Abstract
Conceptual analyses of Newton's use of the Fundamental Theorem of Calculus and of one 7th-grader's understanding of distance traveled while accelerating suggest that concepts of rate of change and infinitesimal change are central to understanding the Fundamental Theorem. Analyses of a teaching experiment with 19 senior and graduate mathematics students suggest that students' difficulties with the Theorem stem from impoverished concepts of rate of change and from poorly-developed and poorly coordinated images of functional covariation and multiplicatively-constructed quantities.
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Anton, H.: 1992, Calculus with Analytic Geometry, John Wiley and Sons, New York.
Baron, M.: 1969, The Origins of Infinitesimal Calculus, Pergamon Press, New York.
Boyd, B. A.: 1992, The Relationship between Mathematics Subject Matter Knowledge and Instruction: A Case Study, Masters Thesis, San Diego.
Boyer, C. B.: 1959, The History of the Calculus and Its Conceptual Development, Dover, New York.
Cobb, P. and von Glasersfeld, E.: 1983, ‘Piaget's scheme and constructivism’, Genetic Epistemology 13, 9–15.
Courant, R.: 1937, Differential and Integral Calculus, Interscience, New York.
Dewey, J.: 1929, The Sources of a Science of Education, Liveright Publishing, New York.
Dubinsky, E.: 1991, ‘Reflective abstraction in advanced mathematical thinking’, in D. Tall (ed.), Advanced Mathematical Thinking, pp. 95–123, Kluwer, Dordrecht, The Netherlands.
Goldenberg, E. P.: 1988, ‘Mathematics, metaphors, and human factors: Mathematical, technical, and pedagogical challenges in the educational use of graphical representation of functions’, Journal of Mathematical Behavior 7, 135–173.
Hayes, J. R.: 1973, ‘On the function of visual imagery in elementary mathematics’, in W. G. Chase (ed.), Visual Information Processing, pp. 177–214, Academic Press, New York.
Johnson, M.: 1987, The Body in the Mind: The Bodily Basis of Meaning, Imagination, and Reason, University of Chicago Press, Chicago, IL.
Kaput, J. J.: in press, ‘Democratizing access to calculus: New routes to old roots’, in A. H. Schoenfeld (ed.), Mathematics and Cognitive Science, Mathematical Association of America, Washington, D.C.
Kieren, T. and Pirie, S.: 1990, April, ‘A recursive theory for mathematical understanding: Some elements and implications’, Paper presented at the Annual Meeting of the American Educational Research Association, Boston, MA.
Kieren, T. and Pirie, S.: 1991, ‘ Recursion and the mathematical experience’, in L. P. Steffe (ed.), Epistemological Foundations of Mathematical Experience, pp. 78–101, Springer-Verlag, New York.
Kieren, T. E.: 1989, ‘Personal knowledge of rational numbers: Its intuitive and formal development’, in J. Hiebert and M. Behr (eds.), Number Concepts and Operations in the Middle Grades, pp. 162–181, National Council of Teachers of Mathematics, Reston, VA.
Kosslyn, S. M.: 1980, Image and Mind, Harvard University Press, Cambridge, MA.
Kuhn, T. S.: 1970, ‘Logic of discovery or psychology of research’, in I. Lakatos and A. Musgrave (eds.), Criticism and the Growth of Knowledge, pp. 1–22, Cambridge University Press, Cambridge, U.K.
Piaget, J.: 1950, The Psychology of Intelligence, Routledge and Kegan-Paul, London.
Piaget, J.: 1967, The Child's Concept of Space, W. W. Norton, New York.
Piaget, J.: 1968, Six Psychological Studies, Vintage Books, New York.
Piaget, J.: 1971, Genetic Epistemology, W. W. Norton, New York.
Piaget, J.: 1976, The Child and Reality, Penguin Books, New York.
Piaget, J.: 1980, Adaptation and Intelligence, University of Chicago Press, Chicago.
Pirie, S. and Kieren, T.: 1991, April, ‘A Dynamic Theory of Mathematical Understanding: Some features and implications’, Paper presented at the Annual Meeting of the American Educational Research Association, Chicago, IL.
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.
Steffe, L. P.: 1991, ‘Operations that generate quantity’, Journal of Learning and Individual Differences 3, 61–82.
Steffe, L. P.: in press, ‘Children's multplying and dividing schemes: An overview’, in G. Harel and J. Confrey (eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics, SUNY Press, Albany, NY.
Swokowski, E. W.: 1991, Calculus, PWS-Kent, Boston, MA.
Tall, D.: 1986, Building and Testing a Cognitive Approach to the Calculus Using Interactive Computer Graphics, Doctoral dissertation, University of Warwick.
Tall, D., Van Blokland, P., and Kok, D.: 1988, A Graphic Approach to the Calculus, Computer Program for IBM and Compatibles, Warwick University, Warwick, U.K.
Tall, D. and Vinner, S.: 1981, ‘ Concept images and concept definitions in mathematics with particular reference to limits and continuity’, Educational Studies in Mathematics 12, 151–169.
Thompson, A. G. and Thompson, P. W.: in press, ‘Talking about rates conceptually: A teacher's struggle’, Journal for Research in Mathematics Education.
Thompson, P. W.: 1985, ‘Experience, problem solving, and learning mathematics: Considerations in developing mathematics curricula’, in E. Silver (ed.), Teaching and Learning Mathematical Problem Solving: Multiple Research Perspectives, pp. 189–243, Erlbaum, Hillsdale, NJ.
Thompson, P. W.: 1991, ‘To experience is to conceptualize: Discussions of epistemology and experience’, in L. P. Steffe(ed.), Epistemological Foundations of Mathematical Experience, pp. 260–281, Springer-Verlag, New York.
Thompson, P. W.: in press a, ‘The development of the concept of speed and its relationship to concepts of rate’, in G. Harel and J. Confrey (eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics, SUNY Press, Albany, NY.
Thompson, P. W.: in press b, ‘Students, functions, and the undergraduate mathematics curriculum’, Research in Collegiate Mathematics Education 1.
Thompson, P. W. and Thompson, A. G.: 1992, April, ‘Images of rate’, Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco, CA.
Vinner, S.: 1987, ‘Continuous functions: Images and reasoning in college students’, in Proceedings of the Annual Meeting of the International Group for the Psychology of Mathematics Education, PME Montréal, Canada.
Vinner, S.: 1989, ‘Avoidance of visual considerations in calculus students’, Journal of Mathematical Behavior 11, 149–156.
Vinner, S.: 1991, ‘The role of definitions in the teaching and learning of mathematics’, in D. Tall (ed.), Advanced Mathematical Thinking, pp. 65–81, Kluwer, Dordrecht, The Netherlands.
Vinner, S.: 1992, ‘The function concept as a prototype for problems in mathematics learning’, in G. Harel and E. Dubinsky (eds.), The Concept of Function: Aspects of Epistemology and Pedagogy, pp. 195–214, Mathematical Association of America, Washington, D.C.
Vinner, S. and Dreyfus, T.: 1989, ‘ Images and definitions for the concept of function’, Journal for Research in Mathematics Education 20, 356–366.
von Glasersfeld, E.: 1978, ‘Radical constructivism and Piaget's concept of knowledge’, in F. B. Murray (ed.), Impact of Piagetian Theory, pp. 109–122, University Park Press, Baltimore.
Wilder, R.: 1967, ‘The role of axiomatics in mathematics’, American Mathematical Monthly 74, 115–127.
Wilder, R.: 1968, Evolution of Mathematical Concepts: An Elementary Study, Wiley, New York.
Wilder, R.: 1981, Mathematics as a Cultural System, Pergamon Press, New York.
Winograd, T. and Flores, F.: 1986, Understanding Computers and Cognition: A New Foundation for Design, Ablex, Norwood, NJ.
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Research reported in this paper was supported by National Science Foundation Grants No. MDR 89-50311 and 90-96275, and by a grant of equipment from Apple Computer, Inc., Office of External Research. Any conclusions or recommendations stated here are those of the author and do not necessarity reflect official positions of NSF or Apple Computer. Also, I wish to thank Paul Cobb and Guershon Harel for their helpful reactions to an earlier draft of this article.
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Thompson, P.W. Images of rate and operational understanding of the fundamental theorem of calculus. Educ Stud Math 26, 229–274 (1994). https://doi.org/10.1007/BF01273664
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DOI: https://doi.org/10.1007/BF01273664