CrossRefGoogle Scholar - Möbius, A.F. (1967).
*Gesammelte Werke*[Collected works] (Vol. 1). Wiesbaden: Martin Sandig. (Original work published 1885 )Google Scholar - Natsoulas, T. (1983). `A selective review of conceptions of consciousness with special reference to behavioristic contributions’.
*Cognition and Brain Theory*, Vol. 6, pp. 417–447.Google Scholar - Newell, A., and Simon, H.A. (1972).
*Human problem solving*. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar - Newman, J.H.C. (1903).
*An essay in aid of a grammar of assent*. London: Longmans, Green. (Original work published 1870 )Google Scholar - Papert, S. (1975). `Teaching children thinking’.
*Journal of Structural Learning*, Vol. 4, pp. 219–229.Google Scholar - Papert, S. (1980).
*Mindstorms: Children*,*computers*,*and powerful ideas*. New York: Basic Books.Google Scholar - Peter, R. (1967).
*Recursive functions*(3rd ed.). New York: Academic Press. Piaget, J. (1956). `Les étages du developpement intellectuel de l’enfant et de l’adolescent’ [The stages of intellectual development in the child and theGoogle Scholar - Adolescent]. In P. Osterrieth et al.,
*Le probleme des stages en psychologie de l’enfant*(pp. 33–41). Paris: Presses Universitaires de FranceGoogle Scholar - Piaget, J. (1971a).
*Science of education and the psychology of the child*(D. Coltman, Trans.). New York: Viking. (Original work published 1969 )Google Scholar - Piaget, J. (1971b). `The theory of stages in cognitive development’. In D.R. Green (Ed.),
*Measurement and Piaget*(pp. 1–11 ). New York: McGraw-Hill.Google Scholar - Piaget, J. (1976).
*The grasp of consciousness: Action and concept in the young child*(S. Wedgwood, Trans.). Cambridge, MA: Harvard University Press. ( Original work published 1974 )Google Scholar - Piaget, J. (1978).
*Success and understanding*(A.J. Pomerans, Trans.). Cambridge, MA: Harvard University Press. ( Original work published 1974 )Google Scholar - Pinard, A., and Laurendeau, M. (1969). “Stage’ in Piaget’s cognitive-developmental theory: Exegesis of a concept’. In D. Elkind and J.H. Flavell (Eds.),
*Studies in cognitive development: Essays in honor of Jean Piaget*(pp. 121–170 ). New York: Oxford University Press.Google Scholar - Polya, G. (1983). `Mathematics promotes the mind’. In M. Zweng, T. Green, J. Kilpatrick, H. Pollak, and M. Suydam (Eds.),
*Proceedings of the Fourth International Congress on Mathematical Education*(p. 1 ). Boston: Birkhauser.CrossRefGoogle Scholar - Rappaport, D. (1957). `Cognitive structures’. In
*Contemporary approaches to cognition: A symposium held at the University of Colorado*(pp. 157–200 ). Cambridge, MA: Harvard University Press.Google Scholar - Rotenstreich, N. (1974). ‘Humboldt’s prolegomena to philosophy of language’.
*Cultural Hermeneutics*, Vol. 2, pp. 211–227.Google Scholar - Schoenfeld, A.H. (1983). `Episodes and executive decisions in mathematical problem solving’. In R. Lesh & M. Landau (Eds.),
*Acquisition of mathematics concepts and processes*(pp. 345–395 ). New York: Academic Press, 1983.Google Scholar - Schoenfeld, A.H. (in press-a). `Beyond the purely cognitive: Belief systems, social cognitions and metacognitions as driving forces in intellectual performance’.
*Cognitive Science*Google Scholar - Schoenfeld, A.H. (in press-b).
*Mathematical problem solving*New York: Academic Press.Google Scholar - Scriven, M. (1980). `Self-referent research’.
*Educational Researcher*, Vol. 9, No. 4, pp. 7–11, and Vol. 9, No. 6, pp. 11–18.Google Scholar - Silver, E.A. (1982a). `Knowledge organization and mathematical problem solving’. In F.K. Lester, Jr., and J. Garofalo (Eds.),
*Mathematical problem solving: Issues in research*(pp. 15–25 ). Philadelphia: Franklin Institute Press, 1982.Google Scholar - Silver, E.A. (1982b).
*Thinking about problem solving: Toward an understanding of metacognitive aspects of mathematical problem solving*. Unpublished manuscript, San Diego State University, Department of Mathematical Sciences, San Diego, CA.Google Scholar - Simon, H.A. (1977). `What computers mean for man and society’.
*Science*, 195. 1186–1191.CrossRefGoogle Scholar - Skemp, R.R. (1979).
*Intelligence*,*learning*,*and action*New York: Wiley. Snow, R.E. (1980). `Aptitude processes’. In R.E. Snow, P.A. Federico, andGoogle Scholar - W.E.Montague (Eds.),
*Aptitude*,*learning*,*and instruction: Vol. 1. Cognitive process analyses of aptitude (pp. 27–63)*Hillsdale, NJ: Erlbaum.Google Scholar - Thomas, R.M. (1984).
*`Mapping meta-territory’*. Educational Researcher, Vol. 13, No. 4, pp. 16–18.Google Scholar - Thompson, A.G. (1984).
*`The relationship of teachers’ conceptions of mathematics and mathematics teaching to instructional practice’*. Educational Studies in Mathematics, Vol. 15, pp. 105–127.Google Scholar - Van Hiele, P.M. and van Hiele-Geldof, D.(1958). `A
*method of initiation into geometry at secondary schools’. In H. Fruedenthal (Ed.)*Report on methods of initiation into geometry (pp. 67–80).*Groningen: J.B. Wolters*Google Scholar - Vergnaud, G.(1983).
*`Why is an epistemological perspective necessary for research in mathematics education?’ In J.C. Bergeron and N. Herscovics (Eds.)*Proceedings of the Fifth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1 pp. 2–20).*Montréal: Université de Montréal, Faculté de Sciences de l’Education*Google Scholar - Von Glasersfeld, E (1983).
*`Learning as a constructive activity’. In J.C. Bergeron and N. Herscovics (Eds.)*Proceedings of the Fifth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1 pp. 41–69).*Montréal: Université de Montréal, Faculté de Sciences de l’Education*Google Scholar - Whitehead, A.N. (1929).
*`The rhythm of education’*The aims of education and other essays.*New York: Macmillan*Google Scholar