Educational Studies in Mathematics

, Volume 23, Issue 3, pp 247–285 | Cite as

Development of the process conception of function

  • Daniel Breidenbach
  • Ed Dubinsky
  • Julie Hawks
  • Devilyna Nichols
Article

Abstract

Our goal in this paper is to make two points. First, college students, even those who have taken a fair number of mathematics courses, do not have much of an understanding of the function concept; and second, an epistemological theory we have been developing points to an instructional treatment, using computers, that results in substantial improvements for many students. They seem to develop a process conception of function and are able to use it to do mathematics. After an introductory section we outline, in Section 2, our theoretical epistemology in general and indicate how it applies to the function concept in particular. In Sections 3, 4, and 5 we provide specific details on this study and describe the development of the function concept that appeared to take place in the students that we are considering. In Section 6 we interpret the results and draw some conclusions.

Keywords

College Student Substantial Improvement Function Concept Specific Detail Process Conception 
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.
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References

  1. AyresT., G.Davis, EdDubinsky, and P.Lewin: 1988, ‘Computer experiences in learning composition of functions’,Journal for Research in Mathematics Education 19(3), 246–259.Google Scholar
  2. BaxterN., EdDubinsky and G.Levin: 1988,Learning Discrete Mathematics with ISETL, Springer-Verlag, New York.Google Scholar
  3. BuckR. C.: 1970, ‘Functions’, in E. G.Begle (ed.),Mathematics Education — The Sixty-ninth Yearbook of the National Society for the Study of Education, The University of Chicago Press, Chicago, pp. 236–259.Google Scholar
  4. DreyfusT. and T. A.Eisenberg: 1983, ‘The function concept in college students: Linearity, smoothness and periodicity’,Focus on Learning Problems in Mathematics 5(3,4), 119–132.Google Scholar
  5. Dubinsky, Ed: 1989, ‘A learning theory approach to calculus’,Proceedings of the St. Olaf Conference on Computer Algebra Systems, Northfield, Minnesota.Google Scholar
  6. DubinskyEd: 1991a, ‘Constructive aspects of reflective abstraction in advanced mathematical thinking’, in L. P.Steffe (ed.),Epistemological Foundations of Mathematical Experience, Springer-Verlag, New York.Google Scholar
  7. Dubinsky, Ed: 1991b, ‘Reflective abstraction in advanced mathematical thinking’, in D. Tall (ed.),Advanced Mathematical Thinking.Google Scholar
  8. Dubinsky, Ed, J. Hawks, and D. Nichols: 1989, ‘Development of the process conception of function in pre-service teachers in a discrete mathematics course’,Proceedings of the Thirteenth International Conference for the Psychology of Mathematics Education, Paris, France.Google Scholar
  9. DubinskyEd, and P.Lewin: 1986, ‘Reflective abstraction and mathematics education: The genetic decomposition of induction and compactness’,The Journal of Mathematical Behavior 5, 55–92.Google Scholar
  10. Even, R.: 1988, ‘Pre-service teacher's conceptions of the relationships between functions and equations’, in A. Borbas (ed.),Proceedings of the Twelfth International Conference for the Psychology of Mathematics Education, OOK, Veszprem, pp. 304–311.Google Scholar
  11. Frendenthal, H.: 1982, ‘Variables and functions’,Proceedings of the Workshop on Functions, Organized by the Foundation for Curriculum Development, Enschede.Google Scholar
  12. GoldenbergP.: 1988, ‘Mathematics, metaphors, and human factors: Mathematical, technical, and pedagogical challenges in the educational use of graphical representations of functions’,Journal of Mathematical Behavior 7, 135–173.Google Scholar
  13. Herscovics, N.: 1982, ‘Problems related to the understanding of functions’, inProceedings of the Workshop on Functions, Organized by the Foundation for Curriculum Development, Enschede.Google Scholar
  14. KaputJ.: 1987, ‘Representation systems and mathematics’, in C.Janvier (ed.),Problems of Representation in the Teaching and Learning of Mathematics, Louvrence Erlbaum, Hillsdale.Google Scholar
  15. LeinhardtG., O.Zaslavsky, and M.Stein: 1990, ‘Functions, graphs and graphing: Tasks, learning and teaching’,Review of Educational Research 60(1), 1–64.Google Scholar
  16. LovellK.: 1971, ‘Some aspects of the growth of the concept of a function’, inPiagetian Cognitive Development Research and Mathematical Education, NCTM, Reston, pp. 12–33.Google Scholar
  17. MarkovitsZ., B.Eylon, and M.Bruckheimer: 1986, ‘Functions today and yesterday’,For the Learning of Mathematics 6(2), 18–24.Google Scholar
  18. Orton, A.: 1970, ‘A cross-sectional study of the development of the mathematical concept of a function in secondary school children of average and above average ability’, unpublished Master's thesis, University of Leeds.Google Scholar
  19. PiagetJ.: 1975, ‘Piaget's theory’, in P. B.Neubauer (ed.),The Process of Child Development, Jason Aronson, New York, pp. 164–212.Google Scholar
  20. PiagetJ.: 1976 (Original published 1974),The Grasp of Consciousness (S. Wedgwood, trans.), Harvard University Press, Cambridge.Google Scholar
  21. PiagetJ.: 1978 (Original published 1974),Success and Understanding (A. J. Pomerans, trans.), Harvard University Press, Cambridge.Google Scholar
  22. PiagetJ., J.-B.Grize, A.Szeminska, and V.Bang: 1977,Epistemology and Psychology of Functions (J. Castellanos and V. Anderson, trans.), Reidel, Dordrecht.Google Scholar
  23. SchoenfeldA. H., J. P.SmithIII, and A.Arcavi: 1990, ‘Learning — the microgenetic analysis of one student's understanding of a complex subject matter domain’, in R.Glaser (ed.),Advances in Instructional Psychology, 4, Erlbaum, Hillsdale.Google Scholar
  24. SfardA.: 1987, ‘Two conceptions of mathematical notions: Operational and structural’, in J. C.Bergeron, N.Herscovics and C.Kieran (eds.),Proceedings of the Eleventh International Conference for the Psychology of Mathematics Education, III, Université, Montreal, pp. 162–169.Google Scholar
  25. Skemp, R. R.: 1976, ‘Relational understanding and instrumental understanding’,Mathematics Teaching, December, 1976.Google Scholar
  26. Thomas, H.: 1969, ‘An analysis of stages in the attainment of a concept of function’, unpublished Doctoral dissertation, Teachers College, Columbia University.Google Scholar
  27. VinnerS. and DreyfusT.: 1989, ‘Images and definitions for the concept of function’,Journal for Research in Mathematics Education 20(4), 356–366.Google Scholar
  28. von Glasersfeld, E.: 1985, ‘Learning as a constructive activity’,Proceedings of the Ninth International Conference of the Psychology of Mathematics Education.Google Scholar
  29. VygotskyL.: 1978,Mind in Society: The Development of Higher Psychological Processes, Harvard University Press, Cambridge.Google Scholar

Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • Daniel Breidenbach
  • Ed Dubinsky
    • 1
  • Julie Hawks
  • Devilyna Nichols
  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA

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