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LIBRARY
Michigan State
University

 

 

 

This is to certify that the
dissertation entitled

INVESTIGATION OF PRESERVICE AND INSERVICE TEACHERS'
MATHEMATICS RELATED BELIEFS IN TURKEY AND THE
PERCEIVED EFFECT OF MIDDLE SCHOOL MATHEMATICS
EDUCATION PROGRAM AND THE SCHOOL CONTEXTS ON
THESE BELIEFS

presented by

Cigdem Haser

has been accepted towards fulﬁllment
of the requirements for the

Ph.D. degree in Teacher Education

 

Awﬂ

// Major Profess‘or’s Signature
I” 6 /;’ /o G

Date

 

MSU is an Afﬁnnative Action/Equal Opportunity Institution

 

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PLACE IN RETURN BOX to remove this checkout from your record.
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DATE DUE

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2/05 p:/ClRC/Date0ue.indd-p.1

 

INVEE
MATHEN‘;
EFFECT OF I

INVESTIGATION OF PRESERVICE AND INSERVICE TEACHERS’
MATHEMATICS RELATED BELIEFS IN TURKEY AND THE PERCEIV ED
EFFECT OF MIDDLE SCHOOL MATHEMATICS EDUCATION PROGRAM AND
THE SCHOOL CONTEXTS ON THESE BELIEFS

By

Ci gdem Haser

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Teacher Education

2006

[NYE 7|
MATHE“
EFFECT OF

TeachC‘
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ABSTRACT

INVESTIGATION OF PRESERVICE AND INSERVICE TEACHERS’
MATHEMATICS RELATED BELIEFS IN TURKEY AND THE PERCEIVED
EFFECT OF MIDDLE SCHOOL MATHEMATICS EDUCATION PROGRAM AND
THE SCHOOL CONTEXTS ON THESE BELIEFS

By

Ci gdem Haser

Teachers’ beliefs have been a major focus of educational research due to their
inﬂuence on teachers’ practice. Understanding teachers’ beliefs and the processes in
which beliefs are formed provides information for teacher training programs. Therefore,
teacher education programs and beginning years in teaching are investigated for their
effect on the preservice and beginning teachers’ beliefs.

This study explored Turkish preservice and inservice (ﬁrst-year) middle school
mathematics teachers’ mathematics related beliefs. Preservice participants were 20
students from 2nd, 3rd, and 4th year levels of a middle school mathematics teacher
education program who were interviewed in 2004. Inservice participants were 12 first-
year teachers who were interviewed both when they graduated from the same teacher
education program in 2003 and after they completed ﬁrst-year teaching in 2004.
Additionally, a professor in the department was interviewed on his perspectives about
program’s impact on preservice teachers’ beliefs.

Participants’ beliefs were investigated in relation to the teacher education program

and inservice teachers’ first-year teaching experiences. A combination of three

frameworks it I"

 

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teachers inter.
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{miffed bell“

frameworks which categorizes the developmental levels of teachers’ mathematics related
beliefs was used in order to detect the belief expressions in preservice and inservice
teachers’ interview data.

The results indicated that teacher education program courses did not generally
have a signiﬁcant impact on participants’ mathematics related beliefs since program
courses did not attempt to change preservice teachers’ beliefs. When a course was
speciﬁcally designed to challenge preservice teachers’ beliefs, there was some impact on
mathematics related beliefs. Program courses lacked sufﬁcient practice in which
preservice teachers would implement their course experiences in the national curriculum
context.

The national curriculum pace and students’ differences in knowledge level were
unexpected for the participants since they were not sufﬁciently prepared for teaching in
real-classroom settings. Although inservice participants had both student-centered and
teacher-centered beliefs, their teaching practices were mostly inﬂuenced by teacher-
centered beliefs. The real-classroom context affected inservice participants’ beliefs and
practices towards teacher-centered teaching.

The study addresses the need for organization of courses that would challenge
preservice teachers’ beliefs, more real-classroom practice in teacher education program

courses, and a better ﬁrst-year mentorship policy to support beginning teachers.

 

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ACKNOWLEGDEMENTS

Having completed my studies at Michigan State University, I would like to
express my appreciation to those who supported me during my Ph.D. and dissertation
studies and made my life in East Lansing easier and enjoyable.

I would not be able to write this dissertation without constant support and
encouragement of my advisor Dr. Jon Star. He challenged me to pursue better ideas,
better analyses, and better writing from the very beginning. He helped me to see and
follow a major idea when I was lost in some other ideas and analyses. I learned a lot ﬁ'om
him about writing (while giving him hard time with my Turkish-English way of writing),
professionalism, and academia. I am deeply grateful.

I extend my gratitude to my committee members Dr. Joan Ferrini-Mundy, Dr.
Dorothea Anagnostopoulos, Dr. Sandra Crespo, and Dr. Helen Featherstone for their
encouragement through the study and constant offers for help, which made me feel cared
and supported. They provided me with their ideas and suggestions in our meetings and
helped me to realize more similarities and differences between Turkish and the US.
contexts. Additionally, Dr. Suzanne Wilson continuously checked me in and offered her
help through my Ph.D. studies. I will miss our conversations about simply everything.

I present my sincere appreciation to all students studying in 2003 and 2004 in the
Elementary Mathematics Education (EME) Program at Middle East Technical
University, Ankara, Turkey. In particular, I am deeply thankful to my participants for
their time and responses. Dr. Erdinc Cakrroglu, Mine Istksal, and Oguzhan Dogan of

EME Program helped me through the data collection process and supported me during

iv

the study. W0!
2005. Thank y.

The sci
I
to pursue my P

Ihaye .
made my East
my PhD. stuc';
my high spirit .
these years. I .1
Lisa. Korma~ a
Emel. Ptnar, 54

their good thou

 

MY Ian-
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mean bCN‘een
me and have a?
Ernraho{llu ca,
Continuous Su;

Final!)
Kemal Alarm],
“Omen and hi:

memoril as he

the study. Words cannot express my gratitude to Mine in helping me with data coding in
2005. Thank you for your time, energy, friendship, and dedication.

The scholarship of Higher Education Council of Turkey made it possible for me
to pursue my Ph.D. studies without worrying for other things.

I have also been blessed with amazing ﬁiends. Zeynep Altmsel and Mary Gebbia,
made my East Lansing life peaceful and supported me with their warmth and care during
my Ph.D. study. Banu has always been my smile, my other eye, my good morning, and
my high spirit and Esen has always been connected to me and encouraged me during all
these years. I always had a working computer because Bugra ﬁxed it. Knowing Kristin,
Lisa, Kurnia, and Marcy made my East Lansing life shiny. My friends from Turkey: Aslr,
Emel, Pinar, Selen, Semra, and Sule always supported me and Asuman and Badrr sent
their good thoughts and energies whenever I needed. Thanks, thanks, thanks!!

My family, Nur-Tuncer-Bahar, suffered from my studies the most. My parents
toOk a step nearly 20 years ago and sacriﬁced the time we would spend together for my
education. I spent most of my life away from them and I also put a continent and an
ocean between us, but I always felt their unconditional love and care. They all believed in
me and have always been patient and encouraging. I love you all. I hope Mual and
Emrahoglu can see what they made possible. And thanks to aunts Giil and Semra for their
continuous supports.

Finally, as a woman, a teacher, and a teacher educator, I am indebted to Mustafa
Kemal Ataturk, the founder of Turkish Republic, for his encouragement for Turkish
women and his belief in education and teachers. This dissertation is dedicated to his

memory, as he addressed teachers as the only key to a nation’s well being.

LIST OF TAB

CFI-‘xPTER l
[\TRODL'C T
Theoretica
My Motix .
Goals and

CHWTER 2
LTTEPATL'RI
Teachers'
Characte
Presen'ir
Beginnir
Summary

 

GAMER3
A COMBINE
Thompsor
Critical
Lindgren‘
Critical
Ernest‘s I
Critical
Combme

CHM’TERa
METHODS
RCSEarCh
Contexts
Th6E1
The E
Panrcrpe
e'Ser

Inna

Pro {ES
‘Umn
Firsr ;

Sen

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................... x
CHAPTER 1
INTRODUCTION .................................................................................... 1
Theoretical Framework ........................................................................... 5
My Motivation for the Study ................................................................... 8
Goals and Outcomes of the Study ............................................................ 11
CHAPTER 2
LITERATURE REVIEW ........................................................................... 13
Teachers’ Beliefs ................................................................................ l6
Characteristics of Teachers’ Beliefs ....................................................... 17
Preservice Teachers’ Beliefs: The Impact of Teacher Education ...................... 21
Beginning Teachers’ Beliefs: The Impact of Real School Contexts .................. 30
Summary and Conclusions ..................................................................... 39
CHAPTER 3
A COMBINED FRAMEWORK OF MATHEMATICS RELATED BELIEFS ........... 44
Thompson’s Framework ........................................................................ 44
Critical Arguments and Points in Thompson’s Framework ............................ 47
Lindgren’s Framework .......................................................................... 48
Critical Arguments and Points in Lindgren’s Framework .............................. 49
Ernest’s Framework ............................................................................ 50
Critical Arguments and Points in Ernest’s Framework ................................. 51
Combined Framework .......................................................................... 52
CHAPTER 4
METHODS .......................................................................................... 56
Research Questions .............................................................................. 57
Contexts .......................................................................................... 58
The Elementary Mathematics Education Program ....................................... 58 '
The Educational System in Turkey and Teaching Institutions ......................... 6O
Participants ....................................................................................... 65
Preservice Teachers .......................................................................... 66
First-Year (Inservice) Teachers ............................................................ 67
Professor in the EME Program ............................................................. 67
Instruments ...................................................................................... 68
First Research Question ..................................................................... 68
Semi-Structured Interview Protocol for Preservice Teachers (2004 Study) ....... 68

vi

 

 
   
     

Addresszr
Beliefs
Beliefs
Beliefs _

Data Colic

2003 Stu.

Detectr
C atego
Decidr
ldentiiyi
Quality of
C redibi
Resea
Natur
Data
Sumr
Reportir.
Organiz

CHAPTER
PRESERY
I’resen
Ye

Summ

CHAPTE
B'SERV]
No-0
Be:

Be}

Be]
01':
Cons
Be
Be
Be
Or

Second Research Question .................................................................. 70

Semi-Structured Interview Protocol for the Graduates (2003 Study) ............... 7O
Semi-Structured Interview Protocol for the First-Year Teachers (2004 Study). . .71
Semi-Structured Interview Protocol for Faculty .......................................... 72
Addressing Beliefs in Interview Protocols ................................................ 72
Beliefs about the Nature of Mathematics ................................................ 73
Beliefs about Teaching Mathematics .................................................... 74
Beliefs about Learning Mathematics .................................................... 75
Data Collection Procedure ..................................................................... 76
2003 Study (Data Collection) ................................................................ 77
2004 Study (Data Collection) ............................................................... 78
Data Analysis Procedure ........................................................................ 79
Identifying Belief Levels ..................................................................... 8O
' Detecting Belief Statements ............................................................... 8O
Categorizing Belief Statements ........................................................... 81
Deciding on Final Belief Levels ......................................................... 82
Identifying Factors ........................................................................... 85
Quality of the Research ......................................................................... 87
Credibility ..................................................................................... 88
Reseacher’s Role ........................................................................... 88
Nature of Data .............................................................................. 91
Data Analysis ............................................................................... 93
Summary .................................................................................... 95
Reporting the Results ........................................................................... 96
Organization of Result Chapters .............................................................. 97
CHAPTER 5
PRESERVICE PARTICIPANTS’ MATHEMATICS RELATED BELIEFS .............. 99
Preservice Participants’ Mathematics Related Beliefs ................................... 100
Year-Level Variances and Possible Factors .......................................... 104
Summary ........................................................................................ 107
CHAPTER 6
INSERVICE PARTICIPANTS’ MATHEMATICS RELATED BELIEFS ............... 109
No-Change Group .............................................................................. l 10
Beliefs about Nature of Mathematics .................................................... 111
Beliefs about Teaching Mathematics ..................................................... 114
Beliefs about Learning Mathematics ..................................................... 117
Other Changes and Conclusions .......................................................... 119
Consistent-Increase Group .................................................................... 121
Beliefs about Nature of Mathematics .................................................... 122
Beliefs about Teaching Mathematics ..................................................... 125
Beliefs about Learning Mathematics ..................................................... 126
Other Changes and Conclusions .......................................................... 127

vii

Slight-De:
Beliefs a
Beliefs a
Beliefs 2
Other C i'
light-lncr:
Beliefs a‘
Beliefs al
Beliefs 2'
Other C I'

Mixed-Ch.
Beliefs a
Beliefs a
Beliefs .1
Other C 1'

Summary

CHAPTER 7

FACTORS A?
Different
Concert. I
Summer;I

CHAPTER 8
DISCL'SSon
Conclusior'
Effectit ;I
COmpkil
SummkI
Firsm'ch
Policy
Contnbuil‘
Teach;-
M." Pr;
RESea‘v‘

BUlelEWQR

APPEVD .-
‘ LX

 

Slight-Decrease Group ........................................................................ 129

Beliefs about Nature of Mathematics .................................................... 131
Beliefs about Teaching Mathematics ..................................................... 133
Beliefs about Learning Mathematics ..................................................... 135
Other Changes and Conclusions .......................................................... 137
Slight-Increase Group ......................................................................... 138
Beliefs about Nature of Mathematics .................................................... 139
Beliefs about Teaching Mathematics ..................................................... 142
Beliefs about Learning Mathematics ..................................................... 143
Other Changes and Conclusions .......................................................... 144
Mixed-Change Group ......................................................................... 145
Beliefs about Nature of Mathematics .................................................... 147
Beliefs about Teaching Mathematics ..................................................... 151
Beliefs about Learning Mathematics ..................................................... 153
Other Changes and Conclusions .......................................................... 155
Summary ........................................................................................ 158
CHAPTER 7
FACTORS AFFECTING INSERVICE PARTICIPANTS’ BELIEFS ..................... 163
Different Teaching Methods in the Teacher-Centered Classrooms .................. 164
Concerns about Students’ in the Teacher-Centered Classrooms ...................... 170
Summary ..................................................................................... 1 75
CHAPTER 8
DISCUSSIONS AND IMPLICATIONS ....................................................... 179
Conclusions, Discussions, and Implications ................................................ 180
Effectiveness of EME Program ........................................................... 180
Comparison of US. and Turkish Preservice Teachers .............................. 187
Summary ................................................................................... l 89
F irst-Year Teaching Contexts and Impacts ............................................. 190
Policy Concerns for Beginning Teachers in Turkey ................................. 195
Contributions of the Study .................................................................... 196
Teacher Education Programs and Initial Years in Teaching ........................ 196
My Practice of Research on Teacher Education ...................................... 198
Research on Teachers’ Beliefs .......................................................... 200
APPENDIX A
FRAMEWORKS .................................................................................. 202
APPENDIX B
ELEMENTARY MATHEMATICS EDUCATION PROGRAM DETAILS ............. 208

viii

APPENDIX C"
NERVIEW

APPENDIX D
SA‘sIPLE DA

APPENDIX Er
A MEMO F Ol
PROGRAM. .

REFERENCE

 

APPENDIX C

INTERVIEW PROTOCOLS OF THE STUDY ............................................... 224
APPENDIX D

SAMPLE DATA CODING ...................................................................... 233
APPENDIX E

A MEMO FOR THE ELEMENTARY MATHEMATICS EDUCATION

PROGRAM ......................................................................................... 244
REFERENCES .................................................................................... 246

ix

   
   

Tahle3l1AS

Table 4.1: The
in the Progran

Table 4.2: The
Participants. . .

Table 4.3: lnte
Questions .....

Table 4.41 Ma

.. ........

Table 4.6: M
Imﬁryiew (2t

Table 4.7: E
Mathemaiic:

Table 4.8: E
Klazhemmc

Table 4.9: I
.‘tlammatic

Tab}

€4.10.
Imp '

aeting I

LIST OF TABLES
Table 3.1: A Summary of the Combined Framework for Analyzing Teachers’ Beliefs...53

Table 4.1: The Number of the PreService Teacher Participants According to Year Level
in the Program and Gender ........................................................................ 66

Table 4.2: The Number, Gender, and the Teaching Contexts of the First-year Teacher
Participants ........................................................................................... 67

Table 4.3: Interview Protocols Implemented in Relation to the Primary Research
Questions ............................................................................................. 68

Table 4.4: Main Concepts in the Preservice Teacher Interview and Selected Questions
(2004) .................................................................................................. 69

Table 4.5: Main Concepts Graduates’ Interview and Main Questions for Each Concept
(2003) ................................................................................................. 70

Table 4.6: Main Concepts and Additional Main Questions in the First-year Teacher
Interview (2004) ..................................................................................... 71

Table 4.7: Examples of Questions in 2004 Study Addressing Beliefs about Nature of
Mathematics .......................................................................................... 73

Table 4.8: Examples of Questions in 2004 Study Addressing Beliefs about Teaching
Mathematics .......................................................................................... 74

Table 4.9: Examples of Questions in 2004 Study Addressing Beliefs about Learning
Mathematics .......................................................................................... 75

Table 4.10: Examples of Questions in 2004 Interview Protocols Targeting Factors
Impacting Beliefs .................................................................................... 86

Table -l.ll: Ey

Table 5.1: Pro
Three Belief I

 

Table 6.1: lnsI
Mathematics t

 

Table 6.2: Ge:
Levels were it

Table 6.32 Ger
lVere lncreasr

Table 6.4: Ge
their Belief L.

Table 65 Ge
hiSBelief Le'

TAIIIE 6.6; G
less in I

Table A. l ; ']
Belief LCVe]

Levels ......

Table 4.11: Examples of Speciﬁc Additions to Participants’ Quotes ........................ 97

Table 5.1: Preservice Participants’ Year in the Program, Gender, and Belief Levels in
Three Belief Areas .................................................................................. 99

Table 6.1: Inservice Teachers’ Belief Levels on Nature of Mathematics (N), Teaching
Mathematics (T), and Learning Mathematics (L) in 2003 and 2004 ........................ 109

Table 6.2: Gender, Type of Teaching Context, Belief Levels of Participants Whose Belief
Levels were not Changed ......................................................................... 111

Table 6.3: Gender, Teaching Context, Belief Levels of Participants Whose Belief Levels
Were Increased Consistently ..................................................................... 122

Table 6.4: Gender, Teaching Context, Belief Levels of Participants Who Had Decrease in
their Belief Levels ................................................................................. 130

Table 6.5: Gender, Teaching Context, Belief Levels of Participant Who Had Increase in
his Belief Levels .................................................................................. 139

Table 6.6: Gender, Teaching Context, Belief Levels of Participants Who Had Mixed
Changes in their Belief Levels ................................................................... 146

Table A1: Thompson’s (1991) Framework that Characterizes Mathematics Related
Belief Levels ................................................................................................ 202

Table A2: Lindgren’s (1996) Framework that Characterizes Mathematics Related Belief
Levels ................................................................................................ 203

Table A3: Ernest’s (1989) Model that Characterizes Mathematics Related Belief
Levels ............................................................................................................................... 204

Table A4: Complete List of Level 0 Beliefs (T: Thompson’s (1991); L: Lindgren’s
(1996); E: Ernest’s (1989)) ....................................................................... 205

xi

Table AS: Co
0996); E: Err.

 

Table as: Co
(1996); E: Err.-

 

Table A5: Complete List of Level 1 Beliefs (T: Thompson’s (1991); L: Lindgren’s
(1996); E: Ernest’s (1989)) ....................................................................... 206

Table A6: Complete List of Level 2 Beliefs (T: Thompson’s (1991); L: Lindgren’s
(1996); E: Ernest’s (1989)) ....................................................................... 207

xii

 

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h
A

CHAPTER 1

INTRODUCTION

Visible teaching, or observable practices that occur in the classroom, is partly a
result of invisible processes that occur in a teacher’s mental space. A teacher is likely to
alter her pedagogical decisions depending on what happens in a classroom moment, what
has happened before, and what she would consider as an effective next step. The
perception of effectiveness of a pedagogical decision depends on how a teacher connects
it to what she has experienced previously or what she anticipates from the change. The
nature of previous experiences and anticipations is likely to change as the teacher
becomes experienced. Teachers’ thought processes, however, are not easy to observe
(Clark & Peterson, 1986).

Teachers have certain ideas, expectations, attitudes, and beliefs about or towards
students, teaching, learning, and the subject matter they are teaching. Among teachers’
personal constructs, beliefs have been an important concern of educational research since
teachers’ beliefs are considered to have an impact on their practice. Teachers orchestrate
multiple variables in highly complex, uncertain, and unexpected classroom environments.
They manage this by using their beliefs, which are mostly formed by previous
experiences (Kagan, 1992; Nespor, 1987). Since beginning teachers lack sufﬁcient
classroom experience, their beliefs become the main guide in their teaching (Feiman-
Nemser, 2001) with a high possibility of limiting their choice of teaching practices and
possibly their effectiveness (Cooney, 1985). Hence, beliefs play a key role in making

sense of the classroom context.

Preser

 

experiences

 

filter for prescl
and expert enc
mostly former
programs (Lat
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Preservice teachers’ beliefs inﬂuence their interpretation of knowledge and
experiences gained in teacher education programs. In other words, beliefs function as a
ﬁlter for preservice teachers. Preservice teachers screen and re-organize new knowledge
and experiences by using their beliefs (Kagan, 1992; Pajares, 1992). These beliefs are
mostly formed during the pre-college education and are carried to teacher education
programs (Lampert, 1990; Schmidt & Kennedy, 1990). To what extent these beliefs are
challenged or changed through the preservice education remains an issue of concern
(Wideen, Mayer-Smith, and Moon, 1998).

Understanding preservice teachers’ beliefs is important in documenting the
effectiveness of teacher education programs in preparing teachers. Most teacher
education programs expect their students to teach differently than how they were taught,
but research shows that this has not been a major achievement so far (Wideen, et al,
1998). Despite the importance of preservice teachers’ beliefs in learning to teach, most
research is on the inservice or student teaching experience. The absence of longitudinal
research about how teachers’ beliefs are formed is a critical gap in teacher education
research. The evolution of teachers’ beliefs and practices over time is an area to be
explored in order to design and re-conceptualize teacher education programs (Kagan,
1992)

In this study, I investigated mathematics related beliefs of Turkish preservice and
inservice (ﬁrst-year) teachers who were in the same teacher education program. I was
particularly interested in exploring their beliefs about the nature of, teaching, and Teaming
mathematics and understanding the impact of certain previous experiences on their

beliefs. I did not only focus on the teacher education program experiences, but also on the

cumulative 5\

system.

 

My int
ditTerenceS 1“
educational r:
their beliefs 0
beliefs using <
exclusryely d:

bets een the g

 

which are like
maintenance I

Turkis
compared to l
the be.iefs art
the teacher er
Turkish SOCII
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pIOIﬁSSlOns 1

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Turkey l det

EDI-“am SC‘T'X

\
i

cumulative experience of being a student and a ﬁrst-year teacher in a national curriculum
system.

My interest in beliefs emerged early in graduate school when I began to notice
differences in educational research, in the US. and in Turkey. Unlike in the United States,
educational research in Turkey has not focused on teachers’ thought processes and on
their beliefs extensively. While U.S. studies investigate inservice and preservice teachers’
beliefs using several research traditions, Turkish studies are few in number and
exclusively designed quantitatively. Moreover, there are considerable differences
between the general educational system and teacher education of Turkey and the US.
which are likely to produce different beliefs and phases of belief development or
maintenance for preservice and inservice teachers.

Turkish preservice and inservice teachers’ beliefs may be different when
compared to US. teachers’ beliefs. The difference could be observed mostly in the ways
the beliefs are developed and centralized within the Turkish culture and the structure of
the teacher education programs that does not support beginning teachers in Turkey.
Turkish society values the teaching and learning of mathematics to a great degree.
Mathematics-based professions, such as engineering, are highly recognized over other
professions such as law. Mathematics is the key area in the national examination, the
University Entrance Examination, which is required for attending at any university in
Turkey (detail explanation of Turkish educational system is provided in Chapter 4). Due
to the importance of the University Entrance Examination in attending a university,
private services for test preparation training emerged and attending to this training have

been widely considered as the only way to pass the national examination in the Turkish

society. The?»

 

tutoring In “

subjfcl area 1"
multiple CPO“
emphasis on I
illustrated as ~
during the it‘s?
mathematical
correct ans“ 0
students inclu

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as a coliectror
especially dur
from high sch
not thinking if
cnculturation

Valuable teacl

society. These services are in the form of Test Preparation Centers (TPC) or private
tutoring, in which many university students including preservice teachers, are involved as
subject area tutors. Test training basically provides students with skills for solving
multiple choice questions in limited time and reduces student involvement and the
emphasis on the nature of mathematical knowledge. Mathematical knowledge is mostly
illustrated as a collection of rules and shortcuts that will work for solving the questions
during the test training by the instructors. Meanings of and connections among
mathematical concepts are not emphasized and often ignored. Therefore, reaching a
correct answer is generally seen as an indicator of Teaming mathematics by high school
students including future preservice teachers.

Despite its high status within Turkish culture, mathematics is mostly considered
as a collection of rules to be used to solve problems and this conception is emphasized
especially during the examination training in TPCs or through private tutoring beginning
from high school. Preservice teachers are also a part of this long enculturation process of
not thinking deeply but just solving questions correctly, but they also continue their
enculturation through tutoring in TPCs or in private settings which they consider as a
valuable teaching experience before formal teaching. Therefore, it might be expected that
the Turkish system not only impacts the ways preservice teachers consider mathematical
knowledge and teaching and Teaming of mathematics, but also provides them with
contexts in which they can practice the general conception of mathematics teaching in the
Turkish culture.

In addition to extensive opportunities for practice of informal examination

teaching before formal teaching at schools, a major difference between the US. and

Turkish teach:
teacher educat
beginning tea;
and the teache
critical reﬂect
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Turkish teachers is the lack of internship year in Turkish system. Turkish elementary
teacher education programs are four-year programs with no internship year that
beginning teachers are supported and monitored by collaborating teachers in their schools
and the teacher education institutions, as conducted in the US. Therefore, they lack
critical reﬂection from experienced teachers and teacher educators on their beliefs and
practices, and they mostly tend to practice considering the national examinations and
curriculum pace.

The difference between the US. and Turkish educational systems in terms of the
conception of mathematics in the Turkish culture through national examinations and the
structure of each country’s teacher education process make U.S. research ﬁndings about
teachers’ beliefs unlikely to be generalized for the Turkish context. Hence, it seems a
necessary step to gather initial data about Turkish inservice and preservice teachers’
beliefs with the help of existing studies in the other contexts.

In this chapter, ﬁrst, a theoretical framework for beliefs as a personal construct,
their importance, and how they should be investigated will be described brieﬂy. This
section also includes some information about how I addressed certain issues in belief
research in the present study. Then, the motivation for the study and the goals and
outcomes of the study will be explored. Finally, a description of chapters of the
dissertation will be given.

Theoretical Framework

Following arguments about the need for understanding teachers’ affective and

cognitive processes beginning from 1980’s (e. g. Clark & Peterson, 1986), studies have

been conducted to analyze teachers’ knowledge and beliefs about subject matter and

various contl“
studiCS was a
Processes and
mathematics :
eletnﬁ’tlIS I“ I‘L-
claimed thiiI 17
Pajares. 1993
AlthOt
common frat:
consensus on
up. with next .:
researchers h..
3032 l. Host 6‘.
(Will claims
and socral cor
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as cited in Pa
of a wide am;

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various components of the teaching and Teaming process (Fang, 1996). Underlying these
studies was a general assumption that there might be a relation between teachers’ taught
processes and teachers’ way of teaching (Clark & Peterson, 1986). In the ﬁeld of
mathematics education, mathematics related beliefs are considered to be one of the key
elements in the practice of teaching mathematics (Ernest, 1988). Hence, it is often
claimed that investigating beliefs is a critical issue in educational research (Kagan, 1992;
Pajares, 1992).

Although beliefs are considered to be important in educational research, a
common framework for their investigation is lacking (Paj ares, 1992). There is no
consensus on the deﬁnition of beliefs in the ﬁeld of education as researchers always come
up with new deﬁnitions for their studies (Pajares, 1992). Even in mathematics education,
researchers have different ideas about what constitutes beliefs (F uringhetti & Pehkonen,
2002). However, considering a mixed literature of anthropology and philosophy, Pajares
(1992) claims there is a general agreement that beliefs are formed through enculturation
and social construction. Similarly in psychology, beliefs are considered to store personal
experiences and episodes of cultural or institutional transitions (Abelson, 1979) or
inferences about self and contexts gathered from early’experiences (N isbett & Ross, 1980
as cited in Pajares, 1992). In the ﬁeld of education, beliefs are considered to be a product
of a wide array of personal experiences (Schmidt & Kennedy, 1990).

The existence of multiple belief deﬁnitions in the literature (e.g., Furinghetti &
Pehkonen, 2002; Pajares, 1992; Thompson, 1992) makes it necessary to specify a
deﬁnition for the concept of beliefs in this study. The notion of beliefs as formed by

previous experiences underlies research about inservice and preservice teachers’ beliefs.

for example:
praCllCCS In [It
Presen'ice teé‘i
students (LEFT
notion of belt:
‘rnental const
concepts" (P-l
employed for
between prese-
u their existir
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For example, Nespor (1987) claimed that experiences as a student impacted teachers’
practices in the classroom. Similarly, teacher education research generally concluded that
preservice teachers’ beliefs are mostly gained during their experiences as pre-college
students (Lampert, 1990; Schmidt & Kennedy, 1990). Building on this experience-driven
notion of beliefs, Sigel’s (1980, as cited in Pajares, 1992) deﬁnition of beliefs, which is
“mental constructions of experience — often condensed and integrated into schemata or
concepts” (p.351) where these constructions have a truth value and guiding function, is
employed for this study. This deﬁnition allows for the framing of the relationship
between preservice and inservice teachers’ beliefs and their previous experiences, as well
as their existing or future experiences. It also allows me to narrow down the belief
literature to the studies that explicitly considered belief as an experience—driven personal
construct, hence, forming a more consistent set of previous studies in order to support my
results.

Nisbett and Ross (1980, as cited in Pajares, 1992) claim that prior experiences
have a strong impact on ﬁnal judgments, where these judgments are actually highly
resistant beliefs. This implies that beliefs gained through earlier experiences are not easy
to change, but those gained at the end of later experiences are perhaps more amenable to
change. Pajares (1992) points that there is a need to explore the impact of early
enculturation on the development of educational beliefs. Thus, I tried to see what kind of
early experiences affect preservice and inservice teachers’ mathematics related beliefs
and what part of new experiences have less impact on those beliefs.

Studying preservice and beginning (ﬁrst-year teachers) is potentially problematic

because their beliefs are situated in a particular context. Teachers’ range of the beliefs is

limited by lhs

researchers \

 

mxestigation »

intestigate: s:

and inservice

   

also asked pa:

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limited by the data gathering methods, the context that the beliefs are studied, and the
researchers’ views about beliefs (Wideen, et a1, 1998). In order to have a more accurate
investigation of individuals’ beliefs, Rokeach (1968) suggests three types of evidence to
investigate: statements, intentions, and behaviors. In this study, I investigated preservice
and inservice teachers’ statements in the form of responses to my interview questions. I
also asked participants about how they would respond in certain teaching scenarios, to get
a sense of their intentions for different teaching situations and dilemmas. I was not able to
observe their behaviors in the classroom; however, I tried to encourage them to talk about
their actual practices and reasons for them.
My Motivation for the Study

The roots of this study go back in my high school education where I studied
mathematics through theorems and proofs for three years with the same teacher. He used
to say that he was not preparing us for the university entrance examination, but he was
preparing us for the midterm examinations at the university. His teaching was direct,
authoritarian, and included hardly any discussion, similar to my middle and elementary
school experiences in mathematics classrooms. However, the type of mathematics we
were Teaming was different than the mathematics taught at other high schools, a
difference which revealed itself clearly when my Calculus professor in college wrote the
deﬁnition of limit on the board and asked “Is there anyone who knows this deﬁnition?”
This particular deﬁnition was one most applicable to several proofs. Among the freshman
students in mathematics and mathematics education programs, only three students,
including me, raised their hands, all from the same type of high school. This episode

convinced me that I had a different view about the nature of mathematics than others. The

nature of this

new for the o:

 

My fr;

mathematics -

 
    

views about t?
content knou _
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student. nor a
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obseryed 1h :1
such as disa
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Whe
my SIUderrt:
8Xaminazi0
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but that So.

X1:

abourtEac

nature of this difference, which was related to the idea of “proof,” was something totally
new for the other students.

My friends and I did not have any idea about ways of teaching and learning in
mathematics other than the teacher-centered practices from our own experiences. My
views about the teaching and Teaming of mathematics were challenged in pedagogical
content knowledge courses in college, especially when my group made a presentation
about co-operative Teaming. My views were, however, completely changed during my
Masters in Mathematics Education in Turkey since I had a different role, neither a
student, nor a mathematics teacher, but a teacher educator. As a graduate student, I had
the opportunity to read and discuss research and I conducted an experimental study on the
effect of collaborative teaching activities in a 5’h grade mathematics classroom. I
observed that none of the teachers in my study taught by using different teaching methods
such as discovery or collaborative groups, or used different teaching materials such as
hands-on manipulatives, although the school had a great mathematics laboratory.

When I became a graduate teaching assistant in Turkey, I observed that many of
my students relied on study habits they had developed during university entrance
examination preparation. My students, who were preservice teachers, initially believed
that if they solved a lot of practice questions for an abstract algebra course instead of
studying and understanding the theorems, they would pass the course. However, it turned
out that solving a lot of questions was not an effective strategy in college level
mathematics.

My students were challenged similar to the way I was challenged for their views

about teaching and Teaming mathematics. In the early years of the program, they were

challenged ft"
procedures at

students. lat.
I

I
a process of c

mathematics
crafted fascir‘
preservice te-
of student tea-
because of n:
THUS. my Stir.
in the classrc
My Si
Cumculum 5;.
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this system sr

 

challenged for their beliefs that characterized mathematics as a set of rules and
procedures and teaching and Teaming as a transmission process from teacher to the
students. Later, as they took more courses in the program, most of them were involved in
a process of change in their beliefs about the nature of, teaching, and Teaming
mathematics towards non-traditional beliefs. From time to time, I observed them as they
crafted fascinating ideas about how a mathematics classroom could look. However, these
preservice teachers were challenged in a different way when they started two semesters
of student teaching. I received many e-mail messages about how disappointed they were
because of not being taken seriously as a student teacher for their non-traditional ideas.
Thus, my students started to believe that what they had Teamed so far would not be useful
in the classroom.

My students and I were all products of the examination-oriented national
curriculum system in Turkey, which affected most of us all in similar ways during our
pre-college education. Now that we are teacher candidates, teachers, or teacher educators,
this system still inﬂuences us. My students had a chance to realize the nature of
mathematical knowledge and how it should be studied, taught, and Teamed during their
studies in the teacher education program. However, the impact of their training might be
less than expected. Four-year teacher education programs try to bring teacher candidates
to a world of mathematics education which they have never experienced as a student,
after 11 years of experience where they were taught in highly traditional ways. At the end
of their training in teacher education programs, we send our students back to their old

planet as a teacher with expectations that they will change their old practices. We

10

generally do r
program Oil I:

In this
in Turkey in t
experiences c
theirbeliefs. 7
they will stud
Turkey rank c
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education is a

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illiterate. and y

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l

generally do not hear from them anymore. The effectiveness of our teacher education
program on teacher candidates’ beliefs and practices remains uninvesti gated.

In this study, I wanted to explore the effectiveness of teacher education programs
in Turkey in the area of mathematics related beliefs. I wanted to have an idea about how
experiences our students had in the past and in the teacher education program related to
their beliefs. Unfortunately, Turkish high school students do not generally know what
they will study in college due to the national examination system. High school students in
Turkey rank college-major pairs and they are placed in one of these pairs based on their
examination scores. Hence, reaching teacher candidates during their high-school
education is almost impossible. So I could not focus on pre-college experiences of
preservice teachers, but I was able to focus on their experiences in the teacher education
program and gather their reﬂections on the pre-college experiences.

Another major motivation for me was my future status after I complete my
doctoral studies. Since I am going to be a member of the faculty in the same department
where I worked before, I conducted this research to help me formulate my ideas about
how our program should be improved and which necessary steps should be taken in order
to challenge our students’ mathematics related beliefs. By analyzing factors impacting
their beliefs, I am hoping to target my students’ beliefs in a better way in my courses.

Goals and Outcomes of the Study

Driven by the literature and my personal motivation, I investigated preservice and
inservice (ﬁrst—year) middle school mathematics teachers’ beliefs about the nature of,
teaching, and learning mathematics. In particular, I tried to determine how the

Elementary Mathematics Education program affects preservice and inservice (ﬁrst-year)

11

 

middle schOO
mathematics.
middle schoo
Mathematics
mathematics I
graduates be
in Turkey as x
The (11
literature abet
related beliefs
3describes th
presen'ice ant
Order to anal y
deSCnptionS 0
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how the tCac‘n

 

their beliefs a,

conclusjOns 0

 

 

middle school mathematics teachers’ beliefs about the nature, teaching, and Teaming of
mathematics. The primary outcomes of the study are the current beliefs of preservice
middle school mathematics teachers studying in or graduated from the Elementary
Mathematics Education program, the effectiveness of certain courses on participants’
mathematics related beliefs, and the impact of the ﬁrst year of teaching on program
graduates’ beliefs. Suggestions for the future practices of mathematics teacher education
in Turkey as well as future research paths will be an additional outcome of this study.

The dissertation is composed of eight chapters. In Chapter 2, I combine the
literature about the beliefs generally and preservice and inservice teachers’ mathematics
related beliefs, and document the need for this research in the ﬁeld of education. Chapter
3 describes three frameworks which help in analyzing mathematics related beliefs of
preservice and inservice teachers and how I combined these three frameworks into one in
order to analyze the data of this study. The methodology used in the study, with
descriptions of the participants, contexts, instruments, and the procedures are given in
detail in Chapter 4. Chapter 5 reports preservice teachers’ mathematics related beliefs and
how the teacher education program impacted on their beliefs. In Chapter 6, inservice
teachers’ mathematics related beliefs are described in detail while the factors inﬂuencing
their beliefs are described in Chapter 7. The discussion of the ﬁndings and the

conclusions of the study are presented in Chapter 8.

12

TIT This

teachers ma:

 

and first-year
teachers' ma'.‘
beginning tea
order to pros ;
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related belie :1

 

CHAPTER 2

LITERATURE REVIEW

In this study, preservice and inservice (ﬁrst-year) middle school mathematics
teachers’ mathematics related beliefs are explored and the possible impact of preservice
and ﬁrst-year experiences are investigated. The study is derived from research on
teachers’ mathematics related beliefs, mathematics teacher education, and experiences of
beginning teachers. I narrow the literature review to the speciﬁc goals of the study in
order to provide major topics and important ﬁndings of research in the ﬁeldrln particular,
I document (i) the importance and characteristics of teachers’ beliefs and mathematics
related beliefs, (ii) the impact of teacher education programs on preservice teachers’
beliefs in general and mathematics related beliefs in particular, and (iii) the inﬂuence of
ﬁrst-year experiences on beginning teachers’ general teaching and Teaming beliefs and
speciﬁcally mathematics related beliefs.

There are different traditions in belief research. Although most studies investigate
beliefs through interviews about and observations of teaching, surveys are also employed
and written works are analyzed. In some of the studies, not only preservice or inservice
teachers are considered as participants, but also administrators and students are involved.
I describe data collection methods and instruments of the studies brieﬂy in order to
provide better information about how the results are reached.

Research documents the nature and ﬁrnction of teachers’ beliefs and emphasizes
their importance in the classroom. Speciﬁcally, teachers of all grade levels hold certain

beliefs about mathematics and its teaching and Teaming. These beliefs are inﬂuenced by

T3

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certain educa
education. te.
teacher deye'.
teachers her
teaching exp.
program and

One c
beliefs on the
beliefs. Stud.
beliefs about
education sh.
mathematica'
Versehar‘i‘ei, y
malhs’matics
Cannot be 1m
8.0%]st Sc
tnlernced by
are COnsiderc,

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teaching and

 

 

certain educational experiences that the teachers have been through, such as pre-college
education, teacher education programs, teaching experiences in the classroom, and
teacher development programs. Since my focus is on pre- and inservice (ﬁrst-year)
teachers’ beliefs in relation to a specific teacher education program and the ﬁrst year of
teaching experience, I will document studies conducted during the teacher education
program and initial teaching years.

One of the major areas of belief study is concerned with the impact of students’
beliefs on their Teaming and the relationship between teachers’ beliefs and students’
beliefs. Students’ beliefs about mathematics are considered to be the collection of their
beliefs about the classroom context, the self, and the ways they believe mathematics
education should be, and the nature, justiﬁcation, sources, and acquisition of
mathematical knowledge in recent reviews (Muis, 2004; Op’t Eynde, de Corte, &
Verschaffel, 2002). Typically, elementary and secondary students believe that
mathematics is composed of basic arithmetic calculation and application of rules, which
cannot be learned alone and need conﬁrmation from an authority (Lampert, 1990;
Stodolsky, Salk, & Glaessner, 1991). Students’ mathematics related beliefs are mostly
inﬂuenced by their teachers’ beliefs and practices (Carter & Norwood, 1997) and beliefs
are considered to have an impact on the way students learn mathematics, problem
solving, as well as on their motivation in the mathematics classroom (Kloosterman,
2002)

The relationship between teachers’ beliefs and students’ Teaming has multiple
non-linear aspects. Teachers hold several beliefs about the nature of the subject matter,

teaching, and Teaming. They have certain beliefs about how students Team the subject

14

matter, how insti
with the students
beliefs inﬂuence
teachers with d”
mathematics tau
traditional belief
knowledge and t
teaching and Tea
students Team in
students (Calder
instruction in III:
consider instruc'
With students in
Importance of ir
areas will not be

mathematics tea

Sacral t
Epistemology cal
hay -

mowing and h-

I‘IerSn-V PIOfcs

teen .
d’hers' belief
5

chapter, I “iii
If:

matter, how instruction in the classroom should occur, and how a teacher should interact
with the students in the classroom (Borko & Putnam, 1996; Thompson, 1992). These
beliefs inﬂuence teachers’ practices. For example, in Thompson’s (1984) study of
teachers with different mathematical beliefs, the teacher with more traditional view of
mathematics taught mathematics in more traditional ways whereas teachers with non-
traditional beliefs about mathematics emphasized the connected nature of mathematical
knowledge and tried to engage students in knowledge generation through several
teaching and learning activities. Similarly, teacher’s beliefs about students and how
students learn impact their teaching preferences and their communication with the
students (Calderhead, 1996). Therefore, teachers’ beliefs impact the way they consider
instruction in the classroom, which inﬂuences the way students learn and how they
consider instruction. Teachers’ beliefs also inﬂuence the ways they interpret experiences
with students in the classroom (Thompson, 1992). Although many studies address the
importance of investigating students’ beliefs and their relation to teachers’ beliefs, these
areas will not be explored here in order to focus on the preservice and beginning
mathematics teachers’ beliefs.

Several different beliefs such as beliefs about self-efﬁcacy and self-esteem,
epistemological beliefs, and the beliefs about teaching and learning in the content area,
have been investigated in the ﬁeld of education at different levels such as elementary
schooling and higher education, with different populations such as students, teachers, and
university professors. The present study is concerned with preservice and ﬁrst-year
teachers’ beliefs about the nature of, teaching, and learning mathematics. Thus, in this

chapter, I will review research on preservice and beginning teachers’ teaching and

15

[gaming beliefs. 4:

education and If?!

 

Veenman ( 198-ll
address first-year
mathematics teac?
teachers. second;

first-year teaching

 

pet'SpCCtiVe.

The notioi
presented in the P
their characteristi
of teacher educ at
teachers‘ general
teaching. and le:
experiences in tl
beginning teach
Then. I describe
ditTerent groups
for presen‘ice a

. I. .
ImpdcatlOtis f0-

L‘nders1

L N
1% bCEn 3 Cha'

learning beliefs, and mathematics related beliefs with respect to impacts of preservice
education and initial years in the ﬁeld. I borrow the term “beginning teacher” from
Veenman (1984) which he uses to refer to teachers in their ﬁrst three years, in order to
address ﬁrst-year inservice teachers. Although this study is about middle school
mathematics teachers and their beliefs, I will be referring to and discussing elementary
teachers, secondary teachers, and the general impact of teacher education programs and
ﬁrst-year teaching on teaching and learning beliefs in order to provide a broad
perspective.

The notion and the deﬁnition of beliefs (experience-driven constructs) was
presented in the previous chapter. Hence, I continue with specifying teacher’s beliefs,
their characteristics, and importance. Then, I review the studies that document the impact
of teacher education programs on preservice teachers’ and preservice mathematics
teachers’ general teaching and learning beliefs and on their beliefs about the nature of,
teaching, and learning mathematics. A review of how these beliefs are inﬂuenced by
experiences in the early years of teaching follows. While documenting preservice and
beginning teachers’ beliefs, I initially present the general ﬁndings of research reviews.
Then, I describe several studies in different contexts, with different interventions, or
different groups of participants. Finally, I summarize and discuss the literature separately
for preservice and beginning teachers, and address important research points or
implications for teacher education based on the studies reviewed.

Teachers’ Beliefs
Understanding the mental mechanisms behind teachers’ actions in the classroom

has been a challenging topic in educational research. The challenge emerges as a result of

16

 

the nature of set:

 

as beliefs. aIIlIUti;
easily obsen'ed 0

different beliefs L

 

Besides la
that people agree
& Pehkonen. 20L!
constructs are no
conceptions incl:
conceptions as a
beliefs. int'estig;

always been 3 cl

LvndC‘YSI;

believes sometl

because there a
context of clas

beliefs is mor
e

the nature of several interrelated personal constructs impacting teachers’ behaviors, such
as beliefs, attitudes, conceptions, emotions, and values. Among them, beliefs are not
easily observed or detected. Two teachers having the same teaching are likely to have
different beliefs underlying that teaching (Kagan, 1992).

Besides lacking sufﬁcient observable indicators, beliefs do not have deﬁnitions
that people agree on, even when mathematics related beliefs are considered (Furinghetti
& Pehkonen, 2002). Moreover, the lines that distinguish beliefs from other personal
constructs are not well-deﬁned. For example, while some researchers claim that
conceptions include beliefs (e.g., Thompson, 1992), other researchers consider
conceptions as a part of beliefs (e. g., Ernest, 1989). Given the complicated nature of
beliefs, investigating teachers’ beliefs and their relation to teaching in the classrooms has
always been a challenge for educational researchers.

Characteristics of Teachers 'Belieﬂs

Understanding teachers’ beliefs is difﬁcult because beliefs do not reveal
themselves easily. Rokeach (1968) claims that even when a person says that he or she
believes something, “he may or may not be representing accurately what he truly believes
because there are often compelling personal and social reasons, conscious and
unconscious, why he will not or cannot tell us” (p.2). Being in a complicated social
context of classroom, fellow teachers, and school, teachers may not claim what their
actual beliefs are. Although a single belief may not be identiﬁed easily, a collection of
beliefs is more likely to be detectable. This collection is often named as “belief system.”

Belief systems can be considered as cognitive structures and be characterized as

dynamic and subject to restructuring when individuals evaluate their beliefs depending on

17

their experiences (Thompson, 1992). Green (1971) identiﬁes three dimensions of belief
systems depending on the relation of beliefs within the system. The ﬁrst dimension is that
a belief is not independent of other beliefs and there is a quasi-logical structure in which
some beliefs are related to others with a relationship similar to that of reasons and
conclusions. There are “primary” beliefs, such as a teacher’s beliefs about the importance
of clarity in presenting mathematics to the students, and “derivative” beliefs, such as a
teacher’s beliefs about the importance of preparing lesson clearly and being ready to
answer the possible questions that students might ask. The second dimension is that
beliefs might vary in their degree of conviction with the other beliefs that are held in the
system. “Central” beliefs are the most strongly held ones where “peripheral” beliefs are
the ones that are most open to any change or examination. Green notes that beliefs may
be logically derivative, yet psychologically central. The third dimension is related to the
nature of the system: that beliefs are held in clusters and these clusters mostly exist in
isolation from each other. Clustered nature of beliefs makes it possible for one to hold
conﬂicting sets of beliefs. Ambrose (2004) gives an example of belief clusters of
creativity and mathematics. While a teacher may believe that students should be given
opportunities to be creative, he/she may connect this belief to his/her beliefs about art and
writing, but not to his/her beliefs about mathematics. Having creativity opportunities in
art and writing during childhood, but lacking similar experience in mathematics may be
effective in forming these clusters.

Similarly, Rokeach (1968) claims three assumptions about belief systems parallel
to Green’s (1971) identiﬁcations. In addition to identifying the importance of beliefs

within a belief system as central and peripheral, he describes more central beliefs as

18

resistant to change. When a more central belief is changed, the impact will be wide in the
rest of the system, but a change in a peripheral belief will not impact the whole belief
system.

Since the nature of beliefs is necessarily subjective, beliefs and belief systems are
often considered different than knowledge and knowledge systems. Thompson (1992)
provides several distinctive features of beliefs and knowledge that are most relevant to
studying teachers’ beliefs depending on wide array of literature. She argues that beliefs
can have varying degrees of certainty. Although there are contradictory issues among
beliefs that people hold, knowledge is true and certain. Evaluation and judgment of
knowledge require certain norms of evidence and inquiry. However, beliefs lack such
evidence for their evaluation and also they often lack agreed upon criteria on which
beliefs can be evaluated or judged. These characteristics are parallel to what Abelson
(1979) addresses as distinguishing features of belief systems.

Pajares (1992) summarizes the function of beliefs at the end of his review about
teachers’ beliefs. Three important and related functions of beliefs seem to emerge from
the review. First, beliefs ﬁlter new information and experience due to their affective,
evaluative, and episodic nature (Abelson, 1979). Existing beliefs organize and deﬁne new
information gathered through learning and inquiry. Similarly, beliefs systems serve as a
base to recognize, categorize, and organize new experiences (Schiebe, 1970). Although
beliefs may be helpful in the organization of new experiences, they are mostly considered
as limiting factors, especially in teacher education. Second, epistemological beliefs
inﬂuence knowledge interpretation because they deﬁne, select, and ﬁlter cognitive

processes. Nespor (1987) claims that the ways domain speciﬁc knowledge is possessed

l9

and recalled f0
and perception
complex situat
goals. and dire
not have clear
Researi
subject matter
teachers and st
such as school
Remillard, 19s,.
the SUblect are
insen'ice ,V'Cars
orEatiizing the
and reasoning
The way teach
Occur in the C},
a Particular CO}

and recalled for reconstruction are inﬂuenced by beliefs. Third, beliefs impact behavior
and perception but they also might misguide them. They help teachers in dealing with
complex situations in schools, and they aid in simplifying classroom life, identifying
goals, and directing them in particular problem situations. However, these beliefs may
not have clear logical relevance to real-world situations (N espor, 1987).

Research indicates that teachers have several interconnected beliefs about the
subject matter they teach, how they should be teaching it, how it is learned, roles of
teachers and students in the teaching and learning process, and other educational issues
such as school contexts and general goals of education (Putnam, Heaton, Prawat &
Remillard, 1992). Since beliefs are experience-driven mental constructs, experiences with
the subject area, previous teachers, and students during their pre-college, preservice, and
inservice years are considered to impact their beliefs. Teachers use these beliefs while
organizing their practices. What teachers believe about the nature of knowledge, skills,
and reasoning in a content area affects the way they determine the concepts to be taught.
The way teachers teach the concepts is inﬂuenced by how they believe teaching should
occur in the classrooms. And ﬁnally, teachers’ beliefs about ways of learning concepts in
a particular content area impact the learning activities they design for students. Thus,
having been inﬂuenced by their beliefs, teachers’ practices affect the way students
experience the content area, develop an idea about teaching, and what and how they
learn. These experiences inﬂuence their students’ beliefs.

The complicated chain reaction of “teachers’ beliefs impacting their practices
which inﬂuence students’ beliefs” becomes a circle when preservice teachers start the

teacher education programs bringing their beliefs from pre-college years. Educational

20

research focused

 

of teaching and l:
of a simple cause
complex ways (F
maintained from
education pro gra
are influenced bj
reviewed here lT
Pr:

Presen-i.
mathematics an.
mathemamc s 10
lenCIpIES‘ me.
their Col-reﬁnes
asSumpllOns ab
EXPCﬁenCeS “in
teacher- lino“ rr
the teacher ask.
ratified by the t
tﬁachers fmm b
Something m m

without dif 13cm

research focused on each of the links in this circle in order to explain the complex nature
of teaching and learning in classrooms. Research showed that each link is not composed
of a simple cause and effect relationship, but of several factors and contexts interacting in
complex ways (Raymond, 1998). How preservice teachers’ beliefs, which are mostly
maintained from pre-college education, impact the extent they beneﬁt from the teacher
education program experiences and how beginning teachers’ practices in the classroom
are inﬂuenced by their preservice beliefs, experiences, and the contextual factors will be
reviewed here mostly in the ﬁeld of mathematics education.
Preservice Teachers ’ Beliefs: The Impact of Teacher Education

Preservice teachers come to teacher education programs with certain beliefs about
mathematics and its teaching and learning (Cooney, 1999). They generally believe
mathematics to be a static body of knowledge, which is learned by memorizing facts,
principles, formulas, and procedures from a teacher who demonstrates them and controls
their correctness (Wilcox, Lanier, Schrarn, & Lappan, 1992). A closer analysis shows that
assumptions about knowing mathematics in most classrooms are shaped by pre-college
experiences “in which doing mathematics means following the rules laid down by the
teacher; knowing mathematics means remembering and applying the correct rule when
the teacher asks a question; and mathematical truth is determined when the answer is
ratiﬁed by the teacher” (Lampert, 1990, p.32, italics in original). For example, preservice
teachers from both elementary and secondary majors generally believe that to know
something in mathematics means to remember rules and to use the standard procedures

without difﬁculty (Ball, 1990).

21

 

\l’ideen v.
programs of tea
preservice teach;
education pregra

mostly causing it

 

(NCSPOR 198“ i
and law whose Sl
Pajares (1992) C]
do not need to W
not shocked by l
and they do not i
experiences.
There an
changed or not 5
education progr-
psychology. me
teachers belief-s
found to be eff e
some courses dc
CXpegt‘ even Vt}:
0f studies With (
Chan gc -

triathi’rnatjCal Cc

Wideen et al. (1998) claim that “the story of how beginning teachers experience
programs of teacher education begins with who they are and what beliefs they bring to
preservice teacher education” (p.141). Preservice teachers’ experiences in the teacher
education programs are ﬁltered and shaped by their existing beliefs (Pajares, 1992),
mostly causing less impact of these programs on the preservice teachers than expected
(Nespor, 1987). Commenting on Schutz’s (1970) work on professions such as medicine
and law whose students are almost new to the medicine and law teaching contexts,

Paj ares (1992) claims that preservice teachers are insiders in teacher education and they
do not need to reforrnulate the context of teacher education as a new student. They are
not shocked by the people, classroom, and practices in the teacher education programs
and they do not tend to change their beliefs despite the (relatively) new information and
experiences.

There are conﬂicting views on whether preservice teachers’ beliefs can be
changed or not since beliefs are resistant to change (Kagan, 1992). The impact of teacher
education program experiences such as a course or a series of courses (e. g. educational
psychology, methods of teaching mathematics) and student teaching on preservice
teachers’ beliefs have been investigated in different content areas. Some experiences are
found to be effective especially when they target preservice teachers’ beliefs. However,
some courses do not result in change in beliefs in the way or extent that researchers
expect, even when the courses are designed speciﬁcally to change the beliefs. Examples
of studies with different levels of impact are given here.

Change in preservice elementary teachers’ beliefs about pedagogical and

mathematical content knowledge and whether these beliefs inﬂuenced their instructional

22

practices were investigated by Foss and Kleinsasser (1996). In a 16-week mathematics
methods course including teaching three lessons in primary schools, 22 preservice
elementary teachers and the instructor of the course were observed in the course. They
were interviewed in the beginning, middle, and end of the course, and were administered
surveys to investigate their mathematics related beliefs. The course, which was based on
constructivism and discovery, was not designed by the researchers but it was observed for
its impact on the beliefs. The study showed that the preservice teachers’ beliefs about
mathematical knowledge did not change through the course. Preservice teachers mostly
mentioned computation, solving problems, manipulation of numbers, theories, and
equations depending on their personal history and beliefs, but not reasoning and
creativity which were emphasized in the course. Preservice teachers characterized good
mathematics teachers as providing ﬁm and enjoyment in the classroom while decreasing
students’ mathematics anxiety and promoting learning. Preservice elementary teachers in
this study mostly disregarded their methods course experiences and described their future
teaching as a collection of drill and memorization with limited use of textbooks and
worksheets where they would build concrete relationships to daily life. They would
employ group work from time to time, but mostly would depend on what they had
experienced and observed as a student in pre-college education. They believed that
students could learn mathematics when they had a certain type of mind.

As a result of their study, Foss and Kleinsasser (1996) claimed that it would be
difﬁcult to change preservice elementary teachers’ beliefs about pedagogical and content
knowledge of mathematics by a methods course. Preservice teachers still relied on their

prior beliefs and personal experiences while formulating their new experiences in the

23

teacher educatic
beliefs affected
education and rr
teachers. maths’
How We
education COWS"
their ovtn belief
interviews in thz
the tasks and W
the course. Amt:
students whom I
when an un fami
mostly interpret:
unfamiliar. The)
their beliefs. Pre
these examples i
researchers argu:
Where teacher ed
and Specific mat:
teacher educators
their beliefs, mm

A

diff
erWe from tr

teacher education program. They concluded that preservice teachers’ mathematics related
beliefs affected their teaching practice but did not promote it. They suggested that teacher
education and mathematics teaching research should consider moving preservice
teachers’ mathematics related beliefs to a conscious level.

How preservice teachers’ beliefs would change in an introductory teacher
education course which was designed to make preservice teachers examine and reﬂect on
their own beliefs, was investigated by Anderson and Bird (1995). They conducted
interviews in the beginning and end of the semester and analysis of written responses to
the tasks and video cases (teaching writing, social studies, and mathematics) presented in
the course. Among 31 students in the class, the researchers presented the case of three
students whom they believed represented the beliefs of the rest of the participants. Even
when an unfamiliar “eye-opening” video case was presented to preservice teachers, they
mostly interpreted it within their beliefs about teaching so that the case did not look
unfamiliar. They ﬁltered the ideas presented in video cases and related papers through
their beliefs. Preservice teachers in this study extracted what they expected to see from
these examples instead of reﬂecting on own teaching beliefs through them. The
researchers argued for the need for pedagogical content knowledge of teacher education
where teacher educators should consider the interaction between the individual student
and speciﬁc materials and activities of teacher education. They concluded that when
teacher educators decide to use video cases in order to help preservice teachers examine
their beliefs, they should consider (i) the history of the cases, emphasizing their

difference from traditional cases; (ii) accompanying texts and assignments in which

24

 

prfsmice teach;

teachers' perccpi

  
   
   

Joram ar
targeted present
identified a total
learning: (it teaci
experience in the
model; (iii) teach
students would l
used metaphors
implemented a s
end of the COUT?
Whether their b
course. Results
actually had a
some Impact c
targeted lhrou

Students and E
Mosr 1
eIDlStemolOgic

conducred bx-

preservice teachers would use the targeted experiences and ideas; and (iii) preservice
teachers’ perceptions of their roles as teachers more carefully.

J oram and Gabriele (1998) describe an educational psychology course which
targeted preservice teachers’ beliefs and had an impact on them. The researchers ﬁrst
identiﬁed a total of four beliefs about teacher education experiences, teaching, and
learning: (i) teacher education courses would not be that relevant when compared to the
experience in the ﬁeld; (ii) teaching could be learned by considering past teachers as a
model; (iii) teaching and learning in the classroom would not be a problem; and (iv)
students would learn easily when classroom management would maintained. Then, they
used metaphors and analogies in order to target these beliefs. The researchers
implemented a survey with some open-ended questions both in the beginning and at the
end of the course to a total of 53 students. Moreover, they asked the preservice teachers
whether their beliefs about teaching and learning had changed since the beginning of the
course. Results showed that when preservice teachers felt a change in their beliefs, they
actually had a change. Thus, the course designed to target preservice teachers’ beliefs had
some impact on their beliefs. They concluded that preservice teachers’ beliefs should be
targeted throughout the program courses so that further changes would occur with more
students and greater impact.

Most recently, a similar intervention that targeted preservice elementary teachers’
epistemological beliefs about mathematics, and teaching and learning in mathematics was
conducted by Gill, Ashton, and Algina (2004) in order to investigate effective ways of
instruction that inﬂuenced beliefs. The researchers prepared a total of eight mathematics

teaching scenarios, four representing constructivist teaching and the other four

25

representing procedural teaching, in order to understand preservice teachers’ implicit (not
depending on self-reports) epistemological mathematics beliefs. They used these texts for
the experimental group and asked the preservice teachers to rate the excellence of
teaching, while they implemented the traditional texts for the control group in a child
development course. A total of 161 preservice teachers attended to the course. They also
implemented an epistemological mathematic beliefs survey and a general epistemological
survey to both experimental and control groups to investigate their explicit (relying on
self-report) beliefs. At the end of their intervention in which they activated and
challenged beliefs, the researchers were able to increase preservice teachers’ awareness
of their epistemological beliefs. The logical counter-arguments they proposed to
preservice teachers’ beliefs persuaded a change in those beliefs, consistent with the goal
of the intervention. The researchers concluded that the general epistemological beliefs
might be a good place to start to develop the type of mathematics related beliefs that
teacher educators would like to promote.

Many studies report no change in preservice teachers’ beliefs because most of the
study participants did not change these beliefs. Nettle (1998) claimed that when there
seemed to be no impact of teacher education programs on preservice teachers’ beliefs,
there was actually some impact which was often underestimated. Nettle investigated the
stability and change in 79 preservice elementary teachers’ beliefs about teaching in
relation to a 3-week period of student teaching. Comparison of pre- and post-
questionnaires showed that majority of participants maintained their beliefs about
teaching after practice teaching, most probably due to the impact of their prior beliefs.

However, there were changes in some of the participants’ beliefs. The change occurred in

26

 

Presert'iCe 163C
and intﬁrpersoI
as prioritlZlng 5
task). Nettle d1:
ineffectiveness
development of
Ambros-
experience on p
preservice teach
addition to a cor
closely with eler
teachers written
mathematics to a
successful in hel;
Preservice teache
“Wiring to the
also believed that

 

the new ones I)
. k.

_VD
\,
S

WOI beliefs mj

program. The kC'

clememaﬂ' SChO(

 

preservice teachers’ beliefs, as affective oriented teaching (such as prioritizing motivation
and interpersonal relations in learning) moved towards more task oriented teaching (such
as prioritizing students’ active involvement and structuring of learning to complete a
task). Nettle discusses that instead of interpreting the stability as an indication of the
ineffectiveness of teacher education programs, considering it as a part of the process of
development of preservice teachers’ beliefs would be a better approach in research.

Ambrose (2004) and her research team investigated the impact of early
experience on preservice teachers’ beliefs. They identiﬁed several beliefs that they hoped
preservice teachers would develop and conducted an experimental methods course in
addition to a content course in which 15 preservice teachers attended and worked in pairs
closely with elementary school students. The results of interviews, surveys, preservice
teachers’ written works and ﬁeld notes showed that the intense experience of teaching
mathematics to an elementary school student and reﬂection on this practice was partly
successful in helping preservice teachers to develop new beliefs. Yet, most of the
preservice teachers maintained their beliefs that teaching mathematics is explaining
everything to the students instead of giving them opportunity to think. Preservice teachers
also believed that mathematics learning would be attainment of symbolic procedures.
Ambrose concluded that preservice teachers maintain their prior beliefs while forming
the new ones. Depending on the effectiveness of the teacher education experiences, these
prior beliefs might become less central as preservice teachers progress through the
program. The key to effectiveness, in this case, was the intense experience with

elementary school students accompanied by reﬂection.

27

 

1n th=

aPPeari F055
6, ,1, 2004) 1‘:
investigated !
While Ambro
integrated a C‘
mimentions ‘
time period. Ct
Single
preservice teac
the teacher edu
addressed in be
their pre-col leg!
teaching (Feima
practice ( C oone.
beliefs might no
In order t
interventions sho

(1993) claim H0

 

In the six studies described in detail above, the ﬁrst three studies (in the order they
appear: Foss & Kleinsasser, 1996; Anderson & Bird, 1995; Joram & Gabriele, 1998; Gill,
et a1, 2004) investigated the impact of a single course over one semester. Nettle (1998)
investigated the impact of 3 weeks ’of practice teaching on preservice teachers’ beliefs
while Ambrose (2004) conducted a semester-long course in which preservice teachers
integrated a content area course and intense teaching practice. Although these
interventions were carefully designed, even the semester-long course is a relatively short
time period, compared to the 4 or 5 year period of teacher education programs.

Single course interventions might be found to be successful in changing
preservice teachers’ beliefs, but what part of changed beliefs is maintained at the end of
the teacher education programs and carried to the initial years of teaching is rarely
addressed in belief studies. Research indicates that inservice teachers generally depend on
their pre-college content knowledge and beliefs in the ﬁrst few years of real-classroom
teaching (Feiman-Nemser, 2001 ), where their beliefs are likely to limit their classroom
practice (Cooney, 1985). Hence, the impact of interventions on preservice teachers’
beliefs might not be long enough to serve them through their initial years in teaching.

In order to have better and longer impact on preservice teachers’ beliefs,
interventions should be consistent through the program courses, as J oram and Gabriele
(1998) claim. However, they also address two related factors that underlie the difﬁculty
in changing preservice teachers’ beliefs about teaching and learning through teacher
education programs. First, preservice teachers’ beliefs are reinforced through observing
other teachers and their own teaching experiences. Practices that might challenge their

beliefs are mostly not present in the context of everyday practice. Second, there is a lack

28

 

of feedback in t.
teachers to char
practice teachir:
likely to lead to

In three

Bird. 1995; Jor;

 

order to challen
that if teacher e
they will be me
first step in this
Although both
this idea the tr
Stlggest the net

presen’iCe [eat

that al

low for
insenice Exm
teaching and i
make it dimer
Although ref].

is also known

change their C

of feedback in the context of teaching and learning which would convince preservice
teachers to change their beliefs. Hence, preservice teachers tend to evaluate their own
practice teaching through their existing beliefs about teaching and learning, which is not
likely to lead to belief change.

In three of the studies described above (in the order they appear: Anderson &
Bird, 1995; Joram & Gabriele, 1998; Gill, et a1, 2004) researchers planned the courses in
order to challenge and inﬂuence preservice teachers’ beliefs. Wideen et al. (1998) suggest
that if teacher education programs are designed considering teacher candidates’ beliefs,
they will be more productive in promoting learning to teach. Wideen et al. address the
ﬁrst step in this process by making preservice teachers reconsider their own beliefs.
Although both the Anderson and Bird (1995) and Gill et al (1998) studies were based on
this idea, the impact was limited, especially in the Anderson and Bird study. This might
suggest the need for more careful planning including feedback or counter-arguments for
preservice teachers’ beliefs, similar to what was done in the Gill et al. study.

In summary, preservice teachers’ beliefs are considered as very important factors
in designing and providing conditions and opportunities in teacher education programs
that allow for the transfer of knowledge and skills from preservice experiences to the
inservice experiences (Cooney, 1985). However, preservice teachers’ existing images of
teaching and learning at schools decrease the impact of teacher education programs and
make it difﬁcult for the teacher educators to estimate the impact of their programs.
Although reﬂection is a key in inducing change in beliefs during preservice education, it
is also known that it is very difﬁcult to convince preservice teachers to examine and

change their own beliefs and there is always a tendency to ﬁlter the new experience based

29

on the existing "

 

should b6 baSCCI
building on the“
Bc‘g

Beginntt

beliefs as referei
teacher. beginni
the first time. T
they have been 1
time. effort. and
hlagliaro. 8; Mt
impact on prese
reduced to their
preexisting belt
some beliefs g:
realize the diff:
Students. cum'c
(Flores & Day;
bt-‘cause of thei
V’eenm
beginning teac
their ﬁFSt three

on the existing ones (Pajares, 1992). Hence, it is often suggested that teacher education
should be based on preservice teachers’ beliefs, not in the sense of changing them, but
building on them (Ambrose, 2004; Wideen, et al, 1998).
Beginning Teachers ’ Beliefs: The Impact of Real School Contexts

Beginning teachers start to teach in schools with their preservice trainings and
beliefs as reference points. Even though they have some teaching experience as a student
teacher, beginning teachers have full responsibility for teaching one or more courses for
the ﬁrst time. They have to deal with two main assignments as a teacher: to teach, which
they have been hired for, and to learn to teach. However, the latter in particular requires
time, effort, and resources which beginning teachers mostly lack (Wildman, Niles,
Magliaro, & McLaughlin, 1989). As studies showed that teacher education has limited
impact on preservice teachers’ beliefs, what beginning teacher can draw upon is mostly
reduced to their student teaching experiences and beliefs ﬁom pre-college years. These
preexisting beliefs are more central and more inﬂuential on their practice compared to
some beliefs gained during the teacher education. Moreover, beginning teachers do not
realize the difﬁculty of simultaneously dealing with various factors and actors such as
students, curriculum, parents, and administrators, until entering the school contexts
(Flores & Day, 2006). The nature and causes of these difﬁculties are described here
because of their relation to beginning teachers’ beliefs carried to the initial years.

Veenman (1984) gives a comprehensive review and analysis of studies about
beginning teachers’ perceived problems. He deﬁnes beginning teachers to be teachers in
their ﬁrst three years and investigates the nature of and reasons for the “reality shock”

and changes in attitudes and behaviors of beginning teachers. In reviewing the studies

30

from different c

 

to carry out the
beginning teacl:
“Reality
formed during t
(Veenman. p_1.‘.
teachers' reality
(1998) reviexye '
teachers expen‘c
not sufficiemry .
and Change of a
it contradicts rh
Change ﬁOm pp
Change in the S;
in Veenman. 19
1" SUmr
b€c0mjng a bet
college S‘Udem

\r

and it

[HIM-emit); ralhc

I' .. .

education Den'
c

from different countries, he identiﬁes difﬁculties that appear while beginning teachers try
to carry out the tasks. These difﬁculties are likely to cause delays in the goals of
beginning teachers.

“Reality shock” is the term used to address the “collapse of the missionary ideals
formed during teacher training by the harsh and rude reality of everyday classroom life”
(V eenman, p.143, 1984). Although the term suggests a short time period, beginning
teachers’ reality shock is continuous and takes longer time. Similarly, Wideen et al.
(1998) reviewed empirical studies about learning to teach and documented that beginning
teachers experience reality shock in their ﬁrst-year of teaching, especially when they are
not sufﬁciently prepared to teach. Reality shock reveals itself through change of behavior
and change of attitudes. Beginning teachers are likely to change their teaching even when
it contradicts their beliefs, due to external pressures. When beginning teachers’ practices
change from progressive teaching to traditional teaching, their attitudes and beliefs
change in the same direction as well (Mﬁller-Fohrbrodt, Cloetta, and Dann, 1978, as cited
in Veenman, 1984).

In summarizing several studies, Veenman (1984) claims that the process of
becoming a begining teacher starts with a rather traditional period of instruction as pre-
college students, continues with a relatively progressive period in a teacher education
program, and is followed by a powerful reformulation of beliefs and attitudes towards the
traditional ones. He cautions that the progressive period may be the general impact of the
university rather than the teacher education program. Moreover, the impact of the
progressive period is likely to differ by personality, nature of the subject matter, teacher

education period, and the conditions in the schools. When beginning teachers have more

31

 

contradictions
the reality of if

The im
they have more
students are m«
addresses seye
The most men‘
students’ indiy
students. Simii
and parents as
Where their ex
experiences, e
Examples of s
beliefs and 3X'

In a St:
mathematics c
methodS Conn
“18 irtVeSuga1
teaching, The
Presenice lea
included ﬁcld

contradictions between their ideals developed through teacher education programs and
the reality of the schools, they tend to develop authoritarian attitudes and beliefs.

The impact of the teacher education courses on beginning teachers decreases as
they have more experience at school, where structure of schools, administrators, and
students are more likely to exert an impact on beliefs, as Veenman (1984) reports. He
addresses several problem areas of beginning teachers’ that are most seriously perceived.
The most mentioned ones, classroom discipline, motivating students, and dealing with
students’ individual differences, are related to the interaction between the teacher and the
students. Similarly, Wildman et al. (1989) point to students, colleagues, school context,
and parents as major sources of inﬂuence on the socialization of beginning teachers,
where their existing beliefs about these populations often contradict with their
experiences, especially in the case of beliefs about and experiences with students.
Examples of studies documenting the impact of contextual factors on beginning teachers’
beliefs and experiences are reported here.

In a study conducted by Wilcox et al. (1992), the impact of a series of three
mathematics courses on number theory, geometry, and probability and statistics, a
methods course, and a curriculum seminar on preservice teachers’ knowledge and beliefs
was investigated in the last two years of teacher education program and the ﬁrst year in
teaching. The courses were developed to give more conceptual knowledge to the
preservice teachers about mathematics and teaching and learning mathematics. Data
included ﬁeld notes and videos of mathematics courses and audio recordings of small
group work in the mathematics content courses and questionnaires, and samples of

preservice teachers’ works. Among the 23 preservice teachers, three were selected for

32

 

more intens
teaching.
The
about the n
experience
responsibil
of instmcti
their belie
mathemat
teaching,
district. a
inﬂuence
leacher (
and not .
differed
SWent
more it
researc
modey
Cgm’st
than 1

Schm

more intensive study and were interviewed and observed during their ﬁrst year of
teaching.

The three beginning teachers in the Wilcox et al. (1992) study had several beliefs
about the nature, teaching, and learning mathematics as a result of their previous
experiences as a student and their preservice education. Although participants had similar
responsibilities such as providing learning activities, creating mathematical tasks, and use
of instructional time, they responded to these responsibilities in different ways due to
their beliefs. These beliefs were about mathematical knowledge, effective ways of
mathematics teaching, and the extent that they viewed the context as limiting their
teaching. Moreover, there were contextual factors such as time, curriculum guides in their
district, and expectations of other teachers and supervisors. These factors had less
inﬂuence when the participants had a well-established set of beliefs. In one case, a
teacher did not have a deﬁnite set of beliefs, which made her accept other teachers’ ideas
and not evaluate their appropriateness for her goals. These other teachers’ practice
differed along several dimensions, including the extent to which teachers engaged
students in problem solving, provided more exercises, or made mathematics classroom
more interesting through different tasks without considering their connectedness. The
researchers concluded that even providing preservice teachers with non-traditional
models of mathematics teaching and learning integrated with their content knowledge
courses might not be sufﬁcient to develop knowledge and beliefs about teaching other
than knowledge transmission. They suggested that the contextual factors existing in

schools where they teach as beginning teachers and how these factors challenge newly

33

formed orienta

 

educators.

The lin‘

 

l

practices is rep
investigated be
impacting their

between the be

 

teachers who w
interviews. cfas
Among
traditional mat
mathematics. 1
collection of r
beliefs about 1
activities ma.
reasoning an
mathematics
teaCher~c ente
quietly follm
EVaanted S IL

panicipant 1'

teaching ma'

formed orientations toward teaching and learning should be considered by teacher
educators.

The limiting impact of contextual factors on beginning teachers’ beliefs and
practices is reported in several studies. Raymond (1997) conducted a study in which she
investigated beginning elementary teachers’ mathematics related beliefs, major factors
impacting their beliefs and mathematics teaching practices, and the degree of consistency
between the beginning teachers’ beliefs and practices. She worked with six beginning
teachers who were graduates of the same teacher education program by conducting
interviews, classroom observations, analysis of written samples, and questionnaires.

Among these beginning teachers, Raymond (1997) focused on one with
traditional mathematics beliefs and non-traditional learning and teaching beliefs about
mathematics. Her traditional beliefs about mathematics characterized mathematics as a
collection of numbers, facts, and rules that should be memorized. She had non-traditional
beliefs about teaching and learning. For her, teachers should provide students with
activities, manipulatives, and different views. Students could learn through discovery,
reasoning, and working in groups. Although her beliefs about teaching and learning
mathematics were mainly non-traditional and student-centered, her teaching practice was
teacher-centered. She depended on questioning rather than discovery where students were
quietly following her teaching. She did not use manipulatives during her teaching and she
evaluated students’ learning through their correct answers on tests and homework. This
participant taught in traditional ways inconsistent with her non-traditional beliefs about

teaching mathematics.

34

ln er
participant:
testing. and
concern anc
(1997) clair
inconsisten
generally. t
experience:
beliefs and
CXperience
mathematit
major factr
Showed th;
and learnir
WhiCh Wag
progranlSt
they Were .

Th.
unitverSal.
3511“}be
teachefsi It
the panicip

COurseS in .

In explaining these inconsistencies between her beliefs and practices, the
participant addressed time concerns, lack of resources, the pressure of standardized
testing, and students’ behaviors as possible reasons. Among them, she ranked time
concern and classroom management as the major causes of inconsistency. Raymond
(1997) claimed that since contextual factors put pressure on the participant, she had
inconsistencies between her non-traditional beliefs and traditional practices. More
generally, beginning teachers in this study claimed that the impact of preservice
experiences on their teaching practice was negligible and the relationship between their
beliefs and preservice training was moderate. Moreover, they indicated that pre-college
experiences were the major factor inﬂuencing their beliefs about the nature of
mathematics, while their teacher education and student teaching experiences were the
major factors impacting their beliefs about teaching and lemming. The results of this study
showed that even when beginning teachers have'non-traditional beliefs about teaching
and learning, they inclined towards traditional practices due to the contextual factors,
which was mentioned by Veenman (1984). Raymond concluded that teacher education
programs would have more indirect impact on teaching practice of beginning teachers if
they were designed to inﬂuence preservice teachers’ beliefs.

The factors beginning teachers face when they start teaching seem almost
universal. Recently, similar results were found in Portugal with 14 beginning teachers in
a study by Flores and Day (2006). The researchers investigated the ways beginning
teachers’ identities were shaped and reshaped during the ﬁrst 2 years of teaching. Most of
the participating teachers were in rural contexts, teaching middle school and high school

courses in areas such as physics, chemistry, biology, mathematics, music, and language,

35

with additiona

 

implemented t.
essays about t't‘
report. and the .
influencing be,
students; (ii) pr

COHICXIS.

 

 

Thepa
yiewed themse
their obsen'ati
situations. Te:
Participants v"
Choice of mo:
PrOgrams did
they exMiter
CIaSSTOQm Te
mlSmatcl-t be
assume as b
beginning U
centered pr;
which Rah
about teach

expeﬁ eiice

with additional school duties. Through semi-structured interviews, questionnaires
implemented to all staff and students in the school contexts participants taught, students’
essays about the change in their (participant) teachers, beginning teachers’ annual formal
report, and their reﬂections about own practice, three main factors were found
inﬂuencing beginning teachers’ identities. They are (i) prior experiences as pre-college
students; (ii) preservice training and student teaching practices; and (iii) teaching
contexts.

The participants considered their former teachers as role models when they
viewed themselves as teachers. They expressed the inﬂuence of their teachers and how
their observations of these teachers helped them in responding to several practical
situations. Teacher education programs had a relatively weak impact on the way
participants viewed themselves as teachers because teaching was not the ﬁrst profession
choice of most of the participants. Beginning teachers believed that the teacher education
programs did not prepare them sufﬁciently for the complexity of school teaching and
they experienced a disparity between their ideals as a preservice teacher and the
classroom reality. When they considered their teaching contexts, they addressed a
mismatch between their initial beliefs about teaching and the roles they were expected to
assume as beginning teachers. Initially, classroom management was a problem for the
beginning teachers. Classroom management concerns made them incline towards teacher-
centered practices even though their initial beliefs were not teacher-centered, a ﬁnding
which Raymond (1997) also addressed. Some of the participants revisited their beliefs
about teaching and teacher identity as a result of the contradiction between what they

experienced and what they actually wanted to experience in the classroom. Although they

36

 

initially menti
(cg. discover:
actual teachin,
on beginning i
contexts w as t
that previous t
factors in con:
professional it

The st:
teachers have
the real teachi
managmcnt.
teaching Come
1971215 cited
Classroom ma]
I“Chm haVe
ieamjng. the}:

0rCler. Draw in

initially mentioned responding to students’ learning and using various teaching methods
(e. g. discovery, manipulatives, group work) as characteristics of good teachers, their
actual teaching did not model these beliefs. Despite the impact of previous experiences
on beginning teachers’ beliefs about their teacher identity, the impact of teaching
contexts was the key in reshaping these identities. Flores and Day (2006) study showed
that previous experiences, teacher education programs, and teaching contexts were the
factors in construction, deconstruction, and reconstruction of beginning teachers’
professional identities.

The studies about beginning teachers summarized above showed that beginning
teachers have to deal with contradictions between their prior beliefs and the constraints of
the real teaching contexts. They have several survival concerns such as classroom
management, students’ learning, building relationship with the students, limitations of
teaching contexts, availability of resources, and effective use of time (Fuller & Brown,
1975, as cited in Veenman, 1984), that they have to deal simultaneously. Among them,
classroom management seems to be the most important problem. Even when beginning
teachers have non-traditional or mostly student-centered beliefs about teaching and
learning, they tend to perform teacher-centered practices in order to maintain classroom
order. Drawing from his study on inservice development programs Guskey (1986, as
cited in Paj ares, 1992) indicates that change in behavior is not a result of change in
beliefs, but is a cause for belief change. Hence, it is likely that beginning teachers’
teacher-centered practices may result in teacher-centered beliefs as they become more

experienced in the ﬁeld.

37

 

Person
make sense 0i
beliefs 31‘0”t 1‘:
However. pets
and beginning
teacher educali
realize the COP"
contexts will V
experieriCe and
beginning ‘53"

Studie.»
and Raymond =
education. In if
school admtnn
beginning teae
participants in
participants b.-
beliefs with critl
txpcnences ur,
years of teachi‘
ripenences in
would inform r

Processes that r.

 

Personal history seems to play an important role in the way beginning teachers
make sense of real teaching contexts. They start teaching with generally well-established
beliefs about their roles, students’ roles, and the way content should be considered.
However, personal history often contradicts the cultural contexts of schools. Preservice
and beginning teacher research has concluded that there is a need for an approach in
teacher education programs by which preservice teachers with different beliefs will
realize the complicated nature of classroom and school contexts and understand that these
contexts will vary even in the same school (Holt-Reynolds, 1992). Opportunities to
experience and reﬂect on personal biography and the contexts of schools may help
beginning teachers deal with possible contradictions better (Flores & Day, 2006).

Studies focusing only on teachers’ initial years, such as Flores and Day (2006)
and Raymond (1997) provide valuable information for teacher induction and teacher
education. In the case of the Flores and Day study, participants’ students, peers and
school administrators were considered as a source of information for investigating
beginning teachers’ difﬁculties in the ﬁeld. However, the preservice experiences of
participants in these two studies remain unexplored. Hence, inferences made for
participants’ beliefs gained during preservice education and for the interaction of these
beliefs with contextual factors in teaching lack information about the nature of
experiences underlying these beliefs. Research that comprises both preservice and initial
years of teaching and investigates changes in preservice teachers’ beliefs through the
experiences in both preservice and inservice contexts, such as Wilcox et al. (1992) study,
would inform research and practice in both teacher education and teacher induction,

processes that should be a continuum (Feiman-Nemser, 2001).

38

Summary and Conclusions

Research on beliefs shows that preservice teachers’ beliefs about teaching,
learning, and content are generally resilient to change. Research on the impact of teacher
education documents that even when the interventions were carefully designed to change
preservice teachers’ beliefs, several factors reduced the impact of these interventions.
Among these factors, the most important ones were preservice teachers’ pre-college
beliefs and their overall perception that teacher education programs are not effective in
learning how to teach. Moreover, any changes that might have occurred in preservice
teachers’ beliefs during time in teacher education programs did not persist beyond the
beginning years. Contextual factors such as classroom management and students’
differences generally did not provide opportunities for beginning teachers who had non-
traditional beliefs to maintain those beliefs and conduct practice reﬂecting them.

As this review suggests, the research literature on teachers’ beliefs still has
missing pieces or uninvestigated areas. First, beliefs of teachers in the Turkish
educational context, which has several features that are different from the educational
context of the US, has not been extensively explored yet. Among the contextual
differences, the ﬁrst one is the informal teaching (which will be explained in the Methods
chapter) that preservice teachers in Turkey are generally involved before they graduate
from teacher education programs. Unlike in the US, preservice teachers in Turkey often
have considerable teaching experiences when they start to teach full-time. Second,
teacher education programs in Turkey do not involve an internship year. The internship
period is the ﬁrst year of teaching in Turkey and it is independent of the teacher

education programs at universities. Although there are mentor teachers ostensibly

39

responsible for
their mentors.
are inspected e
lesson plans a:
previous monzlt
national curric
and presents te
curriculum pa;
educational cor
years, teacher t
Most 0.
different from
and schooling.
ICECl’fers’ math
for Turkey, Th
Preservice and
national Cumct
COHIEXtua] fCtCI
A SeCor
ofthe naIlOnat
addressed in St:
contexrs are m

PiactjCe leg s

 

:sponsible for monitoring beginning teachers, most beginning teachers do not even meet
teir mentors. Elementary school teachers (including content area teachers for grades 6-8)
’6 inspected every year by the Ministry of National Education. Beginning teachers’

:sson plans and practice are generally evaluated only by the inspector, without any
revious monitoring or reﬂection that a mentor could have provided. Finally, there is a
ational cuniculum to follow strictly which does not always address the students’ needs
rd presents teachers with the dilemma of responding to students’ needs or keeping the
miculum pace (Yildirim, 2001). Thus, the national cuniculum and related informal
iucational contexts frame preservice and beginning teachers’ experiences in pre-college
ears, teacher education programs, and initial years in teaching in Turkey.

Most of the studies described here were conducted in the US, which is extremely
ifferent from Turkish context in terms of the structure of teacher education programs
rd schooling. Hence, the ﬁndings of previous research about preservice and beginning
:achers’ mathematics related beliefs cannot be considered to have strong implications
)r Turkey. Therefore, there is a need for Turkish studies in order to investigate
reservice and beginning teachers’ mathematics related beliefs in preservice training and
ational curriculum context. Such studies will address both universal themes in and
)ntextual factors for belief research.

A second unexplored area in the research literature on beliefs concerns the effect
fthe national curriculum on teachers’ beliefs. Teachers’ beliefs are not generally
idressed in studies conducted in national curriculum contexts. National curriculum
)ntexts are mostly studied in terms of student learning (e. g., Strand, 1997), teachers’

ractice (e.g., Silcock, 1992), and issues appearing as a result of policy change such as

40

 

 

assessment (e
for the beginn:
classroom ma:
Johnson. Kant“
the national cu
teachers’ belie
As the
indicates a nee
teachers belt»;
education pro;
beliefs and im
not eitplore th;
Process. Long:
are rare in the .
different Periot
teachers belie:i
their initial he'd
tidUCators in dc
throngh the iny
The int .

afourth nil-351.:

 

SUbjeWVi” du-

through mm

assessment (e.g., Gipps, 1992). Moreover, a national curriculum is perceived as helpful
for the beginning teachers in dealing with difﬁculties such as effective lesson planning,
classroom management, and employing suitable pedagogical approaches (Kauffman,
Johnson, Kardos, Liu, and Peske, 2002). However, to what extent this help is present in
the national curriculum context and how the context impacts and interacts beginning
teachers’ beliefs have not been addressed yet.

As third area that is not represented in the literature, existing research also
indicates a need for extended scope in investigating the change or development of
teachers’ beliefs. Belief studies generally focus on relatively short periods of teacher
education process or initial years while describing preservice and beginning teachers’
beliefs and investigating factors impacting them. However, these snapshots of beliefs do
not explore their developmental process, how they were shaped, and what impacted the
process. Longitudinal studies investigating beliefs of preservice and beginning teachers
are rare in the ﬁeld of education, leaving the impact of different experiences gained at
different periods unexplored. Thus, there is a need for investigating the development of
teachers’ beliefs beginning from early years in the teacher education programs through
their initial years as a teacher. Such research will address critical issues for teacher
educators in designing interventions that start from preservice years and continues
through the initial years.

The inconsistencies among the data analysis procedures in belief research present
a fourth missing piece in the belief literature. Belief research presents a problem of
subjectivity due to the nature of beliefs. Many belief studies investigate teachers’ beliefs

through interviews which are analyzed without any deﬁnite framework. Moreover,

41

researchers ge
as deﬁnition C
beliefs for the
from the his”
consider conSI
researChers dc
methOds and d
researchers gel
verbal data: M
researchers Tl
that would pro

The lack 0f fra

 

educational c0

Consid=
preservice and
a cross-section
longitudinal ar.
contributes to a
First. it int'esti,
beliefs through
literature inyes

overall impact

addresses a mo

researchers generally do not present the degree of consistency of basic assumptions, such
as deﬁnition of beliefs and modes of inquiry, across the studies from which they derive
beliefs for their own studies. Thus, data analysis is based on the beliefs researchers derive
from the literature, depending on what they believe to be important and the degree they
consider consistency. A similar risk of poor data analysis exists in studies where
researchers derive codes from their data where data is inﬂuenced by data collection
methods and derived beliefs are inﬂuenced by researchers’ decisions. This shows that
researchers generally use subjective data analysis tools to infer teachers’ beliefs from
verbal data, which is likely to produce inconsistent results when used by other
researchers. Thus, there seems to be a lack of a common framework in belief research
that would provide more consistent and less idiosyncratic analyses of teachers’ beliefs.
The lack of framework also limits the results of belief research across different
educational contexts or different groups of teachers.

Considering previous studies’ ﬁndings and missing pieces of this literature,
preservice and ﬁrst-year teachers’ mathematics related beliefs were investigated through
a cross-sectional analysis of the impact of a teacher education program in Turkey and a
longitudinal analysis of the impact of ﬁrst-year teaching. The present study addresses and
contributes to studies about inservice and preservice teachers’ beliefs in several ways.
First, it investigates year level differences in preservice teachers’ mathematics related
beliefs through the teacher education program. As documented previously, the existing
literature investigated the inﬂuence of short interventions, but did not investigate the
overall impact of teacher education programs through the years. The present study

addresses a more comprehensive investigation of the teacher education program through

42

 

the year I
inscryice
the teach
few studi
teaching.
inrestiga
program .
presen’ict
pretiousl;
instead of
extensive}
is the (lat;
studies lll‘
from the c
and ll'lt‘Ott

in detail. a

the year level groups. Second, this study explores the impact of ﬁrst-year teaching on
inservice teachers’ mathematics related beliefs by investigating their beliefs at the end of
the teacher education program and at the end of the ﬁrst year. The existing literature has
few studies which explore beginning teachers’ beliefs before and after their ﬁrst-year
teaching. Hence, the present study will contribute to this rarely explored area by
investigating ﬁrst-year teachers’ beliefs when they graduate ﬁom the teacher education
program and after the ﬁrst year of teaching. Third, the present study investigates
preservice and inservice teachers’ beliefs in a national curriculum context. As mentioned
previously, studies in national curriculum contexts were mostly on students’ learning
instead of teachers’ mental processes. Thus, their impact on teachers’ beliefs has not been
extensively explored yet. And ﬁnally, the present study employs a theoretical framework
as the data analysis tool in order to have more consistent results. Unlike the previous
studies investigating teachers’ beliefs by deriving beliefs from the literature or codes
from the data, the study employs a comprehensive framework with consistent empirical
and theoretical fundamentals. The framework, which will be explained in the next chapter

in detail, is potential contribution to the belief research literature.

43

 

A CO)

In it
transcripts c
framework i
proposed by

As rr
analysis fran
ignored and i
do not tend u
number of ex
fTaII‘tCWorks d
assumptions.

Stud-V b." 76’0-

 

 

 

CHAPTER 3

A COMBINED FRAMEWORK OF MATHEMATICS RELATED BELIEFS

In this chapter, the framework that was used to analyze the data (interview
transcripts of preservice and inservice teachers) in this study is described in detail. This
framework integrates three similar models of teachers’ mathematics related beliefs
proposed by Thompson (1991), Lindgren (1996), and Ernest (1989).

As mentioned previously, prior research has problems with inconsistent data
analysis frameworks where the assumptions underlying these frameworks are often
ignored and data analysis tools are highly subjective in nature. Moreover, belief studies
do not tend to use existing data analysis frameworks, but provide more to the high
number of existing frameworks. A contribution of this study is a combination of three of
frameworks derived from the literature considering the consistency across their
assumptions. In other words, a major problem in belief research is eliminated in this
study by recycling existing consistent frameworks.

Thompson’s Framework

Thompson (1991) claims that teachers’ conceptions of mathematics are affected
by personal previous educational and instructional experiences and the way these
experiences were interpreted and internalized by the teachers. She uses “conceptions”
instead of “beliefs” but she also claims elsewhere that the difference between these two
concepts might not be drastic (Thompson, 1992). She formulizes conceptions as bi-
products of experiences (Thompson, 1991 ), which is the way beliefs are considered in

this study. Thus, the conceptions she addressed will be considered as beliefs as well.

44

 

She
teachers‘ Ct
results ofh:
(1991) clair
consistent p
(138). Since
schemes of t
Viability of h

In [hi
conceptions i
Characterizes
mathematics;
eVidcnces of 5
acceptability
are {men belo
APpﬁndix A).

Lm'el (

 

HCUCe teach“

 

The IQLI
classroom and

bﬁCOme a habi

 

She proposes a framework for investigating and analyzing the development of
teachers’ conceptions of mathematics teaching. This framework is developed using the
results of her ﬁve-year work with seven preservice and ﬁve inservice teachers. Thompson
(1991) claims that the framework documents what she has observed as a “fairly
consistent pattern of development of teachers’ conceptions of mathematics teaching”
(p.8). Since her framework is limited to the experiences and the existing conceptual
schemes of the teachers she worked with, she asks other researchers to examine the
viability of her framework.

In this framework, she categorizes development of teachers’ mathematics related
conceptions in three developmental levels: Level 0, Level 1, and Level 2. She
characterizes the levels depending on the conceptions of: (a) mathematics; (b) learning
mathematics; (c) teaching mathematics; (d) roles of teachers and students; and (e)
evidences of student knowledge and criteria” for judging correctness, accuracy, or
acceptability of mathematical results and conclusions. The characterizations of the levels
are given below (the complete list of Thompson’s (1991) characterizations is given in
Appendix A).

Level 0. Mathematics is conceptualized as using arithmetic skills in daily life.
Hence teaching mathematics is focusing on the development of students’ skills in
arithmetic. This is performed through memorization of the mathematical knowledge
which is composed of facts, rules, formulas, and procedures.

The teacher’s role is limited to demonstrating the facts and procedures in the
classroom and the student’s role is to imitate and practice those procedures until they

become a habit. The goal of mathematics teaching and problem solving at this level is to

45

 

implement th
demonstrated
The teacher 0
knowledge.
Level
niles. but the
aresult of the
However. [llla
manipulative
rather increas
SOIVing is see
Separately wt:
mathematics,
Tl'lC lc‘
ll lllCludeg 30‘
“lance in It.
whom}: th
Lel'ej ‘

the nature of.

 

rea“Zed at th«
lQChing {Or Ll

aglllde in Cat

 

implement the correct procedure or obtain the correct answer, usually in the ways
demonstrated in the class. Mental processes are not considered during problem solving.
The teacher or the book is generally considered as the authority for mathematical
knowledge.

Level 1. At this level, mathematics is still considered as a collection of facts and
rules, but the principles behind the rules are realized. This slight shift is considered to be
a result of the use of instructional representations and manipulatives in teaching.
However, this new pedagogical approach to teaching mathematics (such as use of
manipulatives) is not considered as a way of improving conceptual understanding, but
rather increasing the enjoyment of students in the mathematics classroom. Problem
solving is seen as isolated from the mathematical concepts and problems are taught
separately with almost no relation to the concepts. It is not seen as a way to teach
mathematics.

The teacher has similar roles described in Level 0. Student’s role is extended and
it includes some understanding of the principles behind the procedures. Although there is
a change in the way mathematics and mathematics teaching is considered, there is still an
authority who decides on the correctness of mathematical ideas.

Level 2. Thompson does not specifically claim much about Level 2 conceptions of
the nature of mathematics. She only claims that centrality of the mathematical ideas are
realized at this level. Unlike the Level 1 teaching beliefs, using materials and different
methods in mathematics teaching targets conceptual understanding. Mathematics
teaching for understanding includes students’ engagement. Thus, teacher is considered as

a guide in catalyzing students’ thinking. The teacher allows students to express their ideas

46

in order to have a better understanding of their learning process. The student’s role is
increased to understanding the logical connection between the mathematical concepts and
ideas. Students are expected to participate in the mathematics classroom by expressing
their ideas and reasoning. Hence, proving and generalization are seen as a way of
learning mathematics.
Critical Arguments and Points in Thompson ’3 Framework

Thompson (1991) claims that the development of mathematics related
conceptions is not a discrete process. The participating teachers in her study continued to
have higher or lower level conceptions about mathematics simultaneously even if they
were considered to ﬁt a particular level. The teachers who initially had Level 0
conceptions eventually moved to Level 1 conceptions during her studies with them.
However, none of the teachers were able to have fully developed Level 2 beliefs.
Thompson claims that the patterns of movement from one level to the other suggests a
relatively easy move ﬁom Level 0 to Level 1 compared to that of from Level 1 to Level
2. She explains this difference by the nature of restructuring needed in order to achieve
the level change. Moving from Level 0 conceptions to Level 1 conceptions requires no
major structuring of conceptual schemes, but an expansion of or broadening in Level 0
conceptions. However, moving ﬁom Level 1 to Level 2 requires “that a teacher
experience numerous occasions to become aware of and question his deeply rooted ideas
and unexamined assumptions of what it means to know, learn, and teach mathematics”
(p. 14). Within this complicated process of restructtu‘ing, Thompson cautions that

teachers’ resistance to change their conceptions should not be underestimated.

47

Thompson’s (1991) framework appears as a result of a qualitative study with few
participants. In order to have a more accurate and stronger analysis tool, Lindgren’s
(1996) framework, which is a modiﬁcation of Thompson’s framework in Finnish context
through both qualitative and quantitative methods, is additionally considered here.

Lindgren’s Framework

Lindgren (1996) emphasizes the role of previous experiences on the mathematics
related beliefs of preservice teachers. She characterizes mathematical beliefs as implicit
personal mathematical knowledge. For Lindgren, conscious beliefs form conceptions. All
conscious and unconscious beliefs constitute a belief system. She names the conscious or
unconscious beliefs as “views” when the object of the belief system is mathematics or
mathematics teaching and learning.

Lindgren’s (1996) study with preservice teachers in Finland seems to validate
Thompson’s (1991) framework. Her study includes the use of both quantitative (N = 163)
and qualitative (N = 12) methods. She initially uses a Likert-type belief inventory and
then conducts interviews with a selected group of participants. Her study results in a
framework with three partly overlapping categories named Rules and Routines,
Discussion and Games, and Open-Approach, which she claimed corresponded to
Thompson’s Level 0, Level 1, and Level 2, respectively. The model that Lindgren
proposed emphasized the teaching and learning of mathematics, and roles of teachers and
students. The categories and their characterizations are given below (the complete list of

Lindgren’s characterizations is given in Appendix A).

48

 

 

Rules
teaching mat?
teacher and h

DISC“
hating differ-

discussions.

Opett- -

responsibility.

Mathematics

Lind;
(Let'el ll are.
Li”diilelt's st.

ln UP.
and Openness
have beliefs t:
l5 the interset
“'here teacher
might also pr.

 

 

Rules and Routines - RR (Level 0). This category refers to an understanding of
teaching mathematics based on routine procedures that should be demonstrated by the
teacher and be memorized by the students.

Discussion and Games - DG (Level 1). This category characterizes teachers as
having different approaches in teaching such as using games and promoting classroom
discussions.

Open-Approach - 0A (Level 3). This is the category when students have
responsibility for their own learning and where teachers encourage and guide students.
Mathematics is a way of thinking operationalized by problem solving.

Critical Arguments and Points in Lindgren 's Framework

Lindgren (1996) claims that there are sublevels in the Discussion and Games
(Level 1) area where common sub-areas with the other two levels appear. Figure 1 in
Lindgren’s study (1996, p.114) illustrates this structure.

In Lindgren’s (1996) illustration, GR (Games and Rules), GRO (Games, Rules,

and Openness), and GO (Games and Openness) are the intersection areas where teachers

have beliefs from at least two different belief levels. For example, GR (Games and Rules)

is the intersection of Rules and Routines (Level 0) and Discussion and Games (Level 1),
where teachers might believe that facts and rules are the focus of mathematics but they
might also promote class discussion. Lindgren’s (1996) analysis yields a conjoint area of
all three levels (GRO — Games, Rules, and Openness within Level 1) where teachers
simultaneously believe in issues from all three levels. The existence of conjoint areas in
Lindgren’s study suggests that Discussion and Games level (Level 1) is the area of

development of preservice teachers’ beliefs about teaching mathematics.

49

The existence of conjoint areas shows that Lindgren’s (1996) levels are similar to
Thompson’s (1991) levels in terms of continuous structure. As can be seen from the
characterization of categories, Lindgren’s framework is more of a teaching and learning
framework for mathematics related beliefs, as most belief statements directly address
teachers’ or students’ roles but not the nature of mathematical knowledge. When
combined with the Thompson’s framework, Lindgren’s framework brings additional
descriptions for beliefs about teaching and learning, and make Thompson’s framework
stronger in these areas. However, beliefs related to the nature of mathematics are not
mentioned explicitly in Lindgren’s framework. Hence, in order to supplement these two
frameworks with a better description. of beliefs about the nature of mathematics, Ernest’s
(1989) model of mathematics related beliefs is considered as a part of the ﬁnal
framework used here.

Ernest’s Model

Ernest (1989) proposes a model of mathematics teachers’ knowledge, beliefs, and
attitudes; however, he does not give a deﬁnition of beliefs. Rather, he uses “beliefs” and
“conceptions” interchangeably and mentions that teachers’ beliefs about the subject area,
teaching, and learning have a considerable impact on their teaching. He introduces three
views of the nature of mathematics where he claims that teachers may combine different
elements from different views during their practices just as Lindgren (1996) claims. He
does not mention a developmental nature in these views, but they represent a range from
static views to dynamic views. Ernest also links these views to beliefs about the teaching

and learning of mathematics. The description of views of mathematics and their links to

50

 

teaching a
given in A
1715

rules and s
instrument.
problems it
mentioning
Plat

there are str.
Mathematicg
Correct ans“
Concepts ofr
Prob]
expanding ﬁt
seen as a ﬁnis
mﬁlhods and 4

emphaSlZeS {h

Allhop

 

about the Hat;

ThOmpSOn.s ‘

 

C
an be “SOCIE—

teaching and learning are given below (the complete list of Ernest’s characterizations are
given in Appendix A).

Instrumentalist view. Mathematics is a set of tools which include unrelated facts,
rules and skills used in order to reach an external end product. A teacher with
instrumentalist view of mathematics may insist on single correct ways of solving
problems where the mathematics curriculum is about mastery of rules and skills without
mentioning meaningful relationships between these.

Platonist view. Mathematics is a static but combined body of knowledge in which
there are structures and truths connected to each other by logic and meaning.
Mathematics is not created but discovered. A platonist teacher may also insist on single
correct answer while implementing a curriculum that focused on the main and unifying
concepts of mathematics.

Problem-solving view. Mathematics is a dynamic, problem-driven, continually
expanding ﬁeld which includes a process of knowledge generation. Mathematics is not
seen as a ﬁnished product. A teacher with problem-solving view may accept children’s
methods and ways of dealing with mathematical tasks. Mathematics curriculum
emphasizes the procedure that mathematical knowledge is produced.

Critical Arguments and Points in Ernest '5 Framework

Although not developmental in nature, Ernest’s (1989) framework for the beliefs
about the nature of mathematics and their links to teaching and learning seems parallel to
Thompson’s (1991) characterization of levels. Thompson’s Level 0, Level 1, and Level 2

can be associated with Ernest’s instrumentalist, Platonist, and problem-solving views

51

respectively. His framework completes the lack of characterization of beliefs about nature
of mathematics for especially Level 2 beliefs in Thompson’s framework.
Combined Framework

Thompson’s (1991) framework is an overall framework that draws a general
developmental picture of mathematics related beliefs. It is used as the main analysis
framework of the present study. The terminology (Level 0, Level 1, and Level 2) and the
characterization of the levels are used as a starting point. In order to make the
characterizations of the levels richer, Lindgren’s (1996) characterizations are also
inserted into the main framework. These two studies used different methods in order to
frame preservice and inservice teachers’ beliefs: Thompson worked with 12 teachers for
over 5 years, where Lindgren worked with a total of 163 preservice teachers using a
quantitative approach and with 12 of them using a qualitative approach in order to
improve Thompson’s model in the Finnish context. Hence, the combination of these
frameworks provides a richer model especially in terms of teaching and learning beliefs
covering various contexts.

The combination of Thompson’s (1991) and Lindgren’s (1996) frameworks are
supplemented by Ernest’s (1989) model of conceptions of the nature of mathematics.
Table 3.1, below, provides the main points of the “Combined Framework” that will be
used as the analytic framework in the proposed study, arranged in Thompson’s levels.

The complete framework is given in Appendix A.

52

Table 3.1

A Summary of the Combined Framework for Analyzing Teachers ’ Beliefs.

Levels
Level 0

Level 1

Level 2

Main Characteristics

Nature .

0 Mathematics is composed of unchanged facts, rules, and basic
arithmetic, and usage of them in daily life.

0 The goal of mathematics instruction is to obtain single correct/accurate
answer.

0 Problem solving is getting answers to story problems by implementing
“rules of thumbs.”

Teaching

0 Teachers transfer the mathematical knowledge in the form of discrete
rules and procedures without consideration of conceptual
understanding.

Learning

0 Students can learn only by listening to the teacher, memorizing and
imitating teachers’ procedures.

Nature

0 Mathematical knowledge is composed of facts and rules, but there are
connections between these.

0 Problem solving is ﬁnding strategies to problems unrelated to the
content.

Teaching

0 Teachers use pedagogical approaches but do not consider cognitive
outcomes.

0 Mathematics instruction is using concrete representations as a
justiﬁcation of procedures and making students enjoy.

' Learning

0 Students put some effort to understand concepts in mathematics
demonstrated by the teacher. ‘

Nature

0 Mathematical knowledge has a dynamic structure and it is continuously
produced.

0 Problem solving is the way of knowledge construction.

Teaching

0 Teachers guide and provide opportunities for students’ understanding
through appropriate pedagogical approaches.

0 Mathematics instruction is an ongoing teacher guided process of
student-driven knowledge construction.

Learning

0 Students actively engage in mathematical inquiry.

53

The development of mathematics related beliefs is not a discrete process where
one is placed in one of the levels instantly. Rather, it is a continuous process as explained
in Thompson’s (1991) and Lindgren’s (1996) frameworks. Table 3.1 shows that
Lindgren’s and Ernest’s (1989) model supports Thompson’s claim about the differences
among nature of moving from one level to the other. The differences between the
characterizations of Level 0 and Level 1 are not dramatic in the combined framework, but
still distinguishable compared to the differences between Level 1 and Level 2
characterizations, which are quite deﬁnite.

Any collection of mathematics related beliefs in the combined framework can be
considered as an individual belief system as deﬁned in the previous chapter. Since belief
systems are cognitive structures open to reorganization due to the nature of existing and
new experiences (Thompson, 1992), the development of teachers’ mathematics related
beliefs, which corresponds to a change in level in the case of the combined framework, is
a dynamic process. Moreover, the cluster nature of belief systems provides a justiﬁcation
of holding beliefs from different levels of the combined framework.

The belief levels in the framework exemplify belief clusters that Green (1971)
mentioned. The belief clusters can be considered as containers or sets of beliefs within a
belief system and they exist mostly in isolation. Belief clusters allow one to hold several
conﬂicting beliefs together. They also explain Thompson’s (1991) and Lindgren’s (1996)
ﬁndings in which teachers had beliefs from different belief levels at a time. Therefore, the
framework also exempliﬁes and deﬁnes three clusters of mathematics related beliefs for

preservice and inservice teachers.

54

In order to make the data analysis more systematic, the framework is organized
under three belief areas depending on how previous studies categorized preservice and
inservice teachers’ mathematics related beliefs: the nature of mathematics, teaching
mathematics, and learning mathematics. Belieﬁr about the nature of mathematics consist
of beliefs about nature of the mathematical knowledge and what it means to know,
understand and perform in the subject area. Beliefs about mathematics teaching include
beliefs about teachers’ responsibilities, pedagogical content knowledge of mathematics,
and who should control students’ mathematical activity. Finally, belie/Cs about learning
mathematics consider beliefs about learners, learning and nature of mathematical ability.

The combined framework provides a detailed characterization of mathematics
related beliefs and better analysis of the interview transcripts in this study. In the present
study it is used in the Turkish context, which is different than the US. and Finnish
contexts where Thompson (1991) and Lindgren (1996) frameworks were emerged and
tested. Although the combined framework is the main tool to analyze preservice and
inservice teachers’ mathematics related beliefs, this study is also an examination of the

viability of the framework.

55

CHAPTER 4

METHODS

This study explores, identiﬁes, and interprets the mathematics-related beliefs of
12 ﬁrst-year (inservice) and 20 preservice Turkish middle school mathematics teachers
who had graduated or had been studying at the same teacher education program
(Elementary Mathematics Education -— EME) in 2003 and 2004.

The major data source of investigation was one-on-one interviews with the ﬁrst-
year (inservice) and preservice teachers. Each participant’s mathematics-related beliefs
were identiﬁed from their interview data using the framework of mathematics-related
beliefs described in the previous chapter. Inservice and preservice participants’ beliefs
were particularly investigated in relation to their experiences during the speciﬁc teacher
education program and in their teaching contexts (for inservice participants) to
understand if these experiences had an impact on their beliefs.

The method of inquiry will be explained in detail in this section. Aﬁer primary
and secondary research questions are introduced, the contexts of the study will be
described. Next, the participants of the study and the instruments will be explained in
detail followed by the procedures of data collection and data analysis. Since the study is
conducted through qualitative methods, the issues related to the quality of the research
will be addressed in this section as well.

The data for this study was collected in 2003 and 2004 and some of the
participants were involved in both 2003 and 2004 data collections. The data collection

procedures in these two years will be referred to as “2003 study” and “2004 study.”

56

Research Questions
Inservice and preservice middle school mathematics teachers’ mathematics-
related beliefs were investigated through the following primary and secondary research
questions:

1. How does the Elementary Mathematics Education (EME) program affect
the preservice middle school mathematics teachers’ (2"d-, 3'd-, and 4’h-year
students) beliefs about the nature, teaching, and learning of mathematics?

a. What are the beliefs of preservice teachers about the nature,
teaching, and learning of mathematics?

b. What might be the crucial year/course(s)/practice(s) in the EME
program that affects those beliefs? How? Why?

2. How does the Elementary Mathematics Education (EME) program affect
the ﬁrst-year (inservice) middle school mathematics teachers’ beliefs
about the nature, teaching, and learning of mathematics?

a. What are the beliefs of ﬁrst-year teachers about the nature,
teaching, and learning of mathematics?

b. What might be the crucial year/course(s)/practice(s) in the EME
program that affected those beliefs? How? Why?

c. How does the ﬁrst-year of teaching affect the ﬁrst-year teachers’
beliefs about the nature, teaching, and learning mathematics?

Why?

57

The participants, data collection, and the data analysis will be described in
relation to the primary research questions after the contexts of the EME program and the
Turkish educational system are be described.

Contexts

Although the study is related to the Elementary Mathematics Education (EME)
program, it is crucial that the general structure of Turkish Educational System is
introduced. This section will give a picture of the general teaching context of the ﬁrst-
year teachers as well as the participants’ pre-college education context.

The Elementary Mathematics Education (EME) Program

The study was conducted in the Department of Elementary EducationI in the
College of Education in Middle East Technical University (METU) in Ankara, Turkey.
METU is a technology-oriented university, and the language of instruction is English.
New students are administered METU’s own English proﬁciency examination at the
beginning of the fall semester every year in order to determine their skills in English
language. The students who cannot pass the examination are required to study in the
School of Basic English for one year. Those who pass the examination are considered
freshman and start taking departmental courses. The students in the School of Basic
English take the English proﬁciency examination again at the end of the spring semester
and are required to pass this examination in order to qualify as freshman.

The Department of Elementary Education offers B.S. degrees in three
undergraduate programs: Elementary Mathematics Education with a minor in Elementary

Science Education, Elementary Science Education with a minor in Elementary

 

' Note that the program name uses the term ‘Elementary’ in that it trains teachers of grades 1 to 8, which
are collectively referred to in Turkey as the elementary grades. The present study targets what US educators
would refer to as middle school teachers, however.

58

 

at
ill:

Cc

tea
in 1
D31
and
C011:

Mathematics Education and Early Childhood Education. All three programs are four-year
programs that aim “to deve10p teachers with a sound understanding of how children
learn” who are “conﬁdent in using technology; capable in problem-solving; attentive to
human rights, democracy, and ethics” (Middle East Technical University, 2005). The
programs in the department emphasize critical thinking, personal reflection, and
professional development of preservice teachers. The graduates of the program are
qualiﬁed as teachers who can teach in elementary schools (grades 1 to 8) as mathematics
and science teachers and early childhood centers as early childhood educator (Middle
East Technical University, 2005). The programs in the Department of Elementary
Education started to accept its ﬁrst students in 1998. These students studied one year of
English preparation classes and started their freshman year in the 1999 fall semester. The
ﬁrst students of the EME program graduated with B.S. degrees in June 2003. The
department faculty had only one assistant professor with a mathematics education focus
at the time of both the 2003 and the 2004 studies. The EME program was supported by
the faculty in the Department of Secondary Science and Mathematics Education in the
College of Education at METU at that time.

The EME program focuses on mathematics, science, and introductory courses to
teaching and education in the ﬁrst two years, followed by mathematics teaching courses
in the 3rd and 4th years (see Appendix B). The program includes nine courses from the
Department of Mathematics, four courses from the Department of Educational Sciences
and 11 courses from the Department of Elementary Education. There are a total of four
courses in chemistry, biology and physics to support the minor degree in the Elementary

Science Education. Although there is no laboratory room for mathematics activities, the

59

EME program has mathematics laboratory equipment (such as geoboards) to be used in
methods or other courses related to teaching mathematics.

The EME program requires three semesters of teaching practice courses at the
second, seventh, and eighth semesters. Students are ﬁrst assigned to collaborating schools
and then to collaborating teachers in those schools. The ﬁrst teaching practice course is
based mostly on observation of the classroom and teaching context and the lives of
teachers and students. The second and third teaching practice courses are generally based
on both observation and practice. Besides the three semesters of teaching experience,
most of the EME program students teach as private tutors or work at a Test Preparation
Center (TPC)2 during their studies in the program. Some of them work voluntarily at a
student club project where lower socio-economic class students are brought to the METU
campus during weekends and are taught in mathematics, science, and English in order to
support their learning at school and to help them prepare for the national exams. Hence,
students in the education programs at METU generally have some teaching experience
(other than the required practice) either with individual students or in a classroom
context.

The Educational System in Turkey and Teaching Institutions

The Ministry of National Education (MNE) is the institution of authority that

decides on educational policies in Turkey. The MNE decides on the national curriculum

and the examination procedures for all grades, all content areas, and all types of

 

2 Typically, university students in Turkey do not work at a job and study simultaneously. However, private
tutoring is very common among all university students regardless of their major. University students
generally tutor elementary or high school students either to support their learning or to prepare them for the
national exams. The most popular content areas of private tutoring are mathematics, English, and physics.
Some university students also work in Test Preparation Centers as tutors. More information about national
exams and Test Preparation Centers are given in the following section.

60

de‘

Tut

5m.

CF.

bag

hi‘ 7—

elementary and secondary schools. The teachers are placed in the public school system by
the MNE. The process is from top to bottom and there are National Education
Departments in every province that are responsible for implementing the policies at the
province level. Organizationally, the MN E is under the control of the Prime Ministry.

Public and private schools in Turkey generally differ in the size of the classroom,
the available resources (such as science laboratory, computer equipment, software, and
books), improvement opportunities (such as research centers and course material
development projects), and opportunities for extra-curricular activities for students (such
as sports and arts). Note that the terms ‘private’ and ‘public’ mean the same thing in
Turkey as in the US. Most Turkish students attend public (state-supported) schools, but a
small percentage pay for private education.

Turkish elementary education aims to educate students as self-conﬁdent and
environment-conscious individuals with the scientiﬁc thinking skills and the values of the
Turkish culture. Mandatory continuous elementary education in Turkey comprises grades
from one to eight. The ﬁrst phase of elementary education consists of the grades from one
to ﬁve where almost all content areas are taught by primary teachers. The second phase
of elementary education includes the grades from 6 to 8, and content area teachers teach
the courses. All public and private elementary schools implement the national curriculum
in all content areas at the elementary education level, and they are inspected by the MNE
ofﬁcers each year for their curriculum pace and teaching and administrative-related
issues. Instruction in elementary schools in Turkey can be characterized as text-book
based and teacher-directed. Classroom population may vary from 20 to 70 depending on

the population that the school serves.

61

Eighth grade students have to take a national examination (the Secondary
Education Institutions Entrance Examination) which covers the concepts in the sixth,
seventh, and eighth grade curriculum in order to attend better high schools. Secondary
education aims to provide the students with the knowledge and skills to analyze varibus
problems and to think about the solutions in order to participate as productive citizens of
the country and to prepare students for higher education. Secondary education is
considered to have two categories: General high schools (from grades nine to 12) 3 and
vocational technical high schools. General high schools prepare students for the higher
education and contain both examination-track and non-examination-track type high
schools, while vocational technical high schools train students for jobs. Although general
high schools in Turkey implement the national curriculum in all content areas at all grade
levels, they differ in the quality of the education depending on the student population
they serve. Since most of the general high schools take their students from the national
examination administered to the eighth grade students at the end of the school year,
examination-track general high schools are generally considered as superior due to their
student population". Moreover, the depth of the implemented curriculum differs in the
different school types. For example, Science High Schools (an examination-track general
high school type) implement the curriculum with more disciplinary approach and depth,

whereas Regular High Schools (a non-examination-track general high school) lack this

 

3 Secondary education in Turkey has increased from three years to four years beginning from the 2005-
2006 school year. At the time of the study, the high school duration was still three years, comprising grades
from 9 to 11. Hence, the high school context mentioned in this study will refer to three years duration.

‘ Only the 8m grade elementary students who have a certain GPA of 7‘h grade courses are allowed to take
the national examination. Hence a population who is generally successful in school courses takes the
examination.

62

approach and depth. General high schools are inspected by the MNE ofﬁcers every ﬁve
years.

Candidates who want to attend a university in Turkey are required to be a high
school graduate (or be in the senior year) and take a central multiple-choice examination,
the Student Selection Examination, which is held only once a year. The candidates are
placed in the university programs depending on their scores and rankings. Most of the
public universities and some private universities in Turkey have Colleges of Education
that offer four-year or ﬁve-year teacher education programs.

Most of the teachers in Turkey are graduates of education colleges within
universities. The rest are graduates of liberal arts colleges and they become teachers after
completing a certiﬁcation program provided by some of the universities in Turkey. Since
1999, public school teachers have been hired and placed by the MNE based on their
scores in a central examination. Private schools, on the other hand, hire their own
teachers. A high school or a middle school content area teacher generally teaches up to
ﬁve different classes of students per week. Teachers generally do not have their own
classrooms in Turkey, but the students in the same grade level are divided into sections
and these sections have classrooms. In other words, students do not visit the teachers’
classrooms but teachers visit the sections’ classrooms. Although ﬁrst-year teachers in
public and private schools are considered interns, they are fully responsible for teaching
the assigned classes and they have the same responsibilities that other teachers have.
First-year teachers are considered interns and they have to attend several courses on the
schooling system and teaching profession conducted by the MNE. To complete their

internship, they have to pass an examination on the content of each of the MNE courses.

63

 

 

 

 

 

 

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experiencet‘
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collaborati:J
Firs
schools ger.
computers :
schools is tl

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they start te

 

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have a Cert;
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the 36mm
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[080]“ ”ml

Although each ﬁrst-year public school teacher (an intern) is assigned a mentor who is an
experienced teacher in intem’s content area, continuous mentor-intem collaboration is
not common in Turkey. Most ﬁrst-year teachers do not even know or meet their
collaborating mentors.

First-year teachers are generally assigned to the schools in rural areas. Rural area
schools generally lack certain teaching materials and facilities in content areas such as
computers for teachers or science laboratories. Another typical problem in rural area
schools is the continuous teacher flow. Although many beginning teachers are assigned to
rural area schools, they prefer to work in big cities. Thus, they ask for reassignment aﬁer
they start teaching for several reasons such as attending a master’s program or having a
family. Many rural schools lack teachers who work in the same school for more at least
one year. It is very common that rural area schools have ﬁve different teachers in a
semester, one after another, who teach a certain content area. When the school does not
have a certain content area teacher, the principals generally assign another teacher (who
is not certiﬁed in that content) to teach. For example, if the school lacks mathematics
teacher, the principal may assign science teacher or Turkish language teacher to teach
mathematics.

Besides public and private schools, teachers also teach at Test Preparation Centers
in Turkey. These centers prepare students for all sorts of national examinations such as
the Secondary Education Institutions Entrance Examination (at the end of the eighth
grade) and the Student Selection Examination (at the end of the high school or later). The
focus of teaching in TPC is generally not on understanding the concepts but on the skills

to solve multiple-choice questions in a limited time period. TPCs group students on their

64

grade levels and generally provide courses during after-school hours. TPCs are private
institutions and attendance at a TPC depends on the students’ and families’ preference.
Typically, elementary school students start to attend a TPC at the seventh grade level and
continue to attend for more hours at the eighth grade level in order to perform better on
the Secondary Education Institutions Entrance Examination. High school students,
similarly, start to attend a TPC at the beginning of their high school studies (ninth grade

1th grade level) to score higher

level) and continue to attend until the end of high school (1
in the Student Selection Examination. Students who have difﬁculty in any of the content
areas at any level of schooling are generally supported by private tutors who are generally
teachers or university students.
Participants

This study was conducted with two groups of participants who studied or were
studying at the same teacher education program in 2003 and 2004. The ﬁrst group of
participants consisted of a total of 20 preservice teachers who were studying as 2"d-, 3'd-,
and 4‘h-year students at the EME program towards the end of spring semester 2004.
Preservice participants were interviewed only in 2004. The second group was formed by
12 ﬁrst-year (inservice) middle school mathematics teachers who graduated from the
EME program in 2003. These participants were interviewed both in 2003 when they
graduated from the program, and in 2004 when they completed their ﬁrst-year as a formal
teacher. More information about the participants is given in this section.

The two different groups of participants were associated with the two primary

research questions. The ﬁrst primary research question and its associated subquestions

were related to the preservice participants. The second primary question and its

65

associated subquestions were related to the ﬁrst-year participants of the study. The
interviews with the two groups of participants were the data sources to investigate the
research questions. The instruments and the methods of data collection and analysis are
explained in further sections.

In addition, an Assistant Professor from the EME faculty who had been teaching
in the program since 2000 was also interviewed. Although the professor was interviewed,
ﬁndings from his interview data was used to support results derived from the preservice
and inservice participants’ interview data.

Preservice Teachers

Twenty preservice teachers participated in the 2004 study. Among them, 5 were
2"d-year students, 7 were 3'd-year students, and 8 were 4’h-year students. Table 4.1 shows
the number of participants based on year level and gender. The numbers in parentheses
show the total number of students in the entire teacher education program in the speciﬁed
year level.

Table 4.1

The Number of the Preservice Teacher Participants According to Year Level in the

 

 

 

Program and Gender.
2"d-Year 3'd-Year ﬁlth-Year
Female 4 4 7
Male 1 3 1
Number of Participants (Total 5 (37) 7 (38) 8 (38)

Number of Students in Program)

 

These participants were selected among the students who volunteered to

participate in the study based on accessibility and students’ free time.

66

F irst- Year (Inservice) Teachers
Twelve ﬁrst-year teachers (7 female and 5 male) participated in the study ﬁrst,
when they graduated from the EME program in 2003 and then, at the end of their ﬁrst-
year as a teacher in 2004. Table 4.2 shows the number, gender, and the teaching contexts
of the ﬁrst-year teacher participants. The characteristics of the teaching contexts were
described above.
Table 4.2

The Number, Gender, and the Teaching Contexts of the F irst-year Teacher Participants.

 

 

 

Private Schools Public Schools Graduate School
Female 3 4 - 0
Male 1 3 1
Total 4 7 1

 

These 12 participants were selected from the 28 graduates of the EME program in 2003,
who were actually the ﬁrst graduates of the program.

The inservice participants completed their ﬁrst-year as a formal teacher in
different contexts at the time of the dissertation study. Most of the participants who were
placed in public schools were teaching in rural settings or small towns. One inservice
teacher obtained a position as a graduate assistant in the EME program after one semester
of teaching in the public school system.

Professor in the EME Program

Dr. C. (pseudonym) who was a professor in EME program was interviewed in

2004. He was teaching various courses as the only professor with a mathematics

education focus in the program at the time that both 2003 and 2004 studies were

67

 

 

 

 

 

conducted.

studies in t

A o
4.3 shows t
fourth inter
the effect 0
Table 4.3

Interview l:

N

\

lliou
Educ
DIESt
teacl
belit
lcarr
2. Hm
Edut
ﬁrst-
matl
natu

mat'
\

 

lnt

questions

conducted. All participants in this study took at least one course from Dr. C. during their
studies in the EME program.
Instruments

A total of three semi-structured interview protocols were used in this study. Table
4.3 shows the interview protocols used to investigate the primary research questions. A
fourth interview protocol was used to gain information about the faculty’s perspective on
the effect of the EME program on the program students’ beliefs.
Table 4.3

Interview Protocols Implemented in Relation to the Primary Research Questions.

 

Research Question Data Sources

 

l.How does the Elementary Mathematics 0 2004 Semi-Structured Interview
Education (EME) program affect the Protocol for the Preservice
preservice middle school mathematics Teachers

teachers’ (2"d-, 3’d-, and 4‘h-year students)
beliefs about the nature, teaching, and
learning of mathematics?
2. How does the Elementary Mathematics o 2003 Semi-Structured Interview

Education (EME) program affect the Protocol for the Graduates
ﬁrst-year (inservice) middle school

mathematics teachers’ beliefs about the o 2004 Semi-Structured Interview
nature, teaching, and learning 0f Protocol for the First-Year
mathematics? Teachers

 

Interview protocols are introduced below in relation to the primary research
questions they investigated.
First Research Question
Semi-structured Interview Protocol for Preservice Teachers (2004 Study)
This interview protocol was designed to investigate preservice teachers’ beliefs
about the nature, teaching, and learning of mathematics. The preservice teachers were

asked about their ideas on the nature of different types and possible evidences of

68

mathematical understanding, ways of teaching a certain mathematical task, and nature of

mathematics teaching. Moreover, they were asked about the possible effects of previous

policies that they have been exposed to, such as high school type, and the possible

perceived effects of the EME program on their mathematics-related beliefs. The

interview contained 32 main questions. Follow-up questions were also asked to explore

the emerging issues during the interviews. Table 4.4 shows the main concepts and

examples of interview questions related to the main concepts. A more complete

description of the interview protocol is given in the Appendix C.

Table 4.4

Main Concepts in the Preservice Teacher Interview and Selected Questions (2004).

 

 

 

 

 

Main Example Questions
Concepts
Nature 0f 0 What is mathematics for?
Mathematics 0 What does it mean to know mathematics?
0 Can you tell me what a “mathematical concept” is?
o How do you know that a student has understood a mathematical
concept?
Teaching 0 Why do we teach mathematics?
Mathematics 0 What are the types of knowledge that a mathematics teacher should
have?
0 What will be your main purpose in teaching mathematics in the
future?
Attitudes 0 Do you enjoy learning mathematics? When? Why?
Policies 0 Do you think that the knowledge and skills you gained in the

program will be sufﬁcient for your teaching?
0 How would you deﬁne teaching as a profession before you started
studying in this program?

 

Interviews with the preservice teacher participants were generally completed in 90

minutes.

69

Second Research Question
Semi-Structured Interview Protocol for the Graduates (2003 Study)

Semi-structured interviews were conducted in 2003 with the ﬁrst-year teacher
participants of 2004 study when they completed in the 4th (the ﬁnal) year of the program.
The 2003 interview protocol aimed to investigate program graduates’ mathematics-
related beliefs. The effects of EME program and some other factors such as (high school
education and national examination policy) on their beliefs before they started inservice
teaching were also investigated. Table 4.5 shows the main concepts investigated through
the interviews and the main questions under each concept.

Table 4.5

Main Concepts in Graduates 'Interview and Main Questions for Each Concept (2003).

 

 

Main Example Questions
Concepts
Nature of o What is mathematics for?

Mathematics 0 What does it mean to know mathematics?
0 Can you tell me what a “mathematical concept” is?
o How do you know that a student has understood a mathematical
concept?
Teaching 0 Why do we teach mathematics?
Mathematics 0 What are the types of knowledge that a mathematics teacher should
have?
0 What will be your main purpose in teaching mathematics in the
future?
Attitudes 0 Do you enjoy learning mathematics? When? Why?
0 Do you enjoy teaching mathematics? When? Why?
Policies 0 Do you think that the university entrance examination affected your
way of studying mathematics? How?
0 Do you think that the courses you have taken in this program are
useful for your profession? Why?

 

The participants were asked questions about how they perceived mathematical

understanding in relation to the teaching and learning of mathematics. They were also

70

asked questions about the way that the university entrance examination affected their
study habits and about the education that they had in the department. The interview
contained 23 main questions. Follow-up questions were asked to explore the emerging
issues during the interview. Interviews were completed in one hour. The main interview
questions in the 2003 interview protocol are given in the Appendix C.

Semi-Structured Interview Protocol for the F irst- Year Teachers (2004 Study)

The interview protocol for the ﬁrst-year teachers was a modiﬁed version of the
interview protocol for the preservice teachers in 2004. Besides the issues that were
investigated with the preservice interview protocol, the possible effects of one year of
formal teaching experience, school context, classroom context, and the curriculum on the
ﬁrst-year teachers’ beliefs about the nature, teaching, and learning of mathematics were
also investigated. The interview protocol contained 37 main questions with additional
follow-up questions if necessary. Table 4.6 shows examples of additional questions that
appeared in the ﬁrst-year teacher interview protocol under the main concepts. More detail
is given in the Appendix C.

Table 4.6

Main Concepts and Additional Main Questions in the F irst-year Teacher Interview

 

 

(2004).
Main Example Questions
Concepts
Nature of 0 Without mentioning name or any form of student data, can you give

Mathematics examples of three particular students who have rich, medium, and
poor mathematical understanding and related student behaviors?
Teaching 0 Are there any pressures or supports that affect your teaching, such
Mathematics as students, parents, school administrators or curriculum?
0 Which grades would you prefer to teach? Why?

 

 

Interviews with the ﬁrst-year teachers were generally completed in two hours.

71

Semi-Structured Interview Protocol for Faculty
The professor’s view of mathematics teaching, the program’s effectiveness in
educating teachers, and preservice teachers’ beliefs were investigated by an interview
protocol including 15 main questions. Examples of interview questions are given below:
0 Do you think that program effectively prepares students to be mathematics
teachers?
0 Can you describe the characteristics of effective mathematics teachers?
0 Can you describe the beliefs that a mathematics teacher should have (about nature
of mathematics, about teaching and learning mathematics)?
Probing questions were asked during the interview where necessary. The complete
interview protocol is given in the Appendix C.
Addressing Beliefs in Interview Protocols
The three belief areas (beliefs about the nature of mathematics, beliefs about
teaching mathematics, and beliefs about learning mathematics) which were described in
Chapter 2, were addressed through speciﬁc questions in 2003 and 2004 interview
protocols. The 2004 interview protocol is an extended version of 2003 protocols in terms
of the questions addressing belief areas. How each belief area is addressed through the
interview questions in 2004 study will be explained here in detail. The question numbers
are based on the way they appear in the 2004 preservice interview protocol. Additional
questions in the 2004 inservice interview protocol are given with the question number

they appear (see Appendix C).

72

Beliefs about the Nature of Mathematics

In order to address beliefs about nature of mathematics, participants were asked

about how they would consider mathematical knowledge, knowing and understanding

mathematics, and the nature of mathematical problems and problem solving. Examples of

questions that addressed participants’ beliefs about nature of mathematics are given in

Table 4.7

Examples of Questions in 2004 Study Addressing Beliefs about Nature of Mathematics.

 

Questions

 

Preservice
Protocol

Q1. What is mathematics for?

Q2. What does it mean to know mathematics?

Q4. Can you tell me what a “mathematical concept” is depending on
your own ideas?

0 What is a “mathematical concept” for you?

0 Can you given an example of a question that asks a “mathematical
concept”?

Q5. How do you know that one of your students has understood a
mathematical concept?

Q8. Can you tell me what a “mathematical problem” is depending on
your own ideas?

0 What is a “mathematical problem” for you?

0 Can you give an example of a “mathematical problem”?

Q11. How do you know that one of your students has developed a
mathematical understanding?

 

Inservice
Protocol

Q15. Without mentioning name or any form of student data, can you
give examples of three particular students who have rich, medium, and
poor mathematical understanding and related student behaviors?

 

In these questions, the ways participants described the nature of mathematical

knowledge (such as isolated rules or logically connected pieces of knowledge), knowing

mathematics (such as reaching correct answer or building logical connections), and the

nature of mathematical problems (such as questions that could be solved by mathematical

operations or a situation which had no immediate solution) mostly included expressions

73

parallel to the belief statements in the framework. Therefore, it was concluded that
interview questions were mostly useful in detecting participants’ beliefs about nature of
mathematics. Participants’ responses to the questions about the nature of mathematics are
given in Chapter 5 and Chapter 6. i
Beliefs about Teaching Mathematics

Participants’ beliefs about teaching mathematics were addressed through the
questions, which mostly asked about their preferences in teaching mathematics and the
perceived role of the mathematics teacher. The examples for questions addressing beliefs
about teaching mathematics are given in Table 4.8.
Table 4.8

Examples of Questions in 2004 Study Addressing Beliefs about Teaching Mathematics.

 

Questions
Preservice Q13. Why do we teach mathematics?
Protocol Q14. What are the types of knowledge that a mathematics teacher
should have?
0 Can you make a list of / categorize the types of knowledge that a
mathematics teacher should have?
a. Why do you think that a mathematics teacher should have these types
of knowledge?
Q15. What will be your main purpose in teaching mathematics in the
future?
Q17. What are your strengths in teaching mathematics?
Q18. What are your weaknesses in teaching mathematics?
Inservice Q16. Can you plan one or a series of lessons (depending on your
Protocol preference) on a particular topic which reflects your conceptions of
mathematical understanding, without considering other pressures such as
material availability and time allowed for that particular topic in the
curriculum for one of your classes or a class of 30 students?

Participants’ responses to these questions about their preferences in teaching
(such as demonstrating rules or using manipulatives) and how they would consider a

teacher’s role in the mathematics classroom (such as explaining rules and their

74

implementation in detail or guiding students in their learning) indicated their beliefs that
were detectable through using the framework. In other words, the interview questions
were mostly sufﬁcient in capturing participants’ beliefs about teaching mathematics.
Examples of participant responses are presented in Chapter 5 and Chapter 6.
Beliefs about Learning Mathematics

Learning beliefs of participants were mostly detected by the third question in
2004 interview protocols, as given in Table 4.9.
Table 4.9

Examples of Questions in 2004 Study Addressing Beliefs about Learning Mathematics.

 

Questions

 

Preservice Q3. What does it mean to learn mathematics?

& o How is mathematics learned?

Inservrce o How did you learn mathematics?

Protocols 0 How do you think a student should study mathematics in order to
learn it?

 

The responses to this question (such as solving a lot of questions or trying to
relate new knowledge to the existing knowledge) indicated that the question was effective
in identifying their beliefs about learning mathematics.

In addition to Q3, participants’ beliefs about learning mathematics were detected
in their responses for questions about teaching (see Table 4.8). Participants mostly
compared teacher’s role and students’ role in the classroom or mentioned how they
would prefer to teach mathematics in relation to the way students learn as a response to
teaching questions. For example, while responding to the question about her perceived
weaknesses in teaching mathematics (Q25), P7 claimed that “[...] there [were] several

ways to solve a problem and students should be able to see these [ways] " (P7, Q25,

75

 

2004). This indicated that for her, students should be able to see the different ways to
reach a solution, which was parallel to a Level 2 learning belief statement in the
framework. Therefore, questions about teaching mathematics also generated responses
about learning mathematics and were effective in capturing participants’ beliefs about

learning mathematics. Participants’ responses are provided in Chapter 6 and Chapter 7.
. Although the questions in especially 2004 interview protocols were found to be
sufﬁcient in detecting participants’ beliefs, data analysis provided some potentially useful
ideas that might be used in the possible future implementations of the interview
protocols. The interviews would have been more effective in capturing participants’
beliefs if they had included more concrete examples such as Q12 (see Appendix C) on
which participants would work and comment on a student misconception, considering the
classroom context. Concrete examples would have helped participants express their
beliefs about how mathematical knowledge would be organized and how teaching and
learning should occur in classroom contexts.

Data Collection Procedure
The data for this study was collected in Turkey in May-June 2003 and May-July

2004 through interviews. The data collected in 2003 was used to investigate the second
research question whereas the data collected in 2004 was used to investigate both the ﬁrst
and second research questions. All interviews were conducted in Turkish and audiotaped.
Since the language of instruction was English at METU and the students were considered
to have a certain level of English proﬁciency, some questions were also asked in English

in order to reduce the change or loss of meaning in translation. Since a part of 2004 data

76

collection procedure was related to the 2003 data collection, these procedures will be
explained in terms of years they were conducted.’
2003 Study (Data Collection)

The data in 2003 was collected through interviews in with the 4th ~year5 EME
program students. The 4‘h-year students were visited towards the end of the 2003 spring
semester (May 2003) at the end of a class after the professor left the room. They were
informed about the study and asked whether they would like to participate. It was
indicated that participation was voluntary, the results of the study would not be shared
with any of the program faculty, and their participation would have no impact on their
course grade.

Among the 28 4’h'year students, seven female and ﬁve male students were chosen
to be interviewed among the volunteered students based on their accessibility and free
time. The interviews were conducted in May-June 2004 in places where the participants
preferred to be interviewed, and in one-on-one settings. It was emphasized before and
during the interview that there were no right answers for the interview questions and their
beliefs and ideas about mathematical understanding and mathematics teaching and
learning were considered as important. The participants were given as much time as they
asked to think about the questions and were allowed to skip questions if they did not want
to respond at that time. They were also ensured that they could quit the study at any time.
Participants were asked the skipped questions at the end of the interview, and they were
also allowed and encouraged to answer any skipped questions at any point during the

interview.

 

5 Recall that the EME program is a four-year program.

77

2004 Study (Data Collection)

The data in 2004 was collected through interviews in with the 2"d-, 3'd-, and 4’“-
year EME program students, the ﬁrst-year teachers who participated in the 2003 study at
the end of their 4‘h-year in the EME program, and the professor. The program students
were visited in their courses towards the end of the 2004 spring semester (May 2004).
They were informed about the study after the professor left the room and they were asked
if they would like to participate. The students were ensured that the participation was
voluntary, the results of the study would not be shared with any of the program faculty,
nor that their participation would have an impact on their course grade.

Twenty students (15 female and 5 male) were selected among a total of 113
students who were studying in their 2"d-, 3'd-, and 4‘h-year at the EME program (see Table
4.1). The interviews were conducted in May-June 2004 in participants’ preferred one-on-
one settings. The participants were ensured before and during the interview that there
were no right answers for the interview questions and the research’s interest was on their
beliefs and ideas about mathematical understanding and mathematics teaching and
learning.

All of the participants of the 2003 study agreed to participate in the 2004 study at
the end of their ﬁrst-year as a teacher. They were interviewed in J une-J uly 2004 in the
one-on-one settings they preferred. The ﬁrst—year participants were indicated that the
interview questions had no single correct answer and the study’s interest was their beliefs
and experiences regarding the nature, teaching and learning mathematics. They were
ensured that the raw data they provided would not be given to their school administrators

or any MNE ofﬁcer.

78

 

 

The professor in the department was interviewed in 2004 in his preferred one-on-
one setting. He was ensured that his interview data would not be shared with other
program faculty and the study’s interest was his views about the effectiveness of the
program and program students’ and graduates’ beliefs.

All of the participants in 2004 study were given the amount of time they required
to respond to the questions during the interviews. They were ensured that they could quit
the study at any time or skip any questions that they did not want to or did not feel ready
to answer. The questions that they preferred to answer later were asked at the end of the
interview.

Data Analysis Procedure

The data analysis was conducted in order to identify each participant’s
mathematics-related beliefs, levels of these beliefs, and major factors that seemed to
affect participants’ mathematics related beliefs. The change of level of beliefs through the
program and ﬁrst inservice year were investigated in order to detect key experiences in
the program that might have caused change in belief levels.

The data was transcribed and coded by the researcher and a second coder in order
to reduce bias in data analysis and to increase the reliability of the results. The
recruitment and the training of the second coder are explained in the next section (Quality
of the Research). The interview transcripts were prepared so that the coders would not
see the names of the participants. Both coders worked on the same data at the same time
and coded the data using the framework described in Chapter 3.

The professor’s interview transcript was not coded by using the data analysis

framework used for the preservice and inservice interviews. His interview transcript was

79

read several times by the researcher in order to gather the emerging themes such as the
impact of the EME program courses on program students’ beliefs, the nature of the EME
courses, and insufﬁcient experiences in the EME program. These themes were used to
support ﬁndings from the inservice and preservice data.

The two major phases of data analysis, identifying belief levels for each
participant and identifying factors that impacted each participant’s beliefs will be
described in this section in detail.

Identiﬁ/ing Belief Levels

In order to identify the belief levels for each participant, both coders studied the
framework and discussed it in depth to understand the characteristics of the levels. These
discussions prepared the coders for detecting belief statements in transcribed interviews.
During the data coding, ﬁrst, the coders read the interviews and searched for the belief
statements addressed in the framework. Then, in the second step, coders categorized the
belief statements as nature belief, teaching belief, or learning belief. Finally, in the third
step, they decided on the level of belief statements and ﬁnal level of each participant’s
belief levels in different belief areas. These steps are explained below in detail through
examples. Certain difﬁculties appeared during the data analysis and how they were
managed will also be explained here.

Detecting Belief Statements

In this step, ﬁrst, the coders read each participant’s interview transcript and
individually underlined the belief expressions. The major indicators of belief statements
were keywords that appeared in the belief statements of the framework. For example,

“logic” and “connected” in the Level 1 belief statement “Mathematics as a static but

80

combined body of knowledge, in which there are structures and truths connected to each
other by logic and meaning” were keywords for identifying P1 ’5 expression “knowing
mathematics is knowing the meaning of mathematics, which is the logic that we form in
nature or life, the numbers, and the relationship between the numbers ” (Pl, Q2, 2004) as
a belief statement.

Once the coders underlined the belief statements individually, they compared
identiﬁed belief statements to see if the statements were the same or not. When two
coders found different belief statements, coders engaged in a resolution discussion.
Initially, coders discussed the existence of a keyword or keywords in participant’s
expressions. When coders could not ﬁnd any keywords, coders agreed that the expression
should not be considered as a belief statement. This step of data analysis was completed
with full consensus of the coders for each participant.

Categorizing Belief Statements

After coders identiﬁed belief statements for each participant, the next step in data
analysis was to categorize the statements as nature, teaching, or learning beliefs and
deciding on the level of the belief statements. For example, in Pl ’5 expression quoted
above, P1 referred to knowing mathematics. Since knowing mathematics was addressed
under beliefs about nature of mathematics, coders initially considered this statement as a
nature belief statement. Additionally, coders used a Level 1 belief statement in the
framework (as indicated above) as a reference to detect P1 ’3 belief statement. Therefore,
the belief area for his statement was decided as nature of mathematics and the level of

this belief statement was Level 1.

81

Even though coders considered keywords as references to detect belief statements
and then belief levels, coders had some difﬁculties in categorizing participants’ belief
statements. One particular difﬁculty was deciding whether a speciﬁc belief statement in
an interview was addressing teaching belief or learning belief. For example, Pl claimed
that students’ mathematical understanding could be improved by “[providing them] with
situations whose solutions [would not be] familiar to them " (PI, Q1 1, 2004). Coders
considered this claim in two ways. First, coders considered P1 as addressing that students
would be able to learn better when students deal with situations that they have no
immediate solution, mostly because the interview question was asking students’
behaviors. And second, coders considered that P1 might be talking about teachers’ role in
improving students’ understanding. The Level 2 belief statement “One should use, as
often as possible, problems where the student has to think ﬁrst, and where the mastery of
calculation alone will not lead to the solving of the problem” in the framework helped
them to conclude that this was a teaching belief.

Once coders assigned a speciﬁc belief statement in the framework to participant’s
belief statement, coders also identiﬁed the level of participant’s statement. For P1 ’5
teaching belief mentioned in the previous paragraph, the referred belief statement in the
framework was a Level 2 statement. Therefore, the coders concluded that the P1 ’5 belief
statement was a Level 2 belief. The coders completed this step in full consensus.
Deciding on Final Belief Levels

Following the categorization and level assignment of the belief statements, coders
proceeded to the last step, which was reaching a decision about each participant’s ﬁnal

belief levels in three belief areas. Reaching a decision for participants’ ﬁnal belief levels

82

was often a complicated process. Three major decision processes occurred depending on
the frequency of participants’ beliefs in each belief level. These processes appeared when
(i) participants had beliefs mostly from a single level; (ii) participants had almost equal
number of beliefs in a belief area from two different belief areas; and (iii) participants
had no beliefs from a belief area. Decision processes are explained here in detail.

Beliefs from Single Level. The majority of participants had belief statements
almost totally ﬁom a single level. In these cases, coders decided that the most ﬁ'equent
belief level was the ﬁnal level of a belief area. For example, P2 had four teaching belief
statements of Level 1 and one belief statement of Level 2, which led coders to conclude
that she had Level 1 beliefs about teaching mathematics. The ﬁnal belief level decisions
for these participants were completed in full consensus of both coders.

Equal Number of Beliefs. A few participants had almost equal number of belief
statements from different levels. Two main situations exemplify this. First, some
participants had belief statements from all three levels or from Level 0 and Level 2 in a
belief area. In these cases, the coders ﬁnalized the belief level as Level 1, using
Lindgren’s (1996, p.114) ﬁgure that illustrated the intersection of all three levels and the
intersection of Level 0 and Level 2 in Level 1 area. For example, P1 had four Level 2,
two Level 1 and one Level 0 teaching belief statements. Coders concluded that his ﬁnal
teaching belief level was Level 1. 1n the second situation, two participants (P2 and P9)
had almost equal number of beliefs from two consecutive belief levels (see Table 6.1). In
these cases, since both coders could not decide on the ﬁnal level of participants’ learning
beliefs, they indicated both levels for the participants and interpreted that participants

might be in a transition period of moving from one level to the other, considering

83

Lindgren’s ﬁgure and Thompson’s (1991) characterization of teachers’ beliefs that
addressed the existence of beliefs from different levels during a transition period. The
ﬁnal step of data coding in these cases was also completed in full consensus.

No Belief Statements. A few participants did not articulate much in a belief area
so that no belief statements were detected. In these rare cases, the coders discussed how
the participants described the nature of mathematical knowledge, teaching, and learning
mathematics and which level these descriptions might match. An example for this case
might be P10. P10 did not have explicit expressions about the nature of mathematics in
the 2004 interview that would have a complete match with a belief statement in the
framework. But she claimed that knowing mathematics was “solving mathematical
problems. Even when it is not with x ’s and y ’s [...] but it might be by using a piece of
mathematical knowledge and reaching the result” (P10, Q2, 2004). Since beliefs about
problem solving and knowing mathematics were included in beliefs about the nature of
mathematics, coders concluded in the previous steps that this expression was a nature
belief statement. However, the level was not clear for coders as P10’s expression seemed
in between Level 0 and Level 1. When coders looked for additional expressions in PlO’s
transcript to ﬁnalize the level of her expression, they found out that P10’s example of an
elementary school problem addressed simple calculations and she could not articulate on
the nature of a higher level mathematical problem. This helped coders to conclude her
ﬁnal nature belief level as Level 0. Coders decided on the ﬁnal levels of these

participants in full consensus.

84

A sample data coding is given in Appendix D along with a transcribed interview
and the coding sheet. The framework for analyzing the interviews was described in depth
in Chapter 3.

’Identijj/ing Factors

The 2003 study provided some clues about the factors that might have impacted
inservice participants’ mathematics related beliefs when they graduated from the EME
program. In the 2003 study, inservice participants were encouraged to talk about the
impact of some of the courses in the EME program on their views about mathematics
teaching. Inservice participants also discussed the ways they worked with the students in
informal teaching environments. These clues were used to improve inservice and
preservice interview protocols for 2004 study. Additionally, depending on the literature
that addressed the impact of ﬁrst-year teaching contexts on beginning teachers (e.g.
Veenman, 1984), factors such as students and cuniculum pace were addressed in
inservice interview schedule in 2004. During the data analysis, the factors that impacted
participants’ beliefs were mostly detected through their responses to the questions and
were noted on the space provided for factors in the coding sheet (see Appendix D).

Table 4.10 gives the examples of questions in 2004 interview protocols for
preservice and inservice participants about the factors in preservice training and ﬁrst-year

teaching that might have impacted participants’ beliefs.

85

Table 4.10

Examples of Questions in 2004 Interview Protocols Targeting Factors Impacting Beliefs.

 

Interview Questions
Preservice Q16. Do you have any teaching experience? In which ways and for
Training“ how long?
Q27. Do you think that the courses you have taken in this program are
useful for your profession? Why?
a. What are the courses that you would like to have?
b. What are your expectations from the professors in your program
during the courses?
c. Do you think that the courses would be different if you had
another instructor?
F irst-Year Q13. Do you think that your ideas have changed since you graduated
Teaching and started teaching? What has affected your ideas?
Q22. Are there any pressures or supports that affect your teaching,
such as students, parents, school administrators or curriculum?
0 Do you think that you would teach more effectively if any one or
more of these factors had changed? Why and in what ways?
Q35. How would you deﬁne teaching as a profession before you
started teaching as a formal teacher?
0 How have your ideas changed about teaching profession?
* These questions were also included in 2004 inservice interview protocol.

 

 

 

There were differences in the detail that inservice and preservice participants’
responses included. Inservice participants mentioned factors more frequently, explicitly,
and in detail. They were mostly aware of what they have been affected by in the 2004
study, such as curriculum rush and informal teaching experiences, an awareness which
was not common in the 2003 study. Inservice participants also mentioned factors that
emerged from the preservice years and ﬁrst-year teaching in their responses to other
interview questions. For example, while responding to the question about her perceived
weaknesses in teaching mathematics (Q25 in 2004 Inservice Interview Protocol, see
Appendix C), P10 claimed that:

"[Ueaching in a classroom is different. During private tutoring, if you

realize that you don 't really know the concept, you can open [a book]

86

You don 't really have to make [mathematics] enjoyable and give real-life

examples Since you are expected to make [the student] solve many

questions, you can 't really consider [private tutoring as an experience]

Or the classroom presentations [in the EME program, do not really count

as a teaching experience] ” (P10, Q25, 2004).

This claim helped the coders to consider a possible limiting impact of private tutoring on
P10’s beliefs and teaching during her studies in the EME program. Since the study also
investigated the perceived impact of factors on participants’ beliefs, PlO’s awareness was
considered as evidence of the impact. Preservice participants’ responses were mostly not
as detailed as inservice participants’ responses. The factors that impacted their beliefs
were detected through their responses to the speciﬁc interview questions targeting
factors.

All major factors were detected during the data analysis in which participants’
belief levels were identiﬁed. The factors were also detected in consensus with the second
coden

Quality of the Research

Issues that underlie the quality of research seem important to address here in order
to present the reasons for the methods of inquiry used for this study. Miles and Huberrnan
(1994) refer to these issues as the practical standards that help in judging the quality of
the conclusions drawn from the research. There are contrasting views about the
applicability of the quantitative research terminology and methods, such as reliability and
validity, to the qualitative research (for a detailed discussion, see Golafshani, 2003).

While reliability and validity are issues that more typically ensure the quality of

87

quantitative research, the efforts and the skills of the researcher address the quality of
qualitative research. Moreover, reliability and validity are generally not discussed
separately in qualitative research but rather terms such as “credibility” or
“trustworthiness” are suggested in order to address both reliability and validity
(Golafshani, 2003). Similarly, Bogdan and Biklen (1998) do not suggest use of the term
“triangulation” to address the multiplicity of data sources, researchers, or theoretical
approach in order to strengthen the results of the study, but they rather suggest explaining
what has been done to have more accurate data. Considering that terms such as reliability,
validity, and triangulation have several different approaches in the qualitative research
paradigm, the quality of the research will be described under the term “credibility” to
address them all.
Credibility

In this section the researcher’s role and attitude in the data collection and analysis
process, convergence between multiple coders’ accounts, limitations that were caused by
the data analysis framework and how they were handled will be explained. Moreover, the
reasons of having single data type and the measures taken during the data collection
process to reduce this disadvantage will be explained in detail.
Researcher ’s Role

The data of this study is gathered from two different groups of participants:
Preservice teachers studying in the EME program in 2004 and inservice teachers
graduated from the same program in 2003. My relationship to inservice participants was
different then the relationship I had with the preservice participants at the time of the

2004 study. The difference was due to the close contact I had with inservice participants

88

when they were studying at the EME program. Inservice participants and I had
established a friendship through my three years of advising them for their academic
progress in the EME program. I conducted both extracurricular and recreational activities
for them during my assistantship in the Department of Elementary Education. Moreover,
I was a teaching assistant for their pedagogical content knowledge courses in the ﬁrst
three years of their studies in the EME program. This impacted the study in terms of the
depth and width of the interview data they provided for me especially in 2004 study.
Inservice participants were more motivated to provide me with longer and detailed
responses about their preservice and ﬁrst-year practices most probably due to the
friendship we established and their willingness to help me in my study.

My relationship to the preservice participants was, however, not that close and
mostly in a more formal way. I met almost all 2"d-year and 3'd-year participants for the
ﬁrst time during the 2004 study. Fourth-year participants and I knew each other but they
were not close as inservice participants were. Although I tried to comfort preservice
participants before and during the interviews through casual conversations about daily
issues, some participants were not as motivated as others in providing me with responses
to the interview questions most probably due to the lack of sufﬁcient pre-existing
communication. Another reason for short responses might be the lack of experience in the
program especially for the 2"d-year and 3'd-year participants.

In order to comfort all of the participants during the interviews, I tried to ensure
the conﬁdentiality of the data and I repeated several times that there were no correct
answers for the questions. When they asked for approval of their answers, I indicated that

my interest was what they believed or thought and I was not judging their answers. The

89

participants were interviewed where they felt comfortable in terms of context,
transportation, and timing so that their answers would not be in a rush. During the
interviews, I tried to express what I had understood from the participants’ responses to
them, when the responses were not. clear. I asked them whether I understood their point
correctly or not. If they claimed I did not understand it fully or correctly, I asked them to
correct my interpretation. I generally summarized participants’ responses with one or two
sentences before moving to the next question and they generally approved.

The interview schedules for both preservice and ﬁrst-year teachers had questions
related to the course instructors in the program or people in the school contexts (for ﬁrst-
year teachers), where participants might have negative comments. I was expecting these
negative comments as a result of my interaction with especially the inservice participants,
but I did not force them to report to me such information. In situations where there
appeared a need to express the name of a faculty in the college, I suggested to them to
name the course instead of naming the instructor, if they did not feel comfortable in
naming the instructor. Almost all participants named both the instructor and the course,
which might suggest that participants considered my approach as conﬁdential. In other
words, my role in the department did not affect the way that participants answered the
interview questions about the program effect. Rather, the participants perceived the
interviews as an opportunity to anonymously express the issues related to the EME
program or the educational system in general (including the ﬁrst-year participants’
teaching contexts) since they did not want to express them publicly.

The difference in the amount of the data gathered from inservice participants and

that of preservice participants might have impacted the results, conclusions, and

90

 

 

 

interpretations of the present study. Moreover, since I was once an insider in the EME
context and then on the border of being an insider and an outsider, I was aware of
reaching subjective conclusions and interpretations. However, being an insider also
provided me with clues for interview questions and supporting evidences for
investigating participants’ beliefs, and knowledge of general tendencies in the EME
program. The insider knowledge has always been a motivating issue for me and helped
me to organize my results and conclusions.
Nature of Data

Miles and Huberrnan (1994) addressed the main problem in qualitative data
analysis as the “selectivity.” It is impossible to get all data for a study since
instrumentation, data collection procedure, and the form of the data are selective. For
example, in the case of interviews, the transcription eliminates the clues, such as mimics,
from nonverbal data. Beliefs are unavoidably investigated through inferences from what
people say, intend, and do (Pajares, 1992) where even mimics or tone of the voice might
help. However, these are generally eliminated and data sources for investigating beliefs
are considered as either verbal responses to questions with abstract levels of thought
(Thompson, 1992), or written responses to hypothetical tasks or items (Fang, 1996). In
this study, the main data source was audiotaped interviews with the participants, and their
written and mostly verbal responses to a mathematical task and the teaching of that task.
Belief data in the form of verbal responses is selective also in the sense that participants
might not always be willing to talk about their beliefs or represent them (Rokeach, 1968)
probably because expression of beliefs is mostly inﬂuenced by values (Scheibe, 1960).

Thompson (1992) claims that a single data source might result in inconsistency between

91

the beliefs and the practices of the teachers. However, studies with multiple data sources
like classroom observations, stimulated recalls, and think aloud processes also yielded in
inconsistency as well (Fang, 1996).

There were several reasonsifor having a single type of data (i.e. interview
transcripts) in this study. The main reason was the conditions in the study context that
limited the availability of additional data sources. For example, in the beginning of the
2004 study, the ﬁrst-year teachers were asked to provide lesson plans and evaluation
tasks that exempliﬁed their own conceptualization of thoughtful mathematics lessons.
They hesitated, but did not refuse, since their lesson plans did not reﬂect their
conceptualization but reﬂected the certain template given by the MNE that they must
imitate in writing lesson plans". They also claimed that they generally did not follow the
lesson plans in their teaching but conducted their teaching according to the responses
from the students, rendering the lesson plans as not particularly useful data source.
Observing the ﬁrst-year teachers’ classroom, another alternative data source, required
permission from the MN B through a long process of paperwork which must be carried
out for 11 different schools in 10 different cities. Since there was limited available time
period for the study and there were only single available observer during the research,
this option for additional data source was eliminated as well. In the case of preservice
participants, most preservice teachers who participated in the study completed their
student teaching at the time of the study or were not registered in student teaching,

thereby eliminating the possibility of observing their teaching in real-classroom settings.

 

6 The yearly inspection of elementary schools includes inspecting new teachers’ current and previous lesson
plans and observing their teaching. Hence, teachers have to write their lesson plans based on the template
provided by the MNE.

92

 

 

 

The ﬁrst attempt to reduce the disadvantage of having a single data source was
presenting the participants with different situations during the interviews in the 2004
study. Both the ﬁrst-year and preservice participants were asked about their ideas to re-
teach an issue about fractions where students had misconceptions by using actual student
work (Q12 in the both 2004 interview protocols, see Appendix C). This helped in
underStanding the participants’ beliefs about teaching mathematics and beliefs about how
students learn in actual classroom settings. But this does not ﬁrlly reﬂect their practice in
the real-classroom context where students with different levels of knowledge and interest
exist.

There is no doubt that my investigation of teachers’ beliefs would have been
potentially richer had I been able to observe teachers’ practice. Belief research beneﬁts
from a continuous observation of teachers’ practice in order see the potential instantiation
of teachers’ beliefs. Despite the absence of observation data, I feel that my results are
useful and valid; instead of observing their beliefs in action, I asked several proxy
questions to inservice participants about their actual and ideal practices. Although these
interview questions helped me in making better inferences about participants’ beliefs,
they still did not count as real-classroom observations.

Data Analysis

A doctoral candidate in the Secondary Science and Mathematics Education
Program in the College of Education at METU was recruited for conducting the data
analysis in order to increase the inter-rater reliability of the results. The second coder was
a graduate assistant in the EME program, a native speaker of Turkish language, and was

ﬂuent in speaking, writing, and reading English. The second coder was trained about the

93

characterization of beliefs and steps of data analysis by the researcher, who was the ﬁrst
coder. After the data analysis framework was explained to the second coder, the ﬁrst and
the second coder analyzed an interview together to practice coding the beliefs and belief
levels. Another goal of the sample coding was to practice reaching full-consensus
between the two coders on ﬁnal decisions of belief levels during the actual data coding.
During the data coding the coders tried to identify the belief statements and belief levels
based on the data analysis framework. Both coders analyzed the transcribed data with
pseudonyms for the participants in order to eliminate the bias.

Although a second coder was recruited in order to reduce the bias in data analysis,
categorization of mathematics related beliefs as nature, teaching, and learning beliefs had
some limitations for data analysis and interpretation of the results. First, the number of
nature beliefs was less than the number of teaching and learning beliefs listed in
Thompson (1991) and Lindgren (1996) frameworks. In order to reduce this limitation,
Ernest’s (1989) model, which was consistent with Thompson and Lindgren frameworks
in terms of the way beliefs were considered and teaching and learning beliefs were
described, was added to the framework. Still, nature beliefs were not described in detail
compared to the teaching and learning beliefs. Moreover, there is only one Level 1
learning belief described in the framework (see Appendix A). This might have limited the
belief statement that the coders were able to detect in the interview transcripts, as well as
the ﬁnal levels of beliefs.

Second, categorizing beliefs in three belief areas resulted in difﬁculty in
identifying the area of participants’ belief statements. This difﬁculty appeared when the

coders started to code interview data. The coders were more inclined to identify learning

94

belief statements from what seemed to be nature belief or teaching belief statements.
Most of the discussions between the coders occurred when, for example, a participant
believed that obtaining a correct answer was the goal of mathematics and one of the
coders considered that this belief implied the belief that considered reaching a correct
answer as the indication of learning mathematics. Although the coders deve10ped more
rigorous ways of detecting beliefs and avoiding inferences by considering key words. as
explained earlier in this chapter, categorization of beliefs in three belief areas and the lack
of sufﬁcient belief statements in some of these areas might have a limiting impact on
detecting participants’ belief statements and the results of the study.
Summary

In summary, the study is limited by three major factors. First, the amount of
interview data participants provided depending on my interaction with them limited my
inferences for the participants who were more hesitant to talk during the interviews.
Second, the data analysis framework might have excluded some of the mathematics
related beliefs that participants might have expressed during the interviews. And third,
the results are limited to the participants’ verbal data and they are not conﬁrmed by their
practices. Since belief research has a methodological problem with the validity and the
reliability of the self-reported data (Fang, 1996), the results of this study and their
interpretations have limited generalizability. However, using a data analysis framework
which was effectively used in previous studies and the data coding process with two
coders did capture the program students’ and graduates’ major mathematics related

beliefs and the impact of EME program on these beliefs.

95

 

Being a

 

I
observable and ti
(Pajares, 1992).
as being experie

consistent with t

 

procedure. Partic
two coders. How
believe what the
individuals are l
1992l- Hence, it
are not also fun

I use the
the deﬁnition. c
between me am
especially [mm
bElieve“ I also
a SPeCiﬁc belle
neutrality: an
COHSlSIEnCy 0ft

Interpreted the t

Reporting the Results

Being a personal construct with no common deﬁnition, beliefs are not directly
observable and must be inferred from people’s verbal reports, intentions, and behaviors
(Pajares, 1992). For the present study, a deﬁnition for and characteristics of beliefs (such
as being experience-driven) are identiﬁed from the literature and a theoretical framework
consistent with the characteristics is employed in order to guide the data analysis
procedure. Participants’ beliefs are inferred through these guidelines in full consensus of
two coders. However, even when individuals claim they believe something they may not
believe what they say they believe (Rokeach, 1968). In the case of observing a behavior,
individuals are likely to have different beliefs underlying similar behaviors (Kagan,
1992). Hence, it is almost impossible to detect individuals’ actual beliefs of which they
are not also fully aware for the most of the time.

I use the terms “belief” and “believed” for individual participants depending on
the deﬁnition, characteristics, the data analysis framework, and the full consensus
between me and the second coder. However, I try to use these terms with less certainty
especially throughout the results chapter and generally report what participants “seem to
believe.” I also provide extensive exact quotes which I and the second coder inferred for
a speciﬁc belief based on the framework. Additionally, I use the terms “central”,
“centrality”, and “peripheral” that I borrow from the Green (1971) referring to the
consistency of what participants seem to believe over time. Although most of the time I

interpreted the results through exact quotes from the participants, the professor in the

 

7 The yearly inspection of elementary schools includes inspecting new teachers’ current and previous lesson
plans and observing their teaching. Hence, teachers have to write their lesson plans based on the template
provided by the MNE.

96

program, and the literature, I present the concern that my interpretations of the results are
likely to be subjective.
Organization of Result Chapters

The following three chapters document the results of the study mostly through
participants’ paraphrased claims or direct quotes. In order to decrease the loss of meaning
in quotes that might have occurred during the translation, at times additional phrases were
added to some transcripts. By these additions readability and tense adjustment through
the text were also aimed. Table 4.11 shows the examples of additions to participants’
quotes and their usage.
Table 4.11

Examples of Specific Additions to Participants ’ Quotes.

 

 

 

 

 

 

 

Addition Usage
The mathematics [that a teacher] uses [in Completing the meaning of the quote
teaching] is not complex Increasing readability
a question that require[d] a solution Tense adjustment
JDjiscipline is very important Sentence admstment
there are both good (high) students Adding a similar meaning that would clarify
the quote
These are all PCK (in his own My explanation for the quote
abbreviation) for me.
[...] Claims between two statements which are

not included in the quote

 

Pause during the interview

There are a few phrases used in the following results chapters in order to improve
the ﬂow of the text by simplifying long phrases. The simpliﬁed and the original phrases
are “nature beliefs” simplifying “beliefs about the nature of mathematics”; “teaching

beliefs” simplifying “beliefs about teaching mathematics”; and “learning beliefs”

simplifying “beliefs about learning mathematics.”

97

The result chapters are organized according to the order of the research questions.
Chapter 5 answers the ﬁrst research question concerning the preservice participants’
mathematics related beliefs and the crucial experiences during their studies in the EME
program that might have an impact on their beliefs. Chapter 6 answers the ﬁrst
subquestion of the second research question regarding the inservice participants’ beliefs.
In Chapter 7, the possible impact of the EME program experiences and the ﬁrst-year
teaching experiences on inservice participants (the second and the third subquestions) are

documented.

98

CHAPTER 5

PRESERVICE PARTICIPANTS’ MATHEMATICS RELATED BELIEFS

Preservice participants, who were a total of 20 EME program students from the
2"d-, 3’d-, and 4m-year level, were interviewed only in 2004 about their mathematics
related beliefs and possible factors affecting those beliefs. The transcript data were
investigated in order to understand the nature of their beliefs by implementing the belief
framework and to explore the possible changes existing through year levels. Preservice

participants’ year levels, gender, and level of their mathematics related beliefs are given

 

 

 

 

in Table 5.1 .

Table 5.1

Preservice Participants ' Year in the Program, Gender, and Belief Levels in Three Belief

Areas.

Year Gender Nature Teaching Learning

P13 4 F l 0 0
P14 4 F 0 1 1
P15 4 M 0 1 0
P16 4 F O 0 0
Pl 7 4 F l l 0
P1 8 4 F 0 0 0
P19 4 F 0 1 1
P20 4 F 0 0 0
P21 3 M l 0 0
P22 3 F O 0 0
P23 3 M 1 1 1
P24 3 F 0 1 0
P25 3 M l l 0
P26 3 F l 1 1
P27 3 F 0 0 0
P28 2 F l 0 1
P29 2 F 0 0 0
P30 2 F 0 l 0
P3 I 2 F l l 1
P32 2 M l 0 0

 

99

Alter participants’ belief levels were detected, certain factors that might have
affected preservice participants’ beliefs were explored. In this chapter, general
characteristics of preservice participants’ mathematics related beliefs will be explored
ﬁrst. Then, some year-level variances will be addressed through the impact of certain
factors such as EME program courses and informal teaching experiences.

Preservice Participants’ Mathematics Related Beliefs

None of the preservice participants in this study had Level 2 beliefs about the
nature of, teaching, or learning mathematics. Participants had Level 0 and Level 1 beliefs
in the beliefs areas, as can be seen in Table 4.1. The number of participants with Level 0
and Level 1 beliefs were almost equal in the nature and teaching belief areas. Learning
was the area in which only 6 of the 20 preservice participants had Level 1 learning
beliefs.

Analysis of preservice participant’ beliefs showed that there were similarities
among the participants’ beliefs at the same belief level regardless of the year in the
program. In other words, participants from different years had similar Level 1 beliefs and
Level 0 beliefs. The similarities in the same belief levels across the participants are
described below.

Preservice participants’ Level 0 and Level 1 beliefs about the nature of
mathematics differed in the way participants considered the construction of mathematical
knowledge. Participants with Level 0 beliefs about the nature of mathematics claimed
that “[ I] here [were] always determined things in mathematics. [One could] not know
them " (P30 (2"d-year), Q3).” For Level 0 nature belief participants, mathematics was

simply a collection of rules that were determined by an outside authority. Moreover,

100

these rules should be known very well through memorization and be implemented in
daily life. “[0]therwise, there [would be] no meaning in knowing mathematics if [one
could] not transfer in daily life " (P22 (Yd-year), Q2). Participants considered knowing
mathematics as “to solve problems...it might be to perform operations, calculations...to
solve mathematics questions ” (P18 (4’h-year), Q2). On the other hand, Level 1 nature
belief participants considered that pieces of mathematical knowledge were logically
connected and reasons for these connections were important, as P31 illustrated: “To think
logically...why is this in this way? It should not be “This is in this way [...] Do not object
to it! " [...] Now in the proofs that [we] study in college, we ask “Why? (P31 (2”!-
year), Q29). Knowing mathematics was not solving a lot of questions, but trying to
answer the questions about “what [the mathematical concept] means, how it is formed,
and where else this could be implemented ” (P1 7 (4'h-year), Q2). The difference between
Level 0 and Level 1 beliefs were noticeable when participants’ claims about the nature of
mathematical knowledge and the ways to construct this knowledge were considered.
With respect to the beliefs about teaching, all preservice participants stated
teacher-centered practices as effective in teaching mathematics. However, there were
differences in participants’ consideration of the way teaching-centered practices should
take place in the mathematics classroom. Some participants claimed Level 0 beliefs
where they stated that the teacher should decide about what would happen in the
classroom: "The best thing to do is to start [from the beginning] and [teach] until the
end, without confusing [the students]. To say "These should be done... these should be
thought " ” (P28 (I'd-year), Q19). Level 0 teaching belief participants also described

correct-response oriented teaching practices in which thinking skills were not

101

 

 

 

emphasized, as Pl6’s descriptions illustrated: “I mean [I would ask] how many 1/5 does
I have. and they would answer. I would ask, now, in how many were we looking for
[1/5] Maybe they will answer [as] ﬁve " (P16 (4'h-year), Q12). These participants
assumed that students would most probably answer the questions in the way they, as the
teachers, would want them to answer. Participants did not consider that students might
have a different answer or might not answer at all.

In contrast to Level 0 participants who believed in highly teacher-centered
traditional practices, Level 1 teaching belief participants considered using different
approaches in teaching mathematics such as using games and manipulatives, although
they placed the teacher as the only decision maker in the classroom. The different
approaches were, however, not targeting student understanding at ﬁrst, but their
enjoyment in the mathematics class, as P24 claimed: “1 ’11 try to teach mathematics as a
game so that they won 't get bored, and they will understand better if they don 't get
bored ” (P24 (yd-year), Q1 7). Unlike Level 0 participants, Level 1 participants
emphasized the importance of understanding and increasing student thinking in the

mathematics classroom: “Mathematics teacher does not only say 2 x 3 = 6. He/she

should be able to evoke reasoning in children ’s mind. " (P31 (2"d-year), Q19). In general,

Level 0 and Level 1 teaching beliefs in the preservice group differed in the way the
nature of teacher-centered mathematics teaching was described.

Learning was the area in which most of the inservice participants had Level 0
beliefs. Level 0 learning belief participants placed the teacher as the authority who
decided about what and how students should learn, as P18 illustrated: “[Learning at

school] should deﬁnitely be by the teacher, [and] in the way that he/she shows ” (P18

102

 

 

-
-l_

 

(4'h-year), Q3). Student responsibility was limited to completing the tasks that the teacher
assigned, as explained by P21: “I believe that the students should solve a lot of problems.
I believe they should be assigned a lot of homework and exercises ” (P21 (Yd-year), Q3).
Level 1 learning participants also held the teacher mostly responsible for student
learning, but they claimed that students should study to comprehend the logic behind the
mathematical knowledge once the teacher introduced the concepts: “After understanding
[the logic of the concepts that the teacher taught], [the student] has to think about it,
[about] why...[ T he student] has to put that in [his/her] mind by [himself/herself]. ” (P26
(Yd-year), Q3). These participants suggested that students should try to understand the
idea behind the solutions to certain questions instead of solving a lot of questions. Level 0
and Level 1 learning belief participants’ beliefs differed mostly in the way they
considered student responsibilities during the learning process. The teacher was more
authoritarian in Level 0 beliefs as he/she was the only decision maker in the mathematics
classroom. For Level 1 belief participants, the teacher was still the decision maker in the
classroom, but they did assign some responsibility to the students other than solving
many questions.

In general, the difference between the Level 0 and Level 1 beliefs were detectable
but not dramatic. Level 1 belief participants in the teaching and learning belief areas had
similar beliefs with Level 0 participants. However, Level 1 participants believed that
different methods and materials should be used for student enjoyment and learning. Level
1 participants indicated that students should have some responsibility in learning
mathematics other than answering questions correctly. There seemed to be no radical

difference between the content of year-level groups’ beliefs.

103

 

Year-Level Variances and Possible Factors

A closer look at Table 4.1 showed that there was no pattern of increase or
decrease in number of Level 1 or Level 0 beliefs in any of the belief areas as participants
advanced through the program. However, there was a decrease in the number of Level 1
nature belief participants in the 4'h-year group. Only two 4’h-year participants out of eight
had Level 1 beliefs about the nature of mathematics. Moreover, the number of
participants with Level 1 learning beliefs was quite low especially in the 3rd- and 4‘h-year
level groups compared to the number of participants in these groups. In addition to the
lack of signiﬁcant level increase/decrease pattern, the nature of beliefs in Level 0, and
also in Level 1 beliefs across the year levels did not differ much. For example, teaching
beliefs of Level 1 teaching belief participants were similar through year levels in the
EME program.

One interpretation of the similarities between the 2"d-, 3'd-, and 4’h-year students’
beliefs in the belief areas might be that the EME program courses did not seem to have a
signiﬁcant impact on preservice participants’ beliefs. The relatively low number of
participants with Level 0 nature beliefs at the 4'h-year level might suggest that the impact
of the mathematics courses on participants’ nature beliefs even had decreased as the year
level in the program increased. The lack of emphasis on the nature of mathematical
knowledge in methods courses might impact this decrease as well.

Somewhat interestingly, although 3'd-year and 4‘h-year participants were still
taking or had just completed the coursework about methods of mathematics teaching,
some participants mentioned the use of different methods in teaching mathematics (a

recurrent theme in methods courses) while others did not make such claim. In general, the

104

PCK coursework which included methods, curriculum, and educational psychology
courses, did not seem to have a considerable impact on the participants’ beliefs about
teaching mathematics.

Preservice participants’ learning beliefs were generally low through the year-level
groups compared to the beliefs about the nature of and teaching mathematics. Participants
mostly believed that students could learn mathematics only by the help of an authority,
who was the teacher in the case of the mathematics classroom. Students had to solve as
many questions as they could in order to learn better. This ﬁnding, however, was not
completely unexpected. The mathematics education professor in the EME program
claimed that the program was not providing opportunities for the preservice teachers so
that they would develop higher level beliefs about learning. He generalized this to both
preservice and inservice teachers: “They don ’t have any idea about how environments in
which students could learn by themselves could be organized ” (Dr. C., 2004, Q6). In
general, EME program courses did not appear to have an impact on participants’ learning
beliefs.

Preservice participants had different perceived needs in the program as the year
level increased. 2nd-year participants expressed a need for a course in which they would
practice teaching in front of their friends during the course: “I wish we could teach to our
friends [in a course] ” (P27 (I'd-year), Q27). Since 2"d'year participants had not taken the
3'd-year methods course which is organized in this way, they believed that this kind of
course would be a good practice before school experience courses. Having taken this
methods course, 3'd—year participants believed that they needed more practice of various

teaching approaches, as P22 expressed: “Maybe there should be more practice of [things

105

we learned in the methods course]. I haven 't taken the other school experience courses
yet, maybe those courses have what I am looking for, but things should be practiced, they
should be implemented” (P22 (Yd-year), P2 7). It seemed that as the year level increased
in the program, participants expressed needs for experiences that were to appear the
future program courses. Thus, they seemed to be aware of their needs as future teachers.

However, at the end of the program, 4’h-year participants expressed the need for
some courses they already took in the program. This might suggest that participants did
not beneﬁt from the courses as intended. One reason for this seemed to be participants’
beliefs about their and professors’ responsibilities in the EME courses. Interestingly, 4’”-
year participants who already completed all the coursework including the school
experience courses in the EME program believed that the program faculty should have
given them the exact knowledge of “how a certain [mathematical] concept should be
taught ” (P20 (4'h-year), Q27). F ourth-year participants complained that they had to learn
the methods course contents by themselves through the projects and the presentations,
and the professors did not teach sufﬁciently. These participants believed that they needed
more authority in their learning as a college student and they should have been told about
what and how to teach mathematical concepts. Thus, perhaps it is not surprising that the
learning belief level of preservice participants was relatively low compared to the other
belief areas across the year-levels.

Most preservice participants also had informal teaching experiences during their
studies in the EME program. Although these experiences did not seem to have a
considerable impact on 2"d- and 3'd-year participants’ beliefs, it had some inﬂuence on

the way 4’h-year participants considered their teaching practice. Most 4‘h-year Level 0

106

teaching belief participants had two years of private tutoring experience and they tended
to answer questions in terms of single student. These participants claimed that they did
not really know how they would conduct teaching in the actual classroom context. On the
other hand, most Level 1 teaching belief participants in the 4’h-year group. also had
informal teaching experiences, but they mentioned the content of the EME program
courses during the interviews. The existence of participants with long or no informal
teaching experiences in both Level 1 and Level 0 groups suggested that informal teaching
experiences did not have a systematic effect on preservice participants’ teaching beliefs.
Participants’ informal teaching experiences generally started one or two years before they
started to take methods courses. It might be speculated that, when the participants started
to take methods courses, those who were able to connect the course experiences to their
existing teaching experiences had Level 1 beliefs. Preservice participants with Level 0
beliefs seemed to lack this connection and preferred to prioritize informal teaching
experiences.
Summary

The data gathered from the preservice participants indicated that participants’
beliefs about the nature of, teaching, and learning mathematics did not dramatically vary
across the year level groups as well as the belief level groups. Preservice participants’
teaching beliefs were mostly teacher-centered and their learning beliefs were mostly
addressing teacher authority. These beliefs did not have a pattern as the year level in the
program increased, but the relatively lower level of learning beliefs among the
participants through all year levels was noticeable. The program faculty’s comment on

inservice and preservice teachers’ lack of beliefs about student-oriented learning

107

ll

environments was also evident in 4’h-year participants’ beliefs about how they should
have been taught in the EME program.

The courses in the EME program and the informal teaching experiences seemed
to have some impact on participants’ beliefs towards Level 1, but the effect was very
limited as could be seen in the difference between the Level 0 and Level 1 beliefs
especially in teaching and learning areas. Similarly, informal teaching practices had some
very limited effect on the way participants considered teaching and learning mathematics,
moving them towards Level 0 beliefs. However, in both Level 0 and Level 1 belief
groups there were participants with no informal teaching experience or with considerable
private tutoring experiences respectively. This might suggest that the impact of the
experiences in the informal teaching contexts and in the EME program were mixed and
might be related to the way people connect these to their existing experiences. It seemed
that when participants could not build this connection meaningfully, they prioritized the

informal teaching experiences and developed beliefs based on them.

108

CHAPTER 6

INSERVICE PARTICIPANTS’ MATHEMATICS RELATED BELIEFS

This chapter documents the ﬁndings from the inservice middle school
mathematics teachers’ data collected in 2003 and 2004. Inservice participants’ beliefs and
belief levels when they graduated from the Elementary Mathematics Education (EME)
program (in 2003) and when they completed their ﬁrst-year of teaching (in 2004) were
identiﬁed and compared. Table 6.1 gives the belief levels of participants’ 2003 and 2004
beliefs about the nature of mathematics, teaching mathematics, and learning mathematics.
Table 6.1
Inservice Teachers ' Belief Levels on Nature of Mathematics (N), Teaching Mathematics

(T), and Learning Mathematics (L) in 2003 and 2004.

 

N2003 N2004 T2003 T2004 L2003 L2004
Pl 1 1
P2 1 1-2
P3 1
P4 1
P5 0
P6 1
P7 0
0
l
1
1
l

 

H
bi

l
1
O
2
1
1

P8
P9
P10
P11
P12

O-l

--ooo-—-—N-—-o--u—-
v—u-n—v—oomo—n-u—u—
o—-—-—-—-—-—o—~—~—-

ooo~oo~oo-~

l
l
0

 

Some participants had the same pattern of change in the belief areas, as can be
seen on Table 6.1. While some participants’ belief level did not change in any of the
belief areas, some participants had similar changes in the form of increase or decrease in

at least one of the three belief areas. Inservice participants’ beliefs were described and

109

analyzed in the groups that the change in their belief levels suggested. For example, P7
and P8 formed the “Consistent-Increase Group” since their belief levels increased
consistently from Level 0 to Level 1 in all belief areas from 2003 to 2004. In each group,
participants’ beliefs were investigated and interpreted in relation to the speciﬁc teacher
education program (Elementary Mathematics Education (EME) program), informal and
formal teaching experiences during the EME program, and the ﬁrst-year teaching
experiences in order to identify possible common factors affecting the beliefs.

Inservice participants’ mathematics related beliefs were investigated through ﬁve
groups of change pattern: No—change, consistent-increase, slight-decrease, slight-
increase, and mixed-change. In each group, ﬁrst participants’ belief levels and preservice
and inservice teaching experiences are described. Then, their beliefs about the nature of,
teaching, and learning mathematics are explored and analyzed in relation to their
experiences. A detailed summary of ﬁndings across all inservice participants will be
given in the following chapter.

No-Change Group

P1, P2, and P3 maintained the same Level 1 beliefs in 2003 and 2004. P2 revealed
some Level 2 beliefs about learning mathematics, but since she still maintained Level 1
beliefs she was considered in this group. Table 6.2 shows the gender and belief level of

participants with their teaching contexts.

 

8 The nature of informal teaching contexts is described in the Methods chapter.

110

Table 6.2
Gender, Type of Teaching Context, Belief Levels of Participants Whose Belief Levels

were not Changed.

 

Gender Teaching N2003 N2004 T2003 T2004 L2003 L2004

 

Context
P1 M Public 1 1 1 1 1 1
P2 F Private 1 1 1 1 1 1-2
P3 F Private 1 1 1 1 1 1

 

Despite the similarities in their belief levels, participants in the no-change group
were teaching in different school contexts at the time of the 2004 study. Pl taught in a
rural public school for a semester before he was hired by the EME department as a
graduate assistant during his studies in the EME master’s program. While in college, his
teaching experiences were student-club teaching for three years and private tutoring for
one year. P2 taught in a private school in her ﬁrst year of teaching. She had two years of
private tutoring experience while in college. P3’s ﬁrst year of teaching was in a private
school which was opened few years ago and she was the only mathematics teacher in that
school. Her college teaching experiences included private tutoring for two years and
working in a private middle school curriculum project for one year. All no-change group
participants taught middle school mathematics in their school contexts.

Beliefs about Nature of Mathematics
All participants in the no-change group had Level 1 beliefs about the nature of

mathematics. More speciﬁcally, the participants in this group consistently believed that

111

mathematical knowledge included logical thinking and connections between ideas; they
deﬁned knowing mathematics as having systematical thinking and multiple views. The
participants revealed similar beliefs in 2003 and 2004, as Pl illustrated:

“We can say that knowing mathematics is having the logic [...] [It is]

more than knowing the formulas and certain rules... [It is] to have the

systematic thinking construct. ”(P1, 2003, Q2)

“F irst of all, it is to know that logic [...] If we deﬁne mathematics as the

logic that we created in nature or life, numbers, and relationship between

numbers, then [mowing mathematics is knowing these. ” (P1, 2004, Q2)

Similarly, beliefs about the nature of mathematical problems were quite similar in
2003 and 2004 for no-change participants. For example, P1 and P2 believed in 2003 that
reaching a solution through using certain concepts and procedures was the essence of
mathematical problems. After completing the ﬁrst year of teaching in 2004, they believed
that mathematical problems were the questions expressed in numbers where (i)
connections between the numbers could be built (P1, 2004) and (ii) new information
would be reached as a result of existing information in the problem (P2, 2004). P1 and P2
deﬁned a mathematical problem as the context where certain information was connected
to each other, and P3, somewhat similarly, framed her beliefs around the unknown or
“problematic” character of the mathematical problems. She believed in 2003 and 2004
that a mathematical problem had to be " a problem for the kids... how can I ﬁnd, where
should I go... he [had] to find other solutions by himself (P3, 2003, Q7). " For her, “one
should be confused about what to do (P3, 2004, Q8) " when one initially dealt with a

mathematical problem.

112

Although all three no-change participants maintained Level 1 beliefs in 2004,
there were some subtle differences in their beliefs about the nature of mathematical
understanding. In particular, P1 ’5 beliefs could be considered as falling somewhat lower
than P2 and P3’s beliefs, even though all three were ultimately rated as having Level 1
beliefs. Evidence for this determination was found in the ways that participants
characterized a student with rich mathematical understanding. Pl believed that rich
understanding could be seen in a student when he/she expressed interest in mathematics
and responded to questions correctly. In contrast, and falling somewhat higher on the
belief level scale, P2 and P3 believed that rich mathematical understanding could be seen
in a student when he/she expressed knowledge of deﬁnitions and important points and
connections between the ideas.

Similarly, Pl ’5 perception of the usefulness of mathematical content courses in
the program was somewhat different than P2 and P3’s perception in 2004. All no-change
participants believed in 2003 that mathematics courses in the program would be useful
for their profession. However, in 2004, P1 believed that the mathematical thinking he
gained through those courses could not be integrated into his teaching. In contrast, P2 and
P3 believed that they were able to integrate their mathematical understanding in their
teaching as P3 explained:

“Math courses... Discrete course changed my life. I learned how to think

[...] [I use it in my teaching] a lot. I am not using the concepts there, but

that kind of... asking the "why ” question " (P3, 2004, Q31).

It seemed that not all the participants in no-change group connected the

mathematics they studied in the EME program to the mathematics they taught. All of the

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participants believed that mathematical content courses were useful in improving their
mathematical understanding and they all had similar Level 1 beliefs. However, they had
different beliefs about the degree that this mathematical understanding could be
incorporated into teaching middle school mathematics. The difference in P1 ’5 ideas might
be related to his teaching context where he had limited resources. P2 and P3, on the other
hand, had a wide range of resources and materials that were available in their teaching
contexts. This helped them in teaching mathematics on which their mathematical
understanding was reﬂected.

In summary, despite the differences in no-change participants’ college and ﬁrst-
year teaching experiences, participants mostly had similar Level 1 nature beliefs in both
2003 and 2004. One might imagine that participants’ beliefs about the nature of
mathematics might be shaped by mathematical content courses during their studies in the
program. No-change participants’ teaching contexts generally seemed to help them to
maintain Level 1 beliefs.

Beliefs about Teaching Mathematics

All three participants in the no-change group had Level 1 beliefs about teaching
mathematics. Their 2003 beliefs addressed different but closely related issues,
particularly indicating that mathematics teaching should include different pedagogical
approaches (P1, 2003); in a teacher-centered classroom (P2, 2003); where student
differences were considered (P3, 2003). In 2004 these Level 1 beliefs were maintained
and expressed in detail where participants reported that mathematics teaching should

include: (i) relating mathematics to daily life and using concrete materials (P1, 2004); (ii)

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making children reach correct solutions by logical thinking (P2, 2004); and (iii) drawing
visual representations for students with different levels of understanding (P3, 2004).

These somewhat more detailed views of mathematics teaching in 2004 seemed to
have emerged for no-change participants as a result of their ﬁrst-year teaching. The ﬁrst
year of teaching did not change the level of their beliefs about mathematics teaching but
rather seemed to help no-change participants to develop better articulated beliefs in which
they provided more details.

The more detailed beliefs in 2004 were also reﬂected in no—change participants’
aims in teaching mathematics. P1 was concerned about pedagogical approaches to
teaching mathematics that included daily life connections. He aimed to make students
apply mathematics in their everyday lives. Similarly, P2 believed in more teacher-
centered practices while considering students’ logical thinking, which was consistent
with her aim of making students understand the concepts without memorizing them. P3
believed in addressing student differences and she aimed to increase student interest and
enjoyment. In summary, the common element in the 2003 and 2004 beliefs about
teaching mathematics for no-change participants was the characterization of mathematics
teaching as a teacher-centered practice including different pedagogical approaches in the
classroom where student understanding was considered.

No-change participants’ school contexts seemed to play a key role in maintaining
their belief levels and adding more detail to their existing beliefs. Participants claimed in
2004 that they did not feel any pressure from the school administrators regarding their

teaching practice. The administrators supported their teaching and the schools tried to

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provide no-change participants with the resources they asked for, especially in the case of
P2 and P3.

Although there was no pressure from the administrators, curriculum did put
pressure on no-change participants’ teaching. Participants indicated in 2004 that they had
difﬁculty in keeping the same pace with the yearly plan. Part of this difﬁculty was
attributed by participants to the students’ missing prerequisite knowledge. No-change
participants assumed that the students already knew the previous years’ concepts but
discovered that many students lacked such knowledge. Parent support to no-change
participants’ teaching was generally low. Participants claimed that parents only
considered students’ grades and blamed participants for not teaching well enough so that
the students would have higher grades.

Despite the similarities in relationships with administrators and parents,
participants had different sets of ﬁrst-year teaching experiences. P1 ’5 ﬁrst year
experiences started in a rural public school where resources were limited. He might have
had lower Level 1 teaching beliefs by the end of the school year if he had not quit
teaching to work as a graduate assistant for preservice teachers. It seemed that P1 was
able to maintain his teaching belief level and reveal more detailed Level 1 beliefs by
changing his relative position from being a teacher to a teacher educator and graduate
student. P2 and P3, on the other hand, had the support of the administrators and a wide
variety of resources during the year which probably helped them to maintain their belief
levels. No-change participants seemed to compensate for the various negative factors

(such as cuniculum pressure and lack of student knowledge) that might have affected

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their belief levels with different elements (such as administrative support or change of
position) in their contexts.

As a result, it could be concluded that no-change participants’ Level 1 beliefs
about teaching mathematics did not undergo substantive change between the end of the
EME program (2003) and the end of the ﬁrst-year teaching (2004). In fact, these beliefs
became more central through their ﬁrst year experiences.

Beliefs about Learning Mathematics

As was the case with the other two types of beliefs, all no-change participants had
Level 1 beliefs in 2003 and maintained the same level in 2004. No-change participants
believed in 2003 that in order to learn mathematics, students should be able to: (i)
implement their knowledge on different situations (P1, 2003); (ii) connect the existing
knowledge to the new knowledge (P2, 2003); and (iii) comprehend all prerequisite
knowledge (P3, 2003). Similarly, in 2004 P1 emphasized that students should be able to
implement mathematics in any situation, where P2 and P3 believed that comprehending
the logic and the concepts through connecting new knowledge with the existing
knowledge led to learning.

No-change participants’ beliefs about learning mathematics not only remained
consistent between 2003 and 2004 but also continued to be well-articulated. P1 believed
that mathematics could be learned "when [the students] create[d] problems by
themselves and analyze[d] them " (PI, 2004, Q3). P2 and P3 believed that the concepts
should be logically comprehended by the students until all details were clear for the
students. For example, P3 stated that “[a] student must start from the lowest step and

should go through every step. Or if he had missed a step, then he ha[d] to compensate

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that step ” (P3, 2004, Q3). The consistency and wordiness in no-change participants’
learning beliefs could also be related to their beliefs about nature of mathematics, as
discussed above. Recall that no-change participants believed mathematics involved
logical connections between the ideas and they also believed that learning mathematics
was about learning the connections and the logical structure. Given that there was no
change in their beliefs about the nature of mathematics, it is perhaps not surprising that
their learning beliefs also remained the same.

Besides having maintained learning belief levels, no-change participants had
learning beliefs consistent with their teaching beliefs. As stated in the previous section,
no-change participants believed that the teacher was the main responsible actor for
students’ learning. Correspondingly and in terms of learning mathematics, participants
assigned few roles to the students in the classroom. No-change participants believed that
students should learn the connections in mathematics in a teacher-centered classroom. On
the other hand, they were aware in both 2003 and 2004 of the fact that not all students
were identical. P2 noted: “You see many different kinds of students, I mean they are all
different than you can imagine. Something you don ’t expect that they will understand is
understood and something you expect that they will understand is not understood. You
cannot determine it earlier" (P2, 2004, Q13). No-change participants’ beliefs about
student differences, which were maintained from 2003 to 2004, fell into Level 1 as well.

No-change participants’ Level 1 learning beliefs might be attributable to their
students’ lack of knowledge in their teaching contexts. They experienced difﬁculty in
teaching mathematics to their students who often lacked previous knowledge. This

difﬁculty possibly resulted in beliefs that assigned less role and responsibility to the

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ll

students for their own learning. It seemed that although no-change participants were
challenged by the lack of student knowledge in the classrooms, they were able to
maintain their Level 1 beliefs about learning mathematics.

Other Changes and Conclusions

Despite the similarities between no-change participants’ beliefs in all three belief
areas, differences did emerge in participants’ beliefs about the types of knowledge that
were useful in teaching mathematics and the effectiveness of the EME program in
preparing them to teach.

With respect to the knowledge that was useful for teaching, for P1 , mathematical
content knowledge was the least important type of knowledge in both 2003 and 2004.
Rather, Pl emphasized the importance of pedagogical content knowledge:

“The mathematics [that a teacher] uses [in teaching] is not complex [...]

You have the chance of relating elementary school mathematics to the

daily life, not all the time, but most of the time. You have the chance to use

concrete materials. These are all PCK (in his own abbreviation) for me

and mathematical content knowledge comes later in my area " (P1, 2004,

Q1 9).

The possible reason for P1 ’5 emphasis on PCK might be the inﬂuence of his teaching and
research in the EME masters program that focused on teachers’ pedagogical content
knowledge. In contrast, P2 and P3 rated mathematical content knowledge as primary in
both 2003 and 2004 since “[a teacher] could direct the students to different and incorrect
things” (P2, 2004, 19) if she did not have the sufﬁcient content knowledge and “[a

teacher] should think about everything [about a mathematical concept] ” (P3, 2004,

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Q19). These participants referred to their experiences in the classroom where students
asked different and unexpected questions and participants’ mathematical content
knowledge helped them to answer these questions with conﬁdence. No-change
participants’ experiences in their ﬁrst-year teaching contexts seemed to affect their
beliefs about the priority of different types of knowledge for teaching.

No-change participants also had somewhat divergent views on the effects of the
EME program on their preparation to teach. Perhaps not surprisingly, Pl (who indicated
that PCK was the most important knowledge for a teacher) also indicated that PCK
courses had better prepared him for teaching as compared to content knowledge courses,
both in 2003 and 2004. In fact, he indicated that mathematical content courses seemed to
be getting less useful as he completed his ﬁrst year of teaching. And similarly, P2 and P3
(who prioritized mathematical content knowledge over pedagogical content knowledge in
both 2003 and 2004) believed in the usefulness of content courses in the teacher
preparation. process.

No-change participants’ similar beliefs were accompanied by parallel needs for
what they felt that the EME program might have provided to them in 2004. Participants
claimed that they would like to have a course that focused on how to teach the 6’”, 7’",
and 8th grade mathematics curriculum including speciﬁc teaching activities and
manipulatives during their studies in the EME program. Although no-change participants
believed that there were differences among the student population which required
different and perhaps individualized teaching approaches, they also believed that best
practices for teaching middle school mathematics concepts did exist (e.g., “the best

method that ﬁts teaching fractions”) regardless of student differences. These two beliefs

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(student differences exist, but there is one best way to teach a particular concept) seemed
contradictory. A possible reason for the presence of two contradictory beliefs might be
that they existed in two different clusters of beliefs as Green (1971) explained, although
they both seemed to be a result of their experiences in the classroom. Participants
believed that their difﬁculties in dealing with student differences or lack of student
knowledge in the classroom could have been eliminated if they had known these best
practices.

In summary, belief levels of no-change participants (P1, P2, and P3) had not
changed after ﬁrst-year teaching and remained as Level 1 in all belief areas. Participants’
beliefs about the nature of mathematics and teaching mathematics seemed to affect their
beliefs about learning mathematics. They seemed to struggle to ﬁnd ways to teach
mathematics when students lacked previous knowledge and certain pressures, such as
curriculum rush, were effective. No-change participants believed that the way they
conceptualized mathematics could be taught in the classrooms by decreasing student
contribution and participation while increasing and prioritizing teacher decisions and
authority in the classroom.

Consistent-Increase Group

The belief levels of P7 and P8 increased from Level 0 to Level 1 in all three belief

areas after the ﬁrst year of teaching. The gender, belief level and teaching contexts of

participants are given in Table 6.3.

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Table 6.3
Gender, Teaching Context, Belief Levels of Participants Whose Belief Levels Were

Increased Consistently.

 

Gender Teaching N2003 N2004 T2003 T2004 L2003 L2004

 

Context
P7 F Private 0 l 0 1 0 1
P8 M Public 0 1 0 1 0 l

 

Although these two participants in consistent-increase group had the same pattern
of increase in their beliefs, their teaching contexts were different. P7 taught in a private
school in her ﬁrst year. She had one year of private tutoring experience while in college
and completed a double degree program in mathematics during her studies in the EME
program”. P8 taught in a rural public school in his ﬁrst year and he had one year of
private tutoring and 1.5 years of student club teaching experiences in college. P7 taught
middle school mathematics and P8 taught middle school mathematics and high school
physics in their ﬁrst year.

Beliefs about Nature of Mathematics

Consistent-increase participants did not give complete and clear answers to the
main and probing questions about the nature of mathematics in 2003. It could be inferred
from their answers that their 2003 beliefs addressed (i) knowing mathematics was having
sufﬁcient mathematical knowledge (P7, 2003), and (ii) solving many questions was the

most effective way of improving mathematical understanding (P8, 2003). After the ﬁrst

 

9 Students in the EME program can have a second degree which is named as “Double Major Degree" in the
Mathematics Undergraduate Program by completing extra coursework.

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year of teaching, both consistent-increase participants were more articulate on the same
or similar questions and provided clearer belief expressions about nature of mathematics.
For example, P7 believed that mathematics provided abilities that one could use in daily
life. Such abilities included “analytical thinking [and] approaching issues through
different perspectives ” (P7, 2004, Q2). For P8, advanced mathematical knowledge
should be built on fundamental knowledge: “[In math] each concept has a beginning
point. When you know these beginning points, then... level by level... [you] add more on
[this beginning point] depending on your level or your knowledge " (P8, 2004, Q3 ).

The way consistent-increase participants characterized a student with rich
mathematical understanding also reﬂected an increase in their belief levels to Level 1 in
2004. They believed that some of their students had rich level of mathematical
understanding since they were able to comprehend connections between mathematical
ideas, as P8 illustrated: ”After I teach a concept, I ask a question which includes
connection to another concept. Those students [with rich level of mathematical
understanding] can immediately respond in some ways " (P8, 2004, Q15).

The relative increase in consistent-increase participants’ nature belief level was,
however, not reﬂected on their other related beliefs in this category. Both participants had
lower level beliefs about the nature of mathematical problems in both 2003 and 2004.
Participants characterized a mathematical problem in 2003 as “the problem that [could]
be solved by using mathematics ” (P7, 2003, Q7) and ”a question that require[d] a
solution " (P8, 2003,Q7). Similarly in 2004 they believed that mathematical problems
were the contexts where certain operations and procedures could be implemented (a

Level 0 sentiment) rather than as the contexts that provided a challenge that required

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advanced thinking skills (a Level 1 belief). Despite the overall increase in the
sophistication of their beliefs about the nature of mathematics, consistent-increase
participants persisted with Level 0 beliefs about the subcategory concerning the nature of
mathematical problems.

Although consistent-increase participants generally had similar Level 1 beliefs
about mathematical understanding and similar Level 0 beliefs the nature of mathematical
problem in 2004, their beliefs regarding the useﬁrlness of mathematics courses were
somewhat different. Both participants believed that the mathematics courses improved
their thinking skills in 2003 and 2004. However, they had different beliefs about the
usefulness of these thinking skills in teaching mathematics in 2004. P8 did not believe
that the thinking skills learned in mathematics courses could be incorporated into
teaching middle school mathematics, whereas P7 believed in and illustrated the usage of
these thinking skills as: “[N] ow I approach [to the problems] in terms of both how to
teach and how to interpret in a different way [since I took the discrete mathematics
courses] ” (P7, 2004, Q10). This difference in beliefs might be due to the impact of the
double major program in mathematics that P7 completed while in college.

In summary, despite the differences that the two consistent-increase participants
had in their teaching contexts, they had quite similar and more sophisticated beliefs in
2004. The fact that they had lower level beliefs about the nature of a mathematical
problem might suggest that P7 and P8 were still in the process of moving from Level 0
beliefs to Level 1 beliefs about the nature of mathematics at the end of their ﬁrst year in

teaching.

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Beliefs about Teaching Mathematics

Consistent-increase participants’ beliefs about teaching mathematics increased
from Level 0 to Level 1 after the ﬁrst year of teaching. Participants actually had quite
similar beliefs in each year, but the change in the overall belief level seemed to be caused
by a deﬁnite change in their beliefs about students’ role in teaching. Consistent-increase
participants claimed almost no role for the students but held the teacher as the only
responsible person in the mathematics classroom in 2003. For example, P7’s main
concern in the class was “after they understand this, I can give this and give that... Will
they be able to understand this example? ” (P7, 2003, Q1 7) since she located herself as
the main information giver in the classroom. After the ﬁrst year of teaching, however,
both consistent-increase participants considered student input in their teaching as
important. They claimed that they recognized student responses and implemented
different pedagogical approaches to address individual differences in learning
mathematics. Although participants believed in the importance of having more student-
centered practices in 2004, they still placed the teachers as the main authority in the
classroom.

This greater emphasis on student participation seemed to be associated with a
shift from cognitive outcomes to affective outcomes in consistent-increase participants’
aims. In 2003, participants aimed to increase students’ understanding and level of
thinking. However, in 2004, their aim was to make students enjoy mathematics and have
a positive attitude towards mathematics course, as P8’s beliefs illustrated: “[My ﬁrst aim
is] to make them like mathematics... 0r ﬁrst like me as a person, then have an interest in

mathematics, and then to reach a certain knowledge level " (P8, 2004, Q20). It was not

125

the case that participants were less concerned with cognitive outcomes; rather,
participants believed that once the student math anxiety and dislike of math were
eliminated, students could learn better. It seemed that the change in beliefs about how
mathematics should be taught might be related to the change in beliefs about how
students could learn mathematics. This issue will be revisited in the next section.

The participants in the consistent-increase group had supportive administrators in
their teaching contexts. Moreover, unlike P8 who taught in a setting with rather limited
resources and no parent support, P7 had many materials and resources available to use in
the classroom in her school and she generally had support from parents. Both P7 and P8
frequently mentioned student mathematics anxiety as the most important factor that
affected their teaching in the classroom: “[ Students have] a fear for mathematics and a
negative attitude. First, these should be erased from their minds " (P 7, 2004. Q20). This
suggested that dealing with students’ mathematics anxiety possibly caused a change in
participants’ beliefs and aims, and initiated a new belief that prioritized affective
outcomes over cognitive outcomes.

Beliefs about Learning Mathematics

Similar to the results in the other two belief categories, consistent-increase
participants’ learning beliefs also increased from Level 0 to Level 1 after the ﬁrst year of
teaching. Participants’ 2003 beliefs considered students as passive receivers of
knowledge and imitators of question solving procedures that the mathematics teacher had
demonstrated. Whether or not students successﬁrlly received knowledge could be
evaluated through examination results. After the ﬁrst year of teaching, the major change

in consistent-increase participants’ learning beliefs was an increase in the responsibility

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that students should have in their learning and the importance of having positive attitudes
in this process, as mentioned in the previous section. Consistent-increase participants
believed that students should contribute to the learning process by their explanation (P7,
2004) and that students’ effort to reach advanced mathematical knowledge would be an
indication of learning (P8, 2004).

Having positive attitude and motivation were the preconditions for learning
mathematics, as P7’s beliefs illustrated: “You have to have [a] positive attitude [towards
mathematics]. You have to like mathematics. Additionally, math should be interesting for
us... so that we can learn” (P7, 2004, Q3). However, it was teacher responsibility to
motivate the students: “There is a need for at least one person who can motivate or adapt
a student [to mathematics] so that he can learn or understand mathematics " (P8, 2004,
Q3). Participants believed that once students were motivated, they would study the
concepts in order to learn.

In summary, having students with mathematics anxiety appeared to have an
impact on consistent-increase participants’ learning beliefs. Participants believed in 2004
that they were responsible for increasing student attitude towards mathematics and
students were responsible for studying once they had positive attitudes.

Other Factors and Conclusions

The participants in the consistent-increase group had somewhat similar beliefs
about the types of knowledge that were useful for teaching mathematics and the inﬂuence
of the EME program experiences on their teaching.

The two consistent-increase participants agreed about the knowledge needed for

teaching mathematics in both 2003 and 2004. They believed in 2003 that mathematical

127

content knowledge was more important than PCK for teaching mathematics. In 2004,
however, they both believed that PCK was the most important type of knowledge,
followed by content knowledge. There seemed to be a connection between consistent-
increase participants’ beliefs about’the types of knowledge and their aims in teaching.
When participants had cognitive aims in their teaching such as increasing student
thinking in 2003, they prioritized mathematical content knowledge over PCK. In contrast,
when they had more psychological aims in 2004 such as increasing student attitude and
motivation, they prioritized PCK over content knowledge.

The ﬁrst year of teaching did not change much in consistent-increase participants’
beliefs about the inﬂuence of the EME program experiences on their teaching. Both
participants believed in 2003 and in 2004 that PCK courses were useful in their teaching.
However, they believed in 2004 that more PCK courses should have been offered in the
program so that they would conduct better teaching in their classrooms. The need for
more PCK courses might be related to participants’ change in beliefs about how
mathematics should be taught and how students could learn mathematics. Since
consistent-increase participants tried to make mathematics more interesting for their
students (because of a new-found concern for affect and motivation), they needed more
PCK courses that focused on effective mathematics teaching through increasing
motivation. This was also consistent with the way they prioritized pedagogical content
knowledge over mathematical content knowledge in 2004.

The consistent increase in participants’ belief levels seemed to be a result of
meeting the students in actual classroom enviromnents. Participants claimed that being in

the real-classroom environment with a real student population made everything they

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learned different once they started to teach. Participants started to make sense of what
they had learned in the program after they started to teach in the real-classroom context.
They prioritized students’ motivation and positive attitude in the mathematics classroom
instead of a more dominant focus on increasing content knowledge. Despite the radical
difference between the school contexts (in terms of available resources, parents’
involvement, and students’ socio-cultural and socioeconomic background) and the
differences in college teaching experiences, both consistent-increase participants had the
same pattern of change in their belief levels. Their perceived needs were also related to
their ﬁrst-year teaching experiences. Consistent-increase participants considered more
PCK courses as needs in the program, which was consistent with their rankings in 2004.
It seemed that the ﬁrst-year teaching experience had a strong impact on the change of
participants’ belief levels.
Slight-Decrease Group

P9 and P12 were considered to be the slight-decrease group since their belief
levels decreased to a lower level in at least one of the belief areas in 2004. P9’s 2004
belief level in nature of mathematics decreased from Level 1 to Level 0 and her beliefs in
learning mathematics indicated both Level 0 and Level 1 beliefs, suggesting a possible
future decrease in learning belief level. Her teaching belief level was maintained at Level
1. P12’s teaching belief level decreased from Level 1 to Level 0, whereas his nature and
learning belief levels stayed the same, Level 1 and Level 0 respectively. Table 6.4 shows

gender and belief levels of participants with their teaching contexts.

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Table 6.4
Gender, Teaching Context, Belief Levels of Participants Who Had Decrease in their

Belief Levels.

 

Gender Teaching N2003 N2004 T2003 T2004 L2003 L2004

 

 

Context
P9 F Public 1 O l 1 1 1-0
P12 M Private 1 1 1 0 0 O

 

P9 and P12 had somewhat different college and ﬁrst-year teaching contexts but
they had similar experiences with the students in their ﬁrst year, which will be reported in
the Beliefs about Teaching Mathematics section. The analysis of their interviews showed
that P9 and P12 were cases where similar experiences with the student population
resulted in a decrease of belief levels in different belief areas. Hence, it seemed
reasonable to report the change in P9’s and P12’s beliefs separately under the three belief
areas (nature of mathematics, teaching mathematics, and learning mathematics) followed
by a summary section for each area.

P9 taught middle school and high school mathematics in a small town public
school while she was also studying for her master’s degree in the EME Master’s
Program. She had four years of private tutoring experience and one semester of TPC
teaching while in college. P9 took most of the courses in the Mathematics double major
program during her studies in the EME program, but could not complete this degree due
to her commitment to the TPC she was teaching.

P12 taught middle school mathematics in a private school and continued his

studies for a master’s degree in measurement and evaluation at another university. He had

130

one year of TPC and 1.5 years of student club teaching experiences during his studies in
the EME program.
Beliefs about Nature of Mathematics

Although both slight-decrease participants had similar Level 1 beliefs about the
nature of mathematics in 2003, P9’s belief level decreased to Level 0 whereas P12’s
belief level remained the same in 2004.

P9. P9 had Level 1 beliefs about the nature of mathematics indicating connections
and relationships among the pieces of mathematical knowledge in 2003. She mentioned
several times that theorems and their proofs would build relationships between
mathematical ideas. Although P9 characterized mathematics as a language in 2004, she
could not articulate more on this analogy. Furthermore, she believed that telling the
correct step in solving a question was the indication of students’ mathematical
understanding: ”As a simple example, when we teach the equations, [the student should
be able to] tell the order of the operations, perform the operations, and reach the
solution by this way" (P9, 2004, Q6).

The decrease in P9’s belief level was also apparent when her beliefs about the
nature of mathematical problems were considered. She believed in 2003 that
mathematical problems appeared when a new knowledge in mathematics was created
through theorems. However, in 2004, her belief level decreased to Level 0 where she
deﬁned a mathematical problem as ”something that require[d] an operation " (P9, 2004,
Q8). Despite having lower level nature beliefs in 2004, P9 believed that mathematics

courses and especially geometry courses were useful for her teaching.

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P12. In 2003 P12 believed that building relationships between mathematical ideas
and to “indicate where things come from " (P12, 2003, Q3) were crucial in understanding
mathematical concepts, which were Level 1 beliefs. He consistently believed in 2004 that
understanding the logic behind the concepts in reaching mathematical knowledge was
important.

P12 indicated in 2003 that a mathematical problem was the narrative form of
certain mathematical operations that required a student “to symbolize it in order to solve "
(P12, 2003, Q10). He had more elaborated beliefs in 2004 in that he deﬁned a
mathematical problem as a question that students would comment on and discover.

Somewhat interestingly, P12 did not believe that mathematics courses were useful
since he “just memorized and forgot their content” (P12, 2004, Q31). Despite the fact
that P12 recognized connections and logic behind mathematical concepts, he did not
seem to maintain these beliefs when he considered his own education in mathematics.
P12’s beliefs about the usefulness of the mathematics courses and how he learned their
content might suggest a further decrease in his belief level or the peripheral nature of his
existing beliefs, as Green (1971) would suggest.

Summary and Discussion

P9 and P12 had unique patterns of change in their beliefs and belief levels in
2004, although their Level 1 beliefs in 2003 were similar as both participants claimed that
mathematics was a connected web of ideas. While P12 generally had Level 1 beliefs
about the nature of mathematics, mathematical problems, and mathematical
understanding in 2004, P9 mostly had Level 0 beliefs in these subcategories. One reason

might be the scope of the grade levels that the participants were teaching at the time of

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the study. While P12 was teaching two different middle school grades, P9 was teaching a
total of ﬁve different grades in middle school and high school. Preparing lesson plans and
materials for ﬁve different grade levels at a very early phase of her teaching was probably
not easy for P9. It was possible that P9 had less focus on the nature of the mathematics
she was teaching and she underemphasized its connected and logical nature.

Beliefs about Teaching Mathematics

Although P9 and P12 had Level 1 teaching beliefs in 2003, they did not have the
same level of beliefs in 2004. P12’s teaching belief level decreased to Level 0 while P9’s
belief level remained at Level 1.

P9. The teacher was the main responsible person for teaching mathematics for P9
in 2003. However, students also had some roles in the classroom. She claimed that
instead of using already existing teaching materials, she would prefer to produce a
teaching material with the students and use it in the classroom.

P9’s teaching beliefs and belief level did not change after the ﬁrst year of
teaching. She still believed that a teacher-centered classroom in which students
participated in the teaching-leaming process would be an effective teaching context:

“[ The teacher] should be in continuous communication with the students.

[...] If there is a concept which is not understood by a certain way, then

[the teacher] should explain it in a different way. [The teacher] should

always include the students in [the teaching-learning process] ” (P9, 2004,

Q1 9).

P9’s aims for teaching mathematics were consistent in 2003 and 2004. She

believed that relating mathematics to daily life was an important part of teaching,

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because students could learn better if they saw the connection between the
mathematics concepts and the daily life.

P12. For P12, a teacher should have continuous, friendly communication with the
students in 2003 so that their motivation and learning could be increased. Moreover, a
teacher should gain student attention during his teaching: “[ The teacher] would do
activities that would grab students ’ attention and also he would put knowledge in these
[activities] little by little. " (P12, 2003, Q10).

When P12 was asked in 2004 about how he would plan his teaching to correct a
misconception, he described a teacher-centered session in which he told the students
about all the steps they needed to solve a question and to perform certain operations,
indicating Level 0 beliefs. Although he still aimed for increasing student motivation and
having a good communication with the students in 2004, his beliefs became more
teacher-centered.

Summary and Discussion

P9 and P12 had similar experiences with and concerns about students in 2004.
Both participants claimed that having students who lacked prerequisite knowledge in the
classroom was the most discouraging factor in their teaching, as P12 explained:

“T he only thing that discourages me is the students without suﬂicient

knowledge [...] Because there are both good (high achieving) students

and bad (low achieving) students. You come across such a dilemma that

you want to give more to the good students, but you know that the low-

level students won ’t be able to get it, so you stop yourself” (P12, 2004,

Q22).

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Despite the somewhat similar discouragement P9 and P12 felt in their ﬁrst-year teaching,
their beliefs were affected in different ways. P9 had lower level beliefs about the nature
of mathematics, whereas P12 had lower level beliefs about teaching mathematics. The
reason might be related to the centrality of their beliefs in belief areas (Green, 1971). It
was possible that participants’ discouraging experiences (in addition to the other factors
such as teaching many different grade levels) affected the belief area where their beliefs
were less central compared to the beliefs in other areas. For example, P9’s discouraging
experiences with students made her reconsider her teaching skills:

“The baskets come to the classroom full [...] There is no need for me to

say anything [The students] try, do something, [and] the baskets are

full The next day, I don 't know what happens, I think there is a hole on

the bottom of the basket It ’s empty, but believe me there is nothing left,

I cannot even find a sentence [...] I mean, when I see those [incidents

where students cannot even do simple tasks] I start to say there is a

problem [about] me" (P9, 2004, Q28).
However, she maintained her Level 1 beliefs about teaching, probably because those
beliefs were more central compared to her Level 1 beliefs about the nature of
mathematics. Hence, she adapted Level 0 beliefs about the nature of mathematics in order
to increase student learning.

Beliefs about Learning Mathematics
Slight-increase participants had different learning belief levels in both 2003 and

2004. P12’s learning belief level did not change during his ﬁrst year of teaching and

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remained at Level 0. However, P9’s Level 1 learning beliefs in 2003 were not completely
maintained in 2004 where she had both Level 0 and Level 1 beliefs.

P9. P9 had Level 1 learning beliefs in 2003. She considered students’ thoughtful
efforts as an important input in producing teaching and learning materials. She believed
that relating mathematics to daily life would enable students to learn mathematics better.
Since producing teaching materials was actually relating mathematics to daily life for P9,
she considered student involvement in this process as crucial for their learning. Although
she still believed the value of student efforts in both teaching and learning mathematics in
2004, she started to believe that either memorization or repetition might be a way to learn
(a Level 0 belief): “I mean there is repetition Maybe [the student] should try to
memorize the rules but memorization does not work. He has to repeat ” (P9, 2004, Q3 ).

The reason that P9 was simultaneously holding Level 0 and Level 1 learning
beliefs seemed to be discouraging experiences that she had with students. She was not
able to get any response to her questions about the concepts previously covered by
student-centered approaches. Although P9 still believed in the value of students’
thoughtful efforts in learning mathematics, the fact that students’ efforts did not result in
learning, may have made her reveal lower level learning beliefs.

P12. P12 believed in both 2003 and 2004 that solving as many questions as
possible was the only key skill to learn mathematics, which was a Level 0 belief: “Some
people can make it, some people cannot [...] Even if I don 't know anything, I mean
someone who says “I cannot do " can improve himself or herself by solving many
questions ” (P12, 2004, Q3). Moreover, P12 also believed in both 2003 and 2004 that

having higher motivation and a positive attitude would lead to learning. He seemed to

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maintain these two beliefs simultaneously by assigning roles both for the teacher and for
the students in student learning. P12 believed that students would learn mathematics if
teachers would increase student motivation and students would solve as many questions
as they could. Although he seemed to assign a role for the students for their learning, that
role indicated a Level 0 belief that minimized student effort to solving many questions.
Summary and Discussion

The reason for slight-decrease participants’ maintained (for P12) or decreased (for
P9) Level 0 beliefs might be the effect of the student population in their teaching
contexts, as mentioned in the previous section. Participants claimed that they had
difﬁculty in managing students with different knowledge levels in the classroom. It might
be speculated that participants were discouraged when their more student-centered
practices did not work in the classroom because of knowledge gaps among the students.
Hence, they either adopted more Level 0 beliefs and practices besides their Level 1
beliefs (as in the case of P9), or maintained their existing Level 0 learning beliefs (as in
the case ofP12).

Other Changes and Conclusions

In order to understand how the slight-decrease participants’ ﬁrst-year experiences
affected their mathematics related beliefs, they were asked to categorize types of
knowledge needed in teaching mathematics both in 2003 and 2004. P9 and P12 had
different beliefs about the essential types of knowledge, but their beliefs were consistent
with their aims in teaching.

In both 2003 and 2004, P9 believed that a mathematics teacher should have

mathematical content knowledge and should know how to connect this knowledge to

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daily life, which was consistent with her beliefs about the importance of relating
mathematics to daily life in teaching.

Although P12 prioritized content knowledge over good communication skills in
2003, he believed in 2004 that good communication skills were the most important type
of skill that a teacher should have, followed by content knowledge. His 2004 ranking was
somewhat consistent with his aims in teaching, where he prioritized increasing student
attitude and motivation. Having good communication skills were important for P12 since
he believed communication skills were the tools he would use to make students enjoy
mathematics.

The most effective factor on slight-decrease participants’ belief levels could be
stated as the “unexpected” classroom reality. Participants had a student population whose
previous mathematical knowledge was incomplete and unequal. This had different
impacts on each slight-decrease participant’s beliefs and belief levels. P9 had a lower
level of beliefs about the nature of mathematics whereas P12 had a lower level of
teaching beliefs at the end of their ﬁrst year in teaching (2004) compared to their belief
levels when they graduated from the EME program (2003). Although P9 still held mostly
Level 1 beliefs about learning mathematics, she had some Level 0 learning beliefs as
well. Considering that P12 maintained his Level 0 beliefs in 2004, it might be concluded
that discouraging experiences with the students were likely to result in more central Level
0 beliefs about learning mathematics.

Slight-Increase Group
P5 was the only participant who had a slight increase in his belief levels after the

ﬁrst year of teaching. His belief levels in all areas in 2003 were Level 0. In 2004, the

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level of his beliefs about nature of mathematics increased to Level 1, whereas the other
belief levels remained the same. Table 6.5 shows P5’s gender, teaching context, and
belief levels.

Table 6.5

Gender, Teaching Context, Belief Levels of Participant Who Had Increase in his Belief

 

 

Levels.
Gender Teaching N2003 N2004 T2003 T2004 L2003 L2004
Context
P5 M Public 0 1 0 0 0 0

 

P5 taught in a rural public school after teaching at a TPC for one semester in his
ﬁrst year. He had four years of private tutoring and three years of TPC teaching while in
college. He was the 15th teacher for that year who was assigned to teach middle school
mathematics in his public school. Among those 15 teachers, P5 was the only mathematics
teacher.’0 Besides teaching middle school mathematics, P5 was teaching Englishll at the
same school and working as the school counselor.

Beliefs about Nature of Mathematics

P5 did not have much to say for the questions about the nature of mathematics in

2003. Although he mentioned that mathematics improved analytical thinking skills, he

could not provide clear arguments about the nature of mathematical concepts, procedures,

 

1° Recall that the principals generally have the authority to assign teachers to various content areas even if
they are not certiﬁed to teach the content.

“ Since the language of instruction at METU is English, graduates of METU are generally assigned to
teach English even if they are not certiﬁed as English teacher.

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or a mathematical problem. The following explanation of the nature of a mathematical
problem illustrated this:

“[Problem solving] is the method that is followed while questioning or

teaching of a mathematical concept. You convert it to a problem and

connect to daily life ” (P5, 2003, Q7).
Moreover, the way he would evaluate student mathematical understanding was simply by
in-class written examinations.

In 2004, P5 provided clearer explanations for the nature of mathematical
problems:

“I think that it is the place where daily life is completely integrated into

mathematics one to one [..] which provides many drﬂerent thinking

methods to the students. It might be a regular writing, an operation, or

only a logical thinking, demonstrating a method... but it is in the

problem " (P5, 2004, Q8).
He addressed logical thinking as a part of a mathematical problem, where the problem
was derived from daily life. Moreover, P5 believed that reasons for mathematical
procedures should be known by the students. Hence, he made his students always ask
“Why?” questions for everything that they did in the mathematics class. These views of a
mathematical problem and the emphasis on reasons for mathematical procedures
indicated that P5 had an increase in his nature belief level from Level 0 to Level 1.

P5’s perception of the usefulness of the mathematics courses in the program also
reﬂected an increase in the level of his beliefs about the nature of mathematics. P5

believed in 2003 that with the exception of the geometry course, the mathematics courses

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were not useful for teaching middle school mathematics since they were not related to the
middle school content. Moreover, these courses were not useful for improving thinking
skills for P5. In 2004, he believed that the mathematics courses might be useful for his
own thinking, but not for teaching:

“[Mathematics courses] improved my speed in mathematical operations

and improved my questioning... For me, maybe it is useful in this way. But

I did not teach [in middle school mathematics] the rules [I learned] in

those [mathematics] courses. " (P5, 2004, Q31)

Although P5 recognized the thinking skills he gained through the mathematics
courses in the EME program, he still believed that these courses were not useful
for teaching middle school mathematics.

The increase in the level of P5’s beliefs about the nature of mathematics seemed
to be the result of the change in his teaching context. Since P5 had a lot of TPC teaching
experience (3.5 years) prior to his public school teaching, he noticed the difference
between the mathematics taught in TPC (which emphasized solving different types of
questions in a short time) and the mathematics taught in schools (which emphasized
understanding concepts more) when he started to teach in a public school. He claimed
that the students in TPC had never asked and the teachers had never mentioned the
reasons for mathematical principles. Hence, he tried to focus on reasons for the rules and
principles in his public school teaching. However, the effect of TPC teaching on his
beliefs could still be seen when his beliefs about the nature of mathematical
understanding was considered. He believed that giving the desired answer for a question

was evidence of a rich mathematical understanding. This indicated that although P5

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realized differences between TPC teaching and public school teaching, he was still
affected by his TPC teaching. For him, students’ correct answers on questions were an
indication of their mathematical understanding.
Beliefs about Teaching Mathematics

Despite the increase in P5’s belief level about the nature of mathematics, his
teaching beliefs did not change after his ﬁrst year of teaching. He consistently believed
that teaching should be in a classroom where there was an authoritarian environment. He
mentioned no role for students in the class and considered teacher authority as priority, as
his following phrases from 2003 and 2004 interviews illustrated:

"I would like to have discipline [in the classroom] and teach in a

disciplined environment. " (P5, 2003, Q15)

“[Djiscipline is very important for me. In some cases I am really

authoritarian. ” (P5, 2004, Q24)
Interestingly, P5’s aims in teaching in 2003 were to have a student-centered classroom,
where he allowed discovery and his method of evaluation would deﬁnitely not be exams:
“I evaluate [whether a student understood a mathematical problem or not] in a group
work ” (P5, 2003, Q8). His 2003 aims contradicted his teacher-centered responses to
other questions in the same interview where he claimed, “In my way [of teaching] ,' there
would not be so much group work. ” (P5, 2003, Q13). In 2004, he had more consistently
stated aims and beliefs in teaching where he prioritized student learning. He believed that
the teachers should make students generate alternative ways and make them ready for

life; however, he did not claim any contribution from the students in this process.

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P5’s school context was supportive. Although his students lacked considerable
knowledge and skills in mathematics due to the frequent teacher change (recall that P5
was the 15’h mathematics teacher within one year), P5 did not feel discouraged.

In summary, PS’s teaching beliefs did not change from 2003 to 2004 and he
believed that teaching was an authoritarian practice where students had no contribution to
the ongoing classroom activities. P5’s long TPC teaching experience might have a
considerable inﬂuence on his teacher-centered beliefs. Since TPC teaching did not
include any student-centered teaching practices (but focused on question solving skills in
an examination setting), he came to believe that students learned with teacher-centered
approaches. As a result, he maintained these beliefs even after he started to teach in a
public school.

Beliefs about Learning Mathematics

P5’s learning beliefs also did not change from 2003 to 2004 and maintained at
Level 0. He believed that students’ correct responses for the examination questions
showed that they learned the concepts. He articulated more on his beliefs in 2004 and
claimed more details:

“You process something and give it to the kid, and he scans this, imitates

this, and evaluates in his environment and tells ‘here is the result. ’ You

see it in the examination paper" (P5, 2004, Q18).

P5’s 2004 beliefs indicated that he maintained his previous Level 0 beliefs and revealed
more. Learning was a teacher initiated process for P5 where students imitated the
procedures. His beliefs about the role of skills were also consistent with his above

expressions. He believed that students’ only responsibility was to scan the topic of the

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day like reading a novel before coming to the class. He believed that “locating
[mathematical knowledge] " in his mind and making him do his homework” (P5. 2004,
Q3) was a teacher’s responsibility, indicating Level 0 beliefs.

In summary, P5’s 2003 Level 0 learning beliefs were maintained and became
more central in 2004, where he believed the authoritarian teacher-centered approach
would result in student learning and that students were not a part of the learning process.

Other Changes and Conclusions

The major and minor changes in P5’s mathematics related beliefs seemed to be a
result of the change in his teaching context. P5’s beliefs about the types of knowledge
necessary for teaching mathematics reﬂected this change as well.

In 2003, P5 believed that a mathematics teacher should have content knowledge
and knowledge of daily life connections to teach mathematics. When he started to teach
at a rural public school after a considerable period of TPC teaching (about 3.5 years), he
seemed to develop nurturing beliefs in 2004 related to teaching, such as being a model for
the students. This also might be due to his additional duty at the school as the school
counselor.

The unchanged nature of P5’s Level 0 teaching and learning beliefs could be
attributable to teaching at a TPC for a long time. The ways mathematical knowledge,
teaching and learning mathematics were considered in the EME program were quite
different than the ways they were considered at TPCs. It seemed that P5’s 2003 teaching
and learning beliefs were shaped by his practice at the TPC rather than his studies in the

EME program. Although P5 claimed that his ideas related to teaching changed after he

144

started to teach in the public school, he only changed the way he addressed mathematical
knowledge in teaching.

In summary, P5’s long TPC teaching experience provided the context for
maintaining and centralizing his Level 0 teaching and learning beliefs. However, teaching
in a totally different context evoked certain Level 1 beliefs about the nature of
mathematics.

Mixed-Change Group

The ﬁnal group of participants to consider was the mixed change group.
P4, P6, P10, and P11 had mixed increase and decrease in their belief levels in different
areas from 2003 to 2004. Among them, P4, P10, and P11 had the same pattern of change
in their belief levels, while P6 had completely a different pattern of change. P4, P10, and
P1 1’s belief levels in the nature of mathematics decreased from Level 1 to Level 0 after
the ﬁrst-year of teaching, while their teaching belief levels remained at Level 1 and their
learning belief levels increased from Level 0 to Level 1. In contrast, P6’s belief level in
nature of mathematics increased from Level 1 to Level 2, his teaching belief level
decreased from Level 2 to Level 1, and his learning belief level remained at Level 2 after
his ﬁrst-year of teaching. Since the participants in this group except P6 had the same
pattern of change, the analysis of their beliefs will be reported together and P6’s analysis
will be reported separately. Table 6.6 shows gender and belief levels of participants and

their teaching contexts.

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Table 6.6
Gender, Teaching Context, Belief Levels of Participants Who Had Mixed Changes in

their Belief Levels.

 

Gender Teaching N2003 N2004 T2003 T2004 L2003 L2004

 

 

Context
P4 F Public 1 0 l . l 0 1
P10 F Public 1 0 1 l 0 1
P1 1 F Public 1 0 1 l 0 1
P6 M Public 1 2 2 l 2 2

 

P4, P10, and P11. The mixed-change participants P4, P10, and P11 were teaching
in rural public schools at the time of the 2004 study. The participants had a different
range of teaching experiences during their college education in the EME program and
had generally similar experiences in their ﬁrst year of teaching.

P4 had student-club teaching, TPC teaching, and private tutoring experiences for
one year before graduating from the program. In her ﬁrst year, she worked at a TPC for a
semester, and then started to teach middle school mathematics and science as well as high
school mathematics at a rural public school.

P10 had private tutoring experience for one year during college, along with a
short TPC experience. After teaching at a TPC for a semester in her ﬁrst year, she started
to teach at a rural public school as the 5th teacher assigned to teach middle school

mathematics in that school in the 2003 — 2004 academic year.

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Unlike P4 and P10, P11 did not have any informal teaching experiences during
her studies in the EME program12 . In her ﬁrst year, she taught middle school mathematics
at a rural public school. Additionally, she was teaching to prepare students for the
national examination during the weekends at a school-supported TPC”.

P6. P6 was one of the participants who had considerable teaching experience
during his college education. He taught mathematics in the student-club teaching project
for three years and in an orphanage for four years. He taught middle school mathematics
at a public school in his ﬁrst year. In his teachingicontext, he voluntarily conducted
problem-solving sessions for his students either after class or during weekends. In the
problem-solving sessions, he generally assigned a single difﬁcult problem which students
often could not solve completely, and he discussed it with them in detail. His college and
ﬁrst-year teaching experiences could be considered distinct among all other participants
in the study.

Beliefs about Nature of Mathematics

P4, P10, and P1]. The mixed-change participants mainly believed in 2003 that
mathematical knowledge included logical relationships between the ideas. They believed
that mathematics could be seen as a phenomena integrated into daily life, not apart from
it. For example, P4 claimed that mathematical thinking was the tool that one could use in
“comparing things [and] evaluating situations " (P4, 2003, Q1 ). However, in 2004,

mathematics was not considered as a mental tool by the participants, but as a helper for

 

'2 Recall that all participants had three semesters of student teaching in the EME program. The other
teaching experiences (such as TPC private tutoring, and student club teaching) were not organized by the
EME program but were arranged by participants.

'3 In rural areas where there is no TPC, schools provide students with weekend courses where the school
teachers teach in order to prepare students for the national exams. School supported test preparation courses
are organized by the school administrators.

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basic everyday calculation, as P4 illustrated: “[Mathematics is in] [ajt least when we go
to a supermarket... We learned [math] slowly since it is in everything” (P4, 2004, Q3).
They believed that building mathematical knowledge was simply reaching correct
answers of given questions, which was a Level 0 belief.

P4, P10, and P11 did not say much about the nature of mathematical problems in
2003. Their expressions indicated Level 1 beliefs where a mathematical problem was a
question or a situation expressed in a narrative form. They claimed that mathematical
problems needed advanced thinking skills to be solved. Somewhat interestingly, although
participants’ belief levels about the nature of mathematics decreased from Level 1 to
Level 0 in 2004, their Level 1 beliefs about the nature of mathematical problems in 2003
became more detailed in 2004. Participants realized in 2004 that “the thing that [was]
expressed in long sentences was not actually a problem, but a normal operation
question” (P4, 2004, Q8). Moreover, for the participants, a mathematical problem should
be the situation that “[ one] should not see the solution initially” (P11, 2004, Q8).

The participants in the mixed-change group had different beliefs about indicators
of rich mathematical understanding in 2004, such as asking questions, verbalizing one’s
own understanding, and solving questions that no one in the class could solve. Their
beliefs about rich mathematical understanding indicated a belief level between Level 0
and Level 1.

Mixed-change participants had different beliefs about the usefulness of
mathematics courses in the EME program, but their beliefs were consistent from 2003 to
2004. P4 consistently believed that mathematics courses provided her with connections

between mathematics concepts, although she realized this contribution later in her

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education. P10 and P11 believed that the courses were useful in terms of increasing their
mathematical content knowledge but had no effect on teaching middle school
mathematics.

In summary, mixed-change participants P4, P10, and P11 generally had similar
beliefs on subcategories concerning nature of mathematics except on the indicators of
rich mathematical understanding. Only P4 believed that the content courses were useful
in teaching mathematics. Although participants’ 2004 beliefs about the nature of
mathematics were generally at a lower level, their beliefs about the nature of a
mathematical problem were at Level 1. This simultaneous existence of Level 0 and Level
1 beliefs in 2004 might be due to the centrality of their 2003 beliefs. Although
participants’ 2003 beliefs were not fully maintained at Level 1 until 2004, their beliefs
were somewhat less peripheral so that participants still had Level 1 beliefs in some of the
subcategories in 2004.

P6. P6 was the only participant whose beliefs about nature of mathematics
increased ﬁ'om Level 1 to Level 2 from 2003 to 2004. His Level 2 beliefs were consistent
through all sub-areas in the nature of mathematics in 2004. He claimed several times in
2003 interview that “the fundamental of mathematics [was] logical thinking” (P6, 2003,
Q1 ). He believed in 2004 that “a person who comprehend[ ed] mathematics [could]
always interpret [logically]. He [could] build the cause-eﬂect relationship in the
discussions ” (P6, 2004, Q2). Not only were his beliefs more detailed in 2004, but also
they were at a higher level (Level 2) compared to 2003 beliefs (Level 1).

The level of P6’s beliefs about nature of mathematical problems was also at a

higher level in 2004. P6 believed in both 2003 and 2004 that a mathematical problem

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should have a daily life connection. He considered mathematical problems as “the
questions that [took] their fundamentals from the daily life, that [could] be expressed
through mathematical symbols, and that [could] be solved and interpreted in daily life "
(P4, 2003, Q7). He believed in 2004 that “[ a mathematical problem] could not be
answered by routine (not thoughq‘ul) answers " (P6, 2004, Q8). He distinguished a
mathematical problem from a routine question by emphasizing its connection to daily life
and the thinking skills that should be used to solve it.

P6 had Level 2 beliefs about the indicators of rich mathematical understanding in
2004 as well. He believed that a student with a rich level of mathematical understanding
should be able to carry out the solution of a mathematical problem and should be able to
ask for reasons underlying mathematical procedures.

P6 believed in 2003 and 2004 that the mathematical content courses in the EME
program improved his mathematical thinking skills. He consistently claimed that the
content courses provided him with the skills of questioning and understanding the
fundamentals of mathematics. He believed that be reﬂected his own thinking skills on his
teaching by emphasizing proofs of many principles in the classroom:

“I try to address the proofs more or less, I don 't want it to be [an] empty

[can]. I don 't want them to ask “Where these things come from? " I want

them to have the skill [to prove the principles] " (P6, 2004, Q3 I ).

P6 had Level 2 beliefs about the nature of mathematics, mathematical problems,

characterization of rich mathematical understanding, and how he reﬂected his own

mathematical thinking to the students. His beliefs were built on his central belief that

150

mathematical knowledge was a web of logical relationships, which he consistently stated
in both 2003 and 2004 interviews.
Beliefs about Teaching Mathematics

P4, P10, and P11. The mixed-change participants P4, P10, and P11 maintained
the level of their 2003 beliefs (Level 1) about teaching mathematics in 2004. Participants
believed in 2003 that mathematics teaching should be in a teacher-centered classroom
environment, where the teacher “should know dijferent methods, implement these
methods in the class, and should know how to evaluate the results in a better way ” (P4,
2003, Q12). Mixed-change participants had similar beliefs in 2004 and they also believed
that the teacher should “behave in a way that would make students want to learn more ”
(P4, 2004, Q19). Although they believed that the teacher-centered classroom practice
would be effective in teaching mathematics, this practice should include different
methods of teaching mathematics and ways of motivating students. Participants did not
clearly mention the teaching methods they would use, but these methods could be
considered as teacher-centered versions of mathematics teaching methods they have
studied in two methods courses, such as discovery learning and problem solving. Their
aims in teaching mathematics also indicated consistent Level 1 beliefs in both 2003 and
2004. They generally believed that it was the teacher’s responsibility to make the students
enjoy mathematics or to prepare them for further steps of education by sharing their
knowledge. However, they did not mention any role that students might have during this
process.

P4, P10, and P11 had similar ﬁrst-year teaching experiences, especially when

their ﬁnal teaching contexts (rural public schools) were considered. Having students

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without essential prerequisite knowledge and future educational goals was a limiting
factor for participants’ teaching. Moreover, the curriculum was loaded for the participants
especially because they were expected to keep the same pace with the yearly plan despite
being assigned to their schools in the middle of the academic year. These limitations
might be a factor in maintaining teacher-centered Level 1 beliefs since the participants
were able to deal with these factors only through teacher-centered practices.

In summary, mixed-change participants mainly believed that a teacher should
implement different teaching methods of mathematics in a teacher-centered classroom
and should make students enjoy mathematics. Although there were limiting factors in
their school contexts (such as keeping the same pace with the curriculum where the
students were lacking essential knowledge), they were able to maintain their belief levels
at Level 1 from 2003 to 2004.

P6. P6’s Level 2 beliefs about teaching mathematics in 2003 decreased to Level 1
in 2004. He believed in 2003 that mathematics teaching should be organized in the way
that students would “implement, think, interpret, and product results ” (P6, 2003, Q13)
by the help of instructional materials and activities. Being driven by these Level 2 beliefs,
P6 built a mathematics laboratory in his school in the beginning of his ﬁrst year where his
administrator provided all of the resources that he asked for. However, after the ﬁrst-year
teaching (in 2004), he believed that mathematics laboratory should serve to improve
students’ attitude and motivation instead of helping them to construct knowledge:

“It is sometimes a magic for the kids, sometimes something that they

haven ’t seen before. You make kids both like mathematics and respect

you " (P6, 2004, Q3 ).

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His 2004 aim was also at a lower level in that he claimed himself as the main responsible
person who would develop mathematical behaviors in students’ minds. Although he
placed the teacher at the center of teaching, his placement was not the same as that of
other mixed-change participants (P4, P10, and P11). The extra problem solving hours that
P6 organized voluntarily in his school showed that he prioritized increasing students’
mathematical thinking skills by guiding them in their thinking while they were solving
problems. However, limiting factors, such as the curriculum load, seemed to affect his
teaching in the mathematics classroom (during the school hours) and lowered his belief
level. When P6’s teaching practice in the problem solving hours was considered, his
overall Level 1 teaching beliefs might be addressed as less teacher-centered than the
other mixed-change participants’ (P4, P10, and P11) beliefs.
Beliefs about Learning Mathematics

P4, P10, and P11. The mixed-change participants P4, P10, and P11 had Level 0
beliefs about learning mathematics in 2003 and Level 1 beliefs in 2004. They believed in
2003 that solving many questions correctly was the indication of learning mathematics.
Moreover, they did not mention any responsibility that students should take for their own
learning during the 2003 interview. Their beliefs in 2004 indicated that the students
should spend some effort other than solving questions correctly so that they could learn
mathematics:

"[Mathematics] cannot be memorized, or it cannot be taken as a prepared

software. You have to reach mathematics " (P10, 2004, Q3).

“When we talk about learning, there comes... comprehending... You

should know the rules, why you do what you do, generalization... and if

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you are learning generalization, then you will be able to implement it to

specific cases ” (P11, 2004, Q3).

In addition to the Level 1 learning beliefs stated above, participants believed that
having a positive attitude towards mathematics was an important catalyst in learning
mathematics. Although mixed-change participants generally had Level 0 beliefs in 2003,
they believed that students could learn and think about the mathematical concepts in
different ways. They maintained these beliefs in 2004 as well.

In summary, mixed-change participants’ 2003 Level 0 beliefs that reduced
learning to merely solving questions correctly changed in 2004. They believed in 2004
that students should spend effort in comprehending, generalizing, and implementing so
that they would learn mathematics. Having a positive attitude towards mathematics was
an important factor in this process.

P6. P6’s Level 2 beliefs about learning mathematics in 2003 were maintained and
became more detailed in 2004. Initially, he believed that mathematics could be learned
through “interpreting, understanding, [and] thinking" (P6, 2003, Q1 20). His latter
beliefs indicated that students should learn mathematics “through experiencing it [and]
by observing what it works for " (P6, 2004, Q3). Moreover, since students would know
where they could use mathematics by the help of various instructional materials, “[ they
could] learn mathematics as a behavior” (P6, 2004, Q3). Although some other inservice
participants of the study had beliefs about learning mathematics through understanding

the use of mathematical knowledge, they never mentioned converting these experiences

into actions that could be observable, as P6 did.

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P6 claimed in 2004 that having students with different levels of knowledge and
skills was not unexpected for him. Although he had students who lacked prerequisite
knowledge, he still believed that they could learn mathematics by experiencing it through
the instructional materials. In sumntary, P6’s Level 2 beliefs were more detailed in 2004
in which he characterized learning mathematics as a collection of advanced thinking
skills that could be converted into actions that students could practice.

Other Changes and Conclusions

P4, P10, and P1]. The mixed-change participants’ (P4, P10, and P11) beliefs
about the type of knowledge and skills that a mathematics teacher should have and their
perceived needs in the EME program seemed to provide evidence for the effect of their
teaching context on their mathematics related beliefs. P4, P10, and P11 generally
believed in 2003 that a mathematics teacher should have mathematical and pedagogical
content knowledge (PCK) in order to teach mathematics effectively. Somewhat
interestingly, having skills that would increase student enjoyment in mathematics was
more important than having content and pedagogical content knowledge in 2004.
Participants’ beliefs that prioritized increasing students’ attitudes towards mathematics
seemed to be related to their experiences in their teaching contexts. Since participants had
parents and students who typically did not have further educational goals, they had to
deal with students lacking both the essential knowledge of and the interest in
mathematics. Hence, their learning beliefs, aims for teaching mathematics, and beliefs
about the skills essential for teaching mathematics prioritized increasing student attitude

towards mathematics.

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Similarly, mixed-change participants’ perceived needs in the program in 2004
were affected by their beliefs that prioritized increasing student attitude. Participants
believed that there should be more PCK courses and pedagogical knowledge courses in
the EME program. Having more pedagogical knowledge on middle school students was
important for the participants probably because they believed that these courses would
provide them a knowledge base for increasing student interest in mathematics.

Although P4, P10, and P11 had different teaching experiences while in college,
their nature, teaching, and learning beliefs were at the same level when they graduated
from the EME program. This might suggest that participants’ preservice teaching
experiences, such as TPC teaching or private tutoring, did not strongly affect their belief
levels. The mixed-change participants P4, P10, and P11 taught in rural public schools in
their ﬁrst year and they had similar changes in their belief levels. Having the same
changes in belief levels in 2004 might indicate that the similar rural public school context
affected participants’ belief levels in similar ways.

Teaching in rural public schools was not easy for the participants when their
school contexts were considered. Students’ level of knowledge and skills were lower than
what it should be. This was mostly caused by continuous teacher change in participants’
teaching contexts. The participants claimed that their schools did not have mathematics
teachers who taught in the school for more than a semester. Moreover, previous teachers
who were not certiﬁed as mathematics teachers had to teach mathematics in participants’
schools since there was no mathematics teacher. Hence, participants’ students typically

did not have sufﬁcient mathematical knowledge. Moreover, the continuous change of

 

"’ Recall that the principals generally have the authority to assign teachers to various content areas even if
they are not certiﬁed to teach the content.

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mathematics teachers that occurred prior to the participants’ arrival made it very difﬁcult
to analyze what students knew or did not know. Somewhat interestingly, participants’
teaching belief levels remained the same at Level 1 in these contexts, whereas the level of
their beliefs about the nature of mathematics decreased to Level 0 and the level of their
learning beliefs increased to Level 1.

P6. P6’s beliefs about the types of knowledge and skills that a mathematics
teacher should have were different than all other participants of the study, especially in
2004. Although he prioritized being competent in PCK over having a positive attitude
towards teaching in 2003, his ranking was reversed in 2004. P6’s strong desire to be a
teacher since high school possibly inﬂuenced his beliefs about the knowledge and skills
of an effective teacher. He considered having a positive attitude towards teaching as the
most important qualiﬁcation of a teacher. Somewhat interestingly, despite the increase in
his belief level about the nature of mathematics, he ranked content knowledge as the last
type of knowledge needed for teaching in both 2003 and 2004. On the other hand, this did
not mean that he thought content knowledge was unimportant. He claimed that he had
been trying to improve his content knowledge by reading mathematical journals since the
beginning of his studies at theEME program.

In relation to P6’s characterization of the knowledge types and generally higher
level of mathematics related beliefs, he did not claim any need for the PCK or the content
courses in the EME program. He indicated that a course on the educational system and
policies in Turkey and more courses on the psychology, sociology, and philosophy of

education should be added to the EME program in both 2003 and 2004.

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In summary, P6’s strong beliefs about teaching mathematics in 2003 possibly
helped him to create opportunities for himself and his students, such as building a
mathematics laboratory and organizing extra problem solving hours during his ﬁrst-year
teaching. These opportunities probably resulted in an increased belief level concerning
the nature of mathematics (from Level 1 to Level 2) and a maintained belief level in
learning mathematics (Level 2). The responsibilities that the classroom teaching
environment brought, such as the curriculum load and the knowledge level of his
students, inﬂuenced the level of his teaching beliefs in a negative way, although these
factors were not unexpected for him. He claimed he always believed that mathematics
should be reachable and enjoyable for everybody and this could be possible through
developing students’ mathematical behaviors. However, his 2004 teaching beliefs did not
support this claim. P6’s 2003 Level 2 teaching beliefs were probably less central. When
he was challenged by the real-classroom environment, he revealed Level 1 beliefs.

Summary

Presenting ﬁve groups of change patterns for a total of 12 inservice participants
was an important process in terms of detecting the common experiences and factors that
impacted the beliefs and change in beliefs and how these common experiences shaped
inservice participants’ beliefs. The results documented that inservice participants had
many common pre-college and preservice experiences and contextual factors or
conditions in their ﬁrst-year teaching. When the impact of these common experiences
was investigated, it became apparent that common experiences as a student and as a
teacher inﬂuenced individual inservice participants in different ways. As a result of

impact differences, there were little overall commonalities in inservice participants’

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beliefs and belief change patterns. The nature of these common experiences and factors
will be presented here in relation to the differences in belief change patterns.

There were common pre-college, preservice training, and ﬁrst-year experiences
for all inservice participants. All inservice participants studied the same national
curriculum during their high school education and took the same university entrance
examination. While in the EME program, they took the courses in a single section and
they were all teaching the national curriculum in mostly similar grade levels. Moreover,
almost all inservice participants had some informal teaching experiences during their
preservice training. They were teaching in the national curriculum context mostly in rural
schools to students with diverse knowledge levels in the same classroom.

Given the common nature of their preservice and ﬁrst-year experiences, it might
be expected that inservice participants would have similar belief change patterns after the
ﬁrst-year of teaching. However, the results showed that belief change was not a linear
result of single or few factors impacting in similar ways, even when major factors in
inservice participants’ preservice and ﬁrst-year contexts were common. Instead, each
participant’s belief change was a result of a unique combination of different interacting
experiences impacting in complicated ways. For example, the slight-increase participant
P5 had considerable TPC teaching and private tutoring experiences during his EME
program studies, which resulted in low level of beliefs in 2003 and a very slight increase
in 2004. In contrast, P6, who taught in the student club project for a long time while
studying the EME program, developed the highest level of beliefs among the inservice
participants in both 2003 and 2004. Although both P5 and P6’s informal teaching

experiences could be considered as experiences in classroom contexts and in university

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entrance examination preparation, they had very different beliefs and belief change
patterns. The ways P5 and P6 considered these informal classroom teaching experiences
as effective in students’ learning seemed to impact the ways they processed the program
experiences and deve10ped new beliefs. While P5 believed that students could learn
mathematics in the teacher-centered way he was practicing at TPC, P6 did not consider
examination preparation teaching as an effective way of learning and mostly had student-
centered beliefs. Therefore, despite the very common nature of their informal teaching
experiences (in addition to the common EME program experiences), P5 and P6 had
different levels of beliefs in belief areas in both 2003 and 2004, depending on the way
they considered these experiences in relation to other experiences they had.

Results showed that commonalities in major factors and background did not
imply similar beliefs or belief change patterns, as exempliﬁed above. Although there
were common major factors impacting the belief change, some other minor factors, such
as the number of students and different grade levels taught during the ﬁrst year, had a
considerable inﬂuence. For example, while P8 and P9 were teaching in rural schools to
similar range of grade levels, P8 was in consistent-increase group while P9 was in slight-
decrease group. Similarly, P6 and P12 were teaching two grade levels where public
school teacher P6’s student population was three times the population of P12’s students
who was working in a private school. Although P12 might be expected to have higher
level beliefs or an increase pattern due to his less loaded work and easier access to
resources compared to P6, he was placed in slight decrease group with relatively low
level of beliefs where P6 mostly maintained his high level of beliefs at the end of the

ﬁrst—year teaching.

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In some cases, participants in the same change pattern group had different sets of
experiences. A general look into the pattern group results showed that participants in the
no-change and consistent-increase groups had different preservice and ﬁrst-year teaching
experiences but similar change patterns of mathematics related beliefs. Similarly,
participants in mixed change group (except P6) had some differences in their preservice
and informal teaching experiences, but teaching in similar rural contexts in their ﬁrst year
resulted in similar belief change pattern.

One reason for presenting inservice participants’ beliefs in different change
pattern groups was to document the interaction of several factors that resulted in the non-
linear cause-effect relationship. As could be seen in the above emphases, focusing on a
single factor did not provide a complete and accurate explanation for each belief change
pattern. The detailed change descriptions for small number of participants in each change
pattern group demonstrated how major and minor factors interacted in unique ways for
each participant. This detailed picture of several unique change processes helped in
understanding, exploring, and generalizing, as well as to identify the impact of major
factors (to be documented in the next chapter).

In addition, this analysis conﬁrmed the impact of certain other factors on
participants’ mathematics related beliefs such as the impact of pre-college education.
Although EME program courses presented a common context and set of experiences,
almost none of the courses were designed to challenge or make them reﬂect on their
beliefs, which is a feature of interventions that has proved to have an impact on
preservice teachers’ beliefs (Gill, et al., 2004). Moreover, research indicates that

preservice teachers tend to maintain their pre-college mathematics related beliefs though

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their teacher education program experiences (Cooney, 1999). Thus, it could be speculated
that inservice participants ﬁltered these experiences by their unique set of pre-college
beliefs and experiences, probably interacting with other factors. The diversity of
participants’ beliefs about the usefulness of the program courses and experiences when
they graduated ﬁ'om the program highlighted that EME program experiences were
ﬁltered by pre—college and informal teaching experiences and beliefs in different ways for
each participant.

In summary, the change patterns in inservice participants’ mathematics related
beliefs from 2003 to 2004 showed that there was no single factor or a single experience
resulting in belief change. Each participant had a unique personal history composed of
several different sets of unique experiences gained during pre-college education,
preservice training, and ﬁrst-year of teaching. Personal histories impacted participants’
mathematics related beliefs in different ways and resulted in different patterns of belief
change. In other words, inservice participants’ beliefs and changes in beliefs had very.
few common features despite the commonality of major factors in their background and
teaching contexts. The commonalities detected among the factors and resulting belief

changes will be focused in the following chapter.

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CHAPTER 7

FACTORS AFFECTING INSERVICE PARTICIPANTS’ BELIEFS

In the previous chapter, the data indicated that inservice participants mostly
believed in the effectiveness of teacher-centered practices for teaching and learning in the
mathematics classroom. Teacher-centered beliefs were consistently stated in participants’
2003 and 2004 interviews. Participants often emphasized the use of different teaching
methods such as discovery and they considered students’ interest in learning mathematics
as important. However, they tried to address these concerns by teacher-centered practices
in the mathematics classroom. When the inservice participants elaborated on the use of
different methods of teaching mathematics and students’ role in learning, they referred to
certain experiences in the EME program and in the ﬁrst-year of teaching.

The experiences that might have an impact on inservice participants’ teacher-
centered beliefs will be analyzed in detail in this chapter. These experiences were
reported by the inservice participants as the courses and experiences during their studies
in the EME program, curriculum pace, and contextual and systemic factors within their
teaching context such as students’ knowledge level and being impacted by the Ministry.
These experiences often seemed to have a combined inﬂuence on participants’ beliefs.
Participants seemed to re-interpret their experiences and beliefs gained in the EME
program through their ﬁrst-year teaching experience. At the end of their ﬁrst year,
participants seemed to transform their beliefs into more teacher-centered beliefs in which

they considered using different teaching approaches and some roles for the students. This

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chapter will explore the transformation in participants’ beliefs and document the factors
that had a considerable impact on them.
Different Teaching Methods in the Teacher-Centered Classrooms

Inservice participants took several courses on methods of mathematics teaching in
the EME program until they graduated in 2003. These courses emphasized more student-
centered teaching methods such as cooperative learning and discovery in teaching
mathematics and also teaching middle school concepts by using them (see ELE 336 and
ELE 443 course outlines in Appendix B). Participants explored teaching mathematics
through the use of different pedagogical approaches in the methods courses where
student-centered practices were emphasized, prepared lesson plans for teaching middle
school mathematical concepts by implementing these approaches, and presented these
lesson plans to their friends and the professor pretending they were in a middle school
classroom. Such methods courses seemed to have an impact on participants’ beliefs about
mathematics teaching in 2003 when they graduated from the program, as they expressed:

“A ctually, I learned a lot of things [in the methods courses]. I did not

know teaching would be [conducted] in this way before. Moreover, I did

not know teaching mathematics would be [conducted] in this way. My

views changed a lot” (P4, 2003, Q22).

“My views changed directly. I was not aware of [the state of teaching

mathematics] before. Actually, I have so much confidence now " (P10,

2003, Q22).
Similarly, the mathematics education professor, Dr. C., claimed that he observed a

“change” in the program students after they started taking PCK courses. Among these

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courses, Dr. C. claimed that the elective course “Activity-Based Mathematics
Instruction” that he offered only to the 2003 graduates (to inservice participants) was the
most important one in terms of affecting participants’ beliefs. He speciﬁcally targeted
preservice teachers’ beliefs about mathematics teaching in schools in that course: “I was
trying to emphasize that the mathematics teaching when they were students [in the pre-
college education] and the mathematics teaching they would practice should be
different ” (Dr. C., 2004, Q11). His efforts seemed to impact participants’ way of
considering the mathematics teaching, as most participants particularly addressed the
course as eye-opening: "There was a course that made me go towards completely a
different direction, that changed me, which was the Activity-Based Mathematics
Instruction [...] That [course] changed my view. It was something that I have never come
across” (P3, 2003, Q23).

Pedagogical content knowledge courses in the EME program had some impact on
the way the inservice participants considered mathematics education in the classroom, as
indicated by their reference to the courses and different mathematics teaching
approaches. However, the courses and the EME program in general seemed to lack real-
classroom practices. The professor stated that there was a need for practice of the
knowledge and skills learned in the PCK courses, “in order to be integrated into
[program students '] personality as a teacher" (Dr. C., 2004, Q10). He claimed that
since the program lacked sufﬁcient opportunities for practice, the program students could
not implement these knowledge and skills and see how and what they worked for. Most
participants were aware of the lack of practice in the program when they graduated, as P1

illustrated: “I would like to experience the implementation of methods course in the

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classrooms. [It ’s] because you don ’t practice it, you just see it in theory [..] Practice
courses (School Experience and Practice Teaching courses) do not provide [practice of
methods of teaching] that much ” (P1, 2003, Q22). After the ﬁrst year of teaching, almost
all participants claimed that since the program lacked sufﬁcient Opportunities for practice,
they had difﬁculties while teaching in real-classroom settings.

The difﬁculties inservice participants had in their ﬁrst-year were mostly related to
the lack of experience in classroom management and in implementing different teaching
methods and materials in the classroom. When participants were faced with various
components and tensions of the classroom, such as managing the students and the time
simultaneously, they mostly had difﬁculty in teaching mathematics through student-
centered ways. P1 explained this problem after his ﬁrst year of teaching by referring to
the nature of the methods courses where participants practiced teaching a concept through
a speciﬁc method in the regular course hours to their classmates:

”[I don ’t know] how students would construct mathematical concepts in

practice... but in practice. I know it [in the program courses], I teach it

in the practice course, [and] I teach it assuming these university students

were middle school students [..] I don 't really know how [teaching] a

mathematical concept is practiced in any school in any classroom " (P1,

2004, Q25).

For the inservice participants, the way the course content was practiced in the EME
program was far removed from the conditions of the real-classroom settings. However,
most inservice participants were not aware of the extent of this difference until they

started the ﬁrst-year teaching: “We learn [in the EME courses] really ideal things [for

166

teaching mathematics], we implement these ideal things [during the courses], [and] I
think I now have the disappointment [because] of those [ideal] experiences” (P10, Q31,
2004). Given that the participants lacked more realistic teaching experiences during the
EME program, it was difﬁcult for them to implement student-centered methods in the
mathematics classroom although they believed in the effectiveness of these methods in
teaching mathematics. Classroom contexts and curricular concerns were limiting factors
for the participants.

Considering the limitations in real-classroom contexts, participants were inclined
towards believing in the effectiveness of teacher-centered practices in which some
student-centered approaches were integrated in the most time-efﬁcient way. These beliefs
and associated practices seemed to work better in the real-classroom contexts for the
inservice participants. For example, P3 described several practices in which she
conducted teacher-centered version of a student-centered teaching approach, considering
the student differences in her class:

“[...] considering the [students at the] lowest level [of knowledge] and

not really [organizing] everything for them but [explaining mathematical

concepts] in a way that they would catch up works [ in the classroom] [...]

When they measure the perimeter of a circular region and divide it by its

diameter and when I ask for few students to write their results on the

board to compare, students believe that they discovered the Pi number "

(P3, Q16, 2004).

"Sometimes I try to make them discover things by worksheets. For

instance, they count the number of [small] squares in a [large] square to

167

ﬁnd the area of the square, then they do count the [small squares] in a

rectangle" (P3, Q16, 2004).

In the above practices and similar ones that the inservice participants described, the
common issues were using a worksheet on which students would discover principles or
characteristics of mathematical concepts either as a group or individually, mostly by
using everyday manipulatives such as ropes. The teacher was the main decision maker
and explained most of the tasks to the students in the classroom. Students were using the
manipulatives, measuring, and calculating in order to “discover” the nature of the
mathematical concepts and responding to the teacher’s questions. Similar practices were
presented in courses related to the methods of mathematics teaching in the EME program.
Participants generally claimed that they referred to their portfolios or course materials
from time to time while planning learning activities for the students.

It seemed that preservice participants developed beliefs about the effectiveness of
student-centered methods of teaching mathematics through the methods courses in the
EME program. However, they could only implement the surface features of these
methods in which students discover few mathematical ideas in a very teacher directed
way. Although they beneﬁted from the EME program course materials in planning these
short periods of student-centered practices, the main features of these practices in which
students had more responsibilities were not implemented. Participants’ lack of
experiences in practicing student-centered teaching approaches seemed to be a limiting
factor in implementing these approaches in the real-classroom contexts.

The participants had generally more informal teaching experiences in private

tutoring and Test Preparation Center (TPC) settings, as compared to their real-classroom

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experiences in the EME program. The informal teaching experiences were teacher-
centered and were focused on getting the correct answer since most of the informal
teaching settings were examination-preparation oriented. Thus for the participants,
private tutoring and TPC experiences were not helpful in thinking about the concerns of
the real-classroom settings, such as gaining student attention. Therefore, participants
often had to re-evaluate their teaching beliefs and practices gained during the informal
teaching experiences when they were in real-classrooms settings, as they expressed
below:
“[Already] prepared students come to the T PCs and you only teach them

certain things. You don ’t have to gain attention, you don ’t have to make

them enjoy [the class]. They already come [to the T PC] with a goal in

mind, [and] they learn. However, this is not [the case] in public school

teaching" (P4, 2004, Q24).

“Teaching in a classroom is so much different. In private tutoring, you

can [open] a book if you feel like .you don ’t really know [the concept]. You

don ’t need to make [the student] like mathematics, give [him/her] daily

life examples. You are [just] required to make [the student] solve as many

questions as [he/she] can ” (P10, 2004, Q25).
Inservice participants’ private tutoring and TPC teaching experiences seemed to limit the
way they considered mathematics teaching. They realized this limitation when they
started to teach in school contexts in their ﬁrst year. This awareness seemed to help them
re-organize their practice by integrating different methods of teaching mathematics such

as using manipulatives and discovery. Even though participants considered more student-

169

centered approaches in their teaching, students’ role was still very limited in discovering
mathematical ideas and teacher was still directing all classroom activities.

In summary, EME program courses seemed to have some impact on inservice
participants’ beliefs about teaching. mathematics, as they used some of their knowledge
from these courses in their teaching. However, this impact was limited due to insufﬁcient
practice opportunities in the EME program and participants’ teacher-centered informal
teaching experiences during the preservice education. Participants were underprepared in
the EME program to practice student-centered methods in the real-classroom contexts.
Hence, they mostly referred to their informal teaching experiences which were generally
longer than their real-classroom teaching experiences. These factors inﬂuenced their way
of addressing student-centered teaching in the mathematics classroom towards a hybrid
of teacher-centered practices including some student-centered moments.

Real-classroom contexts include several factors impacting teachers’ practices in
complicated ways. When inservice participants started to teach in real-classroom settings,
they had to deal with other contextual and systemic factors that directed them towards
teacher-centered beliefs and practices in which student role in learning was limited.
These factors and related concerns about the students are explained below.

Concerns about Students in the Teacher-Centered Classroom

Inservice participants’ experiences with students as beginning teachers in real-
classroom contexts seemed to be unexpected for them. Certain responsibilities and
pressures appeared to have considerable inﬂuence on participants’ beliefs about the

effective ways of mathematics teaching. The responsibilities and the pressures were

170

related to the tension between the cuniculum pace controlled by the Ministry of National
Education (MNE) and participants’ concerns about student learning.

Most of the inservice participants had classrooms where students had poor and
unequal mathematical knowledge and skills. The participants observed that only a small
number of their students had sufﬁcient knowledge and skills upon which new
mathematical knowledge could be built. However, the inservice participants did not have
the time to complete students’ missing previous knowledge and skills because they had to
be in the same pace with the yearly plan based on the national curriculum, as P12
explained:

"[Let '5 say] I taught the 3'“ unit and passed to the 4m one. But I know that

there are many [students] who could not comprehend many [concepts] in

the 3"! unit [...] I can see this! [...] [Sometimes the students] say that they

did not understand it fully. But since there is a curriculum that I have to

keep a pace with, I skip [re-teaching] [...] While teaching the 4“ unit after

the 3rd one, there are things in the 4'” unit [related to] the 3rd one. This

time, I cannot really [teach] the 4” unit. There is always this

disconnectedness [and] it is going like a chain " (P12, 2004, Q23).

Moreover, many participants who were teaching in rather rural contexts were assigned to
their schools several months after the school year started. Yet, they were expected to
cover the previous concepts in that years’ plan and keep up with the curriculum pace that
required them to teach a speciﬁc concept at a speciﬁc week. Most participants claimed
that it was difﬁcult to cover everything stated in the curriculum at the speciﬁc time period

deﬁned for each concept in the yearly plan, as P11 illustrated: “If it would be left to [my

171

decision], I would teach the fractions for a month [...] [ You must] compress [the
concepts, in order to keep the pace] (P11, 2004. Q22). Moreover, the consistency
between the participants’ pace in teaching and the pace of the yearly plan was inspected
by the MNE which brought additional pressure for the participants:
“I mean, you don ’t have to teach something to a kid. The fact that [the

concept in the yearly plan] had been completed is sufﬁcient for the

curriculum [pace], I mean the person who inspects [the pace of the

teacher] would like to see that. He would take look at the [plan] book and

would conclude that the teacher taught this on this day. I think this is

wrong” (P5, 2004, Q22).
The lesson plans that the participants wrote for the concepts identiﬁed for each week in
the yearly plan were also investigated by the MNE ofﬁcers both for structure and
implementation. However, for the participants, implementing a lesson plan did not really
show that the concept was learned by the students:

”The inspector is not interested in whether you taught sufﬁciently or not,

or you started to teach [in that school] later [ in the semester]. He expects

you to write directly whether you are in the same pace with the yearly plan

or not. I mean [...] this would not be realistic since I have not been

teaching in that class since the beginning of the year " (P10, 2004, Q22).

In addition to the pressure that the curriculum pace put on participants’ teaching,
the lack of consideration of student differences in the curriculum brought extra difﬁculty
for the inservice participants. When participants started to teach in the heterogeneous

classrooms, it appeared that they were also not prepared to do so. Many participants

172

expressed difﬁculties associated with teaching to a group of students having mixed levels
of knowledge under the same curriculum. It seemed that heterogeneous classrooms were
somewhat unexpected and disappointing for the participants, as P3 explained:

"There are things that I am disappointed with. But I was able to ﬁnd 4-5

kids that I was in the same frequency [...] you teach a class, it 's crowded,

but the kid in the front seat asks a question, [he/she] asks something

diﬁ'erent. I immediately answer only that question. Others don 't even hear

it, I mean they don ’t even see (realize) ” (P3, 2004, Q13).

Participants’ disappointments and difﬁculties expressed above and in the previous
chapter suggested that they did not have sufﬁcient experience with heterogeneous
classrooms during their studies at the EME program. It seemed that they mostly
considered homogeneity in the classrooms in terms of student knowledge and skills.
When the participants experienced difﬁculty in dealing with the students and the
curriculum pace at the same time, they seemed to incline towards teacher-centered beliefs
since these beliefs and associated practices were more time-efﬁcient and might work
better in heterogeneous classrooms.

In addition to teaching in heterogeneous classrooms, students’ lack of interest in
learning mathematics was a source of difﬁculty for the participants. The inservice
participants attempted to manage the cuniculum pace, heterogeneous classrooms, and
lack of student interest by trying to increase student attitude and enjoyment in the
mathematics classroom. One way to ensure the enjoyment was to make students like the
teacher as a person: "I mean, the teacher is an important factor. [The teacher] makes

[the students] listen; [the students] should like the teacher. When the course is taught by

173

a teacher that the student does not like, the student becomes [uninterested] in the course "
(P2, 2004, Q19). Most of the participants believed that there should be a good
relationship between the teacher and the students so that the students would adopt a
positive attitude towards learning mathematics.

The other way to increase student interest in the mathematics classroom was to
provide the students with examples of the connection between mathematics and daily life
issues. Inservice participants believed this connection to daily-life was crucial in learning
mathematics, as P4 explained:

“Men I build connections with the daily life, [the student] both
understands and later extracts things from that [connection] [...] For a
student to understand mathematics, [he/she] should be able to understand
the examples I give, [I mean] the daily life examples ” (P4, 2004, Q11).
However, the connection between mathematical knowledge and daily life was rather in
the form of an arithmetical tool for the participants. They claimed that they emphasized
mathematics as a need in daily life skills for their students so that the students would have
an interest in learning mathematics: "[Students] should not see mathematics as a course,
[ rather, I wish they would think] ‘This is useful for me in daily life, I will need this
(P12, 2004, Q20). Similarly, participants mentioned about the use of materials in the
classroom for increasing student interest and enjoyment rather than their learning, as
reported in the previous chapter.

In summary, the national curriculum pace that must be maintained in the
classrooms where students lacked considerable mathematical knowledge seemed to lead

the participants towards more teacher-centered beliefs and practices. However, they still

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considered student learning a priority and emphasized mathematics’ connections to daily

life in their teaching. By prioritizing student interest and enjoyment in teacher-centered

classrooms in the ﬁrst year of teaching, participants seemed to formulate ways to survive

the curriculum pace and heterogeneous classrooms with an emphasis on student learning.
Summary

Inservice participants generally had teacher-centered beliefs both in 2003 and
2004 and the characteristics of these teacher-centered beliefs were quite similar across the
participants in 2004 after the ﬁrst-year of teaching. They generally claimed the
effectiveness of teacher-centered practices in which few student-centered activities would
be used, students’ role is limited, and their interest in learning mathematics was
considered. Two important phases of teaching that were the focus of this study, namely
the teacher education program (EME program) and the ﬁrst-year teaching, seemed to
impact participants’ mathematics related beliefs. This ﬁnding was consistent with
previous studies (Flores & Day, 2006; Raymond, 1997), which found that preservice
education and ﬁrst-year in teaching inﬂuenced beginning teachers’ teaching and learning
beliefs in the contexts of Portugal and the US.

Participants seemed to believe in the effectiveness of students discovering the
nature of mathematical concepts and use of materials in teaching mathematics at the end
of their preservice education in the EME program. The participants claimed they mostly
beneﬁted from the PCK courses in the program and that they tried to use course materials
whenever possible. Although they generally beneﬁted from the program experiences, an

extensive use of knowledge and skills gained in the program was not possible for them.

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Besides lack of experience in real-classroom settings, contextual factors played role in
rare use of program experiences in teaching.

Inservice participants had difﬁculties related to the tensions in classrooms such as
management, a concern which many studies addressed (see Veenman, 1984). Curriculum
pace was another important difﬁculty for the inservice participants since they did not
practice teaching considering the pace and load of the curriculum. The lack of sufﬁcient
practice component in EME program was also effective on inservice participants as they
could not realize the limitations of the curriculum until they started ﬁrst-year teaching.

The impact of informal teaching experiences was an important ﬁnding in this
study. Participants mostly had informal teaching experiences in the form of private
tutoring or TPC teaching in which teacher-centered practices were valued. Moreover,
they experienced especially TPC settings as a hi gh—school student for at least one year
while they were preparing for the university entrance exams, which served them with a
considerable personal history. Informal teaching experiences seemed to inﬂuence
inservice participants towards believing the effectiveness of teacher-centered practices in
the mathematics classroom.

Participants’ teacher-centered beliefs at the end of ﬁrst-year teaching seemed to
be partly formed through their experiences with the student population. Most participants
had students with poor and unequal knowledge and skills in their classrooms. It was
difﬁcult for the participants to teach in such classrooms since they had to keep a pace
with the national cuniculum for which they were inspected. Additionally, they were not
sufﬁciently prepared to teach in the heterogeneous classrooms in the EME program.

Thus, participants were inclined to believe in the effectiveness of teacher-centered

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practices in the classroom since these practices seemed to work in their heterogeneous
classrooms. In addition to having poor mathematical knowledge, participants’ students
did not have interest in learning mathematics. Hence, participants tended to emphasize
mathematics’ connections to daily life in the form of arithmetical tools in their teaching.
They believed that such connection would increase student interest in mathematics,
which they considered as a precondition for learning mathematics.

The participants claimed that emphasizing the connections between mathematics
and the daily life would increase student interest and enjoyment in the mathematics
classroom. Participants seemed to believe that student learning could be increased if their
interest and enjoyment were increased and if students liked the teacher. They prioritized
student enjoyment in the mathematics classroom and tended to use different approaches
in teaching mathematics such as discovery and materials to increase student enjoyment
rather than their understanding.

Inservice participants’ rather unusual teacher-centered beliefs seemed to be a
result of the transformation they experienced when they started to teach in real-classroom
settings. It was difﬁcult for the participants to import their EME program experiences to
real-classroom settings since those experiences were shaped in somewhat artiﬁcial
conditions where curriculum pace and student heterogeneity were not present. Thus, the
participants re-considered these experiences through real-classroom conditions and re-
formulated them in a way that would work both for maintaining curriculum pace and
increasing student learning. In conclusion, participants expressed the impact of methods
courses in the EME program on the way they considered mathematics teaching. These

courses helped them to develop beliefs like effectiveness of using manipulatives in

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teaching and impact implementing different methods of teaching on students’
mathematics learning. However, these beliefs were transformed to beliefs that considered
the effectiveness of more teacher-centered beliefs in which some student-centered
activities were used, teacher assumed the main responsibility, and students had limited
duties while learning mathematics. First-year teaching did not provide contexts for the
inservice participants in which student-centered beliefs would become more central in

their belief systems.

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CHAPTER 8

DISCUSSIONS AND IMPLICATIONS

Conducting this study and presenting the results have been a challenging process
because of the highly subjective nature of beliefs. Although I tried to use the phrase
“believe” with a less certain language, the theoretical framework I used and the
consensus between the second coder and I provided me a safe space to conclude that the
participants had certain beliefs. Our consensus was actually what we inferred from data
within the limits of the theoretical framework.

Inferences from the data, or, in other words, what we concluded that participants
believed, were documented in detail in the previous three results chapters. Results
suggested certain points to emphasize in relation to the impact of EME program and ﬁrst-
year teaching experience. These points will be reviewed and discussed in this chapter
with references to the previous studies. Although the title of the chapter addresses
discussions and implications, this chapter initially documents conclusions,
interpretations, and discussions of the ﬁndings. Some contributions and implications of
the study will be mentioned in the ﬁrst section, as well. Then the main contributions of
the ﬁndings are summarized. The contributions of this study to my future career as a
teacher educator and to educational research investigating teachers’ beliefs are also
explained. Although I give a detailed explanation of the limitations of the study in the

Methods chapter, I will address it through the results of the study here as well.

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Conclusions, Discussions, and Implications

In this section, ﬁrst, EME program impact will be discussed based on the results
from preservice and inservice participants’ interviews. The focus will be on the practice
of teacher education in the EME program, its role in the enculturation process in which
preservice teachers’ mathematics related beliefs are developed, and the differences
between the US. context and the Turkish context in this process. Possible changes or
improvements in the EME program practices are also addressed. Second, the impact of
the ﬁrst-year teaching experience and how it interferes with the EME program
experiences will be discussed. The contributions and the implications of the present study
are also emphasized through this section.

E ﬂectiveness of EME Program

Teacher education programs are generally investigated for their effectiveness on
changing preservice teachers’ beliefs, mostly in terms of single course interventions. The
present research addressed the impact of the program across multiple years to understand
what preservice teachers in different year levels of the EME program believed and if
there was any detectable impact of a certain year level or a certain course. The analysis of
cross-sectional data of 2"d-, 3rd-, and 4’h-year participants showed that preservice
teachers’ beliefs about the nature of, teaching, and learning mathematics might not differ
with respect to the year level in the program. -

The lack of detectable difference in participants’ beliefs through the year levels
showed that EME program experiences might not have a signiﬁcant impact on preservice
teachers’ beliefs. Two interrelated factors seemed to be the reason for the lack of impact.

First, the courses were not speciﬁcally designed to challenge preservice teachers’ beliefs;

180

such a lack of explicit focus on changing beliefs has been found to be ineffective in
challenging preservice teachers’ beliefs (Foss & Kleinsasser, 1996). Pedagogical content
knowledge courses were generally considered by the Department faculty as having a
content which preservice teachers would learn for the ﬁrst time and ﬁnd interesting and
challenging. Different teaching approaches were introduced in these methods courses, but
the courses did not explicitly challenge the preservice teachers’ mathematics related
beliefs and thus did not have a measurable impact on preservice teachers’ beliefs in the
present study.

A second explanation for the lack of effect of EME courses on preservice
teachers’ beliefs might be the practice of teacher education in the EME program. EME
course experiences were mostly group presentations and projects of mathematics teaching
methods or teaching of a certain mathematical concept. However, group presentations
were mainly lectures about the teaching method or mathematical concept, presentation of
few learning activities to the class pretending that it was a real-classroom context, but
with no critical discussion of the presented teaching method or students’ misconceptions
in a speciﬁc mathematical concept. Only a few of the preservice participants referred to
these course experiences during the interviews, although they were taking the course at
the time of the interviews. Preservice participants generally perceived these presentations
as less useful for their future teaching practice. It could be concluded that EME program
course experiences might not have a strong or long-lasting impact on preservice
participants’ beliefs.

Despite the fact that no detectable changes occurred in preserve teachers’ beliefs

across the year levels in the present study, it should be noted that results of belief change

181

studies should be interpreted with caution (Nettle, 1998). Even when there seems to be no
overall impact of the teacher education programs or course interventions on preservice
participants’ beliefs, there might still be some change in few preservice participants’
beliefs. In support of Nettle’s view, the present analysis of preservice teachers’ interviews
did reveal minor changes, particularly when few Level 1 preservice teaching belief
participants spoke (although infrequently) about implementing different mathematics
teaching methods in the classroom.

Interestingly, although only few preservice participants referred to methods
course experiences and contents during the interviews, inservice participants referred
frequently to their program experiences, especially to the ﬁrst methods course on
teaching mathematics and the elective course on activity based teaching of mathematics,
both when they graduated from the program and after their ﬁrst-year of teaching. This
might indicate a difference between preservice and inservice participants’ program
experiences and beliefs about teaching and learning mathematics.

The difference between the impact of EME program experiences on inservice and
preservice participants’ beliefs could be explained by opportunities of challenging course
experiences that inservice participants had but preservice participants did not have due to
the lack of sufﬁcient number of faculty with mathematics education focus. Pajares (1992)
claimed that beliefs were unlikely to change unless they were challenged. Although
inservice participants complained about the lack of challenge in some of the EME
program courses, most of them noted that there was challenge in two of the program
courses: The ﬁrst methods of mathematics teaching course and the elective course which

was speciﬁcally designed to challenge inservice participants’ beliefs about mathematics

182

instruction at schools. Especially the elective course was designed to challenge the
inservice participants and make them believe that mathematics teaching could be
practiced in different ways at schools. Unlike other program PCK courses, the elective
course had a variety of major innovative components such as creating and learning with
mathematical games, using technological tools to understand mathematical concepts, and
realizing mathematical misconceptions. These experiences encouraged inservice
participants to reﬂect on their prior mathematics related beliefs and seemed to initiate a
change. In contrast, preservice participants did not experience a course with these kinds
of challenging components, which might account for the lack of belief change in
preservice participants.

The lack of challenge in PCK courses addresses a more general problem of the
EME program, which inﬂuences preservice teachers’ experiences in the program.
Although inservice participants had challenging course experiences in the EME program,
these experiences were mostly limited to one course. However, belief change in the
teacher education programs should be a continuous process of targeting and challenging
beliefs through courses (J oram & Gabriele, 1998). The EME program lacked continuity
and consistency of course contents and experiences, as addressed by most of the inservice
participants. The inconsistency was mostly due to the change of professor in consecutive
courses and the lack of communication among the professors who were teaching these
courses. The present study showed that the EME program offerings might limit the nature
of course experiences which would impact preservice and inservice teachers’
mathematics related beliefs. The lack of sufﬁcient number of professors in the program

required recruiting faculty from other departments — faculty who were less familiar with

183

the program and its goals and students. There was a lack of communication between the
recruited professors and program faculty, which impacted the PCK course content and
program students’ experiences. This study highlighted the need for an increased number
of faculty and continuous communication and reﬂection of teacher education practices
among the course professors in order to inﬂuence preservice teachers’ program
experiences and the development of their mathematics related beliefs.

In addition to the EME program factors such as lack of continuity of challenge in
courses, preservice teachers’ informal teaching contexts seemed to decrease the impact of
the program on participants’ mathematics related beliefs. Teacher education program
experiences constitute a very important component of the enculturation process in which
preservice teachers’ beliefs are considered to develop (Pajares, 1992). In the case of
Turkey, there are other educational contexts such as TPC teaching and private tutoring,
which impact preservice teachers’ enculturation in addition to the teacher education
programs. Understanding these contexts and experiences is important in understanding
how preservice teachers’ beliefs are formed. In the present study, exploration. of the
impact of informal teaching contexts also showed that these experiences might limit the
enculturation process which the EME program was supposed to provide for the
preservice teachers. Preservice and inservice participants of the present study mostly
attended highly teacher-centered TPCs and private tutoring as pre-college students and
then as tutors before they graduated from the EME program. Thus, teacher-centered
informal teaching contexts continued to impact participants’ mathematics related beliefs

during their studies at the EME program. These contexts might interfer participants’

184

EME program experiences, which tended to emphasize student-centered practices in
teaching mathematics.

The informal teaching experiences might have a large impact on participants’
beliefs due to the lack of opportunities to practice teaching in the EME program courses.
Methods course experiences were not practiced in real-classroom contexts immediately
and practice teaching courses did not provide extensive teaching experience for the
preservice teachers because they basically included observation of the school and
classroom context instead of teaching. Preservice teachers were required to teach only
two classes in the last semester of practice teaching, which could be extended only if the
collaborating teacher offered more hours to teach. It seemed very likely that many
participants in this study substituted their TPC teaching and private tutoring experiences
for the classroom experiences that they felt the EME program should have provided. And
recall that there is no “internship” in Turkey in teacher preparation programs. Being
driven by their prior experiences and beliefs that considered teacher-centered practices as
effective, many participants might have ﬁltered the EME program experiences through
these beliefs which reduced the impact of the program, similar to what Paj ares (1992) has
pointed out.

It has been noted that teacher education program experiences often do not prepare
preservice teachers for the classroom realities (Wideen, et al., 1998). The present study
found similar results. After their ﬁrst-year teaching, inservice participants complained
about the lack of sufﬁcient teaching practice in the EME program. Although inservice
participants attended three semesters of student teaching and almost all of them taught in

informal teaching contexts for many years, participants felt that these teaching

185

experiences were not effective in preparing them for the real-classroom contexts. Ryan
(1979, as cited in Veenman, 1984) claimed that teachers are not necessarily fully trained
in teacher preparation programs for what their work required.

One potential solution to this problem would be to lengthen the period of practice
teaching of preservice teachers. However, longer student teaching might not prepare
EME program students better for teaching at schools unless two important practices in
teacher education would be changed or improved. First, the student teaching period
should include more and continuous opportunities for preservice teachers to teach several
mathematics concepts or a class through the semester, instead of teaching only two
lessons in one semester. This would provide preservice teachers with more realistic
experiences of classroom teaching where they would realize and deal with student
differences, cuniculum rush, and classroom management, compared to mostly irregular
and disconnected classroom teaching in the EME program.

Second, methods courses should include an immediate microteaching component
in which preservice teachers would experience the implementation of several student-
centered approaches in real-classroom contexts throughout the courses. Since EME
program students do not have the opportunity to implement methods course contents in
microteaching immediately after the content is introduced, they tend to implement the
course experiences during a very limited student teaching period which does not give
sufﬁcient insight into the reality of classroom teaching.

Previous studies in the US. showed that a short period of ﬁeld experience during
methods courses in which preservice teachers practiced skills emphasized in the courses

did not change preservice teacher’s beliefs (Foss & Kleinsasser, 1996; Nettle, 1998).

186

However and more recently, Ambrose (2004) provided support for the impact of having
continuous and immediate practice opportunities (such as microteaching) on preservice
teachers’ beliefs. She found that an immediate, intense, and semester-long microteaching
component in a content course for teaching mathematics to elementary school students
helped preservice elementary teachers to develop new student-centered mathematics
related beliefs.

Pedagogical content courses in teacher education programs in Turkey do not
include any microteaching in which preservice teachers would practice student-centered
methods of teaching mathematics in real-classroom contexts. As the Ambrose (2004)
study suggests, an intense microteaching component for pedagogical content knowledge
courses in EME programs could be considered as a promising intervention in impacting
preservice teachers’ beliefs. This type of extended microteaching for teacher education
courses is not a magic bullet for developing higher level mathematics related beliefs.
However, more microteaching of student-centered practices in real classrooms during
preservice training could help Turkish preservice teachers gain more realistic
understanding of the real-classroom contexts that might reduce the impact of reality
shock on their beliefs. Through the microteaching component, Turkish preservice
teachers might develop new student-centered beliefs and might be able to maintain some
of these beliefs in their ﬁrst years of teaching, since the impact of the ﬁrst-year shock
would be reduced by microteaching during the preservice training.

Comparison of US. and Turkish Preservice Teachers
So far, the conclusions, discussions, and implications of the present study have

been presented and compared to previous studies mostly conducted in the US. However,

187

it is useful to also present an explicit comparison of Turkish and US. preservice teachers’
mathematics related beliefs and factors impacting these beliefs. Since there are
considerable differences in the schooling and teacher education between US. and
Turkey, it seems relevant to mention some emerging differences and similarities by
comparing the results of the present study and previous ﬁndings of US. research.

The present study highlighted a difference in how US. and Turkish preservice
teachers might make sense of what they experience in teacher education program
experiences. While U.S. preservice teachers’ prior beliefs are typically developed through
their pre-college experiences as students (Lampert, 1990), Turkish preservice teachers’
prior beliefs are developed both during their pre-college experiences as students and
teaching experiences in TPC and private tutoring contexts as tutors. In other words, U.S.
preservice teachers’ beliefs are mostly observation-driven (based on what has been
observed during time spent as a student), whereas Turkish preservice teachers’ beliefs are
both observation- and experience-driven (based both on observations as students and also
on reﬂections on experiences as a teacher or tutor). Based on Green’s (1971)
characterization, one might argue that Turkish preservice teachers’ prior and mostly
teacher-centered beliefs (which emerge from TPC teaching and tutoring experiences)
might be more central and would be more effective on their practice, since they become a
part of their personal history through experience, as Nisbett and Rose (1980, as cited in
Paj ares, 1992) would suggest. Hence, for Turkish teachers, these teacher-centered beliefs
might essentially ‘ﬁlter’ the teacher education program experiences more extensively
compared to the US. preservice teachers’ beliefs. The comparison is plausible; however,

it is open to empirical investigation as no pattern was detected in preservice participants’

188

beliefs in relation to differing amounts of TPC teaching and private tutoring experiences.
Preservice participants with long versus no informal teaching experiences often had
similar beliefs. Interestingly, inservice participants with longer TPC teaching experiences
realized the impact of these experiences on their beliefs after the ﬁrst-year teaching. In
conclusion, although Turkish preservice teachers’ prior beliefs were likely to be more
central due to the informal teaching practices, the impact of these beliefs 'could not be
generalized.
Summary

Teacher education programs are the contexts in which preservice teachers are
expected to develop certain beliefs that will help them to promote learning for their ﬁrture
students. However, the effectiveness of these programs depends on many conditions such
as the program offerings and the ways that preservice teachers process program
experiences. The existing literature indicated a need for further research in order to
understand the processes in which preservice teachers ﬁltered the teacher education
program experiences (Wideen, 1998). The present study investigated the enculturation
process in which preservice teachers’ mathematics related beliefs were formed in the
context of an Elementary Mathematics Education program. It also explored the impact of
informal teaching experiences on the teacher education program experiences and how
these two sets of experiences affected participants’ beliefs. Through investigating the
beliefs of preservice and inservice (ﬁrst-year) teachers, the present study reported the
conditions and degrees of effectiveness of the EME program experiences in training
middle school mathematics teachers. Considering that the program was initiated in 1998,

this study provides a general initial program evaluation.

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F irst- Year Teaching Contexts and Impacts

There is some evidence in the present study that beliefs impacted practice, a
relationship that prior research has documented (see, Clark & Peterson, 1986; Nespor,
1987). Inservice teachers often expressed student-centered beliefs which considered
students as capable of discovering important issues in mathematics. These student-

' centered beliefs might be attributable to the challenge inservice participants had in some
of the EME program courses, as mentioned in the previous section. The impact of
inservice participants’ student-centered beliefs was evident in the description of their
classroom practice where they allowed students discover mathematical concepts by the
help of manipulatives. However, the underlying nature of such practices was teacher-
centered, inconsistent with their student-centered beliefs.

Student-centered practices that inservice participants claimed to have in their
classrooms were only a surface feature of their main practices. Participants explained
concepts to the students and assumed most of the responsibilities in the classroom, which
was inconsistent to their beliefs about the effectiveness of student-centered ”teaching
practices. There may be three main explanations for this inconsistency: (i) existence of
different clusters of beliefs, (ii) impact of broader schooling context, and (iii) the level of
teachers’ awareness of beliefs. These are explored here.

The ﬁrst explanation concerns the existence of two main clusters of beliefs
besides the belief levels, regarding mathematics instruction in the classroom in inservice
participants’ beliefs systems. One cluster seemed to be composed of the student-centered
beliefs gained through the EME program experiences as explained above. The second

cluster seemed to include teacher-centered beliefs that inservice participants would

190

consider as necessary to survive in the highly complicated classroom context. Although it
might seem unusual to hold both teacher-centered and student-centered beliefs at the
same time, Green (1971) claimed that beliefs are held in clusters, a condition which
enables one to hold conﬂicting beliefs in different clusters. While some inservice
participants seemed to gain many teacher-centered beliefs through informal teaching
experiences during their EME program studies, many of the participants seemed to gain
these teacher-centered beliefs as a result of their attempts to deal with contextual factors.
The two clusters of beliefs might have impacted participants’ practice depending on the
students, curriculum, and available materials. Hence, the present study seemed to conﬁrm
the ﬁndings related to the inconsistencies between beliefs and practices of teachers
because of the complexity of the classroom (Fang, 1996). This leads to the second
explanation which might explain the inconsistencies in a broader context.

The second explanation, given by Ernest (1989), is that the social context
including institutionalized curriculum, system of assessment, and national system of
schooling cause teachers to have practices inconsistent with their beliefs. The cuniculum
is centralized at the national level and assumed to have a single pace for all the schools in
Turkey, which are inspected every year. There are important common themes in
schooling in Turkey such as the curriculum. However, schools, classrooms, teachers, and
students are different in terms of availability of resources, the practice of previous
teachers, and the level of students’ knowledge. The present study showed that the
inservice participants might not be ready to face these factors or limitations when they
started their ﬁrst-year teaching. Hence, they seemed to change their practice in the most

efﬁcient way that would maintain the curriculum pace, consider students’ differences and

191

learning, and use time efﬁciently. As a result, they started to have a hybrid of teacher-
centered practices where they used surface features of student-centered practices.

The third explanation for the inconsistencies between inservice participants’
student-centered beliefs and their mostly teacher-centered practices might be the lack of
inservice participants’ level of awareness of their beliefs, as suggested by Ernest (1989).
The Turkish system, unlike the US. system, does not include an internship year in which
teacher educators and schools monitor beginning teachers’ practice and their learning of
teaching. Although beginning teachers are assigned mentors who are experienced
teachers and are responsible for monitoring and reﬂecting on beginning teachers’
practices, mentors and beginning teachers meet on very rare occasions, or do not meet at
all. Therefore, beginning teachers are mostly alone in their classrooms without any
second opinion on their teaching, for which some participants expressed a need. Inservice
(ﬁrst-year) participants in this study were mostly assigned to rural schools where they
were generally the only mathematics teacher. This might also reduce the possibility of
having a community of practitioners who would form a support group for the inservice
participants in their ﬁrst-year. Inservice participants in the present study mostly lacked
the reﬂection from peers and mentors about their beliefs and practices that previous
studies (e. g., Aitken & Mildon, 1991) addressed as important for beginning teachers. The
lack of constructive reﬂection on inservice participants’ beliefs and practices enabled
them to hold conﬂicting beliefs, a ﬁnding which previous studies addressed (6. g.
Charalambous, Philippou, & Kyriakides, 2002; Schmidth & Kennedy, 1990; Thompson,
1984). Thus, inservice participants might have reﬂected on their own practice through

their existing teacher-centered beliefs, rather than student-centered beliefs since they

192

generally counted teacher-centered practices as effective, a ﬁnding consistent with
previous studies (6. g. Flores & Day, 2006). Hence, inservice participants seemed to
believe that student-centered practices which partly covered the actual teacher-centered
practices were effective in teaching mathematics in the classrooms.

The three main explanations for the inconsistencies between inservice
participants’ student-centered beliefs and teacher-centered practices, which were
clustered nature of beliefs, impact of context, and the awareness of belief, seemed to
address two serious issues for preservice and inservice training. The ﬁrst issue seemed to
be the lack of immediate practice of learned skills and the insufﬁcient student teaching
experiences in the EME program as explained in the previous section. Inservice
participants did not have sufﬁcient opportunities during the EME program that they
would practice their student-centered beliefs in real-classroom contexts. When inservice
participants started to teach, they did not have opportunities which would support the
development of their student-centered beliefs in their teaching-contexts. Therefore, they
might have developed teacher-centered beliefs whose associated practices worked in the
highly complex real-classroom settings. These teacher—centered beliefs about real-
classroom contexts seemed to become more central in the everyday practice of inservice
participants’ maintaining curriculum pace and dealing with students’ differences.

The second issue seemed to emerge as the perceived strength of the challenge
which real-classroom contexts imposed on beginning teachers’ beliefs and practices.
Although inservice participants’ beliefs seemed to be challenged in a program course and
some student-centered beliefs were developed, these beliefs were re-challenged by the

real-classroom contexts. The challenge of real-classroom contexts was perceived strong

193

compared to the challenge of the EME program course perhaps because participants had
to practice in the challenging context. Moreover, inservice participants initially might
have highly unrealistic beliefs about the ways they would teach, as addressed by previous
studies (Weinstein, 1989 and Wilson, 1990, as cited in Pajares, 1992). However, they
seemed to have a reality shock in their classrooms where students lacked previous
knowledge, schools lacked essential materials, and curriculum rush was the main concern
of teaching. These factors seemed to lead a contradiction between inservice participants’
ideal world and the real world (or their student-centered beliefs and teacher-centered
practices), which was a universal concern for beginning teachers (Flores & Day, 2006;
Veenman, 1984).

In summary, the three explanations for the inconsistency between inservice
participants’ student-centered beliefs and mostly teacher-centered practices and the two
issues for the preservice and beginning years seemed to address that individual factors
impacting beginning teachers’ practice in the classrooms were impacted by other factors.
Inservice participants’ student-centered beliefs and unrealistic ideas about ﬁrture teaching
might be formed by the lack of sufﬁcient real-classroom practices in the EME program.
These beliefs and ideals were challenged by unexpected and unprepared context of
schooling and inservice teachers were forced to survive in their ﬁrst-year instead of
practicing the skills they once were prepared and excited to implement. Schooling in
Turkey seemed to promote teacher-centered beliefs for inservice participants, conﬁrming
the dominant impact of institutional factors on beginning teachers’ practice as

documented in prior work (Flores & Day, 2006; Raymond, 1997).

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Policy Concerns for Beginning Teachers in Turkey

The present study highlighted a need for school re-organization in terms of
curriculum, materials, and support groups. Although a new curriculum policy was
initiated in Turkey beginning from 2005 -2006 school year, this policy only considered
decreasing the load (through increasing the high school education from 3 years to 4
years) and providing more time for student-centered practice. Even when the curriculum
load would be reduced, students would likely to have a knowledge gap as a result of
frequent teacher turnover existing at schools. Hence, beginning teachers would still have
to deal with student differences for which they were not prepared in the teacher education
programs. They would still lack self-reﬂection and second opinion opportunities through
mentors and peers since the mentorship policy would not be effectively implemented.

A very important policy concern in teacher education in Turkey was also
addressed in this study: the lack of communication between the teacher education
programs and the Ministry of National Education on beginning teacher mentoring.
Turkish beginning teachers are alone in their initial years. Some schools try to support
beginning teachers through content-area teacher groups. However, this does not
guarantee opportunities for reﬂection on beliefs, practices, and the inconsistencies
between them, since experienced teachers’ educational beliefs are rarely investigated and
are very likely to be traditional. Similarly, even when mentors are interested in
monitoring beginning teacher development, their beliefs, which are also likely to be very
traditional, are not studied. Hence, an effective mentoring system including the
collaboration of teacher education pro grams, schools, and mentors is needed in schooling

in Turkey, in order to make beginning teachers reﬂect on their beliefs and practices and

195

support them in their initial years. Such a mentoring system should start in the last year of
teacher education programs in the form of communicating classroom reality to the
preservice teachers. This is likely to reduce the impact of reality shock on beginning
teachers’ beliefs and practices. In this sense, the study conﬁrms the suggestions presented
in the literature that address 3-year monitoring of beginning teachers ﬁom the last year of
preservice education (Aitken & Mildon, 1991).
Contributions of the Study

The contributions of this study are mostly expressed in the previous sections
blended with discussions and implications. Here, I want to summarize the main points of
the study’s contribution in preservice education and beginning teacher support brieﬂy.
Although the study contributes to the research literature on teachers’ beliefs generally, it
also contributes to my own individual future practice of teacher education in the
department where the study was conducted. I will address these contributions here. The
contribution of the present study to the research on teachers’ beliefs will be presented as
well.

Teacher Education Programs and Initial Years in Teaching

The present study’s main contribution is on the practice of teacher education in
Turkey. It provides feedback from beginning teachers on the effectiveness of EME
programs, which Turkish teacher educators have not attempted to gather through
research. The present study also documents how preservice teachers’ beliefs are
developed in informal teaching contexts and how these beliefs may interfere with the
EME program experiences. Generally, TPC and private tutoring contexts have been

considered as a limiting factor in the development of preservice teachers’ beliefs about

196

learning mathematics as they promote solving a lot of questions correctly as success.
However, prior to the present study, preservice teachers’ informal teaching practices in
especially TPC contexts have not been considered as a limiting factor for the teacher
education program experiences and especially their beliefs about teaching mathematics.
The present study shows that these contexts promote teacher-centered beliefs for
preservice teachers by providing them a place to practice these beliefs. Hence, the study
highlights a previously underestimated factor which decreases the desired impact of
teacher education programs on preservice teachers’ beliefs.

The results from the inservice participants provide a View of beginning teacher
practice in national curriculum context — which has not previously been studied in depth.
Although the national curriculum seems to provide a guide for the teachers and especially
beginning teachers, it also has its own limitations and pressures, as documented in the
present study. The curriculum pace and being inspected have always been concerns for
teachers in Turkey. However, the concerns of beginning teachers seem to be different
from experienced teachers since they are not experienced and they are generally assigned
to rural schools in which teacher turnover was frequent. When beginning teachers start to
teach in these schools, although their students lack essential previous knowledge, they are
still expected to be on the same pace and inspected for that. Thus they face the dilemma
of re-teaching previous concepts that students need to understand the new concepts and
keeping the curriculum pace that they have not sufﬁciently prepared and they are
inspected for. Therefore, the present study indicates that assigning beginning teachers to
schools aﬁer the school year starts does not necessarily guarantee students’ learning or

teachers’ effective teaching.

197

 

 

The present study documented the potential pitfalls of the national curriculum
both for beginning teachers’ teaching and students’ learning, from the beginning
teachers’ point of view. Although the national curriculum provides guidance to the
beginning teachers in the form of clearly stated concepts to teach and the learning
objectives for the students, it brings limitations and difﬁculties due to other schooling and
teacher recruitment policies. The present results are likely to contribute to the recent
curriculum change in Turkey in terms of raising the attention of policy makers to the
existing context and future impact of change in a more global sense including beginning
teachers, students, and school contexts.

One of the most important outcomes of this study is the need for teacher
preparation programs to explicitly challenge beliefs and encourage reﬂection on beliefs,
in both preservice and beginning years. The EME program seems to have more impact on
preservice teachers’ beliefs when the courses challenge their beliefs more. The results of
this study may be used for targeting preservice teachers’ beliefs and organizing course
experiences in a way to challenge those beliefs. Moreover, this study addressed a need for
an effective mentoring system for the beginning teachers. The inservice participants
expressed their need for a more knowledgeable person to address teachers’ concerns
about classroom management, teaching, and student differences. The study emphasizes
that beginning teachers need mentors and opportunities for reﬂecting on their practice.

My Practice of and Research on Teacher Education

My main goal in conducting the present study was to contribute to the teacher

education practice in the EME program, which also includes my practice as a future

teacher educator. The present study provided me insight into the speciﬁc beliefs of

198

 

preservice teachers. Since changing beliefs of preservice teachers is highly unlikely in the
teacher education programs as documented by the present study, I will try to build on my
students’ existing beliefs through the courses that I will offer as Wideen et a1 (1998)
suggests. This will deﬁnitely open a new window of research for me in terms of studying
the effectiveness of my own teaching.

The present study provided me with convincing evidence from the inservice
participants of the present study on the lack of effectiveness of EME courses and
experiences. It conﬁrmed my concern for the ineffectiveness of the general practice of
teacher education in my College of Education. The present study gave me clues about
potentially effective practices, such as making preservice teachers realize and reﬂect on
their beliefs. It also helped me to identify missing or insufﬁcient courses and experiences
in the EME program. I communicated these issues and my suggestions for how these
missing issues may be addressed to the EME program faculty (see Appendix E).
Although the process of establish new courses and practices will take a long time, the
process will provide a new area of research based on the present study’s ﬁndings. The
present study suggests a need for a more extensive longitudinal study concerning
inservice participants’ beliefs and practices. As I am still maintaining communication
with the inservice participants of this research, I am planning to continue investigating
their beliefs and practices as they get more experienced in teaching. Finally, the study
highlighted the importance of effective constructive communication among the program
faculty which I have been trying to establish and strengthen since the beginning of my

doctoral studies.

199

Research on Teachers ' Beliefs

In this study, a combination of three frameworks was used in order to analyze
data and detect preservice and inservice participants’ mathematics related beliefs. Two of
these ﬁ'ameworks were developed empirically, implemented in US. and Finnish contexts
and presented considerable consistency in investigating mathematics related beliefs. For
this study, I added a third theoretical framework to make the ﬁrst two frameworks
stronger and implemented the combined framework in a Turkish context. The present
study conﬁrmed the ﬁndings in US. and Finnish contexts that characterized preservice
and beginning mathematics teachers’ beliefs mostly as Level 1 (Thompson, 1991;
Lindgren, 1996). Hence, the study contributes to educational research by developing a
comprehensive framework that can be used to investigate preservice and beginning
teachers’ mathematics related beliefs.

As mentioned in Chapter 2 (Literature Review), studies investigating teachers’
beliefs state several methods that can be used in research. These methods vary in terms of
what they conceptualize as “beliefs”, how the beliefs are detected, and how they are
justiﬁed. There is a general tendency in belief research to use study-speciﬁc deﬁnitions
and methods which limit the consistency of ﬁndings with the previous studies and across
different contexts as well as their implications. The framework used in the present study
includes three frameworks that are consistent in terms of the general characterization of
beliefs and the speciﬁc list of teachers’ beliefs. Hence, the present study contributes to
belief research in education by presenting a more accurate useful framework of teachers’
beliefs which clariﬁes the process of detecting beliefs and has the potential to provide

consistency across studies.

200

A future direction for the framework will be identifying contextual differences for
Turkey and reorganizing it by adding or eliminating some belief statements. This process
addresses both qualitative and comprehensive quantitative studies which will be based on
the present study.

In smnmary, the implications of the study for the ﬁeld of teacher education
address three major issues. First, teacher education program experiences, which teacher
educators might consider as eye-opening, may not impact the preservice teachers as
intended. Hence, investigating teachers’ beliefs from the beginning of teacher education
programs might enable teacher educators to evaluate the impact of the program
experiences and to plan necessary courses and experiences that would challenge and even
change the beliefs. Second, contextual differences might result in different paths where
preservice and inservice teachers develop mathematics related beliefs. Findings from a
highly decentralized system such as the US. system should not be generalized to the
highly centralized system of Turkey. Moreover, countries have different teacher
education programs and cultures of teaching and learning which are likely to impact
preservice teachers in different ways. This study shows that in order to investigate and
compare certain issues common to all educational systems, such as teachers’ beliefs,
there is a need for an initial investigation of each system’s nature in every level so that
the impact of structural differences would be documented and a discussion base could be
formed. Third, using more deﬁnite tools in different settings for data analysis in
qualitative research of highly subjective constructs might enable researchers to reach a
consensus on a working tool that is open to improvement. This might also address the

impact of cultural and systemic differences that are likely to appear in teachers’ beliefs.

201

 

Table A]

APPENDIX A

MATHEMATICS RELATED BELIEFS FRAMEWORKS

Complete list of Characterizations in Thompson’s (1991) Framework

Thompson ’3 (I 99]) Framework that Characterizes Mathematics Related Belief Levels.

 

Levels

Characterization

 

Level 0

Mathematics is basically the usage of arithmetic skills in daily life.

Mathematical knowledge is composed of facts, rules and procedures.

The goal of the mathematics instruction is to obtain the correct/accurate answer, through
the ways demonstrated in the class.

Mathematics teaching is developing students’ arithmetic skills through memorization of
rules without mentioning relationships among them.

Textbook is followed to teach mathematics without considering relevancy of the
concepts.

0 Teacher’s role is to demonstrate the procedures as mathematical knowledge.

Student’s role is to imitate and extensively practice the procedures the teachers had just
demonstrated.

There is an authority, either the teacher or the text book, for the correctness/accuracy of
mathematical knowledge.

 

Level 1

Mathematics is composed of rules and procedures with the principles behind them.
The distinction between the meanings and skills in the mathematical concepts has just
initiated.

Conceptual analysis of content domains and appreciation of complexity of mathematical
content has just started.

Teaching through manipulatives is associated with attitudinal goals and empirical
justiﬁcation rather than cognitive goals and making connections to mathematical
concepts.

Teaching for conceptual understanding is using pedagogical tasks and instructional
representations to explain isolated sets of concepts and procedures.

Problem solving is taught separately from the mathematical content.

Integrating problems in the content means spreading problems unrelated to the content.
Teaching “about” problem solving (i.e. selection of strategies) takes place rather than
teaching “with” problem solving (i.e. using it as an instructional approach).
Pedagogical decisions are based on perceptions of experts rather than cognitive
effectiveness of instructional practices.

Students put effort to understand the justiﬁcation of the mathematical procedures.
Experts are still an authority for the correctness/accuracy of the mathematical
procedures.

 

Level 2

The importance of various concepts and centrality of various ideas in mathematics are
realized through understanding the relationships between them.

Proof and generalization processes are considered as a part of mathematics teaching and
learning.

Visual and concrete representations are designed to provide students contexts that they
can explore their ideas.

Teachers guide and provide opportunities for students’ understanding through carefully
designed pedagogical approaches.

Student-generated ideas are considered as important.

Students must involve in constructing mathematical ideas in order to understand them
better.

The goal of instruction is develop students’ reasoning through investigating
mathematical ideas.

202

Complete List of Characterizations in Lindgren’s ( 1996) Framework

 

 

 

 

Table A2
Lindgren 's ( I 996) Framework that Characterizes Mathematics Related Belief Levels.
Levels Characterization
Rules and 0 Above all one should teach mathematical knowledge i.e. facts, rules, and statements.
ROUtiDCS - When solving problems, it is most important that the students get the right answers.
(l-é‘veI 0) o In teaching one should use as often as possible such routine problems, where the
correct answer can be achieved by using a familiar method.
0 In learning mathematics, it is most important that pupils practice extensively.
0 Above all, the pupils should learn to master basic calculation.
0 The most immrtant task for the teacher is to maintain good order in the class.
Discussion 0 Above all the teacher should try to promote active class discussions.
and 0 In the math class one has to emphasize individual work.
Games 0 In teaching mathematics, the teacher should let the students use many learning games.
(Level I ) 0 In the teaching process one should promote the pupil’s ability to work with other
pupils.
Open- 0 The student should have the possibility to experience that the same result can be
Approach achieved in different ways.
(LC’VEI 2 )1 I The teacher should encourage the students to ﬁnd different strategies for solving

problems, and to discuss these strategies.

0 The pupils should have a possibility to formulate problems by themselves and then
solve them.

0 The students should use concrete manipulatives as often as possible.

0 In teaching mathematics one should use many verbal problems where the student must
apply his knowledge.

0 One should use, as often as possible, problems where the student has to think ﬁrst, and
where the mastery of calculation alone will not lead to the solving of the problem

0 During math lessons one should emphasize the importance of mathematical thinking.

0 One should consider the possible situations for the use of computers in the teaching of
mathematics.

 

203

 

 

Table A.3

Complete List of Characterizations in Ernest’s (1989) Framework

Ernest 's (I 989) Model that Characterizes Mathematics Related Belief Levels.

 

 

 

 

child’s mastery of skills model
child’s constructed understanding driven model
child’s constructed understanding and interest driven model

Models Levels
Conception of o lnstrumentalist View: Mathematics as a set of tools that include unrelated
Nature of facts, rules and skills used in order to reach an external end product
Mathematics - Platonist View: Mathematics as a static but combined body of knowledge, in
which there are structures and truths connected to each other by logic and
meaning. Mathematics is not created but discovered
- Problem-solving View: Mathematics as a dynamic, problem-driven,
continually expanding ﬁeld in which there is a process of knowledge
generation. Mathematics is not seen as a ﬁnished product
Models of 0 the day to day survival model
Teaching 0 the mastery of skills and facts model
Mathematics 0 the mastery of skills and facts with conceptual understanding model
0 the conceptual understanding model
0 the conceptual understanding enriched with problem-solving model
0 the pure investigational, problem posing and solving model
Models of o child’s complaint behaviour model.
Learning 0 child’s linear progress through curricular scheme model
Mathematics .
O
O
O

child’s exploration and autonomous pursuit of own interests model

204

Combined Framework

Table A.4
Complete List of Level 0 Belieﬂs (T: Thompson ’3 (I 991); L: Lindgren 's ( I 996); E .' Ernest 's (1989)).

 

Nature of 0 Mathematics is basically the usage of arithmetic skills in daily life (T).
Mathematics 0 Mathematical knowledge is composed of facts, rules and procedures (T) (L).
0 The goal of the mathematics and problem solving is to obtain the correct answer,
usually in the ways demonstrated in the class (T).
0 There is an authority, either the teacher or the text book, for the mathematical
knowledge (T).
lnstrumentalist view: Mathematics as a set of tools that include unrelated facts, rules
and skills used in order to reach an external end product (E).
Teaching 0 Mathematics teaching is developing students’ arithmetic skills through
Mathematics memorization of rules without reasoning (T).
0 Textbook is followed to teach mathematics without considering relevancy of the
concepts (T).
o Teacher’s role is to demonstrate the procedures as mathematical knowledge (T).
0 Above all one should teach mathematical knowledge i.e. facts, rules, and statements
(L).
o In teaching one should use as oﬁen as possible such routine problems, where the
correct answer can be achieved by using a familiar method (L).
0 The most important task for the teacher is to maintain good order in the class (L).
Learning 0 Student’s role is to imitate and practice the procedures the teachers had just
Mathematics demonstrated (T).
0 When solving problems, it is most important that the students get the right answers
(L).
0 In learning mathematics, it is most important that pupils practice extensively (L).
0 Above all, the pupils should learn to master basic calculation (L).

205

Table A5

Complete List of Level I Belieﬂs (T: Thompson '5 ( I 991); L: Lindgren 's (I 996); E: Ernest 's (I 989)).

 

Nature of
Mathematics

0 Mathematics is still composed of rules but the principles behind the rules are also
considered (T).

o The distinction between the meanings and skills in the mathematical concepts
has just been realized (T).

0 Integrating problems in the content means spreading problems unrelated to the
content (T).

0 There is an authority for the correctness of the accuracy of the mathematical
procedures (T).

- Platonist view: Mathematics as a static but combined body of knowledge, in
which there are structures and truths connected to each other by logic and
meaning. Mathematics is not created but discovered (E).

 

Teaching
Mathematics

Leaming
Mathematics

0 Mathematics teaching includes the use of instructional representations and
manipulatives (T).

0 Teaching through manipulatives is seen as a way to make mathematics more fun
rather than a way to achieve cognitive goals (T).

0 Teaching for conceptual understanding is using pedagogical tasks to explain
isolated sets of concepts (T).

0 Problem solving is taught separately from the mathematical content (T).

0 Integrating problems in the content means spreading problems unrelated to the
content (T).

0 Teaching “about” problem solving (i.e. understanding of snategies) takes place
rather than teaching “with” problem solving (i.e. using it as an instructional
approach) (T).

0 Role of the teacher is similar to the roles described in Level 0 (T).

0 Above all the teacher should try to promote active class discussions (L).

0 In the math class one has to emphasize individual work (L).

0 In teaching mathematics, the teacher should let the students use many learning
games (L).

o In the teaching process one should promote the pupil’s ability to work with other
pupils (L).

0 Students put effort to understand the justiﬁcation of the mathematical procedures

(T)-

206

Table A6

Complete List of Level 2 Beliefs (T: Thompson 's (199]); L: Lindgren 's (I 996),: E: Ernest 's (1989)).

 

Nature of
Mathematics

The importance of various concepts and centrality of various ideas in mathematics
are realized (T).

Problem-solving view: Mathematics as a dynamic, problem-driven, continually
expanding ﬁeld in which there is a process of knowledge generation. Mathematics
is not seen as a ﬁnished product (E).

 

Teaching
Mathematics

Learning
Mathematics

Mathematics teaching means understanding through doing mathematics where
students are engaged (T).

Proof and generalization processes are considered as a part of mathematics
teaching (T).

Visual and concrete representations are designed to help students explore their
ideas (T).

0 Student-generated procedures are valued (T).
0 Teachers guide students’ thinking and allow students to express their ideas in order

to have a better understanding of their learning process (T).

The teacher should encourage the students to ﬁnd different strategies for solving
problems, and to discuss these strategies (L).

In teaching mathematics one should use many verbal problems where the student
must apply his knowledge (L).

One should use, as often as possible, problems where the student has to think ﬁrst,
and where the mastery of calculation alone will not lead to the solving of the
problem (L).

During math lessons one should emphasize the importance of mathematical
thinking (L).

One should consider the possible situations for the use of computers in the teaching
of mathematics (L).

Concrete manipulatives should be integrated in the teaching as often as possible
(L)-

Students engage in doing mathematics in order to understand (T).

Proof and generalization processes are considered as a part of mathematics learning

(T).

0 Students express their ideas (T).
0 The student should have the possibility to experience that the same result can be

achieved in different ways (L).

The pupils should have a possibility to formulate problems by themselves and then
solve them (L).

The students should use concrete manipulatives as often as possible (L).

 

207

APPENDIX B

ELEMENTARY MATHEMATICS EDUCATION PROGRAM DETAILS

Elementary Mathematics Education Program Courses

The following list gives the Elementary Mathematics Education program and the descriptions of
the courses offered by the Department of Elementary Education. The ﬁrst number in the parenthesis refers
to the class hours and the second number refers to the application or laboratory hours. The number out of
the parenthesis refers to the total credit of the course calculated by adding the class hours with the half of
the application/ laboratory hours.

Courses by Semester

1" Year, First Semester

MATH 111 Fundamentals of Mathematics (3-
0)3

MATH 151 Calculus I (4-2)5

PHYS 181 Basic Physics I (4-2)5

ENG 101 Development of Reading and Writing
Skills I (4-0)4

EDS 119 Introduction to Teaching Profession
(3-0)3

IS 100 Introduction to Information Technologies
and Applications NC

2"d Year, Third Semester

MATH 115 Analytical Geometry (3-0)3

MATH 201 Elementary Geometry (3-0)3
CHEM 283 Introductory General Chemistry (3-
2)4

EDS 221 Development and Learning (3-0)3
ENG 21 1 Academic Oral Presentation Skills (3-
0)3

HIST 2201 Principles of Kemal Ataturk I NC

3“I Year, Fifth Semester

MATH 260 Linear Algebra (3-0)3

ELE 317 Instructional Development and Media
in Mathematics

Education (3-2)4

ELE 331 Laboratory Applications in Science I
(2-2)3

TURK 305 Oral Communication (2-0)2
Elective I (3-0)3, Elective II (3-0)3

4th Year, Seventh Semester

ELE 437 School Experience II (l-4)3

ELE 443 Methods of Mathematics Teaching (2-
2)3

ENG 311 Advanced Communication Skills (3-
0)3

Elective IV (3-0)3, Elective V (3-0)3

1" Year, Second Semester

MATH 112 Introductory Discrete Mathematics
(3-O)3

MATH 152 Calculus II (4-2)5

PHYS 182 Basic Physics II (4-2)5

ENG 102 Development of Reading and Writing
Skills 11 (4-0)4

ELE 132 School Experience I (1-4)3

2"“I Year, Fourth Semester

MATH 116 Basic Algebraic Structures (3-0)3
MATH 255 Introduction to Differential
Equations (4-0)4

BIO 106 General Biology (3-0)3

ELE 224 Instructional Planning and Evaluation
(3-2)4

ELE 300 Computer Applications in Education
(2-2)3

HIST 2202 Principles of Kemal Ataturk 11 NC

3'” Year, Sixth Semester

ELE 240 Probability and Statistics (2-2)3

ELE 332 Laboratory Applications in Science 11
(2-2)3

ELE 336 Methods of Science and Mathematics
Teaching (3-2)4

EDS 304 Classroom Management (2-2)3
TURK 306 Written Communication (2-0)2
Elective III (3-0)3

4'" Year, Eighth Semester

ELE 420 Practice Teaching in Elementary
Education (2-6)5

ELE 448 Textbook Analysis in Mathematics
Education (2-2)3

EDS 424 Guidance (3-0)3

Elective VI (3-0)3

208

Description of Undergraduate Courses Offered by the Department of Elementary Education

ELE 132 School Experience I (l-4)3

Classroom observation including organization and management of school, daily activities in the school,
group activities, a day of a teacher, a day of a student, school-family cooperation, observation of major and
non-major courses, school and related problems, various teaching learning activities, examination of
materials and written sources. ‘

ELE 224 Instructional Planning and Evaluation (3-2)4

Basic concepts in curriculum development and its processes; development of lesson plan, unit plan, annual
and daily plan; selection of content and organization. Teaching methods and strategies; properties of
materials and their selection; measurement and evaluation; approaches in evaluation; types of tests;
developing achievement tests.

ELE 240 Probability and Statistics (2-2) 3

Data collection, sampling, sampling distribution and estimation, permutation and combination, probability,
continuous random variables and distribution, estimated value, data analysis, hypotesis testing, Chi-square
test, regression and correlation.

ELE 300 Computer Applications in Education (2-2)3

Develop proﬁciency with networked workstations and computer applications. Provides experience with
word processors, spreadsheets, presentation software and the Internet. Focuses on the principles and
speciﬁc ideas for appropriate, responsible, and ethical computer use. Promotes self-conﬁdence for on-going
professional development.

ELE 331 Laboratory Applications in Science I (2-2)3

Laboratory experiments in science education. Improvement of skills in setting up experiments in science
courses for elementary schools through 6-8.

ELE 332 Laboratory Applications in Science I] (2-2)3

Continuation of ELE 331

ELE 317 Instructional Development and Media in Math Education (3—2)4

Characteristics of various instructional technologies, the place and use of technologies in instructional
process, development of teaching materials through instructional technologies (worksheets, transparencies,
slides, videotapes, computer-based course materials, etc.), and assessment of the qualities of various
teaching materials.

ELE 336 Methods of Science and Mathematics Teaching (3-2)4

Concepts of methods and teaching strategies in elementary education. Different methods of instruction,
such as expository, inquiry, discovery, demonstration, discussion, problem solving and cooperative
learning as applied to the teaching and learning process in teaching science and mathematics at elementary
schools.

ELE 420 Practice Teaching in Elementary Education (2-6)5

Field experience and teaching practice (minimum 12 weeks) including class observation, adaptation to
classroom condition, planning and preparation for teaching. Guided teaching practice in science in
elementary schools

ELE 437 School Experience 11 (1-4)3

Continuation of ELE 132.

ELE 443 Methods of Mathematics Teaching (2-2)3

Concepts of methods and teaching strategies in elementary education. Different methods of instruction,
such as expository, inquiry, discovery, demonstration, discussion, problem solving and cooperative
learning as applied to the teaching and learning process in teaching mathematics at elementary schools.
ELE 448 Textbook Analysis in Mathematics Education (2-2)3

Examination of subject matter course books and educational programs, approved by Ministry of Education,
in terms of student's level, content, language, format, and contribution to meaningful learning.

209

 

Syllabuses for the Courses Offered by the Department of Elementary Education
ELE 132 School Experience I

Course Description

The school experience gives time for student to assimilate their experience to relate them to the
work being done at the Faculty, and to discuss them with staff and other students. Therefore, it provides
opportunities for an increase in professional competence.

General Rules

0 4 hours attendance to your cooperating school (total 40 hrs)
0 Doing the activities asked by teachers.

0 Attendance to the one hour course in the campus

0 Complete required assignments and reports on time.

Late Assignments

I expect that assignments will be turned in by the announced due dates. I will accept assignments
after the due date, but your grade will decrease by 10% of the allocated points for each calendar day the
assignment is late.

Assignments must be submitted both in printed (hard copy) and electronic (soft copy) form. To
send assignments in electronic form use following e-mail address: elel32@gmail.com
Academic Ethics:

All assignments you hand in should be the result of your effort only. Academic dishonesty,
including any form of cheating and plagiarism will not be tolerated and will result in failure of the course
and/or formal disciplinary proceedings usually resulting in suspension or dismissal. Cheating includes but
is not limited to such acts as; offering or receiving unperrnitted assistance in the exams, using any type of
unauthorized written material during the exams, handing in any part or all of someone else’s work as your
own, copying from the Internet. Plagiarism is a speciﬁc form of cheating. It means using someone else’s
work without giving credit. Plagiarism is a literary theft. Therefore, you have to acknowledge the sources
you use in your assignments.

Grading

Grading will be based on the projects and assignments. The list of the major assignments is given

below. The details of each project/assignment will be provided to you during the semester.

Activity Percent

Report 1 20
Report 2 25
Report 3 20
Report 4 20

J oumal Writing 10
Field Notes & Reﬂection 5

Course Requirements

Attendance: Full attendance is expected to all required classroom visits. The purpose for ﬁeld experience is
to give you an opportunity to observe and reﬂect upon teaching. Successful completion of ELE 132
includes completing a minimum of 4 hours of classroom observation/participation per week.

Field Notes: You are expected to keep a notebook that includes your ﬁeld notes. Your reports should be
based on your ﬁeld notes. Each week, you should show your ﬁeld notes to one of the instructors. At the end
of the semester, you are expected to submit your notebooks.

Journal writing: You should maintain journals of your day to day experiences. The purpose of the journal
is to allow you to reﬂect upon and synthesize your learning experience and your attitudes, keeping in mind
the concepts of “writing for learning”. The following questions may guide you: (Maximum 2 pages—double
spaced, [2 pts; total 5 journals):

What did you observe? (A general description of your observations)

What was the most striking/ interesting classroom event that you observed?

What confused you?

What was new learning to you? / What discoveries are you making?

210

 

o What lessons did you learn from this experience as a prospective teacher?
The due date will be announced. Late reports will not be accepted.
Reﬂection Paper: Based on your observations throughout the semester, prepare a reﬂection paper. This
should be a two to four page synopsis of the overall experience that includes the following three elements:
0 Fully describe one speciﬁc meaningful experience and reﬂect on it’s impact related to
your growth as a teacher in training.
o How has this experience effected your decision to pursue a career as a teacher? Include
information related to working with this grade level.
0 Indicate the grade you feel you earned for this course and why.
Reports: During the semester, you are expected to write 4 different reports. The content and the format of
these report will be explained.

Required Reports for ELE 131
Report 1: Planning for Mathematics Instruction

This report should include a description of how the curriculum is implemented in this school. The

followings are some of the issues that you may discuss in your reports.

0 How school and teacher plan the mathematics instruction? What resources do they use in planning
and preparing the instruction? Are there any additions or revisions that school made in the
curriculum (if yes , what are they, and why they are doing it)? What are the formats for daily and
yearly plans?

0 What materials are available for mathematics instruction (computers, laboratories, library, and
others)? How frequently are they used? For what purpose are they used? How are they used (a
description of the procedure)?

0 What are the major characteristics of the textbooks? How textbooks are selected? What is the
procedure they use in selecting textbooks?

0 Is there any mathematics department in your school? How does it work? What are the teachers’
responsibilities in the department? How they make decisions in the department meetings? etc.

NOTE: Your report should not be limited to the questions given here. You should try to expand these
issues.

Report 2: Mathematics Lessons

This is a report about mathematics lessons. The report should describe the conduct of mathematics
lessons, including a description of the kind of work done, interactions, assessment procedures, habits, and
such major characteristics of a mathematics lesson. The followings are some of the issues that you may
discuss in your reports.

0 How teaching takes place? What is/are the method(s) teacher use in mathematics lessons? How
s/he uses these methods (a description of the procedure)? Are there any traditions/habits of teacher
in mathematics lessons? How does s/he assess student learning? Are there any speciﬁc materials
that teacher use (overhead projector, worksheets)? etc.

0 How learning takes place? What are the expectations from students during, before, and aﬁer the
lessons? What kind of interaction exists among students and between students and the teacher
during math lessons? Are there any common learning problems?

0 How the classroom management handled? What speciﬁc methods does teacher use to handle the
misbehavior? How does s/he manage the orderliness of the lesson during teaching?

NOTE: Your report should not be limited to the questions given here. You should try to expand these
issues.

Report 3: Other Aspects of School
In this report you need to write about the other activities of the school, such as general rules. services,
clubs, sports, and etc.
0 A description of school family cooperation
Student clubs and some of their typical activities
Counseling service and other services
The rules of the school
Infrastructure

211

0 Speciﬁc and unique aspect of the school.
NOTE: Your report should not be limited to the questions given here. You should try to expand these
issues.

Report 4: Mathematics Teacher Interview Report
In this assignment you will interview your mentor teacher. The focus of this interview is to learn

about duties and responsibilities of a mathematics teacher related to teaching and other school tasks.
The following are some of the questions you should ask. You may also add your own questions to the
question list.

0 Why did you select teaching as a career?
What are your teaching responsibilities in this school?
What are the responsibilities towards mathematics department?
What are the responsibilities during the duty on?
What is a typical day for you? Can you describe your daily routine tasks?
What are the difﬁculties of being a mathematics teacher?
What is most exciting about teaching mathematics? .
What kind of roles does teacher have in extra curricular mathematics activities?
What kinds of resources are available for you (both within the school and outside the school)?
What do I most need to know to be an effective classroom teacher?
What do you consider two of the most important characteristics to be a competent teacher?
NOTE: Your report should not be limited to the questions given here. You should try to expand these
issues.

The procedures you should follow during interviews:

Study your questions before you ask to the teacher. Try to anticipate teacher responses and ask
more questions to reveal more information. For instance, if the teacher says “teaching mathematics is
stressful” and doesn’t say anymore, ask to him/her “what makes mathematics teaching stressful?

You need to carefully listen to the teacher’s answers and take notes to your notebooks. During the
interview if you need more time to write down your notes, ask the teacher to wait for a while. It can be also
helpful to you, if you practice an interview with one of your friends.

Report Format
Reports should be written by using MS-Word and submitted in both printed and electronic form.
At the end of the semester you will have 4 reports. The following plan should be followed in your report, in
the given order:
1. Cover page: See sample cover page.
2. Introduction
0 Write about the general purpose of the report.
0 Indicate speciﬁc questions that you are going to address in this report (if applicable)
0 Explain the importance/value of the issue(s).
Include information about how you collected information about this report in detail.
Where did you get the information?
How did you get it?
How long it took to get the information and when did you collect data?
0 Other relevant information related to your data/information collection procedures.
3. Description and Interpretation of collected information/data
0 Include a description of what you observed/obtained from the school about this report. For each
peace of information you present, include your interpretations. In this part you may use sub titles
related to the topic of your report.
4. Discussion, conclusion, and reﬂection
0 Discuss each information/data you obtained. Can you generalize your ﬁndings? Why? Why not?
Reﬂect on how your ﬁndings inﬂuenced your thinking about being a teacher.

212

ELE 224 Instructional Planning and Evaluation

Goals
The purpose of this course is to study on the issues around planning for instruction and assessment

in the elementary school mathematics and science. The course content includes a review of broader
perspectives about curriculum development. Upon the completion of this course, we expect that our
students ,

0 Understand the basic concepts related to curriculum.

0 Understand instructional objectives at various levels of cognitive, affective, and psychomotor domains.

0 Have a critical perspective on the curricular experiences in Turkey.

0 Develop plans for instruction (lesson plans and unit plans).

0 Understand basic concepts related to assessment and measurement.

0 Understand the issues that make a well designed instrument for classroom evaluation.

0 Develop various assessment materials for the classroom use.
Grading and Assignments
Mid-term exam: There will be one mid-term examination. The exam will cover information presented in
class and in the readings.
Final Exam: There will be a comprehensive ﬁnal exam. The ﬁnal will include questions that are covered
during the entire semester.
Project: Details of the project will be announced during the semester.
Presentation of Teaching Methods and Summary paper: You will be asked to make presentation about
the assigned teaching methods in groups of 3 or 4. The presentations will be about 10 -15 minutes. You will
also be asked to prepare a “summary paper” that outlines the major characteristics of teaching methods.
Quizzes: There will be two quizzes during the semester. Date of each quiz will be announced in advance
(a week before the date).

Mid-term 25%
Project 1 5%
Presentation 1 0%
Quizzes l 0%
Final 40%

Academic Ethics:

All assignments you hand in should be the result of your effort only. Academic dishonesty,
including any form of cheating and plagiarism will not be tolerated and will result in failure of the course
and/or formal disciplinary proceedings usually resulting in suspension or dismissal. Cheating includes but
is not limited to such acts as; offering or receiving unpermitted assistance in the exams, using any type of
unauthorized written material during the exams, handing in any part or all of someone else’s work as your
own, copying from the Internet. Plagiarism is a speciﬁc form of cheating. It means using someone else’s
work without giving credit. Plagiarism is a literary theft. Therefore, you have to acknowledge the sources
you use in your assignments.

Required Texts:
Marsch, J. C. & Willis, G. (1999). Curriculum: Alternative Approaches, Ongoing Issues. Upper Saddle

River, NJ: Merrill (Chapter 1 & 2).

Clark, L. H. & Starr, I. S. (1996). Secondary and Middle School Teaching Methods. New Jersey: Prentice

Hall (Chapter 5 & 6).
look, K. (2001), Instructional Design for Classroom Teaching and Learning. Boston: Houghton Mifﬂing

Company. (Chapter 4 & 5).

Linn, R. L. & Gronlund, N. E. (2000). Measurement and Assessment in Teaching. New Jersey: Prentice

Hall (Chapter 4 & 5).

213

 

Tentative schedule

Weeks
Week 1
Week 2

Week 3

Week 4

Week 5

Week 6

Week 7

Week 8
Week 9

Week 10

Week 1 1

Week 12

Week 13

Week 14

Introduction, overview.

Basic concepts of curriculum
Reading I: Marsch & Willis (I999), Chapter 1

Curriculum development process
Reading 2: Marsch & Willis (I 999), Chapter 2

Planning for teaching
Reading 3a: Clark & Starr (I 996), Chapter 5
Reading 3b: Callahan, Clark & Kellough (p. 109-123)

Planning for teaching (continued)

Course unit and lesson planning
Reading 4a: Clark & Starr (1996), Chapter 6
Reading 4b: Tebligler dergisi (in Turkish)

Teaching methods
Reading: Handouts

Student presentations about assigned teaching methods

Principles of classroom assessment
Reading 5 : Zook (200]). Chapter 4

Issues of validity and reliability
Reading 6 & 7: Linn & Gronlund (2000), Chapter 4 & 5

Issues of validity and reliability
Reading 6 & 7: Linn & Gronlund (2000)

Classroom Assessment Strategies
Reading 8: Zook (200]), Chapter 5

Alternative Assessment
Reading: Handouts

Wrap Up

214

 

ELE 317 Instructional Development and Media in Mathematics Education

Course Goals
Upon the completion of this course I expect the students to be able to:
- Engage in the preliminary steps of instructional material development using computer applications.
0 Use basic tools for creating computer-mediated learning modules and be conﬁdent in using them.
0 Use effective message design in the creation of instructional materials. In addition, you are expected to
show sensitivity to ethical issues and concerns.
Method of Instruction

ELE 329 is a lab-based course. The role of the instructor is to guide students through experiences
that allow you to complete the project assignments of the course, and to act as a facilitator and supporter of
your learning. Your success will be directly related to how much time and energy you put into the class,
and your willingness to try new things. Do not expect things to run smoothly: the computers will crash,
ﬁles will be corrupted, and other bad things will happen. Dealing with these situations is part of the class,
and your ability to deal with them will impact your success in the class.

You are expected to take responsibility for your own learning and be willing to experiment with
new technologies; to work with your classmates and the instructor to create a positive learning
environment; to study assigned readings prior to each class session; to participate in class discussions and
learning opportunities; and to complete all assignments on time and with the highest quality possible.
Content Outline

Media, Technology, and Learning

Visual Principles

Technologies for Learning

Systematic Planning for Media Use

Media and Materials

Visuals

Internet and WW
Attendance and Participation

You are expected to demonstrate energetic, high quality performance in all aspects of this course.
Regular attendance is expected and required in the laboratory sessions. Your attendance and participation in
laboratory sessions will be reﬂected in your lab grades.

Late Assignments

I expect that assignments will be turned in by the announced due dates. I will accept assignments
after the due date, but your grade will decrease by 10% of the allocated points for each calendar day the
assignment is late.

Grading

Grading will be based on the projects, assignments, and a midterm examination. The list of the
major assignments is given below. The details of each project/assignment will be provided to you during
the semester.

%

Midterm 30

Poster 1 5

Spreadsheet Project 15
Web Site 20

Reﬂection Paper 5
Lab work (attendance, assignments, etc) 15

215

Week of
Sept, 28

Oct, 5

Oct, 12
Oct, 19

Oct, 26

Nov, 2
Nov, 9

Nov, 16
Nov, 23

Nov, 30
Dec, 7
Dec, 14

Dec, 21
Dec, 28

Topic and Assignments

Course introduction and overview of the objectives.
Media, Technology, and Learning

Reading 1: Heinich (2002), Chapter 1

Visual Principles

Reading 2: Heinich (2002), Chapter 5

Poster design project introduced

Technologies for Learning .

Reading 3: Heinich (2002), Chapter 2

Poster project due

Learning from this project - An open show of posters
Systematic Planning for Media Use

Reading 4: Heinich (2002), Chapter 3

Spreadsheet project introduced

Media and Materials

Reading 5: Heinich (2002), Chapter 4

About the use of spreadsheets in mathematics education.
Midterm will be done in this week. Date will be announced
Holiday

Learning from this project - Presentation of selected projects.
Spreadsheet project due

Visuals

Reading 5: Heinich (2002), Chapter 6

About Internet and WW.

WWW project introduced

Creating HTML documents and publishing in METU servers.
Evaluating content on WWW.

Reﬂections about experiences.

216

ELE 336Methods of Math and Science Teaching

Course Description:
This course is aimed to have preservice teachers develop skills in methods of teaching science
mathematics to 4"'-8‘h grade level students. It focuses on the issues around what can be done to help
young learners understand the science and math concepts. There will be an emphasis on critical
discussion and applications of each of the teaching strategy that are covered.

Course Objectives:_At the end of the semester students should able to;

be aware of speciﬁc teaching methods that can be used in 6-8 mathematics and science instruction.

critically review a given teaching strategy.

engage students in a discourse to help them understand mathematics and science ideas.

select appropriate teaching approach to design an effective learning environment.

communicate about teaching and learning and work cooperatively on projects related to teaching.

Be self-conﬁdent in mathematics/science teaching methods.

Enjoy teaching mathematics/science.

Course Requirements

' Attendance: The nature of the class activities and course objectives make attendance important. For that

reason, attendance is required. Please notify me in case of illness or emergency.

Presentations: Presentations are required. Presentation dates will be assigned. The presentation reports

will also be handed in two weeks after the presentation.

Portfolio: Each group should prepare a portfolio in which all your activities and other group activities will

SQMPPN.“

be kept.
Grading
1. Participation and Attendance: 10%
2. Presentations: 5%
3. Project Report: 20% (10 August, 2005)
4. Midterm Exam: 30%
5. Final Exam: 35%
6. Portfolio: 5% (10 August, 2005)
Topics
1. First Meeting. Information about the course; Video

Factors Inﬂuence how we teach. Being an effective teacher.
2. Personal reﬂections on being an effective teacher.
Communicating about lessons: Planning, goals, and objectives

3. Cooperative Learning

4. Questioning /Discussion

5. Direct Instruction Strategies/Demonstration [Mathematics Laboratory
6. Drama

7. Discovery Learning/Inquiry

8. Problem-Solving

9. Problem Based Instruction/Project Based Learning

10. Concept Mapping

217

References:

Ajewole, GA. (1991) “Effects of discovery and expository instructional methods on the attitude of students
to biology”. J. of Research in Science Teaching, 28, 401-409.

Anderson, L.W. (1989). The Eﬂective Teacher, USA: McGraw-Hill, Inc. (LB1025.2 A6955)
Arends, RI. (1998). Learning to Teach. Boston: McGraw Hill (LB1025.2 .A773)

Bednarz, N., Kieran, C. and Lee, L. (Eds.) (1996). Approaches to Algebra: Perspectives for Research and
Teaching. Dodrecht: Kluwer Academic Publishers.

Bell, A. (Ed.)( 1997). Matematik Ogretimi (Ortaogretim). YOK/Dunya Bankasi Milli Egitirrri Gelistirme
Projesi: Hizmetoncesi Ogretrnen Egitimi.

Brock, RA. (1994). Educational Technology in the Classroom. Englewood Cliffs, NJ: Educational
Technology Publications, Inc.

Busbridge, J. & Ozcelik, DA. (1997). Ilkogretim Matematik Ogretimi. Ankara: YOK/Dunya Bankasi Milli
Egitimi Gelistirme Projesi: Hizmetoncesi Ogretrnen Egitimi.

Carin, A. A.and Sund, R.B. ( 1980). Teaching Modern Science. Columbus, Ohio: Charles E. Merrill Pub.
Co. (LB1585 C27)

Collette, A.T. (1973). Science Teaching in the Secondary School: A guide for Modernizing Instruction.
Boston: Allyn and Bacon. (Q1833 Al C64)

Colletti, AB. (1987). Teaching Methods and Applied Techniques, USA: Keystone Publications, Inc.
(LB1025.2 C38)

Cooney, T.J., Davis, E.J. & Henderson, KB. (1975). Dynamics of Teaching Secondary School
Mathematics. Atlanta, GA: Houghton Mifﬂin.

Cooper, J .M. (1994). Classroom Teaching Skills. Toronto, D.C: Heath and Company. (L81025.3 C57)
Cruicksahnk, DR. (1995). The Act of Teaching. New York: McGraw-Hill.

Dembo, M. H. (1988). Applying Educational Psychology in the Classroom. New York: Longman (LBlOSl
D37).

Edwards, EL. (1990). Algebra for Everyone. Reston, VA.: NCT M, Inc.

Froleich, G.W. (Ed.)(l995). Connecting Mathematics. Reston, VA.: NCTM, Inc.

Gilbert, H. (1950). The Art of Teaching. (LBlOZS H63)

Hirsch, C.R. & Laing, R.A. (Eds) (1993). Activities for Learning and Teaching. Reston, VA.: NCTM, Inc.
Hirsch, C.R. (Ed.) (1992). Data Analysis and Statistics. Reston, VA.: NCTM, Inc.

Jacobsen, D., Eggen, P., Kauchak, D., & Dulaney, C. (1985). Methods for Teaching: A Skill Approach.
Columbus : C.E. Merrill Pub. Co. (LB1025.2 M4478)

Johnson, RT. (1976) “Relationship between inquiry and cooperative learning structure " J. of Research in
Science Teaching I 3, 55-63.

Joyce, B. & Weil, M. (1996). Models of Teaching. Boston: A Simon & Schuster Company. (LB1027.3
J69)

Leonard, W.H. (1983) “An experimental study of a BSCS-style laboratory for university general biology”.
J. of Research in Science Teaching, 20, 807-813.

Leonard, W.H. ( 1983). An experimental study of a BSCS-style laboratory for university general biology.
Journal of Research in Science Teaching, 20, 807-813.

Linden, DD (1990). Teaching: A introduction. New York : McGraw-Hill (L81025.2 D857)
McKeachie, W.J. (1994) “Teaching Tips”,9th ed., Canada: D.C. Heath and Company.
Milli Egitim Bakanligi. (1995). Ilkogretim Okulu Programi. Ankara: Author.

Mintzes,J.J., Wandersee, J .H., & Novak, J .D. (Eds) (1998). Teaching Science for Understanding. San
Diego, CA: Academic Press. (LB1585 T438)

Moore, K.D. (1992). Classroom Teaching Skills. USA: McGraw-Hill, Inc. (LBlOZS.3 M66)

National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School
Mathematics. Reston, VA.: NCT M, Inc.

218

 

Olaitan, 8.0. (1981). Principles of Practice Teaching. Toronto: John Wiley & Sons.
Olaitan,S.O.,(l981) Principles of Practice Teaching, John Wiley & Sons, Toronto.

Orlich, DC. (1990). Teaching Strategies: A Guide to Better Instruction. Lexington: D. C. Heath and
Company, (L81025.2 T253)

Orton, A. & Frobisher, L. (1996). Introduction to Education: Insights into Teaching Mathematics.
LondonzCassel

Popham, W.J. and Baker, EL. (1970). Systemic Instruction. Englewood Cliffs, NJ: Prentice-Hall.
(LBlOZS.2 P63)

Posamentier, A.S., & Stepelman, J. (1990). Teaching Secondary School Mathematics: Techniques and
Enrichment Units. New York, NY: MacMillan Company.

Posner, G.J., Strike, K.A., Hanson, P.W., & Gertzog, W.A. (1982). Accommodation of a scientiﬁc
conception: Toward a theory of conceptual change. Science Education, 66: 211-227.

Reys, B.J. (Ed.) (1996). Developing Number Sense. Reston, VA.: NCT M, Inc.
Rose, HQ (1961). The Instructor and His Job. Chicago: American Technical Society, (LBlOZS R795)

Ryan, K. & Cooper, J.M. (1980). Those Who Can Teach. USA: Houghton Mufﬂin Company (LB1025.2
R9) -

Sharp, V. (1996). Computer Education for Teachers. Madison, erBrown and Benchmark Publishers.
Shoen, L.H. (1986). Estimation and Mental Computation Year Book. Reston, VA.: NCTM, Inc.

Shymansky, J ., Kyle, W.C., & Alport, A. (1983). A meta analysis of discovery and traditional instructional
on attitude to science. Journal of Research in Science Teaching, 20, 387-404.

Sobel, M.A0 & Maletsky, EM. (1991). Teaching Mathematics: Sourcebook of A ids, Activities, and
Strategies. Boston, MA: Allyn and Bacon.

Sund, R. B. and Trowbridge, L.W. ( 1973). Teaching Science by Inquiry in the Secondary School.
Columbus: Ohio, Merrill (Ql81 885 1973)

Tamir, P. (1983). Inquiry and the Science Teacher. Science Education, 6 7, 657-672.

Trowbridge, L. W. & Rodger, B. (1996). Teaching Secondary School Science Strategies for Developing
Scientiﬁc Literacy. Englewood Cliffs, NJ: Merrill.

Trowbridge, L. W., and Bybee, R. W. (1986). Becoming a Secondary School Science Teacher. 4th ed.,
Columbus : Merrill Pub. Co. (0183.3 A1 T76)

Walter, M.I. ( 1970). Boxes, Squares and Other Things. Reston, VA.: NCTM, Inc.

Washton, NS. (1961). Science Teaching in the Secondary School. New York: Harper and Brothers. (Q1 71
W27)

Wolﬁnger, D.M. (1984). Teaching Science in the Elementary School: Content, Process. and Attitude.
Boston: Little, Brown (LB1585 W63)

Zawojewski, J .8. (Ed.) (1997). Dealing with Data and Chance. Reston, VA.: NCTM, Inc.

219

 

ELE 420 Practice Teaching in Elementary Education

Course Description

Field experience and teaching practice including class observation, adaptation to classroom
condition, planning and preparation for teaching. Guided teaching practice in science in elementary
schools.

Course Objectives

Practice teaching is a means of providing opportunities for student teachers, under typical
conditions in selected cooperating schools, to obtain experience in observing and participating actively in
all the diverse educational activities in the school.

At the end of the course students should be able to:

0 Demonstrate knowledge regarding different techniques of teaching mathematics.

0 Develop and implement mathematics lessons for the elementary school students and be familiar with
classroom management techniques.

0 Select and use appropriate instructional strategies and equipment.

0 Design and implement activities which promote the development of concepts and problem solving
skills in mathematics, as well as promote positive attitude toward mathematics.

0 Understand how elementary school students learn mathematics.

0 Be aware of speciﬁc mathematics topics taught in each of the grades 6-8 and know where to gather
resources to aid in the teaching of those topics.

0 Be familiar with how to assess progress of elementary school students who are learning mathematics
and be able to adjust instruction for students with special needs.

0 Use different technological tools to develop elementary school students' understanding of
mathematics concepts.

Online Components

Online components of this course can be accessed from the following address:

httpsz/Ionline.metu.edu.tr/

Log in using your METU Id and passwords. Please do not forget to update your proﬁles (esp. your
e-mail addresses).

Required Texts:

None required.
Academic Ethics:

All assignments you hand in should be the result of your effort only. Academic dishonesty,
including any form of cheating and plagiarism will not be tolerated and will result in failure of the course
and/or formal disciplinary proceedings usually resulting in suspension or dismissal. Cheating includes but
is not limited to such acts as: offering or receiving unpermitted assistance in the exams, using any type of
unauthorized written material during the exams, banding in any part or all of someone else’s work as your
own, copying from the Intemet. Plagiarism is a speciﬁc form of cheating. It means using someone else’s
work without giving credit. Plagiarism is a literary theft. Therefore, you have to acknowledge the sources
you use in your assignments.

Course Requirements

Attendance: Full attendance is expected to all required classroom visits. The purpose for ﬁeld experience is
to give you an opportunity to observe and reﬂect upon teaching. Successful completion of ELE 420
includes completing a minimum of 4 hours of classroom observation/participation per week.

Lesson Plans: You will prepare lesson plans for 4 different teaching practices you will conduct. A lesson
plan format will be introduced to you during semester.

Campus Lesson: You are required to conduct one lesson in Campus sessions. For this lesson you are
expected to submit a lesson plan and a seIf-critique paper.

Teaching in School: You are required to plan, implement and reﬂect on two lessons in school. Only one of
these lessons will be observed and assessed by your instructor. For each lesson, you are expected to prepare
lesson plans and write a self-critique about your teaching. These lessons should involve learning activities
based on “active learning.”

Self Critique Paper: For each of the lesson you conduct, you will write a self-critique that will describe
your opinions about your teaching performance. Self critique should be submitted in electronic form
through METU-Online (2 pages double spaced).

220

Self Improvement Paper: I expect you to identify a problem you want to solve in your own teaching
behaviors. Then, prepare a document that explains your plan about your self improvement on this problem.
Your self improvement paper should include a clear description of the problem that you identiﬁed. Then
include an analysis of this problem I expect you to read literature related to this problem. Then write a plan
of self-improvement for this problem by citing suitable references. (4 pages double spaced)
Your Teaching Philos0phy: You are required to write a paper about your teaching philosophy. Based on
your previous experiences, you are expected to describe your understanding of a “good teacher” and “good
teaching.” As you already know, the teaching phenomenon has various dimensions. In this assignment I
expect you to comment about three basic dimensions: (a) classroom management, (b) interaction with
students, and (c) teaching of mathematics. You should explain your own opinions and what you write
should reﬂect your personal teaching philosophy (the teacher that you want to be). (4 pages double spaced)
Discussion Forum Participation: I expect you to participate in an online Discussion Forum at
(online.metu.edu.tr). The discussion topics may be based on your observations in the schools. Your
participation in discussion forums will be assessed based on the following criteria. The assessment will be
done 4 times during the semester.

0 Write substantive posts and present new ideas on a regular basis.

0 Read others’ postings and make comment about them or respond on them (Interact with a variety of

participants.)

_ 0 You should present your own ideas.

0 Your writings should be clear and free of mechanical errors (complete sentences, well organized,

grammatically correct, and free of spelling errors)

Grading

Activity Percent
Three Lesson Plans ................................ 15
Teaching in School ................................ 20
Campus Teaching ................................. 15
Three Self Critique Papers ..................... 15
Self Improvement Paper ........................ 15
Discussion Forum Participation .............. 5
Your Teaching Philosophy .................... 15

221

 

ELE 443 Methods of Mathematics Teaching

Course Description: Misconceptions/errors in selected mathematical concepts; Teaching these concepts
by emphasizing on active learning.

Course Objectives: At the end of the semester, students should be able to

To understand how to use computers and calculators in mathematics course.

To teach mathematics by using corrrputers and calculators.

To understand the misconceptions on mathematical concepts in Numbers/ Algebra! Geometry/
Probability/Statistics.

To understand the errors on mathematical concepts in Numbers/ Algebra/ Geometry / Probability/Statistics.
To apply the teaching methods to teach Numbers/ Algebra/ Geometry / Probability/Statistics.

To evaluate the effectiveness of their own teaching Itheir classmates teaching.

To be self-conﬁdent in teaching mathematics

To have positive attitude toward teaching mathematics.

Grading:
% 30 Midterm % 15 Group Presentation/ Group Project % 35 Final
% 10 Attendance and Participation %10 Poster % 5 Portfolio
References:
Books:

Bednarz, N., Kieran, C. ve Lee, L. (Eds.) (1996). Approaches to Algebra: Perspectives for Reserach and
Teaching. Dodrecht: Kluwer Academic Publishers.

Bell, A. (Ed.)(l997). Matematik Ogretimi (Ortaogretim). YOK/Dunya Bankasi Milli Egitimi Gelistirme
Projesi: Hizmetoncesi Ogretrnen Egitimi.

Brock, RA. (1994). Educational Technology in the Classroom. Englewood Cliffs, NJ: Educational
Technology Publications, Inc.

Busbridge, J. and Ozcelik, DA. (1997). Ilkogretim Matematik Ogretimi. Ankara: YOKfDunya Bankasi
Milli Egitimi Gelistirme Projesi: Hizmetoncesi Ogretmen Egitimi.

Cooney, T.J., Davis, E.J. & Henderson, KB. (1975). Dynamics of Teaching Secondary School
Mathematics. Atlanta, GA: Houghton Mifﬂin.

Cruicksahnk, DR. (1995). The act of Teaching. New York: McGraw-Hill.

Edwards, EL. (1990). Algebra for Everyone. Reston, VA.: NCT M, Inc.

Froleich, G.W. (Ed.)(l995). Connecting Mathematics. Reston, VA.: NCTM, Inc.

Hirsch, C.R. & Laing, R.A. (Eds.)( 1993). Activities for Learning and Teaching. Reston, VA.: NCT M, Inc.

Hirsch, C.R: (Ed.) (1992). Data Analysis and Statistics. Reston, VA.: NCT M, Inc.

Jacobsen, D. eta] (1985). Methods for Teaching: A Skill Approach. Columbus: Merrill Pub. Co.

Milli Egitim Bakanligi. (1995). Ilkogretim Okulu Programi. Ankara: Author.

National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School
Mathematics. Reston, VA.: NCTM, Inc.

Orton, A. & Frobisher, L. (1996). Introduction to Education: Insights into Teaching Mathematics.
LondonzCassel

Posamentier, A.S., & Stepelman, J. (1990). Teaching Secondary School Mathematics: Techniques and
Enrichment Units. New York, NY: MacMillan Company.

Reys, B.J. (Ed.) (1996). Developing Number Sense. Reston, VA.: NCT M, Inc.

Sharp, V. (1996). Computer Education for Teachers (2“d Ed.). Madison, leBrown and Benchmark
Publishers.

Shoen, L.H. (1986). Estimation and Mental Computation Year Book. Reston, VA.: NCTM, Inc.

Sobel, M.A0 & Maletsky, EM. (1991). Teaching Mathematics: Sourcebook of Aids, Activities, and
Strategies (2"d Ed.). Boston, MA: Allyn and Bacon.

Walter, MI. (1970). Boxes, Squares and Other Things. Reston, VA.: NCT M, Inc.

Zawojewski, J .S. (Ed.) (1997). Dealing with Data'and Chance. Reston, VA.: NCT M, Inc.

Thesis:

Bulut, S. (1994). The Effects of Diﬂerent Teaching Methods and Gender on Probability Achievement

and Attitudes Toward Probability", Unpublished Ph.D. Thesis, Middle East Technical
University, Ankara, Turkey.

222

 

Cankoy, O. (1989). Diﬁ'erence Between Traditional Method and Mathematics Laboratory Instruction
in terms of Achievement Related to A Probability Unit. Unpublished Master’s Thesis, Middle
East Technical University, Ankara, Turkey.

Cankoy, O. (1998). Determining and Overcoming Preservice Elementary Teachers’ Misconceptions
in Interpreting and Applying Decimals. Unpublished Ph.D., Middle East Technical University,
Ankara, Turkey.

Erbas, KB. (1999). An Investigation Students’ Performances Difficulties and Misconceptions in
Elementary Algebra. Unpublished Master’s Thesis, Middle East Technical University, Ankara,
Turkey.

Iseri, A.I. (1997). Diagnosis of Students' Misconceptions in Interpreting and applying Decimals.
Unpublished Master’s Thesis, Middle East Technical University, Ankara, Turkey.

Koc, Y. (1998). The Effect of Cooperative Learning Method on Students’ Mathematical Problem
Solving Skills. Unpublished Master’s Thesis, Middle East Technical University, Ankara,
Turkey.

Journals:

0 Arithmetic Teacher 0 Journal of Educational Psychology

0 Educational Studies in Mathematics 0 Journal of Experimental Education

0 Educational Technology 0 Journal for Research in Mathematics Education

0 International Journal of Mathematics Education in 0 Journal of Mathematical Behavior

Science and Technology
0 Journal of Computers in Mathematics and Science
Teaching

Mathematics Teacher

0 School Science and Mathematics

Content Outline
1. Review the teaching methods (i.e. Cooperative learning, questioning, discussion, discovery
learning & problem solving methods). Use of Calculators and Computer Applications in Mathematics
Education Watching Video Cassettes on Teaching Mathematics (1.-2 Weeks)

2. Teaching of Mathematics, Misconceptions/Errors on Selected Mathematical Concepts (3.-14.
Weeks)
3.1. Numbers (3. - 5. Weeks)
3.2. Algebra (6.-7. Weeks)
3.3. Geometry (8.-10. Weeks)
3.4. Probability (1 l.-l3. Weeks)
3.5. Statistics (14. Week)
4. Poster Presentation
Project:
The project will be on teaching of given subjects. It should cover the followings:

Content

Objectives

Prerequisite knowledge/skills

Terminology

Historical background

Misconceptions

Real life, occupations, other sciences

Teaching of mathematical concepts, by using games/puzzles, text, activity/worksheets, materials and
by giving directions to teachers to use this teaching materials (Each group member should develop 3
activity sheets)

223

APPENDIX C

INTERVIEW PROTOCOLS OF THE STUDY

2004 Interview Protocol for Preservice Teachers

Hi, .

I am Cigdem Haser, and I am a doctoral student in Department of Teacher Education at Michigan
State University in US. I am conducting a study about preservice teachers’ perceptions of mathematical
understanding and various factors that affect their perceptions. This interview that you volunteered is about
my study. Having been volunteered is not a strict condition, and you can quit the interview at any time you
want to.

The questions that I am going to ask you are not a part of your program. The answers that you
provide will not be shared by any program member, and will not have an impact on your course grade. I
will be the only person who has the access to the answers that you presented. In case of using your answer
in any publication, I will not mention your name. However, I may use the demographic data that you
provided in the instruments that you answered before.

There are 29 main questions in this interview and there are no true answers to the questions that I
am going to ask. I am deﬁnitely interested in your own ideas and feelings about issues in mathematical
understanding and mathematics teaching and learning. Thus, I don’t want you to feel stressed about
answering my questions.

I want to audiotape our interview if you allow me. If you do not want this interview to be
audiotaped, please feel free to express. If you want any part of the interview to be audiotaped, please feel
free to express. I respect your privacy and I will turn off the recorder. Although I informed you about this
before (in our e-mail or phone communication), I just want to be sure if you know that this interview is
likely take one to two hours. Have you organized your time for this interview?

Are there any questions that you want me to answer about the interview before we start? Can we
start now? Please tell me if you need a break at any time of the interview.

Thank you.

Interview Questions
Nature of Mathematics
1. What is mathematics for?
2. What does it mean to know mathematics?
3. What does it mean to learn mathematics?
0 How is mathematics learned?
0 How did you learn mathematics?
0 How do you think a student should study mathematics in order to learn it?
4. Can you tell me what a “mathematical concept” is depending on your own ideas?
0 What is a “mathematical concept” for you?
0 Can you given an example of a question that asks a “mathematical concept”?
How do you know that a student has understood a mathematical concept?
6. Can you tell me what a “mathematical procedure” is depending on your own ideas?

0 What is a “mathematical procedure” for you?

0 Can you given an example of a question that asks a “mathematical procedure”?
7. How do you know that a student has understood a mathematical procedure?

8. Can you tell me what a “mathematical problem” is depending on your own ideas?

0 What is a “mathematical problem” for you?

0 Can you give an example of a “mathematical problem”?

9. How do you know that a student has understood a mathematical problem?
10. Can you describe me “mathematical understanding” depending on your own ideas?

0 What is a “mathematical understanding” for you?

Ur

224

0 How did you / Did you develop your own “mathematical understanding”?
11. How do you know that a student has developed a mathematical understanding?
12. Now, I want to learn more about your ideas on a speciﬁc task. Here is a question about fractions:
“How many l_ s are there in 5?”
5
a. Can you solve the question and explain me how you solved it?

i. Can you show me the parts of your solution related to your conceptual
understanding and procedural understanding? (Remind him/her his/her previous
answers about understandings.)

b. How would a 5m grade student solve this problem correctly?

i. Can you show me parts of this solution related to your views about conceptual
understanding and procedural understanding? (Remind him/her his/her previous
answers about understandings.)

c. Here is an answer that 5‘” grade student provided.
“ «one, .L. 5
D $55 5 K‘E-s'g
i. What might he/she have thought while answering this question in this way?
ii. What might be missing in his/her conceptual understanding?
iii. What might be missing in his/her procedural understanding?
iv. What could you do in order to cover those missing points in his/her
understanding?

 

 

 

 

Teaching Mathematics
13. Why do we teach mathematics?
14. What are the types of knowledge that a mathematics teacher should have?
0 Can you make a list of / categorize the types of knowledge that a mathematics teacher
should have?
3. Why do you think that a mathematics teacher should have these types of knowledge?
15. What will be your main purpose in teaching mathematics in the future?
- Students’ understanding, learning, enjoyment, attitude, or the concept itself as a focus.
16. Do you have any teaching experience? In which ways and for how long?
17. What are your strengths in teaching mathematics?
18. What are your weaknesses in teaching mathematics?
19. Which mathematical concepts do you like/not like to teach?
a. Why do you like/ do not like these concepts?
Attitudes towards Mathematics and Teaching and Learning Mathematics
20. Do you enjoy learning mathematics? When? Why?
21. Do you enjoy teaching mathematics? When? Why?
22. In general, do you enjoy dealing with mathematics? When? Why?
23. Do you think that enjoyment is important in teaching and learning mathematics? When? Why?
Policies
24. Do you think that your high school education was effective in the way you study mathematics?
How. Is there a speciﬁc mathematics teacher that affected your ideas about mathematics?
b. How did you study math during high school?
25. Do you think that university entrance examination affected your way of studying mathematics?
How?
26. How would the previous system affect your way of studying mathematics if you were to take the
new exam? Why?
27. Do you think that the courses you have taken in this program are useful for your profession? Why?
a. What are the courses that you would like to have?
b. What are your expectations from the professors in your program during the courses?
c. Do you think that the courses would be different if you had another instructor?
28. What kind of knowledge and skills should a mathematics teacher educator have?
a. What kind of expectations should he have from the preservice teachers?

Other Questions

225

 

 

29.

30.

31.

32.

How can you deﬁne teaching as a profession?
a. What does a teacher do?
b. What does a mathematics teacher do?
Do you think that the knowledge and skills you gained in the program will be sufﬁcient for your
teaching?
a. What can a teacher do to improve his/her knowledge and skills?
0 What are the sources of knowledge for a teacher to improve his/her knowledge and skills?
0 Do you think a teacher can learn from his/her students? How?
How would you deﬁne teaching as a profession before you started studying in this program?
a. How have your ideas changed about teaching profession?
What would you want me to ask in this interview, that I did not ask you?
a. Why is this question important to you?
b. Would you answer it if I ask now?

226

2003 Interview Protocol for the Graduates

Hi,

I am Cigdem Haser, and I am a doctoral student in Department of Teacher Education at Michigan
State University in US. I am conducting a study about prospective teachers’ perceptions of mathematical
understanding and various factors that affect their perceptions. This interview that you volunteered is about
my study. Having been volunteered is not a strict condition, and you can quit the interview at any time you
want to.

The questions that I am going to ask you are not a part of your program The answers that you
provide will not be shared by any program member, and will not have an impact on your course grade. I
will be the only person who has the access to the answers that you presented. In case of using your answer
in any publication, I will not mention your name. However, I may use the demographic data that you
provided in the instruments that you answered before.

There are 23 main questions in this interview and there are no true answers to the questions that I am going
to ask. I am deﬁnitely interested in your own ideas and feelings about issues in mathematical understanding
and mathematics teaching and learning. Thus, I don’t want you to feel stressed about answering my
questions.

I want to audiotape our interview if you allow me. If you do not want this interview to be
audiotaped, please feel free to express. If you want any part of the interview to be audiotaped, please feel
free to express. I respect your privacy and I will turn off the recorder. Although I informed you about this
before (in our e-mail or phone communication), 1 just want to be sure if you know that this interview is
likely take one to two hours. Have you organized your time for this interview?

Are there any questions that you want me to answer about the interview before we start? Can we
start now? Please tell me if you need a break at any time of the interview.

Thank you.

Interview Questions

Nature of Mathematics
1. What is mathematics for?
2. What does it mean to know mathematics?
3. Can you tell me what a “mathematical concept” is depending on your own ideas?

0 What is a “mathematical concept” for you?
4. How do you know that a student has understood a mathematical concept?
5. Can you tell me what a “mathematical procedure” is depending on your own ideas?

0 (What is a “mathematical procedure” for you?)
6. How do you know that a student has understood a mathematical procedure?
7. Can you tell me what a “mathematical problem” is depending on your own ideas?

0 What is a “mathematical problem” for you?

. How do you know that a student has understood a mathematical problem?
9. Can you describe me “mathematical understanding” depending on your own ideas?

0 What is a “mathematical understanding” for you?

0 How did you / Did you develop your own “mathematical understanding”?
10. How do you know that a student has developed a mathematical understanding?
Teaching Mathematics
11. Why do we teach mathematics?
12. What are the types of knowledge that a mathematics teacher should have?

0 Can you make a list of / categorize the types of knowledge that a mathematics

teacher should have?
a. Why do you think that a mathematics teacher should have these types of knowledge?

13. What will be your main purpose in teaching mathematics in the future?)

- Students’ understanding, learning, enjoyment, attitude, or the concept itself as a focus.
14. What are your strengths in teaching mathematics?
15. What are your weaknesses in teaching mathematics?
Attitudes towards Mathematics and Teaching and Learning Mathematics
16. Do you enjoy learning mathematics? When? Why?
17. Do you enjoy teaching mathematics? When? Why?

00

227

 

18. In general, do you enjoy dealing with mathematics? When? Why?
19. Do you think that enjoyment is important in teaching and learning mathematics? When? Why?
Policies
20. Do you think that university entrance examination affected your way of studying mathematics? How?
21. How would the new system affect your way of studying mathematics if you were to take the new
exam? Why?
22. Do you think that the courses you have taken in this program are useful for your profession? Why?
a. What are the courses that you would like to have?
b. Do you think that the courses would be different if you had another instructor?
23. What would you want me to ask in this interview, that I did not ask you?
a. Why is this question important to you?
b. Would you answer it if I ask now?

228

2004 Interview Protocol for First-Year Teachers

Hi,

I am Cigdem Haser, and I am a doctoral student in Department of Teacher Education at Michigan
State University in US. I am conducting a study about inservice teachers’ perceptions of mathematical
understanding and various factors that affect their perceptions. This interview that you volunteered is about
my study. Having been volunteered is not a strict condition, and you can quit the interview at any time you
want to.

The questions that I am going to ask you are not a part of your program. The answers that you
provide will not be shared by any member of the school that you are teaching and will not have an impact
on your reappointment. I will be the only person who has the access to the answers that you presented. In
case of using your answer in any publication, I will not mention your name. However, I may use the
demographic data that you provided in the instruments that you answered before.

There are 35 main questions in this interview and there are no true answers to the questions that I
am going to ask. I am defmitely interested in your own ideas and feelings about issues in mathematical
understanding and mathematics teaching and learning. Thus, I don’t want you to feel stressed about
answering my questions.

I want to audiotape our interview if you allow me. If you do not want this interview to be
audiotaped, please feel free to express. If you want any part of the interview to be audiotaped, please feel
free to express. I respect your privacy and I will turn off the recorder. Although I informed you about this
before (in our e-mail or phone communication), I just want to be sure if you know that this interview is
likely take one to two hours. Have you organized your time for this interview?

Are there any questions that you want me to answer about the interview before we start? Can we
start now? Please tell me if you need a break at any time of the interview.

Thank you.

Interview Questions
Nature of Mathematics

1. What is mathematics for?

2. What does it mean to know mathematics?

3. What does it mean to learn mathematics?

0 How is mathematics learned?
0 How did you learn mathematics?
0 How do you think a student should study mathematics in order to learn it?

4. Can you tell me what a “mathematical concept” is depending on your own ideas?

0 What is a “mathematical concept” for you?

0 Can you given an example of a question that asks a “mathematical concept”?
How do you know that one of your students has understood a mathematical concept?
6. Can you tell me what a “mathematical procedure” is depending on your own ideas?

0 What is a “mathematical procedure” for you?

0 Can you given an example of a question that asks a “mathematical procedure”?
7. How do you know that one of your students has understood a mathematical procedure?
8. Can you tell me what a “mathematical problem” is depending on your own ideas?

0 What is a “mathematical problem” for you?

0 Can you give an example of a “mathematical problem”?

. How do you know that one of your students has understood a mathematical problem?
10. Can you describe me “mathematical understanding” depending on your own ideas?
0 What is a “mathematical understanding” for you?
0 How did you / Did you develop your own “mathematical understanding”?
1 1. How do you know that one of your students has developed a mathematical understanding?
12. Now, I want to learn more about your ideas on a speciﬁc task. Here is a question about fractions:
“How many l_ s are there in 5?”
5
0 Can you solve the question and explain me how you solved it?

£11

229

 

i. Can you show me the parts of your solution related to your conceptual
understanding and procedural understanding? (Remind him/her his/her previous
answers about understandings.)

o How would a 5m grade student solve this problem correctly?

i. Can you show me parts of this solution related to your views about conceptual
understanding and procedural understanding? (Remind him/her his/her previous
answers about understandings.)

- Here is an answer that 5lh grade student provided.

*“ «one» J... 5
5‘ arse 5 '5‘" 55"
i. What might he/she have thought while answering this question in this way?
ii. What might be missing in his/her conceptual understanding?
iii. What might be missing in his/her procedural understanding?
iv. What could you do in order to cover those missing points in his/her
understanding?

13. Do you think that your ideas have changed since you graduated and started teaching? What has
affected your ideas?

14. Can you give examples of your lesson plans each of which reﬂects emphasis on one or more types
of understanding that we have talked about?

15. Without mentioning name or any form of student data, can you give examples of three particular
students who have rich, medium, and poor mathematical understanding and related student
behaviors?

16. Can you plan one or a series of lessons (depending on your preference) on a particular topic which
reﬂects your conceptions of mathematical understanding, without considering other pressures such
as material availability and time allowed for that particular topic in the curriculum for one of your
classes or a class of 30 students?

17. Can you provide me copies of examples of assessment tasks (written exams. quizzes, oral exams,
homework, term projects) that you implement on your students?

0 What kind of understanding do these tasks address?
Teaching Mathematics

18. Why do we teach mathematics?

19. What are the types of knowledge that a mathematics teacher should have?

0 Can you make a list of / categorize the types of knowledge that a mathematics teacher
should have?
0 Why do you think that a mathematics teacher should have these types of knowledge?

20. What is your main purpose while teaching mathematics?

- Students’ understanding, learning, enjoyment, attitude, or the concept itself as a focus.

21. Did you have any teaching experience before you started teaching as a formal teacher? In which
ways and for how long?

22. Are there any pressures or supports that affect your teaching, such as students, parents, school
administrators or curriculum?

0 Do you think that you would teach more effectively if any one or more of these factors had
changed? Why and in what ways?

23. Which grades are you teaching now?

0 Which grades would you prefer to teach? Why?

24. What are your strengths in teaching mathematics?

0 Did you ever realize that these are your strengths before you become a formal teacher?

25. What are your weaknesses in teaching mathematics?

0 Did you ever realize that these are your weaknesses before you become a formal teacher?

26. Which mathematical concepts do you like/not like to teach?

0 Why do you like/ do not like these concepts?
Attitudes towards Mathematics and Teaching and Learning Mathematics

27. Do you enjoy learning mathematics? When? Why?

28. Do you enjoy teaching rrrathematics? When? Why?

 

 

 

 

230

29. In general, do you enjoy dealing with mathematics? When? Why?
30. Do you think that enjoyment is important in teaching and learning mathematics? When? Why?
Policies
31. Do you think that the courses you have taken in the program are useful for your profession? Why?
0 What are the courses that you would like to have but not offered in the program?
0 What were your expectations from the professors in your program during the courses?
32. Are you a student in a master’s program? Where?
0 Why have you decided to pursue a MS. degree?
0 Do you think that it will be useful for your profession as a mathematics teacher? If so, in
what ways?
33. What kind of knowledge and skills should a mathematics teacher educator have?
a. What kind of expectations should he have from the preservice teachers?

Other Questions

34. How can you deﬁne teaching as a profession?
o What does a teacher do?
0 What does a mathematics teacher do?

35. How would you deﬁne teaching as a profession before you started teaching as a formal teacher?
0 How have your ideas changed about teaching profession?

36. Do you think that a teacher should improve his/her knowledge and skills? Why / Why not?
0 What kinds of knowledge and skills should a teacher improve?
0 What are the ways or sources of information that a teacher can improve his/her knowledge

and skills?

0 Do you think a teacher can learn from his/her students? How?

37. What would you want me to ask in this interview, that I did not ask you?
0 Why is this question important to you?
0 Would you answer it if I ask now?

231

Interview Protocol for Faculty

Hi,

I am Cigdem Haser, and I am a doctoral candidate in Department of Teacher Education at
Michigan State University in US. I am conducting a study about prospective teachers’ conceptions of
mathematical understanding and various factors that affect their conceptions. This interview that you
volunteered is about my study. Having been volunteered is not a strict condition, and you can quit the
interview at any time you want to. '

For this study, I need the ideas of the faculty about the program and about certain characteristics
that preservice and inservice mathematics teachers have. The questions that I am going to ask you are
generally about these issues. The answers that you provide will not be shared by any program member. I
will be the only person who has the access to the answers that you presented. In case of using your answer
in any publication, I will not mention your name. However, I may use the demographic data that you
provided.

There are 15 main questions in this interview and there are no true answers to the questions that I
am going to ask. I am deﬁnitely interested in your own ideas about issues in preparation of mathematics
teachers and their beliefs. Thus, I don’t want you to feel stressed about answering my questions.

I want to audiotape our interview if you allow me. If you do not want this interview to be
audiotaped, please feel free to express. If you want any part of the interview to be audiotaped, please feel
free to express. I respect your privacy and I will turn off the recorder. Although I informed you about this
before (in our e-mail or phone communication), I just want to be sure if you know that this interview is
likely take one to two hours. Have you organized your time for this interview?

Are there any questions that you want me to answer about the interview before we start? Can we
start now? Please tell me if you need a break at any time of the interview.

Thank you.

Interview Questions
How long have you been teaching in the program? / How long have you taught in the program?
2. How long have you been teaching in general?
- In another program, at a school

3. Which courses did you offer in the program? Could I get a copy of the syllabuses of the courses
that you offered?

4. Do you think that program effectively prepares students to be mathematics teachers?

a. What are the effective experiences/courses in the program?
b. What are the ineffective experiences/courses in the program?
c. How would the program be improved?

What are your goals in general in teaching a course?

6. Can you describe the characteristics of effective mathematics teachers?

a. What kind of knowledge should mathematics teachers have in order to teach mathematics

effectively?

7. Can you describe the beliefs that a mathematics teacher should have?

- About nature of mathematics, about teaching and learning mathematics

8. Can you describe the beliefs of younger students in the program about nature of mathematics and
teaching and learning of mathematics?

9. Can you describe the beliefs of older students in the program about nature of mathematics and
teaching and learning of mathematics?

10. How do you think these beliefs are/can be changed?

11. Do you speciﬁcally address/target these beliefs in your teaching? How?

12. Do you see any difference in beliefs about and attitudes towards mathematics and teaching and
learning mathematics, and depth of students’ work based on the high school experience?

13. Do you see any difference in beliefs about and attitudes towards mathematics and teaching and
learning mathematics, and depth of students’ work based on the university entrance examination
experience?

14. Which courses do you prefer to teach? Why?

15. What do you expect from your students in your courses in general?

H
e

.V‘

232

 

 

APPENDIX D

SAMPLE DATA CODING

Complete Translation of Dina’s Interview

Notes:
0 IC: (for Cigdem) means that I ask a prompt question]
Italic words mean that the participant spoke in English.
[Brackets generally contain my explanations about what the participant is talking about]
(Parentheses contain the exact translation of a word, which may not ﬁt the meaning in that context.
Another closer meaning is just provided before the parentheses.)
- There is no gender difference in Turkish while one talks about an arbitrary person. So I translated
references to an arbitrary person as “he” if Dina did not use female pronouns.

Interview Questions
Nature of Mathematics
1. What is mathematics for?
Mathematics is for making it easy for people to understand the life. I mean, it also makes their lives
easier. By enabling people to think comprehensively(widely), it [mathematics] is providing an intellectual
view. In this sense, I think it [mathematics] is for enjoying the life.

2. What does it mean to know mathematics?

Knowing mathematics is approaching to the problems that human face in a mathematical way. I mean,
it is inrplementing the solutions by a mathematical view.

[C : What kind of problems are these?|

It can be any_thing that we face in daily life. Such as calculating the money in a shopping or whether
the money will be enough and measuring time (U1). Or, the things that we have difﬁculty in dealing with
such as calculus problems that we study at the university level or producing comprehensive theorems.
Knowing mathematics does not enable us to solve these problems, but it makes it easier for us to solve
these problems.

3. What does it mean to learn mathematics?

Mathematics is actually. cannot be learned all of a sudden... something that can be learned in a way...
It requires too much effort (U1). We are sharing our knowledge(savings) with people ﬁrst in the elementary
level, then in the secondary level, and then at the university, we make people get this |knowledge |. While
doing these, we either explain or showLwe mgke them [people] discover these (U2 1. I mean, people need to
know an essential level of mathematics. If they want to put more on this they move towards mathematics
and continue to the existing studies themselves in some way.

0 How did you learn mathematics?

I. .. was enjoying mathematics at the high school and I actually did not think of being a mathematics
teacher but my father guided(directed) me. He is an elementary school teacher. I actually did not really
know the kind of mathematics that I was going to study in the process of being a teacher. How did I learn
mathematics? Actually, the mathematics that we will teach and the mathematics that we study here do not
overlap that much. But, like every university student, we have to have some qualiﬁcations(things). I
actually enjoyed the mathematics that I studied here. I think that it enhances one [one’s thinking] in terms
of perspectives.

I did not study to the concepts that much but I, for example, used to think that I would not be able to
understand a problem which I don’t know anything about it [the problem]. I am not taking any math
courses in the past three semesters and I took a course on geometry as an elective course. In the courses I
studied, I realizemt when I gl_r_§§dy mgstered the concepts. I gained better practical skills by solving the
questions myself. If there Jappeared anything thgt 1 could not understand during the class, I formed the

233

 

following system: ﬁrst studying the conce_pts, then solving exangles, then new solving questions with
others, and ﬁnally solving the questions and tests at the end of the clmpter LU3).

 

4. Can you tell me what a “mathematical concept” is depending on your own ideas?

0 What is a “mathematical concept” for you?

For me, it is the case that “that thing” can be explained in the mathematical language. For example, if
idiom is a concept in Turkish, then it can only be explained in the Turkish [course]. I think that, that thing,
that object, or whatever the thing that requires a solution is.. if that has an equivalent in the mathematical
language, then it becomes a mathematical concept.

0 Can you give an example of a question that asks a “mathematical concept”?

For example... “What is an angle?” Angle is a mathematical concept.

5. How do you know that a student has understood a mathematical concept?

In relation to this concept [angle]. If he can explain it in the knowledge level, give exanrples in the
comprehensive level [referring to the Bloom’s Taxonomy], implement it to the new questions, and also to
other things such as other things in his life, then we can say that he learned this [concept].

6. Can you tell me what a “mathematical procedure” is depending on your own ideas?

Mathematical procedure, probably is ﬁrst understanding the problem, then tracking the data out of this
[the problem] and producing a solution, implementing them and then going back [to the beginning] and
controlling, and reaching various analysis through these results.

0 Can you give an example of a question that asks a “mathematical procedure”?

Worker, pool problems [famous types of problems in middle school level] might be examples for
these. “If A can do a work in this many days and B can do the same work in this many days, then in how
many days will A and B do the work together?” [can be an example].

7. How do you know that a student has understood a mathematical procedure?

For example, in these question [like the ones above] most of the teachers give formulas in the test
preparation centers and [they understand] whether [the students] have understood the question... If there
are those words in the question [there are certain expressions in those problems which leads to certain
mathematical operations with the given quantities], then they [test preparation center teachers] try to teach
the worker-pool problems by forrnulizing. But the fact that students can use the formula alone does not .
guarantee (show) that they [the students] have understood this [problem]. If he [the teacher] in some way,
can maLke them [the students] explacin, if they [the students] tell the procedures themselves..[such as] there

is the given Idatal, there is the required thing, if we implement this solution we ﬁnd this, and the solution

_tlya,t we hpve gives the work that they [A and B, referring to the problem she mentioned in question 6] do

together, and if they |the students] explain the results in a logical way, then I think that they have
understood (UlL

8. Can you tell me what a “mathematical problem” is depending on your own ideas?

The problem that contains mathematical terms, the problem that [people] use mathematics.
0 Can you give an example of a “mathematical problem”?

For example... triangle question. It can be area question.

9. How do you know that a student has understood a mathematical problem?

Similarly, they [the students] can implement the method in the pervious question [question 7]. m
can use these skills( savings) in different questions... They can enh_ance in some ways.. I mean they can
produce different gigstionsJUl)

10. Can you describe me “mathematical understanding” depending on your own ideas?

To approach(look at) to the problems or the things that they [people, in general] face with the eyes of a
mathematician or by using mathematics. [Giving an arbitrary example that she generates at that moment] It
can be saying “there is a bottle of water”, “there is some water", or “there is 0.5 liters of water”. Using
“0.5” instead of “some” is [an indication of] a mathematical perception or expression for me.

o How did you / Did you develop your own “mathematical understanding?”

234

 

 

Somehow by guidance, by realizing that I somehow enjoy mathematics, and by having chosen this
type of college experience [referring to department]. When you ﬁnd yourself in this track...l mean, mostly
since I wanted. It was developed through the courses that I took and by my own effort.

11. How do you know that a student has developed a mathematical understanding?

Students come to schools with a certain level [of intelligence/skills/knowledge] and they are
implemented certain measurement and evaluation processes [in the process of reaching a level]. These
could be observations or scales/tests. If there appears a positive improvement when they [the students] are
implemented these [evaluation processes], then we can say that there is an improvement. Or if he can
mress th_at, I mean the thing that he understood (U1). . . [and] if you can observe this [expressing his own
understanding] improvement. Or if he can implement this point of view to the things that he faces...

12. Now, I want to learn more about your ideas on a speciﬁc task. Here is a question about fractions:
“How many _1_ s are there in 5?”
5
a. Can you solve the question and explain me how you solved it?

I divide 5 by 1/5. 5 times 5 is 25. Here, [we ﬁnd] how many times does 5 have 1/5 by division.

i. Can you show me the parts of your solution related to your conceptual understanding
and procedural understanding?

For example... in terms of concept, there is division... in terms of procedure...First, what does it ask
after understanding the problem... Just a second. and in terms of procedure. .. we have, for example... We
use multiplication at the same time. I wonder which concept does “how many US 5 are there in 5”
associate with... Just a second. I thought about division, but... yes, there is division, the concept of
division.

b. How would a 5"I grade student solve this problem correctly?

I think, ﬁrst, he can do.. within 1..

 

For instance, this is a whole [the ﬁgure on the top]. And this is
the piece that shows 1/5 [the ﬁgure on the bottom]. He will think
that a whole includes 5 small boxes. In this case for 5, if this is 1
whole, we need 5 of this [1 whole]. Hence, how many are there in
5 wholes? He can think like. how many... he can solve it by
[using] proportion:

1 whole 5 many

5 wholes x many

 

 

 

 

 

 

\ \\\

 

 

 

c. Here is an answer that 5"I grade student provided.

" «on?» . 3.. 5.
D tit 54; a“ 55'

i. What might he/she have thought while answering this question in this way?

1 think he knows multiplication concept and division concept, but maybe. For example, when it is 5. ..
he considered that there are 5 many of this [1/5] here [in l, referring to the ﬁgure the student drew]. But he
could not think that he has to write 5 instead of 1/5 here [the multiplication the student did].

ii. What might be missing in his/her conceptual understanding?
iii. What might be missing in his/her procedural understanding?
iv. What could you do in order to cover those missing points in his/her understanding?

1 think. the student here.. expressed that there is 1/5 in 1 whole nicely. But, for instance, he did not
think much about 5 things side by side. We can make him explain this again. 1 mean, after this operation
[multiplication], at the moment ask about “there are 5 many l/S’s in 1”, he will say “a-ha!” 1 mean, I
thought that there is something missing there. When I take a look at here [the ﬁgure], it is great, yes he can
do correctly alter this, but why did he take 1/5 here [into the multiplication]? If we express this to him.. if
he took 1/5 instead of taking 5.. there is nothing that can be recovered... I mean, I would probably teach in
this way. By asking him to explain and by asking him questions about the issues tlpt he is missing some
points, 1 will be able to mpke him think deeply. Or, if he cannot understand this in this way, then I might
give daily life examples, such as, 5.. oh, we could use small boxes, as well. How many small boxes of 1/5

 

 

 

 

235

 

 

[me sm_a)l boxes phat represent US] in the boxes of 5. . .or I might be able to express this by a balance scale
in some way. How many US should I put |on one side of the weight scale] so that this side will be equal to

the 5 on the other side (U1). This might be... I mean... kilogram measurement and gram measurement. I
mean the numbers.

 

 

 

Teaching Mathematics

13. Why do we teach mathematics?

[For making] people view life much easier, [for making] them view things they face somehow with the
mathematical way.. I mean, not all people know mathematics, but they can live. Actually, people do not
need to lqrow mathematics, university level mathematics, to continue their lives. But this [knowing
mathematics] opens up people’s perceptions and opens up the innovations that can change the world. And
we use them, eventually. We need mathematics for the advancement of humanity and we teach it for this
reason.

14. What are the types of knowledge that a mathematics teacher should have?

First of all, content knowledge-mathematics knowledge is needed. Then, there is a need for educational
knowledge, I mean pedagogical knowledge. Pedagogy includes [knowledge ranging from] classroom
management, [to] understanding students, [to] guiding them, and [to] motivating them. Knowledge of
education and being a friend. becomes. too interrelated. I mean, students should see their mathematics
teacher be close to them so that they enjoy mathematics. We have to make them feel that we are close to
them, too. Actually, all teachers should have an admirable personality [character]. He has to be consistent.
He should not make mathematics perceived as a class, he has to make students feel that mathematics is
used in every stage of life. As an example, in one of the schools that I did my student teaching, the teacher
was not only teaching mathematics. In an article that he brought to the class, there was something that was
about mathematics. He was cutting it [that article, referring to a newspaper article], bringing it to the class,
and reading it to the students. He was both making students say “There is math in other ﬁelds” and making
them motivated.

] C: Can you put them in an order of importance?]

Content area knowledge. It is too much for teaching mathematics to the elementary school students. I
mean, we don’t even teach the amount [of mathematical knowledge] that we know. We know both calculus
and linear [algebra], but we don’t teach them. But having the ability(specialty) to teach them is so
important for me, [such as] to reach [down] to the level of the students, to present them different
examples... A person might know mathematics really good but he might not be able to develop methods to
teach it, there also a need for this kind of intelligence: “Where should I use what [kind of knowledge] so
that students will understand better?” .. I think these two [content knowledge and knowledge of how to
teach (pck)] are equal [in terms of importance]. A teacher should not cause misconceptions as well... but
not that much critical [in terms of importance], as I said. I see them all equal [in terms of importance].

a. Why do you think that a mathematics teacher should have these types of knowledge?

The fact that one is a teacher requires these, I think. Teaching is not only transferring what you know,
but it is transferring whpgou know in the best way (UH I mean, transferring in a way that the students can
progress further. I mean, we don’t only teach 5 times 5 makes 25 to the kids. Moreover, there is a need give
them the enjoyment of learning it [mathematics]. The person who will enable them to learn it regularly
(orderly), to learn it more is the teacher. In this sense, there is a need for educational knowledge and
knowledge of methods, besides the content knowledge. Besides these, we don’t only do instruction at
schools. Education and instruction go together. In order to have people with healthy development [in terms
of growing up], they have to get different things. We have to help them in this development as well. In this
sense, we have to do other things like. being a friend and providing motivation.

15. What will be your main purpose in teaching mathematics in the future?)

Actually, I haven’t generated a way to go through yet, such as “this is my goal, this is my principle.”
But it won’t be only transferring the curriculum In any way.. eventually. there are things that a student
who completed the 50' grade. One of the jobs of the teacher is to Lake the students comprehend these
things, for sure (U1). But it is to make them enjoy mathematics, to have positive ideas towards
mathematics, and to make them curious, make them approach life with a mathematical viewpoint. To make
them enjoy mathematics, though it is still... in so many schools and... although there are too many tools
related to mathematics that are in use, it is still the course that so many students are afraid of. To make

236

them overcome this fear, to tell them that mathematics is more than being a course, it is a perspective that
you have to view life.

16. Do you have any teaching experience? In which ways and for how long?

I tutored few students, for not so long, about a month. All of them were in the elementary level. I
worked in a test preparation center for about a month. (She also taught to lower SES students in a student
club organization for sometime.)

17. What are your strengths in teaching mathematics?

18. What are your weaknesses in teaching mathematics?

Actually, I think that my content knowledge is sufﬁcient, but I have to put more effort in the
motivation issue, on providing students with motivation. I enjoy teaching them something but at the same
time, for example, it is a little difficult task to motivate all students. While making one of them involved in
the class, you can ignore the others. I experienced this [situation] once during the student teaching. When
that happens, it [teaching] does not reach its goal. It is something that improves through experience. It is so
important to have that kind of openness. And one more thing, when you teach for the ﬁrst time you focus
on only one part. Because you are so anxious, you think about the thing that you will teach immediately
after what you are teaching at that moment: “Am I teaching properly, maybe I have to do this there.” In
general course [you can ignore this]... This might be my insufﬁciency. For example, I think that I
comfortably be ﬁiend with the students, I actually do not treat students like I am an adult. But, as I told
you, the point that I am insufﬁcient is the ability to do things that will increase all students’(students as a
class) motivation. I think I can improve this.

19. Which mathematical concepts do you like/not like to teach?
a. Why do you like/ do not like these concepts?

I enjoy geometry a lot. I used to solve the geometry questions with enjoyment when the students asked
them the test preparation center [where she worked for a month]. I mean, I used to teach those as well. But
besides this, I realized during my student teaching that I teach the classeﬁhpt includes materials with a lot
of enjoyment, too. Actuglly, I don’t discrimipaﬁte but teaching a little more abstract classes are likely to be
difﬁcult. For this reason I enioy more the clarsses M student enjoy material use. I mean, the clpssesthLI
realize thpt they understand (U1).

In terms of concepts. Actually, it changes from students to students, it also changes from the age
group to age group. For example, the [concept] that I like the most. angles. binomials is also a nice
concept. But when the students don’t understand... you get bored, too. For this reason, I like the classes
that students understand.

Attitudes towards Mathematics and Teaching and Learning Mathematics

20. Do you enjoy learning mathematics? When? Why?

21. Do you enjoy teaching mathematics? When? Why?

Yes, I enjoy. For example, it can also be a bad thing, but not sitting down by myself and not wondering
about a theorem in somewhere, not saying that I have to know that and not studying that [theorem] is a bad
thing. But, I don’t do this [studying theorems] too much. But, I am making it up, for example, I learn
something new while reading Science and Technology [a monthly, more than a popular science journal
published by Turkish Scientiﬁc and Technical Research Institution, a governmental research institution]:
Something mathematical, something about mathematics, or something about the mathematicians... I enjoy
it a lot. Or I have a roommate. She prepares activities, she ﬁnds a new method, and she shares this with me.
We say “Yes, we can teach this in this way to the students, this is a great method, we can develop this kind
of materials.” I like these kind of things.

I enjoy mathematics both at the time that I am learning about how to teach math and when I learn
about a method. [I enjoy] When I learn something new: something about the history of mathematics, it ca
be a new problem, a puzzle... in Science and Technology [same journal] or in Intelligence Games [another
journal]. things like these.

22. In general, do you enjoy dealing with mathematics? When? Why?
Yes .

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23. Do you think that enjoyment is important in teaching and learning mathematics? When? Why?

For me, it is important. Your facial expression is so important. I mean, it is because, when you say
“uffﬂ !” [an interjection that is generally used to show that you got really really bored] [while teaching], the
person right across you would not enjoy that, too. Same is also [valid] for us. When we were students and
when the teachers were not smiling and were bored, we weren’t enjoying [the class] either. Hence, if we get
their motivation and shine, then we can teach the things that we enjoy, we can make students enjoy it. For
this reason, it is really important to enjoy mathematics. For this reason, it is different to love mathematics
than to enjoy mathematics. ‘

[C : What is the difference?]

A person studies only for mathematics. But not all mathematics teachers are like this. But I think, all
mathematicians and the people who are dealing with mathematics enjoy mathematics. It is about enjoying
mathematics. But the love of mathematics is something different. It would be much better if we love
mathematics.

Policies

24. Do you think that your high school education was effective in the way you study mathematics?

How?

[She studied in a science high school.] Actually, we did not do many things at the 3'” grade [last year
of high school]. If we think about the recent educational system [the situation that cuniculum of 3rd grade
does not appear in the university entrance exam]... But in the Zn and 1" grade, for example in the geometry
classes, the teacher used to write a theorem and used to ask us to prove that. Easy theorems, like “if
something happens, then something happens”, as we know, about triangles, for example, bisector theorem.
He used to tell us to prove that. He did not teach the class like: “This is this, and this is this. Memorize
these to solve the questions.” Since he did nottjrch in th_at way, we used to struggle with it [the theorem],
and we used to enioy thpt (U1). In the following class we solved those [questions]. This [way of
teaching/learning] helped me to enjoy mathematics more. And since I did not memorize I did not feel
difﬁculty lpter (U2). I enjoy geometry for this reason... I enjoyed by this way..

a. Is there a speciﬁc mathematics teacher that affected your ideas about mathematics?
I actually liked my physics teacher. But mathematics teachers were not the teachers that you don’t like.
b. How did you study math during high school?

We were learning mathematics in the class. Since it was not thpt much a memorization class I mean
since there was no memorization of formula and since they |the teachers] taught us until we answered “Has
everyone understood?.... Yes, understood”,we did not need to study a lot (U3). We generally used to solve
more difﬁcult problems in the classes compared to the examinations, hence there was no need to study too
much for the exams, I mean going to the class prepared was more important.

 

 

25. Do you think that university entrance examination affected your way of studying mathematics?

How?

Actually.. in the 3"l grade. It requires a surface level of studying more than it requires the [studying]
essence of the concept. Thus, there is a need to solve a lot of questions in order to improve practical [skills].
I used to like it sometimes, but I got bored sometimes, too. I enjoy studying the concept slowly in the
beginning and then solving questions about it more. My system of studying was in this way. But I realized
in the 3'd grade that, if I study in this way, I would not be able to cover all the concepts. Hence, I started to
solve more questions. For example, although I did not understand some parts of the probability concept, I
scanned these roughly (surface level), by solving questions. In this sense, I don’t think that studying for
university exam is an educative study.

26. How would the previous system affect your way of studying mathematics if you were to take the

new exam? Why?

It would probably be changed. Because people who took that exam used to say that it measured more
about whether you have understood the concepts or not, they used to say that the content of the questions
were in that way. For this reason, it would probably be changed.

IC: Would you be continuing with your way of studying?|

Yes, it would probably be in that way.

238

27. Do you think that the courses you have taken in this program are useful for your profession?

Why?

(about math courses) Math courses are not one-to-one [useful for my profession]. We are not teaching
derivative and integral, or matrix and determinant. However, I think that the 111 and 112 courses [courses
about abstract mathematics] that we took in the ﬁrst year will be useful in [teaching] the concept of
probability. But actually the university education that we have been through is something like that. To
improve people’s perspective and it is a step [for people] to improve themselves. I mean, if we do master’s
and doctoral degree, then maybe we will learn something in the name of improvement. I don’t think that we
are learning to implement one-to-one.

(about the educational coursesz) But, for example, the educational courses I took serve for this purpose.
For example, we take methods course. I think they [the courses from the department] will be useful. It is
not that I don’t think the others [math courses] will be useful.. they just do not reﬂect one-to-one [match].
The educational courses that we take here [in the college of education] are the courses that we can see
equivalent in our classrooms. I will be able to use the things that I presented or the materials that I used in
the methods courses that I took this year or last year. Or as in the case of guidance class, the methods I
studied(saw) there are the things that I said “OK., I will behave in this way now” when I had to deal with
problem students [referring to her student teaching].

a. What are the courses that you would like to have?

There were classes that our professors thought about offering, but they could offer not since there were
not enough students. I would like to take a class. about geometrical concepts. how we teach them. For
example, there is a course that is offered in this semester but I could not take, Sketchpad. I would like to
take that course, too. I think there is a course [about] making mathematical materials. I think this course
could be offered as a must course, not as an elective course. Actually, in general, there are [courses]
offered. There have been more courses offered in the past few years.

b. What are your expectations ﬁom the professors in your program during the courses?

Teaching that class in a good way. The classes that I am not going to say “ufff!” [an expression of
getting bored, or doing something unwanted at]. To teach me things that will be useful when I become a
teacher or during my university life. It is not only the things they teach but it can be anything in general...
to teach something. really... Not to present the course alone, to make us contribute... to leave a question
mark in our minds(us).. In the ﬁrst years, we used to complain a lot, they were giving us too many
assignments. But we realized that we were able to do many things by doing assignments. We would not be
able to gain many things if there were not any assignments. There were no assignments, no other things, but
we were made to present things in some of the courses. I learned less from these.

c. Do you think that the courses would be different if you had another instructor?

Yes. The second methods course. Since there is not so many professors, I cannot corrrpare. But while
taking the second methods course, I said I wish we took that class from the professor who offered the ﬁrst
one [methods course].

28. What kind of knowledge and skills should a mathematics teacher educator have?

First of all he should have great knowledge and skills. For example. I am making up [the situation]
now, we will be teaching in public schools and we won’t be able to use many things [that we learned here].
For example, instruction with media course was not necessary for us, I am making it up. But a mathematics
[teacher] educator should know these. He has to know the current, updated things. He has to know a lot
about the ﬁeld. I think that he has to know a lot about theoretical things at the same time, not only the
practical (in use) things.

a. What kind of expectations should he have from the preservice teachers?

From us, for instance. while writing an essay, our professor once told us, “This [the essay] should be
in the way of a scientiﬁc paper.” He did not say it in the 1" year but in the 4m year. He did not have such an
expectation in the 1St year. Because we were still busy with watching the remainders of our high school
education in the 1" year. The expectations change eventually every semester. But by now, this is a person
who can be a teacher, I mean, the expectation of having the qualifications [of a teacher]. [The student]
reaches to that at the end. Something good.

For instance, in my 1" year, I would not be able to do the things that I can do in the 4th year. But the
situation that these are not expected. I mean it would be a bad thing to expect less in the 4th year. I will be a
teacher in a year or a few months later. Actually the expectations are the things that improve me. For this
reason, these should be higher so that I can improve.

239

 

Other Questions
29. How can you deﬁne teaching as a profession?
a. What does a teacher do?

The tgacher teaches. Teachers, in some way. .. knowledge alone... knowledge. how can I express?..
like digestible wholes, good, and. if we think about a dish, it is not preparing that dish alone and presenting
it to the student, but mpybe it is preparing thgt dish together. To prepare a dish together with the students by
using the ingredients and eating it somehow with the students... it is not only students eat, but eating it with
the students and being full.. and growing up.. mgybe it can be described in this way (U1).

30. Do you think that the knowledge and skills you gained in the program will be sufﬁcient for your
teaching?
I don’t think that it will be sufﬁcient. [I will feel a] need for sure.
a. What can a teacher do to improve his/her knowledge and skills?

I think that improving is a result of expectations and needs. For this reason, although it should not be
connected to only these [expectations and needs], but it will be most probably in the direction of
expectations and needs. In order to improve, I will use current [sources], the sources that I use in the
university, intemet, and different books about the issues. I will continue to use these. [I will be using] The
things on the intemet that the foreign schools and sites have prepared. I will make use of these. I will ‘
beneﬁt from my ﬁiends’ knowledge.

31. How would you deﬁne teaching as a profession before you started studying in this program?

Teaching. an obligation... The life style that people come and somehow teach a class and go, and then
forget the things that they taught. I used to view it this way for a lot of courses. Because, teachers’
classroom behaviors and outside behaviors were inconsistent. And this made me think, “This is probably
what all teachers do, they behave in his way since it is a duty.” For me, this is so wrong. I will also try not
to [do this]. . .actually, maybe there should be something like this, too, but the teachers’...people’s
impression in the classroom and at the outside should be consistent and similar.

IC: Do you think that the program courses have an effect on the change of your ideas?]

Yes, university... I think that the university courses and the university in general changed [my
ideas]. I think in the positive way. I mean my experiences at the university, the student teachings I did, the
educational courses that I took, and professors’ impressions [on me] changed me in this way.

]C: What are your plans for future?|

Actually, while coming to the university I did not so much... Since I became a teacher by my
father’s guidance and since I have never thought of teaching people, being a teacher before, the idea of
teaching kids was so far away from me. It was far away until my student teaching. But [used to think] I
could not deﬁnitely teach to the small kids since I used to think that I would not be able to go down to their
levels and I would not be able to explain [something] again if they would not understand. I was thinking of
staying at the university and teaching to the people at the university level, if this idea had not changed. I
mean. if the conditions were suitable, if the department were hiring grad assistants, or if I would be hired as
a grad assistant, then this was more appealing for me. But after this year, I think that I will be teaching
while enjoying it at the same time. Hence, I don’t know what the conditions will bring, but I want both an
academic career and to get involved in education. I want to study on both educational sciences and
mathematics education. And at the same time, I want to teach to the younger students. I don’t know what I
will do, but I will probably start initially by being a teacher. And, I did not apply to all private schools.
Most of the private schools required experience or I missed the deadline. For this reason, the private school
alternative does not seem to be that much possible for me. I will take KPSS [a national exam for those who
- want to apply a governmental job, the jobs that you are hired as a civil servant]. And if it is [the place she
will teach, ministry assigns] going to be somewhere close to here, I am planning to do both master and
teaching. I don’t want to loose this connection [with the university] in order to study on educational
sciences in the future. Because I like [studying] educational sciences. If I continue [on academic work,
masters degree] and if [the things] will not be in the way I want when the teaching experience is sufﬁcient
for me, then I may transfer to the universities, if it is concluded that I have the sufﬁcient qualiﬁcations.

IC: Do you think that you will be able to improve yourself in public schools?]
Yes, if we don’t be lazy so much. As I told, the needs. I mean, the school’s. actually every school
needs but... For instance, most of the schools do not have computers, mathematics classrooms, and

 

240

materials. But we see that they can actually have [these] by the efforts of the teachers. We may just sit
down since there is no need and no request. But we can also create this need and request. I probably want
this happen in this way. I now feel that I will be working for this to happen this way. .. for this to happen...
It will be, it will be like this. I mean, I think that I will be doing something. At least, if students will not be
motivating me, this will happen this way.

241

Coding Dina’s Interview

The original coding sheet requires line numbers to be indicated. Instead of giving line numbers
here, the expressions indicated in the ﬁrst three sections of original coding sheet are underlined. Each
separate underlined expression is named with U followed by a number. For each response, the underlined
expressions start with U1. No expression is underlined for the 4‘“, 5'”, and 6h sections of the coding sheet
here, only question number is given.

Coding Sheet
Participant #: 19, Year: 4, Coder: Final Coding, Date: April 1", 2005

l. Belief Statements (Write the line number)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Q 1 2 3 4 5 6 7 8 9 10 11 12
Line U1 U1,U2,U3 U1 U1 U1 U1
Q 13 14 15 l6 17 18 19 20 21 22 23
Line U1 U1 U]
Q 24 25 26 27 28 29 30 31 32
Line U1, U2, U3
Notes (if any):

0 Daily life, intellectual view, providing approaches to solve problems (Ql, Q2, Q3, Q6, Q10, Q14)
0 Application of knowledge to new situations (Q5, Q9, Q11)
0 I learn if I enjoy/like my teacher (Q10, Q14, 023)

2. Categories (Write the line number) N = Nature of Mathematics, T = Teaching Mathematics, L =
Learning Mathematics

1 2 10 ll 12
U1

U1

N

T
L U1 U U3
Notes (if any):

 

242

3. Levels (Write the level)

N

T
L l, l l
Resulting Levels:
Nature of Mathematics: 0
Teaching Mathematics: 1
Learning Mathematics: 1
Notes (if any):

 

4. Perceived Effect of Program Courses on Profession (positive (+), negative (-): Q number; Line
Number)

 

 

 

 

 

 

Course/I Specific Prof. Project Presentation Assignment Need Note |
Dept. Course (5)
Math +,Q27,
(111,112) 1
? EME +,Q27 -,Q27 ’ +,Q27 +,QZ7 Q27
(method) -,Q27
EDS +,QZ7
(guidance) j

 

 

 

 

 

 

 

 

Teaching Experience: None

5. School Effect on Ideas about Mathematics (positive (+), negative (-); Q number; Line number)

 

 

 

 

 

School Level Specific Teacher Specific Course Speciﬁc Year Notes
Elementary School
Middle School
High School +, Q24, Geometry +, Q24, Geometry +, Q24; -, Q24

 

 

 

 

 

 

High School Background: Science High School

6. Teaching and Learning Mathematics Concepts

Concepts enjoyed in teaching (or criteria of concepts that are enjoyed in teaching):
Geometry, Q19; Angles and expressions, Q19:
Use of material, Q19; Students, Q19

Concepts not enjoyed in teaching (or criteria of concepts that are enjoyed in teaching):
-, Abstract concepts, Q19

Concepts enjoyed in learning (or criteria of concepts that are enjoyed in learning):
QZO-QZ 1; Geometry, Q24; Teacher enjoyment, Q23

Concepts not enjoyed in learning (or criteria of concepts that are enjoyed in learning):

Quotable quotes:

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APPENDIX E

A MEMO FOR ELEMENTARY MATHEMATICS EDUCATION PROGRAM

I have been informed that the Elementary Mathematics Education (EME) Programs are going to
be evaluated soon for their effectiveness in preparing middle school mathematics teachers. My doctoral
study is about the impact of the METU EME program on preservice and inservice teachers’ mathematics
related beliefs, but it is also partly concerned with its effectiveness in general. Here, I provide a short
summary of my results in relation to students’ program experiences and my suggestions for the changes in
METU EME program, which I believe can be applicable to the other EME programs in Turkey. I also
propose ideas for a broader change in teacher education and beginning teacher mentoring programs. The
evidences for the results and suggestions are presented in my dissertation which will be available for your
review in mid June. I will be happy to have conversations with you in the ﬁrture about new directions to
improve the program

I interviewed 12 inservice participants who are 2003 graduates of EME program both when they
graduated in 2003 and after their ﬁrst-year of teaching in 2004. I refer them as first-year teachers. I
interviewed 20 students from the 2'“, 3m, and 4"1 year of the program in 2004. I refer them as preservice
teachers.

The results and implications for the EME program, the Department of Elementary Education, and
broader educational contexts are provided below.

I. Program Courses and Needs

The results of the study showed that the courses are not impacting preservice teachers’
mathematics related beliefs when they are not speciﬁcally designed to challenge these beliefs. The elective
course “Activity Based Problem Solving” seems to be a good example of how a course would impact
preservice teachers’ beliefs. Most of the ﬁrst-year participants referred to this course (which was offered
only to 2003 graduates) as an eye-opening course which challenged their views about teaching and learning
mathematics. However, only few preservice participants slightly referred to some of the program courses
during the 2004 interviews.

An important experience lacking in the program courses is the sufﬁcient practice of learned skills.
in methods and classroom management courses. Especially ﬁrst-year teachers expressed this in both 2003
and 2004 interviews. F irst-year teachers claimed that although they learned different methods of
mathematics teaching, they had difﬁculty in implementing these skills in the real-classroom settings
because of lack of practice during the methods courses. Clearly, practice teaching course does not give
sufﬁcient teaching experience to the preservice teachers. When ﬁrst-year teachers started to teach in public
schools, they mostly experienced reality shock since they were not prepared to manage contextual factors
and could not implement their skills. In order to reduce the reality shock, (i) managing student differences
in learning, (ii) maintaining national curriculum pace and (iii) maintaining good relationships with school
administrators and parents should be emphasized in the program courses. The course content related to
these issues as well as teaching and learning mathematics should be supported by continuous, immediate,
and critical practice and monitoring in real-classroom settings. This will be revisited in the second section.

I am aware that EME program courses which are not offered by the program faculty are beyond
our control. However, the related program faculty should be contacted and communicated about our
students’ needs in these courses (e.g. Introduction to Teaching Profession and Classroom Management). I
realized that students’ experiences in especially Introduction course is critical to the perception of teaching
as a profession which requires more than dictating some form of knowledge to the students.

Methods courses in EME program emphasize teaching, but do not seem to refer to students’
learning and the nature of mathematical knowledge that should be in the mathematics classrooms. If we
emphasize and challenge them about the nature of mathematical knowledge and how it should be
represented in the classrooms in these courses, this may help our students in developing higher level beliefs
about the nature of mathematical knowledge. Teaching methods of mathematics are not generally
considered in relation to students’ learning. Many ﬁrst-year teachers considered using manipulatives in the

244

mathematics classroom to increase students’ interest, rather than to increase their learning. We should
emphasize and challenge our students about the goal of using different pedagogical approaches.

First-year teachers and some of the preservice teachers claimed an important issue as lacking in
the EME program: The lack of communication among professors who teach the ﬁrst and the second
methods courses. They mostly claimed that it would be better if the second course was offered by the
professor who offered the ﬁrst course Here, I claim that there should be continuous communication across
the program faculty through all courses to make latter course experiences build on the former ones. This
would also make the courses more meaningful for the students and increase their perception of the
usefulness of course experiences.

11. Relationship with the Ministry of National Education and Long Term Policies

The results of the study indicated that there is a lack of communication between EME program
faculty and the Ministry of National Education staff working on teacher education and teacher induction. A
close communication is needed in order to provide preservice teachers with more teaching experiences
during their studies in our program and to support our graduates during their initial years in teaching.

EME programs can be organized better to prepare teachers for initial years more realistically. We
send our graduates to the ﬁeld with high expectations from themselves, their students, school settings, and
the cuniculum which results in a clash of ideals when faced with the real-classroom contexts. In order to
provide EME program students with more real-classroom practice during their program studies, an
increasing number of skillful and helpful teachers should be recruited as mentors. This requires a new
policy for the Ministry and collaborative efforts of both our program and the Ministry staff in training the
mentors and organizing the teaching experiences for students.

Mentoring in the ﬁrst-year teachers in Turkey is not a working system. Most of the ﬁrst-year
participants mentioned the need for a critical eye for their teaching or a feedback from an experienced
teacher. However, ﬁrst-year participants did not even meet their mentors assigned to monitor their teaching.
Ministry should be contacted about the need for an effective mentorship policy that would help beginning
teachers in their initial years. University collaboration in an effective mentorship policy is likely to provide
beginning teachers and mentors with ideas and materials when they need it.

Perhaps, the most important issue that should be communicated to the Ministry is the limiting
impact of beginning teacher recruitment policies on begimring teachers’ teaching. Most of the ﬁrst-year
participants in this study were recruited later in the ﬁrst or second semester of the school year. Our
graduates were generally the ﬁrst mathematics teachers who were recruited since the beginning of the
school year. The previous teachers who taught mathematics in these schools were not matherrratics
teachers. Hence, ﬁrst-year teachers’ students lacked essential mathematical knowledge and skills.
Therefore they had difﬁculties in managing the national curriculum pace and students’ missing knowledge
simultaneously.

Beginning teachers are under the pressure of being inspected by the Ministry for the national
curriculum pace and revisiting previously “taught” concepts that students have no knowledge or skills.
Clearly, a beginning teacher who starts teaching in March at a rural public school (which did not have a
single mathematics teacher since the beginning of the school year) cannot keep the curriculum pace without
ignoring the lack of students’ prior knowledge. The dilemma of keeping the curriculum pace or teaching for
students’ learning which was expressed by most of the ﬁrst-year participants should be communicated
effectively to the Ministry.

245

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