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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
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© 2012
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
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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.
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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
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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
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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
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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
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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
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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
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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.
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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
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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.
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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
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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
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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,
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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
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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
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(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
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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:
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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.
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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
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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
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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
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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.
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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;
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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
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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
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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
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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
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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.
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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
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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.
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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
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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
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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
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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
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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).
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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
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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
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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
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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;
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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).
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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.
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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
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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
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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.
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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
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