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COMPARISON OF NATIVE-ENGLISH AND NATIVE-
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COMPARISON OF NATIVE-ENGLISH AND NATIVE—KOREAN SPEAKING
UNIVERSITY STUDENTS’ DISCOURES ON INFINITY AND LIMIT

By

Dong-Joong Kim

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Mathematics Education

2009

ABSTRACT

COMPARISON OF NATIVE-ENGLISH AND N ATIVE—KOREAN SPEAKING
UNIVERSITY STUDENTS’ DISCOURES ON INFINITY AND LIMIT

By
Dong-Joong Kim

This study investigated and compared how native-English and native-Korean
speaking university students, who received their education respectively in the US. and in
Korea, thought about the concepts of inﬁnity and limit. The primary motivation for this
study was the discontinuity in Korean and the continuity in English between the non-
mathematical and mathematical discourses on inﬁnity and limit: non-mathematical uses
of the mathematical words inﬁnity and limit appear only in English. Based on the
communicational approach to cognition, according to which mathematics is a kind of
discourse, the characteristics of students’ discourse on the topics were identiﬁed. The
participants’ discourse was scrutinized with an eye to the common characteristics as well
as culture- and education-related differences.

The setting for the study was a calculus class for university students in the US.
and Korea. Methodology involved surveys and interviews. A total of 132 English
speakers and 126 Korean speakers participated in the survey. Within each linguistically
distinct group, twenty paired representatives were selected from the survey participants
for follow-up interviews. The detailed discourse analyses of the interview transcripts
generated preliminary hypotheses in the interviewees’ discourse on inﬁnity and limit.
Overall analyses of response patterns involved searching for frequencies and percentages
of the survey participants’ responses and comparing the proportions of the two groups

through chi-square analysis to conﬁrm the emerging hypotheses. Data from both sources

were used to elaborate and verify hypotheses. In order to attribute the speciﬁc source of
differences in mathematical discourses of the two groups, students’ backgrounds which
were likely to influence their mathematical discourses on inﬁnity and limit were also
investigated in this study.

It is concluded from this study that the mathematical discourse of US. English
speakers seemed to be growing continuously from their non-mathematical discourse, but
there was no such continuity in the case of Korean speakers. The use of the mathematical
nouns inﬁnity and limit by English speakers was processual, just as it was in their non-
mathematical, colloquial uses, whereas Korean speakers’ language was more formally
mathematical and structural. This shows in developing their discourses on inﬁnity and
limit, English speakers seemed to build more on their previous everyday experience with
discourses on inﬁnity and limit, whereas Korean speakers had no choice but to rely on the
structural discourse of mathematical textbooks. These differences in the use of the words
inﬁnity and limit between US. and Korean groups seemed to result in differences in
endorsed narratives accepted by the students as true, routines they used in problem
solving, and visual mediators they applied. In general, none of the mathematical
discourses, either of English or Korean speakers, was fully compatible with the canonical
mathematical discourse.

The results of this study provide strong support for the conjecture that non-
mathematical discourse shapes mathematical discourse of students — a fact that should be
kept in mind in both the research on advanced mathematical concepts and teaching

advanced mathematics.

DEDICATION

For my parents, my wife, Noah, and Ian

iv

ACKNOWLEDGMENTS

I would like to acknowledge my thanks to the many people who made this study
possible. Throughout this study and my doctoral work, the love, encouragement and
support of my advisor, Dr. Joan Ferrini-Mundy left an indelible impression on my life. Dr.
Anna Sfard, my co-advisor, patiently guided me throughout the writing process and
continuously provided me insightful comments and an enthusiastic spirit. Thank you, Drs.
Joan Ferrini-Mundy and Anna Sfard, for helping me to grow professionally. Next, I am
grateful to all of my committee members, Drs. Glenda Lappan, Wellington Ow, and Woo-
Hyung Whang, for their support and willingness to help me throughout this process.

Thanks also to my colleague, Dr. Elizabeth Brown at Indiana State University,
for her willingness to read and edit my dissertation. I particularly want to thank Dr. Woo-
Hyung Whang, Dr. In-Chul Jung, and other graduate students at Michigan State
University, Jung-Bun Park, Ga—young Ahn, and Joo-Young Park for helping with
translations from Korean to English of the questionnaire and the interview transcripts. Of
course, my family has also been with me through this process. Mom and Dad, I would
like to thank you for loving and encouraging me. I want to give special thanks to my wife,
Dr. Jeong-il Cho, for loving me. She and our sons, Noah and Ian, have been with me
under both sad and happy circumstances. I could not have completed this study without
them. Noah and Ian, I love you both very much. Lastly, I wish to thank Pastor Won and

the church members in East Lansing for their warm caring and support.

TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................... viii
LIST OF FIGURES .......................................................................................................... xii
CHAPTER I. INTRODUCTION AND RATIONAL .......................................................... 1
CHAPTER II. THEORETICAL BACKGROUND ............................................................. 4
Epistemology and History of the Notions of Inﬁnity and Limit .................................. 4
Previous Research on Student Learning on Inﬁnity and Limit .................................... 5
Previous Research on Use of Colloquial Language in Learning Mathematics ........... 6
Previous Research on Comparative Studies of Language ........................................... 7
Communicational Approach to the Study of Mathematics Learning ........................... 9

The Need for an Additional Approach ................................................................. 9
Communication Approach ................................................................................. 10

The Question to be Answered in This Research ........................................................ 13

The Pilot Study .......................................................................................................... 13
Summary .................................................................................................................... 15
CHAPTER III. DESIGN OF STUDY ............................................................................... 17
Research Questions .................................................................................................... 17
Overview of the Method ............................................................................................ 17
Method ....................................................................................................................... l8
Context ............................................................................................................... 18
Participants ......................................................................................................... 25

Data .................................................................................................................... 26

Analysis ............................................................................................................. 38
CHAPTER IV. ANALYSIS AND FINDINGS .................................................................. 41
Background Data about the Students and their Past and Present Mathematical
Learning ..................................................................................................................... 41
Students' Background ......................................................................................... 42

Students' Current Mathematics Learning ........................................................... 50

Attitudes toward Mathematics ........................................................................... 6O
Comparability of the E-group and K-group ....................................................... 62

Students’ Discourse about Inﬁnity and Limit ............................................................. 64
Discourse on Inﬁnity .......................................................................................... 65
Discourse on Limit ............................................................................................. 98
CHAPTER V. SUMMARY, DISCUSSION, AND CONCLUSIONS ............................. 138
Summary of Findings ............................................................................................... 139
Research Question 1 ........................................................................................ 139
Research Question 2 ........................................................................................ 140
Research Question 3 ........................................................................................ 144

vi

Discussion of Findings ............................................................................................. 151
Reasons for Differences in Mathematical Discourse on Inﬁnity and Limit ....151

Generalizabiltiy ........................................................................................................ 157
Limitations of the Study ........................................................................................... 157
Further Questions ..................................................................................................... 159
Conclusions .............................................................................................................. 160
Theoretical Conclusions ................................................................................... 160

Practical Implications ....................................................................................... 162
APPENDIX A: The Pilot Study Questionnaire ................................................................ 165
APPENDIX B: Others ..................................................................................................... 168
APPENDIX C: Questionnaire .......................................................................................... 179
REFERENCES ................................................................................................................ 204

Vii

LIST OF TABLES

Table 3.1. Colloquial and mathematical uses of infinity and limit in English and Korean18
Table 3.2. Summary of the characteristics of MSU and KU ............................................. 21

Table 3.3. Percentages of Students’ responses to the question about high school textbooks

............................................................................................................................................ 21
Table 3.4. The course and content categories for limit ...................................................... 38
Table 3.5. Theoretical components assessed by each item ................................................ 40
Table 4.1. Demographic characteristics of the survey participants ................................... 43
Table 4.2. Summary of textbooks used in high school ...................................................... 44
Table 4.3. Summary of course series taken in high school ................................................ 44

Table 4.4. Summary of the participants’ recollection of experiences with inﬁnity and limit
............................................................................................................................................ 21

Table 4.5. Grades in which students recalled ﬁrst hearing the words inﬁnity and limit ....47

Table 4.6. Percentages of students reporting having studied speciﬁc topics related to
inﬁnity in high school and the results of chi-square .......................................................... 48

Table 4.7. Percentages of students reporting having studied speciﬁc topics related to limit
in high school and the results of chi-square ....................................................................... 48

Table 4.8. Percentages of students’ responses to the classroom environment items and the
chi-square results ............................................................................................................... 52

Table 4.9. Percentage distributions of extra lessons and homework and the chi-square
results ................................................................................................................................. 53

Table 4.10. Percentages of Students’ responses to the out-of—school activity items and the
chi-square results ............................................................................................................... 55

Table 4.11. Mean (M) and SD results to the extent of time spent out-of—school ............... 56

viii

Table 4.12. Summary of students’ descriptions of how to study calculus ......................... 57

Table 4.13. Students’ views of themselves as mathematics learners and the t-test results of
the comparison of percentage of the Korean and E-speakers’ responses .......................... 61

Table 4.14. The role of mathematics in Students’ lives and the t-test results of the
comparison of percentage of the Korean and E-speakers’ responses ................................ 62

Table 4.15. Summary of interview responses using the word inﬁnite in items I and IX ...66
Table 4.16. Summary of interview responses using the word inﬁnity in items I and IX ...67

Table 4.17. Distribution of the contexts of use of the word inﬁnite and the chi-square

results of the comparison of difference between the E-group and K-group ...................... 68
Table 4.18. Distribution of types of entities described with the word inﬁnite ................... 68
Table 4.19. Distribution of the use of the word inﬁnitely as a descriptor of many ............ 69
Table 4.20. Distribution of sentences not using the noun inﬁnity ...................................... 70
Table 4.21. Distribution of the use of inﬁnity as process ................................................... 70

Table 4.22. Distribution of endorsed narratives for comparing pairs of two 'sets in item H

............................................................................................................................................ 21
Table 4.23. Comparisons between ﬁngers and toes in item II-a ........................................ 73
Table 4.24. Comparisons between odd and even numbers in item II-b ............................. 74
Table 4.25. Comparisons between odd numbers and integers in item II-c ........................ 75
Table 4.26. Distribution of phrase-driven and structural uses of inﬁnity .......................... 76
Table 4.27. Distribution of routines in comparison tasks in item H .................................. 79
Table 4.28. Distribution of contradictory answers in items II-b and H-c .......................... 80
Table 4.29. Distribution of endorsed narratives in item II-c and item 111 ......................... 83
Table 4.30. Distribution of the word use of bigger ............................................................ 83

ix

Table 4.31. Distribution of routines in comparison tasks in items II-c and III .................. 84

Table 4.32. Comparisons between even numbers and integers in item 111 ........................ 86
Table 4.33. Distribution of processual and structural uses of inﬁnity ............................... 87
Table 4.34. Distribution of routines in comparison tasks in items 1H and IV .................... 88
Table 4.35. Distribution of the word use of larger in item IV ........................................... 89
Table 4.36. Comparisons between the two inﬁnite sets A and B in item IV ..................... 90
Table 4.37. Distribution of processual and structural uses of inﬁnity ............................... 91
Table 4.38. Responses to the question in item VIII-a ........................................................ 92
Table 4.39. Use of inﬁnity in a sequence ........................................................................... 93
Table 4.40. Salient properties of E-speakers’ and K-speakers’ discourse on inﬁnity ........ 96

Table 4.41. Summary of interview responses using the word limited in items I and IX ...99
Table 4.42. Summary of interview responses using the word limit in items I and IX ..... 100

Table 4.43. Distribution of the contexts of use of the words limited and the chi-square
results of the comparison of difference between the E-group and K-group .................... 101

Table 4.44. Distribution of the use of the word limiteda,” as an upper-bounded proceslel
Table 4.45. Distribution of the use of limitedKMam as extreme ........................................ 101

Table 4.46. Distribution of the contexts of use of the word limit and the chi-square results

of the comparison of difference between the E-group and K-group ................................ 102
Table 4.47. Distribution of the use of the word limitCou .................................................. 103
Table 4.48. Distribution of the use of and limitKMa,’I as extremity .................................. 103
Table 4.49. The table shown in item V in the questionnaire ............................................ 104
Table 4.50. Distribution of endorsed narratives in item V .............................................. 104

Table 4.51. Responses to the question about the limit of an inﬁnite sequence in item V106

 

Table 4.52. Distribution of the use of the limit of the inﬁnite sequence .......................... 107
Table 4.53. Distribution of the use of two columns ......................................................... 107
Table 4.54. Distribution of routines in item V ................................................................. 109
Table 4.55. Distribution of endorsed narratives in items VI-a and VI-b ......................... 111
Table 4.56. Distribution of endorsed narratives in item V—c ........................................... 111
Table 4.57. Responses to the question about the limit of l in item VI-a ...................... 114
x
Table 4.58. Responses to the question about the limit of Ti in item VI-b ................. 115
+ x
x2
Table 4.59. Responses to the question about the limit of 2 in item VI-c .............. 116
(1+ x)
Table 4.60. Distribution of the use of the contexts in item VI ......................................... 117
Table 4.61. Distribution of routines in item V-a .............................................................. 120
Table 4.62. Distribution of routines in items VI-b and VI-c ............................................ 120
Table 4.63. Distribution of endorsed narratives in item VII ........................................... 124

Table 4.64. Responses to the question about the limit of regular polygons inscribed in a

circle in item VH .............................................................................................................. 126
Table 4.65. Distribution of the use of the limit in the sequence of regular polygons ...... 127
Table 4.66. Distribution of routines in item VII .............................................................. 128
Table 4.67. Responses to the question in item VIII-b ...................................................... 131
Table 4.68. Distribution of the use of the limit P in a sequence ...................................... 132
Table 4.69. Salient properties of E-speakers’ and K-speakers’ discourse on limit .......... 134

xi

LIST OF FIGURES

Figure 1. Questions 1 through 7 from the background questionnaire ................................ 28
Figure 2. Questions 10 and 14 from the background questionnaire .................................. 29
Figure 3. The questions in the ﬁrst item ............................................................................ 31
Figure 4. The questions in the second, third, and fourth items .......................................... 32
Figure 5: The questions in the ﬁfth and sixth items ........................................................... 33
Figure 6: The questions in the seventh and eighth items ................................................... 34
Figure 7. Geometric representations provided in item VII .............................................. 124

xii

CHAPTER I
INTRODUCTION AND RATIONALE

The purpose of this study is to investigate and compare how native-English and
native—Korean speaking university students who received their education in the US. and
in Korea, think about the concepts of inﬁnity and limit. Based on the communicational
approach to cognition, according to which mathematics is a kind of discourse (Sfard,
1998), Sfard’s frame for analyzing discourse is used to investigate students’ thinking
about inﬁnity and limit. Inﬁnity is the conceptual basis for mathematical topics such as
the number line and inﬁnite decimals. Since the 19’h century the concept of limit has been
essential to mathematical analysis. As known to teachers and as conﬁrmed by
researchers, most students have considerable difﬁculty with the notions of inﬁnity and
limit.

Since the mid 19603 large-scale international comparative studies have been carried
out to investigate student learning. These studies show that East Asian students, such as
those in China, Japan, and Korea, outperform their US. counterparts in mathematics.
However, these large-scale studies do not examine the reasons for this disparity in the
results. Nor do they provide a satisfactory method of helping students overcome their
leaming difﬁculties (Wang and Lin, 2005). In order to have a better understanding of
why and how East Asian students outperform US. students, more grounded qualitative
studies are needed. This study is an attempt to investigate students’ thinking about
inﬁnity and limit using a different conceptual framework of learning — not from a
cognitivist, acquisitionist perspective, but rather using a participationist view with

attention to socio-cultural contexts.

The ﬁrst basic assumption in this study is derived from the communicational
approach to cognition. The communicational framework conceives of thinking as
communication with oneself, of mathematics as a form of discourse, and of learning
mathematics as a change in ways of communicating (Sfard, 2000). The second
assumption is that when students are thinking, meta-discursive rules guide them to do it
in a certain patterned way. Throughout discursive activity, routines as repetitive patterns
are implemented with other features of mathematical discourses (uses of words, visual
mediators, and endorsed narratives). The third important assumption is that when
students come to the classroom to learn mathematical concepts they already have a
certain amount of knowledge about these concepts that come from daily experience. This
is based on the view that mathematical learning involves thinking with and through
informal everyday language, generalizing everyday concepts associated with
mathematical words, and abstracting those concepts with schooling.

This study attempts to answer the questions raised by these assumptions. One
speciﬁc question about learning mathematics as changing mathematical discourse is:
What are the leading characteristics of non-mathematical and mathematical uses of
inﬁnity and limit both between and within each of two linguistically distinct groups of
university students? Because mathematical discourse may have culture-speciﬁc
characters, two critical questions derived from the third assumption include: “What are
the similarities and differences between native-English and native-Korean speakers in
their former mathematics education and curricular experience with the words inﬁnity and

limit?” and “What are the similarities and differences between native-English and native—

Korean speakers in their non-mathematical and mathematical discourses on inﬁnity and
limit?”

This study may be of signiﬁcance to the larger ﬁeld of mathematics education for
several reasons. First, discourse analysis may lead to methods for helping students
overcome their difﬁculties with the central concepts of inﬁnity and limit, with important
implications for teacher education and K-16 curriculum. It may provide insight into the
role of language in mathematical thinking in general and into the basis of the common
phenomenon known as misconceptions in particular. Second, such investigations will
address some previously unanswered questions such as “What is the role of the
knowledge that students bring to the classroom from out-of-school settings?” and “Why
do people have different uses for what they learned in different contexts?” Third, the
communicational approach may prove fruitful for investigating advanced mathematical
thinking in other areas. Findings of this study regarding inﬁnity and limit may enlighten
us about the ways that students and teachers deal with other notions such as continuity,
differentiability, and integration in calculus. Finally, applying this approach to culturally
diverse groups of students may shed additional light on an issue of great importance,
vigorously pursued currently by researchers, namely how culture affects student learning

in mathematics.

CHAPTER II
THEORETICAL BACKGROUND

This chapter sets forth epistemology and history of the inﬁnity and limit concepts to
describe the genesis and evolution of both of the notions, as well as the interdependence
between them. This chapter also includes a review of previous research on learning
inﬁnity and limit. In addition, literature reviews on two topics, colloquial use of language
in learning mathematics and comparative studies of language in learning mathematics,
are presented. Next, the communication approach to mathematical learning that guides
this study and contributes to the research design is discussed. Finally, my pre-dissertation
research as a pilot study is summarized.

Epistemology and Histogy of the Notions of Infinity and Limit

The histories of the mathematical concepts of inﬁnity and limit have been interwoven
since their beginning. The story of inﬁnity begins with the ancient Greeks. For the Greeks,
inﬁnity did not exist in actuality, but rather as a potential construct. Although there was
the notion of bounded processes, there was no concept of limit as a concrete bounding
entity.

In the Middle Ages, Christianity came to value inﬁnity as a divine property. With the
developments of astronomy and dynamics in the 16’h century, there was an urgent need to
ﬁnd methods for calculating the area, volume, and length of a curved ﬁgure. In the 17””
century, to ﬁnd the areas of fan-shaped ﬁgures and the volumes of solids such as apples,
Kepler used inﬁnitesimal methods (Boyer, 1949). Throughout the 18’“ century, calculus

lacked ﬁrm conceptual foundations. At the end of the 18’” century, mathematicians

became acutely aware of inconsistencies which plagued the theory of inﬁnitesimal
magnitudes.

Today’s notion of limit emerged gradually in the 19th century as a result of attempts
to remedy the uncertainties existing within mathematical analysis at that time. Cauchy
and Weierstrass were pioneers of the movement toward a rigorous calculus. At this time,
limit turned into an arithmetical rather than geometrical concept, as it was before, in the
context of inﬁnitesimals. Inﬁnity was now actual rather than potential. In order to
complete Weierstrass’ foundations of arithmetic, Dedekind and Cantor developed the
theory of the inﬁnite set.

Previous Research on Student Learning on Infinity and Limit
Various aspects of the learning about inﬁnity and limit have been investigated over
the last few decades. Anchoring their research in the analysis of the mathematical
structure of the notions, Cottrill et a1. (1996) report that there are two reasons for student
difﬁculties with limits. One reason is the need to mentally coordinate two processes:

x —-> a, and f (x) —-> L. The other is the need for a good understanding of quantiﬁcation

related to e and 6. Borasi (1985) suggests several alternative rules about how to compare
inﬁnities based on students’ intuitive notions (within this tradition, see also Cornu, 1992;
Tall, 1992).

Other research has focused on misconceptions and cognitive obstacles related to
inﬁnity and limit. Fischbein, Tirosh, and Hess (1979) and Tall (1992) emphasized the
role of intuition. One source of difﬁculty with the notion of inﬁnity is the belief that a
part must be smaller than the whole. Other researchers (Comu, 1992; Davis and Vinner,

1986) stress the inﬂuence of language. Students may have had many life experiences with

boundaries, speed limits, minimum wages, etc. that involved the word “limit”. These
everyday linguistic uses interfere with students’ mathematical understandings (Davis and
Vinner, 1986). Przenioslo (2004) focuses on the key elements of Students’ concept
images of the limits of functions. Still others have focused on informal models that act as
cognitive obstacles (Fischbein, 2001; Williams, 2001). According to Williams, informal
models based on the notion of actual inﬁnity are a primary cognitive obstacle to students’
learning.

Finally, some researchers address students’ difﬁculties through the lens of the theory
of actions, processes, objects, and schemas (APOS; see Weller, Brown, Dubinsky,
McDonald, and Stenger, 2004). Weller et al. discuss the cognitive mechanisms of
interiorization, encapsulation, and thematization that are used to build and connect
actions, processes, objects, and schemas.

Previous Research on Use of Collguial Language in Learning Mathematics

To investigate the use of language in learning mathematics, many researchers have
examined several mathematical topics on the basis of three different approaches to
everyday language. One of the approaches focuses on the meanings of everyday language
and their effects on learning mathematics. Some research has shown that Students’
misconceptions are influenced by the meanings of everyday language in learning
mathematics (Comu, 1992; Davis and Vinner, 1986; Pimm, 1984; Raiker, 2002;
Schwarzenberger and Tall, 1978). Students continue to rely on their misconceptions
stemming from everyday language because even though the everyday and mathematical
meanings of words are similar, their different uses in mathematics classrooms are

signiﬁcant. In the same vein, other research addresses the use of language in bilingual or

multilingual contexts (Clarkson, 1992; Gutierrez, 2002; Moschkovich, 1996; Setati,
2002). Bilingual students employ not only different uses of concepts, but also different
perspectives on mathematical tasks.

Another approach stresses the effects of the structures of everyday language on
mathematical learning (Bintz and Moore, 2002; Cotter, 2000; Leung, 2005). For instance,
the difﬁculty of the patterns for counting in the English language (e.g., “l-ten 3” in the
Asian number pattern instead of thirteen in English) interferes with American students’
mathematical understandings of counting.

Most previous research on the role of colloquial language in learning mathematics
has been grounded in the meanings and structures of everyday language. Comparative] y,
the approach used in this dissertation views mathematical learning as a discursive activity
that employs mathematical objects including language as well as meta-level factors.
Previous research based on the discursive approach to mathematical learning is not
extensive (Anderson et al., 2004, Barwell, 2003; Ben-Yehuda et al., 2005). Based on the
communicational framework, Ben-Yehuda and colleagues (2005) characterize the
arithmetic discourses of two 18-year old girls with learning difﬁculties in terms of four
features of mathematical discourse including routines as a meta-level factor. They claim
that the use of colloquial arithmetic discourse in learning literate mathematical discourse
is important.

Previous Research on Comparative Studies of Langgage in Learning Mathematics

There is a limited number of comparative studies on language in learning
mathematics. Song and Ginsburg (1988) are among the earliest to conduct a comparative

study between two languages in learning mathematics. They investigate how far

American and Korean preschool children can count. There are two different systems of
number words in the Korean language. One is the everyday system, which is irregular
like the English language number system, and the other is the academic system, which is
regular like the Chinese number system. Song and Ginsburg report that the four year-old
Korean had an initial difficulty with both counting systems (everyday and academic
systems) and their ability to count was less developed than that of the American
counterparts. However, after learning the academic system of counting, the six year-old
Koreans attain a higher level of performance in counting than their US. counterparts. It is
concluded that the everyday number system provides the Korean a cognitive advantage
for understanding a base-ten structure at school (within the comparison of counting
systems, see also Fuson and Kwon, 1992; Miura, Okamoto, Kim, Chang, Steere, and
Fayol, 1994).

Few comparative studies focusing on the Chinese language have in mathematics
been conducted (Han and Ginsburg, 2001; Li and Nuttall, 2001). One representative of
experimental studies is a work by Han and Ginsburg (2001). They ask 48 judges to rate
the clarity of 71 mathematical words in Chinese, English, and “Chinglish” which is an
English equivalent to the Chinese words. Chinese mathematical words are rated higher in
clarity than English mathematical words. Chinglish mathematical words also tend to be
rated higher in clarity than English words. After that, a mathematics test including 71
mathematical words is given to three groups of Chinese-American eighth graders: 33
Chinese-only speakers, 29 bilingual speakers of Chinese and English, and 20 English-
only speakers. The result of the test reveals that the Chinese-only speakers and the

Chinese-English bilinguals outperform the English-only speakers. Therefore, Han and

Ginsburg hypothesize that the clarity of Chinese mathematics language may contribute to

the better performance of the groups of the Chinese-only and bilingual speakers.

Communicational Approach to the Study of Mathematical Learning

The Need for an Additional Approach to Study the Learning of Infinity and limit

In spite of the mutual interdependence of the concepts of inﬁnity and limit in the
history of mathematics, there has been little research to examine Students’ understandings
of and difﬁculties with both concepts simultaneously. For a better understanding of the
notions of inﬁnity and limit and of their relationships, more research about student
learning on the concepts together is needed.

As inﬂuenced by the work of Piaget, mathematical learning has been considered as a
process of active cognitive acquisition. Most previous research on the learning of inﬁnity
and limit has been grounded in the acquisitionist framework. The acquisitionist tenets
have brought important insights (e. g., cognitive structures of inﬁnity and limit, Students’
misconceptions, and the hierarchical relationships between categories in these concepts)
into mathematical learning and teaching on the basis of the three different cognitive
approaches to the learning of inﬁnity and limit. The acquisitionist research, however, has
not led to satisfactory solutions to account for inter-personal and cross-situational
differences and the source of human development (Sfard, 2006). They also underestimate
not only the inherently social nature of student thinking, but also the role of discourse and
communication in learning and in other intellectual activities.

Most previous research on the use of colloquial language in learning mathematics
has not moved beyond attempts to characterize the meanings and structures of everyday

language and their impact on mathematical learning. Some researchers in the discursive

view on mathematics learning reported mathematical object-level and meta-level factors
(such as patterns of attention, and routines) in mathematical discourse. However,
previous discursive research focused only on primary and secondary school mathematics.
Previous comparative studies emphasizing the importance of language in learning
mathematics have also been conducted mainly with primary and secondary mathematics.
Thus, there has been little comparative research on the role of colloquial discourse in
learning mathematics at the post-secondary level. More comparative studies on advanced
mathematical topics using the discursive approach to mathematical learning are
warranted for a better understanding of how students think in general.
Communicational Approach

While Piaget focuses on the individual development, Vygotsky concentrates on the
social activity of the mind as a starting point. From Vygotsky’s point of view, speech for
communicating with others comes before internal speech in children’s internalization of
higher mental processes. The claim is vividly made in the following excerpt:

Every function in the child’s cultural development appears twice: ﬁrst on the social
level, and later, on the individual level; ﬁrst between people (inter-psychological), and
then inside the child (intra-psychological). (Vygotsky, 1978, p.57)

Thinking arises from an individualized version of interpersonal communication
because of the inherently social nature of human activities. Higher mental processes are
developed through the procedures of internalization from public speech to inner private
speech, and are tightly related to tool-mediated activity (Vygotsky, 1986). For instance,
symbols and mathematical language are important tools in mathematics. The properties
of such tools are inseparable from the cognitive processes of the uses of the tools
(Rogoff, 1990). Thus, thought and language are inseparable; so are thought and symbols.

Thought must be transferred through meanings and only then through words as a tool.

10

According to Vygotsky (1986, p.256), “the word is a direct expression of the historical
nature of human consciousness”. In other words, consciousness can be investigated as it
is expressed in words.

In the communication approach, thinking is a special form of communication which
is the process of internalization from public speech to inner private speech. Because
thinking arises from an individualized version of interpersonal communication, it is
simply communication with oneself and thus learning is a change in ways of
communicating (Ben-Yehuda et al., 2005). In particular, learning mathematics is seen as
becoming more skillful and articulate in mathematical discourse (Sfard, 2001a).

The word discourse signiﬁes any type of communicative activities, whether with
others or with oneself, whether verbal or not, whether synchronic (like in a face-to face
conversation) or asynchronous (like in an exchange of emails or reading a book) (Sfard
and Lavie, 2005). A discourse is regarded as mathematical if it deals with mathematical
objects, such as inﬁnity and limits (Ben-Yehuda et al., 2005). When we use mathematical
words such as limit in an everyday context, we call it a colloquial use in colloquial
discourse. Thus, to discover the mechanisms of mathematical thinking, the discourses
that students use in both the contexts of everyday life and mathematics are important
because they can be used not only to express thinking processes, but to reveal what
students already know.

Whenever mathematical discourses are being analyzed, compared, and watched for
changes over time, four distinctive features are often considered: words and their use,

mediators and their use, endorsed narratives, and discursive routines (Ben-Yehuda et al.,

11

2005; Sfard, 2001b; Sfard and Lavie, 2005). In this study, all four features are considered
to identify characteristics of mathematical discourses.

Uses of words are the ways the participants use keywords of colloquial and
mathematical discourse regarding inﬁnity and limit. Word uses are important as a feature
of colloquial and mathematical discourses because they can reveal how students
understand those words. For instance, how students use their mathematical words can be
assessed by the degree of objectiﬁcation. The degree of objectiﬁcation will be assessed
according to the frequency with which students speak about inﬁnity and limit, as if these
words referred to self-sustained, discourse-independent objects. The degree of
objectiﬁcation can indicate, among others, how students make the use of the words
inﬁnity and limit similar to the discourse on material objects such as apples or atoms.

Discursive routines are the patterns of repetitive actions in students’ discourses. For
example, we can observe such patterns in the mathematical discourses of comparing,
counting, and deﬁning. When students mention mathematical use of keywords, certain
discursive routines can also be detected. Therefore, discursive routines are the central
concept in discourse analysis. Routines are a set of meta-rules. When we consider meta-
rules, two important aspects need to be emphasized. One of them is when the routine
procedure is used and the other is how it is implemented. When refers to the cues for
beginning discursive routines. How refers to the kinds of patterns which exist in
discursive routines.

Endorsed narratives are propositions that students accept as true. A narrative
endorsed by an interviewee does not have to be a quotation from the interviewee. It may

be an interpretation of assumptions that tacitly guides him or her. For instance, the

12

expression “Limit is something ...” can be inferred from statements actually made by
students. Thus, through discursive activity, students can produce endorsed narratives.

Visual mediators are means that the participants use for their communication
activity. While colloquial discourses are communicated with the help of concrete objects
or individual images, mathematical discourses are mediated by informal and formal
symbols. For example, to identify inﬁnity in mathematical discourse, students can use
and refer to such mediators as an inﬁnity symbol (00), three dots (. . .), and an “eight
sideways” (oo) in multiple ways.

The uestion to be Answered in This Research

American and Korean students differ considerably in their mathematics
achievements, as shown in large-scale international comparative studies. Difference in
language is one of many cultural differences between these two groups that are relevant
to understanding the relationship between mathematics learning and culture. Therefore,
the difference in relations between colloquial and mathematical discourses on inﬁnity and
limit in Korean-speaking and English-speaking students are explored in this study.

The Pilot Study

A pilot study was conducted in preparation for this study (see Kim, 2006). I
interviewed four American and four Korean students in fall 2004. Each ethnically distinct
group included one elementary school student, one middle school student, one high
school student, and one undergraduate. The participants within the same grade level
(elementary, middle and high schools, and college) were selected based on the criteria of
the same age, grade, and educational institution. Their discourse on inﬁnity and limit was

scrutinized with an eye to common characteristics as well as culture-, age-, and

13

education-related differences. Because the interviews were conducted in English, the four
Korean students who were selected had been living in the United States and attending US
schools for more than three years. The interview questionnaire consisted of 29 questions
in eight categories. The ﬁrst two categories aimed at scrutinizing Students’ colloquial
discourses on inﬁnity and limit, whereas the remaining six categories were targeted at
investigating Students’ mathematical discourses on the topics (see Appendix A for the
full questionnaire). The interviews were audio- and video-taped and then transcribed in
their entirety for further analysis. Data were analyzed so as to identify and describe the
four distinctive features of the respondents’ discourses: word use, discursive routines,
mediators, and endorsed narratives.

One signiﬁcant ﬁnding was that the Korean and American students used inﬁnite and
inﬁnity in different ways, and used limited and limit in similar ways in both colloquial
and mathematical discourses. There were few noticeable differences between the
American and Korean groups in the colloquial and mathematical discourses on inﬁnite
and inﬁnity (e. g., the element-based approach to the word inﬁnity in the Korean group
versus the set-based approach in the American group). However, there was no systematic
difference between the American and Korean students in their discourses on limited and
limit. Another important ﬁnding was that there were no signiﬁcant systematic differences
with age in the colloquial and mathematical discourses on inﬁnity. In contrast, some age-
related characteristics of the Students’ word use and endorsed narratives on limit were
found in the study (e. g., objectiﬁed word use as a particular number and the endorsement

of limit as an unreachable object for the older students).

14

Although the sample size of the pilot study was too small to allow for
generalization, the ﬁndings served as a basis for the hypotheses that were tested in this
study. According to the results of the pilot study, colloquial discourse seems to have an
impact on mathematical discourse as evidenced by certain clear differences between the
mathematical discourse on inﬁnity of the American and the Korean students. These
differences can be ascribed to the fact that only in English do the mathematical words
inﬁnity and inﬁnite (as well as set and element) appear in colloquial discourse. The
absence of systematic differences between the two ethnic groups in their discourses on
limited and limit might be ascribed to the fact that unlike the word inﬁnity, the words
limit and limited are quite commonly used in English colloquial discourses. Due to the
fact that all the Korean students had been living in the United States and attending US
school for more than three years, they may have been inﬂuenced by the colloquial and
mathematical English discourses on inﬁnity and limit. Therefore, in the present study,
native Korean students, who have not experienced English colloquial and literate
discourses on inﬁnity and limit, will be chosen and interviewed about their discourses on
inﬁnity and limit in Korean.

Stunner!

The interconnectedness between inﬁnity and limit in the history of mathematics
justiﬁes this study’s focus on both inﬁnity and limit simultaneously. Because of the
inherently social nature of student thinking, previous cognitive approaches to the learning
difﬁculties of inﬁnity and limit provide justiﬁcation for the need for discourse analysis as
a research methodology, with the promise of answering some previously unanswered

questions within the traditional cognitive framework. Literature reviews on the use of

15

colloquial language and comparative studies of language in learning mathematics justify
the need for comparative studies of language on learning advanced mathematics at the
post-secondary level to better understand how students think in general. As noted earlier,
communication is of principal importance in learning mathematics as a tool-mediated
activity. These different perspectives as well as the pilot study laid the groundwork for

the design of this study.

16

CHAPTER III
DESIGN OF STUDY

This study investigated how native-English and native-Korean speaking university
students thought about the concepts of inﬁnity and limit and compared their colloquial
and mathematical discourses. The method chosen to fulﬁll this purpose was the
communicational approach to cognition, according to which mathematics is a kind of
discourse. This chapter describes the research questions and method. In addition,
connections are drawn between the pilot study and certain methodological choices for
this study.

Research Questions
The purpose of this study, to characterize how students think about inﬁnity and limit,

led to the following research questions:

0 What are the relationships between colloquial and mathematical uses of the English
words inﬁnity and limit? What are the relationships between colloquial and
mathematical uses of the corresponding Korean words?

0 What are the similarities and differences between native-English and native-Korean
speakers in their previous mathematics education experiences, their learning habits
and their curricular experience with the words inﬁnity and limit?

0 What are the similarities and differences between non-mathematical and
mathematical discourses on inﬁnity and limit of native-English and native-Korean
speakers?

Overview of the Method
Two groups of university students (native—English and native-Korean speakers)
participated in the study. The comparisons of these two groups were based on analyses of
Students’ responses to a survey questionnaire and discourse analyses of paired interviews.

After collecting the survey data, I ﬁrst analyzed and categorized 132 English-speaking

and 126 Korean-speaking university Students’ written responses to the survey within each

17

group, and chose their representatives from each group for the interview study. Twenty
representatives in each group were interviewed in pairs, and discourse analyses on those
interview data was conducted on the basis of the communicational framework. Finally,
the written responses to mathematics problems on the survey were reanalyzed to make
conjectures about the generality of characteristics observed in interviews.
Method

Context
Sites

The two sites, the United States and Korea, were chosen to investigate the three
research questions (see Appendix B for an overview of the educational systems at the two
sites). The speciﬁc reason for the selection of native-English and native-Korean speakers
is the discontinuity in Korean and the continuity in English between the colloquial and
mathematical discourses on inﬁnity and limit. For the sake of the clarity in the following
discussion, let us denote the colloquial and mathematical words for inﬁnity and limit as
speciﬁed in Table 3.1.

Table 3.1 Colloquial and mathematical uses of inﬁnity and limit in English and Korean

 

 

English Korean
Inﬁnity Limit Inﬁnity Limit
InﬁnityKCon LimitkCon
Colloquial InﬁnityECo" Limitgco" 391% (han-up-um), XIIEI (foe-hon),
$¢(moo-soo),... 737‘“ (kyung-gae),
InﬁnityKMath LimitKMam

Mathematical InﬁnityEMath LimitEMath 'T'ﬂ' (mu-halt) and 23* (gzlk-han) and
$35" (mu-han—dae) 3331' (grik-han-gab)

As can be seen from the table, the relation between the colloquial and mathematical

words may now be presented as follows:

InﬁnityECO" = InﬁnityEMath, whereas lnﬁnityKCo" at InﬁnityKMam

l8

Limitgco” = LimitEMath, whereas LimitKCo“ ¢ LimitKMath
Here, the equality symbol ‘=’ means that the words sound the same, not that their use is

the same. InfmityKCou and LimitKCou signify the colloquial Korean words that can count
as rough translations of InﬁnityECOH and LimitECOH, respectively.

In English, the words inﬁnity and limit appear in colloquial language and students are
likely to be familiar with the terms before they encounter them in a formal mathematical
context. Even young children are likely to be acquainted with expressions such as
“inﬁnity of problems” or “speed limit” that feature the same words which they will latter
learn in its mathematical version. This is not the case for Korean children. The formal
mathematical words for inﬁnity and limit learned in school or university do not originate

in colloquial discourse and the odds are that when the Korean children hear the Korean
mathematical words for inﬁnity and limit (InﬁnityKMmh and LimitKMath) for the ﬁrst time,

these words (the sounds) are completely new for them. Later, these words may make their
way into their colloquial discourse, but this is irrelevant to this study.

This difference between the Korean and American children’s linguistic experiences
may be consequential. One can hypothesize that the early colloquial uses of the English
words inﬁnity and limit would inﬂuence, even possibly interfere with, the way students
learn the formal mathematical use of these words. The pilot study brought an initial

corroboration of this thesis when it showed, for example, that English speakers tend to

use limitEMath the way mathematicians use the term “upper boundary” — the use that

corresponds to the limitECo", as appearing in the colloquial expressions such as “speed

limit” or “There is a limit to my patience.” Since there is no colloquial counterpart to the

l9

limitKMath, it is reasonable to assume that there will be no such bias in the Korean

students’ use of limitKMam1 (the use of the Korean mathematical counterpart of limitEMam).

This linguistic hypothesis, if it is empirically corroborated, may explain at least some of
the phenomena known as “misconceptions about inﬁnity and limits” and, more generally,
may shed light on the linguistic mechanisms that produce such phenomena. Thus, I tried
to examine the possible impact of the discontinuity in Korean and the continuity in
English on students’ mathematical thinking about inﬁnity and limit.
Research Sites

A major university in the capital city of each site, Michigan and Korea, was selected
for collection of data for this study. The Korean participants were selected from Korea
University (KU). KU is located in Seoul, the capital city of Korea. KU is a private
university with an enrollment of approximately 20,000 undergraduate and graduate
students. Almost all the students in KU are Korean and international students comprise
less than 3% of the total student population. At KU the Department of Mathematics
offers only one calculus course. All freshmen whose major was science took the calculus
course as a requirement. If students took calculus as one of the four elective courses in
high school, they were expected to have some background in calculus. All calculus
classes were taught by faculty and instructors in the Department of Mathematics. Each
class size was under 60 students. Thomas’ Calculus (Thomas, 2005) was used as the
textbook.

The US. participants were selected from Michigan State University (MSU). MSU is
located in East Lansing, three miles east of Michigan’s capitol in Lansing. MSU is a

public university with approximately 45,000 undergraduate and graduate students. More

20

than 7% of the total student body is international students. There were more than 200
programs of study. MSU students came from all 83 counties in Michigan, all 50 states in
the United States, and about 125 other countries’. All students whose major was natural
science or engineering at MSU were required to take the calculus course MTH 132 as a
requirement. If students had taken Pre-calculus as an elective course in high school, they
might come to the university class with a background in inﬁnity or limit. Faculty in the
Department of Mathematics taught all MTH 132 classes. Each class size was under 35
students. In all MTH 132 calculus classes, the same Thomas’ calculus book was used.
The characteristics of the two universities, MSU and KU, are summarized in Table 3.2.

Table 3.2 Summary of the characteristics of MSU and KU

 

 

MSU KU
Location East Lansing, MI Seoul, Korea
Number of students About 45,000 About 20,000
International students More than 7% of the total Less than 3% of the total

. Students whose major is natural Students whose major is

Calculus as a requrrement . . . .

scrence or engineering scrence
Each calculus class size Under 35 students Under 60 students

 

Opportunities for Learning on Mathematical Inﬁnity and Limit

International comparative studies are meaningful and useful when they include a
good alignment of achievement and opportunity to learn (see Wolfe, 1999 for positive
correlations between achievement and opportunity proﬁles in the Second International
Mathematics Study (SIMS) data and the International Assessment of Educational
Progress (IAEP) data). In order to analyze students’ opportunities to learn inﬁnity and
limit, I used student responses about the textbooks they had used for learning inﬁnity and

limit as well as policy documents such as state and national standards.

 

' From the website, http://newsroom.msu.edu/snav/ I 84/page.htm

21

1) Opportunities of native-English speakers in the United States

Characterizing students’ opportunity to learn about inﬁnity and limit is complex in
the United States because different curricula are used across schools and universities and
this implies different experiences. However, Michigan’s Grade Level Content
Expectations (GLCE) for grades K8 and High School Content Expectations (HSCE) in
mathematics can be used to address students’ expected opportunity in general, in
Michiganz. In order to describe what students should know and be able to do in
mathematics, the Michigan’s GLCE addresses content standards categorized into the ﬁve
strands at each grade level. The ﬁve strands consist of the following: 1) number and
operations; 2) algebra; 3) measurement; 4) geometry; and 5) data and probability. The
Michigan’s HSCE describes content standards in each course (Algebra 1, Algebra II,
Geometry, Pre-calculus, and Statistics and Probability) on the basis of the four strands: l)
quantitative literacy and logic; 2) algebra and functions; 3) geometry and trigonometry;
and 4) statistics and probability.

Michigan’s GLCE prescribe that non-terminating and non-repeating decimals are
assessed at grade 8.3 In algebra 1, students are expected to learn about recursively deﬁned
functions, the behavior of a function as x approaches inﬁnity, and asymptotic behaviour

at inﬁnity". In algebra 11, students are taught inﬁnite geometric sequences and asymptotes

 

2 From the website, http://www.michigan.gov/mde/O,l607,7-l40-28753-—-,00.html

3 N. ME.08.03: Understand that in decimal form, rational numbers either terminate or eventually repeat,
and that calculators truncate or round repeating decimals; locate rational numbers on the number line; know
fraction forms of common repeating decimals, e.g., 0,]— = g; 05 = 1..
N. ME.0804: Understand that irrational numbers are those that cannot be expressed as the quotient of two
integers, and cannot be represented by terminating or repeating decimals; approximate the position of

familiar irrational numbers, e.g., J2 , J3 , 7r , on the number line.

" A2.l.5 Recognize that ﬁmctions may be deﬁned recursively. Compute values of and graph simple
recursively deﬁned functions (e.g.,f(0) = 5, and ﬁn) =f(n-l) + 2).

22

of rational functions’. If students take pre-calculus, they would be exposed to an
opportunity to learn about the limit of a function as x approaches a number or inﬁnity as
well as inﬁnite geometric series”.

2) Opportunities of native-Korean speakers in Korea

Unlike in the United States, similar curricula are used across schools in Korea due to

a centralized educational system. The mathematical word 51?." (mu-hon) for inﬁnity

without using the Chinese characters is ﬁrst used in Korean schools to introduce the
concept of inﬁnite sets in the 7’h grade school curriculum (see Appendix B for the general
development of the Korean language). As explained in the appendix, one thing to note is
that Chinese characters show meanings and etymologies of words. In a 7’h grade
textbook, a set is deﬁned as a collection whose objects are clearly delineated. In the same

textbook, an element is then described as a member of a set. After the introduction,

Fri-15%| (mu-soo-hee) as a Korean colloquial word without using Chinese characters is

used to deﬁne an inﬁnite set. In the word mu-soo-hee, the meaning of the ﬁrst character

 

A2.I.6 Identify the zeros of a function and the intervals where the values of a function are positive or
negative. Describe the behavior of a function as x approaches positive or negative inﬁnity, given the
symbolic and graphical representations.

A2.3.l Identify a function as a member of a family of functions based on its symbolic or graphical
representation. Recognize that different families of functions have different asymptotic behavior at inﬁnity
and describe these behaviors.

’ *L2.2.4 Compute sums of inﬁnite geometric sequences.

A2.9.2 Analyze graphs of simple rational functions (e.g., f (x) = 2 x + 1 ; g(x) = x ) and understand
x—l x -4

the relationship between the zeros of the numerator and denominator and the function’s intercepts,

asymptotes, and domain.

6 P1.7 Understand the concept of limit of a ﬁrnction as x approaches a number or inﬁnity. Use the idea of
limit to analyze a graph as it approaches an asymptote. Compute limits of simple functions (e.g., ﬁnd the
limit as x approaches 0 of ﬁx) = l/x) informally.

P8.4 Compute the sums of inﬁnite geometric series. Understand and apply the convergence criterion for
geometric series.

23

(mu) is none or nothing and the second character (soo) means number or count and the
last character is a Korean sufﬁx to make the word mu-soo as an adverb. Thus, an inﬁnite
set is introduced as a set which includes countlessly many elements using a Korean

colloquial word mu-soo-hee. In the 10th grade mathematics curriculum, the other
mathematical term $EI'EH (mu-han-a’ae) for inﬁnity and its symbol (00) are introduced
along with the concept of inﬁnite sequence, as well as the new symbol (n—roo: as ‘n is

getting bigger endlessly’).

As in the case of inﬁnity, there are also two Korean mathematical words for limit

only written with Korean letters: 23 (gtik-han) and 3331 (guk-han-gab). The

meaning of the ﬁrst character (grik) is an extreme and the second character (han) means
bound. The word at (gab) as a pure Korean word means value or amount. Thus the ﬁrst
mathematical word gtik-han for limit means “the utmost boundary.” According to a

Korean dictionary, the Korean word gtik-han is equivalent to the word grik-han-gab. In

the dictionary, the second mathematical word ngk-han-gab, used only in mathematics, is

deﬁned as “a value to which a function approaches, as independent variables approach to

a number endlessly.” The word gzik-han for limit without using Chinese characters is ﬁrst

introduced to signify the limit of a sequence in the 10th grade mathematics curriculum. By
the end of grade 12, students whose major will be science are expected to learn about the
limit of a function, the limit of an inﬁnite sequence, and sums of inﬁnite series, whereas

students who will study liberal arts at a university are not taught those concepts.

24

Opportunities of native-English and native-Korean speakers in the United States and
Korea respectively for learning about inﬁnity and limit are different. For native-English
speakers in the US, the colloquial use of the English words inﬁnity and limit precedes
the mathematical use because native-English speakers ﬁrst experience colloquial
meanings of those words in everyday language and later they experience the
mathematical meanings in high school. However, for native-Korean speakers in Korea,
the mathematical words inﬁnity and limit are ﬁrst introduced at grades 7 and 10
respectively.

Participants

Students from several calculus classes at MSU and KU were recruited, yielding a
sample of 132 English-speaking and 126 Korean-speaking university students for the
study. Only MTH 132 calculus classes at MSU were included in this study because the
content in MTH 132 was equivalent to that of the calculus course at KU in terms of using
the same textbook (Thomas, 2005). All freshmen who majored in science were required
to take the calculus course at KU, whereas most students who studied natural science or
engineering took the MTH 132 calculus class at MSU. In Spring 2007, when the data
were gathered, about 1740 students distributed in 29 sections were taking the calculus
class at KU, whereas 533 students in 16 sections were registered for the MTH 132 class
at MSU. Of those students, almost all the students at KU and about 70 percent of the total
students at MSU were freshmen.

Out of the undergraduate students who volunteered to participate, 20 students in
each site were selected for paired interviews on the basis of their responses to the survey.

The selection process was designed to maximize difference in degrees of exposure to and

25

experience with the concepts of inﬁnity and limit throughout their formal education as
well as to include students whose responses to the questions investigating colloquial and
mathematical discourses on these concepts fell into various categories (see the section of
procedures for collecting data about how to select 20 interviewees in detail). As the
results of the pilot study (Kim, 2006) suggested, native Korean students who have not
studied mathematics in English or any other language were chosen and interviewed in
Korean regarding both their colloquial and literate discourses on inﬁnity and limit. I
selected interview subjects who had no experience in any language other than their native
language in learning mathematics. Data from these native-Korean speakers might help
explain the characteristics of their discourses described in the ﬁrst research question. In a
similar way, native-English speakers, who had not studied mathematics in Korean or any
other language, were chosen and interviewed in English.
M
Questionnaire

The primary instrument for data collection was a questionnaire (see Appendix C for
the entire questionnaire, with both the English and Korean versions). The questionnaire
consisted of two sections: Section I, Background; and Section II, Discourse. Section 1
aimed at examining students’ linguistic background, formal mathematics education, and
curricular experience with the concepts of inﬁnity and limit. Section II was composed of
questions regarding creating sentences with inﬁnity and limit and solving mathematical
problems about these concepts. The discourse questionnaire (Section II) was piloted and
subsequently revised. The questions from the discourse questionnaire were used for

paired interviews to scrutinize students’ non-mathematical and mathematical discourses

26

on inﬁnity and limit. Because the questionnaire was used in English and Korean, three
mathematics educators and one linguist who were all Korean-English bilinguals were
selected to evaluate the Korean translations of the questions into English.

To address the ﬁrst research question (about the leading characteristics of non-
mathematical and mathematical discourses on inﬁnity and limit), the problems for
investigating non-mathematical and mathematical discourses in the discourse
questionnaire were open-ended, providing the participants with an opportunity to freely
express their thinking. Responses to open-ended questions permitted an investigation of
students’ thought processes. Because of the nature of the questions focusing on speciﬁc
features, only a subset of the four features (word use, endorsed narratives, visual
mediators, and routines) in mathematical and non-mathematical discourses was dealt with
within each question. Then, common characteristics (in terms of the four features) in
discourse analysis on the interviews of ten pairs of native-Korean speakers were
compared and contrasted to those of their native-English counterparts.

To answer the second and third research questions (about the differences between
native-English and native-Korean speakers in terms of their colloquial and mathematical
discourses, formal mathematics education, and curricular experience), I hypothesized
about relationships between characteristics of mathematical and colloquial discourses,
formal mathematics education, and curricular experience with inﬁnity and limit. Finally, I
compared and contrasted the characteristics in these relationships between native-English
and native-Korean speakers.

The background questionnaire (Section I) included 21 questions mainly asking about

Students’ linguistic background, formal education, and the textbooks that they used when

27

learning inﬁnity and limit. Questions 1 and 2 ask about students’ year and native
language. Question 5 is aimed at examining whether students have studied mathematics
in any language other than their native language. Questions 6 through 9 ask in which
grade and in which mathematics course students had ﬁrst heard the mathematical words

of inﬁnity and limit. Questions 1 through 9 are shown in Figure 1.

 

l.Year: D Freshman c1 Sophomore 13 Junior 0 Senior

2.Native Language: C] Korean Cl English D Other (Specify):

3. How often do you speak your native language at home?
0 Always Cl Almost always Cl Sometimes c1 Never

4. About how many books are there in your home? (Do not count magazines, newspapers, or your
school books.)
0 None or very few (0-10 books) 0 Enough to ﬁll one shelf (1 l-25 books)
0 Enough to ﬁll one bookcase (26-100 books) 0 Enough to ﬁll two bookcases (101-200 books)
0 Enough to ﬁll three or more bookcases (more than 200 books)

5. Have you studied mathematics in any language other than your native language?
nYw 0N0
If yes, specify the languages in which you learned mathematics:

6. Can you remember the ﬁrst mathematics course in which you heard the word (mathematical) inﬁnity?
0 Yes 0 No

7. If yes, specify in which grade(s) and in which course/class you heard the word (mathematical)
inﬁnity? Grade(s): Course: .

8. Can you remember the ﬁrst mathematics course in which you heard the word (mathematical) limit?
0 Yes 0 No

9. If yes, specify in which grade(s) and in which course/class you heard the word (mathematical) limit?
Grade(s): Course:

 

 

 

 

Figure 1: Questions 1 through 7 from the background questionnaire

Questions 10 through 14 were intended to investigate students’ different levels of
exposure to and experience with inﬁnity and limit in their formal education. Question 10
regarding the textbooks which students used in high school is included. A set of widely
used textbooks was provided to the students. In question 11, the titles of sequences within
each curriculum were used to investigate different levels of courses that students took in
high school. Because students may take advanced mathematics courses other than the
examples in question 11, question 12 was targeted to examine the most advanced
mathematics course they took in high school. Questions 11 and 12 are included to

investigate the topics in which students may have learned about inﬁnity and limit in their

28

 

10.Please specify ﬁ'om which high school textbook(s) series you studied among the examples below. If

you used textbooks other than those in the exam les, lease check the box for Others.
- 1 2) 3) ‘4 —ﬁ

rursokareo
mmrmwcs

 

   
 

     

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(SEE‘HWH‘
: \~‘l " . .
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Core-Plus Mathematics .. I _. . ‘ ’~ Interactive Mathematics SIMMS: Integrated
Project (CPMP) Mathematics Connections program (IMP) ‘ ' ' '
5 6) 7) Others

 

 

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The University ofChicago Mathematics: Modeling iﬁ.‘::.-:+1'i 3U
School Mathematics Our World Prentice Hall

Project (UCSMP)

ll.In high school, which course(s) in the series below did you take? Please check all the boxes that apply

to you (under the textbook serieslou usej).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

CPMP Math Connections IMP Integrated Math
U Course 1 U Algebra U Year 1 U Level 1
U Course 2 U Geometry U Year 2 U Level 2
U Course 3 U Advanced Algebra U Year 3 U Level 3
U Course4 U Year4 U Level4
UCSMP Moxl’ilrhgegﬁrmlzlorl d Prentice Hall Others
U Transition Math U Course 1 U Pre-Algebra
U Algebra U Course 2 U Algebra 1
U Geometry U Course 3 U Geometry
U Advanced Algebra U Algebra 2

 

U Functions, Statistics
..ﬁndIEIEQE‘PIFFU'Y

U Preealculus and
Discrete Math
12.1n high school, what is the most advanced mathematics course you took other than those in #9?
13. Please check the box(es) next to topics which you have learned. Where and when did this learning
occur?

 

 

 

 

 

:1 Inﬁnite processes 0 Inﬁnite decimal fractions

El Inﬁnite sequence or series D The sum of inﬁnite geometric series
0 Comparison of inﬁnite sets C1 Countable and uncountable sets

a Inﬁnitely small or inﬁnitesimal D Other topic (Describe):

14. Please check the box(es) next to topics which you have learned. Where and when did this leaming
occur?

0 Limits to inﬁnity CI The 8-5 (epsilon-delta) deﬁnition of limit

c1 Limit of a sequence or series a Limit of a function numerically

 

 

 

D Limit of a function geometrically u Limit and the deﬁnition of continuity
1:] Limit and the deﬁnition of the derivative :1 Other topic (Describe):

Figure 2: Questions 10 and 14 from the background questionnaire

29

former education. Seven possible responses to each question are organized in a hierarchy
and speciﬁed at several different levels with regard to the degrees of difﬁculties of
inﬁnity and limit. Questions 10 and 14 from the survey are shown in Figure 2.

In addition to examining students’ former mathematics education and curricular
experience with inﬁnity and limit, the background questionnaire was used to investigate
their crurent mathematics learning and attitudes toward mathematics. Questions 15 and
16 were intended to examine students’ view of themselves as mathematics learners and
the role of mathematics in their lives respectively. Questions 17 through 21 were aimed at
investigating students’ self-report on their learning of mathematics, out-of-school
activities, extra lessons and homework (see Appendix C for details).

The discourse questionnaire (Section 1]) consisted of 16 questions organized into
eight items (see the section on data analysis for how the questions connected the four
features in non-mathematical and mathematical discourses). Open-ended questions were
employed so that the participants had an opportunity to freely express their thinking.
Students’ responses to open-ended questions and the method of paired problem solving
reduced interviewer’s effects and biases on interviewee’s responses. The ﬁrst item aimed
at scrutinizing students’ colloquial discourses on inﬁnity and limit. The next seven items
were designed to investigate students’ mathematical discourse on the topic. The
distinction between non-mathematical and mathematical discourses in this study is
important in this investigation of students’ thinking about inﬁnity and limit. Students’
discourse on inﬁnity and limit in response to items 11 through VIII was regarded as
mathematical, not only because the questions framed the discourse as mathematical (and

thus the student volunteers agreed to participate in this discourse), but also because the

30

participants did it by adjusting their colloquial discourse to the rules of school
mathematical discourse.

The descriptions of each questionnaire item are as follows. In the ﬁrst item, students
were asked to create a sentence for a given word to investigate their non-mathematical
discourse on inﬁnity and limit. In the Korean survey questionnaire, some representative
words of inﬁnitchO", inﬁniteKcOu, limithOn, and limitechOn were used. The questions in

the ﬁrst item are shown in Figure 3.

 

I. (First Item) Create a sentence with the following word (term).
1. (Colloquial) Limited 2. (Colloquial) Limit
3. (Colloquial) Inﬁnite 4. (Colloquial) Inﬁnity
Figure 3: The questions in the ﬁrst item

 

 

 

The remaining seven items were designed to investigate students’ mathematical
discourses on inﬁnity and limit. In the second item, students were asked to compare two
sets (i.e., ﬁngers and toes, odd numbers and even numbers, and odd numbers and
integers). The aim of questions in the second item was to examine whether the routines
for comparing ﬁnite sets were the same as the routines for comparing inﬁnite sets. The
inﬁnite cardinalities in the second item are one of the most important concepts in inﬁnity.
Different degrees of inﬁnity have been controversial in the history of mathematics.
Cantor himself could not believe the different cardinalities between rational and real
numbers when he ﬁrst created them. Thus, questions involving comparisons of inﬁnite
sets have been one of the main methods for investigating student thinking about
mathematical inﬁnity (Borasi, 1985; F ischbein, Tirosh, and Hass, 1979; Tsamir and
Dreyfus, 2002).

In the third item, students also were asked to compare integers with even numbers.

Unlike the second item, integers and even numbers were presented horizontally in order

31

to investigate students’ approaches to a contradiction. If students used the part-whole
comparison in the second item, they may have reached a contradiction as they saw no
consistency between part-whole and one-to-one correspondence comparisons in different
representations of inﬁnite sets. The third item was intended to raise a contradiction and
promote student’s awareness of their contradictory conclusions when comparing inﬁnite
sets (Tsamir and Dreyfus, 2002). Thus, the third item could reveal meta-discursive rules
in the ways of viewing a contradiction on the process of mathematical problem solving.
The fourth item was targeted to examine students’ conceptions of the word larger in
comparing inﬁnite sets. Students may use different criteria in order to claim that one
inﬁnite set A is larger than the other inﬁnite set B. As shown in the pilot study, one
criterion is based on a competition (or race) between two running lists of elements within
each inﬁnite set. Another criterion is to compare two amounts of entire sets of elements.
Thus, the fourth item could reveal criteria that students use for comparing two inﬁnite

sets. The questions in the second, third and fourth categories are shown in Figure 4.

 

11. (Second Item) 01’ which are there more? Please check one of the boxes. How do you know?
5. 1] Of your ﬁngers or Cl Of your toes — because
6. El Of odd numbers or 1] Of even numbers — because
7. [:1 Of odd numbers or 1] Of integers — because
111. (Third Item) Which of the two sets A and B is bigger? How do you know?
A = { 1, 2, 3, 4, 5, 6, 7, }
B = { 2, 4, 6, 8, 10, 12, I4, }
IV. (Fourth Item) A student claims that each of the two sets A and B is infinite, but A is larger
than B. What can this mean?

Figure 4: The questions in the second, third, and fourth items

 

 

 

In the history of mathematics, one aspect of students’ learning difﬁculties with the
notion of limit is the failure to link geometry with numbers (Cornu, 1991). In other words,
the geometric interpretation in the concept of limit differs from the understanding of the
notion of a numerical limit. Thus, both arithmetic and geometric contexts often have been

used to investigate students’ difficulties regarding limit (Davis and Vinner; Monaghan,

32

1991, Szydlik, 2000). Speciﬁcally, the limit of a sequence in an arithmetic context and
the limit of a function in a geometric context have been focused on in previous studies on
limit. Thus, the intention of the ﬁfth item was to investigate how students calculated the
limit of a given sequence in arithmetic context. Graphical representations in the sixth
item were utilized to examine students’ mathematical discourse on limit by using spatial
principles in geometric context. Students might use different ways of mathematical
discourse to approach different mathematical contexts. The questions in the ﬁfth and

sixth categories are shown in Figure 5.

 

V. (Fifth Item) What do you think will hap en later in the columns of this table?

 

 

 

x Vx+2 -5
x
1.0 0.099020
0.5 0.099505
0.1 0.099900
0.05 0.099950
0.01 0.099990
0.005 0.099995
0.00 1 0.099999

 

 

 

 

VI. (Sixth Item) What will happen to the curve in #8 as x approaches 0 from left or from right?
What will happen to the curve in #9 and #10 when x goes to positive inﬁnity?
2 2
8. l 9. x 10. x
x l + x (1 +x)2

 

 

 

 

 

 

 

l
l
’0‘ .
' 00
, l
. o
. r
, O9
* v v v w v 7 v v v v . L I
O
" I
l I.
l l
' ‘9
' e
L
l
I l
L .‘Q ‘ .
, .
b

Figure 5: The questions in the ﬁfth and sixth items

 

 

 

In the seventh item, concrete geometric representations in limiting processes were
used in the questions to examine different characteristics of students’ mathematical
discourse on limit from the ﬁfth and sixth categories. The questions in the seventh item

could draw a result which explicitly explains how students think and their approach to the

33

limit in geometric progressing processes of regular polygons inscribed in a circle. The

eighth item was intended to assess how students identity the limit of a sequence as either
inﬁnity or a number. Another purpose of the questions in the eighth item was to evaluate
how students utilize the deﬁnition of limit in their minds in answering the questions. The

questions in the seventh and eighth categories are shown below in Figure 6.

 

VII. (Seventh Item) Examine the sequence of the square, the regular pentagon, hexagon (6 sides),
heptagon (7 sides), ..., which is inscribed in the circle.

”’00 ......

What happens if you continue increasing the number of sides of a regular polygon inscribed in a
circle? Explain your reasoning.
VIII. (Eighth Item)

- A student claims that a sequence a], a2, a3, goes to infinity. What does this mean?

 

 

 

 

- A student claims that the number P is the limit of a sequence a}, a2, a3, What does this
mean?

 

 

 

Figure 6: The questions in the seventh and eighth items

Unlike the English questionnaire, a ninth item was considered in the Korean
questionnaire. With the Korean mathematical words of inﬁnity and limit, the same
question asked in the ﬁrst item was used in the ninth item of the Korean questionnaire to
thoroughly investigate Korean students’ discourse. In summary, the discourse
questionnaire included a total of 16 questions in eight categories to investigate students’
colloquial discourse on inﬁnity and limit and their mathematical discourse about these
concepts in the different mathematical contexts.

Procedures for Collecting Data

During data collection, two main types of data were collected: students’ responses to
the questionnaire and interviews with pairs of students solving the same tasks in the
discourse questionnaire. First, I contacted instructors (e. g., teaching assistants or

professors) who taught calculus classes at MSU and KU to ask about their willingness to

34

allow me to collect data in their classrooms. After obtaining approvals from the
instructors, all the students in each identiﬁed classroom were asked during their classes to
participate in this study during their classes. The native-English participants at MSU were
provided a consent form for soliciting their participation which was a part of the MSU
Institutional Review Board (IRB) requirements. The consent form described the purpose
of this study, its importance, and the procedures involved. The beneﬁts and risks were
also explained (see Appendix C for details). From several calculus classes, 132 English
speakers and 126 Korean speakers were recruited to take the survey on the basis of
instructors’ agreement to data collection in their classroom and students’ willingness to
participate. The volunteers were informed that they might be selected for the further
paired interviews after completing the questionnaire.

Of those volunteers in each university, twenty students were selected for paired
interviews based on their responses to the questionnaire. The following procedures were
used in selecting interview participants. First, I looked at whether the participants had
studied mathematics before in any language other than their native language; any
participant who had done so was not considered as a possible interviewee. Second,
different kinds of students were selected in terms of their responses to the questionnaire
based on their previous curricular experience with inﬁnity and limit. The selection of
those students in terms of their curricular experience improved the validity of the data for
the proposed study (Patton, 2002), because the selection of students from various levels
can be more representative of the population.

To examine students’ intended curricular experience with inﬁnity and limit, the

textbooks students used in high school were analyzed. Table 3.3 summarizes percentages

35

of the English-speaking participants’ and Korean-speaking participants’ written responses
to the question about high school textbooks.

Table 3.3 Percentages of students’ responses to the question about high school textbooks

 

 

 

1.:sz K-participants K-interviewees “35:ka E-participants E-interviewees

K1 9 (7.1%) 2 (10.0%) USl l (0.8%) 0 (0.0%)

K2 12 (9.5%) 2 (10.0%) U82 1 (0.8%) 0 (0.0%)

K3 10 (7.9%) 2 (10.0%) US3 0 (0.0%) 0 (0.0%)

K4 1 (0.8%) 0 (0.0%) U84 14 (10.6%) 4 (20.0%)

K5 9 (7.1%) 2 (10.0%) USS 20 (15.2%) 4 (20.0%)

K6 26 (20.6%) 4 (20.0%) U86 4 (3.0%) 2 (10.0%)

K7 13 (10.3%) 2 (10.0%) US7 24 (18.2%) 4 (20.0%)

Mixed 31 (24.6%) 6 (30.0%) Mixed 7 (5.3%) 0 (0.0%)

Others 9 (7.1%) 0 (0.0%) Others 53 (40.2%) 6 (30.0%)
No remnse 6 (4.8%) 0 (0.0%) No response 8 (6.1%) 0 (0.0%
Total 126 (100%) 20 (100%) I32 (100%) 20 Q00%L

 

In order to ensure that the selected interviewees were representative of each
population in terms of curricular experience, I selected the number of interviewees within
each category on the basis of the ratio of the number of students in each category to the
total number of students. For instance, because the ratio in the K6 category is 21% (the
number of students (n = 26) / the total number of the Korean students), four interviewees
were selected within the K6 category among the total of twenty selected interviewees.
After deciding the number of interviewees within each category based on their curricular
experience, a random selection procedure was used to choose interviewees within the
category. For instance, in the US7 category, 4 students were randomly selected out of the
total of 24 students. If a student declined to participate in paired interview, another
random selection was conducted with all of the students in each category, excluding the
student who declined to participate in the interview.

The following paragraph describes the interview method for this study. The method
of interviewing pairs of students, who solved problems together with minimal

participation of the interviewer, was used because this type of interview may further

36

reveal mechanisms of student thinking (especially in terms of routines in the ﬁrst
research question) rather than interviewing a single student by a researcher. That is,
student thinking processes were monitored through paired problem solving. A pair of
students was able to watch each other’s errors, probe the other student’s understanding,
and help each other to reach a deeper level of comprehension. To ensure the occurrence
of the listed three advantages of the paired problem solving, I helped pairs of students to
continuously verbalize their thoughts aloud and express the major steps they used to
solve each mathematical problem on the questionnaire. I also assisted each pair of
students to clarify his or her thoughts for details whenever the listener was not sure of
how the problem solver was thinking (Whimbey and Lochhead, 1984). The following
possible prompts were used: “Can you explain how to come to your answer step by
step?”, “What do you mean by the word you just used?”, and “Can you explain more
about what you just said?” This was a way to monitor student thinking and reduce
interviewer’s effects and biases on students’ responses.

The interviews, which were conducted in the participants’ ﬁrst language, took as
much time as the participants needed to respond to the problems in the discourse
questionnaire. For pairs of students to concentrate on a given problem, one small card
listing each term or each problem was prepared and shown to interviewees during the
interviews. For instance, the problems in the ﬁrst category were given to interviewees on
a card with only one word written on it, such as inﬁnity and limit. The interviews took
place in a university ofﬁce, or another convenient location. All interviews were audio-
taped, video-taped, and transcribed in their entirety for further analysis. As compensation,

each interviewee received 25 dollars. The Korean-English bilinguals (three mathematics

37

educators and one linguist who helped the Korean translations of the survey questionnaire
into English) assisted in transcribing the Korean interviews into English.
Analysis
Analysis of Former Education

To analyze and categorize students’ written responses to the questions regarding
former education in the background questionnaire, the columns of course and content
were used to search for variations in different levels of previous education. For instance,
the following categories were used for responses to the questions about limit (see Table
3.4). Compared to those university students who did not learn about the concept of limit
in their secondary education, university students who learned about limits when taking
Pro-calculus or Calculus in high school may respond differently to the notion of limit
because of their different degrees of exposure to limit. Similarly, the content column was
used to select different levels of students in terms of different experiences and degrees of
difﬁculties with the concept of limit.

Table 3.4 The course and content categories for limit

 

 

Course Content

Algebra Limits to inﬁnity

Geometry The 8-8 (epsilon-delta) deﬁnition of limit
Advanced Algebra Limit of a sequence or series
Pre-calculus Limit of a function numerically

Calculus Limit of a function geometrically

L Limit and the deﬁnition of continuity
Limit and the deﬁnition of the derivative

 

Analysis of Intended Curricula
Intended curriculum in this study means planned learning experiences in terms of

Standards or frameworks at the national or state level, as well as instructional guides and

38

textbooks. Thus, instructional materials and policy documents were used for intended
curricular emphases. As noted, information about instructional materials was collected on
the basis of students’ written responses to the questions regarding the textbooks that they
used. However, due to the complexity of curricular experience, interviewees were
selected as representative of each population in terms of their curricular experience rather
than analysis of their textbooks.

Discourse Analysis

The transcribed data from paired interviews were used as a primary source to
analyze students’ non-mathematical and mathematical discourses. Through paired
problem solving, students’ non-mathematical and mathematical discourses on inﬁnity and
limit were recorded, transcribed and coded according to the four features of discourse.
Then, interview data were analyzed in order to identify and describe those four features.
Because of the nature of the questions asked, only a subset of the four features was
depicted in each category. At the next stage, the analysis of the data was guided by the
three research questions of this study.

In each of the four features, the representative analysis samples were elicited from
the pilot study (see Appendix B for details). Based on the four features, I deduced
hypotheses to relate characteristics in mathematical discourse to those in non-
mathematical discourse and curricular experience. Finally, I examined similarities and
differences of these hypothesized relationships between native-English and native-
Korean speakers. To decide what to include and focus on from the entirety of the
conversations, my analytic decisions were based on representatives of general themes

through the interview rather than on outliers. Repeated parts were also dropped.

39

Therefore, only a sample of representative utterances was presented in the results. The
theoretical components (the four distinctive features of colloquial and mathematical
discourses) which each item assesses are summarized in Table 3.5.

Table 3.5 Theoretical components assessed by each item

 

 

 

Theoretical components
category Word use Visual mediators Routines Egg:
1 «l 4
ll w/ «I w/
Ill ‘l \l J
IV «I J
V J ~/ «I
VI \/ w/
VII 4 w/
VIII «I ~/

 

 

40

CHAPTER IV
ANALYSIS AND FINDINGS
This chapter is divided into two main sections. The ﬁrst section describes the overall

characteristics of the survey participants’ backgrounds relative to mathematical learning,
using descriptive and inferential statistical results of all the participants’ responses to the
survey questionnaire. The second section provides a characterization of the participants’
colloquial and mathematical discourses, drawing on analyses of all the participants’
written responses to the survey, as well as analyses of the interview data of twenty
representatives in pairs within each linguistically distinct group. The analyses in the
second section are organized to identify the primary features of the participants’
colloquial and mathematical discourses on inﬁnity and limit.

Background Datgpbout the Students and
their Past and Present Mathematical Learning

In this section, I will discuss students’ past and present mathematical learning, and
will present an analysis of the 126 Korean and 132 English speaking students’ written
responses to the background questions in the survey. For the second research question,
this analysis of the background data will generate differences between the Korean and
English speaking groups in their former mathematics education and curricular
experiences with inﬁnity and limit. To investigate whether or not there is a statistically
signiﬁcant background difference between the two groups on categorical variables,
Pearson’s chi-square statistics (with alpha set at .01) in Statistical Package for the Social
Sciences 16.0 for Windows (SPSS Inc., 2007) were used because most of the items on the
background questions measure categorical variables. One of the important assumptions of

chi-square is that the minimum expected cell frequency should be 5 or greater (Pallant,

41

2005). If the minimum expected count violated this assumption, two or three categories
were collapsed into one category to use chi-square analysis. For background items
measuring a continuous variable, an independent samples t test was conducted to
compare the mean score for the two groups. Chi-square analysis compares the
proportions of the two groups rather than their means as in the case of the t test.

The background survey consists of 21 questions. Most of the questions were either
adapted or borrowed from the student questionnaire of the Third International
Mathematics and Science Study (TIMSS, 2003). These 21 questions are aimed at
investigating demographic information about the participants, their attitudes toward
mathematics learning, and characteristics of their formal education related to inﬁnity and
limit. This questionnaire provides not only the information about students’ background,
but also their past and present mathematics learning about inﬁnity and limit.

Students’ Background
Personal background

Table 4.1 presents a summary of the demographic characteristics of the Korean and
English speaking participants (K-participants and E-participants) in the survey study. In
total, the 126 Korean and 132 English speaking university students participated in this
study from three and six classrooms respectively. The 126 Korean speakers (K-speakers)
consisted of 31 (24.6%) females and 95 (75.4%) males. Of the 132 English speakers (E-
speakers), 37 (28.0%) students were female and 95 (72.0%) were male. The grade levels
of the K-speakers were as follows: 123 (97.6%) freshmen, 1 (0.8%) sophomore, and 2
(l .6%) juniors. The E-speakers were composed of 109 (82.6%) freshmen, 15 (11.4%)

sophomores, 6 (4.5%) juniors, and 2 (1.5%) seniors.

42

One hundred percent of the K-participants said that they spoke Korean as their
native language. The majority (84. l %) of the E-participants stated that they had English
as their native language and 15.9% (n=21) participants, including ﬁve Koreans, reported
“other” as their native language. The majority (83.3%) of the K-speakers responded that
they were studying calculus in English due to the use of the English calculus textbook in
their classroom. However, the language of instruction in class was not English, but
Korean. The majority (86.4%) of the E-speakers replied that they had not been exposed to

other languages in learning mathematics. Both their instruction and their textbook was in

 

 

 

 

 

 

English.
Table 4.1 Demographic characteristics of the survey participants
K-speakers E-speakers
(n = 126) (n = 132)
0. Gender Male 31 (24.6%) 37 (28.0%)
Female 95 (75.4%) 95 (72.0%L
1. Year Freshman 123 (97.6%) 109 (82.6%)
Sophomore l (0.8%) 15 (11.4%)
Junior 2 (1.6%) 6 (4.5%)
Senior 0 (0.0%) 2 (1.5%)
2. Native Language Korean 126 (100.0%) 5 (3.8%)
English 0 (0.0%) III (84.1%)
Other 0 (0.0%) 16 (12.1%)
3. How often do you speak Always 120 (95.2%) l 14 (86.4%)
your native language at Almost always 4 (3.2%) 16 (12.1%)
home? Sometimes 2 (1.6%) l (0.8%)
Never 0 (0.0%) 0 (0.0%)
No response 0 (0.0%) l (0.8%)
4. About how many books 0-10 books 7 (5.6%) 4 (3.0%)
are there in your home? 1 l-25 books 13 (10.3%) 10 (7.6%)
(Do not count magazines, 26-100 books 33 (26.2%) 39 (29.5%)
newspapers, or your 101-200 books 28 (22.2%) 33 (25.0%)
school books) More than 200 books 45 (35.7%) 46 (34.8%)

 

On the question regarding how often the participants speak their native language
at home, 95.2% of the Korean speaking group (K-group) and 86.4% of the English
speaking group (E-group) answered “always.” With respect to the question on the

number of books which the participants estimate having at home, this survey indicated

43

that there was no signiﬁcant difference between two linguistically different groups in
terms of the chi-square statistic (x2 = 1.99, p > .01).

Mathematics background

To investigate and categorize variations in different levels of former education,
students’ written responses to the background questions regarding textbooks used and
courses taken in high school were analyzed.

Table 4.2 Summary of textbooks used in higr school

 

 

Korean K-speakers U.S. E-speakers
Textbooks (n = 126) Textbooks Q1 = 132)
K 1 9 (7.1%) US l l (0.8%)
K 2 12 (9.5%) US 2 l (0.8%)
K 3 10 (7.9%) US 3 0 (0.0%)
K 4 1 (0.8%) US 4 14 (10.6%)
K 5 9 (7.1%) US 5 20 (15.2%)
K 6 26 (20.6%) US 6 4 (3 .0%)
K 7 13 (10.3%) US 7 24 (18.2%)
Mixed 31 (24.6%) Mixed 7 (5.3%)
Others 9 (7.1%) Others 53 (40.2%)
No response 6 (4.8%) No response 8 (6.1%)

 

Table 4.2 indicates that about 90% of the K-speakers said that they studied from one
or more of the seven representative textbook examples in the survey, whereas only 53.7%
of the E-speakers answered that they used one of the seven examples in high school (see
Appendix C for the examples). Of the K-speakers, 24.6 % reported that they used

multiple textbooks in the examples rather than one textbook series, and 5.3 % of the E-

 

 

speakers reported the same thing.
Table 4.3 Summary of course series taken in high school
Korean courses Iii-15:63:? U.S. courses E (Zing?)

Math Application 4 (3 .2%) Pre-Algebra 13 (9.9%)
Mathematics 10 l 10 (87.3%) Algebra or Algebra 1 62 (47.0%)
Mathematics 1 115 (91.3%) Geometry 83 (62.9%)
Mathematics 11 114 (90.5%) Advanced Algebra/Algebra 2 91 (68.9%)
Calculus 112 (88.9%) Pre-Calculus/Calculus 114 (86.4%)
Others 0 (0.0%) Others 4 (3.0%)
No response 10 (7.9%) No response 29 (22.0%L

 

44

Table 4.3 demonstrates that about 90% of the K-speakers who responded to the
question about courses taken in high school said that they took courses consistently in the
series from Mathematics 10 to Calculus. In contrast, only 47.0%, 62.9%, and 68.9% of
the E-speakers reported that they took Algebra 1, Geometry, and Algebra 2 respectively,
in high school.

Another question was asked in order to inquire about the most advanced
mathematics course that students took in high school. Of the K-speakers, 90.5 %
answered that they did not take any course other than calculus. However, 88.9% of the K-
speakers reported that they took calculus in high school as part of a course series taken in
high school. Comparatively, 86.4% of the E-speakers claimed that they took either

Calculus or Pro-Calculus in high school as the most advanced mathematics course taken

in high school.

Experience with inﬁnity and limit

To examine the participants’ experience with the mathematical words inﬁnity and
limit, they were asked to recall the ﬁrst mathematics course in which they heard the terms.
Table 4.4 shows a summary of frequencies and percentages of the participants’ responses

to the question about their experiences with the words inﬁnity and limit.

Seventy seven percent of the K-speakers responded that they remembered the ﬁrst
mathematics course in which they heard the word inﬁnity, compared to about 50 % of the
E‘Speakers. Of the K-speakers, 88.9 % reported that they recalled the ﬁrst use of the word
limit in a mathematics course, whereas 84.1 % of the E-speakers said that they
remembered hearing the word in a mathematics class. Chi-square analysis of this

distribution indicated that the K-speakers were signiﬁcantly more likely to report

45

 

 

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remembering the ﬁrst mathematics course in which they heard inﬁnity (x2 = 19.15, p

< .001), but there was no signiﬁcant difference between the K-group and E-group in

remembering the ﬁrst use of limit in their mathematics course (x2 = 1.27, p > .01).

Table 4.4 Summary of the participants’ recollection of experiences with iryinity and limit

 

 

 

 

 

K-speakers E-speakers
(n = 126) (n = 132)
6. Can you remember the ﬁrst math course in Yes 97 (77.0%) 67 (50.8%)
which you heard the word (mathematical) inﬁnity? No 29 (23.0%) 65 (49.2%)
7. If yes, specify in which grade(s) and in which 2 0 (0.0%) 1 (0.8%)
course/class you heard the word (mathematical) 4 4 (3.2%) 2 (1.5%)
inﬁnity. 5 2 (1.6%) 3 (2.3%)
6 5 (4.0%) 3 (2.3%)
7 16 (12.7%) 9 (6.8%)
8 18 (14.3%) 8 (6.1%)
9 7 (5.6%) 7 (5.3%)
10 17 (13.5%) 13 (9.8%)
11 26 (20.6%) 13 (9.8%)
12 0 (0.0%) 7 (5.3%)
14 0 (0.0%) l (0.8%)
No response 31 (24.6%) 65 (49.2%)
8. Can you remember the ﬁrst math course in Yes 1 12 (88.9%) 111 (84.1%)
which you heard the word (mathematical) limit? No 14 (11.1%) 21 (15.9%L
9. If yes, specify in which grade(s) and in which 6 0 (0.0%) l (0.8%)
course/class you heard the word (mathematical) 8 0 (0.0%) 2 (1.5%)
limit. 9 6 (4.8%) 6 (4.5%)
10 29 (23.0%) l6(12.1%)
ll 77 (61.1%) 42 (31.8%)
12 0 (0.0%) 33 (25.0%)
13 0 (0.0%) 10 (7.6%)
14 0 (0.0%) l (0.8%)
No response 14 (11.1%) 21 (15.9%)

 

On the question regarding the grades in which they heard the words inﬁnity and limit,

Overall response rates are the same as the percentage of the participants who answered

“y 68” for the previous question. For instance, about 75% of the K-speakers and 50% of

the E-speakers did specify grades in which they recalled having ﬁrst heard the word

inﬁnity. Among the students who speciﬁed a grade, the range of grades was from 4th to

I l th grade in the K-speakers’ responses. The range of grades of the E-speakers’ responses

was more varied, from 2nd to 14th grade. This survey result also indicated that the

maj ority (84.1%) of the E-speakers’ responses had a variety of grades relative to ﬁrst

46

 

 

hearing the word limit, ranged from 6th to 14’h grade as compared to the majority (88.9%)
of the K-speakers who said that they ﬁrst heard the word limit between 9’“ and 11’” grade.

Table 4.5 Grades in which students recalled ﬁrst hearing the words inﬁnity and limit

 

 

 

Word K-speakers (n=l26) E-speakers (n=l32)
M (SD) M (SD) df t-score
Inﬁnity 8.75 (1.99) 9.01 (2.37) 160 - .78
Limit 10.63 (0.59) 11.15 (1.22) 158.09 - 4.06‘

 

The grade items were analyzed using attest because grade is a continuous outcome
variable. In the case of the question about the participants’ recollection of the grade in
which they were ﬁrst introduced to the word inﬁnity, equal variance was assumed
because the signiﬁcant value of Levene’s test for equality of variances was larger than
0.05. The t test [t (160) = -.78, p > .01] conﬁrmed no signiﬁcant difference between the
K-group and E-group in the range of grades in which they ﬁrst heard inﬁnity. In the case
of the same question about the word limit, equal variance is not assumed because the
result of Levene’s test was signiﬁcant. Interestingly, the t test result [t (158.09) = -4.06, p
< .01] revealed that the mean of the E-speakers’ responses to grade was signiﬁcantly
higher than that of the K-speakers, as shown in Table 4.5.

In order to indentify the topics related to the words inﬁnity and limit which students
studied in high school, seven concept categories within each term were considered.
Tables 4.6 and 4.7 show percentages of students reporting that they had studied particular
topics related to inﬁnity and limit in high school as well as the results of chi-square
comparing the percentages of each concept category in the two groups. Pearson’s chi-
square statistics were employed to compare the percentages of the K-group and the E-

group in each concept category.

47

Table 4.6 Percentages of students reporting having studied speciﬁc topics related to
inﬁnity in highschool and the results of chi-square

 

 

 

13. Please check the box(es) to topic K-speakers E-speakers Chi-square

Wthh you have studied. (n =126) (n = 132) Value Signiﬁcance Level
a Inﬁnite processes 42.9% 47.0% 3.69 .06

b. Inﬁnite decimal fractions 84.9% 45.5% 34.95 .000"

c. Inﬁnite sequence or series 91.3% 61.4% 23.94 .000*

d The sum of inﬁnite geometric series 75.4% 50.8% 8.48 .004“

e. Comparison of inﬁnite sets 46.8% 32.6% 2.00 .16

f Countable and uncountable sets 69.8% 18.9% 58.51 .000"

g. Inﬁnitely small or inﬁnitesimal 46.0% 36.4% .33 .57

 

Note. "‘ The percentage difference is signiﬁcant at the 0.01 level.

In the case of inﬁnity, the chi-square analysis conﬁrmed a signiﬁcant difference
between the K-group and E-group in categories b, c, d, and f. The results revealed that the
percentage of K-speakers who reported having learned each of the four concepts was
signiﬁcantly different from the percentage of E-speakers who reported having learned the

same concept.

Table 4.7 Percentages of students reporting having studied speciﬁc topics related to limit
in high school and the results of chi-square

 

 

 

14. Please check the box(es) to topic which K-speakers E-speakers Chi-square
You have Smdled- (n = 126) (n =l32) Value Sig.
0. Limits to inﬁnity 86.5% 77.3% 5.21 .022
b. The 3-8 deﬁnition of limit 14.3% 31.8% 1 1.22 .001 "
0. Limit of a sequence or series 88.1% 53.0% 44.23 .000“
(1 Limit of a function numerically 61.1% 65.9% .66 .42
e. Limit of a function geometrically 60.3% 52.3% 1.90 .17
f Limit and the deﬁnition of continuity 92.9% 65.2% 38.00 .000"
81.0% 65.2% 9.86 .002“

g. Limit and the deﬁnition of the derivative

 

Note. Sig; = signiﬁcance level. "‘ The percentage difference is signiﬁcant at the 0.01 level.

In the case of limit, the differences in categories b, c, f, and g were large enough to
be statistically signiﬁcant. It is notable that the percentage (31 .8%) of the E-speakers who
reported having studied the 3-5 deﬁnition of limit in high school was signiﬁcantly higher
than that (14.3%) of the K-speakers, whereas the percentages of the E-speakers who
stated that they learned the other topics were less than those of the K-speakers at

statistically signiﬁcant levels.

48

 

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Summary

In summary, most E-speakers and K-speakers in the survey were freshmen and the
ratio of males to females was similar between the E—group and K-group. The majority of
the E-speakers said that they were exposed only to English in learning mathematics, and
the majority of the K—speakers answered that they studied mathematics in Korean with
the use of an English textbook (Thomas, 2005) in their calculus classrooms. However, in
most cases the K-speakers were exposed to the English textbook only recently. This
English textbook was used at the university, whereas high schools in Korea use Korean
mathematics textbooks.

The E-speakers reported that they used a variety of textbooks in high school other
than the representative examples in the survey, whereas the majority of the K-speakers
said that they studied from the seven textbook examples. E-speakers also claimed that
they took courses less consistently in the course sequences than the K-speakers. It is
noteworthy that even though the majority of the E-speakers and K-speakers reported
taking either Pre-calculus or Calculus in high school, K-speakers indicate that the
calculus course is required, whereas E-speakers imply that US. pre-calculus or calculus
is optional. Note that the non-response rate of the E—speakers is more than double that of
the K-speakers on the item about course series taken in high school. This is probably not
only because some of the E-speakers took different course sequences from the Algebra 1
— Geometry - Algebra 2 sequence, perhaps through the use of integrated curricula, but
also because they studied from textbooks other than the survey examples.

As for experience with inﬁnity and limit, it seems to be easier for students to

remember when they ﬁrst heard the word limit in a mathematics course than to remember

49

when they ﬁrst heard the word inﬁnity. More K-speakers responded that they
remembered the ﬁrst mathematics course experience about inﬁnity compared to E-
speakers. However, there was no signiﬁcant difference between the two groups in the
range of grades in which they ﬁrst heard inﬁnity. In the case of limit, there was no
signiﬁcant difference between the two groups in remembering the ﬁrst mathematics
course experience about limit. However, the E-speakers varied signiﬁcantly in their
responses to the question regarding the grade in which they heard the word limit. As for
the topics related to the words inﬁnity and limit which students studied in high school, K-
speakers reported that they had studied most of the seven categories related to the words
inﬁnity and limit at higher rates than the E-speakers. This is probably because the K-
speakers had studied more topics related to inﬁnity and limit in their high school curricula.
Students’ Current Mathematics Learning
Description of the calculus classes in the U.S. and Korea

The calculus courses in the U.S. institution which is the site of this study are 3-credit
classes taught by faculty and post-doctoral instructors. The U.S. class meets on Monday,
Wednesday, and Friday for 50 minutes, and has a class size that is usually under 35
students. Chapters 2 through 5 of Thomas’ calculus book (Thomas, 2005) are covered
during the semester. The U.S. calculus course is planned so that students take four one-
hour tests and a two-hour ﬁnal examination. Each test covers approximately one chapter
in Thomas’ calculus book. The ﬁnal exam is a comprehensive examination which covers
all the content from chapters 2 through 5. A lab course is designed for the U.S. calculus
course. This lab course as a 2-credit class is offered on a voluntary basis for students who

want extra work and more credits. This lab class meets Monday, Wednesday, and Friday

50

and students are required to attend class. In the lab class, students complete worksheets
and solve problems to review the content of the calculus course. In the fall of 2007, about
5 % (30 out of 983 students) of the total number of students in the calculus course took
the lab course.

The calculus class in Korea University is a 4-credit class taught by faculty and post-
doctoral instructors. The Korean class meets on either Monday and Wednesday or
Tuesday and Thursday for one hour and ﬁfteen minutes. The size of each class is
generally under 60 students. Chapters 1 through 12 of Thomas’ calculus book (Thomas,
2005) are covered during the semester. Some sections of the textbook are not covered
during the course, In the calculus class, students are required to take two two-hour tests
and a two-hour ﬁnal examination. Each test covers about four chapters in Thomas’
calculus book. However, the ﬁnal is not a comprehensive examination and covers the
content learned aﬁer the second test. Part of the class also includes a 50- to 80-minute
drill period every week. During the period, students solve problems in the calculus
textbook.

Student ’s self-report on their learning of mathematics

As context for interpreting the particular ﬁndings of this research relative to E-
speakers’ and K-speakers’ understanding of inﬁnity and limit, data were gathered to
determine the speakers’ perception of the calculus instructional setting. In order to
understand how students perceive their learning of mathematics in the classroom, each
classroom environment item was rated with the four categories of frequency in
mathematics lessons (‘Never,’ ‘Some lessons,’ ‘About half the lessons,’ and ‘Every or

almost every lesson’). Table 4.8 shows percentages of the participants’ responses to the

51

questions about their classroom environments and the chi-square results of the
comparison of the E-speakers’ and K-speakers’ responses about the mathematics lesson
items. The lowest expected frequency in any cell should be 5 (about four percent) or
more to use chi-square statistics. If the expected frequency was less than 5, a method of
collapsing two or three categories into one category was used. For the classroom
environment questions with the minimum expected counts less than 5 (e.g., 17-a, 17-b,
and 17-f), for instance, the ‘Never’ category and the ‘Some lessons’ category were
combined together into one category and the ‘Almost every lesson’ category and the
‘About half lessons’ category were collapsed into another category.

Table 4.8 Percentages of students’ responses to the classroom environment items and the
chi-scmare results

 

 

 

 

 

 

 

 

 

 

17. How often do you do these things Almost About Some No Chi-square
in your mathematics lessons‘7 every halfthe lessons Never response -
' lesson lessons df Value 518-
17-21. We work together in small K 3.9 6.4 20.6 68.3 0.8 1 2 32 l 3
groups. B 2.3 3.0 25.8 68.9 0.0 ' '
17-b. We relate what we K 2.4 7.1 45.2 45.2 0.0 1 12 91
learning in math to daily lives. E 4.6 5.3 33.3 56.1 0.8 ' '
. K 2.4 12.7 50.0 34.1 0.8 ,,
I7-c. We explain our answers. E 33.3 25.0 29.5 12.1 0.0 3 59.52 .000
l7-d. We decide on our own K 8.7 34.9 45.2 9.5 1.6 3 25 84 000,,
procedures for solving problems E 9.9 16.7 39.4 33.3 0.8 ' '
. K 15.9 27.8 38.9 16.7 0.8 ,,
17-e. We revuew our homework E 19.7 14.4 26.5 39.4 0.0 3 20.85 .000
17-f. We listen to the teacher’s a K 70.6 19.0 7.9 1.6 0.8 1 2 52 1 l
lecture-style presentation E 90.2 5.3 2.3 213 0.0 ° '
I7-g. We work problems on our K 18.3 37.3 38.1 5.6 0.8 ,
own. E 38.6 18.2 21.2 22.0 0.0 3 36'59 '000
I7-h. We use calculators K 0'8 1'6 27.8 69'0 0'8 3 19.40 .000"

E 9.9 9.1 18.2 62.9 0.0
Note. K = K-speakers; E = E-speakers; Sig. = signiﬁcance level. *The distribution differences are
signiﬁcant (p S .01).

There were statistically signiﬁcant differences between the E-group and K-group in
items 17-c, 17-d, 17-e, 17-g, and 17-h. In classrooms, the K-speakers said that they had
more opportunities to use their own procedures for solving complex problems and spent

more class time to review their homework, as shown in items 17-d and 17-e respectively.

52

In contrast, items 17-c, 17-g, and 17-h show that the E-speakers reported that they had
more opportunities to explain their answers, work on problems on their own, and use
calculators in their classrooms. Note also that there were no signiﬁcant differences in E-
speakers’ and K-speakers’ reports about use of small group work in the calculus
classroom (neither group reports this occurring very much); that both groups report few
opportunities to relate calculus to their daily lives, and that both groups report a similar
prevalence of lecture-based instruction.

Table 4.9 Percentage distributions of extra lessons and homework and the chi-square

 

 

 

 

 

results ,
. . K-speakers E-speakers Chi-square
Ques’w“ categmes (n = 126) (n = 132) df Value Sig.
19. During this school Every or almost every day 0.8 2.3
year, how often have you Once or twice a week 1.6 10.6
had extra lessons or Sometimes 0.0 24.2 I 10.05 .002“
tutoring in math that is not Never or almost never 97.6 62.1
part of your regular class? No response 0.0 0.8
Every day 1.6 60.6
20. How often does your 3 or 4 times a week 0.0 27.3
. . 1 or 2 times a week 97.6 6.1
instructor give you 1 195.59 .000“
homework in math? Less than once a week 0.8 0.8
Never 0.0 4.6
No response 0.0 0.8
Fewer than 15 minutes 2.4 18.9
21. When your instructor 15-30 minutes 2.4 13.6
gives you math homework, 31-60 minutes 25.4 34.1 4 47 78 000,,
about how many minutes 61-90 minutes 37.4 14.4 ' '
are you usually given? More than 90 minutes 32.5 16.7
No response 0.0 2.3

 

Note. Sig. = signiﬁcance level. *The distribution differences are signiﬁcant (p S .01).

Questions on extra lessons in mathematics that were not a part of regular class and
mathematics homework were also asked. Table 4.9 shows the percentage distributions of
extra lessons and homework in mathematics and the chi-square results of the comparison
0f the E-speakers’ and K-speakers’ responses about the extra lessons and mathematics

homework items. Because the expected frequency is less than 5 in the question on extra
lessOns in mathematics, the ﬁrst two categories (‘Every or almost every day’ and ‘Once

or tvVice a week’) were combined together into one category and the other two categories

53

(‘Sometimes’ and ‘Never or almost never’) were collapsed into another category to use
chi-square statistics. In the next question on frequency of mathematics homework,
because the minimum expected count was .98, a method of collapsing categories was to
combine the ﬁrst two categories (‘Every day’ and ‘3 or 4 times a week’) into one
category and the other three categories (‘1 or 2 times a week,’ ‘Less than once a week,’
and ‘Never’) into another category. Finally, in the question on length of mathematics
homework, no collapsing method was considered because the minimum expected count is
bigger than 10.

There was a statistically signiﬁcant association between the E-speakers and extra
lessons. In other words, the E-speakers reported that they had more extra lessons or
tutoring than the K-speakers (x2 = 10.05, p < .01). Seventeen of 132 E-speakers had extra
lessons at least once a week, compared with three of 126 K-speakers. The results of the
chi-square analysis revealed that there were statistically signiﬁcant differences between

the K-group and E-group for frequency and length of mathematics homework. The
statistically signiﬁcant difference (x2 = 195.59, p < .001) was observed in frequency of
mathematics homework. The statistical result shows that the E-speakers were assigned
more homework than the K-speakers. Chi-square analysis of this distribution indicated a
signiﬁcant difference (12 = 47.78, p < .001). The K-speakers were signiﬁcantly more
likely to report spending time doing mathematics homework than the E-speakers.
Out-of-school activities

Nine items were asked to explore possible differences between the two linguistically
distinct groups in the ways that out-of-school time was spent on a typical school day.

Table 4.10 displays a summary of percentages of students’ responses to the questions

54

regarding activities before or after school and the results of chi-square to the extent of
time spent out-of-school activities.

Table 4.10 Percentages of students’ responses to the out-of-school activity items and the
chi-square results

 

 

 

 

 

 

 

 

 

 

 

18. On a normal school day, how No Less 1-2 More No Chi-square
much time do you spend before/after time than 1 hours than 2 res ns e d V 1 8'
school doing each of these things? hour hours p0 f a ue 1g.
18-a. I watch television and K 30.2 43.7 17.5 8.7 0.0 ,,
videos E 10.6 37.1 31.8 20.5 0.0 3 24'28 '000
K 36.5 32.5 19.8 10.3 0.8
l8-b. I play computer games E 54.5 2 5. 8 9.9 9.1 0. 8 3 10.08 .02
18-c. 1 play or talk with K 1.6 31.0 38.1 27.8 1.6 ,,
friends E 0.0 12.1 39.4 48.5 0.0 1 162° °°°°
. K 44.4 44.4 6.4 1.6 3 .2 ,,
18-d. 1d0jobs at home E 27.3 52.3 14.4 6.1 0.0 1 7.66 .006
. . K 73.8 2.4 11.1 12.7 0.0
18-e.Iworkatapa1djob E 64.4 4.6 6.1 25.0 0.0 1 1.70 .19
K 32.5 48.4 15.1 3.2 0.8 ,
’8": I play 31”“ E 22.7 29.5 31.8 15.9 0.0 3 26'6’ '000
18-g. 1 read a book for K 37.3 48.4 10.3 3.2 0.8 1 1 0] 32
enjoyment E 42.4 39.4 15.2 3.0 0.0 ' '
. K 0.8 20.6 47.6 30.2 0.8
18-h. I use the mtemet E 0.0 1 3. 6 3 8. 6 47.7 0.0 l 2.82 .09
. K 3.9 22.2 46.8 27.0 0.0 ,,
18-1. 1 do homework E 0.0 10.6 43.9 45.5 0.0 l 10.51 .001

 

Note. K = K—speakers; E = E-speakers; Sig. = signiﬁcance level. *The distribution differences are
signiﬁcant (p S .01).

Each item on activities before or after school was encoded on the basis of the
following four categories (‘No time,’ ‘Less than 1 hour,’ ‘ 1-2 hours,’ and ‘More than 2
hours’). Because the minimum expected frequencies of items 18-c, 18-d, l8-e, 18—g, 18-h,
and 18-i were less than 5 (about four percent), the ﬁrst two categories (‘No time’ and
‘Less than 1 hour’) and the other two categories (‘ 1-2 hours’ and ‘More than 2 hours’)
were collapsed into one and another categories respectively to use chi-square. The chi-
square results indicate that the E—speakers answered that they used more time than the K-
speakers on accomplishing most activities.

In order to investigate the extent of time spent out-of-school activities on average,

each item on activities before or after school was encoded based on the following scores

55

(0 = ‘no time,’ 1 = ‘less than 1 hour,’ 2 = ‘1-2 hours,’ and 3 = ‘more than 2 hours’). The

mean (M) and the standard deviation (SD) results for each item were summarized in

 

 

Table 4.11.
Table 4.11 Mean (M) and SD results to the extent of time spent out-of-school
Item K I:Ipeakgrls) Elsipeakesrls)
18-a. I watch television and videos. 1.05 0.91 1.62 0.93
18-b. I play computer games. 1.03 1.00 0.73 0.97
18-c. I play or talk with friends. 1.90 0.84 2.36 0.69
18-d. I do jobs at home. 0.62 0.68 0.99 0.82
18-e. I work at a paid job. 0.63 1.1 l 0.92 1.31
18-f. I play sports. 0.88 0.78 1.41 1.01
18-g. I read a book for enjoyment. 0.79 0.76 0.79 0.81
18-h. I use the intemet. 2.06 0.76 2.34 0.71
18-i. I do homework. 1.97 0.81 2.35 0.67

 

The descriptive statistic of item 18-i, for instance, shows that the E-group had a
mean score of 2.35 with a SD of .67 in time spent on doing homework on a usual school
day, which indicates that the E—speakers reported spending about one hour and 20
minutes on average doing homework on a typical school day. In contrast, the K-group
had a mean score of 1.97 with a SD of .81, which reveals that the K-speakers spent about
an hour on average.

Students ’ way of life

The results about students’ engagement in mathematics outside of the course (e.g.,
extra lessons, homework, etc.) reported in Table 4.9 raised additional questions about
whether there were indeed differences in what was happening outside of class with
students in these courses, and what the nature of those differences might be. In order to
gain additional insights about how students were studying mathematics, all English
speaking and Korean speaking interviewees (E-interviewees and K-interviewees) were

asked about the length of time they were spending studying and their study methods, to

56

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provide descriptions of their typical weekday in their typical week via email. In the
emails, the following two questions were asked: 1) Could you please describe how many
hours and in what kinds of format (e. g., group study, tutoring, study alone) you studied
calculus in a typical week last semester? 2) Could you please describe your typical
weekday? Your weekends? And your holidays?

Table 4.12 Summary of students’ descriptions of how to study calculus

 

Students Students’ descriptions of how to study calculus

 

I mainly reviewed contents at night on Monday and Wednesday when I had class. If there
was a test coming up, 1 studied at night on Monday, Tuesday, and Friday. I studied by
looking at the textbook and solving examples. ..I studied alone for about 2 hours at night on
Monday and Wednesday. Because 2 o'clock on Saturday would be the examination time, I
studied alone from about 5 pm when class was over to 12 o'clock and Friday night and
Saturday morning. . .I followed my class schedule during the week.

 

K-speakers In class, I roughly wrote the professor’s notes from the blackboard in my workbook ...And
then I reviewed contents that were covered in class by transcribing them from the workbook
into my notebook after class was over. ..So I worked in this way. I studied contents in the
textbook alone and solved problems related to them. I solved homework problems ﬁrst
alone, and then gathered with ﬁ'iends later and compared answers. If there were problems
which I did not solve alone, I solved them with the help of friends. ..I tried to be faithful to
my school life during the week. (In the given time table, she wrote “reviewing notebook” on
Tuesday and Thursday and “group study” on Friday)

 

1 often make the unwise choice of waiting until the night before a test or quiz to start
studying for it. . .The only way I can learn is if I sit down in a quiet place and read the text
book while doing example problems, or having a tutor sit down next to me and personally
explain the material...l have good intentions to study, but I usually end up wasting a lot of
time sitting around talking to my friends. (In the given time table, he wrote “alone” several
times on Mondgy thrmgh Sunday)

 

On weekdays, I go to class and do whatever homework is due the following day around 9 or
10 at night. On weekends, I ﬁnish all the homework I possibly can for the upcoming week,
and any reading that I missed over the preceding week. (In the given time table, she wrote “1
(hour)” on Monday, Wednesday, and Friday and “alone or Math Learning Center for 4
(hours)” on Sunday)

E-speakers

 

My typical weekday involves going to my classes and then relaxing for a little while once I
get home. I usually do homework before and after dinner... My weekends involve studying
and spending time with ﬁ'iends. (In the given time table, she wrote “study alone” several
times on Monday through Sunday and “study alone or Math Learning Center for “a half
hour” on Thursday)

 

(In the given time table, he just wrote “do problems out of the book” and “2 (hours)” on
Tuesdayand Thursday and “1 (hour)” on Sunday)

Four E-interviewees and two K-interviewees responded to those questions in March
and April 2008 (see Appendix D for their entire responses; notice that K-interviewees
were interviewed in May 2007, compared to E-interviewees in September and October

2007 due to different systems of academic year). Table 4.12 is a summary of the

57

students’ descriptions of how they studied calculus. The students’ responses provide a
more reﬁned view on how the two groups of students are studying. Two K-interviewees
reported that they had everything systematically planned including study methods,
periods of rest and they spent time exactly according to that plan. In contrast, the four E-
interviewees appeared to have planned less and admitted to not being able to follow it too
tightly even if they had a plan. These responses, although from a very small sample of
students, could suggest that the K-speakers are more serious about their studying, more
carefully organized, and devote more thought and more time to it than the E-speakers.
For the K-speakers interviewed here, studying is more central in their lives, compared to
the E-speakers.

Summary

Students in this study reported that they listened to teachers’ lecture-style
presentation in both U.S. and Korean classrooms for more than half of the lessons. In
classrooms, the K-speakers replied that they used their own procedures more ﬂexibly for
problem solving. However, the E-speakers responded that they had more opportunities to
explain their answers and use calculators. According to student responses, K-speakers
may be spending more class time reviewing their homework than the E-speakers,
probably because of the 50- to 80-minute drill period every week.

The E-speakers reported that they spent more time on homework at home on a
normal school day than the K-speakers. However, when talking about mathematics
homework speciﬁcally, the K-speakers replied that they spent about 60 minutes twice a
week on average, while the E-speakers reported that they spent less than 30 minutes three

times a week. The E-speakers seem to be more engaged in homework for several subjects,

58

in addition to mathematics. The frequency of mathematics homework (three times per
week in E-speakers’ responses and twice per week in K-speakers’ responses) seems to be
related to the number of times class meets per week.

As for extra lessons in mathematics that were not a part of regular class, the K-
speakers said that they have almost never had extra lessons or tutoring (see question 19 in
Table 4.9). In contrast, 37.1% of the E-speakers reported that they had extra lessons or
tutoring more than “sometimes.” In fact, some E-speakers said that they received extra
lessons from the Mathematics Learning Center (MLC) offered by the Department of
Mathematics at MSU]. The MLC is staffed primarily by teaching assistants and is open
approximately 40 hours per week for students in lower level mathematics courses. In
terms of students’ descriptions of how they study calculus in their typical week, K-
speakers’ and E-speakers’ responses could be interpreted as an indication that for K-
speakers, learning was more central in their lives because they reported systematically
devoting more time to it, compared with E-speakers.

Regarding students’ activities before or after school, the E-speakers reported that
they were more involved in the out-of-school activities given in the survey than the K-
speakers. The K-speakers may have spent time on other out-of-school activities which
were not listed in the survey examples, such as departmental activities and club activities,

as shown in two K-interviewees’ explanationsz.

 

’ Some interviewees, who said that they had extra lessons or tutoring “sometimes,” were asked how they
studied Calculus in a typical week via email. Two E-interviewees responded to the email. According to
their responses, they used the MLC for extra lessons and tutoring.

2 All Korean and E—interviewees were asked to describe their typical weekday via email. Two K-

interviewees answered the question. One Korean interviewee responded that she participated in many
activities of her department. The other Korean interviewee reported that he did club activities aﬁer classes.

59

Attitudes toward Mathematics
Students’ views of themselves as mathematics learners

Another potential important component of background is in the area of students’
views of themselves as mathematics learners and their attitudes about mathematics
learning. Six items on the written survey addressed these issues. To investigate whether
or not there was a statistically signiﬁcant difference between the K-speakers and E—
speakers in their attitudes toward mathematics learning, an independent samples t-test
was employed. Each item was scored 4 = ‘agree a lot,’ 3 = ‘agree a little,’ 2 = ‘disagree a
little,’ and l = ‘disagree a lot.’ The higher the mean scores are, the more strongly each
group agrees with statements in the attitude items. Table 4.13 presents the descriptive
statistics and t-test results of the survey participants’ response to each item.

K-speakers had a signiﬁcantly lower score (that is, bigger than 2.5) on item 15-a
than the E-speakers, which indicates that the E-speakers expressed a stronger agreement
with the statement that they usually do well in mathematics because a 2.5 indicates a
neutral response. Item 15-e reveals that the K-speakers had a signiﬁcantly higher score
(that is, less than 2.5) than the E-speakers, indicating that the E-speakers had a stronger
disagreement on the statement that “mathematics is not one of my strengths” than the K-
speakers. In other words, the E-speakers were more likely that the K-speakers to claim
that mathematics is one of their strengths. The K-speakers had a signiﬁcantly lower score
(less than 2.5) than the E-speakers in item' 15-d. In other words, the K-speakers had a
stronger disagreement than E-speakers with the statement that when they do not
understand a new topic initially then they will never understand it, suggesting that they

believe that they can eventually understand a topic that is difﬁcult initially.

60

Table 4.13 Students’ views of themselves as mathematics learners and the t-test results of
the comparison of percentage of the Korean and E-speakers’ responses

 

 

 

 

 

 

 

15. How much do you agree with these Agree Agree a Disagree Disagree M SD Sig
statements about learning mathematics? a lot little a little a lot '
. . K 10.3 61.1 23.8 4.8 2.77 .70 *

15-a. I usually do well in mathematics. E 53.0 40.2 6.1 0.8 3.45 . 65 .00
15-b. I would like to take more K 20.6 44.4 27.8 7.1 2.79 .85 91
mathematics in school. 7 E 26.5 40.2 17.4 15.9 2.77 1.01 '
15-c. Mathematics is more difﬁcult for me K 4.8 23.0 54.8 17.5 2.15 .76 29
than for many of my classmates. E 5.3 21.2 46.2 27.3 2.05 .84 '
15-d. When 1 do not initially understand a K 1.6 6.4 24.6 67.5 1.42 .69
new topic in mathematics, I know that I will .00*
never really understand it E 2.3 16.7 36.4 44.7 1.77 .81
15-e. Mathematics is not one of my K 8.7 29.4 44.4 17.5 2.29 .86 00*
strerlgths. E 6.8 13.6 38.6 40.9 1.86 .90 '

. . . . K 7.9 60.3 28.6 3.2 2.73 .65
15-f. I learn things quickly in mathematics. E 18.2 53.8 23.5 4. 6 2.86 .7 6 .15

 

Note. K = K-speakers; E = E-speakers; M = mean; SD = standard deviation; Sig. = signiﬁcance level.
*Differences are signiﬁcant (p _<_ .01).

The role of mathematics in students ' lives

Five items were included to examine students’ sense of the role of mathematics in
their lives. These were scored on the scale 1, 2, 3, and 4 (1 = ‘disagree a lot,’ 2 =
‘disagree a little,’ 3 = ‘agree a little,’ and 4 = ‘agree a lot’). In order to check for group
differences between the K-speakers and E-speakers in their view of the role of
mathematics in their lives, the statistical signiﬁcance of the mean difference was
determined by an independent samples t-test. The descriptive statistics and t-test results
are summarized in Table 4.14.

Overall mean scores of both the K-speakers and E-speakers in the ﬁve attitude items
were higher than a neutral response (that is a 2.5). More than 60% of both the E-speakers
and K-speakers agreed with the statements in the ﬁve items regarding the need of
mathematics in their lives. These descriptive statistics imply that students agreed on the
need for mathematics. However, as shown in items 16-a and 16-b, the E-speakers had a
signiﬁcantly stronger agreement on the need for mathematics in their daily lives as well

as for other school subjects than the K-speakers. There was also a signiﬁcant difference

61

between the two groups in the mean scores of item 16-c, which indicates that the K-
speakers were more convinced about the role of mathematics to get into the university of
their choice.

Table 4.14 The role of mathematics in students’ lives and the t-test results of the
comparison of percentage of the Korean and E-speakers’ responses

 

 

 

 

 

 

 

16. How much do you agree with these Agree Agree a Disagree Disagree M SD Si
statements about mathematics? a lot little a little a lot g.
16-a. I think learning mathematics will K 18.3 47.6 25.4 8.7 2.75 .86 002,,
help me in my daily life. E 34.1 47.0 12.9 6.1 3.09 .84 '
16—b. I need mathematics to learn other K 37.3 33.3 21.4 7.9 3.00 .96 01,,
school subjects. E 43.9 43.2 9.8 3.0 3.28 .77 '
16-c. I need to do well in mathematics to K 77.0 20.6 0.8 1.6 3.73 .56 000,,
get into the university of my choice. B 46.2 41.7 9.8 1.5 3.34 .72 '
16-d. 1 would like ajob that involved K 23.0 36.5 33.3 7.1 2.75 .89 92
using mathematics. E 25.8 38.6 22.0 13.6 2.77 .99 '
l6-e. I need to do well in mathematics to K 43.7 36.5 15.9 4.0 3.20 .85 36
get the job I want. E 40.9 33.3 20.5 5.3 3.10 .91 '

 

Note. K = K-speakers; E = E-speakers; M = mean; SD = standard deviation; Sig. = signiﬁcance level.
*Differences are signiﬁcant (p S .01).

Summary

The E-speakers in this study showed more conﬁdence in their capability of leaming
mathematics than the K-speakers. As for the need of mathematics, there was’a signiﬁcant
difference between the E-group and K-group. The E-speakers agreed more with the
practical need for mathematics in their daily lives, whereas the K-speakers showed a
stronger agreement on the need for mathematics to get into the university of their choice.
Comparability of the E-group and K-group
Similarities and differences between the E—group and K—group

Background analyses showed similarities and differences between the K-group and
E-group in terms of their past experience with inﬁnity and limit, current mathematics
learning, and attitudes toward learning mathematics. In students’ past mathematics
learning regarding inﬁnity and limit, the K-speakers seemed to have had more

opportunities to learn concepts related to inﬁnity and limit in high school than the E-

62

speakers. In addition, the grades in which students ﬁrst heard the word limit were less
varied in the K-group. However, there was no signiﬁcant difference between the E-group
and K-group in the range of grades in relation to ﬁrst hearing inﬁnity.

In students’ present mathematics learning, the E-speakers reported that they were
more involved in out-of-school activities and they devoted less time to learning,
compared with the K-speakers. Data from the follow-up questionnaire suggested that for
K-speakers learning was more important, and thus they systematically devoted more time
to it. In students’ attitudes toward learning mathematics, the E-speakers showed more
conﬁdence in learning mathematics than the K-speakers. As for the role of mathematics
in students’ lives, the E-speakers had more focused on its application to their daily lives
and other school subjects than the K-speakers. However, there was no statistically
signiﬁcant difference between the E-group and K-group in their perceived need of
mathematics for their jobs.

Summary

The primary purpose of this study is to examine potential differences in
mathematical discourse between E-speakers and K-speakers. However, any differences
that might emerge would need to be understood and interpreted with understanding of the
differences in students’ background that may contribute to those differences in
mathematical discourse. While analysing students’ discourses on inﬁnity and limit, all
these differences in the background analyses should be kept in mind as possible

attributions for observed differences in mathematical discourses on inﬁnity and limit.

63

Students’ Discourse about Infinity and Limit

In this section, I shall identify primary features of the participants’ colloquial and
mathematical discourses on inﬁnity and limit. These features were drawn from two
analyses. The ﬁrst analysis is based on the paired interview data of 20 representatives
within each linguistically distinct group. The other is an analysis of all the participants’
written responses (n=25 8) to the discourse questions in each item in the survey.
Differences in characteristics of the Korean speaking group (K-group) and the English
speaking group (E-group) were identiﬁed on the basis of the paired interview results, and
then with the help of chi-square analysis applied to the responses to the written
questionnaire, it was checked whether the ﬁnding could be generalized, that is, whether
the differences were statistically signiﬁcant.

In reporting the data from paired interviews, the following symbolization was
introduced. To refer to interviewee members within the E-group and K-group, the
symbols such as K6a and K6b were used for the ﬁrst pair of the K-interviewees who used
the Korean 6th textbook in the survey. The symbols of Eoc and Eod represent the second
pair of the E-interviewees who used other textbooks. The ﬁrst characters K and E
represent the Korean and English speakers respectively. The second characters 6 and 0
mean the Korean 6th textbook and other English textbooks respectively. The last
characters a and b in K6a and K6b represent the ﬁrst pair of the K-speakers who used the
Korean 6th textbook, whereas the characters c and d in Eoc and Eod refer to the second
pair of the E-speakers who used other English textbooks.

This section is divided into two main subsections: 2.1 Discourse on inﬁnity and 2.2

Discourse on limit. Each subsection presents the properties of the given discourse as

64

reﬂected in the responses to the relevant items in the questionnaire. Each subsection
closes with a summary of characteristics of colloquial and mathematical discourses on
inﬁnity and limit respectively.

Discourse on Infinity
Items 1 and IX

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67

Table 4.17 Distribution of the contexts of use of the word inﬁnite and the chi-square
results of the comparison of difference between the Eggrog) and K-group

 

 

 

 

Non-mathematical Mathematical Chi-square
Students Words context context None df Value Sig.
E-interviewees (n = 20) Inﬁnite}; 45 % 55 % 0 %
' 85 °/ 15 °/ 0 °/
K-interviewees (n = 20) InﬁniteKcOn o o o
InﬁniteKMath 65 °/o IS % 20 °/o
E-participants (n = 132) Infinite}; 34.1 % 52.3 % 13.6 % l 55 69 000*
K- artici ants (n =126) InﬁniteKcon 81.7 % 12.7 % 5.6 % . '
p p InﬁniteKMmjI 36.5 % 38.9 % 24.6 % 1 1.69 .19

 

*Differences are signiﬁcant (p S .01).

A. Inﬁnite and inﬁnitely: the type of discourse. E-speakers used the adjective, but not
the adverb. The use was mainly in non-mathematical discourse. K-speakers used both the
adjective and the adverb. They used the adverb when they were asked to create a sentence
with the noun inﬁnity (either mathematical or colloquial). Their uses of inﬁnitely and
inﬁnite was mainly in non-mathematical context (even more so than in the E-group).

The E-interviewees used the adjective predominantly in non-mathematical contexts
([2], [4], [7], [9], [12], [13], and [14]; 7 responses out of 14 in Table 4.15). Among the K-
interviewees, the use of mathematical contexts was rare even for the mathematical
version of the word inﬁnite (2 responses out of 12 for inﬁniteKCou and 2 out of 13 for
inﬁniteKMau| in Table 4.15). The chi-square results in Table 4.17 show that the distribution
of the word use of the E-participants and K-participants in the survey between

mathematical and non-mathematical contexts is signiﬁcantly different when the word
inﬁnite is colloquial (x2 = 55.69, p = .000), but not signiﬁcantly different when the word
is mathematical (x2 = 1.69, p > .01).

Table 4.18 Distribution of types of entities described with the word inﬁnite

 

 

 

 

Students amount or number of something self-sustained entities Chi-square
described as inﬁnite described by inﬁnite 4f Value Sig_._

E-interviewees (n = 20) 6O % lO %

K-interviewees (n = 20) 5 % 60 %

E-participants (n = 132) 30.3 % 9.9 % 1 67 94 000*

Kjarticipants (n = 126) 0.8 % 48.4 % ' '

 

68

Table 4.19 Distribution of the use of the word inﬁnitely as a descriptor of many
Chi-square

 

 

 

Students Inﬁnitely as a descriptor of many (if Value Si .
E-interviewees (n = 20) 0 %
K-interviewees (n = 20) 6O %
B—participants (n = 132) 0 %

.4 . *
K-participants (n =126) 37.3 % l 60 1 000

 

*Differences are signiﬁcant (p S .01).

B. Inﬁnite and inﬁnitely: the type of use (meaning). E-speakers’ use of the adjective
was in the context of large sets. In their sentences, the adjective inﬁnite appeared in the
expressions “inﬁnite number of” or “inﬁnite amount of,” that is, as a descriptor of a size
of a set. K-speakers used the adjective inﬁnite as descriptors of concrete things known for
their large dimensions or of spiritual matters. They used the adverb inﬁnitely as a
descriptor of many, in the context of sets.

The majority of E—interviewees used inﬁnite as descriptors of amount of or number
of elements in a set such as grains of sand and stars (see [1], [2], [7-8], [10], and [12-14]
in Table 4.15). The set was thus treated as one entity. The majority of K-interviewees,
with just one exception ([27]), applied the adjective inﬁnite, either in its colloquial or
mathematical version, directly to self-sustained entities (concrete objects: sea ocean,
seashore, earth, and universe — 5 out of 12 responses; spiritual matters: sorrow, love,
complains, possibilities, and potentialities in Table 4.15). Only K-interviewees used the
adverb, and they did it when asked to use the noun inﬁnityKCon (they added the sufﬁx of
hee to inﬁnitchOn which turned the noun inﬁnity into the adverb inﬁnitely). They used it
as a descriptor of “many” (see [56-59] and [67-68]). These phenomena were clearly seen
also in the written questionnaire. According to the chi—square results in Table 4.18, there
is a signiﬁcant difference between the E-group and K-group in the percentage of

sentences in which the word inﬁnite is used as describing entities. Chi-square analysis in

69

Table 4.19 indicates a signiﬁcant difference between these two groups (x2 = 60.41 , p

= .000) in the use of inﬁnitely to describe many.

Table 4.20 Distribution of sentences not using the noun inﬁnity

 

 

 

 

. . . Chi-square
Students Sentences not usmg the noun mﬁmty df Value SiL
E-interviewees (n = 20) 0 %
K-interviewees (n = 20) 95 %
E-participants (n = 132) 0 %
1 179.91 .000"‘
K-participants (n =126) 77.8 %

 

*Diﬁ‘erences are signiﬁcant (p S .01).

Table 4.21 Distribution of the use of inﬁnity as process

 

 

 

 

Inﬁnity is unreachable and unpassable end-point of Chi-smiare
Students . .
never-ending process ay Value Sig.

E-interviewees (n = 20) 45 %

K-interviewees (n = 20) 0 %

E-participants (n = 132) 15.2 %

1 17.86 .000“

K-participants (n =126) , 0.8 %

 

*Differences are signiﬁcant (p S .01).

C. Inﬁnity. E-speakers’ use of the word inﬁnity was in association with processes, which
were said to be never ending. The inﬁnity itself was said to be unreachable or impassable.
There was no association of inﬁnity with largeness. K-speakers did not use the noun
inﬁnity, either in its colloquial or mathematical version. When asked to construct a
sentence with the inﬁnityKMa,;,, they converted the noun into adjective (inﬁnite) and when
asked to construct a sentence with the inﬁnityKCou, they converted the noun into adverb
(inﬁnitely).

Only one E-interviewee gave a fully structural response that described inﬁnity as a
number ([52]). Others were talking about it in the context of something in an unending

,9 ‘6

process (see the use of verbs “go on/to,” “pass, reach,” and “approach” - 14 out of 16
responses in Table 4.16). None of the K-interviewees’ responses contained any of these
verbs. The single response of a K—interviewee that did contain the noun inﬁnity featured

the verb diverge (in the expression “sequence diverges to inﬁnity” [73]). For some E-

70

interviewees, inﬁnity was tantamount to a never-ending process ([49], [55]). For others, it

was an object that was an inextricable part of a process. This object was applied

predominantly in the sense of an unreachable endpoint of processes that do not end. The

word inﬁnity appeared in some other object (x, ﬁmction, and numbers) that goes to

inﬁnity (in an expression such as “x approaches inﬁnity” [42]). Table 4.20 shows that

about 78 % of the K-participants used other grammatical forms to create a sentence rather

than using the noun. In contrast, the only grammatical forms employed by the E-

participants were nouns. A chi-square test in Table 4.21 indicated that the E-participants

were signiﬁcantly more likely to report having used the word inﬁnity as an unreachable

and unpassable end-point.

 

 

 

 

 

 

 

 

Item 11
Table 4.22 Distribution of endorsed narratives for comparing pairs of two sets in item 11
. Endorsed narratives Chi-square
Comparisons Students A = B A > B A < B Can’t Cy Value Sig._
E-interviewees 25 % 10 % 60 % 5 %
11-3: Fingers (A) K-interviewees 60 % 40 % 0 % 0 %
and toes (B) E-participants 39.4 % 14.4 % 38.6 % o % 2 47 94 000,
K-participants 56.3 % 28.6 % 3.2 % 0 % ' '
E-interviewees 45 % 25 % 25 % 0 %
II-b: Odd (A) and K-interviewees 30 % 15 % 30 % 20 %
even numbers (3) E-participants 41.7 % 17.4 % 22.0 % 9.1 % 3 I3 53 004'
K-participants 21.4 % 20.6 % 19.8 % 19.8 % ' '
132-interviewees 10 % 10 % 80 % 0 %
21:25:?‘32‘23'5 K-interviewees 5 % 5 % 7o % 20 %
(3) g E-participants 15.2 % 8.3 % 61.4 % 3.8 % 3 15 61 001:
K-participants 6.4 % 1.6 % 73.0 % 10.3 % ' '

 

Note. “A=B” = A is the same as B; “A>B” = A has more than B; “A<B” = B has more than A; “Can’t” = A
and B are incomparable. *Differences are signiﬁcant (p S .01). Missing data were not included in the table.

D. II-a: A = ﬁngers and B = toes. Almost all E-speakers compared the two sets

according to the number of elements, and if they did not choose the option A = B, it was

only because of the fact that they interpreted the word ﬁnger as not referring to the thumb.

The 40% of K-speakers who chose an option other than A = B did so because their

71

comparison was non-numerical. In other words, they responded ﬁngers mainly because
they thought ﬁngers were used more often than toes.

E. II-b: A = odds and B = evens. Only two ﬁfths of E-speakers chose the canonical
response A = B. Other responses were distributed evenly between A < B and A > B. Only
half as many K-speakers chose the canonical option A = B. Other responses were
distributed evenly between A < B, A > B, and “can’t say”. Some E-speakers (17.4 %) and
K-speakers (20.6 %) determined that there were more odd numbers because counting
started at one. Other E-speakers (22.0 %) and K-speakers (19.8 %) concluded that there
were more even numbers because zero as a starting number was an even number.
However, only K-speakers (19.8 %) replied that even numbers could not be compared
with odd numbers because they were both inﬁnite and countless.

F. II-c: A = odds and B = integers. Only 10%-15% of E-speakers chose the canonical
response A = B. An overwhelming majority (80% in the interview and 61.4% in the
survey) chose the option A < B. The percentage of K-speakers who chose the canonical
response A = B was even smaller than that of E-speakers (5% in the interview and 6.4%

in the survey). Approximately 70% chose the option A < B.

72

Table 4.23 Comparisons between ﬁgers and toes in item II-a

 

11. Of which are there more? (a) A: Your ﬁngers, B: Your toes

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

54a [1] Toes. 1 don’t count thumbs as a ﬁngerm Kla [17] They are the same because ﬁngers are

E4b [2] Neither. 1 don’t know if you can do Klb 10 and toes are 10 and the cardinal numbers
that are decided exactly

E4c [3] T968 because thumbs aren t part K2a [18] Same because of ten and ten
technically ﬁngers

E4d [4] Fingers because they’re longer and K2b [19] Fingers because ﬁngers are more
there’s more mass to your ﬁnger sensitive

85a [5] Fingers because they’re longer K3a [20] Fingers because ﬁngers are used more

E5b [6] Numerically they’re the same K3b [121] F mgers have senses and can move on

err own

E5c [7] There are more toes because 112:: [3%] game. ' '31:): agenaturaldmémbers

E5d technically thumbs aren’t ﬁngers E11036 rom w a '5 mg use ’ mgers are

E6a [8] Toes...you only have eight ﬁngers and K6a [24] They are the same.. .the numbers to be

E6b ten toes K6b seen are the same

57a [9] Your toes because thumb doesn’t count K6c [25] Fingers are used more than toes

E7b [10] There are exactly as many ﬁngers or K6d [26] They are both 10 and 10, so they are
toes so there’s neither more the same

E7c [l 1] Toes...Because thumbs don’t count K7a [27] I counted them so that they were the

57d K7b same

:0: [3] ram: Snag ttl’tem arebmore Kma [28] Same...each of them is 10...because

o [ ] c ec e oxes. .. ecause you Kmb they are ﬁnite

have an equal number

Eoc [14] Toes, because thumbs aren’t ﬁngers ch [29] We use ﬁngers much more...there are

, more ﬁngers
Eod [15] Neither...ten ﬁngers and ten toes Kmd [30] Fingers are ten and toes are ten
Kme [31] Their cardinal numbers are the
Eoc , same...there are 10 and 10
Eof [l6] Toes. . .because thumbs don t count Km f [32] Fingers are more highly valued in

 

utilization

 

73

 

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75

Table 4.26 Distribution of phrase-driven and structural uses of inﬁnity

 

 

 

 

 

 

Comparisons Students Phrase-driven use Structural use Chi-square .
df Value Sig.
E-interviewees 55 % 0 °/o
Il-b. K-interviewees 5 % 40 %
Even and odd numbers - ' ' ° °
. E participants 30.3 /o 0.8 /o 2 73.84 .0001,
K-partncnpants 1.6 % 33.3 %
E-interviewees 3O % 5 %
Il-c. K-interviewees O % 35 %
Odd numbers and inte ers - ' ' o o
g E participants 29.5 /o 2.3 /o 2 73.98 .000,
Kjartlgpants O % 34.] %

 

Note. *Differences are signiﬁcant (p s .01).
G. K-speakers’ discourse on inﬁnite sets appeared more structural than that of E-
speakers, notwithstanding the fact that only E-speakers used the noun inﬁnity. K-
speakers explicitly referred to sets and to cardinal numbers associated with sets, whereas
such reference was practically non-existent in the E-speakers’ responses. The E-speakers’
use of the noun inﬁnity was phrase-driven rather than structural. In other words, the noun
inﬁnity appeared in phrases such as “goes [on] to inﬁnity”.

Many E-interviewees spoke about inﬁnite sets in items II-b and II-c in terms of
processes (“goes to inﬁnity”, “goes on and on”, etc.; see [34-35], [38], [44], [46-51], [72-
74], and [77] in Tables 4.24 and 4.25). Such reference was rare among K-interviewees
and when it did appear, it was often formulated in a formal mathematical way (e.g., “I
send 2k and 2k+1 to inﬁnity” [53] or “As n goes to inﬁnity...” [59] in Table 4.24). The
majority of K-interviewees made reference to cardinal numbers associated with sets (see
[53], [65-67], [70], [88], [93], [95], [98], and [103] in Tables 4.24 and 4.25). The chi-
square results in Table 4.26 show that the difference between E-respondents and K-
respondents found in the interview also appeared in written survey responses.

H. Eleven routines for comparing sets were identiﬁed in the interviews and the

survey:

76

1. Comparing by counting: Some interviewees counted elements in A and
elements in B, compared the counting of elements in A with those in B, and then
endorsed that A or B had more elements because the counting of elements for the
set A or B ended up with a larger number (see [52] in Table 4.24).

2. Comparing that involves treating size of infinite sets as subject to the same
rules as that of finite sets (doubling increases, halving makes smaller): The
students calculated the number of elements in the given set from some data. For
example, the students would state that two sets are equal because they are “halves”
of the same inﬁnity. The amount of elements for given inﬁnite sets consists of half
parts of the same inﬁnity (see [53] in Table 4.24 and [72], [81], and [88] in Table
4.25).

3. Comparing the numbers of elements in two sets without specifying how the
numbers were found.

4. Comparing the size of corresponding elements of the two sets: There were
interviewees who created a one-to-one mapping between two sets A and B and
checked whether the number of elements of one set was consistently bigger than the
counterparts from the other set. If the elements of the set A were bigger than those
of B, for instance, they endorsed that A had more, A was bigger, or there were more
elements in A (see [62] and [71] in Table 4.24).

5. Comparing the size of corresponding partial sums of the two sets: Other
interviewees focused on comparing the magnitude of corresponding partial sums.

They interpreted that if the partial sums of one set could be shown to be consistently

77

greater than those of the other set, they endorsed that the set that was greater in
partial sums, had more, or was larger (see [8] in Table 4.32).

6. Checking two sets for inclusion of one of them in the other: Some other
interviewees checked whether one set was included in the other (e. g. by checking
the relative density of the two sets on the number line). If one set included the other,
they concluded that the including set had more or was larger (see [40] in Table 4.24
and [94] in Table 4.25).

7. Comparing the numerical range covered by the elements of the sets: The
students would deem a set of numbers as ‘bigger’ if it spanned a larger segment of
the number line (see [45] in Table 4.24).

8. Constructing one-to-one mapping between the sets: There were interviewees
who built a one-to-one mapping between A and B in any way. If there was a
mapping from A to B, then they endorsed either that A and B were equal if the
mapping was a surjection on B, or that B had more (or B was greater than A) if the
mapping was an injection on B (see [67] in Table 4.24).

9. The use of the assumption that any two infinite sets are equal: Some
interviewees endorsed that A and B were equal if they saw that both sets were
inﬁnite (see [43] in Table 4.24).

10. The use of the assumption that inﬁnite sets are incomparable: Other
interviewees endorsed that A and B could not be compared if they saw that both
sets were inﬁnite (see [61] in Table 4.24).

11. Non-numerical comparisons (see [19] in Table 4.23).

In addition, in several cases we opted for the following categories

78

12. Unidentiﬁed routine: This is the case, for example, when the student says that

one set has ‘more elements’ than the other.

13. No answer

Table 4.27 Distribution of routines in comparison tasks in item 11

 

 

 

 

 

 

E-speakers K-speakers
Routine interview Survey interview survey
lI-b ll-c Il-b lI-c Il-b II-c II-b II-c

1) Comparing by counting 0 % 0 % 0.8 % 2.3 % 5 % 0 % 2.4 % 2.4 %
2) Comparing that involves treating size of

inﬁnite sets as subject to the same rules as 0 % 15 % 0.8 % 7.6 % 5 % 0 % 1.6 % 0.8 %
that of finite sets

3) Comparing the number of elements in

two sets without speci/ying how the 0 % 0 % 0 % 0 % 0 % 0 % 0 % 0 %
numbers were found

4) Comparing the Size of corresponding 0 % 0 % 2.3 % 0 % 20 % O % 7.9 % 0 %
elements of the two sets

5) Comparing the Size of corresponding 0 % 0 % 0 % 0 % 0 % 0 % 0 % 0 %
partial sums of two sets

6) Checking two sets for mcluszon of one of 0 % 75 % 0 % 50.0 % 0 % 80 % 0 % 67.5 %
them in the other

7) Comparing the numerical range covered 40 % 0 % 32.6 % 0 % 25 % 0 % 31.0 % 0 %
by the elements of the sets

8) comma“ ””34”“ "’“PPW o % o % 0.8 % o % 10 % o % 9.5 % 4.0 %
between the sets

9) Th.“ use Of’he ”WWW” ”W “’9’ ’W" 60 % 10 % 44.7 % 16.7 % 5 % 5 % 1.6 % 2.4 %
glimte sets are equal

’0) The “‘9 ”f‘h" “3"“me ’ha’ ’"ﬁm’e o % o % 6.1 % 2.3 % 20 % 15 % 22.2 % 10.3 %
sets are incomparable

ll) Non-numerical comparisons 0 % O % 1.5 % 0 % 0 % 0 % 3.2 % 0.8 %
12) Unidentified routine 0 % 0 % 3.8 % 9.9 % 0 % 0 % 6.4 % 3.2 %
L3)Noanswer 0% 0% 9.1% 11.4% 0% 0% 14.3% 8.7%

 

I. In comparing A = odds and B = evens (II-b), E-speakers tended to claim that two

inﬁnite sets are always equal (routine 9, 44%). Only very few K-speakers used routine 9

(5% and 1.6% in the interview and in the survey, respectively). They relied more strongly

than E-speakers on the endorsed narratives that inﬁnite sets cannot be compared (routine

10; 22.2% for K-speakers versus 6.1% for E-speakers). This is due to the fact that the

Korean word for inﬁnity was countless, that is, “which cannot be counted,” and the

Korean participants simply derived that what cannot be counted cannot be compared.

79

EEC

[v '
:2"

J. In comparing A = odds and B = integers (II-c), both groups relied strongly on the
endorsed narrative “proper subset is smaller/has less elements than the whole set” (or
“inﬁnite set that includes ‘half as much elements’ as another inﬁnite set is smaller)
(routine 6; E-speakers: 50% and K-speakers: 67%)

Table 4.28 Distribution of contradictory answers in items II-b and II-c

 

 

Item E-s eakers K-s akers
II—b: odds (A) and even (B) lI-c: odds (A) and integers (B) p pe
Answer Routine Answer Routine Interview survey Interview survey
Equal 2 A is smaller 6 0 % 0 % 5 % 1.6 %
Equal 8 A is smaller 6 0 % 0 % 5 % 4.0 %
Equal 9 A is smaller 6 40 % 26.5 % 10 % 7.9 %
Incomparable 10 A is smaller 6 0 % 1.5 % 10 % 11.1 %

 

K. In items II-b and II-c, there were respondents who produced two contradictory
answers. The contradiction was usually the result of the fact that the respondent endorsed
two narratives: 1) If A is included in B then A is smaller and 2) If A and B are inﬁnite
then they are equal (or cannot be compared). This contradiction appeared as the result of
applying two different routines of comparison. The respondents’ routines are presented in
Table 4.28.

The following Episodes 1 and 2 are excerpts from the conversations with an English
speaking pair (E-pair) and a Korean speaking pair (K-pair) respectively in comparing
even numbers with odd numbers and odd numbers with integers (see Appendix E for the
Korean original conversation). These two episodes exemplify two contradictory answers,
and interview procedures between the two interviewees and the interviewer indicated by

“.1 99

Episode 1. E-pair in comparing odds with evens and odds with integers

Speaker What was said What was done
38. I [show the card “A: Odd numbers B: Even numbers”| What about, Show the card

you know, odd number and even number?

39. E4c We do technically...1’d believe it’d be even because the numbers go on
for, you know, inﬁnity, and there are...they’re inﬁnite numbers...so
every time you go up one, go down one, etcetera. It’s either an odd

80

number or even number and it’s never ending. So, I’d say that it would
be even because we know that there won’t be one more odd number
than one more even number.

40. E4d Yeah, that’s pretty much it, that’s what I put too, I think.

41. I You ﬁrst responded, you know...

42. E4d [take a look at his questionnaire] Oh, I put even numbers I think. Yeah.

43. DJ Right

44. E4d I don’t know what 1 think...I was just trying to ﬁgure out the answer so
I put even numbers because, uh, this doesn’t even make sense now that I
look at it again. So, it was just, kind of, putting something down, but, I
mean, I couldn’t think of a way that you could have more even numbers
or more odd numbers. So, I think there’s the same too, 1 just thought we
had to check a box, so I checked at box.

45. I But then ﬁrst, you know, when you checked even numbers, did you
think of anything?

46. E4d Uh, just kind of, um, zero is an even number

47. l Uh-huh

48. E4d And so, you start with zero, and then, for some reason 1 was thinking,

the numbers, itjust seems like zero is before all the numbers and so
there would be more even numbers, but that’s not right...1 don’t think.

49. I [show the card “A: Odd numbers B: Integers”| What about odd
number and integers?

50. E40 Uh, I said integers because an odd number can be an integer. So there
are, uh, and then even numbers are also integers

51. E4d - Yes

52. E4c So for every odd number you have, you get an integer anyway, so even
though that they go on, for, um, you know, there...there are inﬁnite
numbers, or inﬁnite integers, it’s still odd numbers are integers. So if
there was a breaking point, there’d be more integers.

53. E4d Yeah, it’s. ..1 have integers too just because every whole number is an
integer, and any, I think, isn’t it? OK. And, uh, I just thought I’d make
sure before I said that, um, and, 1 mean, that’s twice as many as there
are odd numbers if you have the odd and even.

Episode 2. K-pair in comparing odds with evens and odds with integers

45. I Now then [show the card “A: Odd numbers B: Even numbers”| of
which are there more? Please check one of the boxes. How do you
know?

46. K6b I didn’t check this one either

47‘ I Um. ..

48. K6b Because odd numbers and even numbers are not ﬁnite

49. K6a I checked on even numbers.

50' l Um...

51. K6a Just 0 is...in my feeling, 0 seems to be included in even numbers. So, I
feel such a feeling though, but it seems not to be.

52. K6b So, looking back, it seems not to be...

53. 1 Please talk with each other about the problem because you have different
thoughts.

54. K6b Key point is whether or not 0 is an even number.

55. K6a But, if it is seen from a certain point, we seem not to be able to decide
that 0 is only either an even or an odd number. In my feeling, 0 looks
like so...

56. K6b But, when they are expressed with n, even numbers are expressed as 2n

and odd numbers as 2n-1 like these....but if we substitute 0 for

81

Show the card

Show the card

n...then....Uh! It seems to be right. [...] But, my thought is
....and....aﬁer listening to that, it seems to be right....what I
thought...so, we can count ﬁnite numbers, can’t we? Because I thought
that we cannot count inﬁnite things, I answered so.

57. I So, because they both are inﬁnite...
58. K6b Yes, they are inﬁnite.
59' I Um. ..

60. K6b As a result, because there is no end in a single word...1 thought that we
cannot say that one is more than the other.

61' I Um...then [show the card “A: Odd numbers B: Integers”] What do Show the card

you think about the next problem? About odd numbers and integers

62. K6b Oh, integers

63. K6a Me too. Integers

64. I The reason is?

65. K6a I thought that integers are....integers are divided into odd and even
numbers. So, because integers are a bigger range. ..I thought that the
cardinal number of integers is more.

66. K6b Because I thought that the set of odd numbers is included in integers.

67. I Do you have anything else to add?
68. K6b No
Item 111

Item 111 in the questionnaire was very similar to the third question in item II except
for three differences. First, in item 111, odd numbers were replaced with even numbers to
be compared with integers. Second, the sets A (integers) and B (even numbers) in item III
were not described in words, but represented horizontally and in parallel by using the set
builder notation. Finally, the wording of the question in item 111 was different from that in
item 11. In item III, the question, “Which of the two sets A and B is bigger?” was used,
whereas the question, “Of which are there more?” was asked in item 11. The intention of
item III was to check the impact of these technical differences, especially of the different
wording by the use of the adjective “bigger,” which implicates reiﬁcation of the inﬁnite

set, instead of the adverb “more” that is less dependent on reiﬁcation.

82

 

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Table 4.29 Distribution of endorsed narratives in item II-c and item III

 

 

 

 

 

 

. Endorsed narratives Chi-s uare

Compansons Students A = B A > B A < B Can’t df Valucb Sig._
E-interviewees 15 % 30 % 55 % 0 %

III. Integers (A) and K-interviewees 15 % 60 % O °/o 25 %

evens (B) E-participants 23.5 % 19.7 % 47.7 % 4.5 % 3 93 68 000,
K-participants 18.3 % 56.3 % 0.8 % 19.8 % ' °
E-interviewees 10 % 10 % 80 % 0 %

II-c. Odds (A) and K-interviewees 5 % 5 % 70 % 20 %

integers (B) E-participants 15.2 % 8.3 % 61.4 % 3.3 % 3 15 61 001,

K-participants 6.4 % 1.6 % 73.0 % 10.3 %
Note. ‘Differences are signiﬁcant (p S .01). Missing data in item 111 were not included in the table (4.5 %
of the E-participants and 4.8 % of the K-participants).

L. The changes in the question III in comparison to [Le did have an impact on the
responses of E-participants but not on those of K—participants. K-participants’
responses to item 111 showed similar distribution of the four endorsed narratives (Notice
integers (B) in item II-c and integers (A) in item III; 73 % and 56.3% in items II-c and 111
respectively). In other words, a dominant endorsed narrative in the K-group was that
integers were bigger (or more) than evens (or odds). However, a major endorsement in
the E-group had been changed from item II-c to III (Integers; 61.4% in item II-c versus
evens; 47.7% in item III). Many E-participants changed their endorsed narratives
between items II-c and III because of different prompts (more in item II-c and bigger in
item III).

Table 4.30 Distribution of the word use of bigger

 

 

 

 

. Chi-square
Students More numbers I-I1gher values d f Value Sig._
E-interviewees 45 % 50 %
K-interviewees 100 % 0 %
E-participants 38.6 % 46.2 %
2 82.12 .000"
K-pgrticipants 86.5 % 0 %

 

x"Differences are signiﬁcant (p S .01).

M. The change in wording inﬂuenced strongly the way E-participants implemented

the comparison of sets, but it did not change K-participants’ choices. The word

83

L". ..

4 .)r-z-

h "Hr-

i.‘ is:

bigger seemed to guide the E-speakers to two different understandings: more numbers
and higher values. About 50% of the E-participants indicated that the meaning they
ascribed to bigger meant increasing faster. Some of the E-interviewees explicitly
hesitated between the two possible interpretations of bigger in this context. In contrast,
all of the K-interviewees accepted this question as being the same as the previous one in
item II (“Of which are there more?”).

Table 4.31 Distribution of routines in comparison tasks in items II-c and III

 

 

 

E-speakers K-speakers
Routine Interview Survey interview survey
II-c III Il-c III II-c III II—c III
1) Comparing by counting 0 % 10 % 2.3 % 0.8 % 0 % 0 % 2.4 % 10.3 %

2) Comparing that involves treating size of
inﬁnite sets as subject to the same rules as 15 % 0 % 7.6 % 1.5 % 0 % 5 % 0.8 % 2.4 %
that of ﬁnite sets
3) Comparing the number of elements in
two sets without specifying how the 0 % 0 % 0 % 0 % 0 % 0 % 0 % 0 %
numbers were found
4) Comparing the size of corresponding
elements of the two sets
5) Comparing the size of corresponding 0 % 5 % 0 % 2.3 % 0 % 0 % 0 % 0 %
partial sums of two sets
6) Checking two sets for inclusion of one of

. 75 %
them m the other
7) Comparing the numerical range covered
by the elements of the sets
8) Constructing one-to-one mapping
between the sets
9) The use of the assumption that any two
inﬁnite sets are equal
10) The use of the assumption that inﬁnite
sets are incomparable

 

 

 

0% 50% 0% 43.2% 0% 0% 0% 0.8%

20% 50.0% 15.9% 80% 55% 67.5% 42.1%

 

0% 0% 0% 2.3% 0% 0% 0% 0%
0% 0% 0% 0.8% 0% 10% 4.0% 12.7%

10% 15% 16.7% 23.5% 5% 5% 2.4% 3.2%

 

0% 0% 2.3% 3.8% 15% 25% 10.3% 21.4%

I I) Non-numerical comparisons 0 % 0 % 0 % 0 % 0 % 0 % 0.8 % 0 %
12) Unidentiﬁed routine 0 % 0 % 9.9 % 1.5 % 0 % 0 % 3.2 % 2.4 %
13) No answer 0% 0% 11.4% 4.6% 0% 0% 8.7% 4.8%

 

N. A dominant routine in the E-group was changed from routine 6 in item II-c to
routine 4 in item 111 but not in the K-group. In both items II-c and III, a dominant
comparison routine in the K-group was routine 6 (67% and 42% respectively). In contrast,

43 % of the E-participants used routine 4 in item III, compared to routine 6 (50%) in item

84

Epit:

Spea'

llf‘ It
‘1'.

515
59 1

116.1;

II-c. In other words, they employed the routines of “which one gets to inﬁnity faster?”

with the conception of increasing faster in the use of the word bigger.

The following Episodes 3 and 4 exemplify the differences in word use and routine

between the E-group and K—group.

Episode 3. E-pair in comparing integers (A) with evens (B)

Speaker
50. l

51.E7a

52. E7b

53. E71!

54. E7b

55. E73
56. E7b

57. I
58. E7a
59. I
60. E7b

What was said What was done
[show the card about item 111] What about this problem...which of Show the card
the two sets A and B is bigger? How do you know?

Um I said A since um even though the uh set continues um it includes
more integers within uh... a set amount of uh like say if it was like 1 to
100 um.. A would include more integers than B would.

Um this reminded me of the set before it except that I took the opposing
stance this time. I said because B is increasing at a greater rate than A
um... it would just be increasing to inﬁnity just at a faster rate so that’s
why I decided that B uh I thought was larger. ..Mm. . .It depends ‘cause
bigger is such a vague term...like you can...

Yeah it’s. . .1 didn’t really know whether to say like contains more
values or has a greater amount inside of it or...

Mm-hmm. Because they’re the same number of values in each set. . . it’s
just the range on. . .on B is greater except A contains more. . .I

guess. . .precise values or...

Yeah

Or uh smaller interval between values so. ..I guess that just depends on
your uh idea of bigger and I just went in terms of its rate of increase.

So do you want uh say something more-

Um...

About the other’s opinion?

lt’s...it’s really hard to determine like you know differ between
opinions since everything is-in inﬁnity its all subjective to what your
views on inﬁnity are...it’s like um...l mean...l...l mean I can see from
his standpoint how-where he comes from his conclusions so it. ..it’s
logical.

Episode 4. K-pair in comparing integers (A) with evens (B)

111.1
112.1(3b
113. K3a
114.1
115.1(3b

116. K3a

Then [show the card about item III| Show the card
I said it was A.

Me too.

Why did you say it was A?

When explaining, I interpreted it this way. Of course there are
preconditions and even though you don’t know what happens later, if
[numbers are] listed according to this rule, A will be a set of natural
numbers. The set B will be a multiple of 2, which is also a set of even
numbers. Similar to the reasoning above, natural numbers can be divided
into even and odd numbers, so B is included in A. Because of this
reasoning, I thought A was bigger.

Because they are sets whose cardinal numbers cannot be counted, I
thought we could show what is big and small by the method of taking
away if you can’t count them. If you are ﬁguring out who is tall and who
is small, you can ﬁnd out who is tall by a difference when they are

85

standing. Just like this, you can see who is tall without measuring. Based
on this concept, you can take away B from A and you will still have
something left, sol answered that A was 100% bigger.

117. I Do you have any other thoughts about this question?

1 l8. K3a No

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Table 4.33 Distribution of processual and structural uses of inﬁnity

 

 

 

 

. Chi-square

Compansons Students Processual use Structural use df Value Sig.__
E-interviewees 90 % 10 %
K-interviewees 25 % 75 %
III. Integers and evens E art' . ts 46 2 o/ 6 l o/
-p lcnpan . o . o

2 53.32 .000“

K-participants 13.5 % 38.1 %

 

Note. l'Differences are signiﬁcant (p S .01).
O. The phenomenon first observed in [G] returned. Many E-speakers spoke about

,9 ‘6

inﬁnite sets in terms of processes (“goes to inﬁnity , goes on and on”, “continues
forever”, “keeps going on” etc.). Such reference was rare among K-speakers. Once again,
the Korean responses were formulated in a more formal mathematical way (e. g., “there is
one-to-one correspondence” (see [28] and [36])). K-speakers also explicitly referred to
cardinality of sets ([22], [27], [29], and [34]). Such reference did not appear in the E-
speakers’ responses. According to chi-square statistics in Table 4.33, this difference
between the E-group and K-group was also signiﬁcant in the survey participants’ written
responses.
Item IV

In item IV, the relation between two inﬁnite sets was unspeciﬁed. The wording in

item IV was different from those in items 11 and III. The word larger in item IV was used

to compare two inﬁnite sets rather than more in item 11 or bigger in item III.

88

Table 4.34 Distribution of routines in cormarison tasks in items III and IV
E-speakers K-speakers
Routine interview Survey Interview survey
III IV III IV III IV 111 IV

Rejected the possibility of inequality of

inﬁnite sets (claimed inﬁnite sets must be 15 % 20 % 27.3 % 15.2 % 30 % 5 % 24.6 % 7.9 %
equal or incomparable)

1) Comparing by counting 10 % 0 % 0.8 % 3.0 % 0 % 5 % 10.3 % 3.2 %
2) Comparing that involves treating size of

inﬁnite sets as subject to the same rules as 0 % 0 % l.5 % 0.8 % 5 % 0 % 2.4 % 0 %
that of ﬁnite sets

3) Comparing the number of elements in

two sets without specifying how the 0 % 15 % 0 % 13.6 % 0 % 0 % 0 % 1.6 %
numbers were found

4) Comparing the size of corresponding

50% 25% 43.2% 17.4% 0% 0% 0.8%16.7%

elements of the two sets

5) Comparing the size of corresponding
partial sums of two sets

6) “id” “”0 se’sfo’ mam” ofa“ of 20 % IS % 15.9 % 16.7 % 55 % 75 % 42.1 % 48.4 %
them m the other

7) Comparing the numerical range covered
by the elements of the sets

3) comma” one'm'o’w mapping 0 % o % 0.8 % 0.8 % 10 % 5 % 12.7 % 3.2 %

etween the sets

9) The use of the assumption that any two
inﬁnite sets are equal

5% 0% 2.3% 2.3% 0% 0% 0% 0%

0% 25% 2.3% 5.3% 0% 5% 0% 4.8%

15%20% 23.5% 5.3% 5% 0% 3.2% 0%

10) The use ofthe assumption that inﬁnite 0 % 0 % 3.8 % 1.5 % 25 % 5 % 21.4 % 4.8 %
sets are incomparable

l I) Non-numerical comparisons 0 % 0 % 0 % 0 % 0 % 0 % 0 % 0 %
12) Unidentiﬁed routine 0 % 0 % 1.5 % l5.9 % 0 % 0 % 2.4 % 8.8 %
13) No answer or I don ’t know 0 % 0 % 4.6 % 117.4 % 0 % 0 % 4.8 % 8.7 %

 

P. The K-speakers were more open toward the possibility that two inﬁnite sets in
item IV can be unequal. Of the K-participants, 82.5 % reported that they accepted the
endorsement that the two sets A and B were inﬁnite, but A was larger than B, and 67.4 %
of the E-participants reported the same thing. Many E-speakers (23.5%) and K-speakers
(21 .4%) who used routines 9 and 10 in item 111 changed their routines in item IV because
they accepted the claim that the inﬁnite set of A was larger than the inﬁnite set of B (see
5.3 % of the E-speakers in routine 9 and 4.8 % of the K-speakers in routine 10). With the
prompt larger in item IV to compare two inﬁnite sets, only 17.4 % of the E-speakers used
routine 4, compared to 43.2 % of the E-speakers in item 111. Interestingly, the use of

routine 4 in the K-group increased from 0.8% in item III to 16.7 % in item IV.

89

Table 4.35 Distribution of the word use of larger in item IV

 

 

 

 

. Chi-square
Students More numbers Higher values d f Value Sig.—
E-interviewees 75 % 25 %
K-interviewees 100 % 0 %
E-participants 42.4 % 22.7 %
. . 7"
K-participants 65.9 % 13.5 % 2 9 96 00

 

‘Differences are signiﬁcant (p S .01).

Q. Some respondents compared the sets with respect to the size of corresponding
elements rather than with respect to the amount of elements. The majority of E-
speakers (42.4%) and K-speakers (65.9%) reported that the word larger meant that there
were more numbers in A than B. However, about 23 % of the E-speakers considered the
meaning of larger as either increasing faster or larger sum in item IV, compared to

13.5 % of the K-speakers. Many participants often used the example in item III to explain
that A was larger than B. The chi-square analysis of the comparison of the E-participants’
and K-participants’ responses about the word use of larger indicates a signiﬁcant

difference, as shown in Table 4.35.

90

 

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Tabla .

1

form

..i,\.

on“,
~\Hiii

Table 4.37 Distribution of processual and structural uses of inﬁnity

 

 

 

 

. Chi-square
Compansons Students Processual use Structural use df Value Sig._
E-interviewees 75 % 25 %
Even numbers and K-interviewees 20 % 80 %
integers E-participants 35.6 % 2.3 %
2 69.50 .000“
Kjiarticipants 4.8 % 36.5 %

 

Note. ‘Differences are signiﬁcant (p S .01).

R. The same as [G] and [0] above. K-speakers’ discourse on inﬁnite sets was
seemingly more structural than that of E-speakers, in spite of the fact that only the latter
used the noun inﬁnity. K-speakers’ talk was more formal mathematical. The majority of
K-interviewees consistently focused on one-to-one correspondence as well as cardinality
of sets. In contrast, the majority of E-interviewees employed inﬁnity in conjunction with
an inﬁnite process to compare two inﬁnite sets ([1-9], [12-13], [15-16], and [IS-19]). The
chi-square results of the comparison of the E-participants’ and K-participants’ responses
about the use of inﬁnity show a signiﬁcant difference in Table 4.37.

Item VIII-a

92

 

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93

Table 4.39 Use of inﬁnity in a seggence

 

 

 

 

Chi-square
Students Processual use structural use d f Value Si .
E-interviewees 95 % 5 %
K-interviewees 25 % 75 %
E-participants 65.2 % 6.1 % ,
K-participants 11.9 % 57.] % 2 1020 '000

 

’Differences are signiﬁcant (p S .01). Missing data were not included in the table.

S. E-speakers’ responses were relatively consistent and in tune with the canonical
interpretation of the word inﬁnity. K-speakers’ interpretations were more diverse

and often inconsistent with the canonical interpretations. Almost all E-speakers
translated the expression sequence a 1, a2, goes to inﬁnity into utterances saying that the

sequence is in the process of unbounded or never-ending growth (see [1-3], [5], [8-9],
[12], [14] and [18-20]). K-speakers’ responses were much less consistent. Many of them
made no requirements with respect to the size of the elements of the sequence. This use
was clearly phrase-driven (see [25-26], [30], [32-34], [36], and [39]). They also oﬁen
made speciﬁc assumptions about the sequence (e. g., “a regular rule” [2]], “geometric
sequence” [22], and “1, l, 1, 1” [26]).

T. E-speakers provided processual explanations. K-speakers spoke almost
exclusively in structural terms. Almost all E-speakers used processual expressions such

9, ‘6

as “increasing, getting bigger,” “keeps going on,” and “gets higher.” Many K-speakers
did not mention any process, except simply repeating the phrase “goes to inﬁnity,” used

in the formulation of the question. Instead, they spoke in structural terms of “sets,”

9, 66 9, 6‘

“number of elements, sizes, values,” and “last term.” The chi-square results in Table
4.39 show a signiﬁcant difference between the E-group and K-group. One word for a
process that appeared in their talk and was not found in E-speakers’ responses was

“diverge.”

94

U. E-speakers formulated interpretations in their own words. K-speakers used

more formal mathematical formulations, even though some of them were not fully
versed in it. Many K-speakers were careful to distinguish between the index of an
sequence element and the element itself (e. g. [2]). Some K-speakers showed awkward
uses of mathematical formulations (e.g., “A limit; n goes to inﬁnity, an is inﬁnity” [31];

see also [36-37]). They often provided formal language to deﬁne going to infinity such as
“converge” and “limitE” (see [27], [31-33], and [36-37]).

V. In particular, unlike in the previous items, K-speakers used the noun inﬁnity,

but they did it in phrase-driven way, by simply repeating the expression goes to
inﬁnity that appeared in the question. The phrase-driven rather than structural nature of
their use was evident from the fact that the word inﬁnity did not appear except in this
phrase. The K-respondents clearly had difﬁculty providing an equivalent statement
expressed in words that was different from the interpreted phrase itself.

The following Episodes 5 and 6 exemplify the differences in deﬁning that a

sequence a1, a2, a3,. .. goes to inﬁnity between the E-group and K-group.

Episode 5. E-pair in deﬁning that a sequence a], a2, a3,. .. goes to inﬁnity (item VIII-a)

Speaker What was said What was done
238. 1 0k. [show the card about item VIII-a] What about this problem? A show the card
student claims that a sequence goes to inﬁnity. What does this mean?
239. E6b Uh...l wasn’t sure if that one was a trick question because it’s pretty
simple. I mean you just keep adding maybe it’s the uh what are these
things called. Can’t remember the name of them but one of those
sequences where numbers become. . .have the same equation and they
become larger. What is the name of that? I can’t remember. But maybe
its like one and then one plus three and then two I mean you know four
plus three or something like that but. Regardless the sequence continues
forever
240. E6a Keeps on going and going and going getting bigger and bigger the
sequence does not stop. It has no limit.
241. I 0k. what do you mean by you know the sequence is continuous?
242. E6b It means it.. . it doesn’t end. There is no set ending or beginning its
continuous it doesn’t stop.

95

243. l
244. E6a

245. I
246. E6a

247. I
248. E6a
249. I
250. E6a
251. I
252. E6a
253. 1
254. E6b
255. 1
256. E6b
257. I
258. E6b

Do you want to add something else?

Like 1 mean if you had a sequence when. ..where you’re adding
numbers...

Mm-hmm.

like say this was one and that was one plus two and this was one plus
three it just keeps on. . .keeps on adding more...

Mm-hmm.

and more and

Mm-hmm.

More and more actually so you keep on going to inﬁnity so.
Mm-hmm. Anything else?

No.

Do you want to add?

It basically goes out to inﬁnity.

About uh go to inﬁnity?

Well that’s what I meant by continuous. It doesn’t have an ending
Mm-hmm.

You can’t say it stops at a certain point it goes to inﬁnity.

Episode 6. K-pair in deﬁning that a sequence a], a2, a3,. .. goes to inﬁnity (item VIII-a)

159.1

160. K2b

161.K2a
162.1

.163. K2a

164.1
165. K2b

I66. I
167. K2b
168. I

169. K2b

170.1
l7l.K2b

Now then [show the card about item VIII-a] A student claims that a Show the card
sequence goes to inﬁnity. What does this mean?

[...] 1 don’t know well what kind of sequence this is...if list [sequence]
by numbering like this, then here 1. ..when the number of here
[subscript] increases. . . its value is inﬁnity.

[...] l also...same...

When you answered last time, you seemed to write that it means {an} is
an inﬁnite sequence

Then as soon as 1 saw it, I thought so. When this is represented by a set,
I thought that it becomes an inﬁnite set and I solved it. When I heard
what he said, I thought this was not...

Can you explain this more?

So, a1 is a certain number and a2 is some number and like this way. So,
because the sequence is not known, if thought in this way, then a really
big number of a is substituted, when to be listed...really...if listed from
left, if picked up a number way to the right, then there is a certain
number. 1f 1 go to right inﬁnitely, 1 pick up this. This can be expressed
like this [write an]. Because this goes to inﬁnity, this value also has
inﬁnity. Yeah.

How far does it go to the right?

It should go inﬁnitely

Here you wrote the symbol of ago. What is the meaning? Please explain
it concretely

In the case that this is not a ﬁnite sequence which has a limit (Jae-hon),
but an inﬁnite sequence, so. . .there is an increase or decrease or a
regularity which a sequence has. That this [sequence] is inﬁnite means
that it cannot go due to endlessness. If 1 pick [it = ago] up here. ..if I can
pick [it] up, then it says that this [sequence] will become like this. So,
it’s approaching but reach. . .doesn’t seem to reach it. Even though it
[ago] cannot be picked up, if [1] go and pick it up, it’s going to be inﬁnity
when seen within the rule in this. That kind of story...

Anything else to add?

A sequence is the array of numbers which have a rule. Then, 1 don’t

96

know how far a [term] should be. ...if the general term of this sequence
is an, going to inﬁnity means the term itself...the number of term [11]

itself becomes the value of the sequence. So, if an is equal to n, then
going to inﬁnity means value also inﬁnitely....in this way.

Summary: Discourse on inﬁnity

Table 4.40 summarizes salient characteristics of the colloquial and mathematical

discourses on inﬁnity in the E-group and K-group.

Table 4.40 Salient properties of E-speakers’ and K-speakers’ discourse on inﬁnity

 

Item: topic

Aspect E-spea kers K-speakers

 

Used both the adjective and the
adverb. They used the adverb when
they were asked to create a sentence
with the noun inﬁnity (either
mathematical or colloquial). Their uses
of inﬁnitely and inﬁnite was mainly in
non-mathematical context (even more
so than in the US group).

A. Inﬁnite and Used the adjective, but not the
inﬁnitely: type adverb. The use was mainly in
of discourse non-mathematical discourse

 

- _ ' 'v . .
The use was m the context of K speakers used the adjecti e inﬁnite

 

 

 

 

Item I: . as descri tors of concrete thin
Creating [.3 1",]; mte and large “fem. E-speakers used the known ft: their large dimensiogrsis or of
sentences inﬁnitely: i‘dJCCtive '"ﬁ""e.ah208t always , spiritual matters. They used the adverb
with given context of use in ﬁOIljllﬂCthﬂ:Vlth number of inﬁnitely as a descriptor of many, in
words or amount Of' the context of sets.
With just one exception, K-speakers
E-speakers’ P53 ofthe word . did not use the noun inﬁnity, either in
’"ﬁm’y was massocration With its colloquial or mathematical version.
processes,.which were S'fud [0 be When asked to construct a sentence
C. Inﬁnity never ending. The inﬁnity itself with the inﬁnityKCon, they converted
was said to be unreachable or . . . . .
impassable. There was no the noun into adjective (inﬁnite) and
. . . . . when asked to construct a sentence
assocration of inﬁnity wrth , , .
largeness. With the inﬁnityMcOu, they converted
the noun into adverb (inﬁnitely).
Almost all E-speakers compared The 40% of K-speakers who chose an
the two sets according to the option other than A=B did so because
item I" . D 11 . number of elements, and if they their comparison was non-numerical.
Comparing A.- f -a. _ did not choose the option A=B,
pairs of two _ mgers, it was only because of the fact
“‘3 B=toes that they interpreted the word
ﬁnger as not referring to the
thumb. u
Only two ﬁfths of E-speakers Only half as many K-speakers chose
E. ll-b: chose the canonical response the canonical option A=B. Other
A=evens; A=B; other responses were responses were distributed evenly
B=odds distributed evenly between A<B between A<B, A>B, and “can’t say”.
and A>B.
Only as few as 10%-15% of E- The percentage of K-speakers who
speakers chose the canonical chose the canonical response A=B was
F. ll-c: A=odds, response A=B; An even smaller than that of E-speakers
B=integers overwhelming majority (80%...l!!._. (5% in the interview and 6.4% in the

 

 

97

 

the interview and 61.4% in the
survey) chose the option A<B.

survey); approximately 70% chose the
option A<B.

 

G. Discourse on
inﬁnite sets

E-speakers’ use of inﬁnity was
phrase-driven rather than

K-speakers’ discourse on inﬁnite sets
appeared more structural than that of

structural (it appeared in phrases E-speakers

such as “goes [on] to inﬁnity”).

 

H. routine

Eleven routines for comparing sets were identiﬁed in the interviews and

the survey

 

1.. Il-b: A=odds;
B=evens

E-speakers tended to claim that Only very few K-speakers used routine

two inﬁnite sets are always
equal (routine 9, 44%)

9 (5% and 1.6% in the interview and in
the survey, respectively). They relied

more strongly than E-speakers on the
endorsed narratives that inﬁnite sets
cannot be compared (routine 10;
22.2% for K-speakers versus 6.1% E-
speakers).

 

J. ll-c: A=odds,

Both groups relied strongly on the endorsed narrative “proper subset is
smaller/has less elements than the whole set” (or “inﬁnite set that

 

B=integers includes ‘half as much elements’ as another inﬁnite set is smaller)
(routine 6; E-speakers: 50%, E-speakers 67%)
In items lI-b and ll-c there were respondents who produced two

K. lI-b and Il-c contradictory answers as the result of applying two different routines of

L. Changes in
the question 111:

comparison.
The changes in the question 111 in
comparison to ll-c did have an

They did not have an impact on
the responses of K-participants

 

 

 

 

 

 

 

A=integers, impact on the responses of E-
B=evens participants
. The change in wording inﬂuenced It did not change K-participants’
Item "I: M' (Ell-range m strongly the way E-participants choices
Comparing wor mg implemented the comparison of sets
Integers (A) A dominant routine in the E-group A dominant comparison routine in
wrth evens N. Routine was changed from routine 6 in item the K-group was routine 6 in both
(B) ll-c to routine 4 in item [11. item ll-c and 111
O The Many E-speakers spoke about Such reference was rare among K-
phenomenon inﬁnite sets in terms of processes speakers; once again, the Korean
ﬁrst observed in (“goes to inﬁnity”, “goes on and responses were formulated in a
[G] returned on”, ‘contmues forever’, keeps more formal mathematical way.
m on” etc.).
, 67.4 % of the E-participants reported Of the K-participants, 82.5 %
P. W. Two th .
inﬁnite sets A at they accepted the endorsement reported the same thing.
that the two sets A and B were
Item IVE and B inﬁnite, but A was larger than B
Explaining Some respondents compared the sets with respect to the size of
that inﬁnite Q. Use of larger corresponding elements rather than with respect to the amount of
set A is elements.
larger than The majority of E-speakers K-speakers’ discourse on inﬁnite

infinite set B R. The same as

[G] and [0]
above

ltem VIII-a:
Defining that S. Interpretation
a sequence of inﬁnity

8', 32, 83,...

employed inﬁnity in conjunction
with an inﬁnite process to compare
two inﬁnite sets

E-speakers’ responses were

sets was seemingly more
structural than that of E-speakers.
K-speakers’ talk was more formal
mathematical.

K-speakers interpretations were

relatively consistent and in tune with more diverse and often

the canonical interpretation of the
word inﬁnity.

inconsistent with the canonical
interpretations.

 

 

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98

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infinity inﬁnity explanations.

exclusively in structural terms.

 

E-speakers formulated

U. Deﬁnition of interpretations in their own words.

K-speakers used more formal
mathematical formulations, even

 

inﬁnity though some of them were not
fully versed in it.
K-speakers used the noun inﬁnity,
V. Phrase- but they did it in phrase-driven
driven use of way, by simply repeating the
inﬁnity expression goes to inﬁnity that

appeared in the question.

 

Discourse on Limit

Items 1 and

99

 

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101

EOE-I

Table 4.43 Distribution of the contexts of use of the words limited and the chi-square
results of the comparison of difference between the E-grorm and K-group

 

 

 

 

Students Words Non-mathematical Mathematical None Chi-square .
contexts contexts df Value S1g._

E-interviewees LimitedE 65 % 35 °/o

K-interviewees LimitechOn 90 % 10 %

E-panicipants LimitedE 50.0 % 34.1 °/o l5.9 % 8 75 003*

K-participants LimitedKCo" 73.8 % 2| .4 % 4.8 % . .

 

*Differences are signiﬁcant (p S .01).

A. Limited: context of use. Both groups used the word limited in any version mainly in
non-mathematical context. Among K-speakers the phenomenon was stronger than among
E-speakers.

The E-interviewees used the word limited in both non-mathematical and
mathematical contexts. However, the K-interviewees employed the word limitedKCo"
predominantly in non-mathematical contexts (11 responses out of 14 for limitechO" in
Table 4.41). The chi-square statistics in Table 4.43 indicate that the distribution of the
word use of the E-participants and K-participants between non-mathematical and
mathematical contexts is signiﬁcantly different when the word limited is colloquial.

Table 4.44 Distribution of the use of the word limiteda,” as an upper-bounded process
Chi-square

 

 

 

 

Students The use as upper-bounded process (if Value Sig.—
E-interviewees (n = 20) 30 %

_l(-interviewees (n = 20) O %
E-participants (n = 132) 38.4 %

kK-participants (n =126) 41.6 % l .50 .479

 

Bible 4.45 Distribution of the use of limitedKMmh as extreme

 

 

 

 

. . Chi-square
jtudents The use of limited/{Mat}, as extreme df Value Sig.
E-interviewees (n = 20) O %
iinterviewees (n = 20) 95 %
E-participants (n = 132) O %
. 2 . *
j-participants gn =126) 42.1 % 1 63 6 000

 

I"Differences are signiﬁcant (p S .01).

102

ill

SOL .l

:5:
nor.-
lie 1
the L
spea

4.4—l

’her

B. Limited: type of use. Both groups used IimitedCo” in the sense of ﬁnite (not inﬁnite —
something that has an upper bound) or restricted — one that could be larger. K-speakers
used limitedKMau, in the sense of extreme.

The majority of interviewees employed limited as something they could not go over
in their daily experience (“height” [2], “as far as you can go” [3], “ability to learn” [8],
“how much TV you can watch” [16], “food” [17], and money” [22] in Table 4.41). In the
case of the Korean mathematical word limitedKMath, K-interviewees employed it in both
non-mathematical and mathematical contexts. However, in non-mathematical contexts
the word IimitedKMath was only applied to something extreme. This was not observed in
the use of the Korean colloquial words (see [31-32], [34-41], [75], [78], and [80]). The E-
speakers never used the words limited and limit in these ways. A chi-square test in Table
4.44 indicates that there is no signiﬁcant difference between the E-group and K-group in
the number of written survey responses in which the colloquial word limited is employed
to refer to an upper-bounded process. All of the K-participants who employed the words
limitedKMath in non-mathematical contexts used it as the concept of extreme, as shown in
Table 4.45.

Table 4.46 Distribution of the contexts of use of the word limit and the chi-square results
of the comparison of difference between the E-group and K-group

 

 

 

 

Students Words Non-mathematical Mathematical None Chi-square .
_ contexts contexts df Value Sig.
E-interviewees Limit}; 50 % 50 %
' ' 75 ‘V 25 °/
K-interviewees LfmftKColl o o
__‘__ LimItho" 30 % 70 %
E-participants Limit}; 28.8 % 63.6% 7.6 % l 76 l 7 000*
K mid ants LimitKCon 81.7 % 12.7% 5.6 % ' '
P P LimitKCo“ 30.2 % 52.4% l7.5 °/o l .02 .896

 

*Diﬁ‘erences are signiﬁcant (p S .01).

C. Limit: context of use. Both groups used the word, in whatever its version, in both

mathematical and non-mathematical contexts. E-speakers used the word limited, mainly

103

in mathematical contexts, whereas K speakers employed it predominantly in non-
mathematical contexts. The chi-square results in Table 4.46 show that the distribution of

the word use of the E-participants and K-participants between mathematical and non-
mathematical contexts is signiﬁcantly different when the word limit is colloquial (x2 =
76.17, p = .000). However, there is no signiﬁcant difference between the two groups
when the word is mathematical (x2 = .02, p = .89).

Table 4.47 Distribution of the use of the word limited;

 

 

 

 

. . Chi-square
Students The use of limit as an upper bound d f Value Si .
E-interviewees (n = 20) 30 %
K-interviewees (n = 20) 5 %
E-participants (n = 132) 15.6 %
l 6.00 .014
K-participants Q: =126) 15.4 %

 

Table 4.48 Distribution of the use of and limitKMam as extremity

 

 

 

 

. . . Chi-square
Students The use of llmlt as extremity df Value Sig._
E-interviewees (n = 20) 0 %
K-interviewees (n = 20) 30 %
E-participants (n = 132) 0 % .
K-participants (n =126) 30.2 % l 4625 '000

 

D. Limit: type of use. Both groups used limitcO” to denote an upper bound — something
that does not let a thing or process to get larger (see Table 4.47). The usage of IimitKMmh
as the concept of extremity in non-mathematical contexts was also seen in the K-
participants’ written responses (see Table 4.48). However, the majority of K-speakers
used limitKMmh mainly in expressions that were formulated as a textbook instruction to do
something or as textbook deﬁnition. In general, it seemed that K-speakers were not very
proﬁcient in creating sentences with limitKMam. In several cases, the noun was converted
into adjective. When it was used as a noun, the expression was personal (about a person

rather than about mathematical objects) and appeared as if copied from a textbook (see

104

“ﬁnd the limit” [74, 76, 79] for instruction; [73] and [78] for deﬁnition in Table 4.42).
Finally, several pairs did not mange to produce more than one sentence.

Item V

In item V, the following table was shown to students along with the question “What
do you think will happen later in the columns of this table?”

Table 4.49 The table shown in item V in the questionnaire

 

 

x J x + 25 -— 5
x

1.0 0.099020

0.5 0.099505

0.1 0.099900
0.05 0.099950
0.01 0.099990
0.005 0.099995
0.001 0.099999

 

 

 

 

Table 4.50 Distribution of endorsed narratives in item V

 

 

 

Endorsed narratives ﬁ-speakers K-speakers
lntervuew survey lntervrew survey

1) The sequence is increasing/decreasing (without the limit) 30 % 29.5 % 0 % 2.4 %
2) The sequence is getting close to (or approaches) 0. l 30 % 29.5 % 30 % 27.8 %
3) The sequence approaches 0.] but never reach it 30 % 8.3 % 5 % 2.4 %
4) The sequence converges to or becomes 0.] 0 % 3.0 % 15 % 15.1 %
5) The limit of the sequence is 0.1 10 % 12.9 % 50 % 45.2 %
6) Other 0 % 10.6 % 0 % 4.0 %
7)No answer 0 % 6.1 % 0 % 3.2 %

 

Even though some students made a mistake and answered one or 0.01 instead of 0.1
as the limit, their responses were categorized as one of the ﬁve endorsed narratives
because they showed the same idea as one of them. The “other” category in Table 4.50 is
the case, for example, when the student writes 0.1 or any number without providing
further information.

E. A process of the sequence dominated the idea of limit in the E-group, compared
to an object as the limit of the sequence in the K-group. About 30 % of E-speakers

conceptualized the idea of the limit in the sequence as an increasing or decreasing process

105

without the limit. However, in the E-speakers, 29.5 % reported it as an approaching
process to the limit. Both E-speakers (29.5 %) and K-speakers (27.8 %) considered
numbers of the sequence as an approaching process to the limit. The limit of the sequence
as a number (15.1 %,and 45.2 % in categories 4 and 5 respectively in Table 4.50)

dominated the idea of limit in the K-group.

106

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Table 4.52 Distribution of the use of the limit of the inﬁnite sequence

 

 

 

 

 

 

 

 

 

Chi-square
Students Processual use Structural use d f Value Sig_._
E-interviewees 90 % 10 %
K-interviewees 15 % 85 %
E-participants 75.8 % 16.7 % ...
K-participants 33.3 % 63.5 % 2 60'62 '000
*Differences are signiﬁcant (p S .0 I).
Table 4.53 Distribution of the use of two columns

Chi-scniare
Students Separate processes connected processes D f Value Sig.__
E—interviewees 8O % 20 %
K-interviewees 15 % 85 %
E-participants 62.9 % 31.8 %

4 . . "'

gparticipants 23.0 % 73.8 % 2 6 25 000

 

*Differences are signiﬁcant (p 5 .0l).

F. K-speakers’ discourse on the inﬁnite sequence appeared more structural than
that of E speakers. E-speakers used the inﬁnite sequence in terms of inﬁnite processes,
whereas such reference was rare in the K-speakers’ responses. Instead, K-speakers
explicitly referred to mathematical algorithms associated with the function of the inﬁnite
sequence. In other words, the K-speakers’ use of the sequence was structural rather than
processual.

The processual use of the sequence in the E-interviewees was related to either that
the limit is unreachable or that it is rounding up ([1-5], [IO-12], and [14-18] for
unreachability; [5] and [7] for rounding up). In contrast, the inﬁnite sequence seemed to
guide most K-interviewees to focus on ﬁnding its limit value according to known
mathematical algorithms (“a form of 0 over 0” [22], “limitE” [23, 24, 27, 31, 33],
“multiply m +5 ” [28, 30, 32, 34, 36, 38], and “use l'Hopital’s rule” [29, 33, 37]). In
addition, most of the E-interviewees seemed to separate two processes in two columns
(the x and function columns) of the table shown in item V (see [1], [3], [4], [5], [6], [7],

[9], [12], [14], [17], [18], and [19]). In contrast, the task of ﬁnding the limit in the K-

108

 

LASE:

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group seemed to make two processes in these two columns connected ([21-23], [26-29],
[31-32], and [34-36]). Thus, when the question “What will happen later in the columns of
this table?” was asked, the K-speakers associated numbers of the inﬁnite sequence with
mathematical algorithms. The chi-square results in Table 4.52 show a signiﬁcant
difference between the E-group and K-group in the use of the sequence. Chi-square
analysis of the distribution in the use of two columns also indicates a signiﬁcant
difference between these two groups, as shown in Table 4.53.
C. When ﬁnding the limit in a given infinite sequence, ﬁve routines were identiﬁed
according to how to use limiting processes to ﬁnd out the limit:
1. Describing a process by looking at patterns of numbers in the sequence:
There were interviewees who looked at either increasing or decreasing patterns of
numbers in the inﬁnite sequence. They then endorsed that it was increasing or
decreasing. Their discourse seemed to be mediated syntactically on the basis of
changes of numbers in the sequence.
2. Describing a process by characterizing the given function: Some interviewees
focused on a character of the given function, that is, that one divided by a smaller
number was a bigger number. Then they concluded that the function kept increasing
because x got smaller and smaller.
3. Finding the limit by looking at patterns of numbers in the sequence: Other
interviewees checked the value which numbers in the sequence approached. When
there was an approaching number, then they endorsed that the sequence approached

the number.

109

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4. Rationalization: Some other interviewees multiplied m + 5 to both
numerator and denominator to ﬁnd out the limit value in the inﬁnite sequence. Aﬁer
calculating the limit value, they endorsed that it was 0.1, it converged to 0.1, or it got
closer to 0.1.

5. l'H6pital’s rule: There were interviewees who seemed to ﬁrst look at the
function as a form of 0/0 when x approaches 0. Then they used l'Hopital’s rule to ﬁnd
out the limit value by differentiating both numerator and denominator. After getting
the limit, they endorsed that it became 0.1, or that it converged to 0.1, or that it
approached 0.1.

6. Unidentiﬁed: This is the case when students say 0.1 without giving a reason.

7. No answer

Table 4.54 Distribution of routines in item V

 

 

 

Routines . li-speakers . KO-speakers
mtervrew survey InterVIew Survey

I) Describing a process by looking at patterns of numbers m 20 % 27.3 % 0 % 1. 6 %

the sequence

2) Describing a process by characterizing the given function 5 % 6.1 % 0 % 0 %

jiggizg the limit by looking at patterns of numbers in the 70 % 50.0 % 25 % 44.4 %

4) Rationalization 0 % 1.5 % 60 % 41.3 %
5) I'Ho‘pital '3 rule 0 % 0 % 15 % 7.1 %
6) Unidentiﬁed 5 % 10.6 % 0 % 2.4 %
j No answer 0 % 4.5 % 0 % 3.2 %

 

H. E-speakers used routines based on processual use of limit. K-speakers employed
routines on the basis of structural use of limit. About one third of the E-speakers in
routines 1 and 2 of Table 4.54 and many E-speakers (70 % and 50 % in the interview and
in the survey, respectively) in routine 3 appeared to employ their routines based on
processual use of limit. In contrast, 48.4 % of the K-participants in routines 4 and 5 and
many K-speakers in routine 3 used their routines on the basis of structural use of limit by

considering the concept of limit as an object, that is, a number. In routine 3, only 34.8 %

llO

of the E-participants (23 students out of 66) seemed to report two connected processes in

the use of two columns on the table in item V. In contrast, 53.6 % of the K-participants

(30 students out of 56) appeared to present two connected processes in the use of the two

columns.

The following Episodes 7 and 8 are excerpts from the conversations with an E-pair

and a K-pair respectively. These two episodes exemplify their routines and difference in

the use of the inﬁnite sequence.

Episode 7. E-pair in ﬁnding the limit of an inﬁnite sequence

Speaker
I90. 1

l9l.E5a
192.E5b
193.E5a
I94.E5b
l95.E5a
I96.E5b
l97.E5a
l98.B5b
199.E5a

200. ESb

201. l
202. E53

203. ESb
204. E5a

205. ESb

What is said What is done
[show the card with “what do you think will happen later in the columns show the card
of this table”] What about this one? What do you think will happen later
in the columns of this table?

Um...

I said it would just approach point one

Yeah this one would

Yeah and that one would approach zero.

Yeah. I’m guessing because it just keeps getting point 09999

Right. '

It’s just going to keep getting nines.

Yeah.

I think it’s gonna...it’s gonna approach zero from the left side the right
sides gonna approach point one.

Mm-hmm.

Why?

Because it’s just the way the trend is appearing to be this one keeps kind
of moving over one decimal place well like every other one. And this one
you keep adding more nines which means eventually it will have to
run...round up so

Mm-hmm.

Because I don’t think they would make the answer point 099999 forever
[laughing] so it would eventually have to round up to point one.

Yeah [laughing] Sounds good to me. That’s what I though too.

Episode 8. K-pair in ﬁnding the limit of an inﬁnite sequence

71.1

72. Kla
73. Klb

74. [(18
75. Klb
76. [(13

[show the card with “what do you think will happen later in the columns of Show the card
this table”) What do you think will happen later in the columns of this table?

I looked at it only around here and I thought it is getting closer to 0.1

I simply thought. . .This [ﬁrst column] is decreasing by less units from

l...less and less...0.5, 0.1 and once again by a half of l ...0.05 and again

0.01 and so 5, l, 5, l separately like this...following the trend, I thought that

x is getting closer to 0 more and more

To 0

Yeah...so...this from 1 to 0...] thought the next one is going to be 0.0005

Yeah

11]

77. Klb It continues to go on like the rule. . .rule. . .anyway. . .because I thought x is
decreasing in that trend, as a result, x is not zero but is going to get to a

number that is close to 0.
78. Kla Yeah

79. Klb So if x is 0, this answer is going to be in the form of 0 over 0.

80. Kla Yeah

81. Klb As x is getting closer to 0, I thought that root x plus 25 minus 5 over x is

getting closer to 0.01...ah, 0.1
82. Kla Uh
83. Klb Just exactly...

84. Kla Ijust thought...as the values of x are decreasing, [it’s] getting closer to 0.1

85. Klb Wordings seem to be just a little different.

Item VI

Item VI was designed to elicit students’ mathematical discourse on limit by asking

them “What will happen to the curve when x approaches 0 from leﬁ or from right?” in

item VI-a (l) and “What will happen to the curve when x goes to positive inﬁnity?” in
x

2

 

2
items VI-b (1x_) and VI-c ( x ). These three different ﬁanctions and their graphs
+ x

(1+ x)2

were provided in the questionnaire.

Table 4.55 Distribution of endorsed narratives in items VI-a and VI-b

 

 

 

 

 

 

 

 

 

E-speakers K-speakers
Endorsed narratives Interview survey interview Survey
Vl-a VI-b VI-a VI-b Vl-a Vl-b VI-a Vl-b
I) It approaches 0 10 % 0 % 13.6 % 0.8 % 0 % 0 % 0 % 0 %
2) It is increasing (without the limit) 25 % 30 % 9.8 % 31.8 % IO % 0 % 4.0 % 5.6 %
3) It approaches an asymptote 10 % 0 % l 1.4 % 8.3 % 0 % 0 % 10.3 % 11.9 %
4) It goes to inﬁnity 45 % 50 % 30.3 % 28.0 % 25 % 35 % 16.7 % 19.0 %
5) It diverges (to inﬁnity) 0 % 0 % O % 0 % 25 % 20 % 23.8 % 20.6 %
6) (It becomes) inﬁnity 10 % 10 % 26.5 % 17.4 % 35 % 35 % 35.7 % 39.7 %
7) Unidentiﬁed O % 0 % 6.8 % 6.8 % 5 % 0 % 7.1 % 0.8 %
_8)No answer 0% 0% 1.5% 6.8% 0% 0% 2.4% 2.4%
Table 4.56 Distribution of endorsed narratives in item V-c
Endorsed narratives . E-speakers . 116-speakers
interwew survey interwew SurveL
I) It approaches 0 5 % 3.8 % 0 % 0.8 %
2) It gets closer to l 60 % 25.8 % 15 % 20.6 %
3) It converges to (becomes)! 10 % 16.7 % 80 % 68.3 %
ﬂit is increasing (without the limit) - 0 % 10.6 % 0 % 0.8 %
5) It approaches an asymptote 10 % 7.6 % 0 % 5.6 %
6) It goes to infinity 15 % 12.1 % 5 % 0.8 %
7) Unidentified 0 % 7.6 % 0 % 0.8 %
8iNo answer 0 % 15.9 % O % 2.4 %

 

112

I. VI-a: —l- . About one third of E-speakers in the ﬁrst highest percentage responded “it
x

goes to inﬁnity.” In the next highest percentage, the E-speakers (26.5 %) replied “(it
becomes) inﬁnity.” Other responses were distributed evenly among “it approaches 0,” “it
is increasing,” and “it approaches an asymptote.” More than one third of K-speakers in
the ﬁrst highest percentage answered “(it becomes) inﬁnity.” Other Korean responses
were distributed evenly between “it goes to inﬁnity” and “it approaches an asymptote.”

However, only K-speakers (23.8 %) replied “it diverges (to inﬁnity)”

2
J. VI-b: .112. . About one third of E-speakers responded “it is increasing.” In the next
+ x

two highest percentages, 28 % and 17.4 % of E-speakers answered “it goes to inﬁnity”
and “(it becomes) inﬁnity” respectively. Approximately 40 % of K-speakers replied “(it
becomes) inﬁnity.” However, only K-speakers (20.6 %) responded “it diverges (to
inﬁnity)” Other Korean responses were distributed between “it goes to inﬁnity” and “it

approaches an asymptote.”

2
K. VI-c: x 2 . About 26 % of E-speakers answered “it gets closer to 1.” Other
(1 + x)

 

responses were distributed almost evenly among “it converges to 1,” “it is increasing,”
and “it goes to inﬁnity.” The percentage of K-speakers who responded “it gets closer to
1” was smaller than that of E-speakers (15 % in the interview and 20.6 % in the survey).
An overwhelming majority of K-speakers (80 % in the interview and 68.3 % in the
survey) replied “it converges to 1.”

L. The number of “no response” from item VI-a to item VI-c. The number of E-

speakers who did not answer questions increased considerably from item VI-a to item VI-

113

c. Comparatively, the number of K-speakers who did not respond to questions remained

at the same rate.

114

 

 

 

 

 

 

 

 

 

 

 

 

 

  
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Table 4.60 Distribution of the use of the contexts in item VI

 

 

 

 

 

 

 

 

 

Functions Students Processual use Structural use Chi-square .
df Value S'L
E-interviewees 90 % 10 %
Vl-a _ K-interviewees 45 % 55 %
x E-participants 72.7 % 25.8 % l 49.60 .000‘,
K-participants 28.6 % 68.3 %
E-interviewees 95 % 5 %
Vl-b 12— K-interviewees 15 % 85 %
1+ x E—participants 65.9 % 27.3 %
K-participants 31.7 % 65.9 % 1 ”'96 'OOO‘l
E-interviewees 80 % , 20 %
Vl-c x2 2 K-interviewees 5 % 95 %
1+ x E-participants 64.4 % 19.7 %
( ) K-participants 10.3 % 85.7 % 1 102.8 .000"

 

Note. *Differences are signiﬁcant (p S .01).

M. The phenomenon ﬁrst observed in [F] returned. Many E-speakers spoke about the

limit of a function in terms of processes. Once again, the Korean responses were

formulated in a more formal mathematical way with the purpose of ﬁnding the limit.

The contexts, “x approaches 0 from left or from right” or “x goes to positive

inﬁnity,” directed almost all E-interviewees to use either geometric features on the graph

(such as slope and asymptote) or algebraic properties (such as comparison of degrees and

increasing rates between numerator and denominator). These contextual uses led them to

a processual use of the limit. This processual use of the limit in the E-group oﬁen seemed

to be related to unreachability in the limit algebraically or geometrically ([3-6], [8-9],

[10-13], and [18]; [40] and [48]; [75], [80-82], [84-86], [89], and [91-921). Comparatively,

the same contexts seemed to guide the majority of K-interviewees, with just two

exceptions ([68] and [93]), to an algebraic reasoning to ﬁnd the limits of three ﬁmctions

by using different mathematical algorithms ([19-20], [23-25], [27], and [29-33]; [56],

[58], [61], [63-65], [67], and [70-72]; [94-95], [99-101], [104-105], [107], and [109-110]).

When the question “What will happen to the curve?” was asked, almost all K-

118

interviewees took for granted the task of ﬁnding the limits, and then structuralized
limiting processes with the concept of a limit.

This difference between the E-group and K-group in the use of the contexts was also
salient in written survey responses. The chi-square results in Table 4.60 show a
signiﬁcant difference between the E-group and K-group in the use of the contexts.

N. In the task of ﬁnding the limit of a given function with its graph, eight routines
were identiﬁed according to the aspects of limiting process and the limit that were
used.
1. Describing the limit by looking at the graph: There were interviewees who
looked at either increasing or decreasing features on the graph. Based on the graph,
they endorsed that it was increasing or that it went to inﬁnity or that it diverged to
inﬁnity or that it converged to one.
2. Describing the limit by looking at an asymptote: Some interviewees focused on
an asymptote approached by a function or its graph. When there was a vertical
asymptote, then they endorsed that it went to inﬁnity. When there was a horizontal
asymptote, then they endorsed that it approached the asymptote value.
3. Finding the limit by assuming that inﬁnity is a number: Other interviewees
substituted inﬁnity for x to ﬁnd the limit of a given function. Then, they interpreted
inﬁnity as a real number (e. g. 1/0 = 00 or one over an inﬁnitely small number is
inﬁnity). For example, if the result after substitution was inﬁnity squared over inﬁnity,
they concluded that it went to inﬁnity because an inﬁnity in both numerator and

denominator was cancelled out.

119

4. Finding the limit by comparing increasing rate in numerator with that in
denominator: There were interviewees who focused on a comparison between
numerator and denominator to decide which one grew faster than the other. If
numerator grew faster than denominator, then they concluded that it went to inﬁnity.
In the opposite case, they endorsed that it went to zero. In the case of l/x, because the
numerator was ﬁxed as one, they endorsed that it went to inﬁnity when x got smaller.
Also, if increasing rates between numerator and denominator were the same, they
endorsed that it converged to one.

5. Finding the limit by dividing the greatest term in numerator or by
comparing the greatest terms in both numerator and denominator: Some
interviewees divided both numerator and denominator by the greatest term in
denominator or compared the greatest terms in both numerator and denominator.
Then they calculated the limit value of the function by direct substitution.

6. Finding the limit by using l'Hépital’s rule: Other interviewees seemed to ﬁrst
look at the function as a form of 00/00 when x approaches inﬁnity. Then they used
l'Hopital’s rule to ﬁnd the limit value.

7. Finding the limit without specifying the method

8. Undeﬁned: Other interviewees focused on the idea that the limits from the right
and left were not equal and then endorsed that the limit did not exist.

9. Unidentiﬁed: This is the case, for example, when the student reported that it
keeps going or increases with no other explanation.

10. No answer

120

Table 4.61 Distribution of routines in item V-a

 

 

 

 

Routines E-speakers K-speakers
interview Survey interview survey
1) Describing a process by looking at the graph 30 % 18.9 % 45 % 41.3 %
2) Describing the limit by looking at an asymptote 35 % 29.5 % 0 % 11.1 %
3) Finding the limit by assuming that infinity is a number 0 % 1.5 % 10 % 3.2 %
4) F indtng the limit by comparing numerator With 3 5 % 9.8 % 25 % 20. 6 %
denominator
5) F mdmg the limit by dtwdmg the greatest term m 0 % 0 % 0 % 0 %
numerator
6) Finding the limit by using l'Ho‘pital ’s rule 0 % 0 % 0 % 0 %
7) Finding the limit without specifying the method 0 % 12.1 % 0 % 4.8 %
8) Undeﬁned 0 % 6.1% 0 % 0 %
9) Unidentiﬁed 0 % 20.5% 20 % 16.7 %
M No answer 0 % 1.5% 0 % 2.4 %

 

Table 4.62 Distribution of routines in items VI-b and VI-c
E-speakers K-speakers
Routines Interview Survey interview survey
Vl-b Vl-c Vl-b Vl-c VI-b Vl-c Vl-b VI-c
gngfb’bmg bb’bbb” by ’bbk’”g b’ ”be 30 % 25 % 23.5 % 12.9 % 10 % 5 % 16.7 % 5.6 %
2) Describing the limit by looking at an 20 % 40 % 16.7 % 22.0 % 0 % 0 % 4.0 % 0.8 %
asymptote

b) F‘bd'bg ”'3 "m” by “‘“m’bg ”7‘“ 20 % 0 % 1.5 % o % o % o % 0.8 % 2.4 %
infinity ts a number
4) F'”d‘”g me I'm" by bbmbb’mg 15 % 20 % 10.6 % 6.1 % 6o % 3o % 34.9 % 24.6 %

numerator with denominator

5) F‘mbbg ’bbbm" by d'v'd’”g ”"3 0 % 5 % 2.3 % 4.5 % 15 % 45 % 12.7 % 38.1%
greatest term in numerator

6) Finding the limit by using l'Ho‘pital '3
rule

Zefzzgm’he1"""W"”0“’spec’ﬁ”"g’bb 0% 0% 6.1% 10.6% 0% 0% 5.6% 10.3%

0% 0% 6.1% 0% 15% 10% 3.2% 4.0%

8) Undeﬁned 0% 0% 0.8% 0% 0% 0% 0% 0%
9) Unidentified 10% 10% 31.8% 29.5% 0% 10% 19.0% 11.1%
10) Noanswer 0% 0% 6.8% 13.6% 0% 0% 2.4% 2.4%

 

O. In ﬁnding the limitbof l in item VI-a, dominant routines in both groups were based
x

on the geometric representation (i.e., routines 1 and 2 in Table 4.61).

P. In ﬁnding the limits of the more complicated functions in items VI-b and VI-c,
routines of E-speakers were based on the geometric representations, compared to those of
K-speakers on algebraic representations. In other words, the use of geometric
representations in routines 1 and 2 was still dominant in the E—group, whereas the use of

algebraic reasoning (i.e., routines 4 and 5) dominated routines in the K-group.

121

As previously explained, visual mediators are visual means used by students for
communication. In item VI, both algebraic symbols and graphs of three ﬁinctions were
provided for limit-ﬁnding tasks. In calculating the limit of each function, different visual
mediators between the E-group and K-group were employed for the sake of scanning
mathematical context.

Q. Syntactic mode in the E-group and objectiﬁed mode in the K-group. The ﬁrst
attempt in the responses of most E-speakers at scanning the written phrase of “when x
goes to positive inﬁnity,” involved uses of the shape of graphs and geometric mediators
such as asymptotes and slopes. In contrast, the same written phrase was scanned as the
algebraic expression of each function in the responses of about 60 % of K-speakers (see
Table 4.62). The scanned processes in the E-group may be called syntactic mode, as it
requires only knowing the written phrases and given geometric mediators. However,
most K-speakers seem to make a transition from syntactic mode to a different mode for
ﬁnding limits. In other words, they seem to reify both original written phrases and
geometric mediators with the purpose of ﬁnding limits. The ﬂexibility in the scanned
process of the written phrases and given geometric mediators can be described as better
performed than the less ﬂexible syntactic mode because the syntactic mode allows for
very little interpretation of the visual mediators and few predictions.

The following Episodes 9 and 10 exemplify differences between E-group and K-
group in ﬁnding the limits of three functions in item VI.

Episode 9. E-pair in ﬁnding the limits of three functions in item VI

Speaker What is said What is done
146. I So, [show the card about item VI-al what about this one? What show the card
happens to the curve as x approaches 0 from the leﬁ or right?
147. £70 The curve goes to zero.
148. E7d The curve goes to neg... [both say “oh”] [laughing]. I was gonna say
negative inﬁnity, but that’s 0. 1 don’t know.

122

I49.

150.
151.
I52.
153.
154.
155.
156.

157.
158.
159.
160.

161.
162.
163.
164.
165.
166.
167.

168.
169.
170.
171.
172.
173.

174.

175.
176.
I77.
178.
179.
180.

181.
182.
183.
184.
185.
186.

187.
188.
189.
190.

191.
192.
193.
194.
195.

E7c

I
E7c
I
E7c
I
E7d
E7c

E7d
I

E7c
E7d

I
E7d
l
E7d
l
E7c
E7d

I
E7d
l
E7c
l
E7c

E7d

l
E7d
1
E70
E7d
E7c

l
E7c
E7d
l
E7c
l

E7c
l

E7c
E7d

E7c
E7d
l
E7d
1

Well, yeah it’d go, yeah it’d go to negative inﬁnity and this is the
asymptote would be 0. [...] So...

So because?

It depends on if you’re looking at...

Mm—hmm

The y axis or if you’re looking at the x axis.

Mm-hmm

But it says x

Yeah as x approaches 0 from the leﬁ, so as x goes to 0, y will go to
negative inﬁnity.

Mm-hmm

Because?

Because there’s an asymptote at 0 and it doesn’t cross it.

[. . .] It just keeps getting higher and higher and higher but it never touches
0, so

Mm-hmm

Uh, yeah the x coordinate would be...

Mm-hmm

close to 0

Mm-hmm

Mm-hmm

But the y y coordinate is gonna be inﬁnity. It’s like the same, that
right, negative inﬁnity.

Anything else?

No.

[show the card about item VI-b] What about this one? show the card
As x goes to positive inﬁnity, um, y goes to positive inﬁnity.

[..] Because?

Um because here as x goes on to inﬁnity, the y goes up to inﬁnity, and
gets closer to inﬁnity.

[. . . .] Yes, I agree both goes to inﬁnity and because the t0p is higher than
the bottom

Mm-hmm

So, it’s just going to keep becoming a larger number.

OK, [...] anything else? Do you want to add?

Um, it looks like...isn’t there an asymptote here at negative 1?

It looks like.

Yeah, so um but this one for this side um, it doesn’t go to positive
inﬁnity, it goes to negative 1. So, oh never mind [laughing]

But, you know, when x goes to, you know, positive inﬁnity?

But...

Yeah there’s an asymptote at negative 1 and whatever

[....] Do you want to add something?

Oh no

[show the card about item VI-c] What about this one? x squared over 1 show the card
plus x squared?

As x goes to positive inﬁnity

Mm-hmm positive inﬁnity.

1’ goes to 1

That’s what I said, it gets closer to 1, you can see that there would be an
asymptote right here.

Yeah

And that would be at 1

Can you explain why?

Because of the way that these both curved

Mm-hmm

123

196.
197.
198.
199.
' 200.

201 .
202.
203.
204.

205.
206.

E7d
E7c
E7d
I

E7c

I
E7c
l
E7c

E7d
E7c

They’ll end...

[...] Well

And here, there’s an asymptote at one, I think.

Mm-hmm

[ ...... ] That’s...that’s what I have, y equals 1 as x goes to
inﬁnity...positive inﬁnity

Do you want to say why?

Um, basically the same thing that you can tell there’s an asymptote here.
Mm-hmm

And it doesn’t cross it, 1 was looking at the equation but I can’t ﬁgure out
why

Yeah

Yeah

Episode 10. K-pair in ﬁnding the limits of three functions in item VI

191.

192.

193.
194.

195.
196.

197.
198.

199.
200.

201.

202.

203.

204.

205.

206.
207.

208.

I

Kmf

Kme

Then...|show the card about item VI-al let’s solve the next problem. What show the card
happens to the curve as x approaches zero from left or from right?

[.....] When approaching from the right, the curve inﬁnitely upward grow
toward plus inﬁnity...

Um...um...

When approaching from the left, the curve toward minus inﬁnity

inﬁnitely. ..it does not intercept y axis. ..I thought it’s going to that direction.
The reason is. . .?

[...] in 1 over x. . . in 1 over x because it’s not 0 but approaching plus
0...from a number which is a little bigger than 0...so it’s positive...so it’s
growing to the upward direction on x axis...if plugging in 1 over 10 and 1
over 100 and 1 over 1000, it continues to increase...by becoming 10, 100,
and 1000. . .Considering that way, because I thought that it’s becoming
inﬁnity...without intersecting with y axis...because it’s not 0.

I also thought the same way.

So...because it continues to increase by plugging in 1 over 10, 1 over 100,
and 1 over 1000.

Yes

[show the card about item VI-b] When x goes to positive inﬁnity, what show the card
happens to the curve?

[...] I thought that this one is similarly going to become inﬁnity. Here now
what he and I said, speed which is similar to ratio comes out. . .Because the
increasing speed of the numerator is bigger than the increasing speed of the
denominator, the disparity between numerator and denominator is going to
become wider by plugging in [numbers]

Um...um

If this disparity decreases, it’s converging. Because the disparity increases, 1
think that it’s going to diverge.

Thinking so is okay, but...that x grows to positive inﬁnity means that [x]
equal to 0 doesn’t need to be thought. Because x equal to 0 doesn’t need to
be thought,..that...thinking that x is not 0. . . .if dividing this numerator and
denominator by x, the numerator is approaching l inﬁnitely and the
denominator is going to positive inﬁnity because x is positive inﬁnity....if
by doing so, I think that it can be more easily calculated

Instead, are you saying that x goes to inﬁnity?

Yes

[show the card about item Vl-c | When x goes to positive inﬁnity, what show the card
happens to the curve?

As I said before, the difference continues to decrease. The difference
between numerator and denominator is big at ﬁrst, but as x increases, the

124

difference is decreasing.

209. I How did you know that the difference is decreasing?

210. Kmf By plugging in [numbers]...continues to do ...the difference continues to
decrease little by little.

211. I Please tell me concrete numbers?

212. Kmf Yes. At ﬁrst if plugging in 1, it comes out 1 over 4.

213. I Yes

214. Kmf Because it’s 1 over 4, the difference between 1 and 4 is 3
215. I Um. . .um

216. Kmf If plugging in 2, um...it comes out 4 over 9. But ...this one becomes 5.

217. l Um...um

218. Kmf Just if continuing to plug in [numbers], because 1 ﬁnally thought that this
difference is going to decrease little by little...

219. 1 How do you think?

220. Kme As I said before. . . .uh. ...that denominator. . .so in the status that numerator is
ﬁxed, if denominator is increasing inﬁnitely or decreasing inﬁnitely, while
thinking that value is approaching 0, both denominator and numerator were
divided by x square. Then it can be calculated as in the case of the previous

one.
221. I So, what’s your answer?
222. Kmf One
223. Kme One

Item VII
In item VII, the question “What happens if you continue increasing the number of
sides of a regular polygon inscribed in a circle?” was asked with the following concrete

geometric representations in Figure 7.

 

 

 

 

 

Figure 7. Geometric representations provided in item VII

Table 4.63 Distribution of endorsed narratives in item VII

 

 

 

Endorsed narratives E-speakers . I(.-speakers
lntervnew survey 1nterv1ew Survey
I) It approaches the circle 55 % 48.5 % 15 % 38.1 %
2) It gets closer to the circle but never reaches it 25 % 7.6 % 0 % 0 %
3) It converges to (or becomes) the circle 20 % 28.8 % 85 % 57.1 %
4) Unidentiﬁed 0 % 9.1 % 0 % 4.0 %
3N0 answer ' 0% 6.1% 0% 0.8%

 

R. The majority of K-speakers reported that they accepted the endorsement that
the geometric sequence of polygons became the circle, compared to less than one

third of E-speakers. Only 28.8 % of E-speakers reported “it becomes the circle.”

125

However, an overwhelming majority of K-speakers (category 3: 85 % in the interview

57.1 % in the survey) replied “it converges to (or becomes) the circle.”

126

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127

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Table 4.65 Distribution of the use of the limit in the sequence of regglar polygons

 

 

 

 

Chi-square
Students Processual use Structural use d f Value Sig._
E-interviewees 65 % 35 %
K-interviewees 15 % 85 %
E-participants 60.6 % 33.3 %
1 .2 .012
K-participants 48.4 % 50.8 % 6 6

 

*Differences are signiﬁcant (p S .01).

S. The same as [F] and [M] above. In the case of regular polygons inscribed in a circle
in item VII, the context of increasing the number of sides of a regular polygon in the B-
group seemed to be used only for its geometric features. Such uses of the context can be
evidenced by the utterances of either increasing sides or decreasing areas (see “sides” [1 -
4, 6, 10, 11, 13-15, 18] and “lines” [15, 18] for increasing sides; “area” [7, 9, 16-17, 20]
for decreasing areas). These geometric uses of the context appeared to direct them to
think about a processual use of the limit of regular polygons ([1-12], [16-17], [19], and
[20]). This processual use of the limit often was related to unreachability ([1-2] and [18]).
In contrast, almost all K-interviewees continued to employ structural uses of limit. In the
same context, many K-interviewees characterized mathematical features of polygons
such as the lengths of sides, vertexes, and triangles ([27], [35] and [38] for the lengths of
sides; [21-22] and [39] for vertexes; [29-30] and [36] for triangles). This structural use of
the limit was related to reachability, that is, the sequence of regular polygons became the
circle ([22-25], [27], [29-30], [31-35], and [3 7-39]). However, this difference between the
E-group and K-group in the use of regular polygons did not appear clearly in written
survey responses. The chi-square results in Table 4.65 show little difference between the
E-group and K-group.

T. Two routines for ﬁnding the limit in item VII were identified in the interviews

and the survey:

128

SE1

'm

3‘;

1. Describing a process by looking at patterns of regular polygons in the
sequence: Some interviewees focused on the idea that the number of sides was
increasing or that the areas between polygons and circle were decreasing. Then they
endorsed that the sequence of polygons was getting closer to the circle.

2. Finding the limit by looking at algebraic features of regular polygons in the
sequence: Other interviewees considered the concepts of limit with limiting
processes of regular polygons. As for the concept of limit, they used the ideas that the
length of the sides became a point (or zero), or that the number of sides went to
inﬁnity, or that the difference in area converges to zero. Then they concluded that the
sequence of polygons became a circle.

3. Unidentiﬁed: This is the case when students said that it approached the circle or
that it becomes the circle without giving a reason.

4. No answer

Table 4.66 Distribution of routines in item VII

 

 

 

 

Endorsed narratives l3-speakers . K-speakem
lntervnew survey 1nterv1ew Survey
1:)0 ggztbmg a process by lookmg at patterns of regular 0 % 56.1 % 0 % 31.0 %
25:11:33; :Smn by looking at algebraic features of O % 20.5 % 0 % 49.2 %
3) Unidentified O % 17.4 % 0 % 19.0 %
5) No answer 0 % 6.1 % 0 % 0.8 %

 

U. Routines in the E-group were based on syntactic mode of regular polygons,

compared to those based on their objectiﬁed mode in the K-group. The majority of E
speakers scanned the written phrase of “increasing the number of sides of a regular
polygon” and then they used geometric features such as the sides of the polygon and the
areas to give an answer. Comparatively, after the majority of K-speakers scanned the

same written phrase, they focused on algebraic features of geometric mediators such that

129

84

the length of sides converges to zero or that the number of sides went to inﬁnity. The

former discourse in the E-group seemed to be mediated syntactically on the basis of

changes of regular polygons in the sequence, whereas the latter in the K-group appeared

to be based on its objectiﬁed mode.

The following Episodes 11 and 12 exemplify differences in their routines between

E-group and K-group in ﬁnding the limit of the sequence of regular polygons in item VII.

Episode 11. E-pair in ﬁnding the limit of the sequence of regular polygons in item VII

Speaker

301

302.
303.
304.

305.
306.
307.
308.
309.

310.

311.

312.
313.

314.
315.

316

317.

318

Episode 12.

183

184. K5a

.1

Bob

Eob

an
Eob
an
Eob
an

Eob
an

Eob
an

Eob
an
. Bob
I

. Bob

.1

What is said What is done
[show the card about item VIII What about, you know, this one, using Show the card
the sequence, the square, pentagon, hexagon inscribed in the circle,
what happens if you continue... increasing the number of sides of a
regular polygon inscribed in the circle

Alright, 1. ..

Explain your reasoning.

I thought about this one in geometrical terms, which 1 don’t know why I
switched up right here, but I said that the difference in the area between,
like, the area of the square in the circle and the area of, like, pentagon
circle, the area, the difference in it would become smaller. But I don’t
think that’s the...

I said...

direction I should have gone [laughing] in.

I said that...

It doesn’t make...

There would be less area, less, or less area inside of the circle, like, I
meant, like, pretty much exactly what you just said, I just said it a little
differently.

Ok, we both thought about it, like, geometrically

yeah, like, like, this area decreases each time, like, the area between the
outside of the shape, like, outside...

Yeah

of this sided ﬁgure and the circle just decreases, so every time you add
another line so it’s getting closer to becoming...

a circle

the circle

yeah.

[...] So what-what-what’s your ﬁnal answers?

That the area of the ﬁgure and the area of the circle would become
closer and closer to each other, but it would never be the same.

K-pair in ﬁnding the limit of the sequence of regular polygons in item VII

Then [show the card about item Vll] examine the sequence of the Show the card
square, pentagon, hexagon, heptagon..., which is inscribed in the circle.

What happens if you continue increasing the number of sides of a

regular polygon inscribed in a circle? Explain your reasoning.

close to the circle...if it goes to inﬁnity, we can say that it is the same as

130

the circle.

185. 1(5b So, the meaning that the sides are increasing is that the length of the
sides is getting shorter. Later the sides can eventually be seen as a point.
Then, a circle is originally the collection of points in the same distance.
Then, from the center, because the lengths of the inscribed polygons are
same, it becomes a circle.

186. 1(5a Yes.

Item VIII-b

131

 

 

 

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132

Table 4.68 Distribution of the use of the limit P in a sequence

 

 

 

 

Chi-square
Students processual use structural use d f Value Sig:_
E-interviewees 80 % 15 %
K-interviewees 30 % 70 %
E-participants 40.2 % 5.3 % ,,
K-participants 11.9 % 45.2 % 2 62'76 '000

 

*Differences are signiﬁcant (p S .01).

V. E-speakers’ responses were relatively consistent but not in tune with the
canonical interpretation of the word limit. K-speakers’ interpretations were more
diverse but often consistent with the canonical interpretations. The majority of E-
speakers used expressions such as “approaching,” and “getting closer to” to explain the
number P is the limit ofa sequence ([1], [3], [6], [8], [1 1-12], [14—16], and [l8-20]). K-
speakers’ responses were diverse. However, they often made utterances saying that the

sequence an converges to P as n goes to inﬁnity (see [25], [29-34], and [37-38]).

W. E-speakers provided processual explanations. K-speakers spoke about the limit
P in structural terms. In explaining the meaning of “P is the limit of a sequence,” most
E-interviewees’ responses referred to the word limit as processes in their deﬁnitions. This
processual use seemed to be related to only sequence processes of an without considering
processes of subscript n. The processual use often was connected with the idea of an
asymptote ([3], [7], and [8]). In contrast, most of the K-interviewees focused on either a
general term an or a relationship between two terms at ﬁrst. Then either this general term
or a relationship between two terms was characterized by speciﬁc mathematical features.
The chi-square results in Table 4.68 show a signiﬁcant difference between the E-group
and K-group in written survey responses.

X. The idea of reachability in the K-group and the idea of unreachability in the E-

group. This processual use in E-group was often characterized by unreachability to the

133

limit P ([1-3], [5-8], [1 1-12], [14-16], and [l9-20]), that is, “P is the limit as an
unreachable process.” Comparatively, the K-speakers’ ideas of “P is the limit” were
dominated by the idea of reachability ([21-22], [30-32], and [37]).
Y. E-speakers formulated interpretations in their own words. K-speakers used
more formal mathematical formulations. Many K-speakers clearly made a relationship
between the two processes of an and n in the limit of a sequence. In order to deﬁne “P is
the limit of a sequence,” they often provided formal languages such as “converge” and
“limits” (“converge” [21, 22, 31, 32, 37] and “limits” [33, 37]).

The following Episodes 13 and 14 exemplify differences in deﬁning that P is the

limit of a sequence a], a2, a3,. .. between the E-group and K-group.

Episode 13. E-pair in deﬁning that P is the limit of a sequence a], a2, a3,. . .(item VIII-b)

Speaker What is said What is done
145. I [show the card about problem VIII-b] What about this one? A show the card
student claims that a number P is the limit of a sequence al, a2, a3,
What does this mean?
146. Eoe Uh, it, this, uh, personally I interpreted this as, you know as, a1, a2, a3
it just keeps rising to a4, a5, etc., and it’s, slowly, or quickly,
approaching P, whatever P might be...

147. Eof Yeah, no matter what the value of the, of a is going to continue to
approach P ‘cause P is the limit.
149. 1 Is there anything else you want to say?

150. Eoc Not really, I mean that’s-

151. Eof Yeah .

152. Eoe This based, um, you know everything I’ve done on that’s just my
understanding of what a limit is

153. l Um-hmm

154. Eoe You know, numbers rising like that, it just approaches P. lts, it gets
inﬁnitely close to P.

155. 1 So do you, uh, understand the inﬁnitely close to P? What he just said?
156. Eof Yeah, uh-huh.

157. 1 What is the meaning of inﬁnitely close, to you?

158. Eof It means that, like, if you never actually touch P, never actually reach P,

but it’s going to keep approaching it, no matter how big a gets, it’s
going to keep going towards P.

159. I So do you agree with mainly what he said?
160. Eoe Yeah, more or less.
161. 1 OK

134

Episode 14. K-pair in deﬁning that P is the limit of a sequence a], a2, a3,. . .(item VIII-b)

 

175. I Then [show the card about problem VIII-bl A student claims that the Show the card
number P is the limit of a sequence. What does this mean?
176- Kmd In this case, I set up an expression in the same way (lim an = P )
177. I Then what is the meaning? If you explain the meaning more...
178. Kmd Um...so...seeing on the graph...no...rather than in the graph, when this
becomes a... .. when n becomes inﬁnity, I thought the meaning of that it
converges to a number P...ln case that graph is painted like that [root
shape]. ..like the previous problem, it can converge to 1...] thought so ,
[laughing]. . .yes, I thought so... ‘
179. I What does the word “converge” mean concretely?
180. ch lt approaches a certain value.
181. Kmd approaching as close as possible
182. ch If put 0.99999 as 1 equally, like that....the same as approaching
endlessly
183. I How did you explain this?
184. ch I just. . .sequence...in the sequence...in the sequence of an, as n grows, it
approaches a certain value and considered an example.
185. I What kind of example?
186. ch So, if there is the sequence of 1 minus 1 over 11
187. l Um. . .um
188. ch As 11 grows endlessly, the value of sequence approaches 1. So, like that.

Summary: Discourse on limit
Table 4.69 summarizes salient characteristics of the colloquial and mathematical
discourses on limit in the E-group and K-group.

Table 4.69 Salient properties of E-speakers’ and K-speakers’ discourse on limit
Item: topic Aspect E-speakers K-speakers
A Limited' Both groups used the word limited in any version mainly in non-
' ' mathematical context. Among K-speakers the phenomenon was stronger
context of use
than among E-speakers

Both groups used limitedCOH in
the sense of ﬁnite (not inﬁnite —

 

 

 

Sbﬁlgt‘tf: something that has an upper ::::ﬁ::::;: hmuedKMa'h m the
Item 1: bound) or restricted — one that '
Creating could be larger. ..-
sentences C. Limit: Both groups used the word, in whatever its version, in both
With given context of use mathematical and non-mathematical contexts
words Both groups used limita,” to K-speakers used limitKMwh mainly in
denote an upper bound — expressions that were formulated as a
D. Limit: something that does not let a textbook instructionto do something.
type of use thing or process to get larger or as textbook deﬁnition. In general, It
seemed that K-speakers were not very
proﬁcient in creating sentences with
llMltKMafh.
Item V: A process of the sequence An object as the limit of sequence
Finding the E. V: Limit ofa dominated the idea 0f ”"1”- dominated the idea of limit. The limit

limit Of an

sequencer—mm”-

AbOUt 30 % 0f E-speakers of the sequence as a number

 

 

I35

 

conceptualized the idea of the dominated the idea of limit in the K-
limit in the sequence as an group.

increasing or decreasing process

without the limit.

 

E-speakers used the inﬁnite K-speakers explicitly referred to
sequence in terms of processes mathematical algorithms associated

 

 

 

 

 

 

 

inﬁnite F. Use of with the function of the inﬁnite
sequence sequence sequence. In other words, the 1(-
speakers’ use of the sequence was
structural
G Routine When ﬁnding the limit in a given inﬁnite sequence, ﬁve routines were
' identiﬁed according to how to use limiting processes to ﬁnd out the limit
H Use of E-speakers used routines based K-speakers’ routines were based on
' . on processual use of limit structural use of limit by considering
routmes . .
the congpt of limit as a number.
About one third of E-speakers More than one third of K-speakers in
in the ﬁrst highest percentage the ﬁrst highest percentage answered
1. v1-3; l responded “it goes to inﬁnity.” “(it becomes) inﬁnity.” However, only
x In the next highest percentage, K-speakers (23.8 %) replied “it
the E-speakers (26.5 %) replied diverges (to inﬁnity)”
“(it becomes) inﬁnity.”
About one third of E-speakers Approximately 40 % of K-speakers
responded “it is increasing.” In replied “(it becomes) inﬁnity.”
J- V”): the next two highest However, only K-speakers (20.6 %)
x2 percentages, 28 % and 17.4 % responded “it diverges (to inﬁnity).”
1T); of E-speakers answered “it goes
to inﬁnity” and “(it becomes)
inﬁnity” respectively.
About 26 % of E-speakers An overwhelming majority of K-
K Vl-C' answered “it gets closer to 1.” speakers (80 % in the interview and
' 2' Other responses were 68.3 % in the survey) replied “it
x distributed almost evenly converges to I.”
Item VI: (1 + x)2 among “it converges to 1,” “it is
Finding the increasing,” and “it goes to
limit of a mﬁmty.” ,
- The number of E-speakers who The number of K-speakers who dld not
given , , , ,
function L. The rate of dld not answer questions respond to questions stayed in the

“no response”

increased considerably ﬁ'om same rate.

item Vl-a to item Vl-c.

 

 

 

 

 

M. The Korean responses were formulated in a
Many E-speakers spoke about . .
phenomenon . . . . more formal mathematical way With
. the km“ of function m terms of . . .
ﬁrst observed m the purpose of ﬁnding the limit.
[F] returned processes.
In the task of ﬁnding the limit in a given function with its graph, eight
N. Routine routines were identiﬁed according to the aspects of limiting process and
the limit that were used.
0. v1.3: 1 Dominant routines in both groups were based on the geometric
x representation
P. Vl-b and VI- The use ofgeometnc . The use of algebraic reasoning
representations was $1111 . . .
c d . . dominated l'OllthS 1n the K-group.
ommant m the E-group.
Q Visual The scanned processes in the E- Most K-speakers seem to make a
Nfediators group may be called syntactic transition from syntactic mode to a

mode, as it requires only different mode for ﬁnding limits in
knowing the written phrases and items VI-b and Vl-c

 

136

 

 

Item VII:
Finding the
limit of the
sequence of
regular
polygons
inscribed in a
circle

R. Limit of

regular polygons reported “it becomes the circle.

givemmetric mediators.
Only 28 8 % of E- speakers An overwhelming majority of K-
' ,, speakers replied “it converges to (or

becomes) the circle.”

 

S. The same as

Geometric uses of the context

appeared to direct them to think Almost all K-mtervrewees contmued

 

[F] and [M] about a processual use of the to employ structural uses of km“ 111
above . . the same context
limit of regular polygons
. Two routines for ﬁnding the limit in item VlI were identiﬁed in the
T. Routine

interviews and the survey
Those of K-speakers were used on the n:
basis of objectiﬁed mode. In other '

Routines in the E-group were

 

Item VIII-b:
Deﬁning that
the number P
is the limit of
a sequence

a], 82, 83,...

 

U. Use of . words, they seem to reify both original
. based on syntactic mode of . .
routine re ular 0' ons written phrases and geometric
g p y g ' mediators with the purpose of ﬁnding
limits.

E-speakers’ responses were K-speakers’ interpretations were more
V. Interpretation relatively consistent but not in diverse but often consistent with the
of limit tune with the canonical canonical interpretations

W.Use of limit

interpretation of the word limit.
E-speakers provided processual K-speakers spoke about the limit P in

 

 

explanations. structural terms.
This processual use in E-group The K-speakers’ ideas of “P is the
X. Idea of limit was often characterized by limit” were dominated by the idea of
unreachabiligr to the limit P. reachability.
Y. Deﬁnition of E-speakers formulated K-speakers used more formal
limit integpretatlons in their own mathematical formulations.
wor s.

 

137

CHAPTER V
SUMMARY, DISCUSSION AND CONCLUSIONS

This study sought to describe characteristics of university Students’ non-
mathematical and mathematical discourses on inﬁnity and limit using the
communicational analysis framework. The primary motivation for this study was the
difference between Korean and English speakers when it comes to the relation between
their non-mathematical and mathematical discourses on inﬁnity and limit. The
mathematical words for inﬁnity and limit in Korean are different from the Korean
colloquial words that correspond most closely to the English colloquial words inﬁnity and
limit. It is known that colloquial uses of mathematical words inﬂuence students’
mathematical discourses featuring these words. It was thus likely that mathematical
discourse of E-speakers would be inﬂuenced by their colloquial discourse, whereas it was
likely there would be no comparable inﬂuence of the colloquial discourse in the case of
K-speakers. In other words, it was expected that there would be differences between K-
speakers’ and E-speakers’ mathematical discourses that could be ascribed to this
difference in colloquial inﬂuences.

In order to better understand possibilities about the speciﬁc sources of differences in
mathematical discourses of the two groups, students’ backgrounds which were likely to
inﬂuence their mathematical discourses on inﬁnity and limit were also investigated in this
study. The research questions were as follows:

0 QUESTION 1: What are the relationships between colloquial and mathematical
uses of the English words inﬁnity and limit?; What are the relationships

between colloquial and mathematical uses of the corresponding Korean words?

0 QUESTION 2: What are the similarities and differences between native-English
and native-Korean speakers in their previous mathematics education

138

experiences, their learning habits and curricular experience with the words
inﬁnity and limit?

0 QUESTION 3: What are the similarities and differences between non-

mathematical and mathematical discourses on inﬁnity and limit of native-
English and native-Korean speakers?

The setting for the study was a calculus class for university students in the U.S. and
Korea. Methodology involved surveys and interviews. The results of the study derived
from two types of analyses: analysis of response patterns of the survey participants and
detailed discourse analysis of interview transcripts. The survey responses were then re-
examined to help validate ﬁndings from the interviews. A total of 132 English speakers
and 126 Korean speakers participated in the survey. Within each linguistically distinct
group, twenty representatives were selected from the survey participants for follow-up
interviews in pairs. The detailed discourse analyses of the interview transcripts generated
preliminary hypotheses in the interviewees’ discourse on inﬁnity and limit. Overall
analyses of response patterns involved searching for frequencies and percentages of the
survey participants’ responses and comparing the proportions of the two groups through
chi-square analysis to conﬁrm the emerging hypotheses. Data from both sources were
used to elaborate and verify hypotheses. Detailed ﬁndings are found in chapter 4. This

chapter presents a summary of the detailed ﬁndings.

Summagy of Findings

Questionl: What are the relationships between colloquial and mathematical uses of
the English words inﬁnig and limit?; What are the relationships between colloquial
and mathematical uses of the corresponding Korean words?

As explained in Chapter 3, the relation between the colloquial and mathematical
words in the two languages can be presented as follows:

Inﬁnitygcon = InﬁnityEMath, Whereas InﬁnitchOn #5 InﬁnityKMam

139

 

LimitEcOu = LimltEMam, whereas Limitxcon i LimitKMmh

The equality symbol ‘=’ means that the words sound the same. InﬁnitchOn and
LimitKCOu signify the colloquial Korean words that can count as rough translations of the
colloquial English words InﬁnityEcOu and Limjtgcw, respectively. IrrﬁnityKMaﬂI and
LimitKMath do not have any previous meaning in colloquial Korean — they are words of
Chinese origin. Therefore, there is a discontinuity in Korean and a continuity in English

between the colloquial and mathematical discourses on inﬁnity and limit. K-speakers

 

have little experience with the colloquial Korean use of the mathematical terms inﬁnity
and limit, whereas E-speakers do have experience with the colloquial use of the English
words inﬁnity and limit.

uestion 2: What are the similarities and differences between native-E lish and

native-Korean speakers in their previous mathematics education experiences, their

learnin habits and curricular ex erience with the words in ni and limit?

The second research question about students’ backgrounds is: What are the
similarities and differences between native-English and native-Korean speakers in their
former mathematics education and curricular experience with the words inﬁnity and
limit?

Similarities between the two groups relative to previous mathematics education
experiences, learning habits and curricular experience with inﬁnity and limit

To analyze different levels of Students’ former education with inﬁnity and limit,
course and content variables were considered. In the course variable, students’
recollections about their ﬁrst experience with inﬁnity and limit in a mathematics course
were analyzed. Even though some E-speakers and K-speakers reported that they ﬁrst
heard the words inﬁnity and limit before 7th grade, the majority of the participants

recalled their ﬁrst experience with inﬁnity and limit in a mathematics course between 7th

140

and 12m grade. According to the results of t-tests, there was no signiﬁcant difference
between the E-group and K-group in the range of grades for ﬁrst experience with inﬁnity
in a mathematics course. This result may indicate that students in both groups ﬁrst
learned about inﬁnity during similar time periods.

The survey included a list of seven topics related to inﬁnity and limit and asked
students to indicate whether they had learned about these topics. A chi square statistic

was used to test for possible differences in the percentage of students who reported

 

having learned the seven topics. According to the chi square analysis, there was no ‘
signiﬁcant difference between the E-group and K-group in their reports about having 7
learned the following six topics: ‘inﬁnite process,’ comparison of inﬁnite sets,’ and
‘inﬁnitely small or inﬁnitesimal’ in the case of inﬁnity; ‘limits to inﬁnity,’ ‘limit of a
function numerically,’ ‘and limit of a ﬁmction geometrically’ in the case of limit. These
results (especially with the comparison of both inﬁnite sets and limit of a ﬁmction used in
the survey) can support comparability of these two groups.
In terms of cunicular experience with inﬁnity and limit, the majority of students in
both groups (88.9 % of the K-speakers and 86.4% of the E-speakers) took either a pre-
calculus or calculus course in high school. Students in both groups also reported that they
listened to teachers’ lecture-style presentation in their classrooms for more than half of
the lessons. These are important similarities between the E-group and K-group for
comparability of these two groups.
In addition to examining students’ former mathematics education and curricular
experience with the words inﬁnity and limit, other aspects that were likely to inﬂuence

their mathematical discourse on inﬁnity and limit also were investigated in the survey.

141

These aspects are Students’ self report on their experiences of learning mathematics and
attitudes toward mathematics. Both the K-speakers and E—speakers reported that they
listened to teachers’ lecture-style presentation for more than half of their lessons. Neither
group reported small group work occurring very much.

Overall mean scores of both the K-speakers and E—speakers in their attitudes about
learning mathematics and the role of mathematics in their lives were higher than a neutral
response. These results imply that students in both groups showed positive attitudes about
learning mathematics and they also agreed on the need for mathematics in their lives.

Diﬂerwces between the two groups relative to previous mathematics education
experiences, learning habits and curricular experience with inﬁnity and limit

The survey asked about when students were ﬁrst introduced to the concept of limit
and inﬁnity. The mean of the E-speakers’ responses about the grade level at which they
were ﬁrst introduced to the concept was signiﬁcantly higher than that of the K-speakers
in the case of limit, indicating they were ﬁrst introduced to limit later than the K-speakers.
As for course sequences taken in high school, almost all K-speakers reported having
taken Mathematics 10 (for 10‘h grade), Mathematics I (for 11th grade), and Mathematics
11 (for lZ‘h grade) course sequence through integrated curricula. In contrast, most E-
speakers reported having taken a course sequence in which algebra and geometry are
separated.

There was a signiﬁcant difference between the E-group and K-group in self-reports
about having learned the following four topics related to inﬁnity: ‘inﬁnite decimal
fractions,’ ‘inﬁnite sequence or series,’ ‘sums of inﬁnite geometric series,’ and
‘countable and uncountable sets.’ The percentage of the K-speakers who reported having

learned each of the four topics was signiﬁcantly higher than that of the E-speakers. In the

142

case of limit, the percentage of the K-speakers reporting having learned related topics
was signiﬁcantly higher that that of the E—speakers in the following three topics: ‘lirnit of
a sequence or series,’ ‘limit and the deﬁnition of continuity,’ and ‘limit and the deﬁnition
of derivative.’ Interestingly, the percentage of the E-speakers, however, is signiﬁcantly
higher than that of the K-speakers in the topic of ‘the 8-8 deﬁnition of limit.’ These
results indicate that the K-speakers seemed to have had generally more opportunities to

learn concepts related to inﬁnity and limit in high school than the E-speakers.

 

E-speakers reported that they used a variety of textbooks in high school, beyond the
representative examples provided in the survey, whereas many K-speakers replied that
they studied from one of the seven textbook examples, and in some cases more than two.
That is, there was considerable variation in the textbooks that the E-speakers used in high
school, in comparision to consistency in the textbooks used by the K-speakers. Due to
this complexity, it was clear that analysis of textbooks, as a possiblemeans of
illuminating the different discourse experiences of students, would not be a productive
endeavor., student representatives were selected and interviewed in terms of their high
school textbooks instead of analyzing their textbooks.

There were differences in the two grops in their descriptions of how they studied
calculus. During interviews, K-speakers reported that they had systematically planned
study methods and periods of rest and followed that plan. In contrast, E-speakers seemed
to have planned less and did not follow it too tightly even if they had a plan. Even if these
responses were from a very small sample of students, they suggest that K-speakers more

carefully organized their studying and they devoted more time to it than E—speakers.

143

lust

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and
mar

5177.
dis

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of l
res]
eler
con
[mi
subs

rtpc

Question 3: What are the similarities and differences between non-mathematical
and mathematical discourses on infinity and limit of native-English and native-
Korean smakers?

The third research question about students’ discourse is: What are the similarities
and differences between native-English and native-Korean speakers in their non-
mathematical and mathematical discourses on inﬁnity and limit?

Similarities between the two groups relative to non-mathematical and mathematical
discourses an inﬁnity

a. Discourse on inﬁnity

- Non-mathematical uses of inﬁnite, inﬁnity, and inﬁnitely

No systematic similarities were found in the non-mathematical discourse on inﬁnity
of both groups — Only E-group actively used the noun inﬁnity.

- Mathematical uses of inﬁnity

The mathematical discourse on inﬁnity of both groups was not yet well developed,
in that it was quite different from the canonical mathematical discourse on inﬁnity. In the
mathematical discourse on inﬁnity, the following common characteristics in routines and
endorsed narratives were observed in both groups.
Set-comparison routines: Many respondents in both groups made non-canonical choices
of routines for comparison. Some respondents in both groups compared the sets with
respect to the size of corresponding elements rather than with respect to the amount of
elements. In addition, there were respondents in both groups who produced two
contradicting answers as the result of applying two different routines of comparison.
Endorsed narratives: Many respondents in both groups endorsed the narrative “A proper
subset is smaller/has less elements than the whole set”. K-speakers were more likely to

report that they accepted non-canonical properties of comparison.

144

 

b. Discourse on limit

- Non-mathematical uses of limit and limited

Both groups used the colloquial words limited and limit in both mathematical and
non-mathematical contexts. The adjective limited was used in the sense of ﬁnite or
restricted to describe something that has an upper bound or one that could be larger. The
noun limit was employed to denote an upper bound — something that does not let a thing
or process to get larger.

- Mathematical uses of limit

In the mathematical discourse on limit, the following common characteristics in
routines and endorsed narratives were observed in both groups.
Limit-finding routines: Many respondents in both groups made non-canonical choices of
routines for ﬁnding limits by looking at their graph. There were also some respondents in
both groups who found limits by assuming that inﬁnity is a number.
Endorsed narratives: Some respondents in both groups endorsed the narrative “limit is an
upper boundary”. E-speakers were statistically signiﬁcantly more likely to report the non-
canonical property of unreachability.

Dzﬂerences between the two groups relative to non-mathematical and mathematical
discourses

a. Discourse on infinity
- Non-mathematical uses of inﬁnite, inﬁnity, and inﬁnitely
E-speakers had a relatively well developed non-mathematical discourse on inﬁnity,
which seemed to be building on their non-mathematical uses of the words inﬁnite and
inﬁnity. The adjective inﬁnite appeared mainly in the phrases “inﬁnite number/amount

of. . .,” as a descriptor of sets, signaling these sets’ large dimensions (see the property [B]

145

of inﬁnity in Chapter 4). The noun inﬁnity appeared mainly in the context of processes
that do not end, as the target-point which these processes never reach (see the property
[C] of inﬁnity in Chapter 4).

K-speakers’ non-mathematical discourse on inﬁnity seemed less developed. They
used the adjective inﬁniteKMmh in non-mathematical discourse as descriptors of objects
known for their large size or of spiritual matters — not of sets or processes (see the
property [B] of inﬁnity in Chapter 4). They did not use the noun inﬁnityKMam at all. When
asked to use the non-mathematical version of the noun, they converted it to the adjective
inﬁnite. When given the mathematical word for inﬁnity, they turned it into adverb
inﬁnitely and used it in conjunction with many (inﬁnitely many), as a descriptor the
cardinality of sets (see the property [C] of inﬁnity in Chapter 4).

- Mathematical uses of inﬁnity

There were several differences between mathematical discourses of the two groups
in terms of use of words, routines, and endorsed narratives.

Use of words: There was not much difference between E-speakers’ mathematical and
non-mathematical use of the words inﬁnite and inﬁnity — they used the adjective with
relation to sets and they used the noun within the same phrases reporting on unending
processes: “goes to,” “continues to,” etc (see the properties [G], [O], and [R] of inﬁnity in
Chapter 4). In contrast, K-speakers’ mathematical discourse on inﬁnity was quite
different from their non-mathematical discourse. In their mathematical talk inﬁnitelwmh,
which appeared not very frequently, was used as a descriptor of cardinal numbers. The
word inﬁniO/KMaﬁ; appeared only when explicitly mentioned in the question, and it then

was used within the phrase ‘goes to inﬁnity’, which also appeared in the question. It was

146

 

not elaborated. In K-speakers’ mathematical discourse, there was almost never any
reference to processes. The discourse used mathematical vocabulary and was structural
rather than processual (see the property [T] of inﬁnity in Chapter 4).

Routines of comparison: One of the ﬁndings was that E-speakers’ choice of comparison

routine was sensitive to the wording of the question and, more speciﬁcally, it depended

 

f
on whether they were asked which set was bigger or which had more elements (see the .i
property [N] of inﬁnity in Chapter 4). K-speakers’ choice of routine did not change with
wording (see the property [M] of inﬁnity in Chapter 4). Another ﬁnding was that the '.

majority of E-speakers employed inﬁnity in conjunction with an inﬁnite process to
compare two inﬁnite sets. K-speakers’ talk about inﬁniO’KMam in their comparison routine
was more formally mathematical (see the property [R] of inﬁnity in Chapter 4). The other
ﬁnding was that E-speakers tended to claim that two inﬁnite sets are always equal. K-
speakers relied more strongly than E-speakers on the narrative that inﬁnite sets are
incomparable.
Endorsed narratives: E-speakers formulated interpretations about what inﬁnity is in their
own words. Their responses were relatively consistent and in tune with the canonical
interpretation of the word inﬁnity. In contrast, K-speakers used more formal
mathematical formulations about the concept of inﬁnity. Their interpretations were more
diverse and often inconsistent with the canonical interpretations. They spoke ahnost
exclusively in structural terms (see the property [S] of inﬁnity in Chapter 4).

To sum up, E-speakers seemed to be quite proﬁcient in the non-mathematical and
mathematical discourse on inﬁnity (e.g, they were able to answer questions using their

own formulations, different ﬁom that in the question itself). Their mathematical use of

I47

 

di

for

be:

Com

the words inﬁnite and inﬁnity was not much different from their non-mathematical use.
Routines of E-speakers were sensitive to the wording of the question. The majority of E-
speakers employed inﬁnity in conjunction with an inﬁnite process to compare two inﬁnite
sets. Their endorsed narratives were relatively consistent with the canonical
interpretation of the word inﬁnity.

In contrast, K-speakers’ discourse on inﬁnity was more formally mathematical and

structural, but they were not yet very proﬁcient in this discourse — in their responses, they

 

Ir . . ~._ I

tended to use phrases that appeared in the question. The mathematical discourse on
inﬁnity of K-speakers was quite different from their non-mathematical discourse on
inﬁnity. In the use of word inﬁnity, there were few references to processes. Their routines
did not change with wording. Comparison routines of K-speakers were more formally
mathematical. They endorsed narratives exclusively in structural terms.
b. Discourse on limit

- Non-mathematical uses of limit and limited

K-speakers’ non-mathematical discourse in limitedmam and limitmam seemed less
developed than that of E-speakers’. They used the adjective limitedKMmh in non-
mathematical discourse only in the sense of extreme which was not observed in the use of
limitechOu. They used the noun limitKMad, mainly in expressions similar to a textbook
formulation (see the property [D] of limit in Chapter 4). Generally, K-speakers seemed to
be not very proﬁcient in creating sentences with limitedKMam and limitguam.

- Mathematical uses of limit

The mathematical discourse on limit of E-speakers was. not yet well developed. In

contrast, K-speakers appeared to provide more developed canonical mathematical

148

discourse on limit (see the property [V] of limit in Chapter 4). There were several
differences between the mathematical discourses of the two groups in terms of use of
words, routines, endorsed narratives, and visual mediators.

Use of words: There was little difference between E-speakers’ mathematical and

colloquial use of the words limited and limit — they used the adjective in the sense of

ﬁnite and they used the noun to denote an upper bound. E-speakers’ use was processual

RJIA

 

and they often spoke about limit as unreachable. K-speakers, in contrast, spoke about

limit in structural terms. This structural use of limit was often related to the idea of limit

P.’

as a number (see the properties [F], [M], [S], and [W] of limit in Chapter 4). In general,
they used more formal mathematical formulations.

Limit-ﬁnding routines: E-speakers’ choice of limit-ﬁnding routine was based on a
processual use of limit. K-speakers’ choice of routine was based on a structural use of
limit by considering the concept of limit as a number. Routines in the E-group were
employed on the basis of syntactic mode of context, as it requires only knowing the
written phrases and given geometric mediators. Routines of K-speakers were used on the
basis of objectiﬁed mode (see the property [U] of limit). Finally, the majority of E-
speakers used geometric representations to ﬁnd the limits of more complicated functions,
whereas K-speakers employed algebraic representations with algebraic reasoning in their
routines (see the property of [P] of limit).

Endorsed narratives: E-speakers formulated the idea of limit in their own words. Their
responses were relatively consistent but not consistent with the canonical interpretation of
the word limit. K-speakers interpreted the idea of liMitKMath as an object by utilizing

formal language as well as formal mathematical representation. Their interpretations

149

were more diverse but often consistent with the canonical interpretations. They used
more formal mathematical formulations about the idea of limit (see the property [V] of
limit in Chapter 4).

Visual mediators: E-speakers used a syntactic mode as they scanned processes for ﬁnding

the limit of the sequence of regular polygons inscribed in a circle. K-speakers seemed to

make a transition from syntactic mode to a different mode for ﬁnding the limit (see the
property [U] of limit in Chapter 4).

To sum up, E-speakers did not develop canonical mathematical discourse on limit. i

 

They provided processual explanations of the noun limit and seemed to be quite
proﬁcient in this discourse (e. g, they formulated interpretations in their own words).
They often spoke about limit as unreachable. There was little difference between E-
speakers’ mathematical and colloquial uses of the words limited and limit as a process.
Routines of E-speakers in limit-ﬁnding tasks were based on a processual use of limit.
These routines were often employed on the basis of geometric representations. E-
speakers endorsed narratives in their own words. Their endorsed narratives were not
consistent with the canonical interpretation of the word limit. Visual mediators of E-
speakers in ﬁnding the limit of the sequence of regular polygons were used syntactically.
In contrast, K-speakers used limitmam in non-mathematical discourse only in the
sense of extremity which was not observed in the E-group. In the mathematical discourse
on limit, they spoke in expressions similar to a textbook formulation. They appeared to
provide more developed canonical mathematical discourse on limit than E-speakers. K-
speakers’ choice of limit-ﬁnding routine was based on a structural use of limit. These

routines were often employed with algebraic representations. Their endorsed narratives

150

were more diverse but often consistent with the canonical interpretations. Visual
mediators of K-speakers in ﬁnding the limit of the sequence of regular polygons were

changed ﬁ'om syntactic mode to a different mode.

Discussion of Findings
. Reasons for Differences in Mathematical Discourse on Inﬁnipg and Limit

There is one main difference between the E-group and K-group in their discourses
on inﬁnity and limit. The main difference is in their processual and structural uses of
inﬁnity and limit in the E-group and K-group respectively. In the E-group, the use of
inﬁnity and limit as process seems to be related to the processual use of the words inﬁnity
and limit in mathematical discourse. Due to the processual nature of the colloquial uses of
inﬁnity and limit in English, the mathematical English use also seems be processual. This
may be evidence for a connection between colloquial and mathematical discourses on
inﬁnity and limit. It seems that E-speakers build on their everyday experience with
colloquial discourse in developing their mathematical discourse on inﬁnity and limit.

In contrast, K-speakers’ non-mathematical discourse in the Korean mathematical
words inﬁnity and limit is rare and less developed in their own formulations. Their uses
appear to be formulated from textbook treatments? or deﬁnition. In the mathematical
discourse on inﬁnity and limit, their language is more formally mathematical and
structural. Their mathematical discourse on inﬁnity and limit is more consistent with, and.
possibly developed from, the structural discourse of mathematical textbooks, perhaps due
to lack of proﬁciency or experience with these words in their non-mathematical discourse.
As a’matter of fact, the nouns inﬁnity and limit in their mathematical versions are rarely

used as themselves in colloquial Korean. In addition, the more rigorous structure of their

151

 

deﬁnitions may be additional evidence that these students were introduced to the
discourse on inﬁnity and limit through mathematical deﬁnitions taught at school rather
than through non-mathematical use. There is therefore a clear disconnection between K-
speakers’ mathematical discourse on inﬁnity and limit and their non-mathematical
discourse. Their discourse on inﬁnity and limit is consistent with the structural discourse
of mathematical textbooks, and does not appear to draw on their previous everyday

experience with discourse on inﬁnity/1m,” and limithO”.

 

 

K-speakers exhibited the tendency to use mathematical-objectiﬁed discourse on l
inﬁnity despite the fact that they did not have a good mastery of this discourse. One
possible reason is the lack of alternatives. In other words, unlike E—speakers, K-speakers
did not demonstrate a well developed non-mathematical discourse on inﬁnity and thus
kept using words from the mathematical discourse on inﬁnity by employing phrases that
appeared in the question. Another possible reason is they might have a stronger tendency
than the E-speakers to remain in the discourse introduced by their interlocutor
(interviewer). The results of item I (about creating a sentence) seem to conﬁrm this. The
question in item I framed the discourse as being about language or literature (note the
request to create sentences) and the K-speakers’ answers were mame non-mathematical
— much more so than those of E-speakers. The same may explain the results of II-a,
which was about ﬁngers and toes, where the routines of comparison chosen by K-
speakers were often non-mathematical.

There are other reasons to speculate on a connection between E-speakers’
mathematical and non-mathematical discourses on inﬁnity and limit and a lack of

connection in the K-speakers’. The change in wording for comparing two inﬁnite sets

152

all‘Ol

 

strongly inﬂuenced the way E-speakers implemented the comparison of sets, whereas it
did not inﬂuence K-speakers’ choices. This difference reinforces the claim about a
greater disconnection between the non-mathematical and mathematical discourse in the
K-group. It seems that the K-speakers, rather than gradually proceeding from more
processual non-mathematical discourse on inﬁnite processes to a structural discourse on
inﬁnity, have been directly exposed in their mathematics classes to the structural formal

mathematical version. E-speakers’ sensitivity to the wording of the question may be

unu-—.__.._ 'l '0

evidence for a connection between colloquial and mathematical discourses on inﬁnity and

 

limit.

In limit-ﬁnding tasks, the use of geometric representations was dominant in the B-
group, compared to the use of algebraic representations in the K-group. The reasons for
this difference are likely to be multifaceted. They may include discourse—related
differences in the non-mathematical use of the word limit. Another may be differences in
educational experiences prior to studying calculus. And, it is possible that the curricular
emphasis on asymptotic behavior in the U.S. curriculum and the algebraic approach in
Korea curriculum lead to a disposition to the use of a geometric approach that is more
prominent in E-speakers. Another possible reason is the processual uses of E-speakers in
non-mathematical discourse on limit and the structural uses of K-speakers in
mathematical discourse. Additional research would be needed to understand this more
fully.

Students’ routines and endorsed narratives, as well as their visual mediators seem to
be grounded in the characteristics of their word use (i.e., processual and structural uses of

the words inﬁnity and limit). For instance, E-speakers’ canonical interpretation of the

153

word inﬁnity is consistent with processual uses of inﬁnity. Their routines were employed
on the basis of a syntactic mode of context due to a processual use of limit. In contrast,
routines of K—speakers were used on the basis of objectiﬁed mode with a structural use of
limit. K-speakers often reiﬁed limit-ﬁnding tasks with the idea of limit as a number.
E-speakers’ responses were consistent with the canonical interpretation of inﬁnity
with processual explanations, and K-speakers’ interpretations were more consistent with l
the canonical interpretation of limit. The interplay between non-mathematical and

mathematical discourses on inﬁnity can be helpful for E—speakers to develop their i

 

canonical interpretations of inﬁnity. However, the connection between non-mathematical
and mathematical discourses on limit can be problematic because the non-mathematical
and mathematical uses of the word limit are different and sometimes conﬂicting (i.e., The
processual use was often characterized by unreachability). K-speakers seemed to
differentiate mathematical discourse from non-mathematical discourse on both inﬁnity
and limit. This may help them develop the rigorous deﬁnitions and complicated structures
that underlie the mathematical concept of limit without conﬂicts. However, the difference
between non-mathematical and mathematical discourses on inﬁnity may make it difﬁcult
for K-speakers to reify and conceive of inﬁnity as a process, unlike E-speakers.

A salient property in routines is that many respondents in both the E-group and the
K-group used different routines for the different cases. This may be explained on the
basis of the fact that learning (transfer) does not happen automatically, but takes time and
practice. One may conjecture that students use different routines in different

mathematical contexts. In other words, routines seem to be highly context—dependent.

154

The results of this study can also provide insights about the relationship between
students’ self-concept in mathematics and their efforts in mathematics learning. Higher
self-concept in mathematics may be related to more effort and better performance in
some students, but may not in others. The E-speakers reported that they were more
conﬁdent than the K-speakers about their capability of learning mathematics. This
ﬁnding was consistent with previous studies (Chen & Stevenson, 1995; Stevenson, Chen,
& Lee, 1993) that Chinese students were less conﬁdent than U.S. students about their
mathematics learning. When students are working on mathematics homework, the
Korean students’ lower conﬁdence in mathematics may cause them to devote more time
and effort to it, as other studies (e. g., Fuligni & Stevenson, 1995) have suggested. In
addition, the result from the follow-up questionnaire suggests that for K-speakers
learning was more central because they reported systematically devoting more thought
and more time to it compared with E-speakers.

In addition to the continuity in English and the discontinuity in Korean between the
non-mathematical and mathematical discourses on inﬁnity and limit, differences in the
discourse opportunities between U.S. and Korean classrooms may be a possible reason
which is likely to account for the differences in students’ mathematical discourse on these
words between the E-group and K-group. Because the students in this study reported that
their calculus instructors used lecture-style presentation for more than half of their
lessons, one may conjecture that due to the processual nature of E-teachers’ mathematical
talk about inﬁnity and limit in their classrooms, E-students’ mathematical discourse on
these words may be processual. In contrast, K—students’ mathematical discourse on the

topics may be formally mathematical and structural perhaps because of K—teachers’

155

 

mathematical and structural language in their classrooms. An investigation of differences
in the nature of classroom discourse between the two groups may provide more thorough
information about reasons for differences in their mathematical discourse on inﬁnity and
limit.

The claim about the differences in the mathematics discourse on inﬁnity and limit as
caused by differences in the colloquial uses of these words, however, is supported with
process-type causality argument rather than regularity-type causality. In other words,
rather than trying to look for a statistical evidence for the co-appearance of the
phenomena that are claimed to be in causal relation (similar uses of words inﬁnity in
limit in non-mathematical and mathematical discourses), I argued for the causal
dependence by explaining how the non-mathematical discourse on inﬁnity and limit leads
to the mathematical discourse on these words. The results of this study conﬁrmed the
analytic argument: some English colloquial uses of inﬁnity and limit reappeared in
mathematical uses. In other words, E-speakers continued to use the words in the way they
did so far. The absence of this type of use in Korean mathematical discourse on inﬁnity
and limit was also in accord with the claim about intra-discursive mechanisms: The
unwanted types of use of the words inﬁnity and limit which, in the English mathematical
discourse, stemmed from the colloquial uses of these words did not appear in the K-
speakers mathematical discourse. Considering all this, additional information on the
classroom discourse in U.S. and Korea, although important, is of only secondary
signiﬁcance when it comes to substantiating the claim about the impact of colloquial
discourses on their mathematical counterparts; The classroom discourse can strengthen or

weaken this impact, but it is not its primary source.

156

 

Generalizabili
The ﬁndings of this study may be generalizable to native-English speakers in the

U.S. and native-Korean speakers in Korea who start to learn calculus in a university
setting. However, generalizations from the present ﬁndings should be made with
appropriate caution for several reasons. First, different geographical areas can be related
to differences in students’ former education and curricular experience with inﬁnity and
limit as important factors in mathematical discourse. In addition, characteristics in
mathematical discourse can be determined by various other factors such as
socioeconomic status (SES) and university contexts. Since SES was not taken into
account, and university differences were not considered in this study, the ﬁndings in this
study did not reﬂect this potential inﬂuence of additional variables.
Limitations of the Study

There are several limitations of this study. First, the 132 U.S. and 126 Korean survey
participants were not randomly selected out of the total population in the research sites.
They were recruited from several calculus classes on a voluntary basis and thus area
sample of convenience. Thus the ﬁndings from this work cannot be generalized beyond
this group of students, although they provide important focused hypotheses for
subsequent work. Second, there are some limitations in the survey and interview
instruments. The chi-square results in the survey questions should be interpreted with
caution due to the possible ambiguous nature of some questions to the students. For

instance, the interpretation of “extra lessons” in question 193 could be interpreted

3 Question 19: During this school year, how often have you had extra lessons or tutoring in mathematics
that is not part of your regular class?

157

differently, especially to the E-speakers. The meaning of “every day” in question 204 was
also problematic because of the fact that the U.S. and Korean classes met three days and
twice per week respectively.

Third, follow-up interviews, with 20 % of the survey participants, were conducted in
pairs. A possible limitation of this approach is that different pairings such as ﬁiendship
pairings and acquaintance pairings may cause different levels of performance in
mathematical discourse (e.g., Kumick & Kington, 2005). Variations in interviewees’
friendships were not controlled in this study, even though one or two pairs were
randomly selected within each setting. It may be useful to include the relationship
between different pairings and their mathematical performances in a further study.

A fourth limitation is that the timing of data collection between in the U.S. and in
Korea was different The Korean data of this study were collected in May 2007, the
second half of the ﬁrst semester for university freshmen in Korea, compared to the U.S.
data in September or October 2007, the ﬁrst half of the ﬁrst semester for university
freshmen in the United States. Because of the differences in timing, it could be expected
that K-speakers had more opportunities to learn about limit in their calculus class.

Finally, the K-speakers were exposed to English through the use of an English
textbook in their calculus classrooms in college settings, even though the college
lecturing was in Korean. However, the impact of the English calculus textbook may not

be signiﬁcant in terms of the effects of English on the K-speakers’ mathematics learning
because mathematics in precollege settings in Korea is taught in Korean and its textbooks
are in Korean.

g

4 How often does your instructor give you homework in mathematics?

158

Further Questions

One question that this study has raised concerns about is the generalizability of this
study. Another question is about additional important variables which can affect
mathematical discourse, such as socioeconomic status (SES) and social and
organizational contexts. In the narrow sense of generalizability, the present study can be
expanded by collecting data from other groups of students in other U.S. and Korean
1miversities. Additional comparisons of relationships between non-mathematical and
mathematical discourses on inﬁnity and limit could provide more comprehensive
information about differences and similarities.

In the broader sense of generalizability, the ﬁndings in the present study may not be
generalizable to students in other East Asian countries, even though these East Asian
languages share many common characteristics with Korean in terms of how mathematical
words are represented and used. Further research could include students from different
East Asian countries such as China, Japan, and Singapore, and could compare the results
from the ﬁndings in those studies with those of the present study. In addition, groups of
native speakers in East Asian countries could be compared with those of other native
speakers in English-speaking countries. These comparisons could provide more thorough
information about the phenomena caused by both continuity in English and discontinuity
in East Asian languages between non-mathematical and mathematical discourses on
inﬁnity and limit.

As for other important variables related to mathematical discourse, many further
questions are still interesting. For example, given the belief that social and organizational

contexts are relevant to mathematical discourse, one might ask, do students from different

159

university contexts differ in their non-mathematical and mathematical discourses?
Examining how students’ SES is related to their mathematics performance, one might
ask, do students from different SES levels differ in their non-mathematical and
mathematical discourses? In addition, one might ask, is there a gender difference in non-
mathematical and mathematical discourses?

Mm
Theoretical Conclusions

For the last several decades, most previous research on how students learn
mathematics has been based on cognitive psychological perspectives, called cognitive
theories. These cognitive theories are implicitly grounded in absolutism, that is, the
assumptions they hold about the nature of mathematics are context-independent. In
cognitive theories inﬂuenced by the work of Piaget, mathematical learning has also been
considered as a process of active cognitive acquisition. The acquisitionists who rely on
the acquisition of mathematical concepts as mathematics learning have explained that the
idea of cognitive processes is based on uniform and orderly forms of rule-following
(Harre' & Gillett, 1995).

Static constructs in mathematics education such as concept images and concept
structures are useful descriptions in mathematical discourse. For a better understanding of
how and why students think mathematical concepts, however, there needs to be a new
emphasis on dynamic variables in learning mathematics. The ﬁndings of this study add
two new perspectives to theoretical perspectives on learning mathematics. First,
mathematical thinking is words-based thinking due to the sensitivity of word use in such '

thinking. We notice that different students could decipher different meanings and thus

160

engage in different thinking, even with respect to the same word, inﬁnity. Second,
mathematical thinking is also a continuously dynamic process, rather than a one-way
directional internalization from others to the individual learner because of the contextual
dependence of routines as interwoven processes with word use. To become more skillful
and articulate in a certain mathematical discourse, students should continuously and
dynamically interpret what others who are more discursively advanced in mathematics
say, word by word, with routines due to dynamic processes of routines with word use. If
students misunderstand words or misuse routines in these communicative activities, their
learning could be inapplicable and even wrong in the mode of learning.

Thus, it is warranted to move beyond the cognitive perspective and to examine
continuous changes in the learning context through the study of word use and of routines.
By not examining word use and routines in mathematical discourse, researchers may
have a distorted view. To reveal situated learning difﬁculties, researchers need to
investigate thinking in context rather than thought because of the sensitivity of word use
and dynamic processes of routines based on word use. Thinking is an on-going process,
whereas thought is the end product of thinking. Research methods that are sensitive to
word use and dynamic processes of routines (by using visual mediators and endorsed
narratives) need to be employed in order to explore questions about both how students
know and why.

One difﬁculty in the discursive approach to student learning is the requirement for a
Common systematic and analytical framework. However, with a common framework of
analyzing discourse that is well organized, discourse analysis is a theory-based research

method including contextual sensitivity as well as addressing continuing processes in

161

learning mathematics. To improve mathematical learning in practice requires that
research aids us not only in deepening our fundamental understanding, but also by
indicating pragmatic processes for resolving student learning diﬁ'rculties. Discourse
analysis as a multi-lateral approach is a promising method to reveal the complex
mechanisms of mathematical learning in contexts because it can be sensitive to word use
as well as to the dynamic processes of routines as essential components in mathematical
discourse.
Practical Implications
The results of the current study offer several implications for teaching and learning
about inﬁnity and limit. The ﬁndings indicate that how to use words may be the most
fundamental and signiﬁcant factor in determining the content students try to learn.
Word use
This study suggests that learning is highly words-dependent. Even though a teacher
provides the same word several times on the same concept in class, students could grasp
its unintended meaning because they decipher it from their own point of view. Often their
interpretation might differ from what the teacher intended to explain. Thus, explicit
dynamic dialogues between the teacher and students seem to be a promising pedagogical
option to improve student learning difﬁculties. The most basic and critical perspective on
teaching mathematics is to know the words which students use to communicate the
meanings of mathematical concepts. Students’ learning difﬁculties may come from
packed words which explain mathematical concepts and structures. Reifying the dense

Words and their complex structures could assist students to more easily access

162

mathematical discourse ﬁ'om their own point of view. It seems that mathematical learning
starts with mathematical words and continues to develop through them.

Using problem solving task for developing proﬁcient word use and endorsed narratives

Problem solving in calculus should continue to address how to use words, but it
should also attempt to ascertain Students’ endorsed narratives. In order to assess
students’ development in word use and endorsed narratives, instructors may need to use
alternative assessment techniques such as oral exams and short-answer questions that ask
students to discuss theirunderstanding of mathematical words. Problem solving itself
may not improve Students’ word use and endorsed narratives without explicitly
mentioning the relations between them. For instance, discussing why the limit of the
sequence 1,1,1, is one might help students shift their word use of limit ﬁom
rmreachability to reachability. With explicit clariﬁcations of the relations between
problem solving and word use (including endorsed narratives), each problem solving
episode can be used to extend students’ word use and endorsed narratives from non-
canonical to canonical interpretations.

Making routines more precise

The importance of routines in promoting mathematical learning is established in the
mathematics education literature. In addition to the emphasis on routines in mathematical
discourse, it isdesirable to make routines more precise. In other words, not only how to
implement routines, but when to use those routines need to be explicitly expressed. The
ﬁnding of this study — routines are highly context-dependent — supports such
recommendations. In addition, differences between routines should be emphasized in

terms of when and how to apply them for student understanding. Decompressing routines

163

to when and how to apply them step-by-step and differences between the routines would
assist Students’ access to mathematical discourse.

Thought and language are inseparable in the process of human developmental
pathways (Vygotsky, 1986). Mathematical learning involves thinking with and through
everyday language, generalizing everyday concepts associated with mathematical words,
and abstracting those concepts with schooling. According to the results of the current
study, non-mathematical discourse seems to have an impact on mathematical discourse
due to certain clear differences between the mathematical discourses on inﬁnity and limit
of the E-speakers and those of the K-speakers. These differences can be ascribed to the
fact that only in English do the mathematical words inﬁnity and limit appear in non-
mathematical discourse.

Although some researchers (Cornu, 1992; Davis & Vinner, 1986) addressed the
issue of everyday language conceptions on limit as a cognitive obstacle, they did not deal
with how those conceptions were related to mathematical discourse on limit. Students’
non-mathematical discourse may have an impact not only on the students’ later use of the
mathematical keywords, but also on other aspects of their mathematical discourse, such
as endorsed narratives, routines, and visual mediators — a fact that should be kept in mind
in both the research on advanced mathematical concepts and teaching advanced
mathematics. I hope the practical and theoretical conclusions presented here will
inﬂuence research in advanced mathematical thinking and improve Students’ learning of

the notions of inﬁnity and limit.

164

APPENDIX A

THE PILOT STUDY QUESTIONNAIRE

165

The Pilot Stug Questionnaire

1. Create a sentence with the following word (term).

Small
Large

1 5 8

—2
Triangle
Triangular
Limited
Limit

. Diagonal
lO. Inﬁnite

l 1. Inﬁnity

°P°SP~SAPP°N€

11. Say the same thing without using the underlined word.

12. Some arrow points have triangular forms.

13. Eyeglasses are for people with limited eyesight.
14. There are inﬁnitely too many lawyers.

15. He has inﬁnite potential.

16. My love for you is inﬁnite.

 

17. The limit of -1- is 0 as x approaches inﬁnigy.
——' x

18. In box A, there are more matches than in box B.

III. Which is a greater amount and how do you know?

19. A: Your ﬁngers B: Your toes
20. A: Odd numbers B: Even numbers
21. A: Grains of sand in the world B: Size of the sky
22. A: Odd numbers B: Integers
l 2 3

IV. —= 0.25, —= 0.25, -—= 0.25,
4 8 12

How many such equalities can you write?

166

 

V. What do you think will happen later in this table? How do you know?

 

 

x Jx+25—5
x
1.0 0.099020
0.5 0.099505
0.1 0.099900
0.05 0.099950
0.01 0.099990
0.005 0.099995
0.001 0.099999

 

 

 

 

VI. What is the limit of the following when it goes to inﬁnity?
1

23. —
x

x2

24. —
l+x
x2

25.
(1+ x)2

 

VII. Read aloud

x + 3x ___ %. Explain what it says.

 

VIII. What is limit?
What is inﬁnity?

167

 

APPENDIX B

OTHERS

Overview of Education Systems in the United States and Korea
Development of Korean Words

Representative Analysis Examples (in terms of word use, endorsed narratives, visual

mediators, and routines)

168

 

Qverview of Education Systems in the United States and Korea

Korea has a centralized educational system Compulsory education in Korea begins
with elementary education, when children are six years old. Elementary education has a
duration of six years. Secondary education is divided into two stages: middle and high
school education. Middle school lasts three years. High school education includes two
separate programs: vocational and general education. Vocational education lasts three
years and leads students to have a certiﬁcate for the labor market. General education also
lasts three years and leads to the university. Postsecondary education in Korea includes
both vocational and academic higher education. Vocational higher education usually lasts
two years and results in an associate’s degree. Academic higher education includes the
four university programs that lead to a bachelor’s, a master’s, a professional degree in
medicine, and a doctoral degree respectively. In all universities, semester system is used
during the academic year. Spring semester, which is the ﬁrst semester of the academic
year, goes from March to June, whereas fall semester period goes from September to
December.

Mathematics curricula in Korea are controlled by the Ministry of Education. The 7‘h
School Curriculum is the latest published document (Korea Ministry of Education, 1997).
According to the document, all students from grade I to 10 are required to take
mathematics. From grade 11, students take mathematics elective courses. There are six
elective courses: Mathematics 1, Mathematics 11, Applied Mathematics, Discrete
mathematics, Calculus, and Probability and Statistics. Students who are going to study
social science in the university are required to take only Mathematics 1. In contrast,

students who want to study science in the university take both Mathematics 1 and

169

Mathematics 11 as requirements. They are also required to take one of the last four
elective courses.

The United States has a decentralized educational system. In other words, education
is primarily a State and local responsibility (U.S. Department of Education, 2006).
Primary education begins at age six in elementary schools and lasts for ﬁve or six years.
Secondary education is divided into two levels. Lower secondary education has three or

four years and is offered in grades six or seven to nine in middle school or junior high

 

schools. Upper secondary education lasts three or four years and includes grades nine or L
ten to twelve in high schools. Postsecondary education in the United States is offered in
community colleges and includes vocational certiﬁcation programs. Vocational higher
education is typically about two years in length and results in an associate’s degree.
Academic higher education includes four university programs that lead to the award of a
bachelor’s degree, a master’s degree, a professional degree in ﬁelds such as medicine and
law, and a Ph. D. Some universities in the U.S. are in a semester system, whereas other
universities are in a quarter system. In a semester system, there are fall and spring
semesters implemented as the ﬁrst and second periods respectively. In a quarter system,
there are four academic periods during the academic year.

Mathematics curricula in the U.S. are each state’s responsibility. Thus, the standards
and expectations to guide mathematics curriculum in Michigan have been developed by
the Michigan Department of Education. The Michigan Department of Education’s Ofﬁce
of School Improvement led the developments of both the K-8 Mathematics Grade Level
Content Expectations (GLCE) in 2004 and High School Content Expectations (HSCE) in

2006. The HSCE extends the GLCE as appropriate for grades 9-12. According to the

170

GLCE, all students from grade 1 to 8 are required to take mathematics. Based on

Michigan’s HSCE, all high school students take four courses as requirements out of ﬁve

choices. The three courses (Algebra I, Algebra II, and Geometry) are required and both

Pre-calculus and Statistics and Probability are optional. The two different educational

systems in the United States and Korea are summarized in Table I.

Table 1: Overview of educational systems in the United States and Korea

 

 

 

 

 

 

 

 

U. S. Korea
System Decentralized Centralized
Compulsory Elementary, middle school, Elementary, middle school, high
education high school educations school educations
Postsecondary Vocational and academic Vocational and academic higher
Education higher educations educations
For grades 1 _ 8, mathematics For grades 1 — 10, mathematics as a
. requirement, For grades 11 and 12,
as a requirement. For grades 9 . . . .
. . . srx elective courses. Mathematics 1,
Mathematics — 12, ﬁve elective courses. . .
. Mathematics 11, Applied
curriculum Algebra 1, Algebra II, . . .
Mathematics, Discrete Mathematics,
Geme‘ry’ P‘e'calculus" and Calculus and Probabili and
Statistics and Probability Statistic; ‘y

 

 

171

' 'm‘ _u'3q

Development of Korean Words

Korean has borrowed a huge number of Chinese characters. The Korean lexicon
consists mainly of pure Korean words and Korean words built on Chinese characters:
about 30 percent native Korean words and 65 percent Chinese character words (Sohn,
2006). Those borrowed words and Chinese characters have become integral parts of the
Korean language. Ancient Koreans learned both the Chinese characters and language.

However, they also explored ways of recording their native language with the Chinese

 

characters. In the middle of the 15th century, Hunminjungeum (Hangtil - the Korean
alphabet) was invented in order to bring literacy to the common people who were not
intellectually elite and proﬁcient in Literary Chinese (Lee & Ramsey, 2000). Until this
time, there was no common way to read Chinese characters.

From this time, many words in Korean were constructed as terms parallel to the
words conveyed by Chinese characters and numerous words have also been created by
Koreans represented with Chinese characters. Chinese characters are usually associated
with the meanings of the words in Korean. Pronunciations and writing systems of
borrowed Chinese-character words in Korean have evolved independently in Korea.
Nowadays, use of Hangﬁl spellings has increased in newspapers and even scholarly
books, and the use of Chinese characters is considerably limited (Sohn, 1999). To show
meanings and etymologies of words more clearly, however, some professional writings,
such as in law and in Korean history, continue to be written in Korean mixed with
Chinese characters.

As Korean language, culture, and society were inﬂuenced by Chinese language and

culture in the past, they have currently been inﬂuenced by their American counterparts

172

since 1945. Most borrowed words after 1945 are words from English with American
socio-cultural concepts. For instance, over ninety percent of some 20,000 borrowed
words currently used. in Korean are from English, such as coffee, café, and hotel (Sohn,
2006)

Most mathematical terminologies in Korean (e. g., inﬁnite, inﬁnity, limit, element,
and set) were borrowed directly from the Chinese characters, but in school mathematics

they are written in native Korean spellings without explaining the words with Chinese

 

 

characters. For example, the word 91'— ?! in Korean terminology (without using the
Chinese character of half: which means boundless or boundlessness) is written and read

as mu-han in Korean pronunciations in a 7th grade mathematics textbook According to a

Korean dictionary, there are two Korean mathematical words for inﬁnity: 3?— Ei (mu-han)
and 1:35;! EH (mu-han-dae). The ﬁrst Chinese character (flit - mu) of the two mathematical
words means none and the meaning of the second character (FE - han) is bound and the
last character (ii-doe) means bigness. Thus, the word mu-han means “boundless or

boundlessness,” whereas the meaning of the word mu-han-dae, used only in mathematics,
is “boundless greatness.”

A Korean word comprises one or more syllables. Most Korean syllables borrowed
from Chinese characters are used colloquially and mathematically. Mathematical and
colloquial uses of Korean words are determined by cultural and historical uses of the
words, rather than their syllables or characters. The existence of sharing the same syllable
or character between two words does not mean necessarily their colloquial uses or
mathematical uses. Whether each syllable is used as part of either a colloquial word or a

mathematical word is decided on the basis of whether the Korean word (including the

173

syllable) is either a colloquial term or a mathematical term. Korean colloquial words for
inﬁnity and limit are often used in everyday language rather than the mathematical words
of inﬁnity and limit. For instance, 9— 4‘- 8| (mu-su-hee, countlessly) or if 81 0| (hon-up-
see, endlessly) are used for “inﬁnitely” in colloquial Korean, rather than the mathematical

word L?- 53 3| (mu-han-hee, inﬁnitely). Instead of the mathematical word it 5i (gﬁk-han)
for limit, the Korean colloquial words HI if (Jae-hon, limitation) and 23" 3‘11 (kyung-gae,

boundary) are often employed. The Korean mathematical words for inﬁnity and limit are
summarized in Table 2.

Table 2: Summary of the Korean mathematical words for inﬁnity and limit

 

 

 

 

 

 

 

 

 

. Korean
English _ . ,
Words Pronuncratron Chinese characters Meaning
. gr" ‘3’ mu-han m BE Boundless or boundlessness
Inﬁnity ‘
5?- PJ CH mu-han-dae 315?: 7t Boundless greatness
2 it gﬁk.han @131? . Utmost boundary
Limit A value to which a function
:1 at F ' - - t
m _ 3A guk han gab limit approaches

 

 

 

 

174

 

 

Representative Anaysis Examples
In the ﬁrst example of the pilot study, students were asked to deﬁne limit. The

deﬁnitions of limit given by the students are presented in Table 3.

Table 3: The summary of deﬁnitions of limit

 

 

 

 

 

 

 

 

 

Students What is limit?
A5 [1] Limit can go on and will stop at some point.
A7 [2] The limit is the value of a number F
> A10 [3] Limit is something that numbers can’t go past. They have to stop at a 3,
a certarn pomt. z,-
:::. [4] As not in mathematics, limit is the absolute ending of something like no 5
g more. In a mathematical sense, it would always be the ending of. . .like an i
AU answer, but sense... numbers are inﬁnite. You can get to the answer but J _.
never reach the limit because there’s inﬁnite numbers. . .kind of opposite in
a way because there isn’t... there is no limit in inﬁnity.
K4 [5] Limit means like the maximum.
W K7 [6] Limit is the furthest something to .go. .: like where it stops. . .like .
g boundary you cannot go on. . .I think inﬁnity IS the opposrte of the hunt.
g Km [7] Limit is a certain number that you can’t reach but you get very close to.
K [8] Limit is a special number. . .for instance if I say limit four, then there is
U one, two, three, and four. Not over four.
1)‘ Word use

Obiectiﬁcation: There is a considerable difference with age in ways of deﬁning the word
limit. In the group of the younger participants in both ethnic groups, the word limit was
used without explicitly relating it to numbers. In contrast, the older participants speciﬁed
limit with numbers. Evidence of such a use (as an objectiﬁcation) can be seen only in the
utterances of the 10th graders and university students. In all such cases the entity was
presented as a particular number or related to numbers (“something that numbers can’t go
past” [3], “the answer” [4], “a certain number” [7], and “a special number” [8]; the status
of the utterance [2] is unwarranted, in the present context).

2) Endorsed narratives

The students’ deﬁnitions of limit revealed the following endorsed narrative.

175

 

Limit is an upper boundary; In this endorsement, limit is used to describe something
that limits the process from above. Some of the students held this view, as evidenced
by expressions such as: “limit can go on and will stop at some point” [1], “numbers
can’t go past” [3], ‘you never reach the limit” [4, 7], “limit means like the
maximum” [5], the “limit is the furthest something to go” [6], and “one, two, three,

and four” [8].

. . 1
In the second example, students were asked to ﬁnd the ill’lllt of — when x goes to
x

 

inﬁnity. The summaries of students’ responses are presented in Table 4.

Table 4: Summary of responses about the limit of l

 

 

 

 

 

x
1
Students VI. (a) What is the limit of - when it goes to inﬁnity?
x
A [9] This one is zero. . .because last time we looked at it. One (problem) equals
5 zero. . .x equals. . .inﬁnity approaches... it’s like zero.
A [10] I don’t know what the limit is... That’s gonna be one over inﬁnity.
g 7 When x goes on forever, one over forever.
g- A [11] The limit is always one over inﬁnity. . .inﬁnity does have no
g '0 limit. . .something keep going on and on. . .I am not sure what the limit is.
[12] One (problem) would be zero because the bigger x gets. . .it just be
AU smaller and smaller decimal. It’s a sense of going inﬁnity, there is no ending
which is gonna be very, very tiny.
K4 [13] I don’t get what it means by when x goes to inﬁnity...
K [14] This would become zero. . .because the number one, you know one. . .this
7 will be close thirty. . .It won’t really be anything.
g K [15] It will get close to zero. . .because x is two. . .point ﬁve. . .x. . .ten. . .it will
'53 1° get to keep smaller and...
g [16] Zero or one. . .I don’t remember exactly why. But I know it’s gonna be
one or zero because I learned it. One over x goes to inﬁnity. . .then I can say
KU like. . .then I can think like one over inﬁnity. . .if I divided one. . .one over
one. . .but one over two is point ﬁve. . .that meaning is zero because inﬁnity
means too large number.

 

 

176

3) Visual mediators
For the purpose of this analysis, I differentiated between three visual mediators from my
observations. The words inﬁnity and limit in the task can be syntactic mediators used for
the sake of scanning mathematical contexts. The second type is concrete mediators which
students used in order to perform operations related to the task of ﬁnding the limit. The
third type of mediators is objectiﬁed, which are used to reify those operations with the
purpose of ﬁnding limits. As explained below, these mediators can be employed in

different ways within the process of calculating the same limit. The American 7th and 10th

graders scanned the word limit in a deﬁned way. To calculate the limit of l , the word
x

inﬁnity was scanned as a process (“forever” [10], “keep going on and on” [11]) and the
variable x in each function was replaced by inﬁnity. In order to give an answer, they (A7
and A10) used algebraic expressions as mediators (“one over inﬁnity” [11], [12]). This
scanned process may be called syntactic mode, as it requires only knowing the words
inﬁnity and limit and algebraic expressions.

In contrast, the word limit was used in different ways in the utterances of the American
undergraduate and the Korean students (K7, K10, and KU). They (AU, K7, K10, and KU)
manipulated concrete input values to calculate the limit of each function. Their ﬁrst
attempt at scanning the context of ‘x goes to inﬁnity’ involved substituting a small
number for x, performing the required operations (“you know one” [14], “two. . .point
ﬁve” [15], “one over one” [16], AU showed “the bigger x gets” [12] as the process of
substituting a small nmnber for x in the next problem) and then increasing the input
values. In order to ﬁnd the limit, they looked at the substituted values of each function

and decided whether these values are increasing or decreasing to a number (approaching

177

 

 

a value). They turned spontaneously to concrete mode and then made a transition from
concrete to objectiﬁed mode regarding the concept of limit as approaching a value.
4) Routines
The routines of the American 7th and 10‘h graders were to substitute inﬁnity for x by
alluding to an operational use of inﬁnity (“when x goes on forever” [10] and “something
keep going on and on” [1 1]). They seemed to be using just one mediational mode .7
(syntactic) with little ﬂexibility. This syntactic mode allows for very little interpretation

of ﬁnding limits and no predictions.

 

In contrast, the Koreans (K7, K10, and K0) and the American undergraduate
substituted several increasing numbers for x (“the bigger x gets” [12], “the number one,
you know one...this will be close thirty” [14], “x is two...point ﬁve...x...ten. . [15],

and “if I divided one. . .but one over two. . .inﬁnity means too large number” [16]) and

1 . . . . . .
checked whether the values of — were increasmg or decreasrng to determine the limit.
x

In the case of -1— , they (K7, K10, KU, and Au) made a few transitions from one mediational
x

mode to another; from syntactic to concrete mode and then from concrete to obj ectiﬁed

 

mode. Their ﬂexibility from one mediational mode to another in the process of ﬁnding
limits can be described as objectiﬁed discourse. This mediational ﬂexibility provides
more interpretations and predications regarding the concept of limit than only one

mediational mode.

178

 

APPENDIX C

QUESTIONNAIRE

179

 

QUESTIONNAIRE

The purpose of this study is to investigate and compare how native-English and native-
Korean speaking university students, who received their education respectively in the U.S.
and in Korea, think about the concepts of inﬁnity and limit.

Participation is voluntary, and you are free to discontinue your participation at any time.
If you agree to participate in the study, you can agree to participate in Part I, or Parts I
and II.

Part I: You will be asked several questions on this questionnaire. The questionnaire
consists of two sections: Section I, Background; and Section II, Discourse. The questions
on the questionnaire are dealing with your background regarding the mathematical
concepts of inﬁnity and limit, and questions about colloquial and mathematical
discourses on these concepts. Based on your responses to these questions, you may be
invited for an interview.

Part 11: Some students will be asked to participate in a follow-up interview. The
interviewer will ask pairs of students to think aloud about solving the same tasks
presented in Part I. If you participate in the research study, you will be audio-taped,
video-taped and transcribed in their entirely for further analysis. As compensation, each
interviewee will receive 25 dollars.

There are no risks associated with the survey and-interview. Any information obtained
from this research will be kept strictly conﬁdential. Only researchers will have access to
these data. As soon as the investigators complete analysis, all video and audio data will
be erased.

You are free to ask any questions concerning the procedure. If you have any questions
about this study, please contact the investigators (DJ Kim, 884-1481, kimdon14@msu.edu;
Dr. Joan F errini-Munay, 432-1 4 90, jferrini@msu. edg; Dr. Anna Sfard, 353-0881,
annasfar@math.msu.edu). If you have questions or concerns about your rights as a
research participant, please feel free to contact Peter Vasilenko, Ph.D., Director of the
Human Subject Protection Programs at Michigan State University: (51 7) 355-2180, fax:
(51 7) 432-4503, irb@msu.edu, or regular mail: 202 Olds Hall, East Lansing, MI 48824.

Your signature below indicates your voluntary agreement to participate in this study to
increase our understanding of student conceptions of inﬁnity and limit.

 

 

Participant's Printed Name PID

 

 

Participant's Signature Date

180

 

Section I: Background
About You

1. Year:
’4 Freshman
413 Sophomore
J Junior
J Senior

2. Native Language:
-J English
3 Korean
J Other (Specify):

 

3. How often do you speak your native language at home?
’41 Always
Ll Almost always
J Sometimes
J Never

4. About how many books are there in your home? (Do not count magazines,
newspapers, or your school books.)
'43 None or very few (0-10 books)
J Enough to ﬁll one shelf (1 1-25 books)
'43 Enough to ﬁll one bookcase (26-100 books)
11 Enough to ﬁll two bookcases (101-200 books)
'-J Enough to ﬁll three or more bookcases (more than 200 books)

Mathematics in School
5. Have you studied mathematics in any language other than your native language?

L1 Yes
—3 No

If yes, specify the languages in which you learned mathematics:

6. Can you remember the ﬁrst mathematics course in which you heard the word
(mathematical) inﬁnity?

43 Yes
J No

181

 

7. If yes, specify in which grade(s) and in which course/class you heard the word
(mathematical) inﬁnity.

Grade(s): Course:

8. Can you remember the ﬁrst mathematics course in which you heard the word
(mathematical) limit?
3 Yes
J No

9. If yes, specify in which grade(s) and in which course/class you heard the word
(mathematical) limit.

Grade(s): Course:

10. Please specify from which high school textbook(s) series you studied among the
examples below. If you used textbooks other than those in the examples, please
check the box for Others.

 

 

         

Core-Plus

Math . Pr _ t Key Curriculum
emanc s 016C Mathematics Press McDou a1 Littell
(CPMP) Connections g

 

 

5) 6) Others

   
   

 

 

The University of
Chicago School
M th mat' Pr ' t

a fucsnifrp) 016° H0“ Prentice Hall

 

 

 

 

 

 

 

182

11. In high school, which course(s) in the series below did you take? Please check all
the boxes that apply to you (under the textbook series you used).

 

 

 

 

 

 

 

 

CPMP Math Connections Key curriculum McDougal Littell
J Coursel 3 Math connections] J Algebra 3 Algebral
"JESSIEEEWM"7.3"it'42iﬁ'c'éiiiééii3rié'é """ 365;.95'"""""3"6é3}£12t}§ """"
.-::E?E€:ii-.-----.-if-.¥3f¥ii?&‘35€fi?3i?.-_-Ea??§ff‘fi------iésitiéf .......
J Course 4
UCSMP Holt Prentice Hall Others
J Transition Math lJ Algebral L1 Pre-Algebra
"Sign?"m""""'f'3"6é'c;;1}'é££§'"m"""Yieii'g'éﬁii'i """""
"3"a.;'.;.;;.;;y"'"""'"'"fringing;""'"""Sagas; """""
"ZJ'XAIAQEJIigéBQW """""""""""""" B'Ai'g'éiirlé """""

 

 

 

12. In high school, what is the most advanced mathematics course you took other than

those in #11?
Course:

 

13. Please check the box(es) next to topics which you have studied. When and where
did this study occur?

When

Where

 

[ELLLLFL‘LLL‘

Inﬁnite processes

Inﬁnite decimal fractions
Inﬁnite sequence or series
The sum of inﬁnite geometric series
Comparison of inﬁnite sets
Countable and uncountable sets
Inﬁnitely small or inﬁnitesimal

Other topic (Describe): ‘

 

 

 

 

 

 

 

 

 

 

183

 

14. Please check the box(es) next to topics which you have studied. When and where

did this study occur?

When

Where

 

Limits to inﬁnity

 

The 8-6 (epsilon-delta) deﬁnition of limit

 

Limit of a sequence or series

 

 

Limit of a function numerically

 

Limit of a function geometrically

 

Limit and the deﬁnition of continuity

 

Limit and the deﬁnition of the derivative

 

L’LL'LL‘L'LQL.

Other topic (Describe):

 

 

 

15. How much do you agree with these statements about learning mathematics?

a lot

.—

a) I usually do well in mathematics. _.;

I would like to take more mathematics in a
school. “
Mathematics is more difﬁcult for me than

for many of my classmates.

Sometimes, when I do not initially

d) understand a new topic in mathematics, I _'
know what I will never really understand it.

e) Mathematics is not one of my strengths.

0
V
L]

f) I learn things quickly in mathematics. __:

Check one box for each line
Agree Agree Disagree Disagree

i—V

‘.

...-i

t
t
.._J

.—

._J

l I

ll [-1

16. How much do you agree with these statements about mathematics?

a little a little a lot

[3

[‘l

w

[3

Agree Agree Disagree Disagree

a lot
a) I think learning mathematics will help me n
in my daily life. T
b) I need mathematics to learn other school
subjects. ”’
c) I need to do well in mathematics to get into ,—.
the university of my choice. T
d) I would like a job that involved using 1
mathematics. T
e) I need to do well in mathematics to get the _:
job I want. _

184

[J

.ﬁ

a little a little a lot

I:

17. How often do you do these things in your mathematics lessons?

Every or About Some Never
almost half the lessons

every lessons
lesson
a) We work together in small groups. I! D I L
b) We relate what we are learning in n D _. L
mathematics to our daily lives. 7‘ ' 7
c) We explain our answers. 1 D L”; L
We decide on our own procedures for .—. _
d) . _ D _ L
solvmg complex problems.
e) We review our homework. I B I [I
t) We listen to the teacher give a lecture- +7 E .. D
style presentation. 7 ’ "
g) We work problems on our own. I E I L
h) We use calculators. : D I L
Things You Do Outside of School
18. On a normal school day, how much time do you spend before or after school
doing each of these things?
No time Less than 1-2 More than
1 hour hours 2 hours
a) I watch television and videos. I: E : I
b) I play computer games. D [3 L _
c) I play or talk with ﬁiends. C: D Z :
d) I do jobs at home. [I L 2 i:
e) I work at a paid job. I: L: I I
f) I play sports. L E :3 ”Z
g) I read a book for enjoyment. L C : 2
h) I use the intemet. E E I _
i) I do homework. I: E _ :

19. During this school year, how often have you had extra lessons or tutoring in
mathematics that is not part of your regular class?

J Every or almost every day
'—J Once or twice a week

41 Sometimes

'—3 Never or almost never

185

20. How often does your instructor give you homework in mathematics?

J Every day

J 3 or 4 times a week

J l or 2 times a week

“—1 Less than once a week
4—1 Never

21. When your instructor gives you mathematics homework, about how many
minutes are you usually given?
ll Fewer than 15 minutes
3 15-30 minutes
11 31-60 minutes
—3 61-90 minutes
J More than 90 minutes

186

Section II: Discourse

1. Create a sentence with the following word (term).

 

 

 

1. Limited:
2. Limit:

3. Inﬁnite:
4. Inﬁnity:

 

11. Of which are there more? Please check one of the boxes. How do you know?

 

 

5. :1 Of your ﬁngers or [3 Of your toes
Because:

6. 2 Of odd numbers or E Of even numbers
Because:

7. 2 Of odd numbers or [3 Of integers
Because:

 

111. Which of the two sets A and B is bigger? How do you know?

 

A = { l, 2, 3, 4, 5, 6, 7, 8, }
B = { 2, 4, 6, 8, 10, 12, 14, 16, }
Answer:
Because:

 

187

IV. A student claims that each of the two sets A and B is inﬁnite, but A is larger
than B. What can this mean?

Response:

V. What do you think will happen later in the columns of this table?

 

 

x Jx+25—5
x
1 .0 0.099020
0.5 0.099505
0. 1 0.099900
0.05 0.099950
0.01 0.099990
0.005 0.099995
0.001 0.099999

 

 

 

 

Answer:

Because:__v__w___ ..., 7 7, .7 .7 .. _. .-.,

188

VI. What happens to the curve as it approaches 0 from left or from right?

3.1
x

 

 

 

 

Answer:

 

Because:

 

What will happen to the curve in #9 and in # 10 when it goes to positive infinity?

x2.

1+x

9.

 

 

 

Answer:

 

Because:

 

189

10

 

' (1+x)2

 

 

 

 

t T
I n '1
L.

17

 

Answer:

 

Because:

 

VII. Examine the sequence of the square, the regular pentagon, hexagon (6 sides),
heptagon (7 sides), ..., which is inscribed in the circle.

6
What happens if you continue increasing the number of sides of a regular
polygon inscribed in a circle? Explain your reasoning.

 

 

 

 

 

Answer:

 

Because:

 

190

VIII. A student claims that a sequence a1, a2, a3, goes to inﬁnity. What does this
mean?
Response: »

A student claims that the number P is the limit of a sequence a1, a2, a3, What
does this mean?
Response:

mhn'~--. -..,...__._. .. ..L, --H L .. Viﬂ-.,.. _ - _____-. _ . _... -....- _. ._... .... .. .. .__-. _.... ..._ .-..L - .. __ .. ”77...... .. .__.._........ .-.-.. ..., . . . ‘_.. - ﬂ .. 7+ .

191

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