Reinventing the formal definition of limit: The case of Amy and Mike

Journal of Mathematical Behavior

Volumn 30, Issue 2, 2011, Pages 93-114

  Swinyard, Craig ^a  

^a UNIVERSITY OF PORTLAND   (United States)

Author keywords

Calculus; Defining; Limit; Realistic mathematics education; Reinvention; Undergraduate mathematics education

Indexed keywords

EID: 79954701321     PISSN: 07323123     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jmathb.2011.01.001     Document Type: Article

References (42)

1. Artigue M. Teaching and learning calculus: What can be learned from education research and curricular changes in France?. Research in collegiate mathematics education IV 2000, Vol. 8:1-15. American mathematical society, Providence, RI.
2. Bezuidenhout J. Limits and continuity: Some conceptions of first-year students. International Journal of Mathematical Education in Science and Technology 2001, 32(4):487-500.
3. Carlson M., Larsen S., Jacobs S. An investigation of covariational reasoning and its role in learning the concepts of limit and accumulation. Proceedings of the Psychology of Mathematics Education North American Chapter, 23 2001, 145-153.
4. Cobb P. Conducting teaching experiments in collaboration with teachers. Research design in mathematics and science education 2000, 307-333. Erlbaum, Hillsdale, NJ. R. Lesh, A.E. Kelly (Eds.).
5. Cornu B. Limits. Advanced mathematical thinking 1991, 153-166. Kluwer Academic Publishers, Dordrecht, The Netherlands. D. Tall (Ed.).
6. Cottrill J., Dubinsky E., Nichols D., Schwinngendorf K., Thomas K., Vidakovic D. Understanding the limit concept: Beginning with a coordinated process schema. Journal of Mathematical Behavior 1996, 15:167-192.
7. Davis R., Vinner S. The notion of limit: Some seemingly unavoidable misconception stages. Journal of Mathematical Behavior 1986, 5:281-303.
8. Dorier J. Meta level in the teaching of unifying and generalizing concepts in mathematics. Educational Studies in Mathematics 1995, 29:175-197.
9. Dubinsky E., Elterman F., Gong C. The student's construction of quantification. For the Learning of Mathematics - An International Journal of Mathematics Education 1988, 8:44-51.
10. Ervynck G. Conceptual difficulties for first year university students in the acquisition of limit of a function. Proceedings of the Psychology of Mathematics Education, 5 1981, 330-333.
11. Fernandez E. The students' take on the epsilon-delta definition of a limit. Primus 2004, 14(1):43-54.
12. Ferrini-Mundy J., Lauten D. Teaching and learning calculus. Research ideas for the classroom: High school mathematics 1993, 155-176. Macmillan, New York. P. Wilson (Ed.).
13. Freudenthal H. Mathematics as an educational task 1973, Reidel, Dordrecht, The Netherlands.
14. Gass F. Limits via graphing technology. Primus 1992, 2(1):9-15.
15. Gravemeijer, K. (1998). Developmental research as a research method. In J. Kilpatrick & A. Sierpinska (Eds.), Mathematics education as a research domain: A search for identity (ICMI study publication) (Book 2, pp. 277-297). Dordrecht, The Netherlands: Kluwer.
16. Gravemeijer K. How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning 1999, 1:155-177.
17. Gravemeijer K., Cobb P., Bowers J., Whitenack J. Symbolizing, modeling and instructional design. Symbolizing and communicating in mathematics classrooms 2000, 225-273. Erlbaum, Mahwah, NJ. P. Cobb, E. Yackel, K. McClain (Eds.).
18. Harel G. The development of mathematical induction as a proof scheme: A model for DNR-based instruction. The learning and teaching of number theory 2001, 185-212. Kluwer, Dordrecht, The Netherlands. S. Campbell, R. Zazkis (Eds.).
19. Harel G. A perspective on " concept image and concept definition in mathematics with particular reference to limits and continuity." Classics in mathematics education research 2004, 98. T. Carpenter, J. Dossey, L. Koehler (Eds.).
20. Knuth E. Student understanding of the Cartesian connection: An exploratory study. Journal for Research in Mathematics Education 2000, 31(4):500-507.
21. Lakatos I. Proofs and refutations 1976, Cambridge University Press, Cambridge.
22. Larsen, S. (2001). Understanding the formal definition of limit. Unpublished manuscript, Arizona State University.
23. Larsen S. Reinventing the concepts of group and isomorphism. Journal of Mathematical Behavior 2009, 28(2-3):119-137.
24. Larsen S., Zandieh M. Proofs and refutations in the undergraduate mathematics classroom. Educational Studies in Mathematics 2007, 67(3):205-216.
25. Marrongelle K., Rasmussen C. Pedagogical content tools: Integrating student reasoning and mathematics in instruction. Journal for Research in Mathematics Education 2006, 37(5):388-420.
26. Monaghan J. Problems with the language of limits. For the Learning of Mathematics 1991, 11(3):20-24.
27. Oehrtman M. Approximation as a foundation for understanding limit concepts. Proceedings of the Psychology of Mathematics Education North American Chapter, 23 2004, 95-102.
28. Oehrtman M.C., Carlson M.P., Thompson P.W. Foundational reasoning abilities that promote coherence in students' understanding of function. Making the connection: Research and practice in undergraduate mathematics 2008, Mathematical Association of America, Washington, DC. M.P. Carlson, C. Rasmussen (Eds.).
29. Orton A. Students' understanding of integration. Educational Studies in Mathematics 1983, 14:1-18.
30. Steffe L.P., Thompson P.W. Teaching experiment methodology: Underlying principles and essential elements. Research design in mathematics and science education 2000, 267-307. Hillsdale, NJ, Erlbaum. R. Lesh, A.E. Kelly (Eds.).
31. Steinmetz A. The limit can be understood. MATYC Journal 1977, 11(2):114-121.
32. Stewart J. Calculus: Concepts and contexts 2001, Brooks/Cole, Australia. 2nd ed.
33. Swinyard, C. (2008). Students' reasoning about the concept of limit in the context of reinventing the formal definition. Unpublished doctoral dissertation, Portland State University.
34. Swinyard, C., & Larsen, S. (2010). What Does it Mean to Understand the Formal Definition of Limit?: Insights Gained from Engaging Students in Reinvention. Journal for Research in Mathematics Education. Manuscript submitted for publication.
35. Tall D.O. The transition to advanced mathematical thinking: Functions, limits, infinity, and proof. Handbook of research on mathematics teaching and learning 1992, 495-511. Macmillan, New York. D.A. Grouws (Ed.).
36. Tall D.O., Vinner S. Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics 1981, 12:151-169.
37. Vinner S. The role of definitions in the teaching and learning of mathematics. Advanced mathematical thinking 1991, 65-81. Kluwer, Boston. D. Tall (Ed.).
38. von Glasersfeld E. Radical constructivism: A way of knowing and learning 1995, Falmer Press, London.
39. Williams S. Models of limit held by college calculus students. Journal for Research in Mathematics Education 1991, 22(3):219-236.
40. Zandieh M. A theoretical framework for analyzing student understanding of the concept of derivative. Research in collegiate mathematics education IV 2000, 103-127. American Mathematical Society, Providence, RI. E. Dubinsky, A.H. Schoenfeld, J. Kaput (Eds.).
41. Zandieh M., Rasmussen C. Defining as a mathematical activity: A framework for characterizing progress from informal to more formal ways of reasoning. Journal of Mathematical Behavior 2010.
42. Zaslavsky O., Shir K. Students' conceptions of a mathematical definition. Journal for Research in Mathematics Education 2005, 36(4):317-346.