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LEARNING THROUGH NEGOTIATION:
AN ANALYSIS OF STUDENT-INITIATED DISCOURSE
IN THE COLLEGE MATHEMATICS CLASS

presented by

Mary Anne Loewe

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LEARNING THROUGH NEGOTIATION:
AN ANALYSIS OF STUDENT-INITIATED DISCOURSE
IN THE COLLEGE MATHEMATICS CLASS

BY

Mary Anne Loewe

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of English

1994

ABSTRACT
LEARNING THROUGH NEGOTIATION:

AN ANALYSIS OF STUDENT-INITIATED DISCOURSE
IN THE COLLEGE MATHEMATICS CLASSROOM

BY

Mary Anne Loewe

The use of international graduate students to teach
American students in the university has been the subject of
some debate. Lhdversities need the international teaching
assistants (ITAs) to teach the rising undergraduate enrollment
in freshman courses, while parents complain that their
children are not receiving an adequate education. Studies
have shown that~ most of these protests center in the
mathematics departments, where the students complain of ITAs
with poor English and an inability to adequately answer
student questions.

This study examines the way American students negotiate
the content of the course in what is apparently a difficult
speech event. It expands upon already existing research in
conversational analysis to describe a particular classroom
discourse pattern -- the student-initiated interaction
sequence -- and provide a possible framework for future
analysis of classroom discourse.

One hundred twenty three student-initiated sequences were
isolated from 18 hours of observation and audiotaping of six
American and Chinese first year mathematics teaching

assistants. These sequences were analyzed for (a) overall

structure and function, and (b) amount and function of their
embedded negotiation structures. The appearance of these
sequences and their embedded structures was contrasted in the
American and international TA classes in order to provide a
description of the negotiation strategies used by students in
the classes of international TAs.

The findings revealed no differences in the American and
international TA data in terms of the overall structure of the
student-initiated sequences or the number of turns taken to
successfully negotiate a resolution to the students' question.
Differences did emerge, however, in terms of the functions of
the student initiations and the kinds of embedded negotiation
structures. The students corrected and disagreed with the
international TAs less, made less use of alignment talk such
as accounts and conversational repair, and made less repair
involving course content in the ITA classes.

These results have ramifications for both ITA and
freshman orientation programs and raise questions regarding
what is considered successful interaction in the college

mathematics class.

Copyright by
MARY ANNE LOEWE
1994

ACKNOWLEDGMENTS

As with any dissertation, there are many people who contributed to the success of
this project. A special note of appreciation is extended to the members of dissertation
committee for their consistent support throughout this process: to the chairperson of the
committee, Dr. James Stalker, for his advice and encouragement on this project as well
as throughout my graduate program; to Dr. Mary Bresnahan and Dr. Paul Munsell for
their helpful critiques and support; to Dr. Michael Lopez for also agreeing to be a part
of this process.

I am also grateful to Dr. Gerald Ludden, Director of Undergraduate Studies,
Barbara Miller, Administrative Assistant, and to the teaching assistants and students in
the Department of Mathematics at Michigan State University for their cooperation in this
and previous studies.

Finally, I would like to thank my husband, Tom, for his patience, especially during

the final preparations.

TABLE OF CONTENTS

LIST OF TABLES ................................................... ix

TRANSCRIPTION NOTATION ............................................ X

INTRODUCTION ...................................................... 1
CHAPTER

I. Students’ Questioning as a Guide to Classroom

Processes ............................................... 4

Factors Contributing to Students' Complaints ............ 4

Pedagogical Factors .................................. 4

Cultural Factors ..................................... 8

Linguistic Factors ................................... 9

Pertinent Research .................................. 11

Students' Questions as Guides to Classroom Processes. . . 13
Questions as Guides to Students’ Thought Processes. .14
Questions as Guides to Nature of Student/TA

Relationship ........................................ 14
Pertinent Research .................................. 16

II . Theoretical Framework .................................. 19
Conversational Analysis ................................ l9
Question/Answer’SequencerStructure .................. l9
Alignment Talk ...................................... 20
Classroom Interaction .................................. 25
Question/Answer SequenceeStructure .................. 25
Alignment Talk ...................................... 26

III . Methods ................................................ 2 9
Subjects ............................................... 29
Observations ........................................... 30
Analysis ............................................... 33
Tabulations ............................................ 37

vi

TABLE OF CONTENTS (CONT’D)

IV. Findings ............................................... 40
Structure of Student-Initiated Sequences ............... 42
Demand Ticket ....................................... 42
Presequence ......................................... 45
Resolution Sequence ................................. 46
Frequency and Type of Sequence ......................... 47
Requests-to-do—Problems ............................. 47
Assertions .......................................... 51
Requests ............................................ 54

Phatic Language ..................................... 60

Use of Embedded Sequences .............................. 63
Types of Negotiation ................................ 66

Types of Repair ..................................... 71

Use of Other Alignment Talk ......................... 74

Summary ............................................. 83

V. Discussion of Findings ................................. 86
The Kind of Interaction in Mathematics Classes ......... 86
Structure of Sequences .............................. 86
Function of Sequences ............................... 88

Use of Negotiation” ................................. 88
Comparison of ATA and ITA Classes ...................... 89
Structure of Sequences .............................. 89
Function of Sequences ............................... 90

Use of Negotiation .................................. 92
Strategies Employed by Students ........................ 93

VI . Conclusion ............................................. 96
Ramifications for ITA Training ......................... 97
Areas for Further Study ................................ 99
NOTES ON CHAPTERS ............................................... 101
APPENDIX A: SAMPLE DOCUMENTS FROM FRESHMAN MATH ................ 102
APPENDIX B: ANNOTATED INTERNATIONAL TA BIBLIOGRAPHY ............ 106

vii

TABLE OF CONTENTS (CONT’D)

APPENDIX C: TA AND STUDENT CONSENT FORMS ....................... 114
APPENDIX D: PROFILES OF PARTICIPATING TEACHING ASSISTANTS ...... 116
APPENDIX E: SAMPLE PAGE OF NOTES ............................... 119
APPENDIX F: LIST OF REPAIR SEQUENCES ........................... 120
TRANSCRIPT ...................................................... 121
BIBLIOGRAPHY .................................................... 162

viii

LIST OF TABLES

ELBE BEE
1 Sections of Entry Level Math Courses .............. 5
2 Average Attendance per TA Over Three Day

Observation ....................................... 41
3 Average Number of RP Sequences .................... 48
4 Average Number of Sequences by Function ........... 50
5 Types of Repair ................................... S9
6 Frequency of Repair ............................... 7O
7 Types of TA—Initiated Repair ...................... 72
8 Types of Student-Initiated Repair ................. 73
9 Use of Alignment Talk ............................. 82

ix

TRANSCRIPTION NOTAT ION

(...) Pauses are marked with a series of dots. The
length of the pause is not strictly relevant to
this study, so the measurement has not been
included. Long pauses such as those for writing on
the board are marked as such.

[ ] Utterances that were not audible are marked with
square brackets. Attempts to fill in those blanks
will have the approximations within the bracket.

( ) Empty parentheses indicate the observer’s
interjections or alternate descriptions such as
nonverbal behavior.

[ Single brackets indicate overlapping speech or
interruption.

St: Indicates student is speaker.

TA: Indicates teaching assistant is speaker.

Stzz Use of a subscript indicates that the speaker is
not the same student as the one initiating the
interaction.

Other

Transcripts follow generally accepted punctuation rules
where possible. A period indicates a fall in intonation with
a brief pause as would generally be considered a sentence
ending. A.question mark indicates a rise in intonation at
utterance end, in generally accepted question intonation.

Stutters, repetitions and pause fillers such as uh and ah
are represented. However, standard spelling for pronunciation
is used throughout. Contractions such as can't and don't are
represented in the way the students used them.

The complete explanations of problems are not written out
if there was no continuing discussion of the problem.

Samples of transcribed<discourse taken from.other sources
follow those sources’ notations.

Introduction

Many freshman mathematics students contend that it is
difficult to learn the material and earn high grades in
classes that are taught by international teaching assistants
(ITAs). Bailey (1984) refers to a "groundswell of complaints
from students and parents" (p. 6). Guneskera’s research
findings (1988) suggest that perhaps as many as half of the
American students may hold negative opinions of their
international TAs. Students cite linguistic, cultural, and
teaching problems as contributing to the difficulty they
experience in class.

The students' complaints call into question the
relationship the students have with their teaching assistants
and therefore the quality of the education they receive.
These complaints indicate that students' expectations are not
met when they enter classes taught by international teaching
assistants, that the course content is less successfully
learned in those classes, and that, as a result, students may
choose different means by which to learn mathematics rather
than negotiate successful communication with their TAs.

When the interaction in conversations becomes
problematic, the participants normally negotiate both the

content and the meaning of the message by making use of

2

various types of alignment talk such.as conversational repair,
accounts, and other metatalk (Stokes and Hewitt 1976). It has
been suggested. that in any' problematic interaction, the
responsibility for the success of the interaction falls to the
person with the more power. For example, in parent/child
interaction, more repair is conducted by the parent; in
teacher/student interaction, more repair is conducted by the
teacher; in NS/NNS interaction, more repair is conducted by
the native speaker (McHoul 1990). This raises the question of
who will take this responsibility when there 18' a
contradiction in power, as in a classroom when the native
speaker is a student.

This study examines how American students negotiate
content and meaning with the TAs in their mathematics classes.
It is a description of the students' uses of questions and
repair strategies over a three-day period in the mathematics
recitation sections of ‘three .American and ‘three Chinese
teaching assistants. Secondary analyses (such as attendance
patterns) are also discussed as necessary to clarify the data.
One hundred twenty-three student-teacher audiotaped
question/answer sequences were taken from the 18 hours of
observation and analyzed for the following features:

1. overall structure and function
2. number and function of negotiation features

The first goal was to describe the overall structure and
function of the question/answer sequences and their embedded

negotiation structures. The second goal was to determine the

3

students' negotiation strategies by contrasting the appearance
of these sequences and their embedded structures in American
(ATA) and international TA (ITA) classes. Because
communication is assumed to be more difficult in ITA classes,
the expectations were that (a) students would ask different
kinds of questions of the international TAs and (b) more
negotiation of the students' questions would take place in the
ITA classes.

The findings confirmed that the students in the
international teaching assistants' classes ciui participate
differently in the negotiation of content and meaning than did
those in American TA classes. Attendance was lower in two of
the ITA classes; students made use of fewer assertions to
correct or disagree with the ITAs; the ITAs and their students
employed less repair, and it appeared to be at the surface
level, to reference and utterance, rather than to the
presuppositions underlying the students’ questions. Chapter
1 discusses the basis of the students’ complaints about ITAs
and offers a rationale for the use of students' questions as
guides to classroom processes; Chapter 2 reviews the classroom
discourse analysis literature which provides the theoretical
framework for the study; Chapter 3 describes the methodology;
the findings are discussed in Chapters 4 and 5; and Chapter 6
concludes with suggestions both for further research and for

ITA and undergraduate training programs.

Chapter 1: Students’ Questioning as a Guide
to Classroom Processes

Factors Contributing to Students’ Complaints

When freshmen enter a large university system, they enter
a school situation that may be very different from the one
they were accustomed to in high school. The class sizes are
larger; the work load is heavier; the students carry more
responsibility for their learning. In the mathematics
department, the students face additional challenges when they
find themselves in classes with tightly controlled syllabi,
taught by teaching assistants who often have limited teaching
experience. Often these teaching assistants are from foreign
countries. Many freshman mathematics students contend that it
is difficult to learn the material and earn high grades in the
classes taught by the international teaching assistants
(ITAs). The source of trouble most often cited by the
students is the ITAs’ difficulty with English, but there are
also other contributing factors. The international TA
"problem” is more complex, having roots in pedagogical and
cultural as well as linguistic factors.

Pedagogical Factors

While every'department hiring international TAs has seen

student complaints, in the departments of mathematics, the

5
number of students and ITAs, as well as the number of students
with failing grades, further contributes to the students’
complaints about international teaching assistants (Orth 1983,
Bailey 1984).

Michigan State University’ 5 mathematics department serves
over 20,000 students (10,000 in the freshman mathematics
series) each fall. Table 1.1 below indicates the number of
sections offered for each course during 1989-1990, a typical

academic year.

Table 1.1

Sections of Entry Level Math Courses

 

 

 

 

Semesters

Math Course Fall 89 Winter 90 Spring 90
Elementary Math

Level 1 9 4 1

Level 2 37 16 6
College Math

Level 1 78 44 22

Level 2 11 39 22
Applied Math

Level 1 ll 22 20

Level 2 58 11 6

Total # of Sections 204 136 77

 

6

Typically, when students arrived as freshmen, they were
required to take a mathematics placement exam. Those who did
not achieve a passing score were placed in Elementary Math 1
or 2, the high school refresher courses. They were required
to take that course regardless of the requirements of their
major. Most students, however, placed into College Math.

Students who entered the system at Elementary Math 1 may
have had to take Elementary Math 2 also before they entered
College Math. Once they completed the remedial requirement,
or if their placement score was high enough, they may have
opted for either of the two-term freshman mathematics series,
Applied Math 1 and 2 or College Math 1 and 2. The College
Math courses taught Algebra and Trigometry; Applied Math dealt
mainly with the application of mathematical principles in
business. It was often taken by business majors who needed a
single term of mathematics. .Applied.Math.2 was an accelerated
course covering the same material in one term that College
Math 1 and 2 covered in two. It was also possible for someone
to take one term of College Math 1 and continue on to Applied
Math 2 instead of College Math 2.

While the Department of iMathematics offers an
undergraduate major, most of the students enrolled in
mathematics need only the freshman level courses to satisfy
the requirements of their majors. Business, nursing, and
engineering are among the majors having freshman mathematics
as one of their prerequisites, requiring a course grade of at

least 2.5 for successful completion. The average grade earned

7
by students in the College Math series at the time of the
study was under 2.0. The failure rate was approximately 20%
(Rittenberg 1988).

According to Jacobs (1988) , 45% of mathematics recitation
sections nationally are taught by TAs, and of those, 1/3 are
taught by international graduate students. As many as 1/3 of
the doctoral degrees awarded nationwide in mathematics and
computer science go to foreign students (Byrd 1988). At
Michigan State University in Fall 1988, the mathematics
department offered assistantships to 12 new international
graduate students out of the 86 hired overall at the
university*. Many of the international graduate students who
are assigned teaching duties have never taught before. Of
nine ITAs interviewed by the Michigan State University’s ITA
Program in 1988, four had previous experience; of the three
international participants in the current study, only one had
previous experience?. These TAs have the same difficulties as
any new teachers in presenting theory. The international
teachers, however, also need to know the terms and
explanations for the mathematical concepts and functions in
English and cannot draw on their past educational experiences
as the American TAs do to help them meet the students'
expectations. (Wieferich 1989).

In addition to lacking teaching experience, the
international teaching assistants are placed in a teaching
situation that restricts their freedom and time to experiment

with their skills. Each freshman math course is supervised by

8
a faculty member who is responsible for course content and
for the teaching assistants teaching in tflmmn Of the 76
sections of the College Math series which were offered in the
quarter during which this study began, 4 were taught by
advanced teaching assistants or faculty members. The other 72
were taught by 9 faculty members who supervised 8 teaching
assistants each. Each faculty member taught 8 sections in a
large lecture which met 2 days each week; 2 days each week the
lecture broke into 8 recitation sections which were taught by
teaching assistants. The syllabi for these mathematics
recitations and all the tests were determined by the faculty
members (See Appendix A for a sample syllabus, test and
homework assignment sheet.) The teaching assistants only had
to review the lecture material with the students in the
recitation, and answer questions. IEach.TA.taught 2 recitation
sections, each under the supervision of a different faculty
member. The TAs followed the sometimes conflicting
instructions of the supervising faculty, and the time
constraints limited any choices they may have had in
conducting their classes. These TAs were rated by the
students as if they had full control over what transpired in
their classroom.
Cultural factors

The international teaching assistants and students have
different expectations regarding the role of the teacher,
regarding not only the teaching methods used, but also the

relative power relationship that is established between the

9

student and teacher. In Asian countries such as the Peoples'
Republic of China, the authority of the teacher is primary.
Students rarely ask questions in class when they do not
understand the material (Paine 1986; Gass et a1. 1988), nor do
they question the validity of the information they receive.
This is quite different from the American classroom where
students routinely interrupt the instructor for clarification
or discussion. Additionally, in the United States, students
are taught critical thinking skills from a young age
(Angeletti 1990; Strother 1989), and.i¢: is acceptable for
students to challenge their teachers on the material
presented. While international graduate students are told of
the cultural differences they will encounter in the American
classroom, their previous school experiences make the
transition difficult. Some make this transition more easily
than others, depending in part on how well they tolerate
ambiguity and how flexible they are in defining their roles
(Smith 1993, Smith and Simpson 1993, Ruben and Kealy 1979).

Linguistic factors ‘

Both students and the ITAs themselves report that the
ITAs suffer from fluency and pronunciation problems and have
difficulty understanding the colloquial English used by the
undergraduate population (Bailey 1984; Rittenberg and
Wieferich 1988; Anderson 1990; Bresnahan 1993). Some studies
center on discourse organization as a source of trouble,
suggesting that the misuse of discourse level markers that

signal the ordering or relative importance of information adds

10

to the difficulty in comprehending (Rounds 1987; Tyler, 1988
and. 1992). Others focus (n1 appropriate lecturing' style
(Rounds 1988; Douglas and Myers 1989 and 1990; Byrd 1992).
Student complaints are supported by the fact that a large
percentage (the MSU’Office of Planning and Budget reported 62%
percent at Michigan State University in 1990) of the
international teaching assistants originate from Asian
countries (Bresnahan 1993). The emphasis in the English
classes in their home countries is placed on grammar and
reading rather than on speaking and listening skills. Their
difficulty with speaking and listening skills may account for
Asians describing themselves as introverted and preferring to
deal with concepts instead of people (Torkelson 1992).

International graduate teaching assistants dealing with
the above issues meet American students who are new to college
life in general and to the work load in particular. These
undergraduate students generally have limited experience with
people from ethnic backgrounds different from their own and so
are not accustomed.to taking responsibility for the success of
an. NS-NNS interactioni. Because of the great need for
teachers at the undergraduate level, the international
graduate students often have been placed directly into the
classroom upon arriving in the United States, without the
benefit of preparatory training‘. First-time international
teachers in an irregular teaching situation have been
encountering first time college students and neither have been

adequately prepared for the experience.

11

Pertinent Research

Research regarding students’ claims that subject mastery
is more difficult in ITA classes than in American TA classes
is sparse, although there appears to be a correlation between
students' performance and the ITA’s oral proficiency. Jacobs
(1988) concluded that the oral proficiency of the ITA made a
difference in the performance of the students, although her
attempt to correlate students’ end of semester test outcomes
with the nationality of the teaching assistant proved
inconclusive.

Research by Williams and Marenghi suggest that ITAs’
perceived unintelligibility actually may be the result of
native speaker inability to process nonnative speaker speech.
In an examination of the planned and unplanned speech of
Mandarin and Kbrean speaking teaching assistants, Williams
(1990) found more marking in the planned speech of the ITAs
than in their unplanned speech; she also found, however, that
native speakers were understood better than the nonnative
speakers even though they did not make more use of discourse
markers. Marenghi (1986) compared the questions asked by
international and.American students in a linguistics lecture.
She found that clipped forms of questions such as those
commonly used by American students were not understood when
used by international students.

Some researchers have looked.int0>other potential reasons
for the students' negative reactions to international teaching

assistants. IBailey (1982) found that students not sharing the

12

major field of study with the ITAs gave them a poorer end of
term evaluation than those students who shared a major field
with the international TA. Ainsworth (1986) and Sardokie—
Mensah (1991) both discuss the ITA/student relationship in
terms of understanding and responding to cultural differences
in behavior and expectations in the college classroom.
Finally, Bresnahan (1993) attempted to draw a relationship
between the negative messages American students receive
regarding international TAs and the attitudes of these
students toward ITAs. She determined, however, that while
positive messages regarding foreigners did have an impact on
the attitudes of Americans, negative messages did not.
Eisenstein (1983) and.Ryan (1983) both.correlated.the negative
judgments held by native speakers regarding non-native speaker
phonology, syntax, lexicon and intonation to judgments about
the non-native speaker's socio-economic class or ethnic group.

The other research notwithstanding, most of the concern
surrounding international teaching assistants has been focused
on the development of programs to help international teaching
assistants become more intelligible to American undergraduate
students, although a major complaint has been that second
language acquisition research has not sufficiently measured
the level of oral proficiency that should be attained by the
ITA in order to be considered an effective teacher (Ard 1987,
1989). Mellor (1988) offered suggestions on ways for ITAs to
help improve their spoken English with lttle or no extra time

spent in practice. Stevens (1989) suggested using drama to

13

improve segmental and suprasegmental features of ITA
pronunciation. Others are more hi-tech, offering suggestions
for the use of video (Axelson 1990) and computer-assisted
pronunciation work (Stenson 1992). Discipline-specific
research has been conducted in the language used by teachers
in chemistry and mathematics (Anderson-Hsieh 1990). Douglas
and. Myers (1989, 1990) have researched. the language of
chemistry lab teachers, offering an analysis of errors in the
language used by chemistry ITAs as well as suggestions for
improvement.

This study contributes to discourse level research such
as the NS-NNS research conducted by Tyler (1988 and 1992),
Blum-Kulka (1989), and Gumperz (1977) who consider discourse
level features such as the appropriate use of discourse
markers, the perception of speech acts such as requests, and
the interference of the NS system of conversational inference
as factors that contribute to communication problems.
Students’ Questions as Guides to Classroom Processes

The situation in which the ITAs and students find
themselves makes the achievement of quality education
difficult. Several questions present themselves:

1. What is the nature of the interaction between the
TAs and their students?

2. How does student participation in ITA classes
differ from that taking place in classes taught by
their American counterparts?

3. What, if any, conclusions can be drawn regarding
the strategies the students employ to understand
the ITAs’ presentation?

14

This study addresses these concerns by looking at the
questions students ask in class. Even though studies suggest
that as much as 80% of classroom interaction is initiated by
the teacher (Dillon 1990), an analySis of students’ questions
offers an insight into the students’ learning processes and
the instructor/student relationship> by' giving' information
regarding the success of the communication process.

Questions as Guides to Students' Thought Processes

As guides into the students’ learning processes,
questions allow the teacher to determine the point at which
students have arrived in their knowledge of the subject matter
and to adjust the teaching accordingly -- before formal
evaluation affects their grades. As Dillon (1986) notes, any
student question involves an underlying set of presuppositions
that gave rise to the question (p. 333). The experienced
teacher attends to these presuppositions first. If the
presuppositions are invalid, revealing some underlying
misunderstanding of the material, the specific question goes
unanswered in favor of remedying the error in. the
presupposition. If the presuppositions are found valid, the
question is answered, possibly giving rise to another
question.. Through these question/answer sequences, the
teacher follows the student’s thought processes.

Questions as Guides to the Nature of the TA-Student
Relationshi

The teacher’s treatment of student questions can affect

the teacher/student relationship because of the importance the

15

student places on the negotiation process. How well teachers
answer their students' questions may be determined by the
level of the teachers’ subject matter knowledge. .According to
a study conducted by Carlson (1988), subject matter knowledge
affected. teachers’ (a) length of responses, (b) factual
content of responses, (c) relating of material to students’
personal lives, (d) use of humor, and (e) use of metatalk.
Analyzing student questions effectively for indications of
student misconceptions and using that knowledge in teaching is
addressed by Dillon (1986), Flannery (1989), and at the
college level, Aldridge (1989). In a study conducted by the
ITA Program at Michigan State University, the second most
cited source of trouble for the students (after the TAs’
English) was the teachers' inability to explain problems
adequately (Rittenberg and Wieferich 1988). They indicated
that ITAs either discouraged clarification questions, or they
simply recopied the problem rather than address the student’s
specific concerns. These practices created frustration,
supporting the students' negative views of the process as a
whole, the ITAs in particular, and possibly their fears of
being able to master the material.

The number of student questions may signal the level of
involvement achieved by the students in class. If students
lose the courage to ask questions in class because of
inappropriate responses, their level of involvement with the
material decreases (Marzano 1989) and the communication

process breaks down. Dillon (1986) points out that if

16

students lose confidence in their ability to learn the
material in the context of the class, then their desire
lessens, the feeling of needing to know -- at least in that
situation -- is lessened (p. 336). Students could undertake
to master the material, if at all, through other means such as
friends or help room, rather than attendance in class. As the
students quit participating, perhaps even attending, the
teacher-student relationship is damaged. The teaching
assistant cannot determine the extent to which the students
have mastered the material, and therefore cannot adjust class
time accordingly. The communication process has been
undermined.

Pertinent Research

Pertinent classroom interaction research focuses mostly
on the teachers’ use of questions and those studies focusing
on student interaction tend to be limited to classes in
preschool to 12th grade. While, at the college level,
emotional and. educational maturity may account for some
differences in the interaction patterns, this research in
elementary and high school does offer some insight into the
functions of student-generated questions, their importance for
the students, and their role in curriculum planning.

Ralajthy (1984) holds that students can improve their
retention of expository materials through self-questioning.
Kenzie (1991) describes the development and improvement of
study skills of handicapped students through the use of

student-generated questions. Gillespie (1991) summarizes

17
other research into student-generated questions in secondary
level content area reading classes.

Several studies suggest that classroom atmosphere in
large part determines whether the students feel comfortable in
asking questions in class (Ortiz 1988). Pizzini (1991)
suggests that small-group, problem solving formats might allow
for more student-generated questions than teacher-directed
activities do. At the college level, shyness and gender have
been cited as factors. Beins (1988) describes how shy
students were encouraged to ask questions in freshman
psychology classes by being allowed to write them on paper.
Pearson (1990, 1991) found females asked fewer questions in
classes taught by males.

Finally, it has been suggested that the international
status of the TA has an effect on the types and number of
questions asked in class. Katchen (1984) studied student-
initiated question/answer sequences similar to those appearing
in this study. She found that in comparison with American
TAs, students asked fewer questions of the international
teaching assistants but asked more Yes-No questions of them.
Her conclusions imply that the students accommodated to the
situation by requiring less interaction from the ITAs in
class.

Much more work is needed at the college level to
determine the type of interaction that takes place between the
faculty and students in the various disciplines. This study

is a beginning look at the interaction in freshman

18
mathematics. Only after a complete examination is completed
will ITA trainers know the discourse standards in these
classes and really be able to assist the international
teaching assistants in meeting their students' needs and
expectations.

In summary, as students enter a freshman mathematics
course, they appear-to'be entering a novel situation, not only
because it departs so much from their high school experience,
but because they often face teachers whose language use and
classroom expectations differ from their own. The freshman
mathematics system in the Department of Mathematics is highly
structured.and tightly controlled, leaving little room for the
teaching assistants to learn their craft and adding fuel to
the complaints the students already had about the
international teaching assistants. The analysis of the
students’ use of questions and negotiation strategies serves
to define, in part, the kind of interaction taking place in
mathematics classes. From this it is possible to determine
how students negotiate content and meaning in classes where
communication is considered to be difficult, i.e. in those

taught by international teaching assistants.

Chapter 2: Theoretical Framework

Conversation Analysis
QuestiongAnswer Seggence Structure

This study follows along' the line of Sinclair and
Coulthard (1975), McHoul (1990) and others who use
conversational analysis as a basis for classroom interaction
research. The question/answer sequences presented here stem
firmrthe basic structural unit of the conversation, termed the
adjacency pair. The adjacency pair sets up the turn-taking
process by selecting the next speaker and arranging for
transference of the floor. It is two utterances long, each
one produced by separate successive speakers (Sacks 1972).
The first utterance belongs to a group of first pair parts and
requires as a response the second utterance, belonging to the
group of second pair parts. Examples of adjacency pairs are
Greeting-Greeting, Question-Answer, Complaint-Justification or
Apology.

Tsui (1989) proposes a basic conversational unit that is
tripartite, in which the initiator’s second turn may be
eliminated. This occurs in interaction where the two
participants are especially close or where the initiator does
not accept the response but does not wish to continue with a

disagreement, as in the following where A does not wish to

19

20

debate politics with B.

A: I hope Smith gets in.

B: I voted for Jones.

A: silence

Demand Ticket

Nofsinger (1975) expands the adjacency pair structure by
postulating a "demand ticket" which can (a) get the attention
of the intended partner, (b) give the speaker’s role to the
person who uttered it, (c) switch the speaker’s role from one
to another, and which (d) obligates the initiator to make some
statement and obligates the acceptor to listen (p. 3). This
demand ticket is a type of conversational "ticket," such as a
comment about the weather or the conversational partner’s
name, that Sacks postulated is used to open conversations.
It differs somewhat from Sacks’ ticket in that it is coercive
in returning the floor to the speaker:

(2.1) A: Yuh know something?

B: What?
C: It’s time for lunch. (Nofsinger p. 2)

In the above, person A’s "Yuh know something?" requires the
response "What?" from person B, which in turn requires the
return of the floor to person A.
Alignment Talk

Within these sequences other interaction sequences are

often inserted, serving to expand upon the initial utterance

(Schegloff 1972). For example,

(2.2) P1: How do I get to your place? (1)
P2: Where are you now? (2)
P1: I’m at home. (3)

E5: Well, take your road to . . . (4)

21

These embedded or insertion sequences can last a number of
turns before an answer is finally given. Linde and Labov
(1975) discuss the distribution of a proposition across more
than one turn in these sequences. In the above example, the
embedded sequence appears in lines 2-3. In order to give an
adequate answer, P2 needs clarification before giving the
answer.

Jefferson (1972) proposes the misapprehension-

clarification side sequence as a type of embedded sequence.

(2.3) P1: If Percy goes with Nixon, (1)
I’d sure like that. (2)

P2: Who? (3)

P1: Percy. That young fella that uh - (4)

his daughter was murdered. (1.0) (5)

P2: Oh ye:ah. Yeah. (6)

Lines 3—5 contain the embedded sequence in this case.
The misapprehension results in the question on line 3 and is
clarified in the answer beginning on line 4. Both of the
above are examples of alignment talk (Stokes and Hewitt 1976).
When conversations do not go as planned, people attempt to
determine why and how things have not worked out. They
attempt to sort out exactly where in the interaction the
misalignment occurred.

Repair

Conversational repair (under which category side
sequences fall) is one type of alignment talk. Repair can be
made to the proposition or to the surface structure and can be

handled in a number of ways (Schegloff 1972):

22

(2.4) self-initiate/ P1: Turn right . er . (1)
self-repair I mean left at the (2)
next corner. (3)
(2.5) self-initiate/ P1: Turn left . uh . (1)
other-repair I mean (2)
P2: right? (3)
P1: Yea. At the next (4)
corner. (5)
(2.6) other-initiate/ P1: Turn right at the (1)
self—repair next corner (2)
P2: Right? You sure? (3)
P1: I mean left at the (4)
next corner. Yea. (5)
(2.7) other-initiate/ P1: Turn right at the (1)
other-repair next corner. (2)
P2: You mean turn left (3)
at the next corner. (4)
P1: Yea. That's right. (5)
(2.8) third party P1: Turn right at the (1)
repair Shoprite. (2)
P2: Is that after the (3)
Sunoco? (4)
P1: What? (5)
P3: He wants to know (6)
where the Shoprite (7)
is. (8)
P1: Oh, yea. It's right (9)
there after the (10)
Sunoco Station. (11)

In (2.4), the speaker realizes a misstatement and corrects it.
In (2.5), the bearer fills in the correct word after the
speaker has initiated the repair. In (2.6) and (2.7), the
repair is initiated by the second person in the conversation
and carried out either by the original speaker (2.6) or the
originator of the repair (2.7). The last is an example of a
third party offering assistance to the other two principle
participants and was discussed by Schegloff as a type of
other-initiate/other-repair. Self-repair was considered to be

preferred over other-repair because of the greater frequency

23

of occurrence. Other-initiate repair generally yields self-
repair.

Accounts and Metatalk

In addition to repair, other types of alignment talk
pertinent to this study include the use of accounts and
metatalk such as formulations and framing devices. Accounts
include such structures as<disclaimers, apologies, excuses and
justifications (Potter and Wetherall 1987). Disclaimers are
pre-initiation utterances (pre-accounts) that are stated to

ward off an anticipated negative reaction. Statements such as

"I’m no sexist, but . . . " followed by a negative statement
about women would. fall into this category (p. 77).
Apologizing is self explanatory. Studies conducted by

Schlenker and Darby in 1981 indicate that apologies were the
preferred response to a situation where some offense was
given“ The more serious the transgression, the more elaborate
the apology that was expected (Potter and Wetherall 1987).
Excuses and justifications assign responsibility for an
action that was considered wrong or bad. The user of an
excuse admits that the action or previous utterance was wrong,
but denies responsibility by blaming some external agent for
the behavior. The user of justification does not deny
responsibility for the action or the utterance, but denies
that it was wrong in this particular situation. .An example of
an excuse for being late for work might be that the alarm did
not go off, while a justification for hitting someone's car in

the parking lot might be that the wronged person had parked

24
incorrectly (Potter and Wetherall 1987).

Other alignment talk includes the use of metatalk such as
formulations and linguistic framing devices. JHeritage and
Watson (1979) define formulations as the "practice of saying-
in-so-many-words-what-we-are-doing" (p. 124). That is,
formulations summarize, explicate, or furnish the gist of a
previous utterance in a conversation in progress, as in the
following interview excerpt (Ragan 1983):

(2.9) I: When are you available for employment? (1)

A: I’m available for employment uh in town (2)
I could start sooner because I wouldn’t (3)
have to move. (4)

I: Yeah. (5)

A: Out of town I could start as soon as (6)
June (7)

I: Okay. (8)

A: But it would be fine with me if I could (9)
start in the middle of July or the (10)
beginning of August. (11)

I: Want to take some time off, is that (12)

A: That’s right. I’ve been going to school (13)
for about five or six years and I'd (14)
like to take a little vacation before (15)
starting full time. (p. 162) (16)

The interviewer (I) uses the formulation on line 12 to
draw a conclusion about A's motive in giving a date of mid
July to begin work. According to Ragan, those in the power
situation, such as I above, use more formulations.

Linguistic framing devices are utterances that introduce
or clarify specific properties of speech. Utterances such as
"I just mean . . . " or "I said . . . " fit into this
category, as well as utterances that refer to the process
itself such as "That's it for now . . ." indicating that the

process has ended.

25
Classroom Interaction
QuestionZAnswer Seggence Structure
Exchange structures similar to the adjacency pair have
been found in classroom interaction, although generally as
they describe teacher—initiated interaction. Sinclair and
Coulthard (1975) postulate a tripartite structure:
(2.10) T: Initiation
St: Response
T: Feedback (p. 21)
The teacher asks a question, gets a response, and then
evaluates the response, as in the following.
(2.11) T: What is the capital of Michigan?
St: Lansing.
T: Right.
In some instances, however, such as lecture classes or during
explanations, students are not required to respond. This

would modify the above sequences in this manner:

(2.12) T: Initiation
St: (Response)

Berry (1981) suggests that a teacher's feedback.may not always
be required following a student's response, further modifying
the tripartite structure in this way:
(2.13) T: Initiation
St: (Response)
T: (Feedback)
Pupil-elicit exchanges were analyzed by Sinclair and

Coulthard (1975) to have the simple structure

(2.14) St: Initiation
T: Response

Berry (1981) also supports this analysis, discussing the

dipartite structure in terms of moves taking place between a

26
primary and seconday knower.

(2.15) St: What is the capital of Spain again?
T: Madrid.

While students, who are secondary knowers in the class
situation, might respond to the teacher’s answer with "Oh,"
it is not obligatory. They would not necessarily evaluate a
teacher's response.

Alignment Talk

Kasper (1985) adapted Schegloff's (1972) analysis of
repair use to classroom interactiona Using the repair
structure of (a) self-initiated/self—completed repair, (b)
self-initiated/other-completed repair, (c) other-
initiated/other-completed repair, and (d) other-
initiated/self-completed repair, she found that the use of
repair was dependent on the type of course in which the
interaction was taking place. The repair in language courses
was overwhelmingly to the learner's utterances. There were no
instances of self— or other-initiated/other completed repair
of the teachers’ utterances at all. Any self—initiation of
the teachers' utterances were self-completed. The preference
was for other-initiated/self-repair (p. 203).

In content courses, as in non-educational discourse, the
preference. was for self-initiated/self-completed repair by
both the teacher and learner. The preponderance of self-
repair conducted by the teacher made it unnecessary for other-
initiated repair of the teachers’ utterances (p. 213).

McHoul (1990) used the same repair schema to analyze the

27
use of repair in classroom talk. He found the preponderance
of the repair to be other-initiated/self-conducted repair on
the student’s talk. That is, the teacher not only initiated
and completed repair on the students' talk, but more often,
initiated repair and witheld correction, giving the student
space instead in which to make the correction.

Two studies have focused on how student signal non-
comprehension to the teacher. Kendrick and Darling (1990)
concluded that how students signalled incomprehensibility was
dependent on class size and class type. Jordan and Fuller
(1975) found two main strategies: a focused directive which
gives the teacher a yes/no choice or requests a specific short
answer, or a focused non-directed question which focuses the
teacher but does not require a specific answer. It is the
difference between a and b below:

(2.16) a. When you say that you’re talking about
World War One aren’t you? or How many
stages did you say there were in the Dewey

model?

b. I don’t understand what you mean by author
intention. (Kendrick p. 935)

In summary, this report expands upon already existing
research in conversational analysis by Schegloff (1972),
Sinclair and Coulthard (1975), Nofsinger (1975), Stokes and
Hewitt (1976), Heritage and Watson (1979), Ragan (1983), and
and in classroom interaction by Kasper (1985) and McHoul
(1990) to describe a particular classroom discourse pattern --
the student-initiated question/answer sequence -- and.provide

a possible framework for future analysis of classroom

28
interaction. At the same time it contributes to the overall
knowledge base regarding the functions of student questions at
the college level and the repair strategies employed to ensure

successful communication.

CHAPTER 3: Methodology

Subjects

The purpose of this study is to describe student
negotiation patterns in mathematics recitation sections. The
information was taken from interviews, departmental documents,
observation and audiotape transcript.

Preliminary interviews with the Director of Undergraduate
Studies and the administrative assistant yielded all of the
information regarding the structure of the department and its
policies for hiring and training of teaching assistants. The
names of prospective participants were chosen from lists of
current TAs furnished by the department and were approached
based on the following criteria:

1. All should be newly hired first- or second-term
teaching assistants. This was based on the assumption that
more complaints would arise from classes taught by newly
hired, less experienced TAs, and that these were the TAs who
most often met college freshmen in freshman mathematics.

2. All should be assigned to freshman mathematics
recitation sections. Newly hired TAs who passed the
Department of Mathematics interview were most often placed
into a freshman mathematics recitation section. TAs who were

assigned to their own section had substantial teaching

29

30
experience, either as a TA within the department or as a
teacher outside of the universityu .Additionally, the freshman
mathematics sequence was the most student-populated sequence
in the department. The assumption was that most of the
complaints arose from this course sequence due to the large
student population.

3. There should be an equal number of American and
Chinese teaching assistants. The largest ITA population in
the department was from China, and the Chinese had the
reputation.of being less proficient in oral English due to the
methods of language teaching employed in their native
countries.

The study was limited to six teaching assistants, three
American and three Chinese who met the above criteria, signed
consent forms (see Appendix C) and provided information of
their educational and professional backgrounds, goals, and
strengths and weaknesses. Their profiles appear in Appendix
D. Each TA's class was visited four times. The first was to
explain the study and obtain the students’ consent; the rest
were for observation and.audiotapingu .Any students who missed
the initial explanation received one during the subsequent
visits.

Observations

The observations were all scheduled at approximately the
same time during the second half of the term in order to
control to some extent for type of classroom activity.

Students received the results of tests in each of the TAs'

31
classes. Most TAs also reviewed for tests during one
observation. During one class, however, the last twenty
minutes was preserved for a quiz that only that TA (MATA 3)
gave.

Each of the participating TAs received code names at the
outset to assure anonymity. These code names consisted of an
M or F denoting the gender of the participant, ATA or ITA
denoting international status and a number assigned solely to
create a numerical order in which to list the participants.
Therefore, for example, FATA 1 indicated a female American TA
designated as number 1 and MITA 5 indicated a male
international TA designated as number 5.

The classrooms themselves were typical American
classrooms. Each room had seven to eight rows of
approximately eight. moveable desks each” The audiotape
equipment included three portable cassette tape recorders with
internal microphones. One was placed on the teacher’s table,
the microphone supplemented by a PZM conference microphone to
allow for maximum movement by the teaching assistant. The
other two recorders were placed on either side of the room,
approximately in row two seat two and row six seat six. This
guaranteed a complete class recording. The only microphones
used for these were the recorders' internal microphones. The
observations took place from two locations within the class:
in a front seat on one side and.a back seat on the other side.
This was to reduce any interference with the concentration of

any particular students in the class. During each visit, the

32
notes reflected attendance patterns, all the teacher and
student initiated interaction, and boardwork (see Appendix E
for a sample page of notes).

Three main reasons motivated the decision to observe and
audiotape rather than videotape. First, audiotaping provided
for accurate, complete data on the interaction between the
teaching assistant and the students and also allowed for a
more detailed linguistic analysis after the taping session.
Observation allowed for focus on non-verbal behavior and
boardwork during the class period.

Second, a primary concern was the comfort of the teaching
assistant and.the students, in order to limit the interruption
of the class’ routine. In order for videotaping to be
maximally effective, the class would have had to have been
moved temporarily to a classroom already set up for that
purpose, with equipment run by a person trained to do so.
Because the three observational visits per class accounted for
roughly one week’s worth of meeting times for each, it was not
only difficult to correlate the TAs’ schedules with the
schedule of availability of the room, but the disruption
caused to the class proceedings by relocating the students to
a new site would have been excessive. All of the TAs were
first year Ph.D.. students. None had .American teaching
experience, and only one had taught in his home country. None
were accustomed to being observed while teaching.

Another concern was the much discussed "observer’s

paradox" (Labov 1972). This is the term coined to explain the

33
phenomenon where the quality of the data is somehow affected
by the presence ofthe observer. During videotaping sessions
conducted by the staff of the International TA Program,
students tended to act self consciously in front of the
camera, even when notified that the teacher was the main
focus. Some turned.to see the reaction of the camera operator
or to see if something had been captured on tape. They also
whispered their answers and limited the number of questions
they asked in class. After class, there was a line of
students wishing to ask their questions individually and off
camera to the TA. Repeated experiences of this kind
contributed to the decision to keep intrusive equipment to a
minimum.
Analysis

The tape for each class was transcribed and reviewed
first to isolate the student-initiated exchanges. These
exchanges were considered to begin when a student made any
utterance directed at the teacher, either interrupting the
teacher’s explanation or in response to a request for
questions; they were considered to be complete when the
question or utterance was answered and resolved" The analyzed
data do not include questions and sequences such as (3.1) and
(3.2), both of which were considered to be TA-initiated

sequences.
(3.1) TA: . . . Next I want some of you to put your
solutions [approaches students pointing and

giving problems numbers].

St: What do you want me to do? Write it up there?
TA: Yea. Write up here and uh this one.

34

(3.2) [TA approaches to have student put a problem on

the board.]

St: I didn’t get it right.

TA: Ok.
The first appeared to be an insertion sequence with a student
attempting to repair an interaction initiated by the TA” This
differed from some of the interaction analyzed here because
this interaction was only between the TA and one student, as
opposed to being between the TA and the entire class with one
student making a request or assertion. The second appeared to
be a response to a non—verbal request made by the teacher.
Neither of them reflected student-initiated interaction.

Each of the 123 isolated student-initiated sequences
received a code number reflecting the code name of the TA and
the numerical order in which it appeared over the three day
observation period. For example, [FATA 1.1] indicates
sequence number one appearing in the classes taught by FATA 1,
while [MITA 5.1] refers to sequence number one in the classes
taught by MITA.5n These codes serve as a numbering system for
the sequences and appear at the end of each of the examples
cited in this report to identify the origin of the example.
The framework for the principal analysis was adapted from

the framework already existing in conversational analysis and
in classroom interaction research as discussed in Chapter 2.
Each sequence was coded for function as either a request-to-
do-problem (RP), assertion, request for confirmation or

information, or phatic language. The first utterance of the

35

sequence, excluding demand tickets and presequences determined
the category' assignments. 131 cases where ‘the student's
utterance alone did not indicate its function, the surrounding
context, including the TA’s answer and any subsequent
exchange, provided the additional clues. These sequences are
discussed in terms of the frequency of their appearance in the
data in Chapters 4 and 5.

A sequence was considered to be a request-to-do-problem
sequence if the student only asked for a problem from the
homework or test to be solved on the board, without any other
specific indications of trouble. These requests elicited
problem solutions that were performed for the entire class,
rather than one person.

Assertion sequences began with a student's statement of
opinion. They functioned.as corrections or disagreements. In
the initialstatement in the request sequences the student
asks either requests confirmation of an understanding or
answer or requests an explanation. Finally, phatic language
was defined as language used to preserve or change the
TA/student relationship. The jokes and challenges were placed
into this category. Jokes alerted the TA to the students’
attitude in class and eased tensions. The sequences
categorized as challenges contained questions that, because
of their nature, were difficult to answer. They often began
with a phrase such as "How am I supposed to know...?" These
questions, again, apparently alerted the TA of the students’

displeasure with some aspect of the course rather than request

36
any specific information or disagree with a particular answer
or process.
The framework for the analysis of the repair structures
was adapted from the repair organization suggested by
Schegloff (1972) and later used by Kasper (1985) and McHoul

(1990) to the following five part organization:

1. teacher-initiated/teacher repair: the teacher
initiated and completed repair on the student’s
utterance.

2. teacher-initiated/student repair: the teacher

initiated a repair on the student’s utterance, but
the student carried it out.

3. student-initiated/student repair: the student
initiated and conducted a repair to the teacher's
utterance.

4. student-initiated/teacher repair: the student

initiated a repair to the teacher’s utterance, but
the teacher carried the repair out.

5. third party repair: repair initiated or carried
out by a person other than the student involved in
the interaction.

Although the argument can be made that all student
assertions and requests are forms of repair, only the repair
structures that were embedded within the main sequences were
analyzed in this manner. This analysis gives information
regarding the negotiation necessary for the success of these
sequences, rather than for the success of the class
interaction as a whole.

The focus of this five component organization structure
is on other-repair, that is, repair conducted or initiated by

someone other than the speaker. The overwhelming preference

for self-initiated/self-completion of repair by both the

37
teacher and the student was assumed. In describing the
negotiation of this interaction, the interest was in the
aspects which caused trouble for the other person in the
interaction and how this trouble was managed.

The framework for the analysis of the other alignment
talk came directly from ‘the theoretical base which. was
discussed Chapter 2. The definition of formulation is the
same as that used by Heritage and Watson (1979); the
definition of linguistic framing devices is the same as that
used by Tyler (1988) in her study of NNS intelligibility to
NS.

Tabulations

Chapter 4 is devoted to the tabulation of the frequency
and function of student-initiated sequences and their embedded
structures. The totals for each feature appear in both raw
numbers and percentages wherever possible and are contrasted
in the American and international teaching assistant classes.
The initial results appear in the first three tables which
deal with the frequency and function of the sequences in
general. Table 4.2 gives the average number of sequences per
class and the percentage of sequences that functioned as
request-to-do-problems. This table separates the RP sequences
from the rest of the data. The references to the number of
sequences in the subsequent discussions do not include these
sequences. The frequency of use of the three main types of
sequences is computed in Table 4.3, and the frequency of use

of the four types of requests appears in Table 4.4.

38

The second principal analysis involved the interior
negotiation structures of these student-initiated sequences.
Based on the conversational analysis framework already
discussed, the embedded sequence analysis arose from the data.
These sequences were also examined in terms of function and
frequency of appearance in ATA and ITA classes. This
negotiation (or alignment talk) was further categorized into
repair (of utterance, reference, and content), accounts
(including disclaimers, justifications, and excuses), and
other metatalk (formulations and framing devices).

Four tables are devoted to the description of the
frequency and type of alignment talk used by the students and
TAs. Pertinent tabulations are again presented in averages
and percentages for ATAs and ITAs. Table 4.5 analyzes the
use of repair according'to the repair responsibility discussed
earlier. The instances of repair were counted and the totals
are presented in percentages out of the total number of
repairs.

The next two tables (Table 4.6 and Table 4.7) describe
the use of the three types of repair (to question, to
reference, and to content) by initiator. The ATA and ITA
totals appear in both raw numbers and percentages out of the
total number of their respective instances of repair.

The final tabulations reflect the use of other types of
alignment talk employed by the TA.and.the student (Table 4.8).
All of the figures appear in raw numbers with the percentages

of their appearance out of the total number of sequences.

39

A.secondary tabulation, discussed first, presented itself
as it became apparent that it may have a bearing on the data.
This was of the attendance patterns in each of the classes
observed. (see Table 14.1). The wide fluctuation. of 'the
attendance numbers in the data forced the discussion of the
relationship between attendance, the use of alignment talk,
and the students’ strategies for learning mathematics.

In summary, this is a qualitative study conducted to
determine differences in the patterns of student negotiation
in the classes of American and international teaching
assistants. There were six participating TAs, three American
and three Chinese. IEach were observed and audiotaped on three
separate occasions, yielding 18 hours of data. Two primary
analyses were conducted: the function and frequency of
appearance of student-initiated sequences and that of their
embedded alignment talk sequences. One secondary analysis is
discussed (attendance) because of its relevance for the rest
of the data. The framework for these analyses arose from the
collected data and is based on previous conversational
analysis work as discussed in Chapter 2. All the tabulations
are presented and discussed in the following chapters in both

raw numbers and percentages.

Chapter 4: Findings

In completing a description of student-initiated
negotiation patterns in mathematics recitation sections, two
components of the speech event are taken into account here.
The discussion begins with a account of the attendance
patterns. In this set of data, there is evidence that the
attendance may have a bearing on the amount and kind of
interaction taking place in class. Second, this chapter
includes an analysis of the appearance of interaction
sequences according to structure and function. Along with
this discussion is an examination of the use of embedded
sequences in terms of repair and other forms of alignment
talk. Finally, comparing the occurrences of these structures
in American TA and international TA classes illustrates the
difference in the kind of negotiation the students engaged in
in these classes.

Attendance

Mathematics recitation sections met three times per
week. Students who registered for a math lecture/recitation
class offered at a given time were assigned to a recitation
section, and their names appeared on the class list for that
section. Generally, the TAs did not take attendance in a

recitation, and although approximately 30 students were

40

41
assigned to a given section, the actual number of people
attending varied from 0 to over 30 as students chose alternate
sections to attend. Attendance patterns in a given class
might be illuminating; often if students are not attending a
particular section, they have either chosen not to participate
in the classroom content negotiation for that subject area or
they may' be "voting with their feet," 'expressing
dissatisfaction with a particular TA’s instructional method or
ability. Attendance levels, then, might provide clues as to
the level of communicative success in that section, the
relationship that has evolved between the teaching assistant
and the students, and the students' attitudes toward that
particular recitation section. They may also have a bearing
on the kind of interaction that takes place between the TA.and

the students who do attend (see Table 4.1).

Table 4.1

Average Attendance per TA over Three Day Observation

 

 

 

Observations
First Second 1 Third Average
ATA 18 21 20 19
ITA 14 12 12 12

 

In Table 4.1, attendance is given in averages for the

ATAs and ITAs for each day of the observation period. The

42
three American teaching assistants showed similar attendance
records, with an average attendance of 19 students. There was
wider' variation. in .attendance among ‘the students in 'the
international teaching assistants’ classes. The attendance
levels of MITA 5 and MITA 6 were substantially lower than the
rest of the TAs (MITA 5’s most attended class had only 9
students), making the attendance average for the ITA classes
only 12 students. While it is not possible to conclude from
these data that attendance is consistently lower in all ITA
classes, the lower attendance may have a bearing on the number
and function of student-initiated sequences.
Structure of Student-Initiated Seguences
The overall structure of student-initiated sequences

contains smaller sequences that serve to gain the floor for
the speaker, focus the direction of the conversation, repair
any misunderstandings or mitigate any conflict in the
interaction. The model that emerges from the collected data is
the following sequence.

demand ticket sequence

(presequence)

initiation

(embedded alignment sequences)

response

(resolution sequence)
Demand ticket

Because of the size of the group and the conflicting

demands on the teacher’s attention, students generally began

any interaction with an attention getter or demand ticket.

The demand ticket as it appears in the classroom is similar to

43
that proposed by Nofsinger (1975) in that it requires a
response from the teacher which in turn requires a further
move by the student. Demand tickets may be nonverbal as in a
raised hand or change in posture, or verbal as in
(4.1) St: Wait...When you graph that, what’d you (1)
how’d you plug it into the calculator? (2)
[FATA 1.16] (3)

In line 1 above, the wait served as a demand ticket and

 

was accompanied by a half—raised hand. In this case, it was
incorporated into the student’s main question with a pause
between it and.the sequence initiation. In others, the demand

tickets formed their own sequences.

(4.2) St: Say that again? (1)

TA: No solution. Because you see b should (2)

be bigger than a. From this graph b (3)

should be bigger than a. Right? (4)

St: Right. (5)

TA: All right. So we [see] there no (6)

solution. You have to check formally, (7)

uh, to to do that to do that thing (8)

right. Now. How to do it formally. (9)

St: I’m just confused because in the book (10)

it says b is less than h which is a (11)

sine beta then there is no solution. (12)
[MITA 5. 4]

The initial question the student asked in (4.2)
functioned as the demand ticket, getting the teacher’s
attention and switching the speaker’s role to the student.
The TA responded to the literal interpretation of the question
and repeated and clarified the last part of his explanation.
He would have been within normal discourse expectations just
to focus on the student and ask him what the area of confusion
was. As it was, the student asked his real question after the

explanation, in line 10.

(I)

44
Evidence for the existence of a nonverbal demand ticket
can be taken from the data where there is linguistic
confirmation of its existence.
(4.3) St: I just wanted to say that uh he’s going (1)
kinda quick through this stuff in lecture (2)
you know and we’re trying to keep up (3)
[writing it down] and not catching a lot (4)
of this stuff so I hope I learn a lot in (5)
here (laugh) SERIOUSLY! [FATA 1.2] (6)

(4.4) St: I just didn't know where you got it from. (1)
[MATA 2.4]

The students’ use of the word m; in the above two
examples indicated that they had both been acknowledged
nonverbally, in these cases because of raised hands. In other
cases, teacher acknowledgment may result from changes in
posture or facial expression.

Evidence that the demand ticket might be mandatory can be
seen in sequences where there were no apparent verbal or
nonverbal occurrences.

(4.5) St: You can also take the sixth root, can’t (1)

you? (2)

TA: Pardon me? [FITA 4.11] (3)

(4.6) St: What was the high? (1)
TA: What? [MITA 5.13] (2)

When demand tickets are not used, confusion may follow
and the students may be required to ask their questions a
second time.

The teacher is required to respond, verbally or
nonverbally, to the demand ticket, even if the student does
not always wait for the acknowledgment as in (4.1). When

students' hands go unacknowledged, the students feel

45

uncomfortable, just as the participant in.a conversation feels
discomfort if a greeting goes unacknowledged.
Preseggence

The presequence is an optional component of the exchange
sequence. Similar to the conversational presequence, it is
a way of opening an interaction. The presequence differs from
the demand ticket in that it not only gains the floor for the
student, but also (a) orients the teacher to a particular area
of misunderstanding that the student has or (b) allows the
student to solidify his or her question by talking through the
problem. The existence of this feature may explain the
confusion that teachers sometimes feel over what the students
are actually asking; iEvidence for the presequence can be seen

in the following two sequences observed in the data:

(4.7) St: I’m lost. What're you trying to figure (1)
out? (2)
TA: Ok. I’m trying to find out what c is. (3)
St: Why don’t you just use the law of sines (4)
to find angle a then use the law of (5)
cosines? [FATA 1.19] (6)
(4.8) St: I don't understand. Isn’t that a? (1)
TA: This is this is a and this angle from (2)
here to here approximately. Oh don’t let (3)
my drawing confuse you. Taking the two (4)
St: You said b didn't overlap a (5)
[MATA 3.27]

In each of the above sequences, the student's main
question came after the presequence was completed. In the
first example, the student had.either really lost the train of
a rather lengthy explanation or was expressing impatience with
it. First she established the focus of the discussion, then

offered her correction to the TA’s procedure. In the second,

46

the student first alerted the TA to the reference point in
question, then disagreed with the procedure.
Resolugion Seggence

The appearance of a resolution sequence, at the end of
the question/answer sequence, indicates that the problem that
had precipitated the initial question has been successfully
addressed. They are sequences in that there often is an
exchange of concluding remarks between the TA and student as
the explanation draws to a close, and possibly a ’helpful
hint’ or concluding remark from the TA after the student has
acknowledged understanding. In these data the sequences were
both verbal and nonverbal, sometimes containing only a check
for understanding by the TA (0k?) Examples of both nonverbal

and verbal resolutions appear in the following excerpts from

exchanges:

(4.9) St: You know the k minus one? Where’d you (1)
get the minus one right there? Is it (2)
one minus one? (3)
TA: K times one. Then thirty seven. Ok. (4)
(points and underlines answer on the (5)
board -- student nods and writes (6)
something down.) [FITA 4.10] (7)
(4.10) TA: Yea. You can do that. (1)
St: Comes out the same. (2)
TA: Yea. You can do that. X minus one, will (3)
be . . . Yea you can do that. You have (4)
answer? .Qk? That’s a good idea.. (5)

[FITA 4. 11]
(4.11) TA: Here? Yea. I will multiply by two plus i (l)
squared and this two together. I feel (2)
this will be easier. (3)
St: Oh. All right. (4)

TA: 9);? [FITA 4.16] (5)

47

The first example, (4.9), illustrates a nonverbal
acknowledgment on the part of the student in the form of a
nod. Other acknowledgments may be as subtle as a questioning
look from the TA. The second and third examples (4.10 and
4.11) were both.verbal resolution sequences. In (4.10), the TA
was checking for understanding; in (4.11) both were
participating verbally. Resolution sequences appeared in 44%
of the 102 question/answer sequences (not counting request-to-
do-problem sequences): the American TAs used them 42% of the
time and the international TAs 44% of the time.

Fre enc and T e of Se ence

Reggests-to-do-Problems

The results of the tabulations of frequency and types of
student-initiated Q/A sequences begin with Table 4.2. It
presents for both.the American (ATA) and the international TAs
(ITA) the average number of student initiated sequences
recorded per class and the percentage of these sequences that
were requests for the TA to explain specific mathematical
problems from the homework or quiz. The total number of
request-to-do-problem sequences was subtracted subsequently
from the rest of the data, and all other tabulations are
computed based on the balance or 102 sequences.

These request-to-do problem (RP) sequences were separated
from the data for two reasons. First, the initiators were
generally written on the board at the beginning of the class
period and therefore were not part of the verbal interaction

of the class. Those asked during class were not treated as

48
question/answer sequences by the TA nor by the student, but
rather as topic initiations. Sometimes the number of the
problem was added to the already existing list; other times
the problem was immediately discussed. Either way, once the
explanation was begun, it ceased to be an explanation given
for a particular student- ‘Unless a lengthy discussion evolved
regarding a certain process, the teacher did not check the
clarity of the explanation with the one who had requested the
problem. It became a class lesson. Second, because these
were class lessons, it is possible that the explanation
structure is different from the structure used to explain the
process to one student. It is possible, for example, to
explain a given mathematical problem simply by writing the
procedure on the board, saying very little aloud to the class.
A fine analysis of’ the difference in group explanation
structure and individual explanation structure is beyond the

scope of this study.

Table 4.2

Averagg Number of RP Seggences

 

Sequences

 

Average Request-to-do-Problem

 

ATA 24 13

ITA 17 24

 

49

The overall number of student-initiated.sequences for the
three observations was fairly uniform across five of the
teaching assistants. The range for those five classes
extended from 22 to 27 sequences, averaging 24 sequences per
class for the American TAs and 17 sequences per class for the
international TAs. The most obvious deviation was in the
class of MITA.6 where students initiated interaction only five
times during the entire observation period.

The percentage of interaction that was request-do-problem
varied widely among the teaching assistants, from 0 in the
classes of FATA 1 to 35% in the class of MITA 5. On the
average, 20% of the student questions were requests-to-do—
problems. The comparison of the American TAs with the
international TAs in Table 4.2 shows that interaction in the
classes of international TAs included a higher percentage of
requests to do routine problem solving than it did in the
American TAs’ classes (24% to 13%). This supports Katchen’s
(1984) findings that students ask more request-to-do-problem
questions of international TAs and suggests that the students
may have been accomodating the ITAs by giving them topic
initiations rather than negotiating one on one with them.

The balance of the student-initiated question/answer
sequences had.three main functions: for the student to assert
his or her own ideas regarding the mathematical concepts or
procedures (assertions), for the student to request further
information or clarification (requests), or for the student to

establish or change the TA/student relationship (phatic

 

50
language). The relative appearance of these sequences in the
ATA and ITA classes is presented in Table 4.3. Figures are
given for ATA and ITA classes, by function, in both raw
numbers (n) and percentages (%) out of their respective number

of sequences.

Table 4.3

Average Number of Seggences by Function

 

Function of Sequences

 

 

Assertions Requests Phatic

n % n % n %

ATA ' 18 28 44 69 2 3
ITA 3 8 31 82 4 10

 

Out of the 102 sequences that form the basis for this
analysis, the overwhelming majority (75) in both sets of data
were request sequences. Phatic or social language appeared
the least in both sets of data. ‘There were differences in the
use of the structures in the ATA and ITA classes. Students in
the ATA classes used assertions more often (28% of the 64 ATA
sequences to only 8% of the 38 ITA sequences) and the students
in the ITA classes used requests (82% to 69%) and phatic
language (10% to 3%) more often. These differences may be

attributed to the functions of these sequences.

 

51

Assertions

The assertion sequences found in the data fell into two
categories: corrections and disagreements. The determination
was based on the first utterance of the sequence, unless it
involved a verbal demand ticket or presequence, in which case
the first student utterance after the TA’s response to the
demand ticket/presequence was used.

Assergions functioning as corrections

(4.14)
FATA 1

1.5 Would the hypotenuse be the square root of two?

1.11 Isn’t that the other way around?

1.18 Wouldn’t the reason why eight, why it has to be
less than eight is cuz if it was longer than
eight and had two value it would [could] be on
this side, and then if it were longer than that it
would be on this side, and you can’t have it?

MATA 2

2.12 They wanted us to do it by graphics.

2.13 You’re not going to do forty three?

2.14 They have two pi.

2.15 Is that the square root of nine?

2.19 Wouldn’t that be a right triangle?

2.20 That's not on there.

MATA 3

3.15 You mean that’s forty five degrees.

FITA 4

4.12 We didn’t have to do that one.

These assertions took both interrogative ("Isn’t that the
other way around?") and declarative ("They wanted us to do it
by graphics.") form. The interrogatives generally were in the

form of negative questions, indicating perhaps a polite or

formal structure, except for number MATA 2.15 ("Is that the

52

square root of nine?"). The possible illocutionary force of
this utterance could have been (a) correction, (b) request for
clarification of writing on board, (c) confirmation of an
answer, or (d) expressed surprise over an unexpected answer.
Looking at the context for clarification, in this instance,
the indicated reference point on the board was not the square
root of nine. A paraphrase might be "Isn’t that supposed to
be the square root of nine?" This was taken as a correction
by the teacher ("I’m sorry.") and, it seems, was intended as
such by the student. There was no indication that the student
was experiencing any confusion and the reference was not to
any answer, but was to an intermediary step in the problem
solution.

All of these corrections addressed some utterance or
procedure made by the instructor; Either the TA was doing the
wrong problem, skipping a problem, adding wrong, or getting
the wrong answer. They were immediately recognized. as
corrections.

Assertions functioning as disagreements

(4.15)
FATA 1

1.19 Why don't you just use the law of sines to find
angle a then use the law of cosines?

MATA 2

2.8 Ok, the the one plus tan of x is equal to secant
two over four. It doesn't apply when everything’s
squared cuz I

2.16 It looks like you’re adding two pi to me.

2.18 Wouldn’t that be a right triangle?

2.21 You could've just took half of eighty nine degrees
to get that angle

53

MATA 3
3.11 In the middle one, wouldn’t x be squared, too,
then? Because
3.26 Just real quick. You can use a quadratic on trig
function, but you can’t distribute them? That
doesn't make much sense.
3.27 You said b didn't overlap a.
PITA 4
4.19 Couldn't you, um, couldn’t you just like graph it?
MITA 6
6.2 Wouldn’t it be from negative one to zero instead
of from negative infinity? Cuz how would you have
negative infinity if it's square
These utterances also took both interrogative and
declarative fornn Numbers MATA 3.11 and.MITA 6.2 are examples
of the interrogatives. Again, the use of "WbuldnLt it be
. . ," as a jpolite form appears. They' both can. be
paraphrased.easily into declaratives (MATA 3.11 "In the middle
one, x should be squared, too, then." and MITA 6.2 "It should
be from negative one to zero instead of from. negative
infinity.") while keeping the original illocutionary force.
The declaratives were often two utterances long (see
numbers MATA 2.8 and MATA 3.26). The first indicated the
reference, the second indicated the disagreement ("It doesn’t
apply . . . " and "It doesn’t make sense..") There were two
main referenceqpoints for these utterances: disagreements over
an explanation that the TA was giving that the student had
completed a different way (MATA 2.8 and MATA 2.22) and a

general disagreement over the procedure (FATA 1.9, MATA 2.16,

54

MATA 3.11, MATA 3.26, FITA 4.19, MITA 6.2).

Reggests
The vast majority of student initiated sequences were

requests. They fell into two main categories based on

preferred response: confirmation and request for information.
The preferred response for the requests for confirmation is
agreement which can be delivered with a one word answer. The

dispreferred response, however, requires an explanation. In

cases where the function of the utterance was not immediately
apparent, the context of the sequence was considered. There
were three types of information questions, those requesting

yes-no, one word answers, and longer or fuller explanations.

Reggests for confirmation
(4.16)
FATA 1

1.6 You don’t have to take it any farther than that do

you? Cuz [ ]

1.12 So you don’t want an exact answer?

1.13 You use radiens mode?

1.17 The ten was given?

1.20 If you just used the law of sines, would it be
wrong?

MATA 2

2.1 Can you just use eight then?

2.5 So on the test, if we just wrote down cotangent of
-theta and then

2.7 Can you always do that? When you have an equation
like that?

2.17 That's all I have to write is pi k?

MATA 3

3.8 That’s in feet?

3.12 Can you multiply one plus sine x over cosine x by

l ]? Cuz . . . It’ll cancel

out.

Oh.

55

3.13 Is that the answer then?

3.17 Would it work if you just put the top sixty minus
forty five. Where could just do one half minus
one over square root of two and just solve that?
Instead of doing that long

3.22 Would you start off with cosine two, you know?

FITA 4

4.4 Can you just factor it though?

4.7 So you do that with all the complex factors?

4.8 So are you allowed to do it like the first part or
no?

4.9 So if that was like on a test we could put x plus
one cubed and get it right?

4.11 You can also take the sixth root, can’t you?

4.16 Does it matter that you multiply those two
together as opposed to like uh two plus i squared

[ ]?
MITA 5
5.6 This one right here? When b is less that a then
there’s no x right?
5.9 To get side c, uh, you use alpha I mean a squared
plus b squared equals c squared?
5.18 Could we, uhm, just just take r r one times r two
and take the absolute value of 2 one
These utterances were requests for confirmation of (a) a
procedure that the student had completed that was different
from that demonstrated on the board (FATA 1.20, MATA 3.12,
MATA 3.17, MATA 3.22, FITA 4.4, FITA 4.8, FITA 4.11, FITA
4.16, MITA 5.6), (b) a method or answer that the student
wanted to confirm as correctly heard or understood (FATA 1.6,
FATA 1.13, FATA 1.17, MATA 2.1, MATA 3.8, MITA 5.9, MITA
5.18), and.(c) the completeness of the explanation (FATA.1.12,
MATA 2.5, MATA.2.7, MATA 2.17, MATA 3.13, FITA 4.7, FITA 4.9).

In each of the utterances, the students had methods or answers

in mind for which they wanted confirmation.

 

56

Reggests for Information

(4.17)

Yes/No Questions

FATA

1.23

MATA
2.23
MATA
3.1

MITA
5.5

5.11
5.23
MITA

6.3

1

When we [do the test] will he give us a diagram do
you think?

2

Do you think the test’ll have uh proofs of uh

3

Is it curved?

5

Is this on the test?

He curving any [of these]?

Is the eight root of fifty the same as the square
root of fifty to the one fourth?

6

Are you going to give us an example?

£3hort Answer Questions

FATA

NNI—‘bH

HPJF‘HPJ
NHO

2.6

MATA

3.14

1

What’re we supposed to put down for six?

Can you just give us the answer for number four?
What mode should we be in for our calculator?
What’s 3?

What number was that?

2

So when we solve these we going to use um
equations or you want graphics?

3

What’s the average?
What was two?

 

FITA

57

4

4.2 What was the range you used?
4.15 What section are we on?

MITA

5

5.7 What was the average?
5.13 What was the high?

5.14

FATA

1.3

H

.7

3.18
3.20
3.21
3.23
3.24

In our section or overall?

Longer Explanation

1

Is there a couple of different ways to go about
doing this? Cuz I didn’t do it that way.

How do you know that? Cosine theta equals cosine
two minus theta?

Explain where you get the three pi over two again?
Where’d you get the five pi over three again?
Why’d you choose radiens?

Do you graph, um, the top equation you have or do
you graph sine three x equal y?

Wait. When you graph that what’d you how’d you
plug it into the calculator?

2

I just didn’t know where you got it from.

I lost you over one point sine x one plus sine x
over sine x. There right there.

How do you know that, that one angle is forty five
degrees?

3

When you give partial credit, is there some
specific thing that you
Wait. How’d you get the five point two four.

[ ] where you came up with sixty over four. I
can understand how you got the fifteen after that.
You got pi over three times one fourth.

I got it, but where’d you get negative the square
root of [ ]?

I don’t understand how you got yea, that
right there. That’s the part I don’t understand.
How can you

Can you finish that one?

How do you know when you’re when you’ve made it to
the last step?

Why?

value.

3.25

FITA

4.1

4.3

4.6
4.10

4.17
4.18
4.20

58

On number fifty-nine, how come in the back of the
book it says it’s the arctangent of fifty over i?
I just seem to get tangent of fifty over i.

4

Um. For my graph on number fifteen, I have two
places greater than five hundred and you only put
only one point down.

I don’t understand how you used you used ten not
twelve.

I don’t understand what you’re doing over there.
You know the k minus one? Where'd you get the
minus one right there? Is it one minus one?

How’s there two complex?

How'd you know when it’ll be a double double root?
What if it was degree three?

4.23 Why’d you use the five times? Five times thirty
five?

MITA 5

5.4 I'm just confused because in the book it says b

5.19

5.20

less than h which is a sine beta. Then there is
no solution.

On number twenty nine, I got the right answer, but
I don’t understand why the second one is square
root of thirteen and the cosine of what angle
equals two k pi.

What would you do if it said find exactly theta?

5.22 Ok. Now graph the now graph the root.
MITA 6
6.1 How did you get the one third?

These questions were divided into three main categories:

those seeking a yes/no answer,
requesting longer explanations.
questions.

information such as a certain number,

In many instances

a one word answer, and those
There were seven yes/no
The short answer requests focused on specific
range, or variable

(see numbers MATA 2.6, FITA 4.8,

MITA 5.23) the TA was given a choice of answers ("Do you want
x or y?”) stressing the interest in a short answer.

The balance of the questions were requests for longer

59
explanations. They asked how to, what if, and why. Table 4.4
contains a comparison of the kinds of requests made in

American and international TA classes.

Table 4.4

Types of Regpests

 

Types of Requests

 

 

Confirm Yes/No SA LA

n % n % n % n %
ATA 14 32 3 7 8 18 19 43
ITA 9 29 4 13 5 16 13 42

 

Out of the 75 request sequences in these data, 44
occurred in the ATA classes and 31 in the ITA. Table 4.4
contains the breakdown of the requests based on the students’
preferred responses: confirmation, yes/no, short answer (SA)
and long answer (LA). The figures are given for ATAs and ITAs
in both raw numbers (n) and percentages (%) out of their
respective number of requests. By far the type of request
appearing most frequently was the request for longer
explanations (LA). Nearly half of the requests in both the
ATA and ITA classes were of this type. The next most widely
used type of request was the request for confirmation, which
appeared in nearly 1/3 of the request sequences for both the

ITA. and .ATA. classes. These data do not support the

60

hypothesis that students ask questions of the ITAs that
require shorter answers. They made fewer requests overall of
the ITAs, but within the request category, there was
surprising consistency in the frequency of appearance of the
different types. The one exception was the yes/no answer
request, which was used.with the ITAs in 13% of their requests
to only 7% with the ATAs. These questions dealt largely with
procedural matters such as how tests were graded and it may be
the case that students ask more procedural questions of ITAs
than they do of ATAs.
Phatic Language

The rest of the utterances appearing in the data were
assigned the comprehensive label of phatic language. This
classroom dialogue between the teacher and student serves a
social function to maintain or change the relative power
relationship that has been established“ In these data it took

the forms of complaints, jokes, and challenges.

Complaints

(4.19)
FATA 1
1.2 St: I just wanted to say that uh he’s going (1)
kinda quick through this stuff in (2)
lecture, you know, and we’re trying to (3)
keep up [writing it down] and not (4)

catching a lot of this stuff. So I hope (5)
I learn a lot in here. (laugh) SERIOUSLY(6)
TA: I hear everything in terms of sine and (7)

cosine. You see in this figure this (8)
tangent of u plus v formula? I that’ll (9)
show up in.... (10)

There was only one complaint over the three days. This

61

complaint was against the professor for the course.
Complaining in this manner in the recitation section served
two purposes. It (a) alerted the TA to the extent of the
student’s confusion, and. (b) put the responsibility for
clarifying the material on the TA. There was no sense that
the student expected that the professor would or could change
anything to meet the needs of this student. The TA addressed
both purposes. The answer that was elicited was a very basic
"rule of thumb." By responding in this manner, the TA
acknowledged that the student was deeply confused and accepted
responsibility for helping the student succeed. This created
an "ingroup" situation between the TA and recitation class
against the professor, putting the responsibility for the
student’s success on the TA, but also acknowledging that the
TA could fulfill the student’s expectations. It is not
unusual for the TA and student in recitation to form an
ingroup against the supervising professor (see Rounds 1987 for
a discussion of the use of certain linguistic features to form
an ingroup in mathematics recitations).

Challenges
(4.20)

FITA.4

4.5 Isn't there an easier way to do it than using u?

MITA 5

5.8 How did we how’re supposed to know we were

supposed to put that down. I mean it didn't
really say, you know, put both solutions down.

62
MITA 6
6.5 Now like on twenty one where you got the negative
one point seven one, you said to use the [trace
key]. Now

Three questions functioned as challenges in the data.
They were categorized as such because they were not
disagreements over a particular answer or procedure, but
rather were complaints over a seeming impossibility. In
number FITA 4.5, the student felt that the procedure proposed
was too difficult to apply. In the next two examples, the
students questioned that possibility of ever getting the
correct answer with the information given by the TA. In
number MITA 5.8, there was not enough information given on the
test for the student to succeed, and in MATA 6.5 the student
felt that using the calculator key recommended by the teaching
assistant would not yield the correct answer.

Challenges such as these are difficult for the TA to
address. They aren’t assertions; the student in number FITA
4.5, for example, did not know there was an easier way to
solve the problem. These were open ended questions that did
not indicate a specific area of misunderstanding but put the
responsibility on the TA for the student’s failure. They
challenge the TA's authority or ability to help the student
succeed. The international teaching assistants were the only

ones challenged in this manner.

63
M
(4.21)
MATA 2

2.2 St: Can we have that in writing?
TA: After the exam.

MITA 5

5.12 St: What's he gonna drop a test?
TA: (Laugh)

There were two jokes over the three days of observation.
The first occurred during the explanation of the upcoming
test. The TA had assured the class that a certain procedure
would be followed and the student asked for the assurance in
writing. He wanted to be sure the TA could be held to that
promise. Everyone in class treated this utterance as a joke,
yet it revealed the students’ level of anxiety over the test.
By answering the way he did, the TA was playing along with the
joke.

The second example was treated as a joke only by the TA.
This was because the question was rhetorical. The students
all knew the answer. This student had performed very poorly
on the test and the joke was a way of indicating his anxiety
over his overall grade in the course. None of the students
laughed at the question -- they snickered and murmured --
revealing that they all shared the anxiety and were perhaps
hoping that there might be some recourse available to them to
improve their grades. By laughing, the TA ignored the
sarcasm, avoided explaining to the students departmental

policy that they already knew, and avoided creating more of a

64
conflict than there already was.
Use of Embedded Segpences

The students' initiations were often responded to within
one turn or one turn followed by student acknowledgment.

(4.22) St: Wogld the hypotenuse be the square root of

TA: 5:2. Thank you. [FATA 1.5]

(4.23) St: Where'd you get the five pi over three
again?

TA: Uhm. This equals two pi minus pi over
three and that's six pi over three. It’s
always nice to connect this in terms of
three so I have six pi over three and it’s
really straightforward.

St: Oh. 0k. [FATA 1.9]

The number of turns taken to complete a given Q/A
sequence in these data ranged to 18. This suggests that
students did. negotiate with their TAs ‘toward successful
communication. The average number of turns per sequence per
TA.ranged from 3.7 to 6.2 with the overall average for the TAs
being 4.6 turns. Both the high and low end of the continuum
appeared in the international TA sequences with the average
number of turns per sequence nearly identical (4.1 to 4.5) in
the ATA and ITA classes. The range also crossed genders and
experience levelsu The simple two turn question/answer
sequence does not exist in the mathematics classroom.
Negotiation was consistently conducted between the student and
the teaching assistant regardless of the national status of
the teaching assistants. Example (4.24) on the following page

illustrates the level of complexity possible in student-TA

interaction sequences.

65

Within this larger sequence, there were five sequences,
one of which was embedded within other embedded sequences.
The main sequence began on line 1 with the student's vague
statement about being lost. As the TA attempted to address
the student’s concern, he began to walk the student through
the problem and initiated the the first embedded sequence
(lines 6-8). During the second embedded sequence of this type
(line 9), the student initiated.a sequence to clarify the TA's
question (lines 10-11) before she finally answered (line 12)
and that sequence was resolved (line 14). The student’s
question had not been answered, though, and she began a new
sequence (line 15) to clarify her reference point for the TA.
This sequence ended when the question was answered (line 33)

and the issue resolved (lines 34-36)

(4.24) St: I lost you over one point sine x one ( 1 )
plus sine x over sine x. There right (2)
there. (3)

TA: Ok. This piece right here sine x over (4)

sine squared x. There’s a sine of x (5)

in each of them. How many sine x's are (6)

in this one? (7)

St: One. (8)

TA: How many sine of x’s are in this one? (9)
St: How many whats? (10)
TA: Sine x's. (11)
St: Oh. Sine x. But how’d you (12)
[ (13)
TA: Right. (14)
St: But how'd you get rid of the one from (15)
the from the one before that? (16)
TA: Cuz we have a one plus sine x and we (17)

have a one plus sine down here. These (18)
two things are multiplied. This whole (19)

[quantity] is multiplied. (20)
St: Up. The one above. (21)
St,: The one plus sine x minus. Yea that (22)
one right there. (23)

St: How'd you get rid of the one plus sine (24)

66

x? (25)

TA: Oh. I really I really didn’t get rid of (26)
this one. I combined this one and (27)

this one. One minus cosine squared. (28)

[ (29)

St: Oh. To get the (30)
sine. (31)

[ (32)

TA: And this one came along for the ride. (33)
St: Oh, oh. Now I see. (34)
TA: All right. So I should be a little (35)
more explicit? [MATA 2.10] (36)

T es of Ne otiation

Several kinds of alignment talk appeared.in the classroom
interaction data. Many took the form of embedded
interactional repair sequences. Initiated.by either the TA.or
the student, these embedded sequences requested clarification
of an unheard or not understood utterance (question), board
reference (reference), or the mathematical concept or
procedure (content). Note the following examples of repair

initiated by the teaching assistant:

TA-initiated repair

 

(4.25) Repair to St: Can you finish that one? (1)
Question TA: app? (2)

St: Can you finish that one? (3)

TA: Can I? (4)

St: Yea. (5)

TA: 0k. Sure. This is a (completes (6)

problem). [MATA 3.21] ‘ (7)

(4.26) Repair to St: I just didn't know where you (1)
Reference got it from. (2)

TA: Where I got this thing from? (3)

St: Yea. (4)

TA: Pulled it out of the book. So (5)

we (writes on board) and (6)

that’s what we had to come up (7)
with. [MATA 2.4] (8)

67

(4.27) Repair to St: What're we supposed to put (1)
Content down for six? (2)
TA: On six? It was pi over three (3)

plus k pi. Cuz if the . . . (4)

the tangent has a period of (5)

uh pi. You put down two (6)

p_i_? (7)

St: No, I didn’t. (8)

TA: Yea, cuz I . . . You got (9)

credit for putting down the (10)

right answer and then there (11)

was four points. The half (12)

of it was for getting the (13)

right period of pi of (14)

tangent so . . . If (15)

there are questions of the (16)

[ ] [FATA 1.1 (17)

The above examples illustrate the kinds of TA—initiated
repair that occurred. In (4.25), the TA had begun to explain
part of a problem to aid in the explanation of the student-
requested problem. Once the concept had been illustrated, the
TA.intended.to discontinue the explanation of the new'problem.
As he dropped it in favor of a new unit, a student requested
that he continue the explanation. The TA hadn’t been ready
for the question. Perhaps because there had been no demand
ticket, or because the reference had.been unclear, or because
his mind had already left the current explanation, the TA
needed the student to repeat the question in order for him to
(understand what was being requested.

In example (4.26), the TA needed the student to indicate
the number on the board to which she was referring. There was
no obvious antecedent for her use of "it" so the TA did not
know at what point to begin the explanation.

Example (4.27) is one in which the TA attempted to

determine where in the student’s process for solving the

68

problem lay the misunderstanding. In this example, although
the student asked a fairly straighforward question (“What’re
we supposed to put down for six?"), by virtue of the fact that
the question was being asked, the TA knew that the student
must have had the wrong answer. In order to completely clear
up any misunderstanding, the TA needed to determine where the
student’s reasoning fell short, hence the question, "You put
down two pi?" A number of students had probably put that
answer down, so the TA was interested in addressing that
particular misunderstanding.

In negotiating' the content in class, students made
repairs regarding previous utterances (question), the board
reference (reference), and the course content (content). The

following are examples of student-initiated repair.

Student-Initiated Repair

(4.28) Repair to St: Do you graph, um, the top (1)
Question equation you have or do you (2)

graph sin three x equal y? (3)

TA: It’s your choice. I mean if (4)

[ (5)

St: Which (6)

one'd be simpler? (7)

TA: Hmm? (8)

St: Which one’d be more simple? (9)

TA: This is this is going to tell (10)
you where it crosses the axis, (11)
and this is going to give you (12)

intersecting points . . . . (13)
[FATA 1.15]

(4.29) Repair to St: You’re not going to do forty (1)
Reference three? (2)

TA: Oh. I need to do forty three? (3)

Thank you. Oh that’s a good (4)

one. Can I erase this one? (5)

Forty three. We have sine of (6)

the inverse tangent of x. So (7)
[ (8)

69

St: Um. (9)
Are you on eight point three? (10)

TA: Ha. Yes I know what I’m doing. (11)
Sine squared of x minus one. (12)
So how we going to solve (13)
this? [MATA 2 . 13]

(4.30) Repair to St: Would it work if you just put (1)
Content the top sixty minus forty five (2)

where could just do one half (3)

minus one over square root of (4)

 

 

two and just solve that? (5)
Instead of doing that long (6)
[ (7)
TA: Instead of (8)
splitting it apart? (9)
St: Yea. (10)
TA: I'm sorry. Wait. Co . . . I’m (11)
sorry. (12)
St: Well, the cosine of sixty is (13)
one half. Right? (14)
TA: Yea. (15)
St: The cosine of forty five is (16)
one over sqggre root of two. (17)
The cosine of (18)
[ (19)
TA: Are are you doing this? (20)
Are you using like a (21)

distributive property? Are you (22)
saying that this is equal to (23)

this? (24)
St: Right. (25)
TA: No. That’s not true. (26)
St: Why not? (27)
TA: Because this, um, no it’s not (28)
even true in this special (29)
case...[MATA 3.18] (30)

Similar to the repair initiated by the teaching assistants,
the students offered repairs to their questions, to the board
reference and to the content of the explanation. In (4.28)
above, the student did not receive an answer to her question.
She had wanted the TA to choose a method for the student to
follow. When the student was told it was a matter of choice,
she amended the question in an attempt to get a definite

answer .

70

In example (4.29), the teaching assistant had skipped a
listed homework problem, so the student asked if he were
intending to skip it. When he began the explanation, it was
of a problem with the same number in a different section of
the text. The student clarified the section reference and.the
explanation continued.

Example (4.30) is one in which.the student volunteered.to
walk the TA through her thought processes as the TA had been
so confused by the question as to not be able to answer at
all. The student led the TA through until they came to an
area that the TA identifies as problematic. The TA confirmed
what he considered the student’s process to be and cleared up

the misconceptions.

Table 4.5

Fregpency of Repair

 

Repair Responsibility

 

 

TA Student
initiated carried out initiated carried out
n % n % n % n %
ATA 25 55 11 24 20 45 34 76
ITA 24 75 10 31 8 25 22 69

 

Table 4.5 give the figures for the use of repair in both

raw numbers (n) and percentages (%) out of their respective

71
number of repairs in ATA and ITA classes, according to the
person assuming responsibility for the repair. There were 45
instances of the repair in the American TA sequences and 38
instances in the international TA sequences; the percentages
in the ATA data are out of 45 instances, and those in the ITA
data are out of 38 instances. In both the ATA and ITA data,
the tendency was for the teaching assistant to initiate the
repair (55% of the time for the ATAs and 75% of the time for
the ITAs), and for the student to carry out the repair (76% of
the time in the ATA classes and 69% of the time in ITA
classes) regardless of who initiated it. The ITAs, however,
both initiated and carried out a greater percentage of the
repair in their classes than their American counterparts did,
and the students both initiated and carried out a greater
percentage of the repair in the American classes than they did
in the international TA classes.

The appearance of ‘third. party or other repair' was
relatively rare in these data. It occurred in sequence
numbers MATA 2.10, FITA 4.5, and FITA 4.12 only. Because its
occurrence was so rare, it was difficult to assess whether it
was significant that two sequences appeared in the ITA data to
only one in the ATA data. It may be significant that pply_two
instances of third party repair occurred in the ITA data.
This is an area in need of further study.

Types of Repairs
The American and international teaching assistant classes

differed in the types of repairs initiated by the TAs and the

72
students. Table 4.6 gives the frequency figures for the three
types of repair (to question, reference, and content)
initiated by the TA. The figures appear in raw numbers and
percentages for the ATAs and ITAs, out of their respective

number of repair instances.

 

 

 

Table 4.6
Typgg of TA-initiated Repair
Types of Repair
Question Reference Content
n % n % n %
ATA 9 36 3 12 13 52
ITA 13 52 8 .33 3 9

 

An examination of types of repair shows substantial
differences the interaction in the ATA and ITA classes. Out
of the 25 instances of TA-initiated repair in the ATA classes,
52% was conducted to the content of the course. This means
that the TAs were walking the students through problems by
asking them questions. The second most frequent type of
repair was to the question (36%), where the TA clarified the
students’ questions. In the ITA classes, it appears that the
type of repair initiated least by the TA was repair to course

content (9%). The overwhelming amount of repair initiated was

 

73
to the students' questions (52%) or to the board reference
(33%).

Similarly, Table 4.7 gives the figures for the frequency
of appearance of the three types of student-initiated repair.
The figures for ATA and ITA classes again appear in raw
numbers (n) and percentages (%) out of their respective number
of repair instances. There were 20 instances of student-
initiated repair in American TA.classes and.8 instances in the
international TA classes; the percentages given are out of 20
and 8 respectively. In both sets of classes the majority of
the repair occurred to a previous utterance (question), with
the second most frequent type of repair being to course
content. In the American TA classes there was less repair to
question (60%) than in the international TA classes (88%) and

more repair to course content (25% to 12%).

Table 4.7

Types of Student-initiated Repair

 

Types of Repair

 

 

Question Reference Content
n % n % n %
ATA 12 60 3 15 5 25

ITA 7 88 0 0 1 12

 

 

74

The tabulations from Table 4.6 and Table 4.7 suggest that
the main concern in the ITA classes was with getting the
message across. There was less concern for this in the
American TA classes where there appeared to be more
negotiation of course content.

Usg of Other Alignment Talk

In these data, there were 186 instances of the use of other
alignment talk (in the forms of accounts, formulations and
framing devices) on the parts of the TA and the students.

In conversations, accounts, including disclaimers,
apologies, justifications, and excuses, are used to explain
why a certain behavior is at variance with expected or
preferred behavior. In the classes observed for this study,
the teachers used them when (a) giving more of an explanation
than the students felt was necessary, (b) being corrected by
the students, and (c) arguing for their methods of solving a
problem. They were used by students when (a) asking a
question they felt may not be of interest to the rest of the
class, and (b) arguing for their own solution to a problem.

Note the following examples of accounts used by the TAs:

Disclaimer
(4.31) St: Can you just give us the answer for ( 1 )
number four? (2)

TA: Number four. It’s, uh, cotangent. So, (3)
this is number four. Let me just...The (4)
cotangent of theta plus two pi equals (5)
cotangent of theta and you can use this (6)

as...let me just set it up. This is (7)
one over tan theta plus two pi. That (8)
works out, or have cosine theta plus (9)
two pi over sine theta plus two pi. (10)

Both of those methods work out. (11)

75

 

Apology
(4.32) St: Is that the square root of nine? (1)
TA: Oh. sorry. So this equals forty five (2)
sine of theta. [MATA 2.15] (3)
Justification

(4.33) St: Can you multiply one plus sine x over (1)
cosine x by [ ]? Cuz...Oh. It’ll (2)

cancel out. (3)

TA: Oh yea. It’s good that you’re thinking,(4)

Uhm. I have this written in my notes. (5)

So, I will finish this way. But you're (6)
right. In fact, that’s what I told the (7)

 

 

afternoon class. Go home and see if (8)
you can gind an easier way to do this. (9)
Nobody who did it, everybody pretty (10)
much did it this way. which surprised (11)
me cuz I thought I wgp...[MATA 3.12] (12)

 

In example (4.31), the TA mitigated.any possible conflict
caused by her decision to answer the student's question with
a longer explanation than requested by saying she was just
going to "set it up" (lines 4 and 7). The TA in (4.32)
apologized for making an error that the student caught. Two
justifications appear in (4.33). The TA justified his
explanation by saying both that he had previously worked
through the problem (lines 5-6, "I have this written in my
notes.") and.that no one in the other class could come up with
an easier solution (lines 7-12).

The TAs’ use of these accounts reflects the kind of
relationship being established in the recitation sections.
The use of accounts reduces the distance between the TA and
the student and lessens the TA’s power. It would be unlikely
that these accounts would be found to any degree in the speech

of the supervising professors, unless they, also, were trying

76
to establish closer relationships with the students.

In conversations, the hearer uses formulations to check
for understanding of the message and framing devices orient
the hearer to aspects of the speaker’s remarks that will
clarify the message. The teachers’ uses of formulations and
framing devices ensure success in the transmission of the
intended message in class. Formulations are used to summarize
or explicate an utterance previously stated by the student.

(4.34) St: Wouldn't the reason why eight, why it has (1)

to be less than eight is cuz if it was (2)
longer than eight and had two values (3)

it would [could] be on this side, and (4)
then if it were longer than that it would (5)

be on this side, and you can’t have it? (6)

TA: So you’re |going intol the definition, (7)
whole theory behind the triangle (8)

[ (9)
St: I’m just saying, you (10)
didn't know where it came from. Isn’t (11)
that why? . . . [FATA 1.18] (12)

The TA/s response in line 7 to the student's question
indicates where the TA feels she should. begin with an
explanation. The student is then free to agree with the TAs’
understanding of the question, or can then initiate a
clarification. In this case, the student follows it with a
formulation of her own (line 10), repeating the TA’s previous
utterance about not knowing why a certain answer was
necessary.

In non-educational discourse, the speaker’s use of
framing devices orients the hearer to aspects of the speaker’s
remarks, that will clarify the message, such as "I'm just

saying . . . '" or "I mean . . . ," are the most important

77

parts of the message, such as "This is important . . . " or
"Remember . . . ," or frame the process, such as "We have time
. . ’ or " . . . everything works out." The TA’s use of

framing devices are seen in the following excerpt.

(4.35) TA: Ok. I'm going to do y equals sine (1)
parenthesis three x close parenthesis (2)
plus cosine x, and it should work put (3)
like that. Like that. Cuz you got to (4)
put the [parenthesis around it], or you (5)
could do y equals sine three x colon y (6)
equals cosine x. Uhm. Let me make one (7)
other gpick note. We've got some time. (8)
I have no clue what this one looks like. (9)
This is, uh. Give an example. Sine three (10)
x sine squared of three x plus cosine x (11)
equals zero. You do the same thing. You (12)
graph it on your calculator as y equals (13)

parenthesis sine three x, um, hit the x y (14)
button, here, ok, um, ok. If I have the (15)

square here, put parenthesis around the (16)
sine or whatever, square it, that’s yea. (17)
That’s how they [ l on the calculator, (18)
and so that’s how you treat those. So, (19)

 

that's the simplest way of doing it. [ l (20)
and then everything works out. (pause) (21)
Other gpestions on that? [FATA 1.16] (22)

There are a number of framing devices in the above example.

 

Three refer to what the TA is going to say next: "I’m going
to.."; "Let me make one other quick note . . . "; "Give an
example . . ." The other three signal the end to a portion of
the explanation.

In summary, accounts, formulations, and linguistic
framing' devices are tools in 'the TA/student negotiation
process. The TA's use of accounts mitigates conflict in the
class and reduces the distance between the teaching assistant
and the student by changing the relative power held by the

parties involved. A balance needs to be struck in their use

78
because excessive use of this type of talk can cause the TA to
lose too much power and thus the respect of the students. Not
enough use, perhaps, makes the TA seem less approachable and
the content less negotiable.

By the same token, the TAs’ use of other alignment talk
such as formulations and framing devices facilitates the
students’ receipt of the message. With the formulations, the
TA tells the student the conclusion the TA has drawn from the
student’s questiona The student then indicates if ‘the
conclusion was correct or incorrect, and the TA's explanation
can begin. With the framing devices, the TA signals the
beginning and end of the explanations, the appearance of
examples, and the relative importance of the information.

Students also use accounts in class, in the form of
disclaimers, excuses and justifications.

Disclaimer

(4.36) St: This is my gpestion. So. if no one else (1)

 

wants to hear then. Can you just use (2)
eight then? (3)
TA: Right. (4)
St: Ok. . (5)
TA: Still, do it exactly. Well, first of all (6)
figure out how many feet per rotation. (7)

[MATA 2.1]

Excuse

(4.37) TA: . . . Sine of x, and then I have x plus (1)
y. So I replace this one with y and this (2)
one with y so I have cosine of this, sine (3)
of this. (4)
St: Ok. Ok. (5)
TA: Sine of this cosine of this. (6)
St: Ok. I was doing it differently. (7)
TA: That’s ok. Grill me. (8)
St: I was using the distributive property. (9)

I (10)

79

 

TA: Oh god. I was (11)
wondering . . . [MATA 3.21] (12)
Justification
(4.38) St: You can also take the sixth root, can’t (I)
you? (2)
TA: Pardon me? (3)
St: You can also take the sixth root? (4)
TA: You mean (5)
[ (6)
St: That’s how they did it in the other (7)
class and that's how (8)
TA: Yea. You can do that. (9)
St: Comes out the same. (10)
TA: Yea. You can do that. X minus one, will (11)
be . . . Yea you can do that. You have (12)

Answer? Ok? That’s good idea.[FITA 4.11] (13)

In (4.36), the student used a disclaimer to ward off any
potential dispreferred responses by claiming the question that
had been written on the board.as her own and giving permission
not to listen to it. Example (4.37) is an illustration of a
student giving an excuse for the question that she had asked.
Once the TA explained the problem to the student’s
satisfaction, the student offered an explanation for her
confusion to the TA. "I was using the distributive property."
This functioned as an excuse as there was an acknowledgment
that the TA had the right procedure, while offering an
explanation that seemed plausible for the way the student had
completed the problem.

Example (4.38) served to justify the student’s question
to the TA. Before the TA had the opportunity to answer the
question, the student justified her procedure with "That's how
they did it another other class." Justifications also appear

when the student is negotiating at length with the TA over the

80
correct procedure or answer as a means of saving face.

The use of formulations and framing devices in student
discourse again ensured that the speaker and hearer shared the
intended meaning of the message. With formulations, the
student told.the TA.what conclusion the student had drawn from

the TA's statement.

 

 

(4.39) TA: . . . I hear everything in terms of sine (1)
and cosine. You see in this figure this (2)
tangent of u plus v formula? I treat (3)
everything the easiest way sine and (4)
cosine so that’ll show up in (5)

[ (6)
St: So the object ( 7 )
of a lot of this stg_£ (8)
[ (9)
TA: Pardon? (10)
St: The object of a lot of this stuff pg (11)
[finding the easiest thing to do?] (12)
TA: Yea, that’s all you do. (13)
St: You pretty much have to memorize (14)
[FATA 1.2]

Here, the student summarized what he believed to be the main
point of the TA's answer. He told the TA what he understood
from her remarks and allowed her to agree or to clarify. In
this case, she agreed.

The use of framing devices served the same purpose for
the students as they did for the TA. They referred both to

the speech and to the process.

(4.40) TA: . . . To get secant of x you kind of (1)
took the square root. It’d be it would (2)

not be one plus tangent of x. (3)

St: What I’m saying is that I just assumed (4)

since you could do it with squared you (5)

could do it with not being squared, so I (6)
said one one plus tangent x equals secant (7)
x . . . [MATA 2.8] (8)

or

81

(4.41) St: I got the next step where you break down (1)
the sine of theta or cosine. I get that (2)

part. (3)
TA: Ok. Let me show you . . . Yea I was able (4)
to get it into uh a form similar to this (5)
and then I got bored with the problem (6)
Ok. So cosine two theta. . . [MATA 3.22] (7)

In (4.40) the student signalled that the TA had not understood
her reasons for choosing a particular method for solving the
problem. Her clarification was marked with "What I’m saying
is . . . ." In the second example, (4.41), the student was
referring to the process, indicating that the TA did not need
to continue with the problem solution. That the TA understood
this is reflected in his response "Ok. Let me show you .

." He understood but, for whatever reason, he decided to go
ahead with a fuller explanation.

In summary, the students' uses of accounts reflect the
type of relationship that exists between the students and the
TA. Too much use can indicate a problem in the student-TA
relationship as students become argumentative or whiny; not
enough. might signal a lack of the negotiation that is
reflective of a productive teacher-student relationship. By
the same token, student use of the other types of alignment
talk such as formulations and framing devices may be an
indication of the students' level of involvement as they make
an effort to understand and be understood by the teacher.

The teachers’ and students’ uses of alignment talk other
than repair differ in the classes of the American and

international teaching assistants (see Table 4.8).

 

 

82
Table 4.8

Use of Alignment Talk

 

Alignment Talk

 

 

 

Accounts Other Metatalk
n % n %
ATA
TA use 11 17 101 158
Std use 7 11 26 41
ITA
TA use 2 5 27 71
Std use 1 3 11 29
Table 4 . 8 summarizes the use of alignment talk . The

figures are given for the ATA and ITA classes in raw numbers
(n) and percentages (%) of appearance in their respective
number of sequences. The percentages in the ATA data are out
of 64 sequences; similarly, the percentages in the ITA data
are out of 38. The tabulations for the appearance of accounts
contains all the figures for disclaimers, apologies, excuses,
and justifications. The other metatalk includes formulations
and linguistic framing devices. Many sequences contained more
than one kind of alignment talk. In MATA 2.21, the TA used a
disclaimer and the student a justification in the course of
their negotiation, in MATA 3.12 both the TA and the student
used justifications, and.in FATA 1.15 there were six instances
of framing devices used by the TA in the course of her

explanation.

83

Overall, the teaching assistants made more use of both
kinds of alignment talk than the students, and the American
teaching assistants made more use of it than their
international counterparts. Similarly, the students in the
American classes used alignment talk more than the students in
the ITA classes.
Summary

This chapter analyzes student negotiation patterns in
terms of attendance, the function and frequency of the
student-initiation sequences, and function and frequency of
the sequences within those initiation sequences. Attendance
patterns differed in the classes taught by international
teaching assistants. Two of the ITAs had substantially lower
attendance figures (affecting one third of the classes
studied). Each was a class in which other communication
irregularities occurred.

The exchange sequences initiated by the student contained
a number of shorter sequences that combined.to form the larger
sequence. The overall structure appeared to include, in
addition to the traditional Q/A sequence, an obligatory demand
ticket/acknowledgment sequence, an optional presequence that
served to orient both the TA and the student to the
problematic area. to be discussed, a series of optional
negotiation sequences, and a final optional resolution
sequence that marked the completion of the explanation.

The main sequences of. the exchange functioned as

requests-to-do-problems, requests for information, assertions

84

or phatic language. Requests-to-do-problems (the type
appearing the most often) were separated from the rest of the
data because their nature suggests a TA/group interaction
rather than TA/student interaction that is beyond the scope of
this study. For similar reasons, the discussion of phatic
language is limited. The balance of the data was analyzed in
terms of assertions and requests. Students made more use of
requests in all the classes; in the American TA classes they
used assertions to correct or disagree with the TA.more often
than they did in the international TA classes. Differences
also appeared.in students’ use of requests. The international
TAs received.a higher percentage of yes/no requests than their
American counterparts.

Within the main initiation/response [sequence appeared
layers of embedded sequences, allowing the students and the
TAs to negotiate a successful conclusion to the interaction.
These embedded sequences serve as alignment talk and take the
form of conversational repair, disclaimers, apologies,
justifications, excuses, and framing devices that focus the
attention of the listener; These embedded sequences appear in
both.mmerican and international TA classes, with the number of
turns taken to complete an exchange sequence nearly identical
(4.1 to 4.5). Likewise, the initiation of these instances of
alignment talk is similar, with teaching assistants initiating
more use of all kinds of alignment talk than students and
students carrying out more of the repair. .A comparison of the

ATA and ITA classes, however, revealed that the international

85
TAs initiated and carried out more repair than the American
TAs and the students in the American TA classes initiated and
carried out more repair than their counterparts in ITA
classes.

Differences between American and international teaching
assistants also arose in the types of alignment talk used in
class. In the American TA classes, the TAs initiated more
repair to the course content than to the previous utterance or
to the board reference. In the ITA classes the TAs initiated
more repair to the previous utterance and to the board
reference than they did to course content. The students in
both the international and American TA classes initiated
repair to a previous utterance the majority of the time, but
in the American TA classes, they initiated repair to course
content approximately 25% of the time, to only 12% in the
international TA classes.

Similarly, there was a significant difference in the
appearance of other types of alignment talk including
disclaimers, justifications, excuses, formulations, and
framing devices in the two sets of data. Overall, the TAs
made more use of accounts and other metatalk than the
students, and the American TAs and the students in their
classes made more use of them than the international TAs and

their students did.

Chapter 5: Discussion of Findings

The mathematics recitation class functions as a
counterpart to the mathematics lecture. Students must
register for both the lecture class and the recitation in
order to receive credit for freshman mathematics. While the
lecture is a large and completely teacher-controlled
environment, the recitation is designed to help the students
get answers to the difficulties they face in attempting to
solve the homework problems. The relatively small classes (30
compared to the 200 or more attendance at the lecture), the
largely student-controlled problem solving format, and the use
of teaching assistants are supposed to contribute to an
atmosphere in which the students feel comfortable asking
difficult questions and negotiating with the instructor until
they understand the material. In the ideal recitation
section, as in any class, students would attend regularly and
participate fully, i.e. ask questions. Therefore the
recitation section is a suitable environment in which to study
student-initiated interaction patterns.

The Kind of Interaction in Mathematics Classes

Structure of Segpences

This study refutes the analysis of student-initiated

86

87

interaction sequences as being di- or even tri-partite
question/answer structures (Sinclair and Coulthard 1975), at
least in the college mathematics class. The exchange is
better described as a complex series of exchange sequences
embedded within each other. This analysis suggests the
existence of four possible exchange sequences (demand ticket,
presequence, negotiation, and resolution) in addition to the
main sequence. They are sequences in that each separate
initiation requires a response from the hearer. Two of these
sequences can be initiated and/or responded to verbally or
non-verbally: the demand ticket sequence and the resolution
sequence.

Of these five possible sequences, perhaps the most
deserving of further study is the presequence. It has two
functions: (a) to orient both the student and teacher to the
problematic area and (b) to allow the student to elicit an
explanation from the TA when the student does not know how to
word the question. It’s appearance may be a contributing
factor in some teaching assistants’ difficulty in
understanding what the student is asking.

The value in describing student-TA interaction in its
complexity lies in TA training; ‘New teachers need.to be aware
of the kind of interaction to expect in the classroom. For
the international TAs the reason is obvious: most come from
backgrounds that value a different kind of interaction in the
class. The Americans can benefit also, however, because even

though they have experienced this interaction, few know the

88

structure -- unless the interaction is really unexpected.
They have been attending to the communication itself rather
than to the type of discourse that occurs. Knowing how
students will address them will ease some of the tension the
TAs experience during the early stages of their professional
development.
Function of Segpences

Sinclair and Coulthard’s reference to the student-
initiations as "pupil-elicit exchanges" rather than
question/answer exchanges appears to be accurate. The
students’ initiations take a declarative structure as well as
interrogative. In addition to requesting information,
confirmation, or problem solutions, student interaction uses
a substantial amount of assertions and phatic language. The
students in this study corrected.or disagreed.with the TA, but
also tried to maintain or change the relationship by joking,
challenging, or complaining. In order to get a full
description of the interaction in any given class, the balance
that is achieved among the three kinds of initiations should
be taken into account.
Use of Negotiation

The students negotiated both the content and meaning
within the interaction sequences using a number of different
forms of alignment talk. The most frequently used alignment
talk was conversational repair. The students and the TAs
initiated and carried out three main types of repair: to a

previous utterance that had been unclear (question), to a

89

vague board reference (reference), and to a misunderstanding
of the course content itself (content). This study supports
McHoul’s (1990) findings that the teacher initiates most of
the repair allowing the students to carry it out. The
students also carried out most of the repair they initiated.

Of the various other types of alignment talk, the most
widely used was the framing device. The importance of NS use
of framing devices in NS-NNS interaction is the subject of
several studies (Tyler 1987, Williams 1990). There is
currently no agreement regarding their value in assisting
native speakers to understand NNS discourse; however,
researchers agree that more framing devices appear in NS
discourse. This study confirms those findings.

Disclaimers and other accounts appeared less frequently
in the data. The TAs used all of them more often than did
their students. The most likely purpose was to reduce the

distance between the TA and the student.

Comparison of ATA and ITA Classes
Structure of Segpences

No difference was noted in the structure of the complex
student-initiation sequences discussed above. Each of the
components appeared in the data of each of the classes. The
number of turns taken to resolve a given interaction was
surprisingly similar in the two sets of classes. It did not
take any longer to resolve a student’s question in an ITA

class than it did in an ATA class.

90
Function of Sequences

Of the three main functions taken by the student
initiations, the overwhelming majority'in.both sets of classes
was for request-to-do-problems. The ITA data contained more
of these, however, than the American TA data (including a 35%
appearance in the class of MITA 5). Since RP initiations are,
in essence, topic initiations, it may be that the students
accomodated to the international TAs by asking for the entire
problem to be solved.on the board rather than negotiate one on
one with the TA. This supports the findings of Arthur et al.
(1980) who found that in NS-NNS discourse the native speaker
attempts to lighten the interactional burden of the non-native
speaker.) This may have been the case in the class of MITA 5,
whose students complained of his having problems with English.

His profile appears in Appendix D. In routine problem
solving, less English is required and there is more non-verbal
support (in the form of boardwork) for the TAs.

Similarly, the students initiated less interaction
overall in the ITA classes than they did in the ATA classes,
only 38 sequences appeared there to the 64 in the ATA data.
In addition to the previously discussed ITA class which had
35% of the interaction being request-to-do-problems, another
ITA (MITA 6) class only had five questions total over the
three day period of observation. It is most likely that MITA
6 discouraged students’ questions. :n: was his first term
teaching, his English pronunciation was admittedly poor, and

he had six years of teaching experience under a different

91

educational philosophy. He wrote long theoretical texts on
the board and followed them with written explanations of
problem solutions. While writing, his back was turned toward
the students, with no pauses for interruptions until the
writing was completed. His request for questions was not
accompanied by eye contact. This style served him in two
ways: it compensated for his English, and it was possibly
closer to the style of teaching to which he was accustomed.
The students responded by taking copious notes and asking few
questions. Either the compensations strategies worked and
there were no questions, or the students felt ill at ease with
the interaction.

In the ATA data, the students used more assertions, while
the students in the ITA classes made more requests and used
more phatic language. The main distinction may between the
use of assertions and phatic language. The students corrected
and disagreed with the American TAs more, but they challenged
the international TAs more. This might be an indication of
the students’ attitude toward the TA or their impatience with
the interaction in the class.

The function of the students' requests (for confirmation
and information) was based on the kind of answer the students
wanted, hence the question "Can you jpgp give the answer to
number 4?" Based on this distinction, no really meaningful
differences emerged from these data. It may be that this
distinction is the wrong one. Even though the ITAs received

more yes/no questions, the yes/no questions in these data.were

92

almost exclusively procedural. It may be more appropriate to
ask what topics the questions involve. If the students
discuss complex issues less often with the ITAs, perhaps they
indeed are attempting to lighten the interactional burden as
proposed by Arthur et al. This is an area for further study.

The use of the various kinds of alignment talk was the
last difference noted in the data of American and
international teaching assistants. A41 of the TAs showed the
tendencies discussed above in their initiations of repair;
however, the repair that was initiated by the American TAs was
more balanced among repair to question, reference, and
content. Repair to content appeared 52% of the time. This
was not true in the international TA data where very little
repair was initiated toward.the content of the course (52% was
devoted to repair to question, instead). The American TAs,
then, were walking the students through the problems with.more
frequency than the international TAs, and the main concern
with the ITAs was in understanding what the students were
saying. This is an expected result for the ITAs for two
reasons: the ITAs report having more difficulty in
understanding the students’ English, and often the ITAs have
been advised to use repair strategies to "buy time” in
answering the students’ questions.

Another contrasting feature in the ATA and ITA data was
the students’ initiation and completion of repair. The
students in the American TA classes initiated and carried out

more repair than their counterparts. This is an unexpected

93
finding because in interaction that is viewed as problematic
the assumption is that more repair would be taking place.
This apparently was not the main strategy employed by the
students in the ITA classes. The repair the students did
initiate and carry out was overwhelmingly to question; there
was none to reference and only one to content. The students
appear to have been concerned largely the receipt of the
message rather than with negotiating their understanding of
the content. Again, this suggests that the students limited
their interaction to accomodate the ITAs.
Strategies Employed by Students

Because the numbers are small, any conclusions regarding
the students’ interactional strategies must be drawn
cautiously. In mathematics classes in general, students use
a variety of strategies including several functions of
initiations and repair.

In the ITA classes, the students’ strategies may include
limiting the interaction. This may be true particularly in
the ITA classes in which the communication patterns diverged
most from the American or expected patterns, such as those of
MITA 5 and MITA 6.

In both of these classes, the attendance numbers were
lower. The largest class size for MITA 5 was only 9, and the
largest for MITA 6 was 18, the lowest number among the
American TA classes. Bailey (1984) also draws a connection
between attendance and classroom interaction. The "mechanical

problem solver" in her typology had lower attendance numbers.

94

In classes such as these where attendance is not required,
students manipulate their schedules according to what they
perceive will benefit them the most. Attendance varies
according to the importance the planned activities hold for
the students. None of the TAs that were observed had full
attendance during any class period” If the interaction is too
difficult, then, for the students, the probability is that
some of the students will choose to get the information other
places.

The number of student initiations was lower in both of
those classes. Students asked for problem solutions 35% of
the time in the class of MITA 5 and only asked 5 questions
total in the class of MITA 6.

Within the interaction sequences, students in the ITA
classes appeared more concerned with just understanding the
message than arguing content. They corrected and disagreed
less with the ITAs, they made more repair to question, and
used fewer accounts to justify, excuse, or disclaim their
procedures.

Because the study was conducted late in the semester, it
is possible that limiting the interaction was not the
students’ initial strategy. It may be that previous attempts
at verbal accomodation did not work and so they opted out of
the communication process. It is also possible, at least in
the case of MITA.6, that the attending students were responded
to what they perceived to be the preferred interaction in the

class. The teacher generally sets the tone of the

95
communication for the course, so that if he discourages
interaction, for whatever reason, the students may respond
accordingly.

It is clear that the patterns of negotiation in
mathematics recitation classes differed depending on whether
the TA was American or international. Less negotiation took
place in the classes taught by international teaching
assistants: attendance was lower, there were more request-to-
do-problem sequences, students made fewer assertions to
correct or disagree with the ITAs, they asked more yes-no
questions, both.the ITAs and students initiated less alignment

talk.

Chapter 6: Conclusion

The national debate over the use of international
graduate students to teach American undergraduate students
continues, as first time international teachers with limited
preparation clash with first time college students who have
limited experience with diverse populations. Students
complain that they'have trouble'understanding’thelexplanations
given by ITAs, or that the explanations are inadequate and.the
students’ questions go unanswered. Students maintain that it
is more difficult to learn the material in class and that
their grades suffer as a result.

This study is an attempt to shed more light on the
differences in the students’ participation patterns in ITA.and
American classes. From the nature of the students’
complaints, it would be expected that the pattern of their
interaction in mathematics classes taught by ITAs should
contain more negotiation of both meaning and content as both
sides struggle to succeed in this communication process.
Indeed there is more negotiation of meaning in ITA classes,
even after the ITAs receive training and orientation, but less
negotiation of content. There still may be discrepancies
between the students’ expectations in these classes and the

actual interaction taking place.

96

97

As speech events, mathematics recitation classes function
in much the same way as non-educational discourse. As with
conversations, classroom interaction entails shared
intentionality and expectations on the parts of the
participants as well as negotiation. When entering into a
conversation, the participants share a sense of purpose for
that conversation. They expect, for example, that the other
person will listen and participate, and respond to questions
and statements appropriately, that is, according to the social
roles they assume. By the same token, teachers and students
should share a jpurpose for that interaction. They have
expectations of each other: they expect each other to listen
and participate; they expect each other to be clear in their
questions and statements; they expect the language usage to
reflect the social roles assumed.

Negotiation in a conversation takes place on two levels,
meaning and content. In mathematics recitation classes, as in
non-educational discourse, the students and teacher negotiate
both the meaning and the content of the interaction. The
examination of the number and types of questions, attendance
patterns, and use of alignment talk in the classroom offers a
preliminary account of the type of strategies that students
use to facilitate communication when their expectations are
not met.

Ramifications for ITA Training
.As new teachers from different educational philosophies,

the ITAs need to be informed of the complex nature of the

98
interaction that will take place in their classes. Special
attention should be paid to the identification of the
functions of students’ questions as well as to the
identification of the presequence in the students' discourse.
This will help reduce some of the anxiety the ITAs feel over
not knowing exactly what the student is asking.

The apparent importance of negotiation patterns in
classroom interaction suggests that ITA trainers may want to
include more practice in negotiation language in their
training programs. International teaching assistants should
be given practice explaining real mathematics problems to real
students, preferably one on cnmn The general practice has
been to emphasize microteaching at the class level, which
entails explaining a problem and asking for questions. Less
importance has been placed on fielding those questions due to
the difficulty in modelling real student negotiation
discourse.

The relatively high occurrence of repair to students’
questions indicates that ITAs need more practice processing
students’ questions quickly and analyzing them for indications
of misunderstanding. Currently, ITA.trainers propose the use
of utterance or reference repair as an interactional strategy
designed to give the ITA more time to formulate an answer in
English. Trainers may wish to provide other "buying time"
strategies for the international TAs such as the use of
formulations to check for understanding or the use of higher

order questions to address to students who need help

99

determining for themselves where their misunderstanding
occurs. Encouraging the ITA to state to the students that he
or she needs a moment to process an answer is preferrable to
Suggesting the current strategy that leads the students to
believe that language is more of’a problem than it is. Again,
more one on one interaction with students is needed to give
the ITAs the appropriate practice.

Argas for further study

This study proposes a complex structure for student-TA
sequences in college mathematics classes that diverges from
previous analyses. Each of the proposed components of this
structure is in need of further study. In particular, study
of the form and function of the presequence would help new
teachers in their attempt to understand. their students'
questions.

Additionally, this report suggests that both.students and
teachers initiate and complete repair to question, reference,
and content. The next logical step in this area of study is
the determination of the source of trouble that motivates the
repair as well as the structure of the repair sequence itself.

The corpus of data for this study comes from a very small
‘population, making it difficult to generalize these results to
include all international teaching assistants. What can be
concluded from these results is that in the classes taught by
these three international TAs the interaction varies
dramatically from that in the American TA classes, each class

in a somewhat different way. Given that these TAs are all

100
assigned to two recitation sections, this affects 180 American
students who may consider themselves to have had an unexpected
experience with an ITA. That alone is significant, but in
order to gain a more complete description of the variations
within the ITA classes, larger ITA populations need to be
studied.

Second, the data arising from the American TA classes
were considered the standard from which to set parameters for
normal student/TA interaction. This is not meant to imply
that the American TAs are better teachers. They are also
first year, inexperienced TAs. This study raises questions
regarding what is considered good teaching in the university
level recitation class. More background research should
be conducted in successful American classes in order to form
a clearer basis upon which to compare the interaction taking
place in ITA classrooms. After this background information
has been collected, then work with larger and more diverse
populations of students and international teaching assistants
will yield a more generalizable account of classroom

interaction patterns and the factors that contribute to them.

NOTES ON CHAPTERS

Circumstances have changed in the Department of Mathematics
since the data for this study were collected. For the
academic year 1993—1994, only nine international teaching
assistants were hired by the department. For the academic
year 1994-1995, only one new international student has been
offered an assistantship. .As a :result of the current
economic climate which is making it difficult for the
students to find employment, fewer students are leaving the
department, so fewer are being hired.

This information. has also changed since this study' was
initiated. This year, of the nine international teaching
assistants hired by the Department of Mathematics, seven
have teaching experience. The qualifications of the ITAs
have improved.

There is little real information substantiating this
impressionistic statement. In a preliminary survey
conducted in conjunction with this study, only three out of
seventy-nine students claimed to have had no experience with

people from other backgrounds. Students cited exchange
students at school and in their homes, as well as people in
their neighborhood. The students, then, may disagree with

the assumption that they do not know much about
international people.

According' to IMonoson. and. Thomas (1993) seventeen states
currently have legal requirements for ITA language training.
From their survey of two hundred forty institutions, it was
clear that in the absence of a legislated. mandate few
institutions initiated ITA training.

101

APPENDICES

APPENDIX A

SAMPLE DOCUMENTS FROM FRESHMAN MATH

COURSE OUTLINE
Sections 5-12

 

TEXT. Hestenes & Hill, Algebra and Trigonometry, 2nd Ed.

LECTURER. ; ‘ ~- .

LECTURES. _ . -._ ‘

OFFICE HOURS. Mo., Ue., Fr. 9:00-10:00 a.m.

RECITATIONS. Tu. and Th., according to the Schedule of Classes.

HELP. Help Room for " located in C-108, . _ ' .. will start
functioning on Tuesday, April 4; it will be open from 10:10 a.m. to
2:50 p.m., Monday through Thursday and from 10:10 a.m. to 12:30 p.m. on
Friday. When seeing your instructor or Help Room staff, you are expected
to have specific questions (and not to ask for an entire lecture or part
of one to be repeated to you). If your,question concerns a specific
exercise, be sure to bring your attempts to solve it. As immediately
before an exam the number of students seeking help is much higher than
at other times, you should try to ask for help as soon as you need it.
If you wish to become acquainted with more problems. you might try to
View tapes (with solved problems) that can be obtained in both the main
and math library (ask for tapes for MATH , then select the one with
the section desired; note that not all sections are available).

CALCULATOR. You need a "scientific calculator“ that has keys for log, in and
exp functions (trig functions as well if you plan to take .). It will
be to your advantage if your calculator also has an e key instead of an
INV key). YOU ARE FULLY RESPONSIBLE FOR HAVING A SCIENTIFIC CALCULATOR
FOR ALL EXAMS AND FOR KNONINC HOW TO USE IT. If you forget to bring your
calculator to an exam, you will have to work without it.

HDNENORK. On the sheet attached, you will find a day-by-day schedule of the
course as well as a list of problems from the text that you are expected
to solve on your own (solutions are at the end of the text). Solutions
of those problems are not expected to be turned in.

TESTS. You will have four one-hour tests and the final exam. Three of the
one-hour tests will be held during the recitation periods (see the
day-by-day schedule). The midterm one-hour test will be held on
Thursday, April 27, from 5:20 to 6:10 pxm. (locations will be
announced). The final exam will be held on Monday, June 5, from 10 a.m.
to 12 noon. There will be no make-ups for one-hour tests. A Justifiable
conflict with the time of the final exam will excuse you from that exam;
if such is the case, arrangements for the (common) make-up .final exam
have to be made through the Mathematics Department office located in
A-212 ' 'r“

GRADING. Every one-hour test counts 100 points and the final exam counts 200
points. You can determine your grade for any one-hour test by using the
following scale:

90 to 100 - 4.0 73 to 78 - 2.5 55 to S9 - 1.0
85 to 89 - 3.5 65 to 72 - 2.0 0 to S4 - 0.0
79 to 84 - 3.0 60 to 64‘- 1.5

102

Math

Thursday. 1/5
Friday, 1/6
Monday. 1/9
Tuesday, 1/10
Wednesday, 1/11
Thursday, 1/12
Friday, 1/13
Monday, 1/16
Tuesday, 1/17
Wednesday, 1/18
Thursday. 1/19

Friday, 1/20
[lined-v; 1/23
Tuesday. 1/24
vhdneeday, 1/25
..aradev. 1/26
Friday, 1/27
Monday. 1/30
Tuesday, 1/31
Wednesday, 2/1
Thursday. 2/2

103

M

6.7
6.8
6.8
7.1
7.2
7.3
7.4
7.5
8.1
Review
Test 1
6.7, 6.8, 7.1 - 7.5
8.2
8.3
8.5
8.6
8.6
8.7
8.8
8.8
Review
Review

W 7:00 - 8:30 Polo
Eben-*will be announced.

Monday. 2/ 13
Tuesday, 2/14
Wedneaday, 2/15
Thursdu. 2/16
Friday, 2/17
Monday, 2/20
Tuesday. 2/21

Nedneaday, 2/22
Thursday. 2/23
Friday, 2/24
Monday, 2/27
Tue-div. 2/28
Wednesday, 3/1
Thursday, 3/2
Friday, 3/3
m. 3/6
Md”. 3’7

9.6
9.6
9.7
9.8
9.8
Review
Test 3
9.1 - 9.8
10.1
10.2
11.1
11.2
12.1
12.2
12.3
12.3
Review
Test 4

1001. 10.2. 1101'
11.2. 12e1 '7 12.3

M, am
Thursday, 3/9
Friday, 3/10

Friday, 2/3 9.1
Monday, 2/6 9.2
Last day to dropltha coureelwith no grade
Tuesday. 2/7 9.3
Wedneeday, 2/8 9.4
Thursday. 2/9 9.4
Friday. 2/10 9.5

W
Roo- will be announced later in the term.

104
MATH DAY-BY-DAY SCHEDULE .

The following is the projected day-by-day schedule of the course, with
the exercises that you are expected to solve on your own (imediately after
the corresponding lecture):

Ue., March 29: 2.1 p. 52 7.9.11,19,21,23.27,33,39.48.52.55.

Fr., March 31: 2.2 p. 60 7.9.15,17,19,23,29,31,33,35,37,39.

No., April 3: 2.3 p. 69 1.5.7.9,15,19,23,27,33,37,39.41.45.49,51,
55, 59.

Ne., April 5: 2.4 p. 76 1,9,13,15,17,19,29,31,33,35,37.43.47.53,
67.

Fr., April 7: 2.5 p. 84 1.5.13,17,23,25,29,31,35,39,41,47,53,SS,
57. -

No., April 10: Review

Ue., April 12: 2.6 p. 90 3,9,19,21,25,29,31,35,37,41.

Th., April 13: TEST 1 (will cover sections 2.1 through 2.5: knowledge of
Chapter 1 is understood)

~Fr., April 14: 2.7 p. 96 1 - 35 odd, 39.
Ho., April 17: 3.1 p. 106 1.5.9.13,17,19,21,22,23,25,27,29,31.
3.2 p. 116 3,9,11,13,19 - 29 odd, 35 - 45 odd, 51,59.
61,63,65,71,72,73.
Tu., April 18: LAST DAY TO DROP BACK TO
He., April 19: 3.2 CONTINUED
Fr., April 21: 3.3 p. 124 1,5,7,9,13,21 - 29 odd, 33,37,39,41,43.
Ho., April 24: Review
He., April 26: 3.4 p. 140 3, 7 - 15 odd, 21,27.31,33.

Th., April 27: 5:20-6:10 p.m.: UNIFORM NIDTERH TEST (will concentrate on
sections 2.6-2.7 and 3.1-3.3, but will have
one or two problems from the old material)
(Locations will be announced)

Fr., April 28: 3.5 p. 148 3,7,15,19 - 31 odd, 35,41 - 51 odd, 55.

”o. . Hay 1: 3.6 p. 154 1,3.7.1s.17,21.25.27,29.
usrmvmnaopnmcmmsrvrmuocms.

SECTION

Review
2.1
2.2
2.3
2.4
2.5
2.6
2.7
3.1
3.2

3.3
3.4
3.5
3.6
3.7
4.1
4.4

5.1
5.2
5.4
5.5
5.7
6.1
6.2
6.3. 6.4

PAGE

44
52
60
69
76

90

106
116

124
140
148
154
165
177
198

218

233

248

256

263
269

105

MATH

H 0 H E H 0 R K

HOHEHORK

7.15,19,21.23.25.35,39.41.4S.47.49.51,55,56.
7,9,11,19.21.23.27.33.39.48.52.55.
7.9.15.17.19.23.29.31.33.35.37.39.
1.5.7.9.15.19.23.27,33,37.39.41.45.49.51.55.59.
1.9.13.15.i7.19.29.31.33.35.37.43.47.53.67.
1,5.13.17.23.25.29.31,35.39.41.47.53.55.57.
3.9,19.21.25.29.31.35.37.41.

1 - 35 odd. 39.
1.5.9.13.17.19.21.22.23.25.27.29.31.
3.9.11.13.19 - 29 odd. 35 - 45 odd.51.59.61.63.65.
71.72.73.

1.5.7.9.13.21 - 29 odd. 33.37.39.41.43.

3. 7 - 15 odd. 21.27.31.33.

3.7.15. 19 - 31 odd. 35. 41 - 51 odd. 55.
i.3.7,15.17.21.25.27.29.

1.3.5 13 - 51 odd. 5? - 65 odd.

1 - 43 odd.

1 - 29 odd: also equations of asymptotes of a
hyperbole.

1.3.5.9.13.25.27.29.35.37.41.

1 - 19 odd. 25.31.33.37.

1 - 55 odd.

3.5.7.11.15.17.19.

1 - 37 odd.

1.15.19.21.23.29.

5.7.9.11.21.25. 33 - 41 odd.

Solve problems 1-15 odd.19.25.31-41 odd from 6.3
by using the method from 6.4.

APPENDIX B

ANNOTATED INTERNATIONAL TA BIBLIOGRAPHY

APPENDIX B

ANNOTATED INTERNATIONAL TA BIBLIOGRAPHY
Complete Volumes

TESOL Newsletter 20, 1986
English for Specific Purposes 8(2), 1989
Innovative Higher Education 17(3), Spr 1993

Literature Reviews

Briggs, S., Hyon, S., Aldridge, P., & Swales, J. (1990). The
ITA: An annotated critical bibliography. Ann Arbor:
English Language Institute Publications. 323852.

Contains a critical annotation of 137 items in three
sections: 1) papers, presentations and reports, 2)
dissertations, 3) manuals, textbooks and videos.

Nelson,<3. (1990, Mardh). International teaching assistants:
A review of research. Paper presented at 24th Annual
TESOL Convention, San Francisco. ED321535

Reviews current state of research in ITA pronunciation,
effective ITA teaching behavior, and intercultural

communication.
Articles/Papers
Anderson-Hsieh, J. (1990). Teaching suprasegmentals to

international teaching assistants using field-specific
methods. English for Specific Purposes 9(3), 195—214.

Provides a rationale for teaching pronunciation using
cognitive-based field specific methods. Gives the
results of using such a method with Chinese and Korean
Chemistry teaching assistants.

Axelson, E., & Madden, C. (1990). Video-based materials for
communicative ITA training. IDEAL, _5_, 1-11. EJ465580.

Discusses the use of video-based materials to help ITAs
improve their use of certain classroom discourse
features.

Barnes, G. (1990, March). A bill of rights for international
teaching assistants. Paper presented at the 24th Annual
TESOL Convention, San Francisco. ED323792.

106

107

Lists 10 basic ITA rights and discusses how these rights
protect the ITA population. Included in the list: the
right to practice their own culture and teaching styles;
the right to a safe wrok environment; the right to be
recognized as an employee.

Boyd, F. (1989). Developing presentation skills: A
perspective derived from. professional education.
English for Specific Purposes, 8(2), 195-203. EJ392302.

Offers course materials and description of course design
of training program at Columbia University.

Byrd, P. (1991 December). Funding to attend graduate school
in the United States: An update. Paper presented at the
OSEAS European Conference, La Grande Motte, France.
ED350890.

Report discusses the types of assistantships available,
the funding in the form of tuition waiver or reduction,
and the requirements for English-proficiency testing.

Byrd, P. & Constantinides, J. (1992). The language of
teaching mathematics: Implications for training ITAs.
TESOL Quarterly 26(1), 163-167. EJ443057.

Reports on the language used by regular faculty in
mathematics.

Constantinides, J. (1989). ITA training programs. New
Directions for Teaching and Learning, 32, 71-77.

Review of existing ITA programs suggests that the key to
program success is the staff members who conduct the
program.

Council of Graduate Students in the United States, Washington,

DC. International graduate students: A guide for
qgaduate deansrffaculty and administrators. ED331418

Discusses academic, administrative, and social issues in
working with ITAs including standardized testing,
immigration and sponsorship.

Davies, C. (1989). Face—to-face with English speakers: An
advanced training class for international teaching
assistants. English for Specific PurposesL 8(2), 139-
153. EJ392298.

Describes the use of one-on-one and small group
interaction in the ITA program at University of Florida.
Also discusses the use of videotape analysis.

108

Dick, R. & Robinson, B. (1993, July). Oral English
proficiency requirements for ITAs in U.S. colleges and
universities: An issue in speech communication. Paper
presented at the Biennial Convention of the World
Communication Association. ED360653.

Suggests that more work should be done to improve ITAs
pedagogical skills and cultural knowledge in addition to
improving speaking skills.

Douglas, D. & Myers, C. (1989). TAs on TV: Demonstrating
communication strategies for international teaching
assistants. English for Specific Purposes 8(2), 169-

 

179. EJ392300.

Describes a technique for videotaping native and
international TAs to teach TAs about specific, definable
language skills.

Douglas, D. (Ed.). (1990). English language testing in U.S.
colleges and universities. Washington, D.C.: NAFSA.

Collection of essays and research reports on the testing
of ESL among foreign students in U.S. universities.
Includes addresses for use in obtaining information about
English language testing.

Gokcora, D. (1989 November). A. descriptive study of
communication and teaching strategies used by two types
of international teaching assistants at the University
of Minnesota and their cultural perceptions of teaching
and teachers. Paper presented at the National Conference
on the Training and Employment of Teaching Assistants,
Seattle. ED351730.

The results of a two part study at the University of
Minnesota examining the use of communication strategies
of ITAs and the cultural perceptions of teaching and
students. Results showed that TAs who encouraged
students to ask questions in class and stimulated
students to talk also asked more comprehension questions.
They also showed that students and ITAs both considered
reliability and encouragement to be the most important
concepts in defining a good teacher.

109

Gokcora, D. (1992 March). The SPEAK test: International
teaching assistants' and instructors’ affective
reactions. Paper presented at the Annual Meeting of the
American Association for Applied Linguistics, Seattle.
ED351731.

Results of a study conducted to determine the perceptions
of ITAs and instructors toward the SPEAK Test. Some
instructors voiced.concern.over lack Of face validity and
the difficulty of judging the ITAs’ overall
comprehensibility; There ‘was no difference in the
reactions of the ITAs.

Hill, L. (1992, October). Preparing international teaching
assistants: Intercultural training from a genetics
perspective. Paper presented at the Annual Meeting of

the Speech Communication Association, Chicago. ED354557 .

Describes an intercultural training program for ITAs at
the University of Oklahoma that is designed to accomodate
the diversity of the students. The framework for this
program is based on an analogy of the genetic process in
the combination of chromosomes.

Hoekje, B. & Williams, J. (1992). Communicative competence
and. the» dilemma. of international teaching' assistant
education. TESOL Quarterly. 2§(2), 243-69. EJ448698.

Suggests that ITAs would be better prepared for teaching
is communicative competence were stressed in ITA training
programs.

Hoekje, B. (1994). Authenticity in language testing:
evaluating spoken language tests for international
teaching assistants. TESOL Quarterly, 28(1), 103-126.

Jain, N. (1988, February). International teaching assistant
training seminar at Arizona State University. Paper
presented at Annual Meeting of the Western Speech
Communication Association, San Diego. ED297605.

Description of ASU’s ITA training program. Class meets 3
hours once a week and focuses on language improvement,
cultural issues and teaching strategies.

Johncock, P. (1991). International teaching assistants tests
and testing policies at U.S. Universities. College and
University, §§(3), 129-137. EJ27341.

Gives the result of a survey of 100 universities
regarding' test types, cut-off scores, processes for
testing language proficiency and performance of
international teaching assistants.

110

Kaplan, R. (1989). The life and times of ITA programs.
English for Specific Purposes, 8(2), 109-124. EJ392296.

Discusses the basis of complaints against ITAs and
describes the ITA program at the University of Southern
California.

Monoson, P., & Thomas, C. (1993). Oral English proficiency
policies for faculty in‘U.S. higher education” RevieW'of
Higher Education, l§(2), 127-140. EJ457693.

A survey of 240 institutions indicated that without a
state mandate, institutions typically did not develop
policies to certify the language proficiency of faculty.
Seventeen states have such mandates.

Nyquist, J. (Ed.). (1991). Preparing the professoriate to
teach. Selected readings in TA training. Dubuque, IA:
Kendall/Hunt Publishing Co. ED332635.

Collection of 56 papers devoted to the training of

 

 

teaching assistants. One section devoted to ITA
training.

Rice, D. (1979). A description. of a model program. for
orienting the new foreign teaching assistant. Washington,
DC: TESOL.
Describes a model orientation program with three
components: 1) oral/aural - to help develop
communication skills, 2) techniques for increasing
reading comprehension, and 3) a cross-cultural
orientation to the US university system. Includes a

sample course outline.

Rubin, D. (1993). The other half of international teaching
assistant training: Classroom communication workshops
for international students. Innovative Higher Education,
1113), 183-193. EJ462784.

Designed to help the ITAs balance their roles as TA and
student. Focuses on communication for participative
learning and dealing with academic advisors.

Sardokie-Mensah, K. (1991). The international student as TA:
A beat from a foreign drummer. College Teaching, 39(3),
115-116. EJ431474.

Domestic and foreign students should interact more to
develop a reciprocal understanding of behaviors and
expectations.

lll

Sequeiro, D. & Costantino, M. (1989). Issues in ITA training
programs. New Directions for Teaching and Learning, 39,
79-86. EJ396824.

All aspects of ITA training are discussed, including
course versus ongoing training programs, the ITA as
employee or visiting scholar, and the ITA.as a teacher of
minority students.

Smith, J. (1989). Topic and variation in ITA oral
proficiency: SPEAK and field—specific oral tests.
English for Specific Purposes, 812), 155—167. EJ392299.

An analysis of the performance of 38 ITAs on field

specific and general topic SPEAK tests. ffiue pass or

fail recommendations for 8 of the TAs depended on which
test was used.

Smith, K. (1993). A case study on the successful development
of' an, international teaching' assistant“ Innovative
Higher Education. 11(3), 149-163. EJ462782.

Case study of one successful ITA, focuses on the process
of becoming an effective instructor in an undergraduate
classroom. Emphasizes the need to clarify the student's
individual linguistic, cultural, social and professional

goals.
Smith, K. & Simpson, R. (1993). Becoming successful as an
international teaching assistant. Review of Higher

 

Education. 1_6_(6), 483-497. EJ469058.

This is a multicase study that suggests that the ability
to redefine personal goals to become more compatible with
department imposed conditions is a factor in having a
successful experience as a teaching assistant.

Smith, R. (1992). Crossing' pedagogical oceans:
International teaching assistants in U.S. undergraduate
education. ED358810.

Discusses the problem of using, training and assessing
the English of international TAs. Suggests areas for
more communication research. 141 pages.

Stenson, N. (1992). The effectiveness of computer-assisted
pronunication training. CALICO Journal, 9(4), 5-19.
EJ464135.

112

Examination of the effectiveness of the IBM SpeechViewer
indicated that while the students and instructors had

enthusiasm for it as an instructional tool, the
pronunciation of ITAs did not significantly improve with
its use.

Stevens, S. (1989). A dramatic approach to improving the

intelligibiligy of ITAs. English for Specific Purposes
.§(2), 181—194. EJ392301.

Describes the drama based approach to communicative
competence at the University of Delaware.

Thomas, C. (1993). Oral English language proficiency of ITAs:
Policy, implementation and contributing factors.
Innovative Higher Education, 11(3), 195-209. EJ462785.

Discusses the reponse of institutions to state mandates
for oral language proficiency testing.

Torkelson, K. (1992 March). Using imagination to encourage
ITAs to take risks. Paper presented at the 26th Annual
TESOL Convention, Vancouver, Canada. ED349898.

Results of two personality tests administered to 35 Asian
TAs: over 65% appeared introverted and over 90%
preferred to work with concepts than with people. Gives
ramifications for ITA training programs

von Saal, D. (1988). A ‘University-wide assessment and
training program for international teaching assistants.
Journal of Agronomic Education, 11(2), 68—72. EJ382851.

Description of TA training program and the followup

program from the director’s and instructor's
perspectives.
Williams, J. (1990 March). Evaluating ITA preparation

programs: intensive versus concurrent. Paper presented
at the 24th Annual TESOL Convention, San Francisco.
ED332502.

Evaluation of the relative pedagogical and cost
effectiveness of ITA training programs conducted before
and within the academic year. Concluded that given
limited resources, a concurrent program is preferrable
with a possible addition of a limited summer course
geared toward social and cultural orientation.

Young, Richard. (1990). Curriculum renewal in training
programs for international teaching assistants.
ED317067.

113

Reviews the history of ITA progranldesign since the 19703
for changes in the system as a whole, in program purpose
and mission, in measures of the system’s performance, in
resource allocation, and in system boundaries.

Yule, G. & Hoffman, P. (1990). Predicting success for
international teaching assistants in U.S. universities.
TESOL Quarterly, 2412), 227-243. EJ416671.

Report of a study correlating TOEFL verbal scores of 233
ITAs with.their performance reports. ‘Those‘with.negative
reports had lower TOEFL verbal scores.

Yule, G. and Hoffman, P. (1993). Enlisting the help of U.S.
undergraduates in evaluating international teaching
assistants. TESOL Quarterly, 2112), 323—327. EJ468896.

Discusses the value of using groups of undergraduate
students to vote on the readiness of an ITA to assume
instructional duties. Includes an example of a formal
evaluation sheet.

APPENDIX C

TA AND STUDENT CONSENT FORMS

APPENDIX C

TA AND STUDENT CONSENT FORMS

TA Consent for Classroom Visitation

The goals and.procedures of my participation in the project on
classroom discourse have been fully explained to me.

I understand that:

The researcher will visit my class three times during the
quarter and audiotape the interaction.

All data collected will be kept strictly confidential. The
results of the study will not be released to any faculty
member in my academic or employing department. My identity
will be known only to the researcher; any identifying
information will be disguised in the final report.

The data collected will be used for Ms. Wieferich's
dissertation and may be used in articles, presentations and
instruction along with other data collected on classroom
discourse.

I am under no obligation to participate in this study; I have
the right to stop participation in the study at any time

and/or have any part of the research data deleted without
penalty.

PLEASE SIGN BELOW

I agree to participate in the study on classroom discourse.

NAME DATE

 

ADDRESS PHONE

 

114

115

STUDENT CONSENT FORM

The goals and procedures of the classroom discourse project
have been fully explained to me. ‘

I understand that
I do not have to participate in this study.

The class will be visited and audiotaped three times. The
focus of the study will be on the patterns of communication,
not on any individual student. My identity will be completely
protected in the final report. Any identifying information
will be disguised.

All research data will be kept strictly confidential and will
not be used by my department or others to evaluate me as a
student.

I have the right to stop participating in the study at any
time and/or request that any part of the research data be
deleted from the study without penalty.

PLEASE SIGN BELOW

I agree to participate in the project on classroom discourse.

NAME DATE

ADDRESS PHONE

 

APPENDIX D

PROFILES OF PARTICIPATING TEACHING ASSISTANTS

APPENDIX D

PROFILES OF PARTICIPATING TEACHING ASSISTANTS

Bailey (1984) proposed a typology of teaching assistants based on
her observations of chemistry and physics TAs. She arrived at five
types: active unintelligible, mechanical problem solver,
knowledgeable helper, entertaining allies, and inspiring
cheerleaders. While she cautions that not all TAs fit into these
categories, it is useful to use these as a departure point from
which to profile the TAs participating in this study.

FATA l

FATA 1 was an enthusiastic American woman who made it a point
to know each of the students by name, and frequently used their
names during her explanations. She walked the class through the
problems, asking questions at each step of the solution process.
She knew which individuals performed well on particular quiz items
and asked them to help with the explanations. She corresponds to
the knowledgeable helper/casual friend type of TA.

This TA was popular with her students. Her attendance rate
was consistently high. There was no evidence of discomfort or
frustration among the students, although she had difficulty
explaining theory to the students. During the observation period
she had to abandon explanations several times and let the students
know that she found it difficult to discuss the reasons behind
certain procedures.

FATA 1 cited difficulty in explaining theoretical applications
as her main weakness; her strength was that she came from a
teaching'backgroundu She agreed to submit class. evaluations to me
and to the department, but failed to do so.

MATA 2

MATA 2 was a male, American TA who joked frequently with the
class. He walked students through explanations, asking questions
at each step of the solution process. He did extra work with the
students, giving review sheets for tests and conducting review
sessions during the evening before the test. He corresponds to the
inspiring cheerleader type of TA.

This TA was very popular with the students. He received the
highest TA evaluations of the six TAs. Twenty-three out of twenty-
five students rated.him.excellent or above averageu They stated he
was well prepared, helpful, enthusiastic, and a great teacher. He
cited as his strength his willingness to do extra for the students;
his weakness was his dwindling strength.-— he had taken on tOO‘mUCh
work.

116

117
MATA 3

MATA 3 was a male, American TA who also joked with the
students in class. His teaching style was interactive, involving
the students as often as possible. He also provided some extra
help for the students outside of class. His goal was to be an
inspiring cheerleader for the class.

This TA was very shy. EMIing his first term teaching, he
spoke mostly to the board. Even during the term of this study he
was nervous and his students mentioned this on the evaluations.

He received good evaluations from his class. Fifteen out of
19 evaluations rated him as above average or excellent. Students
cited helpfulness and preparedness as what they appreciated the
most.

FITA 4
FITA 4 was a native of China. Her teaching style was
interactive. She encouraged questions and had no difficulty with
explanations. She was friendly in class, but always maintained

authority. She did not do any work with them outside of class at
all. She joked only once during the observation period. This TA
does not correspond exactly with any of the TAs proposed by Bailey.
She was active, but spoke at an intelligible speed so that
communication was not hindered. She had good compensation
strategies, such as clear board work.

FITA 4 often made use of repair strategies when answering
students’ questions. A number of q/a sequences lasted more turns
because she was clarifying the students’ utterances. It appeared
that these clarifications were strategies she used to either buy
time to process an answer or to be sure of what the student was
asking.

Thirteen out of 25 students rated FITA 4 as above average or
excellent. They mentioned that she was difficult to understand,
but that she was knowledgeable and helpful.

118
MITA 5

This TA was a native of China. His style approaches Bailey’s
mechanical problem solver, although he attempted to become more
interactive. IHis chief behavior was solving problems on the board.
His board work was clear and easy to follow. His English was very
accented and the syntax was sometimes difficult to follow.

He had the lowest attendance of all the TAs. The class began
with as few as two students and the rest would come in throughout
the class period. It is likely that the attendance is linked to
his teaching style. The attending students were angry on the first
day of the observation and indicated that they felt he was poor in
English and unprepared in class.

No one rated this TA as above average. Out of the 13
evaluations completed, 8 rated him as average.

MITA 6

This TA was a native of Taiwan, and was the only TA with
teaching experience in his home country. He taught the recitation
by writing down long theoretical explanations on the board and then
reading and explaining them. He followed this by the customary
problem solving activities. He was very well prepared.

MITA 6 spoke very quietly and was very accented. He had
noticeable pronunication difficulties with technical terms and
numbers. He received the fewest number of questions of the six,
although the students indicated that they felt free to ask them.
His teaching style was most likely a combination of insecurity over
his speaking ability and cultural expectations.

Out of 11 evaluations he received at the end of the term, 3
rated him as above average, 7 as average. The students liked the
level of detail the TA had in his explanations and his helpfulness,
but indicated that he did not understand their questions.

APPENDIX E

SAMPLE PAGE OF NOTES

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APPENDIX F

LIST OF REPAIR SEQUENCES

TA Initiated
TA Repaired

TA Initiated
Student Repaired

APPENDIX F

List of Repair Sequences

Student Initiated
7 Student Repaired

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.27 TA Repair

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mmpwwwwwwwwwmmpt—IHH

mmtbobole-‘l-‘HH

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TRANSCRIPT
INTERACTION SEQUENCES PER TA

FATA 1

What’re we supposed to put down for six?

On six? It was pi over three plus k pi. Cuz if
the...the tangent has a period of uh pi. You put down
two pi?

No, I didn’t.

Yea, cuz I...You got credit for putting down the right
answer and then there was four points. The half of it
was for getting the right period of pi of tangent
so...If there are questions on the [ ]

I just wanted to say that uh he’s going kinda quick
through this stuff in lecture you know and we’re trying
to keep up [writing it down] and not catching a lot of
this stuff so I hope I learn a lot in here. (laugh)
SERIOUSLY!

I hear everything in terms of sine and cosine. You see
in this figure this tangent of u plus v formula? I
that’ll show up in

[
So the object of a lot of this stuff

[

Pardon?

The object of a lot of this stuff is [finding the easiest
thing to do?]

Yea, that's all you do.
You pretty much have to memorize

Well, if you know like two
formulas and the sine and cosine of addition ones,
you’re all set.

Really?

Yea. So it should work out.

121

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122

Is there a couple of different ways to go about doing
this? Cuz I didn’t do it that way.

Yea. I’m I’m sure there is. What did you do? Convert
this in terms of tangent? One over tangent

[

Yea, one
over tangent x, then I got

[

Ok, hang on a second. You have
one plus tan of x over one plus tan of x and ah you
clear. You combined this, right?

Yea, I got the common denominator.

One plus tan of x ah excuse me, tan x plus one over tan
of x and this is my division line. Yea, it’ll work the
same way cuz you get one plus tan of x over one times tan
of x over one plus tangent of x and they cancel.

Can you just give us the answer for number four?

Number four. It’s, uh, cotangent. So, this is number
four. Let me just..The cotangent of theta plus two pi
equals cotangent of theta and you can use this as...let
me just set it up. This is one over tan theta plus two
pi. That works out, or have cosine theta plus two pi
over sine theta plus two pi. Both of those methods work
out.

Would the hypotenuse be the square root of two?

Yea. Thank you. (completes problem)

You don't have to take it any farther than that do you?
Cuz [ ]

Actually, I do. Because I have to find all answers
between zero and two pi.

Oh, ok.

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123

How do you know that? Cosine theta equals cosine two
minus theta?

Ok that comes from

[
Why?

Hmmm?
Why’d you do that?

Ok. The...I might as well do both of them. Do you do
you know that sine of x equals sine of pi minus x?

Yes.

Ok. The same the same concept holds when I’m looking at
cosine. Write this a little bigger than usual.

Well, how come you didn't do it on number five [ ]?
Know what I mean?

Ok. Uhm. When when I draw my line, ok, I draw my line
through to find my points. Now I if I can [find]
original cosine original sine curve and I draw my line,
you know it only intersects once, so this is really the
same as uh one eighty minus pi over two. .And.I still get
still ninety degrees the same way. Now in this case,
uhm, this is pi, this is two pi. Now I sort of draw some
lines in through here, choose some points. NOW“With this
point and this point they're the same they’re the same
function. Because the distance from here to here is the

same distance as from here to here. And all I do, in
this case, is I'm just gonna [ ]. If that was sine
half I would have I would've done it. Uh. But

when...that’s why I went to the original cosine curve the
original sine curve. It only intersected once so I was
ok. Uhm. You can give decimal representations to
these. It’s not really inportant, cuz you you sort of
um you just add pi, so it you want to figure out you know
you can add one, three point one four to get the answer
but it’s not it’s not essential. Ok. Thirteen.

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124

Explain where you get the three pi over two again?
Please.

Ok. Here’s uhm, I want to know when it’s zero. Ok.
It's zero. This is ninety degrees. Ok. Cuz if there’s
a zero here so ninety plus two plus one eighty, cuz it's
also zero down here, gives me two seventy. But that's
represented as three pi over two cuz this is uh this here
is pi over two. This is pi, and this angle is pi plus pi
over two, so I get three pi over two. And so it’s uhm,
it’s easy to think of these in terms of breaking this up
into four parts on this graph. This middle is pi, this
is pi over two this is three pi over two. Ok. That's
general. Now see this I think is a much easier method.of
looking at it. Just breaking it up into four cuz you can
get sine the same type of a thing cuz it’s zero at pi,
the high point is pi over two the low point is at three
pi over two, and then it's at two pi so these are the two
types of methods. That that will help on this. Rm:me
I can see this a lot easier using this type of a of a
diagram, but it you look at the unit circle and the

[ ] the same. If anything is zero and ones this is
this is nice sort of. Ya gotta get used to it depending
on in high school you got it one or the other ways, and
I did it this way in high school. That’s why I'm very
familiar with this. Some people use the other one, so
different methods [ ] Can I go on?

Where’d you get the five pi over three again?

Uhm. This equals two pi minus pi over three and that’s
six pi over three. It's always nice to connect this in
terms of three. So I have six pi over three. And it’s
really straightforward.

Oh. Ok.

What mode should we be in for our calculator?

We're gonna be in degree mode. It’s gonna be mode four.

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125

Isn’t that the other way around?
Yea.
B and c?

You’re right. Sorry about that. See I want to put a c
here. Gamma. I just get confused with gamma. I'm
sorry. Uhm.

So you don’t want an exact answer?

Well, see when you trace, you’re gonna you’re gonna if
you’re close enough on tracing, because this is such a
small interval, that usually you’re fairly close on
stuff. You could find something like one point eight six
and maybe the correct answer is one point eight one you
know and you're gonna run out of if you spend a lot of
time on.a problem like this. On the test you’re going to
uh, you’re not gonna really have time to find out
precision in here and I’ve been fairly lenient on, I
mean, if it’s close, I've been giving credit because if
you can get this far, you know how to do it. And the
question’s really do you know how to solve this problem
is really what (names supervising professor) is looking
for. And so with a scale like this in radiens, you
should be able to come extremely close uh at just. You
know if you want to find out when there is negative and
when it’s positive but...I think I think I’d just graph
it. Because the times when you can't when you
substitute. You have something equal to a number, like
equal to a constant, but if you have a this equal x, I
mean I know how to substitute to do stuff, I’m going to
graph it. So unless it’s uh like something that we know
like sine of two x cosine of two x, stuff like that, uh,
I’m just not going to spend the time trying to do it
algebraically. Because if I have a cosine to the first,
anything to the first power means I can usually get a
quadratic or a cubic of that term without sines or
cosines or other things around and I can't do that
here. So I can’t do a nice simple substitution, so I'm
going to kind of quickly graph it. Try to get as close
as I can.

You use radiens mode?

Yep.

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126

Why’d you choose radiens?

Because this is, uh, these numbers are more accurate
because they’re smaller numbers, and.generally the answer
you want...This is going to be periodic and you’re going
to have...I’m not sure what the graph looks like. But if
I graph these two what’s going to happen is I [ ].
Out here’s the first line curve, the second one, the
third one. This answer this sine cosine [ ]. I'm
going to have a period I'm gonna have a periodic with pi
and so just radiens is a lot simpler.

Do you graph, um, the top equation you have or do you
graph sine three x equal y?

It’s your choice. I mean if

[

Which one’d be simpler?
Hmm?
Which one'd be more simple?

This is this is going to tell you where it crosses the
axis, and this is going to give you intersecting points.
So this one is simpler cuz what's going to happen, uh let
me call this f of x. Ok what’s going to happen is that
f of x at one point. I mean this is the, uh, the shift x
y button...Give me y guys. Ok, None of these is going
to be less than zero. The next [ ] is greater than
zero and I know I can just take a number between them. So
that this is easier cuz I can tell when it's zero. Ok.
But if you want to find intersection points, that's ok,
but you can’t go wrong finding out when is it positive
when it's negative and it’s zero someplace in between.
And so, that’s that’s another way. But there’s anything
like, I'll just write something up, three sine two x
minus x squared equals zero, you have an x squared in
there and.you can’t do anything with that just graph it.
You know graph either three sine two x equals x squared,
you can do that, or go with the positive negative. But,
this is actually pretty [ ], so it’s [ ] fairly
straightforward.

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127

Wait. When you graph that what’d you how’d you plug it
into the calculator?

You want to graph this one or this one?
The top one.

Ok. I'm going to do y equals sine parenthesis three x
close parenthesis plus cosine x, and it should work out
like that. Like that. Cuz you got to put the
[parenthesis around it], or you could do y equals sine
three x colon y equals cosine x. Uhm. Let me make one
other quick note. We’ve got some time. I have no clue
what this one looks like. This is, uh, give an example.
Sine three x sine squared of three ijlus cosine x equals
zero. You do the same thing. You graph it on on your
calculator as y equal parenthesis sine three x, um, hit
the x y button, here, ok, um, ok. If I have the square
here, put parenthesis around the sine or‘whatever, square

it, that’s yea. That's how they [ ] on t 11 e
calculator, and so that’s how you treat those. So,
that's the simplest way of doing it. [ ] and then

everything works out. (pause) Other questions on that?

The ten was given?

No, I just chose it. I arbitrarily chose it. I didn’t
I didn’t know anything, so I just chose a number. Such
as ten. Nice simple numbers and so when I cross
multiply...(finishes problem).

Wouldn't the reason why eight, why it has to be less than
eight is cuz if it was longer than eight and had two
values it would [could] be on this side, and then if it
were longer than that it would be on this side, and you
can't have it?

So you’re [going into] the definition, whole theory
behind the triangle
[

I'm just saying, you didn’t know where it came
from. Isn’t that why?

So I have something like this

[

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128

Cuz if it were longer than that
one side you couldn’t put in two spots. Cuz it would
bring it back to the other side of that line

[

Here and. I can’t bring it this way.
Right. It has, that’s right. You’re swinging your lines
but I'm not. I really am not sure how to explain this
thoroughly. (looks at student, both nod)

I’m lost. What're you trying to figure out?
Ok. I’m trying to find out what c is.

Why don't you just use law of sines to find angle a then
use the law of cosines?

Uhn. Yea. That would work. very very easily. The
chapter was on law of cosine, so I tried to apply
everything to law of cosine first.

If you just used the law of sines would it be wrong?

Uh. No, no. I didn’t even see it like that, (person’s
name). Can you

[

Well, I just used the law of sines to find
angle a.

It just happens that the top just doesn't work anyway.
Hang on. It just so happens that alpha equals sine

negative one point three four. This can't happen. See,
I never did learn this the easy way.

What’s 5?

S is the, half of the perimeter. This is when you know
only three sides.

Oh.

What number was that?

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129
Uhm. Eight five number nineteen. It’s talking about

distances east of north and west of north. There’s a
nice diagram of it. That’s important.

When we [do the test] will he give us a diagram do you
think?

Maybe not.
Maybe not?
Just remember that angle of declination is measured down

from the horizontal and angle of inclination is measured
up from the horizontal.

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130

MATA 2

This is my question. So, if no one else wants to hear
then. Can you just use eight then?

Right.
Ok.

Still, do it exactly. Well, first of all figure out how
many feet per rotation.

Can we have that is writing?

After the exam.

How about thirteen?

[no answer..TA goes on to next section]

I just didn’t know where you got it from.
Where I got this thing from?
Yea.

Pulled it out of the book. So we (writes on board) and
that’s what we had to come up with.

So on the test if we just wrote down cotangent of theta
and then

[

If you had a satisfactory explanation, such as
simply the cotangent of theta.had period pi plus shifting
everything horizontal to left two pi shifts everything
down exactly two cycles two period and.you’re right back
on top of where you started, something like that, yea.
So we have (continues problem)

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131

So when we solve these we going to use uhm equations, or
you want graphics?

Uhm. I’m saying, you could probably do either way. You
could, I mean this is absolutely correct way to think of
it. Uhm. In the book I think they’re really trying to
make you use the identityu Use the equations. Uhnn But
this, but this graphic approach helps you think of it,
helps you come up with an answer cuz in the other book,
I mean of a test, it’s something that might say simplify,
but you don’t know what you're supposed to be aiming at.
Here we have an idea graphically of what we're supposed
to be aiming at.

Can you always do that?

Yep, you cross multiply.

Ok, the the one plus tan of x is equal to secant two over
four. It doesn’t apply when everything's squared cuz I
[

Y o u
mean you want this as secant?

I started out that one plus tangent x and I made that
secant x. You know how that's one plus tangent x squared
equals secant squared x.

Right.

I tried it like that. I don't know if that's right or
not, but I still come up with tangent x.

To get secant of x you kind of took the square root.
It’d be it would not be one plus tangent of x.

What I’m.saying'is that I just assumed since you could do
it with squared you could do it with not being squared,
so I said one on plus tangent x equals secant x.

Well, this is the identity that we have. That's the
identity in the book, true know solved true everything.
But the secant of x which would be the square root of
this number, so this is not an identity. This is not
true.

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132

Ok.

Actually, a good way to do this (completes problem).

Do number six?

Number six? (does problem)

I lost you over one point sine x one plus sine x over
sine x. There right there.

Ok. This piece right here sine x over sine squared x.
There's a sine of x in each of them. How many sine x’s
are in this one?

One.
How many sine of x’s are in this one?
How many what’s?
Sine x’s.
Oh. Sine x. But how'd you
Right.

But how'd you get rid of the one from the from the one
before that?

Cuz we have a one plus sine x and we have a one plus sine
down here. These two things are multiplied. This whole
[quantity] is multiplied.

Up. The one above.
The one plus sine x minus. Yea that one right there.
How’d you get rid of the one plus sine x?

Oh. I really, I really didn't get rid of this one. I
combined this one and this one. One minus cosine
squared.

[

01 1b
get the sine.

[

And this one came along for the ride.

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133
Oh, Ok. Now I see.

All right. So I should be a little more explicit?

Can you do seventeen?

Seventeen. Ah, we’ll do fifteen, which is a fast one.

They wanted us to do it by graphics.

Oh they did? Ok, in that case, stick it into your
calculator.

That’s all I had to do it?

Yea. Graph the first one sine t minus cosine t over
cosine t plus one. That’s your first function colon.
Graph y equals tangent of x.

All right.

[We] did it this way.

You’re not going to do forty three?

Oh. I need to do forty three? Thank you. Oh that's a
good one. Can I erase this one? Forty three. We have
sine of the inverse tangent of x. So

[

Um. Are you on eight point
three?

Ha. Yes I know'what I'm doing . Sine squared of x minus
one. So how we going to solve this?

They have two pi.

What they have is actually equivalent to this and uh
after class if anyone wants to know how they got that
I’ll go over it. We just have like five or so minutes
before I have to give you the quiz and uh sixty two to
sixty six. See how many of those we can do.

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134

Is that the square root?

Oh, sorry. So this equals forty five sine of theta.

It looks like you’re adding two pi to me.
I’m adding. No I’m just adding pi.

Ok. You started from the first point [ ] to the
third point. Isn’t there two pi between there?

No. This, see here’s the period is pi so from here from
here to here is moving down one pi. And this is pi. And
that’s going to two pi and that’s zero it's going to pi.
So everything moving exactly onejperiod” Exactly one pi.

I see what you’re saying.

That’s all we have to write is pi k?

That’s, yea. Because that's correct. You see why it’s
correct?

Cuz if we put two in for k we get two pi.

We put one in for k would be this one. Zero in for k to
get this one. This is just in the first. This is just
on the first cycle. Already have a graph over here.
Here's pi two pi zero. Keep going. Three pi, four pi,
come back this way. That’s where zero, cuz we're solving
this thing right here. Ok.

How do you know that, that one angle is forty five
degrees?

That, that’s a good question. Uhm. Just basically
because we have to use a little bit of geometry of a
baseball diamond that the pitching mound is is, uh,
directly (draws) in line (draws) between uh

[

second and home

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135

Yea. Second and home. Thank you. And uh we know this.
See this is a square. It's given in the beginning sixty
degrees sixty degrees sixty degrees sixty degrees. And
so this line bisects it. So it bisects these angles.
Yea. I did say that. Thank you for pointing it out.

Wouldn't that be a right triangle?

That's the thing. We’re not sure. This is a right
triangle. It may just be the way I’ve drawn it. I mean
forty feet away from home base could be like this. I
mean I’m not sure that forty foot mark is dead center. I
mean we have enough information using law of cosine.

That’s not on there.

Not on there? Ok.

You could've just took half of eighty nine degrees to
get that angle.

Again, I’m not sure that thing bisects jig Maybe it
does. Maybe it does cut it in half. Uhm. like this
one. I, uh, ok. Maybe it does but I don’t remember my
geometry well enough to know that this cuts it. It
probably does, but we don’t really need it for this
problem. Does anybody know for sure does it bisect it?
To satisfy our own curiosity?

I tried it and came up with the same answer.

Ok. Let's let’s do it this way. We [ ] do it your
way. So for this problem again then we just use the law
of...All right a and b law of cosine c squared equals
eighteen squared plus twenty six cosine of one forty one.
One hundred forty one degrees. .And now it's just use the
calculator to figure out

[
Oh. It's wrong. I did it wrong.

Oh, ok. And you get forty one point fifty six.

Do you think the test’ll have uh proofs of uh
[

identities

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136
Yea.

Yea. That'll be like the first thing on there.

MATA 3

Is it curved?

I don’t know the answer to that question” Ah: I believe
the highest grade (interrupted by 3.2).

What’s the average?

Ninety eight. I don't know if this is going to be curved
any differently. I do not know that. So, ah, check the
adding, ok? Check the adding. Make sure I added yp the
points. Look closely at each problem. Make sure you
agree with. my grading. I’m not going to be too
cooperative changing grades cuz I did spend.a lot of time
going over each. problem to see if there was some
semblance of understanding. Ok. I know there are
certain certain things everyone had problems on.

Number three? Can you do that one?

Sure. (Answers 3.4 first.) Any questions of that
problem? A lot of people messed up on what the angle of
depression was.

When you give partial credit is there some specific thing
that you

[

I’ll tell you, [name], when I grade these when I
grade these papers, uh yea, he gives me a piece of paper
that tells me specifically what partial credit is. Now,
I will usually give more partial credit and I’m being
totally serious. I give more partial credit than I’m
allowed to on paper. That’s because I can see lot of
times what you’re trying to do and I know what we’ve
covered in class. That’s why you want to come to me if
you disagree with the grading of a problem. You know.
come to me first and I will look at it and I will show

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137

you exactly what kind of credit I gave you. Ok.
Sometimes I’ll give you a couple extra points if it looks
like you’re doing ok. This...All right problem three.

Could you do number six please?

Ok. The first thing I did was draw a picture. I wanted
to see what the sine the sine of x looked like (completes
problem). One last stepu Ok. 'You you were asking about
partial credit. This is worth something, this is worth
something, and then writing down this and this is worth
something, and then writing down this and.tw01pi is worth
something. This is worth something. And these two are
worth somethingu Ok: That’s hOW'he tells me what to do.
Then I look at the whole problem and say, well they know
what they’re doing and give them a few extra points so
take some time to memorize on page five forty eight
figure eight point two point four. This triangle and
another triangle to memorize.

Yea. Do number seven please?

All right. When you see the words "within point zero one
accuracy" that should tell you you can use your
calculator. (completes problem) That’s six point two
eight minus one point eight eight. So it'll come out
four point four one.

Wait. How’d you get the five point two four?

How did I get this number? Oh, ok. I'll [explain it]
better. This is uh this is two pi times the number of
times the radius in inches. So this is going to be sixty
two inches. Ok. That’s how big around it is. And then
what I did was I said, I should do this.. I said, how
many feet is that?

That’s in feet?

Cuz this is in feet. Yea. That's all I did. Be careful
with that kind of stuff.

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138

Can you do number nine?

Number nine? Ok.

You do eight?

Nine and eight? Let me make a choice between those two.
For number nine, uh, the sine...Here’s how you do it.

In the middle one, wouldn’t x be squared, too, then?
Because

[

No. No. When I write this, actually, all I’m doing is
taking...Ok, this bit of notation here means take the
cosine of something and raise it to the fourth power.
Ok, in other words, ok, I’m taking, all right, I’m taking
the cosine of x to the fourth and I’m just rewriting it
as this. Cosine t the fourth of x. Ok?

[
Uh, ok.

I’m not touching the x’s. They're [locked inside the

sine and cosine] ok. Then there I just expanded it out
using that first rule. Ok. Now...

Can you multiply one plus sine x over cosine x by [
]? Cuz...Oh, it'll cancel out.

Oh, yea. It's good that you’re thinking. Uhm. I have

this written in my notes. So, I will finish this way.
But you’re right. In fact that's what I told the
afternoon class. Go home and see if you can find an
easier way to do this. Nobody who did it, everybody

pretty much did it this way, which surprised me cuz I
thought I was
[

I did. it a
different way.

Ok. Good. Wait on it. And if it works, then that's
good. Ok, my next step would be.

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139

Is that the answer then?

Ah. I would, this looks pretty darn correct to me.
(laughter) I mean the sine squared of anything plus the
cosine squared of anything is equal to one. That's a
verification that that identity is true. So, yea. yea
that’s an even better way; This is what I'm looking for,
the shortest way possible. The first way is never the
shortest almost. Ok. Ahm.

What was two?

Two was the tangent of x.

You mean that’s forty five degrees.

Uh. Yea, sorry. (changes board)

I [ ] where you came up with sixty over four. I can
understand.howryou go the fifteen after that. You got pi
over three times one fourth.

Ok. (pause) It came from here. Watch. Um. I'm just
rewriting this. I didn’t start with pi over three. I
started with pi over twelve. And that’s, uh, that's pi
over three times one fourth, ok? So, pi over three is
the sixty. So it’s sixty over four and then that's the
fifteen. And I wouldn't've, you know, even thought to
use this triangles if I hadn’t looked at the examples.
So it's a special type problem that 1H1 is cooked Lu)
especially for an exercise. Any other questions?

Would it work if yo just put the top sixty minus forty-
five where could just do one half minus one over square
root of two and just solve that? Instead of doing that
long

[
Instead of splitting it apart?
Yea.

I'm sorry. Wait. Co...I’m sorry.

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140
Well, the cosine of sixty is one half. Right?
Yea.

The cosine of forty—five is one over square root of
two. The cosine of fifteen

[

Are are you doing this? Are you using
like a distributive property? Are you saying that this
is equal to this?

Right.
No. That’s not true.
Why not?

Because this , um, uh, no it's not even true in this
special case. Basically it's just...ok, if I could use
distributive then I could say cosine theta plus beta is
equal to cosine theta plus cosine beta. Ok. That would
also be true

[

Thmfis
...we have the other identities.

And not I got to I can't do that. I have to memorize
this long thing. They’re they’re just not equal that’s
all. Um yea. It’s just a rule. What I thought you were
going to say, and this is another alternative, you
could've also used, uh, I think thirty and forty—five and
then you would’ve gotten nice nice angles that you can
deal with.

I got it, but where’d.you get negative the square root of
[ l?

I got it. Ok that's that's important to know because I
hate it when I disagree with the back of the book but I
double and triple checked it.

I got positive [ ].

Well, well the thing is is that they say you can can tell
by looking at the at the preamble of the problem. Let's
say if the sine of x is two thirds we know'what that does
for a unit circle. It puts the sine right about here and
the angle here and.here. .And then they say x is between
pi over two and pi. Ok.

Oh.

141
TA: Yea. See what I'm saying? if x is in this quadrant
there aint no way you can have a positive cosine. I’m
pretty sure that it’s supposed to be minus. Ok.
St: Ok.
3.19
St: Can you do thirty?

TA: Do which?

St: Thirty seven. (several people).
TA: Thirty seven. Sure. (pause) Oh yea that is a long one.
3.20

St: I don’t understand how you got...yea that right there.
That’s the part I don't understand. How can you

[
TA: Wait. I haven’t written this part down yet.

St: Yea. [ ] Right over there on the left. How

[
TA: Here?

St: Yea. Where’d you get that?

TA: The expansion. Ok, well let’s see sine of two theta,
sine of two x, sine of two of anything is equal to
(pause) sine x cosine x. Yea. I'm pretty sure this is
correct. Cosine x.

St: All right.

TA: Sine of x, and then I have x plus y. So I replace this
one with.y and this one with.y so I have cosine of this,
sine of this.

St: Oh, Ok.

TA: Sine of this cosine of this.

St: Ok. I was doing it differently.

TA: That's ok. Grill me.

St: I was using the distributive property.

[
TA: Oh god. I was wondering.

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142

Sine x two theta and then taking the two theta and then
expanding instead of doing that.

Oh. Ok. Um. Alls I can say is very very loudly, don’t
distribute these things. I mean that's part of the part
of the pain of trig is you have to literally memorize how
to expand this. Ok. Uh, so for instance when you first
learned algebra they said that this (pause) was equal to
this. And they didn’t tell you why. They just said it
is. Well, now they're telling you that this is equal to
this and they’re not telling you why. Just telling you
it is. Yea.

Can you finish?

Huh?

Can you finish that one?
Can I?

Yea.

Ok. Ah. Sure. This is a (completes problem).

Would you start off with cosine two, you know.

Well, I take this

[

Right now yea I understand that. But
would you start off with cosine two theta of sine two
theta there? Where you 're at where you have cosine two
theta, should it should it say that or...cuz before I
think we had sine two theta there first.

Yea.

It doesn’t matter. I mean, as long as I only have them
once. Ok. I mean you got your commutative and and your
associative for your commutative you can rewrite it in
any order. Just made sure you don't have two of
anything. Ok. You know there’s no set way to write it
down. Do you still need me to finish it or or was that
your question?

I I get the next step. After that I have problems.

[
Mm Hmm.

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143
combining things.

combining them and getting it to look like the back of
the book?

I got the next step where you break down the sine of
theta or cosine. I get that part.

Ok. Let me show you...Yea I was able to get it into uh
a form similar to this and then I got bored with the
problem. Ok. So cosine two theta...

How do you know when you’re when you’ve made it to the
last step?

When it matches the back of the book. (laughter)

Yea. But like on a test.

[

Literally. Oh, uh, you mean on a test.

Yea.

On the test there's no reason to even worry about that
because they will always tell you when to stop. In this

Oh.

problem, well this problem wouldn't be assigned because
you have to know the answer before you knowaou’re done.
Literally. Otherwise you could do this forever. You
could go backwards and rebuild that and say wow look how
I've simplified. You know? (laugh) So, so the point is
don’t worry about when you’re done. Ok. Uh if they say
simplify any expressions involving only sine theta that
means get anything that looks like a two theta out of
there. Ok. 'Uh. So actually, see what I should say is if
this if this problem said simplify this in terms of sine
theta cosine theta, I would be done after this step.
Literally. I mean that would be the end of the test
problem. Because I would have no I would have no cosine
two theta sine two theta I would be done. I, uh, I
wasn’t done in this case becauseeI didn’t match the book.

Ok.

Why?

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144

Uhm: The simplest answer I can give is because it's just
the difinition of the arctangent. For an arctangent we
plus in a number and it spits out an angle. Ok. So it’s
just the backwards. We started with the angle and took
the tangent. It’s just I’m working backwards. Ok?

On number fifty nine, how come in the back of the book it
says it's the arctangent of fifty over i. I just seem to
get tangent of fifty over i.

Ahm, because what you end up with is this. Well, let me
ask you if this is what you ended up with. Ok, when you
write the triangle down, I’m just doing this for the
people that have trouble doing the problem, the reason
the answer looked like it did was because it said
"express theta as a function of 1." So you want theta
by itself. That’s the reason. So they just write theta
as equal to the arctangent of fifty over 1. Theta is now
a function of 1. Plug in different 1's and get theta.
Other than that it's just like the previous section.

Just real quick. You can use a quadratic on trig
functions but you can’t distribute them? That doesn’t
make sense.

Oh, ok. Let me see if I can explain what’s going on
[
I know what you did. I
just don’t understand
[
Ok. I can I can shOW'you the diffrence. Uhm.
When I. . .that’s a good question. There’s an answer
forming in the depths of my mind. I thought about that
quite a bit because somebody in the afternoon asked me
the exact same question. The same thing. Ok. This is
just a god-given law of trig. Ok. It's just the way it
is. Ahm. If you write them out, if you write the cosine
as angle, if you write it like this (pause) basically a
plus b is some angle. So I take an angle b from here to
here and this is a plus b. It’s the angle a plus b.
Not, I haven’t overlapped them I've written a then b.
Now if I take the cosine of the a plus b, that’s this
number, this length. Now for reasons I don’t know off
the top of my head, this number this cosine of a plus b
is not equal. Yea it kind of makes sense too because
here's angle a and here’s the cosine of the angel a.
Ok. Pretty big number; And here's b. IHere I overlapped
b. I added them here I’m just going to written what b
was from a standard position. Its cosine is this line

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145

two when I add those cosine together.

I don't understand. Isn’t that a?

This is this is a and this angle from here to here
approximately. Ok don’t let my drawing confuse you.
Taking the two

[

You said b didn't overlap a.
Huh?

You said b didn’t overlap a.

In this picture in this picture they do not. In this
picture I am rotating it. In this picture I’ve very
badly drawn an a and a bigger b. Ok. The point is is
here’s your cosine a and here’s your cosine b. When you
add them together like this you get some huge number
bigger than one. By the definition of cosine. If but this
can be greater than one so that’s just a reason why it
can be greater than one so that's just a reason why it
wouldn’t work. Now back to your question. I am using no
that that’s an intrinsic. That's an inherent property of
the cosine. That I can't do that. What I was doing
before was just taking taking that, and notice all I’m
doing is I’m saying instead of writing cosine )L I’m
going to write the letter c. Ok. Now I’m just I’m going
just back to this then, so I’ll substitute back. ‘You see

I’m not really using any property of the cosine. I'm
just throwing it around. I’m not taking it apart and
distributing it and placing it in places. Does that

help? That's a good question. Uhm. I'm not sure how
to answer more clearly than this. I’m not really using
property of the cosine. I'm just writing the cosine as
one thing. Ok? Here here’s an example. What you might
be thinking of is this (explains a related example on
board). See what I'm saying? There’s a distinction
between messing with the rules of trig and using them
algebraically.

Ok.

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146

FITA 4

Um. For my graph on number fifteen, I have two places
greater than five hundred and you only put one point
down.

You mean the part B number fifteen?

Uh huh.

Number fifteen part A. Most of people got part A. Most
of people miss part B

What was the range you used?

Range? The range of the graph or of this?
Yea.

The graph?

Yea. The graph.

The graph [ ] so we will continue goes down adn
here we continue goes up.

I don’t understand how you used you used ten not twelve.
Here?

Yea.

This thirteen.

Oh. Thirteen. How’d you know how to use ten instead of
thirteen?

Ok. We need twenty six minus two x greater than zero.
But we have twenty six greater than [this two x] divided

by two divide by two who is thirteen. X less than
thirteen. Oh we have to check this condition and this
condition. Ok? The domain should satisfy this three

condition. And so this [why] we get the solution from
zero to ten or we can draw the graph. We can draw graph
here (completes graph)- Ok? Any other questions?

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147

Can you just factor it though?
If...You cannot factor it. You can.<k> the formula.
Quadratic formula. Two times two, negative one, one [

] If you don’t know how to factor it, you can use the
formula. Yea?

Isn’t there an easier way to do it than using u?
Pardon me?
Can you do it a different way than using u?
isn’t there a different way than
letting us equal x squared?

This is the easiest way. ‘You think there is another easy
way?

Got to be.
Pardon me?

It it’s just confusing.

Confusing. Ok. Or the other way is [ is u] you can
[ ] by this x square over square x square x
square. Ok minus fifteen. Then you have to think

about...This one is for for one unit. For unit. So this
become a quadratic equation. Then you can do the
factoring faster. This is to the power four and we don't
have any formula.

Can't you just use two x squared minus five and x squared
plus three?

Yea. 'You can do that but, why I show guarantee everybody
can solve for u but, if you do two x square minus five,
x squared plus three, not everybody can do that.

Ok.

You are very familiar with the factoring you can do this
one. But a lot of you are just beginning. [Haven’t]
learn how to do the factoring right, ok? But we will get
we will find that u...this is one plus...If you are very
good you can do this way. If you are beginner you can
follow this way. This guarantee you can solve it.

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148

I don’t understand what you’re doing over here.
Here?
Yea.

The root of this equation will be this one. Right?

Mmm Hmm.

Understand this one? Ok. And from there we know the
root of complex: It’s not real. Since it's complex, the
the means the inside the [discriminamy] is negative.
This is the only criterion. IflflL So this one, we
require this one. Actually it’s four cubed less than
zero. So we can write our equation. Our polynomial has
this form. And the only criteria is p squred minus four

cubed. Like if I pick up p is one then q is one.
It’ll be ok. Ok? P is one now pick up q is two. It'll
be ok. It’ll satisfy this [discriminant] is less than
zero.

So you do that with all the complex factors?
Yea. This one here you, uh, do the factoring, you find

our [it's true] it’s complex. So there are infinite many
solutions.

So are you allowed to do it like the first part or no?
This two.
Number one. Is that right also?

I think. Yea this one also in your book. Only give you
one solution.

Mmm hmm. But there are more ways to do it?

Actually there for this one for real there is only one
case. For this one there are infinite many. Yea.

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149

So if that was like on a test we could put x plus one
cubed and get it right?

Which one?

The first one.

Yea.

And it’d be right?

It’d be right.

There’s nothing to it then.

Pardon me?

This is for more complex.

[

Why go through step two when you can just scratch that
off right there and be done?

Yea. But I just want to explain, uh, since I don’t think
your book give you give you this solution: The book give
you one of the solutions conditions. The second case,
that’s why I explain, and sometimes they will uh tell you
the only one real [true] complex then you have to know

how to find out the complex one if they give you more
condition. Ok?

You know the k minus one? Where’d you get the minus one
right there? Is it one minus one?

K times one» Then thirty seven ok (points and underlines

answer on board - student nods and writes something
down).

You can also take the sixth root, can't you?

Pardon me?

You can also take the sixth root?

YOU mean

 

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150

[

That’s how they did it in another class and that’s

how
Yea. You can do that.

Comes out the same.

Yea. You can do that. X minus one, will be...Yea you
can do that. You have answer? Ok? That's good idea.

We didn’t have to do that one.

Pardon me?

3 This wasn’t on our homework.

(louder) We didn't have to do thirty nine.

We didn’t have to?

[

It only went up to thirty eight.
Do I have to do?

It wasn't on our homework.

No? Ok. Then we go on to something else.

five.

Twenty five.
Twenty five?

Twenty five.

You do seventeen before that?

Ok.

What section are we on?

Four five page two eighty three.

We go to four

 

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151

Does it matter that you multiply those two together as
opposed to like uh two plus i squared [ ]?

Here? Yea. I will multiply by two plus i squared and
this two together. I feel this will be easier.

Oh. All right.

Ok?

How’s there two complex?

(along with a student) Since we have three three zeroes.
We have three zeroes.

Ok. I got it.

Ok?

How’d you know when it’ll be a double double root?
Double root?
Double root.

Ok. If this is a double root, they must [ ] triple
root.

They must what?

Ok. If this is double root?

Mmm hmm.

Ok. This means there two zeroes here. Uh two real here.
Then I have another one. Must have three zero. It's
impossible. Since in in this case we can factor in its
[ ] form but we cannot. And this if you have a
[ ] here. Actually the complex happen here.
Right.

Ok. Here and here. So here is a. Here is...just is
conjugate. Ok?

Ok.

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152

What?

What? Complex zeroes. So we have one integer, two
complex non-real. Ok. That’s on problem twenty nine.

Couldn’t you, um, couldn’t you just like graph it? Find
the one, and then you know there’s going to be three [
] for the other two.

Uh. Yea. You can do that. From the graph you know only
one [cross] here. Only one. Real. And since uh the [
] cross x axis once so you have to you will know one for
real, two for complex. Ok.

What if it was degree three?
Degree three?
Yea.

If...in this case degree three. Ok. We have this two
complex, ok. We have two two complex then the third one
will be either real or complex. Do you think it will be
complex?

No.

Should be two. It’s a pair so the other one will be
real.

So how do you find that you multiply those two together
and then and then and then you factor it or

[

Yea.

factor it. Multiply those two together then what are
they?

Yes, well. The other way is...You know this way is real.
Mmm hmm.

If you know this one then you can.uh.have...Since we have
no any other information, we assume this c ok. So we

have another x-c. Ok. And can be any number any real
number. Ok?

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153

Number seven?

Number seven. Ok.

Fifteen.

Fifteen? Ok.

Why’d you use the five times? Five times thirty five?

[
Yea. I'm just checking

you.
Ok.

Ok? (laugh) Thank you. It’s five times the initial.

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154

MITA 5

Thirteen?

(lists on board)

Eighteen?
Eighteen?
Eighteen.

(lists on board)

I’ve got a problem way back in eight point two, number
11?

Eleven. (lists on board) Ok. Uh Afterward we covered
this section we’ll do the other the other thing.

Say that again?

No solution. Because you see b should be bigger than a.
From this graph b should be bigger than a. Right?

Right.

All right. So we [see] there no solution. You have to
check formally, uh, to to do that to do that thing right.
Now. How to do it formally.

I’m just confused because in the book it says b is less
than h which is a sine beta. Then there is no solution.

B is less than h. H is [ ] here.
Right.

But even b is less than h, even b is bigger than h but
less than a it still gives you no solution: What in this
situation even if b bigger than.h but less than a. Still
no solution, right? If b is bigger than h but is is less
than a, if b is here but the angle has to be here. This
one. The reason is b is less than a not less than h. I

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155

guess b is less than a. The reason is b is less than a,
not b is less than h. Let’s compute uh h. Can tell you
more reason. Let’s computed h yea. H equals a sine one
sixteen degree. That's one sixteen degree that eleven
on. (reading’ in Zbook) angle, sine of, one is ten,
that’s bigger than that's less than ten. Less than b.
Less than b. B is bigger than h.

That’s why I’m confused. Because it says that b has to
be smaller than h for there to be no solution.

Oh. No no. Forget about this. The reason is not this.

Ok. Yea. Different from the textbook. The criterion
says compare b with h but here is different.

Is this on the test?
Hm?

About the test. Is it like going to have story problems
on it? Like these things?

The previous test you shouldn’t have such a kind of
problem on.i:n. The problems are usually [ ] by
formula, ok?

This one right here? When b is less than a then there’s
no x right?

B less than a. 'You can tell (reads in book) when beta is

acute b less than a you have two solutions. So these
things right here only so for acute angles.

What was the average?

Forty eight.

Forty eight?

(nods) We had some people did very good, uh, excellent.

Ninety six or uh ninety five. So some students did very
good.

 

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156

How did we how’re we supposed to know'we were supposed to
put that down. I mean it didn’t really say you know put
both solutions down.

You compare b with uh, first you compute h. Fine. This
is h. You compare b even with h. You compare h you
can compute h compare with b. You know h is less than
b. Ok. If h is less than b and here b is less than a
right. B is less than a. So you have you have two
solutions you have two solutions. There is there is a
criterion in the textbook. If this condition satisfied
we have two solutions.

Yes, but it it just said solve the triangle. I mean be
more specific. You want two or you just want one.
[
Yea, solve the triangle. But but in this problem
[
IfIIgotcme
answer that was right, would we almost get the next one,
the second solution right?

Uh.
Why take off so many points?

Just from the law of sine, right, you compute sine alpha
but you have no no reason to say sine alpha a is this.
Just this. It it can be this. Right. Just from this
law of sine you can’t you can’t say that only this one [
]. So if from this you only get one answer it mean you
lose the other one. The other one. Right. So just
from here you get two alphas, and uh yea there is a
criterion in the but there is also you can do this. You
get two alpha [ ] like this. So there is alpha,
there is.also alpha. Anyway this section expecially there
are there are criterions to understand how how many
solutions you have. So according to the grading policy
you give one answer you just get sixty six points.

To get side c, uh, c you use alpha I mean a squared plus
b squared equals c squared?

No. It’s not a right triangle. Instead of right
triangle have the formula Pythagorean law. ZUL. Formula
right? I think and if and, uh, in the triangle this
triangle is not right. So instead you use the law of
cosine. Right. That’s the uh uh case of the Pythagorean
law. Not a right triangle. It’s the laws of cosine.

St:
TA:
St:

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5.12
St:

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5.14
St:
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157
The law of cosine special case of the law of cosine is
when alpha is ninety degree so so uh cosine alpha is zero
so you get just uh the right hand side. You get a
squared. A squared the left left hand side. You get a
squared the right hand side. A squared b squared plus c
squared right.? Minus something minus something is zero
so you get Pythagorean law. It’s a special case of the

law of cosine. And next I want some of you to put your
solutions.

Do five?
Pardon?
Five?

(lists on board)

He curving any [of these]?

Yea. I guess should be curved.

What’s he gonna drop a test?

(laugh)

What was the high?
What?
What was the high grade?

High score is what six ninety sixg What'd I say? Ninety
six.

In our section or overall?
Pardon?
In our section or overall?

In our section. [That seem] pretty high.

St:

TA:

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158

Number eleven.

(lists on board).

Twenty one.

Ok. (lists on board).

Thirty seven?

Ok. (lists on board)

Could we uhm just take r r one times r two and take the
absolute value of each one because the absolute value of
2 one

[

Right. Right. That's another way to do this to do this
is to to compute to compute 2 one absolute value from
this formula squared and compute 2 two 2 two absolute
value from this similar formula. Right. And multiply
that and.you know that theorem one cuz theorem one tells
you that to get uh absolute value of z one times 2 two
just multiply their their absolute values. Multiply
their ablsolute values so that's another method. (writes
on board) It’s this formula so you can compute the these
two these two absolute values first.

On number twenty nine, I got the right answer, but I
don't understand why the second one is square root of
thirteen and the cosine of what angle equals 2 k pi.

Oh. Yea. There are two problems in this group. The
first is writing the trig formula for theta between zero
and 2 pi. Right. The next is general algebra for every
angle. [ ] the first you got angle theta in, uh,
zero. Three sixty in this interval. The next [ ]
all the possible answers so that uh so uh twenty nine.

What would you do if it said find exactly theta?

TA:

.21

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.22

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5

.23

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159

No. No this time you can’t find you can’t find exact
solution. Why? cuz, uh, there is not a special angle.
Right. So you cannot find exact solution. You just have
theta over here.

You do twenty one?

(lists on board)

Ok. Now graph the now graph the root.

Ok. Now root, uh, so this x this number and you know
that z is uh this number two cuz this is one of the cube
root of three. Right. So three is just cubed. Ok. So
x is this CIS cubed. And you just [ theorem] by [
theorem] two cubed and CIS. Here you just multiply this
angle by three» By this exponent multiply this by three.
So that’s (pause to write) this is just eight. Right.
CIS this angle is pi (writes) Ok. Trig formula of 2.

Is the eighth root of fifty the same as the square root
of fifty to the one fourth?

Yea. That’s the same thing. You can...this this is
equal to eight. You just multiply and have many forms.
The the inside just the inside is different. Suppose
have one half this whole thing is one half the fourth
root is supposed one fourth. You can multiply these two
things. You get many forms you can multiply. This is,
uh, there is probably for this formula right. There's
just power four that’s right.

But how about if you have square root of fifty to one
fourth? Is that the same?

Yea. Just the same. Just different forms. You can put
a square up here or you can put a squre right here.

6.

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160

MITA 6

How did you get the one third?
Huh?
How’d you get the one third?

Just you can graph this function because the graph for
the function is through this point through this point.
Because is hard to for you to decide. Because you gotta
find x is equal to one third. Is hard to find but you
can use the the graph for this function to find the to
find the x intercept. X equals one third. Just graph
this function.

Wouldn’t it be from negative one to zero instead of from
negative infinity? Cuz how would you have negative
infinity if its square
[
You.mean.negative square here but [I mean] x is
very close to two but less than two. Suppose x

[
Hmm

squared equals to negative three equals to two (writes)
two (writes) minus...This is (writes) in this case ok.
In this case 2 squared equals to two minus one third.
Right. That means x squared minus two equals negative
one third. Right. So the reciprocal z squared minus two
equals to one negative one third. So is equals to
negative three. If this is negative one n, this here is
negative one n over :1. There is n or it would be
negative infinity, ok? So this interval, because this x
squared could be very near close to could be very close
to two x squared. Very close to two in the axis. This
means here is two x squared would be very close to two
very close to two. And the reciprocal. So x squared
minus two very close to zero. So the reciprocal one
over x squared minus two very, very, uh, is really
negative infinity. Any other problem? Just notice that
notice this one because 2 squared could be very close to
two from right side or left side. If it is from left
side that means the reciprocal would be go to positive
infinity; Here 2 squared.go to two goes to two left side
goes to two then the range would be from from uh left
side to to negative infinity. Go zero 9“) to negative
infinity.

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161

Are you going to give us an example?

Yea. I will give you an example in the exercise.

Can you twenty three, too, please?

Oh sure. (lists on boand) X power five minus two x
squared plus four

[

That’s twenty one.

Oh: X power four minus three x square minus four x minus
two over x ndnus three is greater than equal to two.
Right? (completes problem)

Now like on twenty one where you got the negative one
point seven one, you said to use the [trace key]. Now
how’s it supposed to get the exact value if it never

[

It's hard to find
exact value
Just estimate?

Yea, yea. Just estimate.

 

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