Contemplating problems taken from the history of limits as a

Contemplating problems taken from thehistory of limits as a way to improve

students’ understanding of the limit concept

Andisheh Kamali Sarvestani

Supervised by:Dr. Wolter Kaper

Universiteit van Amsterdam

May, 2011

Dedicated to my lovely sons, Ilia and Aria

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Acknowledgements

Now here is the place to thank several people whom without their help andsupport, I could not finish my study. In the first place I gratefully acknowl-edge my supervisor Dr. Wolter Kaper for his supervision, advice, severalbrainstorming discussions, and guidance from the very early stage of thisresearch as well as going out of his way and contributing to my research ac-tively as a teacher which made him a backbone of this research. He providedme unflinching encouragement and support in various ways. I am indebtedto him more than he knows. I gratefully acknowledge Harm Houwing, mycooperating teacher who gave me the opportunity to conduct my researchproject in his school. I would particularly like to thank the coordinator ofthe master program, Dr. Mary Beth Key for supporting and inspiring me allthe time and also for being a friend who let me share my happy and sad mo-ments with her. My sincere thanks to Drs. Andre Heck for his kind supportsand advices. I am also grateful to members of my thesis committee whohave accepted to read this and attend my defense session, I hope they find itworthwhile. I also would like to thank my fellow colleagues for providing mea warm and friendly atmosphere during my study. My special thanks go toLilia Ekimova for letting me count on her help and support and always beingthere when I needed her. Most of all I wish to express my sincere and deepestgratitude to my husband Saeed for his devotion, love, encouragement, andpatience. Finally, I thank my parents whom I miss so much for the pricelessgift of their unconditional love and support.

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Abstract

The limit concept as the origin of calculus has always been one of the mostdifficult concepts for students to grasp. In this research, I have tried totackle this dilemma by pursuing the process in which this concept is formedduring several centuries with the students via oriented discussions. I hadthree students who were 4th VWO graders and had taken the WIskunde Dprofile as an optional course. As expected, I had to choose just a few signifi-cant historical problems which involved the concepts of infinity, infinitesimal,tending, and finally the limit notion, and try to carry the students throughthese problems at least to the correct informal concept. However, arrivingat the formal definition was my ambition. For this purpose, three historicalturning points including paradoxes, regarding the limit notion were selected.Whereas students were introduced with the limit no sooner than the end ofthe paradoxes. Up to this point, the informal knowledge of students wastested and analyzed with respect to the models of limits which are usuallyheld by students as distinguished by Williams (Williams, 1991). For teachingthe formal knowledge, a bridge was needed from the informal knowledge tothe formal one. From here on, the historical process was no more applied.Instead, this bridge was designed as practicing on the geometrical interpre-tations of the intervals and neighborhood concept. The results at the endof the three paradoxes, showed that students were encountering some of themisconceptions every now and then while some other misconceptions wereevaded more successfully. In finding the formal definition of limit, studentscould not succeeded to construct it by themselves. However, later in thesession on continuity, they themselves could find out the formal definition ofcontinuity by manipulating the limit definition.

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Contents

1 Introduction 1

2 Theoretical Framework 22.1 Students’ Common Misconceptions & Concept Images . . . . . 22.2 Limit phenomena . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Choosing a didactic view of limits . . . . . . . . . . . . . . . . 92.4 Three Worlds of Mathematics . . . . . . . . . . . . . . . . . . 122.5 APOS Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 What is Intuition . . . . . . . . . . . . . . . . . . . . . . . . . 152.7 Choosing relevant episodes from the history of calculus . . . . 16

3 Research Questions 23

4 Methodology 244.1 Teaching Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.1 Choice of Teaching Goals . . . . . . . . . . . . . . . . . 254.1.2 General Remarks about Teaching Strategy . . . . . . . 264.1.3 Zeno’s paradoxes . . . . . . . . . . . . . . . . . . . . . 274.1.4 The Quadrature of the Parabola . . . . . . . . . . . . . 284.1.5 Derivative Activity . . . . . . . . . . . . . . . . . . . . 294.1.6 Neighborhood . . . . . . . . . . . . . . . . . . . . . . . 324.1.7 Working Towards Formal Definition . . . . . . . . . . . 334.1.8 Epsilon-delta proof & Mathematica . . . . . . . . . . . 344.1.9 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Comparing our plan to APOS genetic decomposition . . . . . 394.3 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.4 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4.1 The Test and the interviews . . . . . . . . . . . . . . . 414.5 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5 Results 435.1 Lesson preparation and time planning . . . . . . . . . . . . . . 445.2 Zeno Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 Archimedes Activity . . . . . . . . . . . . . . . . . . . . . . . 505.4 Derivative Activity . . . . . . . . . . . . . . . . . . . . . . . . 555.5 Neighborhood . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

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5.6 Working Towards Formal Definition . . . . . . . . . . . . . . . 675.7 Epsilon-Delta Proof & Mathematica . . . . . . . . . . . . . . . 725.8 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.8.1 Formal Definition of Continuity . . . . . . . . . . . . . 815.8.2 Investigating the Continuity of some functions . . . . . 83

5.9 Test and Interviews . . . . . . . . . . . . . . . . . . . . . . . . 875.9.1 Question 1 . . . . . . . . . . . . . . . . . . . . . . . . . 885.9.2 Question 2 . . . . . . . . . . . . . . . . . . . . . . . . . 905.9.3 Question 3 . . . . . . . . . . . . . . . . . . . . . . . . . 925.9.4 Question 4 . . . . . . . . . . . . . . . . . . . . . . . . . 935.9.5 Question 5 . . . . . . . . . . . . . . . . . . . . . . . . . 945.9.6 Question 6 . . . . . . . . . . . . . . . . . . . . . . . . . 955.9.7 Question 7 . . . . . . . . . . . . . . . . . . . . . . . . . 965.9.8 Question 8 . . . . . . . . . . . . . . . . . . . . . . . . . 975.9.9 Question 9 . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.10 Interview About the Paradoxes . . . . . . . . . . . . . . . . . 1035.10.1 Interview with John . . . . . . . . . . . . . . . . . . . 1035.10.2 Interview with Otto . . . . . . . . . . . . . . . . . . . . 1045.10.3 Interview with Dave . . . . . . . . . . . . . . . . . . . 106

6 Conclusions and Discussions 1086.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.1.1 Research Question 1 . . . . . . . . . . . . . . . . . . . 1086.1.2 Research Question 2 . . . . . . . . . . . . . . . . . . . 1086.1.3 Research Question 3 . . . . . . . . . . . . . . . . . . . 114

6.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.2.1 Comparison . . . . . . . . . . . . . . . . . . . . . . . . 117

6.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7 References 131

8 Appendices 1348.1 Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

8.1.1 Appendix 1, Zeno’s Paradoxes . . . . . . . . . . . . . . 1348.1.2 Appendix 2, The Quadrature of the Parabola . . . . . 1348.1.3 Appendix 3, Derivative Activity . . . . . . . . . . . . . 1388.1.4 Appendix 4, Neighborhood Activity . . . . . . . . . . . 1418.1.5 Appendix 5, Epsilon-Delta Proof . . . . . . . . . . . . 143

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8.1.6 Appendix 6, Mathematica . . . . . . . . . . . . . . . . 1488.2 Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1488.3 Transcripts (on the attached CD) . . . . . . . . . . . . . . . . 151

8.3.1 Zeno’s Paradox . . . . . . . . . . . . . . . . . . . . . . 1518.3.2 The Quadrature of the Parabola . . . . . . . . . . . . . 1518.3.3 Derivative Activity . . . . . . . . . . . . . . . . . . . . 1518.3.4 Neighborhood . . . . . . . . . . . . . . . . . . . . . . . 1518.3.5 Working Towards Formal Definition . . . . . . . . . . . 1518.3.6 Epsilon-Delta Proof & Mathematica . . . . . . . . . . 1518.3.7 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . 1518.3.8 Last Interviews . . . . . . . . . . . . . . . . . . . . . . 151

1 Introduction

The aim of this research is probing ways to improve students understandingof the concept of limit as a fundamental notion in calculus. What is proposedin this research is to investigate the process of finding limits and students’perceptions of this concept in a school context.

Limit is usually just taught as a mathematical rule with its formulas,whereas it is not regarded as a concept generated through centuries whichis not just made of formulas and rules. But, where does it come from orthe necessity of its existence or what does it represent and the answer tomany other conceptual questions are remained unanswered in the books. Itis applied in calculus for instance in the definition of derivatives withoutdrawing pupils’ attention to its origin. The question is, will students reallyunderstand the rest of calculus without deep understanding of limit? Myanswer is no! The Students after teaching, usually know the techniques ofsolving problems about limits and continuity, but they do not know what dothese concepts mean and represent.

Many researches have shown that students have problems in gaining adeep and correct idea about the concept of limit (Bagni, 2005; Tall, 1990;Tall and Vinner, 1981; Williams, 1991). My goal in this research is to try totackle this dilemma.

My idea for improving is trying to discover this concept with studentsvia controlled and oriented discussions. My suggestion is to follow the mainstages in evolution of the limit concept in class. With respect to the limitnotion, there is a possible parallelism between history and cognitive growth.It would require a specific theory of knowledge to completely spell out acomparison of the students’ growth of knowledge and the historical develop-ment of the concepts (Bagni, 2005). Although such a theory of knowledgedoes not yet exist in complete form, we want to try out this possible par-allelism. Possible theories to back it up could be Piaget (Piaget, 1968) andVan Hiele, as cited by Kaper and Goedhart (Kaper and Goedhart, 2001).We agree with Bagni that these theories are incomplete, but they do provideexplanations why some steps in the history of a concept might parallel somesteps observed in student learning. This method of recapitulating historicalsteps would have some benefits as well as some limitations. Therefore, thediscussions in the class must be controlled in a way that students move frommisunderstandings to deep understandings.

Students’ misconceptions about the limit concept are so tricky and deep.

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They are partly caused by the language used with respect to limits: “tendsto”, “limit approaches”, “converges”. “these words are not equivalent asregard their everyday meanings and students hardly recognize that such ex-pression have the same mathematical meaning”1 (Cornu, 1980, Davis andVinner, 1986, pages 298-300; Monaghan, 1991, pages 23-24, cited by Bagni(Bagni, 2005)).

Their misconceptions are also partly influenced by the dynamic spirit ofthe usual talk about limit. This dynamic character of limit provokes the ideaof movement or tending but never reaching, etc. While the formal definitionof limit is quite hard for students and even teachers to grasp, my hypothesisis that it is possible to guide students to be less affected by the now commonmisconceptions by letting them know and experience why and how did limithappen to enter into mathematics.

2 Theoretical Framework

Regarding the importance of calculus as a branch of mathematics wheregeometry, algebra, and other mathematical topics also other sciences meeteach other, also with respect to the limit concept as the fundamental conceptof calculus, a deep and correct understanding of limit and continuity willsupport students’ further mathematical knowledge. A review of differenttheories investigating this problem are presented in this chapter. In thischapter first an overview of students’ common misconceptions (Williams,1991), and concept image and concept conflict introduced by Tall and Vinner(Tall and Vinner, 1981) will be presented and compared. The dynamic andstatic view of limit concept will be discussed. Two studies presenting thescales for understanding limit concept are reviewed. And finally the relevantepisodes from history are selected and explained.

2.1 Students’ Common Misconceptions & Concept Im-ages

Students often have intuitive images of the mathematical notions in theirminds which are influenced by many factors such as language and daily life.Such images are known as concept image. The term concept image describes

1Literal citations are marked by Italics!

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the total cognitive structure that is associated with the concept, which includesall the mental pictures and associated properties and processes (Tall and Vin-ner, 1981, page 152). Such images in pupils’ minds may be in conflict withthe formal image or formal definition of a concept which represents the mostaccurate description of the concept. Even when the concept is presented toor constructed by an individual, he could have his personal concept definitionwhich can differ from the formal concept definition.

The conflicts between students’ concept image and the “formal definition”of that concepts are known as cognitive conflicts (Tall and Vinner, 1981, page151). These images are highly affected by the characteristic, syntax andlanguage used with respect to the concepts. Limit and continuity conceptsalso are reported and known as concepts whose individual’s concept imagesare different with their formal images, thus are accompanied by the cognitiveconflicts (Tall and Vinner, 1981, page 151). In this research I try out away to cope with such conflicts. As Bagni mentions “The limit process isintuitive from the mathematical point of view but not from the cognitive one,so sometimes cognitive images conflict with the formal definition of limit(Bagni, 2005, page 2). I find this, “cognitive images conflict with the formaldefinition”, a key problem in learning this concept.

The formal definition of the limit of a function was written by Weierstrassin the nineteenth century as below:

limx→a

f(x) = L

means

∀ε,∃δ; 0 < |x− a| < δ ⇒ |f(x)− L| < ε

This definition requires quit a high degree of knowledge of logic and syntax.But this definition has informal interpretations which are easier to under-stand. An informal definition of limit can be stated as below:

A limit is a number that the y-values of a function can becomeas close to as we want, by choosing x within an x-interval that isas small as needed.

Williams in his investigations to find models of limits held by students, dis-tinguished 5 models most often believed by students (Williams, 1991).

1. A limit describes how a function moves as x moves towarda certain point. (dynamic theoretical)

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2. A limit is determined by plugging in numbers closer andcloser to a given number until the limit is reached. (dynamicpotential)

3. A limit is a number or point the function gets close to butnever reaches. (unreachable)

4. A limit is a number or point past which a function can notgo. (acting as a boundary)

5. A limit is an approximation that can be made as accurate asyou wish. (limit as approximation)

In his research, students could express agreement or disagreement with theabove statements and choose more than one sentence as the correct descrip-tion for the limit concept. The expressions between the brackets are thenames that Williams has given to each model of limit.

These models or images are fitting with the concept images mentioned byTall and Vinner (Tall and Vinner, 1981). Therefore, in the rest of this study,these models are taken equivalent with students’ concept images of the limitnotion. Williams’ list of misconceptions about limits can be understood asright descriptions of true mathematical phenomena which are linked to thelimit concept. It’s just that neither of these phenomena fully catches all thatis said in the definition of limits, and that’s why these good conceptions arein fact misconceptions of the limit concept. We agree with Williams thatthese patterns are wrong. However, to be clear we find it necessary to stateour reasons for finding them wrong:

1. A limit is a number or point on the y-axis (if it exists) and a point cannot by itself describe how a function moves. If we know that the limitof a function is for instance zero, we can imagine different scenarios;it can be a linear movement toward the origin like f(x) = x or withinfinitely many oscillations near the origin such as f(x) = xsin 1

x.

2. This misconception has in common with number 5 that the limit isinitially approximated by plugging in x-values closer and closer to thefinal value x0. In contrast with 5 (which holds that the result willalways be an approximation) now we go one step further and plug inthe final value x0 itself, and indeed we get an f(x0) that seems sosmoothly fit into the sequence. The mathematical phenomenon is atrue one, it is called “continuity”. However, limits are often applied to

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points of discontinuity, and in that case this limit concept would notbe of use. In other applications of limits, we might want to prove thecontinuity of a function. However, this definition assumes it, and istherefore not fit for that purpose either.

3. This misconception is problematic for the opposite reason as (2): forcontinuous functions the limit value is “reached” in all points. Theunderlying phenomenon is not true because simply the limit value isattained for many values in which a function is continuous and/or forthe values in which it does not have asymptotes. The underlying phe-nomenon or picture might be the same as the 4th misconception. It’sjust a different attitude towards “infinity”: do we allow it as a hypo-thetical point on our number line, or not.

4. For monotonous functions having a horizontal asymptote this is a trueconception. Indeed when calculating f(x) for a higher and higher valuesof x, there is a definite point on the y-axis that we can approach butnot pass. Depending on our attitude towards “infinity” as a number,we can either say that we reach it (like in 2) or not reach it (as in 3).

As a general concept of limit this is not usable because non-monotonousfunctions, like frac1x sinx, also “approach” a certain value for x→∞,but in such a case the value is “passed” or even “reached” many times.

5. This misconception holds that a limit can only be known inaccurately.Likely it is based on the experience of plugging in numbers closer andcloser to a given number and seeing that a certain value is approached(like in 2). However, now we are aware that the given number x0 mightbe a trouble maker, e.g. a point where the function is defined, so wedare not plug this number in. Instead we conclude that a limit cannotbe totally sure or totally accurate, because we have to keep a distance(however small) from the number we would like to plug in.

The phenomenon experienced is real. Also, this phenomenon posesa problem that the limit concept is designed to solve. The solutionthat the limit concept allows us, however, is ,much better than anapproximation.

From Williams’(Williams, 1991) five models, the dynamic models (the firsttwo models) are very resistant to change. Williams states that “the reason

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of this resistance could be students’ belief in the a priori existence of graphs,their prior experiences with graphs of simple functions, the value they put onconceptually simple and practically useful models, and their tendency to viewanomalous problems as minor exceptions to rules” (Williams, 1991).

Now, we know the most common inaccurate patterns in pupils’ minds.and we know more or less where these patterns are generated from.

There are two possible ways of coping with conflict: one is tobe prepared to face the cognitive conflicts when they occur. Thesecond is to give a richer conceptualization from the start to re-duce the later conflict, or at least give the experience to set it incontext. (Tall, 1990)

In this approach we choose to try the second mentioned way by followingthe historical process which ended with the formal definition of limit. Inother words, to discover this concept with students, a possible way is passingstudents through the path of the evolution of the concept and letting studentsexperience the problems and conflicts that led to the development of limit.Such a method is then inherently inspired by the history of mathematics.

2.2 Limit phenomena

When looking at Williams’ list of misconceptions, we suggested that mostof them might be called true conceptions of real mathematical phenomena,although none of them pictures rightly the mathematical concept of limit.Below is is an incomplete catalogue of such phenomena. We call them “limitphenomena”, because clearly the limit concept seems designed with thesephenomena in mind. All of these phenomena can be explained (though per-haps not accurately described) without knowing the word or the concept oflimit, as we will demonstrate in the below descriptions in which the wordlimit is not used 1.

1. Special points phenomenon. Some values of x can be special. For in-stance, the functions f(x) = cos( 1

x) and g(x) = x cos( 1

x) both behave

1We did not consult any particular source, except the Dutch school book “Getal enRuimte”, but any other school book or calculus syllabus would probably suffice to showthese phenomena as applications of the limit concept.

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quite specially near x = 0.

What is going on? Of course, the function is moving up and down, asexpected of a cosine. But when we approach x = 0, we see the functionmoving up and down “faster and faster” (with increased frequency).We can use software to “zoom in” and then at each magnification new“moves” (periods) show up. Near x = 0, there seem to exist an unlim-ited number of them! The special points phenomenon is a superclass: itcontains the below “gap phenomenon” and possibly other phenomena(the “jump phenomenon”) as special cases. The special points, andespecially the more spectacular ones clearly pose a problem: how canwe describe the “movement” near the special point? We suggest thisphenomenon might explain Williams’ misconception 1.

2. The simple horizontal asymptote phenomenon. A monotonous increas-ing or decreasing function may have a horizontal asymptote. If thefunction has N as its domain, then it is a sequence. The Zeno sequence1

2nis just such a function. Zeno’s paradox says we cannot reach our

goal because the remaining distance will not reach the value 0 for anyvalue of n in N . Let alone it will ever get into the negative values: wecan’t get past the zero point!

The horizontal asymptote phenomenon shows itself as a value that we

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can approach but not reach or pass. This phenomenon is involved inZeno’s paradoxes, as well as in Williams’ misconceptions 3 and 4.

3. Meandering horizontal asymptote phenomenon. A function like ( 1x) sinx

shows this phenomenon: The function keeps moving up and down, butit moves in a smaller and smaller range as we look at x-intervals furtherand further to the right. It can be argued that the range will get assmall as we want as we move further to the right.

Hypothetically, this could be a cure for Williams’ misconceptions 3and 4, because it might show the need of a new concept that does notdepend on “not reaching” or “not passing” (because we see there is stilla special value, but we reach and pass it a lot of times).

4. Gap phenomenon. An otherwise continuous function can have a gap:a value x = x0 for which f(x) is not defined, while f(x) is defined forall other x-values in the neighborhood of x0.

The gap phenomenon shows itself in that we cannot calculate f(x0),but we can setup a sequence of x-values, x1, x2, x3, ... approaching x0

(for instance halving the distance each time) and calculating the cor-responding f(xi). The result will be that subsequent f(xi) values willbe closer and closer together. We cannot reach certainty about whichsingle f(x) value is being approached , but we can make guesses orapproximations of such a number: Williams’ misconception 5 is trueas long as we just have this experience, and no conception of limit yet.In cases of monotonous functions such a value is approached but notpassed or reached: hence Williams’ misconceptions 3 and 4 will also betrue in such simple cases.

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The gap phenomenon is involved in Newton’s problem of determiningan instantaneous velocity, because this threatens to result in a divisionby zero.

5. Changing slope, or, “velocity” phenomenon. On the one hand the graphsuggests that there is a different slope for each value of x. On the otherhand attempts to calculate such a “instantaneous velocity” result indivision by zero. We therefore have an instance of the gap phenomenon.This gap does not exist in the original function f(x) but it exists in the

difference quotient, q(o) = y(x+o)−y(x)o

when seen as a function of o, fora fixed value of x.

6. Continuity phenomenon. Even if a function does not have any gaps orspecial points, we can still act like it could have a gap at x0, and so wecould calculate f(x) for a sequence of x-values, x1, x2, x3, ... approachingx0 (for instance halving the distance each time). But now we can finishby calculating f(x0) itself and conclude that it fits smoothly into thesequence of calculated f(xi). This is just Williams’ misconception 2:for continuous functions it is a true description.

7. ...

The list is not meant to be exhaustive, for instance a “jump phenomenon”and some “vertical asymptote” phenomena should probably be added atleast, but they did not play their due role in our work yet.

In our teaching setup, the above phenomena 2 and 5 (and indirectly 4)are introduced to the students before the word limit is mentioned. There-fore, they cannot yet have misconceptions about limits. However, they canexperience the phenomena and react intuitively to the paradoxes thatthey seem to pose. We are also interested in how far these intuitive reactionswill already “match” possible later (mis)conceptions of limits, and we willlook for such suspected matches.

2.3 Choosing a didactic view of limits

The historical approach in teaching the limit concept functions as a guidelinefor me not as a detailed recipe. It is mainly used for constructing knowledgewith the students. Therefore, I have to mention some advantages and somelimitations of such an approach to teaching. Some of the difficulties as Kro-nfellner (Kronfellner, 2000) says are as below:

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• Mathematical concepts are developed through several years and some-times several centuries and it is impossible to follow such developmentsstep by step, including all details. In teaching it would take very long.

• Historical methods as well as the other constructive methods take muchtime and produce difficulties for teachers in time planning.

• Pupils do not necessarily think like the mathematicians of the past.“pupils do not necessarily feel a lack of rigor, whereas the great math-ematicians had a more subtle perspective” (Kronfellner, 2000).

Some main advantages of a historical approach are mentioned by Lakoma:

• Creating a didactic approach to mathematics teaching whichtakes account of the students’ cognitive development.

• Recognizing the student’s ways of arguments as correspond-ing with past problems, and encouraging their responses toreal situations similar to those known from the history ofmathematics.

• Organizing the process of learning mathematics according tothe student’s actual abilities. (Lakoma, 2000)

Juter has investigated students’ conceptual development of limits notion incomparing with the historical development of the concept (Juter, 2006). Theresults demonstrated that although there are some similarities between stu-dents and past mathematicians in struggling with rigor and attainability,not all the students arrive to the most developed versions of the concept.Therefore, most of the students remain in the intuitive level of understand-ing which are often in conflict with the concept definition. She mentionsthat such a comparison is yet complicated since the historical developmenthas taken place during several years . Also considering that history and phi-losophy are two different domains, there are theoretical and methodologicalcomplications in such a comparison.

Some researchers like Bagni (Bagni, 2005) believe that students’ miscon-ceptions are rooted in the concept of the infinitesimal. I agree with himbecause calculus is a branch of mathematics that studies changes. Changescan be studied by contemplating infinitesimals.

Bagni then articulates different views of limit and classifies them intodynamic and static views. He considers the misconceptions in the dynamic

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views to be based on potential infinitesimals. He believes that the formaldefinition -ε, δ definition- presupposes a static view and the teachers shouldguide students from dynamic views to the static one. Here is where I havea problem with his interpretation of the formal definition of limit.I do notinterpret the formal definition as static. I think that the formal definition canbe interpreted as fitting in a dynamic view. In the ε, δ definition, we shouldfind a relation between ε and δ for all the ε and those δ which fulfill thatrelation. We can consider both ε and δ as dynamic, changing quantities, andwe can imagine a process in which ε gets smaller and smaller, without bound.We should assure ourselves that for each of the infinity of possible ε-valuesthere is a fitting delta. As a conclusion, I do not need to follow Bagni’sopinion (Bagni, 2005) that a static view is needed. Bergsten (Bergsten,2008) analyzes the theoretical aspects of didactical researches on limits offunctions. He classifies the different researches into a cognitive, a social, andan epistemological approach. One of the treatises of the cognitive approachis in particular how it treats the reconciliation of the dichotomy between“dynamic or process conceptions of limits and static or formal conceptions(Bergsten, 2008). He refers to Cottrill et al. 1996, as the researchers whoseresearches are categorized in the cognitive approach in this issue

[...] dynamic conception of limit is much more complicated than aprocess that is captured by the interiorization of an action” (page190), and that a strong such conception is needed to move to aformal conception of limit, which is not static “but instead is avery complex schema with important dynamic aspects and requiresstudents to have constructed strong conceptions of quantification”(page 190). Cottrill et al. 1996, cited by Bergsten (Bergsten,2008)

This dynamic view of limit as mentioned before is based on the conceptof potential infinitesimals. Potential infinitesimals represent the idea of avariable quantity getting smaller and smaller. Newton wrote about such“evanescent quantities” as well as Euler and some ancient mathematicians.Bagni then proposes a solution for overcoming the misconceptions and that ispassing from potential infinitesimals to actual infinitesimal. Actual infinites-imal on the other hand is defined in 20th century by A. Robinson. 1996,“non-standard Analysis”, Referred by Bagni (Bagni, 2005) though implicitlyknown from ancient time by the Greeks.1 Actual infinitesimals are applied

1Archimedes is known for having given a proof that actual infinitesimals do not exist.

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in nonstandard analysis which is taught at the university level.So, I would like just to pick Bagni’s ideas about the dynamic view in

respect of historical process. The common misconceptions in the limit notionare rooted in this dynamic character that limit inherits. My hypothesis isthat it is possible to lead students to get the accurate and formal image oflimit in spite of thinking of infinitesimals as evanescent quantities (potentialinfinitesimal) and consequently a dynamic view of limit. That is to say,although students’ misconceptions are due to an idea of motion which isembedded in the limit concept, the formal image also could be gained bysuch a characteristic of limits including the ε, δ definition.

2.4 Three Worlds of Mathematics

David Tall has introduced the idea of three worlds of mathematics and de-scribes it as a theory of three distinct but interrelated worlds of mathematicalthinking each with its own sequence of development of sophistication, andits own sequence of developing warrants for truth, that in total spans therange of growth from the mathematics of new-born babies to the mathemat-ics of research mathematicians (Tall, 2004, page 281).” This idea as well asother ideas of cognitive development such as Burner’s, Fischbein’s, Skemp’s,van Hiele’s, Dubinsky’s, and Sfard’s theories of development are inspired byPiaget’s successive stages of development as having a tripartite theory ofabstraction embedded in it. The three theories as Piaget explains them arenamed empirical abstraction focusing on how the child constructs meaning forthe properties of objects, pseudo-empirical abstraction, focusing on construc-tion of meaning for the properties of actions on objects, and reflective abstrac-tion, focusing on the idea of how “actions and operations become thematizedobjects of thought or assimilation” (Tall, 2004, pages 282-283pages 282-283).Tall categorizes the cognitive development into three separate but integrat-ing developments, named as: conceptual-embodied world, formal-axiomaticworld, and Proceptual-symbolic world.

The conceptual-embodied world or embodied-world which is applied onany mathematical concepts which are perceived via the senses, such as ge-ometry (Tall, 2004, page 285). The individuals construct mental conceptionsof mathematical concepts by using their physical perceptions (Juter, 2006,page 411).

The second world, Proceptual-symbolic world is a world of ability to ap-ply symbols used in mathematics. In fact the symbols are the tools which

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switches the to do mathematics to think mathematics (Tall, 2004, page 285).In this world, individuals with procedural actions on mental conceptionswhich are acquired from the first world. The symbols represent both theprocesses and the concepts (Juter, 2006, page 412).

The third world is termed formal-axiomatic world. This world is basedon properties expressed in terms of formal definitions that are used as axiomsto a mathematical structure (Tall, 2004, page 285). Individuals arrive thisworld while their mental representations of concepts are formed and are ableto encapsulate the processes as objects and are able to go beyond the proceptsto the formal definitions (Juter, 2006, page 412). Other properties could bethen concluded by the formal proofs and a sequence of theorems could bederived (Tall, 2004, page 285).

2.5 APOS Theory

APOS, Action- Process- Object- Schema, theory is originated from Piaget’stheory of constructivism and addresses the mental construction of mathemat-ical knowledge. Piaget (Piaget, 1977) describes operations as actions whichare internalizable, reversible, and coordinated into systems characterized bylaws which apply to the system as a whole (page 456). He (Piaget, 1968) be-lieves that, “the operative aspect of thought deals not with states but withtransformations from one state to another” (page 8). Such operative pro-cesses, so called reflective abstraction by Piaget, have four characteristics; 1.They are internalized actions, 2. They are reversible, 3. They refer to somevariant, 4. they are part of a system of operations. Reflective abstractioninvolves action on an object and moves from action to operation and refersto “reorganization at the level of thought itself which is based on the coor-dination of actions and not only on an individual action” (page 9). In otherwords;

• an Action is a transformation of a mathematical object according tosome algorithm.

• mind’s reflecting on the action, creates an internal operation which isnamed as Process.

• when the mind distinguishes the process as a totality and becomesaware of it, a new Object is constructed.

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• Schema is a collection of Action, process, Object.

Regarding the concept of limits, these steps are worked out by Dubinsky(Cottril et al., 1996) for the concept of limit in what he calls a revised geneticdecomposition, as follows:

1. The action of evaluating the function f at a single point x that is con-sidered to be close to, or even equal to, a.

2. The action of evaluating the function f at a few points, each successivepoint closer to a than was the previous point.

3. Construction of a coordinated schema as follows.

(a) Interiorization of the action of step 2 to construct a domain pro-cess in which x approaches a.

(b) Construction of a range process in which y approaches L.

(c) Coordination of (a), (b) via f . That is, the function f is appliedto the process of x approaching a to obtain the process of f(x)approaching L.

4. Perform actions on the limit concept by talking about, for example,limits of combinations of functions. In this way, the schema of 3 isencapsulated to become an object.

5. Reconstruct the processes of 3(c) in terms of intervals and inequalities.This is done by introducing numerical estimates of the closeness ofapproach, in symbols, 0 < |x− a| < δ and |f(x)− L| < ε.

6. Apply to quantification schema to connect the reconstructed process ofthe previous step to obtain the formal definition of a limit.

7. A completed ε− δ conception applied to specific situations.

The outcome of considering infinitesimals as quantities which become smallerand smaller infinitely many times, versus the static view which considers theinfinitesimal a very small number, is a process of tending to a number whichis indeed the limit of the function. In this research, our stress is on thedynamic view of infinitesimals and consequently limits. Therefore, APOStheory could provide us a method for analyzing our qualitative data andverify if the students have arrived at the last step and if not why.

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2.6 What is Intuition

Since intuitively learning is discussed occasionally in this research, it wouldbe useful to investigate what is meant by intuition and what is meant byintuitive learning. The term intuition has several definitions which are notalways in accordance with each other but all of them have in common a kindof immediacy. Therefore, it is known as a non clearly defined term. Thereare some terms which are closely related to intuition such as insight, commonsense, self evidence, natural thinking. Fischbein (Fishbein, 1987) describesintuitions as cognitions which appear subjectively to be self evident, imme-diate, certain, global, and coercieve. Different interpretations of intuitionaccording to Fischbein (Fishbein, 1987) are:

• source of every true knowledge (Spinoza, Descartes)

• potentially misleading every quest for truth.

• a method or a sort of mental strategy which is able to reach the essenceof phenomena (Bergson). It is through intuition that we are able tograsp the very essence of living and changing phenomena.

• “a category of cognition which is directly grasped without or prior to anyneed for explicit justification or interpretation”. Spatial and temporal,empirical and operational, pure intuitions classified by Piaget.

• the faculty through which objects are directly grasped in distinctionto the faculty of understanding through which we achieve conceptualknowledge (Kant). He classifies intuition into intellectual and sensibleintuitions, but in the end he concluded that intellectual intuition doesnot in fact exist.

• a global guess for which an individual is not able to offer a clear andcomplete justification.

• an elementary common sense, popular, primitive form of knowledge asopposed to scientific conceptions and interpretations.

• (in pedagogy) is often related to sensorial knowledge as the first neces-sary basis for a further intellectual education. In this sense it is almostequivalent to perceptual knowledge

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Despite the fact that mathematics has a pure logical structure we cannotavoid the intervention of intuitive interpretations when coping with math-ematical situations. Intuitive representations and interpretations are activewhen we are striving to understand, to solve, to memorize or to create in thefield of mathematics. Therefore, we must suppose that many difficulties inlearning mathematics are generated by a lack of adequate intuitive represen-tations or by the distorting effect of poor, incorrect or inadequate intuitiveinterpretations.

The various descriptions of intuition are used as the sources, the methods,or the type of cognition (Fishbein, 1987, pages 3, 4). Intuition or intuitiveknowledge as a type of cognition is our desired interpretation of intuition.

As a definition we choose Piaget “a category of cognition which are di-rectly grasped without or prior to any need for explicit justification or in-terpretation (Fishbein, 1987). ” Intuitive knowledge could be a process inwhich students are having experiences (originated by a teacher or not) andnot yet putting it into words. We call the learning “more than intuitive” ifit is aiming at agreed formulations and consistency (like solving an apparentparadox).

Intuitions are not always correct knowledge. As suggested by us (2.1),just experiencing limit phenomena can bring various intuitions into students’minds which could turn into misconceptions of the limit concept as describedby Williams (Williams, 1991), unless these phenomena are explained “morethan intuitively”.

In the history of science, there was an endeavor towards rigor, which re-vealed in the positive and negative that the implications of intuitive knowl-edge. The fundamental effects of intuitive mechanisms appeared to the sci-entists and philosophers when trying to build the formal and deductive struc-ture of science.

2.7 Choosing relevant episodes from the history of cal-culus

Mathematicians from ancient times up to now have tried to match the mathe-matical thoughts with the physical world 1. Motion as one of the simplest andmost fundamental phenomena had been studied from different perspectives.

1The historical arguments in this chapter are inferred from: Boyer, (Boyer, 1959) “theHistory of the Calculus and Its Conceptual Development”

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Such inquiries usually raised problems, either philosophical or mathematical.One of the oldest of such problems and the most challenging one of its owntime was Zeno’s paradox. This paradox includes a set of problems whichcreates doubt about motion. Is there any motion at all? And if so, how? Dowe ever reach our destination?

Zeno addressed the contradictions in his famous set of problems (knownas the paradox of motion) that represented the same idea in different forms.Some of his famous problems are:

1. Achilles and the tortoise

In a race, the quickest runner can never overtake the slow-est, since the pursuer must first reach the point whence thepursued started, so that the slower must always hold a lead.Aristotle, Physics VI:9, 239b15, (Aristotle, 0 BC)

The quickest runner is indeed Achilles and the slowest one is the tor-toise. Achilles and a tortoise have a race. Achilles, being much faster,allows the tortoise a 10 meter head start. They start, and Achillessoon covers the distance to the point where the tortoise started from.At that time, the tortoise will have moved ahead by half this distance,to a new point. At the time Achilles reaches this point, the tortoisewill again have moved ahead by half this distance, and by the time hereaches that point the tortoise will have moved ahead by half that dis-tance, and so on and so on. Therefore, according to Zeno, Achilles nevercatches up with the tortoise because each time he reaches one of thesepoints where the tortoise was, the tortoise has moved ahead. However,the tortoise is no longer there, its continued ahead and Achilles mustagain catch up. Having always to catch up this distance, Achilles willnever catch the tortoise!

2. Arrow paradox

If everything when it occupies an equal space is at rest, andif that which is in locomotion is always occupying such aspace at any moment, the flying arrow is therefore motion-less. Aristotle, Physics VI:9, 239b5, (Aristotle, 0 BC)

Zeno’s arrow paradox appears to show that motion is impossible. Inthe arrow paradox, Zeno cuts the time into infinitely small pieces. He

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splits the arrow’s movement into infinitely many still pictures takenwith a camera. He states that for motion to be occurring, an objectmust change the position which it occupies. He gives an example ofan arrow in flight. He states that in any one instant of time, for thearrow to be moving it must either move to where it is, or it mustmove to where it is not. It cannot move to where it is not, becausethis is a single instant, and it cannot move to where it is because itis already there. In other words, in any instant of time there is nomotion occurring, because an instant is a snapshot. Therefore, if itcannot move in a single instant it cannot move in any instant, makingany motion impossible. This paradox is also known as the fletcher’sparadox as fletcher being a maker of arrows.

3. Dichotomy paradox

That which is in locomotion must arrive at the half-way stagebefore it arrives at the goal. Aristotle, Physics VI:9, 239b10,(Aristotle, 0 BC)

In a classroom, the guided questions about the half-way paradox could makethe students first doubt about motion just like the way the ancients did, thenencourage them to explain it. The challenge that this question provokes instudents’ minds could stimulate them to develop an intuition of concepts likecontinuity, infinity, infinitesimals, and converging.

Although these paradoxes have a philosophical side, the modern answeris based on the mathematical limit concept. For several centuries peoplemainly gave philosophical answers to Zeno’s challenges. The current mathe-matical answer was only found recently in the period starting with Newtonand ending with Cauchy, Weierstrass, Dedekind, etc. In motion we are deal-ing with continuous change, while the numbers were known to be discreteor discontinuous. Zeno’s statements of the paradoxes were first provided byAristotle, and thus studied by him in his physica which is known as an actualtreatise on the differential calculus (Boyer, 1959, page 43). In physica, he as-serts that “nothing can be in motion in a present ... Nor can anything beat rest in a present”. Indeed Aristotle denied the existence of instantaneousvelocity because it was not perceptible for him. Instead Aristotle believedthat the world is known through the senses and he regarded mathematics asa pattern of this world. If only he had been able to see the “pattern” of aconverging infinite sequence then instantaneous velocity could have fitted his

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views. But not having the concept of limit, this pattern could not be seen inany exact sense and was explicitly denied.

Starting from Zeno, the next episodes recognizable as steps towards mod-ern calculus are realized by Archimedes and Newton. They can be regarded asthe inventors of the methods of exhaustion and fluxions respectively. Whereexhaustion was the initial stage of integration, fluxions was the essence ofthe modern method of differentiation and integration. The concept of limitis the fundamental notion implicit in both of them as the main parts of thecalculus.

Regarding the development of calculus, Archimedes was the greatestmathematician of antiquity. He proved various results about areas and vol-umes that we today would prove using the integral calculus. He was mainlyinvolved with finding the areas and volumes. What made him distinguished,was his different approach for giving a deductive geometrical demonstration.He found the results by mechanical methods in advance. For instance hecalculated a surface as the sum of the polygons inscribed in that surface, byhis mechanical law of lever (Heat, 1912)1. For instance he knew through me-chanics that the area of a parabolic segment is 4

3rd of the triangle having the

same base and vertex. So, by knowing what he should find before hand, heapplied his geometrical method of exhaustion. To find the area of a curvedfigure, eg. a circle, using the method of exhaustion, in most cases, a regularpolygon with a known area was inscribed inside and the same polygon withdifferent size was circumscribed on the curved figure. Then by increasing thesides of the inscribed and circumscribed polygons, their areas converged fromboth sides to a specific number which is indeed our desired area. Because thenumbers found by this method were based on a finite number of approximat-ing steps, these numbers are approximations of the real area. Archimedesused this method for finding the area of the circle.

Figure 1: Method of exhaustion for finding the area of circle

1This is supposed to be part of general scientific knowledge, therefore, is not explainedhere.

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For finding the area of a parabolic segment, Archimedes used the samemethod of exhaustion but slightly different known as “the quadrature of theparabola”. This time, he only inscribed a specific polygon (a triangle) insidethe parabolic segment and by increasing the number of triangles, he createda sequence of triangles. The sum of this sequence gave an approximate num-ber for the area of the parabolic segment.

Figure 2: Quadrature of the parabola

What he did resembled the technique of integration in the modern cal-culus but yet quite different. As a similarity, we can mention that both inRiemann integration and in Archimedes method of exhaustion, our desiredanswer is found by the sum of the sequence of areas. The sequences are theareas of the rectangles in the first and the areas of the triangles in the secondmethod. However in Archimedes method, the concept of limit is embeddedin the sum of the surfaces. It is not correct to assert that what he did was in-tegrating. Because in integrating, the definite integral is defined as the limitof the sum of infinite sequences (which are the areas of the Riemann rect-angles) and not the sum of the infinite sequences of the surfaces (which arethe areas of the triangles in this case) (page 50). In Riemann integration allthe rectangles get smaller, while in Archimedes method of exhaustion of thequadrature of the parabola, all (triangles) can stay as they are, only smallerand smaller ones are added to them. His method made a fundamental de-velopment in the limit concept though. So, setting an activity based on hismethod of exhaustion could help us give a better intuition of the concepts ofinfinity, infinitesimals, and converging.

Calculus is rather a new discipline in mathematics in comparison with theother disciplines such as geometry, algebra, etc. The development of the con-cept of limit remained more or less at the same level until 18th century whenNewton and Leibnitz introduced calculus or calculus of infinitesimals. Duringall these centuries, mathematics developed and fertilized a lot but more in its

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other disciplines. For instance the analytical geometry of Descartes in 17thwas a fundamental improvement for the whole of mathematics. Also Fermatand Barrow made important contributions but yet insufficient to create amain step towards calculus.

Newton and Leibnitz are both known as the inventors of the calculus.Although they arrived both at the same conclusion, they had two differ-ent approaches. Newton followed the kinematic method pursuing Plato,Archimedes, Galileo, Cavalieri, and Barrow. Leibnitz followed the atom-istic method, pursuing Democritus, Kepler, Fermat, Pascal, and Huygens.What made Newton different was his use of infinite series in his method offluxions.

He was the first man to give a generally applicable procedure for de-termining an instantaneous rate of change and to invert this in the case ofproblems involving summations. The method of exhaustion which was usedbefore him had the opposite direction of the method of fluxion that Newtonused. For instance, the mathematicians before Newton, found the area as awhole through the addition of the infinitely small elements. While Newtonfound the area from its rate of change at a single point (Boyer, 1959, page193). He seemed to be a follower of Galileo in accepting the concept of ve-locity as a continuous motion of points. The concept of infinitesimal enteredpersistently after Newton’s contributions to calculus. According to Newton,variable quantities are made up of the continuous motion of points, lines,and planes, rather than as the sum of infinitesimal elements. This fact is stillin debate between the mathematicians and has caused the different dynamicand static views of the limit notion. The dynamic view of infinitesimal whichcomes from Galileo’s moment, describes a small quantity as a small quantitywhich becomes smaller than any other small quantity but at the same time,never becomes zero. The static view of infinitesimal which comes from thestatic form of Cavalieri’s indivisible, describes an infinitely small quantity asa fixed small quantity which is very small but not zero (Boyer, 1959).

Before Newton, infinite series were studied in connection with the geo-metric representation of variability or in infinite progressions. In developingthe method of fluxions, Newton used infinitely small, both geometrically andanalytically. With such a dynamic view of infinitesimals, Newton found theinstantaneous velocity of a mass by finding the rate of change of its position.He divided the time into small intervals and found the average velocity forsuch time intervals. By considering the “ultimate ratio” when such intervalgets smaller and smaller, we become closer and closer to the instantaneous

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velocity. Eventually he neglected the infinitely small term in the calculations.Newton himself seemed to feel some need for the limit concept(Boyer).

In this method, he used infinitely small rectangles with the abscissa xand ordinate y and the function y = ax

mn , where m,n ∈ N . He also knew

that the area under this function is z = ( nm+n

)axm+n

n before hand. Whatmade his method fundamental, was to arrive from the area to the functionitself. For this purpose, he assumed that the abscissa increases a moment o,so it becomes x + o. Then according to such an increase, the area will bez+oy = ( n

m+n)a(x+o)

m+nn . He then applied the binomial theorem and ignores

the terms containing o. The result is that when the area is z = ( nm+n

)axm+n

n ,

the curve is y = axmn or conversely. In contrast to previous mathematicians,

he did not add the infinitely small areas and the limit of their sum to findthe area. But, he found the area by finding its rate of change at any momentwhat we now call the indefinite integral of the function. In fact he gave ageneral method for determining the instantaneous rate of change and also toinvert such problems to the problems involving summation. In this method,given the relation of quantities, we can find the relation between them whenone of them changes. Limit appears in this method when the change which isalways a ratio, becomes infinitely small and approaches zero. Infinitely smallsfor him were the small quantities which are neither finite nor yet preciselyzero. Although Newton’s method suffers from some mathematical problemssuch as omitting the terms containing o and was not complete (as he himselfwas aware of it), it is worthwhile to be mentioned and probed with studentsas a fundamental improvement in calculus.

Although the role of Leibnitz in developing calculus created the sameresults as Newton, their different approach made us to choose one of theirmethods. Leibnitz was one of the founders of formalisms and opposed to theintuitionism which is found in Newton. He believed that if he formulated therules clearly and appropriately, the result would be reasonable and correct.His view about infinitesimals was different with that of Newton’s. He believedthat the infinitely small quantities resemble the infinitely many of animalculeswhich the microscope had disclosed. Therefore, his views can be seen as apredecessor of A. Robinsons (Bagni, 2005) non-standard analysis, in whichinfinitesimals are thought of as fixed static numbers. On the other hand,Newtons views are close to the ε− δ definition of limits in which both ε andδ are variables that we can easily imagine as getting smaller. Newton as ascientist was satisfied with the science which he had found in the motion and

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the notion of velocity. While Leibnitz as a theologian and scientist preferredto find the mathematics in the differential.

To summarize, the steps chosen from history are:

1. Zeno’s Paradox

2. Archimedes’s quadrature of parabola

3. Derivative activity inspired by Newton’s effort to find the instantaneousvelocity

3 Research Questions

Our diagnoses of current students understanding of limits (2.1) led us tolook towards history as a source of steps to use in the teaching of limits,which was one of the new aspects of this research. Particularly, it led usto consider a dynamic interpretation of infinitesimals, as an important stepforwards understanding the modern definition of limits. Another new aspectis the comparison of the common models of limit held by students (recognizedby Williams (Williams, 1991)) with the cognitive conflicts (studied by Talland Vinner (Tall and Vinner, 1981)) and combining them together. Also toinvestigate the situations when such models could be interpreted as correctmodels. Another new aspect in this research is to lead students to reach tothe necessity of a new concept which later is named limit concept. Apart fromthe steps taken from history, an emphasis on the geometrical interpretation ofneighborhoods was used to lead students towards understanding the modernlimit concept (see sections 4.1.2 and 4.1.6 for explanation) . We thereforeask the following research questions:

1. What steps seen in the history of the limit concept are worth repeatingin a guided discussion if we want to lead students to the modern conceptof limit?

2. Will an emphasis on understanding the notion of infinitesimals, usingthe steps found in question 1, help students overcome the commonconflicting images, regarding the limit notion

(a) Which of the common misconceptions are more/better tackledvia the above suggested method and which of them still appear toexist in students’ minds after teaching and why?

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3. Will an emphasis on understanding the notion of infinitesimals, usingthe steps found in question 1, and the emphasis on geometric interpre-tation of neighborhood, help students reach the informal and formalconceptions of limits?

(a) Do students show understanding of a geometric interpretation ofthe ε and δ- neighborhood?

(b) Do they understand the (in)formal definition?

i. Can they formulate the definition of limit in their own wordsafter some weeks?

ii. Can they recognize the correct definition of limit?

iii. Can the students apply the formal and/or informal definitionsof limit in the new concept, continuity?

(c) Can students use the notion of infinitesimal to reason about thespecial cases?

A tentative answer to question 1 has already been given, based on ourunderstanding of the history of calculus (2.2). However, this answer stillneeds to be tried out in a teaching experiment. We therefore state it as ahypothesis:

We estimate that repeating the following steps from history in a guideddiscussion will help students to reach the modern conception of limits: 1)Zeno’s paradox of motion, 2) Archimedes’ paradox in quadrature of parabola,3) Newton’s conflict in finding the instantaneous velocity.

Question 3 contains a second hypothesis namely that a geometrical inter-pretation of neighborhoods will help students reach to the (formal) concep-tion of limit. This idea has not been introduced in chapter 2 but details canbe found in 4.1.2 and 4. 1. 6.

4 Methodology

To answer the above questions and test the hypotheses, I picked some criticalproblems which were important in the development of the limit notion andwere described in (2.2) (Boyer, 1959; Burn, 2005). Then some activities weredeveloped in a way to satisfy my pedagogical expectations. I would try tohelp students build the concept in their minds with guiding questions. The

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questions must deal with students’ common misconceptions and must aim atthe desired conceptualization of limit. These lesson plans are applicable togive a better intuition of the limit concept.

The intervention made in this research is explained in the next parts(4.1). At the same time it will be explained how the chosen phases from theevolution of the limit notion were made to fit into the teaching plans andteaching goals.

The collected data during and after implementing these activities will beexplained in chapter 4.4. In 4.5, we will describe how we checked whether ornot, and to what extent, this teaching intervention is useful for students toget a more accurate and deeper knowledge of limit concept.

Some researchers like Barbin (Barbin, 2000) and Lakoma (Lakoma, 2000)believe that the nature of such investigations fits only with qualitative re-searches. “For assessing the effectiveness of using history of mathematics inmathematics education, a qualitative analysis seems to be more useful than aquantitative approach” (Lakoma, 2000). This study will be a case study giventhat only 3 students participated in the intervention. This small number iscaused by the chosen setting: an optional course, aimed at mathematicallyinclined students from one school (see section 4.3). So, a qualitative researchis inevitable.

4.1 Teaching Plan

In accordance with the teaching goals (4.1.1) and teaching strategies (4.1.2),and also regarding the historical steps which are recognized as the essentialones in developing the concept of the limit, some lesson activities are designedfor students.

These critical problems include three questions; Zeno’s paradox (Ap-pendix 1), the quadrature of the parabola (Appendix 2), and the paradoxthat Newton met while calculating the instantaneous velocity (Appendix 3).

The use of these critical problems will be explained in 4.1.3 to 4.1.5, andafter that we explain how we arrived at the modern limit concept on thisbasis in 4.1.6 and 4.1.7.

4.1.1 Choice of Teaching Goals

We choose an informal understanding of limits as our main goal. The aimedat informal understanding of limit can be defined as follows:

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A limit is a number that the y-values of a function can becomeas close to as we want, by choosing x within an x-interval that isas small as needed

In comparison, the formal definition is as follows:

limx→a

f(x) = L

means

∀ε,∃δ; 0 < |x− a| < δ ⇒ |f(x)− L| < ε

Although understanding the formal definition of limit is not the main goal ofthis research, it is quite important for us to check how does giving a deeperintuition helps to understand the formal definition. Informally knowing theconcept of limit is necessary but not sufficient for the students who wouldchoose mathematics as their main academic discipline later. What makesthe conceptualization more complete is understanding the formal definitionas well.

4.1.2 General Remarks about Teaching Strategy

After building an intuitive understanding based on history, the step towardsthe modern definition could proposedly be made by achieving a geometricalunderstanding of the prerequisites of the so called Weierstrass definition oflimit. These geometrical understandings include the geometrical interpre-tation of absolute value and neighborhood on the real line and Cartesiancoordinate system. The formal definition of limit could then be a consequentof the neighborhood approach (4.1.6).

In designing the preparatory activities and questions taken from history,it is important that limit be not directly or indirectly suggested. The mainaim is to help students feel the lack of something which is then finally called“limit”. This teaching method is also named Problem-Posing education byKlaassen (Klaassen and Lijnse, 1996).

The interactive dynamic software Geogebra is often used in the activitiesin order to help us and the students to visualize the problems and simplifythem. Although, it is not really useful for teaching the concept of limit itself,it will be used for illustrating the questions. Software is naturally not appli-cable for understanding the limit notion itself because tending to a number

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does not have the same meaning in such software that it has in mathemat-ics. The software approximates the numbers at some point with a specificaccuracy which is exactly one of the expected students’ misconceptions (see2.1). So Geogebra was mainly used for understanding the problem but notfor representing the limit notion as the solution.

4.1.3 Zeno’s paradoxes

Zeno’s argument touches concepts of continuity, infinity and infinitesimals(Cajori, 1919). Main purpose of using this paradox is creating a challengein students’ minds. The problem involves an unlimited division of a limitedcontinuous quantity. This division is presented in such a way that it becomeslogical to expect that the result of dividing a limited quantity infinitely manytimes would become zero, while it never becomes zero according to mathe-matics.

The activity designed in this research is based on the dichotomy problem.That is walking from point A to point B with the condition that with eachstep, we pass half of the remaining distance. If we take AB = 1, then theremaining distance after the nth step would be 1

2n. By increasing the number

of steps, the remaining distance decreases and becomes very close to zero, butit never becomes zero. But, we know that when we start from point A towardpoint B, when we walk normally, we will reach point B and our remainingdistance becomes really zero. But then, walking normally, we should havepassed an infinite number of intervals, each being half of the previous one.So, how can we explain this situation?

This is the first created challenge in students’ minds. We can say thatthe remaining distance becomes eventually (its limit becomes) zero. But itis supposed that in the activities and discussions in the class, there would beno pointing to limit in order to give this choice to students to feel the lackof it.

Some of the concept images of the limit notion such as: a limit is anumber or point the function gets close to but never reaches, or, a limit isan approximation that can be made as accurate as you wish, are traceable inthis challenge. Questions 4, 5, and 6 of the Zeno activity (Appendix 1) aredirectly designed for tackling these misconceptions. Before checking how thisidea contributes to getting rid of the misconceptions about the modern limitconcept, one issue is important and should be mentioned. All the William’slist of misconceptions are made from student’s mistakes about the meaning

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of the limit of a function while in Zeno we are speaking about a series. Aseries is defined from N to R; N → R. For instance 1

2nwhere n can be

1, 2, 3, .... Despite of this difference, Zeno is still an appropriate questionbecause our aim is to create the question mark in students’ minds togetherwith addressing some of the common conflicting concept images. But, assaid before, the main purpose of this activity is creating a question mark inpupils’ minds!

4.1.4 The Quadrature of the Parabola

Following the history, one of the concerns of the ancient mathematiciansinvolved finding areas and volumes. The method used for finding areas wasthe “method of exhaustion” as introduced in 2.2.

The quadrature of the parabola is preferred by us to the quadratureof the circle because in the second one, we would become involved withtrigonometric ratios. While in designing the activities, We decide to focus onproblems related to the limit concept. Therefore we decide to evade unrelateddifficulties that might be caused by trigonometry.

The method of exhaustion used in the quadrature of the parabola (Ap-pendix 2) is to find the area of a parabolic segment by inscribing a specifictriangle in it and then increasing the vertices and sides of the triangle to apolygon with a potential infinity of sides and vertices (see 2.2). Here againwe are confronting infinitely small quantities which never become zero: theremaining area between the polygon and the parabola. On the other handArchimedes has proved by geometrical proofs that the area of the parabolicsegment is exactly 4

3rd of the area of the first triangle (see 2.2). It means that

we know that the remaining area between the polygon and the parabolic seg-ment should become absolutely zero while on the other hand it should not.Again we need to know what is infinitely small and how does it influence ouranswer. Infinitesimal is the concept that later, after Newton and Leibnitz,became the base of calculus.

Although Archimedes method of exhaustion is one of the main pointsin the evolution of the limit concept, relying on it for teaching the limitconcept is not sufficient (Boyer, 1959). The reason is that in this problem,Archimedes and generally ancient Greeks did not know infinity as we knownow. He was applying the sum of a finite sequence in the quadrature of theparabola. The limit concept requires the sum of an infinite series. Someresearchers like Bagni (Bagni, 2005) believe that using only the method of

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exhaustion is a superficial way of teaching limit. I must say that I agree withhim. The method of exhaustion is part of our designed route to the limitconcept, not all of it.

This problem again creates a challenge in students’ minds when on theone hand, by calculation they find the area between the parabolic segmentand the polygon, and they see that this difference does not become zero. Onthe other hand this difference should be zero in the end. Geogebra is veryhelpful in this activity because students can inscribe the triangle and increaseits vertices and sides and observe that the difference between the polygon andthe parabola does not become zero. By zooming in on the graph, they canobserve that there is always a remaining distance between the parabola andthe polygon. Although, as mentioned before, it does not work the samefor the very big numbers, in other words, when the number of vertices istending to infinity: in Geogebra you don’t come close to infinity. So are wespeaking about approximating or are we missing something that helps usexplain this situation? This is the challenge for the students. Some of thestudents’ common conflicting concept images are traceable in this activitysuch as: a limit is determined by plugging in numbers closer and closer toa given number until the limit is reached, or, a limit is a number or pointthe function gets close to but never reaches, or, a limit is an approximationthat can be made as accurate as you wish. Questions 7, 8, and 9 are directlyaddressing these misconceptions.

4.1.5 Derivative Activity

The next important step to try with the students is Newton’s contributionto calculus. Newton’s problem has even less philosophical sides than thequadrature of the parabola and Zeno. Newton used the method of fluxions(see 2.2). Although his method in finding the instantaneous velocity is dif-ferent with ours, both of the methods end up in the same paradox and havethe same essence, ignoring the moment and counting it as 0. In his method,he investigated the instantaneous velocity by knowing the distance traveledS(t), which is the area under the curve of the velocity function V (t) = S ′(t),and then finding the function V (t) itself. He knew that the instantaneousvelocity is the slope of the tangent line to the curve of the motion, S(t). Healso knew that this could be found by finding the derivative of the function.That is, he knew the integration of the function, and he knew also that suchan integration is created by a rate of change at any moment, then by ap-

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plying the algebraic calculations, he arrived at the main function itself. Hehad thus an equation, in which he ignored an infinitely small number fromone side of it. The problem was solved numerically correct but somethingwas missing and he himself knew that something should be wrong. We didnot follow his method strictly mainly because we were hesitating to speakabout the integration which is an after “limit” and “differentiation” subjectfor the students. Students had already seen the limits symbol in applyingderivatives, without being told its exact meaning. But, the main question,finding the instantaneous velocity, and the paradox are the same as Newton’s.We changed the problem a little bit in order to, firstly, not to speak aboutdifferentiation and integration before speaking about limit and secondly, tosimplify the problem as much as possible.

Our activity based on Newton’s method for finding the instantaneous ve-locity is designed to be as simple as possible (Appendix 3), and as close aspossible to what students already know about differentiation. In fact it is ashort refresher on derivatives, but with emphasis on the problematic part: thelimit concept. It begins with finding the slope of some lines passing throughtwo points. The main purpose of these questions is to review the concept ofsteepness of a line which represents the ratio of rise

runof any line. Then the

students will be asked to answer some questions about a falling mass. A masswhich falls from the height 16km from the earth and according to the functionf(t) = 16− t2

200. Where t is the variable, time, and f(t) shows the height of the

mass at a given time which is indeed a parabola. The question is to find theslope of a given line that intersects the (height-time) graph of the mass in twopoints and then calculate its slope. Students have already understood thatthe slope of such a line is ∆f(t)

∆t. From physics they know that the slope of this

line represents the average velocity of the mass. For this specific question,the main purpose is finding the instantaneous velocity in the 20th second.So, the questions are finding the slope of the line (average velocity) betweent = 20, t = 40, and then between t = 20, t = 25, 21, 20.02; 20.001. The aver-age velocity becomes consequently −0.3,−0.225,−0.205,−0.201,−0.200005.Students might observe that by becoming closer and closer to the 20th sec-ond, the average velocity (the slope of the line), becomes closer and closer to−0.2. So following the Derivative Activity, to find the instantaneous velocity,the students are asked to find the average velocity between two points thatdiffer by only a moment h. As it is mentioned before, it is the slope of theline passing from t = 20, t = 20 + h. The answer becomes −40×h−h2

200×h . Forsimplifying this fraction and dividing its nominator and denominator to h,

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we should put the constraint h 6= 0. So, this is part of the paradox, h can notbecome zero. On the other hand we know that in a movement, any momenthas its specific velocity which might or might not differ from the velocity atthe other moments. In other words, daily intuition says that instantaneousvelocity exists at any moment. This fact completes the other side of thisparadox. That is, we do not have any constraint for tending to t = 20 asmuch as we want, so indeed h could become 0. Here again students meeta paradox that brings the idea of some misconceptions such as approximat-ing or tending and not reaching, in students’ minds. The average velocitybetween two points while one of them is tending to the other one but cannot reach it, is dynamic. If such a point could “eventually” reach the otherpoint, then the average velocity between them would become equal to theinstantaneous velocity at that point! How can we explain this situation? Theanswer to this question might need pretty much discussions in the class. Inconstructing this activity, the dynamic nature of Geogebra is again useful.The students can move the line which represents the average velocity andobserve that it tends to become the tangent line to the curve at the t = 20sec.

Questions 13, 14, and 15 are designed for discussions about whether wecan reach t = 20sec, is our final answer an approximation? etc. Afterthis activity, we will start discussing about limit and infinitesimal for thefirst time. The questions in this activity are designed to lead the studentsfrom simple concepts like the slope of the line and average velocity to thecomplicated concepts such as limit and infinitesimals.

The dynamic nature of the points tending toward each other, the waystudents plug in numbers for finding the average velocities in the questions,etc, seems to be almost appropriate for eliciting students’ common miscon-ceptions.

After this activity, students observe that the average velocity betweentwo points becomes eventually, or its limit becomes, equal to the momentaryvelocity. In the previous activities, they have seen the concept of tending.But our approach which is inspired by Newton’ approach seems to be moreintuitive because it could be more tangible for students. They see that thelimit concept is independent of having a constraint for approaching to a point.It is possible that we can’t reach a point but the function can have a limit atthat point. This activity is supposed to give an intuitive perception of limitwhich might seem to be enough for the secondary level. After introducingthe “limit” notion, we will turn back to the previous activities and will try

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to answer them with this new explanation. The main difference between thisproblem and the two previous problems is that Newton was challenged bylimits concerning position as a function of time. But Zeno and Archimedeswere challenged by the limits of sequences. In the first two activities, we cannot pass the limit values: we can not pass point B in the first activity andwe can not pass the area of the parabolic segment in the second one. Whilein Newton, we have a motion and we can get past the limit of the functionat a moment (cf. Wiliams’ misconception nr. 4 as cited in 2.1).

Below is a summary of the different cases of limit of functions and limitof sequences:

Independent variables Function f(x), x ∈ R Series f(n), n ∈ Nn or x→ a Newton case Can not existn or x→+

− ∞ Can exist but we don’thave it.

Zeno and Archimedescases.

4.1.6 Neighborhood

But in order to transform this intuitive idea of a limit notion to a formalone, we need some more mathematical knowledge about properties of realnumbers. What we need more, is the geometrical interpretations of intervals,absolute values, and neighborhood. The main purpose of the following ques-tion is to lead students to understand what do |x−x0| < δ and |f(x)−L| < εmean geometrically as a preparation for understanding the formal definitionof limit. Therefore one session is allocated to practicing and refreshing theseconcepts (Appendix 4). Students are first asked to answer some preliminaryquestions about the geometrical interpretation of absolute values on the realline that each correspond to the distance of a point to the origin 0. Ques-tions of this activity are again set in a simple to less simple order. Therefore,the questions are started by finding the set of points on the real line whosedistance from the origin, 0, is a. Students should first distinguish how towrite this sentence in mathematics terms, |x| = a, and then recognize allsuch points on the real line. The next questions involve finding the set ofpoints on the real line whose distance from a is b which is |x − a| = b. Af-terwards, the questions comprise inequalities. For instance; find the set ofpoints on the real line whose distance from 1 is less than 4 which algebraicallyis: |x− 1| < 4. In such questions, they practise to illustrate the inequalities

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on the real line, the intervals. They are then introduced with the new symbolBr(x0) = {x||x − x0| ≤ r}, the neighborhood which corresponds to all thepoints whose distance from x0 is less than or equal to r. Understanding theformal definition of limit requires mastering the application of the absolutevalue and inequality symbols. Therefore, they will be asked more questionsabout neighborhood, but this time in the cartesian coordinate system, notjust on the real line. That is, a relation between x and y will be given, andits geometrical interpretation would be asked. For instance: How do youinterpret the following relation in term of distances?

y ∈ Bε(3)⇒ x ∈ Bδ(1)

which corresponds to

|f(x)− 3| < ε⇒ |x− 1| < δ

and then checking how do δs vary when the εs change. It means ε is given,and δ are asked. In these questions the neighborhood on the y axis is givenand the neighborhood on the x axis is asked. The reason is that the valueson the y axis are the dependent variables and in the formal definition oflimit, indeed we search for the x values or independent variables which fulfilthe formal definition. So, however in the formal definition, the premise is|x−a| < δ, and the desired conclusion is that |f(x)−L| < ε, in fact we mustfind the δ neighborhood which legitimates the truth of the ε neighborhood.That is, in the formal definition, we have ∀ε, ∃δ such that 0 < |x − a| <δ ⇒ |f(x)− L| < ε, indeed we are searching for the x values which fulfil therelation. Therefore, the neighborhood questions are asking directly how do xneighborhood change when y neighborhood changes. For instance, “find thebiggest neighborhood of 2 that catch all the x values that have their f(x) inBε(1)”, which in neighborhood symbols is: y ∈ Bε(1)⇒ x ∈ Bδ(2).

Then they are asked to check δ, when ε changes; Working on these ques-tions, students are expected to gain the geometrical knowledge which isprerequisite for understanding the limit notion. The neighborhood activityguides students to one step before the formal definition of limit.

4.1.7 Working Towards Formal Definition

The final activity is introducing the formal definition of limit which is in-deed connected to the neighborhood questions together with what we had

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before learned about tending and approaching in the previous sessions. Be-fore starting to find the relation between εs and δs, we plan to review all theprevious three activities and provoke discussions about them with the newconcept, limit. Starting to find the relation between εs and δs, a very simplefunction such as f(x) = 2x+ 1 when x→ 1 seems preferable from the pointof view of illustrating the difficult definition in the easiest way possible. Thequestions start by giving a neighborhood for y and asking the correspond-ing neighborhood for x. That is, in the relation y ∈ Bεf(x) ⇒ x ∈ Bδ(x),finding the δs for given εs. For instance, when y ∈ B0.2(3), then x needs tobe in B0.9(1). Or when y ∈ B0.1(3), then x needs to be in B1.05(1) and soforth. But, this process can not last for ever and we need to write a logicalrelation that guarantees the association between εs and δs like: δ = 1

2ε in this

case. Now is the time to try to elicit the formal definition of limit togetherwith students. In the same process also the informal definition of limit wasformulated. Although it is not expected that students be able to write thedefinition themselves, it is expected that they understand all that is writtenin the definition.

4.1.8 Epsilon-delta proof & Mathematica

This workshop took one afternoon and had the following design 1:

1. Recapitulation of the problems (Zeno, Archimedes, Newton) that thelimit concept is designed to solve, and of the definition of limit (informaland formal) that should solve those problems.

2. An example application of the formal (epsilon-delta) definition to anobvious case:limx→2(2x) = 4Demonstration by the teacher with student participation.

3. Introduction to the Mathematica software.

4. Application of the definition of limit to various problematic cases, bythe students, partly on paper, partly with the software (they couldchoose a case).

Below we’ll elaborate on the first two of these steps.It was planned to connect the Zeno problem to the informal definition of

1See appendix 5 for the detailed lesson plan.

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limit as part of step 1 (see Figure 1):

Distance walked Σnk=1

12n

= 12

+ 14

+ 18

+ ...+ 12n

= 1− 12n

“the remaining distance is halved each time”

there is a number that we are getting closer and closer to, we can get asclose as we want by increasing n, but we can not reach it in this case

limx→∞

(1− 1

2n) =? 1 ← exactly?

is there another number, close to 1, about which we could say the samething?- Imagine a number a bit more to the right... can we get as close as wewant...?- Imagine a number to the left ...Question for today is: can we prove such a limit?

• can we prove that there is exactly one number that we can getas close to as we want, by making n closer to infinity.

Figure 1: lesson plan: connecting the Zeno problem to the informaldefinition of limit.A dialogue was planned, the idea was to challenge the “limit is an approx-

imation” idea: is there another number close to 1 about which we can say thesame? It was expected that by trying various possibilities (1+ 1/1000, 1+ amillionth, ) students would come to the conclusion that no number except 1can be approached “as close as we want” by the Zeno sequence.

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The formal and informal definitions presented to the students are asshown in Figure 2. They are almost equal to the definitions as shown insection 2.1, except:

• Wolter omitted quantifier symbols and wrote “for every epsilon thereis a delta” in words

• Layout was used to make the versions easily comparable

Informal Definition

- The limit of a function f(x) for x→ a exists and is = LIf and only ifwe can make the difference between f(x) and L as small as we wantby making x as close to a as we need.

Formal Definition, Version 1.

- The limit of a function f(x) for x→ a exists and is = LIf and only iffor every neighborhood Bε(L) there is a corresponding Bδ(a) suchthat:x in Bδ(a)⇒ (“ensures”) f(x) in Bε(L)

Formal Definition, Version 2.

The limit of a function f(x) for x→ a exists and is = LIf and only iffor every ε > 0 there is a δ such that:0 < |x− a| < δ ⇒ (“ensures”) |f(x)− L| < ε

Figure 2: Lesson plan: Informal and formal definitions of limit (someconnecting reasonings left out).

With the definitions on the board, a first epsilon delta proof would bedone by the teacher with student participation. The plan for this part wasas follows:

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• Introduce the case, and emphasize that this is not one of the problemcases like Zeno. Lets see if the complicated definition can cover alsothis obvious case.

• Ask for proposals to calculate a fitting delta, for a given epsilon.

• Ask whether a bigger or a smaller delta would also do.

• Write the choices made on the board in the format: “assumptions” and“to prove”

• Ask for proposals how to get from the assumptions to the desired con-clusion.

The completed proof would look like shown in Figure 3. It can equally wellbe done with a smaller delta (e.g. δ = ε

3), and one of these variants would

be tried.

Assuming:

0 < |x− 2| < δδ = ε

2

⇒ 0 < |x− 2| < ε2

To Prove:

|f(x)− L| < ε|2x− 4| < ε

Proof:

|x− 2| < ε2

we assumed it2|x− 2| < ε why is it good?|2x− 4| < ε why is it good?We proved that for all ε > 0 we can make this distance smaller than ε.

Figure 3: Lesson plan: A first epsilon-delta proof

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After this preparation, it was expected that students could try variousmore difficult epsilon-delta proofs, partly relying on the software, but relyingon themselves in planning and setting up the proof.

Among the cases to try were non-obvious ones. There would be too littletime to try all the prepared cases, so students could choose a case.

4.1.9 Continuity

In the beginning the conditions for a function to be continuous, having equallimits from left and right equal with the function value in that point, areintroduced. In this activity, the formal mathematical language of limit ismore often applied in order to let the students be more used to it and beencouraged to apply it. At the start some general properties of limit areintroduced; the general rule for the sum, subtraction, product, and quotientof the limits of two functions. Then the general form of the polynomial isreviewed and the continuity of the polynomial of the first degree is provedformally by the teacher.

The next step is to solve the problems that usually include the indetermi-nate or indefinite cases which are to be studied in their discontinuous pointsor intervals. The l’Hopital rule which is introduced for finding the limit of theindeterminate problems is like a simple key which solves all such functions.It could prevent students to understand the core of what they do during theprocess of finding the limit of such functions. Therefore, a thorough discus-sion about the difference between infinitesimals and absolute zero is planned.The discussions are about five fraction form functions by focussing on theconcepts of infinitesimal and absolute zero. The five mentioned cases are:

• Defined, for example: limx→1x2+2x+1

• Undefined, for example: limx↓0x

[x]

• Absolute zero, for example: limx↓0[x]x

• Infinity, for example: limx→∞x2−1x−1

=+− ∞

• Indeterminate, for example: limx→1x2−1x−1

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Where [x] is the “Floor Function”1, R → Z that adds to each real numberthe biggest whole number that’s below it. (e.g. [0.6] = 0, [1.7] = 1, [-2.1] =-3, etc.) .

4.2 Comparing our plan to APOS genetic decomposi-tion

Comparing my teaching plans’ objectives with Dubinsky and Mcdonald’s(Dubinsky and Mcdonald, 2001) objectives in establishing the levels of cog-nition, following comparisons would be possible.

In the first three activities, Zeno, Archimedes, and Newton activities(Appendices 1, 2, 3 respectively), students are asked to find how the givenfunction value changes when the independent variable is tending to a spe-cific number (infinity, in 2 of the 3 cases). The function is the function ofremaining distance in Zeno, remaining area in Archimedes, and average ve-locity in our Newton activity respectively. Therefore, our action is the sameas Dubinsky’s “level 1” action which is evaluating the function f at somepoints. The result of this series of actions is represented by a process of f(x)approaching L as x approaches a. Such a tending is explicitly addressed inthe 15th question in the Newton activity though. In the first 2 activities, xindeed approaches infinity but it is not mentioned in the questions explic-itly. Yet, Dubinsky’s representation of Process fits with our description ofProcess. What I did not find in our designed learning process, is addressingDubinsky’s 3rd step in the genetic decomposition. That is taking limit asan object by itself and applying another action to it in the way they havementioned. In our study, and in teaching continuity, limit could be regardedas an object and proving continuity is to be a new action. The process thencould be applying the limit concept during the new action until limit be-comes an object itself. Our second phase in the teaching plan was movingfrom the intuitive idea of the limit concept which was described in the first3 activities to the geometric illustration of it on the coordinate system. Thistransition was done by emphasizing the real line, intervals, and neighborhoodconcepts. This is the same as what Dubinsky mentions as this 4th step in

1This function was mistakenly named as step function during the teaching. While thestep function is a piecewise constant function which has only finite pieces according to itsdefinition, the integer-part function, either roof or ceiling function have infinite number ofintervals.

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his genetic decomposition. In the next activity which is named Weierstrass,the questions are set to lead students to the formal definition of limit. TheMathematica questions mainly lead students to find the relation between εand δ and finding the inequalities for which the relation is guaranteed. Level5 and 6 of Dubinsky’s model of learning limits seem similar to these two lastactivities. Although we did not set out to follow Dubinskys plan, our planhas much in common with his, the big exception being his step 3: treatinglimit as an object. In our setup something like this was planned to happenonly after introducing the formal definition of limits.

4.3 Setting

The teaching plan discussed above has been tried out in a small group of 3students. One of them is 16 and the other two are 17 years old. They are allin the 4th VWO grade. All of them took this course as an optional course ofthe D profile, although one of them (the younger one) is taking this courseas an extra course. The D profile is designed for those students who aremore interested in science and mathematics and would like to continue theiracademic career in science. Teaching took place in a whole block of 9 weeks,starting at 28th October 2008 and ending at 6th January 2009. 3 hoursper week was allocated to this class which was set in one afternoon of theweek. The first four weeks were allocated to teaching the research materials.The second four weeks were allocated to teaching some draft chapters of thebook (Getal en Ruimte, VWO D, published in 2010) including: Continuity,Differentiability, and standard limits 1. The last week included their finalexam, and interview. In the middle of the block, a workshop was designedand organized by Wolter in order to introduce the Mathematica software andhow to use it for delivering ”ε− δ” proofs (Appendix 5,6).

The students who participate in my research have not previously beenintroduced with the concept of limit specifically. They have just seen thesymbol in the definition of the derivatives without being told about whatit is. This is an advantage for me because as I mentioned before, the maineffort of this research is to give a richer conceptualization from the beginning

1These lessons from Getal en Ruimte were part of the deal struck with students andcooperative teacher. Together all lessons should make up a valid block of work thatstudents could get credits for. To keep the amount of work within bounds, however thesesecond four weeks won’t be reported on, except for the interview that took place at theend.

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to reduce the later misconceptions.

4.4 Data Collection

The data is collected during the whole block of 9 weeks. Collected data areprovided via:

• Video capturing the teaching sessions which includes students’ answersto the questions and their discussions in the class. The videos aretranscribed for further analysis.

• Taking a test which was done in the 5th meeting after teaching theexperimental lessons and before starting the subject of continuity.

• Interviewing students at the end of the block. This interview containedasking those questions in the test which were answered wrong at theprevious time. Also asking students’ general ideas about the activitiesand how knowing about these historical steps have helped them learnthe limit notion according to them. How they described the concept oflimit at the end of the block.

We will now discuss in more detail the selection of questions for the test(4.4.1), which also determined the interviews.

4.4.1 The Test and the interviews

The questions of the test (Appendix 6) are already asked several times fromother students in other studies (Tall, 1977; Williams, 1991; Tall and Vinner,1981; Jordaan, 2005) in the last few decades. More in detail, the first questionis William’s famous question which asks about the best sentence among 5sentences which best describes the limit concept. The second question also isadopted from William’s research questions and asks students to describe ina few sentences what they understand a limit to be. That is, describe whatit means to say that the limit of a function f as x → s is some number L(Williams, 1991). Our aim with these two questions was to check whetherstudents have the misconceptions that are known from the literature. In orderto make our results comparable to William’s, we asked the same questionsin the same order! The third question asks the limit of a constant function,f(x) = 3 when the variable x, tends to a given number, 2. This question

41

was asked because students often have difficulty to apply the limit conceptin a case where the function is continuous, because then the limit value canbe reached by the function - in contradiction to one known misconception.We took this question from (Williams, 1991) The fourth question is mainlychosen to check one of the common misconceptions; “a limit is not determinedbut approximated by plugging in numbers.” The question is designed byWilliams to draw students’ attention to the fact that by tending to a numberfrom right and left (by plugging in numbers) we can not necessarily concludethat the limit of that function is equal to that number. Question 5 asksto draw the graph of the function f(x) = x2−9

x−3and check its limit when x

tends to the number 3 which is the root of the denominator. This questionaims to check if the students have understood that a function could have alimit at a point while it is not defined at that point. The 6th question is anordinary limit question, except its last part which asks the limit of a sequence,limn→∞(1 + 9

10+ 9

100+ ... + 9

10n). This question is common in checking the

concept of infinitesimal with question 4 and 8, but this time with the limitsymbol. The 7th question is chosen to verify if the students can interpret thelimit of a function by having only its graph. The 8th question is also a famousquestion which is chosen to check the concept of infinitesimal. It asks what isbetween 0.999... when the 9s repeat and 1. Question 9 was mainly chosen tocheck how much the students have learned continuity without being taughtand only by understanding the limit concept.

4.5 Data Analysis

For analyzing the data, both William’s common misconceptions and theirrelated limit phenomena listed in chapter 2.2 are used. This will help uswith analyzing our data about the three paradox activities. During theseactivities, students are confronted with “limit phenomena” without know-ing about limits. We will study their contributions to the discussions usingWilliams misconceptions and our interpretations of them as categories. Inorder to analyze our data concerning the designed final steps towards theformal definition, the sub-questions of the 3rd research question are consid-ered. William’s 5 misconceptions will help us to classify students’ commonmistakes in interpreting the limit concept. They cover almost all the pos-sible misunderstandings of the limit notion. As long as the word “limit” isnot introduced, we will try out using our interpretation of William’s “mis-conceptions” as right conceptions of the limit phenomena that students are

42

confronted with.The Zeno, Archimedes, and Newton activities each include a paradox.

First we want to check if the students have understood both sides of theparadox. That is, we selected the discussions which show that the studentsare challenged. Then their conceptions in each of the problems is tried to bedistinguished. In the neighborhood activity which comes next, there is noparadox. We have selected those parts of discussions which show whether thestudents have understood the geometrical interpretations of absolute valuesand intervals or not. Also those parts of discussions which address students’understanding of dependent and independent variables are selected. TheEpsilon-Delta proof is the next chapter in which the formal definition oflimit is introduced and again we have tried to trace students’ misconceptionsduring the review of the paradoxes. Also we checked if they have understoodthe relation between epsilon neighborhood and and delta neighborhood. Weselected data from the continuity lesson to check how the limit concept isunderstood and applied by students.

5 Results

In this chapter, some of students’ discussions in each of the lessons is selectedand analyzed according to our analysis plan. It includes:

• Zeno’s Paradox Activity

• The Quadrature of Parabola Activity

• Derivative Activity (Newton)

• Neighborhood Activity

• Working towards Formal Definition Activity

• ε, δ Proof Activity

• Continuity Activity

• Test and Interviews

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5.1 Lesson preparation and time planning

The following analysis will focus on the five lessons (5×3 hours) whose designis described in chapter 4. Plans for these five lessons were partly developedwhile lessons had already started. For the first two lessons there was amplepreparation time. However, as the sequence progressed, preparation time gotshorter. For the third lesson a plan was ready shortly before the meeting.For the fourth lesson, the one about informal and formal definition of limit, aplan was not ready and the lesson was largely improvised. Despite not beingtotally satisfied about that lesson there are some interesting results in it thatwill be shortly discussed (see 5.5). The fifth lesson, delivered by Wolter, wasmeant as an “extra” and started with some recapitulation of what had goneon before. However, detailed information about what had happened in lessonfour was missing at that time.

5.2 Zeno Activity

Zeno’s paradox describes a one dimensional movement which pops the ques-tion; do we ever reach our destination? Our main focus here is on tendingand giving an idea about infinitely smalls while infinity is inevitably involvedas well. In fact we confront a series which becomes 0 at infinity.

The challenge in this problem is presented as a paradox. One side of thisparadox is moving from point A to point B. We all know by experience thatthis movement is doable and we can reach point B. The other side of thisparadox claims that we can not reach the destination since there remainsalways a small distance which does not allow us to reach the destination. Inorder to imagine this side of paradox, we need to walk strangely, each stephalf of the previous step. This strange way of walking might mislead studentsthat the paradox is originally wrong because we can never walk with this for-mat. They need to that see the core of the paradox is about motion itself,no matter how the movement is. In other words, the challenge is createdwhen they experience that on the one hand, no matter how they walk, nor-mal or strange, they won’t be able to reach to their destination, on the otherhand they know by practice that we can reach to point B. Transcript 1 shows:

Transcript 1

A. Suppose you start walking from the point A and your destination

44

is point B. Also suppose that with each step, you can go only halfof the remaining way. Just think about it.

J. You never reach it (...).

A. (...) Why is it called a paradox?5

O. You never end!

J. The theory is that you never reach the end.

O. You are never on the beach!

A. But you reach!

D. You never go to exactly 0, you go closer and closer (...).10

A. (...) So, you say that you approximately reach to the point. Younever reach to it?

J. According to the rule, yeah.

A. No, according to reality.

O. In reality you come there.15

D. In reality you’d be there.

A. So is it an approximation?

O. Yes.

D, J. No, you are really there. You really are there.

O. Okay.20

J. searched for the word verzinnen in English. They explained it forme as something that comes into your mind but does not exist.

J. When you come from point A to point B and you think there is apoint C and go there. Then you can reach point B.

A. Yeah it’s strange.25

D. (Laughing) You are here (pointing a point on his paper) and thenyou are here (pointing the successive point on his paper)

A. You know, I know it’s a strange walking you can not put your footlike this and this (walking strangely by following the rule). You can’tfollow this rule in reality but in reality when for instance a ball is.30

I shoot you a ball, not speaking about this half half half, the ballis continuously coming to you. Theory again says that it does notreach to you. So, now we are not speaking about this strange typeof walking. This is a ball which is continuously going, but neverreaches you.35

45

O. Why?

A. Because always there is a distance, a remaining distance.

O. Oh, yes,

D. But the time would be 0 by then. You spoke about time. it goesthat small.40

A. What about infinite time? You saw here, you spoke about infinity(pointing to the previous discussion about infinity). Here we saythat n can be a very big number. But speaking about the time,can we say that we need a very big amount of time for passing frompoint A to point B?45

J. Yes.

A. Can we?

J. Yes, you can say you ... infinity time before ....

A. So it means that you die, your grand children die, so on, so on andthen still it is not reached the destination?50

J. Yes.

A. But it’s not true. We don’t have time infinity. It’s not true (...).

J. (...) Always it remains a little piece of it. You never reach the end.

A. So, you never pass from A to B?

O. No!55

A. No? So, how did you arrive to school today from your home?

O. On the bike!

A. Okay, but your bike arrived here (...).

As we can see in the discussions, students have seen both sides of the paradox.One side of the paradox which mentions that the destination is not accessibleis traceable in Lines 2 to 8. All of them assert that we can not reach theend (point B). The other side of the paradox which addresses the fact thatwe can reach the destination is identified in lines 12 to 16. It is clear forthem that by real experience we can arrive point B. John but mentions thestrange way of walking as a barrier in the reality of the walking. Here it wasimportant to make it clear that the core of the paradox is about the motionitself not about the different ways of passing. Even if we walk normally, wewill pass all those points which are passed by strange walking. It is not clearif they accept this part of our message. Maybe they don’t. John mentions

46

this issue in line 21 and 22 by calling the walk imaginary. He also pointsout that the paradox is a consequence of the strange way of walking: if wedo the Zeno problem with point C as our chosen end point (with C furtherthan B) then we can get past point B. The teacher (line 28) wants to showthat the problem also exists in case of continuous movements (the intendedargument is that still you have to cross the infinite number of intervals, andeach will cost you some time). Dave (line 39) correctly interprets that thetime intervals needed to cross the space intervals will also get smaller, whichis of course part of the ideas needed to solve the paradox. The teacher(line 41) draws attention to the infinite number of intervals. Following hersuggestion, John (lines 46-50) thinks infinite time will be needed to get tothe endpoint. Seeing that students see one side of the paradox, the teacherthen (line 54) presses them to see the other side.

For the week after, the students were asked to find more paradoxes fromZeno (from Wikipedia, etc) which did not include strange walking, such asthe paradox of the arrow. There they noticed better that the question isabout motion itself. Here we can see that they have noticed the paradox isindependent of the movement form:

Transcript 2

D. I found the other ones (he is speaking about the other Zeno’s para-doxes), the other paradoxes, eh one is about an arrow, it can’t movebecause it was eh, (showing in front of his face with his two hands)it was here, here split up in small parts when the arrow was in theend, in that part it didn’t move so it couldn’t move in total.5

A. Okay and did you find any answer for these paradoxes?

D. eh (thinking) for this one I found some thing but eh they said itwas, you can see what’s wrong when you look to the time when it’scollapsing, because the time when you are almost there and standingstill, the time is almost 0, so that’s impossible so something like10

that and there was a mathematical explanation but that I didn’tunderstand.

A. Okay thank you. (to Otto) did you find something?

O. Yeah, the arrow, because Zeno didn’t say there is, you, the arrowis at time 1 second is he in the time as also as speed. Zeno forgot15

that it has speed.

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A. Okay, can you explain more?

O. eh, when you are, when you have the arrow, at time 1, it is some-where. It is somewhere is true but it has also some speed at thattime, that was what Zeno had forgot.20

A. Yeah okay it has speed, but,

D. Split it up in points not in periods, so (...).

Dave sees the movement (line 1-3) as a collection of infinitely small move-ments which occur when we split the movement into infinitely many intervals.Since motion is not possible in each of the intervals, then motion can not ex-ist at all. Otto on the other hand talks about the speed (lines 14-16) and thefact that Zeno had ignored the speed of the arrow. Unfortunately John doesnot discuss the arrow paradox. He just mentions Achilles and turtles anddoes not have any clear explanation for it. Dave seems to have understoodthe main purpose of the paradox; the motion is questioned no matter how isthe movement.

Meanwhile, when we were busy with the challenge of the paradox, someof the students’ intuitive reactions appeared which linked to some of our socalled “limit phenomena” (chapter 2.2). Mainly, those which are based onthe concept of tending and infinitesimals. For instance our so called ”simplehorizontal asymptote phenomenon” which in comparing with William’s listof models held by students, could be linked to: ”a limit is a number or pointthe function gets close to but never reaches”. What we mention here areindeed students’ intuitive reactions to the challenges. The three of thembelieved at some point that we can become very close to the point B withoutreaching it completely (Transcript 1, lines 2-8). This idea could be caused bya lack of enough perception about the concept of approaching/tending andinfinitesimals. This belief strongly remained and was dominant in John’smind although he knew it was wrong.

The other “limit phenomenon” which is again associated with the conceptof tending and discussed with students is our so called ”Gap Phenomenon”.Regarding William’s list of misconceptions, it is addressing : “a limit is anapproximation that can be made as accurate as you wish”. In order to bemore precise about this idea, questions 4 and 5 were asked to promote stu-dents to get involved with it. In these questions, the aim was to indicate thatthe remaining distance can never become 0 unlike in approximation that onecan say for instance 10−4 is approximately 0. The teacher explains whatis meant by approximation also what is meant by fixing a number with a

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specific precision. For instance if the precision of a calculator or computer is2−60, then the machine considers all smaller numbers as equal to zero. Theyare asked if the very small remaining distance in going from point A to pointB should be regarded as 0 just like machines fix a small number as 0. Or weare approximately at point B. Transcripts show the discussions about thisquestion.

Transcript 3

A. (...) Yes, so when you pass your 60th step, your remaining distanceis 2−60. It’s a very small number, but it’s not 0. So, what is it?(...).

A. (...) Is it an approximation? When we reach to B, do we say thatthe remaining distance is approximately 0?5

D. No, because you are exactly there (...).

A. (...) Is it the same as machine or how do you think?

D. eh, yes, it is so small, yes it is the same.

O. It becomes 0.

A. Could you write it mathematically? We are speaking about n.10

D. It’s infinity.

A. But what is infinity? Is it a number?

O. It’s very big.

A. What is very big?

J. It’s a theory, we were trying to prove a theory and now we have15

another theory! (...).

O. (...) When you walk, it’s a precision. And when you have this one,this series, you have approximation.

J. I think it’s just contrary.

A. So, you didn’t choose non.20

D. I think they are both precisions.

A. You can have your individual ideas!

D. (Laughing) Alles is anders.

J. I think this rule is just precision, and if you take steps it’s justapproximation.25

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In these arguments, students are trying to describe how the paradox could beexplained. Their argument show that they are puzzled by the paradox andthey don’t know what is the best explanation for it. Each of them sees thisproblem different from the others. Otto distinguishes the two sides of theparadox from the view point of approximation. He thinks that in walkingnormally, we will reach to the destination but in Zeno’s designed case wearrive at the point B approximately (line 17). Probably he has not thoughtabout the concept of infinity and infinitesimal adequately at that moment.Dave mentions infinity and seems the puzzle is solved for him from both sides,either in the strange way of walking or in the normal walking (Transcript 3,lines 11, 21). John on the other hand thinks that approximation explainsjust one side of paradox, walking normally. He thinks that in Zeno’s series;1

2n, we arrive exactly to 0 but in walking normally we arrive approximately

to point B and not exactly (Transcript 1, line 17, also Transcript 3, lines 19).The reason could be again embedded in the dynamic process of tending andthe fact that we get very very close to the destination. We can then ignorethe very very small difference and say we are in point B. In this case alsoteacher and students need to know more about infinitely smalls to help themunderstand the limit notion better.

5.3 Archimedes Activity

In this problem, the quadrature of the parabola, the challenge is originatedfrom a paradox as well, investigating the area of a parabolic segment. Inorder to find the area of a parabolic segment, we inscribe a specific trianglefollowing Archimedes. By increasing the vertices and sides of the triangles wewill have a polygon with infinitely many vertices and sides. The difference inarea between this polygon and the parabolic segment must be 0 at infinity.Archimedes claims that the area of a parabolic segment is 4

3rd of the area

of a triangle which is inscribed in it and has some specific properties. Forinstance, one of such properties is that this triangle has the maximum area.This problem has less philosophical sides and therefore is more illustrative.The quadrature of parabola together with Zeno make a package which ad-dress the concept of approaching in the content of sequences. If in Zeno’sparadox, students kept believing that the problem is with the strange wayof walking, with this question we expected them to see that the inscribedpolygon increases sides in a normal way without any limitation and at in-finity it becomes exactly the parabolic segment. The Dynamic mathematics

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software, Geogebra, helped us a lot to illustrate the problem for students.One side of this paradox claims that the difference between the area of theparabolic segment and the polygon never becomes 0. Our discussions in classshow that this side of the paradox was clear for students (Transcript 4, lines1-5, lines 34-39). The other side of the paradox on the other hand revealsthat this difference can become 0. Following our conversations in the class,students also believed that this difference becomes exactly 0 (lines 46-67).A lot of side knowledge in calculations was involved in our attempt to per-ceive the problem. Thus we had to allocate part of our time to work onsuch calculations (see 8.3.2). Since two of the students did not know aboutthe geometric series beforehand, the teacher needs to introduce the formulafor the sum of a geometric series. In the remaining calculations, they weremainly left on their own to make mistakes and notice their mistakes aftergoing to further steps. Illustrating the problem with Geogebra and lettingthem verify the dynamic process of increasing the sides, needed some timealso. Therefore, we had to spend pretty much time on “side problems”.

Transcript 4

A. (...) How close can we get to that number by following this sequenceof numbers or areas?

O. Very close.

A. Very close,

J. But not exactly ...5

A. (...) You see by increasing the number of steps, the polygon sidesincrease also. And what happens to this difference when n increaseswhat happens to this difference?

O. It’s eh, close to the zero (others also mumbling the same thing)... .I zoomed in on the very small parts between parabolic segment and10

the side of the 4th or 5th polygon to show them that there exists abig space between them although it seems nothing without zoomingin and it depends on the scale, it could be very big what seems verysmall. I told them that we are finding the area of this space.

D. I think it’s −4... (he showed me the expression including the sub-15

traction of 43

and the area of the nth polygon.)

A. Yeah, but what does it become? Dave tried to ignore this differencefor n more than 5 but I reminded him the applet and how big was

51

this difference if we just zoom in. So it is not 0! and they continuedcalculating. After they all agreed about their answer, Otto said:20

O. (−13)n.

A. Waarom? They started explaining but their explanation was notclear.

A. Actually it should be positive and you should subtract the smallerarea from the bigger area. They continued calculating and also no-25

ticed that they had missed one of the terms (...) Finally we arrivedat 4

3−apolyn = 4

3− 4

3(1−(1

4)n) and consequently 4

3−apolyn = 4

3× 1

4n...

A. What happens to this difference when n increases? It becomes closeto 0, but does it become 0?

They reply no.30

A. If this difference wants to become 0, then the difference which is43× 1

4nshould become 0 but is it possible?

They reply No.

A. No because it means 4 = 0, it’s not possible. So, here mathematicstells us that this difference can not become 0 but it’s getting close35

to 0. What happens? Is it like the machine? Is it like the machinethat after a very small number we say okay from now on we decideto put it 0 because it’s so close to 0.

D. Yes (...).

A. You think it’s like this? Let’s see. In the machine, it’s called40

accuracy, okay. How many steps should be taken that our accu-racy be this number? It means that (I write on the board probably2−60

3= 4

3× 1

4n).

J. I said that!

A. Actually I have,45

J. 4 divided by 4, yeah I solve the equation and the answer is n = 60.

A. So you mean that for instance after 31 steps (here I had made mis-take in thinking that I made a mistake. I ignored John’s answerand continued with my own answer which happens to be right. Inspeaking I said 31 instead of 61! ), you can say this difference is 0?50

They watch me and the board as if they are puzzled.

A. This is wrong.? So you don’t agree that this is an approximation?

D. Yeah.

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A. So you don’t agree that this is an approximation.

J. Can you repeat that?55

A. this question says that after the 31st step, I mean 32nd step . Thisdifference is 0.

O. No, It’s not approximation.

A. Do you agree that it’s true? When the difference becomes verysmall, we say it’s 0. Dave still says yes.60

A. Do you agree also (to John)?

J. (laughing) You are asking me all the time!

A. No I ask all of you. Because I should know all of your answers. Idon’t agree!

J. But you are saying now that after 31 steps, this difference becomes65

0? But you said that it never become 0.

A. Yes it never becomes 0, I don’t agree. There is a problem, let’s see.(we move to the 8th question) Remember that in theory, you canincrease “n” as much as you want there is no limitation. You canhave 31st step, 32nd step also. You can have as many steps as you70

want. There is no limitation for you. You are free to go, your mindis not like the machine. the machine says it’s not defined for me,after the 31st step, I put it 0, because no one has programmed itfor me. But our mind says no you can go as long as you want. Youcan always have a very small number but it’s not 0.75

J. Yeah.

A. So do you think that by doing this process without stop we can getan approximation of the area of the parabolic segment? I think yousaid yes (to Dave), no (to John), what do you think (to Otto)?

O. I think we get 0.80

A. Approximately or exactly?

O. Exactly.

A. Exactly! How? we said here that it’s not 0, it never becomes 0.(pointing to the equation 4

3× 1

4n= 0 and eventually 4 = 0) If it

becomes 0, then 4 is 0.85

O. Because you can always go one more, one more.

A. How? How much?

O. Infinity!

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D. Without stop!

A. Yeah, but we should have something to describe it mathematically.90

We can not use more or you know? We need something to describeit mathematically.

A. Yeah but I am sorry but what the hell is going here? It seems thatthis non stop process gives us really 4

3. We really reach to 4

3when we

increase these polygons. Finally we get it but these relations show95

us that yeah there is a problem some where. Yeah there should beprobably some magic word or magic concept?

O. Yeah I have a magic, I am a magician.

A. You are magician? What is this word according to you?

O. Abra cadabra.100

A. Yeah exactly and what does this word do? It makes this difference0?

O. yeah I think it makes the difference 0. Yeah.

A. So, there is a word that explains us that yes this difference becomes0, but not with putting this difference 0 because 4 = 0 is wrong!105

There should be something else that helps us to say that this dif-ference becomes really, exactly 0. ha?

The ”Gap Phenomenon” (chapter 2.1), which is linked to the idea of ap-proximation : A limit is an approximation that can be made as accurate asyou wish., interpreted as students’ intuitive reactions in our case, is visiblein this problem also. Questions 9 and 10 are particularly designed to checkstudents’ ideas if the value is approached and approximately arrived or is aprecise number. The question asks if by continuing the process and increas-ing the number of steps, the area of the polygon will approximately becomeequal to the area of the parabolic segment or they become exactly equal.Dave again believes that there is no approximation although once he tendsto ignore the remaining area (lines 15-17). He mentions the dynamic processof tending to 0 without stop and believes that since we are tending to infinity,our remaining area becomes exactly 0 (lines 50-53 and 89). But he does notmention the word limit a fact that did not surprise us. Otto, notices theconcept of infinity better than before (see 5.1) in this problem. He refutesapproximation for both sides of the paradox (lines 58, 80). He also mentionsthe dynamic process of tending as the reason of arriving exactly to 0 as thearea of the remaining distance (lines 82, 86, and 88). John does not seem yet

54

to agree that by increasing the sides of the polygon, the remaining distancebecomes exactly 0. He still thinks that by this process, a small area willalways remain (lines 5, 76) 1.

5.4 Derivative Activity

In this activity, students had to find the instantaneous velocity by findingthe average velocity and make it tend to the instantaneous velocity. Doingthis process step by step, we arrived at some point at the average veloc-ity between a second a momentum h after that second. The h had to befactored out using the condition of never being 0 from the nominator anddenominator. The conflict happened at this step where h could not be 0 butcalculations showed that it eventually will become 0. Understanding the twosides of the paradox was not difficult for students this time (see transcript 5).

Transcript 5

A. (...)(writing −h2+40×h200×h

2 on the board) So with 1 condition we cando this [factor out h]. What is that condition? When can we factorout a,

J. When we have an equation.

A. No.5

O. It can’t be 0?

A. It can or can’t?

O. Can not.

A. It can not, it should not be 0, why?

D. Because then you got divided by 0 (John and Otto accompanied him10

on “divided by 0”) (...).

A. (...) h is tending to 0 and not becomes 0. So what about m 3? Weguess that m is this (0.2). But how? Is m really this number or isit close to this number?

O. It is this number but the formula goes close to this number.15

1As long as the paradox is not explicitly solved, everybody has a right to remaincautious

2This is the slope of the line AB. Line AB is now a line which is very close to thetangent line.

3m is the slope of the line AB.

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D. I think m is really that number.

J. Yeah.

D. Because we calculated it with two points very close to each other.

A. Yeah but how? The slope of that line which is very close to A. Butit has h and h can not be 0.20

D. That’s because you have space between A to B. When you got m,you don’t get A to B.

A. So you think m becomes this number or becomes close to this num-ber?

D. Yes, it becomes this number.25

J. With this formula, it doesn’t (...).

A. (...) It means that h is tending to 0. There is always one numberbetween h and 0. Is it true? This is what I told you before, buthow? If this is equal to this (−40+h

200= −40

200) then h should be 0. But

h can not be 0. So is it true?30

O. Yeah.

A. Why? You know when h tends to 0, or it means that we are tending,closing and closing to A, is it true?

J. The equation, it is not true. h couldn’t be 0. How could it be true?

D. In reality yes.35

A. But how? We should find something. (...)

We can notice that they are in challenge by both sides of this paradox intranscript 5. In lines (6-10) Otto and Dave express (John also agrees withthem) that h can not be zero because then we can not factor it out fromthe nominator and denominator of a fraction. On the other hand, in line 16,when Dave mentions that m is exactly equal to 0.2, he is in fact putting hequal to 0. He is aware that this will create a calculation conflict but stillhe believes that h must be 0 in spite of our previous condition (lines 19-25).Otto and John (lines 15, 26 and 34) still believe that mathematically it iswrong to put h equal to 0, so maybe the formula has to be changed. But inthis fraction h should remain always bigger than 0. At the same time theydo believe that the slope of the line AB is exactly equal to 0.2 which is theslope of the tangent line (lines 15 and 17).

Students noticed that they are experiencing the same phenomena of ap-proaching a number as they had experienced previously in Zeno and Archimedes

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(see transcript 6). We also discussed the similarities and differences ofthis case and Zeno’s case by addressing infinity. In fact this time they al-ready know what is going to be the question by comparing the situation byArchimedes and Zeno and discovering the paradox.

Transcript 6

A. (...) yeah you see, when B becomes closer to A then this line be-comes closer and closer and closer to the tangent line.

J. This is in fact what Archimedes said with the parabola, isn’t it?

A. yeah a kind of similarity, yeah.

J. Point B every time closer to point A,5

O. Yeah.

J. Not really,

A. What did Archimedes really,

J. Not exactly 0 because then they become 1 point and you can’t drawa line through 1 point. But to do it very close to each other. (...)10

John noticed the similarity in lines 3-10 when he predicts the paradox bycomparing this situation of dynamically approaching point B to what hesaw in the Zeno and Archimedes problems where a value (e.g the area) wasapproached in an infinite number of steps.

As it is traceable in the three activities, we were indeed plugging in num-bers closer and closer to a given number (here: h = 0), and the discussionpoint is whether we may “reach” it (i.e. plugging in h = 0 itself). Thisin fact provokes one of the common models or concept images distinguishedby Williams (Williams, 1991) which even showed up in Otto’s calculationsclearly. Apparently he had seen the word limit before and he knew thatwhen he wants to find the limit of an expression, he just puts that num-ber as the variable in the expression and the limit is found. Our plugging innumbers, made him do the same and mention it even explicitly (transcript 7).

Transcript 7

A. (...) So B was 40 at first, now we make it 25, then 21, then 20.02,then 20.001. And see what happens to the slope of the line. Okay?

J. Yeah.

57

A. So continue for these 4 numbers also. We checked that mAB =−0.225,−0.205,−0.2001,−0.200005 respectively between the times5

, t = 20 and t = 25, 21, 20.02, 20.001

A. Some machines probably calculate these numbers different accord-ing to their accuracy. You see, to which number are they gettingclose?

D. -0.2?10

A. yeah. So you see that when we are getting closer and closer, veryvery close to A, the tangent line, the slope of the tangent line, orour average velocity is getting very very very close to -0.2. But,how long can we continue this process? Can we do this forever?Because we have infinitely many numbers on the real line. We can15

continue it till ever, but is it doable? Is it possible? No!

O. It’s time, it’s time, time is not infinity. So you can’t do it infinitytimes. At the same time I hear something like “limit” in the wordsthat John is saying to Dave in Dutch (4:32 20th video).(...)

A. (...) Yeah we are close to limit. We can not continue this process20

till ever. (...)

A. (...) You found out a number?

O. Yeah.

A. Oh you put limit and you found out a number, aaaaaaa!

O. Yeah we have learned by mathematic.25

A. You have already the limit.

O. Yeah.

A. yeah, okay.

O. I don’t know what but we have limit.

A. You don’t know what it is but you have learned it.30

O. Yeah, well h is going to nul. So when we have only h zero, then youhave 0.2.

A. Yeah, but I don’t accept it. You should explain completely for whatever you say. mathematically, you are going to become mathemati-cians I think and here I don’t understand what you have exactly35

written. (...)

Otto already noticed that what we were doing was close to finding the limitof an expression. This was the first time I heard about limit from them,

58

although he did not know what he is doing. Fortunately in the next sessionswe did plan to excavate the different cases and address the cases of continuousand discontinuous functions as well (continuity and asymptotes).

So at this time, they have experienced a paradox which occurs in thephenomena of tending continuously and infinitely many times either to anumber or to infinity. They feel the lack of some mathematical explanationfor this paradox (transcript 5), therefore, we may perhaps expect that theyare ready to be introduced with the words limit and infinitesimal and theircontribution in solving the paradoxes. This introduction gives them the key-word Limit which seems just to be a word at first hand (transcript 8, lines20, 21). The idea of approximation was students’ sudden reaction after beingintroduced with limits. John’s first reaction is that we are fixing a numberclose to 0 but not 0 and we are approximating the answer and my impressionwas that Dave agreed with him (transcript 8, lines 10-12).

Transcript 8

A. (...) It means that eventually, eventually, at the end these 2 be-come equal to each other. The limit of this (writing on the board):limh→0

−40+h200

= limh→0−40200

= −0.2. Yeah? Finally a mathematicalexplanation can explain it. (...)

J. (...) But still why? Why is it possible now?5

A. Yeah, because it’s the limit of ....

J. But what does that say?

A. It says that the limit of the line, the limit of the slope of the line,the line between A and B is tending to instantaneous velocity. Johnand Dave were searching for a word in the web dictionaries and I10

asked them what is the word. The word was fixing.

J. Do they just fix a number to 0?

A. No, here exactly comes the infinitesimal. Infinitesimal is infinitelysmall, it’s not 0 but it’s smaller than any small number you canimagine and this is what is happening in Zeno, in parabola and15

here in Newton. (...)

J. (...) The limit is very close to 0?

A. Its limit is exactly 0. It really becomes 0. Limit doesn’t tend to 0.The limit becomes 0.

D. It’s just a word they found to explain it just.20

59

J. Just,

A. But limit has a definition. (...) About what you said, it’s veryimportant. Keep it in your mind because our next activity is relatedto what you said and then I’ll explain it again. (...)

It seems that the idea of fixing a number and approximating the answer wasdominant in their minds for some while. In line 17, after all of our discussionsin the last three sessions, John asks if the limit is exactly a number or it isclose to a number. Intuitively, the idea of fixing a number and approximationwas preferred by students in the three paradox cases except in the Archimedesactivity where Dave and Otto refuted the idea of approximation.

5.5 Neighborhood

In this session students were not challenged by a paradox or a mathematicalconflict. Instead, the questions were designed to make the students thinkabout the real line and imagine intervals and neighborhoods. This was in facta preparation for the formal definition of limit. The main aim in teaching thischapter, was to guide the students to refresh their geometrical interpretationsof the absolute values as the distances, intervals and finally neighborhoods.They were occasionally asked to find out the δ neighborhood (on the x-axis)corresponding to a given ε neighborhood (on the y-axis) for a given function.If the function is not linear, the pre-image of the ε neighborhood will bean asymmetric interval in the symmetric δ-neighborhood. Therefore, weallocated almost 2 sessions with many discussions to this part. The followinganalysis of these discussions has two aims:

• To check if the neighborhood concept is understood with specific at-tention to the role of prerequisite concepts like the geometrical inter-pretation of intervals and absolute values.

• To check if students have understood how does δ change when ε changesin case of a linear function (pre-image is symmetric around a1).

• To check if students have understood how does δ change when ε changesin case of a non linear function (pre-image is asymmetric with respectto a).

1Where a is the x-value that’s being approached.

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In the beginning, we started with what we guessed are easy questions: find-ing the set of points on the real line whose distance from the origin is a. Thequestions became harder gradually until they had to find the set of pointswhich show a line segment (transcript 9). In fact the representation of in-equalities on the real line.

Transcript 9

A. (...) [Find] the set of points on real line whose distance from 1 isless than 4.I was doubting if they have payed attention that this time the answershows a line not just 2 points

J. -3 and 5?5

A. So how many points are in this area?

J. Very much.

D. Infinity.

A. Infinitely many (I drew on the board and I explained them a bit aboutit). How do we show it mathematically?10

O. Yeah, |x− 1| < 4? (...)

A. (Writing: Bε(x0)) New symbol. From now on we can use this symbolwhich shows the neighborhood.15

J. Epsilon.

A. Yeah. Epsilon is not new for you, neighborhood I mean; Bε(x0) ={x| |x−x0| < ε}. This shows the points whose distance from x0 isless than ε. Epsilon is a number usually known as a small numberand is shown like this: ε on the real line (showing the neighborhood20

on the real line). Do you have a problem?a discussion followed about the notation {x| |x − x0| < ε}. Thenwe had the first Geogebra applet.

Finding such points was not new for them. It was refreshing their previousknowledge by specific emphasis on its geometric meaning. But the fact thatthey made the link between the real line and infinity was interesting for us(transcript 9, lines 5 - 11). Students can adequately translate a statement

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in words (line 1-2) into a set of points on the number line (lines 5-9). Also,they can translate this set of points into an inequality (line 11). Additionalevidence is to be found in transcript 10. From this we can conclude thatJohn, Dave and most probably also Otto understand the geometric meaningof inequalities.

Our second step was guiding students to find how does the neighborhoodon the y-axis change when the neighborhood on the x-axis changes (Tran-script 10). For this purpose we first drew their attention to the dependentand independent variables. Then illustrating this relation using a Geogebraapplet. We started with a linear function so that the intervals on both axesbe symmetric. Our example function was f(x) = 2x+ 1

Transcript 10

A. (...) In all the previous questions, we had just one axis. But we canmove on both axes. We have x and y. What is x? What kind ofvariable is x?Dave said something which is not clear in the film.

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A. it’s the independent variable.5

J. Oh yeah.

A. It’s free.

O. Why not.

A. x is a free variable and y is the variable which is dependent on x. Butx is free to move. (showing the applet) Now let’s see for instance in10

that applet that you have, what does it mean? It means that whenx is in the neighborhood of 2, we say it belongs to Bδ(2). It meansthat x is between 2− δ and 2 + δ. When x is in this neighborhood,in the neighborhood of δ, then y is in the neighborhood of ε. y ismoving on a neighborhood also.15

D. Oh,Yeah.

A. Now let’s see when δ is 0.1, what is ε? Check it on the applet whenthe ε is 0.1. (...)

A. (...) How do you interpret the following relation in terms of dis-tance?

y ∈ Bε(3)⇒ x ∈ Bδ(1)

(...) This time Dave went to the board and wrote but I don’t knowwhat exactly20

A. (to Dave) Can you explain for him (for Otto)?

D. This is y − 3 eh,

A. What does it mean?

D. that 3 is in the middle and we start at 3, then you can go ε up andε down.25

J. Yeah but why 12? (then they have a short discussion in Ducth and

ask if there should be < or 5, Otto still had problem and this timeJohn went to the board and drew the real line and reviewed whatdoes absolute value mean on the real line and then Otto noticed hisproblem. His problem was that he was not realising that absolute30

value demonstrates a symmetric area on the real line.)

A. So what does it mean now in terms of distance (the main question)?

D. When y is between −ε and +ε, then the distance x is between, whatis this sign?

A. δ.35

D. δ is between,

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J. But then if you have this (drawing the rectangle which shows thecommon graph area which full fills both inequalities (one for y andone for x)) does that mean that. Do you mean this area with thatformula?40

A. Yeah, that’s right. Do you agree with him?

D. eh, (...)We were engaged with changing εs and δs numerically and checkingthe above relation for them. While John realized the general relationbetween ε and δ:45

J. Isn’t that δ is always the half of the ε? In this case.

A. Do you agree with him?

D. (to John) Could you repeat?

J. In this case, is δ always the half of ε?

D. (thinking) Yes if you look to the formula you can see it.50

A. And in Geogebra? Did you see it?

D. Yes, yes. (he explains in Dutch for John)

A. When ε is half, δ is a quarter. When ε is 2, δ is 0.1 and when ε is0.1, δ is 0.05. Let’s see another applet. (...)

Solving this type of problems was not difficult for these students althoughoccasionally they made some mistakes. For instance Otto still needed sometime to see the neighborhood as a symmetric interval. Dave explains thequestion one more time for Otto and during his explanation, I notice thathe has understood the neighborhood concept (lines 26-31). Although I amnot sure to what extent he has realized the relation between the changingof the two variables. What was remarkable was John’s interpretation of thisquestion. He distinguished the rectangle in the graph that fulfils the relationy ∈ Bε(3) ⇒ x ∈ Bδ(1) (transcript 10, lines 37-39). He also realized thegeneral relation between the ε and δ (transcript 10, lines 46-49 ). From ourevidences, it seems that the three of them are able to find how does δ changewhen ε changes in case of a linear function. Such a general relation betweenε and δ is needed to prove a limit as we planned to do it in the next meeting.

The next step was going to the non linear functions and finding the δwhen ε changes. Also finding the biggest value for δ which gives us all thevalues in a specific neighborhood for ε. Our example function this time is

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f(x) = 14x2

Transcript 11

A. (...) Find the biggest neighborhood of 2 that catch all the x val-ues that have their f(x) in Bε(1). This neighborhood here, see?(pointing to the applet on the screen indicating Bε(1).)

J. Ah the biggest neighborhood?

A. the biggest neighborhood.5

J. It means from G to E.

A. If you take G to E as δ, then what happens?

J. yeah, but this is the biggest neighborhood.

A. Yeah, it’s the biggest neighborhood but the biggest neighborhoodthat gives me, that their image be in this neighborhood (pointing10

to Bε(1) on y axis).

J. Oh, yeah (others also moving their head). (...)

A. (...) (They watched me blank) But the [other] way around, if I takethis one as δ (pointing to GF ), then what happens?

J. then you miss a, you miss a field that,15

A. Yeah but it’s not in conflict with my question.

O. Oh. (...)

O. (...) So you say δ is G to F .

A. I mean do you agree with my idea?

65

J. Yeah.20

A. So,... the minimum of these 2 differences, then we are safe. then forsure the image of this neighborhood (on the x-axis) is for sure inthis neighborhood (on the y-axis). Then I can write (writing on theboard) |f(x)−1| < ε for instance now it is 0.3. Speaking just aboutthis case, if |f(x) − 1| < 0.3, then |x − 2| is less than what? You25

can check the numbers here (pointing to the left bar of the Geogebrapage).

J. 2.7?

A. F is 2.28

O. Yeah, 2.2830

A. Do you agree?

J. Oh yeah.

A. Is less than 2.28 (it’s a mistake, δ is 0.28 not 2.28 ). Do you agree?(I had written |f(x)− 1| < 0.3, then|x− 2| < 2.28). So when I havea line I can easily write that if f(x) is in the neighborhood of ε,35

then x is in the neighborhood of δ but now that we don’t have aline anymore, the safest solution is to take the smaller distance andmake our neighborhood with this distance. so this is true. Let’s goto the next question (which is still 10th question). Let’s find outsome of these δs yourself. Find the δ when ε is half. You can see it40

on Geogebra.

O. That’s,

J. 0.45?

O. Yeah, 0.45.

J. But, so it’s always eh, oh wait, eh, I did that with 0.1 as ε, and45

then you get δ is also 0.1.

A. yeah it is.

J. And if you do ε is 0.2, then it comes 0.9,...

A. you have a asymmetric neighborhood then.

J. No I mean (he was trying to find the general relation between ε and50

δ).

A. You can do it, but I don’t want to go to it now because it will take alot of not a little time. But it’s very interesting that you are findingthis relation. (in the rest of the session, we were manipulating ε and

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δ numerically and checking how do the two neighborhoods change.)55

(...)

It was one of the most difficult parts of the questions. John chooses the biggerdistance GE as the radius of the neighborhood (transcript 11, line 6) withoutpaying attention to what the question asks. I found it necessary to emphasizethe part of the question that we overlooked (line 9-10) and when it seemed thechallenge was still too big for them (line 12) I made a suggestion (line 13-14)and asked for consequences. John was not immediately convinced (line 15)and it is not totally sure if they understand this choice (line 15-20). Studentswere able to apply the suggestions (taking the smallest distance of the two)to new cases (line 30, 44) but it could be a trick for them. Unfortunately, wehave no data to show whether they understood this choice.

It is interesting that John again tries to find a general relation between εand δ (line 50). Something he did earlier for a linear function. The teacherknows this is much more difficult now and urges him to try it out withconcrete numbers for ε and δ first. In the rest of the transcript, the samerelation is investigated by allocating the numbers instead of ε and δ.

• The concept of neighborhood seems to be understood by all three ofthem.

• In case of a linear function, it was doable for them to find out howdoes δ change when ε changes. They even found the general relationbetween ε and δ.

• In case of a non linear function, the two created distances were asym-metric and we don’t know whether choosing the smallest distance asthe appropriate radius for the neighborhood was understandable forour students.

5.6 Working Towards Formal Definition

In this session, we want to check in how far does the neighborhood conceptallow students to follow the arguments building up to the formal definitionof limits. Also to investigate which difficulties do they have with followingthese arguments.

The session started by taking a function formula, f(x) = 3x+ 1, and ap-plying the neighborhood concept to it. Different ε values were given and the

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corresponding δs were asked. But this time we did not use a pre-programmedGeogebra applet, instead just the formula was given to show the relation be-tween epsilon and delta on the graph of the function.

Transcript 12

A. For instance f(x) = 3x− 1, then when y ∈ Bε(2) then x ∈ Bδ(?) ofwhich neighborhood? (...)

D. (...) 1? A short discussion between students in Dutch.

A. Dave is right,

J. That’s 1.5

O. Oh yeah.

A. Given different εs, for instance when ε = 0.7, it’s a line so theneighborhood is symmetric, now find the δ. We don’t have Geoge-bra, now we have just the function. What does it mean first ofall?1

10

Students seem to be able to apply the neighborhood concept in this newsituation as can be seen from the preceding transcript. More evidence isgiven in transcripts 13, 14, and 15. So, the Geogebra activities analyzed in5.5 seemed to be useful as a stepping stone.

Then we reviewed the paradox in the derivative activity and went backto a simple linear example, limx→1 f(x) = 2x + 1 = 3. Applying limit forthe linear function, raised the question for some students that this is justplugging in numbers and the limit concept does not make any difference.This question made me give a discontinuous example to show them that weare not always able to plug in the number in the function in order to findthe limit. From transcript 13, we can see if this was convincing for Dave.

Transcript 13

D. (...) In this case you can make it 3 so why did you use limit? Itcould be 1 itself.

A. can I rephrase your question and tell me if it’s not the same as yourquestion. Dave says that in this question, in this example, the limitof the function is equal to the value of the function in this point. It5

1For further examples see 8.3.5

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means that if we remove x and put 1 instead of it, it becomes whatwe call limit. So what is the value of limit? It’s the value of y inthis point. Is it your question?

D. Yes.

A. My answer is yes in this example they are equal but if this was10

what we do always, how do you reply to this (writing on the boardlimx→0

1x)?

D. Exist eh,

A. I want to tell you that it’s not always equal to the function in thepoint. Can x be 0?15

D. No.

A. But x can tend to 0 and the limit has an answer.

J. Yeah, but x can be 1 in this.

A. It can be 1, but now I’m speaking about the limit of this (pointingto x→ 0).20

D. So it’s just an example to show what you can do ... in case it’s moredifficult.

A. It’s an example to show not always the limit of the function is equalto the value of the function in the point.

D. I mean the other one is just to explain about limits.25

A. Yeah, you are right in this question, yes it’s natural that you askthis question.

J. But in the (he meant in 1x) x can never be 0.

A. But can tend to 0.

J. It can not be 0.30

A. Exactly, so we don’t have ...

J. I mean, why x can tend to 0 but never be 0. But x in this case,tend to 1 but it also it’s 1.

A. Yeah.

J. But can you still write this limit?35

A. Yeah the limit in this case is equal to f(1). But can you guess,

J. Can we say x can never be 1? Don’t you?

A. Here x is tending to 1, yeah,

J. But it never can be,

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O. (to John) No.40

A. (I changed the denominator from x to 2x because in the previouscase both x and y were 1 and it could create confusion. So thefunction became f(x) = 1

2x.

Dave recognizes the pedagogical intuition of the teacher (line 21, 25). Hesees a limit as needed only in case the number can not be plugged in (line1), but he accepts for the sake of explanation, that you can use limits also ina case where they are not needed (line 25). John on the other hand, is notsure if you can write limit in such a case (line 35) and if yes, if x is allowedto become equal to the approached value (line 37-39). It was the first timethey were asked for the inequalities for ε and δ for a given limit problem.Then, we tried to translate the straight line problem into the neighborhoodconcept as practised before and we will check that students can participatein completing this translation.

Transcript 14

A. (...) You say that limx→1 2x + 1 = 3. Let’s translate it. It meansthat when our distance of f(x) from 3 is less than any arbitrary ε,do you know what is arbitrary? John said something in Dutch toOtto.

A. Given, any given, then make it simpler it becomes, instead of δ let’s5

write,

J. Can we write x− 1?

A. yes but then it becomes ..., agree?

J. Yes.

A. This is 2x+ 1 we say that its limit is 3, it means that the distance10

of 2x+ 1− 3 is less than for instance 0.1 or any other number, anyother small number.

J. eh, how did you say that 0.1?

A. It’s a given ε, arbitrary. But what does it mean?

O. it is 1.05?15

A. Yeah but what does it mean? It means that all the xs whose distancefrom 1 is less than this number. So what do we have now? We saidthat when we have this relation, we reach to this relation and we saw

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it on Geogebra also. It means that all the point whose their distancefrom 1 is less than this number and when we make it smaller and20

smaller and let’s make a table here and make it smaller.Then I started to make a table in which ε is given and δ should befound (...)

John comes up with part of the inequality for the δ-neighborhood (line 7).The teacher adds some necessary elements. Like the absolute value signs (line8-9). Otto estimates correctly the upper boundary of the δ-neighborhoodthat fits the given epsilon (line 15).

In the end we tried to extract the formal definition of limits out of theepsilon-delta game in the neighborhood relation.

Transcript 15

A. How can we finish this discussion that we have started from thebeginning? How can I write what I am saying in general (...)?

A. (...) yeah, I say whatever ε I give you, you should find a δ to giveme otherwise it doesn’t have a limit. I mean the limit doesn’t exist.If the limit exists, for an ε I give you, you should be able to give5

me a δ. Do you know this symbol ∀?O. No.

A. Oh, it means for every. All, it means all. They are the logic symbols.

J. The A from all is,

A. Probably. All εs, ∀ε, what is the limitation that ε has? The only10

limitation that ε has.

J. it can’t be negative?

A. Yes, for each ε which is bigger than 0, there must exist a δ. mustexist is shown with this symbol: ∃. Exists δ, again delta is positivesuch that, such that what we talked about today. Such that |y−3| <15

ε⇒ |x− 1| < δ1. It means that we can get close and close and wedon’t have any limitation to get closer. We have different types oflimits like the limits of infinity or limit at infinity. We will discusslater.

1Here I made a mistake and I didn’t notice at that time. One main reason for such amistake was our lots of discussions in neighborhood topic, while ε was given and δ wasasked. This mistake made me write the formal definition wrong for the first time, althoughit was corrected later in the later sessions.

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D. What type of limit is this?20

A. This is the limit of function when x tends to a number. (...)

As it is traceable in the transcripts, we cannot see yet if students were able tofollow the step from trying particular values of epsilon and delta to arguingabout “all” values of epsilon.

The geometric interpretation of absolute values seems to be needed andthere is ample evidence (transcripts 12, 14, 15 line 12) that students can useit. They are interested in the earlier paradoxes and they are keen in pointingout that we don’t follow a very straight route in solving them (transcript13). The pedagogic use of “continuous functions first” might need somerethinking.

5.7 Epsilon-Delta Proof & Mathematica

After a short recapitulation of the problems we wanted to solve (Zeno,Archimedes, Newton) and a recapitulation of various formal and informaldefinitions of limit (see 4.1.8), a first epsilon-delta proof was tried, as an ap-plication of the limit concept to an “obvious” case: limx→2(2x). The teachingplan for this interactive demonstration was as follows:

1. Introduce the case limx→2(2x), and emphasize that this is not one ofthe problem cases like Zeno. Draw the graph of f(x) = 2x and statethe suspected limit value: 4. Let’s see if the complicated definition cancover also this obvious case.

2. Ask for proposals to calculate a fitting delta, for a given epsilon.

3. Ask whether a bigger or a smaller delta would also do.

4. Write the choices made on the board in the format: “assumptions” and“to prove”

5. Ask for proposals how to get from the assumptions to the desired con-clusion.

The discussion regarding steps 2 and 3 went as shown in Transcript 16. Whilethese discussions occurred, it is important that the definitions of limit (see4.1.8) are still on the board.

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Transcript 16

W. (...) How should I choose my δ? What calculation to ε should I doto get the δ (on the board: δ = ε

?)?

D. divided by half?

O. By 2.

D. By 2.5

W. (writing on the board and dividing ε to 2 ) δ = ε2

that seems safeisn’t it? Would this one: δ = ε

4also be right?

D. No not with this formula.

W. So then it wouldn’t fit exactly in this place that I’m allowed (point-ing to the interval (2 − δ, 2 + δ)) it would be much more. I would10

choose this δ.

O. But it’s also right.

D. Ah.

W. It’s also right, it ensures that my f(x) fits with the chosen ε. So2 is right, 4 is right, 3 is of course also right (the he changed the15

denominator to 1.5 ).

O. No wrong.

W. Yeah I agree. So let’s take just one of the right ones and see if wecan do the proof with that because the definition doesn’t requirethat we come up with an exactly fitting δ environment. (...)20

The teacher suggests a calculation, and the students are able to completethe suggestion. At first, Dave seems to think that delta is to be chosen suchthat the corresponding values of f(x) totally fill the corresponding epsilon-neighborhood. Otto (line 12) sees that this is not required by the definitionof limit, and when the teacher suggests a delta that’s too big, he recognizesthis as being wrong.

After this discussion, the teacher writes on the board what’s to be provedand what can be assumed, as shown in Figure 4.

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Assuming:0 < |x− a| < δδ = ε

3

(⇒ |x− a| < ε3)

To prove:|f(x)− L| < ε

After this, the discussion happens as shown in Transcript 17.

Transcript 17

W. (...) And in this case it means we have to prove: |f(x) − L| < ε.Any ideas? (silence) Can I get from here (assumption) to there(proof )? (silence)

O. You can say by the δ, x− a. Then you can prove it.

W. Oh I had to fill numbers here (then he replace a by 2 and L by 4 ).5

Then I can do it? How?

O. eh, 2x− 4 is the same as x− 2. Or not?

W. you multiplied this one (|x− 2|) by 2?

O. Yeah.

W. So 2× |x− 2| = I say 2× δ or I say,10

O. 2× ε eh,

W. = 23ε isn’t it?

O. Can you do the 2 inside the absolute value? To get |2x− 4|.W. I think you can, I think you can. Let’s look at it critically after-

wards. Then I have this one: |2x− 4| < 23ε Am I ready now?15

O. Now you have now the same as,

W. I am sure that when the absolute value is allowed then I am surethat this one: |2x − 4| is smaller than 2

3ε. Is it then also smaller

than ε?

O. Yes.20

W. You are sure of it, isn’t it? And this step that we take here is onlyallowed because ε is positive.

J. How did you do the last step?

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W. If ε is a positive number, then taking 23rd of it makes it always

smaller.25

J. Yeah, okay.

W. So, also I know for sure that this is smaller than that: |f(x)−4| < ε.

J. Yeah.

W. So now I am ready. I did prove what I wanted to prove. Then it’sstill interesting to look at this step why is it allowed and is it always30

allowed, does there exist a rule that says a × |b| is always equal to|ab| because that’s the law we have been using. Is this law alwaysright?

O. I tried it with,

W. But is it always right?35

D. No. It isn’t. Because when you get a negative a and you get anegative answer for the, the other ones is always positive.

W. So for negative a it’s not good. For positive a I am quit sure aboutit. b can be anything. It can be positive or negative, ..., verygenerally it is true if: |a| × |b| = |a× b|. if I don’t know what a is, I40

can safely take the absolute value of it and then they will be equal.

W. What did we prove? We didn’t say what ε was. I did that onpurpose, I didn’t start talking about now ε is 1

100, now ε is 1

10no

word about that. For all ε however small they go, we proved thatwe can find fitting δs for it because this [δ = ε

3] was our guess and45

our guess proved to be right. It proved to guarantee that the f(x)is within the ε environment. So we can really get as close as wewant.

From the assumptions, this one is taken: |x − 2| < ε3. Following Otto’s

suggestion, the left side is multiplied by 2, but then the teacher insists theright side is also multiplied by 2 (line 12). Otto inquires whether the 2can be taken inside the absolute value (the teacher had not done that yet).This suggestion is followed (discussion whether it’s allowed is postponed: ithappens in lines 29-41). As a third and final step, it is discussed if from|2x − 4| < (2

3)ε we can or cannot conclude that it’s also < ε? Otto thinks

yes but did not give an argument. The teacher (perhaps too quickly) fills infor him.

From these discussions we can conclude that the students can follow theargument enough to be able to fill in some details, which is not the same

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as to say they understand the whole of it. The fact that Dave and Ottocan contribute to specifying a recipe for calculating fitting deltas for a givenepsilon (Transcript 12, and 16) is interpreted as a sign that they can read thedefinition of limit well enough to draw this consequence from it. John hadearlier formed such a recipe all by himself (see transcript 10, lines 46-54).

Of course, this rather simple example should not have convinced thestudents that limits or epsilon-delta proofs are very useful. The next discus-sion (Transcripts 14 and 15) again shows some doubts that students hadabout artificialness of our examples (compare section 5.6), and it showsthe teachers busy explaining that some of it is actually close to reality.However, the whole discussion started off with a detail of the definition:0 < |x− a| < δ ⇒ |f(x)− L| < ε.Does the first part, before the arrow, mean that x can not be a?

This question is then linked to various cases of discontinuity, e.g. whencalculating an instantaneous velocity, the case of zero divided by zero.

Transcript 18

A. I have a question. Does it mean here that x can not be a? Becausehere h can not becomes 0. (A asked this question because I knewthat John still has question here) Here h can not become 0, butthere x can be a. So, how?

W. The definition of limit is very cautious, it assumes that we might5

be considering the problematic cases. So it requires that we keepx distant from a. You may apply it to a function where there isno problem then for that function, x can become equal to a but assoon as we start considering the limit, we leave out the case wherex is a. So there is a whole neighborhood around a that we consider10

but with a hole in the middle.

J. Yeah, yeah.

W. We can get δ to the right or δ to the left but the middle is notregarded.

A. But here the problem is that it’s a fraction. h can not be 0 be-15

cause it’s a fraction and we can not have 0 as the denominator, or00. Okay? but here in this line we don’t have any problem to put x

equal to 2. We don’t remove it. (to John) DO you remember youasked me x can be 1 last week? It was 1

x,

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Harm entered at this moment and joined our discussions. I ex-20

plained John what had been his question and when he asked hisquestion again from Wolter in Dutch also.

W. That really means that we are not considering that they becomeequal. When you look at the definition, I consider the whole neigh-borhood, the distance must be larger than 0. So x and a must not25

be equal. It may be the case that the function does not forbid thatthey are equal but when I try to find the limit, I must disregardthat fact.John continued asking the question in Dutch but his problem wasthat how is it possible when x can not become the number to which30

it tends to (like in the cases when the function forbids us to do so),the limit of the function becomes exactly a number. In other word,when x can not become equal to a, how does the limit of f(x) be-comes equal to L? (We will see later that it remained as a resistantmisconception for John)35

W. Yeah the limit is exactly 4 because limit is a special number thatwe can approach as close as we want. And we just proved there isonly one such number, it’s 4.

So, this was mainly a discussion between two teachers, while a third teacherjust dropped in. The answer is that, when looking for the limit, the casex = a is disregarded, to take care of cases like 0

0. We don’t know what the

students think of this, because they did not join yet.Then John asks the question: if x is not allowed to become equal to the

number to which it tends, how is it then possible that the limit of f(x) be-comes equal to L? Transcript 15 shows the discussion about this question.

Transcript 19

H. (to john) Do you understand? When the x is coming very close to2, the function is becoming very close to 4. So when the differencebetween x and 2 is smaller, then the difference between the 4 andthe function value is also smaller. And when you want to make thedifference between the function value and 4 as small as you want,5

there is always a neighborhood to find.

J. yeah but it’s not precise.

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H. But you don’t say the function value is 4, you say the limit, he limitis 4. ( John says something in Dutch, I think he says that this is theword which makes everything work or something like this, actually10

the same idea that he had at the end of Newton day)

A. John asked last week that in this case 1x, x can not become 0, it can

tend to 0. but if x tends to 1, then what is the problem? Is thereany problem?

H. When x becomes 1, the limit goes to 1. It’s a very easy question.15

it’s 1. When x tends to 1, function value tends to also 1.

J. Yeah that’s just because you can’t divide it by 0.

H. But when you can’t divide by 0,

W. In many cases when you have x approaching to a number, the func-tion value approaches to,20

J. But why would you say it’s approaching? Because if it’s 1, whycan’t you just say it is 1.

W. Yeah you are right.

H. It is 1, ... when the function is continuous. So take this function: x2

x

but x is not 0 because it’s forbidden. And 2− x is 0. You can take25

every number you want, when you make a graph of this function,the graph is like this: just a line, a hole, for every number you takex2

xso when you have 4, it’s 16

4= 4. When you have 5, the answer is

5. When you have 0, the answer is not 0 because 00

does not exist.So you say you have this function for when it is not 0 and 2 for30

when it is 0.He wrote on the board:

f(x) =

{x2

x; when x 6= 0

2 ; when x = 0

J. This is an extra function so. (He means the f(0) = 2)

H. This is the function.

J. Yeah, okay.

H. And for every x there is a value. When you make the graph of this35

function, you have this graph. There is a hole because it doesn’texist by x = 0. A hole in line and there is a point, one extra point,2 on the y axis. Okay? And when you have to look the functionx→ 0 in this function, what’s the limit?

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O. It’s 0? I think it’s 0.40

H. And why isn’t the limit 2?

O. Because you don’t do 0.

H. Because it’s forbidden to take 0 in the function.

W. And because the definition of limit prohibits them to be equal. Weneed to look at the neighborhood.45

H. So the limit is 0. And the function value is 2. And when you lookat the graph, when you examine your animal and you are walkingon the graph toward 0, the close you are to 0 for x, the closer thefunction value becomes to 0.

J. Yeah, okay.50

H. And the function value never tends to 2.

D. But 2 is then just an optional number, you can choose everything,

W. When I define a function,

H. You can define a function just like you want, just in some system.

W. Often in secondary school, you define it by just one formula, but55

you can write a sentence to define a function. Doesn’t need to be aformula.John showed Harm his calculator with a function given to it andwhen Harm had a look at it he said:

H. For this one, for 0, you should have a look at the table, and when60

you have a look at the table, you can do trace but you also can havea look at the table. In that case, it gives an error.

J. Oh, yeah.

H. There is a possibility but it’s quit difficult to also put this (pointingto something on the board) on your graphic calculator and for this65

only one point it gives a hole but it’s only one point it’s so small. Itgives a difficulty. And only it’s such a small hole you can’t see it inyour graphic calculator. But the hole exists. You can think thereare crazy mathematicians who can make such functions but youcan see them by cruise control in the car, then you have a function70

like this: (drawing a graph on the board) That’s the distance andthis is speed: (drawing another graph on the board). Then try toimagine what’s happening to the acceleration? The distance you’vedriven, this is the moment you put your cruise control on. Thisis the speed is going up then you put your cruise control on (he75

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showed a constant speed by his hand movement). Acceleration is:(drawing its graph on the board), and then it makes jump and thenwhen you have the limit from the left or you have the limit from thetwo at this time acceleration doesn’t exist. When you take a stillball, and you drop it on the floor, try to imagine what’s happening80

to acceleration, to the speed. Speed is going down and it is on thefloor, it’s going up. Is that a continuous function?

D. It’s increasing and then,

H. A steel ball bouncing on the floor that gives this kind of function.And there is interesting the cruise control. Everything in industry,85

when something is completely changing, in a machine, in fabric, inthat case you have such functions.

W. And also in the car when you are just driving and suddenly yousee someone jumping over the road and you press the breaks veryhard then from acceleration 0 because constant speed suddenly the90

acceleration becomes negative. Very negative, when you press thebreak.

H. That’s the time when you are sitting in the train when the trainstops, the very last one hundredth of seconds you feel a movementin train.95

J. Because if the acceleration would be 0, then you have a constantvelocity.

W. Jumpy functions.

H. Functions that jump, functions that go from one value to an otherand they are not, it’s not possible to draw them with the pencil100

without putting your hand up. (...)

From the fact that x needs to be different from a, John concludes that thelimit will be an approximation, it will not be precise (line 7). Andi brings inthe case when there is no problem at x = a (line 16). John says that in sucha case the “approaching” terminology is overkill (line 21), you could just aswell calculate f(a) directly, no limit needed.

Nobody disagrees with that, Harm goes back to a case where there is aproblem at x = a (line 24), he shows that his example function,f(x) = x2

x,

has a hole in its domain at x = 0, and he plugs the hole by arbitrarily definingf(0) = 2. Asked about the limit limx→0 f(x) Otto rightly guesses 0, although(as Harm stresses) the function value is 2.

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Dirk rightly remarks that f(0) = 2 is an arbitrary choice (line 52), itlooks like he regards the function given by Harm as artificial. This starts theteachers to come up with examples where discontinuities happen in real life(lines 68-101).

This discussion may have been useful to answer doubts about artificial-ness of given examples, however, it does not go very deep into John’s (line7) argument that a limit is an approximation. We cannot see here if weconvinced him, but from the test and a final interview it appears that wedid not. Again, our “simple function first” approach proved problematic asalready noted in section 5.6.

5.8 Continuity

Our analysis of students discussions on the continuity topic is divided intotwo sections (see the plan in 4.1.9);

• Do students understand the formal definition of continuity well enoughto contribute to a proof of the continuity of polynomials of the firstdegree?

• Are students able to check the continuity of some functions, and indoing so, distinguish between zero and infinitesimal in an adequateway, resulting in adequate distinction between the various cases whereexpressions are defined, undefined, zero, infinite, or indeterminate?

5.8.1 Formal Definition of Continuity

In this session, the formal definition of limit was applied in formulating theconditions of a continuous function in the formal language. That is, thelimit from the right must be equal to the limit from the left and they mustbe equal to the function value at that point. Using ε-δ language: ∀ε,∃δ,0 < |x− x0| < δ, implies |f(x)− l| < ε. In this definition, l was replaced bythe function value at x0 according to the definition and we arrived at thisdefinition for continuity: ∀ε,∃δ, 0 < |x− x0| < δ, implies |f(x)− f(x0)| < ε(appendix 8.3.7). Then we proved the continuity of the polynomials of thefirst degree in the general form.

Unfortunately the video of finding the formal definition of continuity ismissing and the evidence in this paragraph is only the teacher’s memory.

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After reviewing the definition of continuity and the conditions of continu-ity, Otto mentioned that for finding the definition of limit we should simplyreplace l by f(x0). We conclude from Otto’s contribution that the formaldefinition of continuity was easier to grasp using the formal definition of limit.

Transcript 20

A. (...) I want to first give you another definition of continuity whichis the same but in another way. So if the limit of f(x) when x istending to x0 is L, see this is a limit I write the definition of limit forthis (pointing to the slides) I want to prove that the limit of ax+ bis L and is equal to this function value, f(x0). Recalling from the5

definition of limit, we first take an ε near the L, so you see (drawingon the board the line ax+ b and taking a neighborhood ε around L)so that |f(x)− L| is less than ε. What is the next step? Find ...,

O. δ.

A. Give me a neighborhood of δ near x0 such that that δ guaranties10

that our f(x) lies in Bεl. The definition of limit; for each ε youshould find a δ such that if we have this neighborhood, (pointingto the neighborhood on the x axis), then we have this neighborhood(pointing to the neighborhood on the y axis, also the definition ofthe limit on the screen). L should be equal to f(x0) so can we put15

f(x0) instead of L and it could be the definition of continuity. Is ittrue? So if the function is continuous, here L is exchanged by f(x0)then the function is continuous. Because it has limit and the limitis equal to its value at that point (...).

A. (...) I want to prove that the polynomial of the first degree, let20

me show it with f(x) = ax + b, is continuous (...), so what are theconditions for function to be continuous.

O. The limit from the left must be equal to the limit from the rightand the function value also.

A. The function value also should be equal to the limit of the right and25

left. So let’s see if it for the polynomial of the first degree (...).

A. (...) First step, take any ε, but for that ε you should be able tofind a δ. I put δ red here because I don’t know what it is and I’mgoing to find it. I’m going to find a specific δ which guaranties thisrelation (writing: Step 1: take any ε, Step 2: Find δ such that if30

0 < |x− x0| < δ ⇒ |ax+ b− ax0 − b| < ε) (...).

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J. (...) Is this −ε? Then it can not be bigger than 0.

A. What do you mean by bigger than 0? ε is a value, a value cannot be negative. The proof is like this (pointing to the slide: 0 <|x− x0| < δ ⇒ |ax+ b− ax0 − b| < ε),35

O. Then you can factor them out.

A. Can you write it on the board? Otto multiplied the left side by aand wrote the relation but he didn’t know what to do with it. ThenJohn went to the board and we had discussion about the propertiesof inequalities. then I followed the proof ’s steps as below:

0 < |x− x0| < δ ⇒ |a(x− x0)| < ε

0 < |x− x0| < δ ⇒ |a||x− x0| < ε

0 < |x− x0| < δ ⇒ |x− x0| <ε

|a|and finally the last step,

|δ| 6 ε

|a|⇒ |x− x0| < δ ⇒ |x− x0| <

ε

|a|

And we had some discussions about it. John explained it in Ducthfor Dave and he noticed the meaning by reinterpreting it. We fin-ished with reviewing continuity and I started asking some question.

Students participate in proving the continuity of f(x) = ax + b. Otto andJohn are contributing to the proof process while Dave is more following thediscussions (transcript 20 lines 36-40). The absolute value in the definitionof continuity: ∀ε,∃δ, 0 < |x − x0| < δ ⇒ |f(x) − f(x0)| < ε makes Johnthink that ε could be also negative (line 32). Dave and John seemed to befollowing the proof although they do not speak too much, so our evidencethat they understood it is scanty.

5.8.2 Investigating the Continuity of some functions

In the next part, investigating the continuity of different functions, we probedthe differences between zero and infinitesimals. In lines 10 - 30, we discussthe cases when the function is absolutely zero and the floor function is intro-duced. Then in lines 31 - 40, the undefined rational functions which have 0 in

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their denominator and the fact that they do not have a limit was discussed.Then in lines 48 - 65, the example of the determined rational functions whichhave an infinitesimal in their denominator was discussed. In lines 67 - 76,we discuss the limit of the rational functions which become absolutely zero.Finally in lines 82-112, we discussed the limit of the rational functions whichare indeterminate. In lines 89 - 94, the reason why they are called indefiniteor indeterminate was discussed.

Transcript 21

A. What is number0

?

O. impossible, error by dividing by 0.The other two had the same idea that it is not possible.

A. Now let’s think more mathematically, it is undefined, but what is0? Let’s speak like this: when I say x→1, 1

x−1, do you think the5

denominator is 0?

D. No.

A. What is then?

J. Very close to 0.

A. Yeah infinitely small or infinitesimal. This is not 0 let’s ask this10

question, when do we have really 0? in the limit of which functionwe can have really 0 not infinitesimal?

D. When there is 0 in it, when there is no x in the denominator?

A. When do I have 0 in the denominator?

O. When -1 is very close to,15

A. But I want just exactly 0. Not very close to 0 but 0. Well one is todefine a function 0 in a point (I wrote on the board an example of acase function with value 0 for one of the cases). Then it is 0.

J. Also you can have x divided by 0 then you have 0.

A. have you ever seen x0? It doesn’t have any meaning.20

J. Yeah.

A. But there is another function that gives you exactly 0 at somepoints. Do you know the step function? (I explained floor func-tion for them, Otto knew about it but the rest learned about it there(probably). My next question was what is limx→0[x]. We had to25

divide it into 2 cases, one when x tends from right and one when

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x tends from left to 0. Then when tending from right, the functionvalue was absolute 0 not close to 0 and when tending from left itwas -1. Dave asked for repeating the definition of the floor function.I explained again.)30

A. Do you remember what did we do last week and also in your home-work? We had the limit of some functions that in the nominatorand denominator you had to factor out something that made themzero, but were they really zero? No, they were infinitesimals. Theywere not really zero. When we have fractions whose denominator35

is zero, I mean exactly zero, it’s undefined. Do not search for itslimit, it’s undefined. The cases that we are searching, the denomi-nators are infinitesimals. I will give you different examples of eachone. Then I wrote three examples of the cases where the function isdefined in the point that x is tending to.40

A. Then I wrote limx↓1x2+1[x−1]

, what does it become when is tending to1 from right?

J. A little bit bigger than zero?

A. No,

O. Zero.45

A. John noticed that he had not paid attention to the [] which showsthe floor function. So it’s 2

0, isn’t it? It’s undefined. Don’t try to

find a limit for it. But the same function when it’s tending to 1from the other side, it’s not zero. The third example, (John askedabout the same example when x tends from left to 1 and I explained50

that it will be -2 ) limx→1x2+1x−1

. Then it’s a number over what? It’s2?

J. It can be from left or it can be from right. If it tends from right side,it would be positive denominator, if it is from left side, it would benegative denominator.55

A. (First, I did not notice the point that John was making but after ashort discussion about checking the functions from left and right, Inoticed his point. He was speaking about the limit of function notonly about the denominator. We noticed his point when he wrote iton the board.) Do you agree if I write +

−infintesimal in the denom-60

inator? What happens when I divide 2 by a very small number?

O. It becomes infinity or minus infinity.

85

A. So the answer for this is plus or minus infinity. So each of thesecases are different (pointing to the first example), this is defined.(pointing to the second example), this is undefined. (pointing to the65

third example), this is dividing a number to an infinitesimal. It

could be plus or minus infinity. The fourth example is limx↓1[x−1]x−1

.What is this step function when tending from right?

O. Zero.

A. Do you agree? It’s exactly zero. And what about the denominator?70

J. It’s infinitesimal but plus.

A. So it’s just plus infinitesimal. (writing on the board: 0+infinitesimal

),Ok what is the answer?

O. It’s zero?

A. Zero.75

D. But it was plus and minus if it was a normal function?

A. (I explained the same example for the case x tends to 1 from left.)The upper part becomes minus one and the lower part becomesminus infinitesimal and it becomes plus infinity. It is very importantfor me that when you see a limit, make distinguish between the80

zeros, the infinitesimals and zero. And the last one that we workedon it last week, is this example; limx→1

x2−1x−1

. What does it become?

O. It’s x− 1.

A. Before factoring out, before doing anything. It is = limx→1 x2−1limx→1 x−1

whichbecomes,85

J. Zero.

A. What is zero? (pointing to the nominator) Does it become zero?

J. I think so. Oh wait, no, wait. yeah, well!

A. Isn’t it infinitesimal? = lim infinitesimallim infinitesimal

but we don’t know which kindof infinitesimal is the upper part and which kind of infinitesimal is90

the lower part. In these cases that both of the nominator anddenominator are infinitesimal, the English word for it is indefinite.It means that we don’t know, both of them are infinitesimal butwhat would be the answer? This was the case that we worked withlast week. We factored out what made the fraction zero, then we95

removed them together, this is what we did last week. For instancehere, = limx→1

(x−1)(x+1)x−1

= limx→1(x+ 1) = 2. It seems very simple

86

but now you know what is behind it. This is not 00, this is not

right for mathematicians. But we usually do that. We usually showinfinitesimals as zero, but for you, you should make distinguish when100

it is infinitesimal and when it is zero.

O. So, the answer will be two?

A. Yeah. So, all what we do in finding the limit is in this case, cause inthe other cases, we don’t have any problem. This is the case whichis indefinite.105

J. So for (moving his hands to show the graph of a function),

A. 1x,

J. Yeas, it’s again plus or minus (moving his hand to draw the symbolof infinity),

A. Infinity? Yeah for John, we will reach to it later.110

J. That was the answer to one of the last test?

A. I don’t remember the order, what was it? (...)

In discussing the third example, number divided by infinitesimal, John dis-covered that, the left and the right limits must be evaluated separately be-cause they are different and it influences the final answer dramatically (lines53-54). In line 62, we can see that Otto also joins the discussion. In thefourth example which is absolute zero divided by infinitesimal, Otto andJohn mention that it is zero (lines 70-74). While Dave is thinking about thelimit from both sides not just from one side (line 76). In explaining the fifthexample which is about an infinitesimal divided by another infinitesimal, in-determinate case, John connects it to one of the questions of the test whichwas indeed a defined asymptote case, 1

x(line 106-111).

We conclude that students’s attention was drawn to the various specialcases and also that they understood and were able to handle those cases: thecase when a function does not exist in a point, the cases where either thenumerator or the denominator, or both, are absolutely zero or are infinitelysmall and the case of one sided continuous functions. In handling these cases,students show understanding of the difference between zero and infinitesimal.

5.9 Test and Interviews

Each of the questions and students’ answers to them are described separately.The test was asked two times, once right after the end of teaching the formal

87

definition. The next time was in the end of the block as an interview 3 weekslater. Only some of those questions which were answered wrong in the firsttime were asked in the interview. So the questions asked in the second timeare not the same ones from each of the students.

5.9.1 Question 1

1. Which of these sentences best describes a limit as you understand it?

(a) A limit describes how a function moves as x moves toward a cer-tain point.

(b) A limit is a number or point past which a function can not go.

(c) A limit is a number that the y-values of a function can be made asclose as we want by choosing the x-intervals as small as needed.

(d) A limit is a number or point the function gets closer to but neverreaches.

(e) A limit is an approximation that can be made as accurate as youwish.

(f) A limit is determined by plugging in numbers closer and closer toa given number until the limit is reached.

The first question aimed to check students’ understanding of the limit con-cept. All except one of the sentences are students’ common misunderstand-ings of the limit concept. Choosing the right sentence which describes thelimit concept correctly does not guarantee that they have learned this con-cept. Choosing the wrong answer could be an indicator that the concept isnot properly understood though. In our case studies, Dave and Otto chosethe correct answer, while John chose the wrong one. John was asked to ex-plain why he chose to describe the limit as a number or point the functiongets closer to but never reaches. He explained that he was thinking aboutchoice (c) which is the right one as well (lines 3). But what had made himchoose (d) was dividing by zero (lines 7-10). The discontinuous cases suchas while the variable is tending to the root of the denominator of a fraction.His selected sentence is true for some phenomena but not true as a definitionfor limit. He noticed his mistake in the end of our discussion (lines 11-26).

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William’s models of limit Dave Otto Johna. A limit describes how a function moves as xmoves toward a certain point.b. A limit is a number or point past which a func-tion can not go.c. A limit is a number that the y-values of a func-tion can be made as close as we want by choosingthe x-intervals as small as needed.

Yes Yes

d. A limit is a number or point the function getscloser to but never reaches.

Yes

e. A limit is an approximation that can be madeas accurate as you wish.f. A limit is determined by plugging in numberscloser and closer to a given number until the limitis reached.

This conversation (transcript 22) took place right after handing in thetest:

Transcript 22

A. You chose (d) : A limit is a number or point the function gets closerto but never reaches.

J. Yeah, I also thought about (c).

A. But why did you answer (d)? I mean it says that a number orpoint the function gets, it means f(x), becomes closer and closer5

but doesn’t reach to it. Can you explain why?

J. I think I am wrong because that’s only when you divide by 0 isn’tit?

A. You mean in this case: limx→a1

x−a?

J. Yeah. (...)10

J. eh, even denk hoor! I am thinking about an example. I just thoughtabout, I think I messed it a little bit up with instantaneous velocity.Two points and bringing them closer and closer.

A. But even in that case, we saw that the average velocity becomesexactly equal to instantaneous velocity when h is tending to 0. h15

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couldn’t become 0 but the limit of the function becomes exactlyequal to the,

J. The limit of the function can be?

A. Yeah. So here is a limit, limit is a number or point the functiongets closer to but never reaches. You see?20

J. But this can be true, can’t it? Like with the, even kijk hoor. Likewith a gap in the middle. (...)

J. (...) But you say here that the function can never reach that pointbut the limit can. That’s how I interpreted.

A. But is it a definition for a limit?25

J. No it’s just for one case, or not? I thought about (c).

The same question was asked from John once again in the last interview atthe end of the block.

Transcript 23

J. It’s the test we made long time ago?

A. Yeah. Please think about the first question and see if you give thesame answer.

J. Yeah why?

J. That’s what I think it’s a limit (He is pointing at answer (c)).5

A. I remember that you chose (d) can you say that why did you changeyour mind?

J. Well I thought then the limit couldn’t reach the point but now Iknow the limit can reach the point even if the function does notexist at that point.10

5.9.2 Question 2

2. Please describe in a few sentences what you understand a limit to be.That is, describe what it means to say that the limit of a function f asx→ s is some number L.

The second question could clarify better how they have learned the limitconcept since they have to describe the limit (L) of a function (f) when thevariable (x) is tending to a specific number (s).

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Dave When x→ s then f(x) goes to point, this point is L and it is thelimit of f(x).

Otto When you normally goes with a x to s, you will come closer andcloser to L. When you take the limit of x → s, you go infinitetimes closer and closer with x to s but never reach it, you will getL.

John A limit you can find an instantaneous slope just by bringing a pointB as close as you want to a point A. Now you can also draw a linethrough these points, or just use ∆y

∆x. If f(x) hasn’t got an answer,

the limit has got the answer.

Dave’s description does not clarify what he has learned since he is repeat-ing the same sentence which is in the question by using symbols. Althoughthe fact that he believed the limit of a function is the value of the functionin the point (transcript 13) is not actively contradicted. We had not studieddiscontinuous cases at the time they are answering these questions.

Otto seems to be thinking of some kind of neighborhood concept. Hementions tending by mentioning that when x is moving towards s, we (f)becomes close to L and at infinity the limit becomes L.

John on the other hand links it with the instantaneous velocity and triesto link it to the slope of the tangent line to the curve. The concept ofneighborhood could be traced in his words by using the words “bringing apoint B as close as you want to a point A”. John emphasizes that if f(x) or∆y∆x

is undefined at the point A, then the limit could exist.This question was asked once more in the end of the whole teachings but

this time orally in an interview. Their responses are below:

Transcript 24

Interview with Dave:

D. (...) You can choose a (thinking), you can choose a number for xand as close to a as we want, such that f(x) comes close to L.

A. What is important in the definition of limit?

D. eh,5

A. the same thing that you said but in another word. You say thatwhen we come closer to a we can be able to be closer to L so whatdoes it mean?

91

D. Neighborhood?

A. yes neighborhood so,10

D. You can choose a number in the neighborhood of a such that f(x)is closer to L (in the neighborhood of L).

A. So if you be able to find such numbers, then this limit is equal toL. (...)

Interview with Otto:15

A. (...) I mean okay x is tending to s and f(x) becomes L. What doesit mean?

O. When you go choose a point, infinity space between points and xand the limit,

A. The neighborhood you mean?20

O. Yes the neighborhood is infinity small, then you get the limit. (...)

Interview with John:

A. (...) Okay, can you give a definition for the limit? For instancewhen we say that the limit of a function when the variable tends toa number is L, how do you describe it?25

J. Even kijk hoor, what was this L?

A. The limit of the function.

J. Yeah the limit is (we were interrupted) by decreasing the x thenyou come to a certain point this is a limit. (...)

Students’ oral responses to this question does not reveal any improvement.Their weakness in speaking spontaneously about the definition could havetwo reasons:

1. They have forgotten the definition.

2. The definition requires writing down and needs some time to thinkabout all the elements of the definition and sort them in the correctorder.

5.9.3 Question 3

Given a function f(x) = 3, evaluate the limit of f(x) as x→ 2.All of them answered this question correct. This question was asked

to check if they have difficulty to apply the limit concept in a case where

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the function is continuous since it is then in contradiction to some knownmisconceptions such as tending and not reaching, etc. Apparently the factthat the function is continuous and constant had not made them to thinkthat the limit of the function can not reach the point. So, our “simplefunction first” approach caused the evading one of the so called William’smisconceptions.

5.9.4 Question 4

4. A student was given a function F and asked to find the limit of F as xapproaches 0. He plugged in numbers on each side of 0 and made thefollowing table:

What can you conclude about the limit of function F as x approaches0?

Students often have the idea that a limit can be determined by calculatingf(x) for a limited number of x-values. This question is used to check thatidea. To make the wrong answer maximally believable, the question givesnumerical data that approach the suggested limit value from the left as wellas from the right. Both Dave and John have concluded that the limit of thefunction is 1. But Otto is more careful and he writes: ”The limit x → 0 isprobably 1, but it can also being 1,000...001” or 0,99999999999.” It is clearthat his first reaction is doubting if the limit is 1. Depending on how tointerpret the ... (dots), his reasoning might not be totally right. e.g. ifby ... (dots), he means an infinite number of zeroes, then the remainingone would indicate an “infinitesimal” difference. And standard mathematicswould consider their number as equal to 1. We conclude that two of our threestudents show evidence of the misconception that a limit can be determinedby plugging in numbers.

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5.9.5 Question 5

5. Sketch the graph of the function f(x) = x2−9x−3

Answer the following questions:

(a) What happens at the point x = 3?

(b) Does the limit of f(x) exist at x = 3?If yes, what is this limit?

(c) What is the value of the function at x = 3?

This question was asked to check if the students have understood that afunction could have a limit at a point while it is not defined at that point.They had to first draw the graph of it. It was unexpected for me to see thatOtto and John could not sketch the graph because it had a hole at point 3.In response to what happens at point x = 3, they answer:

Dave a. The function is divided by 0, there is no point there.b. yes, it’s limx→3 f(x) = 6.

c. It doesn’t exist, because you can’t solve f(3) = (32−9)3−3

= 00

Otto a. That is impossible, because you can’t divide by 0.b. The limit exists because you never reach 3. So limx→3 f(x) = 6.c. The function doesn’t have a value at 3.

John a. There is a gap: you can’t divide by 0.b. yes. limit is 6.c. none

Since I did not expect that they can not sketch the graph, i asked Ottoone more time in the last interview to draw the graph. He started calculatingpoints using the calculator:

Transcript 25

A. (...) How do you sketch the graph of this function x2−9x−3

?He started to draw the graph by plotting a line through some points.I opened the fraction as (x−3)(x+3)

x−3= x+ 3.

O. Oh yes.

A. So you could draw just this.5

He then drew the line and said that in -3 there is nothing here(pointing on the corresponding point on the graph). (...)

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From the answers we could say that probably they have noticed that a func-tion can have limit in a point which is not in its domain. But, Otto’s answerto part (b) is remarkable. he says: “the limit exists because you never reach3. So limx→3 f(x) = 6”. Otto’s expression could have two interpretation. Itcould be in accordance with the formal definition of limits: “for all εs thereis a δ, for which 0 < |x− a| < δ implies ...”. What he says fits with the factthat in 0 < |x−a|, x is not allowed to become equal to a when we are lookingfor the limit. But it also could be interpreted as the concept of tending mightstill be vague for him. It seems that for him the limit for x → 3 only existsif the function is undefined for x = 3.

5.9.6 Question 6

6. Find the following limits:

(a) limx→2 3x+ 8

(b) limx→∞ sin(x)

(c) limn→∞(1 + 910

+ 9100

+ ...+ 910n

)

part (a) asks for the limit of a continuous function. Part (b) is the limitof a trigonometric function, sin(x), at infinity, which does not exist. Thisquestion was asked because the limit of the trigonometric functions was partof the book which was taught afterwards. So, it was just to check how dothey think about the limit of such functions before they learn about them.

Part (c) is the limit of a sequence and aims to check the concept ofinfinitesimal by using the limit symbol. The responses are given in tablebelow:

Dave a. limx→2 3x+ 8 = 14.b. limx→∞ sin(x) = 0.c. limn→∞(1 + 9

10+ 9

100+ 9

100n) = 2

Otto a. limx→2 3x+ 8 = 14.b. limx→∞ sin(x) isn’t possible.c. The limit is 2.

John a. 14b. -1c. 2

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Although they all answered part (a) simply correct, part (b) was morechallenging. Apparently Dave and John thought that this limit exists, whileOtto sees that it doesn’t. Part (c) seems to be clear for them since they allhave answered right. This could point to the fact that when limit symbolexists, students do not have problem in considering 9

10+ 9

100+ ...+ 9

100nequal

to 1. This is what they don’t do in question 8 when they don’t see the limitsymbol but they should pay attention to the same concept embedded in thequestion. Probably they do not consider 9

10+ 9

102+ ...+ 9

10nequal to 1! But

they consider its limit equal to 1. They see the sum itself as an infiniteprocess just like 0.999... endlessly growing! The reason could also be soughtin the meaning of three dots (...) for students. Three dots in this questioncould mean an arbitrarily chosen number of times. In this case infinity is notinvolved in the meaning of the dots itself. While the three dots in question8 represent the idea of infinity.

5.9.7 Question 7

7. Answer the following questions referring to the function graphed bel-low:

(a) limx→2 f(x)

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(b) limx→−∞ f(x)

(c) limx→1 f(x)

(d) limx→5 f(x)

This question was asked to check if students are able to interpret the limitof a function only by having its graph. Students’ responses are:

Dave a. |∞| b. 0 c. 2 d. 212

Otto a. ∞ (from the right side) b. 0 c. 2 d. f(2)John a. - b. 0 c. 2 d. 3

Dave puts ∞ in the absolute value symbol which might show that he isseeing ∞ as a number. We might interpret that he wants to indicate thatthe limit might be +∞ or −∞, depending on the direction of approach.However, we can not be sure that this is his meaning. In any case |x| is nota standard way of indicating this. Otto mentions only the approach fromright. However, from the right the answer should be −∞ not +∞. Maybehe is not thinking of the sign. It seems that they did not have much problemin interpreting the graph of a function with asymptotes, except John in thefirst question while x = 2 is the vertical asymptote and x → 2. Only Johngives the standard right answer: the limit does not exist, because the leftlimit and the right limit are different from each other, (we don’t know if thatwas his reasoning behind the “-”, though). Question b, c, and d were notdifferent for our three students.

5.9.8 Question 8

8. What is between 0, 999... (The nines repeat) and 1?

(a) Nothing because 0, 999... = 1.

(b) An infinitely small distance because 0, 999... < 1.

(c) You can’t really answer because 0, 999... keeps going forever andnever finishes.

(d) If you don’t agree with any of the above, circle (d) and give yourown answer.

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Students answered this question by choosing simply an answer among 4choices without bringing any reason and explanation. So, most of our dis-cussions about this question are in our last interviews. Students are askedthe question separately and their answers are presented in the transcript 26,27, and 28.

Dave bOtto In the limit 0.999... = 1. But when you write it down: 0.999... < 1.

This is like the Zeno paradox.John b

From our discussions with the three students except in one case, it seemsthat the idea that 0.9̇ is not equal to 1 but infinitely smaller than 1 is adominant idea. From these discussions, we can investigate the concept ofinfinity, either potential or actual infinity. In the potential view, infinity is aprocess of adding 9s which never stops. While the actual infinity representsa very very big number which is infinity1. Our stress in this research hadalways been on the process or dynamic view of infinitesimal and consequentlyinfinity since this process is more tangible and is experienced in the real life.On the other hand the actual infinity is a more abstract idea which does notshow up often in students’ minds (Monaghan, 2001; Fischbein et al., 1981).In spite of this fact, I had not to ignore the cases when a student might thinkof infinity as a very very big number which exists far from us. Since myown view was always a process based view, I had an unconscious promotionto such a view, as it could be seen in my questions posed to students. Thestudents also seem to have the same view, but it is not clearly checked inthis research.

Their responses to this question could show that they all have noticedinfinity, but they have not noticed the concept of limit in it. In discussingwith Dave, I tried to lead him to distinguish the limit notion by putting adifference between the cases where the 9s are a lot but finite and infinite 9s(lines 4-6). He recognizes the idea of limit quickly (line 8)2. After my trial torelate infinity with limit again, he noticed the concept better (lines 12,13).We discuss the notion of infinitesimal (lines 16-19 ), but it does not convince

1In this case, the 9s are decimal digits.2This limit is the limit of infinity which was taught in part 4 chapter 16.2 of the

book (mentioned in 4.3) which investigates asymptotes. The formal definition was notmentioned there for this case.

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him that infinite number of 9s after the . do necessarily stand for the limitconcept. But, its difference with 1 is an infinitesimal which always remainsan infinitesimal but not 0 without explicitly mentioning the limit (line 34).

Transcript 26

A. (pointing to the 8th question) Just read it again and see what isyour idea?

D. I think (b).

A. Okay, let’s discuss about it. I say that there are infinite 9s here andhere there are ten 9s. What is the difference between that case and5

this case?

D. The distance between this (0.999...) and 1 becomes smaller tillinfinitely small. So in limit they are the same.

A. Yeah in limit they are the same but when i say that they are infinite9s doesn’t it give you the sense of something in infinity?10

D. Can you repeat it?

A. I mean that when they are infinite 9s,

D. You mean that it shows, it is a limit more or less,We were interrupted by John who wanted to hand in his exam an-swer sheet.15

A. I just want you to think about the difference between this case andthe case 9s are infinite.

D. The difference is that you don’t know how small is the difference.(again a short repeating conversation)

A. When the difference becomes infinitely small, which of these answers20

better can be correct?

A. The calculators fix the number somewhere here because they do notunderstand infinity and infinitely small but they do approximation.So 0.99999 u 1 but we are not calculators so what is the relationbetween 0.999... and 1?25

D. I think still the same.

A. So if you are still the same you never reach to 1. Is it true?

D. Yes in theory it’s true.

A. In Zeno paradox, when we were walking, we knew that we reachto the point practically. But how did we answer the paradox in30

theory?

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D. With the limit?

A. With the limit. So,

D. So you should answer (d) and then start with the limit.

A. No because when I say there are infinite 9s here, infinite is not a35

number like 10100 is infinity, no. Infinity is not a number, so whenI am speaking about infinity indeed I am speaking about limit. Sowe don’t really need to say limit and pronounce it, I say eventuallyagain means limit. I say that there are infinitely many 9s here andeventually it becomes 1. So we don’t need to use the word limit.40

Otto on the other hand believes first that the difference between ˙0.9 and 1is nothing (line 2). And he thinks that infinitely small is equal to 0 (line4), but in choosing, he chose part (b). The idea that ˙0.9 is infinitely smallerthan 1 seems to be not so strong though. Since after the same reason that Ibrought for Dave, he said nothing again (line 13). He also mentions that thelimit of infinitesimal is 0 (lines 15-18).

Transcript 27

A. What is the difference between this number 0.999... and 1?

O. Nothing.

A. What do you mean nothing?

O. Infinitely small.

A. So which of them do you choose?5

He chose (b).

A. Okay I tell you that there are infinite 9s here, now I tell you thatthey are not infinite but they are 0.9999999999. Then what do yousay?He said that he would choose (b).10

A. So what is the difference between this case and the case that theseare infinite?

O. yeah, oh yes nothing.

A. Can you say something about the infinitesimal?

O. It’s like infinity you can never reach it. It is there still always. It is15

never 0.

A. It is never 0 but,

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O. In the limit it is 0.

John also believed that this difference is infinitely smaller than 1. In order todraw his attention to the concept of infinitesimal, I with him the Zeno casewhich is the same situation as in this question (lines 1-20). This similaritybetween the situations made him say that he does not believe that in theZeno case we arrive at the destination without applying limit notion (lines20,21). He then mentioned that by applying the limit the difference wouldbe 0. But still he was very strong that ˙0.9 does not necessarily show the limitnotion although he knew that he is wrong (line 30).

Transcript 28

A. What is your answer to this question now (8th question)?

J. I think (b).

A. If (b) is correct then what is the difference between this case andthis case?

J. The difference between these 9s and infinite 9s?5

A. uhum.

J. This one tends to 1 very very close and the other one.

A. What is the meaning of very very close?

J. infinitesimal.

A. But when the 9s become more and more then the difference becomes10

less and less with 1. Do you see any similarity between this andZeno?

J. With the steps?

A. Yes they became smaller and smaller and we saw later that yeah weare in the point. We didn’t say that we don’t reach to B. Eventually15

we proved that when we put infinitely many steps then we eventuallyreach to B. 1

2nwhich is the remaining distance does not only tend

to 0 but eventually becomes 0. Here isn’t it the same? The 9s areincreasing so what do you think?

J. Yeah again it’s the same as Zeno and infinite steps and infinite 9s20

and I would think it will never reach to 1.

A. I understand what you say but when we speak about infinity, whathappens at infinity?

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J. The limit will reach 1?

A. It becomes 1 but do we really need to use the word limit? Don’t25

you see the concept of limit here? At infinity it eventually becomes1.

J. Yeah then we don’t have to use the word limit.

A. Okay now you think that it can be equal?

J. Yeah but my heart says it’s (b) but my answer is (a).30

The remarkable point of this question is that the limit notion is implicit andrecognizing it requires a higher level of understanding this concept. Relatingit to the APOS theory, this question is not addressing the limit concept itself,but a new process in which limit functions as a prerequisite. In other words,maybe to conclude that a limit concept must be involved in 0.9̇, it is neededthat the process has already become an object. But in our case studies,probably it has not yet become an object (except possibly for Otto).

5.9.9 Question 9

9. Sketch the graph of the function f(x) = x−2x−2

(a) What is the value of f(2)?

(b) Does f(x) have a limit at x = 2?

(c) If yes, what is the limit?

Question 9 asks students to draw the graph of a function which has a hole.This is meant for checking how much the students have learned about discon-tinuities without being taught and only by understanding the limit concept.Their answers are:

Dave a. He drew the graph correctly. He writes: “it doesn’t exist, be-cause f(0) = (0−2)

0−2= 0

0and that’s impossible.” b + c. yes, it is

limx→ 2f(x) = 1Otto a. He drew the graph without putting a hole at x = 2 and writes:

“see 5a, there is no value.” b. Yes the limit of x → 2 = 1. c. see9.b

John a. He did not draw the graph and writes: “none”. b. yes c. 1

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Generally, the critical questions of the test which could help us draw/ hy-pothesize some conclusions were question 4, 6.c, and 8. All of these questionswere common in investigating students’ ideas about infinitesimals and limitnotion. Although they had some other aims as well. Checking the idea ofplugging in numbers in the 4th question does reveal that this idea was in factdominant in their minds. It was a weaker idea for Otto though. 8th questionhas the most focuss on the concept of infinitesimal and how it relates to thelimit concept1. Although this question was more or less the same case of6.c which was explicitly the limit of sequence, students did not recognize itssimilarity with 8th question with the limit concept embedded in it.

5.10 Interview About the Paradoxes

The interviews mainly had two themes; discussing those questions of thetest which were answered wrong (see 5.9); discussing and reviewing some ofthe paradoxes. The questions about the paradoxes and possible connectionbetween them and the test questions are given (some parts might be repeated)below.

5.10.1 Interview with John

Most of the interview with John is brought in explaining the test questions.Therefore, in reviewing the paradoxes in the end of the interview, he doesnot reveal any new point.

Transcript 29

A. Okay can you give me some explanation about Zeno and Archimedes?

J. Yeah well Zeno was also when he wants to go from A to B tak-ing the steps each time half the distance, every time smaller andsmaller steps and also by Archimedes the parabola also decreasingthe remaining distance to fill out the gaps and,5

A. Can you find any similarity between these 2 and Newton case?

J. Newton had 2 points and brought them very close to each other.

1Although it was just about the case when the independent variable tends to infinity,limit of sequence, and not about the case of the limit of a function when the independentvariable tends to a number.

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A. And he was finding the instantaneous velocity by finding the averagevelocity and make it smaller and smaller (...)

5.10.2 Interview with Otto

In reviewing Zeno’s paradox with Otto, he mentioned both sides of the para-dox and refers to limit as a solution for it (lines 1-11). He then brings thesame reason as a solution for Archimedes’ paradox as well. Since in bothof these cases the functions were continuous, also he had answered the 9thquestion wrong, more questions were asked to check what he thinks about thedifference between the continuous and discontinuous functions. Although hisresponses did not seem to be quick and fluent, he finally mentioned the pointthat in the discontinuous functions the function value could be not equal tothe limit of the function (lines 26-52).

Transcript 30

A. Can you tell me whatever you remember from Zeno?

O. When you, the walk paradox, when you walk the distance half andthe half and the half and you never reach it. But at home I cometo 1 with the limit, I solved it with the limit.

A. The limit of what?5

O. The limit of the sum, the sum of 12n

.

A. But you see in the Zeno we had (drawing a line and determining Aand B) A and B and assume this is 1 meter (showing AB). So westart walking first we come here then here then here and it neverstops. So Zeno says that we never reach to the end.10

O. Yes when you take the limit you reach it.

A. What about Archimedes?

O. Archimedes was parabola. The problem was that the space wasalways left. In the limit there was nothing more.

A. How can you relate what we said with Zeno and Archimedes? Are15

there any common, (...)

O. (...) They both try to fill the ....

A. (drawing the figure of the parabola with the inscribed triangles in it)The area of this parabolic segment is 4

3rd of the area of this triangle.

I reviewed the problem again and repeated my question again.20

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O. they both decrease.

A. Can they become 0?

O. In the limit it is 0.

A. So when it tends to 0 but it is not 0 what it is?

O. Infinitesimal?25

A. Infinitesimal yes, so in this question (9th question) what is the valueof the function at 2 (f(x) = x−2

x−2)?

O. (thinking) You can say 0 I think but I don’t know.

A. We are speaking now about the function value not about the limitso it is undefined but does the function have a limit at this point?30

O. Yes.

A. So.

O. can I write?

A. Yes please.

O. Infinity or minus infinity.35

A. Why?

O. when you take 2 point 0.000001, it’s always it goes to 0 and,

A. What about denominator?

O. Oh here also, eh it’s 1 yes yeah.

A. So what can you say about these two questions?40

O. the limit is the same as the function?

A. But you see here that the we don’t have any function value here.

O. Oh yes, what do you mean?

A. I mean in which cases the function value is equal to the limit?

O. Straight line,45

A. Or, see here I have (I took the pen to draw other continuous func-tions).

O. Or a parabola.

A. parabola or see this, this is not a parabola (I drew another graph).

O. When it’s continuous.50

A. Yeah. So why here the function value is not equal to the limit?

O. Because it’s not continuous. (...)

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5.10.3 Interview with Dave

In reviewing Zeno’s paradox, Dave seemed to be quite aware of both sidesof the paradox and also how to solve it. That is why the teacher found itan appropriate time to relate it’s similarity with the 8th question which wasanswered wrong by Dave1 (lines 4-15). He then explains the Archimedes’paradox and there, in describing the similarities between the two paradoxes,he mentioned that in theory, the remained area or distance never reacheszero, except at infinity (lines 15-26). He then reviewed derivative activity bythe same reasoning lines (29-31).

Transcript 31

A. Can you speak about Zeno and Archimedes and their similarity?

D. Zeno had a paradox about turtle and Achilles, oh no,

A. This is also one of the paradoxes.

D. I prefer the other one. When you have points A and B and younever reach point B. Because when you take half and half and half5

steps then you don’t get to the point. It sounds really clear youreach but he didn’t know how to prove it.

A. But how can you prove it?

D. Again with limit.

A. But we can even not use the word limit but if you like you can use10

it.

D. We say that he is infinitely close to it so he is there.

A. It’s exactly this question isn’t it (pointing to the 8th question)?We are going infinitely close to 1. So we are in 1. What aboutarchimedes?15

D. Archimedes had a parabola and he wanted to know this area so heused this triangle and he put another one and he went on and ontill the polygon is almost as big as the parabola.

A. And what is the similarity between this and this?

D. They both in theory it reach not the whole area and it reach not20

the point,

A. But in theory also we reach why you say we don’t reach?

1More discussions about the 8th question are given in 5.9.8

106

D. eh, no, no, but (thinking) they both took small and small till theyboth reach it.

A. When do they reach it?25

D. At infinity.

A. At infinity they reach it. So can you find the same thing in Newtonactivity?

D. Yes Newton had a graph, a parabola, (drawing on then paper) andthis point gets closer and closer and at infinity it reaches it also.30

You also took smaller and smaller steps.

A. So we became as close as we wanted to the x and we saw that ygets closer and closer to a number. (...)

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6 Conclusions and Discussions

6.1 Conclusions

6.1.1 Research Question 1

1. What steps seen in the history of the limit concept are worth repeatingin a guided discussion if we want to lead students to the modern conceptof limit?

Considering our discussions in chapter 2, two main steps are chosen to beapplied in designing three activities.

1. Zeno and Archimedes’ paradoxes

2. Newton’s approach to find the instantaneous velocity

In chapter 5, the research questions 2 and 3 have been answered. Below,these answers will be summarized. Since the first research research questionwas of a theoretical kind, it was already answered in chapter 2 (see # for asummary of the answer).

This summary section is divided to three parts;

• The three paradoxes which create a challenge for the students beforebeing introduced with limit notion (research question 2),

• Towards the limit concept and being introduced with the limit notion(research question 3.a),

• Potential misconceptions after being introduced with the limit notion(research question 3.b).


2. Will an emphasis on understanding the notion of infinitesimals, us-ing the steps found in question 1, help students overcome the commonpotential conflicts, regarding the limit notion?

(a) Which of the common misconceptions are more/better tackled viathe above suggested method and which of them still appear to existin students’ minds after teaching and why?

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In order to address this question, a summary of each paradox is given.

6.1.1.1 Zeno

Since our students have not learned limits yet, we can not really interprettheir ideas as conflict conflicts. Their ideas could be mentioned as theirintuitive reaction to the challenge which could be linked to some of the limitphenomena mentioned in 2.1. Since each paradox has two different sides,two different intuitive reactions ,which are not necessarily wrong, to thephenomenon involved in the paradox could be expected.

In Zeno’s paradox, the simple1 horizontal asymptote phenomena” men-tioned in 2.1 which can be linked to ”a limit is a number or point the functiongets close to but never reaches (Williams, 1991)” , is traceable in our dis-cussions. The two sides of this paradoxes are: reaching the destination atinfinity or never reaching. The three of them believed at some point that wecan come very close to point B without reaching it completely. This beliefstrongly remained and was dominated in John’s mind later, although, heknew it was wrong.

The teacher promoted the students getting involved with the paradox andthe idea of approximation. Therefore, some questions were asked to checkstudents’ ideas about approximation as an explanation. In the Limit phenom-ena mentioned in chapter 2, this idea is called Gap phenomena which couldturn into the misconception: ”a limit is an approximation that can be madeas accurate as you wish (Williams, 1991). The question investigated that ingoing from point A to point B, what happens to the very small remainingdistance. should it be regarded as 0 just like machines fix a small numberas 0, or we arrive approximately at point B? Otto appeared to think thatapproximation explains only one side of the paradox which is Zeno’s strangeway of walking. He believed that in that situation, we will arrive to thedestination approximately. John also seemed to believe that approximationexplains just one side of the paradox. But, likewise Otto, he believed that inthe real walking, arriving to the destination is an approximation and Zeno’sseries; 1

2n, becomes exactly zero when n grows infinitely big. Dave does not

seem to have such an intuition for explaining any sides of the paradox.

1simple because the function is monotonous.

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6.1.1.2 Archimedes

The two sides of this paradox are; having equal areas when n ∈ N growsbigger and bigger, getting to infinity or having unequal areas for all stepsn ∈ N . In this session and in discussing this paradox, only one of the”Limit Phenomena”, ”Gap Phenomena” was mentioned and discussed. Incomparing with the common images of limit concept, it is linked to ”a limitis an approximation that can be made as accurate as you wish” (Williams,1991). Dave and Otto refuted the idea of approximation for both sides ofthe paradox by mentioning the dynamic process of tending to zero infinitelymany times which makes the difference between the area of the polygon andthe parabolic segment exactly zero. Although, Dave happened to ignore theremaining area once in his discussions. John believed on the other hand thatthe remaining area will never become zero.

6.1.1.3 Derivative

In this activity, students had to find the instantaneous velocity by find-ing the average velocity between a second and a moment h after that second.The two sides of this paradox are; the instantaneous velocity does not existor the instantaneous velocity must exist. Of course we know by experiencethat the momentary velocity exists. But on the other hand, for determin-ing it, we calculated the distance traveled in a certain period. The periodincludes at least two moments which are different with each other. Other-wise we will get zero divided by zero which is not possible mathematically.During the calculations, students were promoted to factor out momentumh by the condition of not being zero from the nominator and denominatorof a fraction. This created a calculation conflict for them that they had tofind an explanation for. Their first intuition was close to our ”ContinuityPhenomena” mentioned in chapter 2, in the list of our ”Limit Phenomena”,addressing one of the misconceptions listed by Williams (Williams, 1991);a limit is determined by plugging in numbers closer and closer to a givennumber until the limit is reached. In answering the questions, Otto pluggedin the number in the expression of the function regardless the condition forh. He had found the relation of what we were doing with what is done infinding the limit of a function.

The approximation was their spontaneous reaction for explaining the cal-culation conflict when they were introduced with the limits. The idea ofapproximation in this problem could be linked to the ”Velocity phenomena”

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explained in chapter 2. John and probably Dave were believing that we arefixing a number close to zero but not zero and are approximating the answer.

6.1.1.4 Summary for RQ 2.

These activities were designed to let students explore both sides of theparadoxes and recognize the problem, no matter which side they took, in thehope that the students would not be fixed on one side and their intuitivereaction does not turn into a misconception after being introduced with thelimit concept.

With respect to a paradox, the following stances seem possible:

a. Choose one side, defend it, no evidence of seeing a problem.

b. Choose one side, defend it, notice there is a problem.

c. Notice the problem, no evidence of taking sides (maybe even refusingto take sides)

d. No evidence (not speaking out)

Students stances on the three paradoxes can be summarized as in the follow-ing table.

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Students’ intuitive reactions to limit phenomena

Phenomena Activity John Otto Dave EvidenceSimple Zeno b: not

reachingeven atinfinity

b b 1-2

Horizontal Archimedes d d d 4Asymptote Derivative a: no

momen-taryvelocity

c: rec-ognizingconflictbetweenformulaandreality

c: rec-ognizingconflictbetweenformulaandreality

5-6

Gap Zeno b: ap-prox.whenwalkingstrange

b: ap-prox.whenwalkingnormal

e:reach-ing atinfinity

3

approx./exact Archimedes a: un-equalareaseven atinfinity

a: equalareas atinfinity

a: equalareas atinfinity

4

Velocity Derivative 1 b: ap-prox.zero

d b: ap-prox.zero

8

Continuity, Zeno d d d -plugging in Archimedes d d d -

numbers Derivative d b: plug-ging innumber

d 7

With respect to our evidences, the RQ 2 could briefly be answered asbelow:

• The emphasis on understanding the concept of infinitesimal has thecapacity to help students overcome some of the conflicting images.

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• The different misconceptions appeared for different students. But, theemphasis on investigating the idea of approximation in three paradoxes,made them aware that it is a misconception. The idea of tending andnot reaching was the strongest misconception which was more difficultto be removed.

6.1.1.5 Interview

In the interview, the main aim was to review the paradoxes and also thewrong answered questions. Since the interview was done at the end of theteaching, students’ responses could help us trace the common misconceptionsin their discussions. However, two of so the called William’s misconceptionswere not discussed in our activities, the other three were sometimes trace-able. I must indicate that the three sentences were not asked from studentsexplicitly except for the first question, thus, in filling the table, I have in-terpreted their ideas according to the similarity between what they said andthe misconceptions. For instance, our evidence for claiming that ”A limitis determined by plugging in numbers closer and closer to a given numberuntil the limit is reached” is students’ responses in the 4th question of thetest and also their discussions in the interview. John and Dave believe thatthe limit of the sequence of the numbers is 1 while Otto doubts it. For ”Alimit is a number or point the function gets close to but never reaches”, ourevidences could be found in the students’ answers to the 8th test question,also in the transcript 19 when John has a discussion with Wolter, and Harm.For ”A limit is an approximation that can be made as accurate as you wish”,it was a bit different. In the activities, some questions were asked to checkthe idea of approximation, but, after teaching and in the test or interview,this idea was not asked explicitly. Therefore we had to interpret their ideasin answering question 8 and also in reviewing the paradoxes.

The third item probes the corresponding common misconceptions as for-mulated and listed by Williams et al. after limit was taught. They couldbe investigated in the test, interview and all the discussions after the limitwas taught. Although not all of them are addressed and discussed in ouractivities.

A summary of the results is given in table below:

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Students’ possible misconceptions

cognitive conflicts Activity J O D Evid.

2. A limit is determined by Zenoplugging in numbers closer Archimedesand closer to a given num- Derivativeber until the limit is reached. Test/Inter. X deny X Q.4

3. A limit is a Zenonumber or point Archimedesthe function gets Derivativeclose to but never reaches. Test/Inter. X X Q.8/int/19

4. A limit is an Zenoapproximation that Archimedescan be made as Derivative

accurate as you wish. Test/Inter.


3. Will an emphasis on understanding the notion of infinitesimals, usingthe steps found in question 1, and the emphasis on geometric inter-pretation of neighborhood, help students reach the informal and formalconceptions of limits?

(a) Do students show understanding of a geometric interpretation ofthe ε and δ- neighborhood?

(b) Do they understand the (in)formal definition?

i. Can they formulate the definition of limit in their own wordsafter some weeks?

ii. Can they recognize the correct definition of limit?

iii. Can the students apply the formal and/or informal definitionsof limit in the new concept, continuity?

(c) Can students use the notion of infinitesimal to reason about thespecial cases?

In order to address this question, a summary of the neighborhood, prepara-tion of ε-δ definition of limit and the definition is given.

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The geometric interpretation of neighborhood was divided into threeparts,

• neighborhood concept on the real line. That is, geometric interpreta-tion of absolute values and intervals. Also introducing the neighbor-hood symbol.

• neighborhood concept on the plane in case of linear functions. Thatis, the geometric interpretation of two variables, changing when thefunction is linear.

• Checking how do variables change on the plane in case of non linearfunction.

The first series of questions were more refreshing students’ previous knowl-edge and the three of them were participating actively in the discussions.The second series of questions were a bit more difficult. They also neededtime to understand the concept and get used to apply it in different ques-tions. The more evidence for understanding the geometric interpretationof the ε and δ- neighborhood appeared in the two later sessions; Workingtowards formal definition and epsilon-delta proof, while they were applyingthe neighborhood concept in a new concept, limit. In these two sessions, themain focus is partly in applying neighborhood and partly in introducing thelimit concept. The evidences in our transcripts of these two sessions, showthat students were applying neighborhood with its geometric interpretationto express how do the changing of one variable, changes the other variable(ε and δ).

Formulating the formal or informal definition of limit was more difficult.The transcripts show that they were able to follow the teachers’ formulationsbut they could not do it themselves in the beginning. In the test which wastaken right after teaching the limit, two questions were asked to check thisissue (transcript, the first two questions ). The answers show that they wereable to choose the correct choice among the other choices (list of William’scommon misconceptions). But, they were not able to formulate the formaldefinition quit accurate. After some weeks, in the final interview the samequestion was asked and again they did not show any improvement in beingable to give an accurate definition.

Although, in the next sessions which we were busy with the new concept,continuity, they seemed to be able to understand the limit concept by apply-ing it in formulating the continuity concept. Otto seemed easier applying the

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symbols and participating more in finding the formal definition of continuity.A summary of the results is given in the table below:

Towards the Formal Definition of Limit

Question Activity John Otto Dave Evidenceε and δ real line yes yes yes 9

neighborhood? plane yes yes yes 10(in)formal formulating no no no 15

definition?choosing no/yes yes yes Q. 1applying yes yes yes 20

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6.2 Discussion

6.2.1 Comparison

In comparing this research with other studies, three types of comparisons arepossible:

1. comparison of theories behind the teaching approaches,

2. comparison of the teaching approaches,

3. comparison of the results.

Comparison of the theories is already done in chapter 2 where differentideas about the limit concept and the process of learning it are discussed.

William’s (Williams, 1991), Cottrill et al’s (Cottril et al., 1996), Tall’s(Tall, 1977), and Juter’s (Juter, 2006) share with our study the aim of dis-tinguishing, measuring and possibly reducing or removing students’ miscon-ceptions about limits. Therefore, we will compare the teaching approachesused in these studies to ours below. In comparing the teaching approaches,three main categories which sometimes overlap each other were distinguished:

1. Traditional teaching

2. Experimental teaching

3. Remedial teaching

We define these three categories as follows: The traditional teaching startswith teaching limits and introducing its formal definition to students fromthe very beginning. This approach ignores students’ misconceptions. Theremedial teaching leaves the introduction of the concept to the traditionalmethod, but then follows up or runs in parallel to the traditional teachingsome additional activities to adjust the undesired outcomes, the misconcep-tions. It does not include a new method of introducing the limit concept,however, it attempts to connect to students’ own ideas. The experimen-tal teaching attempts to a new method of introducing the limit concept (orpreparing students for the introduction) that connects to students’ ideas andbeliefs, to distinguish their misunderstanding and adjust them.

Juter’s study (Juter, 2006) is aiming at deciding the learning results oftraditional teaching of limits. She compares students’ development to the

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historical development of the limit concept, while these students are attend-ing a traditional first-level university course in mathematics, including thelimit concept. She uses Tall’s (Tall, 2004) three worlds of mathematics toschematize the history of the limit concept to make a comparison possible.The students had two lectures and two sessions for task solving devoted tolimits. The teaching was based on a textbook. In the first lecture on lim-its, the formal definitions and theorems on indefinite and definite limits offunctions and limits of monotonic functions as x tends to infinity were pre-sented. As the author mentions, the text book had an intuitive approachin the beginning and a strictly formal approach in the rest. In the secondlecture the standard limit values, number e, ε-δ definitions as x tends to anumber, continuity and some theorems on it were presented. It is mentionedthat some students were critical about the speed of the teaching. In the thirdlecture the theorems taught in the previous lecture and continuity were re-viewed and derivatives were introduced. The rest of the lectures were dealingwith derivatives and integrals. The major difference between Juter’s (Juter,2006) research and this one seems to be in their aims. In Juter’s study, themain aim is to measure students’ understanding of the limit concept by usingTall’s three worlds of mathematics as a scale. Since the aim of the author isto measure the knowledge, less stress is put on how this knowledge is triedto be transfered to students. For us, the main aim was to lead the studentsto first the more accurate intuition of the concept without mentioning it ex-plicitly and then to lead them to the formal definition by trying to let themelicit the concept. Comparing with Juter’s research, there is more interestin letting students find out the new notion and less interest in continuouslymeasuring students’ knowledge during their learning. Probably it could besummarized that her research is diagnostic while our research is a teachingexperiment.

Williams’ study (Williams, 1991) in distinguishing and classifying stu-dents’ misconceptions about limits involves a remedial teaching approach.Just like Juter’s, his research was aimed at a traditional course, but thistime activities are developed in parallel to that course with the aim of chang-ing/ correcting students’ ideas about limits. The research had two phases,in the first phase a short questionnaire was asked from 341 second-semestercalculus university students. One of the questions included the six differentviews about the limit definition among which one was the formal view andthe others were interpreted as limit as bound, limit as approximation, limitas unreachable, limit as dynamic. These questions were developed to check

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students’ different beliefs about limit which the previous research of Williamssuggested to be held by students.

Volunteers were asked and from those, based on the questionnaire results,10 students were finally selected for the second phase, the remediation ex-periment: 7 males and 3 females. From the selected students, 4 were judgedto have a dynamic view, 4 had unreachable view, 1 had the limit as a boundview, and 1 had the limit as an approximation view. None of the studentswho chose the formal view were selected to participate in this study. These10 students participated in the second phase. This second phase took placein 5 sessions during a period of 7 weeks. Efforts were made to elicit themodels for limit from the subjects using different approaches. Then variousquestions were posed to challenge students models of limits. These questionshad two aims: 1. to help students establish the implications of the opposingviewpoints. 2. to encourage students to change their viewpoints by clarify-ing these difficulties with the informal viewpoints. Sample problems given tostudents are as below:

2. 1. Given the function f(x) = 3, evaluate the limit of f(x) as x→ 2.

2. 2. Define a function f(x) by letting f(x) be the distance from a certaintrain to the station at time x, where x is measured in hours after 12:00 noonon March 1, 1989. At exactly 2:00 p.m. on that day, the train arrives andcomes to a complete stop at the station. Discuss the limit of f(x) as x→ 2.

sample problem of session 3 are as below:

3.1. A student was given a function F and asked to find the limit of Fas x approached 0. He plugged in numbers on each side of 0 and made the

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following table:

What can you conclude about the limit of function F as x approaches 0?

3.2. Given F (x) = x+ 1 + 11020x

, evaluate limF (x) as x→ 0.

Comparing Williams’ (Williams, 1991) teaching approach to ours, like inJuter’s (Juter, 2006) case, again a big difference is that in our case studentswere not told the formal definition of limits right at the beginning, like it hap-pened in the traditional course that these students attended. Still Williams’aim is to change students’ incorrect intuitions and this aim makes their workcloser to ours. the fact that their main aim is to change students’ incorrectintuitions make their work closer to ours. The six models recognized andcategorized by them are now used to check students’ ideas in most of theresearches about students’ understanding of the limit concept as well as inthis one. The questions posed to students with respect to the six models,are so carefully designed that they have turned into a kind of standard ques-tions for other researches to use. Although Williams’ work provides a goodstarting point, I would be critical if a remedial teaching approach in parallelto a traditional course were seen as a final solution. Given the large scalemisunderstandings, as found by Williams and others, caused by traditionalteaching of limits, it would be strange if we were content that these misunder-standings can only be cured after the initial teaching. It would be preferableif these misunderstandings could be evaded by changing the method of in-troducing the limit concept. employs an experimental teaching approach,like ours. From a theoretical perspective, their aim is to design and studythe genetic decomposition of learning the limit concept. This genetic decom-position is formed first by designing a first version, and then implementingit in a teaching experiment. Then by analyzing subjects’ responses to the

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teaching materials a new version of the genetic decomposition is proposedand tried out in a next cycle. This theory helps them to suggest specificmental constructions in learning the concept to be considered in designingthe instruction. The teaching materials are aimed at getting students tomake constructions on a computer which according to the authors could leadthe students to make corresponding constructions in their minds. Then byindicating a particular mathematical point which is seemed to be understoodby some students and not understood by others, they adjust the genetic de-composition regarding their data assessment according to two criteria: (a)There are indications in the evidences that show those who have succeededwith that point, have interiorized a particular action to a process or haveencasulated a certain process into an object. (b) Such indications are notpresent in the data of those students who did not succeed in understandingthe point. If both criteria are met, the authors have added their observedconstruction to the genetic decomposition. So there is a mutual interactionbetween the results of their data and their theoretical perspective.

The treatment took place during the first 6 to 7 weeks of a first-semesterexperimental calculus course. Students worked on teams of 3, 4, or 5. Thelimit concept was reconsidered throughout the students’ subsequent studyof calculus, in particular with derivatives, integrals, and sequences. Thecomputer activities had five major themes:

1. Investigating approximations to calculate a speed: students wrote com-puter code to compute the changing of a variable with respect to forexample time.

2. Graphical investigations of the limit concept in which a question includ-ing a changing component in a function was posed and the studentswere asked to find the answer first by drawing the graph of the functionand then by finding a formula for it, using a computer algebra system.For instance to find the slope of the tangent to the curve of a function.

3. Computer constructions of a value approaching a limit in which thestudents were asked to find the values increasingly close to the limitpoint by constructing the computer code. This theme aimed at helpingstudents interiorize the action of plugging in numbers closer and closerto a limit point.

4. Computer constructions of the limit concept, in which students were

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asked to study and modify a program which was provided by the in-structor which approximated the values of the limits. The programwas applied to the same examples which were asked before. They wereasked to make the same modifications for limits from the left and fromthe right.

5. investigations of ε− δ windows: a function and vertical dimensions forthe window were given and students were asked to find the horizontaldimensions such that the function fits in the window and does notexceed it from any upper or lower sides.

In comparing the teaching approach of this study with ours, some dif-ferences and similarities are remarkable. The students in their study arealready introduced with a limit concept. That is, their aim is not to teachthem this concept for the first time, but, to study their understanding of thisconcept. These students are older than our students. Their main aim is tocheck and revise the genetic decomposition with the hope to find out duringwhich process do students learn this concept. Although checking our stu-dents’ understanding of this notion by their 5 themes was interesting for usas well, it was not our main aim to find out the process itself. The fact thatboth of the two studies have chosen the dynamic view of limit has influencedsimilarities between the activities especially the Newton and neighborhoodactivities.

David Tall’s study (Tall, 1977) in comparing the cognitive developmentof students with a model of a dynamic system is a fourth piece of researchwhich is aiming at deciding students learning in a traditional course. Hementions data collected about 36 first year mathematics students at WarwickUniversity. There is not enough information about the teaching approach inhis study. Therefore, there is not much for comparison at this point.

This research like Cottril’s (Cottril et al., 1996) is set up as a teachingexperiment. That is, it tries a new approach in introducing the concept oflimit instead of only diagnosing the problems or remedying them afterwards.Teaching is designed in a way to first challenge the students by the para-doxes involving the limit concept. Then the limit concept and its formaldefinition are taught. The difference between this study and the rest regard-ing the teaching approaches is that we did not mention the limit concept atthe beginning of the teaching period, instead, the students were promotedthrough the activities to see the need for a limit concept themselves before

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we taught it to them. Another difference was the emphasis on the conceptsof “infinitesimal” and “approaching” as the key problems in the activities.

In comparing the results we will consider the same four studies as men-tioned in the previous part: Juter (Juter, 2006), Williams (Williams, 1991),Cotril (Cottril et al., 1996), and Tall (Tall, 1977). Approaches take placerange from qualitative, explorative to quantitative. This might complicatea comparison. The results more or less support each other with respect tostudents’ ideas and misconceptions about limit concept.

The collected data in Juter’s research were provided through interviews,students’ solutions to limit tasks and responses to attitudinal queries. Thestudents were confronted with tasks which were of increasing level of difficultyat five different times (Stage A to Stage E) during the semester. Among the38 volunteers participating in the research, 18 were selected after individualinterviews and according to their resemblance to the whole group. Amongthe 18 students, 15 have attended all the stages and interviews till the endof the study. Students were categorized into three groups based on theirfinal scores as follows: 1. failed students 2. students that barely passed 3.very good students. The author mentions that according to her analysis ofthe data, the historical progression through the three worlds of mathematics(Tall, 2004) matched only the high achieving students’ progressions. Amongthe others, there is sometimes no progress, and the third world seems tobe unreachable for many of these students. The students with the averagelevel when entering university were in the first world of mathematics withan intuitive perception of the limit concept. The formal definition as it wastaught in this course was not encapsulated in most students’ concept imagesand they did not use it properly in solving the problems (page 429). Shealso mentions that students need to evaluate their perceptions repeatedlyduring mathematics education to adjust their mental representations. Tasks,discussions, and other stimulating activities help them to find critical partsof their concept images (page 427).

Comparing Juter’s results to ours, then, if we use Tall’s scale to checkand measure our students’ understanding of the limit concept, we can prob-ably assert that our three students had passed the first world. That is, ifwe refer to Piaget’s definition of intuition as ”a category of cognition whichare directly grasped without or prior to any need for explicit justification orinterpretation”. Or ”Intuitive knowledge could be a proces in which studentsare having experiences (originated by a teacher or not) and not yet puttingit into words” (chapter 2.6), and compare it with Tall’s meaning of the first

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world which is perception of concepts via the senses, we can conclude thathaving such an intuitive knowledge is equivalent to pass Tall’s first world ofmathematics. The three challenges of the paradoxes could be referred to asproviding such intuitive knowledge! Tall’s second world involves the integra-tion of the procedural actions on the conceptions and its symbolic knowledge.I must assert that this world contains a wide range of understanding a notionand Tall’s categorization seems to be too vague to describe such a wide range.For instance we miss a clear distinction between world 2 on the one hand andfollowing ill-understood prescriptions on the other. The integration of theprocedural actions on the limit concept according to us is better describedin APOS theory, thus, this comparison will be done later with the APOStheory. Regarding the symbolic knowledge, it is difficult to assert accuratelyhow far each of our students had passed this world. Although they had somedifficulties with applying the symbols in the beginning, with respect to theevidences in the later sessions such as continuity, this problem was largelyovercome. Therefore, we could claim that our students had passed Tall’s sec-ond world as well. According to Tall, arriving at the third world means beingable to encapsulate the process as objects and being able to go beyond theprocepts to the formal definitions. Taking this definition into consideration,the evidence about our attempt to let students formulate the formal defini-tion themselves shows that they had not arrived at this world. Nevertheless,after attending our introduction of the formal definition of limits, studentswere able to apply this in formulating a definition for continuity. In any case,the three students in our research could be set among the highly achievinggroups of students in Juter’s research according to their own final results.Then, they belong to the category of students whose cognitive development,according to Juter, is in accordance with the historical steps in the evolutionof the limit concept, and for whom it is possible to reach Tall’s third world.

In comparing her results with ours, we can use the question that both weand she took from Williams (see our section 2.1, and 5.9.1). The questionasks which of the models of limit describe the limit concept better accordingto students. She mentions that statement 4, was selected most frequentlyamong failed or barely passed students to be the statement most similarto their own perception of limits of functions. Statement 3 was selectedmore frequently among the students with high marks in the exam. In ourstudy, two of the students chose statement 3 and one chose statement 4 inthe beginning and changed his mind soon afterwards. This changing ideaseemed to be not consistent in this one student’s mind though. They were

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all quite interested students and probably in Juter’s categories, they wouldbe among the very good students. Still this comparison seems a bit fruitlessto me since it does not shed any light on the path of those who would like toameliorate student’s difficulties in learning this concept. She also mentionsthat there was a great confusion about functions abilities to attain the limitvalues. This is in fact the fourth limit model which we also experienced as astrong idea among some of our students during the class discussions as well.The reason according to us was distinguished as the lack of enough stresson the concepts of dependent and independent variables. The independentvariable, x is free to move or tend as long as the f(x) does not put someconstraints on it. The importance of behavior of f(x) and analyzing it issomething which is less stressed and less distinguished from the behavior ofx.

According to Jutter, many students treated limit as being unreachable intheoretical discussions and reachable while solving the tasks (page 425). Inour class discussions while presenting the paradoxes, we also had the sameexperience with the students in a different way. For instance while beingchallenged by the Zeno’s paradox, students sometimes were convinced thatwe never arrive to the destination exactly in practice or the other way aroundin theory. But this situation was continuing mainly during the paradoxdiscussions and not so much after that.

William’s results are elicited from the interviews done in his fifth session.They are analyzed with respect to two models of limits: 1. what it meansfor a function to never reach its limit, and 2. what it means for a functionto approach a limit.

Regarding the changes in the concept of reaching a limit, 1 of the stu-dents had more or less standard view of limit but 9 of them had chosen 4thstatement as true in the initial questionnaire. That is they believed that thelimit is unreachable. The research however, showed that their interpretationof this idea was different with each other. Among them, 3 had changed theirmind after answering to the first question in the second session. They sawthat the limit of the constant function is obviously reachable and their alter-native choice as the standard view of limit then. 4 of the other 6 studentschanged their mind in the last session and believed that the idea of limit asunreachable is false. Still 2 of them were doubting that the limit could bereached even in the last session.

Regarding the changes in concept of approaching a limit, no real changehad occurred in the students’ dynamic view of limit. In choosing the best

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idea of the two hypothetical calculus students, they mainly chose the firstidea:

• Students 1: describing limit as a process by which values of the functionmove closer and closer to the limit as x moves closer and closer to apoint of interest.

• Student 2: supporting the idea that f(x) can be made as close as youwant to the limit by making x close enough to the point of interest.

6 of the students chose first student’s statement. 3 of them chose part of eachstatement or did not choose any of them. 1 of them chose second student’sstatement.

In comparing our results with Juter’s regarding the students’ ideas aboutthe six models of limit, as it is already mentioned, one of the students hadthe idea that the limit is unreachable. He changed his mind later but still hementioned that he had understood that the limit is reachable but his hearttells him that it is unreachable. It showed a strong wrong intuitive perceptionwhich had given its place to a weak and inconsistent right perception. Theother two did not have this idea and chose the third statement from the be-ginning. They also did not show strong evidence of such a misunderstandingduring the class discussions.

In our activities and discussions, we were not dealing explicitly with theidea of “a limit describes how a function moves as x moves towards a certainpoint” which is named dynamic-theoretical. Instead, we investigated the ideaof “a limit is determined by plugging in numbers closer and closer to a givennumber until the limit is reached are named ” which is named as dynamic-practical. The results showed that two students had this idea although theydid not choose it as the best statement among the others. one of them evenhad chosen the correct statement. But in answering the fourth questionof the test, his answer was interpreted as representing such an idea. Thisproblem probably could be cured by letting students compare both the limitof continuous and discontinuous functions.

The results in Cotril et al.’s study are meant to help them revise the pre-liminary genetic decomposition. Therefore, students responses, are classifiedinto five levels corresponding to the five levels of the genetic decomposition.If there were some answers which did not fit with those levels, they added onelevel or adjusted them to be able to represent such answers of students. Thosestudents who believed that the limit at a point a is the same as the value

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of the function at point a if existing, had extremely static notion. Hence,they changed the first level of the genetic decomposition from ”‘the actionof evaluating the function f at a few points, each successive point closer toa than was the previous point” to ”The action of evaluating f at a singlepoint x that is considered to be close to, or even equal to, a”. Those studentswho evaluated the limit by evaluating some successive points close to thatpoint in the second level. According to them, such students are strugglingbetween a static evaluation and the beginning of a more dynamic action ofseveral evaluations (Cottril et al., 1996, page 180). The next step in thepreliminary decomposition was the interiorization of the previous step to asingle process in which f(x) approaches L as x approaches a. While theyfind three other levels before this level be achieved according to the analysisof their data. They mention that first students must be able to construct theprocess in which x approaches a, then they must be able to construct theprocess in which y approaches L. Then we could conclude that this level ofgenetic decomposition is met. For the next level, students must be able tothink about the limit as an object and apply the arithmetical actions on it.Finally, he mentions that only a few students indicated the evidences of goingbeyond these four levels. Indeed what they knew were some vague notion ofthe standard inequalities involved in the ε − δ description of limit (Cottrilet al., 1996, page 186). According to them non of the students were in thelast step of the decomposition therefore they did not have enough data tochange or adjust the last 2 steps of the preliminary genetic decomposition.

Since our main aim was not to find the process of students’ learning ofthe limit concept, there is not much to compare. However, there are stillseveral similar points in both researches which are mentioned here. In ourstudy, we first build up a need for the limit concept and then we go throughthe steps studied by Cotril et al., which is one of the differences of the twostudies. We can not evaluate our data completely in accordance with theirgenetic decomposition since our teaching approach and goals were different.Yet, to begin comparing the similarities, the focus on distinguishing the dif-ferences between the approaches of the independent and dependent variablesin their revised genetic decomposition in steps 1 to 3(b) is a remarkable sim-ilarity. Our data for steps 1 to 3(b) could be traced in the neighborhoodactivities while the teacher and students were comprehensively studying theapproach of x and the approach of f(x) by using the Geogebra software.The evidences show that our three students had passed steps 1 to 3(b). Ourdoubt is on the step 3(c) where the function f is applied to the process of x

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approaching a to obtain the process of f(x) approaching L. Our evidencesdo not approve such an understanding of the students before the continuitywas taught. Their fourth step in performing action on the limit concept isinterpreted completely different with ours. performing actions on the limitconcept is interpreted as the application of the arithmetic operations on thelimits while we performed such actions all in the last sessions and duringthe teaching of continuity. Therefore, our fourth step would be different ifwe wanted to revise the genetic decomposition according to our data. Incomparing our data with their 5th step, reconstructing the step 3(c), ourdata in neighborhood activities show that students were able to numericallyestimate the closeness of approach in symbols. Although our students couldnot apply the quantification schema to connect the reconstructed process ofthe previous steps and obtain the formal definition of the limit. Therefore,probably we could conclude that our students followed APOS revised geneticdecomposition’s steps up to the 5th step.

Their interpretation of those students who evaluate the limit by eval-uating some successive points close to that point as struggling in a staticevaluation and the beginning of a dynamic evaluation is not in accordancewith ours though. If we interpret such students as those who plug in suc-cessive numbers in order to find the limit, then, our evidences show that wecould not agree with them. Our students had dynamic view of the limit andinfinitesimals but still chose to plugin numbers in the fourth question of thetest. Regarding their fifth and sixth step of their genetic decomposition, wealso did not have strong evidence that our students were there!

David Tall points out a series of four questions about the limit of se-quences. One of the questions asks if the students have been taught theconcept of the limit of a sequence. Then other questions are asked. 10 ofthem claimed that they had learned a precise definition but only 7 of thesecould give an adequate definition. On the other hand 21 claimed they areintroduced with this notion informally, and from these 21 only 1 could givea precise definition. Most of his discussions are focussed on 2 questions, oneasking to evaluate the limit below:

limn→∞

(1 +9

10+

9

100+ ...+

9

10n)

Another one asks:Is ˙0.9 (nought point nine recurring) equal to one, or is it just less than

one? Explain the reason behind your answer.

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In answering the first question, most of the students evaluate the limitcorrectly equal to 2. Students answers to the second question were morediverse. He mentions that fourteen students have answered that ˙0.9 = 1, but2 of them change their mind later. 20 replied that ˙0.9 < 1. Among the 7students who gave a precise definition of limit, only 1 of them answered that˙0.9 = 1. According to him, they had lack of knowledge about infinitesimals

and infinity. Here there is a new concept, limit of sequence. And the conflictis not understanding the core of infinitesimals. His approach to remove thisconflict was to point out the decimal form of the fractions. Afterwards, mostof the students, 24, claimed that ˙0.9 = 1 (page 12). I must say that hissolution to solve this problem does not seem helpful to be able to add someknowledge of limit concept to students. it seems like a magic that changessomething which was strongly believed as smaller than one, suddenly intoone. There is not any infinitesimal concept involved in this solution.

In comparing our results with his, we also experienced the same ideas.two of our three students believed that ˙0.9 is less than one. One of them firstmentioned that they are equal but then changed his mind and again changedhis mind that they are equal. But, in finding the limit, they all evaluated itequal to 2.

The evidences in this research shows that our students as well, shared thesame ideas and difficulties in encapsulating the abstract ideas such as infinityand infinitesimal. However 2 out of the 3 students selected the correct limitmodel in the test and the other one selected the correct one later, the twomain ideas of limit as unreachable and limit as a number which is achievedby plugging in successive numbers were distinguished as the dominant ideasamong our 3 students. T

Students’ ideas could not be evaluated as is evaluated in the previousresearches, one representing a better understanding than the other. Theyseemed to function parallel in students’ minds. For instance, Dave was thecase which showed the least misunderstanding among the rest. He chose thecorrect limit model from the beginning and was more able to represent themore accurate ideas. But, still in deciding what happens when a number isapproached by plugging in some numbers (Test, question 4), he chose thewrong answer. This idea resembled one of the students’ common ideas aboutlimit which said the limit is determined by plugging in numbers closer andcloser to a given number until the limit is reached distinguished by WIlliamset al.. Therefore wrong and right ideas could sometimes exist parallel to eachother, although, the right idea is preferred when choosing. Later, while we

129

were busy with the continuity, differentiability, etc., he seemed to be moreadvanced and stable about the limit notion as well.

6.3 Limitations

This research has te following limitations, which at the same time are sug-gestions for future research:

• Within the negotiated time limits for the course we were not able tofollow up on all of students wrong models of limit, although doing sowould have fitted well in our approach.

It would be desirable to let students experience and practice all thosephenomena in which their wrong chosen model of limit is right (seesection 2.2). Also, it would be good to let them experience and practicethose cases (like the meandering asymptote) where the various wrongmodels of limit show their limitations.

• In the developed teaching setup there is still a big jump between thehistorically motivated activities on the one hand and the introductionof the modern limit concept on the other. The historical activities suc-ceeded in convincing students that there’s a problem to be solved. Wehave evidence that they saw the various paradoxes and were interestedin their solution. However we have no evidence that they were con-vinced that the limit concept solves these problems. Indeed, regardingdiscovering the limit concept, we could not provide a suitable situationto let students produce the new concept themselves.

To make the process of inventing the definition of limit smoother, possi-bly more steps from the history of the limit concept could be included.Although history itself was not smooth and sometimes contained bigtime periods of no significant change, still following more of the his-torically important steps could help us to achieve our goals better.Specifically, the bridge between intuitive understanding and formal un-derstanding with all of those difficult syntax and language involved,which means the time period between Newton and Leibniz on the onehand and Cauchy and Weierstrass on the other. By studying this pe-riod we could gain ideas for a smoother evolution of the limit conceptin education.

130

7 References

References

Aristotle (350 B.C.). Physics, http://classics.mit.edu/aristotle/physics.6.vi.html,urlddate: March, 2011. 1, 2, 3

Bagni, C. (2005). Historical roots of limit notion, developments of its repre-sentation registers and cognitive development. Canadian Journal of Sci-ence, Mathematics and Technology Education, pages 453–468. 1, 2.1, 2.3,2.7, 4.1.4

Barbin, E. (2000). Integrating history: Research perspectives. In Fauvel, J.and Maanen, J., editors, History in Mathematics Education, pages 63–99.Dordrecht: Kluwer Cademic Piblishers. 4

Bergsten, C. (2008). How do theories influence the research on teaching andlearning limits of functions? ZDM Mathematics Education, 40:189–199.2.3

Boyer, C. B. (1959). The History of the Calculus and Its Conceptual Devel-opment. Number Reprint. Courier Dover Publications, New York. 1, 2.7,2.7, 4, 4.1.4

Burn, B. (2005). The vice: Some historically inspired and proof-generatedsteps to limits of sequences. Educational Studies in Mathematics, 60:269–295. 4

Cajori, F. (1919). A History of Mathematics. AMS Chelsea Publishing. 4.1.3

Cottril, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K., andVidakovic, D. (1996). Understanding the limit concept: Beginning with acoordinated process schema. Journal of Mathematical Behavior, 15:167–192. 2.5, 6.2.1, 6.2.1

Dubinsky, E. and Mcdonald, A. (2001). Apos: A constructivist theory oflearning in undergraduate mathematics education research. The Teachingand Learning of Mathematics at University Level: An ICMI Study. Series:New ICMI Study Series, 7:273–280. 4.2

131

Fischbein, E., Tirosh, D., and Melamed, U. (1981). Is it possible to measurethe intuitive acceptance of a mathematical statement? Educational Studiesin Mathematics, 12:491–512. 5.9.8

Fishbein, E. (1987). Intuition in Science and Mathematics. D. Reidel Pub-lishing Company, Dordrecht, Holland. 2.6

Heat, T. (1912). The Works of Archimedes With the Method of Archimedes.Dover Publications, Inc. 2.7

Jordaan, T. (2005). Misconceptions Of the Limit Concept In a MathematicsCourse For Engineering Students. PhD thesis, University of South Africa.4.4.1

Juter, K. (2006). Limits of functions as they developed through time and asstudents learn them today. Mathematical Thinking and Learning, 8(4):407–431. 2.3, 2.4, 6.2.1, 6.2.1, 6.2.1

Kaper, W. H. and Goedhart, M. (2001). Van hiele level research, equivalentor complementary to phenomenology? presented at the 3rd E.S.E.R.Ainternational conference in Thessaloniki. 1

Klaassen, C. W. J. M. and Lijnse, P. L. (1996). Interpreting students’ dis-course in science classes: An underestimated problem? Journal of Researchin Science Teaching, 33(2):115–134. 4.1.2

Kronfellner, M. (2000). The indirect genetic approach to calculus. In Fauvel,J. and Maanen, J., editors, History in Mathematics Education, the ICMIStudy, pages 71–74. Dordrecht: Kluwer Cademic Piblishers. 2.3

Lakoma, E. (2000). Stochastics teaching and cognitive developmen. In Fau-vel, J. and Maanen, J., editors, History in Mathematics Education, theICMI Study, pages 74–77. Dordrecht: Kluwer Cademic Piblishers. 2.3, 4

Monaghan, J. (2001). Young people’s idea of infinity. Educational Studies inMathematics, 48:239–257. 5.9.8

Piaget, J. (1968). Genetic epistemology. Columbia University Press. 1, 2.5

Piaget, J. (1977). The Essential Piaget, chapter Logic and Psychology, pages445–477. Basic Books, Inc., Publisher newyork., 1st edition edition. 2.5

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Tall, D. (1977). Limits of functions as they developed through time andas students learn them today. Mathematical Education for Teaching, 2,4:2–18. 4.4.1, 6.2.1, 6.2.1

Tall, D. (1990). Inconsistencies in the learning of calculus and analysis. Focuson Learning Problems in Mathematics, 12:49–63. 1, 2.1

Tall, D. (2004). Thinking through three worlds of mathematics. Proceedingof the 28th Conference of the International Group for the Psychology ofMathematics Education, 4:281–288. 2.4, 6.2.1, 6.2.1

Tall, D. and Vinner, S. (1981). Concept image and concept definition inmathematics with particular reference to limits and continuity. EducationalStudies in Mathematics, 12:151–169. 1, 2, 2.1, 2.1, 3, 4.4.1

Williams, S. R. (1991). Models of limit held by college calculus. Journal forresearch in Mathematics Education, 22(3):219–236. (document), 1, 2, 2.1,2.1, 2.6, 3, 4.4.1, 5.4, 6.1.2, 6.2.1, 6.2.1, 6.2.1

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8 Appendices

8.1 Activities

8.1.1 Appendix 1, Zeno’s Paradoxes

Suppose you start walking from the point A and your destination is pointB. Also suppose that with each step, you can go only half of the remainingway.

1. Give an arbitrary distance to AB and write how much of the distanceis remaining after the 1st step.

2. How much is remaining in the 2nd, 3rd, and4th step?

3. Continue measuring the remaining distance and find out how much isremaining in the nth step.

1

2n

4. Determine how many steps should be taken, in order to make the re-maining distance as small as 2−60?

5. Due to the machine precision, 2−60 is considered as 0. Indeed, this isan approximation which can be accepted and is fixed in practice. Canyou find a big enough N in which the remaining distance becomes 0?

6. However, do you think in theory we have any limitation for increasingthe steps and consequently decreasing the remaining distance? “in-creasing step without stop”, what happens to the remaining distance?

8.1.2 Appendix 2, The Quadrature of the Parabola

Suppose we have a parabola. Draw a chord of it “aB” and make aparabolic segment.

How do you think we can find out this area?Archimedes had a special method; He considered a certain inscribed tri-

angle.Continue the same process with parabolic segments A1 and B1;

134

This triangle has another property also;This triangle has the Maximum Area among other inscribed triangles.Archimedes claimed that area of each triangle generated in the 2nd step

is 18

of the 1st triangle.If we continue the process, in every step, the number of triangles are

doubled and the area of each of them is 18

of the previous triangle.Do you think these properties and proportions are true for other curves

also?for instance for an ellipse, a circle, etc.

1. After 2, 3, 4, and n steps, how many triangles do we have? How manysides does the generated polygon have after 2, 3, 4, and n steps? steps?

number of triangles:

2ndstep⇒ 3triangles

3rdstep⇒ 7triangles

4thstep⇒ 15triangles

nthstep⇒ (2n − 1)triangles

number of polygon sides

2ndstep⇒ 5sides

3rdstep⇒ 9sides

4thstep⇒ 17sides

nthstep⇒ (2n + 1)sides

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Suppose that the area of the 1st triangle is 1. What’s the area of eachof the triangles generated after the 2nd, 3rd, 4th and nth steps?

Let’s denote the area of triangles in each step by atrin ;

atri2 =1

8

atri3 =1

82

atri4 =1

83

atrin =1

8n−1

2. What’s the area of the polygon generated in the 2nd, 3rd, and 4th?

3. Let’s denote the area of polygons in each step by apolyn ;

apoly1 = 1

apoly2 = 1 + 2× 1

8

apoly3 = 1 + 2× 1

8+ 22 × 1

82

apoly4 = 1 + 2× 1

8+ 22 × 1

82+ 23 × 1

83

4. What’s the area of the polygon generated after N steps?

apolyN = 1 + 2× 1

8+ 22 × 1

82+ 23 × 1

83+ ...+ 2N−1 × 1

8N−1

apolyN = 1 +1

4+

1

42+

1

43+ ...+

1

4N−1

136

apolyN =N∑n=1

1

4n−1

Geometric series:N∑n=1

qn−1 =qN − 1

q − 1

apolyN =4

3

(1− (

1

4)N)

.

apolyN =4

3

(1− (

1

4)N)

5. Now guess what is the area of the parabolic segment?

6. How close can we get to that number by following this sequence ofnumbers or areas?

7. Archimedes found out that if the area of the 1st triangle is 1 then thearea of the parabolic segment is exactly 4

3

8. Now what’s the difference between the area of parabolic segment andthe area of the 1st, 2nd, 3rd, 4th, and nth generated polygon?

4

3− apol1 =

4

3− 1

4

3− apol2 =

4

3− 1− 2× 1

8

4

3− apol3 =

4

3− 1− 2× 1

8− 22 × 1

82

4

3− apol4 =

4

3− 1− 2× 1

8− 22 × 1

82− 23 × 1

83

43− apoln = 4

3− 4

3

(1− (1

4)N)

137

43− apolN = 4

3× 1

4N

9. It’s clear that by increasing the number of steps and consequently thepolygon’s sides, this difference that we might call it an approximationerror, decreases. If we want to set the accuracy to be 2−60

3, how many

steps should be taken?

10. Remember that in theory, you can increase “n” as much as you want.Do you think that by doing this process without stop we get an ap-proximation of the area of the parabolic segment? Does this accuracyeventually becomes 0?

11. What the hell is going here? It seems that this non stop process givesthe actual area. So we are missing some thing here. Is there any magicword or concept?

8.1.3 Appendix 3, Derivative Activity

1. How many lines pass through two points A(x1, y1), B(x2, y2)?

• Draw the line passing from A(1, 3), B(4, 9);

2. If we call the increment ∆x = xC−xA, and ∆y = yB−yC , respectivelyrun and rise, what is your interpretation from rise

run?

3. Which characteristic of the line does this proportion represent?

4. If each of your steps is a unit, and you move with constant velocity,what is the slope of your movement if for each 1

2step forward, you move

2 steps up?

m = 4

5. What if for each 3 steps forward, you move 1 step down?

m =−1

3

138

6. What does riserun

represent when speaking about the motion from timetA to time tB and respectively from position xA to position xB?

mAB =rise

run=

∆x

∆t: average velocity

• The slope of the line height-time (that shows the changing of thedistance in time), demonstrates the average velocity of that par-ticle.

Now, let’s go to a physic problem!

• Assume that a mass - for instance a cannon ball - is falling straightdown from a height of 16 Km according to the formula: f(t) =16− t2/200.

• http://staff.science.uva.nl/~kaper/limits/newton.html

• Mark point A which shows the mass’s position in the 20th second;

7. How many lines pass through this point?

8. Which one is the tangent line?

9. Can we draw a line by having just one point of it?

What is the tangent line of the curve in point A?

• Let’s choose an arbitrary point B on the intersect of lines passingfrom A and the parabola.As you see, rise in this graph is negative while run can not becomenegative in a motion!

mAB =∆x

∆t

10. what is the physical interpretation of

mAB =rise

run

• It demonstrates the average velocity between time A and time B.

• As you see, when B becomes closer and closer to A, AB becomescloser and closer to the tangent line.

139

http://staff.science.uva.nl/~kaper/limits/newton.html

• Let’s make B closer and closer to A numerically;

11. Find out the average velocity of this mass between 20th and 40th sec-onds.

mAB =rise

run=f(40)− f(20)

40− 20=−402

200+ 202

200

20= −0.3

12. Make B closer to A; find out the average velocities between

t = 20 and t = 25, 21, 20.02, 20.001

mAB =f(25)− f(20)

25− 20=−252

200+ 202

200

25− 20= −0.225

mAB =f(21)− f(20)

21− 20=−212

200+ 202

200

21− 20= −0.205

mAB =f(20.02)− f(20)

20.02− 20=−20.022

200+ 202

200

20.02− 20= −0.2.01

mAB =f(20.001)− f(20)

20.001− 20=−20.0012

200+ 202

200

20.001− 20= −0.200005

• As you see, we can not continue this process till ever. So, let’s findout the slope of line A(20,14) and B(20+h, f(20+h))and find outthe average velocity between 20th second and (20 + h)th second.

mAB =−(20+h)2+202

200

h

mAB =−40× h− h2

200× h

Concerning the condition that h can not be 0, we can factor it out fromnominator and denominator; h 6= 0

mAB =−40− h

200

140

• This is the slope of the average velocity between tA and t(A+h).Therefore it’s clear that if h = 0 then, we can get the instantaneousvelocity.

13. Do you think if we can do that?

14. Is there any other constraint on h apart from h 6= 0?

15. If we let h get closer and closer to 0 without stop, recall the previ-ous activity and explain why it’s not an approximation or accuracyestimation.

• In this case, we say that h approaches 0, or

h→0

16. Is the following argument true? as h→0 or t(A+h) → tA,

−40 + h

200=−40

200

• Let’s use the symbol ” lim ” which is the abbreviation of “limit”,to explain this situation!

limh→0

−40− h200

= −0.2

• Our lost magic words were “infinitesimal” and “limit”.

8.1.4 Appendix 4, Neighborhood Activity

1. Find the set of points on real line whose distance from origin; 0, is 2.

|x| = 2

2. Find the set of points on real line whose distance from origin; 0, is a.

|x| = a

141

3. Find the set of points on real line whose distance from 2 is 3.The distance of x = −1 and x = 5 from 2 is 3

• Set the origin at 2 by transferring the real line. Now, introduce anew variableX = x−2 and formulate the question in an equivalentway in terms of new variable.

• The distance of x− 2 or X from 0 is 3

• This is equivalent to |X| = 3 or |x− 2| = 3

4. Find the set of points on real line whose distance from 1 is less than 4.

|x− 1| < 4

5. Find the set of points on real line whose distance from a is b.

|x− a| = b

6. Find the set of points on real line whose distance from -1 is more than2.

|x− (−1)| > 2⇒ |x+ 1| > 2

• New SymbolBr(x0) = {x||x− x0| < r}

http://staff.science.uva.nl/~kaper/limits/neighboorhood1.html

7. How do you interpret the following relation in term of distances?

y ∈ Bε(3)⇒ x ∈ Bδ(1)

|f(x)− 3| < ε⇒ |x− 1| < δ

8. As you see in this function,y ∈ Bε(3) ⇒ x ∈ Bδ(1). Check δ when εchanges. For instance it changes from 0.5 to 0.2, and 0.1;

ε = 0.5⇒ δ = 0.25

ε = 0.2⇒ δ = 0.1

ε = 0.1⇒ δ = 0.05


142



9. Given a neighborhood Bε(1) on the y-axis (images), color all the x-values on the x-axis (origins) that have their f(x)in Bε(1)

10. Find the biggest neighborhood of 2 that catch all the x values thathave their f(x) in Bε(1).

11. As you see in this function, y ∈ Bε(1) ⇒ x ∈ Bδ(2), check δ, when εchanges;

ε = 0.5⇒ δ = 0.45 or smaller

ε = 0.2⇒ δ = 0.19 or smaller

ε = 0.1⇒ δ = 0.1 or smaller

8.1.5 Appendix 5, Epsilon-Delta Proof

November 2008 1

Wolter Kaper

A script for the paper part of the Mathematica workshop

[ I was asked to organize a Mathematica workshop for Andi’s students.It would be the first meeting after Andi had introduced the formal definitionof limits to them. Therefore I planned a short recapitulation of the wholesequence: Zeno - Archimedes - Newton - formal definition.After that, I would demonstrate the use of the formal definition in doing aproof on the board. This would be an intro for doing more difficult proofsusing Mathematica.

The full session was 4 hours, of which it was planned:

1. 1 hour: recapitulation and interactive demo of a proof on the board

2. 1 hour: getting to know Mathematica.

3. 2 hours: trying some interesting limit proofs with Mathematica or onpaper

11 Typed out: November 2009! Remarks in brackets were added during typing, so, ayear after the facts.

143

Of these, part (1) took more time and was more interesting than expected 1.These were the notes I prepared the evening before the workshop. A sequenceof three dots means: improvise, elaborate, etcetera.]I am... One of my jobs is educational adviser... but I also teach about theuse of computers in mathematics to first year...We have a master program on Mathematics and science education, that I amalso involved in... Andi is studying in that master.You are probably not interested yet in becoming a math teacher but probablyyou are interested in mathematics....

(Let’s do some mathematics then) Andi told you about the mathematicaldefinition of limits.

Limits are designed to solve various problems that you encountered inthe previous lessons with Andi:

• Zeno: a story about a strange guy who takes shorter and shorter steps,each step is half of the previous one.

• Distance walked Σnk=1

12n

= 12

+ 14

+ 18

+ ...+ 12n

= 1− 12n

“the remaining distance is halved each time”

1E.g. Harm, the usual teacher of the students dropped in, and joined our discussion thatstarted with the question by Dave and Andi: what use are limits of the type limx→1(2x+1), where there is no problem: 1 can be directly substituted for x...?

144

there is a number that we are getting closer and closer to, we can get as closeas we want by increasing n, but we can not reach it in this case

limx→∞

(1− 1

2n) =? 1 ← exactly?

is there another number, close to 1, about which we could say the samething?- Imagine a number a bit more to the right... can we get as close as wewant...?- Imagine a number to the left ...

Question for today is: can we prove such a limit?

• can we prove that there is exactly one number that we can getas close to as we want, by making n closer to infinity.

Andi also showed you 2 other applications of limits(2 pictures)

• The sums of the areas of all the triangles ... gets close to a certainnumber: 4

3.

• If we want to know the velocity at 20 secs. ... between 20 and 30 secs.the velocity is changing so this is just an average if we make the timeperiod we are looking at shorter and shorter, the slope will get close to...And we hope there is just one number for the slope that we canget as close to as we want by making the time period shorter and shorter...so, we hope there is a limit here.

Can we be sure that there is a limit in these 3 cases?

Program for the rest of the meeting:- definition of limit Andi told you about- try to use it on paper on 1 or 2 cases to prove that a limit exists- use Mathematica for 3 other cases [They could choose which cases to do.]

145

Informal definition:

• The limit of a function f(x) for x→ a exists and is = L

if and only if:

we can make the difference between f(x) and L as small as we wantby making x as close to a as we need.

Formal definition [2 versions]:

• ...

if and only if:

for every neighborhood Bε(L) there is a corresponding Bδ(a) suchthat:

x in Bδ(a)⇒ (“ensures”)f(x) in Bε(L)

x in Bδ(a) means:

a− δ < x < a+ δ and x 6= a or:

0 < |x− a| < δ it has a hole! Why?...

f(x) in Bδ(L) means:

L− ε < f(x) < L+ ε or:

|f(x)− L| < ε

Note that f(x) may be = L for some x! Not required. [So, no holehere.]

[final version of the definition:]

• ...

if and only if

for every ε > 0 there is a δ such that:

0 < |x− a| < δ ⇒ (“ensures”) |f(x)− L| < ε

146

Let’s see if we can use the definition on an easy case:

f(x) = 2x limx→2

2x =? 4

(graph)given an epsilon, what delta would you choose?δ= ... (formula with epsilon) ...δ = ε

2

but every smaller δ is also right!δ = ε

3it still ensures that if x in Bδ,

then surely f(x) is in Bε

Now we made our choice:δ = ε2

Now we have to prove that the choice is right.We already know it is right:We see it in the picture, but mathematicians want a proof!Why...? [...the paradoxes...]Let’s try to do the proof. First step: write down

• what you may assume

• what you want to prove

Assuming:

• 0 < |x− 2| < δ

• δ = ε2

(⇒ 0 < |x− 2| < ε2)

To prove:

• |f(x)− L| < ε

|2x− 4| < ε

We won’t replace these epsilons with a number,because: we want to prove it for all epsilons. 1

1[Most sentences in my notes were meant to be said rather than written. Most formu-las’s were meant to be written.]

147

Proof:|x− 2| < ε

2we assumed it

2|x− 2| < ε why is it good ...?|2x−4| < ε why is it good? which rule do we use here ...?

|a| × |b| = |a× b| ...is it always good...?a× |b| = |a× b| ...??only if a is positive

What did we prove...?For all ε > 0 we can make this distance smaller than ε!This proof didn’t tell us anything that we didn’t know.

We already knew the limit is 4 in this case. [Because: no discontinuity]We tried if the difficult definition also covers an obvious case, and indeed, itdoes.

The nice thing is: the same method also works for the more difficult cases!

8.1.6 Appendix 6, Mathematica

This appendix has been moved to the attached CD!

8.2 Test

1. Which of these sentences best describes a limit as you understand it?

• A limit describes how a function moves as x moves toward a certainpoint.

• A limit is a number or point past which a function can not go.

• A limit is a number that the y-values of a function can be madeas close as we want by choosing the x-intervals as small as needed.

• A limit is a number or point the function gets closer to but neverreaches.

• A limit is an approximation that can be made as accurate as youwish.

• A limit is determined by plugging in numbers closer and closer toa given number until the limit is reached.

148

2. Please describe in a few sentences what you understand a limit to be.That is, describe what it means to say that the limit of a function f asx→ s is some number L .

3. Given a function f(x) = 3 , evaluate the limit of f(x) as x→ 2.

4. A student was given a function F and asked to find the limit of F as xapproaches 0. He plugged in numbers on each side of 0 and made thefollowing table:

What can you conclude about the limit of function F as x approaches0?

5. Sketch the graph of the function f(x) = x2−9x−3

Answer the following questions:

• What happens at the point x = 3?

• Does the limit of f(x) exist at x = 3?If yes, what is this limit?

• What is the value of the function at x = 3?

6. Find the following limits:

• limx→2 3x+ 8

• limx→∞ sin(x)

• limn→∞(1 + 910

+ 9100

+ ...+ 910n

)

7. Answer the following questions referring to the function graphed bel-low:

149

• limx → 2f(x)

• limx → −∞• limx → 1f(x)

• limx → 5f(x)

8. What is between 0, 999... (The nines repeat) and 1?

• Nothing because 0, 999... = 1.

• An infinity small distance because 0, 999... < 1.

• You can’t really answer because 0, 999... keeps going forever andnever finishes.

• If you don’t agree with any of the above, circle (d) and give yourown answer

9. Sketch the graph of the function f(x) = x−2x−2

• What is the value of f(2)?

• Does f(x) have a limit at x = 2?

• If yes, what is the limit?

150

8.3 Transcripts (on the attached CD)

Transcripts are published on the attached CD. The following transcripts willbe found there:

8.3.1 Zeno’s Paradox

8.3.2 The Quadrature of the Parabola

8.3.3 Derivative Activity

8.3.4 Neighborhood

8.3.5 Working Towards Formal Definition

8.3.6 Epsilon-Delta Proof & Mathematica

8.3.7 Continuity

8.3.8 Last Interviews

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Contemplating problems taken from the history of limits as a

Documents

Transcript of Contemplating problems taken from the history of limits as a