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l LIBRARY
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This is to certify that the
dissertation entitled

DEVELOPMENT OF DISCOURSE ON LIMITS:
CONNECTING HISTORY AND CLASSROOM PRACTICE
THROUGH A COMMUNICATIONAL APPROACH TO
LEARNING

presented by

Beste Gfloler

has been accepted towards fulfillment
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PhD. degree in Mathematics Education

 

 

 

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DEVELOPMENT OF DISCOURSE ON LIMITS: CONNECTING HISTORY AND
CLASSROOM PRACTICE THROUGH A COMMUNICATIONAL APPROACH TO
LEARNING
By

Beste Gficler

A DISSERTATION

Submitted to
Michigan State University
in pattial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Mathematics Education

2010

 

DIVELU
CLASSF

ABSTRACT
DEVELOPMENT OF DISCOURSE ON LIMITS: CONNECTING HISTORY AND
CLASSROOM PRACTICE THROUGH A COMMUNICATIONAL APPROACH
TO LEARNING
by
Beste Giicler
The notion of limit is considered to be the building block of many calculus
concepts such as continuity, derivative and integral. On the other hand, the concept
presents students with many challenges. This study views mathematics learning as
initiation to the historically established mathematical discourse and uses a
communicational approach developed by Sfard (2008) to explore the conceptual
obstacles in learning limits. One of the goals of this study is to investigate how the
discourse on limit and its underlying concepts is generated over history. This exploration
goes in conjunction with the discursive analysis of the historical junctures that led to
particular changes in the discourse on limits as mathematicians encountered conceptual
obstacles. The study then focuses on one college-level calculus classroom to explore how
the discourse on limits is generated by the instructor. This is followed by an investigation
of students’ discourse on limits at the end of their instruction. Finally, possible
connections between the discourse on limits as generated over history and as generated in
the classroom are explored to examine whether the communicational approach is useful
to gain further insights about learning of limits.
The study revealed that the consideration of limit as a distinct object of
mathematics (a number) obtained at the end of a process was challenging for

mathematicians over history. The students in the study had difficulties distinguishing the

process aspect of limits from the realization of the concept as an end-state (a number),
which is consistent with the historical development of the concept. Opportunities for
addressing the differences between the consideration of limit as a process and limit as a
number were present in the instructor’s discourse. However, the distinct assumptions

underlying each realization of limit remained implicit for students.

C0pyright by
BESTE GUCLER
2010

To my mother and father...

ACKNOWLEDGMENTS

I am thankful for the help, encouragement, and guidance of many individuals
throughout this arduous journey. I thank my father for showing enormous effort to
understand the characteristics of a doctoral program and supporting me through my ups
and downs. I thank my friends, especially Irini and Violeta, with whom we shared many
ideas, emotions, experiences, and stories along the way.

I am grateful to Raven McCrory for all the support she provided during the
dissertation process. Without her guidance, it would be impossible to bring this study to
life. I also thank my committee members, Suzanne Wilson, Joan Ferrini-Mundy, and
Vince Melfi, for their valuable input and support. I thank Anna Sfard for her help during
the initial stages of my work and for the beautiful framework she developed. I thank
Sharon Senk for always being there when I needed her guidance. For all their help,
kindness and support, thanks to Jean Beland, Margaret Iding, Lisa Keller, and Jim Keller.

I also owe great debt to the participants of my study. Thanks to the instructor and
the students who gave me access to their classroom to help me conduct my study.
Without them, I would have no story to write about.

I am forever grateful to music and my favorite musicians for making me feel alive
and giving me the strength to write. Last but not least, [thank my mother. Although you
are long gone, you are always with me. It is because of you I embrace people and their
stories with curiosity and love; it is because of you I can “feel” music and life; it is

because of you I believe in education; it is because of you I am who I am. Thank You...

vi

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TABLE OF CONTENTS

 

 

 

 

 

 

LIST OF TABLES x
LIST OF FIGURES xiii
CHAPTER I
INTRODUCTION 1
CHAPTER II
THE CONTEXT OF THE STUDY 4
2.1 Limit: What is it and what do we know about its learning? ................................. 5
2.1.1. The notion of limit ............................................................................................ 5
2.1.2. Research on learning about limits ..................................................................... 6
2.1.3. Difficulties about the underlying concepts of limits ....................................... 20
2.1.4. Textbooks, teaching and attitudes towards mathematics ................................ 24
2.1.5. Possible links between historical development and student learning
of limits ........................................................................................................... 26
2.2 Commognitive framework ...................................................................................... 28
2.2.1. The general tenets ........................................................................................... 28
2.2.2. Metarules and metaphors in discourse development ...................................... 34
Summary ....................................................................................................................... 37
CHAPTER III
HISTORICAL DEVELOPMENT OF THE DISCOURSE ON INFINITY,
INFINITESIMALS AND LIMITS 39
3.1. The notion of infinity ............................................................................................. 40
Summary ................................................................................................................... 46
3.2. The notion of infinitesimals ................................................................................... 47
Summary ................................................................................................................... 52
3.3. The notion of limit ................................................................................................. 54
Summary ................................................................................................................... 59
3.4. Historical junctures in the development of discourse on infinity,
infinitesimals and limits ........................................................................................ 60
Summary ................................................................................................................... 62
CHAPTER IV
DESIGN OF THE STUDY 64
4.1. Specific research questions .................................................................................... 64
4.2. The participants ...................................................................................................... 64
4.3. Data collection ....................................................................................................... 66
4.4. Data analysis .......................................................................................................... 72
4.4.1. Analysis of the instructor’s discourse ............................................................. 72

vii

4.4.2. Analysis of the diagnostic survey and student interviews with

 

 

respect to the instructor’s discourse ................................................................ 78
4.4.3. Analysis of the classroom discourse with respect to the
historical development of limits ..................................................................... 82
CHAPTER V
THE INSTRUCTOR’S DISCOURSE ON LIMITS 84
5.1. Word use ................................................................................................................ 87
5.1.1 Colloquial word use ......................................................................................... 90
5.1.2 Operational word use ....................................................................................... 90
5.1.3 Objectified word use ........................................................................................ 94
5.1.4 Operational and objectified word use .............................................................. 97
5.1.5 Mathematical context of use .......................................................................... 100
Summary of word use ............................................................................................. 129
5.2. Visual mediators .................................................................................................. 131
5.2.1 . Written words ............................................................................................... 131
5.2.2. Drawn pictures of geometric shapes ............................................................. 133
5.2.3. Graphs ........................................................................................................... 133
5.2.4. Symbolic notation ......................................................................................... 137
Summary of visual mediators ................................................................................. 139
5.3. Routines ............................................................................................................... 140
5.3.1 Algebra-based routines .................................................................................. 141
5.3.2 Geometry-based routine (Graphing) .............................................................. 144
5.3.3 Using the metaphor of continuous motion ..................................................... 146
5.3.4 Using the metaphor of discreteness ............................................................... 148
Summary of routines ............................................................................................... 149
5.4. Endorsed narratives .............................................................................................. 150
Summary of endorsed narratives ............................................................................ 153
Summary of Jason’s discourse on limits ..................................................................... 154
CHAPTER VI
STUDENT DISCOURSE ON LIMITS IN RELATION TO THE IN STRUCTOR’S
DISCOURSE 160
6.1. Responses to the diagnostic survey ..................................................................... 161
6.1.1. Questions I and II .......................................................................................... 162
6.1.2. Questions 111 and IV ..................................................................................... 172
Summary ..................................................................................................................... 178
6.2. Responses to the individual interview sessions ................................................... 180
6.2.1. Students’ discourse with respect to the instructor’s discourse on limits ...... 183
6.2.2. Students’ discourse with respect to student difficulties indicated
by research on learning about limits ............................................................. 202
Summary ..................................................................................................................... 212

viii

 

 

APPI
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CHAPTER VII
CLASSROOM DISCOURSE ON LIMITS IN RELATION TO THE HISTORICAL
DEVELOPMENT OF LIMIT RELATED CONCEPTS THROUGH THE

 

 

 

 

COMMOGNITIVE LENS 218
7.1. The instructor’s discourse with respect to the historical development of
limit related concepts .......................................................................................... 220
7.2. Students’ discourse with respect to the historical development of
limit related concepts .......................................................................................... 226
Summary ..................................................................................................................... 230
CHAPTER VIII
SUMMARY, DISCUSSION AND CONCLUSIONS 234
8.1. Summary of findings ........................................................................................... 235
8.2. Discussion ............................................................................................................ 240
8.3. Conclusions .......................................................................................................... 245
8.3.1. Implications of the study ............................................................................... 246
8.3.2. Limitations of the study ................................................................................ 248
8.3.3. Further questions ........................................................................................... 248
APPENDIX A 251
APPENDIX B 252
REFERENCES 256

 

ix

LIST OF TABLES

Table 2.1: Difficulties mentioned by the research on student thinking about limits ........ 16
Table 3.1: Timeline of the mathematicians discussed in this chapter over history .......... 40

Table 3.2: Comparison of the historical development of calculus with Cauchy’s
framework ....................................................................................................... 56

Table 3.3: Historical junctures in the development of discourse on infinity, infinitesimals
and limits over history .................................................................................... 61

Table 4.1: Distribution of students across their majors .................................................... 66

Table 4.2: Classification of student responses for Question 1 and Question 2 of the

survey .............................................................................................................. 69
Table 5.1: Topic outline for Jason’s lessons ..................................................................... 86
Table 5.1 cont’d: Topic outline for Jason’s lessons .......................................................... 87
Table 5.2: Examples of Jason's operational word use ...................................................... 92
Table 5.3: Examples of Jason's objectified word use ....................................................... 95
Table 5.4: Examples of Jason's operational and objectified word use .............................. 98
Table 5.5: Categorization of Jason’s limit related utterances in four categories .............. 99
Table 5.6: Jason's utterances about the informal definition of limit ............................... 102
Table 5.7: The rationale for the precise definition of limit ............................................. 104
Table 5.8: Jason's precise definition of limit .................................................................. 105
Table 5.9: Jason’s utterances in the context of computing 11:12 :1—7. .......................... 109

Table 5.10: Jason’s utterances in the context of computing lim f (x) from a graph. 110
x—>l

Table 5.11: Jason's utterances when computing lim 1. .............................................. 113
x—-)oo x

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Table 5.12: Jason's utterances about the x values when he computed limits at infinity. 115

 

Table 5.13: Jason's utterances when he computed lim srn x ........................................ 116
x—‘ND x
Table 5.14: Jason’s utterances about infinity in the context of infinite limits ................ 1 18

Table 5.14 cont’d: Jason’s utterances about infinity in the context of infinite limits ..... 119

Table 5.15: Jason's infinitesimal related word use in the context of infinite limits ........ 122
Table 5.16: Jason's utterances about the intuitive definition of continuity ..................... 124
Table 5.17: Jason's utterances about the precise definition of continuity ....................... 125

Table 5.18: The relationship between computing the limit of a continuous
function and plugging in ............................................................................... 127

Table 5.19: Types of visual mediators used in the context of computing a limit ........... 134

Table 5.20: Examples of Jason’s hand-drawn graphs besides the context of computing

limits ............................................................................................................. 136
Table 5.21: Algebra-based routines in Jason's discourse ................................................ 142
Table 5.22: Graphing as a geometry-based routine in Jason’s discourse ....................... 145
Table 5.23: Jason’s routine of using the metaphor of continuous motion ...................... 147
Table 5.24: Jason’s routine of using the metaphor of discreteness ................................ 148

Table 6.1: Statements in the first question of the diagnostic survey and
the corresponding views of limit (Williams, 1991; 2001) ............................ 162

Table 6.2: Students’ responses to the first question of the diagnostic survey ................ 163

Table 6.3: Cross-comparison of student responses with respect to the
statements they chose as true ........................................................................ 163

Table 6.4: Cross-comparison of students’ responses with respect to the

statements they chose as best describing their realizations of limit ............. 166

Table 6.5: Students’ responses to the third question of the diagnostic survey ............... 173

Table 6.6: Students’ responses to the fourth question of the diagnostic survey ............. 177
xi

 

lite 6.'

Table 6.!

Table 6.7: General information about the students participating in the
interview sessions ......................................................................................... 183

Table 6.8: Some examples of students’ word use when referring to the limit notation. 195

xii

Figure 2.1:
Figure 2.2:
Figure 2.3:
Figure 2.4:
Figure 2.5:
Figure 2.6:
Figure 2.7:
Figure 5.1:
Figure 5.2:

Figure 5.3:

Figure 5.4:
Figure 5.5:

Figure 5.6:

Figure 5.7:

Figure 5.8:
Figure 5.9:

Figure 6.1:

Figure 6.2:

Figure 6.3:

LIST OF FIGURES

An example where the difficulty limit implies continuity can occur ............... 7
An example where the difficulty limit as the function’s value can occur ........ 8
An example where the difficulty limit as bound can occur .............................. 9
An example where the difficulty limit as monotonic can occur .................... 10
An example where the difficulty limit as monotonic can occur .................... l 1
An example where the difficulty limit as approximation can occur .............. 13
An example where the difficulty limit as unreachable can occur .................. 15
Common words used in Jason’s discourse on limits ...................................... 88
Jason's hand-written example for finding limits from a given graph ............. 94

Categorization of Jason’s limit related utterances as operational,

objectified or both in lessons 2-8. ................................................................. 100
Jason’s informal definition of limit (hand-written) ...................................... 102
Jason's precise definition of limit (hand-written) ......................................... 104
Computing limits from a given graph (hand-written) .................................. 110
Jason's hand-drawn graph of f (x) = l ....................................................... 113
x
Jason's precise definition of continuity (hand-written) ................................ 125
Jason’s introduction of the intermediate value theorem (hand-written) ...... 146

Cross-comparison of student responses with respect to the statements

they chose as true .......................................................................................... 164
Cross-comparison of student responses with respect to the statements

they chose as best describing their realizations of limit ............................... 166
Interview session problems .......................................................................... 182

xiii

CHAPTER I
INTRODUCTION

Starting with the calculus reform movement initiated in mid-eighties, research on
undergraduate mathematics education has focused on improving curricular and
pedagogical approaches to beginning calculus. Today, students in majors such as
economics, engineering and physics are expected to be competent in various
mathematical domains and calculus is one of the main courses required of students in all
these majors. As a result, there also exists a relatively rich research base at the
undergraduate level on student thinking about the content of elementary calculus
concepts such as function, slope and derivative (Carlson, 1998; Monk, 1987, 1994; Monk
& Nemirovsky, 1994; White & Mitchelmore, 1996; Zandieh, 2000).

The concept of limit has also been of particular interest for researchers since it is
considered to be the building block of many fundamental calculus concepts such as
continuity, derivative and integral. The notion of limit, however, presents major
difficulties for students (Bezuidenhout, 2001; Cottrill et al., 1996; Tall & Vinner, 1981;
White & Mitchelmore, 1996; Williams, 1991). These studies highlight that the formal
understanding of the concept is unlikely to occur unless students first have an intuitive
understanding of the concept. However, they also argue that the intuitive understanding
of the concept relies heavily on the idea of continuous motion, which might hinder
understanding of the other aspects of limit. In that respect, some of the representational
tools (verbal, visual and symbolic) used by students while thinking about limit lead to
additional difficulties (Bagni, 2004, Bezuidenhout, 2001; Cottrill et al., 1996; Williams,

1991). Therefore, the concept of limit presents the student with two challenges: the need

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to make the transition from intuitive to formal understanding and the need to cope with
the issue of compatibility of the conceptual and representational tools within the
intuitively understood aspects of limit.

Considering mathematics learning as initiation to the historically established,
patterned activity of doing mathematics, one of the goals of this study is to explore how
the discourse on limit and its underlying concepts as generated over history. This
exploration will go in conjunction with the discursive analysis of the historical junctures
that led to particular changes in the discourse on limits as mathematicians encountered
conceptual obstacles. In this study, such changes will be will be defined through elements
of a communicational framework developed by Sfard (2008). Sfard’s framework will
then be used to explore how the discourse on limits is generated by the instructor in a
college-level calculus classroom. This will be followed by an investigation of students’
discourse on limits at the end of their instruction and what conceptual obstacles they
encounter. Finally, possible connections between the discourse on limits as generated
over history and as generated in classrooms will be explored to examine whether the
communicational approach, in general, and a discursive analysis of historical junctures, in
particular, help us gain further insights about learning of limits. More specifically, the
study addresses the following questions: 1) How is the discourse on limits generated by
the instructor in a beginning college-level calculus classroom? 2) Given the instructor’s
discourse on limits, how do students talk about limits in a beginning college-level
calculus course? and 3) How do the elements of discourse on limits as generated over
history compare and contrast with the discourse on limits generated in a beginning-level

calculus course?

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The contribution of the study to educational research is two-fold: First, although
research has identified many of the conceptual obstacles students have about limits,
explanation of the nature of those obstacles remain incomplete. Some similarities
between mathematicians’ and students’ struggles is implied by research but there is no
elaboration on the principles underlying the transitions learners need to go through as
they attend to different aspects of the limit notion. This study approaches the same
problem by means of a different lens that emphasizes communication to examine whether
it can provide further insights about the conceptual obstacles in learning of limits. I will
use elements of Sfard’s (2008) framework to investigate the historical development of
discourse on limits with respect to possible roots of the conceptual obstacles faced by
mathematicians. The study will then explore whether Sfard’s (2008) framework can be a
useful lens to gain more information on students’ discourse on limits.

Second, although research about limits suggests that the intuitive aspects of the
notion are perpetuated in calculus classrooms, there is no analysis of instruction in order
to justify this claim. In this work, I will analyze one instructor’s discourse on limits in a
college-level calculus classroom and investigate the possible impacts of the instructor’s
discourse on students’ thinking about limits. In this respect, the study is an attempt to fill

out an important gap in the literature about teaching of limits at the undergraduate level.

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CHAPTER II
THE CONTEXT OF THE STUDY

This chapter starts with a brief description of the limit concept followed by a
literature review on learning about limits. Next, the theoretical fi'amework for the study is
introduced and explained in terms of its main characteristics. Finally, particular elements
of the theoretical framework that I use for the analysis of the historical development of
limits are described.

Axioms, definitions, theorems and proofs presented in their formal
representations are among the final products of mathematics. The processes with which
mathematicians and learners of mathematics initially think about mathematical concepts,
on the other hand, are informal and intuitive. Although such processes may be invisible in
the mathematics curricula, the transition from the informal aspects of mathematical
concepts to their rigorous formulations is by no means trivial. The historical development
of a concept is a valuable tool in providing clues about where to look for the obstacles
learners may face and the transitions they may go through as they tackle different aspects
of the concept. The nature of those transitions and conceptual difficulties, however, may
not be found in history itself but can be identified by means of a focused analysis of the
historical development. This study uses elements of the commognitive framework (Sfard,
2008) for the analysis of the historical development of limit and its underlying concepts.
In what follows, the notion of limit is briefly discussed and a summary of the research on
its learning is given. Then the commognitive framework is introduced as a potentially
useful lens that can help us gain more insights about the nature of the conceptual

obstacles in the learning of limits.

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2.] Limit: What is it and what do we know about its learning?

2.1.]. The notion of limit

The concept of limit is the foundation on which fundamental concepts of calculus
are based. Limits are used to define the tangent to a curve, which leads to the notions of
the derivative of a function and instantaneous rate of change. In that sense, limits are used
to determine how functions vary. When used to define the behavior of Riemann sums at
infinity, the notion leads to the concept of the integral of a fimction. In particular, limits
are also used in determining whether sequences and series converge. Given these, it is
impossible to talk about the essential concepts of calculus without limits. The informal
definition of limit is ofien given in some form similar to the following:

Let [a function] f (x) be defined on an open interval about [the point] x0 ,
except possibly for x0 itself If f (x) gets arbitrarily close to L (as close as
we like) for all x sufficiently close to x0 , we say that f approaches the

limit L as x approaches x0 , and write lim f (x) = L , which is read ‘the
x—)x0

limit of f (x) as x approaches x0 is L ’. (Thomas, Weir, Hass &

Giordano, 2008, p. 77)
A typical calculus-level formal definition, on the other hand, would be similar to the
following:

Let [a function] f (x) be defined on an open interval about [the point] x0 ,
except possibly for x0 itself. We say that the limit of f (x) as x

approaches x0 is the numberL , and write lim f (x) = L if for every
x—>x0

numbers > 0 , there exists a corresponding number 5 > 0 such that for
allx,0 < Ix—xol < 6 :> [f(x) —L| < e . (Thomas et al., 2008, p. 92)

At the undergraduate level, students are introduced to the notion of limit in their

preliminary calculus courses]. The textbooks designed for these courses often outline the
content of calculus starting from reviews of basic notions such as real numbers, number
line, functions and types of functions. Then the notions of limit and continuity are
introduced. This is followed by discussions on derivative and then the integral. Finally,
the notions of sequences and series are introduced with a particular focus on their
behavior at infinity, which form the basis of the discussions about convergence and
divergencez.

2.1.2. Research on learning about limits

In what follows, the difficulties associated with the limit concept as pointed out by
research on student learning will be explained.

Limit implies continuity. Bezuidenhout (2001) argues that this refers to the incorrect
assumption that the existence of limit of a function at a given point is a sufficient
condition for the continuity of the function at that point. The students who have this
difficulty believe that if a function has a limit at a given point, then it must also be
continuous at that point. For example, such students would think the limit does not exist

at the point x=3 for the function in Figure 2.1 since it is not continuous at 3.

 

1 Students can also have familiarity with the limit concept from their high school courses such as AP
Calculus and precalculus.

2 See Thomas et a1. (2008) and Hughes-Hallett et a1. (2008) for their content outline for calculus.
6

 

 

 

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Figure 2.1: An example where the difficulty limit implies continuity can occur

Limit as the function ’s value. Bezuidenhout (2001) notes that the incorrect assumption
limit implies continuity “may also originate from another misconception, namely,

lim f (x) = 3 implies f (2) = 3 , so that f is then continuous at x=2” (p. 494). In this
x—-)2

respect, students could view lim f (x) and f (2) as the same thing. This view of limits is
x—>2

called “limit as the frmction’s value” and it corresponds to the belief that “the limit of a
function at a point means the value of the function at that point” (Cottrill et a1, 1996, p.
178) Students having this difficulty would give ‘limit does not exist’ as the answer
whenever the function is not defined at the point where the limit is taken. Their strategy
while finding the limit of a fiinction at a point is to evaluate the fimction’s value at that
point and give the result as the limit value. For example, for the function in Figure 2, such

students would give 3 as the answer for the limit of the function at x=2.

 

 

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Figure 2.2: An example where the difficulty limit as the function ’s value can occur

Bezuidenhout (2001) argues that the procedures used in the calculation of limits such as
the method of substitution “may sell the idea to some students that the value of the
function at the point concerned is of primary importance, rather than the behavior of
values of the function about the point” (p. 496).

Limit as a bound. Cornu (1991) mentions that limits can sometimes be interpreted as “an
impassable limit which is reachable”; “a higher (or lower) limit”; “a maximum or
minimum”; “a constraint, a ban, a rule”; “the end, the finish”. (pp. 154-155). These
utterances emphasize limit as a boundary. So limit as a bound refers to the idea that “limit
is a number or point past which the function cannot go” (Williams, 1991, p. 221).
Students who have this difficulty think that a function is bounded by a specific limit
value or think of the absolute maximum/minimum values of the fimction, if they exist, as
the limit for the function. They would also have difficulty working with horizontal
asymptotes where the limit of a function at positive or negative infinity can be a number
past which the function, in its whole domain, can go. For example, for the function in

Figure 2.3, the limit at positive and negative infinity is equal to zero but the function also

attains values greater than zero.

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Figure 2.3: An example where the difficulty limit as bound can occur

Limit as monotonic. There are other interpretations of limits such as “monotonic and
dynamic monotonic” which are based on formal teaching (Cornu, 1991, p. 155).
Utterances such as “a convergent sequence is an increasing sequence bounded above (or
decreasing bounded below)” and “a convergent sequence is an increasing (or decreasing)
sequence which approaches a limit” (Cornu, 1991, p. 155) might lead to the expectation
of monotonic behavior from the function in order to find its limit. This difficulty is
related to the expectation of ‘nice behavior’ from the fiinction. Research indicates that if
a function is strictly increasing or strictly decreasing, students can more easily find the
limit at a given point. This difficulty becomes evident when working with constant and
piecewise functions (See Figures 2.4 and 2.5). The function in Figure 2.4 remains
constant in its entire domain and so students could have difficulty considering it as
increasing or decreasing. As a result, they might not be comfortable with thelidea that the
limit of that function at every point in its domain is equal to the same number 2. For the

graph illustrated in Figure 2.5, the function is increasing but not strictly. For every a, b

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e N, the interval (a, b) is constant. Moreover, for every a e N, the function jumps rather
than increasing smoothly. Students who expect strictly increasing or strictly decreasing
behavior fi'om a function might have difficulties finding the limit of the function in
Figure 2.5 at the points which lie between (a, b), where the function remains a constant
value. They also might think the limit at a given point a e N equals to both a and a+1 due
to the discontinuous nature of the function at those points. In fact, Tall and Vinner (1981)
report a similar finding for the case of sequences. They argue that given a sequence
{1,0,1,0,l,0,. . . }students might think there are two separate sequences there instead of
one. Therefore, students who perceive “limit as monotonic” might also have difficulty
determining the convergence of a sequence where the subsequences can have different

patterns fi'om each other.

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Figure 2.4: An example where the difficulty limit as monotonic can occur

10

 

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Figure 2.5: An example where the difficulty limit as monotonic can occur

Difi‘iculties resulting from the dynamic approach to limit. The dynamic approach refers to
using the intuitive idea of motion when working on limit related problems. Although it
implies motion, the dynamic approach to limit is not necessarily a graphical approach.
The common phrases such as approaches, tends to, getting close to all indicate motion-
related processes and are considered among the verbal representations of the dynamic
approach (Bagni, 2004). In his study on models of limit held by students in a second-
sernester calculus class, Williams (1991) found that

In general, the words approaching or getting close were interpreted in one

or both of two ways: as describing the physical process of evaluating a

function at different numbers, which are chosen over time to be closer and

closer to the value s [for lim f (x) ], or as describing the mental process of
x—>s

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imagining the points on the graph moving closer and closer to the limit
point. (p. 228)

Tall and Vinner (1981) also note that limits of functions of the form lim f (x) = c are
x—>a

“often considered as a dynamic process, where x approaches a , causing f(x) to get close
to c” (p. 160, italics in original). They argue that this intuitive approach to limits “is often
so strong that the feeling of the student is a dynamic one: as x approaches a , so f(x)
approaches c, with a definite feeling of motion” (Tall & Vinner, 1981, p. 161, italics in
original).

Research has identified two difficulties resulting from the dynamic view of limits:
limit as an approximation and limit as unreachable. In what follows, these terms will be
explained and also exemplified.

Limit as approximation. One of the possible ways of using the dynamic approach to

limits for lim f (x) is investigating the behavior of a function around the limit point
x—>a

(as x ——) a ) by substituting successive x-values that are closer to the pointa . In this case,
responses such as ‘when x approaches a , the values of f (x) approach the limit L ’ often
points to

a confusion of the limiting process and the product resulting from that

process. Such a confusion may go hand in hand with the erroneous View of

limit as an approximation. One can also sense a dynamic character of the

limit in the student’s motivation. (Bezuidenhout, 2001, p. 492)
This approach is dynamic since it involves the movement of the points closer to the limit
point and looking at the function’s values near that point. It resembles the tabular

representation of the function near the limit point. Some students see this approach as

12

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sufficient to determine the limit of a function at a given point without realizing that
although the function can seem to be approaching or getting close to or tending to a limit
value for those points, there are still infinitely many points near the limit point that

remained unchecked (see Figure 2.6).

 

 

x f(x)
-0.1 0.9
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-0.0001 0.9999
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0.1 1.1

 

Figure 2.6: An example where the difficulty limit as approximation can occur

Students having the dynamic view and who think of limit as an approximation argue that

lim f (x) is equal to 1 since the firnction values approach 1 as the x values approach 0.
x—)0

On the other hand, it is not possible to conclude that the limit of this function is 1 since
we do not have enough information about the behavior of the function between the points
x = —0.0001 and x = 0 as well as between x = 0.0001 and x = 0.

Some students who consider limit as an approximation also round off the
function’s values at the successive points close to the limit point and claim it is reaching a
value, which they would think is the limit value (Tall & Schwarzenberger, 1978). For
example, for the function in Figure 2.6, students can just check the function values at the
x values close to the limit value and round them to 1 to claim the limit value at x = 0 is

equal to 1. “We surmise that at the root of such a misconception is the practice, both

13

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inside and out of the classroom, of approximating numbers to convenient significant
digits” (Parameswaran, 2007, p. 210).

Limit as unreachable. The second idea related to the dynamic approach is mostly used
when determining the limit of the function from a graph. Here the movement feature
comes from the visualization of the graph. The basic logic behind this approach is to ask

where the function approaches as x approaches the limit point. (Does f (x) —) L

as x —-> a ?) Here, some students confuse the idea ‘x never reaches a’ with ‘f(x) never
reaches L’. So, they think the limit is unreachable. These students have difficulty working
with constant functions where the limit value is taken by the function at the limit point.
They also have difficulty working with the continuous functions. Williams (1991) notes
that although this view might suggest students’ awareness of the irrelevance of the
fimction value at the limit point, it might also suggest the consideration of taking the limit
of continuous function as inappropriate. As a result, some students cannot accept
continuous functions as having limits. He argues that “in general, reachability is not a
characteristic of limits, but rather is a matter of continuity” (Williams, 1991, p. 228).
According to Tall and Schwarzenberger (1978), the colloquial use of the words such as
‘close’ implies getting near to but not being coincident with. Given this, they argue that
the informal notion of limit may carry for students the assumption that one can get close

to the limit value but cannot reach it.

14

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Figure 2.7: An example where the difficulty limit as unreachable can occur

For the graph in Figure 2.7, students who have the difficulty “limit as unreachable”
would distinguish the function value at 0, which is f(0), from the function’s value as it

approaches x=0 . Therefore, they would be uncomfortable saying lim f (x) is equal to -l
x—)0

since the fimction attains the value -1 at x = 0.

It is important to note that the consideration of “limit as unreachable” is not
restricted to situations that involve graphing. The realization of limit as unreachable
signals the separation of the process of approaching and the number that is approached
(which can, in fact, be the limit). This distinction might also lead to confusion between
plugging in and approaching the number. As a result, students might be comfortable

x2 —25

 

computing lim by cancelling the common factor (x-S) since the function is not

x—+5 x -
defined at x = 5 and so the values of x cannot reach that point. On the other hand, they

might have difficulty understanding why they can plug 5 into the function when finding

15

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x—>5

summarizes the student difficulties and the related assumptions about limit mentioned in
the literature.

Table 2.1: Difficulties mentioned by the research on student thinking about limits

 

 

Student difficulties about limit Assumption underlying the views
Limit implies continuity If a function has a limit at a point, then it must be continuous at
(Bezuidenhout, 2001) that point.
Limit as the function’s value When finding the limit of a fiinction at a given point, it is
(Bezuidenhout, 2001) enough to look at the function’s value at that point.
Limit as a bound A limit is a value past which the fimction cannot go. A limit is
(Cornu, 1991; Williams, 1991) the absolute maximum (or minimum) value of a function.
Limit as monotonic A function (or sequence) has to be strictly increasing or strictly
(Comu, 1991; Tall & Vinner, 1981) decreasing in order to have a limit.
Limit as approximation In order to find the limit of a function at a point, it is sufficient
(Bezuidenhout, 2001; Parameswaran, to look at the behavior of the function at points successively

2007; Tall & Schwarzenberger, 1978) closer to the limit point.

Limit as unreachable A limit is a value that is approached but never reached.
(Tall & Schwarzenberger, 1978;
Williams, 1991)

These student difficulties about limits are mostly identified by empirical studies,
which generally include surveys followed by individual student interviews. For example,
Bezuidenhout’s (2001) study focuses on first year students’ understanding of limit of a
function and continuity of a function at a point. He selected 100 students as the sample
fiom a much larger population in three South Afiican universities. The students were in
engineering, physical sciences and in other majors that required service calculus courses.

Among the 100 students who responded to an initial survey, 15 students were selected to

 

3 See Szydlik (2000, p. 276) for the details of such student assumptions.

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participate in interviews. The difficulties “limit implies continuity” and “limit as the
function’s value” were mentioned frequently by the students who were interviewed.

In Williams’ (1991) study, 341 students from two second-semester calculus
classes were given a questionnaire about limits. Based on the student responses to the
survey, students were selected for in-depth questioning. The first question of the survey
included six statements to categorize students’ views of limits as dynamic-theoretical;
boundary; formal; unreachable; approximation; and dynamic-practical, respectively (See
Appendix A for the survey). Among the 341 students, 36% stated that the statement that
considered “limit as unreacheable” described best how they thought of limits. 30% chose
the statement about the dynamic-theoretical characterization of limit as the best way they
made sense of the concept. The statement about the formal definition was chosen as best
by 19% of the students. The statements about “limit as boundary”, “limit as
approximation” and the dynamic-practical aspect were selected by 3%, 4%, and 5% of
the students, respectively.

From 341 students, Williams (1991) then classified 50 volunteers in terms of the
models of limit they held. He classified 24 students as having the view limit as dynamic;

20 students as having the view limit as unreachable; three students as having the view

limit as a bound and one student as having the view limit as an approximation". From
those 50 students, he selected 10 students for treatment that consisted of five sessions

over a period of seven weeks. He mentions that:

 

4 Williams (1991) couldn’t classify two of the student responses since they were ambiguous.

17

f

All 10 students in the study expressed at some point a view of limit as dynamic,
that is, involving motion along the graph, and plugging in points that over time get closer

and closer to the value s as x approaches in the limit [for lim f (x) ]. (Williams, 1991, p.
x—>s

229)
He further notes that the problems students worked on in the treatment sessions targeted
student difficulties through situations that could lead to cognitive conflicts. However, no
real change occurred in students’ views of limit as dynamic after treatment. “The stage
was set for cognitive conflict, and in fact, some conflict did occur. What did not occur
was real cognitive change” (Williams, 1991, p. 229). By the end of the final session, “no
student was willing to give up the view that plugging in a finite number of values was
essentially correct or that moving along the graph was a good way to view a limit
problem” (Williams, 1991, p. 230).

Limits tend to be seen as processes performed on functions, an idealized

form of evaluating the function at a series of points successively closer to

a given value. The dynamic element here is clear, and because the actual

value of the function at the point of interest is irrelevant, the limit is never

reached. The paradigm picture seems to be the classical geometric

progression involved in walking halfway to a wall, then half the remaining

distance, and so forth; students seem to be willing to accept the fact that

we never reach the wall even though we may know exactly where the wall

is. To most of the students, this was a compelling metaphor for limit...

Still, it has been suggested that, although perhaps necessary, such a view

18

of limit does present a cognitive obstacle to further understanding.

(Williams, 1991, p. 230)

Szydlik’s (2000) study focused on calculus students’ beliefs about mathematics
and the role of those beliefs in their understanding of the limit concept. She initially gave
577 second-semester calculus students a questionnaire about their convictions and beliefs
about calculus and limits. Based on this questionnaire, she identified four groups of
students, 27 of whom participated in a structured interview about limits. “All 27 students
used a definition of limit as motion to explain their work on the limit problems; a few
students used either a static or an infinitesimal definition as well” (Szydlik, 2000, p. 271).
Nine of the 27 students agreed with statements that implied limit as a bound. “The
majority of the students who held this conception believed that the limit is a value the
function cannot exceed. They appeared to hold this View globally, often drawing or
describing a horizontal asymptote.” (Szydlik, 2000, p. 271). On the other hand, three
students in the study thought of limit as a local bound. Students having this idea
“believed that within a certain tolerance of the limiting value, the limit acts as a
boundary; however, they did not think of the limit as a global boundary”. (Szydlik, 2000,
p.271)

In Szydlik’s (2000) study, students who view calculus as a collection of facts to be
memorized and who do not follow the theory behind those facts

often cannot give a coherent definition of limit of a function or explain

why the formulas and procedures that they use to solve limit problems are

valid. Many hold misconceptions of limit as a bound that cannot be

crossed or as unreachable. (p. 273)

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The students who see calculus as consistent and logical, on the other hand, can have
“access to formal definitions, power to solve limit problems, and concept images free of
major inconsistencies” (Szydlik, 2000, p. 273). It is important to note, however, that
Szydlik mentions that not all students holding these beliefs towards calculus have
complete understanding of the limit notion. These findings are in conjunction with
Sierpir'rska’s (1987) findings which imply that “the students’ attitudes towards
knowledge, and mathematical knowledge, in particular, have a strong impact on their
intuitions of infinity and limits” (p. 382).
2.1.3. Dzfliculties about the underlying concepts of limits

Some of these difficulties about limit come from the difficulties students have of
the underlying concepts related to limit. According to Sierpir'rska (1987), the obstacles
related to the notions of scientific knowledge, infinity, function and real number form the
basis of the epistemological obstacles about limits. Carlson (1998) and Vinner & Dreyfus
(1989) argue that some students perceive functions merely consisting of algebraic rules.
Similarly, Bezuidenhout (2001) and Tall & Vinner (1981) found that students mostly rely
on rules and graphs of functions while trying to find the limit of the function at a given
point. Williams (1991) and Szydlik (2000) also mention students’ faith in graphing as a
means of understanding the behavior of firnctions when finding limits

Students’ notions of infinitely small and infinitely large can also play roles in
their understanding of the limit concept. According to Pararneswaran (2007), “it is
common practice in real life as well as in classroom that one often ‘ignores’ negligible
quantities and rounds off numbers to convenient significant digits” (p. 194). He

investigated how such practices affect students’ understanding of the limit concept. He

20

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used questions involving infinitely large and infinitely small quantities (infinitesimals) to
explore the student difficulties such as “limit as approximation” and the dynamic view of
limit. In order to do this, he investigated students’ experiences in arithmetic where they
represent decimals and do finite approximations such as rounding off real numbers to '
relevant decimal numbers. Pararneswaran (2007) found that students use approximations
before taking the limit of the function if the function includes infinitesimal quantities.
Another strategy students used when working on such limit problems was to round the
very small quantities to zero.

...the students identify what they perceive as ‘large numbers’ with infinity

and ‘small numbers’ with zero. Also, in our experiments, they unwittingly

rounded off very small parameter values occurring in the definition of the

function in question to a convenient number close to it... Most of the

students in our samples view limiting as a process of approximation when

very minute quantities are involved in the definition of the function. . .They

tend to approximate the given fimction by changing or ignoring quantities

appearing in its definition which they perceive as ‘small’ constants to zero.

(Pararneswaran, 2007, p. 209, italics in original)
In her study of 31 pre—calculus students Sierpir'rska (1987) found that some students
think of infinity as a large finite number. Tall and Schwarzenberger’s (1978) note that
the students in their study had idiosyncratic views of infinity. Some thought about
infinity as a symbol that represents what is unreachable; some defined it as the biggest

number that exists; and others thought about it as the endpoint of the real numbers.

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Continuity is also one of the problematic terms that cause difficulties in students’
understanding of the limit concept. Daily uses of the term in phrases such as “the rail is
continuously welded” or “it rained continuously all day” ofien result in associating
continuity with having no gaps or breaks (Tall & Vinner, 1981, p.164). Such uses,
together with the initial uses of the term continuous functions result in a “reinforcement
of the intuitive idea that the graph has ‘no gaps’ and may be drawn freely without lifting
the pencil from the paper” (Tall & Vinner, 1981, p. 165).

This viewpoint is ofien reinforced by teacher’s attempts to give a simple

insight into the notion of continuity by speaking of the graph “being in one

piece” or “drawn without taking the pencil off the paper”, thereby

confusing the mathematical notions of continuity and connectedness.

(Cornu, 1991, pp. 156-157)

Tall and Vinner (1981) gave 41 students a questionnaire that involved different functions.
The students were asked to decide which of the given functions were continuous and to
explain their reasoning. They found that most of the students who gave the correct
answers gave those for the incorrect reasons. For example, students might say a given
function is continuous because “it was given by a single formula” or “it is all in one
piece” (Tall & Vinner, 1981, p. 167). Similarly, students might say a given function is
discontinuous because “the graph is not in one piece” or “it is not given by a single
formula” (Tall & Vinner, 1981, p. 167).

There is further evidence that students’ early notions of continuity, geometric
motion and very small quantities (infinitesimals) can lead to problems with respect to

their notions of function, infinity and limit (Carlson, 1998; Cottrill et al., 1996; Tall,

22

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1980; Vinner & Dreyfus, 1989). It is possibly due to the interplay of all these notions that
there are many different colloquial and idiosyncratic interpretations of the word limit
(Cornu, 1991). As a result, everyday uses of the terms limit, approaching, or tending to
can be problematic for students since they are used differently in the theory of limits

(Bagni, 2004). In addition to this, mathematical notations like “ f (x) —> c as x —> a ”,
which is verbalized as “ f (x) approaches c as x approaches a ” entails a feeling of motion

and hinders students’ understanding of the formal definition (Tall & Vinner, 1981,
p.155). The main difficulty for students when dealing with the e — 6 definition results
from the static character of the formal theory and the dynamic character of the intuitive
approach (Bagni, 2004). Bezuidenhout (2001) argues that the formal approach is
extremely difficult to understand for students who have the difficulties mentioned above.
Similarly, Williams (1991) argues that the dynamic approach to limit is likely to hinder
student understanding of the formal approach where the idea is to make f(x) as close to
the limit value as we want by making x close enough to the limit point.

These researchers consider the formal definition of limits as an important element
of understanding limits conceptually. Although they acknowledge that the dynamic view
seems to be usefiil, and perhaps inevitable, in making sense of the limit concept
intuitively at the initial stages of learning, they also highlight that the lack of familiarity

with the assumptions of the formal theory results with difficulties with respect to

particular applications of the concept5 (Bezuidenhout, 2001; Tall, 1980; Tall &

 

5 On the other hand, Parameswaran (2007) mentions that the introductory calculus courses should be
informal and intuitive and notes that “the precise, formal definition of the concept of limit is so complex

23

 

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Schwarzenberger, 1978; Williams, 1991). “Limiting processes are not always intuitive,
and so a formal framework provides a powerful tool for thinking about and evaluating
some limits” (Szydlik, 2000, p. 259).

2.1.4. Textbooks, teaching and attitudes towards mathematics

Some researchers would argue that the intuitive aspects of limit are perpetuated in
teaching and textbooks with their emphases on the visual representations of functions as
graphs or numerical approximations, which are based on the natural perception of
continuity. Cornu (1991) mentions that “in teaching mathematics, certain aspects of the
limit concept are given greater emphasis which are revealed by a review of the
curriculum, the textbooks and examinations” (p. 153). According to him, instead of
focusing on limit conceptually, textbooks may overemphasize “equalities, the notion of
absolute value, the idea of sufficient condition and, above all, on operations: the limit of
a sum, a product, and so on” (Comu, 1991, p. 153, italics in original). He argues that the
textbooks’ focus on algebra and calculation reflects a bias favoring operations and
procedures over the analytical aspects of the limit concept.

Similarly, Bezuidenhout’ (2001) argues that students’ understanding of limits and
its underlying concepts seems to be based on isolated procedures and the conceptual link
among those are missing.

Such a situation may be mainly due to a learning and teaching approach

that emphasizes to a large extent the procedural aspects of the calculus,

and neglects a solid grounding in the understanding of the conceptual

 

and counterintuitive that it fails to bring out readily the simple and intuitively obvious ideas which led to it
in the first place” (p. 194).

24

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underpinnings of the calculus. Moreover, the stereotyped exercises that are

a feature of several calculus texts ofien encourage an instrumental

approach, rather than a relational understanding of the calculus concepts.

Taking into account the procedure-oriented nature of some calculus texts,

it should not be considered as strange if a student confuses manipulative

skills with a real understanding of calculus content. (p. 498)
Williams (1991) considers students’ attitudes toward practicality and mathematical truth,
which hinder their appreciation of formal thinking, as another element that can affect
learning about limits. He mentions that attitudes toward practicality can result from
students’ classroom experiences and the current curriculum provides little motivation
with respect to formal thinking and notes that the students in his study

...often considered the ease and practicality of a model of limit more

important than mathematical formality. This is particularly true in the

sense that models of limit that allow them to deal with the realities of

limits in the classroom, the kind they see on tests, tend to be seen as

sufficient for the purposes of most students. It was noted by several

students that neither formal or dynamic models of limit figure heavily in

the procedures students use to work problems from their calculus class;

their procedural knowledge (e.g., substituting values into continuous

functions, factoring and cancelling, using conjugates, employing

L’Hopital’s rule) is largely separate from their conceptual knowledge.

(Williams, 1991, p. 233)

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Pararneswaran (2007) states that “it is typical for calculus books to motivate the notion of
limits graphically” (p. 212). He then talks about an example in a typical calculus textbook

sin x

 

in which lim
x—-)0 x

is first motivated by the tabulating some values of the function around

x=0 and then drawing the graph of the function. He points out that although the textbook
author mentions explicitly that the table only allows one to guess the limit value but not
to prove it, “the students seem to develop the idea that limit is no different from a process
of approximation” (p. 213).
2.1.5. Possible links between historical development and student learning of limits

Cornu (1991) distinguishes between didactical and cognitive obstacles in learning
mathematics. Didactical obstacles “occur because of the nature of the teaching and the
teacher”, whereas epistemological obstacles “occur because of the nature of mathematical
concepts themselves” (Cornu, 1991, p. 158). He identifies four epistemological obstacles
in the historical development of the limit concept: the failure to link geometry with
numbers, the notion of infinitely large and infinitely small, the metaphysical aspect of the
notion of limit, and the question of whether the limit is attained or not. By considering
some of the conceptual obstacles about limits as epistemological, he acknowledges that
the difficulties faced by students might also result from the nature of the limit concept
besides the teaching approaches. In fact, some of the conceptual obstacles students face
as they learn the concept may be identical to those mathematicians faced over the
historical development of limits. For example, Williams (1991) mentions “limit as
boundary” and “limit as unreachable” as common student difficulties about limit (See
Section 2.1.2). These views of limit are mathematically incorrect and were also

problematized by Lagrange and other mathematicians when they debated “whether a

26

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his could hel

students dete

variable can go beyond the limit and whether a variable can definitely reach the limit”
(Schubring, 2005, p. 293). Given this, some researchers consider bringing some elements
of the historical development of limits to classrooms as potentially useful in student
learning. For example, given the findings of his study, Williams (1991) comments that

Just as students’ informal limit models tend to parallel those of the

mathematical community prior to Cauchy, it is possible that only by

appreciating the sorts of problems that motivated Cauchy’s work will

students be motivated to understand its implications. Perhaps this is to say

that the very historical and cultural contexts that lent vitality to the original

work are the best medium through which to approach the understanding of

that work. (p. 23 5)

Similarly, Bagrri (2004) highlights that the historical development of visual, verbal and
symbolic representations of limits might parallel those of students’. He then notes that
this could help design teaching to overcome some of the conceptual obstacles and to help
students develop the different registers required by the static and the dynamic views of
limit6.

At this point, it is also important to note that research on learning about limits and
infinity is, so far, directed mainly by a cognitivist framework and primarily focuses on
the notion of misconceptions. According to Sfard (2001), the cognitivist framework is
based on the metaphor learning as acquisition, which considers learning “as the storage

of information in the form of mental representations” (p.20). It considers understanding

27

as relating new knowledge to prior knowledge by refining the existing mental
representations. By doing so, it highlights the individual nature of learning, viewing it as
the acquisition of the necessary mental schemes. Although this fi'amework significantly
enhanced our understanding of the conceptual obstacles associated with the learning of
limit, another outlook on the issue might be needed since we cannot have direct access to
abstract constructs such as mental schemes, intuition and (mis)conceptions. Therefore, a
cognitive framework may not offer sufficient tools of analysis with which we can explore
how learners make sense of the limit concept. This study uses the commognitive
framework (Sfard, 2008), which is based on the metaphor learning as participation and
views learning as becoming a participant in a discourse. Basing learning processes on
social foundations, this framework considers discourse as its central unit of analysis in
which “the language of mental schemes, misconceptions, and cognitive conflict seems to
be giving way to a discourse on activities, patterns of interaction and communicational
failures” (Sfard, Forman & Kieran, 2001, p. 1, italics in original).
2.2 Commognitive framework

2. 2.1. The general tenets

One of the highlights of the commognitive framework is the interrelationship
between communication and thinking. By defining thinking as the individualized form of
communication, Sfard (2008) argues that the “cognitive processes and interpersonal
communication processes are thus but different manifestations of basically the same

phenomenon” (p. 83). Given this, the term commognitive entails the combination of the

 

6 On the other hand, Bagni (2004) warns that the introduction of the problems faced by mathematicians in
history would not necessarily help students with their difficulties. In this respect, he seems to also consider
the epistemological nature of some of the conceptual obstacles associated with the limit concept.

28

terms cognitive and communicational. From this perspective, developmental
transformations are “the result of two complementary processes, that of individualization
of the collective and that of communalization of the individual” (Sfard, 2008, p. 80, italics
in original). That the communal and individual aspects of discourse development are
intertwined processes can be explained as follows: Individual learning, which is defined
as participation in a discourse (e. g., mathematical discourse) whose rules are communally
agreed upon, is an example of individualization of the collective. On the other hand, as
individually formed ideas get accepted by the larger discourse community, it is also
possible for the individual to affect the development of a discourse on a broader range.
This is an example of communalization of the individual. By being processes rather than
static entities, discourses construct and reconstruct themselves in the interplay of
individualization and communalization, and are thus the “medium and the carrier of both
continuity and developmental change” (Sfard, 2008, p. 118). Therefore, the study of
human development can be considered equivalent to the study of the development of
discourses, where discourses are defined as “the different types of communication set
apart by their objects, the kinds of mediators used, and the rules followed by participants
and thus defining different communities of communicating actors” (Sfard, 2008, p. 93).
The commognitive fiamework views mathematics as a particular type of
discourse which is distinguishable by its word use, visual mediators, routines, and
narratives. Sfard (2008) differentiates between two types of discourse: colloquial
discourses are non-specialized, everyday discourses, and literate discourses are the
“discourses mediated mainly by symbolic artifacts created specifically for the sake of

communication” (p. 299). Number or quantity related words can frequently be found in

29

colloquial discourses but “mathematical discourses as practiced in schools or in academia
dictate their own, more disciplined uses of these words” (Sfard, 2008, p. 133). Unlike
colloquial discourses, the objects of the mathematical discourse “are featured as
something that can perhaps be ‘represented’ with visual means, but never really shown”
(Sfard, 2008, p.135). Given the abstract nature of mathematical concepts and that
students are expected to participate in the literate mathematical discourse, word use is a
critical element of the discourse because possible differences in participants’ use of those
words can hinder mathematical communication.

Visual mediators refer to the visible objects created and operated upon for the
sake of communication. Colloquial discourses are “ofien mediated by the images of
concrete objects” whereas scientific and mathematical discourses are primarily mediated
by symbolic artifacts (Sfard, 2008, p. 147). In mathematics, such artifacts consist of
algebraic symbols as well as the conventionally or idiosyncratically created diagrams,
graphs, tables and icons.

Routines refer to the set of metarules7 that define repetitive patterns in a
discourse. Routines can be idiosyncratic. For example, a student’s repetitive patterns
while doing mathematics might differ from that of another student or mathematicians.
The routines that are accepted as valid and enacted extensively by the experts of the
community are called norms. Although both routines and norms consist of metarules that
characterize repetitive patterns in a discourse, not every metarule that is enacted or

endorsed can be considered a norm. A metarule must satisfy the following in order to be

 

7 Metarules refer to the rules that characterize the patterns in the activity of the participants of a discourse.
See Section 2.2.2 for a more detailed discussion of the notion.

30

a norm: It must be enacted widely within the community of discourse, and it must be
endorsed by the majority of the community, “especially by those within the community
who count as experts” (Sfard, 2008, p. 204).

Narrative is “any sequence of utterances framed as a description of objects, of
relations between objects, or of processes with or by objects, that is subject to
endorsement or rejection with the help of discourse-specific substantiation procedures”
(Sfard, 2008, p. 134, italics in original). Axioms, definitions and theorems are among the
endorsed narratives of mathematics. Narratives of a given discourse that are endorsed by
the majority of the discourse community, in particular by “experts”, are considered as
“true”. The endorsed narratives of an individual, however, can be different than those
endorsed by the mathematics community. For the analysis of an individual’s discourse,
an endorsed narrative refers to what one considers to be true in relation to the routines
one uses to substantiate those narratives. The idiosyncratic nature of endorsed narratives
results from the idiosyncratic nature of the routines. Mathematical learning takes place as
the endorsed narratives and routines of an individual become compatible with those of
the experts.

Sfard (2008) notes that “mathematics begins where the tangible real-life objects
end and where reflection on our own discourse about these objects begins” (p.129, italics
in original) and so it is “a multilayered recursive structure of discourses-about-discourse,
and its objects therefore are, in themselves, discursive constructs” (p. 161). The
generative power of mathematical discourses, like any other discourse, is obtained
through recursion with which “we can turn one discursive act into the object of another”

(Sfard, 2008, p. 103) and create metastatements, that is, statements about statements.

31

Sfard (2008) argues that discourses develop in terms of expansions and compressions.
Discursive expansions are considered as endogenous “when there is an increase in the
amount of complexity of discursive routines” and as exogenous “when there is a
proliferation of new discourses” (Sfard, 2008, p. 119, italics in original). Discursive
compressions, on the other hand, result from reaching to a metalevel by means of
objectification. Objectification occurs through reification and alienation. Reification “is
the act of replacing sentences about processes and actions with propositions about states
and objects” (Sfard, 2008, p. 44), whereas alienation refers to “using discursive forms
that present phenomena in an impersonal way, as if they were occurring of themselves,
without the participation of human beings” (Sfard, 2008, p. 295). Through
objectification, we identify the commonalities between different processes within a
discourse and we unify many lower-level phenomena under one name. This new
metalevel discourse subsumes the lower-level, independently existing discourses and
“enables us to express in the new language everything that can be said in any of the
original discourses with their own signifiers” (Sfard, 2008, p. 122). Therefore, the
objectified discourse is more abstract than any single one of the discourses it subsumes.
Objectification increases the effectiveness of our communication and is also a means of
formalization of the mathematical discourse, especially through the use of symbolic

artifacts. However, since objectification replaces the talk about processes with the talk

about objects in an impersonal ways, it hides the discursive layers that constitute the

objects and also the metaphorical nature of the objects we speak about. By doing so, it

 

8 Note that changing the talk about processes to the talk about objects necessitates changes in the word use
of a discourse. Therefore, Sfard (2008) considers degree of objectification as crucial factor in the analysis

32

also blurs the differences between the lower-level discourses that are subsumed, which
can be of significant importance, especially at the beginning stages of learning. This is
probably the reason why the insiders of the mathematical discourse (e.g., mathematicians,
mathematics teachers) “lose the ability to see as different what children cannot see as the
same” (Sfard, 2008, p. 59).

When human development is considered equivalent to the development of
discourses, the study of the historical development of mathematical concepts become
equivalent to the study of the evolution of the mathematical discourse about the concepts.
Since mathematics is a patterned, historically established metadiscursive activity, such
evolution consists of the development of thinking about the objects. Therefore, although
the historical processes of object creation follows a different sequence than students’
individualization of those objects, the communal aspect of mathematizing contains the
narratives and the routines students need to adapt to as they become participants in the
mathematical discourse. Moreover, the historical development of mathematical concepts
includes the junctures that enable the grth of the discourse through the interplay of
expansion and compression, which can be useful in the exploration of the conceptual
obstacles associated with the concepts. In this study, I will discuss the conceptual
obstacles associated with limit and its underlying concepts (infinitesimals and infinity)
through the analysis of the historical junctures that necessitated particular changes in the
previous mathematical discourse about the concepts. Next, I describe the elements of the

commognitive fiamework I will use to analyze those historical junctures in Chapter III.

 

of word use. The details of how to analyze of word use, visual mediators, routines and endorsed narratives
will be discussed in Chapter IV (design of the study).

33

2. 2. 2. Metarules and metaphors in discourse development

It was mentioned in Section 2.2.1 that discourses develop through the interplay of
expansions and compressions. More often than not, such junctures in the development of
a discourse result in changes in the metarules, also called metadiscursive rules, of the
existing discourse in order to extend it firrther. Unlike object-level rules, which “take the
form of narratives on the objects of the discourse”, metarules “define patterns in the
activity of the discursants trying to produce and substantiate object-level narratives”
(Sfard, 2008, p. 201). “A metarule in one mathematical discourse will give rise to an
object-level rule as soon as the present metadiscoruse turns into a full-fledged part of the
mathematics itself” (Sfard, 2008, p. 202). For example, one of the metarules of

arithmetic ‘If we add an even number with another even number, the sum is an even
number’, becomes an object-level rule ‘ 2n + 2m = 2(n + m) for all n, m e R ’9 in the

algebraic discourse when expressed as the relation between the algebraic objects n and m .
The tacit nature of the metarules of mathematics is amplified by objectification and
symbol use. Through reification and alienation, mathematical statements reach their
timeless forms, making it seem like mathematics exists independently of the creators of
those statements. Therefore, among the reasons metarules are mostly tacit are the
metadiscursive and metaphorical nature of mathematical objects. By being discourses
about discourses, metadiscursive statements hide the discursive layers they consist of and

the metaphors they are based on. Metaphors help us create new discourses through usage

 

9 Note that the metarule and the object-level rule mentioned here are justified by the experts of
mathematics as true. As a result, they are among the endorsed narratives of mathematics. The former is a
meta—level narrative whereas the latter is an obj ect—level narrative. Said differently, one can talk about rules
as well as narratives of a discourse as being meta-level and object-level.

34

of words from familiar contexts when making sense of an unfamiliar context. Thus, the
mechanism of metaphor can be thought of as “the action of ‘transplanting’ words from
one discourse to another” (Sfard, 2008, p. 39). Although metaphors are crucial
mechanisms with which we build and expand discourses, “words that have been
transferred from one discourse to another cannot be incorporated to the new discourse
without some bending of the old rules” (Sfard, 2008, p. 75). Given this, the exploration of

the metaphors that govern different layers of a mathematical discourse becomes a central

part of the exploration of metarules10 in the development of the discourse. The analysis of
junctures in the development of a discourse with respect to the changes in the metarules
can give us information regarding the transitions learners need to go through as they
participate in the extended discourse. In what follows, an example of such a juncture will
be given in the domain of arithmetic.

When we work with positive integers, the metaphor underlying multiplication is
repeated addition. We make sense of multiplication by means of addition and consider
the product 2 x 3 as 2 + 2 + 2 (two added to itself three times) or 3 + 3 (three added to
itself two times). Therefore, in this case, ‘we multiply by adding repeatedly’ is the
metarule of multiplication. This metarule also leads to another metarule, that
‘multiplication always makes bigger’. This means that whenever we multiply two
positive integers different than 1, the product is bigger than either of those integers. When

we multiply two positive rational numbers, on the other hand, the discourse on

multiplication has to go through some change. Considering %x 3 as adding gto itself

 

‘0 Note that using metaphors consistently in a given discourse is a type of metarule.

35

3 trmes does not make sense. Given this, the metarule, that multrplrcatron rs repeated

addition, has to change. We can think of multiplication of positive rational numbers by

considering each number as the side length of a rectangle and think about multiplication
in terms of the area of that rectangle. Moreover, since %x :1; is equal to% , which is a

smaller number than both of the numbers we multiply, the metarule ‘multiplication
always makes bigger’ also needs to be abandoned. The visualization of multiplication as
hopping on the number line, which works for the case of positive integers, needs to be
replaced by the visualizations of multiplication as area of rectangles1 1. It is these types of
junctures that will be elaborated on in this study since they require changes in the
metarules of the previously existing discourse to extend the discourse further. It should be
noted that the discourse on multiplication of positive rational numbers subsumes the
discourse on the multiplication of positive integers. Given this, the metarules, visual
mediators, and endorsed narratives, as well as the obj ect-level rules that are valid for the
former are also valid for the latter. In contrast, not all the metarules, visual mediators and
endorsed narratives of the discourse on multiplication of positive integers are necessarily
valid for the more general, subsuming discourse on the multiplication of positive rational
numbers.

By being based on the tacit metarules and metaphors governing the discourse,

developmental junctures eventually require changes in the endorsed narratives of the

 

l l . . . . . . .
Of course, the discourse about multiplication goes through yet other transrtrons when negative numbers
get into the picture.

36

discourse. In that sense, they lead to the changes that are essential in order for the learner

to participate in the new aspects of the extended mathematical discourse.

Summary

In this chapter, I described the limit concept and summarized the literature about its
learning. Existing research on the learning of limits mostly characterizes the difficulties
students have about limits through misconceptions based on the assumptions of a
cognitivist framework. From this perspective, students need to change and refine their
mental schemes in order to overcome the conceptual obstacles they have about limits. I
argued that we do not have direct access to constructs such as (mis)conceptions and
mental schema. In other words, a cognitive framework may not offer sufficient tools of
analysis with which we can explore how learners make sense of the limit concept. I then
introduced Sfard’s (2008) commognitive framework as an alternative lens to investigate
development of discourse on limits and student difficulties associated with the concept.
From this perspective, students need to change elements of their discourse on limits in
order to overcome their conceptual obstacles. Sfard (2008) highlights word use, visual
mediators, routines, and narratives as the main tools of analyses to learn more about one’s
mathematical discourse. In this study, I will investigate whether such an analysis
enhances our knowledge of student learning about limits.

Existing research on limits also points to some possible links between the
historical development of limits and student learning. However, it does not elaborate on
the principles underlying the transitions learners need to go through as they attend to

different aspects of the limit notion. By viewing developmental processes as resulting

37

from individualization of the communal and communalization of the individual, I will
concentrate on historical, and therefore communal, development of discourse on infinity,
infinitesimals, and limit in the following chapter. More specifically, I will explore the
historical development of limit related concepts through the commognitive framework,
with a particular focus on the junctures that resulted in changes in the metarules and
metaphors of the previously existing discourse on limits. The purpose of this
investigation is to gain more information about the nature of the conceptual obstacles
related to limits over history. Later in the study (Chapter VII), I will examine whether
such a consideration of historical development can be useful to learn more about

_ individual student learning.

38

CHAPTER III

HISTORICAL DEVELOPMENT OF THE DISCOURSE ON INFINITY,
INF INITESIMALS AND LIMITS

In this chapter, I explore the historical development of infinity, infinitesimals and
limits. While doing so, I rely on historical documents as well as research on the historical
development of those concepts. When investigating the historical development of
discourse about those concepts, I pay attention to particular elements of the
commognitive framework, namely word use (objectification), metarules and metaphors1 ,
that characterized particular realizations of infinity, infinitesimals and limits. I also use
these elements to identify the historical junctures that led to changes in the
metadiscursive rules as the discourse on these notions extended.

Although the historical developments of the concepts of infinity, infinitesimals
and limits are intertwined, I will first focus on them separately. I will then argue that
there were two types of historical junctures that led to changes in the metadiscursive rules
of the discourse on these concepts. The first type led to the objectification of the ideas
about infinity, infinitesimals and limits. The second type, which occurred in the
development of discourse on limits, led to an alternative realization of limit and
ultimately resulted in the elimination of motion as an idea underlying this notion. I will
argue that these two types of junctures form the bases of the changes in the

metadiscursive rules in the historical development of discourse on these three concepts.

 

l .
Recall that using metaphors when making sense of a mathematical concept rs a type of metarule (See
Section 2.2.2)

39

In order for the reader to follow the historical development of these concepts
clearly, I present the timeline for the mathematicians who will be addressed in this

chapter in Table 3.1.

Table 3.1: Timeline of the mathematicians discussed in this chapter over history

 

 

Ancient times Renaissance period 17th and 18th centuries 19th and 20th centuries
Eudoxus (405-355 BC.) Viete (1540-1603) Newton (1643-1727) Weierstrass (1815-1897)
Aristotle (384-322 BC.) Descartes (1596-1650) Leibniz (1646-1716) Dedekind (1831-1916)
Euclid (325-265 BC.) Cavalieri (1598-1647) Euler (1707-1783) Cantor (1845-1918)
Archimedes (287-212 Fermat (1601-1665) Lagrange (1736-1813) Robinson (1918-1974)
BC.) Wallis (1616-1703) Cauchy (1789-1857)

 

3.1. The notion of infinity

Historically, the notions of potential infinity and actual infinity have been of
interest for philosophers, mathematicians and scientists since ancient times. Lakoff and
Nunez (2000) note that “outside mathematics, a process is seen as infinite if it continues
(iterates) indefinitely without stopping” (p. 156). On the other hand, there can be no
direct experience with the notion of infinity in real-life since our environment is restricted
by finiteness. Therefore, Aristotle (384-322 BC) considered infinity only as potential:
“the non-limited possibility to increase an interval or to divide it” (Fischbein, Tirosh &
Hess, 1979, p. 3) and rejected the notion of actual infinity to avoid Zeno’s paradoxes2
and the inconsistencies in the existing discourse about potential infinity. Fischbein
(2001) notes that, in the case of potential infinity,

we deal with a dynamic form of infinity when we consider processes,

which are, at every moment, finite, but continue endlessly. We cannot

conceive the entire set of natural numbers, but we can conceive the idea

40

that after every natural number, no matter how big, there is another natural

number. (p. 3 10)
The dynamic nature of visualizing infinite continuation without an endpoint is also
mentioned by Lakoff and Nunez (2000). They note that we make sense of such
continuation by motion that goes on and on forever. With this metaphor, we talk about
continuous processes without end by thinking of them as “infinite iterative processes,
processes that iterate without end but in which each iteration has an endpoint and a
result” (Lakoff & Nunez, 2000, p. 157). Thus, this metaphor enables us make sense of
infinite processes by means of infinitely many step-by step processes that are discrete.
This is the metaphor with which we realize potential infinity. In mathematics, we use the

notion of potential infinity whenever we write down the elements of a given sequence,

-;—,—;—,...or when we write the decimals of J3 in terms of ones, tenths,

for example% , as 1,
hundredths, and so on. Similarly, Tall (1992) argues that using words such as ‘ x tending
to infinity’ for the notations like ‘ x —> 00 ’also represent infinity as a potentiality.
Therefore, mathematics makes frequent use of the notion of potential infinity through
word use and symbolization. At this point, it is important to note that, despite its common
use in the mathematical discourse, potential infinity is not obj ectified. When we think
about infinity as potential, we talk about it as a process but not as a distinct mathematical
object. Its use in the mathematical discourse is equivalent to its use in the colloquial

discourses, “which are also known as everyday or spontaneous because they often

develop as if by themselves, as a by-product of repetitive day-to-day actions” (Sfard,

 

For the description of Zeno’s paradox and its variations, see (Fischbein, 2001 ).

41

2008, p. 132). Aristotle’s notion of potential infinity continued for centuries and it was
not until the 19th century that the mathematical distinction between potential and actual

infinity was made explicit: “In XVIII century. . .the difference among a very big number
and on ‘infinity’ was neglected and it seemed self-evident that a theorem true for every n
was true for n infinite, too” (Kline, 1991, I, p. 506 as cited in Bagni, 1997, p. 210).

Cantor’s (1845-1918) work in the 19th century replaced the notion of potential

infinity -— the idea that infinity resides beyond any given number (positive or negative) -
with the notion of actual infinity, realized as an accumulation point. Using Cantor’s
words:
Mathematical infinity. . .is crescent beyond every limit or

indefinitely decrescent, and it is a quantity that remains finite. I call it

improper infinity [potential infinity]. Moreover, recently, another kind of

infinity. . .took place. . .By that. . .the infinity is considered as concentrated

in a certain point. When infinity occurs in this form, I call it proper infinity

[actual infinity]. (Bottazzini, Freguglia & Toti Rigatelli, 1992, p. 428 as

cited in Bagni, 1997, p. 210).
Here Cantor talks about improper, or potential, infinity similar to Fischbein’s (2001)
arguments that although the process continues endlessly, it is considered as finite at every
given moment. Cantor uses the word proper, or actual, infinity, on the other hand, to refer
to an end-state. Given this, Cantor realized actual infinity as a distinct entity but not as a
process and therefore obj ectified the notion of infinity through reification. This, on the
other hand, required changes in both the existing discourse about potential infinity and

also in the metarules of mathematical activities such as counting and measuring. “The

42

shift of attention, from the natural numbers as a potentially infinite collection going on
and on, to a single entity the setN , given axiomatically, leads to considering the
relationships with other infinite sets” (Tall, 2001 , p. 212, italics in original). By
introducing the notion of one-to-one correspondence in determining the equivalence of
the cardinality of sets, Cantor eventually transformed the metarules of counting: “If we
have to compare two infinite sets, we should not count their elements as we count finite
groups of objects. We have to determine the equivalence-or non-equivalence-of these sets
by formal means” (F ischbein, 2001, p. 310). The acceptance of infinity as actually
existing required the acceptance of “the strange proposition that the whole may be
equivalent to some of its parts” (F ischbein, Tirosh & Hess, 1979, p.4). For example, we
can come up with a one-to-one correspondence between the set of positive even numbers
and natural numbers, which in Cantorian terms, means that the cardinality of these two
sets are equal to each other. This, on the other hand, also means that the set of positive
even numbers, which is contained in the set of natural numbers, has equal cardinality
with the set of which it is a subset. Therefore, a previous metarule of counting that “if we
find the cardinality of a proper subset of a given set, the result is a number less than the
cardinality of the larger set” had to change in the case of infinite sets. Moreover, Cantor
also showed that the cardinality of the set of natural numbers and the cardinality of the set
of points on a number line are not equivalent to each other since there could be no one-to-
one correspondence between these two sets. Therefore, although both sets contained
infinitely many numbers, they had distinct cardinalities. This meant that more than one
type of infinity existed. In that sense, although infinity was objectified as an ultimate

state and an existing ‘number’ 00 , the symbolic equation 5 = 5 , which made perfect sense

43

for the number 5 as an object of mathematics, was meaningless for the case of infinity:

infinity was not equal to infinity! Instead, we used a family of notations N.- to denote

different infinities. Lakoff and Nunez (2000) also mention that the famous mathematician
Hardy warns us not to consider 00 as a number in the usual sense “because
mathematicians have devised notions and ways of thinking, talking, writing, in which

co is a number with respect to enumeration, though not calculation” (p. 165).

Besides counting, the notion of actual infinity also required changing some of the
metarules of arithmetic. Tall (2001), highlights that neither the subtraction nor the
division of the infinite cardinals can be uniquely defined. On the other hand, the
introduction of the notion of actual infinity results in a proliferation of other

mathematical discourses. Tall (1992) considers cardinal infinity, ordinal infinity and non-

standard (measuring) infinity among the three notions of infinity3 used in mathematics
today. The fact that “natural numbers are not only used for counting, but also for putting
a set into an order” (Tall, 2001, p. 216) eventually led to the definition of order
relationships and the creation of ordinal numbers for which addition was not
commutative. As a result, the metarule of adding quantities regardless of their order,
which is valid for real numbers and the cardinal numbers does not hold for ordinal
numbers. “So strange were these ideas to the mathematics community when first
announced that Kronecker prevented the initial publication of Cantor’s theory of infinite

cardinals...” (Tall, 2001, p. 218).

 

3 Cardinal infinity extends counting by means of comparison of sets. Ordinal infinity is conceptualized in
terms of comparison of ordered sets. Measuring infinity extends measuring from real numbers to larger
ordered field (Tall, 1992).

44

Fischbein, Tirosh and Hess (1979) argue that “the contradictory nature of infinity
can be pushed to higher levels but cannot be completely eliminated. . .even with the most
sophisticated mathematical tools” (p.4). Lakoff and Nunez (2000) seem to agree with this
since they claim that the metaphor with which we realize actual infinity is a special “case
of a single general conceptual metaphor in which processes that go on indefinitely are
conceptualized as having an end and an ultimate result” (p. 158). They call this metaphor
the Basic Metaphor of Infinity and argue that it turns potential infinity into actual infinity
in tenns of a largest ‘number’ 00. By means of this metaphor, proof by induction “needs
“0 longer be considered as potentially infinite process” but becomes a three-step
Procedure following Peano’s axiom4 (Tall, 2001, p. 210). As a result, actual infinity also
Changes the metarules of proving.

It is probably true that even the symbolization of the notion of infinity in
m"=llil'tematics cannot keep us away from the conceptual obstacles associated with it.

The common mathematical notion for infinity-‘ . . . ’— as in the sequence
‘ l + 2 + 4 + ’— does not even distinguish between potential and actual

infinity. If it is potential infinity, the sum only gives an endless sequence
of partial sums always less than 2; if it is actual infinity, the sum is exactly
2. (Lakoff & Nunez, 2000, p. 180)

On the other hand, actual infinity is now a meaningful, “non-contradictory concept,

consistent with the totality of the other mathematical concepts” (Fischbein, Tirosh &

A

I

4 T all (2001, p. 211) mentions that to prove a statement P(n) by induction, it is enough to (a) show P(1) is
true, (b) show the truth of P(k) implies the truth of P(lt+ I ), and (0) quote the induction axiom, which states
that if (a) and (b) are true, then P(n) is true for all n in the set of natural numbers.

45

Hess, 1979, p. 3). Therefore, we accept the notion’s mathematical reality. In that respect,
the mathematical answer for l + g + i + is exactly 2. The difficulty Lakoff and Nunez

addresses above, therefore, results from the conceptual but not the mathematical aspect of
infinity.
Summary

The objectification of infinity as a distinct mathematical entity did not occur till
the 19th century. This required changing the talk (and word use) about potential infinity,

which is a process, to the talk about actual infinity, which is an accumulation point. Such
a transition necessitated changes in the metarules of counting, measuring, arithmetic and
proving. The notion of actual infinity also gave rise to new types of ‘numbers’, namely,
cardinal, ordinal and measuring (non-standard) infinities. Therefore, the development of
actual infinity led to an exogenous expansion in the mathematical discourse since it
ignited the proliferation of different mathematical discourses. Fischbein’s (2001) and
Tall’s (2001) arguments support the idea that the discursive expansion about the notion of
cardinals and ordinals took place only after, not before, the objectification and
formalization of actual infinity by means of an axiomatic system.

Although the notions of potential and actual infinity are distinct from each other

mathematically, they are both used frequently in the mathematical discourses. Lakoff and

Nunez (2000) argue that the metaphors we use for potential and actual infinity are

 

5 It should be noted, however, that the notion of actual infinity created quite a chaos in the mathematics
community before its mathematical existence was accepted (Bagni, 1997; Fischbein, Tirosh & Hess, 1979;
Tall, 2001).

46

variations of each other and are primarily based on our realization of the infinite
processes through iterative processes that have an end and an ultimate state.
3. 2. The notion of infinitesimals

The concept of infinity, which is mostly associated with the notion of infinitely
large, also brings with it the notion of infinitely small. The term infinitesimal entails
infinitely small quantities, generally negligible, that are not real constants. Infinitesimals
are also called indivisibles, differentials, evanescent quantities and infinitely small
magnitudes (Kleiner, 2001). As mentioned in the preceding section, the discussions
about infinity go back to Aristotle and the first integral-like approach given to area
related problems was given by Eudoxus (405-355 BC). Eudoxus suggested an approach
that seemed to have characteristics similar to an infinite process. This approach was
called the method of exhaustion, which led to the implicit notion of infinitesimals. The
method assumed infinite divisibility of magnitudes and was primarily based on the
proposition

If fi'om any magnitude there be subtracted a part not less than its half,

from the remainder another part not less than its half and so on, there will

at length remain a magnitude less than any preassigned magnitude of the

same kind. (Eves, 1983, p. 289)
A magnitude being “less than any preassigned magnitude of the same kind” formed the
initial underpinnings of an infinitesimal quantity. Although this method was handled by
Euclid (about 325-265 BC.) and Archimedes (287-212 BC.) later, it was Archimedes
who tied the problems of finding area to the explicit use of the concept of infinity. He

initiated the idea that an area could be composed of infinitely many geometrical lines but

47

he could not form a solid explanation of the vaguely defined concept of infinitesimals.
The mathematics of this period is considered as static since it lacked the consideration of
motion and change and was based on the axiomatic structure of geometry. Although the
initial underpinnings of infinitesimals were present in this era (around 300 BC), they
could not be justified with the geometric foundation. Since the idea of infinitesimals
relied too much on intuition and was not based on the solid foundation of mathematics,
which was geometry with its axiomatic/deductive structure in this period, it was
considered unsound. Therefore, it was discarded by the ancient Greek mathematicians
and was not emphasized assertively by Archimedes.

In 1635, Cavalieri (1598-1647) applied indivisibles or fixed infinitesimals
successfully to problems in the mensuration of areas and volumes which brought
infinitesimals back into discussion (Boyer, 1970). He used the geometrical approach,
which still dominated the renaissance mathematics, and found the integrals of nth-degree
polynomials accurately. Yet, a revolution was about to come as mathematicians like
Viete, Descartes, Wallis and F ermat recognized the use of algebra as an aid to geometry.
This led to the tendency towards the symbolic-algebraic over geometric by the end of the
renaissance period. Descartes’ (1596-1650) work, published shortly after Cavalieri’s,

changed the course of infinitesimal analysis once again and initiated the period called the

arithmetization of geometry6.

 

6 This, together with the discovery of non-Euclidean geometry, marked the stagnation of geometry for
about a century and a half.

48

The mathematics of the late renaissance and 17th century relied on the dynamic7

approach and overshadowed the static approach given to mathematics since the ancient
Greeks. These changes enabled the reevaluation and reanalysis of the infinitesimal
calculus into what we now know as calculus. The use of infinite-series expansions for the
generalization of analysis not only to polynomial functions but also to rational, irrational,
algebraic and transcendental functions required a new infinite analysis. This was
recognized first by Newton in 1665-66 and then independently by Leibniz in 1673-76
(Boyer, 1970).

Newton’s main contribution was to justify that the infinite processes were as
respectable as the algebraic ones. His approach to infinite processes was mainly dynamic
and relied on the notion of incremental change because of the physical nature of his
problems at hand. However, since mathematicians have been deeply skeptical of the

concept of infinitesimals due to their intuitive and non-rigorous characteristics, Newton

avoided using ‘infinitely little’ but used the term ‘evanescent’ while discussing fluxionsg.
Historically, the strongest criticism of Newton’s calculus came from Berkeley in 1734.
Berkeley’s reaction was primarily based on Newton’s implicit use of infinitesimals in
calculus: “And what are these same evanescent Increments? They are neither finite
Quantities, nor Quantities infinitely small, nor yet nothing. May we not call them the
Ghosts of departed Quantities?” (Berkeley, 1734, as cited in Jesseph, 1993, p. 199).
Leibniz, on the other hand, gave importance to appropriating notations and was

able to give the correct rule for differentiation for the product of two quantities. Yet, he

 

7 . . . . . .
The consrderatron of motion and change in mathematics through physrcal problems.

49

did not consider these quantities as variables one of which depended on the other, so he
missed the idea of differentiation of a variable with respect to another variable. Leibniz’
proof of the product rule was as follows:

d (xy) = (x + dx)( y + dy) — xy where dx and dy are the differentials or
infinitely small differences of x and y . So Leibniz states that “the

quantity dxdy ...is infinitely small in comparison with the rest, and hence

can be disregarded” (Edwards, 1979, p.255).
Boyer (1970) argues that Leibniz and his disciples could not make clear what they meant
by infinitely small change and could not justify the elimination of quantities that were
infinitely small compared to others. According to him, Leibniz’ calculus was a failure
compared to Newton’s from this logical viewpoint. It should be noted that both Newton
and Leibniz worked on the calculus of instantaneous change. Therefore, they were
implicitly using limit as a process, but not as an explicitly defined concept, as they
obtained the tangent line at a point through a sequence of the secant lines passing from
that point (Lakoff & Nunez, 2000). On the other hand, Newton used a geometric

approach in the process whereas Leibniz relied more on arithmetic.
Euler (1707-1783) shone in 18th century with his work on the possibilities

inherent in the infinite power series. By means of symbol manipulation, Euler showed
that “what is true for convergent series is true for divergent series, what is true for finite
quantities is true for infinitely large and infinitely small quantities” (Kleiner, 1991, p.

295). To Euler, the transition from finite differences to the limit method (limit as a

 

8
Newton referred to a varying (flowing) quantity as a fluent. He used the term fluxion to refer to the
instantaneous rate of change of a fluid.

50

 

process) was straightforward and was based on his consideration of continuity of real
numbers:

Both infinitely small and infinitely large quantities often occur in series of

numbers. Since there are finite numbers mixed in these series, it is clearer

than daylight how, according to the laws of continuity, one passes fi'om

finite quantities to infinitely small and to infinitely large quantities (Euler,

2000,p.90)
On the other hand, infinitely small quantities were intentionally banned from Lagrange’s
lectures since he considered them as concepts that lacked adequate foundational basis
(Schubring, 2005, p. 290).

Infinitesimals went on being under attack till the beginning of the 20th centuryg.
On the other hand, “they still continued to flourish in the practical world of engineering
and science ..., representing not a fixed infinitesimal quantity, but as a variable that could
become ‘arbitrarily small’” (Tall & Tirosh, 2001, p. 130). In mid 20th century, Abraham
Robinson (1918-1974) introduced his theory of non-standard analysis, in which
“infinitesimals were formulated on a logical basis” (Tall & Tirosh, 2001, p. 130). This
still did not solve the debate as to whether infinitesimals can be considered as logically

sound mathematical objects since the new formulation of infinitesimals brought its

relevant incompatibilities with the existing mathematical discourses. This debate

continues into the 21 5’ century.

 

See the next section about limit for further details.

51

It was mentioned before that the introduction of different types of infinities
changes the metarules of counting and measuring. Cardinal numbers, for example,
require a different way of counting for infinite sets than finite sets through the notion of
one-to-one correspondence. Recognition of non-standard infinity, which Tall (1992)
refers to as the measuring infinity, marks a change in the notion of infinity and the
metarules of measuring and is based on the non-standard analysis of Robinson. “To
explain such a theory requires the formal interpretation of the notion of an infinitesimal”
(Tall, 1980a, p. 274). Measuring infinity considers a point as a ball of infinitesimal size,
by which “we discover a theory that allows both the indivisibility of ‘points’ and also
infinite divisibility of a line” (Tall, 1980a, p. 274). In that sense, a point which is
considered dimensionless is now a ball with an infinitesimal radius. Tall (1980b) also
mentions that non-standard analysis, through the reformulation of infinitesimals, leads to
new types of numbers, namely hyperreal or superreal numbers.

Summary
Although the objectification of infinitesimals is absent before the 20th century, the

period that precedes Robinson’s non-standard analysis highlights the conceptual
obstacles regarding the acceptance of infinitesimals as mathematically justifiable objects.
Sfard (2008) notes that “not every metarule, whether enacted or endorsed, is a norm. In
order for a rule to be a norm, it must be widely enacted within a community” and it must
also “be endorsed by everybody [as true], and especially by those within the community
who count as experts” (p. 204). Given this, although the uses of infinitesimals had been
present, the narratives about them were not endorsed by the majority of the mathematics

community for a long time.

52

Initially, the reason that hindered the acceptance of infinitesimals as valid
mathematical objects resulted from the almost mystical nature of the utterances about
infinitesimals (e. g., evanescent quantities or vanishing increments). Moreover, the use of
geometry by its dynamic representations, such as curves and graphs, but not by means of
its formal-deductive structure (Euclidean geometry) entailed continuous motion in space,
which was considered intuitive and sensuous. A more important reason, however, was
the way infinitesimals contradicted the endorsed narratives about real numbers. Lakoff
and Nunez (2000) mention that infinitesimals do not obey the arithmetic rules of real
numbers since they violate the Archimedean Principle10 and commensurabilityl 1. Did
Robinson’s objectification of infinitesimals as particular types of numbers solve these
problems?

To this day the debate continues. Although the infinite cardinals are

generally accepted by the mathematical community, there are

mathematicians who fully embrace the theory of infinitesimals in non-

standard analysis, those who deny their existence and assert the pre-

eminence of standard analysis, and even a greater number who do not

agonise over the foundational problems and simply get on using

mathematics for practical purposes. (Tall & Tirosh, 2001, p. 130)

 

10 “Archimedean Principle: Given numbers A and B (where A is less than B) corresponding to the
magnitudes of two line segments, there is some natural number n such that A times it is greater than B”
(Lakoff& Nunez, 200, p. 298).

11 “Those magnitudes are said to be commensurable which are measured by the same measure, and those
incommensurable which cannot have any common measure” (Heath, 1956, p. 10). In modern sense, two
non-zero quantities A and B are commensurable if there exists a quantity C such that A=mC and B=nC for
non-zero whole numbers m and n.

53

Interestingly, the reformulation of infinitesimals by means of logic “invoked the axiom of
choice to assert that such entities existed without being able to give a specific finite
construction” (Tall & Tirosh, 2001, p. 130). That is to say, the formal theory of
infinitesimals assumes the existence of the concept but cannot prove or construct their
existence. In that respect, it seems impossible to justify the notion based on the solid
foundations of mathematics. However, the discourse on infinitesimals continues to lead
to the proliferation of new mathematical discourses such as measming infinities,
hyperreal and superreal numbers.
3.3. The notion oflimit

Being the founders of calculus, Newton and Leibniz both used infinitesimals in
their theories as they worked on incremental change. By obtaining the tangent line at a
point through the use of a sequence of secant lines, they were using the notion of limit as
a process. The dynamic mathematics influencing the renaissance period relied heavily on
the metaphor of continuous motion. On the other hand, the symbolism of arithmetic and
its use for expanding the discourse on functions made arithmetic a better candidate for
‘the’ foundation of mathematics than geometry. Given these, it is not surprising that
Newton was criticized more than Leibniz or Euler. Although they all referred to
infinitesimals in their work, Newton also used geometry in the dynamic, and hence the
most intuitive, manner. Leibniz, by relying on arithmetic, probably avoided some
criticism.

Using the notion of limit as a process is referred to as the limit method in the

historical documents. After Newton and Leibniz, mathematicians such as MacLaurin and

54

d’Alembert kept on using this method on their problems. Lagrange opposed them on their
use of the limit method:
MacLaurin and d’Lambert used the idea of limits; but one can
observe the subtangent is not strictly the limit of subsecants, because there
is nothing to prevent the subsecants from further increasing when it has
become a subtangent. True limits... are quantities which one cannot go
beyond, although they can be approached as close as one wishes.

(Lagrange, 1799, as cited in Schubring, 2005, p. 293)

Lagrange’s arguments were primarily based on
the lacking of the concept of absolute value. . .so that it seems as if the
variable goes beyond the limit; the criticism is also at the problem, which
has always remained controversial, whether a variable can definitely reach
the limit or is only allowed to come close to it at any rate (Schubring,
2005, p. 293).
Although Lagrange uses words like “true limits. . .are quantities”, it was not until Cauchy
( 1 789-1857) that the notion of limit was objectified. Lagrange seems to talk about limit
as a “subtangent”, which is “the limit of subsecants”. In that sense, he considers limits as
the quantities obtained through the limit process and does not explicitly define them. It
Sl'lould also be noted that Lagrange’s word use “[true limits]...can be approached as close
as One wishes” entail very small or infinitely small increments as well as motion.

Therefore, the limit method makes use of the notion of infinitesimals and is based on the

[ 1etaphor of continuous motion.

55

The discourse of calculus went through a fundamental change with Cauchy. He
realized the necessity of establishing a theory of limits, which required the explicit
definition of the concept. He selected the fundamental concepts of calculus like limit,
convergence, derivative and integral and created the grand design of calculus where limit
became the concept on which the others were based and the concept of derivative came
before the concept of integral (Kleiner, 1991, 2001). The college level calculus that we
teach today mostly follows the outline of Cauchy. Table 3.2 shows the comparison of the
historical development of the calculus concepts with Cauchy’s design.

Table 3.2: Comparison of the historical development of calculus with Cauchy’s

fiarnework

 

 

_ Historical development Design of Cauchy
Area and integral Infinity
Infinity and Infinitesimals Limit
Series (finite, infinite) and sequences Derivative
Derivative Integral
Limit Series and sequences

 

 

One of the reasons underlying Cauchy’s revolutionary departure from the established
Practice was his opposition to Lagrange, whose foundation of calculus was based on
algebra. Cauchy wanted to eliminate algebra as a basis of calculus and wanted his

IIlethods to have the rigor demanded in geometry (Kleiner, 1991 , 2001). Cauchy defined
limit as follows:
When the values successively attributed to the same variable approach
indefinitely a fixed value, eventually differing form it by as little as one
could wish, that fixed value is called the limit of all the others (Kitcher,
1983, p.247).

IiS definition of infinitesimals was:

56

When the successive absolute values of a variable decrease indefinitely in
such a way as to become less then any given quantity, that variable
becomes what is called an infinitesimal. Such a variable has zero for its
limit (Kitcher, 1983, p.247).
An analysis of Cauchy’s word use reveals that he obj ectified the notion of limit through
reification by referring to limit as a “fixed value”, that is a distinct mathematical object.
Note also that he uses the word “approaching”, which is based on the metaphor of
continuous motion. Finally, the phrases “absolute values of a variable decrease
indefinitely” and “differing from [a value] as little as one could wish” entail the use of
infinitely small quantities, namely, infinitesimals. Therefore, Cauchy’s definition of limit
Was based on infinitesimals and the continuous motion metaphor, which were both
problematic for mathematicians of his time. The dynamic interpretation of limit was
Considered intuitive by the community since terms like tending to have a “connotation of
desire, of aspiration. Numbers do not tend” (Fischbein, 1994, p. 239).
Since Cauchy based all of his calculus on the concept of limit, a precise definition
0f limit became of crucial importance. Weierstrass (1815-1897) and Dedekind 12(1831-
1 9 1 6) attempted to ‘remedy’ Cauchy’s definitions by finding “a purely arithmetic and
p erfectly rigorous foundation for the principles of infinitesimal analysis” (Dedekind,
1 963, p.1 as cited in Kleiner, 1991). These mathematicians wanted to replace Cauchy’s
l(irletnatic approach with the algebraic-arithmetic approach. The goal was to
ITeQOrrceptualize calculus as arithmetic by eliminating spatial intuition. In order to do this,

\

1 2
Bolzano and Hilbert were also among the mathematicians of 19th century who favored arithmetization of

lalysis.

 

57

Natural continuity had to be eliminated from the concepts of space, planes,
lines, curves, and geometric figures. Geometry had to be reconceptualized
in terms of sets of discrete points, which were in turn to be conceptualized
purely in terms of numbers: points on a line as individual numbers. . .The
idea of a firnction as a curve in terms of the motion of a point had to be
completely replaced. There could be no motion, no direction, no
approaching a point. All these ideas had to be reconceptualized in purely
static terms using only real numbers. The geometric idea of approaching a
limit had to be replaced by static constraints on numbers alone, with no
geometry and no motion. This is necessary for characterizing calculus
purely in terms of arithmetic. (Lakoff & Nunez, 2000, p. 308)
Wei erstrass accomplished this full agenda by considering space as sets of points. This led
to file consideration of the points on a line as numbers, which then led to the realization
of “Continuity for a function as the preservation of closeness” (Lakoff & Nunez, 2000, p.
322) - In that sense, the distance between points in motion was replaced by the distance
bet"Ween numbers. The result was Weierstrass’ ultimate definition of limit:
Let a firnction f be defined on an open interval containing a, except

possibly for a itself, and let L be a number. Then lim f (x) = L if and only

x—>a
if for any number 8 > 0 there exists a 6 > 0 such that if
0 <lx—al <§then|f(x)—L| < 8.
I‘Iaving eliminated the metaphor of continuous motion associated with infinitesimals and

geometry, and being a logical-deductive system that had arithmetic as its foundation, this

58

definition seemed to provide the precision mathematicians were looking for”. Secondly,
the definition was strong enough to explain anomalous cases that violated the geometric
and dynamic conceptions of functions as curves. By doing so, it was believed that this
new paradigm was generalizable to a broader number of situations”. On the other hand,
some mathematicians argued that this definition wiped out all the intuitive tools with
which we make sense of the concept. Note that the formal definition of limit is not
constructive since it does not enable us to find what the limit of a function is but to prove
that the limit we initially hypothesize is indeed the limit of the firnction at a particular
Point. That may be why the dynamic approach is still widely used both by
mathematicians and the students as they make sense of the notion.
Sum mary
The objectification of the notion of limit initiated by Cauchy requires the consideration of
liITlit as a thing: a particular value obtained from the limiting process. Hence, Cauchy
giVes a definition of limit by reificationls. His definition of limit is based on the metaphor
of c()ntinuous motion and infinitesimals. Weierstrass and Dedekind’s attempts to
Tremedy’ Cauchy’s definition result from the incompatibility of these two notions with
tl‘le discourse on previously existing concepts of mathematics that can be described
I1131‘er by means of algebra. The formal definition of limit Weierstrass introduces

e112lnges the metaphor of natural continuity to the metaphor of discreteness. By

\

1 3
l 4 At least till the collapse of the search for foundations (See Hersh, 1997; Lakatos, 1976).
However, Lakoff and Nunez (2000) argue that, since natural continuity uses a different conceptual

metaphor than the continuity of Weierstrass, the latter is neither a formalization nor a generalization of the
‘osl‘lner.
Note that reification is part of objectification and changes the talk about processes to the talk about

pl’Oducts (See Section 2.2.1).
59

considering numbers as sets of discrete points on the number line, he eliminates natural
continuity and therefore geometric motion and time from the discourse on limits. It is
imPortant to point out that such a shift also requires a change in the definition of
functions as curves as well as the phrases like ‘tending to’ or ‘approaching’ since these
utterances entail continuous motion. It is by means of arithmetization, and thus the

di scretization, of calculus that function becomes a type of correspondence between two
sets and that the distance between two points in space becomes the absolute value of the

difference between two numbers.

3- 4- Historical junctures in the development of discourse on infinity, infinitesimals and
limits

In the previous sections about infinity, infinitesimals and limit, the historical
development of these concepts were discussed with a focus on some elements of the
conunognitive framework such as word use (objectification), metarules and metaphors. In
this section, I identify the historical junctures in the development of discourse on these
c0Ilenepts that resulted in changes in the metarules of the discourse in order to extend it

fill'tl'ler. It was mentioned that the exploration of metaphors that govern different layers
of a discourse is a central part of the exploration of the metarules in discourse
development (See Section 2.2.2). Table 3.3 shows the junctures that transformed the
tIlet'c‘lrules in the development of infinity, infinitesimals and limits over history.
I highlight two types of junctures in the historical development of discourse on
t119$: three concepts: one led to the objectification of each concept; and one led to an
alternative realization of the limit concept by the elimination of motion in space. Note
that for infinity and limits, the objectification initially took place by means of reification,

that is, by changing the talk about a process to the talk about a product.

60

Table 3.3: Historical junctures in the development of discourse on infinity, infinitesimals
and limits over history

 

 

 

ficept Infinity Infinitesimals Limits
J uncture(s) Cantor’s objectification of Robinson’s formulation of [1]Cauchy’s objectification of
potential infinity as actual infmitesimals on a logical limit
infinity basis [2] Weierstrass’ introduction of

the formal definition of limit

Changing Realization of infinity as an Realization of [l]Realization of limit as a
metaphor(s) indefinite process through infmitesimals as process is changed to
continuous motion is evanescent or diminishing realization of limit as a fixed
changed to realization of quantities is changed to value obtained as a result of
infinity as an end-state, realization of the process.
ultimate result or an infinitesimals as variables [2]Continuous/spatial motion
accumulation point. that can be made and infinitesimals in
arbitrarily small. Cauchy’s definition are
eliminated. Motion is
replaced by the distance
between discrete numbers.
Changing Counting, measuring, Measuring (a point is a Representing functions (as
metarules proving (proof by induction) ball with an infinitesimal algebraic rules but not as graphs
and properties of arithmetic radius) of curves), points in space (as
such as addition and discrete points on the number
division. line) and geometrical objects
\ (arithmetization of geometry).

 

Potential infinity and the limit method were originally realized as processes in their
historical development. Actual infinity and limit, however, were realized as end-states or
nurl‘lbers obtained at the end of the processes of going on forever and limit method,

respectively. The objectification of infinitesimals was slightly different than infinity and
limits in that the mathematical justification of the concept is still under scrutiny.
I\’Iolf‘eover, their initial realization was not in terms of a process but in terms of very small
qua-ntities that could be eliminated. I consider Robinson’s approach to the concept
quUgh logic a historical juncture since it enabled a formal theory of infinitesimals as
Objects of mathematics. However, the existence of infinitesimals cannot still be justified
through constructive methods of mathematics but rather is assumed.
I argue that the second type of historical juncture took place in the historical
dGVelopment of discourse on limits. By revising Cauchy’s definition of limit, Weierstrass

61

also changed centuries-held metaphors about motion and continuity. Weierstrass’ formal
definition of limit eliminated infinitesimals from the previously existing discourse on
limits and replaced the metaphor of natural continuity with the metaphor of discreteness.
He discarded motion and time from his discourse on limits and geometry and offered
realizations of points as numbers on the number line and distances between points in
space as the distances between numbers.

As discourses develop and expand, the metarules and metaphors underlying the
mathematical concepts can change. Although an expanded discourse on a mathematical
c>01‘lcept subsumes the preceding discourse and enables generalization, some aspects of
the former version of the concept are lost during such transitions. The metaphor that

charlges might be the most natural attribute of the concept with which learners initially
make sense of the concept. Therefore, the junctures in discourse development leading to
tral'lsfonnations in the metarules can be of significant importance for learners and also
Ini.gl'rrt explain some of their difficulties. Changes in metaphors and metarules eventually
I.eslllt in changes in the word use and the endorsed narratives of the discourse as the
Irlet<'=1discursive rules take the form of object-level rules (See Section 2.2.2). As a result,
me)? enable the proliferation of discourses. For example, the objectification of infinity
resIlllted in a proliferation of discourses leading to concepts such as cardinal, ordinal and
IIleasuring infinities. Therefore, historical junctures may highlight some of the transitions
lea:l‘rrers need to go through as they participate in the expanded mathematical discourse.
Surnmary
In this chapter, the development of discourse on infinity, infinitesimals and limits

0\r'er history was explored by means of some elements of the commognitive fiamework.

62

The exploration revealed the objectification of the notions as important milestones in
discourse development. Historically, objectification of mathematical concepts changed
the talk about processes to the talk about products or end states. Such changes also
resulted in transformations in particular metaphors and metarules of mathematics.

It was mentioned in Chapter II that, according to Sfard (2008), developmental
changes take place in the interplay of two processes: individualization of the communal
and communalization of the individual. The historical development of limit reveals how a
mathematical idea generated by an individual mathematician gains acceptance
COllectively (e. g., Cantor’s objectification of infinity as actual infinity). From the

Commognitive lens, this is an example of communalization of the individual. One
inStance where the second process, individualization of the communal, takes place is
Stu(lent learning. Note that Sfard (2008) considers learning as participation in the
communally agreed upon discourse on mathematics. Therefore, a question that will be
pursued later in the study is how and whether the investigation of historical development
of limits through the commognitive lens is useful to gain more information about
teaelling and learning of limits. In particular, can the historical junctures that led to
cIl'C‘lIlges in the metarules of the discourse on limits be useful to explain some of the

hallsmons students go through as they learn about limits? These issues will be addressed

In Chapter VII of the study.

63

CHAPTER IV
DESIGN OF THE STUDY

In this chapter, I first present the specific research questions for the study. I then

describe the overall design of the study including the participants, data collection and
data analysis methods.
4.1. Specific research questions
The study addresses the following questions: 1) How is the discourse on limits
generated by the instructor in a beginning college-level calculus classroom? 2) Given the
instructor’s discourse on limits how do students talk about limits in a beginning college-
level calculus course? and 3) How do the elements of discourse on limits as generated
over history compare and contrast with the discourse on limits generated in a beginning-
level calculus course?
4. 2. The participants
Participants for this study consisted of one calculus instructor and his section of
und ergraduate students who were taking a beginning-level calculus course in a large
IVIiCleestern University. The course addresses the fundamental concepts of calculus such

as lillrits, differentiation and integrals. The course is fast-paced and loaded in terms of the

I7“-‘I?I11)er of topics covered. It is structured to focus more on the concepts and their

applications than proofs.
While selecting the classroom to observe, I initially formed a complete list of

11'1St1’uctors who were teaching the course in Spring 09. I then selected the instructors

' U 1lose teaching schedules enabled me to observe their classrooms. I sent five instructors,

who were randomly selected, an e-mail that briefly described my research interests and

asked whether they would like to participate in my study. One instructor responded to the
e-mail and expressed his willingness to let me observe his classroom. He wanted to learn
how I planned to conduct my study so we met before the beginning of the semester.
During that meeting, he gave me some information about the syllabus, the textbook and
the students enrolled in his classroom. I told him I was interested in both the teaching and
learning of limits. Initially, I suggested only audio-taping the class not to disturb the flow
of the lessons. However, the instructor suggested video-taping. I informed him about the
diagnostic survey I wanted to give students at the end of the unit on limits. He provided a
schedule for finishing the discussion on limits and suggested giving the surveys at the end
of the last lesson, a review session for the first exam. During the meeting, the instructor
also mentioned he planned to go over the formal definition of limit1 as well as proofs of

some basic theorems and facts about limit in the classroom to motivate the students who

migllt take higher-level calculus classes in the future.
There were 31 students registered to the instructor’s section. During the period of

Classroom observations, the number of students attending the class ranged between 17
and 2. 3. The class was very diverse in terms of the majors of students. Table 4.1 shows
the distribution of students across their majors. There were 18 first-year; nine second-
yea-1‘; three third-year students and one fourth-year student enrolled in the class. The

Whole section was asked to take a diagnostic survey at the end of the unit on limits.

 

\

1
I did not give the instructor any directives about what to teach and did not in any way intervene in his
= aching method or topics covered.

65

Table 4.1: Distribution of students across their majors

 

Department Number of students Department Number of students

 

Computer engineering 2 Statistics 1
Electrical engineering 1 Accounting 1
Civil engineering 2 Finance 4
Engineering Biochemistry and
2 . 2
(no preference) molecular biology
Computer Science 2 GBA—Prelaw l
Premedical 3 Marketing 2
Residential College 1 Economics 2
Mathematics 2 Ecological scrences and l
agriculture
Physics 1 Asian langiage l

 

Based on the responses given to the survey, I interviewed four students to further explore

their discourse on limitsz. Three of the students who participated in the individual
interview session had not taken calculus before; this was the first time they were
introduced to the limit concept. One student took a calculus class during high-school.
Having prior knowledge about limits was not necessarily problematic for the study since
Stuclents’ familiarity with colloquial or some of the literate aspects of limits was
anti cipated, though not required. The focus of the empirical part of the study was, given
their previous knowledge and also how the notion was introduced in the classroom, how
Stud ems worked on and talked about particular limit problems.
4. 3. Data collection
The primary sources of data for this study consisted of field notes as well as video
tElpes that were taken during the classroom observations; responses to a diagnostic survey

gven to students; and task-based interviews including students’ written work. The

te)‘ltbook students used in their class (Thomas ’ Calculus, 11th edition) and informal

c118Cussions with the instructor with respect to his mediation of students’ use of the

\
:2

I‘Iow the students were recruited will be discussed in Section 4.3.

66

textbook (e. g., assigned readings or homework problems) were considered as

supplementary data.
The data for the classroom observations were collected in eight days over a period

of two and a half weeks3. I video-taped the classes in which the instructor talked about
limits and continuity starting fi'om the first week of Spring 09 semester. Overall, the
observation data consisted of eight SO-minute lessons. While video-taping, I only focused
on the instructor and did not video-tape the students. I took field notes during classroom
observations and used them to keep track of the number of students attending the class as
well as the questions they asked to the instructor. I also used the field notes to keep a
record of definitions and problems the instructor presented in the class. My role in the
classroom was a participant observer. I did not interfere with the flow of the class during
instruction. However, I helped some students before and after class if they asked me
some questions about limits. Since I wanted to interview some of the students at the end
0f the unit on limits, providing such help was useful to establish a relationship with them
and get ideas about their difficulties about limits. I transcribed the video-taped lessons
both with respect to what the instructor said and what he did in the classroom. Therefore,
for the analysis of the instructor’s discourse, my data consisted of video-tapes of eight

lesSons and their transcripts as well as the field notes taken during classroom

0b Servations.
On the last day of the classroom observations, I gave all students a diagnostic

S‘ll‘Vey. That lesson was the review session for the exam and by that time, the instructor

 

\

3
The class met three times a week.

67

had completed his discussions on limits and continuity. He gave me 10 minutes at the end
of the class to administer the survey to the students. There were 23 students in the class
that day and all of them agreed to take the survey. I used the diagnostic survey (a) to

select the students for an individual interview session, and (b) to analyze student

responses with respect to the instructor’s discourse4. The questions in the diagnostic
survey (See Appendix A) were taken from Williams (2001) since his classification of
Views related to limit is widely endorsed by research on student learning about limit. The
first question of the survey included six statements about limits and asked students to
decide whether the statements were true or false. Each of the six statements in the survey
was related to a different view of limit, some of which were difficulties addressed by
research on student learning. The second question then asked them which of the six
statements best described their understanding of limits. The third question asked students
to describe what they understood a limit to be. The final question asked students to give a
rig<>rous (formal) definition of limit, if possible.

While selecting the students for interview sessions, I focused on their responses to
the first and the second questions of the survey. I initially grouped all students with
reSIDect to the six statements they chose as best for the second question. I planned to
interview one student for each of the six different views of limit they chose as best
describing how they thought about limits. Since I wanted variety in terms of students’

Vi ers of limit, I then recorded the number of correct responses students in each group

gave for Question 1. Table 4.2 is a cross tabulation of questions 1 and 2, and shows the

\

<1.
In this section, I only talk about how the students for the interview session were selected. The details of

e analysis of the diagnostic survey in regard to the instructor’s discourse will be discussed in Section 4.4.

68

classification of responses for the selection of students for the interview session. Its use is

explained below.

Table 4.2: Classification of student responses for Question I and Question 2 of the survey
Number of students selecting the statements as best in Question 2
Total number of students = 23

Number
IV. V. VI. VII.

of correct I. II. 111.
responses Statement] Statement2 Statement3 Statement4 Statements Statement6 "None”

given to (N=11) (N=0) (N=1) (N=6) (N=1) (N=3) (N=1)

 

   

 

 

 

@estionl
All correct 2 0 O O 0 0 0
5 correct " .5.. 0 if: 1113 0 0 f// 0
4 correct .4 O 0 O 1 '_ 0
3 correct 0 O O 2 0' fight:
2 correct 0 O O 3 :sLeumi 0
1 correct 0 O 0 is“ 1 O 0
0 correct 0 0 O 0 O O
3 Initial emailing F inal selection

 

As an example, Table 4.2 shows that of the 11 students who selected statement 1
(“A limit describes how a fimction moves as x moves toward a certain point”) as the best
definition of limit, two categorized correctly all 6 statements in question 1, while 4
Categorized four correctly and two incorrectly. I initially e-mailed six students from
Columns 1, III, IV, V, VI and VII, respectivelys (See Table 4.2). The students I e-mailed
are fiom the shaded cell in each column. The final selection of students is shown with

l1aSl‘1-marks in the relevant cells. The selection process proceeded as follows.

In the e-mails I briefly described my study, provided some information about the
itlterview sessions and offered a $25 gift card from a bookstore for their participation.
only one student, from Column III, agreed to be interviewed. The other students did not

9 V ant to participate (participation was voluntary). I then sent another set of e-mails to six

\

8
Note that none of the students chose Statement 2 as best.

69

students: two students with four correct answers from Column I, two students with three
correct answers from Column IV, and the two remaining students from Column VI (See
Table 4.2). Only the student from Column V1 with five correct responses agreed to be
interviewed. At that point, I had one student selected from Column III and one from
Column VI. The students who were in Columns V and VII did not want to participate and
there was no student in Column 11.

I repeated the procedure again, this time e-mailing all of the remaining students
from Columns I and IV. One of the students from Column I with five correct responses
agreed to participate in the interview session. At that stage, I had three students who
considered three different statements related to limits as best describing their realization

of l imit. On the other hand, the students’ responses to the six statements were mostly
accurate6. In order to find candidates who would likely have many of the difficulties
indi cated by the literature, I went to the classroom and looked for volunteers. Together
With the instructor, we encouraged students to participate since the experience could
cOl’ltlibute to their learning. As a further motivation (besides the gift card), I also
Suggested tutoring the students at their convenience (such as before the exam). Only one
S1“-ICIent agreed to participate. She was from Column I and had five correct responses to
QUestion 1 (See Table 4.2). I wrote my e-mail on the board and asked students to contact

1Tie in case they wanted to be interviewed but did not receive any response.

Getting students’ acceptance to volunteer for the study lasted for about two and a

1It11fweeks. In the end, I decided to go with the four who volunteered. Unfortunately, I

70

could not recruit the students with many incorrect responses to the survey to participate in
the study. However, the interview sessions revealed that some of the four students who
seemed to have grasped the idea behind limits had many of the difficulties indicated by
research when they worked on tasks that targeted those obstacles7.
The questions in the interview session were designed to probe students’
realizations of limits and also investigate in further detail the conceptual obstacles

addressed by research on learning about limits. For the interview sessions, I initially

 

formed a pool of problems, which consisted of ten questions. Some of those questions
were taken directly from research on student learning on limits and limit related concepts;
the others I developed considering the instructor’s discourse on limits. For example, the
instructor told me he planned to go over the formal definition so I added problems to the
P001 that were about the formal view of limit. By the time I interviewed the students, I
1‘ educed the number of questions to six (See Appendix B). There were two reasons for the
elimination of those tasks: (a) It was unlikely for students to work on all of the problems
in about an hour, and (b) given the instructor’s discourse in the class, some of the
Problems turned out to be redundant or completely unfamiliar to students. For example,
one of the questions I eliminated was about computing limits of a variety of functions
represented algebraically. The instructor did not only go through similar examples in the
c1 ass but he also assigned these types of problems as homework. Therefore, it seemed

that this problem was redundant since it was going to assess students’ computational

\

6

It should be noted the term “accuracy” is considered here only in terms of students’ responses to the six

tements in the diagnostic survey and does not necessarily imply students’ realizations of limits were
790mm in general.

The details of the student discourse on limits will be discussed in Chapter VI.

71

skills more than their mathematical reasoning. Another problem I eliminated was taken
fi'om research on student learning about infinity and required familiarity with some early
underpinnings of sequences. Since the instructor did not talk about infinity as a separate
concept in the class and since he did not give any elementary examples of sequences, I
concluded students would be too unfamiliar with the question. The other two questions I
eliminated were different versions of similar problems that were already in the remaining
six questions.
I conducted the interviews with the participants individually over a one week
period. The interviews lasted between 53 and 76 minutes. I audio-taped the interviews
and transcribed them with respect to what the students said and did. In order to keep track
of what the students did, I took notes during the interviews as students worked on the
problems. Besides the transcriptions, interview data also consisted of students’ written
Work, To sum up, for the analysis of students’ discourse on limits, my data consisted of

res13cmses to a diagnostic survey as well as four audio-taped interview sessions,

tral’ls cripts of the sessions and written student work.

4. 4. Data analysis

4- 4- I . Analysis of the instructor’s discourse
I used the transcribed classroom observations to analyze the discourse of the

instructor with respect to the four elements of discourse from the commognitive
fi"i-"lll'rework: word use, visual mediators, routines, and narratives. During the process of
trailscription, I created a list of common words the instructor used and categorized them
as Signifying functions, infinity, infinitesimals, limits, motion and proximity. When the

h anscription of the eight lessons was completed in terms of what the instructor said and

72

what he did, I color-coded the words I initially identified depending on their
categorization. Every category was coded with a different color in the transcribed text.
The words were color-coded based only on their mathematical use in the class. For
example, I coded the word at as signifying proximity in the instructor’s discourse (e.g.,
‘ff is continuous if you can find the limit at a by plugging in”) but I did not code it when
he used it in a colloquial manner (e. g., “now, let us look at this example”).

The next step was to identify the utterances in which the instructor talked about
limits. In order to do that, I first pulled out all the sentences including the word “limit”,
Whi ch was already color-coded, from the transcripts of the video tapes. However, there
Were also utterances about limit in which the instructor did not explicitly utter the word
“1th”. In some of the cases where the instructor did not utter the word “limit”, he
described the behavior of function values as the x values approached the limit point. In
some others, he referred to a previously mentioned limit (e. g., “It [the limit] does not
eXist”; “L [the limit] is equal to three”; “The answer [the limit] is one”). I considered
SUCh utterances as related to limits as well. An utterance about limit was formed by a
Sentence, part of a sentence or multiple sentences that conveyed a particular idea about
linlit that could be interpreted just by reading. About 85% of the 775 utterances I
identified as related to limits consisted of a single sentence. The remaining utterances
eitl'ler consisted of less than (about 2%) or more than (about 13%) a whole sentence

depending on the purposes of the study. The only context in which an utterance consisted
of Some part of a sentence was when the instructor attended to multiple limit notations in
a Single sentence. During the transcription process, I noted that the instructor used a

Val‘iety of words when he attended to the limit notation although he and the textbook

73

suggested using a single word, namely “approaches” when reading the notation. The
existence of a family of words associated with the same notation in the instructor’s
discourse led me consider each instance in which he referred to the limit notation as a
distinct utterance in order to look for the patterns in his word use. In these cases, an
utterance could be less than one sentence if the instructor addressed more than one limit
notation. For example, the following explanation was considered as consisting of three
Utterances since the instructor referred to three different limit notations.

“If the limit as x approaches c from the right is L [first utterance

about lim f (x) = L] and the limit as x approaches c from the left is K
+
x—)c

[second utterance about lim f (x) = K ], and if K is not equal to L, then
x—)c_

the limit as x approaches 0 of f of x does not exist [third utterance

about lim f (x) =does not exist]”.
x—->c

The only contexts in which an utterance was considered as consisting of more
than one sentence were (a) when the instructor described the behavior of the function
vallues in relation to the x values, (b) when he asked what a given limit was equal to and
in‘lInediately gave the answer following his question, and (c) when the idea the instructor
c0l'nrnunicated about limits could be understood only together with the sentences

preceding his conclusive statement about limits. While identifying an utterance in these
coIltexts, I looked whether the utterance conveyed a complete idea about limits. Each of
the following examples was considered as a single utterance about limits although they

c0Ilsisted of two sentences:

74

Example 1: “So really what this [the function] is trying to do, it is
approaching a number very very small. A small negative or a small
positive”.

Example 2: “What is the limit of f of x as x approaches six? You get an
undefined”.

Example 3: “As x approaches, say c, of the function one. What is that
limit?”

Example 4: “When x approaches one, what do the function values do?
They get closer to two”.

Example 5: “What I want to do is to talk about another tool that is useful
for computing limits. This is called the sandwich theorem”.

Example 6: “What is our conclusion that we want here? Here is our
conclusion: f of x minus L [the limit] is less than epsilon”.

Once the utterances about limit were identified, they were coded into four

Cate gories with respect to the degree of objectification: colloquial, operational,

Obj ectified, and both operational and objectifiedg. The identification of these categories in

Word use was based on the commognitive framework (degree of objectification) (See

Chapter III). Colloquial word use referred to talking about limits in everyday sense.
ODerational word use referred to talking about limits as a process, whereas objectified
Word use referred to talking about limit as an end result of the limit process or as a
Illltlrber. There were also utterances in which the instructor talked about limits both in an

\

8
‘ The details of these categories will be firrther discussed in the word use section of the chapter on
l“Structor’s discourse.

75

operational and objectified manner. These four categories (colloquial, operational,

ob j ectified and both operational and obj ectified) and color-coding were used to explore
the patterns of word use in the instructor’s discourse. The exploration of patterns in the
instructor’s word use was then used to identify the contexts in which he altered his word
use from one categorization to the other or where his word use remained consistent, (e.g.,
strictly objectified) without any shifts. As a result, there were four contexts in which the
instructor’s word use was analyzed in detail: (a) informal definition of limit, (b) formal
definition of limit, (0) computing limits, and (d) continuity.

For the analysis of visual mediators, an inventory of all the visual mediators the
instructor used was created from the transcripts, which also included the snapshots of
everything he wrote and drew on the board. Those mediators were then classified in four
Categories: written words; drawn pictures of geometric shapes; graphs; and symbolic
representation.

Routines correspond to the set of metarules that describe repetitive actions of the
discursants (Sfard, 2008). Note that the repetitive nature of routines requires them to be
applied consistently in similar situations. Therefore, not all actions count as routines
unless they are consistently used in analogous contexts. For the investigation of routines,
I mainly focused on what the instructor did9 throughout the eight lessons. There were

ITlalny possible routines that could be elaborated on over the entire observation period

(e-g., assigning homework at the end of the class). For the purposes of the study,

hoVvever, only the routines emerging fiom the transcripts that were most relevant to the

\

 

9
Note that the transcripts of the instructor’s discourse included both his words and actions in the

QIElssroom.
76

analyses of the word use and visual mediators were reported. Some of the routines in the
instructor’s discourse consisted of repeated mathematical procedures (algebra-based
routines) he utilized in the classroom in regard to limits. Some other routines, such as
graphing, were repeated actions emerging also from the analysis of word use and visual
mediators. The main focus during the analysis of routines was to identify the when and
the how of a routine. How of a routine can be thought of as the “course of action or
procedure”, whereas when of a routine refers to the instances “in which the discursants
Would deem this performance as appropriate” (Sfard, 2008, p. 208, italics in original).
The when of a routine embodies the applicability and closure conditions (Sfard, 2008).
Applicability conditions enable the exploration of the situations that trigger the
application of a particular routine. Closure conditions characterize the circumstances
under which a performer considers her routine as successfully completed. Since routines
are used to substantiate mathematical narratives, closure conditions mark the end of a
routine and are followed by the closing statements afier the implementation the routine.
Narrative is “any sequence of utterances framed as a description of objects, of
rel ations between objects, or of processes with or by objects, that is subject to
endorsement or rejection with the help of discourse-specific substantiation procedures”
(Sfard, 2008, p. 134, italics in original). It was mentioned in Section 2.2.2 that “a
metarule in one mathematical discourse will give rise to an object-level rule as soon as
file present metadiscourse turns into a full-fledged part of the mathematics itself” (Sfard,
2008, p. 202). Object-level rules, once endorsed by the community, form the object-level
narratives of mathematical discourse and are known as mathematical facts. The meta-

1eVel narratives, on the other hand, characterize the metarules related to the obj ect-level

77

narratives. For the instructor’s discourse, most of the endorsed narratives were object-
level in the form of a definition, theorem or rule about limits. For example, he endorsed
the following object-level narrative about the limit of the sum of two functions:

If lim f(x)=L and lim g(x)=M,whereLandM arereal
x—+a x—>a

numbers, then lim (f (x) + g(x)) = L + M .
x——)a

In this study, the instructor’s object-level narratives were not reported because this would
result in restating all the facts about limits that are widely endorsed by the mathematical
Community that can also be found in a beginning calculus textbook. Instead, the focus
Was on the meta-level narratives that were most relevant to the instructor’s word use,
visual mediators and routines in the classroom (e. g., “limit is a number”, and “limit is a
process”, etc.).
4- 4- 2. Analysis of the diagnostic survey and student interviews with respect to the
instructor ’s discourse
The diagnostic survey taken from Williams (2001) was used to select students for
the interview sessions and to gain information on students’ discourse on limits at the end
of tlleir lessons (See Appendix A). How the survey was used to select students for the
interviews was discussed in Section 4.3. Here, I talk about the analysis of the survey with
re Spect to the instructor’s discourse.
Recall that the first question of the survey consisted of six statements about limits
Stlldents chose as true or false. In the second question, students picked one of those
StEltements as best describing their view of limit. For the third question, they provided

their own definitions of what a limit is. For the fourth and final question, they provided a

rigorous (formal) definition oflimit.
78

According to Williams (1991), the six statements in the first question of the
survey correspond to six views about limit. He categorizes these views as dynamic-
theoretical; boundary; formal; unreachable; approximation; and dynamic-practical,
respectively. The statements describing limit as dynamic-theoretical, formal, and
dynamic-practical (Statements 1, 3, and 6, respectively) give general information whether
the students’ realizations of limit is based on motion or not. The statements describing
limit as boundary, unreachable, and approximation (Statements 2, 4, 5, respectively), on
the other hand, are used whether students have some of the difficulties identified by
research on student learning about limits (See Section 2.1.2, Table 2.1).

For the analysis of the first two questions of the survey, I initially recorded the
responses students gave for each of these statements as true or false. I then made a cross-
comparison of the responses with respect to the statements students chose as true. For
example, I looked at how many of the students marking Statement 1 as true also marked
Statements 2, 3, 4, 5, 6 as true. The purpose here was to look at the range of responses as
well as how and whether students considered different views of limit as related to each
other. I then made a cross-comparison of the statements students chose as true in regard
to the statements students chose as best describing their views of limit. For example, I
looked at how many of the students marking Statement 1 as true considered it as the best
statement; how many of those students considered Statement 2 as the best statement, etc.
The purpose of this part was to gain information about the views that dominated students’
realizations of limit and what other views of limit were connected to those realizations.

I then analyzed students’ responses for each of the six statements (some of which were

difficulties addressed by research) in relation to the instructor’s discourse. The purpose of

79

this was to examine how instructor’s word use, visual mediators, routines, and endorsed

narratives compared and contrasted with the view of limit indicated by the statements.
For the third and fourth questions, I assigned numbers corresponding to each

student’s response. I also included which of the six statements students chose as best

were closest to their descriptions. I then explored student responses in relation to their

word use, visual mediators and endorsed narratives”). While doing so, I investigated
whether students referred to limit as a process (operational word use) or as an end-state
(objectified word use). I also looked whether students’ word use was dynamic (based on
motion) or static (based on proximity by means of distance). If students’ word use was
operational, they endorsed the narrative “limit is a process”; if their word use was
objectified, they endorsed the narrative “limit is a number”. In case students used any
visual mediators in their descriptions, such as the limit notation, I focused on their word
use when talking about the notation and whether it was based on dynamic or static word
use. I then compared and contrasted students’ word use, visual mediators and endorsed
narratives with the instructor’s. In cases instructor’s word use when talking about a visual
mediator was reflected in students’ discourse, I also examined how the instructor used
that visual mediator (such as graphs or symbolic notation).

Missing in the diagnostic survey were student difficulties such as “limit implies
continuity”, “limit as the function’s value”, and “limit as monotonic” (See Section 2.1.2,
Table 2.1). The interview questions were designed to address all of the difficulties

mentioned by research and also provide more information about students’ discourse on

 

lO . . . . , . .
The diagnostic survey was not a context In which students routmes could be analyzed smce such
analysis requires the exploration of repetitive patterns in a discourse. In order for such patterns to emerge,

80

limits. I analyzed students’ discourse in the interviews, again, in terms of (a) elements of
the instructor’s discourse on limits (word use, visual mediators, routines, and endorsed
narratives), and (b) the difficulties indicated by research on student learning about limits.

While analyzing student responses in relation to the elements of the instructor’s
discourse, I focused on five mathematical contexts that emerged from the analysis of the
instructor’s discourse in relation to his word use. Those contexts were: (a) dynamic
aspect of limits, (b) formal aspect of limits, (c) limit notation, (d) infinity, and (d)
continuity. I used the transcripts of what the students said and did to explore each
instance of talk about these five contexts. I then compiled all the utterances for each
student in regard to these contexts in a separate document. The document also included
each student’s routines (e. g., graphing, plugging in the limit point to the fimction, etc)
and visual mediators (e. g., graphs and symbolic notation) that emerged from the
transcripts. I then looked at the general characteristics of each student’s word use (degree
of objectification as well as use of dynamic and static vocabulary), visual mediators,
routines, and endorsed narratives. I then compared and contrasted these four elements of
students’ discourse with those of the instructor’s.

While analyzing students’ discourse with respect to research on learning about
limits, I looked for the instances in the interview transcripts where students showed signs
of having the difficulties mentioned in the literature (See Section 2.1.2, Table 2.1). I
compiled each student’s utterances in such instances in another document, categorizing

them with respect to the six difficulties identified by the literature (See Table 2.1).

 

students’ discourse over a period of time needs to be observed in multiple contexts. This element of
students’ discourse was investigated during the interview sessions.

81

After this, I focused on the similarities and differences between the elements of students’
and instructor’s discourse and explored how and whether the instructor’s discourse was

reflected in students’ discourse.

4. 4.3. Analysis of the classroom discourse with respect to the historical development of

limits
In Chapter III, the historical development of infinity, infinitesimals and limit was

described with a focus on particular elements of the commognitive framework: word use
(objectification), metarules and metaphors. In the same chapter, I also identified the
historical junctures in the development of discourse on these concepts that resulted in
changes in the metarules in order to extend it further (See Section 3.4). One of the goals
of this study was to explore whether the historical development of limits through the
commognitive lens could help us gain more information about student learning in today’s
calculus classrooms. In order to address this question, I examined the instructor’s and
Students’ discourse on limits in relation to the historical development of limit related
concepts.
While analyzing the instructor’s discourse with respect to the development of
disCourse on limits over history, I compared the ordering and introduction of topics
related to limits with the historical development of the related concepts. Besides this, the
focus of the analysis of instructor’s discourse in terms of the historical development was
to c(>mpare and contrast the word use (objectification), metarules and metaphors in the
instrlrctor’s discourse with those in the discourse on limits as generated over history.
While examining students’ discourse in regard to the historical development of
limit ' ' ’
related concepts, I first explored the contexts 1n which the experts conceptual

b8tacles were srmrlar to or drfferent from those of the students’ 1n the study. I then

82

 

focused on stu
as infinity and
metaphors in 5
history. Finallj
commognitire
limits.

in \tha
discourse on ii
the end of thei

relation to the

focused on students’ realizations of the informal and the formal definition of limit as well
as infinity and compared and contrasted the word use (objectification), metarules, and
metaphors in students’ discourse with those in the discourse on limits as generated over
history. Finally, I looked at whether the historical junctures I identified through the
commognitive framework were reflected in the instructor’s and the students’ discourse on
limits.
In what follows, I will first present the findings of the analysis of the instructor’s
discourse on limits (Chapter V). Next, I will talk about students’ discourse on limits at
the end of their instruction (Chapter VI). Last, I will examine the classroom discourse in

relation to the historical development of limit related concepts (Chapter VII).

83

the instr
nanln

wraps

txplortn
hit-its. l
Operatio
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CHAPTER V
THE INSTRUCTOR’S DISCOURSE ON LIMITS
In this section, I first describe the general characteristics of the lessons taught by
the instructor. I then analyze the instructor’s discourse on limits with respect to word use,
visual mediators, routines, and endorsed narratives using the commognitive framework. I
use a pseudonym for the instructor and refer to him as Jason throughout the study.

A big portion of this chapter will be devoted to the analysis of word use. While
exploring Jason’s word use, I focus on the degree of objectification in his utterances on
limits. The degrees of objectification in his word use are classified as colloquial,
operational, objectified, and both operational and objectified. I then concentrate on
particular mathematical contexts to elaborate further on Jason’s word use. Those contexts

are the informal and the formal definition of limit, computing limits (limits at a point,

limits at infinity and infinite limits), and continuity.
The second element of Jason’s discourse I attend to is the visual mediators he

used in the class. I talk about four types of visual mediators: written words, drawn
Pi Ctln‘es of geometric shapes, graphs and symbolic notation. The discussion about the

VisIlla] mediators is followed by the discussion of the routines (metarules that underlie the

relDetitive patterns) in Jason’s discourse. I talk only about the routines that are most

r I I O 0
e1 ated to hrs word use and vrsual mediators: algebra-based routrnes, geometry-based
I‘ .. , . . . .
O utlnes, usrng contrnuous motion as a metaphor, and usrng discreteness as a metaphor.

11125llly, I describe the narratives that Jason endorsed in the classroom based on the

la~1yses of the previous elements of his discourse.

84

Jason covered limits during eight SO-minute lessons, one of which was a review
session for the exam. There were 31 students registered for the course. Jason’s mode of
teaching was lecture and there was little, if any, discussion among students. He started
the limit chapter on the first day of the semester and did not review or cover functions
and basic algebra. In the first lesson, he gave students information about homework,
textbook, syllabus and also exams. He told students they did not have to know calculus
for the course but they needed to have solid algebra knowledge. He mentioned this
consistently throughout the following lessons. He put a lot of emphasis on homework,
which he expected students to submit every week. He considered homework as the most
important part of the class in order for students to keep the pace of the class and also get

feedback on their work. He firrther noted that the exam problems would be directly from
or slight variations of the homework problems.

Jason assigned the homework problems directly from the textbook (Thomas ’

Calculus, 2008, 11th edition). He considered the textbook as the most important resource
for the class. He mentioned that students could use either the 5th or the 11th edition of the

bOOk. He said that the media upgrade, which is the 11th edition, contained CDs that had

l'Bctllres and mentioned this as possibly useful for some of the students. He also noted that
he would sometimes follow the textbook very closely and sometimes he would deviate
from it to give students extra examples. It should be noted, however, that Jason did not
allow calculators for the exam; did not use any technology in the classroom and did not
assign any problems that would require students to use technology as homework

I)‘rg‘blems. Table 5.1 shows the topics of each of the eight lessons and the corresponding

‘1
e)itbook section.

85

Table 5.1: Topic outline for Jason’s lessons

 

Lesson
number
and date

Topics covered

Textbook
section

 

Lessonl
01/12/09

Introduction: Homework, exam, attendance, calculator policy; syllabus
and textbook information

Rates of change and limits

Average rates of change (average speed)

Instantaneous rate of change (instantaneous speed)

Limit (as the instantaneous rate of change)

Section 2.1

 

Lesson2
01/14/09

Formal definition of rate of change

Geometrical interpretation (slope, secant lines, tangent line)
Instantaneous rate of change

What is a limit? Informal/intuitive definition

Computing limits (polynomials, rational functions, constant functions)
Factor theorem for polynomials

Findingthe limits from a given graph

Section 2.1

 

Lesson3
O 1/ 1 6/09

Limit laws

Computing limits by using the limit laws

Rule (Theorem 2 in the book): You can find the limits of polynomials
by plugging in.

(Theorem 3 in the book) We can find the limit of rational functions by
plugging in as long as the denominator is not zero.

Some applications of the theorems: computing limits

Section 2.2

 

Lesson4
O l /2 l /09

 

Long division (initiated from a homework problem)

Sandwich theorem

An application of the sandwich theorem: computing the limit of a
function sandwiched in between two functions

The precise/ formal definition of a limit

One proof problem f(x)=2x-l; a=2; L=3 (algebraic)

Geometric explanation of the proof problem

Section 2.2 and
Section 2.3

 

[‘eSSOnS
0 1 /23/09

Refining the concept of limit and more about the sandwich theorem
sin 6 _ 1

 

Proof using sandwich theorem: lim
6—>O 6

sin 6

 

Computing other limits derived fiom combinations of

Finding limits from graphs of functions (as an introduction to two-sided
limits)

Theorem (Theorem 6 in the book) If the right hand limit and the left
hand limit exist at a point, then the limit exists at that point.

Limits at infinity

Computing limits at infinity (rational functions)

Horizontal asymptotes (definition and some applications in the form of
computing limits of firnctions at infinity)

86

Section 2.4

Table 5.1 cont’d: Topic outline for Jason’s lessons

 

Lesson
number
and date

Topics covered Textbook
section

 

Lesson6
01/26/09

Homework problem about one-sided limits that include absolute value

Horizontal asymptote applications: finding horizontal asymptotes of

given functions

Oblique/slanted asymptotes (Long division of polynomials) Section 2.4 and
Graphing functions focusing on their behavior at positive and negative 3606011 2-5
infinity; right-hand-side and left-hand-side of a vertical asymptote

(Book’s terminology: dominant terms)

Vertical asymptote (definition and applications)

 

Lesson7
01/28/09

Continuous functions

Intuitive definition of continuous functions

Mathematical definition of continuous functions

Examples of continuous functions

Showing that a firnction is continuous (two examples)

Properties of continuous fiinctions

Theorem (Example 6 in the book): Polynomials are continuous. Section 2-6
Rational functions are continuous when the denominator isn’t equal to
zero. -without proof

Theorem( Theorem 10 in the book): Continuity of a composite function-
without proof

Fact(not in the book): Sin and Cos are continuous-without proof
Continuous extension/maximal continuous extension examples

 

Lesson8
01 /30/09

Intermediate value theorem (won’t be on the exam)-theoretical

definition and an example

Computing limits of various functions (rational, trigonometric,

involving absolute value, limits at infinity)

Graphing a rational firnction using dominant terms (oblique asymptote, . _
vertical asymptote —computing these limits) RCVleW 535510“
Continuous extension/maximal continuous extension of a rational

firnction and a trigonometric function

Discussion: Can a function have an oblique asymptote and a horizontal

asymptote at the same time?

5.1. Word use

The words Jason used when talking about limit of a function mainly consisted of the

words related to motion, proximity, infinitesimals, infinity, and continuity. Figure 5.1
S . . . . . .
hOWs the list of common words Jason used in his discourse on limits and the related

noti ons those words signify. The list includes the words that Jason used most frequently

When referring to limit related concepts and was generated during the process of

bra-Ilscfibing the video-taped classroom sessions.

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The analysis of Jason’s discourse revealed that the degree of objectification in his
discourse varied depending on the context of word use. His utterances were classified as
colloquial, (mathematical) operational and (mathematical) objectified. Jason’s utterances
were considered colloquial if he used limits in everyday sense; operational if he referred

to limit as a process and based his arguments mainly on dynamic motion; and objectified
(or structural) if he referred to limit as a number, that is, as a distinct mathematical object

obtained at the end of the limiting process.

The classification of a particular utterance as operational or objectified turned out

to be complex in the context of reading the notation lim f (x) = L 1. Mathematically, we
x—>a

read lim f (x) = L as “the function f(x) approaches the limit L as x approaches a ”
x—)a

(Hughes-Hallett et al., 2008; Thomas et al., 2008). In this respect, verbalizing x -—> a
using the word approaches is a part of the obj ectified literate discourse on limits and
although the word approaches signifies motion, its use in this context is not necessarily
based on motion; it is how we read the notation. Therefore, Jason’s utterances when
reading the notation in cases where he used the word approaches were classified as
Obj ectified if he referred to L as a number at the end of the notation. His use of the word
approaches was classified as operational when he used it outside the context of reading

the limit notation. His utterances when reading the notation were classified as operational

If: (a) he used different words that signify motion for verbalizing x —> a , for example, “as

3": gets closer and closer to a ”; “as x becomes larger and larger” or “as x becomes a ”,

 

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and (b) he used the word “approaches” while reading x —+ a but did not refer L as a
number that is obtained at the end of the limiting process. Jason sometimes read

lim f (x) = L as “the limit of f of x at a is [the number] L”. Such utterances were

x—>a
classified as objectified since they were not based on dynamic motion and the end

product, which is the limit L, was used as a distinct number.
Overall, there were 775 utterances about limit throughout the eight lessons. Given

this large number of utterances, it is not feasible to discuss every utterance. However,

Jason’s colloquial, operational and objectified uses of limit related words will be

exemplified during the discussions that follow.

5. 1.1 Colloquial word use
Jason used limits in the colloquial sense only twice and he did so in the first

lesson where he talked about average and instantaneous rate of change. When addressing
average rate of change in terms of speed, he asked “What is the speed limit on Grand

River [Ave]? Probably twenty five miles per hour” (Jason, 12 January, 2009, Lesson 1).

Later in the lesson, when he wanted to connect the notion of limit to instantaneous rate of
change, he mentioned the title of the lesson and said “I am trying to get to the second
Word that we had in the title today. Rates of change. . .and the second word was limits”.
mi S was considered a colloquial use of the word limit since Jason referred to limit as a
Word he wrote on the board but not as a concept. Besides these, he did not use limits in
evel‘yday sense but used the term mathematically.

5
‘ I ~ 2 Operational word use
Operational use of limits results from the consideration of limit as a process,

' mch is consistent with the dynamic view of limit based on continuous motion. In most

90

of these cases, Jason did not use the word limit but described the behavior of function
values (f(x) approaching L) as x approaches a . In this respect, he referred to the limiting
process instead of referring to limit as a fixed value obtained from that process. While
doing so, he frequently used words that signify dynamic motion (see Figure 5.1, words
related to motion). 127 of Jason’s 775 utterances about limit were classified as
operational.
Operational word use contains two elements: use of words that signify motion. and
description of the process of obtaining the limit rather than referring to the limit as a
number. Table 5.2 shows some examples of Jason’s operational word use. Jason

explicitly mentioned once that the mathematical way to deal with the “process of getting

closer and closer” (See Table 5.2, [1])2 is limits. In the context of continuity, while
determining the continuity of a function at a particular point from a given graph, he
looked at the function values on the left hand side and the right hand side of the limit
point and compared them with the function’s actual value at the limit point. He referred
to this as a “limiting process” (Table 5.2, [16]). In these utterances, Jason used the word
liI'Iflit but referred to it as a process rather than the end result of the process.

When Jason used words operationally, he referred to x —> a using words that
Signify motion such as “x get(s) closer and closer to” (Table 5.2, [3], [4]); “x goes to”
(Table 5.2, [7], [13]); “x gets/becomes smaller and smaller/larger and larger” (Table 5.2,

[1 1], [14]); and “x becomes very very large” (Table 5.2, [15]). He also mentioned
“Walking on” (Table 5.2, [12]) and “moving along” (Table 5.2, [17]) the x-axis while

determining the related function values in his discussions about particular limits.

91

 

Table 5.2: Examples of Jason's operational word use
Utterance

 

Context of use

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 First of all, this process of getting closer and closer, mathematically the way you Instantaneous
[ . . .
want to deal With this IS limits. rate of change
[2] When x is zero point nine nine nine, then f of x is one point nine nine nine; and x2 _1
when x is one point zero zero zero zero zero zero zero zero one, then we get two lim
point zero zero zero, very close to two. x->l x "1
[3] We have some kind of function and what we are looking for is what happens to 1. 3x2 -1
this expression here when x gets closer and closer to three. x1313 x
[4] If I get closer and closer, as x gets closer and closer to five, this quantity gets lim x + 1
closer and closer to six. x—->5
[5] If we are less than six, the function tries to get smaller and smaller [referring to lim f(x)
negative infinity]. x—>6
[6] If we are a little bigger than six, it tries to get larger and larger [referring to lim f(x)
positive infinity]. x—>6
- 2x2 — 3
[7] So as x goes to infinity, the numerator goes to two. 1'
x—>oo 7x + 4
. l
[8] And if x is smaller than one, it will look like it will tend to negative infinity. 11ml ——1
x—> x-
. l
[9] So as x approaches one, it becomes really large if x is larger than one. 11ml —1
x—> x—
[1 0] So what are we doing here? Denominator is going to zero; how about the 1. x2 - 5
numerator? It is going twenty five minus five and it approaches twenty. :35 x _ 5
[1 1] So as I graph this, as x becomes smaller and smaller [referring to 0], one over x lim sin(—)
becomes larger and larger. x_,0 x
[1 2] So I just keep walking on this axis [referring to the x-axis] the fiinction values, in lim 1
this case, they get closer and closer to zero. x 900 x
. . 2x2 — 3
[I 3 ] If x goes to mfimty, then seven over x approaches zero. 1'
\ x—>oo 7x + 4
. sin x
[I 4] Well, if we consider this, what happens as x becomes larger and larger? 11m
x-—)oo x
[1 5 ] What I want to do is I want, just like here, try to describe what the firnction does as lim (2 + sin x )
x becomes very very large. x900 x
[1 6] If I start drawing like this and I approach say here [referring to the left hand side
of the limit point] I have to make sure actually I can get to that point [the limit Continuity
\ point]. So this here, drawing like this, you can think of it as a limiting process.
1&1 I move along the x axis along my function values. ContinuitL
[1 8] So I want to end up, I want to get closer and closer to the function value at this Continuity

2
(Table 5.2, [1]) refers to utterance number 1 in Table 5.2.

92

There were cases in which Jason’s word use was operational but did not necessarily
include words signifying motion. For example, in [2] (See Table 5.2), although he didn’t
use a word signifying motion, be assigned successive values for x and investigated the
behavior of the function at those points to obtain the limit, which is described as the
dynamic view of limit in literature.

Another element that characterizes operational word use is whether an utterance
describes the process of obtaining a limit or the end result of the process, in which case

limit is a distinct value. If it is the former, the word use is operational, if it is the latter,

2

, Jason

 

the word use is objectified. For example, in the context of computing lim
x—)l x—

noted that the function values got “very close to two” (Table 5.2, [2]) but did not mention
the limit as being equal to the number two in that utterance. Similarly, in the context

0f lim x +1, he said “this quantity gets closer and closer to six” (Table 5.2, [4]) but this
.x—)5

Utterance does not consider the limit of the quantity as equal to six. In the context of

finding lim f (x) from a given graph (see Figure 5.2), Jason talked about the function
x—>6 .

Values as getting “smaller and smaller” [5] when x is less than six and as getting “larger
and larger” [6] when x is greater than six. However, he did not objectify the process of
getting “smaller and smaller” and “larger and larger” with negative infinity and positive

itlfinity, respectively. Therefore, these two utterances describe the process of obtaining

the limit rather than the end result of that process.

93

 

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Figure 5.2: J ason's hand-written example for finding limits from a given graph
S ymbolically, the operational word use seems to describe the processes of x values
approaching a (x —> a) and the function values approaching the limit ( f (x) —> L). In this
respect, the limit L is approached but is not referred to as a distinct value that is obtained.
Jason’s utterances in which the function “get(s) closer and closer to” [4], [12], [18];
“becomes really large” [9]; “tends to” [8]; “is going to” [10]; “goes to” [7] and
“approaches” [10], [l 3] highlight that his focus is on the process of the function values
approaching L rather than the limit being equal to L.
5- 1.3 Objectified word use
An utterance about limits is considered objectified if (a) the word limit is uttered
explicitly to signify a mathematical object or a number that is obtained at the end of the
1ilimiting process or (b) the word limit is not explicitly uttered but the behavior of the
function around the limit point is described through words that signify proximity or
distance, which is consistent with the formal definition of limits. Throughout the eight
1essons, 634 of Jason’s 775 utterances about limit were classified as objectified. Table 5.3
Shows some examples of Jason’s objectified word use on limits.
In Jason’s objectified discourse, limits were particular mathematical objects that

could be found (See Table 5.3, [6], [19]) and computed (Table 5.3, [4], [6], [13], [23])
94

 

having some “properties” (Table 5.3, [3]) and “rules” (Table 5.3, [7]). Moreover, they

could also help define other mathematical concepts such as continuity (Table 5.3, [20]).

Table 5.3: Examples of J ason's objectified word use

 

 

 

 

 

 

 

 

 

 

 

 

Utterance Context of use
[ l ] What does it mean that the limit of the function as x approaches x zero is lim f ( x) = L
L? x->a
[2 ] This limit is one. . 1
11m
x—-)2 x — l
[3 ] So what I want to do is to look at some properties of limits. Limit laws
[4] I want to compute limits from limits that I know already. Limit laws
[ 5] x approaches say c of the function one... What is that limit? One. lim 1
x—+c
[6] If p of x is a polynomial, then you can find the limits by plugging in. lim P( x)
x—>c
[ 7] ...we can use our limit rules and compute a number and we know the When does a limit not
limit exists. exist?
[8] ...and suppose I know that the limit as x approaches c of h of x is L. lim h( x) = L
x —)C
[9] Well if the function values of f get arbitrarily close to L as long as x is Informal definition of
sufficientlflose to x zero. limit
[ l O] This means x is sufficiently close to x zero and this means f of x is Formal definition of limit

arbitrarilyclose to L.
[ l l ] We want to say that the function values should be no further than epsilon
away from the limit.

Formal definition of limit

 

 

 

 

 

 

 

 

 

[ l 2 ] The limit...the function values should be close to the limit. I f( x) _. Li < 3
[1‘3] You want to compute the limit as x approaches two of f of x and that lim f(x) = 3
\ should be equal to three. x—)2

I: 1 4] I say I want to be within one over two hundred close to three [the limit], lim f (x) = 3

how close do we have to be to two in order to insure that? x—)2

[ 1 5 ] What is the limit as x approaches one of the firnction? That is one. lim f( x)
\ x—-)l

[ 1 6] So if the left hand and the right hand limit exists but are not equal, then Right hand and left hand

[ the limit does not exist. limits

1 7 . . . o o o o
] So. the horizontal asymptote just Simply means that the limit at mfimty Horizontal asymptote
\ exrsts and equals a number b. ..
[ 1 8] Infinity is not really a number. So technically, this limit doesn’t exist. lim 1
x—->0+ x

 

\
[ 1 9] Let’s look at this. f is continuous if you can find the limit at a by

\ plugging in.
Q] Well the definition of a continuous firnction is given in terms of the limit.

xl'gna f (I) = f (61)

Continuous functions

 

 

\

 

[2 1] What is this limit? It’s one; we have computed this. 1 sin x
\ x—>O x
I22] So the limit x approaches two fi'om the left. .. What is the answer? 1, 2x — 4
N ti tw . ‘m
.g. v. o .-z-Ix-zi
[23] If I need to compute limits, all I need to know is what the function does (2 + x)2 _ 4
near when x is equal to zero. uni) —x—
x—)

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Jason’s consideration of limit as an object was also apparent when he referred to limits as
the “answer” of a limiting process (Table 5.3, [22]) and when he uttered phrases like
“what is this limit?” (Table 5.3, [S], [21]) Once objectified, the limit also leads to other
mathematical objects such as “the left hand and the right hand limit” (Table 5.3, [16])
that could again be computed.

Jason frequently talked about limits being equal to a number in his objectified
discourse (Table 5.3, [2], [5], [13], [15], [17]). He noted that the limit “exists” if it is
equal to a number (Table 5.3, [7], [17]); it “does not exist” if the right hand limit is not
equal to the left hand limit (Table 5 .3, [16]) or the limit is equal to infinity (Table 5.3,

[ 1 8]). The instances where Jason read the limit notation (Table 5.3, [1], [5], [8], [15])
were also considered a part of his objectified discourse as long as he referred to limit as
an end product (a number or infinity) of the limiting process.

When talking about limits in an objectified manner, Jason’s words that signify
motion gave way to words signifying proximity in terms of distance (see Figure 5.1,
Words related to proximity). When describing the behavior of the function around the
litnit point, he started talking about the function values being “arbitrarily close to L (the
limit)” as the x values are “sufficiently close to” (Table 5.3, [9], [10]) or “near” x0 (Table

5 -3, [23]) . Jason also mentioned the function values being “no further than epsilon away
from the limit” (Table 5.3, [11]); “close to the limit” (Table 5.3, [12]) or “within one over
two hundred (epsilon) close to three (the limit value)” (Table 5.3, [14]) instead of using

Words signifying motion such as “closer and closer to” (See Table 5.2, [2], [3]) or

"become” (Table 5.2, [14]).

96

The only case when Jason used a word signifying motion in his objectified
discourse on limits was when he referred to the limit notation. In these cases, he used the

vvord “approaches” (Table 5.3 [l], [5], [8], [15]) while reading notations of the

form lim f (x) = L. It was mentioned before that his use of the word approaches was
x—>a

considered as objectified since this was how Jason and the textbook for the course
described how to read the limit notation. Moreover, he talked about the limit as a number

when he read the notation whereas he did not explicitly utter the word “limit” when he

used approaches in an operational manner.

5- 1.4 Operational and objectified word use

There were twelve instances where Jason’s utterances about limits were classified
as both operational and objectified. In such cases, Jason used a combination of words that
signify motion, which is consistent with the operational use of limit, together with words
that signify limit as objectified (See Table 5.4).

For example, he mentioned that if the function values get “closer and closer to
Some number L, then we call that the limit” (Table 5.4, [1]). Here, he used “closer and
Closer to”, which be explicitly identified as a process before (see Table 5.2, [1]), together
With the end result of that process - “the limit”. On the other hand, in this utterance, the
1iInit value is not obtained; the function values get “closer and closer” to it. Except this,
all his utterances that were categorized as both operational and objectified include the
Word “goes to” (Table 5.4, [2-12]) in which he used the word limit as a distinct
mathematical object or a number but considered x —) a as a process. For instance, in [7]

(See Table 5.4), Jason explicitly uttered the word limit and stated two as the answer of

97

the limit problem. However, he used the word “goes to” instead of “approaches” while

reading the limit notation.

Table 5.4: Examples of J ason's operational and objectified word use

[1]

Utterance
There was a little if in our definition if you read it carefully, and it says if it
gets closer and closer to some number L, then we call that the limit. If it
doesn’t, we say it is not defined.

Context of use

Informal definition of
limit

 

 

 

 

 

[2] l\iVell, if we have some sort of expressron or function, we can try to take a What is a limit?
mrt, say when x goes to zero

. . sin6?

[3 ] Now I want to compute the hmit as theta goes to zero. I —
6—>0

[4] Let’s compute the limits as theta goes to zero of those two outside functions 1 i m Sing
here. g_,o

5x2 — 3x +1

 

 

 

 

 

 

 

 

 

 

[5 ] So what is the limit as x goes to infinity of the numerator? lim
Hoe 2x2 -5
[6] Well, how do we do it? What is a horizontal asymptote? We say that this Horizontal asymptote
function has a horizontal asymptote y equals b if the limit as x goes to plus f(x) = 2 + srn x
infinity or as x goes to negative infinity of this quantity is b. x
[ 7] What is the limit of two as x goes to infinity? Two. , sin x
11m (2 + —)
‘ x—)oo x
, sin x
[ 8 ] The limit of sine x over x as x goes to infinity? One. Inn (2 + )
x—>oo x
. sin —t
[9] I converted a limit from negative infinity to a limit where t goes to infinity. 2 + lim ——(t—)
t—roo -
[ 1 0] So the limit, to make it precise here, as x goes to infinity is zero of the 1' 2x2 — 3
\ denominator. x1120 7x + 4
[ 1 1 ] How about the limit as x goes to negative infinity of two x squared minus x lim 2x 2 - x + 1

plus one divided by three x plus x squared minus five.

\
[ 1 2] Because in order to have an oblique asymptote, the limit as goes to infinity is
\ what? It is plus or minus infinity

xe-°°3x+x2 —5

Oblique asymptote

At first, it seems that “goes to” and “approaches” are similar phrases that can be

Used while reading the limit notation. However, the textbook and Jason mentioned the

Ilotation lim f (x) = L is read as the limit as x approaches a is L and eight of the eleven

x-)a

utterances in which Jason used “goes to” took place in the context of reading the limit

notation when x approached infinity. These observations suggest that Jason did not use

these two phrases as synonyms. Jason used “goes to” only three times in his utterances

98

 

about limit at a point (Table 5.4, [2-4]). In contrast, he used the phrase eight times when

he talked about limits at infinity. Such operational word use was also consistent with his

overall utterances in the context of limit at infinity and is likely to result from Jason’s

consideration of infinity as potential infinity instead of actual infinity3.

Table 5.5 and Figure 5.3 summarize the categorization of Jason’s utterances

throughout the eight lessons. Table 5.5 shows the count and percentage for each of the

four categories across all the lessons. Figure 5.3 excludes the utterances in Lesson 1, and

thus all the colloquial utterances, because of the sparseness of limit-related discourse in

Lesson 1.

Table 5.5: Categorization of Jason’s limit related utterances in four categories

 

 

 

 

 

 

 

 

 

 

 

Lesson number L1 L2 L3 L4 L5 L6 L7 L8 Total
Colloquial 2 0 0 0 0 0 0 O 2
Limit related Operational 3 21 I 15 2 20 47 3 16 127
utterances Obj ectified 3 44 l 17 87 125 107 87 64 634
ational and
001326 fled 0 1 1 o 3 5 o 2 12
\ Total 8 66 133 89 148 159 90 82 775
Colloquial 25.00 0.00 0.00 0.00 . 0.00 0.00 0.00 0.00 0.26
Operational 37.50 31.82 11.28 2.25 13.51 29.56 3.33 19.51 16.39
Percentages Objectified 37.50 66.67 87.97 97.75 84.46 67.30 96.67 78.05 81.81
OPcraiiml and 0.00 1.52 0.75 0.00 2.03 3.14 0.00 2.44 1.55
\ objectrfied

 

3
This will be discussed more in detail in Section 1.5.3.2 (limit at infinity) and Section 1.5.3.3 (infinite

limits).

99

 

Categorization of Jason's limit related utterances

 

percentage

El Operational
l Objectified
I Operational and objectifiedl

 

 

 

Lesson Number

 

Figure 5.3: Categorization of Jason’s limit related utterances as operational, objectified or
both in lessons 2-8.

5. 1 - 5 Mathematical context of use

It is important to note that Jason often used a combination of operational and obj ectified
utterances in the same mathematical context. In this section, some of his word use in
different mathematical contexts will be described in further detail. For the sake of
Simplicity, I use “context” rather than the longer “mathematical context” in the remainder
of tl‘lis section. The contexts that are fundamental to the realizations of limit were
identified as: the informal definition of limit; the formal definition of limit; computing a
lil‘Ilit (at a point and at infinity); infinite limits; and continuity. The purposes of this
Se=<>tion can be summarized as follows:

0 To exemplify the existence of shifts in word use (objectified, operational and/or

both) in Jason’s discourse in the same context.

0 To describe Jason’s informal and formal definition of limit and to explore the

differences in his word use in these contexts.

100

0 To describe Jason’s word use when he talked about limits at infinity and infinite

limits and to draw attention to his going back and forth between potential and
actual infinity in these contexts.

- To explore Jason’s use of infinitesimal related words in these contexts.

0 To describe Jason’s informal and precise definition of continuity and to explore

the differences in his word use in these contexts.
5.1.5.1 Informal definition of limit

Jason introduced the informal definition of limit in the second lesson, right after his

di scussions of average and instantaneous rate of change. He initially defined
instantaneous rate of change as the limit of average rate of changes “over smaller and
SmaJler intervals”. He then defined what a limit is and gave the informal definition of the
concept (See Table 5.6 and Figure 5.4). Although his discussions of instantaneous rate of
cillange contained words that signify infinitesimals, such as “smaller and smaller” or
“Very small” intervals, he did not use infinitesimals in the informal definition of limit.
During his definition of the informal aspect of limit, Jason referred to the behavior
of the function both in an operational and an objectified manner. He said that the function

VahJes “should get closer and closer to L” as the x values get “closer and closer to” x0

(Table 5.6, [4]), which describes the limiting process by means of continuous motion.
Note that although Jason verbally mentioned this process, what he wrote on the board

Was the objectified version of the function’s behavior: “we say that the limit is L” (Table

5-6, [8]) if the function values get “arbitrarily close to L for all x sufficiently close to x0”

(Table 5.6, [6]).

101

Table

[l] loll

 

Fl
‘gllre

Table 5.6: J ason‘s utterances about the informal definition of limit

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

What is said What is done Type of utterance
He writes “f is defined on an interval
[ l ] What is a limit? about x0 ” on the board and starts Objectified
draws a graph-
2) x zero is here. I have some function
and I want to make sure it is
defined at least near x zero.
b) So if I take a small interval here, He draws an open interval around x0 .
this function is defined.
, _ He shows the point on the left of
c) I don t care what it does x0 where the function has an
somewhere away from x zero.
‘15an-
d) So we don’t ask it to be defined at x He writes “except possibly at x0 ” on
zero but at least nearby. the board.
[2 ] Then we say as x approaches x zero of . .
the function f of x equals some He writes x132 f (x) and then Ob'ectifie d
number L if...(does not finish his 0 J
sentence) pauses.
[3 ] What does it mean that the limit of the He shows the notation and turns back Ob'ecti fie (1
function as x approaches x zero is L? to the m he drew. J
4 . .
I ] It means that the function value, if 1 He puts L on the y-axis in the graph he .
get closer and closer to x zero, it drew (See Fi 5 4) Operational
should approach some number L. gure ' '
[5 I It should get closer and closer to L. Operational
[6] I want to say it gets arbitrarily close to He writes these on the board (See . .
. . Objectified
_ L for all x sufficrently close to x zero. Figure 5.4).
[7] So this is what I would want to call the Ob’ec ti fie d
intuitive definition ofa limit. 1
[8] So we say the limit is L ifI can make
the function values to be arbitrarily Objecti fie d

close to L if I choose my values of x

\ sufficiently close to x zero.

 

 

 

Figure 5.4: Jason’s informal definition of limit (hand-written)

102

lns

llL'C

limi

lirm'

(ll
‘JI

Jfii

8X3

pro
mm
are'

mal

In such utterances, he did not base his arguments on continuous motion but on proximity

through words that signify distance, which is consistent with the formal definition of
limit. Therefore, it seems that Jason used elements from both definitions to introduce the
l imit notion.

Only two of his eight utterances about the informal definition of limit were
operational. However, his utterances following this definition about evaluating the limits
of particular functions and finding limits from a given graph (see Table 5.1 for the topic
outline for this lesson) were often operational. In fact, Lesson 2 is the lesson that contains

the highest proportion of operational utterances throughout the eight lessons (See Table
5 - S and Figure 5.3).

5.1.5.2 Formal definition of limit
Jason introduced the formal definition of limit in the fourth lesson and also worked on an

ex ample where he proved lim 2x —1 = 3. Before talking about the formal definition, he
x—-)2

Provided the rationale for the need for a precise definition of a limit (see Table 5.7). Jason
mentioned that the ways they computed limits “at least convinced us what these limits

31‘ e” (Table 5.7, [4]) but noted that they would need a “precise definition” in order to

make sure those techniques work (Table 5.7, [5-5a]). After this, he said that students
Were going to have some homework problems about the formal definition but the topic
Was not going to be in the exam. He wanted students to consider this as a challenge and
erlcouraged them to try to do the related homework problems. He also told students at the
e11d of the class that the homework problems about the precise definition would not be

graded. In fact, he mentioned this definition being very abstract while working on the

Proof example.

103

 

 

5.5).

 

Table 5.7: The rationale for the precise definition of limit

 

 

What is said What is done Type of utterance
[ l ] I want to spend the rest ofthe time in
this class today by looking at the Objectified

formal definition of the limit...
[2] Sowhathanttodohereistantto . “ . .. . .,,
write down the precise definition of a He writes precrse definition thmlt Objectified
limit. on the board.
[3 J Why would we need such a thing as . .
recise definition of a limit? Objectified
[4] Well, we kind of said what a limit is in
words and it helped us to at least
convince us what these limits are that Objectified
we have been computing in class, say

 

of pplflomials or things like this.
[5 ] It kind of made sense those laws of

limits that I wrote down but to really . .
make sure these work in mathematics, Objectified

we have to prove those things.

 

a) In order to prove something, we
need a precise definition...
[6] So that is why we need the precise All his comments up to here are . .
definition ofa limit verbal. Oblect‘fied

Table 5.8 shows Jason’s utterances about the formal definition of limit (See also Figure

5-5).

   

Figure 5.5: Jason's precise definition of limit (hand-written)

 

 

 

[8] We at
be def
before
delimit.
a) L

pi
e:
inlets:

[i0] Tncn
a pm

ll IJAnd 5.
“me
3mm

[111% 5e
beforc
of l a
x is St

mi 30 a l
langu
than ;
Of the
0m fi
@331:
Playl

Table 5.8: J ason's precise definition of limit

_—f

What is said

T7] So how do we do a precise definition of

a limit?

We need a function f and we need it to

be defined in...my words that 1 said

before when we used for the loose
definition of the limit I said nearx zero.

3) Let’s make this maybe more

precise. On an interval about x zero
except possibly for x zero.

So we have seen before limits are very

interesting if our function is not defined

at the mint.

[1 OJThen we say that the limit offofx as x
approaches x zero is L.

[1 l ] And let’s see in math language how do 1
write this? I say the limit as x
approaches x zero of f of x equals L.

[ l. 2 ] We say that is the case if. ..in my words
before I said well if the fiinction values
of f are arbitrarily close to L as long as
x is sufficiently close to x zero.

[1 3 J So a little more formally, in math
language, I say if for all epsilon greater
than zero, this is going to play the role
of measuring how close we are to L in
our function values, there exists a delta
greater than zero, the delta is going to
play the role to measure saying that we

are sufficiently close to x zero.

[1 4] So let’s put that in words... whenever x
is sufficiently close to x zero so that
means that the difference is no more
than delta then the function values

\should be close to L.

[1 5] So how do we write this? The difference
of the fitnction values from L should be
less than epsilon.

\

[ 1 6] Now this looks somewhat complicated.
So let me maybe decipher this. This
means x is sufficiently close to x zero
and this means f of x is arbitrarily close

[8]

[9]

 

.\to L.
[ 1 7] For any epsilon, I should be able to do

this and for any one of them then there
\exists this delta.

[1 8] That means if x is sufficiently close,
delta close to x zero, then the limit. . .the
function values should be close to the

\ limit.
[19] So let’s maybe do one example and

\ prove that a limit exists in this way.

What is done

He emphasizes the word “near”.

He writes these on the board.

He writes these on the board.

He writes these symbolically on the
board (See Figure 5.5).

He emphasizes the word
“sufficiently”.

He writes Ix - on < 5 and then

If(x) -LI < 8 on the board.

He writes “sufficiently close to x0”
under Ix - xOI < 5 on the board.

He writes “f(x) arbitrarily close to L”
under I f (x) - LI < 8 on the board.
(See Figure 5.5)

He first shows Ix -— xOI < 5 and then

If(x)-L|<8

He shows 5 > 0

He first shows Ix — xOI < 5 and then

|f(x)—L|<e

105

Dpe of utterance

Obj ectified

Objectified

Obj ectified

Obj ectified

Objectified

Objectified

Obj ectified

Objectified

Objectified

Objectified

Objectified

Objectified

Objectified

Oterall. I
categoriz
definitior
in two as
proximitj
associate
13mm H

Jason
"10036" (_
0D contir.
and close
5-6‘ l4];
definlllol
definitiol
abSOlute
are to L i
Sulllcien

Overall, there were 19 utterances about the formal definition of limit and all of them were
categorized as obj ectified. The words Jason used when talking about the precise
definition of limit differed from those he used when talking about the informal definition
in two aspects. First, his words signifying motion were replaced with words that signify

proximity. Second, the presence of symbolism in the formal definition required Jason to
associate symbolic representations with particular words. In what follows, these two
aspects will be described in more detail.

Jason’s words about the informal, or in his words “intuitive” (Table 5.6, [7]) and
“loose” (Table 5.8, [8]), definition of limit addressed the limiting process and were based
on continuous motion. Utterances in which the function values “approach” or “get closer
and closer to L” (Table 5.6, [4], [5]) as the x values “get closer and closer to x0 ” (Table

5 -6, [4]) are examples of Jason’s operational word use when he talked about the informal
definition of limit. In contrast, his words about the formal or “precise” (Table 5.8, [7])
definition addressed proximity and were based on distance measured by means of
absolute values. He mentioned that 6‘ would “play the role of measuring how close we
are to L in our function values” and 6 would “play the role to measure saying that we are

Sufi‘iciently close to x zero” (Table 5.8, [13]). He also talked about the closeness of the x

Values to x0; and the f(x) values to L in terms of the “difference” Ix — xOI being no bigger

than 5 [l4] and the “difference” I f (x) — LI being less than 8 [15], respectively. In

Sl-lllnmary, the operational terminology such as getting closer and closer to (Table 5.2,

[1]), approaching (Table 5.2, [10]), becoming (Table 5.2, [9]) and going (Table 5.2, [7])

106

were re;
close" (f

lnde
objectiti
where J 2
Which he
dit‘l‘erenc
address.

he menti

were replaced by being “sufficiently close” (Table 5.8, [13], [14], [16], [18]), “arbitrarily
close” (Table 5.8, [12], [16]) and “close” (Table 5.8, [18]) in the formal definition4.
Indeed, Lesson 4 stands out as the lesson which had the highest proportion of
Objectified utterances about limits (see Figure 5.3). This is in contrast with Lesson 2,
where Jason introduced the informal definition of limit and worked on some examples,
which had the highest proportion of operational utterances about limits. This signals a
di flerence in word use in these two contexts. However, this shifi in vocabulary was not

addressed by Jason in the classroom. Instead, he tried to connect these definitions when

he mentioned Ix — xOI < 6 can be read as “x is sufficiently close to x0” and

If (x) — LI < a can be read as “f(x) is arbitrarily close to L” (Table 5.8, [16] and Table 5.6,

[8] ). This brings us to the second aspect of the word use in the formal definition.

The difference in word use between the informal and the formal language also
I‘eSulted from the translation of symbolic representations to words. Jason went back and
forth between the “math language” (Table 5.8, [11], [13]) and “words” (Table 5.8, [12],
[1 4]) more frequently while discussing the formal definition than the informal definition.

The informal definition contains lim f (x) = L , which is expressed as “the limit of the
x—>x0

t“-lnction as x approaches x zero is L” (Table 5.6, [3]) in words. The formal definition, on
the other hand, also requires the explanation of what 8 and 6 refer to and what

Ix ~ xOI < 6 and I f (x) — LI < 8 mean. Jason considered this translation as complicated and

\

Jason used sufficiently close and arbitrarily close in the informal definition of limit as well (Figure 5.4)
1.3m his word use was not consistent with this terminology when he worked on examples following the
Informal definition. When describing the behavior of the fimction in those examples, he often referred to
the limiting process using words signifying motion as illustrated in Table 5.2.

107

therefore wanted to “decipher” (Table 5.8, [16]) the formal definition by using elements
of the informal definition.

Jason did not utter any infinitesimals related word in the context of the formal
definition of limit.

5.1.5.3 Computing a limit
Throughout the eight lessons, Jason computed 64 different limits. Among those, 43 were
limits computed at a point; 10 were limits computed at infinity; and 11 were infinite
limits. Infinite limits refer to the limits computed at a point or infinity whose result is
positive or negative infinity rather than a real number L. The reason infinite limits were
considered as a distinct category of computing limits results from Jason’s different word
use between infinite limits and limits at infinity when referring to infinity.

It was mentioned before that, although Jason’s word use about the informal and
formal definition of limit was mostly objectified, his description of a function’s behavior
while computing limits was mainly operational. In what follows, his discourse in three
different contexts will be examined in detail to both justify the existence of mixed
utterance use and to explore Jason’s discourse about infinity and infinitesimals.

5.1.5.3.] Limit at apoint
The majority of the limit computation problems Jason worked on were computing the
limit of a function at a given point. For the purposes of this study, the limits at a point for
which the limit is equal to L, where L is a real number were considered separately from
the limits at a point for which the limit is equal to plus or negative infinity (infinite limits)

since Jason’s word use differed significantly in these contexts.

108

his W0

35011 1

combii

When addressing limit at a point, Jason often talked about limit as objectified, but

his word use on the behavior of the function was primarily operational. Table 5.9 shows

. 1 , . .
Jason’s utterances when he computed lim ———1. In this computation, Jason used a
x——)2 x -—

combination of objectified and operational utterances. He considered the limit of the
function as a distinct number obtained at the end of the limiting process when he said

“this limit is one” (Table 5.9, [5]).

. . . 1
Table 5.9: Jason’s utterances 1n the context of computing lirn —.

 

 

 

x->2 x -1
What is said What is done Type of utterance
[1] What is this limit?
He writes lim on the board.
x—>2 x -1 Objectified

(Some students say it is one.)

 

[2] Let’s see. If x gets closer and

 

 

 

 

 

closer to two, this quantity gets He shows x —> 2 and then shows
closer and closer to one over two , x f Operational
minus one. He says these verbally and doesn t write
anything on the board.
13] It is very close to one over one. Operational
[4] So the closer x gets to two, the .
closer this will get to one. He ShOWS 1 Operational
x _
[5] ms limit rs one. Slrzxnntes 1 near the question; no graph 1S Objecti fie d

 

When he talked about the function’s behavior near the limit point, on the other hand, he
mainly used words signifying continuous motion and considered limit as a process. He
referred to x —> 2 as x getting “closer” (Table 5.9, [4]) and “closer and closer” to two
(Table 5.9, [2]) instead of the notational language of “x approaches two” (See Table 5.6,

[2]). He also talked about f (x) —-) L in an operational manner when he said the function

values get “closer” (Table 5.9, [4]) and “closer and closer to” (Table 5.9, [2]) one.
Jason worked on many of the limit computation problems in similar ways, although he

sometimes used different words signifying motion when describing the behavior of the

109

functic
5.] am
above
talked

when 1

Table

 

9, 6‘

fimction such as “becoming , moving”, and getting “smaller and smaller” (See Figure
5.1 and Table 5.2 for motion related/operational word use). What is not visible in the
above example is his use of infinitesimal related words, which often took place when he
talked about right hand and the left hand limits. Table 5.10 shows Jason’s utterances

when he computed lim f (x) from a given graph.
x—>1

Table 5.10: Jason’s utterances in the context of computing lim f (x) from a graph.
x—>l

 

What is said What is done Type of utterance
[1] The limit as x approaches one of f Objecti fie d
of x. I claim it doesn t eXist.

 

 

3) Why would that be? Let’s
think about the definition.

 

[2] What we want is that we get
closer and closer to one, it should He shows x=1 on the graph. Operational
approach some number.

[3] But if it is a little less than one, He puts a dot to the left hand side of x=l
then it looks like we are getting (See Figure 5.6) and shows the point Operational
close to two. f(x)=2.

 

[4] If we are a little bigger than one, it
looks like the function wants to be He puts a dot on the right hand side of
equal to one. x=l and shows fix)=1 on the graph
a) It can’t make up its mind.
[5] So there is no single number that
this function gets closer and closer Operational
t0.

Objectified

 

[6] So what we say here, the limit is He writes lim f(x) = undefined. Objectified
undefined x—>l

 

 

Figure 5.6: Computing limits fiom a given graph (hand-written)

110

Here, we see that Jason talked about the x values getting “closer and closer to” one (Table
5.10, [2]) and he looked at what number the function values were approaching (Table
5.10, [2]) or getting “closer and closer to” (Table 5.10, [5]). In this respect, he again used
the dynamic view of limit when explaining how the function behaved near the limit point.
Besides this, he drew two points very close to the left hand side and the right hand side of

the limit point x=1 (See Figure 5.6) and mentioned being “a little less than one” [3] and

“a little bigger than one” [4], respectively. This suggests infinitesimal related word use5
since he used these phrases to refer to the points that are very close to the limit point,
which are of very small proximity to the limit point. Jason consistently used infinitesimal
related words when he talked about the right hand and the left hand limits at a given
point. In the context of limit at a point for which the answer is a real number, Jason used

infinitesimals for finding lim f (x) from a graph (See Table 5.10), when he made the
x—>l

transition from average rate of change to instantaneous rate of change in Lesson 1, and

«lice—1)

I II in Lesson 6. During his introduction to
x _

when he computed lim
x—)l

instantaneous rate of change, he mentioned the time intervals getting “smaller and

smaller” and being “a very small number” (Jason, January 12, 2009, Lesson 1) when he

 

referred to h —> 0 in f (x + h}: - f(h) . When looking at the right hand and the left hand

t/Eoc-I)

limits of the function
Ix -II

at 1, he mentioned the x values being “a little larger than

one” and “a little smaller/less than one” (Jason, January 26, 2009, Lesson 6). He did not

 

5 Infinitesimals refer to very small quantities that are often negligible (See Chapter 111, Section 3.2).

111

use in

Furtht'

swim

\.'V‘

use infinitesimal related words in the context of limit at a point besides these instances.

Further details about his infinitesimal related word use6 will be discussed in the following
sections about infinite limits and limits at infinity.

5.1.5.3.2 Limit at infinity
Limit at infinity is a limit computed at positive or negative infinity. Throughout the eight
lessons, all limits Jason computed at infinity had a real number L as the answer except for
one. For one infinite limit Jason computed, the answer was equal to infinity. This
problem was categorized as “infinite limit” rather than “limit at infinity” because how
Jason referred to infinity changed between these two contexts.

Jason introduced the notion in Lesson 5 under the title limits at infinity. After this,

he started computing lim -1- (See Table 5.11). Note that the title, limit at infinity
xawx

(Figure 5.7 and Table 5.11 [1]) signifies that the distance from the x values to infinity is
zero; that is, it treats infinity as a distinct entity that can be reached. This view of infinity
is referred to as the actual infinity and was discussed previously in Chapter 111. Reading
part of the limit notation x —> 00 as x approaches infinity (Table 5.11, [2]), however, talks
about potential infinity that cannot be reached. The only reason this particular utterance
was considered objectified is because both Jason and the textbook defined this as how to
read the notation. Jason’s consideration of infinity as potential in the context of limit at
infinity, on the other hand, is not restricted to how he read the notation. He associated

x —> 00 with also the x values getting “larger and larger” (Table 5.11, [3]) and with

 

See Figure 5.1 for a complete list of mfimtesrmal related words in Jason’s discourse.

112

“mitt

drew It

Table 2'

5

[1} SO]
ti:
tool
3)

l'i \\ h
War
lim
3)
that

lay}

 

“walking on this axis [x-axis]” (Table 5.11, [4]) as he extended the x-axis he initially

drew to represent infinity.

Table 5.11: Jason's utterances when computing lim 1.

 

 

 

 

X —>oo x
What is said What is done Type of utterance

[1] So now, I want to talk about some

other types of limits. I can also He writes “limits at 00”. Objectified

look at limits at infinity.

8) So this horizontal eight, it He writes “infinity” near 00.

means infinigy.

[2] What is the idea behind this? I

“I’m.“ '° 53" ”mething. like.” He writes lim 1. Objectified

limit as x approaches infinity of x—>oo x

say the function one over x.

 

a) What should that be? Does

that make any sense? Let’s just

graph the function one over x to

get some idea.

b) This is what the function one He starts drawing the graph of the

over x looks like if I gaph it. function.

 

[3] So, if x approaches infinity, what
should that mean? That means if I He puts his hand on a point on the x axis

take x and make x larger and and moves along the x axis towards the Operational
larger. . .(does not finish his right with his hand.
sentence)

 

He extends the initial graph he drew
towards positive infinity and also extends

[4 the x axis (See Figure 5.7).

I—l

So I just keep walking on this
axis, the function values, in this

 

Operational
:3: they get closer and closer to He then moves his hands along the y
' values on the graph and then writes 0 as
the answer for the limit.
[5] So this [the limit], I want to say, is He shows 11m __ Objectified

equal to zero. x —>oo x

 

 

Figure 5.7: Jason's hand-drawn graph of f (x) = —1—
x

113

These ut
process
along th

the proc

motion I
Said that
that pror
limits at

(
Some 01
ftittetion
about th

I
Phrase g
motion 1
35 Wet)
IaIter at
Erppmftcl
x a, a: (
infinity l
1imitt at i

“he“ he

These utterances are consistent with the view of infinity as potential: a continuous
process that goes on indefinitely. His bodily gestures of extending his arms and moving
along the function with his bands can also be thought of as further evidence of expressing
the process of approaching infinity by dynamic means. (See Table 5.11, what is done)

In this example, Jason described the behavior of the function in dynamic and
motion related words, which refers to limit as a process (Table 5.11, [4]). Yet, he also
said that the limit “is equal to zero” (Table 5.11, [5]), which refers to limit as the result of
that process. Therefore, we can still see his use of mixed utterances in the context of
limits at infinity.

Overall, Jason worked on ten problems about computing the limit at infinity.
Some of those were tied to the idea of finding the horizontal asymptotes of a given
function. Table 5.12 shows the limits Jason computed at infinity and also how he talked
about the x values approaching infinity in the limit notation.

Besides the word approaches, Jason frequently talked about x —> 00 using the
phrase goes to (Table 5.12, [8], [10], [13], [19-21]). Note that “going to” infinity invokes
motion towards infinity, which is not reached, and is compatible with the view of infinity
as potential. Moreover, Jason’s other utterances in which the x values made/get/become
larger and larger (Table 5.12, [2], [5], [16], [18]) fiirther support this dynamic process of
approaching infinity. On the other hand, Jason also used the word at when talking about
x —> 00 (Table 5.12, [6], [12], [15]), which is more compatible with the view of actual
infinity (See Section 3.1) since it refers to infinity as a discrete point. In the context of
limit at infinity, Jason uttered the word number referring to (negative) infinity only once,

when he said “as x becomes larger and larger but a large negative number” (Table 5.12,

114

[16])
instzn
than .

ohjec

Lem

Lessc

L‘ss

Leg,

[16]). Except for this instance in which Jason talked about infinity as a number and the
instances he used the word at, he talked about infinity as a potentiality (process) rather
than a distinct entity. Therefore, his word use about infinity was mainly operational than

objectified.

Table 5.12: J ason's utterances about the x values when he computed limits at infinity

 

Limits at infinity

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Lesson Number Context of use How x —) 00 is referred to
1i m _1_ [1] Limit as x approaches infinity...
x—>oo x [2] If I take x and make x larger and larger...
. 1
11m [3] Limit as x approaches infinity...
x—)oo x - 5
. 1
Lesson 5 11m — + 5 No infinity related word is uttered.
x—>oo x
l' 5 [4] Limit as x approaches infinity...
x1120 [5] ...when x gets larger and larger...
[6] ...the limit at infinity...
lim 5x2 — 3x +1 [7] Limit as x approaches infinity...
x600 2x2 _ 5 [8] Limit as x goes to infinity...
. [9] Limit as x approaches infinity...
lim (2 + sm x) [10] Limit as x goes to infinity...
x_,w x [1 1] We know what it [the limit] is at infinity.
[12] Limit at positive infinity...
mnt as x goes to negative m mlty
[l 3] L’ . . . f‘ .
- [14] Limit as x approaches negative infinity...
. sm x
Lesson 6 11m (2 + ) [15] Limit at negative infinity...
x"'°° x [16] As x becomes larger and larger but a large
negative number...
. sin x [17] Limit as x approaches infinity
11m —— [18] ...as x becomes larger and larger...
x-m x [19] ...asxgoes to infinity...
2
. - + . . . . .
11m ____2x 2x 1 [20] leIt as x goes to negative Infinity. ..
Lesson 8 x 3x + x - 5
x2 —1
lim [21] Limit as x goes to infinity...
x—)oo x + 2

 

In the context of limits at infinity, Jason used infinitesimals only once when he

computed lim s_1n_x. Table 5.13 shows his utterances when he worked on this limit.
x—>oo x

115

Table 5

 

[l] Hov
app:

x?
[3} “cl

 

[4L Sot

Table 5.13: Jason's utterances when be computed lim

sin x

 

 

 

 

 

 

 

 

 

 

 

 

 

x-—)oo x
What is said What is done Type of utterance
[1] How about the limit as x . sin x
approaches infinity of sine x over He writes 11m Objectified
x? x —) m x
[2] Well, if we consider this, what
happens as x becomes larger and He shows when he says “this”. Operational
larger? x
a) The sine is a number that
stays between negative one
and one.
' x
[3] The larger x gets, the smaller this He shows when he says “this”. Operational
W111 be for sure. x
. . . . . . Sin x ' ‘
[4] So this 11m1t 18 zero. He wntes 11m = 0 Objectified
x—)oo x
. “ ,, sin x
Here, Jason mentloned that as the x values become larger and larger [2], gets

“smaller” [3]. He used the word smaller to talk about small quantities that are very close

. . . . sin x . .
to zero smce he then said the 11m1t of —— would be zero [4]. It was mentloned 1n

x

Chapter III that Cauchy defined infinitesimals as follows:

Jason’s utterance about the process of

When the successive absolute values of a variable decrease indefinitely in

such a way as to become less then any given quantity, that variable

becomes what is called an infinitesimal. Such a variable has zero for its

limit (Kitcher, 1983, p.247).

sin x

 

getting “smaller” as the x values getting

“larger” [3] clearly describes the fimction as decreasing indefinitely and having zero as

its limit. Therefore, Jason used infinitesimals in this particular problem. He did not use

116

 

any oth

is equa

Infinite
infinity

worked

shows ]

functim

G

Klfip.’ .
“It 1”]

any other infinitesimal related utterance in the context of limit at infinity7 where the limit
is equal to a real number L.

5.1.5.3.3 Infinite limits
Infinite limit is a limit computed either at a point or infinity that has plus or minus
infinity as the answer. Although Jason introduced vertical asymptotes in Lesson 6, he
worked on some problems about infinite limits in the preceding lessons (See Table 5 .14).

Overall, there were 11 cases in which Jason computed infinite limits. Table 5.14
shows how he referred to infinity and the limit in each of those cases. He mentioned
function values being/getting/becoming a (very) large positive or negative number (Table
5.14, [5], [7], [15-17], [20]); and being/becoming an arbitrarin large (negative) number
(Table 5.14, [l l], [13]) when he talked about infinite limits. In this respect, he referred to
infinity as a number more fi'equently than he did in the context of limits at infinity. When
referring to infinity, Jason also mentioned the function values getting larger and
larger/smaller and smaller (Table 5.14, [1-2]); becoming larger and larger (Table 5.14,
[10]); becoming really/very/arbitrarily large (Table 5.14, [3], [6], [20-21]); getting
large/really large (Table 5.14, [14-15]); tending to/ going to/approaching to plus or
negative infinity (Table 5.14, [4], [20-22]); becoming arbitrarily large (Table 5.14, [12]);

and getting closer and closer to a line (slanted asymptote) (Table 5.14, [9]).

 

7 2 __
Jason also used infinitesimals when he worked on [im 2x 3 but this limit was considered as an
x—>oo 7x + 4

 

infinite limit in this study.

117

 

 

lawn
lmmhn

lamnl

Lawn}

Table 5.14: Jason’s utterances about infinity in the context of infinite limits

 

Infinite limits

 

 

 

 

 

 

 

 

 

 

 

 

 

Lesson Context of How f (x) —) 00 is referred to How lim f (x) = :00 is

Number use x—)a

referred to
[1] If we are a little bigger than six, it
lim f ( x) [the function ] tries to get larger and [23] The limit is not defined.
Lesson 2 x—)6 larger [24] . . .the limit does not have
from a graph [2] If we are less than six, the function to exist.
tries to get smaller and smaller.
3 ...it the function becomes reall . .

. 1 [ ] large[if x is larger] than one. y [25] ltJuSt does not turn to
11m — . . . any fixed number. So the
x—+l x _ 1 [4] ...and ifx is smaller than one... It limit is undefined

will tend to negative infinity. '
[5] we could get a very large positive
number if x is just a little bigger than
Lesson 3 five. . . . .
2 [6] ...and lf 1 lelde this [twenty] by a
. x - 5 very small number, I get something [26] So this [the limit] is
Ji—+5 x _ 5 really large. So this [the function] undefined.
will become very large.
[7] ...choose x to be a little smaller than
five... it [the function] will be a very
large negative number.
[8] The denominator? If x goes to [27] So the limit...as x goes to

. 2x2 — 3 infinity, then seven over x infinity of the

$3100 7x + 4 approaches zero. denominator is zero.
[9] It [the function] will get closer and [28] The limit [of the function]
closer to a line. does not exist.
[10] one over x, becomes larger and
larger. [29] It [the limit] is undefined.
[11] It’s an arbitrarily large number. [30] The [right hand] limit
[12] ...as x approaches zero from the equals plus infinity.
lim _1_ right, f of x becomes arbitrarily large [31] Infinity is not really a
x_,0 x and is positive. number. So technically,
[13] If we look to the lefi...it becomes an this limit doesn’t exist.
arbitrarily large negative number. [32] The [left hand] limit is just
[14] [The fimction] gets very large and negative infinity.
positive or very large and negative.
Lesson 6 [33] This [right hand] limit will
[15] (approaching from the right of be plus infinity or in other

lim 1 two)... we get a positive number that words, this quantity will

x——>2 x _ is very large. get arbitrarily large.
[34] It [left hand limit] is going
to be negative infinity.
[16] We have a negative number, we
fl divide it by a very small positive [35] This [right hand] will be
lim 49 number, it makes it a very large negative infinity.
4 7x + 4 negative number. [36] So this [left hand] limit in
x"“‘ [17] If x approaches negative four over the end will be plus
seven from the left. . .I get a large infinity.
positive number.
lim x +1 [18] It [the function] is going to become [37] So we say it [the limit] is
2 very large; arbitrarily large plus infinity.

118

 

lahle

 

Les-so
Numb

lessor

Lessor.

ln all
since
based
nlimb
elplic
to ten

tht‘fret
“Until
(Tabtf

1*!h en

mirth

Table 5.14 cont’d: Jason’s utterances about infinity in the context of infinite limits

 

Infinite limits

 

 

 

Lesson Context of How f (x) —) 00 is referred to How lim f (x) = too is referred
Number use x->a
to
11m — No infinity related word 18 uttered. [38] The limit does not exrst.
x—>O x
Lesson 7 [39] So this [limit] does not
x + I exist.

 

No infinity related word is uttered.

 

 

x—->1 x2 _ 1 [40] At plus one, the limit does
not exist.
[19] When x is close to negative two, this [41] ...the limit doesn’t exist.
thlng becomes really large. . . . .
. It 18 plus or minus infinity
[20] So as the denominator gets a small .
. . if I come from the left or
negative number, this [the the right
. x2 —l remainder] becomes a “FY large [42] So the answer [left hand
Lesson 8 11m negative number. So it Will approach . . . . .
x—>—2 x + 2 negative infinity. llmlt] is Just negative

infinity.

[43] Similarly here [right hand
limit]... you will get plus
infinity.

[21] ...as x approaches two from the Iefi,
my function goes to negative infinity.

[22] If I come negative two from the
right, it goes to plus infinity.

 

In all these instances where he described f (x) —> oo , Jason’s word use was operational

since he was talking about limit as a process. His arguments about infinity were mainly
based on potential infinity except possibly for the cases where he referred to it as a
number. It should also be mentioned that in these cases, he rarely uttered the word infinity
explicitly and when he did (Table 5.14, [4], [20-22]), the filnction values were considered
to tend/ go to/approach infinity, suggesting continuous motion towards infinity and
therefore, potential infinity.

When talking about limit in an objectified manner, that is, as the result of the
limiting process, we see Jason talking about the limit as being plus or negative infinity
(Table 5.14, [30], [32-3 7], [42], [43]). Therefore, he uttered the word infinity more often
when talking about the limit as the end of the limiting process. He was more likely to
think about actual infinity rather than potential infinity in these instances where he

mentioned limit as being plus or negative infinity since then infinity was signified as the

119

endr

Inn
.I-Nl

nnat
thatl
35 "11

exht

hnnt
reaM}
hone
large
MSlM
about

Regan

end result of the process. This can further be supported with the fact that Jason wrote

lim f (x) = ioo for each of these problems when representing the limit using the
x -)a

notation. By doing so, the limit was considered as equal to infinity but not as a process
that went on and on as his earlier utterances suggested. Jason considered an infinite limit
as “not defined” (Table 5.14, [23]); “undefined” (Table 5.14, [25-26, 29]); and “does not

exist” (Table 5.14, [24], [28], [31], [38-39]). So for him, lim f (x) = ioo implied that the

x—>a
limit did not exist or was undefined. This was also clear when he said “infinity is not
really a number. So technically, this limit does not exist” (Table 5.14, [31]). Note,
however, that he used utterances that signified infinity as a large/very large/arbitrarily
large number during his discussions on the behavior of the fimction values. Therefore,
his word use about infinity was not consistent between the contexts in which he talked
about the function values approaching infinity and the limit being equal to plus or
negative infinity.

In summary, Jason used a combination of operational and objectified utterances in
the same context when he addressed limits at infinity and infinite limits. Moreover, his
consideration of infinity as operational (potential) and obj ectified (actual) depended on
whether he was addressing the behavior of the x values as well as the function values or
the limit of the function.

Infinite limits turned out to be the most important context for gaining information

about Jason’s word use on infinitesimals. It was mentioned before that Jason used

120

infinitesime
limits at int
infinitesimz
Jason's utte

\lh?
(positive ‘ne
Values appr
"slightly bi
their Values
Smaller that
be "someth
lumber)" ‘
‘Smaller an

M
pang Ofa fi
2% FOT e:
P0111125“) 2
i350“ mtint

Yer), small

infinitesimals three times in the context of limit at a point8 and once in the context of

limits at infinityg. Except these, all remaining seven cases in which Jason used
infinitesimal related words took place in the context of infinite limits. Table 5.15 shows
Jason’s utterances about infinitesimals in the context of infinite limits.

While working on infinite limits, Jason frequently mentioned “a very small
(positive/negative) number (Table 5.15, [1], [5-7], [12], [14-17]). When talking about x
values approaching the limit point from the right, he mentioned being “a little” or
“slightly bigger than” the limit point (Table 5.15, [1], [4], [13]). When the talked about
the x values approaching the limit point from the left, he mentioned being “smaller/a little
smaller than” the limit point (Table 5.15, [2], [7]). He also said the function values could
be “something very very small” (Table 5.15, [3]); a “small negative or small positive
(number)” (Table 5.15, [3], [18]); “quite a small number” (Table 5.15, [9]); and could get
“smaller and smaller” (Table 5.15, [10]).

When Jason used an infinitesimal related word for the function values or specific
parts of a function such as the denominator, he considered the values to be very close to
zero. For example, he said “This [the denominator] will be a very small number, say zero
point zero zero zero one” (Table 5.15, [5]). Such quantities had zero for their limits.
Jason mentioned this explicitly when he said “. . .dividing seven by a very large number is
very small. The limit will be zero” (Table 5.15, [8]) and “So this limit is zero” (Table

5.13, [4]).

 

8 When he computed lim f (x) from a graph (See Table 5.10); when he computed lim m ; and
x—>1 x - l
x—>l

during the transition fi'om average rate of change to instantaneous rate of change.

121

 

lahle
lesson
\utnloe

 

 

lesson

1433011

/

L330“

/

Winn

l
Mm

Table 5.15: J ason's infinitesimal related word use in the context of infinite limits

 

 

 

 

 

 

 

 

Lesson Context of use Infinitesimals related utterances
Number
[1] Well, if x is a little larger than one, then this quantity is one over a very
lim __1_ small positive number.
x —->1 x _] [2] and if x is smaller than one, then this denominator here will be a
really small negative number.

[3] So really what this is trying to do, it is approaching to a number twenty
over something very very small. Maybe a small negative or a small
positive.

[4] we could get a very large positive number if x is just a little bigger

x2 _ 5 than five
lim [5] So x is a little bigger than five then what is the denominator? This will
x—>5 x - 5 be a very small number, say zero point zero zero zero one.

[6] ...and if 1 divide this [twenty] by a very small number, I get something

Lesson 3 really large.

[7] ...choose x to be a little smaller than five... this [the denominator will
be a very small negative number.

[8] Seven over x approaches zero because x becomes a larger and larger
number, dividing seven by a very large number is very small. The

2x 2 _ 3 limit will be zero. .
lim —————. [9] Similarly, here, four over x squared as x becomes larger and larger...
Alt->00 7x + 4 that is quite a small number.

[10] It [4/x3] gets smaller and smaller.

[11] When x is very large... this one [the remainder] will be very small.

1 [12] What do we have in the denominator? We have a positive number that
x—>2 x _ 2 is very small.

[13] If x approaches negative four over seven, but it is slightly bigger than

-115 that...
76— [14] So if the denominator is positive but very small number it means we
lim — divide b 3 ve small ositiv numb r the numerator is ne ative.
y TY P e C i g
x_,_fl 7x + 4 [15] We have a negative number, we divide it by a very small positive

Lesson 6 7 number, it makes it a very large negative number
[16] If x approaches negative four over seven from the left... I have a
negative number, and I divide it by a verxsmall negative number.

 

lim x +1 [17] So this will approach a number that is two divided by a very small
positive number.

 

 

x2 -1 [18] So as the denominator gets a small negative number, this becomes a

Lesson 8 11m very large negative number.

x——>—2 X+2

 

 

 

2
Except for lim 2" - 3 , all infinitesimal related words uttered in the context of

x—>oo 7x+ 4

infinite limits took place when determining the function’s behavior on the right hand side

 

sin x

 

9 When be computed lim (See Table 5.13).

x—-)oo x

122

and the it
from the
Instead. 1

little sma

smallerc

limit is z
context (
fifty ter
156011 US]

e“(lual to

of instan

onhn1:
[1‘50

“litre l‘lt

L“Tar

and the left hand side of the limit point. Jason did not use the terminology of approaching
from the left/right until Lesson 6, where he introduced right hand and the left hand limits.
Instead, he mentioned the x values being a little larger/bigger than or being smaller/a
little smaller than the limit point (Table 5.15, [1-2], [5], [7]).

The analysis of word use also revealed that Jason used the same term, getting
smaller and smaller, for both infinitesimal quantities that are negligibly small (whose

limit is zero) and for negative infinity depending on the context. For example, in the

 

, he said that , i which he obtained after dividing

2
context of computing lim 2x -3 2 i
x

x—)oo 7x+ 4

every term by the highest power of x, “gets smaller and smaller” (Table 5.15, [10]). Here,

Jason used the term to refer to an mfimtesrmal quantity srnce the 11m1t of ——2— at infinity is
x

equal to zero. Similarly, he referred to infinitesimals when he talked about the “time
intervals getting smaller and smaller” (Jason, January 12, 2009, Lesson 1) in the context

of instantaneous rate of change to talk about the h values approaching 0 when working

 

on lirn
h-—>O

f (x + h; _ f (x) . In the context of computing limits from a graph (See Figure 5.2)

where he talked about lim f (x) , however, he also said “if we are a little less than six,
x—>6

the filnction tries to get smaller and smaller” (Table 5.14, [2]). Here, he used smaller and

smaller to talk about the left hand limit of the function at the point 6, which is negative

infinity. Besides this instance, he did not use smaller and smaller to signify negative

infinity but used “a very large negative number” (Table 5.14, [5], [16], [20]) or

“arbitrarily large negative number” (Table 5.14, [13]) instead.

123

Jason i
mentio
nonon.
answer

ifid f

5.1.5.4 Continuity
Jason introduced continuity in Lesson 7 (See Table 5.1 for the topic outline). He
mentioned that he was going to give one intuitive and one mathematical definition of the
notion. He then told students that they would need to use the mathematical definition to
answer questions in the exam.

Table 5.16: J ason's utterances about the intuitive definition of continuity

 

What is said What is done]0

 

[1] So what is a continuous function? He writes continuous functions on the board.

 

[2] A continuous function is a function, and I
am just going to say it in words, that I can
graph without taking the chalk off the
board.

 

[3] So this is really all that we need to know,
from an intuitive perspective, about
continuous functions.

 

[4] Continuous just means I can graph without

taking the chalk off the board. He states these verbally.

 

After giving students an intuitive definition of continuity, he drew graphs of arbitrary
functions and discussed their continuity based on whether he could graph them “without
taking the chalk off the board” (Table 5.16, [2], [4]). While doing so, he moved his hand
along the graphs of the functions and mentioned that approaching and graphing like this
could be thought as a “limiting process” (See Table 5.2, [16]). Therefore, although he did
not utter any limit related word during his discussions of the intuitive definition of
continuity, he verbally stated that he relied on the limiting process as well as whether he
was taking the chalk off the board when determining the continuity of a given function
from a graph. He then introduced the “precise” definition of continuity (See Table 5.17

and Figure 5.8).

124

 

 

 

 

“here;
limit it:
”Unit
Given 1
Were In

pTClpOn.

so, u

Table 5.17: J ason‘s utterances about the precise definition of continuity

 

What is said

What is done

Type of utterance

 

[1] So let’s try to make this precise. A
function f of x is continuous at a
point x equals a if...

[2] So what I want to say is what does
it mean to be continuous at this

int?

He writes these on the board but does not

finish his sentence.

 

[3] The limit has to exist and it has to

 

 

 

 

equal to the function value. Objectified
[4] So I need the limit to exist. Objectified
[5] So the limit as x approaches a of f He writes lim f(x) = f (a) on the
of x has to exist and equal to f of x-ya Objectified
a. board (See Iiggre 5.8).
[6] Just writing this already implies
that it [the limit] exists because it Objectified
uals a number.
[7] So in particular, f 18 defined at a He wntes these besrdes the definition in Objec ti fie d

and the limit exists.

parenthesis.

 

 

Figure 5.8: Jason's precise definition of continuity (hand-written)

Whereas Jason’s informal definition of continuity was based on dynamic motion where

limit was considered as a process, his precise definition of the notion considered limit as

a number (Table 5.17, [6]), which is equal to the function value (Table 5.17, [3], [5]).

Given this, in context of the precisell definition of continuity, his utterances about limit

were mainly objectified. In fact, Lesson 7 was the lesson that had the second highest

proportion of objectified utterances about limit (See Figure 5.3).

 

10 . . .
Note that Jason dld not use any limit related utterance here. Therefore, the types of utterances were not

reported.

125

prior to
motion.

COllIlCCl

It should be noted that Jason did not talk about continuity in any of the lessons
prior to Lesson 7 although his operational word use about limits relied on continuous
motion. The precise definition of continuity was the first context in which he explicitly
connected these two concepts. Later in the class, he also said

How can I tell if a function is continuous or not? A question about

continuity is a question about limits. If I ask to see if a function is

continuous, we need to check that for each point a the limit exists and

equals to the function value. (Jason, 28 January, 2009, Lesson 7)

After this, he talked about properties of continuous functions, which he noted would be
the same as limit laws because “the definition of a continuous function is given in terms
of the limit” (Jason, 28 January, 2009, Lesson 7).

Jason talked about plugging in12 as one of the possible ways of computing a limit
in the lessons prior to Lesson 7. In Lesson 7, however, he seemed to put extra emphasis
on plugging in as a means of computing the limit of a continuous function. For example,

x+
x2+1

 

while showing f (x) = is continuous, he showed for every numbera ,

lim f (x) = 612+] = f (a) and mentioned that he found that limit by plugging in. He then

x—M a +1

 

turned back to the precise definition of continuity and described the relationship between

continuous functions and plugging in as illustrated in Table 5.18.

 

11 Jason did not introduce the formal definition of continuity that involves 6‘ and 5 . Therefore, the
mathematical definition he gave was referred to as “precise” instead of “formal” to be compatible with
Jason’s own description of the definition (See Table 5.17, [1]).

126

 

1m

m Co:

”able

Table 5.18: The relationship between computing the limit of a continuous function and

 

 

 

 

 

 

 

 

pluggingin
What is said What is done Type of
utterance
' ° ° . a + l
[1] In fact, we found this limit by He shows 11m f(x) 2 2 = f( a) Objec ti fie d
plugging 111- x—m a + 1
[2] A continuous function is a
fimction where we can always Objectified
find limits by pluggi_ng in.
a) That is really what this He goes back to the precise definition of
definition says. limit he wrote on the board (See Figure 8xx)
[3] f is continuous if you can find the He til-St shows lim f (x) and then f(a) in . .
limit at a b lu in in x—aa Objectlfied
y p gg g ° the definition.
a) That is exactly what the
definition says.
[4] In other words, function f is
contrnuous at a If we can find the He writes these on the board. Objectified

limit of f at x equals a by plugging
in.

 

a) These are great functions.

 

[5] Continuous functions are exactly
the ones that we can find the limit Objectified

by plugggg in.
Jason did not mention plugging in when he introduced the precise definition of continuity

x+l
x2+

 

(see Table 5.17). However, after showing that the function f (x) = is continuous,

he revisited this definition and explained how plugging in was embedded in the definition
(Table 5.18, [1-5]). Some of his utterances in this context also exemplified Jason’s word
use when referring to the limit notation in the context of continuity. He used “limit at a ”

(Table 5.18, [3]) or limit of the function “at x equals a ” (Table 5.18, [4]) when referring

to lim f (x) . In the context of continuity, Jason only used the words approaches (the
x —>a

limit as x approaches a of a filnction) and at (the limit of a function at the point a) when

 

12 . . . - . . . .
In the context of evaluatrng a limit 11m f (x) , plugging in refers to evaluating the fiinctlon value at the
x—>a

point a, which would be equal to f(a). So plugging in can be thought of as inserting the number a into the

127

talking at
”getting c
discussed
first need
the limit ‘
hehaxior
hith laso
found by
lllt limit 1
in the lim
in Sectior
CQHIanit}

checking

He.

; n _ .
T+JL JC) D "f. b
i: J v4}

SI tr]
L‘Eents o

talking about the limit notation. He did not refer to x —> a in the limit notation as x values
“getting closer and closer to”; “going to” or “becoming” a as he did in other contexts
discussed before. Note that to determine the continuity of a function at a given point, one
first needs to find the limit of the function as x ‘approaches’ that point and then check if
the limit value is equal to the function value ‘at’ that point. In this respect, the function’s
behavior at the limit point is relevant to the discussions about continuity. This, together
with Jason’s explanation about continuous functions as the functions whose limits can be
found by plugging in might be the reason why he used approaches, which is how to read
the limit notation, together with the word at in his discussions about continuity. Plugging
in the limit value to the function is one of Jason’s routines and will be further discussed
in Section 5.3.1. The student difficulties “limit as the function’s value” and “limit implies
continuity” (See Table 2.1) are tied to the routine of plugging in since they result in

checking the function value at the limit point to determine its limit. Students might

generalize Jason’s routine of plugging in for continuous functions to functions in

general”. However, Jason also explicitly mentioned that, in general, the function’s value
at the limit value is not related to the limit of the function at that point both in this lesson
and in the preceding lessons:

To compute the limit as x approaches zero, what the function does at the

point x equals zero is irrelevant. What only matters is what happens

nearby. (Jason, 28 January, 2009, Lesson 7)

 

function f(x). .
Students’ discourse with respect to the instructor’s discourse will be discussed in Chapter VI].

128

Jasor

Sllmr.

l'tter
ultim
Intuit
reach

Consi

Word
insrn
line,
limit

“but

The function value is something completely different. And here we see

something again that the actual value of the function at the point where we

are looking at has very little to do with the limit. In fact, it has nothing to

do with the limit. (Jason, 23 January, 2009, Lesson 5)

Jason did not use any infinitesimal related word in the context of continuity.
Summary of word use

The limit, if it exists, is an obtained value at the end of the limiting process.
Utterances that consider limit as a process, on the other hand, do not emphasize this
ultimate result as the limit. Instead, through the use of words that signify continuous
motion, it treats limit as a process that goes on and on and therefore never actually
reached. Commognitive framework enables the exploration of this issue through
considering objectification in the instructor’s word use.

The analysis of Jason’s discourse on limits reveals the existence of shifts in
objectified and operational word use in the contexts of informal definition of limit;
computing limits, infinity and continuity. The context in which his utterances were
consistently objectified was the formal definition of limit. Although the majority of
Jason’s utterances about limits were objectified (See Table 5.5 and Figure 5.3), shifts in
word use in a given limit related context might be problematic for students”. The
instructor can move flexibly among these utterances and distinguish their similarities and
differences depending on context but students might not yet be able to participate in the
limit related discourse in similar ways. Therefore, switching between the operational and

objectified word use, and treating limits as end products and as processes in the same

129

context
unreacl
Values

can get

u‘hethe
nonon

instruct
HOWeV
such m
Potenri.
“Want
infinity
We SEE
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Contribl

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context, could contribute to one of the common student difficulties with limit, “limit as
unreachable” (See Table 2.1). Talking about limit as a process in which the function
values “approach”, “get closer and closer to” or “become” a number L suggests that one
can get close to the limit but cannot reach it (See Section 2.1.2).

A similar issue arises in Jason’s discourse on infinity. We see that, depending on
whether he computes a limit at infinity or he works on infinite limits, how he refers to the
notion changes between actual (end-state) and potential infinity (process). Again, as the
instructor, he is able to work with different realizations of infinity depending on context.
However, students are unlikely to do that since they are relatively new participants to
such mathematical discourse. On one hand, we see Jason talking about infinity as
potential, especially in the context of referring to x —> oo in the limit notation. His
utterances about the x values “getting larger and larger”, “going to” or “becoming”
infinity consider infinity as an ongoing process without an end state. On the other hand,
we see him talking about infinity as actual in the context of infinite limits and when he
writes the limit of a particular function as being equal to infinity. His utterances about the
f(x) values being an “arbitrarily large number”, “very large number” are examples in
which he talks about infinity as an actual number. The instructor’s shifts in word use in
the context of infinity may result in students’ realization of infinity as a process and
contribute to their realization of “limit as unreachable”. It is also possible that, based on
Jason’s utterances on limit where he talks about infinity as a number might result in

students’ consideration of infinity as a finite number similar to the findings of Sierpir’rska

(1987) (See Section 2.1.3). It should be noted, however, that Jason explicitly mentioned

 

l4 . . . . . . . . . . .
Student thlnklng about llmlts in relation to the lnstructor’s discourse Will be dlscussed in Chapter VI.

130

infinity

is undel

sake of
manipl
Visual r
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Limbo

53'me
limit]

Wise;

infinity is not a number, and stated that if the limit of a function is infinity, then the limit
is undefined.
5. 2. Visual mediators

Visual mediators refer to the visible objects created and operated upon for the
sake of communication. In mathematics, visual mediators can take many forms such as
manipulatives, tables, figures, diagrams, graphs, symbols, etc. There were four types of
visual mediators identified in Jason’s discourse as he communicated with his students: (a)
written words, (b) drawn pictures of geometrical shapes, (c) graphs, and (d) mathematical
symbols. Among these, graphs and symbolic notations were more common means of
visual communication.

5.2.1. Written words

Written words correspond to what Jason wrote on the board besides mathematical
symbols when he talked about limits. The analysis of his word use was conducted in the
previous section. What will be highlighted here is the difference between his written and
spoken words in particular contexts.

It was mentioned in the preceding sections that Jason used a combination of
operational and objectified utterances in the same context (such as the informal definition
of limit and computing limits). The analysis also revealed that his operational utterances
mostly occurred when he used words only verbally whereas his objectified utterances
often occurred when he wrote words on the board. For example, in the context of
informal definition of limit, Jason wrote the following on the board:

f is defined on an interval about x0 (except possibly at x0) then

131

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lim f (x) = L if f (x) gets arbitrarily close to L for all x sufficiently close
x—->x0

to x0 (See Figure 5.4).

Here, Jason mentioned the function values getting “arbitrarily close” to the limit value as
the x values get “sufficiently close” to the limit point, and so used the words in an
objectified manner. When he explained what this definition meant verbally, on the other
hand, he mentioned the function values “approaching” or getting “closer and closer” to
the limit value (Table 5.6, [4], [5]) as the x values got “closer and closer to” the limit
point (Table 5.6, [4]). By doing so, he used the words in an operational manner. In fact,
this difference between the words uttered and written can also be seen in the context of
computing limits (See Tables 5.9, 5.10, 5.11 and 5.13, what is said and what is done) and
in the context of continuity (See table 5.16, [2]). In those instances, Jason used words
operationally when he communicated his ideas verbally and when he talked about the
graphs that he drew, whereas he used them in an objectified way when he wrote the final
arguments on the board. The context where his written and spoken words were most
consistent with each other was the formal definition of limit. Note that, in that case, he
wrote and spoke at the same time (See Table 5.8, what is said, what is done) and his word
use was objectified throughout his explanations.

In summary, there seems to be a difference in word use depending on the means
of communication Jason chose. When he presented his ideas visually, writing words on
the board, he was more formal. When he further explained ideas related to limit verbally,
he was less precise and less formal. Moreover, he rarely, if ever, wrote on the board any
of his operational utterances. This signaled a difference in the mode of endorsement of

narratives about limits, which was likely to be a result of Jason’s alternating positions in

132

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mathematie
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the classroom. On one hand, being a mathematician, he was careful presenting the
mathematical ideas accurately. On the other hand, as an instructor, he was aware when
students did not follow his arguments. It is often in these instances that he was more
flexible in his word use and relied more on informal utterances (as well as graphs) to help
his students.
5. 2. 2. Drawn pictures of geometric shapes

There were only three occasions where Jason drew pictures of geometric shapes
to explain the related concepts. In Lesson 1, he drew a falling rock when he talked about
average rate of change. In Lesson 5, he drew the unit circle and formed triangles within
the circle. He later used the similarity of these triangles to find the boundary functions for

sin (9

 

before computing (Slim %€. Finally, in Lesson 7, he drew the unit circle again,
—>0

this time to talk about the fact that sin 9 and cos6 are continuous. Besides these, he did
not draw any diagrams or figures but frequently used graphs of fimctions.
5. 2. 3. Graphs

Jason used graphs of functions as be communicated his ideas about limits. He
drew graphs in three different settings: (a) when he computed the limit of a fimction, (b)
when he explained a particular definition, theorem or fact about limits, and (c) when he
solved a problem that specifically asked to draw the graph of a given fimction.

In the context of computing a limit, Jason utilized graphs in two ways. He
sometimes determined limits of functions directly fiom the graphs that he drew without
utilizing any algebra-based technique to compute the limit. He also used them for further
explanation or clarification after computing a limit algebraically. When Jason only used
graphs to determine the limit of a function, his approach was considered as graphical.

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When he initially solved a limit problem by means of algebraic manipulations and then
drew the graph to support his arguments, his approach was considered both algebraic and
graphical. When Jason only relied on algebraic manipulations when computing a limit,
his approach was identified as algebraic. Table 5.19 summarizes these approaches

throughout the eight lessons in the context of computing limits.

Table 5.19: Types of visual mediators used in the context of computing a limit

 

 

 

 

 

 

 

 

 

15 Graphical Algebraic Algebraic and Number of graphs

Lesson Number approach approach graphical approach drawn
Lesson 2 6 3 l 3

Lesson 3 l 10 O 1

Lesson 4 0 l 2 2

Lesson 5 8 6 l 3

Lesson 6 l 9 1 2

Lesson 7 O 6 O 0

Lesson 8 0 8 O 0

Total 16 43 5 12

 

Note: The number of graphs Jason drew in the context of computing a limit is less than the number of
instances be relied on a graphical approach while computing limits. This results from the fact that Jason
sometimes computed multiple limits using a single graph he drew. Note also that the nonexistence of
graphs in Lesson 7 and Lesson 8 in the context of computing limits does not mean Jason did not use any
graphs in those lessons; it just means he did not use graphs while computing limits.

Jason used a graphical approach for 16 of the 64 limit computation problems that he
worked on. He used a combination of algebraic and graphical approach for five problems.
In those cases, the graphs were drawn only after the limits were initially computed by an
algebraic method. The remaining 43 problems were solved using only an algebraic
approach”. Therefore, in the context of computing a limit, Jason’s primary visual

mediators were symbolic rather than graphic.

 

Jason did not compute any limits in Lesson 1 (See Table 5.1 for the topic outline)

l6 . . . . . .
Some of the common algebraic approaches Jason used Will be discussed in the section about routines.

134

theorem
graph of
througho
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terms.

limits su
introduc
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ideas he

Besides computing limits, Jason also used graphs when explaining a definition,

theorem or fact about limits; and when solving a problem that specifically asked for the

graph of a function using dominant terms1 7. There were 21 such graphs identified
throughout the eight lessons. 20 of those were utilized to explain limit related facts and
one was drawn as the solution of a problem about graphing a function using dominant
terms.

In the cases where Jason utilized graphs to explain the endorsed narratives about
limits such as definitions and theorems, the graphs were mostly drawn when he
introduced them for the first time. In the remaining instances, Jason drew graphs for
further elaboration after realizing the students were not clear about the mathematical
ideas he communicated previously. Table 5.20 shows some examples of the graphs drawn
in the context of introducing/explaining/elaborating on a limit related idea.

Most important for the purposes of the study is the graph Jason drew when he

proved lim 2x —1 = 3 (See Table 5.20, the last example). He worked on this problem

x—>2
right after he introduced the formal definition of limit and initially solved the problem
using only the algebraic approach that involved a and 6 . The students were mostly
silent and asked some clarifying questions about the algebraic solution of the problem. It
was only then Jason started drawing the graph as “another way that we can play this

game about limits” (Jason, January 21, 2009, Lesson 4).

 

17 Graphing a function using dominant terms refers to graphing the fianction’s behavior at positive and
negative infinity (horizontal asymptotes) as well as its behavior at the points where it is undefined (vertical
asymptotes). This requires the evaluation of the limit of the function at positive and negative infinity as
well as at the points where the function is undefined while graphing. Jason told students they do not have to
draw the graph accurately except for the dominant terms since drawing a precise graph of a function would
be discussed during the chapter on derivatives.

135

Table 5
limits

 

Table 5.20: Examples of Jason’s hand-drawn graphs besides the context of computing

limits

 

Context of use

  

Graph drawn

Average rate of
change

Purpose

“To think about the
[difference] quotient
geometrically" (Jason,
January 12, 2009).

“To represent speed as the
slope of the secant line"
(Jason, January 12, 2009).

 

Informal definition
of limit (See Figure
5.4)

To explain what it means
for a function to be defined
near x0.

 

Sandwich theorem

To explain the theorem
visually.

 

Intermediate value
theorem

To explain the theorem
visually.

 

Continuity

To introduce the
mathematical definition of
continuity based on the
intuitive aspects of
continuity.

 

Proving that
lim 2x —1 = 3
x—)2

To elaborate on students”
confusion about the
algebraic solution of the
problem. Jason presented
this as another way to think
about the problem.

 

 

Note th
(See Pi,
ti eorre
most us
limit cc
5.3.4. 5
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Note that Jason did not represent the formal definition of limit visually when be defined it
(See Figure 5.5), so this was the only instance in which he addressed what a and
5 corresponded to on a graph. As will be discussed later, this graph turned out to be the
most useful mediator with which one of the participants in the study made sense of the
limit concept.
5. 2. 4. Symbolic notation

Mathematical symbols were the primary visual mediators Jason used both in the
context of computing limits (See Table 5.19) and also in other limit related contexts. He
solved most of the limit computation problems by means of algebraic manipulations and
represented definitions, theorems and facts about limits using mathematical notations
consisting of symbols (as well as written words). The characteristics of the most common
algebraic approaches Jason used will be discussed in the following section about routines.
Here, the limit notation and how Jason addressed the notation will be examined.

It was mentioned in the section about word use that Jason introduced the notation

lim f (x) = L which he addressed as the limit as x approaches a of f of x equals some
x—>a

number L (see Table 5.6, [2]). In fact, he used the word approaches for about 70% of the
197 instances where he attended to the notation throughout the lessons. It was noted,
however, that he also used other words besides approaches when he talked about the
limit notation. During the discussions on Jason’s operational word use (see Section
5.1.2), he was reported to refer to x —> a in the notation also as x “gets closer and closer .
to”a (Table 5.2, [3], [4]) and as x “goes to” a, (Table 5.4, [2-4]) when a was a real

number. Jason also used the word “at” ten times when he attended to the limit notation,

137

where a a
continuity

ln
”goes to"
positite‘n
hand. he I

5.14).

where a was a real number. Five of those utterances took place in the context of
continuity (See Table 5.18).

In the context of limits at infinity, he referred to x —> oo in the limit notation as x
“goes to” infinity18;x values “getting/becoming larger and larger” as well as “limit at

positive/negative infinity” (See Table 5.12). When he wrote lim f (x) = ioo , on the other
x-)a

hand, he talked about the limit as “equal to” or “being” plus/minus infinity (See Table

5.14).

In both x —) 00 and lim f (x) = ioo , the symbol 00 signified infinity. In Jason’s

x—>a
discourse, however, 00 in x —-> 00 was often associated with potential infinity whereas 00

in lim f (x) = too was often associated with actual infinity (see Section 5.1.5.3.2 and
x——)a

Section 5.1.5.3.3). So the same symbol 00 was used to signify two different aspects of the
concept of infinity. In the context of limits at infinity, the symbol indicated an infinite
process; in the context of infinite limits, the same symbol indicated the end result of the
process.

To summarize, although both Jason and the textbook read the notation using the

word approaches“), the arrow in the notation lim f (x) = L signified a family of words
x—->a

9, 66 99 6‘

such as “approaches , goes to ,

99 6‘

at”, “becoming , getting closer and closer to” in

Jason’s discourse on limits.

 

18 . . -
In Jason’s discourse, eight of the eleven occurrences of the phrase “goes to” took place In the context of
limit at infinity (see Section 1.4, Table 5.4).

9 , . ,
Mathematically, we read 11m f (x) = L as the functronflx) approaches the limit L as x approaches a
x —) a

(Hughes-Hallett et al., 2008; Thomas et al., 2008).
138

Summary

ll
discourse
notation a
signifies '
word “211'
function
talueL c
relire-sent
that DIOC
the prodt

H

li‘Oh ref

Simbol i

Summary of visual mediators

The different ways Jason addressed the limit notation is another aspect of his
discourse that‘might impact student learning. Referring to the arrow sign in the limit
notation as “approaching”, “going to” and “getting closer and closer to” the limit point
signifies the process of moving towards the limit point without reaching it whereas the
word “at” would signify reaching the limit point. In addition to this, talking about the
function values “getting closer and closer to”, “approaching” or “becoming” the limit
value L considers limit as an ongoing process whereas the limit of the function being
represented as equal to L in the notation considers it as a number obtained at the end of
that process. Given this, it might be difficult for students to distinguish the process from
the product in cases when Jason attended to the limit notation.

Having a family of words associated with the same symbol also occurred when
Jason referred to infinity in the limit notation. In the context of limits at infinity, the 00
symbol in the limit notation signified potential infinity, that is infinity as a process, when

,9 “

Jason addressed it as the x values “getting/becoming larger and larger , approaching” or
“going to” infinity. In the context of infinite limits, however, the same symbol was
associated with actual infinity, an end-state, when he talked about limit as being “equal
to” infinity. Therefore, the way Jason attended to the limit notation contributed to his use
of mixed utterances about limits in the same context.

Jason did not rely mainly on graphs when he determined the limits of functions as
the literature on learning about limits suggested (See Section 2.1.3). He used graphs more

often to introduce a definition, theorem or fact about limits than to compute limits of

functions. In fact, he only drew 12 graphs for the 64 limit computation problems that he

139

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worked on. He used graphs as an aid for teaching the main ideas related to limit and for
addressing students’ confusions about the algebraic solutions of the problems than
making sense of every function by means of graphs, but his primary mode of
representation was symbolic, not graphical.

Jason used a variety of functions in his discussions on limits and drew a variety of
graphs for those functions. In this respect, he represented functions both algebraically and
graphically but he did not use a tabular representation. He gave examples of continuous
functions, constant functions as well as functions with removable and jump
discontinuities. Moreover, he used not only polynomials but also rational fimctions and
trigonometric fimctions as he computed limits. This finding is not consistent with the
research on student learning arguing that instruction heavily relies on continuous
fimctions (like polynomials) and graphs while teaching limits (Bezuidenhout, 2001;
Parameswaran, 2007).

5.3. Routines

It was mentioned in Section 2.2.1 that metarules describe the patterns of the
discursants as they construct and substantiate object-level narratives, that is, narratives
about the objects of mathematics. Routines refer to the set of metarules that describe
repetitive actions of the discursants. Sfard (2008) distinguishes between how of a routine
from when of a routine. How of a routine can be thought of as the “course of action or
procedure”, whereas when of a routine refers to the instances “in which the discursants
would deem this performance as appropriate” (Sfard, 2008, p. 208, italics in original).
The when of a routine can be fiirther split into two conditions: applicability and closure.

Applicability conditions characterize the “circumstances in which the routine course of

140

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action is likely to take place” (Sfard, 2008, p. 209). Applicability conditions also help
determine the routine prompts, which refer to particular aspects of situations that are
likely to trigger the application of a routine. Closure conditions characterize the
“circumstances that the performer is likely to interpret as signaling a completion of
performance” (Sfard, 2008, p. 209).

In what follows, the routines identified in Jason’s discourse that are relevant to his
word use and visual mediators will be presented with a focus on the prompt(s) as well as
the how and the when of the routines. It should be noted that reporting a thorough list of
the instances in which a particular routine is likely to appear (when of a routine) is
complex, “if not altogether unworkable” (Sfard, 2008, p. 209). Given this, when of the
routines described in the study should, by no means, be considered as a complete list of
the circumstances in which the routine takes place. Instead, they represent the most
clearly identified instances when Jason performed a specific course of action. There were
four types of routines prominent in Jason’s discourse on limits: algebra-based routines,
geometry-based routines, using the metaphor of continuous motion, and using the
metaphor of discreteness.

5. 3. I Algebra-based routines

Algebra-based routines refer to the algebraic techniques Jason utilized when
computing the limit of a function. They constituted the main procedures with which he
substantiated the narrative that limit is a specific number, if it exits. Table 5.21 shows the

list of the algebra-based routines.

141

 

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Table 5.21: Algebra-based routines in J ason's discourse

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Prompt Routine How When (Applicability) When (Closure)
The denominator is non- . , .
The function’s zero at the 11m1t pOint. l‘hhe 111:!“ :18 equal to L.

[l] Plugging in value at the limit The fiinction is continuous. exiZt/is utn (1:312?

PO“1t 15 evaluated. The limit is computed at a '
point.

[2] Cancelling Elbe numerator and The numerator and the The limit is equal to L.

e denominator are . . . .

out the . . . denominator of a rational The lirmt does not

Simplified and their . . .
common function are both zero at exrst/rs undefined.
factor common factors are the limit oint

cancelled. p '

The function is composed
. . of basic expressions whose

Laws for the lrmrt limits are computed in the The limit is equal to L.

[3] Using limit 0.” sum, class The limit does not

1 difference, product ' . . .
aws . . Can be applied only when exrst/is undefined.
or quotient is . . .
. the limit of each basrc
applied. .
expressron that make up
the function exists.
ihednumerator and Egrggiggogfcggfiéfilan Expression is of the
[4] Multiplying e .enominator are form aZ-bflx)
b the multiplied by the (a i (lbf(x)) and the . . . '
Y _ conjugate of the The limit is equal to L.
coniugate - - numerator and the The limit does not
expression In the denominator are both zero . t/' d fi (1
Compute denominator. at the limit point. exrs ls un e ne '
the limit The expression .
ofa [5] Getting rid |u(x)| is convened to The function whose limit is “some value 5‘9“
. . are removed.
function of the u(x) or —u(x) taken contains an . . .
. . . The limit is equal to L.
absolute depending on the expressron in absolute . .
value sign sign of u(x) near the value. “1.8 limit does not
limit point exrst/is undefined.
The numerator and
_ _ the denominator Of The limit is computed at , , ,

[6] Divrding by a rational function infinity and the rational The limit is equal to L.
the largest are div1ded by the . _ 00 The limit does not
power largest power of the function 13 0f the form —- exist/is undefined.

polynomial term in °°
the function.
A limit at negative infinity
[7] Substitution A different variable 150::?::i;t:gnthya 1mm at The limit is equal to L.
c an is assrgne or some . . e 1mit oes not
h ging - . df IWhenworkin with Th1. id
variables) part of the function. . . g . exist/ is undefined.
trigonometric functions
such as the sine function.
Long division is The limit is computed at
performed on a infinity. . . .
rational function The degree of the $th limit ls equal to I"
. . e 11m1t does not
[8] Long where the degree of numerator is higher than . .
. . . . exrst/is undefined.
lelSlOIl the numerator ‘5 the degree 0f the The function ets close
higher than the denominator. to a line g
degree of the When finding the oblique '
denominator. asymptote of a function.

 

142

 

Howe
funct
absol
by ca
piobl
prob]
they .

limit

dld n
limit
0f pa

dEnQ

plug
26m.
a Sin

Sim}:

Them

There were eight distinct routines that were classified as algebra-based routines.
However, this does not mean Jason only relied on one while determining the limit of a
fimction. For example, routines such as multiplying by the conjugate, getting rid of the
absolute value sign, dividing by the largest power, and substitution were often followed
by cancelling out the common factor and/or plugging in for the same limit computation
problem. Therefore, Jason sometimes used a combination of routines depending on the
problem. Since all those routines took place in the context of computing a limit, however,
they all had the same prompt and similar closures such as ‘the limit is a number’ or ‘the
limit does not exist/ is undefined’ (Table 5.21).

The algebra-based routines [2-8] (Table 5.21) often occurred when plugging in
did not work. When computing limits in the class, Jason’s first attempt was to plug in the
limit point to the functions formulated algebraically. He also mentioned that in the case
of particular functions, such as continuous functions and for the functions whose

denominators are non-zero at the limit point, one could compute the limit directly by

2

he said “the denominator is not

 

plugging in. For example, when computing lim
x—)1 x—

zero. So we can just find the limit by plugging in” (Jason, 16 January, 2009, Lesson 3). In

a similar fashion, for (Slim Eli , he said “so here, we cannot find the limit by plugging in
—->0

simply because both the numerator and the denominator are zero. So the denominator is
zero; that means we cannot find limits by plugging in” (Jason, 23 January, 2009, Lesson

5). Although Jason considered infinity not as a number (see Section 5.1.5.3.3), he also

2x2 -3x3 +5

2 3 at infinity: “What this

 

mentioned plugging in when he was finding lim

x—>°°4x +3x—x

143

mean
why
J anu
routi

limii

(J.
W

means is we want to formally plug in infinity here. We cannot really do this. So that is
why we come up with some way to write the limit as x approaches infinity” (Jason, 23
January, 2009, Lesson 5). In the context of computing a limit, many of the algebraic
routines (Table 5.21, [2-8]), therefore, were introduced as alternative procedures to find a
limit when the initial routine plugging in did not work.

5.3.2 Geometry-based routine (Graphing)

It was mentioned in Section 5.2 (visual mediators) that Jason used drawn pictures
of geometric shapes and graphs as visual mediators, which both convey a limit-related
idea geometrically. For the purposes of this section, however, drawing pictures of
geometric shapes was not considered a routine since it occurred only three times (see
Section 5.2.2) throughout the eight lessons and their use did not seem to follow a
repetitive pattern. Thus, only graphing will be considered as a geometry-based routine in
Jason’s discourse, whose characteristics in terms of when and how are summarized in
Table 5.22.

The somewhat apparent purpose of graphs is to provide visual aids for the
communication of mathematical ideas. Graphing, as a routine however, helps us gain
more information about the instances where Jason needed such aid in his discourse on
limits. In that respect, the when and how of graphing also inform us about the diverse
reasons Jason used graphs.

Unlike the al gebra-based routines, graphing was a routine that was initiated by

various promptszo. Besides its utilization in the context of computing the limit of a given

function, it was also used to determine the behavior of a function, introduce or explain

144

theor:
classr
panic
situai

graph
the di

Table
\P

\

Comp,
buntt
functi.

Deteit
helm
mach

\

Gram
lull] g

Add.
midi
cont

3.

.ii

theorems and definitions about limits and to address students’ confusions in the

classroom (Table 5.22). There were also instances in which the problem Jason worked on

particularly asked that graphs of a function be drawn. Although the closures of all these

situations usually ended with the graph that was intended to be drawn (Table 5.22),

graphing was used to substantiate a variety of narratives rather than a single one due to

the different prompts that triggered the application of this routine.

Table 5.22: Graphing as a geometry-based routine in Jason’s discourse

 

 

 

Prompt How When (Applicability) When (Closure)
Graphs of arbitrary The limits are to be
. determined only from the . . .
Compute the functions are drawn. graph drawn The limit is equal to L. .
. . . ' The limit does not exrst/is
11m1t ofa Graphs ofparticular The limit is com uted at undefined
function functions formulated p '

algebraically are
drawn.

infinity or the function is a
trigonometric firnction.

 

Determining the

Graphs of particular
functions formulated

The function is a

 

bfifnhgggf ofa algebraically are trigonometric fiinction. The graphs are drawn.
drawn as visual aids.
Graphs of functions The graphs are drawn with a
. are drawn after When the question particular focus on the
5:12}; h (1:21.122? determining their explicitly asks a fiinction functions’ behavior at
horizontal, vertical formulated algebraically to positive/negative infinity and

EMS

 

 

 

 

and oblique be graphed. the points where the function is
asymptotes. undefined.
Informal definition of limit,
. sandwich theorem,
Egocfiugffairfin intermediate value theorem. The graphs are drawn and the
theorems/p g Graphs of arbitrary The graphs are either drawn theorems/definitions are
definitions about functions are drawn. before or in conjunction completely stated or written on
limits with the symbolic notation the board.
used for the
theorem/definition.
Graphsof arbi . 3 Proving an explanation that
. or specrfic functions . . . The graphs are drawn.
Addressrng are drawn to a limit 18 equal to a number.
students’ . .
confusions exemplify the . When the students are Sllent Students confirm they follow
instructor 5 preVIous or ask questions afier an . ,
. the instructor 3 arguments.
arguments. explanation.

 

 

20
See Section 5.2.3 for further details about the contexts in which Jason utilized graphs.

I45

For ex

for I:

x-
hincf
exits.
and E

13501

 

For example, when graphing was used to compute the limit of a function, as was the case
for 11m — = 0 (see Figure 5.7), it was used to substantiate the narrative ‘the limit of the
x—>oo x

firnction one over x as x approaches infinity is equal to zero’ or possibly ‘the limit (if it
exits) is a nurnber’ in general. On the other hand, when graphing was used to introduce
and explain a theorem about limits, say the intermediate value theorem, the narrative that
Jason endorsed was follows:

If f is continuous on [a, b] then f assumes every value between flu) and

f(b). In other words, if yo is between f(a) and f(b), then there is a c in [a, b]

so that f(c) =y. (See Figure 5.9)

 

Figure 5.9: Jason’s introduction of the intermediate value theorem (hand-written)

5.3.3 Using the metaphor of continuous motion

This routine refers to Jason’s use of the metaphor of continuous motionZI in his
discourse. His utilization of this metarule remained implicit in the class since he did not
explicitly mention he was using a particular metaphor when talking about limits. Instead,

this routine emerged from his use of words signifying motion (See Figure 5.1) in the

146

cont

rout

rela

135:

I851

(ml

the

h—a

context of informal definition of limit, computing limits, and continuity as well as his

routine of graphing. Note that using words signifying motion when talking about limit

related concepts support their realization as a process and, therefore, is closely related to

Jason’s operational word use. Another reason this routine remained implicit for students

results from the fact that Jason’s operational utterances occurred when he used words

only verbally whereas his obj ectified utterances took place when he wrote the words on

the board (See Section 5.2.1). In other words, he did not make his words signifying

motion visible to students. Table 5.23 shows the characteristics of the routine of using

dynamic motion as a metaphor in Jason’s discourse.

Table 5.23: Jason’s routine of using the metaphor of continuous motion

 

 

 

 

 

 

Prompt How When (Applicabiljy) When (Closure)
Words signifying motion Determining the behavior
(See Figure 5.1) when of a function at the limit
referring to x values, point before writing its . . .
infinity and function limit as equal to L on the (Referring to hunt) The
. firnction values approach/get
values are uttered (not board.
Compute the written) closer and closer to/tend to L.
“mm” ' Rfrrin or and
function e e. . g “1.th . . (Referring to infinity) The
. . . . the lirnit value in the limit .
Words Signifying motion . function values get/become
. notation
are uttered (not written) larger and larger/ smaller and
as/aliecr) 3:11;”ng {gen When the students are smaller.
gr ap c o ' silent or ask questions after
an explanation.
Words signifying motion “The function value, if I get
when referring to x and . . . . closer and closer to x zero, it
. Informal definition of ham
Intr d . the function values are should approach some number
mom” uttered (not written). L” (Table 5.6, [41).
as ect of Words signifying motion Informal definition of “A continuous function is a
l “:1 ts are uttered (not written) continuity function, and I am just going to
as/afier drawing the When the students are say it in words, that I can graph

graph of arbitrary
functions.

silent or ask questions after
an explanation.

without taking the chalk off the
boar ”(Table 5.16, [2]).

 

 

2]

Using a metaphor consistently in a given discourse is a type of metarule (See Section 2.2.2).

147

5.3.4 Using the metaphor of discreteness

This routine took place only when Jason talked about the formal definition of limit22 and
a proof problem and refers to the elimination of words signifying motion in his discourse.
In the metaphor of discreteness, he used words signifying proximity (See Figure 5.1)
through distance. Recall that Jason’s word use was consistently objectified in the context
of the formal definition of limit (See Section 5.1.5.2). His utilization of this metarule also
remained implicit in the class since he did not explicitly mention he was using a
metaphor when talking about the formal definition. Table 5.24 shows the characteristics

of the routine of using discreteness as a metaphor in Jason’s discourse.

Table 5.24: Jason’s routine of using the metaphor of discreteness

 

 

 

Prompt How When (Applicability) When (Closure)
Words signifying “. ..whenever x is sufficiently
proximity (See Figure F ormal definition of close to x zero so that means that
5.1) when referring to the limit the difference is no more than
Intro ducin x and firnction values are delta then the function values
g uttered (and written) (See Proving that the limit of should be close to L” (Table 5.8,
formal aspect . . .
. . Figure 5.5). a function is equal to a [14]).
of limits ...... .
number usrng the formal
Symbols instead of definition of limit “. . .if x is sufficiently close, delta
graphs that signify motion close to x zero,. . .the function
are used. values should be close to the

limit” (Table 5.8, [18]).

 

In general, the metaphor underlying the informal definition of limit and the behavior of
the function values near the limit point is based on continuous motion, which considers
limit as a process. The metaphor behind the formal definition, however, is based on
discreteness and the elimination of motion, and therefore time, fi'om the informal

definition of limit (See Chapter III). Using distinct metaphors was present in Jason’s

 

22 . . .
Although Jason relied on this metarule in only one context, he repeatedly used the metaphor of
discreteness in that context. Therefore, his (implicit) use of the metaphor was considered as a routine.

148

di

Ct

discourse as metarules when he talked about different aspects of limits. However, these
metarules remained implicit and he did not draw students’ attention to his shifts in word
use and also on the distinct metaphors related to realizations of limit.

Summary of routines

In Jason’s discourse, algebra-based routines often resulted in objectified word use
whereas graphing and the verbal statements about limit as a process resulted in
operational word use. Jason’s routines were used to substantiate the narratives that limit
is a number; limit is a process as well as other narratives of the form of definitions,
theorems and rules.

Jason relied more on algebra-based routines than graphing in the context of
computing a limit (see Table 5.19). Being based on algebraic manipulations and symbolic
notation, algebra-based routines (Table 5.21) also helped determine the instances in
which Jason’s objectified word use took place when he computed limits. The geometric-
based routine, which is identified in the study as graphing, was more often used for
explaining a definition, theorem or fact about limits. Although Jason clearly mentioned
that the limit of a function at a point may be distinct from the fiinction’s value at that
point, his frequent use of plugging in as the initial attempt to compute the limit of a
function links the function value with the limit value in a way that could easily contribute
to confusion of the two.

Jason implicitly used distinct metaphors as metarules in his discourse on limits.
One was based on dynamic motion whereas the other was based on the static aspect of
limit. These different metaphors also supported the realization of limit as a process or a

number. Jason’s shifts in word use and his utilization of different metarules remained

149

hnpl

hifir

rehn
hecx
“rot

thei

clas

rele
end
(Sf
am

off

implicit for his students since he did not explicitly attend to these aspects of his discourse
in the class.

The analysis of Jason’s routines also revealed that his word use was closely
related to his means of communication. His operational word use took place only when
he communicated his ideas verbally whereas his obj ectified word use took place when he
wrote his ideas on the board. It was discussed in Section 5.2.1 that such a difference in
the means of communication might also have resulted from Jason’s dual positions in the

classroom (being a mathematician and being a teacher).

5. 4. Endorsed narratives

Endorsed narratives are the last discursive feature under consideration in this
study. Narrative is “any sequence of utterances framed as a description of objects, of
relations between objects, or of processes with or by objects, that is subject to
endorsement or rejection with the help of discourse-specific substantiation procedures”
(Sfard, 2008, p. 134, italics in original). Once endorsed, narratives are considered as true
and ultimately are known as mathematical facts. Some examples of endorsed narratives
of mathematics are axioms, definitions and theorems. The construction and substantiation
of narratives, however, are not uniquely defined. There is a variety of ways in which
narratives can be substantiated depending on the context and also on a person’s
familiarity with the mathematical discourse. For example, in the mathematics
community, narratives are often endorsed by means of proofs, which are mainly based on
deductive reasoning. In contrast, empirical evidence and routines such as trial and error
can be used to endorse narratives by students, especially at their initial stages of learning.

Therefore, with respect to an individual’s discourse, endorsed narratives refer to what

150

that

sub

the

We

01’

C02

VJ
b—O I
(IO

lht'

Li

of

:7.)

that person utters as true/not true or is/is not after applying particular routines when
substantiating her narrative.

In his discourse, Jason endorsed many narratives both related to limit and also to
algebraic properties such as exponent rules and performing long division. Given this,
there were many definitions, theorems and rules he addressed in the classroom. There
were also many instances in which he explicitly designated a particular sentence as true
or untrue about limits, infinity and continuity. The goal of this section is not to give a
comprehensive list of every endorsed narrative but to focus on the ones that were most
significant based on the information gathered on Jason’s word use, visual mediators and
the routines with which he substantiated those narratives.

Limit is a number

The narrative Jason most fiequently endorsed in his discourse was the description
of limit as a number. This narrative was often endorsed in the context of computing limits
as well as the formal definition of limit and was mainly substantiated by algebra-based
and geometry-based routines discussed in Section 5.3.2. The limit of a given function
existed when it was equal to a number. In fact, Jason explicitly mentioned that the limit
“exists” if it is equal to a number (Table 5.3, [7], [17]); it “does not exist” if the right
hand limit is not equal to the left hand limit (Table 5.3, [16]) or the limit is equal to
infinity (Table 5.3, [18]).

Limit is a process

Limit is a process was another narrative endorsed by Jason, though not as

frequently as limit is a number. It was mentioned in the analyses of his operational word

use (Section 5.1.2) and routines (Section 5.3) that Jason relied on dynamic motion,

151

 

nan
def]
suci
bars
to t
ex;
end
des

ht

I‘€\
Set
Illll
hit
the

DU

511

1h

graphs and verbal statements while substantiating this narrative. Jason endorsed this
narrative in the context of instantaneous rate of change; computing a limit; informal
definition of limit and also continuity. While substantiating this narrative, he used words
such as “getting closer and closer to”, “goes to” and “becoming” (See Table 5.2) that are
based on continuous motion and sometimes used graphs of functions as visual mediators
to talk about the limit as a process. There were only two instances, however, where he
explicitly mentioned the word “process” (See Table 5.2, [l], [16]). Therefore, this
endorsed narrative was mainly inferred through the elements of his discourse in which he
described a process when finding the limit of a function.
Infinity is not a number

Jason’s analysis of word use in the context of limits at infinity and infinite limits
revealed that he talked about infinity as a potentiality and as actual interchangeably (See
Sections 5.1.5.3.2 and 5.1.5.3.3). It was also mentioned that he referred to infinity as a
number in the context of infinite limits as he considered it as the end product of an
indefinite process (See Table 5.14). Moreover, he talked about “plugging in” infinity as
the initial attempt to work on limits and infinity, treating it as a distinct entity that, like a
number, can be put in the function (See Section 5.3.1). It was noted, however, that he
then mentioned one could not do that since infinity is not a number (Table 5.14, [31]).
This narrative was further endorsed when he considered infinite limits as non-existing.
Therefore, although be explicitly endorsed the narrative that infinity is not a number and
substantiated it in the context of limits at infinity, some of his word use and routines in

the context of infinite limits seemed to suggest otherwise.

152

 

llze

COll

that

eqt

exr

tha

dis

ICT

the

$14

The fimction ’s value at the limit point is irrelevant to the limit value

Plugging in was a common routine in Jason’s discourse in the context of
computing limits and continuity (See Sections 5.1.5.3 and 5.1.5.4) He often emphasized
that the limits of continuous functions could be found by plugging in, which would be
equal to the function’s value by the definition of continuity. On the other hand, he
explicitly stated that, in general, the function value is “something completely different”
than the limit value, and “has nothing to do with the limit” (See Section 5.1.5.4). In his
discourse, he substantiated this narrative through graphs and examples of functions with
removable and jump discontinuities for which the limit value, if it existed, was different
than the function value.
Summary of endorsed narratives

In all the contexts he worked on, the primary narrative Jason endorsed was that
limit is a number. His word use when talking about the behavior of the x and f(x) values,
on the other hand, considered limit as a process. Jason explicitly endorsed limit as a
process only twice in his discourse on limits. However, many of the limit computation
problems in which he described the behavior of the function values implicitly endorsed
limit as process. Therefore, although in different frequency, Jason endorsed limit both as
a number and as a process.

Similarly, Jason explicitly endorsed the narrative that infinity is not a number. On
the other hand, his word use in the context of infinite limits also showed that he referred

to infinity as a number (5.1.5.3.3). He also treated infinity as a number when he said we

2x2 —3x3 +5
2 3

 

would “want to formally plug in infinity” to find lirn (See Section 5.3.1).

x->°°4x +3x—x

153

limit

and]

by fr
Rese
Stud:

that

Prue

lflfii

11nd

Although he then said we cannot do that, he implicitly endorsed the narrative that infinity
is a number in the context of infinite limits.

To Stun up, Jason endorsed different narratives in his discourse on infinity and
limit depending on the context, which can lead to confusion whether to consider infinity
and limit as a process or as an end state.

Summary of Jason 's discourse on limits

In this chapter, I investigated one instructor’s discourse on limit related concepts
by focusing on his word use, visual mediators, routines, and endorsed narratives.
Research on learning about limits brings forward many of the conceptual obstacles
students have about limits and discusses the possible links between those obstacles and
the teaching of calculus (See Section 2.1.4). This research adds to the body of literature

by particularly focusing on the teaching of limits. In what follows, I summarize in

general terms what we have learned about one instructor’s discourse23 on limits while
teaching the notion to beginning-level calculus students.

Jason’s word use on limit and infinity revealed that he referred to them as both a
process and an end state depending on the context. Although he flexibly uses limit and
infinity as a process or product and distinguishes the characteristics of each realization

depending on the context, students might be unlikely to notice the characteristics

underlying these differences”. This might result from two factors: (a) Jason did not make

the instances where he shifted his word use from operational to objectified (and vice

 

2 . . . . . .
3 For more detailed results in regard to word use, Visual mediators, routines and endorsed narratives, see
the summaries at the end of the sections 5.1, 5.2, 5.3, and 5.4, respectively.

24 . . . . . . . .
The results of the diagnostic survey and the mtervrews given to students at the end of their instruction
will be discussed in the next chapter.

154

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under
hunts

reahz

possi
niau

consi

COng
dEVe

Ufilfi

versa) explicit in the classroom, and (b) given that this was the first time the majority of
students were introduced to the limit notion, they were relatively new to the discourse on
limits and did not have the experience to notice the important subtleties underlying each
realization of limit and infinity.

The contexts in which Jason shifted his word use seem to support the existence of

possible links between the instructor’s discourse and the historical development of limit

related concepts”. Note that Jason consistently used words signifying motion with a
consideration of limit as a process in the context of informal definition of limit and
computing limits (See Sections 5.1.5.1 and 5.1.5.3). On the other hand, he did not use any
motion related word and did not mention limit as a process when he talked about the
formal definition of limit (See Section 5.1.5.2). Moreover, he used the metaphor of
dynamic motion in the context of the informal definition of limit whereas he used the
metaphor of discreteness in the context of the formal definition of limit. These are

consistent with the realizations of the informal and formal definitions of limits as

developed over history26 (See Chapter 111). Similar to his shifts in word use, however, the
utilization of these metaphors remained implicit for the students as he did not address it
in the class.

The analysis of the visual mediators Jason used in the class showed that he did not
rely too much on graphs in the context of determining the limit of a fimction. However,

he used graphs fi'equently when explaining a limit related definition, theorem or fact.

 

25 Instructor’s discourse in relation to the historical development of limit related concepts will be examined
in Chapter VII.

26 A detailed analysis of the instructor’s discourse with respect to the historical development of limit
related concepts will be given in Chapter VII.

155

 

The
bet
inti

ICV

lor

SH

lii

Therefore, rather than considering graphs as a basic tool with which to make sense of the
behavior of a given function, Jason mainly used them as teaching aids during the
introduction of limit related ideas. The analysis of written words as a visual mediator
revealed that Jason’s word use was always objectified when he wrote on the board. His
operational word use only took place when he communicated his ideas verbally. This
suggests a degree of precision depending on the context. Jason was more precise and
careful when he wrote ideas on the board. He was less precise and less formal when he
addressed limit related ideas about which the students were likely to be confused. Such a
shift seems to signal Jason’s mode of endorsement of limit related narratives depending
on his position in the classroom. Being a mathematician, he wanted to make sure he
conveyed the mathematical ideas correctly. Being a teacher, on the other hand, he
lowered the degree of precision and talked about limits in an intuitive manner to enhance
student learning.

Another important visual mediator Jason used in the classroom was the symbolic
limit notation. In fact, this aspect of his discourse, together with his word use, provided
significant information how Jason talked about limit and infinity. When talking about the

arrow in the limit notation lim f (x) = L , Jason used a family of words such as
x —>a

99 66 99 6‘

“approaches , goes to , gets closer and closer to”, which refer to the process of the x
values moving toward the limit point a without reaching it. This is ambivalent in some
cases, since the word “approaches” is the canonical, endorsed way of reading the symbol
even in the formal definition. On the other hand, he also used the word “at” when

referring to the arrow, which considered the x values as reaching the limit point.

Similarly, when talking about the function values, Jason used “approach”, “become”,

156

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hov
“'01
the
3P?
185
val
ofi
Wh
1%

C01

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ma
1’OL

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Lhat

“get closer and closer to”, which are about the process of moving towards the limit L
without reaching it. When he completed writing the limit notation using the equal sign,
however, he referred to the limit as reachable (the limit is equal to L). Therefore, Jason’s
word use in regard to the limit notation might inevitably lead to confusion with respect to
the consideration of limit as a process or as a product (number). The same issue was
apparent also for the symbol 00 signifying infinity. In the context of limits at infinity,
Jason talked about infinity as a process, or potential infinity, when he mentioned the x
values “getting/becoming larger and larger”, “goes to”, and “approaches”. In the context
of infinite limits he associated the same symbol with an end result, or actual infinity,
when he talked about the limit of a function being “equal to” infinity. Such elements of
Jason’s discourse are likely to trigger the dynamic view of limit that is based on
continuous motion as well as the incorrect realization of limit such as “limit as
unreachable” (See Section 2.1.2).

The analysis of routines in Jason’s discourse showed that he often utilized
algebra-based routines while computing the limit of a function (See Table 5.21). He used
many algebra-based routines but most of those routines took place in cases when the
routine plugging in, that is plugging the limit point a into the function f(x), did not work.
Said differently, plugging in, if applicable, was the initial routine Jason encouraged
students to use when computing a limit. There were many instances in which this routine
did not work. Jason used a variety of functions whose limits could not be found by
plugging in. Moreover, Jason explicitly addressed the relationship between the function

Value at the limit point and the limit value when he talked about continuity. He mentioned

that the value a function attains at the limit point is irrelevant to its limit value (See

157

Section 5.1.5.4). On the other hand, he often emphasized that the limit of a continuous
function could be found just by plugging in. As a result, although Jason provided
opportunities for students to distinguish between the function value at the limit point and
the limit value, some of his word use and his routine of plugging in might support the
student difficulty “limit as the fiinction’s value” (See Section 2.1.2).

In all of the contexts he worked on, “limit is a number” was the main narrative
Jason endorsed most explicitly and frequently. The shifts in the word use and some of his
routines, such as talking about limit as a process only verbally, however, also led to the
endorsement of the narrative “limit is a process”. Jason referred to limit as a process
when describing the behavior of a function near the limit point. Although he referred to
limit as a number at the end of every such description, it might not be clear for students
when the process gives way to the end result of the process in the context of limits.
Similarly, although Jason explicitly endorsed that “infinity is not a number”, his
treatment of the notion as a number in the context of infinite limits might lead to a similar
confusion in students’ realizations of infinity in terms of its realization as a process or an
end state.

Given the possible implications of Jason’s discourse on student learning, it is
important to investigate how students talk about limits at the end of their instruction. The
classroom observations did not provide much information about students’ discourse on
limits since there were very few occasions in which students talked in the classroom. The
format of instruction was lecture and students talked in the class only to ask clarifying
questions and to correct some computational mistakes Jason made. Therefore, the

diagiostic survey and individual interview sessions were the means by which I had some

158

access to the students’ discourse on limits. In the following chapter, I will discuss

students’ discourse on limits in relation to the instructor’s discourse.

159

CHAPTER VI
STUDENT DISCOURSE ON LIMITS IN RELATION TO THE INSTRUCTOR’S
DISCOURSE
In this chapter, I explore students’ discourse on limits at the end of their instruction. I first
present the results of the diagnostic survey I gave to 23 students. I then focus on four
students who participated in an individual interview session in which they worked on
questions targeting the conceptual obstacles in learning of limits. While reporting on the
survey and the interview sessions, I mainly focus on how and whether the students’
discourse on limits is similar to or different from the instructor’s discourse. For the
interview sessions, I only elaborate on the instances which highlight students’ discourse
on limits in relation to the instructor’s discourse. The purpose of this chapter is to explore
the links between students’ and the instructor’s discourse on limits.

It was not possible to gain much information on students’ discourse on limits
based on the video-taped classroom observations since the instructor’s mode of teaching
was lectures. During the period of eight lessons, there was no student-student interaction
and few instances of student-teacher interaction. Although the instructor encouraged
students to ask questions in the classroom, he did not facilitate any student discussion.
Students interacted with the instructor (Jason) when they asked clarifying questions,
when they did not follow Jason’s explanations and when they corrected a few
computational mistakes the instructor made in the class. Therefore, the diagnostic survey
and the individual interview sessions were the means by which I had access to how

students made sense of the limit concept.

160

 

SC

Sf

6.1. Responses to the diagnostic survey

I gave the diagnostic survey to Jason’s class at the end of the unit on limits. In
seven of the eight observed lessons, Jason covered ideas related to limits and continuity.
The eighth lesson was a review session before the exam. I administered the survey to
students during the last ten minutes of the eighth lesson. It was mentioned before that,
although 31 students enrolled in the class, the number of students attending the class
ranged between 17 and 23 in the period of the classroom observations. There were 23
students present in the class on the day I gave the diagnostic survey and all of the
students agreed to take the survey. The purpose of the survey was two-fold. First, I
wanted information on students’ discourse on limits at the end of their instruction.
Second, I used the survey to select the students for the individual interview sessions.

The questions in the diagnostic survey (See Appendix A) were taken fi'om
Williams (2001) since his classification of views related to limit is widely endorsed in
research on student learning. The first question of the survey included six statements (See
Table 6.1) about limits and asked students to decide whether the statements were true or
false. The second question then asked them which of the six statements best described
their understanding of limits. The third question asked students to describe what they
understood a limit to be. The final question asked students to give a rigorous (formal)
definition of limit, if possible.

Note that the views of limit such as boundary, unreachable and approximation
(See Table 6.1) are among the conceptual obstacles students have when thinking about
limits (See Section 2.1.2, Table 2.1). Hence, the diagnostic survey was useful for

assessing whether participants in the study were likely to have those difficulties.

161

 

 

 

 

Tabl
Vim
State
Ere

State
State

State
Stall

Star

\‘a

W;

re

Ht

:1;

Table 6.1: Statements in the first question of the diagnostic survey and the corresponding
views of limit (Williams, 1991; 2001)

 

 

Statements Description View of limit

Statement 1 A limit describes how a function moves as x moves toward a certain Dynamic-theoretical
point.

Statement 2 A limit is a number or point past which the function cannot go. Boundary

Statement 3 A limit is a number that the y-values of a firnction can be made Formal
arbitrarily close to by restricting x-values.

Statement 4 A limit is a number or point the function gets close to but never Unreachable
reaches.

Statement 5 A limit is an approximation that can be made as accurate as you Approximation
wish.

Statement 6 A limit is determined by plugging in numbers closer and closer to a Dynamic-practical
given number until the limit is reached.

 

The difficulties not addressed by the diagnostic survey but mentioned by research on
learning about limits were intended to be examined during the individual interview
sessions. Those difficulties were “limit implies continuity”, “limit as the function’s
value”, and “limit as monotonic” (See Section 2.1.2, Table 2.1). The diagnostic survey
was also useful for identifying if students’ realizations of limits were mainly dynamic,
that is based on motion, or static, that is based on the. assumptions of the formal definition
of limit. The identification of students’ realizations of limits was based not only on the
responses they gave to Questions I and II but also on how they described limit in their
own words (Question 111, See Appendix A). In what follows, I elaborate on student
responses for each question in the survey

6.1.1. Questions I and II

There were six statements in the first question of the survey describing a different view of
limit (See Table 6.1). The students were asked to decide whether the statements were true
or false. Although each statement focused on a distinct view of limit, the views were not
necessarily non-overlapping. For example, views of limit as unreachable and as an

approximation are identified as difficulties resulting from the dynamic view of limit by

162

the research on student learning (See Section 2.1.2). Table 6.2 shows the responses
students gave to the first question of the survey. Table 6.3 and Figure 6.1 show the cross-
comparison of student responses given to the six statements in Question 1 with respect to

the statements they chose as true.

Table 6.2: Students’ responses to the first question of the diagnostic survey

 

 

 

Number of student
Statements View of limit responses (N=23)
True False
l. A limit describes how a function moves as x moves Dynamic- 20 3
toward a certain point. theoretical
2. A limit is a number or point past which the function Boundary 6 17
cannot go.
3. A limit is a number that the y-values of a firnction can Formal l6 7
be made arbitrarily close to by restricting x-values.
4. A limit is a number or point the function gets close to Unreachable 13 10
but never reaches.
5. A limit is an approximation that can be made as Approximation 12 l l
accurate as you wish.
6. A limit is determined by plugging in numbers closer Dynamic- 9 l4

 

and closer to a given number until the limit is reached. practical

Table 6.3: Cross-comparison of student responses with respect to the statements they
chose as true

 

Number of students choosing the statements as true

 

 

Statement] Statement2 Statement3 Statement4 StatementS Statement6

 

($1) (32) (S3) (S4) (85) (S6)
N=20 N=6 N=16 N=13 N=12 N=9
Statement] 20 4 13 10 10 8
N=20
Statement2 4 6 4 6 3 3
Number N: 6
:idems Sage—inesnts 13 4 l6 8 7 6
3:08“ Statement4 10 6 8 i3 7 4
statements N=13
as true StatementS lO 3 7 7 12 5
N=12
Statement6 8 3 6 4 5 9
N=9

 

163

 

Fig]

(

Stat

Stat

the
star

lim

0f
1m

me

 

Cross-comparison of student responses with respect to the statements

 

 

 

 

 

 

 

 

they chose as true

20 1
15 _ I Statement1
l I Statement2
Number of students a Statement3
chasing the 10 . Stat nt4

statements as true - eme

5 4 t3 Statements
I Statement6

o l

 

 

Statements students chose as true

 

 

 

Figure 6.1: Cross-comparison of student responses with respect to the statements they
chose as true

Statement 1 was considered as true by 20 of the students and was the most popular
statement students considered as best describing their realizations of limit (See Table
6.2). The statement clearly entailed a dynamic view of limit since it involved the
movement of the function values as the x values moved toward the limit point. That this
Statement addresses limit as a process can further be supported by the fact that it views
limit as a means for describing the behavior of a function rather than talking about limit
as a specific number. Such a view is consistent with Jason’s explanations of the behavior
0f filnctions in the context of computing limits (See Section 5.1.5.3) and his routine of
using the metaphor of continuous motion in that context (See Section 5.3.4). It was
mentioned in Chapter V that Jason’s description of a function’s behavior while

computing limits was mainly based on operational word use than objectified]. Therefore,

\

1
Operational word use refers to talking about limit as a process whereas objectified word use refers to

talking about limit as a number obtained at the end of that process (See Sections 5.1.2 and 5.1.3).
164

 

 

as if.
limi
dill]

fun:

true
as l
85p
it i:
Ste
Sta
ex;
de
tal
Se

HL

8)

Sta

given the context of the classroom, I did not consider responding to this statement as true
as incorrect since this was how Jason investigated the behavior of a function near the
limit point before reaching a conclusion about the limit of the function at that point. Said
differently, the statement had theoretical validity when exploring the behavior of a
fimction near the limit point.

Statement 3, which corresponds to the formal view of limits, was considered as
true by 16 of the 23 students and was the second most popular statement students chose
as true in the survey (See Table 6.2). In fact, given that Statement 3 is based on the static
aspect of limit in contrast to Statement 1, which is based on the dynamic aspect of limit,
it is interesting that 13 of the 20 students who marked Statement 1 as true also marked
Statement 3 as true (See Table 6.3 and Figure 6.1). Note also that, unlike the first
statement, third statement talks about limit as a number but not as a process. Jason
explicitly referred to limit as a number while introducing the informal and the formal
definition of limit. He also frequently referred to limits as numbers (if they existed) when
talking about the final answers of the problems in the context of computing a limit (See
Section 5.3.1, Table 5.21, closure conditions). Jason’s persistent referral to limits as
numbers (except when he talked about the function values near the limit point) and
students’ familiarity with the formal definition of limit (See Figure 5.4 and 5.5) support
students’ consideration of Statement 3 as true. Table 6.4 and Figure 6.2 show the cross-
comparison of student responses to the second question in the survey with respect to the

statements students chose as best describing their realizations of limit.

165

 

Table 6.4: Cross-comparison of students’ responses with respect to the statements they
chose as best describing their realizations of limit

 

Number of students choosing the statements as best describing
their realizations of limit

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SI 82 S3 S4 SS S6 None
N=1 1 N=0 N=l N=6 N=1 N=3 N=1
Statementl l l O 1 4 1 3 0
N=20
Number State:16ent2 l 0 0 4 O l O
of
students StalthEntB 9 O l 2 O 3 l
$00311“; Statement4 5 o 0 6 o 1 1
e
statements N=13
as true Statement5 3 O 1 4 l 2 1
N=12
Statement6 2 O O 3 l 3 0
=9
Cross-comparison of student responses with respect to the
statements they chose as best describing their realizations of limit
12
10 4 I Statement1
_ I Statement2
Number of students D St at em ent3
choosnng the 6 4
statements as true 4 I Statement4
’ _ : E3 Statement5
2 ~ 2- .” 7 3;. Statement6
0 , , . . .
S1 82 83 S4 85 S6 None
Statements students chose as best

 

 

 

Figure 6.2: Cross-comparison of student responses with respect to the statements they
chose as best describing their realizations of limit

9 out of 13 students who marked Statement 3 as true selected Statement 1 as best
describing their views of limit (See Table 6.4). Only one student who marked Statement 1
as true considered Statement 3 as the best statement. 11 of the 20 students who chose
Statement 1 as true considered it as the best statement with which they make sense of

limits (See Table 6.4 and Figure 6.2). Therefore, student responses given for these

166

 

statements revealed that, although the instructor can clearly distinguish between limit as a
process and limit as a number depending on the context, students considered both

realizations as true about limits with a clear preference for the dynamic (process) View.

Statement 6 is the other statement in the survey2 that describes limit as a process.
Similar to Statement 1, Statement 6 describes how a limit is obtained rather than what the
limit is and so does not refer to limit as a number. The dynamic aspect in the statement
comes fiom plugging successive numbers to the function as the numbers get closer and
closer to the limit point. At first glance, one would expect majority of students to
consider this statement as true since this statement has similar characteristics with
Statement l-the most popular choice as best describing students’ views of limit (See
Table 6.4). However, nine of the 23 students marked Statement 6 as true (See Table 6.2).
This is consistent with the fact that plugging in values to the function as they get
successively closer to the limit point was a strategy Jason did not utilize in the classroom.
There was only one instance during the classroom observations in which Jason mentioned
plugging points closer and closer to the limit value (See Table 5.2, [2]) when finding the
limit of a function. While doing so, however, all his discussion was verbal; he did not
actually plug in points and compute the function values on the board. Except for this
instance, he did not employ the dynamic procedure as described in Statement 6 while
working on limits. In addition to this, Jason did not use any tabular representation of
functions throughout the eight lessons. Such representation of functions is likely to

trigger the idea that one can find the limit of a function by just looking at the values that

 

2 Statement 6: A limit is determined by plugging in numbers closer and closer to a given number until the
limit is reached (Williams, 1991; 2001).

167

 

are close to the limit point. There was no evidence in Jason’s discourse to support such an
approach for finding a limit. The three students who chose Statement 6 as best describing
their views of limit (See Table 6.4 and Figure 6.2) were likely to focus on the dynamic
language of the statement and the process of “getting closer and closer to” the limit point
since 8 out of 9 students who chose Statement 6 as true also marked Statement 1 as true
(See Table 6.3 and Figure 6.1).

Research on learning about limits highlights the close relationship between the

dynamic view of limit as realized through a description similar to Statement 6 and the

student difficulty “limit as approximation”3 (Bezuidenhout, 2001). In other words,

students who consider plugging in values successively closer to the limit point can view

limit as an approximation. Statement 5 of the diagnostic survey4 considers limit as an
approximation and the statement was marked as true by 12 students (See Table 6.2).
Although five of these students also marked Statement 6 as true, ten marked Statement 1
as true (See Table 6.3). Therefore, some form dynamic view of limits was related to the
View “limit as approximation”. That the majority of students thinking about “limit as
approximation” chose Statement 1 but not Statement 6 as true might again result from the
fact that Jason did not utilize the procedure implied by Statement 6 in the class. Only one
student chose Statement 5 as best describing his realization of limits (See Table 6.4 and

Figure 6.2).

 

3 See Section 2.1.2 for the details of the student difficulty “limit as approximation”.

4 . . . . .
Statement 5: A 11m1t is an approxrmatron that can be made as accurate as you wish (Williams, 1991;
2001).

168

 

“Limit as unreachable” is another student difficulty resulting from the dynamic

view of limits5 indicated by the literature (Tall & Vinner, 1981; Williams, 1991).
According to Tall and Schwarzenberger (1978), the colloquial use of the words such as
close to implies getting near to but not being coincident with. Given this, they argue that
the informal notion of limit may carry for students the assumption that one can get close
to the limit value but cannot reach it. Statement 4 of the diagnostic survey6 described

limit as a number that cannot be reached and was considered as true by 13 students (See

 

Table 6.2). 10 of these students also marked Statement 1 as true, suggesting that “limit as
unreachable” is a difficulty based on dynamic view (See Table 6.3). This can further be
supported by the fact that the 13 students who chose Statement 4 as true chose Statement
4 (six students) and Statement 1 (five students) as best describing their realizations of
limits (See Table 6.4 and Figure 6.2).

Statement 1 is based on the idea that limit is a process and talks about limit as a
descriptor of how a function moves as the x values move toward the limit point (See
Table 6.1). Such consideration of limit is consistent with the informal aspect of limit
Jason employed in the classroom when be investigated the behavior of a function near the
limit point. Jason never endorsed the narrative “limit is unreachable” but some
characteristics of his discourse on limits supported this view of limit. It was in the context
of computing limits that he switched between operational and objectified word use
treating limits (and also infinity) as processes and as end products (numbers) depending

on the problems that he worked on (See Sections 5.1.5.3.] and 5.1.5.3.2 ). When

 

5 See Section 2.1.2 for the details of the student difficulty “limit as unreachable”.

169

discussing the behavior of a function near the limit point, Jason mainly explored what
value f(x) approached as the x values approached the limit point. While doing so, he also
used other words besides “approaches” such as “getting closer and closer to”, “tending
to”, and “becoming” (See Section 5.1.2). Talking about limit as a process in which the
function values “approach”, “get closer and closer to” or “become” a number L carries an
implication that one can get close to the limit but cannot reach it.

The different ways Jason addressed the limit notation is another aspect of his

I’m
{-

discourse that supports a view of “limit as unreachable”. Throughout the eight lessons,

Jason referred to the arrow sign in the limit notation lim f (x) = L as the x values
x —>a

“approaching”, “going to”, and “getting closer and closer to” the limit point, which
signifies the process of moving towards the limit point without reaching it (See Section
5.2.4). This is consistent with some students’ views that the x values do not reach the
limit point with the idea that the filnction values can never reach the limit value. In the
context of continuity (See Section 5. 1 .5 .4), Jason also mentioned that the function value
at the limit point is irrelevant to the limit value; “what only matters is what happens
nearby [the limit point]” (Jason, 28 January, 2009). This might be another reason for
students’ conclusion that the function values cannot be reached while determining the
limit.

To sum up, student responses to the fourth statement in the survey reveal the

Connection between the view “limit as unreachable” and the dynamic-theoretical aspect

of limit. The instances in which Jason shifted his word use in the context of computing

Statement 4: A limit is a number or point the firnction gets close to but never reaches (Williams, 1991;

2001).
170

limits support such a realization of limit. In those instances, Jason investigated the
behavior of the function near the limit point and, hence, considered limit as a process.
Although at the end of all such processes he referred to limit as a number, he did not
explicitly address process and product aspects of limits in his discourse leaving an
explanation for the shifts in word use in the classroom.

The final statement that will be elaborated on in Question I of the survey is

Statement 2, which describes limit as a bound7. Students with this view think that a
function is bounded by a specific limit value or that the absolute maximum/minimum
values of the function are the limits for the function (See Section 2.1.2). This statement
was marked as true by six students in the classroom (See Table 6.2). Compared to the
other statements in the survey, Statement 2 was selected by the least number of students
as true and was not selected by any of the students as best describing how they make
sense of limits (See Table 6.4 and Figure 6.2). It is difficult to contemplate whether
Jason’s discourse on limits played any role in students’ consideration of limit as a bound.
This view of limit is often based on the colloquial use of the word limit than the
mathematical aspects of the concept. Everyday uses of the word in phrases such as “the
speed limit is 25mph”, “we reached the city limit”, and “we have to limit our expenses”
could result in students’ realization of limit as a constraint or a boundary. There was no
evidence in Jason’s discourse of support for this view of limit. Note, however, that all of
the students who marked Statement 2 as true also marked Statement 4 (“limit as
unreachable”) as true (See Table 6.3 and Figure 6.1) and four of the six students who

marked Statement 2 as true selected Statement 4 as the best statement to describe their

171

views of limit (See Table 6.4 and Figure 6.2). Therefore, these students were likely

thinking of “limit as bound” as connected to the view “limit as unreachable”, making this

a difficulty related to a dynamic viewg.

6.1.2. Questions 111 and I V

The third question in the survey asked students to describe what they understood a limit
to be using their own words. Table 6.5 shows the student responses for Question 111.

One student did not give a response for Question III (Table 6.5, [5]) and one mentioned

(

that limit was not a clear idea for him (Table 6.5, [8]). One student did not mention limit
or the limit value L and used an arrow to represent the x values approaching the number
one requires (Table 6.5, [7]). However, this student did not use the word “approaches”
but just relied on the arrow to communicate her ideas about limit. Therefore, although it
seemed that she was describing x values approaching the limit point, her description did
not provide clear evidence about her view of limit. These responses (Table 6.5, [5], [7],
[8]) were not descriptive enough to gain information with respect to the students’
realizations of limit.

Three students used elements of the formal view of limit as indicated by the
survey (See Table 6.1, Statement 3) using words such as “arbitrarily close” (Table 6.5,
[6], [10], [15]). The first student also mentioned x values “sufficiently approaching” s
(Table 6.5, [6]). In the formal view of limit, motion is eliminated and the x values are

sufficiently close to 3 rather than approaching s.

 

7 Statement 2: A limit is a number or point past which the function cannot go (Williams, 1991; 2001).

8 The converse of the argument was not necessarily true. Only 6 of the 13 students who marked Statement
4 as true marked Statement 2 as true (See Table 6.3 and Figure 6.1) and none of those 13 students selected
Statement 2 as best describing their realizations of limit (See Table 6.4 and Figure 6.2).

172

Table 6.5: Students’ responses to the third question of the diagnostic survey9

 

 

Response Describe what it means to say that the limit of a function f as Best statement describing
No. x—vs is some number L. students’ view of limit
(N=23) (See Table 6.1)
[1] As x gets closer and closer to the value 3 then y values of f(x) get Statement 3
closer and closer to L
[2] When x approaches to s, the relative value on y-axis will be close Statement 1
to L
[3] That the function won’t reach L as x approaches 3. Statement 4
[4] Limit is while x approaches to an initial interval of the fiinction, “None”
it may be closer to the value at the function at that point. it’s
approximately close; however, never reaches.
[5] No response is given. Statement 6
[6] If x sufficiently approaches 3, f arbitrarily close to L. Statement 6
[7] A number x—’ the number you require Statement 5
[8] A limit is maybe not so clear of an idea to me. Statement 4
[9] Functions have limits. Limits are the point or number the f(x) can Statement 4
never reach but can get as closer as it can.
[10] Limit is when x approaches arbitrarily close to a given number. Statement 4
There is a y—values counterpart with it.
[ l 1] What value in the y coordinate is approached when x approaches Statement 6
s.
[ 12] L would be the point of the y value of the function f as it Statement I
approached the points.
[13] A limit describes a function as it gets closer and closer to a point. Statement 1
As x approaches a number y approaches the limit.
[14] This means that as x approaches some number s the limit is some Statement 1
number L.
[15] As x approaches 5 it will be arbitrarily close to L but never Statement 4
reaches L.
[16] A limit is some number L that a fiinction can get really close to Statement 4
but never actually reaches. If you say lim f( x) = L you are
X“)S
saying as you approach 5 on f(x) you get L.
[17] A limit is a description of what a filnction comes as it approaches Statement 1
certain values, i.e. lim 2x _ 4 = 00
x—-)O
[18] I understand limit to be the y-value a function gets close to as the Statement 1
x value approaches a number.
[19] A limit is a value that the fimction approaches as x approaches a Statement 1
certain value.
[20] To me a limit is just a point that a fimction approaches at a given Statement 1
x-value.
[21] As x gets closer to s the # [number] gets closer to L. Statement 1
[22] As x approaches s the y values get close to L. Statement 1
[23] f( x) = L means that as the function’s x values become closer Statement 1

x—bs
and closer to s, the y value of the function becomes closer and
closer to L.

 

Any grammatical, symbolic or mathematical errors in the sentences are preserved in order to keep the

173

This student used a combination of words from both the formal and the dynamic aspects
of limit and chose Statement 5 (“limit as approximation”) as the best statement describing
her view of the concept (Table 6.5, [6]).

The second student using elements of the formal aspect considered the x values
(but not the function values) “arbitrarily” approaching a given number and mainly
described limit as a process (Table 6.5, [10]). Similar to the first student, she used both
the formal and dynamic views related to limit since she did not talk about being
arbitrarily close but “arbitrarily approaching”. She did not consider limit as a number but
as something that happens “when x approaches” a number (Table 6.5, [10], emphasis
added). As a result, despite her attempt to use the formal aspect of limit, this student’s
description of limit was based on the realization of limit as a process.

The third student who used elements of the formal aspect of limit mentioned the
value L, without referring to it as a number (Table 6.5, [15]). He also used the word
“approaches” when describing the behavior of the x values. Therefore, there were
elements of both the formal and the dynamic aspects of limit in his description. The
existence of the dynamic view of limit in his response can filrther be supported by the
fact that he considered the value L as unreachable. In fact, this student and the second
student chose Statement 4 (“limit as unreachable”) (See Table 6.5, [10], [15]) as best
describing their view of limit and they both marked Statement 3 (formal view of limit) as
false in the survey. On the other hand, all three students marked Statement 1, the
dynamic-theoretical view (See Table 6.1), as true. Hence, student responses [6], [10],

[15] (Table 6.5) were considered as dynamic but not formal.

 

originality of student responses.
174

 

 

The six students mentioned above were the cases in which it was hard to infer
realizations of limit directly fiom definitions either due to a lack of response or a mixture
of word use. All of the remaining 17 students clearly used some version of the dynamic
view when describing their realizations of limit. The only student who marked Statement
3 (formal view of limit) as best describing his View of limit was among those 17 students
(See Table 6.4 and Table 6.5, [1]). Only four of the 17 students referred to limit as a
“number” or “the y value of the fimction” (Table 6.5, [9], [12], [14], [16]). The remaining
13 students described limit as a process. Overall, there were four students referring to
limit as objectified (number), 16 students'0 referring to limit as operational (process), and
three students whose descriptions of limit could not be classified.

Some of the 16 students referring to limit as operational described the process of
the fimction values approaching the limit value as x values got closer and closer to the
limit point. Some others considered limit as a descriptor of how a function behaves (e.g.,
Table 6.5, [13], [17]). Therefore, although Jason’s word use in the classroom was mainly
obj ectified rather than operational, students adopted his operational word use. Jason
referred to limit as a process only verbally, not writing any of his operational word use on
the board (See Section 5.3.3). On the other hand, student responses for the third question
of the survey show that the instructor’s verbal comments played a dominant role in
students’ descriptions of limit. Jason’s word use shifted from operational to objectified
(and vice versa) in the contexts of informal definition of limit, computing a limit, and the

limit notation. It was argued in Chapter V that students were unlikely to distinguish the

 

10 The total number of students viewing limit as a process was 16, including the three students who used a
combination of formal and dynamic view in their description.

175

 

process aspect of limit from the product aspect in these contexts since such shifts
remained implicit in Jason’s discourse

Another aspect of Jason’s discourse that supported the realization of limit as a
process and remained implicit for students was his routine of using the metaphor of
continuous motion in the context of computing limits and the informal aspect of limits
(See Section 5.3.4). Although he also used the metaphor of discreteness when talking
about the formal aspect of limits (See Section 5.3.5), students in the study only relied on
the metaphor of continuous motion in their discourse.

Jason’s shifts in word use and his utilization of the metaphor of continuous
motion are consistent with the fact that, no matter how frequently and explicitly he
endorsed the narrative limit is a number (See Section 5.4), students mainly endorsed the
narrative limit is a process in their responses to the survey. Throughout the eight lessons,
Jason explicitly endorsed the narrative limit is a process only twice (See Section 5.4).
However, he endorsed it implicitly every time be computed the limit of a fitnction by
investigating the function’s behavior near the limit point. Despite his referral to limit as a
number (if it existed) at the end of every computation, the notion mainly remained as a
process for students.

Responses to Question 111 also revealed five students’ consideration of “limit as
unreachable” (Table 6.5, [3-4], [9], [15-16]). These students explicitly mentioned that the
function values cannot reach limit value L and all of them described limit using elements
of the dynamic view. Moreover, all five students marked Statement 4 (See Table 6.1),

which described “limit as unreachable”, as true and four of them selected it as the best

176

 

 

statement with which they made sense of limits (Table 6.5). One student chose none of
the statements as best describing her view of limits1 I.

The fourth question in the survey asked students to give a rigorous definition of
limits, if possible. While administering the survey, I told students that a rigorous
definition of limit is what Jason referred to as the precise (or formal) definition of limit in
the class. In spite of this, majority of the students left this question unanswered. Only five

of the 23 students provided a response for Question IV of the survey. Table 6.6 shows

student responses for the question.

Table 6.6: Students’ responses to the fourth question of the diagnostic survey12

 

 

Response If possible, write down a rigorous definition of limit.

No. (N=23)

[1] f (x) = L
x—)s

[2] “limit” is what is used to describe the number that we would try our best to get on y-axis when
x approaches to another number.

[3] Limit is approaching a value that usually cannot be defined or = in a normal equation.

[4] As x approaches a number, y approaches a number

[5] A limit is a value that a function can never reach, however, it only comes arbitrarily close to it.

 

One of the students considered limit as a “value a function can never reach” (Table 6.6,
[5]) when providing a rigorous definition of limit. Three students talked about limit as a
process (Table 6.6, [2-4]) and one gave the symbolic notation as a rigorous definition of
limit (Table 6.6, [1]). This question did not provide additional information in terms of
students’ realizations of limit both because there were very few student responses and

also the descriptions students provided had similar characteristics to those they provided

 

H This student marked “none” for Question I] of the survey (See Table 6.4).

2 . . . . .
Any grammatlcal, symbolic or mathematical errors 1n the sentences are preserved in order to keep the
originality of student responses.

177

“

 

for Question III. The dynamic view (Table 6.6, [2-5]) and the consideration of “limit as
unreachable” (Table 6.6, [5]) were again common themes in students’ definitions.

Students’ lack of responses for this question could result from their unfamiliarity
with the word “rigorous”. Jason used the term “precise” when talking about the formal
definition of limit in the class. Although I mentioned that those two terms mean the same
thing while administering the survey, it is possible students did not realize these terms as
the same. Another reason this question was left unanswered by the majority of students is
that Jason presented the formal definition of limit as optional, a personal challenge rather
than a required topic for the exam (See Section 5.1.5.2). Therefore, it is possible that
students did not attend to the formal aspect of limit carefully or forgot the complicated
statements including a and 6 as well as the existential quantifiers such as “for all” and
“there exists” in the formal definition.

Summary

The responses given to the diagnostic survey revealed that the students in the study had a
dynamic view of limits at the end of their instruction. 16 of 20 students who had the
dynamic view of limit described limit as a process”. Four students mentioned limit as a
number but still used the dynamic view in their descriptions. This result was consistent
with the fact that the statement describing limit by dynamic-theoretical means (Statement
1 in Question I, see Table 6.2) was marked as true by 20 of the 23 students taking the
survey. Dynamic view of limit refers to the consideration of the concept by means of

continuous motion and results in the realization of limit as a process. The instructor’s

 

l3 . . . . . .
Three students’ explanations of llmlt were not explanatory enough to conclude about their descriptlons
of limit (See Section 6.1.2).

178

 

 

 

routines, and shifts in word use when computing limits, referring to the limit notation and
talking about the informal aspects are consistent with students’ realization of limit as a
process.

The statement which described the formal view of limit was the next popular
statement students chose as true (Statement 3 in Question I, see Table 6.2). The three
students who incorporated elements of the formal definition to their description of limit
did so inaccurately since they mainly relied on the dynamic aspect of limit rather than the
discrete or static aspect in their explanations. Said differently, the formal view of limit
was considered as true by the majority of students but was not employed in their
descriptions of limit. They had seen the formal (precise) definition which Jason
introduced in the classroom, but perhaps did not grasp it implications, instead relying on
motion in their descriptions of limit. In the end, only four of the students mentioned limit
as a number, the obj ectified view that dominated Jason’s discourse.

Statement 4 and Statement 5 in the first question of the diagnostic survey were
also chosen as true by the majority of students (See Table 6.2). Statement 4 assessed
students’ view of “limit as unreachable” whereas Statement 5 assessed the view of “limit
as approximation”. These views are addressed by research on learning limits as student
difficulties resulting from the dynamic aspect of limit (Tall & Schwarzenberger, 1978;
Williams, 1991). Students’ responses to these statements, together with their definitions
of limit are in accord with their consideration of limit as a process.

Some of Jason’s discourse on limits possibly had a direct impact on student
learning. Consistent with Jason’s discourse, the majority of the students marked the

dynamic-practical aspect of limit as false (Statement 6 in Question I, see Table, 6.2).

179

 

 

Recall that Jason did not find limits of functions by plugging in points successively close
to the limit value. Similarly, his introduction of the precise or the formal definition of
limit led to some familiarity with the words associated with the formal aspect of limit.
Such familiarity supports students’ consideration of the statement describing the formal

view as true. The view “limit as boundary” did not seem to be related to Jason’s

discourse on limits in the classroom. Instead, this realization of limit was more closely

related to the colloquial use of the word limit (See Section 6.1.1).

Some other elements of Jason’s discourse on limit, however, support students’
consideration of limit as a process. His routine of using the metaphor of continuous
motion and switching between the operational and objectified word use in the contexts of
informal definition of limit, computing limits, and referring to the limit notation are
consistent with students’ realizations of limit as a process based on dynamic motion.
Jason talked about limit as a process when he investigated the behavior of functions near
the limit point. Although he referred to limit as a number after determining the behavior

of the functions, students mentioned the process but not the end result of the process in

their own descriptions of limit.
6. 2. Responses to the individual interview sessions

In this section, I explore four students’ discourse on limits by focusing on (a) elements of
the instructor’s discourse on limits, and (b) the difficulties indicated by research on
student learning about limits. I use pseudonyms — Amy, Jessica, Harry, and Keith — for
the students’ names. Among these students, Amy, Harry and Keith were from the list of
students I initially identified to interview. Jessica volunteered and I interviewed her as

well (See Section 4.2). All four students responded correctly to five of the six statements

180

 

in Question I of the diagnostic survey (See Table 6.1). However, many of the difficulties
identified by research on learning about limits were found in students’ discourse during
the interview sessions.

The questions in the interview session were designed to probe students’
realizations of limits. The diagnostic survey did not address student difficulties such as
“limit implies continuity”, “limit as the function’s value”, and “limit as monotonic” (See
Section 2.1.2, Table 2.1) nor did it provide information regarding students’ views of
continuity and infinity. The questions in the interview sessions were useful to focus on
ideas not addressed by the survey. The questions also provided contexts in which student
difiiculties “limit as unreachable”, “limit as boundary”, and “limit as approximation”

(See Section 2.1.2, Table 2.1) were investigated in further detail. Figure 6.3 shows the

problems in the interview sessions]4

I started the interview sessions asking some general questions to students in terms
of their background as well as how and whether they utilized the textbook in their
learning. Table 6.7 provides some general information about the students obtained both

through the diagnostic survey and the interview sessions. Note that none of the students

used their textbook for reviewing material or preparing for the exam1 5

 

14 The interview session problems as shown in Figure 6.3 are also included in Appendix B.

15 This was the primary reason the analysis of the textbook with respect to its discourse on limits was
considered beyond the scope of this study.

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Table 6.7: General information about the students participating in the interview sessions

 

 

Amy Jessica Harry Keith
Statement best describing Statement 1 Statement 1 Statement 6 Statement 3
their view of limit in the
survey (See Table 6.1)

Sociology- Finance Computer Computer
Major Residential engineering science
College

Year in the university First year First year Second year Second year
Did you take calculus in No Yes No No

high school?

Did you take calculus before No No No Yes (Once)
during your undergraduate

study?

Do you use your textbook Not at all Never Never No
besides the homework

problems?

Do you use the lecture notes No Yes No No

for reviewing material or

preparing for the exam?

What sources do you use for Reviews the Uses lecture Watches Math TV Reviews the

. . . homework notes from YouTube and homework

rev1ew1ng materlal or .

preparing for the exam? problems and asks frlends problems

asks fi'lends

 

They only used the textbook to work on the assigned homework problems. Only one of
the students used the lecture notes whereas the other students relied on other sources,
such as friends and intemet-based lessons, while studying for the course.
6. 2. I . Students’ discourse with respect to the instructor ’s discourse on limits

In this section, I investigate the students’ discourse in relation to the instructor’s
discourse on limits. The contexts in which the students’ discourse will be discussed in
this section emerged from the analysis of Jason’s discourse with respect to his word use
(See Chapter V). Those contexts are (a) informal aspect of limits, (b) formal aspect of
limits, (c) limit notation, (d) infinity, and (e) continuity.

6.2.1.1. Informal aspect of limits
All of the students participating in the interviews made frequent use of the dynamic view

of limit as they worked on the limit computation problems and talked about limits. Two

183

of the students, Amy and Jessica, considered the statement in the diagnostic survey that
described limit as a process as the best statement describing their view of limits]6 (See
Table 6.7). Harry picked the statement that described limit as an approximation as the
best statement, also a dynamic View of limit. Keith was the only student who considered
the formal view of limit as the best statement with which he made sense of limits. During
the interview, however, he talked about limit as a process and relied on the informal
aspects of limit.

During the interviews, students computed limits and wrote those limits as equal to
particular numbers. On the other hand, none of the students explicitly referred to limit as
a number. Instead, they described the behavior of the function values approaching the
limit value as the x values approached the limit point. This was consistent with Jason’s
operational word use and his routine of using continuous motion in the class. Jason’s
discussions of limit as a process (operational word use) only took place when he
communicated his ideas verbally (See Section 5.2.1). He did not write any of his
operational utterances on the board. The students, however, talked about the instructor’s
investigation of the behavior of the function as x approached the limit point as a
definition of limit. In fact, Amy thought that finding limits in this manner “fit the
definition he [the instructor] presented to us in class” (Amy, 3 March, 2009). Note that
the instructor’s informal definition of limit (See Table 5.6 and Figure 5.4) referred to

limit as a number (objectified word use). His treatment of limit as a process mainly took

 

1 .
6 During the interview, Jessica mentioned that it was a combination of the dynamic View and the formal
view that she made sense of limits. She said she chose Statement 1 (See table 6.1) because it was simpler.

184

place when he computed limits. Therefore, he did not technically present the process
aspect of limit as a definition to students in the class.

When investigating the behavior of a function near the limit point, students’
terminology was similar to the instructor’s, frequently using words signifying motion:

Amy: Coming from the right, it [the function value] is approaching zero

and coming in from the left, it is approaching zero.

Jessica: As x moves towards zero, the y values change.

Han'y: It [the function value] is approaching; it gets closer and closer to

one.

Keith: ...the limit is a value as it is approaching a certain number.
Note that although Keith uttered the word “number”, he did not refer to limit as being
equal to the number. Instead, he mentioned limit as it was approaching that number,
which was the function’s y value. Jessica explicitly mentioned that limit was not just a
number and used the word “proceed” to describe limit as a process.

Jessica: The limit is not only a particular number.

Researcher: The limit is not a particular number. What is it then?

Jessica: It is a moving proceed [sic] I think.

Researcher: When you say proceed, what do you mean?

Jessica: x approaches c; it is not equal to c.
Students talked about limit as a process also when they attended to the right hand and the
left hand side of functions. All of the students computed right hand and left hand limits at
some point during the interview but did not utter the word “limit”. Instead, they

mentioned “approaching from the left/right”. The frequency with which they utilized this

185

approach depended on the context. Keith and Jessica consistently used left and right
limits throughout the questions whereas Amy and Harry did not attend to both sides when
the function was represented algebraically.

Graphing was the primary routine three students utilized when investigating the
behavior of a given function. The only problem involving a graph in the interview session
was Question 3 (See Figure 6.3). Keith and Jessica drew graphs while working on every
problem in the interview. Harry drew graphs for Questions 1, 2 and 4 (part c) of the
interview (See Figure 6.3). These students used graphs not only to compute the limit of a
function at a particular point but also to communicate their thinking about limits and to
provide examples or counter-examples to some of the arguments in the questions. On the
other hand, they expressed a dynamic view of limit (using the metaphor of continuous
motion) and treated limit as a process every time they talked about their graphs. Amy did
not draw any graphs during the interview due to a particular view of limit (“limit implies
continuity”), which will be discussed later in the chapter. She mainly attended to the
function’s value while computing limits and did not use graphing as a routine to
communicate her ideas.

In summary, all of the students’ word use about limit was mainly operational
rather than objectified since they talked about limit as a process but not as a number. This
was in contrast with Jason’s discourse since his word use was mainly objectified. The
students considered the x values approaching the limit point and the function values
approaching the limit as elements of the dynamic view, which was based on continuous
motion. The way they explored the behavior of a function near the limit point was

consistent with the instructor’s routine of computing limits (See Section 5.3.3). On the

186

other hand, unlike the instructor, students could not refer to limit as a number at the end
of the limiting process during the interview. It was mentioned in Section 5.4 that Jason
explicitly endorsed the narrative limit is a process only twice whereas he endorsed the
narrative limit is a number consistently in the classroom. The students in the interview
sessions, however, endorsed the former narrative about limits.

6.2.1.2. Formal aspect of limits

The formal definition of limit Jason introduced in the classroom provided a
context in which to highlight the static aspect of limit. During his discussions on the
formal definition, his word use about limits was consistently objectified. Moreover, he
did not utter words signifying motion but used words signifying proximity instead (See
Section 5.1.5.2). While doing so, he used the metaphor of discreteness as a routine (See
Section 5.3.4). Question 5 (part c) and Question 6 of the interview session problems (See
Figure 6.3) were used to gain information on students’ view of the formal definition of
limit and how they talked about this definition as being similar to or different fi'om the
informal definition of limit Jason introduced in the classroom. In Question 6, I used the
textbook’s definition of formal definition of limit, which was similar to Jason’s formal
definition of limit (See Figure 5.5), since the textbook’s definition refers explicitly to a
and 5 as numbers.

All of the four students marked the statement in the diagnostic survey that
described limit by formal means as true (See Statement 3 in Table 6.1). However, they all
relied on operational word use while talking about the formal definition of limit in the
context of Question 6 of the interview session. After reading the formal definition in

Question 6 (See Figure 6.3), Amy split it into three parts as follows:

187

Part 1. Let a function f (x) be defined on an open interval about [the point] x0 , except
possibly for x0 itself.
Part 2. We say that the limit of f (x) as x approaches to x0 is the numberL , and write

lim f(x) = L

x—)x0
Part 3. if for every numbera > O , there exists a corresponding number 6 > 0 such that

for allx,0 < |x—x0| < 5 :> |f(x) —L| < e . (Thomas et al., 2008, p. 91)

Amy: [Referring to Part 1] So it says like it has to be defined which makes

me think that it has to be continuous which kind of goes along with what I

already know about limits. . . [Referring to Part 2] So this part. . .it goes

along with my informal definition like really well. . . [Referring to Part 3]

For every number. . .there exists a corresponding delta such that. . .for me

goes along with that proof which it is kind of like ‘whatever’.
Amy considered the first part of the formal definition as describing the continuity of the
function at the limit point. The second part, which basically described how to read and
symbolically represent the limit notation, matched her dynamic or informal view of limit.
The third part, on the other hand, represented the part of the definition that she ignored.
When she was asked to explain why Part 3 did not make sense to her, she said

When we did the proofs of the limits, he [the instructor] was not going to

test us on it and he said that we would never really have to do them. So I

didn’t pay attention... I completely disregarded it after that homework

assignment. The way he presented the material to us by saying that we

188

 

would never have to know it for a test just makes me not pay any close

attention to it. (Amy, 3 March, 2009)
Jason’s mentioning of the formal definition as a personal challenge and his explicit
statement that the homework problems would not be graded resulted in Amy’s ignorance
of the formal definition as a relevant part of the course. When she was asked to explain
how she thought of the informal and formal definitions as similar or different, she
mentioned that Part 2 “is the informal definition” (Amy, 3 March, 2009) and did not say
more. In Amy’s discourse, the static aspect of limit was missing. Dynamic view of limit
through the assumption of continuity was how she talked about the formal definition.

Harry realized Question 5 (part c) in the interview session (See Figure 6.3) as
related to the formal definition but did not want to work on it saying he “hated those
symbols” (Harry, 27 February, 2009). He showed some effort to work on the definition in
Question 6.

So what I recall from the epsilon and delta is that difference between

epsilon is proportional to the delta. So if there is a difference of let’s say

two and an x value, it will be proportional to the y value. If there was like

two x, the y value would be four. If it was four x, the y value would be

eight. It is like proportional. (Harry, 27 February, 2009)
Note that he used the term “difference”, a word signifying proximity instead of motion,
and also referred to some sort of dependence. The dependence he described by means of

proportionality seemed, however, to express y values as dependent on the x values rather

189

than 8 being dependent on s1 7. When describing how he thought about this definition as
similar to or different from the informal definition, Harry only commented on the
sentence where the limit notation and how to read the notation was introduced. He said
“they are actually very similar. When Jason [the instructor] did it, he said the limit as x
approaches a of f of x equal to L. . .He explained that L was the y value as x is
approaching a” (Harry, 27 February, 2009). Harry did not comment on the part of the
definition including 8 and 5. Harry realized the formal definition of limit as indicating a
type of dependence but, similar to Amy, he relied on the dynamic view of limit while
talking about the formal definition. Amy and Hany both talked about the part of the
definition that introduced how to read and write the limit notation as related to the
informal definition of limit. As a result, they concluded that the two definitions were
similar.

Unlike Amy and Harry, Jessica and Keith used graphs as visual mediators when
explaining how they made sense of the formal definition. They both drew correct graphs
representing the relationships between a and 8. They both referred to s as related to the
difference between the function values and the limit value, and 6 as related to the
difference between the x values and the limit point. They also mentioned that the graph
the instructor drew when he worked on a proof problem (See Table 5.20, the last graph)
was very useful for them while thinking about limits. Jessica split the definition into four

parts as follows.

 

‘7 In the formal definition, 5 (which is related to the difference between x values and the limit point)

depends on a (which is related to the difference between the function values and the limit value).

190

Part 1. Let a function f (x) be defined on an open interval about [the point] x0 , except
possibly for x0 itself.
Part 2. We say that the limit of f (x) as x approaches to x0 is the numberL , and write

lim f(x) = L

x—->x0
Part 3. if for every numbera > O , there exists a corresponding number 5 > 0

Part 4. such that for allx,0 <|x—x0| < 5 :> [f(x)—Ll < 8 . (Thomas et al., 2008, p.91)

She said Part 1 of the definition “is like the requirement. . .like it should be an open
interval and except for x zero” (Jessica, 5 March, 2009). She wrote “moving” near the
second part of the definition. She described Part 3 of the definition as relation and said “it
shows that there is a relationship between. . .what happened on the x axis and what will
happen on the y axis. This is the relation part” (Jessica, 5 March, 2009). She referred to

Part 4 of the definition “this part just wrote words into another form. But when you see

here [shows Ix — xol ]...it tells you. . .it should. . .move both sides” (Jessica, 5 March,

2009). When asked how the informal definition was similar to or different from the
formal definition, Jessica said that they both mention approaching particular points but
that the formal definition was more confusing since “there are three things to figure out:
L, epsilon and delta” (Jessica, 5 March, 2009). Note that she connected these two
definitions by means of motion. Still, Jessica considered the formal view, together with

the informal view of limit as best describing how she made sense of the concept.

191

 

Keith was the only student who considered the statement that described limit by
means of a formal view (See Statement 3 in Table 6.1) as best describing his realization
of limits in the diagnostic survey. During the interview, he said “the informal approach
helped in give me the right answers and working through it and like you know the quick
methods. . .But I thought that the formal definition provided a much better understanding
of what the actual definition was” (Keith, 3 March, 2009). However, Keith was primarily
referring to the graph the instructor drew for the proof problem as the most useful tool to
make sense of the formal definition. He said he followed the arguments about the formal
definition visually but was initially confused about the symbolism included in the
definition when he worked on Question 5 (part c) (See Figure 6.3).

When we went over the formal definition, we didn’t really discuss what

these symbols were [shows a and 8]. . .When we went over the definition,

like I got it visually. I got it visually what we are doing and it helped

immensely. But when we were using. . .like I don’t really remember what

these symbols are. . .(Keith, 3 March, 2009)

On the other hand, in the context of Question 6 of the interview session, Keith showed a
lot of effort to make more sense of the symbolism. Without any guidance and using only
the graph he initially drew for the formal definition, he was able to realize a in relation to

the difference between the function values and the limit value, and 8 in relation to the

difference between the x values and the limit point. However, he defined |x - x0] as equal

to 8 and | f (x) — L] as equal to a: “it almost seems like x minus x zero should be equal to

sigma [he means delta] and f of x minus the limit should be equal to epsilon” (Keith, 3

192

March, 2009). When asked how he thought the formal and the informal definitions as
similar to or different from each other, Keith said

[Referring to the formal definition] It is similar because it is kind of saying

the same thing. It is different in the fact that it has all these symbols and

it’s much more. . .Using the visualization, that the informal definition and

formal definition I believe should be taught simultaneously kind of

together. Because the formal definition allows me, for me anyways, to

visualize it and see. (Keith, 3 March, 2009)

Jessica and Keith were more elaborate in their responses in relation to the formal
definition of limit. However, the presence of the dynamic view of limit was apparent both
in their word use and also in their routine of graphing throughout the interview. It was
mentioned in Section 5.2.3 that the instructor drew a graph representing the formal
definition of limit only after noticing that students did not follow his discussion on the
proof problem that he worked on. That graph seemed to have enhanced Jessica’s and
Keith’s realization of the formal aspect of limit.

Overall, all of the four students’ discourse on the formal definition of limit was
consistent with the instructor’s in that they talked about the formal definition as similar to
the informal definition. Recall that both the textbook’s and Jason’s definitions of limit
included an explanation of how to read and represent the limit notation. It was because of
the existence of the word approaches that all students related the formal view of limit to
the informal one. The students’ discourse on the formal definition differed from the
instructor’s in that students’ word use about limit was not objectified and they used the

metaphor of continuous motion instead of the metaphor of discreteness. Unlike their

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instructor, students did not use the phrases arbitrarily close and sufliciently close. Jason’s
word use (See Section 5.1.5.2) was most consistent in the context of the formal definition
of limit in which he did not utter any motion related word and used words in an
objectified way. Jessica and Keith were able to talk about the diflerence between values
in the context of the formal definition. However, they connected difference with motion
but not with discreteness. Therefore, although the context of formal definition gave
students an opportunity to attend to the static aspect of limit, their realizations remained
dynamic.

6.2.1.3. Limit notation
All of the students used words signifying motion when attending to the limit notation,
which was similar to how the instructor talked about the notation (See Section 5.2.4). At
_ some point during the interview, Jessica wrote x —> a on the paper and said “this arrow
means getting closer and closer to that particular point” (Jessica, 5 March, 2009). On the
other hand, unlike the instructor, the ways students talked about the notation were often
incomplete or inaccurate. Table 6.8 shows some examples of students’ word use when
addressing particular limit notations during the interviews.

Students used words such as “approaches” (Table 6.8, [1], [4], [6], [8], [IO-11]),
“goes to” (Table 6.8, [2]) and “at” (Table 6.7, [3], [7]) when referring to the arrow in the

limit notations of the form lim f (x) = L. There were few occasions in which they talked
x —+a

about the function values or limit after describing the x values approaching a (Table 6.8,
[5], [7], [11]). In the remaining cases, students mentioned the behavior of the x values
and directly wrote the limit values on the paper without explicitly talking about them as

numbers.

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Table 6.8: Some examples of students’ word use when referring to the limit notation

 

 

(Seeclggirertt 6. 3) Notation Word use
Amy Question 1 lim f( x) [1] “Limit as x approaches zero...”
x—)O
Question 3 lim f( x) [2] “x goes to negative five...”
x—>—5
Question 3 lim f(x) [3] “The limit at x equals two”
x—>2
Jessica Question 1 lim f( x) [4] “When x approach [sic] zero...”
x—)O
Question 4 lim F( x) = 1 [5] “x negative to zero, it [the limit] will be
x—>O— one”.
Harry Question 2 lim f(x) [6] “The limit as x approaches negative
x—>-l one...”
Question 3 lim f (x) = —2 [7] “At negative five, it [the function value]
x 9'5 is negative two”.
Question 3 lim f (x) [8] “x approaches infinity...”
x—>°° [9] “Whenx is infinity...”
Keith Question 2 lim f( x) [10] “As the limit approaches negative one
x—->-l_ from the negative side...”
Question 3 lim f( x) [l 1] “As x approaches negative infinity, it [the
x—)—oo function value] would equal two”.

 

It was mentioned in Section 5.1 about word use that the instructor introduced the notation

lim f (x) = L which he addressed as “the limit as x approaches a of f of x equals some

x—>a

number L ” (see Table 5.6, [2]). The students did not refer to the notation in a similar

manner since (a) they rarely uttered the word “limit” when talking about the notation and

(b) they rarely referred to limit as a number. There were also some instances where the

students confirsed which quantities were approaching what or they thought the x values

could reach the limit point. For example, Keith once stated that the “limit approaches

negative one [the limit point] from the negative side” (Table 6.8, [11]) instead of talking

about the x values approaching the limit point. Amy and Harry mentioned the x values

being equal to the limit point when they said “limit at x equals two” (Table 6.8, [3]) and

“when x is infinity” (Table 6.8, [9]), respectively.

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Overall, students talked about the notation for x approaching the limit point in
ways similar to the instructor’s talk. However, many of the students’ utterances regarding
the notation were incomplete or inaccurate. Compared to the frequency with which they
determined limits during the interview, they rarely verbalized the limit notation. They
also did not talk about limit as a number both in the context of the limit notation and the
limit computation problems although many of them consistently wrote the answer as a
number.

6.2.1.4. Infinity
None of the students attempted to plug in infinity, treating infinity as a number, when
they worked on Question 1 and Question 2 of the interview sessions (See Figure 6.3). On
the other hand, some of their explanations and routines suggested otherwise. When I
asked Amy why she did not plug infinity into the function value for the first question, she
said “because infinity is not a number. It is not a defined value. So you can’t just plug
infinity... It cannot give me any answer to anything” (Amy, 3 March, 2009). When
determining the limit at infinity18 for the second question, however, she mentioned
approaching infinity from the right and the left side as if it were a number.

Jessica explicitly talked about infinity as a number. She used the word

“unlimited” for infinity. When working on Question 4 (part c) of the interview problems

(See Figure 6.3), she incorrectly19 attempted to use the limit law for the

 

sum lim x + lim . She said the first limit would be zero and the second limit

x—-)O x——)01() x

 

l8 . . . . . . . . . . . .
ernrt at infinity rs a lrmrt computed at posrtrve or negative mfimty.

19 In order to apply the limit law for a sum, both limits have to exist and equal a number.

196

would be “unlimited”. She wrote 0 for the former limit and +00 for the second limit and
concluded that the answer would be +00.

Jessica: Because zero is a number and you add another number, it will be

the number itself.

Researcher: Okay. Is positive infinity 3 number?

Jessica: Yeah. You can see it as a number. . .Zero add any number like

five, it will be five itself. So when it [zero] is added to positive unlimited,

it will be unlimited itself.

Researcher: So what does this symbol represent? [1 show 00] How do you

think about infinity?

Jessica: Infinity? Unlimited.

Researcher: Is unlimited a number?

Jessica: Yes. You always can find another number larger than this number

[she shows +00] and in this one [she shows -oo], you can always find a

number smaller than this number.

Researcher: Okay but you still think about it as a number in itself?

Jessica: Yes.
Since Jessica thought of infinity as a number, she also considered infinite limits20 as
existing limits. Although the instructor wrote +00 or -00 as the answer to the limit
computation problems, he considered infinite limits as “undefined” or “does not exist”
(See Table 5.14). Jessica’s view of limit as a number suggested that she considered

infinity as an end result rather than a process. However, she extended the graphs she drew

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and also the graph in Question 3 (See Figure 6.3) when determining the limits of
functions at infinity. The instructor talked about infinity as potential, an ongoing process
that never ended, and used the metaphor of continuous motion in the context of
computing limits at infinity (See Sections 5.1.5.3.2 and 5.3.3). He talked about infinity as
an end state in the context of infinite limits (See Section 5.1.5.3.3). For the latter type of
limits, Jason indeed uttered the word “number” when referring to infinity (See Table
5.14). Jessica’s extension of the graphs with the assumption of continuous motion when
determining limits at infinity suggested that she realized infinity as potential in those
instances. Her reference to infinity as a number in the context of infinite limits, however,
showed that she considered it as an end state. Therefore, Jessica’s word use about infinity
resembled the instructor’s discourse on the notion.

Harry also extended the graphs of functions, including Question 3, when he
worked on limits at infinity. Unlike Jessica, he mentioned that infinity is not a number:
“infinity is not a number because infinity is like... you can’t define infinity; it’s an
endless... it can’t begin somewhere and it can’t finish somewhere” (Harry, 27 February,
2009). Although it seemed that Harry was about to call infinity an “endless proCess”, he
did not complete that sentence. His extension of the graphs for limits at infinity signaled
the consideration of infinity as a potential but he seemed to relate infinity to being
undefined or a state of no beginning and no ending.

Keith extended graphs saying that “it [the fitnction] goes ofl” (Keith, 3 March,

2009). While doing so, he thought of infinity as potential. On the other hand, he

 

2

198

Infimte hmrt rs a lrmrt computed either at a pornt or mfimty that has plus or minus mfimty as the answer.

expressed confusion whether to consider an infinite limit as undefined or as being equal
to infinity.

Keith: I would think that they are different. I am still not entirely sure... I

see them as different because undefined and infinity I view as two

different things. Infinity would be continuously going and going and

going; undefined would be closer to like something divided by zero or you

know something that mathematically just doesn’t make sense and so. . .I

guess I am leaned more towards infinity.

Researcher: So when you get infinity as an answer for the limit, do you

think the limit exists overall or not?

Keith: I view them as completely different. In undefined, I would say that

the limit does not exist but in infinity 1 would say the limit exists as

infinity.

Keith’s consideration of infinity as potential and use of dynamic motion was apparent
when he described it as “continuously going and going and going”. Yet, he also'described
an infinite limit as existing. This was similar to Jessica’s view of infinity and was
different than the instructor’s discourse on infinite limits.

During the interviews, students’ word use and routine of extending graphs in the
context of limit at infinity were in accord with the instructor’s word use (talking about
infinity as potential) and routine of using the metaphor of continuous motion. Their word
use was different from the instructor’s in that none of the students referred to an infinite
limit as undefined or nonexistent. Only Jessica mentioned infinity as a number, which

suggested that she viewed it as an end state rather than a process in the context of infinite

199

 

 

 

limits. The instructor’s reference to infinity as a number (See Table 5.14) is consistent
with Jessica’s view of infinite limits. Amy and Harry were explicit not to consider
infinity as a number. However, Amy explored the right and side and left hand side of
infinity, and thus treated it as a number, in the context of limit at infinity. In the
interview, students worked on some limit computation problems which they answered as
“does not exist” but none of the students wrote “does not exist” or “undefined” for
infinite limits. When working on infinite limits in the classroom, Jason consistently

wrote lim f (x) = ioo on the board. At the same time, he verbally mentioned that the
x—>a

limit did not exist or was undefined. In fact, he explicitly endorsed the narrative that
infinity is not a number in the class (See Section 5.4). However, his use of the symbol 00
as being equal to the limit seemed to lead some confusion for students whether to
consider an infinite limit as an existing or an undefined limit.

6.2.1.5. Continuity
When talking about continuity of functions at particular points, none of the students
uttered the word “limit” during the interviews. Amy talked about points of removable
discontinuity as “jumps” and attended to the “open circles”, or holes in graphs when
exploring continuity of functions: “it [the function] is discontinuous because it has an
open circle [referring to the y value where the function does not attain its limit value],
which means that the function jumps essentially” (Amy, 3 March, 2009).

Jessica’s arguments about continuity were in close relationship to connectedness.
When talking about why she thought the function in Question 2 (See Figure 6.3) was
discontinuous at x = —1 , she said “you just need to focus on the point [she shows f(-1)]

that connects both continuous functions”. Similarly, when she explained why the function

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in Question 3 (See Figure 6.3) was not continuous atx = —5 , she said “it [the function] is
discontinuous because we cannot find a connection or link between these two graphs [she
shows the graph on the right hand and the left hand side of x = —5 ]” (Jessica, 5 March,
2009). Besides her arguments about connectedness, she also attended to the instances
where the function jumped.

Harry also used the word “jump” when talking about continuity of a function at a
given point: “it is not continuous because the value [y value] jumps”. He later mentioned
taking his hand off the paper: “so the way I see if a function is continuous. . .Am I taking
my hand off the paper? I had to take it off now so it is discontinuous” (Harry, 27
February, 2009). He also looked whether the function was defined at the limit point.

Keith’s main routine when determining the continuity of a function was to find
the instances where he took his hand off the graphs of the functions: “It [the function] is
continuous because you would not have to pick up your pencil. I know there is a more
mathematical reason but I can’t remember it now”. (Keith, 3 March, 2009). Keith was
aware that this explanation lacked mathematical precision but could not think of a
different way than the intuitive approach of tracing graphs and reporting the points where
he picked up his pencil as points of discontinuity. Besides this, he also talked about
jumps: “It [the function] is not continuous because there is a jump” (Keith, 3 March,
2009)

Students’ views of continuity were in accord with the instructor’s intuitive
approach. Amy attended to the holes in a graph of a function whereas Harry and Keith
mentioned taking their hands off the graph. Jessica’s arguments about continuity showed

that she was relating the notion to connectedness. All these students relied on motion and

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operational word use as they traced the graphs with their hands or pencils. It was
mentioned in Section 5.1.5.4 that Jason determined functions’ continuity based on
whether he could graph them “without taking the chalk off the board” (Table 5.16, [2],
[4]). Therefore, similar to the students, he used the metaphor of continuous motion (See
Section 5.3.3). Although he also introduced the precise definition of continuity based on
limits and introduced properties of continuous functions via their properties of limit, none
of the students talked about continuity in relation to limits during the interview.
6. 2. 2. Students’ discourse with respect to student difi’iculties indicated by research on
learning about limits

In this section, I talk about the responses students gave to the interview problems
with respect to the difficulties identified by research on student learning about limits.
Those difficulties were discussed in detail in Section 2.1.2 and were summarized in Table
2.1. In what follows, I focus on each of the six difficulties addressed by the literature and
whether the students in the study showed signs of those difficulties during the interview
sessions. While doing so, I again pay attention to the instances in which the students’
discourse on limits is similar to or different from the instructor’s discourse.

6.2.2.1. Limit implies continuity

This difficulty refers to the idea that if a firnction has a limit at a point, then it
must be continuous at that point (Bezuidenhout, 2001). The possible instances of this
view of limit are found in Questions 2, 3 and 5 in the interview (See Figure 6.3). Among
the four students, only Amy had this difficulty during the interview. Her main routine

when finding the limit of a function at a given point was to plug in the limit point to the

202

 

function. In case the firnction value was not defined or there was a hole in the graph, she
said the limit did not exist at that point.

[When working on lim f (x) in Question 3] The fact that it is an open
x—)—l

hole [shows the graph of the function at -l] is throwing me off because it
means the function is not continuous. Limits are only defined for
continuous functions Ibelieve.

[When working on lim f (x) in Question 3] I am going to say it [the limit]
x—>3

does not exit because the point is open so it is not continuous. (Amy, 3

March, 2009)
Amy also marked part (a) of Question 5 (See Figure 6.3) as true mentioning that “so what
I know about limits is that they can only exist if the function is continuous at that
particular point [the limit point]” (Amy, 3 March, 2009). Therefore, Amy clearly thought
that limit implies continuity. The other students did not show any sign of the difficulty
throughout the interview. It was mentioned in Section 5.3.1 that the instructor used
algebra-based routines fiequently when computing limits. His first approach was to plug
the limit point into the function. In case this approach did not provide information about
the limit, he then utilized other algebraic routines (See Table 5.21). However, Amy
applied plugging in as the sole routine when computing limits throughout the interview.

6.2.2.2. Limit as the function ’5 value
This view of limit refers to the idea that when finding the limit of a function at a given
point, it is enough to look at the function’s value at that point (Bezuidenhout, 2001). This
view of limit is in close relation to the difficulty “limit implies continuity” and the routine

of plugging in the limit point to the function to find the limit of the function at that point.

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Possible instances of this view of limit were in Questions 1, 2, 3 and 5 (See Figure 6.3).
Amy also had this difficulty. Her initial and only routine when computing limits was to
compute the function value at a limit point. If she obtained a number, she reported it as
the limit value; if she did not obtain a value or if the function did not attain its limit value,
she mentioned that the limit did not exist. She said “every time I see a number, all I want
to do is plug it in. That is all I can remember doing with numbers. . .like doing the
substitution value” (Amy, 3 March, 2009).

During the interview, Harry also attended frequently to the functions’ y values
when finding limits of continuous firnctions. Yet, he seemed to differentiate between the

limit value and the function’s value at the limit point. “[When working on lim f (x) in
x—)3

Question 3] f of three would be equal to two but if you did a limit as x approaches three,
it [the limit] would be one” (Harry, 27 February, 2009). Therefore, he did not talk about
limit as the function’s value.

Jessica and Keith consistently attended to the right hand and the left hand limits
and whether they were equal to each other when finding limits. Even if they utilized
plugging in, they did not generalize this routine as applicable to all functions and showed
no sign of considering limit as the function’s value.

One of the endorsed narratives in the instructor’s discourse was that the function ’5
value at a limit point is irrelevant to the limit value (See Section 5.4). Jason said the
function values would have nothing to do with the limit value in the context of continuity
(See Section 5.1.5.4) but he also frequently mentioned that one could find the limit of

continuous functions by plugging in. Although Jason used different function types for

204

which the routine plugging in could not be used when determining limits, the routine
supported Amy’s and Harry’s focus on the function values when computing limits.

6.2.2.3. Limit as a bound

Considering limit as a bound refers to the idea that “limit is a number or point past
which the function cannot go” (Williams, 1991, p. 221). It was mentioned in Section
6.1.2 that this view of limit is somewhat independent than the other views of limit in that
it seems to be based on the colloquial use of the word limit than the mathematical aspects
of the concept. Everyday use of the term limit might result in realizations of limit as a
constraint or a bound. As a result, students can report the absolute maximum or minimum
value of a function near the limit point as the limit value. None of the students showed
signs of this difficulty during the interview since they did not refer to limit as a boundary
or a constraint, and did not report the maximum or minimum value of fimctions as the
limit values.

6.2.2.4. Limit as monotonic

This view of limit is based on formal teaching (Comu, 1991) and is based on the
expectation of ‘nice behavior’ fiom a function. In other words, students might assume
that a firnction has to be strictly increasing or strictly decreasing in order to have a limit.

Students having this conceptual obstacle could have difficulty finding limits of constant

functions, piecewise fimctions and also sequences21 in which the subsequences converge
to different values (Tall and Vinner, 1981). Questions 2 and 3 (See Figure 6.3) were sites
for revealing this view. None of the four students showed any sign of this difficulty

during the interview. The instructor used different types of functions when computing

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limits giving students an opportunity to consider limits of constant and piecewise
functions. Student responses to Question 1 and 2 were consistent with the instructor’s
discourse.

6.2.2.5. Limit as unreachable

This view considers limit as a value that is approached but not reached and is
based on the dynamic view of limit (Tall & Schwarzenberger, 1978; Williams, 1991).
Students who have this view of limit have difficulty finding the limits of continuous
functions where the functions attain their limit value at the limit point. Possible instances
in which this view of limit could be observed included Questions 1, 3 and 4 of the
interview sessions.

It was mentioned that Amy had the difficulties “limit implies continuity” and
“limit as the function’s value” during the interview. Since she said only continuous
functions would have limits and applied the routine of plugging in the limit points to
functions when computing limits, it was unlikely that she would consider a limit as
unreachable. However, she talked about asymptotes as exceptions to “normal graphs or
regular limit problems” (Amy, 3 March, 2009). According to her, in the case of the
asymptotes, a function’s y value and the limit would not be reached.

Jessica and Harry clearly stated that a function could attain its limit value and did
not show any sign of the difficulty during the interview. Keith initially had this difficulty
when he talked about limit as approaching a number (but not being equal to that number)
and when he determined the limit of the function in Question 3 at the limit point x = -5.

The function was continuous at that limit point and so attained its limit value. Keith was

 

l . . .
2 errts of sequences are beyond the scope of thrs study.
206

uncomfortable when reporting the limit value for this problem since “it looks so much
like plugging in the number. It looks like you are evaluating f of x at negative five but I
know that is not true” (Keith, 3 March, 2009).

Keith did not consider the routine of plugging in as suitable when finding a limit
of a function since he was aware that functions did not have to attain their limit values,
which was a narrative the instructor endorsed in the classroom (See Section 5.4). Indeed,
Keith immediately determined the limit of the function in Question 3 at the limit point x
= -1 , where the function had a limit but did not attain its limit value.

Researcher: So what bothers you about this limit [I show lim f (x) = ? in
x—)—5

Question 3]?
Keith: When it [the function] gets the value, it feels like you are just

plugging that in and I know that is not correct. It [ lirn f (x)] was
x—-)—5

throwing me off because the point was actually defined.
Researcher: You were quick when finding this limit though [I

show lim f (x) = ? in Question 3].
x—)—l

Keith: In that case you are not really plugging negative one, you are

plugging in a number very very close to negative one so it is going to

approach this value [the limit value]. When it [the firnction value] is not

there, then I don’t feel like plugging in anymore”.
Keith confused the idea that x values approach the limit point without reaching it with the
idea that the function values approach the limit value without reaching it. Therefore, he

talked about limit as unreachable at the initial stage of working on Question 3. When

207

finding the limit of the function in the same question at the limit point x = 2 (See Figure
6.3), on the other hand, he gave the function’s value as the limit value (the function was
constant near x = 2). When I asked him why he was not uncomfortable that the function
attained its limit value at that point, he mentioned that he remembered the instructor
giving examples of constant functions, where the limit value was equal to the function’s
value at the limit point. This provided an opportunity for Keith to think more about his
initial response for limits of continuous functions. In the end, Keith realized focusing
only on the right hand and the left hand limits and whether they are equal at the limit
point when computing limits. He did not have any difficulty working with continuous
functions after this instance.

Note that, when referring to the limit notation lim f (x) = L , the instructor used
x —>a

phrases such as “gets closer and closer to”, “approaches”, “goes to” and “becomes” when
talking about x -> a (See Section 5.2.4), which suggested that the x values approach the
limit point a but never reach a. Similarly, when talking about the behavior of a function
near the limit point, the instructor mentioned function values “approaching”, “getting
closer and closer to”, and “going to” the limit value (See Section 5.1.2). Such operational
word use for the firnction values is in accord with Keith’s view that the function values
approach the limit value without reaching it. The instructor’s operational word use in the
context of computing limits and the limit notation might, in fact, have a larger impact

given that 13 of the 23 students in the study marked the statement about “limit as

unreachable” as true in the diagnostic survey (See Table 6.2).

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6. 2. 2. 6. Limit as approximation

The assumption underlying this view of limit is that in order to find the limit of a
function at a point, it is sufficient to look at the behavior of the firnction at points
successively closer to the limit point (Bezuidenhout, 2001; Tall & Schwarzenberger,
1978). Pararneswaran (2007) argues that considering limit as an approximation might
also result from the common classroom practice of rounding numbers to convenient
significant digits. Similar to “limit as unreachable”, this realization of limit is based on
the dynamic view of limit. Question 4 of the interview session (See Figure 6.3) provided
the most useful context to gain more information about the view “limit as
approximation”.

When working on part (a) of Question 4, all of the students initially stated that the
firnction values approached the value 1. Part (a) of the question included one student’s
method of plugging in successive values into the function to find its limit at the limit
point. The student represented the related x and y values in tabular form.

Amy: It [the function] gets closer and closer to one.

Jessica: It [the function] approaches one from both sides.

Harry: It [the function] is approaching. . .it gets closer and closer to one. It

is a continuous function.

Keith: I would say that it [the limit] would approach one. When he takes

the numbers below zero, they get closer and closer to one; as he takes the

numbers above zero, they get closer and closer to one. Yeah, in both cases

they get closer and closer to one. So that is why I would say one.

209

All students used dynamic views based on the metaphor of continuous motion and used
words operationally when talking about the limit in part (a). Note that Harry directly
assumed continuity of the function from its tabular representation and that the students
talked about limit as a process rather than a number.

When working on part (b) of the question, which asked whether the student’s
method could be used to find the limit, Amy and Keith were skeptical of the student’s
method “since we never did it in the class” (Amy, 3 March, 2009), “because I was never
taught it” (Keith, 3 March, 2009). Jessica referred to the student’s method as follows:

We can’t use this method to prove the result of that limit but we can find

the limit and check the answers. I mean you have plan one, plan two. And

you can solve the problem by plan one and if you want to check the

answer of plan one, it is right. You can use this way [plugging in

successive points into the function] to check it. (Jessica, 5 March, 2009)

Harry chose the statement which considered limit as an approximation as best describing
his view of limit in the diagnostic survey (See Statement 5 in Table 6.1 and Table 6.6).
He said that the student’s method was correct but was ineffective since there were too
many numbers plugged into the function and the numbers looked complicated. He

explicitly mentioned that “the student tried to approximate” (Harry, 27 February, 2009)
and if the student worked with “easier” numbers”, the method would be more effective.

In fact, Harry used a simpler version of the method when finding lim3 f (x) in Question 3
x—>

 

22 Easier numbers referred to plugging in numbers such as -l, -0.5, 0.5, 1 rather than -0.00001 or
0.0000000] into the function when computing its limit at x =1 (See Question 4 in Figure 6.3). Harry’s
comments about such easy numbers might be connected to Parameswaran’s (2007) idea of rounding
numbers to their convenient significant digits.

210

(See Figure 6.3). In that particular case, he looked at the function’s behavior from the
right and the left side of the limit point by plugging in only one value from each side.
Besides that instance, however, he did not utilize this method during the interview.

When working on part (c) of Question 4, which included a firnction represented
algebraically and which had exactly the same y values for the x values plugged into the
function in part (a), all of the students plugged the limit point into the function and
mentioned that the function’s limit was infinity. Therefore, they obtained two different
limit values for a function: its tabular representation suggested that its limit was 1 but its
algebraic representation suggested that the limit was infinity. All of the students realized
this conflict during the interview. Keith was skeptical of the student’s method in part (a)
but did not choose one answer over the other although he said “I can see how the
student’s method can be wrong” (Keith, 3 March, 2009). Amy and Harry thought their
computations of the limit in part (c) might have been wrong and concluded that the limit
was one. Jessica was initially undecided about which answer to choose but then
remembered that she considered the student’s method as a checking mechanism and
concluded that the limit was not defined for the function in part (c). At that point, she
referred to the student’s method in part (a) as “just showing the tendency. It does not
show what really happens [at the limit point]” (Jessica, 5 March, 2009).

’ Except for Harry, students did not refer to limit as an approximation. Amy and
Keith explicitly mentioned that finding limits by plugging in successive values into a
function was a method the instructor did not use in the classroom. As a result, these
students were skeptical of this method. Despite such skepticism, however, students’

consideration of limit as a process by means of dynamic motion was so strong that when

211

_ _‘.-

dealing with the conflict presented to them in part (c) of Question 4, they were more
suspicious of their own algebraic computations than how the limit was determined in part
(a) of the question. It was mentioned in Sections 5.2 that the instructor did not use tabular
representations of functions during the classroom observations. Except for one instance
(See Table 5.2, [2]), he also did not mention plugging in successive values into the
function to compute its limit at the limit point”. Students’ confusion whether to consider
limit as an approximation or not did not seem to result from the approximation aspect of
the difficulty (except for Harry) but from the dynamic view of limit and the consideration
of limit as a process.
Summary

The results of the interview sessions, in conjunction with the findings of the
diagnostic survey, suggest that students primarily used the dynamic view of limit and
considered limit as a process. Their discourse on the informal aspect of limit was similar
to the instructor’s since students used words operationally when investigating functions’
behavior near the limit points. In addition to this, students used the metaphor of
continuous motion as a metarule for realizing limits, which was consistent with the
instructor’s routine in the context of informal aspect of limit (See Section 5.3.3). On the
other hand, students’ discourse was different from the instructor’s with respect to the
endorsed narratives. Although Jason endorsed the narrative limit is a number at the end of

each limit computation problem, students endorsed the narrative limit is a process since

 

23 . . . . . . .
In that one case, hrs comments about mvestrgatrng the behavror of the function were only verbal; he did

not actually plug in those points into the function.
2 12

they rarely talked about limit as a number during the diagnostic survey and the interview
sessions.

The formal definition Jason introduced in the classroom provided an opportunity
for students to learn about the static aspect of limit, which supported the consideration of
limit as an end state. Although students accepted statements about a formal view of limit
as true in the diagnostic survey and the interview sessions, their word use about the
formal definition remained operational than obj ectified. The four students who
participated in the interview session talked about the similarity of the formal definition
with the informal view of limit, paying attention to the part of the definition that
introduced how to read and write the limit notation. For them, the notation entailed
movement, and connected well with their informal realization of limit. Jason’s word use
was consistently obj ectified in the context of the formal definition. Moreover, he used
words signifying proximity instead of motion (See Section 5.1.5.2) and used the
metaphor of discreteness (See Section 5.3.4) instead of the metaphor of continuous
motion. Students, however, used words operationally and relied on motion when talking
about the formal definition during the interview.

Students used graphs as visual mediators throughout the interview to make sense
of functions when computing limits whereas Jason often used them to introduce a
definition, theorem or fact about limits (See Section 5.2.3). His primary mode of
representing functions was symbolic, not graphical. The limit notation was another
context in which students’ discourse was analyzed. Similar to Jason, students used a
family of words such as “approaches” “goes to”, and “at” when talking about the arrow in

the limit notation. Unlike the instructor, students’ utterances regarding the notation were

213

often incomplete or inaccurate and they did not refer to limit as a number despite their

use of the equal sign to write lim f (x) = L.
x-)a

Computing limits, referring to the limit notation and the informal definition of
limit were the contexts in which Jason shifted his word use between operational and
objectified. Although he did not write the operational words he uttered on the board (See
Section 5.2.1), students in the study relied on the spoken rather than written aspects of his
word use when forming their realizations of limit. Jason’s investigation of the behavior of
functions near the limit and his operational word use in that context were consistent with
students’ realizations of limit as a process.

Students’ confusions about process and the end result of a process were also
apparent when they talked about infinity. During the interview sessions, the students who
endorsed the narrative infinity is not a number treated it as a number depending on the
context. When working on limits at infinity, students considered infinity as potential, that
is, a process that goes on and on, which was similar to how Jason talked about infinity in
the same context (See Section 5.1.5.3.2). Students’ routine of extending graphs of
functions at positive and negative infinity supported their realization of infinity as
potential. Extending graphs of functions was a routine Jason did not utilize in the
classroom but his word use was consistent with students’ use of the metaphor of
continuous motion in the context of limits at infinity. When working with infinite limits,
students talked about infinity as an end state, similar to Jason’s discourse. Jason talked
about infinity as a “number” or an end state in the context of infinite limits (See Section
5.1.5.3.3). His word use differed from students’ in that he considered an infinite limit as

undefined whereas students considered it as being equal to infinity. The way Jason used

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the limit notation as a visual mediator might have played some role in students’
realization of an infinite limit as an existing limit. In the context of infinite limits, he first
wrote the limit as being equal to i 00 and then said that “the limit does not exist” or “it is
undefined” (See Table 5.14).

Students’ responses in the interview sessions showed that they focused on the
intuitive but not the precise definition of continuity the instructor introduced in the class.
Unlike Jason, none of the students talked about continuity in relation to limits. However,
they seemed to adopt the phrases such as “jump” and “taking the hand off the graph” that
Jason used in the classroom when talking about continuity (See Section 5.1.5.4). Jason
mentioned that one could find limits of continuous firnctions by plugging in although he
explicitly endorsed the narrative the function ’s value at the limit point is irrelevant to the
limit value (See Section 5.4). Amy, however, considered Jason’s routine of plugging in as
the main method to compute a limit and considered it as the fimction’s value. She also
thought that in order to have a limit, a function had to be continuous. As a result, she had
the difficulties “limit implies continuity” and “limit as the function’s value” during the
interview. Plugging in was a routine Jason utilized in the classroom as the first method of
finding the limit of a function presented algebraically (See Section 5.3.1). On the other
hand, he also used different algebra-based routines in case plugging in did not work.
Given the dominance of the dynamic view of limit in students’ realizations, however, it is
likely that some students would consider plugging in as the only means of computing a
limit.

The majority of the students agreed with the view “limit as unreachable” in the

diagnostic survey (See Statement 4, Table 6.2). The instructor’s shifts in word use when

215

referring to the limit notation as well as his routine of using continuous motion in the
context of computing limits were consistent with students’ realization of limit as
unreachable. The phrases “approaches”, “gets closer and closer to”, “becomes”, and
“goes to” suggest moving towards the limit value without reaching it. As a result,
students could confirse the idea that x values approach the limit point without reaching it
with the idea that function values approach the limit value without reaching it. During the
interview sessions, Amy and Keith showed signs of the difficulty “limit as unreachable”.
Amy stated that, for the case of asymptotes, a function cannot reach its limit value
whereas Keith talked about limit as approaching a value and was initially uncomfortable
when finding the limits of continuous functions.

Jason did not use tabular representations of functions formed by plugging in
successive values to the function when computing limits. That contributed to students’
skepticism about such a method as being suitable for determining limits. Despite the
skepticism, Harry had the difficulty “limit as approximation” at some point of the
interview. The instructor’s use of different types of functions, such as piecewise,
discontinuous, and trigonometric functions, helped students work comfortably with
functions other than polynomials during the interview. Consistent with Jason’s discourse,
the participants in the interviews did not have the view “limit as bound” and “limit as
monotonic”.

Overall, the contexts in which students struggled with limits coincided with the
contexts in which the instructor shifted his word use. Recall that Jason’s utterances about
limits were mainly objectified (See Table 5.5) and his operational word use took place

when he communicated his ideas just verbally, without writing on the board (See Section

216

5.2.1). Although Jason’s operational utterances constituted only about 16% of his total
utterances (See Table 5.5), the dominance of operational word use in students’ discourse
suggest that they adopted the spoken aspects of Jason’s discourse in which he was less
formal and less precise. This can further be supported by the fact that students talked
about both limit and continuity through the intuitive definitions Jason provided in spoken
discourse rather than the precise definitions he wrote on the board.

Intuitive aspects of limits result from the dynamic view based on the metaphor of
continuous motion. Students’ frequent use of this metaphor as a metarule for realizing

limits supports their consideration of limit as a process. In fact, limit was not objectified

till 17th century by mathematicians and the metaphor of continuous motion was also

present in their discourse till 19th century (See Section 3.3). Therefore, it is possible that

learners intuitively realize the concept as a process at the early stages of their learning. In
the following chapter, I explore the classroom discourse on limits in relation to the
historical development of limits to examine how the instructor’s and the students’

discourse compare and contrast with those of the mathematicians’ over history.

217

CHAPTER VII
CLASSROOM DISCOURSE ON LIMITS IN RELATION TO THE HISTORICAL
DEVELOPMENT OF LIMIT RELATED CONCEPTS THROUGH THE
COMMOGNIT IV E LENS
In this chapter, I explore the classroom discourse on limits with respect to the
historical development of infinity, infinitesimals and limits. In Chapter III, the historical
development of these concepts was described with a focus on particular elements of the

commognitive framework: word use (objectification), metarules and metaphors. I then

identified the historical junctures in the development of discourse on limits that resulted

in changes in the metarules], also called metadiscursive rules, of the existing discourse in
order to extend it further (See Sections 2.2.2 and 3.4).

The commognitive framework considers developmental transformations as
resulting from the interplay of individualization and communalization since

...mathematical discourse is a historically established activity practiced

and extended by one generation after another and taught in schools for the

sake of further continuation. Mathematics students are this supposed to

join this activity rather than invent their own, idiosyncratic version. (Sfard,

2008,p.203)
Therefore, we can think of learning mathematics as the individualization of the
communal activity of doing mathematics. Teachers are among the sources that enable the
communication of historically established mathematical discourse to students and play

important roles in the process of individualization of the communal.

 

l I also argued that the exploration of metaphors is an important part of the exploration of metarules since
using metaphors for the realizations of mathematical concepts is a type of metarule (See Chapter IH).

218

It should be noted that “the order of things in the processes of discourse
individualization is different from that in historical processes of object-creation” (Sfard,

2008, p. 177). For example, in this study, students were immediately introduced to limit

in an objectified manner (as a number) although the objectification2 of the concept took
centuries for mathematicians. On the other hand, the findings of the study suggested that,
just because they were exposed to an already obj ectified mathematical concept did not
necessarily mean that students could objectify the concept in their discourse. The
participants in the study had difficulties considering limit as a number, a distinct
mathematical object, at the end of their instruction (See Chapter VI). Thus, although the
order of individualization processes could differ from historical processes of object
creation, the realization of limit as an object of mathematics seems to be challenging for
both students and, historically, for mathematicians.

For the case of limit and infinity, objectification historically resulted in changing

the discourse about processes to discourse about end-states. It was not till 17th century

that Cauchy objectified limit as a number and it was not till 19th century that Cantor

objectified infinity as an end state (See Sections 3.1 and 3.3). The realization of
infinitesimals as objects of mathematics is still under question (See Section 3.2). One of
the benefits of using commognitive framework as a lens to focus on the conditions and
assumptions of object creation over history is that it enables us to acknowledge
objectification as quite a complex phenomenon in the development of mathematical
discourse. I considered objectification of infinity, infinitesimals and limit as among

important junctures in the development of discourse on limits over history because

 

2 Objectification of a concept changes the talk about a process to the talk about a product or an end state

219

 

objectification of these concepts also resulted in changes in the metaphors and metarules
of the previously existing discourse (See Section 3.4). A question that will be pursued
later in the chapter is whether these junctures can also be important in the development of
students’ discourse on limits.

In what follows, I examine the instructor’s and students’ discourse on limits with
respect to the historical development of limit related concepts. While doing so, I compare
and contrast the metarules and metaphors leading to the objectification of the concepts
over history with those reflected in the instructor’s and students’ discourse. I also look for
similarities and differences between the order and characteristics of object creation over
history and in this beginning-level calculus course.

7.]. The instructor 's discourse with respect to the historical development
of limit related concepts

If we look at the order in which the instructor (Jason) introduced the concepts
during the eight observed lessons, we see some elements of the historical development of
limit related notions playing out in the classroom. Jason started his discussions of limits
by first introducing average rate of change and then defining instantaneous rate of change
as the limit of average rate of changes of a function as the time intervals got smaller and
smaller (See Table 5.1 and Section 5.1.5.1). While doing so, he also used graphs to
represent the average rate of changes with secant lines and the instantaneous rate of
change with a tangent line, which was the limit of the secant lines. This resembles closely
Newton (1643-1727) and Leibniz’s (1646-1716) approaches to the limit concept since

they both considered limit as a process when they obtained a tangent line at a point

 

through reification (See Section 2.2.1).

220

 

through the use of secant lines (See Sections 3.2 and 3.3). Besides this, the sequencing of

the calculus concepts in both the curriculum3 and the classroom teaching (See Table 5.1)
followed Cauchy’s outline of calculus concepts. In Cauchy’s design, the notion of limit
precedes derivatives and integrals as well as series and sequences (See Table 3.2).
Although Cauchy objectified “limit”, referring to it as a number obtained at the end of the
limit process, his definition of the concept relied on dynamic motion and infinitesimal use
(See Section 3.3). Jason’s introduction of limits, together with his operational word use
when he described the behavior of functions near the limit points suggest that his focus
was more on the intuitive aspects of the limit notion than the formal aspects. On the other
hand, Jason also chose to introduce the formal definition of limit and solved a proof
problem, which might be uncommon for a beginning-level calculus course. He
considered the formal definition as optional but noted that if students took further
mathematics courses, like analysis, this definition would come up again (See Section
5.1.5.2).

In what follows, I examine how and whether the metarules (in particular, using
specific metaphors) related to infinity and limit4 in the historical junctures I identified in
Chapter III (See Table 3.3) were reflected in the instructor’s discourse.

Historically, potential infinity is consistent with Aristotle’s (3 84-322 B.C.) view

of infinity as an indefinite process. Actual infinity, however, is consistent with Cantor’s

 

3 The order Jason presented the topics was also the order they were presented in the textbook (See Thomas
et al., 2008 for their content outline for calculus).

4 Jason could not possibly address, and was not expected to address, the metaphors and metarules related to
infinitesimals in his discourse since mathematically sophisticated version of the theory is much beyond the
scope of a beginning calculus course. However, since he used infinitesimals in his discourse on limits, this
signaled that be relied on the intuitive and dynamic aspect of limit in those cases than the formal aspect of

221

 

 

(1845-1918) view of the concept as an end state (See Section 3.1).

In Chapter III, I considered Cantor’s objectification of infinity as the main historical
juncture in the historical development of the concept (See Table 3.3). This juncture led to
changes in the previous discourse on infinity and changed the metarules of counting,
measuring, and some properties of arithmetic such as addition and division. Moreover,
using the metaphor of indefinitely continuous processes, a metarule for realizing infinity,
was changed to using the metaphor of infinitely iterative steps, each with an end result
(See Section 3.1 and Table 3.3).

Jason talked about infinity as potential and actual depending on the context. In the
context of limits at infinity, he talked about infinity as a continuous process that went on
and on (See Section 5.1.5.3.2, Table 5.11); in the context of infinite limits, he talked
about it as an end state (See Section 5.1.5.3.3, Table 5.14). That he referred to infinity as
an end state was also visible when he wrote limits as equal to plus or minus infinity (See
Section 5.1.5.3.3). Recall, however that he explicitly endorsed the narrative infinity is not
a number in the class (See Section 5.4). Jason talked about infinity consistent with both
Aristotle’s and Cantor’s realizations of the concept but he did not call students’ attention
to the shifts in his word use and did not address using distinct metaphors as different
metarules when making sense of infinity. In other words, although objectification of
infinity was present in his discourse, it remained implicit for students.

I have identified two historical junctures in the development of discourse on limit
over history. The first juncture was the objectification of limit by Cauchy (See Table 3.3).

In his definition, Cauchy described limit as a number obtained at the end of the limiting

 

the concept. His use of infinitesimals took place only when he talked about limits in an informal manner,
consistent with their use in the historical development of the limit concept (See Section 3.2).

222

 

process. On the other hand, his description of the concept was dynamic and based on the
assumption of continuous motion (See Section 3.3). The second juncture was
Weierstrass’ introduction of the formal definition of limit which eliminated spatial
motion and continuity from the previously existing discourse on limits (See Table 3.3).
By arithmetization of geometry, Weierstrass replaced using the metaphor of continuous
motion, a metarule for realizing limit, with using the metaphor of discreteness (distance
between discrete numbers) (See Section 3.3). The second juncture transformed some
other metarules of mathematics. For example, elimination of motion also resulted in
realizations of functions as algebraic rules but not as graphs of curves.

Jason was carefirl and accurate when he described the informal and the formal
definitions of limit. His definition of the informal aspect of limit corresponded to the
formal aspect of the concept in that he did not use words signifying motion but used
words signifying proximity such as “arbitrarily close” and “sufficiently close” (See
Figure 5.4). When he utilized the informal definition to compute limits, however, his
word use was operational. His descriptions of the behavior of a given function near the
limit point made fiequent use of continuous motion and treated limit as a process. At the
end of every limit computation problem, however, Jason talked about limit as a number.
The instructor’s realization of limit as a number together with his routine of using
continuous motion as a metaphor for the process aspect of the concept are consistent with
Cauchy’s realization of limit. Similar to infinity, objectification of limit was present in
Jason’s discourse but it remained implicit for the students since Jason did not address

when the process aspect of limit ends and the product aspect begins.

223

When talking about the formal definition of limit, Jason’s word use was
consistently obj ectified and his words signifying motion gave way to words signifying
proximity in terms of distance (See Section 5.1.5.2), which is consistent with
Weierstrass’ realization of limit. As a mathematician, Jason was aware of the metarule of
using the metaphor of discreteness since motion was eliminated in his discourse when he
talked about the formal definition. On the other hand, he did not draw students’ attention
to the changing metarules and his shift to objectified word use in this context. Therefore,
similar to infinity and informal aspect of limit, the metarules remained tacit for students
in the classroom. Moreover, rather than drawing students’ attention to the diflerences
between these two aspects of limits, he tried to find ways to talk about how they were
similar (See Section 5.1.5.2 for how Jason connected the formal definition to the informal
definition of limit).

In terms of metarules related to the uses of the definitions, Jason was more
explicit. He mentioned that the informal definition “convinced us what these limits are
that we have been computing in the class...” (See Table 5.7, [4]) and he made explicit
that “to really make sure these [limit computation methods] work in mathematics, we
have to prove those things. In order to prove something, we need a precise definition”
(Table 5.7, [5-5a]). Therefore, Jason clearly mentioned that the informal definition was

different than the precise or formal definition of limit in that, one could not prove the

existence of a limit with the formers.

 

5 However, he did not address that in order to prove the limit of a given function as a number L, one needs
to first hypothesize that number. Said differently, the formal definition of limit does not enable a
constructive proof; one can only prove or disprove an already guessed number L to be the limit of a
function with the precise definition.

224

 

In summary, the historical junctures changing the metarules of infinity and limit
coincided with the contexts in which Jason shifted his word use between operational and
obj ectified in his discourse. He talked about infinity and limit as processes or end states
depending on the context. In each of the contexts, his reasons for utilization or exclusion
of the metaphor of continuous motion remained implicit for students, supporting the tacit
nature of metarules in his discourse. It was mentioned in Chapters V and VI that the
formal definition of limit provided a context in which Jason could attend to the static
aspect of limit, which supported its realization as an end state (a number). Although
Cauchy’s realization of limit also provides a context in which it is possible to talk about
limit as a number, as Jason did, the findings of the diagnostic survey and interview

9

sessions indicated that the dynamic element in the informal definition supported students

consideration of limit as a process6. Jason’s consistent referral to limits as objectified and
his elimination of motion related words in his discourse on the formal definition showed
some evidence that he was aware of the metarules related to the formal realization of
limit. In fact, he used the metaphor of continuous motion in the context of informal
definition and the metaphor of discreteness in the context of the formal definition (See
Sections 5.3.3. and 5.3.4). However, there was no evidence in his discourse that he
considered attending to these distinct metarules as pedagogically relevant in the
classroom. Instead, he talked about how the definitions were similar to each other.

Whether calling students’ attention to the changing metarules related to infinity and limit

 

6 Recall that students associated informal aspect of limit with motion and rarely referred to limit as a
number in the diagnostic survey and during the interview sessions (See Chapter VI).

225

 

 

could be useful for student learning will be discussed in the next section where I examine
students’ discourse in relation to the historical development of the concepts over history.

7. 2. Students’ discourse with respect to the historical development of
limit related concepts

Students may develop many idiosyncratic ways to justify their narratives of
mathematics, which are not necessarily compatible with the narratives endorsed by the
mathematical community. For the case of limit and infinity, however, there are
similarities between some of the difficulties addressed by research on learning about
limits and the historical development of limit. Note that “limit as bound”, and “limit as
unreachable” are among the conceptual obstacles students have about limits (See Section
2.1.2, Table 2.1). It was mentioned in Chapter III that Lagrange (1736-1813) opposed
some mathematicians’ use of the limit method:

MacLaurin and d’Lambert used the idea of limits; but one can observe the

subtangent is not strictly the limit of subsecants, because there is nothing

to prevent the subsecants from further increasing when it has become a

subtangent. True limits... are quantities which one cannot go beyond,

although they can be approached as close as one wishes. (Lagrange, 1799,

as cited in Schubring, 2005, p. 293)

Schubring (2005) notes that Lagrange’s arguments were based on

the lacking of the concept of absolute value. . .so that it seems as if the

variable goes beyond the limit; the criticism is also at the problem, which

has always remained controversial, whether a variable can definitely reach

the limit or is only allowed to come close to it at any rate (p. 293).

226

In these arguments, we see Lagrange problematizing whether limit is a bound when he
says “true limits. . .are quantities which cannot go beyond...” and also the elements of the
debate whether a limit is reachable. Therefore, the conceptual obstacles “limit as bound”
and “limit as unreachable” were also present in the development of discourse on limits
over history. In this study, students’ responses to the diagnostic survey and the interview
problems did not reveal any difficulties in terms of the realization of limit as a bound
(See Section 6.2.2.3). The responses, on the other hand, provided evidence in terms of the

realization of limit as unreachable, which is based on the dynamic view of limit (See

 

Sections 6.1.1, 6.1.2, and 6.2.2.5).

It was discussed in Chapter VI that the students in the study frequently used the
dynamic view of limit based on the metaphor of continuous motion. Such realization of
limits is consistent with the historical development of limit in that (a) the initial

underpinnings of the concept emerged fi'om physical problems that were based on
motion, and (b) it was not till 19th century that spatial motion was eliminated fi'om the

existing discourse on limits. Said differently, similar to the students in the study, the
dynamic view of limit was also the intuitive and initial view with which mathematicians
made sense of the concept historically. Using continuous motion was also a metarule that
shaped mathematicians’ realization of infinity as an indefinite process for centuries.
Students in this study showed similar views of infinity when they talked about it as
potential in the context of limit at infinity (See Section 6.2.1.4). Therefore, objectification
of limit and infinity was challenging both for students in the study and mathematicians
over history. Once objectified, mathematicians were able to separate process from

product, which was not necessarily the case for the students in the study.

227

It was argued in Section 5.5 that, although the instructor can flexibly use limit and
infinity as a process or product and distinguish the characteristics of each realization
depending on the context, students might be unlikely to notice those characteristics or
realize the differences and similarities underlying these concepts. The findings of the
diagnostic survey and student interviews support this claim for the participants in this
study. Both in the diagnostic survey and interviews, students talked about limit as a
process or a descriptor of how a function behaves and rarely talked about it as a number
(See Sections 6.1.2 and 6.2.1). In other words, they did not objectify limit in their
discourse. At the end of their instruction, these students did not make the transition from
the realization of limit as a process to its realization as a number as introduced by
Cauchy, which is one of the historical junctures I identified in the historical development
of limit (See Table 3.3). The students also did not move fiom a discourse on limits based
on the metarule of using dynamic motion to one based on the metarule of using a static
and discrete realization of the concept, which was the contribution of Weierstrass. I
identified his introduction of the formal definition of limit as a second juncture in the
historical development (See Table 3.3).

The interview sessions provided some information in regard to students’
realizations of infinity. Students talked about infinity as a process and an end result

depending on the context. When finding limits at infinity, they talked about the concept

as an ongoing process7 (See Section 6.2.1.4), which is consistent with Aristotle’s view of
infinity as potential. When working on infinite limits, they talked about infinity as an end

state but none of the students considered an infinite limit as undefined. This signaled that

228

 

 

 

they treated infinity as an existing entity, although two of the participants explicitly said
infinity is not a number. One participant talked about infinity as a number in the context
of infinite limits and one mentioned the limit existed as infinity (See Section 6.2.1.4). I
identified Cantor’s objectification of infinity as an accumulation point as a historical
juncture in the development of discourse on infinity. Students in the study seemed to
have some aspects of this transition present in their discourse. Although they talked about
infinity as a process when the x values approached infinity, they treated the concept
differently when they obtained infinity as the limit of a given fimction. Recall that the
instructor considered an infinite limit as undefined or not existing (See Section 5.1.5.3.3)
but the students in the study were not comfortable considering an infinite limit as
undefined. Besides this difference, their discourse on infinity was similar to the
instructor’s in that they talked about it as potential in the context of limits at infinity
(consistent with Aristotle’s realization) and as an end state in the context of infinite limits
(consistent with Cantor’s realization).

The contexts in which students struggled most during the interview sessions about
infinity and limit coincided with the contexts in which the instructor’s word shifts took
place (See Section 5.1). Those contexts also coincided with the historical junctures that
changed the existing metarules about infinity and limit to extend the discourse on limits.
Changes in the metarules historically led to the objectification of limit related concepts
and eventually to the changes in the endorsed narratives of calculus.

Both the informal and the formal definition of limit are contexts in which it is

possible to talk about limit as objectified (a number). The first definition is compatible

 

7 Limit at infinity was a context in which students extended graphs of functions at infinity and implicitly

229

 

 

with Cauchy’s realization whereas the second one is compatible with Weierstrass’

realization of limit8. It was mentioned in Chapter VI that the metarule of using
continuous motion dominated students’ discourse on limits and supported their
realization of limit as a process. In other words, students focused on the process element
in the informal aspect of limit more than the end result of the process. Therefore, it
seemed necessary that, in order for them to realize limit as a distinct object of
mathematics (a number), their assumption of dynamic motion needed to be challenged.
The formal definition of limit was a context in which it was possible for Jason to
highlight the elimination of motion fi'om the discourse on limits. On the other hand, his
word use and the metarules, which were consistent with the historical junctures in the
development of limit, remained implicit for the students in the class.

Similar to limit, the historical juncture leading to the objectification of infinity
with the distinct metarules related to different realizations of the concept were reflected
in Jason’s discourse. However, the word use, metarules and metaphors remained implicit
for the students since he did not address when the process aspect of infinity ends and the
product aspect begins.

Summary

More often than not, the junctures resulting in discursive transformations require

changes in the metadiscursive rules of defining, substantiating and recording narratives of

mathematics. Note that since learners of mathematics are expected to join in the

 

used the metaphor of continuous motion (See Section 6.2.1.4).

Most importantly, since they describe the same mathematical concept, both definitions are also
compatible with each other. They are similar in that they characterize limit as a number. They differ,
however, in their description of the “process” aspect of limits. Cauchy acknowledges the process through
motion whereas Weierstrass eliminates it and uses absolute values of differences between particular
numbers instead.

230

 

 

historically established activity of mathematics, “gradual modification of metarules that
govern the students’ mathematical discourse is one of the goals of school learning”
(Sfard, 2008, p. 2008). What makes such modification difficult for students is that
“metarules are mostly tacit” (Sfard, 2008, p. 221). In Chapter III, I focused on some
elements of the commognitive framework such as word use, metarules and metaphors
while investigating the historical development of limit related concepts to identify the
tacit metarules explicitly as the realizations of infinitesimals, infinity and limit evolved

through history. I identified the historical junctures in the development of discourse when

they led to changes in the metarules of the existing mathematical discourseg. Such
junctures often resulted in the objectification of limit related concepts.

The exploration of the instructor’s discourse in terms of the historical junctures I
identified in Chapter 111 (See Table 3.3) revealed that the contexts in which Jason shifted
his word use coincided with the junctures that changed the metarules of previously
existing discourse on infinity and limit over history. The instructor talked about the
informal and formal definition of limit in different ways, which were compatible with the
historical development of limit, but did not draw attention to his change of word use and
did not address the metarules these definitions are based on in the classroom. Similarly,
he talked about infinity as potential and actual depending on the context but did not
highlight how those contexts and the distinct realizations of infinity differed from each
other.

The examination of the students’ discourse in terms of the historical junctures

revealed that some aspects of limits and infinity students struggled with were consistent

 

In this study, I only rdentrfied the junctures changing the metarules of exrstrng discourse on lrmrts over

231

 

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with the transitions mathematicians went through over history. Students could not
distinguish process from product and they also were not aware of the elimination of
dynamic motion in the formal definition of limit. As a result, they struggled with some of
the interview session problems. Therefore, the junctures in the development of discourse
on limits over history also seemed to be critical in the development of students’
discourse.

It is important to note that I do not consider the historical transitions as identical
to those of students’ since students may have many other realizations of limits that are
not present in the development of discourse over history. Students are introduced to the
elements of the discourse on limits in a very compact manner and in a short period of
time so they do not have as much chance to reflect and realize different aspects of the
concepts. On the other hand, some of the difficulties about limits seem to be common for
both mathematicians over history and students. The objectification of infinity and limit,
the dominance of the dynamic view as well as the realization of “limit as bound” and
“limit as unreachable” are present in both the experts’ and the students’ discourse on
limits as conceptual obstacles.

Limit related concepts inherently present learners of mathematics with difficulties
both over history and in today’s classrooms. It is due to this fact that the analysis of
historical development of limits through the commognitive framework has some useful
implications for classroom discourse on limits. Word use, metaphors and metarules are
some of the elements of mathematical discourse highlighted by the commognitive
framework that might remain tacit both in the historical development and also in the

instructor’s discourse on limits. Students could benefit from those elements of discourse

 

history but one can also talk about such junctures in the development of an individual’s discourse.

232

 

 

 

being explicitly addressed in the classroom since the historical contexts in which experts
struggled about limits were compatible with the contexts in which students in the study
struggled when working with limit related concepts. Therefore, it is likely that the
historical junctures during which mathematicians changed and revised their previously
existing realizations of limits might be useful for student learning. Said differently,
historical junctures changing the metarules of previously existing discourse on limits can
form some of the junctures students have to go through as they participate in the

discourse on limits.

233

 

 

CHAPTER VIII

SUMMARY, DISCUSSION AND CONCLUSIONS

In this chapter, I return to the questions that motivated this study and guided my
analyses. Some educational researchers interested in the concept of limit have
investigated the historical development of limit related concepts. Others have focused on
student learning and investigated the conceptual obstacles students face as they learn
about limits. A missing link in this chain in literature is the teaching of limits at the
undergraduate level. Therefore, one of the contributions of this study is to provide
information about teaching of limits by focusing on one instructor’s discourse.

The literature on historical development of limits and student learning is primarily
based on the assumptions of a cognitive framework that highlight the nature of
difficulties about limits in terms of misconceptions. According to Sfard (2001), the
cognitivist framework is based on the metaphor learning as acquisition, which considers
learning “as the storage of information in the form of mental representations” (p.20).
From the point of cognitivist framework, understanding is defined as relating new
knowledge to prior knowledge by refining the existing mental representations. By doing
so, this view highlights the individual nature of learning and views it as the acquisition of
the necessary mental schemes one either possesses or not. In this study, I used the
commognitive framework developed by Sfard (2008) to explore teaching and learning of
limits. There were two reasons I used this different lens to work on the issues related to
learning of limits: (a) By viewing thinking as an individualized form of communication,
this fiamework highlights the importance of social constructions in individuals’ thinking.

Mathematics learning is considered as participation in mathematical discourse, whose

234

rules are determined communally throughout history. As a result, obstacles learners face
may be attributed to communicational rather than cognitive difficulties. (b) From this
perspective, the three elements considered in the study —— historical development,
teaching, and learning of mathematics — are viewed as interrelated and connected
elements of discourse development. Given these presuppositions, I explored students’
discourse on limits and the historical development of limit related concepts through the
commognitive framework. Therefore, another contribution of this study to educational
research is that it pilots an analytical tool with which to explore students’ discourse on
and the historical development of limits.

. I pursued three specific research questions in this study: 1) How is the discourse
on limits generated by the instructor in a beginning college-level calculus classroom? 2)
Given the instructor’s discourse on limits, how do students talk about limits in a
beginning college-level calculus course? and 3) How do the elements of discourse on
limits as generated over history compare and contrast with the discourse on limits
generated in a beginning-level calculus course? I addressed these questions in Chapters
V, VI, and VII, respectively. Here, I provide a brief summary for each question to set up
my later comments for discussion.

8.1. Summary of findings
For my first research question, I analyzed one instructor’s discourse on limits by

using the four elements of the commognitive framework: word use, visual mediators,
routines, and endorsed narratives. The findings for this question indicated that the
instructor shifted his word use depending on particular mathematical contexts about

limits. Those contexts were identified as the informal and formal definition of limit,

235

 

 

computing limits and symbolic notation. The instructor’s alternating word use signified
infinity and limits as processes or as end states in his discourse. The instructor did not
address such shifts in his word use in the classroom, and did not make specific the
differences between the realizations of limit as a process and as a number.

The visual mediators the instructor utilized in the classroom mainly consisted of
graphs of functions and symbolic notation. The relationship between the limit notation

lim f (x) = L and the instructor’s word use provided interesting results in regard to
x —) a

realizations of infinity and limits as processes or as products. Although he talked about L
as a number, his referral to the arrow in the notation using words such as “approaches”,

9’ ‘6 99 66

“gets closer and closer to , goes to , becomes” supported the realization of limit as a
process. Similarly, when referring to the symbol oo in the limit notation, he talked about
infinity as potential (3 process) in the context of limits at infinity whereas he talked about
it as actual (an end state) in the context of infinite limits.

The instructor referred to limit as a process when he investigated the behavior of a
given firnction near the limit point. On the other hand, each time he talked about infinity
and limits as processes, his utterances were spoken rather than written. The primary
narrative the instructor endorsed in the classroom was limit is a number. He explicitly
endorsed the narrative limit is a process only twice during the eight observed classroom

periods and he did so only verbally. His language, however, along with the conventional

reading of the symbol lim f (x) = L that includes the word “approaches”, suggested an
x —)a

implicit endorsement of the narrative limit is a process.
The only context in which the instructor’s word use, visual mediators, routines

and endorsed narratives about limits were most consistent with each other was the formal

236

definition of limit. In this context, he did not utter any words signifying motion but
uttered words signifying proximity. By doing so, he changed his use of the metaphor of
continuous motion to the metaphor of discreteness. However, similar to his shifts in word
use, the instructor’s utilization of distinct metaphors remained implicit for students since
he did not mention the metaphors he used in the classroom.

For my second research question, I explored how students in the instructor’s
section talked about limits at the end of the unit on limits. I analyzed students’ discourse
with respect to (a) the instructor’s discourse on limits, and (b) research on student
learning about limits. The findings of this part of the study indicated that the contexts in
which the instructor shifted his word use coincided with the contexts in which students
had difficulty talking about limits. When making sense of the limit concept, students
mainly adopted the instructor’s spoken words than written ones. They frequently
endorsed the narrative limit is a process, consistent with the instructor’s discourse when
investigating the behavior of the function near the limit point. On the other hand, students
rarely, if ever, talked about limit as a number. The findings also indicated that students
used graphs as visual mediators more frequently than the instructor when computing
limits. In terms of symbolic notation, students’ utterances were either inaccurate or
incomplete. Similar to the instructor, they used the metaphor of continuous motion when
referring to the limit notation and investigating the behavior of functions near the limit
points. Unlike the instructor, they did not use the metaphor of discreteness in their
discourse on limits. As a result, they could not talk about the differences between the

informal and the formal aspects of limits.

237

When analyzing students’ discourse in relation to research on student learning
about limits, I first investigated whether students showed signs of the difficulties
indicated by the literature. I then explored whether elements of the instructor’s discourse
were reflected in students’ realizations of limits. The findings for this part of the study
indicated that some elements of the instructor’s discourse (e.g., using a variety of
functions, introducing the formal definition, not computing limits by plugging in
successive values near the limit point) supported students’ accurate responses for
different realizations of limit. This was also reflected in the fact that students did not hold
some of the difficulties reported in the literature (e.g., “limit as bound” and “limit as
monotonic”) and these were the very ideas absent from the instructor’s discourse. Yet,
some other elements of his discourse (shifts in word use and routines) supported students’
realization of limit as a process rather than a number.

For the third question, I examined the instructor’s and the students’ discourse on
limits in relation to the historical development of limit related concepts through the
commognitive lens. In particular, I explored whether the historical junctures I identified
in the development of discourse over history were reflected in classroom discourse on
limits. The findings here indicated that the contexts in which the instructor shifted his
word use coincided with the historical junctures that resulted in changes in the metarules
of the discourse on limits. Those contexts were the informal and the formal definition of
limit as well as infinity. The changing metarules related to the concepts over history were
reflected in the instructor’s use of words and different metaphors. However, he did not

explicitly address the metaphors and did not attend to the metarules underlying different

238

 

 

 

 

realizations of limit and infinity. Therefore, the study provided evidence with respect to
the tacit nature of metarules in the classroom.

The exploration of students’ discourse with respect to the historical development
of limits indicated that some of the difficulties addressed by research on student learning
were also present in experts’ discourse over history. Those difficulties were “limit as
bound”, and “limit as unreachable”. In addition, similar to students’ discourse, the
discourse on limits over history was based on the metaphor of continuous motion for
centuries before limits were considered as distinct objects of mathematics. Objectification
of limits were challenging both for the mathematicians over history and for the students
in the study.

Most of the junctures I identified in the historical development of limit related
concepts signaled objectification as the milestones leading to the expansion of the
previously existing discourse on limits. I identified the elimination of spatial motion and
continuity fiom the discourse on limits as another type of historical juncture in the
historical development of discourse. The students in the study did not show signs of
objectification and elimination of motion in their discourse on limits. They talked about
limit as a process but not as a distinct mathematical object (a number). Moreover, their
view of limit was based on the metaphor of continuous motion and not discreteness. On
the other hand, the students who participated in the interview sessions showed signs of
objectification of infinity as an end-state in their discourse.

In sum, the contexts in which students struggled most in the diagnostic survey and
the interviews coincided with the historical junctures that changed the metarules about

limits. Those contexts also coincided with those in which the instructor’s shifts in word

239

 

use took place. Therefore, the junctures in the development of discourse on limits over
history also seemed to be critical for the development of students’ discourse in the study.
8. 2. Discussion

The study illustrates the usefulness of the commognitive framework to analyze
features of mathematical discourse. The exploration of discursive patterns, which could
not be revealed through a cognitive approach, enables the analysis of the connection and
conflicts among the elements of an individual’s discourse (e.g., connection between
words and symbolic notation; words and routines; routines and endorsed narratives;
written and spoken words, etc). For example, although the instructor in the study
endorsed the narrative limit is a number, some of his routines when investigating the
behavior of functions near the limit points supported the narrative limit is a process.

The commognitive framework also enables the identification of the
communicational failures through the analysis and comparison of word use, visual
mediators, routines and endorsed narratives of multiple discursants. For example,
although the instructor’s word use on limits was predominantly objectified, students’
word use was predominantly operational. Similarly, although the instructor often used
graphs as visual mediators to explain definitions and theorems about limits, students
often used graphs to identify the behavior of functions.

The use of commognitive framework highlighted an important and complex
relation between the limit notation and word use. Students in the study used dynamic

motion when making sense of the limit notation lim f (x) = L and connected it to the
x —>a

informal definition of limit. They did not focus on the equal sign and the limit L, a

distinct number, as much as they focused on the x values approaching a and the y values

240

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approaching L. Note that although the notation supports talking about the limit as a
number, it does not necessarily support the elimination of motion. The conventional way
to read the notation includes the word “approaches”, a word signifying motion.
Therefore, even in a formal definition of limit, the symbolic notation and the words
associated with it might result in conflicts with respect to the consideration of limit as a
number or process. In this study, the instructor and students used a family of words
besides “approaches” that signified motion (e.g., “goes to”, “gets closer and closer to”,
etc) when talking about the limit notation. Given that the students’ utterances about the
limit notation were often inaccurate or incomplete, it seems crucial that the relationship
between the symbols and the words associated with those symbols are made clear in the
classroom.

The relation between the written and spoken words in the classroom discourse
was another finding of the study revealed through the commognitive fi'amework.
Although the instructor’s operational word use corresponded to a small portion of his
overall utterances about limits, the students in the study primarily focused on operational
word use in the instructor’s spoken discourse and disregarded his written discourse when
forming their realizations of limit. Students’ lack of attention to the written aspects of the
classroom discourse was also consistent with their approach to the textbook. The students
participating in the interview sessions mentioned that they never used the textbook for
reviewing material. Three of those four students did not even use the lecture notes while
reviewing material or preparing for the exam. This suggests that students relied on what

they heard, saw and recalled from the classroom while forming their realizations of limit.

241

It is possible that students did not utilize the written aspects of the instructor’s and
the textbook’s discourse due to their inexperience with the material. Only four of the
students attending to the instructor’s section took calculus in high school; this was the
first time the remaining students learned about the topic. In addition, students learned the
ideas related to limits in only a two and a half week period over eight lessons. As a result,
it might be hard for them to follow the precise and objectified word use about limits in
the textbook and the instructor’s written discourse. Instead, they focused on the informal
and intuitive aspects of limits in the instructor’s spoken discourse. This signals that they
only attended to the elements of the instructor’s discourse that helped them cope with the
requirements of the course.

Students’ coping strategies were clearer when they talked about how they
prepared for the exam. Although they did not use the textbook and the lecture notes, all of
the students in the interview session stated they revisited the homework problems the
instructor assigned from the textbook when preparing for the exam. One participant
(Amy) explicitly stated that she ignored the instructor’s explanations unless that
particular material would be tested on the exam. Therefore, students seemed to focus on
learning how to solve particular problems for the exam than learning about the
conceptual meanings of limits.

The parallels between the development of discourse on limits over history and
students’ discourse support the idea that it might be inevitable for students to realize limit
as a process at the early stages of their learning. Williams (1991) notes that

Just as students’ informal limit models tend to parallel those of the

mathematical community prior to Cauchy, it is possible that only by

242

appreciating the sorts of problems that motivated Cauchy’s work will

students be motivated to understand its implications. Perhaps this is to say

that the very historical and cultural contexts that lent vitality to the original

work are the best medium through which to approach the understanding of

that work. (p. 23 5)

An important issue that remains to be addressed is whether or at what point
students need to make the historical transitions in their discourse on limits. The study
does not suggest that the dynamic view of limit should be discarded from the classroom

discourse on limits since it can be the most useful tool with which to initially make sense

of the concept. In fact, both the informal and the formal definition of limit1 are contexts
in which it is possible to talk about limit as objectified (a number). However, in this
study, students’ reliance on dynamic motion did not support their realization of limit as a
number. Instead, they talked about limit as a process. Therefore, it seemed necessary that
their metarule of using continuous motion while making sense of limit needed to be
challenged at some point in order to support the objectification of limit (as a number) in
their discourse.

The formal definition of limit was a context in which it was possible for the
instructor to highlight the elimination of motion from the discourse on limits and
challenge students’ assumptions of continuous motion. On the other hand, his shifts in
word use (signifying proximity instead of motion) and use of the metaphor of
discreteness remained implicit for the students in the class. It seems important that the

metarules behind the historical junctures, which were reflected in the instructor’s

243

discourse when talking about the informal and the formal definition, are explicitly
mentioned in the classroom in order for students to distinguish the process aspect of limit
from the product aspect. Although students might not need to know about the existential
quantifiers, the symbolism, and proofs associated with the formal definition, familiarity
with the metarules and metaphors of the formal theory might enhance their realization of
limit as a number but not as a process or “happening”.

Note that the course was designed to address informal aspects of the concepts and
their applications more than their formal aspects and justifications. One possible reason
for this structure might be due to the diversity of students required to take the course with
respect to their majors. In this study, the 31 students enrolled in the classroom were from
18 different majors. 18 of those students were first-year; nine of them second-year; three
of them third-year and one of them was a fourth-year student. Although such diversity

and inexperienced body of students make it difficult for the instructor to attend to all

aspects of calculus concepts in detailz, these characteristics of the classroom also make it
vital that the main ideas about limits are communicated effectively.

Among the most striking results of this study is how the instructor’s accurate use
of words, his endorsed narrative that limit is a number, and his utilization of different
metaphors as metarules for realizing limits remained invisible for the students. Being a
mathematician, the instructor communicated the limit related ideas, which were

compatible with the development of the discourse on limits over history, precisely.

 

The former definition rs consrstent wrth Cauchy’s approach whereas the latter IS consrstent wrth
Weierstrass’ approach (See Section 3.3).

In fact, it IS possrble that the instructor’s introduction of the formal definition was atypical for this
particular course.

244

On a pedagogical dimension, however, his communication with his students did not result
in shared meanings about limit related concepts. The existence of such gaps in the
mathematical communication between the instructor and the students might have resulted
from the participation structure in the class. In the period of observations, the “talk” in
the classroom was unidirectional: from instructor to students. There were few instances
of student-teacher interaction and no student-student interaction. This hindered students’
active involvement in meaning making in the classroom.

A second possible reason contributing to the communicational difficulties can be
tied to the instructor’s level of explicitness in his discourse. It was mentioned that
students learned most of the ideas about limit in a compact manner and in a short period
of time. In order for students to “decipher” the obj ectified discourse on limits in the
required time, it might be necessary for the instructor to be explicit about his word use,
visual mediators, endorsed narratives, routines (including use of metaphors), and, most
importantly, the connections among these elements of his discourse. That the contexts in
which the instructor shifted his word use coincided with the contexts in which students
struggled most in the study (which also coincided with the historical junctures changing
the metarules and metaphors of limits) highlights the importance of explicitness
regarding the elements of the mathematical discourse in the classroom.

8.3. Conclusions
In this section, I discuss the implications and limitations of the study. I then

provide a list of further questions and ideas this study encourages us to think more about.

245

 

8.3.1. Implications of the study

This study contributes to educational research in several ways. It pilots an
analytical lens, the commognitive framework (Sfard, 2008), for analyzing classroom
discourse in a beginning-level calculus course by focusing on the limit concept. The
study also takes a step to fill an important gap in the literature by exploring the teaching
of calculus at the undergraduate level. In addition, commognitive framework offers tools,
namely word use (through objectification), metarules and metaphors, to interpret
classroom discourse in relation to the historical development of mathematical concepts.

The utilization of the commognitive framework provided revealing information
with respect to the patterns in word use, visual mediators, routines, and endorsed
narratives of mathematical discourse on limits. The fiamework also pointed to the
dynamic relationship among these four elements of discourse and how the relationships
are formed and reformed depending on the mathematical context. In particular, the
commognitive framework highlighted the significance of word use in mathematical
communication. A co gnitivist framework would not help us recognize the discursive
patterns and the importance of mathematical contexts in the development of
mathematical discourse since its approach to learning is mostly context-independent.
Therefore, one of the implications of the study is that the commognitive framework is a
promising lens to investigate the teaching and learning of other mathematical concepts.

Besides these theoretical implications, the study also has some practical
implications for the teaching and learning of limits. First, although it took mathematicians
centuries to develop and obj ectify mathematical ideas, students can be immediately

presented with the objectified version of the notions. Since objectification hides the

246

layers of mathematical discourse, it may be important for teachers to unpack the
objectified discourse so that students can have access to the main ideas. The unpacking of
the objectified mathematical discourse can be made possible by an explicit focus on the
word use, visual mediators, routines, endorsed narratives of the discourse and their
relation to one another. Word use and routines, the metarules characterizing the repetitive
actions of participants, seem to be the main elements of discourse that can remain tacit in
the classroom. Therefore, it might be important for teachers to talk about when a routine
is implemented (e. g., mentioning that we use the metaphor of continuous motion when
realizing limits informally) as much as how a routine is implemented (e. g., uttering words
signifying motion when talking about limits informally).

Second, since mathematics learning is participating in the mathematical discourse,
active interaction can promote the discussion and exploration of different uses of
mathematical words and routines in the classroom. Such an approach can give teachers
and students chances to compare and contrast their individual realizations with the
communally agreed upon realizations of the mathematical concepts. By doing so, it can
also contribute to the unpacking of words and metarules related to the concepts.

Third, the junctures in the historical development changing the metarules of
mathematical concepts might be useful for teaching. In this study, such junctures seemed
to be critical for students since the contexts in which mathematicians tackled with limits
over history coincided with the contexts students struggled. In case there are such

parallels between student learning and historical development of other mathematical

247

 

concepts3, awareness of the historical junctures can be quite important for teachers to
recognize and address some of the student difficulties in their classrooms.
8.3.2. Limitations of the study

The study is conducted as a case study. It is an attempt to investigate the teaching
and learning of limits at the undergraduate level by focusing on one classroom.
Therefore, the findings of the study may not be applicable beyond this context. In other
words, I do not claim that it is possible to generalize these findings to a population of
undergraduate calculus classrooms. Note also that the application of a particular lens, in
this case the commognitive framework, brings with it many presuppositions leading to
that decision (e.g., consideration of mathematical knowledge as socially constructed,
consideration of thinking as individualized form of communication, acknowledging the
communicational nature of student difficulties, etc.) Since educational research is
interpretive, I also acknowledge that another researcher using a different lens (e. g., who
operates from a cognitivist framework) could see the classroom in different ways than I
did.
8. 3. 3. Further questions

The study raises additional questions and ideas for further research. First, the
analysis of curricular materials in terms of their discourse on limits and infinity is needed
to complement how the mathematical ideas developed throughout history are reflected in
today’s classrooms. In the context of undergraduate mathematics classrooms, textbooks
are among the main curricular sources with which instructors organize their lessons. For

the utilization of the textbook by students, the study raises important questions: How can

 

3 . . . . . . ,
In fact, there is also evrdence there are Similarities between some of students conceptual obstacles about

248

we encourage students to interact with the textbook except for end-of-section problems?
How can the instructors find ways of incorporating textbooks so that students think of the
textbook as a significant resource for learning? The textbook authors have more chances
to think about and revise their word use, visual mediators, routines and narratives of
mathematics. As a result, the textbook’s discourse can be more consistent than
instructors’ given that the instructors of undergraduate mathematics are not necessarily
trained in pedagogy of mathematics.

The study is an initial attempt to explore the teaching of calculus at the
undergraduate level. More research on the teaching of mathematics in undergraduate
setting is crucial since calculus is a requirement for many majors (e.g., economics,
engineering, physics, and mathematics), and students might not be introduced to any of
the calculus concepts prior to their university courses.

Finally, since the implementation of the commognitive framework for limits
proved to be a useful lens to gain more information about student learning, repetition of
the study in other undergraduate classrooms and for other mathematical concepts seems
to be a promising way to enhance our knowledge of classroom discourse.

Additional questions raised by the study include: Would the investigation of
historical junctures resulting in changes in the metarules of other concepts of
mathematics besides limits be useful to gain information about the conceptual obstacles
students face as they learn about those concepts? Would a teaching approach that
explicitly focuses on word use, metarules and metaphors related to the mathematical

concepts make a difference in students’ realizations of the concepts? What are the

 

firnctions and the historical development of functions (S fard, 1992).

249

conditions under which students seem to be “ready” for making discursive changes as

reflected in the historical development of mathematical concepts?

250

APPENDIX A
DIAGNOSTIC SURVEY

(Williams, 2001, p. 348)

1. Please mark the following six statements about limits as being true or false:

1. T F A limit describes how a function moves as x moves toward a
certain point.

2. T F A limit is a number or point past which a function cannot go.

3. T F A limit is a number that the y-values of a function can be made
arbitrarily close to by restricting x-values.

4 T F A limit is a number or point the function gets close to but never
reaches.

5. T F A limit is an approximation that can be made as accurate as you
wish.

6. T F A limit is determined by plugging in numbers closer and closer
to a given number until the limit is reached.
II. Which of the above statements best describes a limit as you understand it? (Circle one)
1 2 3 4 5 6 None

111. 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 as x—rs is some number L.

IV. If possible, write down a rigorous definition of limit.

251

APPENDIX B

INTERVIEW SESSION PROBLEMS

x when x 2 0 , ,
1) If f (x) = 3 , what can you say about lrm f (x) , 11m f (x) ,
x when x < 0 x—rO x—r—oo
lim f (x) ?
x—M'D

Is the function continuous at 0? How about negative and positive infinity?

7- , _
2) If f (x) = x +1 when x S 1, what can you say about lim f (x) lim f (x) ,
x—l when x > —1 x—r—l x-r—oo
lim f(x) ?
x—roo

Is the firnction continuous at — l ? How about negative and positive infinity?

252

 

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253

4) A student was given a function F and asked to find the limit of F as x approached 0.

He plugged in numbers on each side of 0 and made the following table:

x

-.1

-.01

—.001
-.0001
-.00001
-.000001
-.0000001
-.00000001

. l

.01

.001

.0001
.00001
.000001
.0000001
.00000001

a) What can you say about lim F (x) ?
x—rO

F (x)

0.9

0.99

0.999
0.9999
0.99999
0.999999
0.9999999
0.99999999

1.1

1.01

1.001
1.0001
1.00001
1.000001
1.0000001
1.0000000]

b) What do you think about the student’s method of finding this limit? Is he correct?
Can we find the limit of functions using this method? Why/Why not?

c) Given F (x) = x +1 + , what can you say about lirn F (x) ?

1020x x~+0

(Williams, 1991, p. 224)

254

5) Which statements below must be true if f is a function for which lim f (x) = 3 ?
x—-)2

a) f is continuous at the point x = 2
b) f is defined at x = 2

C) f (2) = 3

d) lim{f(2+h)—3}= 0
h—rO

e) For every real numbers > O , there exists a real number 6 > 0 such that if
0<|x—2| <6,then|f(x)—3| <3

f) None of the above mentioned statements

6) Let a function f (x) be defined on an open interval about [the point] x0 , except
possibly for x0 itself. We say that the limit of f (x) as x approaches to x0 is the

numberL , and write lim f (x) = L if for every numbers > 0 , there exists a
x—rxo

corresponding number 6 > 0 such that for all x , 0 < Ix - x0] < 6 :> I f (x) — Ll < e .
(Thomas et al., 2008, p. 91)

a) Please explain in your words what this definition means for you.

b) How is this definition similar/different than the informal definition of limit you
used in your class?

c) In your opinion, what is the purpose of this definition?

255

 

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