Contributed by:

What is good mathematics teaching? The answer depends on whom you are asking. Teachers, researchers, policymakers, administrators, and parents usually provide their own views on what they consider is good mathematics teaching and what is not. The purpose of this study was to determine how beginning teachers define good mathematics teaching and what they report as being the most important attributes at the secondary level. This research explored whether there was a relationship between the demographics of the participants and the attributes of good teaching. In addition, factors that influence the understanding of good mathematics teaching were explored.

1.
GOOD MATHEMATICS TEACHING:

PERSPECTIVES OF BEGINNING SECONDARY TEACHERS

by

Kwan Eu Leong

Submitted in partial fulfillment of the requirements for the

degree of Doctor of Philosophy

under the Executive Committee of the Graduate School of

Arts and Sciences

COLUMBIA UNIVERSITY

2012

PERSPECTIVES OF BEGINNING SECONDARY TEACHERS

by

Kwan Eu Leong

Submitted in partial fulfillment of the requirements for the

degree of Doctor of Philosophy

under the Executive Committee of the Graduate School of

Arts and Sciences

COLUMBIA UNIVERSITY

2012

2.
© 2012

Kwan Eu Leong

All Rights Reserved

Kwan Eu Leong

All Rights Reserved

3.
ABSTRACT

GOOD MATHEMATICS TEACHING:

PERSPECTIVES OF BEGINNING SECONDARY TEACHERS

Kwan Eu Leong

What is good mathematics teaching? The answer depends on whom you are asking.

Teachers, researchers, policymakers, administrators, and parents usually provide their own view

on what they consider is good mathematics teaching and what is not. The purpose of this study

was to determine how beginning teachers define good mathematics teaching and what they

report as being the most important attributes at the secondary level. This research explored

whether there was a relationship between the demographics of the participants and the attributes

of good teaching. In addition, factors that influence the understanding of good mathematics

teaching were explored.

A mixed methodology was used to gather information from the research participants

regarding their beliefs and classroom practices of good mathematics teaching. The two research

instruments used in this study were the survey questionnaire and a semi-structured interview.

Thirty-three respondents who had one to two years of classroom experience comprised the study

sample. They had graduated from a school of education in an eastern state and had obtained their

teacher certification upon completing their studies.

The beginning mathematics teachers selected these four definitions of good teaching as

their top choices: 1) have High Expectations that all students are capable of learning; 2) have

strong content knowledge (Subject Matter Knowledge); 3) create a Learning Environment that

fosters the development of mathematical power; and 4) bring Enthusiasm and excitement to

GOOD MATHEMATICS TEACHING:

PERSPECTIVES OF BEGINNING SECONDARY TEACHERS

Kwan Eu Leong

What is good mathematics teaching? The answer depends on whom you are asking.

Teachers, researchers, policymakers, administrators, and parents usually provide their own view

on what they consider is good mathematics teaching and what is not. The purpose of this study

was to determine how beginning teachers define good mathematics teaching and what they

report as being the most important attributes at the secondary level. This research explored

whether there was a relationship between the demographics of the participants and the attributes

of good teaching. In addition, factors that influence the understanding of good mathematics

teaching were explored.

A mixed methodology was used to gather information from the research participants

regarding their beliefs and classroom practices of good mathematics teaching. The two research

instruments used in this study were the survey questionnaire and a semi-structured interview.

Thirty-three respondents who had one to two years of classroom experience comprised the study

sample. They had graduated from a school of education in an eastern state and had obtained their

teacher certification upon completing their studies.

The beginning mathematics teachers selected these four definitions of good teaching as

their top choices: 1) have High Expectations that all students are capable of learning; 2) have

strong content knowledge (Subject Matter Knowledge); 3) create a Learning Environment that

fosters the development of mathematical power; and 4) bring Enthusiasm and excitement to

4.
classroom. The three most important attributes in good teaching were: Classroom Management,

Motivation, and Strong in Content Knowledge.

One interesting finding was the discovery of four groups of beginning teachers and how

they were associated with specific attributes of good mathematics teaching according to their

demographics. Beginning teachers selected Immediate Classroom Situation, Mathematical

Beliefs, Pedagogical Content Knowledge, and Colleagues as the top four factors from the survey

analysis that influenced their understanding of good mathematics teaching. The study’s results

have implications on investigating specific mathematical content knowledge that is important for

classroom instruction at the secondary level. Teacher education programs should provide more

opportunities for their students to investigate and practice classroom management skills in the

real classroom settings.

Motivation, and Strong in Content Knowledge.

One interesting finding was the discovery of four groups of beginning teachers and how

they were associated with specific attributes of good mathematics teaching according to their

demographics. Beginning teachers selected Immediate Classroom Situation, Mathematical

Beliefs, Pedagogical Content Knowledge, and Colleagues as the top four factors from the survey

analysis that influenced their understanding of good mathematics teaching. The study’s results

have implications on investigating specific mathematical content knowledge that is important for

classroom instruction at the secondary level. Teacher education programs should provide more

opportunities for their students to investigate and practice classroom management skills in the

real classroom settings.

5.
TABLE OF CONTENTS

1 INTRODUCTION ..................................................................................................... 1

Need for the Study ..................................................................................................... 1

Purpose of Study ........................................................................................................ 6

Conceptual Framework .............................................................................................. 7

Procedures of the Study ............................................................................................. 10

Participants ......................................................................................................... 10

Research Instruments ......................................................................................... 11

2 LITERATURE REVIEW .......................................................................................... 13

Good Mathematics Teaching ..................................................................................... 13

Summary of National Council of Teachers of Mathematics Professional

Standards ...................................................................................................... 16

Worthwhile Mathematical Tasks ................................................................. 17

Teacher and Student Discourse .................................................................... 21

Mathematical Content Knowledge .............................................................. 22

Pedagogical Content Knowledge ................................................................. 25

Attributes of Good Mathematics Teaching ................................................................ 26

Reflection ........................................................................................................... 28

Teachers’ Beliefs ....................................................................................................... 29

3 METHODOLOGY .................................................................................................... 32

Research Questions .................................................................................................... 32

Research Methodology .............................................................................................. 33

Survey Participants .................................................................................................... 35

Interview Participants ................................................................................................ 36

Research Instrument: Description of Survey Questionnaire ...................................... 37

Development of Survey Questionnaire .............................................................. 40

Reliability of the Survey Questionnaire ............................................................. 42

Validity of the Survey Questionnaire ................................................................ 43

Survey Variables ................................................................................................ 45

Research Instrument: Semi-Structured Interview ...................................................... 46

Development of Qualitative Semi-Structured Interview ................................... 47

Reliability and Validity ...................................................................................... 48

Data Collection .......................................................................................................... 49

Data Analysis ............................................................................................................. 51

i

1 INTRODUCTION ..................................................................................................... 1

Need for the Study ..................................................................................................... 1

Purpose of Study ........................................................................................................ 6

Conceptual Framework .............................................................................................. 7

Procedures of the Study ............................................................................................. 10

Participants ......................................................................................................... 10

Research Instruments ......................................................................................... 11

2 LITERATURE REVIEW .......................................................................................... 13

Good Mathematics Teaching ..................................................................................... 13

Summary of National Council of Teachers of Mathematics Professional

Standards ...................................................................................................... 16

Worthwhile Mathematical Tasks ................................................................. 17

Teacher and Student Discourse .................................................................... 21

Mathematical Content Knowledge .............................................................. 22

Pedagogical Content Knowledge ................................................................. 25

Attributes of Good Mathematics Teaching ................................................................ 26

Reflection ........................................................................................................... 28

Teachers’ Beliefs ....................................................................................................... 29

3 METHODOLOGY .................................................................................................... 32

Research Questions .................................................................................................... 32

Research Methodology .............................................................................................. 33

Survey Participants .................................................................................................... 35

Interview Participants ................................................................................................ 36

Research Instrument: Description of Survey Questionnaire ...................................... 37

Development of Survey Questionnaire .............................................................. 40

Reliability of the Survey Questionnaire ............................................................. 42

Validity of the Survey Questionnaire ................................................................ 43

Survey Variables ................................................................................................ 45

Research Instrument: Semi-Structured Interview ...................................................... 46

Development of Qualitative Semi-Structured Interview ................................... 47

Reliability and Validity ...................................................................................... 48

Data Collection .......................................................................................................... 49

Data Analysis ............................................................................................................. 51

i

6.
IV RESULTS AND ANALYSIS .................................................................................... 55

Preliminary Survey Data Analysis ............................................................................. 56

Research Question 1 .................................................................................................. 59

Results for Quantitative Analysis ...................................................................... 59

Preference Test and Binomial Test .................................................................... 68

Analysis of Preference Test and Binomial Test ........................................... 69

Qualitative Analysis ........................................................................................... 73

Research Question 2 .................................................................................................. 82

Quantitative Analysis ........................................................................................ 83

Qualitative Analysis ........................................................................................... 87

Research Question 3 .................................................................................................. 96

Cluster Analysis ................................................................................................. 97

Discriminant Analysis....................................................................................... 101

Research Question 4 ..................................................................................................112

Quantitative Analysis ...................................................................................... 113

Qualitative Analysis ...........................................................................................114

V SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS.............................124

Summary ....................................................................................................................124

Conclusions ................................................................................................................126

Recommendations ......................................................................................................134

REFERENCES ..........................................................................................................137

A Survey Questionnaire .................................................................................................148

B Survey Variables ........................................................................................................155

C Strand of Survey Variables .......................................................................................157

D Interview ...................................................................................................................158

E Descriptive Statistics for Research Question 2..........................................................160

F Table for Agglomeration Schedule............................................................................161

G ANOVA Table for Discriminant Analysis............................................................... 162

H Percentage of Responses .......................................................................................... 163

I Sample of Interview Transcript................................................................................ 168

ii

Preliminary Survey Data Analysis ............................................................................. 56

Research Question 1 .................................................................................................. 59

Results for Quantitative Analysis ...................................................................... 59

Preference Test and Binomial Test .................................................................... 68

Analysis of Preference Test and Binomial Test ........................................... 69

Qualitative Analysis ........................................................................................... 73

Research Question 2 .................................................................................................. 82

Quantitative Analysis ........................................................................................ 83

Qualitative Analysis ........................................................................................... 87

Research Question 3 .................................................................................................. 96

Cluster Analysis ................................................................................................. 97

Discriminant Analysis....................................................................................... 101

Research Question 4 ..................................................................................................112

Quantitative Analysis ...................................................................................... 113

Qualitative Analysis ...........................................................................................114

V SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS.............................124

Summary ....................................................................................................................124

Conclusions ................................................................................................................126

Recommendations ......................................................................................................134

REFERENCES ..........................................................................................................137

A Survey Questionnaire .................................................................................................148

B Survey Variables ........................................................................................................155

C Strand of Survey Variables .......................................................................................157

D Interview ...................................................................................................................158

E Descriptive Statistics for Research Question 2..........................................................160

F Table for Agglomeration Schedule............................................................................161

G ANOVA Table for Discriminant Analysis............................................................... 162

H Percentage of Responses .......................................................................................... 163

I Sample of Interview Transcript................................................................................ 168

ii

7.
LIST OF TABLES

3.1 Mathematics Grade Point Average (GPA) of Interview Participants ........................ 36

3.2 Types of Survey Items ............................................................................................... 38

3.3 Cronbach’s Alpha Results.......................................................................................... 43

3.4 Validation Plan for Survey Questions........................................................................ 44

4.1 Class Distribution of Survey Participants .................................................................. 56

4.2 Ethnic Distribution of Survey Participants ................................................................ 56

4.3 Age Distribution of Survey Participants .................................................................... 57

4.4 Gender Distribution of Survey Participants ............................................................... 57

4.5 Overall Grade Point Average (GPA) of Survey Participants..................................... 58

4.6 Math Grade Point Average of Survey Participants .................................................... 58

4.7 Descriptive Statistics for Survey Quetion 2 ............................................................... 60

4.8 Correlation of Responses to Survey Question 1 Definitions ..................................... 64

4.9 Correlation of Demographic Variables ...................................................................... 65

4.10 Partial Correlation of Responses to Items 1a and 1g ................................................. 66

4.11 Partial Correlation of Responses to Items 1e and 1i .................................................. 67

4.12 Correlation between Math GPA and Item 1i ............................................................. 68

4.13 A 10x10 Preference Matrix........................................................................................ 69

4.14 Binomial Tests of Significance .................................................................................. 71

4.15 Coded Interview Reasons for Subject Matter Knowledge in Question 1 .................. 74

4.16 Coded Interview Reasons for Mathematical Discourse in Question 1 ...................... 76

4.17 Coded Interview Reasons for Student Achievement in Question 1 ........................... 76

iii

3.1 Mathematics Grade Point Average (GPA) of Interview Participants ........................ 36

3.2 Types of Survey Items ............................................................................................... 38

3.3 Cronbach’s Alpha Results.......................................................................................... 43

3.4 Validation Plan for Survey Questions........................................................................ 44

4.1 Class Distribution of Survey Participants .................................................................. 56

4.2 Ethnic Distribution of Survey Participants ................................................................ 56

4.3 Age Distribution of Survey Participants .................................................................... 57

4.4 Gender Distribution of Survey Participants ............................................................... 57

4.5 Overall Grade Point Average (GPA) of Survey Participants..................................... 58

4.6 Math Grade Point Average of Survey Participants .................................................... 58

4.7 Descriptive Statistics for Survey Quetion 2 ............................................................... 60

4.8 Correlation of Responses to Survey Question 1 Definitions ..................................... 64

4.9 Correlation of Demographic Variables ...................................................................... 65

4.10 Partial Correlation of Responses to Items 1a and 1g ................................................. 66

4.11 Partial Correlation of Responses to Items 1e and 1i .................................................. 67

4.12 Correlation between Math GPA and Item 1i ............................................................. 68

4.13 A 10x10 Preference Matrix........................................................................................ 69

4.14 Binomial Tests of Significance .................................................................................. 71

4.15 Coded Interview Reasons for Subject Matter Knowledge in Question 1 .................. 74

4.16 Coded Interview Reasons for Mathematical Discourse in Question 1 ...................... 76

4.17 Coded Interview Reasons for Student Achievement in Question 1 ........................... 76

iii

8.
4.18 Coded Interview Reasons for High Expectations of Learning in Question 1 ............ 77

4.19 Coded Interview Reasons for Learning Environment in Question 1 ......................... 78

4.20 Coded Interview Reasons for Enthusiasm in Question 1 .......................................... 79

4.21 Coded Interview Reasons for Rapport in Question 1 ................................................ 80

4.22 Coded Interview Alternative Definitions in Question 1 ............................................ 81

4.23 Descriptive Statistics of the Items on Attributes of Good Mathematics Teaching .... 84

4.24 Teaching Model Most Suitable in Middle School ..................................................... 86

4.25 Coded Interview Reasons for Good Classroom Management in Question 2 ............ 88

4.26 Coded Interview Reasons for Clear Explanation in Question 2 ................................ 89

4.27 Coded Interview Reasons for Emphasizing Mathematical Concepts in Question 2 . 91

4.28 Coded Interview Reasons for Posing Questions in Question 2 ................................. 92

4.29 Coded Interview Reasons for Strong in Content Knowledge in Question 2 ............. 93

4.30 Coded Interview Reasons for Mathematical Discourse in Question 2 ...................... 95

4.31 Coded Interview Reasons for How Students Learn Mathematics in Question 2 ....... 96

4.32 Agglomeration Schedule ............................................................................................ 97

4.33 Re-formed Agglomeration Table ............................................................................... 97

4.34 ANOVA Table ........................................................................................................... 99

4.35 Tukey Post-hoc Test .............................................................................................. 100

4.36 Multivariate Tests ................................................................................................... 102

4.37 Eigenvalues and Wilks Lambda .............................................................................. 102

4.38 Strands for 23 items on Survey .............................................................................. 103

4.39 Standardized Canonical Discriminant Function Coefficient ................................. 104

iv

4.19 Coded Interview Reasons for Learning Environment in Question 1 ......................... 78

4.20 Coded Interview Reasons for Enthusiasm in Question 1 .......................................... 79

4.21 Coded Interview Reasons for Rapport in Question 1 ................................................ 80

4.22 Coded Interview Alternative Definitions in Question 1 ............................................ 81

4.23 Descriptive Statistics of the Items on Attributes of Good Mathematics Teaching .... 84

4.24 Teaching Model Most Suitable in Middle School ..................................................... 86

4.25 Coded Interview Reasons for Good Classroom Management in Question 2 ............ 88

4.26 Coded Interview Reasons for Clear Explanation in Question 2 ................................ 89

4.27 Coded Interview Reasons for Emphasizing Mathematical Concepts in Question 2 . 91

4.28 Coded Interview Reasons for Posing Questions in Question 2 ................................. 92

4.29 Coded Interview Reasons for Strong in Content Knowledge in Question 2 ............. 93

4.30 Coded Interview Reasons for Mathematical Discourse in Question 2 ...................... 95

4.31 Coded Interview Reasons for How Students Learn Mathematics in Question 2 ....... 96

4.32 Agglomeration Schedule ............................................................................................ 97

4.33 Re-formed Agglomeration Table ............................................................................... 97

4.34 ANOVA Table ........................................................................................................... 99

4.35 Tukey Post-hoc Test .............................................................................................. 100

4.36 Multivariate Tests ................................................................................................... 102

4.37 Eigenvalues and Wilks Lambda .............................................................................. 102

4.38 Strands for 23 items on Survey .............................................................................. 103

4.39 Standardized Canonical Discriminant Function Coefficient ................................. 104

iv

9.
4.40 Functions at Group Centroids ...................................................................................106

4.41 Classification Results ............................................................................................... 106

4.42 ANOVA .................................................................................................................. 108

4.43 Variables and Culster Means .................................................................................. 109

4.44 Post Hoc Tukey Test .............................................................................................. 110

4.45 Descriptive Statistics for Research Question 3 ..........................................................112

4.46 Percentage of Responses for Research Question 3 ....................................................113

4.47 Coded Interview Reasons for Immediate Classroom Situation in Question 3...........116

4.48 Coded Interview Reasons for Colleagues in Question 3 ...........................................118

4.49 Coded Interview Reasons for Personality and Experiences Growing Up in

Question 3 ..................................................................................................................120

4.50 Coded Interview Reasons for Mathematical Background in Question 3 ...................121

4.51 Coded Interview Reasons for Teacher Education in Question 3 ...............................122

4.52 Coded Interview Reasons for Mathematical Beliefs in Question 3 ...........................123

v

4.41 Classification Results ............................................................................................... 106

4.42 ANOVA .................................................................................................................. 108

4.43 Variables and Culster Means .................................................................................. 109

4.44 Post Hoc Tukey Test .............................................................................................. 110

4.45 Descriptive Statistics for Research Question 3 ..........................................................112

4.46 Percentage of Responses for Research Question 3 ....................................................113

4.47 Coded Interview Reasons for Immediate Classroom Situation in Question 3...........116

4.48 Coded Interview Reasons for Colleagues in Question 3 ...........................................118

4.49 Coded Interview Reasons for Personality and Experiences Growing Up in

Question 3 ..................................................................................................................120

4.50 Coded Interview Reasons for Mathematical Background in Question 3 ...................121

4.51 Coded Interview Reasons for Teacher Education in Question 3 ...............................122

4.52 Coded Interview Reasons for Mathematical Beliefs in Question 3 ...........................123

v

10.
LIST OF FIGURES

2.1 The Matched Task Framework .................................................................................. 18

3.1 Research Participants and Instruments ...................................................................... 47

4.1 Distribution of Responses to Survey Question 1 ....................................................... 61

4.2 Four Teaching Models ............................................................................................... 85

4.3 Dendogram Using Ward Method ............................................................................... 98

4.4 Scatterplot of Discriminant Functions .................................................................... 107

vi

2.1 The Matched Task Framework .................................................................................. 18

3.1 Research Participants and Instruments ...................................................................... 47

4.1 Distribution of Responses to Survey Question 1 ....................................................... 61

4.2 Four Teaching Models ............................................................................................... 85

4.3 Dendogram Using Ward Method ............................................................................... 98

4.4 Scatterplot of Discriminant Functions .................................................................... 107

vi

11.
ACKNOWLEDGMENTS

First of all, I am very grateful to God for his blessings. This enabled me to carry out the

research with zest and zeal. I am highly indebted to my sponsor, Professor Alexander Karp, for

guiding me throughout the wonderful journey of my doctoral program. I also thank Professor

Karp for his valuable guidance in defining and refining the research problem and suggesting

appropriate data analysis procedures, and for his valuable comments. Thank you very much for

being generous with your time and knowledge.

My deepest gratitude to Professor Bruce Vogeli, my advisor and the second reader, for

encouraging me when I started my doctoral studies. I will remain grateful to Professor Vogeli for

his emotional and constant encouragement during the doctoral program.

I also wish to thank my Ph.D. defense committee members Professor Erica Walker,

Professor Matthew Johnson and Professor Patrick Gallagher, for their valuable contributions that

helped improve my dissertation. I also owe my thanks to all the faculty members, Professor

Stuart Weinberg, Professor Peter Garrity and Professor Henry Pollak who enriched my

experiences during the course of my studies at Teachers College. I also wish to appreciate Ms.

Krystle Hecker, program academic secretary, for assisting me during many occasions while

dealing with the administrative stuff.

My sincere thanks are due to Professor Noraini Idris from the Facuty of Education,

University of Malaya, for encouraging me to pursue my doctoral studies. Without her

encouragement and support, I would not have had the opportunity to experience this journey.

Professor Nik Azis Nik Pa and Professor Siow Heng Loke, from University of Malaya, thank

you for your guidance and advice.

vii

First of all, I am very grateful to God for his blessings. This enabled me to carry out the

research with zest and zeal. I am highly indebted to my sponsor, Professor Alexander Karp, for

guiding me throughout the wonderful journey of my doctoral program. I also thank Professor

Karp for his valuable guidance in defining and refining the research problem and suggesting

appropriate data analysis procedures, and for his valuable comments. Thank you very much for

being generous with your time and knowledge.

My deepest gratitude to Professor Bruce Vogeli, my advisor and the second reader, for

encouraging me when I started my doctoral studies. I will remain grateful to Professor Vogeli for

his emotional and constant encouragement during the doctoral program.

I also wish to thank my Ph.D. defense committee members Professor Erica Walker,

Professor Matthew Johnson and Professor Patrick Gallagher, for their valuable contributions that

helped improve my dissertation. I also owe my thanks to all the faculty members, Professor

Stuart Weinberg, Professor Peter Garrity and Professor Henry Pollak who enriched my

experiences during the course of my studies at Teachers College. I also wish to appreciate Ms.

Krystle Hecker, program academic secretary, for assisting me during many occasions while

dealing with the administrative stuff.

My sincere thanks are due to Professor Noraini Idris from the Facuty of Education,

University of Malaya, for encouraging me to pursue my doctoral studies. Without her

encouragement and support, I would not have had the opportunity to experience this journey.

Professor Nik Azis Nik Pa and Professor Siow Heng Loke, from University of Malaya, thank

you for your guidance and advice.

vii

12.
I thank the Ministry of Higher Education Malaysia and University of Malaya for

providing funding to complete my doctoral program at Columbia University, Teachers College.

I would also like to thank the administrative staff at the Chancellery and Treasury at Univerity of

Malaya for their support.

I acknowledge the help rendered by my friends, Nicholas Wasserman, Edward Ham,

Lydia Jo, Intzar Butt, Mark Causapin, Katherine Rocco and Kai Chung Tam, for assisting me in

one way or another while completing this dissertation. This journey may have not been so

smooth without their contribution and commitment. Thank you also for making me feel at home

and making things fun during my stay in New York.

Last but not the least; I earnestly thank my parents David Leong and Cindy Khoo for

their support and believing that I could succeed in this tough journey. To my brother, Kwan

Seng, and my friend, Jing Yi, for all the social and emotional support that they rendered above

any doubt.

Ronny Kwan Eu Leong

viii

providing funding to complete my doctoral program at Columbia University, Teachers College.

I would also like to thank the administrative staff at the Chancellery and Treasury at Univerity of

Malaya for their support.

I acknowledge the help rendered by my friends, Nicholas Wasserman, Edward Ham,

Lydia Jo, Intzar Butt, Mark Causapin, Katherine Rocco and Kai Chung Tam, for assisting me in

one way or another while completing this dissertation. This journey may have not been so

smooth without their contribution and commitment. Thank you also for making me feel at home

and making things fun during my stay in New York.

Last but not the least; I earnestly thank my parents David Leong and Cindy Khoo for

their support and believing that I could succeed in this tough journey. To my brother, Kwan

Seng, and my friend, Jing Yi, for all the social and emotional support that they rendered above

any doubt.

Ronny Kwan Eu Leong

viii

13.
DEDICATION

This dissertation is dedicated to my parents,

David Leong and Cindy Khoo and

my younger brother, Kwan Seng,

for always supporting, motivating, and inspiring me

to complete my dissertation.

Without their guidance and prayers,

none of this would have been possible.

ix

This dissertation is dedicated to my parents,

David Leong and Cindy Khoo and

my younger brother, Kwan Seng,

for always supporting, motivating, and inspiring me

to complete my dissertation.

Without their guidance and prayers,

none of this would have been possible.

ix

14.
1

Chapter 1

INTRODUCTION

Good mathematics teaching has been described by many educators for decades (Cooney,

2005; Krainer, 2005; Murphy, 2004; Thompson, 1992). The National Council of Mathematics

Teachers (NCTM) (1989, 1991, 1995, 2000) has discussed good mathematics teaching in

numerous documents. NCTM (2000) asserts in its teaching principle that “effective mathematics

teaching requires understanding what students know and need to learn and then challenging and

supporting them to learn it well” (p. 10). Three requirements of effective teaching provided by

NCTM were: a) knowing and understanding mathematics, students as learners, and pedagogical

strategies; b) a challenging and supportive classroom learning environment; and c) continually

seeking improvement (NCTM, 2000). One way of developing good mathematics teaching is by

promoting the five strands of mathematical proficiency set by the National Research Council

(NRC) (2001) that includes: conceptual understanding, procedural fluency, strategic competence,

adaptive reasoning, and productive disposition (p. 116). More studies are needed to understand

the meaning and development of good mathematics teaching (Cooney, 2005; Krainer, 2005).

Need for the Study

The main goal of a teacher education program is to produce effective mathematics

teachers, but researchers have found that many teachers feel a disconnect between what they

have learned and what really happens in the actual classroom (Brown & Borko, 1992; Cooney,

2005). In the United States, there are different ways to obtain teacher certification for the

secondary level and no uniform body regulates the number of mathematics courses a prospective

teacher should take (Stacey, 2008). Teachers can receive certification by following the traditional

Chapter 1

INTRODUCTION

Good mathematics teaching has been described by many educators for decades (Cooney,

2005; Krainer, 2005; Murphy, 2004; Thompson, 1992). The National Council of Mathematics

Teachers (NCTM) (1989, 1991, 1995, 2000) has discussed good mathematics teaching in

numerous documents. NCTM (2000) asserts in its teaching principle that “effective mathematics

teaching requires understanding what students know and need to learn and then challenging and

supporting them to learn it well” (p. 10). Three requirements of effective teaching provided by

NCTM were: a) knowing and understanding mathematics, students as learners, and pedagogical

strategies; b) a challenging and supportive classroom learning environment; and c) continually

seeking improvement (NCTM, 2000). One way of developing good mathematics teaching is by

promoting the five strands of mathematical proficiency set by the National Research Council

(NRC) (2001) that includes: conceptual understanding, procedural fluency, strategic competence,

adaptive reasoning, and productive disposition (p. 116). More studies are needed to understand

the meaning and development of good mathematics teaching (Cooney, 2005; Krainer, 2005).

Need for the Study

The main goal of a teacher education program is to produce effective mathematics

teachers, but researchers have found that many teachers feel a disconnect between what they

have learned and what really happens in the actual classroom (Brown & Borko, 1992; Cooney,

2005). In the United States, there are different ways to obtain teacher certification for the

secondary level and no uniform body regulates the number of mathematics courses a prospective

teacher should take (Stacey, 2008). Teachers can receive certification by following the traditional

15.
2

route which requires teachers to take educational courses in colleges or universities and the

alternative certification program, where teachers generally have a short training of one to two

months before teaching and completing the certification requirements. Traditional certification

programs generally require teachers to have a longer studying and training time, while alternative

certification programs are intended to reduce the teacher shortage and produce qualified teachers

in a shorter time compared to the traditional certification route (Zumwalt & Craig, 2005).

The depth and number of mathematics and pedagogy courses vary according to

institutions or programs (Stacey, 2008). To assist certification programs in producing good

teachers, the Conference Board of the Mathematical Sciences (CBMS) (2001) suggests four

recommendations about mathematics courses for pre-service teachers: a) deep understanding of

mathematics; b) quality of mathematics preparation is more important than quantity; c) develop

basic mathematical ideas through reasoning and solving problems; and d) develop habits of

mathematical thinking and flexible teaching style. CBMS stresses that for good mathematics

teaching to happen, “we need more content in school mathematics instruction than most realize,

content that teachers need to understand well” (p. 3). This indicates the importance of teachers

understanding subject matter knowledge in order for good teaching to happen.

Carroll (2005) found that only one fourth of a total of 108 teachers attributed their

development as effective mathematics teachers to the teacher education they received. Most

beginning teachers do not adopt what they learn in teacher preparation programs, but “continue

to adopt the instructional practices of their cooperating teachers, many of whom still model and

encourage traditional, direct instruction” (Frykholm, 1999, p. 24). Frykholm concludes that

further investigation should be done on the connection between method courses and field

route which requires teachers to take educational courses in colleges or universities and the

alternative certification program, where teachers generally have a short training of one to two

months before teaching and completing the certification requirements. Traditional certification

programs generally require teachers to have a longer studying and training time, while alternative

certification programs are intended to reduce the teacher shortage and produce qualified teachers

in a shorter time compared to the traditional certification route (Zumwalt & Craig, 2005).

The depth and number of mathematics and pedagogy courses vary according to

institutions or programs (Stacey, 2008). To assist certification programs in producing good

teachers, the Conference Board of the Mathematical Sciences (CBMS) (2001) suggests four

recommendations about mathematics courses for pre-service teachers: a) deep understanding of

mathematics; b) quality of mathematics preparation is more important than quantity; c) develop

basic mathematical ideas through reasoning and solving problems; and d) develop habits of

mathematical thinking and flexible teaching style. CBMS stresses that for good mathematics

teaching to happen, “we need more content in school mathematics instruction than most realize,

content that teachers need to understand well” (p. 3). This indicates the importance of teachers

understanding subject matter knowledge in order for good teaching to happen.

Carroll (2005) found that only one fourth of a total of 108 teachers attributed their

development as effective mathematics teachers to the teacher education they received. Most

beginning teachers do not adopt what they learn in teacher preparation programs, but “continue

to adopt the instructional practices of their cooperating teachers, many of whom still model and

encourage traditional, direct instruction” (Frykholm, 1999, p. 24). Frykholm concludes that

further investigation should be done on the connection between method courses and field

16.
3

experiences and how this develops the thinking of successful beginning teachers in their

classroom practices.

Teacher education programs also aim to have more reflective and adaptive approaches in

educating pre-service teachers, but some are rather unsuccessful, only providing pre-service

teachers with the knowledge of specific teaching methods (Cooney, 2001). Pre-service teachers

also look at the NCTM Standards as a content guide rather than as a philosophy of teaching

(Frykholm, 1999). The disconnect between what was learned in teacher education programs and

classroom practice was further discussed in that study. More studies on pedagogical content

knowledge (Shulman, 1986) and mathematics knowledge for teaching (Ball, 2005) are needed to

provide teacher education programs with more ways to address the pedagogy of good

mathematics teaching.

What are the attributes of being an effective mathematics teacher? An effective

mathematics teacher is able to stimulate student learning of mathematics, as described in the

NCTM (1991) Professional Standards on Teaching Mathematics. In addition, teachers should

focus on mathematical reasoning, problem-solving, communication, and connections, and not on

memorization and manipulation of symbols, and computational algorithms. Studies of the

characteristics of good teaching have been done at the elementary level that: a) compare pre-

service and in-service teachers’ beliefs (Murphy, 2004); b) investigate the views of primary

teachers on the factors that contribute to their good mathematics teaching (Caroll, 2004); and

c) examine factors that influence a group of pre-service teachers to construct their ideas about

teaching mathematics (Ridener, 1995). Studies done at the secondary level have focused on:

a) the relationship between teachers’ beliefs and classroom practices (Brown, 1986; Cooney,

1985; Thompson, 1992); and b) the factors that develop good mathematics teaching (Cooney,

experiences and how this develops the thinking of successful beginning teachers in their

classroom practices.

Teacher education programs also aim to have more reflective and adaptive approaches in

educating pre-service teachers, but some are rather unsuccessful, only providing pre-service

teachers with the knowledge of specific teaching methods (Cooney, 2001). Pre-service teachers

also look at the NCTM Standards as a content guide rather than as a philosophy of teaching

(Frykholm, 1999). The disconnect between what was learned in teacher education programs and

classroom practice was further discussed in that study. More studies on pedagogical content

knowledge (Shulman, 1986) and mathematics knowledge for teaching (Ball, 2005) are needed to

provide teacher education programs with more ways to address the pedagogy of good

mathematics teaching.

What are the attributes of being an effective mathematics teacher? An effective

mathematics teacher is able to stimulate student learning of mathematics, as described in the

NCTM (1991) Professional Standards on Teaching Mathematics. In addition, teachers should

focus on mathematical reasoning, problem-solving, communication, and connections, and not on

memorization and manipulation of symbols, and computational algorithms. Studies of the

characteristics of good teaching have been done at the elementary level that: a) compare pre-

service and in-service teachers’ beliefs (Murphy, 2004); b) investigate the views of primary

teachers on the factors that contribute to their good mathematics teaching (Caroll, 2004); and

c) examine factors that influence a group of pre-service teachers to construct their ideas about

teaching mathematics (Ridener, 1995). Studies done at the secondary level have focused on:

a) the relationship between teachers’ beliefs and classroom practices (Brown, 1986; Cooney,

1985; Thompson, 1992); and b) the factors that develop good mathematics teaching (Cooney,

17.
4

2005, Ham, 2011). There is a need for more studies at the secondary level to obtain more

information on the attributes of good mathematics teaching (Ham, 2011; Wasserman, 2011).

Polya (1962) asserts that “If the teacher is bored by what he is teaching, it is a certainty

that all his students will be too” (p. 60). It is important for teachers to know their subject matter,

as explained by Polya (1981) in his Ten Commandments for Teachers. Thom (1973) asserts that

the conception of mathematics influences how one perceives the preferred way of teaching and

learning of mathematics. Thompson (1992) points out that there are not many common

definitions of what constitutes good teaching. Shulman (2001) explains that good teaching

“relies on whether teachers have a deep and flexible understanding of what they are teaching”

(p. 1). Cooney (2005) fills in the gap on what constitutes good teaching. The study used the

perspectives of nine experienced mathematics teachers.

Teachers without adequate content knowledge spend more time learning the content

instead of planning the lesson to enhance student understanding (Brown & Borko, 1992). These

authors add that teachers with strong content knowledge are able to explain the concepts instead

of just the mathematical procedures. Caroll (2007) and Nickson (1998) have different views and

argue that how the content knowledge was acquired makes the difference and not the level of

content knowledge. Another study found that even though content knowledge is important,

teachers also require knowledge of students and learning to be effective (Shulman, 1986).

Several studies have also connected students’ conception of learning and descriptions of

good teaching in college (Marton & Saljo, 1984; Rossum & Taylor, 1987). Rossum and Taylor

(1987) mention that the perception of college students on good teaching is “presenting the

subject matter in such a way that those who were already interested remain so, or become more

so” (p. 18). Looking at studies done by Brown and Borko (1992), Sowder (2007), and Wilson et

2005, Ham, 2011). There is a need for more studies at the secondary level to obtain more

information on the attributes of good mathematics teaching (Ham, 2011; Wasserman, 2011).

Polya (1962) asserts that “If the teacher is bored by what he is teaching, it is a certainty

that all his students will be too” (p. 60). It is important for teachers to know their subject matter,

as explained by Polya (1981) in his Ten Commandments for Teachers. Thom (1973) asserts that

the conception of mathematics influences how one perceives the preferred way of teaching and

learning of mathematics. Thompson (1992) points out that there are not many common

definitions of what constitutes good teaching. Shulman (2001) explains that good teaching

“relies on whether teachers have a deep and flexible understanding of what they are teaching”

(p. 1). Cooney (2005) fills in the gap on what constitutes good teaching. The study used the

perspectives of nine experienced mathematics teachers.

Teachers without adequate content knowledge spend more time learning the content

instead of planning the lesson to enhance student understanding (Brown & Borko, 1992). These

authors add that teachers with strong content knowledge are able to explain the concepts instead

of just the mathematical procedures. Caroll (2007) and Nickson (1998) have different views and

argue that how the content knowledge was acquired makes the difference and not the level of

content knowledge. Another study found that even though content knowledge is important,

teachers also require knowledge of students and learning to be effective (Shulman, 1986).

Several studies have also connected students’ conception of learning and descriptions of

good teaching in college (Marton & Saljo, 1984; Rossum & Taylor, 1987). Rossum and Taylor

(1987) mention that the perception of college students on good teaching is “presenting the

subject matter in such a way that those who were already interested remain so, or become more

so” (p. 18). Looking at studies done by Brown and Borko (1992), Sowder (2007), and Wilson et

18.
5

al. (2005) can inform educators on what is good teaching and how it is developed along the main

theme of teacher education. A greater understanding of “good mathematics teaching” and beliefs

of high school teachers will add to the body of literature.

Arbaugh (2011) asserts that the two attributes of mathematics teachers that are essential

to student learning are the teacher’s knowledge of teaching and the teacher’s belief about

teaching and learning mathematics. With these attributes, classroom teaching is better. What

kinds of knowledge are important for effective mathematics teaching? Studies indicate that

identifying kinds of knowledge is pertinent for the mathematics education community, especially

for professors training pre-service teachers and in-service teachers at the university (Arbaugh,

2009, 2010; Lampert, 2002).

Wasserman (2011) examined how beginning secondary mathematics teachers defined

success and the attributes of good teaching. The sample was from a traditional certification

program. Ham (2011) conducted a similar study using a sample from an alternative certification

program. Both studies identified several important attributes of good mathematics teaching and

when success was acquired. More studies on the definition and attributes of good mathematics

teaching from the perspectives of beginning secondary teachers would contribute to the body of

literature on teacher education.

Teachers’ content knowledge is important in the teaching of mathematics, but other

forms of knowledge such as pedagogical content knowledge (Shulman, 1986) also are pertinent.

Other studies of teacher education have shown that different kinds of knowledge are needed by

teachers to be effective such as: 1) theory of knowledge (Schoenfeld, 1999); 2) teacher

knowledge and its impact (Fennema & Franke, 1992); and 3) mathematics knowledge for

teaching (Ball & Bass, 2004) and for elementary school teachers and their content knowledge

al. (2005) can inform educators on what is good teaching and how it is developed along the main

theme of teacher education. A greater understanding of “good mathematics teaching” and beliefs

of high school teachers will add to the body of literature.

Arbaugh (2011) asserts that the two attributes of mathematics teachers that are essential

to student learning are the teacher’s knowledge of teaching and the teacher’s belief about

teaching and learning mathematics. With these attributes, classroom teaching is better. What

kinds of knowledge are important for effective mathematics teaching? Studies indicate that

identifying kinds of knowledge is pertinent for the mathematics education community, especially

for professors training pre-service teachers and in-service teachers at the university (Arbaugh,

2009, 2010; Lampert, 2002).

Wasserman (2011) examined how beginning secondary mathematics teachers defined

success and the attributes of good teaching. The sample was from a traditional certification

program. Ham (2011) conducted a similar study using a sample from an alternative certification

program. Both studies identified several important attributes of good mathematics teaching and

when success was acquired. More studies on the definition and attributes of good mathematics

teaching from the perspectives of beginning secondary teachers would contribute to the body of

literature on teacher education.

Teachers’ content knowledge is important in the teaching of mathematics, but other

forms of knowledge such as pedagogical content knowledge (Shulman, 1986) also are pertinent.

Other studies of teacher education have shown that different kinds of knowledge are needed by

teachers to be effective such as: 1) theory of knowledge (Schoenfeld, 1999); 2) teacher

knowledge and its impact (Fennema & Franke, 1992); and 3) mathematics knowledge for

teaching (Ball & Bass, 2004) and for elementary school teachers and their content knowledge

19.
6

(Ball, 2004, 2007; Brown & Borko, 1992; Ma, 1999). Much research has focused on content

knowledge, but little is known about the the connection between pedagogical content knowledge

and good mathematics teaching (Chamberlin, 2005). Ball (2007) conducted many studies on this

issue and then developed the concept of Mathematics Knowledge for Teaching (MKT), which is

defined as “mathematical knowledge needed to carry out the work of teaching mathematics”

(Ball et al., 2009, p. 96) to bridge the gap in good teaching. The researcher divided MKT into

subject matter knowledge and pedagogical content knowledge.

Murphy (2004) explored beliefs about the characteristics of good teaching. The study was

carried out on pre-service teachers, in-service teachers, and second graders using a combination

of survey, drawing diagrams, and interviews. Beginning secondary school teachers’ perception

of good mathematics teaching and some connections with content knowledge have also been

studied (Murphy, 2004; Sowder, 2005).

One interesting area to investigate in good mathematics teaching is understanding how

this concept is influenced by the age and mathematics background of the teacher. It is also

important to know how beginning teachers acquire and develop good mathematics teaching.

Many studies have been done to investigate the connection between effective teaching and

teachers’ knowledge of mathematics (Ball, 2005; Brown & Borko, 1992; Conney, 2005). The

better the understanding of how good mathematics teaching is developed, the more teacher

education programs and in-service training can be improved (Cooney, 2001; Frykholm, 1999;

Shulman, 2001).

Purpose of Study

The purpose of this study was to determine how beginning teachers define good

mathematics teaching and what they report to be the most important attributes at the secondary

(Ball, 2004, 2007; Brown & Borko, 1992; Ma, 1999). Much research has focused on content

knowledge, but little is known about the the connection between pedagogical content knowledge

and good mathematics teaching (Chamberlin, 2005). Ball (2007) conducted many studies on this

issue and then developed the concept of Mathematics Knowledge for Teaching (MKT), which is

defined as “mathematical knowledge needed to carry out the work of teaching mathematics”

(Ball et al., 2009, p. 96) to bridge the gap in good teaching. The researcher divided MKT into

subject matter knowledge and pedagogical content knowledge.

Murphy (2004) explored beliefs about the characteristics of good teaching. The study was

carried out on pre-service teachers, in-service teachers, and second graders using a combination

of survey, drawing diagrams, and interviews. Beginning secondary school teachers’ perception

of good mathematics teaching and some connections with content knowledge have also been

studied (Murphy, 2004; Sowder, 2005).

One interesting area to investigate in good mathematics teaching is understanding how

this concept is influenced by the age and mathematics background of the teacher. It is also

important to know how beginning teachers acquire and develop good mathematics teaching.

Many studies have been done to investigate the connection between effective teaching and

teachers’ knowledge of mathematics (Ball, 2005; Brown & Borko, 1992; Conney, 2005). The

better the understanding of how good mathematics teaching is developed, the more teacher

education programs and in-service training can be improved (Cooney, 2001; Frykholm, 1999;

Shulman, 2001).

Purpose of Study

The purpose of this study was to determine how beginning teachers define good

mathematics teaching and what they report to be the most important attributes at the secondary

20.
7

level. This research explored whether there was a relationship between demographics of the

participants and their perspectives about the attributes of good teaching. In addition, factors that

influence the understanding of good mathematics teaching were explored. This study looked at

where to attribute the elements of good mathematics teaching: to some personality trait, to

teacher’s classroom behavior, to teacher’s mathematical knowledge or to teacher education

This study sought to answer the following research questions:

1. How do beginning teachers define “good mathematics teaching”?

2. How do beginning teachers describe “good mathematics teaching” in middle school

and high school? What are the important attributes of good mathematics teaching?

3. Is there any relationship between demographics (e.g., Age, Math GPA, Overall GPA)

and descriptions of “good mathematics teaching” attributes?

4. What are the factors that influence beginning teachers’ understanding of good

mathematics teaching?

Conceptual Framework

The NCTM (1991) Professional Standards for Teaching Mathematics was a document

intended to guide educators to develop professionalism in mathematics teaching. The Standards

emphasized the important decisions that a teacher needs to make in a mathematics lesson to

reach the teaching goals. In order for good mathematics teaching to work, NCTM proposed six

Standards for the teaching of mathematics organized under four categories. The four main

categories were: Tasks, Discourse, Environment, and Analysis of Teaching and Learning. They

also indicated what each category means:

level. This research explored whether there was a relationship between demographics of the

participants and their perspectives about the attributes of good teaching. In addition, factors that

influence the understanding of good mathematics teaching were explored. This study looked at

where to attribute the elements of good mathematics teaching: to some personality trait, to

teacher’s classroom behavior, to teacher’s mathematical knowledge or to teacher education

This study sought to answer the following research questions:

1. How do beginning teachers define “good mathematics teaching”?

2. How do beginning teachers describe “good mathematics teaching” in middle school

and high school? What are the important attributes of good mathematics teaching?

3. Is there any relationship between demographics (e.g., Age, Math GPA, Overall GPA)

and descriptions of “good mathematics teaching” attributes?

4. What are the factors that influence beginning teachers’ understanding of good

mathematics teaching?

Conceptual Framework

The NCTM (1991) Professional Standards for Teaching Mathematics was a document

intended to guide educators to develop professionalism in mathematics teaching. The Standards

emphasized the important decisions that a teacher needs to make in a mathematics lesson to

reach the teaching goals. In order for good mathematics teaching to work, NCTM proposed six

Standards for the teaching of mathematics organized under four categories. The four main

categories were: Tasks, Discourse, Environment, and Analysis of Teaching and Learning. They

also indicated what each category means:

21.
8

a) Tasks: projects, questions, problems, constructions, applications, and exercises in

which students engage.

b) Discourse: ways of representing, thinking, talking, and agreeing and disagreeing that

teachers and students use to engage in those tasks.

c) Environment: the setting or learning; it is the context in which the tasks and discourse

are embedded.

d) Analysis: the systematic reflection in which the teachers engage; entails the ongoing

monitoring of classroom life—how well the tasks, discourse, and environment foster

the development of every student’s mathematical literacy and power (NCTM, 1991,

p. 22).

Under the main category Tasks, the Standard was Worthwhile Mathematical Tasks.

Posing tasks that elicit students’ knowledge and experiences in mathematics should be one of the

teacher’s main responsibilities. Tasks should be based on the different ways students learn

mathematics which would be beneficial in the teaching process. Teachers plan classroom

activities involving students engaging in tasks that encourage reasoning and connecting

mathematical ideas.

Discourse is divided into three types: Teacher’s Role in Discourse, which includes posing

questions that elicit students’ thinking and reasoning in mathematics; Student’s Role in

Discourse, which involves the teacher promoting classroom discourse in which students question

teachers and make conjectures about mathematical ideas; and Tools for Enhancing Discourse,

which focuses on the use of technology, concrete materials, and enhancing explanations and

arguments of mathematical concepts.

a) Tasks: projects, questions, problems, constructions, applications, and exercises in

which students engage.

b) Discourse: ways of representing, thinking, talking, and agreeing and disagreeing that

teachers and students use to engage in those tasks.

c) Environment: the setting or learning; it is the context in which the tasks and discourse

are embedded.

d) Analysis: the systematic reflection in which the teachers engage; entails the ongoing

monitoring of classroom life—how well the tasks, discourse, and environment foster

the development of every student’s mathematical literacy and power (NCTM, 1991,

p. 22).

Under the main category Tasks, the Standard was Worthwhile Mathematical Tasks.

Posing tasks that elicit students’ knowledge and experiences in mathematics should be one of the

teacher’s main responsibilities. Tasks should be based on the different ways students learn

mathematics which would be beneficial in the teaching process. Teachers plan classroom

activities involving students engaging in tasks that encourage reasoning and connecting

mathematical ideas.

Discourse is divided into three types: Teacher’s Role in Discourse, which includes posing

questions that elicit students’ thinking and reasoning in mathematics; Student’s Role in

Discourse, which involves the teacher promoting classroom discourse in which students question

teachers and make conjectures about mathematical ideas; and Tools for Enhancing Discourse,

which focuses on the use of technology, concrete materials, and enhancing explanations and

arguments of mathematical concepts.

22.
9

Learning Environment is a Standard under the Environment category. It explains how

teachers should create a learning environment that encourages the development of students’

mathematical power. This can be done by structuring the lesson, valuing students’ ideas, and

using materials to enhance the learning of mathematics. The Analysis of Teaching and Learning

highlights the analysis of students’ learning by observing and listening to gauge students’

learning. This could be done by examining the effects of the tasks, discourses, and learning

environment of students’ mathematical knowledge.

Cooney et al.’s (2005) study revealed what constitutes good mathematics teaching and

how it develops from the perspective of experienced high school teachers. Good mathematics

teaching requires prerequisite teacher knowledge, promotes mathematical understanding,

engages and motivates students, and requires effective management skills. Prerequisite

knowledge refers to teachers’ mathematical knowledge as well as knowledge of students’

mathematics so that they can teach well. Promoting mathematical understanding emphasizes the

goal of teachers for their students to understand the mathematics in the classroom. This

mathematical understanding could be procedural, conceptual or connected to the nature of

mathematics. In addition, visualizing mathematics with learning tools like computers and

calculators; connecting mathematical topics; refraining from speaking so that teachers do not

provide information that requires students memorizing formulas; and the importance of assessing

students’ understanding were important attributes that promoted mathematical understanding.

Good mathematics teaching engages and motivates students. This can be done by using

various pedagogical approaches in classrooms like group work, technology, writing

mathematics, and hands-on activities. Students physically moving in the classroom during

activities was another technique to engage students. Another way of doing this was by

Learning Environment is a Standard under the Environment category. It explains how

teachers should create a learning environment that encourages the development of students’

mathematical power. This can be done by structuring the lesson, valuing students’ ideas, and

using materials to enhance the learning of mathematics. The Analysis of Teaching and Learning

highlights the analysis of students’ learning by observing and listening to gauge students’

learning. This could be done by examining the effects of the tasks, discourses, and learning

environment of students’ mathematical knowledge.

Cooney et al.’s (2005) study revealed what constitutes good mathematics teaching and

how it develops from the perspective of experienced high school teachers. Good mathematics

teaching requires prerequisite teacher knowledge, promotes mathematical understanding,

engages and motivates students, and requires effective management skills. Prerequisite

knowledge refers to teachers’ mathematical knowledge as well as knowledge of students’

mathematics so that they can teach well. Promoting mathematical understanding emphasizes the

goal of teachers for their students to understand the mathematics in the classroom. This

mathematical understanding could be procedural, conceptual or connected to the nature of

mathematics. In addition, visualizing mathematics with learning tools like computers and

calculators; connecting mathematical topics; refraining from speaking so that teachers do not

provide information that requires students memorizing formulas; and the importance of assessing

students’ understanding were important attributes that promoted mathematical understanding.

Good mathematics teaching engages and motivates students. This can be done by using

various pedagogical approaches in classrooms like group work, technology, writing

mathematics, and hands-on activities. Students physically moving in the classroom during

activities was another technique to engage students. Another way of doing this was by

23.
10

challenging students at their mathematical level, even though they might feel uncomfortable.

Effective management involved keeping students under control so that the lesson could proceed

smoothly. This requires certain skills such as flexibility in the pedagogical content knowledge

that utilizes a variety of approaches.

The researcher based the concepts of good mathematics teaching on NCTM’s (1991)

Professional Standards and Cooney et al.’s (2005) study. Furthermore, the development of the

survey questionnaire was based on the concepts provided by both of these studies.

Procedures of the Study

Generally, teachers can receive certification in two ways: a traditional certification

program, through which potential teachers enroll in college or university-based education

courses; or an alternative certification program, where potential teachers gain their certification

through programs other than traditional four-year undergraduate education programs. The

traditional certification program offered by this graduate school of education was intended for

two fundamental purposes: producing teachers certified in the state for teaching secondary

mathematics and equipping teachers with strong content in mathematics. The participants of this

study were selected from a traditional certification program from a mathematics education

program. The certification program was part of the master’s program offered by a graduate

school of education, located in an eastern state in the United States. The participants of this study

graduated from the master’s program in mathematics education and were also certified to teach

secondary mathematics in this eastern state. As this study focused on the factors and attributes of

good mathematics teaching, being a beginning mathematics teacher required the teacher to have

challenging students at their mathematical level, even though they might feel uncomfortable.

Effective management involved keeping students under control so that the lesson could proceed

smoothly. This requires certain skills such as flexibility in the pedagogical content knowledge

that utilizes a variety of approaches.

The researcher based the concepts of good mathematics teaching on NCTM’s (1991)

Professional Standards and Cooney et al.’s (2005) study. Furthermore, the development of the

survey questionnaire was based on the concepts provided by both of these studies.

Procedures of the Study

Generally, teachers can receive certification in two ways: a traditional certification

program, through which potential teachers enroll in college or university-based education

courses; or an alternative certification program, where potential teachers gain their certification

through programs other than traditional four-year undergraduate education programs. The

traditional certification program offered by this graduate school of education was intended for

two fundamental purposes: producing teachers certified in the state for teaching secondary

mathematics and equipping teachers with strong content in mathematics. The participants of this

study were selected from a traditional certification program from a mathematics education

program. The certification program was part of the master’s program offered by a graduate

school of education, located in an eastern state in the United States. The participants of this study

graduated from the master’s program in mathematics education and were also certified to teach

secondary mathematics in this eastern state. As this study focused on the factors and attributes of

good mathematics teaching, being a beginning mathematics teacher required the teacher to have

24.
11

just completed the first or second year of classroom teaching mathematics at the secondary

school level.

As the samples of teachers were obtained from one graduate school of education, there

needed to be a distinction to identify teachers with good mathematics teaching. This helped to

obtain answers for the research questions on the attributes or factors that influence the

understanding of good mathematics teaching. Candidates for this study were beginning

mathematics teachers with strong content knowledge, as reflected by their college mathematics

GPA, pedagogical content knowledge, and recommendations by college professors. The subjects

were graduate students in this program who had at least a bachelor’s degree in college

mathematics and a strong background in the subject, having taken at least 24 credits in

mathematics content courses (including two semesters of calculus) and earned a good Grade

Point Average (GPA) score at the undergraduate level. As a requirement, the teachers had to

complete 100 hours of class observation and also have 120 hours of classroom teaching. The

teachers also received 12 months of intensive pedagogical and content instruction while

completing an intensive student teaching experience. All the participants from two cohorts who

graduated from this program were invited to respond to the survey. Thirty-three of the beginning

teachers from the two cohorts participated in this study. Ten beginning teachers from the 33 who

participated in the survey were randomly selected for interviews.

Research Instruments

Two instruments were developed for this study. The first instrument was a survey

questionnaire developed by the investigator and based upon relevant literatures. The survey

questionnaire consisted of five sections. In the first section, the emphasis was on ranking the

definitions of good mathematics teaching—this addressed the first research question. The second

just completed the first or second year of classroom teaching mathematics at the secondary

school level.

As the samples of teachers were obtained from one graduate school of education, there

needed to be a distinction to identify teachers with good mathematics teaching. This helped to

obtain answers for the research questions on the attributes or factors that influence the

understanding of good mathematics teaching. Candidates for this study were beginning

mathematics teachers with strong content knowledge, as reflected by their college mathematics

GPA, pedagogical content knowledge, and recommendations by college professors. The subjects

were graduate students in this program who had at least a bachelor’s degree in college

mathematics and a strong background in the subject, having taken at least 24 credits in

mathematics content courses (including two semesters of calculus) and earned a good Grade

Point Average (GPA) score at the undergraduate level. As a requirement, the teachers had to

complete 100 hours of class observation and also have 120 hours of classroom teaching. The

teachers also received 12 months of intensive pedagogical and content instruction while

completing an intensive student teaching experience. All the participants from two cohorts who

graduated from this program were invited to respond to the survey. Thirty-three of the beginning

teachers from the two cohorts participated in this study. Ten beginning teachers from the 33 who

participated in the survey were randomly selected for interviews.

Research Instruments

Two instruments were developed for this study. The first instrument was a survey

questionnaire developed by the investigator and based upon relevant literatures. The survey

questionnaire consisted of five sections. In the first section, the emphasis was on ranking the

definitions of good mathematics teaching—this addressed the first research question. The second

25.
12

section of the survey focused on the important attributes of good teaching—this answered the

second research question. Section two of the survey also contained the description of four

models of good mathematics teaching. Participants had to select the teaching model they

believed was the best model for teaching mathematics in both middle school and high school.

For the third section, the focus was on the classroom practices of beginning teachers—this

addressed the first and second research questions. The fourth section of the survey addressed the

beliefs of beginning teachers about good mathematics teaching. The fifth section provided

answers to the fourth research question relating to what influences the understanding of good

mathematics teaching and when are good mathematics teaching attributes developed. This web-

based survey was available online for the participants to respond to.

The second instrument was a semi-structured qualitative interview schedule. Beginning

teachers with strong mathematics background were interviewed to complement their responses

from the initial survey. The interview sections was divided into four main areas: a) in-depth

explanation of the definition and important characteristics of good mathematics teaching; b)

reasons for selecting the important attributes of good mathematics teaching; c) teachers’

classroom practices and beliefs; and d) factors that influence beginning teachers’ understanding

of good mathematics teaching. Interviews complemented and extended the data collected via the

first instrument. Data from the survey and interview were used to determine the subjects’

definition of good mathematics teaching, perceptions of the important attributes of good

teaching, the relationship between the demographics of the partcipants and the attributes of good

teaching, and what influenced the understanding of good mathematics teaching.

section of the survey focused on the important attributes of good teaching—this answered the

second research question. Section two of the survey also contained the description of four

models of good mathematics teaching. Participants had to select the teaching model they

believed was the best model for teaching mathematics in both middle school and high school.

For the third section, the focus was on the classroom practices of beginning teachers—this

addressed the first and second research questions. The fourth section of the survey addressed the

beliefs of beginning teachers about good mathematics teaching. The fifth section provided

answers to the fourth research question relating to what influences the understanding of good

mathematics teaching and when are good mathematics teaching attributes developed. This web-

based survey was available online for the participants to respond to.

The second instrument was a semi-structured qualitative interview schedule. Beginning

teachers with strong mathematics background were interviewed to complement their responses

from the initial survey. The interview sections was divided into four main areas: a) in-depth

explanation of the definition and important characteristics of good mathematics teaching; b)

reasons for selecting the important attributes of good mathematics teaching; c) teachers’

classroom practices and beliefs; and d) factors that influence beginning teachers’ understanding

of good mathematics teaching. Interviews complemented and extended the data collected via the

first instrument. Data from the survey and interview were used to determine the subjects’

definition of good mathematics teaching, perceptions of the important attributes of good

teaching, the relationship between the demographics of the partcipants and the attributes of good

teaching, and what influenced the understanding of good mathematics teaching.

26.
13

Chapter 2

LITERATURE REVIEW

This chapter provides relevant research and theoretical perspectives as a background for

this study. It also discusses previous studies and investigation results that were useful for this

research. In the first section of this study, the investigator explores the definition of good

mathematics teaching based on previous literature. The next four sections explore the National

Council of Teachers of Mathematics (NCTM) (1991) Professional Standards on good

mathematics teaching, the attributes of good mathematics teaching, the development of the

attributes of good mathematics teaching, and teachers’ beliefs about good mathematics teaching.

Good Mathematics Teaching

What is good mathematics teaching? The answer depends on who one asks. Teachers,

researchers, policymakers, administrators, and parents will provide their own view of what they

consider good mathematics teaching and what is not. The notion of good teaching is pertinent as

it “strongly influences our decisions on designing and investigating teaching” (Krainer, 2005,

p. 75).

At the college level, Cashin (1989) suggests that good teaching is defined as how the

instructor’s behavior helps in the students’ learning of the materials. Latterell(2008) chose to

study what constitutes good mathematics teaching through three sources: research, student

evaluations, and comments on RateMyProfessors.com. The researcher found that few attributes

were similar in all the three sources. Ultimately, it was concluded that the five main features

describing good mathematics teaching require the professors to: be available to students;

Chapter 2

LITERATURE REVIEW

This chapter provides relevant research and theoretical perspectives as a background for

this study. It also discusses previous studies and investigation results that were useful for this

research. In the first section of this study, the investigator explores the definition of good

mathematics teaching based on previous literature. The next four sections explore the National

Council of Teachers of Mathematics (NCTM) (1991) Professional Standards on good

mathematics teaching, the attributes of good mathematics teaching, the development of the

attributes of good mathematics teaching, and teachers’ beliefs about good mathematics teaching.

Good Mathematics Teaching

What is good mathematics teaching? The answer depends on who one asks. Teachers,

researchers, policymakers, administrators, and parents will provide their own view of what they

consider good mathematics teaching and what is not. The notion of good teaching is pertinent as

it “strongly influences our decisions on designing and investigating teaching” (Krainer, 2005,

p. 75).

At the college level, Cashin (1989) suggests that good teaching is defined as how the

instructor’s behavior helps in the students’ learning of the materials. Latterell(2008) chose to

study what constitutes good mathematics teaching through three sources: research, student

evaluations, and comments on RateMyProfessors.com. The researcher found that few attributes

were similar in all the three sources. Ultimately, it was concluded that the five main features

describing good mathematics teaching require the professors to: be available to students;

27.
14

encourage student-faculty communication; explain lessons well, even using “little steps”; be fair

in grading; and give prompt feedback.

The five themes that characterize good teaching in this study can be summarized by the

willingness of the instructor to devote time to become an effective teacher. A certain pedagogical

approach might be useful in good teaching at the college level. Interestingly, Latterell (2008)

discovered the opposite approach, that a “certain pedagogical approach is not necessary to ensure

good teaching” (p. 10). The author also asserts that “‘the professor is enthusiastic’ is a common

question on evaluations but seems of little interest to students on RateMyProfessors.com and it

does not seem to be a major variable discussed in research” (p. 10). This suggests that an

instructor’s enthusiasm for a subject is not necessarily a good indicator of teaching, as viewed by

college students.

At the secondary level, Cooney et al. (2005) conducted a study on what constitutes good

mathematics teaching and how it develops, not from the researcher’s point of view but instead

focusing on the perspectives of nine high school teachers. The study sought to find out whether

the views of the teachers were similar to the Standards suggested by the NCTM (1989, 1991)

documents. The findings indicated that the teachers’ perspectives of good mathematics teaching

were consistent with the NCTM Standards and the pedagogy prescribed in the NCTM

documents. The study also concluded that good mathematics teaching requires prerequisite

knowledge, promotes mathematical understanding, and requires effective management.

One might conclude that the views of the teachers were towards a more student-centered

classroom as per what they learned during their teacher preparation program. Surprisingly, this

was not the case, however, as the teachers mentioned that they were more comfortable with the

teacher-centered approach “as long as their instruction styles could exercise different ways of

encourage student-faculty communication; explain lessons well, even using “little steps”; be fair

in grading; and give prompt feedback.

The five themes that characterize good teaching in this study can be summarized by the

willingness of the instructor to devote time to become an effective teacher. A certain pedagogical

approach might be useful in good teaching at the college level. Interestingly, Latterell (2008)

discovered the opposite approach, that a “certain pedagogical approach is not necessary to ensure

good teaching” (p. 10). The author also asserts that “‘the professor is enthusiastic’ is a common

question on evaluations but seems of little interest to students on RateMyProfessors.com and it

does not seem to be a major variable discussed in research” (p. 10). This suggests that an

instructor’s enthusiasm for a subject is not necessarily a good indicator of teaching, as viewed by

college students.

At the secondary level, Cooney et al. (2005) conducted a study on what constitutes good

mathematics teaching and how it develops, not from the researcher’s point of view but instead

focusing on the perspectives of nine high school teachers. The study sought to find out whether

the views of the teachers were similar to the Standards suggested by the NCTM (1989, 1991)

documents. The findings indicated that the teachers’ perspectives of good mathematics teaching

were consistent with the NCTM Standards and the pedagogy prescribed in the NCTM

documents. The study also concluded that good mathematics teaching requires prerequisite

knowledge, promotes mathematical understanding, and requires effective management.

One might conclude that the views of the teachers were towards a more student-centered

classroom as per what they learned during their teacher preparation program. Surprisingly, this

was not the case, however, as the teachers mentioned that they were more comfortable with the

teacher-centered approach “as long as their instruction styles could exercise different ways of

28.
15

reaching out to students” (Cooney et al., 2005, p. 105). Is the knowledge gained from teacher

preparation programs sufficient for good mathematics teaching? According to the participants in

Cooney et al.’s study, “knowledge learned at the university was important but that it had to be

tempered with more important knowledge gained from classroom experience” (p. 99). Another

dimension of the notion of good mathematics is connected more with the teachers’ experience

rather than with being a student in a teacher preparation program (Cooney et al., 2005).

At the elementary level, a good mathematics teacher requires certain attributes that are

related to his or her view of the nature of mathematics (Pietila, 2002). The study above

concluded that views of mathematics include knowledge, beliefs, conceptions, attitudes, and

emotions. Is the view of mathematics alone enough to define good mathematics teaching? Pietila

(2001) argues that good mathematics teachers also need sufficient knowledge of mathematics,

sufficient knowledge of mathematics teaching and learning teaching, additional pedagogical

knowledge to arrange successful learning situations, flexible beliefs and conceptions, and a

positive attitude towards learning and teaching mathematics. Furthermore, it is important that for

good mathematics teaching to occur, a teacher usually “gets pupils to understand the topics

presented and to be enthusiastic about mathematics” (Pietila, 2003, p. 12).

The Trends in International Mathematics and Science Study (TIMMS), an international

comparative assessment of fourth and eigth graders, defined effective teaching as a “complex

endeavor requiring knowledge about the subject matter of mathematics, the way the students

learn and effective pedagogy in mathematics” (Beaton et al., 1996, p. 131). The study also

suggested that good teaching can be enhanced through institutional support and adequate

The most important ideas asserted on good teaching is that teachers can support

each other in planning instructional strategies, devising real-world applications of

reaching out to students” (Cooney et al., 2005, p. 105). Is the knowledge gained from teacher

preparation programs sufficient for good mathematics teaching? According to the participants in

Cooney et al.’s study, “knowledge learned at the university was important but that it had to be

tempered with more important knowledge gained from classroom experience” (p. 99). Another

dimension of the notion of good mathematics is connected more with the teachers’ experience

rather than with being a student in a teacher preparation program (Cooney et al., 2005).

At the elementary level, a good mathematics teacher requires certain attributes that are

related to his or her view of the nature of mathematics (Pietila, 2002). The study above

concluded that views of mathematics include knowledge, beliefs, conceptions, attitudes, and

emotions. Is the view of mathematics alone enough to define good mathematics teaching? Pietila

(2001) argues that good mathematics teachers also need sufficient knowledge of mathematics,

sufficient knowledge of mathematics teaching and learning teaching, additional pedagogical

knowledge to arrange successful learning situations, flexible beliefs and conceptions, and a

positive attitude towards learning and teaching mathematics. Furthermore, it is important that for

good mathematics teaching to occur, a teacher usually “gets pupils to understand the topics

presented and to be enthusiastic about mathematics” (Pietila, 2003, p. 12).

The Trends in International Mathematics and Science Study (TIMMS), an international

comparative assessment of fourth and eigth graders, defined effective teaching as a “complex

endeavor requiring knowledge about the subject matter of mathematics, the way the students

learn and effective pedagogy in mathematics” (Beaton et al., 1996, p. 131). The study also

suggested that good teaching can be enhanced through institutional support and adequate

The most important ideas asserted on good teaching is that teachers can support

each other in planning instructional strategies, devising real-world applications of

29.
16

mathematical concepts, and developing sequences that move students from

concrete tasks to the ability to think for themselves and explore mathematical

theories. (Beaton et al., 1996, p. 131)

The Missouri Mathematics Program in the 1970s was considered an intervention program

to model effective teaching based on the large-scale observation studies of teachers’ behavior in

large classrooms during the 1960s (Reynolds & Muljs, 1999). This model of effective teaching

contains primarily of six themes: students have many opportunities to learn; teachers are

academically-oriented; teachers manage the classroom well; teachers have high expectations of

their pupils; students do not spend much time on their own; and teaching is heavily interactive.

Looking at all these themes, one gains an overall perspective of the role of teachers and students

in effective teaching. What is interesting here is that the attributes of effective teaching

mentioned by this 1970 project were consistent with Cooney (2005) and the NCTM (1991)

Standards of good teaching. The Standards and Cooney’s (2005) definition of good mathematics

teaching covered the effective teaching attributes in the Missouri project.

Teacher and student discourse describes the role of teachers and students; effective

management is similar to teachers managing the classroom well; worthwhile mathematical tasks

are part of the students’ opportunities to learn; prerequisite knowledge is needed as teachers are

academically-oriented; and the classroom environment that motivates learning is part of the

teachers’ high expectation of their students and interactive learning. Given such a broad

definition that covers most of the scope of good teaching, our discussion has thus been very

useful in thoroughly defining this concept of teaching and learning.

Summary of National Council of Teachers of Mathematics Professional Standards

The National Council of Teachers of Mathematics (NCTM) has been one of the

prominent bodies in the mathematics education field that produces Standards for the teaching

mathematical concepts, and developing sequences that move students from

concrete tasks to the ability to think for themselves and explore mathematical

theories. (Beaton et al., 1996, p. 131)

The Missouri Mathematics Program in the 1970s was considered an intervention program

to model effective teaching based on the large-scale observation studies of teachers’ behavior in

large classrooms during the 1960s (Reynolds & Muljs, 1999). This model of effective teaching

contains primarily of six themes: students have many opportunities to learn; teachers are

academically-oriented; teachers manage the classroom well; teachers have high expectations of

their pupils; students do not spend much time on their own; and teaching is heavily interactive.

Looking at all these themes, one gains an overall perspective of the role of teachers and students

in effective teaching. What is interesting here is that the attributes of effective teaching

mentioned by this 1970 project were consistent with Cooney (2005) and the NCTM (1991)

Standards of good teaching. The Standards and Cooney’s (2005) definition of good mathematics

teaching covered the effective teaching attributes in the Missouri project.

Teacher and student discourse describes the role of teachers and students; effective

management is similar to teachers managing the classroom well; worthwhile mathematical tasks

are part of the students’ opportunities to learn; prerequisite knowledge is needed as teachers are

academically-oriented; and the classroom environment that motivates learning is part of the

teachers’ high expectation of their students and interactive learning. Given such a broad

definition that covers most of the scope of good teaching, our discussion has thus been very

useful in thoroughly defining this concept of teaching and learning.

Summary of National Council of Teachers of Mathematics Professional Standards

The National Council of Teachers of Mathematics (NCTM) has been one of the

prominent bodies in the mathematics education field that produces Standards for the teaching

30.
17

and learning of mathematics in North America. Since the NCTM Professional Standards for

Teaching Mathematics were introduced in 1991, many researchers have referred to the NCTM

Standards and agree implicitly that they represent good mathematics teaching (Perrin-Glorian et

al., 2008). The NCTM Standards consist of worthwhile mathematical tasks, the teacher’s and

student’s role in discourse, tools and technology, learning environment, and analysis of teaching

and learning.

NCTM came up with the Principles and Standards in Mathematics in the year 2000. One

of the principles given prominence was the Teaching Principle. In the document, effective

mathematics teaching was defined as “requires understanding what students know and need to

learn and then challenging and supporting them to learn it well.” The principle also asserts that

effective teaching: requires knowing and understanding mathematics, students as learners, and

pedagogical strategies; requires a challenging and supportive classroom learning environment;

and requires continually seeking improvement.

Worthwhile Mathematical Tasks. Generally, tasks are problems, exercises, projects,

practice sheets, puzzles, and manipulative materials that teachers select for lessons in the

classroom. The NCTM (1991) document notes that it is important for teachers to select quality

mathematical tasks that engage students. The document adds that the selection, generating or

adapting of the tasks should depend on the mathematical content, the students, and the ways in

which students learn mathematics. Teachers need to address three issues in a mathematical task.

First, teachers need to consider not only the mathematical content of a task, but also how it is

related to concepts and procedures and connections with other ideas. Second, teachers should

deliberate whether the task fits the intended lessons. Third, teachers should consider how the task

assists in the development of a particular mathematical topic. In addition, teachers are

and learning of mathematics in North America. Since the NCTM Professional Standards for

Teaching Mathematics were introduced in 1991, many researchers have referred to the NCTM

Standards and agree implicitly that they represent good mathematics teaching (Perrin-Glorian et

al., 2008). The NCTM Standards consist of worthwhile mathematical tasks, the teacher’s and

student’s role in discourse, tools and technology, learning environment, and analysis of teaching

and learning.

NCTM came up with the Principles and Standards in Mathematics in the year 2000. One

of the principles given prominence was the Teaching Principle. In the document, effective

mathematics teaching was defined as “requires understanding what students know and need to

learn and then challenging and supporting them to learn it well.” The principle also asserts that

effective teaching: requires knowing and understanding mathematics, students as learners, and

pedagogical strategies; requires a challenging and supportive classroom learning environment;

and requires continually seeking improvement.

Worthwhile Mathematical Tasks. Generally, tasks are problems, exercises, projects,

practice sheets, puzzles, and manipulative materials that teachers select for lessons in the

classroom. The NCTM (1991) document notes that it is important for teachers to select quality

mathematical tasks that engage students. The document adds that the selection, generating or

adapting of the tasks should depend on the mathematical content, the students, and the ways in

which students learn mathematics. Teachers need to address three issues in a mathematical task.

First, teachers need to consider not only the mathematical content of a task, but also how it is

related to concepts and procedures and connections with other ideas. Second, teachers should

deliberate whether the task fits the intended lessons. Third, teachers should consider how the task

assists in the development of a particular mathematical topic. In addition, teachers are

31.
18

encouraged to develop and select tasks that can “promote the development of students’

understanding of concepts and procedures in a way that also fosters their ability to solve

problems and to reason and communicate mathematically” (NCTM, 1991, p. 10). What are good

mathematical tasks? Good tasks are “ones that do not separate mathematical thinking from

mathematical concepts or skills, that capture students’ curiosity, and that invite them to speculate

and to pursue their hunches” (p.10).

The selection of demanding tasks is not easy for a teacher. Stein Grover and

Henningstein (1996) recognized the important role of tasks in the academic setting and

introduced the Mathematical Task Framework (MTF). MTF emphasizes the role that

mathematics tasks “play in influencing students’ learning opportunities in ways they unfold

during classroom instruction” (Silver & Herbst, 2007, p. 55).

Tasks as they Tasks as set up Tasks as

appear in by teachers implemented by

curricular teacher and Student

materials students Learning

Figure 2.1. The Mathematical Tasks Framework

What is unique about the MTF is that it goes through a sequence of phases, beginning

with the tasks given in the curricular materials, to the task set up by teachers. Next, it proceeds to

task implementation by the teachers, with the students participating, involving interaction

between the tasks and the learners that leads to student learning. This framework stresses the

crucial role of mathematical tasks that provides students with learning opportunities.

encouraged to develop and select tasks that can “promote the development of students’

understanding of concepts and procedures in a way that also fosters their ability to solve

problems and to reason and communicate mathematically” (NCTM, 1991, p. 10). What are good

mathematical tasks? Good tasks are “ones that do not separate mathematical thinking from

mathematical concepts or skills, that capture students’ curiosity, and that invite them to speculate

and to pursue their hunches” (p.10).

The selection of demanding tasks is not easy for a teacher. Stein Grover and

Henningstein (1996) recognized the important role of tasks in the academic setting and

introduced the Mathematical Task Framework (MTF). MTF emphasizes the role that

mathematics tasks “play in influencing students’ learning opportunities in ways they unfold

during classroom instruction” (Silver & Herbst, 2007, p. 55).

Tasks as they Tasks as set up Tasks as

appear in by teachers implemented by

curricular teacher and Student

materials students Learning

Figure 2.1. The Mathematical Tasks Framework

What is unique about the MTF is that it goes through a sequence of phases, beginning

with the tasks given in the curricular materials, to the task set up by teachers. Next, it proceeds to

task implementation by the teachers, with the students participating, involving interaction

between the tasks and the learners that leads to student learning. This framework stresses the

crucial role of mathematical tasks that provides students with learning opportunities.

32.
19

Furthermore, it points out that the teacher’s decision in selecting tasks influences students’

interaction with challenging tasks that provide opportunities while working on such tasks. MTF

is also used as a tool that considers the challenges of teachers utilizing complex tasks in their

mathematics lessons.

The MTF research suggests that teachers need to learn to orchestrate the work of students

while resisting the persistent urge to tell students precisely what to do. This then removes the

opportunity for thoughtful engagement by responding to student queries and requests for

information in ways that support students’ thinking (Silver & Herbst, 2007, p. 55).

This informs us that frameworks such as MTF might pave a way to mediate the

connection between research and practice. Another similar framework that plays this role is

Cognitively Guided Instruction (CGI), which organizes addition and subtraction tasks by using

the present structure that classifies them according to operations. CGI has been extensively

applied by teachers and teacher educators in the field (Silver & Herbst, 2007).

An analysis of the instruction in eighth grade classrooms in the United States shows that

the emphasis is on low-level tasks like memorization and recalling instead of high-level thinking

tasks that involve reasoning and problem-solving (Silver, 1998). This leads to an understanding

of mathematics in a simplistic way instead of a meaningful understanding of the concepts. What

is lacking are tasks that engage students in critical thinking instead of simply doing mathematical

Tasks that only require memorization without understanding are not worthwhile

mathematical tasks. They are in fact considered low-level tasks because they apply the “drill and

practice” method that requires only memorized facts without much application (Tanner & Jones,

2000). Students who only have the knowledge of memorized facts face difficulties in solving

Furthermore, it points out that the teacher’s decision in selecting tasks influences students’

interaction with challenging tasks that provide opportunities while working on such tasks. MTF

is also used as a tool that considers the challenges of teachers utilizing complex tasks in their

mathematics lessons.

The MTF research suggests that teachers need to learn to orchestrate the work of students

while resisting the persistent urge to tell students precisely what to do. This then removes the

opportunity for thoughtful engagement by responding to student queries and requests for

information in ways that support students’ thinking (Silver & Herbst, 2007, p. 55).

This informs us that frameworks such as MTF might pave a way to mediate the

connection between research and practice. Another similar framework that plays this role is

Cognitively Guided Instruction (CGI), which organizes addition and subtraction tasks by using

the present structure that classifies them according to operations. CGI has been extensively

applied by teachers and teacher educators in the field (Silver & Herbst, 2007).

An analysis of the instruction in eighth grade classrooms in the United States shows that

the emphasis is on low-level tasks like memorization and recalling instead of high-level thinking

tasks that involve reasoning and problem-solving (Silver, 1998). This leads to an understanding

of mathematics in a simplistic way instead of a meaningful understanding of the concepts. What

is lacking are tasks that engage students in critical thinking instead of simply doing mathematical

Tasks that only require memorization without understanding are not worthwhile

mathematical tasks. They are in fact considered low-level tasks because they apply the “drill and

practice” method that requires only memorized facts without much application (Tanner & Jones,

2000). Students who only have the knowledge of memorized facts face difficulties in solving

33.
20

mathematical problems in real life that requires adaptability to new tasks. Mathematical tasks

that apply real-world problems and concrete experiences develop an understanding of concepts

(Cai et al., 2009). Teachers view that “ an indicator of mathematical understanding is the flexible

application of what has been learned to problem situations that require the students to use what

they have learned in different ways” (p. 11).

By utilizing effective tasks, teachers are able to assist students in problem-solving

techniques that involve understanding and analyzing problems in order to obtain the solutions.

Less attention will be given to tasks that require only memorization of algorithms and methods

(NCTM, 1991). The role of the teacher in helping students is crucial in solving problems. Too

much or too little help in solving a problem is not suitable as the students need to “have a

reasonable share of work” (Polya, 1985). Schonfeld (1985) agrees with Polya, but suggests his

own heuristics for dealing with the problems that are similar to Polya’s steps except for the step

of using of different methods but varying them for possible solutions.

Studies describe how mathematical tasks can “give students something to talk about”

(Silver, 1996; Stein & Lane, 1996). It has also been reported that the highest learning gains are

usually achieved by setting up tasks that engage students and higher-order thinking like

reasoning (Stein & Lane, 1996). Beginning with a good task provides students with opportunities

to develop their mathematical thinking (Franke, Kazemi, & Battey, 2008). This also allows the

“teacher to engage students in sharing their thinking, comparing different approaches, making

conjectures and generalizing” (Silver & Smith, 1996, as cited by Franke, Kazemi, & Battey,

2008, p. 234). The engagement of teachers in lessons enriches the mathematical lesson and the

students’ experiences.

mathematical problems in real life that requires adaptability to new tasks. Mathematical tasks

that apply real-world problems and concrete experiences develop an understanding of concepts

(Cai et al., 2009). Teachers view that “ an indicator of mathematical understanding is the flexible

application of what has been learned to problem situations that require the students to use what

they have learned in different ways” (p. 11).

By utilizing effective tasks, teachers are able to assist students in problem-solving

techniques that involve understanding and analyzing problems in order to obtain the solutions.

Less attention will be given to tasks that require only memorization of algorithms and methods

(NCTM, 1991). The role of the teacher in helping students is crucial in solving problems. Too

much or too little help in solving a problem is not suitable as the students need to “have a

reasonable share of work” (Polya, 1985). Schonfeld (1985) agrees with Polya, but suggests his

own heuristics for dealing with the problems that are similar to Polya’s steps except for the step

of using of different methods but varying them for possible solutions.

Studies describe how mathematical tasks can “give students something to talk about”

(Silver, 1996; Stein & Lane, 1996). It has also been reported that the highest learning gains are

usually achieved by setting up tasks that engage students and higher-order thinking like

reasoning (Stein & Lane, 1996). Beginning with a good task provides students with opportunities

to develop their mathematical thinking (Franke, Kazemi, & Battey, 2008). This also allows the

“teacher to engage students in sharing their thinking, comparing different approaches, making

conjectures and generalizing” (Silver & Smith, 1996, as cited by Franke, Kazemi, & Battey,

2008, p. 234). The engagement of teachers in lessons enriches the mathematical lesson and the

students’ experiences.

34.
21

Teacher’s and Student’s Role in Discourse. Classroom discourse between students and

teachers plays a role in the teaching and learning of mathematics. The interaction that happens

develops mathematical understanding as students are able to share their solutions, explain their

solutions and make conjectures, prove how their solutions work, reason their answers, and make

generalizations. Even though research has shown that the teacher’s role is essential in the success

of classroom discourse, not much is known about how teachers can support this process (Franke,

Kazemi & Battey, 2008). One example would be whether students arguing to prove their solution

right is beneficial to students’ development of mathematical understanding.

One of the most used classroom discourse patterns has been the IRE model, which begins

with a teacher-initiated question followed by student response and teacher evaluation (Cazden,

2001; Doyle; 1985; Mehan, 1985). This model is a well-documented study of classrooms in the

United States, especially for students from lower socio-economic status (Silver, Smith, & Nelson

1995). Typically, students will listen to the teacher and answer the questions asked. Not much

time is given for students to explain their ideas, make conjectures or understand a mathematical

concept. As Spilane and Zeuli (1999) discovered “in their study of reform-minded teachers…

they predominantly engaged in procedure-bound discourse, rarely asked students to do more than

provide correct answers, focused on procedural rather than conceptual knowledge and engaged

students in memorization procedures to calculate answers” (in Franke, Kazemi & Battey, 2008,

p. 231). Teachers who engaged students in a good discourse would usually focus on the

conceptual knowledge to develop mathematical thinking.

How can teachers improve classroom discourse? Lampert (2001) suggests that giving

attention to students who are participating, how they participate, the mathematical concept being

explored, students’ mathematical background, what students currently understand, and the

Teacher’s and Student’s Role in Discourse. Classroom discourse between students and

teachers plays a role in the teaching and learning of mathematics. The interaction that happens

develops mathematical understanding as students are able to share their solutions, explain their

solutions and make conjectures, prove how their solutions work, reason their answers, and make

generalizations. Even though research has shown that the teacher’s role is essential in the success

of classroom discourse, not much is known about how teachers can support this process (Franke,

Kazemi & Battey, 2008). One example would be whether students arguing to prove their solution

right is beneficial to students’ development of mathematical understanding.

One of the most used classroom discourse patterns has been the IRE model, which begins

with a teacher-initiated question followed by student response and teacher evaluation (Cazden,

2001; Doyle; 1985; Mehan, 1985). This model is a well-documented study of classrooms in the

United States, especially for students from lower socio-economic status (Silver, Smith, & Nelson

1995). Typically, students will listen to the teacher and answer the questions asked. Not much

time is given for students to explain their ideas, make conjectures or understand a mathematical

concept. As Spilane and Zeuli (1999) discovered “in their study of reform-minded teachers…

they predominantly engaged in procedure-bound discourse, rarely asked students to do more than

provide correct answers, focused on procedural rather than conceptual knowledge and engaged

students in memorization procedures to calculate answers” (in Franke, Kazemi & Battey, 2008,

p. 231). Teachers who engaged students in a good discourse would usually focus on the

conceptual knowledge to develop mathematical thinking.

How can teachers improve classroom discourse? Lampert (2001) suggests that giving

attention to students who are participating, how they participate, the mathematical concept being

explored, students’ mathematical background, what students currently understand, and the

35.
22

attitude of the students during conversation are several steps that can be implemented. As Ball

(1993) summarizes, the teacher’s role is not only concerned with student learning of mathematics

but also with creating a discourse environment that encourages probing and exploring new

mathematical ideas.

The ability to engage students in classroom conversation mostly comes from cooperative

learning groups. Teachers can play a role by asking specific questions and assisting where

necessary to encourage student learning. As Kieran (2002) states:

It is the way we make our thoughts available that is critical, it is not just about

making them “available”, it is how….utterances that were neither complete nor

ever expanded upon seemed much less conducive to the emergence of

mathematical thought for both participants. (p. 219)

Teachers can take this pedagogical chance to support this mathematical discourse by giving

explanations and also asking probing questions that engage students’ learning.

One of the major findings in creating mathematical conversations is that teachers

systematically create this wonderful opportunity (Franke et al., 2008). Kieran and colleagues

suggest that teachers need to facilitate and manage discourse using the mathematical ideas that

come from students’ learning (Kieran & Dreyfus, 1998).

Mathematical Content Knowledge. To be a good mathematics teacher, one needs to

know the mathematics content well. Most teachers have heard of this saying during their teacher

education courses. NCTM (1991) asserts that “To be effective, teachers must know and

understand deeply the mathematics they are teaching and be able to draw on that knowledge with

flexibility in their teaching tasks” (p. 5).

Many studies have been done to define teachers’ content knowledge. Theories developed

in this field include the role of mathematics content knowledge (Kahan, 2003), mathematical

knowledge for teaching (Ball & Bass, 2004), and theory of knowledge that includes content

attitude of the students during conversation are several steps that can be implemented. As Ball

(1993) summarizes, the teacher’s role is not only concerned with student learning of mathematics

but also with creating a discourse environment that encourages probing and exploring new

mathematical ideas.

The ability to engage students in classroom conversation mostly comes from cooperative

learning groups. Teachers can play a role by asking specific questions and assisting where

necessary to encourage student learning. As Kieran (2002) states:

It is the way we make our thoughts available that is critical, it is not just about

making them “available”, it is how….utterances that were neither complete nor

ever expanded upon seemed much less conducive to the emergence of

mathematical thought for both participants. (p. 219)

Teachers can take this pedagogical chance to support this mathematical discourse by giving

explanations and also asking probing questions that engage students’ learning.

One of the major findings in creating mathematical conversations is that teachers

systematically create this wonderful opportunity (Franke et al., 2008). Kieran and colleagues

suggest that teachers need to facilitate and manage discourse using the mathematical ideas that

come from students’ learning (Kieran & Dreyfus, 1998).

Mathematical Content Knowledge. To be a good mathematics teacher, one needs to

know the mathematics content well. Most teachers have heard of this saying during their teacher

education courses. NCTM (1991) asserts that “To be effective, teachers must know and

understand deeply the mathematics they are teaching and be able to draw on that knowledge with

flexibility in their teaching tasks” (p. 5).

Many studies have been done to define teachers’ content knowledge. Theories developed

in this field include the role of mathematics content knowledge (Kahan, 2003), mathematical

knowledge for teaching (Ball & Bass, 2004), and theory of knowledge that includes content

36.
23

knowledge (Schonfeld, 1998) and subject matter knowledge (Borko & Putnam, 1996; Brown &

Borko, 1992) and others.

Ball (2007) developed the concept of Mathematics Knowledge for Teaching (MKT),

which is defined as “mathematical knowledge needed to carry out the work of teaching

mathematics” (Ball et al., 2009, p. 96). The researcher divided MKT into subject matter

knowledge and pedagogical content knowledge. Under subject matter knowledge, the knowledge

was partitioned into three categories, namely common content knowledge (CCK), specialized

content knowledge (SCK), and horizon content knowledge (HCK).

Mathematical content knowledge is essential and requires three elements: a) a deep

foundation of factual knowledge, b) understanding of the “facts and ideas in the context of a

conceptual framework,” and c) organization of the knowledge “in ways that facilitate retrieval

and application” (Brown & Cocking, 2000, p. 16, as cited in Kahan, 2003). Schonfeld (1998)

gives a unique perspective of mathematics content knowledge in his theory of knowledge. The

knowledge a teacher brings into the class is divided into two areas: knowledge inventory and

organization. Knowledge inventory is the base knowledge of an individual, while organization

refers to the accessibility and usage of the base knowledge.

Then, what is subject matter knowledge for mathematics? According to Borko and

Putnam (1996),

What is essential to recognize is the argument that teachers need to know more

than just the facts, terms, and concepts of a discipline. Their knowledge of the

organizing ideas, connections among ideas, ways of thinking and arguing, and

knowledge growth within the discipline is an important factor in how they will

teach the subject. (p. 676)

It should consist of “mathematical facts, concepts, and computational algorithms;

syntactic knowledge encompasses an understanding of the methods of mathematical proof and

knowledge (Schonfeld, 1998) and subject matter knowledge (Borko & Putnam, 1996; Brown &

Borko, 1992) and others.

Ball (2007) developed the concept of Mathematics Knowledge for Teaching (MKT),

which is defined as “mathematical knowledge needed to carry out the work of teaching

mathematics” (Ball et al., 2009, p. 96). The researcher divided MKT into subject matter

knowledge and pedagogical content knowledge. Under subject matter knowledge, the knowledge

was partitioned into three categories, namely common content knowledge (CCK), specialized

content knowledge (SCK), and horizon content knowledge (HCK).

Mathematical content knowledge is essential and requires three elements: a) a deep

foundation of factual knowledge, b) understanding of the “facts and ideas in the context of a

conceptual framework,” and c) organization of the knowledge “in ways that facilitate retrieval

and application” (Brown & Cocking, 2000, p. 16, as cited in Kahan, 2003). Schonfeld (1998)

gives a unique perspective of mathematics content knowledge in his theory of knowledge. The

knowledge a teacher brings into the class is divided into two areas: knowledge inventory and

organization. Knowledge inventory is the base knowledge of an individual, while organization

refers to the accessibility and usage of the base knowledge.

Then, what is subject matter knowledge for mathematics? According to Borko and

Putnam (1996),

What is essential to recognize is the argument that teachers need to know more

than just the facts, terms, and concepts of a discipline. Their knowledge of the

organizing ideas, connections among ideas, ways of thinking and arguing, and

knowledge growth within the discipline is an important factor in how they will

teach the subject. (p. 676)

It should consist of “mathematical facts, concepts, and computational algorithms;

syntactic knowledge encompasses an understanding of the methods of mathematical proof and

37.
24

other forms of argument used by mathematicians” (Brown & Borko, 1992). The study also found

that without adequate content knowledge, teachers spend more time planning to learn the content

rather than using the time to plan lessons that engage students’ understanding. Teachers with

strong content knowledge are able to explain concepts well instead of just doing the procedures.

They are also likely to be more flexible in their teaching and can identify and pose worthwhile

mathematical tasks (Brown & Borko, 1992). Interestingly, teachers with strong content

knowledge are also confident in their classroom, compared with teachers without adequate

knowledge. However, studies by Caroll (2007) and Nickson (1988) showed that the level of

formal mathematics simply does not determine confidence in a class, but rather how the nature of

the content knowledge was acquired makes the difference. Strong teachers are able to connect

the links between concepts and offer an alternative representation or meaning (Caroll, 2007). To

teach effectively, Ball (1990) suggests that teachers must have the conceptual understanding of

the mathematics knowledge to explain the procedures and ability to connect rules, definitions,

and topics.

Studies have been conducted that link the relatioships between teachers’ mathematical

knowledge and students’ achievement (NMAP, 2008). Acording to the NMAP report, most

studies do not explicitly specify the mathematical knowledge needed for effective teaching. A

study done on this issue found that teachers’ mathematical content knowledge predicts the

improvement of mathematics achievement for first and third graders (Hill, Rowan, & Ball,

2005). This study also concluded that teachers’ content knowledge is also important in teaching

fundamental concepts at the elementary level.

Having strong mathematics knowledge is an advantage, but it does not guarantee that

effective teaching happens (Fennema & Franke, 1992; Shulman, 1986). Much more is needed as

other forms of argument used by mathematicians” (Brown & Borko, 1992). The study also found

that without adequate content knowledge, teachers spend more time planning to learn the content

rather than using the time to plan lessons that engage students’ understanding. Teachers with

strong content knowledge are able to explain concepts well instead of just doing the procedures.

They are also likely to be more flexible in their teaching and can identify and pose worthwhile

mathematical tasks (Brown & Borko, 1992). Interestingly, teachers with strong content

knowledge are also confident in their classroom, compared with teachers without adequate

knowledge. However, studies by Caroll (2007) and Nickson (1988) showed that the level of

formal mathematics simply does not determine confidence in a class, but rather how the nature of

the content knowledge was acquired makes the difference. Strong teachers are able to connect

the links between concepts and offer an alternative representation or meaning (Caroll, 2007). To

teach effectively, Ball (1990) suggests that teachers must have the conceptual understanding of

the mathematics knowledge to explain the procedures and ability to connect rules, definitions,

and topics.

Studies have been conducted that link the relatioships between teachers’ mathematical

knowledge and students’ achievement (NMAP, 2008). Acording to the NMAP report, most

studies do not explicitly specify the mathematical knowledge needed for effective teaching. A

study done on this issue found that teachers’ mathematical content knowledge predicts the

improvement of mathematics achievement for first and third graders (Hill, Rowan, & Ball,

2005). This study also concluded that teachers’ content knowledge is also important in teaching

fundamental concepts at the elementary level.

Having strong mathematics knowledge is an advantage, but it does not guarantee that

effective teaching happens (Fennema & Franke, 1992; Shulman, 1986). Much more is needed as

38.
25

teachers also require knowledge about students and pedagogy to produce good lessons (Shulman,

1986). The idea that more mathematics content translates into better teaching is an interesting

issue to explore.

Pedagogical Content Knowledge. The term pedagogical content knowledge (PCK) or

knowledge of subject matter for teaching was introduced in 1986 by Lee Shulman. Since then,

many researchers in the field of mathematics education have given meaning to this term (Ball,

2009; Borko & Putnam, 1996, 1992; Ma, 1999; Schonfeld, 1998). PCK is the ability to represent

ideas in ways that are understandable to students (Shulman, 1986). It includes “an understanding

of what makes the learning of specific topics easy or difficult: the conceptions and

preconceptions that students of different ages and backgrounds bring with them to learning

(Shulman, 1986, p. 9). Ball (2009) defines pedagogical content knowledge as knowledge of

content and students (KCS), knowledge of content and teaching (KCT), and knowledge of

curriculum (KC), all of which encompass NCTM’s attributes of good teaching: teacher and

student’s role in discourse, tools and technology, learning environment, and analysis of teaching

and learning.

Schonfeld (1998) uses the definition of PCK by Borko and Putnam (1996) that divides

the concept into four major components: 1) “the teacher’s overarching conception of the

purposes for teaching a subject matter...the nature of the subject and what is important for

students to learn”; 2) “knowledge of students’ understandings and potential misunderstandings of

a subject area...[including] preconceptions, misconceptions, and alternative conceptions about

topics such as division of fractions, negative numbers”; 3) “knowledge of curriculum and

curricular materials”; and 4) “knowledge of strategies and representations for teaching particular

topics” (pp. 676-677).

teachers also require knowledge about students and pedagogy to produce good lessons (Shulman,

1986). The idea that more mathematics content translates into better teaching is an interesting

issue to explore.

Pedagogical Content Knowledge. The term pedagogical content knowledge (PCK) or

knowledge of subject matter for teaching was introduced in 1986 by Lee Shulman. Since then,

many researchers in the field of mathematics education have given meaning to this term (Ball,

2009; Borko & Putnam, 1996, 1992; Ma, 1999; Schonfeld, 1998). PCK is the ability to represent

ideas in ways that are understandable to students (Shulman, 1986). It includes “an understanding

of what makes the learning of specific topics easy or difficult: the conceptions and

preconceptions that students of different ages and backgrounds bring with them to learning

(Shulman, 1986, p. 9). Ball (2009) defines pedagogical content knowledge as knowledge of

content and students (KCS), knowledge of content and teaching (KCT), and knowledge of

curriculum (KC), all of which encompass NCTM’s attributes of good teaching: teacher and

student’s role in discourse, tools and technology, learning environment, and analysis of teaching

and learning.

Schonfeld (1998) uses the definition of PCK by Borko and Putnam (1996) that divides

the concept into four major components: 1) “the teacher’s overarching conception of the

purposes for teaching a subject matter...the nature of the subject and what is important for

students to learn”; 2) “knowledge of students’ understandings and potential misunderstandings of

a subject area...[including] preconceptions, misconceptions, and alternative conceptions about

topics such as division of fractions, negative numbers”; 3) “knowledge of curriculum and

curricular materials”; and 4) “knowledge of strategies and representations for teaching particular

topics” (pp. 676-677).

39.
26

Ma (1999) conducted a study to compare the mathematical knowledge of elementary

teachers in the United States and China. The study focused on such concepts as subtraction,

perimeter and area, division by fractions, and multi-digit multiplication. In discussing how

teachers acquire a profound understanding of fundamental mathematics (PUFM), Ma (1999)

found that Chinese teachers were able to acquire deeper understanding of mathematics than their

American counterparts through communicating with their colleagues, learning from students,

doing problems themselves, and teaching with and studying teaching materials like textbooks

extensively. The study also identified four elements of understanding: basic ideas,

connectedness, multiple representations, and longitudinal coherence. What is unique on PUFM is

that it forms a connection between mathematical content knowledge and pedagogical content

knowledge. The process of “what is it” that refers to the content and “how to teach it” that is the

pedagogy provides teachers with a sufficient knowledge of school mathematics.

Ball and Bass (2000) examined the connection between content knowledge and

pedagogy. The study also summarized the three problems faced by mathematics teachers: what

teachers need to know, how they have to know it, and how to help them learn to use it. Teachers

need to make the distinction between how to do mathematics and knowing how to use it in

practice. This distinction is essential in understanding the role of content knowledge in good

teaching. In conclusion, having content knowledge is required, but delivering the lesson by using

the right pedagogy makes the lesson a successful one.

Attributes of Good Mathematics Teaching

Studies on characteristics of good teaching either usually look at the qualities of a good

teacher from the view of content knowledge, instruction, personality or practice. Porter and

Ma (1999) conducted a study to compare the mathematical knowledge of elementary

teachers in the United States and China. The study focused on such concepts as subtraction,

perimeter and area, division by fractions, and multi-digit multiplication. In discussing how

teachers acquire a profound understanding of fundamental mathematics (PUFM), Ma (1999)

found that Chinese teachers were able to acquire deeper understanding of mathematics than their

American counterparts through communicating with their colleagues, learning from students,

doing problems themselves, and teaching with and studying teaching materials like textbooks

extensively. The study also identified four elements of understanding: basic ideas,

connectedness, multiple representations, and longitudinal coherence. What is unique on PUFM is

that it forms a connection between mathematical content knowledge and pedagogical content

knowledge. The process of “what is it” that refers to the content and “how to teach it” that is the

pedagogy provides teachers with a sufficient knowledge of school mathematics.

Ball and Bass (2000) examined the connection between content knowledge and

pedagogy. The study also summarized the three problems faced by mathematics teachers: what

teachers need to know, how they have to know it, and how to help them learn to use it. Teachers

need to make the distinction between how to do mathematics and knowing how to use it in

practice. This distinction is essential in understanding the role of content knowledge in good

teaching. In conclusion, having content knowledge is required, but delivering the lesson by using

the right pedagogy makes the lesson a successful one.

Attributes of Good Mathematics Teaching

Studies on characteristics of good teaching either usually look at the qualities of a good

teacher from the view of content knowledge, instruction, personality or practice. Porter and

40.
27

Brophy (1988) discussed four attributes of effective instruction in their article “Synthesis of

Research on Good Teaching.” They classifed the attributes with regard to promoting learning by

a) commutating to their students what is expected and why

b) providing their students with strategies for monitoring and improving their

own learning efforts and with structured opportunities for independent

learning activities

c) effective teachers not only know the subject matter they intend their students

to learn but also know the misconceptions their students bring to the

classroom will interfere with their learning of that subject matter

d) published instructional materials contribute to instructional quality. (pp. 5-6)

In an attempt to include the attributes of good teaching, Porter and Brophy (1988) summarized

the set of goals an effective teacher wants to achieve in practice, including:

a) is clear on the goals of instruction

b) provides students with metacognition strategies

c) creates new learning situations

d) continuously monitors students’ understanding

e) integrates instruction across discipline

f) self-evaluation and reflection of practice (p. 8)

Van de Walle (2001) suggested seven attribues for effective teaching in practice. The list

includes creating a mathematical environment; posing worthwhile mathematical tasks; using

cooperative learning groups; using models and calculators as thinking tools; encouraging

discourse and writing; requiring justification of student responses; and listening actively by

paying attention to the instructional process instead of considering other factors. This list of

attributes is similar to the NCTM (1991) Standards. Hamachek (1999) offered a teaching

philosophy that “Consciously, we teach what we know; unconsciously, we teach who we are”

(p. 209).

Oruc (2008) conducted a study on the qualities of good teaching at the university level.

What is interesting here is that the study compared the perceptions of European students and

Turkish students from a Faculty of Education. The analysis of the data found that both groups of

Brophy (1988) discussed four attributes of effective instruction in their article “Synthesis of

Research on Good Teaching.” They classifed the attributes with regard to promoting learning by

a) commutating to their students what is expected and why

b) providing their students with strategies for monitoring and improving their

own learning efforts and with structured opportunities for independent

learning activities

c) effective teachers not only know the subject matter they intend their students

to learn but also know the misconceptions their students bring to the

classroom will interfere with their learning of that subject matter

d) published instructional materials contribute to instructional quality. (pp. 5-6)

In an attempt to include the attributes of good teaching, Porter and Brophy (1988) summarized

the set of goals an effective teacher wants to achieve in practice, including:

a) is clear on the goals of instruction

b) provides students with metacognition strategies

c) creates new learning situations

d) continuously monitors students’ understanding

e) integrates instruction across discipline

f) self-evaluation and reflection of practice (p. 8)

Van de Walle (2001) suggested seven attribues for effective teaching in practice. The list

includes creating a mathematical environment; posing worthwhile mathematical tasks; using

cooperative learning groups; using models and calculators as thinking tools; encouraging

discourse and writing; requiring justification of student responses; and listening actively by

paying attention to the instructional process instead of considering other factors. This list of

attributes is similar to the NCTM (1991) Standards. Hamachek (1999) offered a teaching

philosophy that “Consciously, we teach what we know; unconsciously, we teach who we are”

(p. 209).

Oruc (2008) conducted a study on the qualities of good teaching at the university level.

What is interesting here is that the study compared the perceptions of European students and

Turkish students from a Faculty of Education. The analysis of the data found that both groups of

41.
28

students had similar views on good teaching. They defined good teaching as the ability to inspire

learners, and be an excellent communicator and well organized. What about the most important

attributes of good teaching? The study concluded that knowledge of subject matter, enjoys and

respects students, motivates and inspires, is creative and innovative, is enthusiastic about

teaching, and manages behavior well were key attributes. Most of the attributes came from the

personality and behavior of the professor, even though some were from the subject knowledge

and professional skills area. The research concluded that both the Turkish and European students

agreed on the attributes of good teaching. Culture can be a factor in defining good teaching.

Timmering (2009) analyzed the perception of teachers and student teachers in Europe on

the qualities of an effective teacher. The study identified 300 attributes of a good teacher. What

is interesting here was that the attributes differed among countries in Northern Europe and

Southern Europe. One reason for this might be that different cultures perceive different attributes

as essential. Surprisingly, in most European countries, personality traits were ranked higher than

the knowledge, skills, and attitude category.

Teachers’ subject matter knowledge is usually relevant to their teaching. Ma (1999)

revealed that to deepen their content knowledge, teachers should reflect on the process of

preparing lessons and teaching the intended material. It is important for effective teachers to

ponder their classroom practice. Doing self-evaluation and reflection helps teachers to gauge

their own instruction (Porter & Brophy, 1988). Taking these steps ensures that teachers are doing

worthwhile tasks during instruction while guiding students’ learning patterns and behavior.

As Thompson (1992) suggests, teacher educators need to “explore ways to help teachers

examine their beliefs and practices, develop intrinsic motivations for considering alternatives to

students had similar views on good teaching. They defined good teaching as the ability to inspire

learners, and be an excellent communicator and well organized. What about the most important

attributes of good teaching? The study concluded that knowledge of subject matter, enjoys and

respects students, motivates and inspires, is creative and innovative, is enthusiastic about

teaching, and manages behavior well were key attributes. Most of the attributes came from the

personality and behavior of the professor, even though some were from the subject knowledge

and professional skills area. The research concluded that both the Turkish and European students

agreed on the attributes of good teaching. Culture can be a factor in defining good teaching.

Timmering (2009) analyzed the perception of teachers and student teachers in Europe on

the qualities of an effective teacher. The study identified 300 attributes of a good teacher. What

is interesting here was that the attributes differed among countries in Northern Europe and

Southern Europe. One reason for this might be that different cultures perceive different attributes

as essential. Surprisingly, in most European countries, personality traits were ranked higher than

the knowledge, skills, and attitude category.

Teachers’ subject matter knowledge is usually relevant to their teaching. Ma (1999)

revealed that to deepen their content knowledge, teachers should reflect on the process of

preparing lessons and teaching the intended material. It is important for effective teachers to

ponder their classroom practice. Doing self-evaluation and reflection helps teachers to gauge

their own instruction (Porter & Brophy, 1988). Taking these steps ensures that teachers are doing

worthwhile tasks during instruction while guiding students’ learning patterns and behavior.

As Thompson (1992) suggests, teacher educators need to “explore ways to help teachers

examine their beliefs and practices, develop intrinsic motivations for considering alternatives to

42.
29

their current practices and develop personal reasons for justifying their actions” (p. 143). This

kind of reflection is an essential attribute of good mathematics instruction. Dewey (1933) also

discusses the role of reflection as a way to respond to problems in teaching. Grant (1984) argues

that reflection also may include other issues like the role of a teacher in reflection and teachers’

beliefs about good teaching and classroom practice. The inclusion of one’s understanding of the

subject may be reflected in practice.

Teachers need to reflect on their experiences as learners in order to be effective. As

Schon (1987) points out, the teacher’s ability to respond to students’ actions by listening,

reflecting, and conversing with them helps the teacher reason out her actions and thus learning

happens. This can occur many times until the teacher is comfortable with the teaching

methodology. Schon summarizes that it is not how much knowledge the teacher has, but how

effectively the teacher engages the students in mathematics. Moreover, this can be achieved by

teachers reflecting upon their teaching. Nickson (1988) adds that an effective teacher needs

continuous evaluation and flexibility in certain content and methods.

Teachers’ Beliefs

A study by Wilson and Cooney (2002) found a relationship between teachers’ beliefs and

student learning. With this knowledge, researchers should consider teachers’ belief as a strong

influence on students’ mathematics learning.

Kuhs and Ball (1986) found four perspectives of how mathematics should be taught:

a) Learner-focused: mathematics teaching that focuses on the learner’s personal

construction of mathematical knowledge;

b) Content-foused with emphasis on conceptual understanding: mathematics

teaching that is driven by the content itself, but emphasizes conceptual

understanding;

their current practices and develop personal reasons for justifying their actions” (p. 143). This

kind of reflection is an essential attribute of good mathematics instruction. Dewey (1933) also

discusses the role of reflection as a way to respond to problems in teaching. Grant (1984) argues

that reflection also may include other issues like the role of a teacher in reflection and teachers’

beliefs about good teaching and classroom practice. The inclusion of one’s understanding of the

subject may be reflected in practice.

Teachers need to reflect on their experiences as learners in order to be effective. As

Schon (1987) points out, the teacher’s ability to respond to students’ actions by listening,

reflecting, and conversing with them helps the teacher reason out her actions and thus learning

happens. This can occur many times until the teacher is comfortable with the teaching

methodology. Schon summarizes that it is not how much knowledge the teacher has, but how

effectively the teacher engages the students in mathematics. Moreover, this can be achieved by

teachers reflecting upon their teaching. Nickson (1988) adds that an effective teacher needs

continuous evaluation and flexibility in certain content and methods.

Teachers’ Beliefs

A study by Wilson and Cooney (2002) found a relationship between teachers’ beliefs and

student learning. With this knowledge, researchers should consider teachers’ belief as a strong

influence on students’ mathematics learning.

Kuhs and Ball (1986) found four perspectives of how mathematics should be taught:

a) Learner-focused: mathematics teaching that focuses on the learner’s personal

construction of mathematical knowledge;

b) Content-foused with emphasis on conceptual understanding: mathematics

teaching that is driven by the content itself, but emphasizes conceptual

understanding;

43.
30

c) Content-focused with an emphasis on performance: mathematics teaching that

emphasizes student performance and mastery of mathematics rules and

procedural; and

d) Classroom-focused: mathematics teaching based on knowledge about

effective classrooms. (p. 2)

How a teacher’s change in belief might influence classroom practice is an important

aspect to be considered (Grant, 1984). Raymond (1987) studied the inconsistency between

teachers’ mathematics beliefs and classroom practice, focusing on six novice teachers at the

elementary level. These data were analyzed using four categories: teachers’ beliefs about

teaching mathematics, teachers’ beliefs about teaching mathematics, teachers’ beliefs about the

nature of mathematics, and teachers’ classroom practices. Raymond used a mixture of research

methods to collect her data, including phone interviews, audiotaped interviews, classroom

observations, concept mapping of mathematics beliefs and practice, and self-report questionnaire

on the factors that influence classroom teaching. The results showed that most teachers held

traditional beliefs in all four areas. One of the teachers, however, indicated a traditional belief for

the nature of mathematics and a non-traditional belief for the teaching and learning of

If teachers believe that their students can learn from their instruction, research has shown

that when the teaching process does not work out, teachers and students need to do some

correction (Brophy & Advertson, 1976). As Porter et al. (1988) discovered, teachers’ beliefs are

important because “teachers who accept responsibilty for student outcomes are more effective

than teachers who see their students as solely responsible for what they learn and how ther

behave.” Studies have shown that pre-service teachers’ beliefs about good mathematics teaching

are heavily influenced by the way they were taught mathematics in schools and are formed from

their schooling experience (Ball, 1988; Owens, 1987).

c) Content-focused with an emphasis on performance: mathematics teaching that

emphasizes student performance and mastery of mathematics rules and

procedural; and

d) Classroom-focused: mathematics teaching based on knowledge about

effective classrooms. (p. 2)

How a teacher’s change in belief might influence classroom practice is an important

aspect to be considered (Grant, 1984). Raymond (1987) studied the inconsistency between

teachers’ mathematics beliefs and classroom practice, focusing on six novice teachers at the

elementary level. These data were analyzed using four categories: teachers’ beliefs about

teaching mathematics, teachers’ beliefs about teaching mathematics, teachers’ beliefs about the

nature of mathematics, and teachers’ classroom practices. Raymond used a mixture of research

methods to collect her data, including phone interviews, audiotaped interviews, classroom

observations, concept mapping of mathematics beliefs and practice, and self-report questionnaire

on the factors that influence classroom teaching. The results showed that most teachers held

traditional beliefs in all four areas. One of the teachers, however, indicated a traditional belief for

the nature of mathematics and a non-traditional belief for the teaching and learning of

If teachers believe that their students can learn from their instruction, research has shown

that when the teaching process does not work out, teachers and students need to do some

correction (Brophy & Advertson, 1976). As Porter et al. (1988) discovered, teachers’ beliefs are

important because “teachers who accept responsibilty for student outcomes are more effective

than teachers who see their students as solely responsible for what they learn and how ther

behave.” Studies have shown that pre-service teachers’ beliefs about good mathematics teaching

are heavily influenced by the way they were taught mathematics in schools and are formed from

their schooling experience (Ball, 1988; Owens, 1987).

44.
31

Most mathematics educators have the same opinion that good teaching is not only about

“telling,” as stated in the NCTM (1989, 2000) documents. Studies done by Chazan and Ball

(1999) found that it was not easy teaching a high school algebra lesson or a third grade class

without “telling.” The challenges faced during the lessons were described in the study.

Philipp (2008) found that most teachers’ beliefs were that “teaching mathematics requires

telling, or providing clear, step-by-step demonstrations of these procedures and students learn by

listening to teachers’ demonstration and practicing these procedures” (p. 281). What is more

important to not just measuring teachers’ beliefs, but rather when do beliefs change and does this

affect classroom teaching. Smith (1996) discovered that for teachers to change their beliefs about

good teaching to suit the reform by the NCTM Standards, two events must happen. First,

teachers must know the benefits of the reform in order to change their classroom practice.

Second, teachers’ “success in making the changes to their practice must bring about their

reconceptualization of their senses of efficacy” (Philipp, 2008, p. 281).

Numerous studies have shown that teachers’ beliefs about mathematics influence good

teaching (Cooney & Wigel, 2003). There is no agreement on the best way to teach mathematics

because of differing views, depending on the level of mathematics and what is important. Kendal

and Stacey (2001) investigated how two teachers with different beliefs about mathematics used a

computer algebra system in their calculus lesson. Even though both classes achieved almost

similar student achievement, the pedagogy was different. One teacher emphasized the conceptual

understanding of calculus while using technology to support the ideas. Another teacher focused

on mathematical procedures while integrating the technology into the lessons. Both methods had

their own strength, but good teaching may have differed depending on individual beliefs.

Most mathematics educators have the same opinion that good teaching is not only about

“telling,” as stated in the NCTM (1989, 2000) documents. Studies done by Chazan and Ball

(1999) found that it was not easy teaching a high school algebra lesson or a third grade class

without “telling.” The challenges faced during the lessons were described in the study.

Philipp (2008) found that most teachers’ beliefs were that “teaching mathematics requires

telling, or providing clear, step-by-step demonstrations of these procedures and students learn by

listening to teachers’ demonstration and practicing these procedures” (p. 281). What is more

important to not just measuring teachers’ beliefs, but rather when do beliefs change and does this

affect classroom teaching. Smith (1996) discovered that for teachers to change their beliefs about

good teaching to suit the reform by the NCTM Standards, two events must happen. First,

teachers must know the benefits of the reform in order to change their classroom practice.

Second, teachers’ “success in making the changes to their practice must bring about their

reconceptualization of their senses of efficacy” (Philipp, 2008, p. 281).

Numerous studies have shown that teachers’ beliefs about mathematics influence good

teaching (Cooney & Wigel, 2003). There is no agreement on the best way to teach mathematics

because of differing views, depending on the level of mathematics and what is important. Kendal

and Stacey (2001) investigated how two teachers with different beliefs about mathematics used a

computer algebra system in their calculus lesson. Even though both classes achieved almost

similar student achievement, the pedagogy was different. One teacher emphasized the conceptual

understanding of calculus while using technology to support the ideas. Another teacher focused

on mathematical procedures while integrating the technology into the lessons. Both methods had

their own strength, but good teaching may have differed depending on individual beliefs.

45.
32

Chapter 3

METHODOLOGY

This chapter gives a detailed description of the methodology of this study. First, the

research questions of this study are described. Second, the overall research methodology of this

study is elaborated in detail. Next, the participants of the study, all of whom had taken a survey

questionnaire, are discussed. Fourth, the two research instruments used in this study are

described; the first is a web-based survey questionnaire while the second is a semi-structured

interview. The development of the survey questionnaire is also discussed. Next, the reliability

and validity of the instruments are described, followed by the development of the interview

questions and their reliability and validity. Following are the details of the data collection

procedures. Finally, the analysis of both quantitative and qualitative data collected to answer the

research questions of the study is described.

Research Questions

This study investigated how beginning teachers define “good mathematics teaching” and

what these teachers deem the most important attributes of good mathematics teaching at the

secondary level. This research also explored whether there was a relationship between the

demographics of the participants and the attributes of good teaching. Furthermore, this study

explored the factors that influence the understanding of good mathematics teaching and when

good mathematics teaching attributes are developed. The study also looked at where one might

attribute the elements of good mathematics teaching: to some personality trait, to teacher’s

classroom behavior, to teacher’s mathematical knowledge or to teacher education programs.

Chapter 3

METHODOLOGY

This chapter gives a detailed description of the methodology of this study. First, the

research questions of this study are described. Second, the overall research methodology of this

study is elaborated in detail. Next, the participants of the study, all of whom had taken a survey

questionnaire, are discussed. Fourth, the two research instruments used in this study are

described; the first is a web-based survey questionnaire while the second is a semi-structured

interview. The development of the survey questionnaire is also discussed. Next, the reliability

and validity of the instruments are described, followed by the development of the interview

questions and their reliability and validity. Following are the details of the data collection

procedures. Finally, the analysis of both quantitative and qualitative data collected to answer the

research questions of the study is described.

Research Questions

This study investigated how beginning teachers define “good mathematics teaching” and

what these teachers deem the most important attributes of good mathematics teaching at the

secondary level. This research also explored whether there was a relationship between the

demographics of the participants and the attributes of good teaching. Furthermore, this study

explored the factors that influence the understanding of good mathematics teaching and when

good mathematics teaching attributes are developed. The study also looked at where one might

attribute the elements of good mathematics teaching: to some personality trait, to teacher’s

classroom behavior, to teacher’s mathematical knowledge or to teacher education programs.

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The research questions for this study were:

1. How do beginning teachers define “good mathematics teaching”?

2. How do beginning teachers describe “good mathematics teaching” in middle school

and high school? What were the important attributes of “good mathematics

teaching”?

3. Is there any relationship between demographics (e.g., Age, MathGPA, Overall GPA)

and descriptions of “good mathematics teaching” attributes?

4. What were the factors that influenced beginning teachers’ understanding of good

mathematics teaching?

Research Methodology

This study applied a mixed methods design to answer the research questions because

using both quantitative and qualitative research methods provides a better understanding of good

mathematics teaching (Creswell, 2009). Using a mixed methods approach was definitely an

advantageous approach for surveying a large group of individuals, followed by an interview with

a smaller sub-sample of individuals to obtain their specific beliefs. Another reason why mixed

methods was applied here was that “mixed methods research as the third research paradigm also

helps bridge the schism between quantitative and qualitative research” (Onwuegbuzie& Leech,

2004, p. 3), and in “utilizing quantitative and qualitative techniques within the same framework,

mixed methods research can incorporate the strengths of both methodologies” (Johnson &

Onwuegbuzie, 2004, p. 11).

Chatterji (2010) asserts that a mixed methods design allows flexibility in the purpose of

the investigation by combining evidence. Results can then be obtained from the quantitative part

to answer the research questions, while descriptions to probe further into the research questions

The research questions for this study were:

1. How do beginning teachers define “good mathematics teaching”?

2. How do beginning teachers describe “good mathematics teaching” in middle school

and high school? What were the important attributes of “good mathematics

teaching”?

3. Is there any relationship between demographics (e.g., Age, MathGPA, Overall GPA)

and descriptions of “good mathematics teaching” attributes?

4. What were the factors that influenced beginning teachers’ understanding of good

mathematics teaching?

Research Methodology

This study applied a mixed methods design to answer the research questions because

using both quantitative and qualitative research methods provides a better understanding of good

mathematics teaching (Creswell, 2009). Using a mixed methods approach was definitely an

advantageous approach for surveying a large group of individuals, followed by an interview with

a smaller sub-sample of individuals to obtain their specific beliefs. Another reason why mixed

methods was applied here was that “mixed methods research as the third research paradigm also

helps bridge the schism between quantitative and qualitative research” (Onwuegbuzie& Leech,

2004, p. 3), and in “utilizing quantitative and qualitative techniques within the same framework,

mixed methods research can incorporate the strengths of both methodologies” (Johnson &

Onwuegbuzie, 2004, p. 11).

Chatterji (2010) asserts that a mixed methods design allows flexibility in the purpose of

the investigation by combining evidence. Results can then be obtained from the quantitative part

to answer the research questions, while descriptions to probe further into the research questions

47.
34

can be supported by the qualitative section. In this case, answers to what constitutes good

mathematics teaching and how is it influenced by several factors can be found using the

quantitative design of a selected sample from a teacher education program, while the reasons for

the selection of good attributes and how they are achieved can be investigated using the

qualitative design of a semi-structured interview.

This study used a mixed methods approach that combined the quantitative technique

using the survey questionnaire and the qualitative technique using semi-structured interviews.

The survey questionnaire from the entire sample was analyzed using the statistical software

SPSS to obtain descriptive statistics and correlations and to do cluster analysis. After analyzing

the data, the researcher anticipated obtaining answers to the definition of good mathematics

teaching and the most important attributes of good teaching from the perspective of a beginning

teacher. In addition, cluster analysis was performed to determine whether the beginning teachers

had groupings and whether there were any relationships with demographic variables such as

Age, Overall GPA, and Math GPA. This exploratory analysis informed the researcher of the

cluster of attributes that was associated with teachers in different cluster groups. The cluster

analysis determined whether there was a relationship between the demographic variables and the

attributes of good mathematics teaching.

With the interview, the reasons for selecting the definition of good mathematics teaching

would be obtained from the participating beginning teachers. Similarly, the reasons for selecting

attributes of good teaching were also investigated. This provided a clearer picture of the reasons

for the meaning of good mathematics teaching. After the interviews, the researcher analyzed the

responses using the qualitative analysis software NVivo. This software assisted in identifying

similar patterns from the responses of the randomly selected participants and enriched the

can be supported by the qualitative section. In this case, answers to what constitutes good

mathematics teaching and how is it influenced by several factors can be found using the

quantitative design of a selected sample from a teacher education program, while the reasons for

the selection of good attributes and how they are achieved can be investigated using the

qualitative design of a semi-structured interview.

This study used a mixed methods approach that combined the quantitative technique

using the survey questionnaire and the qualitative technique using semi-structured interviews.

The survey questionnaire from the entire sample was analyzed using the statistical software

SPSS to obtain descriptive statistics and correlations and to do cluster analysis. After analyzing

the data, the researcher anticipated obtaining answers to the definition of good mathematics

teaching and the most important attributes of good teaching from the perspective of a beginning

teacher. In addition, cluster analysis was performed to determine whether the beginning teachers

had groupings and whether there were any relationships with demographic variables such as

Age, Overall GPA, and Math GPA. This exploratory analysis informed the researcher of the

cluster of attributes that was associated with teachers in different cluster groups. The cluster

analysis determined whether there was a relationship between the demographic variables and the

attributes of good mathematics teaching.

With the interview, the reasons for selecting the definition of good mathematics teaching

would be obtained from the participating beginning teachers. Similarly, the reasons for selecting

attributes of good teaching were also investigated. This provided a clearer picture of the reasons

for the meaning of good mathematics teaching. After the interviews, the researcher analyzed the

responses using the qualitative analysis software NVivo. This software assisted in identifying

similar patterns from the responses of the randomly selected participants and enriched the

48.
35

answers to the research questions. The researcher applied the “coding” method that would detect

and describe the common themes among the participants’ responses (Gibson & Brown, 2009).

Words that had similar meanings were coded under themes created by the researcher. For

example, terms mentioned by the interview participants, such as good in mathematics, have

strong content knowledge, and have sufficient mathematical knowledge, were grouped under the

theme “Subject Matter Knowledge.”

Survey Participants

The traditional certification program offered by this graduate school of education was

intended for aspiring mathematics teachers who had obtained a bachelor’s degree in

mathematics. Since its inception, it has been successful in producing and training outstanding

mathematics educators, with a strong emphasis on mathematical content. With that in mind, the

participants of this study were selected from the mathematics education program from this

graduate school of education. The participants were required to complete a minimum of 36

points through full-time study in one academic year and a summer term.

At the end of the program, participants obtained state certification in teaching

mathematics at the secondary level. Most of the teachers who graduated from the program

continued to teach in schools around the city in this Eastern state. This city was selected as it is

one the largest school district in the United States, with approximately over 50,000 teachers in

more than 1000 schools educating thousand of students. Because this study focusing on what

constitutes good mathematics teaching and what are the important attributes of good teaching as

perceived by beginning teachers, the participants were teachers who have been teaching for one

or two years.

answers to the research questions. The researcher applied the “coding” method that would detect

and describe the common themes among the participants’ responses (Gibson & Brown, 2009).

Words that had similar meanings were coded under themes created by the researcher. For

example, terms mentioned by the interview participants, such as good in mathematics, have

strong content knowledge, and have sufficient mathematical knowledge, were grouped under the

theme “Subject Matter Knowledge.”

Survey Participants

The traditional certification program offered by this graduate school of education was

intended for aspiring mathematics teachers who had obtained a bachelor’s degree in

mathematics. Since its inception, it has been successful in producing and training outstanding

mathematics educators, with a strong emphasis on mathematical content. With that in mind, the

participants of this study were selected from the mathematics education program from this

graduate school of education. The participants were required to complete a minimum of 36

points through full-time study in one academic year and a summer term.

At the end of the program, participants obtained state certification in teaching

mathematics at the secondary level. Most of the teachers who graduated from the program

continued to teach in schools around the city in this Eastern state. This city was selected as it is

one the largest school district in the United States, with approximately over 50,000 teachers in

more than 1000 schools educating thousand of students. Because this study focusing on what

constitutes good mathematics teaching and what are the important attributes of good teaching as

perceived by beginning teachers, the participants were teachers who have been teaching for one

or two years.

49.
36

Using the number of graduates from previous years, the cohort of the participants invited

to participate in the study was approximately 80 beginning teachers; 33 of them responded and

became the subjects of this study. All the participants who joined this study answered the web-

based survey questionnaire. The web-based method was selected as opposed to a paper-based

survey as the method of responding proved accessible to the participants. The survey

questionnaire could be completed anytime at the participants’ convenience.

Interview Participants

The interview participants were randomly selected. The technique used was a simple

random sampling of the 33 participants who participated in this study. This means that each

participant of the study had an equal chance of being selected. All of the beginning teachers who

were randomly selected had a strong mathematics background. This subsample was selected

randomly based on their participation in the survey questionnaire. Based on the criteria for

selecting the subsample for the interview, the researcher randomly selected 10 beginning

teachers. The beginning teachers were then interviewed to probe further their reasons for

selecting certain definitions and attributes of good teaching. In addition, the factors that

influenced their understanding of good mathematics teaching were investigated.

Table 3.1 provides a summary of the Math GPA of the interviewed participants.

Table 3.1

Mathematics Grade Point Average (GPA) of Interview Participants

Math GPA Frequency Percent

2.5 to 3.0 1 10

3.0 to 3.5 5 50

3.5 to 4.0 4 40

Using the number of graduates from previous years, the cohort of the participants invited

to participate in the study was approximately 80 beginning teachers; 33 of them responded and

became the subjects of this study. All the participants who joined this study answered the web-

based survey questionnaire. The web-based method was selected as opposed to a paper-based

survey as the method of responding proved accessible to the participants. The survey

questionnaire could be completed anytime at the participants’ convenience.

Interview Participants

The interview participants were randomly selected. The technique used was a simple

random sampling of the 33 participants who participated in this study. This means that each

participant of the study had an equal chance of being selected. All of the beginning teachers who

were randomly selected had a strong mathematics background. This subsample was selected

randomly based on their participation in the survey questionnaire. Based on the criteria for

selecting the subsample for the interview, the researcher randomly selected 10 beginning

teachers. The beginning teachers were then interviewed to probe further their reasons for

selecting certain definitions and attributes of good teaching. In addition, the factors that

influenced their understanding of good mathematics teaching were investigated.

Table 3.1 provides a summary of the Math GPA of the interviewed participants.

Table 3.1

Mathematics Grade Point Average (GPA) of Interview Participants

Math GPA Frequency Percent

2.5 to 3.0 1 10

3.0 to 3.5 5 50

3.5 to 4.0 4 40

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Teachers who could produce good mathematics teaching need to master and understand

the content knowledge (Ball, 2005; Shulman, 2001). That is why the beginning teachers in this

study generally had a good grade point average (GPA), indicating that they had strong content

knowledge in mathematics. Research has also shown that there is a positive correlation between

teachers’ content knowledge and students’ achievement gains (Ball, 1990; Carpenter, 1989; Hill,

Research Instrument: Description of Survey Questionnaire

This exploratory and empirical research employed a mixed method that utilized both

quantitative and qualitative data. Two instruments were developed for this study. The first

instrument was a survey questionnaire while the second instrument was a semi-structured

qualitative interview. All 33 participants in this study took the web-based survey questionnaire.

The quantitative instrument was a web-based survey. Items for the survey was created

based on studies and a literature review. This included a study done by Cooney (2005) on good

mathematics teaching and by the National Council of Teachers of Mathematics (NCTM) (1991)

Professional Standards of Teaching Mathematics. The five categories outlined by NCTM were:

worthwhile mathematical tasks, teacher and student roles in classroom, classroom teaching

environment, tools for enhancing discourse, and analysis of teaching (NCTM, 1991). In addition,

the 10 characteristics of effective pedagogy in mathematics classrooms done by the International

Bureau of Education in 2009 (Anthony & Walshaw, 1999) included these strategies: an ethic of

care, arranging for learning, building on students’ thinking, worthwhile mathematical tasks,

making connections, assessment for learning, mathematical communication, mathematical

language, tools and representations and teacher knowledge. Some items of the survey

Teachers who could produce good mathematics teaching need to master and understand

the content knowledge (Ball, 2005; Shulman, 2001). That is why the beginning teachers in this

study generally had a good grade point average (GPA), indicating that they had strong content

knowledge in mathematics. Research has also shown that there is a positive correlation between

teachers’ content knowledge and students’ achievement gains (Ball, 1990; Carpenter, 1989; Hill,

Research Instrument: Description of Survey Questionnaire

This exploratory and empirical research employed a mixed method that utilized both

quantitative and qualitative data. Two instruments were developed for this study. The first

instrument was a survey questionnaire while the second instrument was a semi-structured

qualitative interview. All 33 participants in this study took the web-based survey questionnaire.

The quantitative instrument was a web-based survey. Items for the survey was created

based on studies and a literature review. This included a study done by Cooney (2005) on good

mathematics teaching and by the National Council of Teachers of Mathematics (NCTM) (1991)

Professional Standards of Teaching Mathematics. The five categories outlined by NCTM were:

worthwhile mathematical tasks, teacher and student roles in classroom, classroom teaching

environment, tools for enhancing discourse, and analysis of teaching (NCTM, 1991). In addition,

the 10 characteristics of effective pedagogy in mathematics classrooms done by the International

Bureau of Education in 2009 (Anthony & Walshaw, 1999) included these strategies: an ethic of

care, arranging for learning, building on students’ thinking, worthwhile mathematical tasks,

making connections, assessment for learning, mathematical communication, mathematical

language, tools and representations and teacher knowledge. Some items of the survey