Nonstandard student conceptions about infinitesimals

Journal for Research in Mathematics Education

Volumn 41, Issue 2, 2010, Pages 117-146

  Ely, Robert ^a  

^a UNIVERSITY OF IDAHO   (United States)

Author keywords

Advanced mathematical thinking; College university; Constructivism; Historical analysis; Learning theories

Indexed keywords

EID: 77950283093     PISSN: 00218251     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (51)

1. Albeverio, S., Fenstad, J. E., Hoegh-Kjohn, R., & Lindstrom, T. (1986). Nonstandard methods in stochastic analysis and mathematical physics. Orlando, FL: Academic Press.
2. Bos, H. J. M. (1974). Differentials, higher-order differentials, and the derivative in Leibnizian calculus. Archive for History of Exact Sciences, 14,1-90.
3. Bressoud, D. (1994). A radical approach to real analysis. Washington, DC: Mathematical Association of America.
4. Brousseau, G. (1997). Theory of didactical situations in mathematics: Didactique des mathématiques, 1970-1990. Dordrecht, The Netherlands: Kluwer.
5. Carpenter, T. P., & Moser, J. M. (1984). The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education, 15,179-202.
6. Chomsky, N. (1967). Review of Skinner's Verbal behavior. In J. J. Jenkins (Series Ed.), L. A. Jakobovits & M. S. Miron (Vol. Eds.), Prentice-Hall psychology series: Readings in the psychology of language (pp. 142-171). Englewood Cliffs, NJ: Prentice-Hall.
7. Confrey, J. (1991). Learning to listen: A student's understanding of powers often. In E. von Glasersfeld (Ed.), Radical constructivism in mathematics education (pp. 111-138). Dordrecht, The Netherlands: Kluwer.
8. Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced mathematical thinking (pp. 153-166). Dordrecht, The Netherlands: Kluwer.
9. Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K., & Vidakovic, D. (1996). Understanding the limit concept: Beginning with a coordinated process scheme. Journal of Mathematical Behavior, 15, 167-192.
10. Davis, R. B., & Vinner, S. (1986). The notion of limit: Some seemingly unavoidable misconception stages. Journal of Mathematical Behavior, 5, 281-303.
11. Denzin, N. K. (1989). The research act: A theoretical introduction to sociological methods (3rd ed.). Englewood Cliffs, NJ: Prentice Hall.
12. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 95-126). Dordrecht, The Netherlands: Kluwer.
13. Dubinsky, E., Weiler, K., McDonald, M. A., & Brown, A. (2005). Some historical issues and paradoxes regarding the concept of infinity: An APOS-based analysis, Part 1. Educational Studies in Mathematics, 58,335-359.
14. Ely, R. (2007). Student obstacles and historical obstacles to foundational concepts of calculus. Dissertation Abstracts International, 68(8), 3317A. (UMI No. 3278849)
15. Furinghetti, F., & Radford, L. (2008). Contraste and oblique connections between historical conceptual developments and classroom learning in mathematics. In L. D. English (Ed.), Handbook of international research in mathematics education (2nd ed., pp. 626-655). New York: Routledge.
16. Fuson, K. C. (2003). Developing mathematical power in whole number operations. In X Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to Principles and standards for school mathematics (pp. 68-94). Reston, VA: National Council of Teachers of Mathematics.
17. Grabiner, J. V. (1981). The origins of Cauchy's rigorous calculus. Cambridge, MA: MIT Press.
18. Grattan-Guinness, I. (1970). The development of the foundations of mathematical analysis from Euler to Riemann. Cambridge, MA: MIT Press.
19. Gravemeijer, K., Cobb, P., Bowers, J., & Whitenack, J. (2000). Symbolizing, modeling and instructional design. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms. Perspectives on discourse, tools, and instructional design (pp. 225-273). Mahwah, NJ: Erlbaum.
20. Henle, J. M., & Klienberg, E. M. (1979). Infinitesimal calculus. Cambridge MA: MIT Press.
21. Hutchins, E. (1993). Learning to navigate. In S. Chaiklin & J. Lave (Eds.), Understanding practice: Perspectives on activity and context (pp. 35-63). Cambridge, UK: Cambridge University Press.
22. Kaput, X (1994). Democratizing access to calculus: New routes to old roots. In A. Schoenfeld (Ed.), Studies in mathematical thinking and learning: Mathematical thinking and problem solving (pp. 77-156). Hillsdale, NJ: Erlbaum.
23. Keisler, H. J. (1986). Elementary calculus: An infinitesimal approach (2nd ed.). Boston, MA: Prindle, Weber & Schmidt.
24. Keisler, H. J. (2007). Foundations of infinitesimal calculus. Retrieved June 20,2008, from www.math wisc.edu/~keisler/foundations.html
25. Kleiner, I. (2001 ). History of the infinitely small and the infinitely large in calculus. Educational Studies in Mathematics, 48,137-174.
26. Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery (J. Worrell & F. Zahar, Eds.). Cambridge, UK: Cambridge University Press.
27. Leibniz, G. W. (1966). Philosophische Schriften: Sechste Reihe, zweiter Band. Sämtliche Schriften und Briefe [Philosophical writings: Ser. 6, Vol. 2. All essays and letters] (W. Kabitz & H. Schepers, Eds.). Berlin: Akademie-Verlag. (Original works published 1663-1672)
28. Liebniz, G. W. (1969). Nova methodus pro maximus et minimus. In D. J. Struik, A source book in mathematics, 1200-1800 (pp. 271-281). Cambridge, MA: Harvard University Press. (Original work published 1684)
29. Mamolo, A., & Zazkis, R. (2008). Paradoxes as a window to infinity. Research in Mathematics Education. 10,167-182.
30. Mancosu, P. (1996). Philosophy of mathematics and mathematical practice in the seventeenth century. New York: Oxford University Press.
31. Matz, M. (1982). Towards a process model for high school algebra errors. In D. Sleeman & J. S. Brown (Eds.), Intelligent tutoring systems (pp. 25-50). New York: Academic Press.
32. Piaget, J., & Garcia, R. (1989). Psychogenesis and the history of science (H. Feider, Trans.). New York: Columbia University Press. (Original work published 1983)
33. Przenioslo, M. (2004). Images of the limit of function formed in the course of mathematical studies at the university. Educational Studies in Mathematics, 55,103-132.
34. Radford, L. (2000). Signs and meanings in students' emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42,237-268.
35. Robinson, A. (1996). Princeton landmarks in mathematics and physics: Non-standard analysis (Rev. ed.). Princeton, NJ: Princeton University Press.
36. Ross, G. M. (1992). Are there real infinitesimals in Leibniz's metaphysics? In A. Lamarre (Ed.), Simposio Lessico Intellettuale Europeo: Vol.LII. L'infinito in Leibniz: Problemi e terminologia (pp. 125-141). Rome: Edizioni dell' Ateneo.
37. Schwarzenberger, R. L. E., & Tall, D. O. (1978). Conflicts in the learning of real numbers and limits. Mathematics Teaching, 82, 44-49.
38. 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.
39. Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification: The case of function. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 59-84). Washington, DC: Mathematical Association of America.
40. Sierpinska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18,371-397.
41. Skolem, T. A. (1934). Ober die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzahlbar unendlich vieler Aussagen mit ausschließlich Zahlenvariablen [On the noncharacterizability of the natural numbers by means of finitely many or countably infinitely many statements with exclusively number-variables]. Fundamenta Mathematicae, 23, 150-161.
42. Smith, J. P., III, diSessa, A. A., & Roschelle, X (1993). Misconceptions reconceived: A constructivist analysis of knowledge in transition. Journal of the Learning Sciences, 3, 115-163.
43. Strauss, A., & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park, CA: Sage.
44. Stake, R. E. (2000). Case studies. In N. K. Denzin & Y. S. Lincoln (Eds.), Handbook of qualitative research (2nd ed., pp. 435-454). Thousand Oaks, CA: Sage.
45. Szydlik, J. E. (2000). Mathematical beliefs and conceptual understanding of the limit of a function. Journal for Research in Mathematics Education, 31,258-276.
46. Tall, D. O., & Vinner, S. (1981). Concept image and concept definition in mathematics, with particular reference to limits and continuity. Educational Studies in Mathematics, 12,151-169.
47. von Glasersfeld, E. (1990). An exposition of constructivism: Why some like it radical. In R. B. Davis, C. A. Maher, & N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics (pp. 19-29). Reston, VA: National Council of Teachers of Mathematics.
48. von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. London: Falmer Press.
49. Vygotsky, L. S. (1962). Thought and language (E. Hanfmann & G. Vakar, Trans. & Eds.). Cambridge, MA: MIT Press. (Original work published 1934)
50. Weiler, K., Brown, A., Dubinsky, E., McDonald, M., & Stenger, C. (2004). Intimations of infinity. Notices of the American Mathematical Society, 51,741-750.
51. Williams, S. R. (1991). Models of limit held by college calculus students. Journal for Research in Mathematics Education, 22,219-236.