Mathematics Teaching-Research Journal Online

Volume 2 - Issue 2 January 2008

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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 contexts, created several related concept maps (Novak, 1990) and undergone clinical interviews at the end of the semester. While the full analysis of the data is in progress, here we will discuss excerpts from several pertinent essays. The majority of excerpts come from, Essay #1: “In your essay you will explain how the precise definition of the limit can be understood with the help of the conceptual definition. You can use examples of sequences we discussed in the class to support and to explain your point. Among others explain the meaning of phrases “make the terms as close to zero as one wishes”, “…for n sufficiently large…” and “make the terms as close to zero as one wishes for n sufficiently large”.

The intent of the question was to stimulate written reflection upon both definitions and to create the required coordination between corresponding symbols and words. The essay was given as a home assignment after the class during which both definitions were discussed, together with the algorithm for calculating the limit and proving its fulfilment of the precise definition. The central stumbling block turned out to be the phrase “..for n sufficiently large..”

In the following fragment we see the student quite correctly identifying ε, epsilon, as the measure of closeness for the terms (values of the sequence), displays familiarity and understanding of the role of, what later will be called, epsilon-neighborhood, but totally fumbling on the relationship of n with ε:

S1 : “Making the terms as close to zero as one wishes” in the example, we have zero as limit, a person can pick a number, epsilon, as close to zero as one wants and still find an infinite number of values closer to zero. N is the number into the sequence that a person picks as the epsilon.

The last sentence “N is the number into the sequence that a person picks as the epsilon” betrays the confusion about the relationship in question. Clearly, N is not the number picked as the epsilon, and although one can second guess what the student intended to say, that will not detract from the reality of misunderstanding.

S2 : When it says “if we can make the terms of the sequence as close to zero as we wish for n sufficiently large”, it is talking about a sequence like this one: 4n ! 3 , 5n + 2 where as the n is made larger, the sequence begins to get smaller and closer to zero, but never reaches it.”

This student displays one of the more common confusions; he starts with the correct statement of the conceptual definition but interprets it not via the Wierstrassian ε-N definition, but via the dynamic definition, changing at the same time the direction of the relationship between ε and N. Note the phrase “…but never reaches it…”, a trace of a known epistemological obstacle referred to by several authors (Orton, 1983, Cornu, 1991 and Sierpinska, 1987). The student displays here the concern for the last term of the sequence, the very same concern that the Wesierstrassian definition was designed to eliminate since it leads to murky ambiguities (Sierpinska, 1987, Czarnocha, et.al, 2001b) especially in relation to understanding the definite integral. What is also important is that the phrase “for n sufficiently large” is fully left out of the picture.

Finally, the student S3 states directly the source of students’ confusion when he says: Conceptual definitions are a way of telling the “non-math” mathematical world what your proofs are and where they are coming from…. For n sufficiently large? If one would be speaking to an individual without much math experience, that individual would still be lost…It is hard to explain what for n sufficiently large is…a better explanation in English could have been, the larger n becomes, the closer to zero the value becomes. From this student we find out not only the correct identification of the difficulty in understanding the phrase (It is hard to explain what for n sufficiently large is), but also we get implicit suggestions that this spontaneous change of the phrase into the dynamic definition - the larger n becomes, the closer to zero the value becomes may be motivated precisely by the absence of clarity as to “what for n is sufficiently large”, that is precisely by the absence of the n – epsilon coordination. The subtlety of this coordination is revealed by the last student, S4, one of only two amongst 30+ students participating in three different sites of the experiment, who understood correctly, in our opinion, the basic construction of the limit concept (The excerpt is taken from the subsequent essay where students were asked for a direct comparison of the dynamic and precise definitions of the limit). S4 “Also, the “n” in question also tells you what number it has, at least, go to in order to [get] a number that is inside e-neighborhood. The idea that is not taken in consideration by the first statement, that the number has to only increase to certain point, or the point sufficiently large to satisfy that an will be small enough to be inside of the e-neighborhood that you set.

Indeed, n is sufficiently large for an to be closer to the limit than the epsilon set a priori, or, in the student’s words, to be “in the e-neighborhood that you set” to begin with. The phrase “..for n sufficiently large..” is central in the formulation of the conceptual definition because it creates, if correctly understood, a very tight coordination between the distance which one can choose “as small as one wishes” and the appropriate term in the sequence beyond which all the terms have distance to the limit smaller than the chosen a-priori distance, ε. The phrase “…as small as one wishes…” provides a good intuitive meaning for the arbitrariness of that choice (“for every ε > 0”). The coordination of the choice with the position of the term of the sequence beyond which all the terms have the required property is one of the main didactical goals in teaching the notion of the limit of a sequence. The student observation that “ the number has to increase only to a certain point, or to the point sufficiently large..” indicates that the coordination manifested here might enable students to realize the control of N by epsilon.

As we see from the misconceptions contained in the majority of the essays, this is cognitively, an arduous task, which we think, cannot be helped merely by a change to the language-based conceptual approach. Therefore What is to be done? is the question. How should student reflection be stimulated to enable an adequate coordination of the involved concepts? The teaching experiment, Introducing Indivisibles into Calculus instruction, which has been discussed here is based on the teaching-research cycles, which allow for rapid introduction of the results of the data analysis back into instruction. It seems from the presented excerpts, and from other essays, that the emphasis on the visual nature of the concept of an ε-neighborhood might provide a mediating role between the verbal, the symbolic and the numeric components of the limit, increasing the coherence of the limit schema constructed by a student

An ε- neighborhood around the limit L of a sequence an is an open interval around L. It provides a visual image, which conveys a framework of discussion much closer to the ideas expressed in the precise definition. Moreover it allows to re-focus the attention of the student from the process motivated attention to the motion of the variable N, to a more static, and hence object-like-thinking idea of the number of terms of the sequence within the neighborhood:

S5: “you need to look at certain interval around zero which can be called the e- neighborhood. Look to see how many terms are in that interval. You can solve it algebraically by solving the function with an inequality” Note the ease of coordination with the algebraic representation.

S6: “The precise definition includes epsilon (E), which is an area that is arbitrarily chosen to see which values of n lie within this epsilon neighborhood. The N value which is one that lies right on the line of the epsilon neighborhood so that when any term, n, is greater than N, its an value falls within the epsilon neighborhood.” A full understanding of the epsilon-numeric coordination of the definition of the limit uses e-neighborhood (the second of the two students).

S7: “An epsilon neighborhood can be created to better show the limit. An epsilon neighborhood is an extremely small area above and below the zero axis. The epsilon neighborhood can be within the value .001 on the y axis if you wish it to. For a true limit there will be only finite number of terms outside of this neighborhood. Once the sequence crosses into the neighborhood, it will never leave it again. That is where the phrase “for a sufficiently large” becomes important”. And the epsilon-verbal coordination for the phrase “for a sufficiently large”.

Clearly ε-neighborhood suggests itself as the tool, which can mediate the integration of the different components of the schema of the limit concept into a coherent whole.

END-NOTE: DEFINITIONS OF THE LIMIT (1) Weierstrass definition of the limit of a sequence (precise definition)

The sequence an has a limit L if for every e>0 there exists N such that for every n>N we have |an – L | < e. (2)Conceptual definition (modeled on Stewart textbook)

The sequence an has a limit L if the terms an can be made as close to L as one wishes for n sufficiently large. (3) Dynamic definition (mentioned in Davis and Vinner)

Sequence an has a limit L if as n is increasing infinitely, the terms of the sequence are approaching the number L closer and closer. (4) Heine definition of the limit of a function A function f(x) has the limit L at x=a if for every convergent sequence with

lim xn= a, we have lim f(xn) = L. n ! n!

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