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

Artigue, 2000, Teaching and learning calculus: What can be learned from education research and curricular changes in France?, Vol. 8, 1 Bezuidenhout, 2001, Limits and continuity: Some conceptions of first-year students, International Journal of Mathematical Education in Science and Technology, 32, 487, 10.1080/00207390010022590 Carlson, 2001, An investigation of covariational reasoning and its role in learning the concepts of limit and accumulation, 145 Cobb, 2000, Conducting teaching experiments in collaboration with teachers, 307 Cornu, 1991, Limits, 153 Cottrill, 1996, Understanding the limit concept: Beginning with a coordinated process schema, Journal of Mathematical Behavior, 15, 167, 10.1016/S0732-3123(96)90015-2 Davis, 1986, The notion of limit: Some seemingly unavoidable misconception stages, Journal of Mathematical Behavior, 5, 281 Dorier, 1995, Meta level in the teaching of unifying and generalizing concepts in mathematics, Educational Studies in Mathematics, 29, 175, 10.1007/BF01274212 Dubinsky, 1988, The student's construction of quantification, For the Learning of Mathematics – An International Journal of Mathematics Education, 8, 44 Ervynck, 1981, Conceptual difficulties for first year university students in the acquisition of limit of a function, 330 Fernandez, 2004, The students’ take on the epsilon-delta definition of a limit, Primus, 14, 43, 10.1080/10511970408984076 Ferrini-Mundy, 1993, Teaching and learning calculus, 155 Freudenthal, 1973 Gass, 1992, Limits via graphing technology, Primus, 2, 9, 10.1080/10511979208965644 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. Gravemeijer, 1999, How emergent models may foster the constitution of formal mathematics, Mathematical Thinking and Learning, 1, 155, 10.1207/s15327833mtl0102_4 Gravemeijer, 2000, Symbolizing, modeling and instructional design, 225 Harel, 2001, The development of mathematical induction as a proof scheme: A model for DNR-based instruction, 185 Harel, 2004, A perspective on “concept image and concept definition in mathematics with particular reference to limits and continuity.”, 98 Knuth, 2000, Student understanding of the Cartesian connection: An exploratory study, Journal for Research in Mathematics Education, 31, 500, 10.2307/749655 Lakatos, 1976 Larsen, S. (2001). Understanding the formal definition of limit. Unpublished manuscript, Arizona State University. Larsen, 2009, Reinventing the concepts of group and isomorphism, Journal of Mathematical Behavior, 28, 119, 10.1016/j.jmathb.2009.06.001 Larsen, 2007, Proofs and refutations in the undergraduate mathematics classroom, Educational Studies in Mathematics, 67, 205, 10.1007/s10649-007-9106-0 Marrongelle, 2006, Pedagogical content tools: Integrating student reasoning and mathematics in instruction, Journal for Research in Mathematics Education, 37, 388 Monaghan, 1991, Problems with the language of limits, For the Learning of Mathematics, 11, 20 Oehrtman, 2004, Approximation as a foundation for understanding limit concepts, 95 Oehrtman, 2008, Foundational reasoning abilities that promote coherence in students’ understanding of function Orton, 1983, Students’ understanding of integration, Educational Studies in Mathematics, 14, 1, 10.1007/BF00704699 Steffe, 2000, Teaching experiment methodology: Underlying principles and essential elements, 267 Steinmetz, 1977, The limit can be understood, MATYC Journal, 11, 114 Stewart, 2001 Swinyard, C. (2008). Students’ reasoning about the concept of limit in the context of reinventing the formal definition. Unpublished doctoral dissertation, Portland State University. 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 Tall, 1992, The transition to advanced mathematical thinking: Functions, limits, infinity, and proof, 495 Tall, 1981, Concept image and concept definition in mathematics with particular reference to limits and continuity, Educational Studies in Mathematics, 12, 151, 10.1007/BF00305619 Vinner, 1991, The role of definitions in the teaching and learning of mathematics, 65 von Glasersfeld, 1995 Williams, 1991, Models of limit held by college calculus students, Journal for Research in Mathematics Education, 22, 219, 10.2307/749075 Zandieh, 2000, A theoretical framework for analyzing student understanding of the concept of derivative, 103 Zandieh, 2010, Defining as a mathematical activity: A framework for characterizing progress from informal to more formal ways of reasoning, Journal of Mathematical Behavior, 10.1016/j.jmathb.2010.01.001 Zaslavsky, 2005, Students’ conceptions of a mathematical definition, Journal for Research in Mathematics Education, 36, 317