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Limits of Sequences MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE VOLUME 2 - ISSUE 2 January 2008 Readers are free to copy, display, and distribute this ar- ticle, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York. Visit the Mathematics Teaching-Research Journal Online at http://199.219.158.116/~vrundaprabhu/TRJ/site Limits of sequences - foundations of understanding in calculus Bronislaw Czarnocha (Hostos Community College, CUNY, NYC), Jose Giraldo ( Texas A & M, Corpus Christi, Texas), Vrunda Prabhu, Bronx Community College, CUNY, NYC Issues in understanding the limit of a converging sequence are discussed in the context of 4 different definitions representing different approaches. The coordination between the concept of nearness (! ) and the discreteness (N) of the sequence is seen as fundamental in effecting a higher level understanding. The concept of ! - neighborhood is suggested as holding the key for the coherent understanding of the limit.. INTRODUCTION In this article we present the qualitative results of the first teaching-research cycle of the teaching experiment conducted in Fall 2002, supported by NSF-ROLE #0126141- Introducing Indivisibles into Calculus Instruction conducted at 3 different sites across the country. The teaching experiment was designed on the basis of new results of investigations reported in (Czarnocha, et. al., 2000a). The authors of this work inform about the re-discovery of the intuition of indivisibles (as distinct from infinitesimals) in calculus students clinical interviews excerpts. The intuition of a significant number of students turned out to be similar to that of Archimedes (Archimedes, 1957), Cavalieri (Cavalieri, 1635) and Wallis (Wallis, 1655). The instruction based on the formulations of Cavalieri and Wallis (Cavalieri-Wallis construction (Czarnocha and Prabhu, 2002)) was included in the syllabus of calculus classes at all participating sites to supplement the Riemann technique. The goal was investigating the effectiveness of the newly re-discovered method of calculating the area under an irregular curve upon students’ understanding of the definite integral. Since both, the Riemann construction, the basis of the definite integral, and the Cavalieri-Wallis method (Czarnocha and Prabhu, 2002) rely on the concept of the limit of an infinite sequence, the first phase of the teaching experiment is devoted primarily to the re-investigation of students’ knowledge of limits and to the formulation of the successful strategy of instruction, which will promote the development of this difficult concept to an object level of mastery (Asiala, et.al, 1996), (Czarnocha, et. al, 1999), (Davis, et. al, 2000), necessary for an adequate understanding of the area under an irregular curve. This goal requires a reorganization of the standard syllabus of Calculus in order to develop an understanding of sequences and their limits in anticipation of the definite integral, and not after, as the majority of national curricula propose (Edwards and Penny, 1998), (Hughes-Hallett, et.al., 1992), (Ostbee and Zorn, 1997). The proposed curriculum introduces Weierstrass’ definition of the limit of a sequence early in the semester, devotes significant class time to its discussion and understanding and leads students toward the limits of functions via the sequential Heine definition of the limit of a function (Aguilar., et. al., 2001). Understanding of the definition is reinforced through the coordination of epsilon with the notion of error for the approximating Riemann sums. On the basis of preliminary analysis of the first cycle of the on-going teaching experiment, we have come to the following conclusions: 1.We assert that a conceptual approach relying primarily on the verbal formulation of the precise definition of the limit concept proposed by reform oriented-textbooks such as (Stewart, 2005) is not enough to promote a correct understanding of the limit. In fact, we demonstrate how an absence of clarity of the relationship between the conceptual and precise mathematical definition of the convergence of a sequence may lead students astray toward known misconceptions. Furthermore, care needs to be exerted in the process of conceptualization of topics in Calculus, as it is not appropriate to rely on the conceptual approach in opposition to mathematical rigor. Conceptual clarity is reached precisely in enabling students to see the relationship between these two different modes of knowledge. We point out to another instance of the essential role played by the spatial notion of nearness or proximity, and of the discreteness of the sequence in understanding the concept of the limit. The importance of this spatial-discrete coordination on the understanding of the fundamental concepts of calculus, leads us, once again to underline the importance of the Central Conceptual Structure introduced by Case (Case, 1992) in his analysis of early child development and reflecting spatial-discrete coordination. We also suggest the notion of an ε-neighborhood as very helpful in this coordination process through the mediating role it can play between different types of intuitions and representations participating in the definition of the limit. THE LIMIT OF A CONVERGING SEQUENCE The emphasis on the limit of the sequence rather than on the limit of a function has several advantages. It focuses student attention of the coordination between the discrete and continuous components of the concepts represented, for example, in the Weierstrassian definition of the limit, with epsilon representing the concept of closeness (proximity) and N representing the discreteness of the sequence. This coordination seems simpler to students than the continuous-continuous coordination characteristic for ε -δ definition of the limit. Since the Heine definition of the limit of a function is sequence-based, while being equivalent to the known epsilon-delta definition of Cauchy, it provides a natural stepping stone to this difficult concept. It is interesting to note that certain contemporary textbooks such as (Edwards and Penny, 1998) use sequences to develop the intuition of, for example, difference quotients of a continuous function, without however providing a necessary theoretical background. Our work intends to provide the theoretical/didactical/curricular framework in which these new approaches can be successfully situated. The difficulties in students’ understanding of limits of sequences and functions are well known ((Davis and Vinner, 1986, Cornu, 1991, Sierpinska, 1991, Cottrill, et.al., 1996), among others). Davis and Vinner (Davis and Vinner, 1986) pointed out to an important confusion displayed by their students between, what has become known as the dynamic definition of the limit (see End-Note). In very simple terms, using ε, N of the latter definition, one can say that the distinction between the two definitions can be characterized as the difference between ε(N) [closeness as a function of N] in the case of the dynamical definition, and N(ε), [position or location in the sequence as a function of the closeness] in the Weierstrassian definition. In the language of APOS theory, the difference between the two definitions can be seen as the difference between a process level and an object level of understanding, while in Aristotelian terms, it can be seen as the difference between potential and actual infinity (Hitt, 2001). In symbolic terms, given the sequence an and its limit L, the difficulty corresponds to the difference between the statement “an L as n∞” on one hand, and the statement “L = ” on the other. limn"! an The relationship between the first two definitions has been analyzed in (Aguilar, et,al., 2001); the authors suggest that the clarification and mastery of the limit of sequences by students rests in the clarification of the meaning of that difference between the two definitions and in the ability to move freely between them with understanding. The difficult question is how should this actually be brought about in the classroom, what pedagogical approach should be taken to promote the required clarification and mastery of understanding. Conceptual approach. Some textbooks such as (Stewart, 2005) and, to certain degree, (Hughes-Hallett, 2002) suggest an emphasis on the conceptual formulation as the path toward the understanding of the limit of sequences and functions. We show that the attempts to substitute the formal definition of the limit by the conceptual one alone does not promote mathematically correct understanding. The excerpts from students’ essays indicate that this well - meaning approach, can nonetheless lead students toward standard misconceptions. What’s more, we will demonstrate that the reliance on the conceptual formulation breaks down precisely in the most sensitive part of the definition and that is the proper coordination between numerical and continuous aspects of the limit concept. Description of the data collection and its analysis The teaching experiment, Introducing Indivisibles into Calculus instruction uses a variety of techniques to assess its results: during the course of the semester, the students of Calculus 1 have written 4 essays focused on the explication of the meaning of the limit of a sequence in different mathematical
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