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An examination of professional development in algebra for fifth-grade teachers

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An examination of professional development in algebra for fifth-grade teachers

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AN EXAMINATION OF PROFESSIONAL DEVELOPMENT IN
ALGEBRA FOR FIFTH-GRADE TEACHERS

by

Eduardo Morales

A Dissertation Presented to the
FACULTY OF THE USC ROSSIER SCHOOL OF EDUCATION
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF EDUCATION

May 2011

Copyright 2011 Eduardo Morales
ii
DEDICATION
Le doy gracias de todo corazón y le dedico mi título de doctorado a mi Dios
Padre Celestial, a mi Señor Jesucristo, y a todos los Santos por haberme dado la
bendición de graduarme de la Universidad una vez más. Gracias por Su amor,
paciencia, y sabiduría que siempre me brindan.
También le dedico mi título de doctorado a toda mi familia quien siempre
me ha apoyado. A mis papas Máximo y Nieves les quiero dar las gracias con todo
mi corazón y amor por siempre haberme inculcado la importancia de recibir una
educación universitaria y por todos los sacrificios que hicieron para que mis
hermanas y yo saliéramos adelante en la vida. Gracias porque siempre han estado
dispuestos a brindarme su ayuda incondicionalmente. Sin el apoyo de ustedes no
hubiera alcanzado mis metas. Siempre voy acordarme de las noches cuando empecé
la Universidad hace mas de diez años y me esperaban durante la medianoche para
abrirme la puerta de la entrada de la casa y darme de cenar. A mi hermanita Jessica
le quiero dar las gracias por traerme una gran felicidad, por su apoyo y por
enseñarme a ver la vida mas positivamente. A mi hermanito Jesusito, que en paz
descanse, le doy las gracias por sus oraciónes a nuestro Dios por nuestra familia. Te
extrañamos mucho y te queremos con todo nuestro corazón hermanito. A mi
hermana Li y a mi sobrina Monse les doy las gracias por cuidarme y por las alegrias
y consejos que me dan. A mi gran esposa Celina, nuestra hija Isabella y nuestro
bebe que viene en camino, les estoy muy agradecido por todo su amor, sus
atenciones, paciencia, apoyo, y comprensión. Gracias amor por compartir tu vida

iii
conmigo. Siempre recordare las noches cuando les compartía mis ideas de mis
estudios y me escuchaban atentamente y luego me daban sus consejos. Gracias a
Dios, a la Santísima Trinidad, a la Sagrada Familia, a todos los Santos, a mi esposa,
mis hijos, mis papas, mis hermanas, mi hermanito y mi sobrinita por todo el amor
que siempre me han dado. Los quiero con todo mi corazón!

iv
ACKNOWLEDGEMENTS
This journey of earning my doctorate degree has been a dream of mine since
I was a child. I am wholeheartedly grateful to God, my Father Almighty, and to my
Lord Jesus Christ who made this all possible with the many blessings I have
received. It is through the love, support, patience, and wisdom of my God and Lord
Jesus Christ that I have been successful with my achievements. Thank you God, my
Lord Jesus Christ and all the Saints!
Since I was a child my parents have always reminded me how valuable it is
to receive a college education. To them I give my deepest gratitude. Thank you
mom and dad for your love, encouragement, guidance, patience, and for the
happiness you always bring me! You always have words of wisdom for me. To my
younger sister Jessica I will always be grateful for the many ways in which she has
enriched my life and has made me value more all the marvelous things that life has
to offer. Thank you Jessi for encouraging me to pursue my dreams and for always
bringing me happiness! You always know how to make me smile. To my younger
brother Jesusito, may you rest in peace, thank you for all your prayers and for always
looking after us from heaven! We love you and miss you!! To my older sister and
niece I will always be thankful for looking after me and giving me valuable advice.
Thank you Li and Monse for your encouragement and smiles! To my beautiful wife
Celina, daughter Isabella and our baby boy on the way I must say thank you! Thank
you for your love, patience, understanding, for cheering me on every day as I found
myself reading and writing many months for long hours and for those moments we

v
found ourselves in the middle of night laughing and eating a snack after a long night
of work. Thank you for listening to my thoughts and for sharing your ideas. I will
always remember the nights when our daughter Isabella would call me in the middle
of the night to let me know that it was time for me to go to bed, but most
importantly, that it was time for me to prepare her milk, and will also remember the
nights when our baby boy was kicking to let us know that he was celebrating our
triumphs with us! To my two grandmothers I owe them my heart for always wishing
me the best and for their prayers, thank you! To my mother-in-law and father-in-law
who have given me their support and words of encouragement, thank you! To all my
family members I give them all my love!
Thank you to my advisor, Dr. Garica, and committee members, Dr. Castruita
and Dr. Rousseau, for their genuine interest in my work, encouragement, expertise,
patience, time, and valuable feedback! Without their assistance I would not have
completed my dissertation.
To Dr. Berg, who has always believed in me ever since he was my professor
as an undergraduate student, I say thank you for your encouragement, for sharing
your wisdom with me, and for your unconditional support. Thank you for being my
friend, mentor, and teacher! You have always made time to share a part of your life
with me.
To Dr. Hollins, Dr. Slayton, Dr. Kasabian, Dr. Bennish, Professor Tummers-
Stocum, Dr. Goldberg, Professor Georgevich, Professor Perinetti, Professor L. Ho,
Professor Ryan, Professor Boerger, Professor A. Seyedin, Dr. M. Seyedin, Dr.

vi
Glucksman, Dr. Hamza, and all my colleagues who continue to pursue excellence as
educators I say thank you for sharing your wisdom and knowledge. Thank you for
caring about me as a person, as an educator, and as a graduate student pursuing a
doctorate degree.
To my friends Patty C., Maricela M., Aracelia and Jose R., Lessette M.,
Sheila S., Steve P. (Huey), Mike Y. (Minkowski), Edgar H., Alberto A., Angela B.,
everyone in my cohort, and other friends who I owe my sincerest gratitude for their
support and encouragement! Thank you for being there for me.
Thank you to Janeen Steel and the team at LRLC for their encouragement
and the marvelous work that they do for the children! You continue to be my
inspiration.
To all the fantastic teachers from elementary, middle and high school, and
college faculty that gave me the foundation that I needed to succeed academically in
college and in life, thank you!
Thank you to the University of Southern California and the Rossier School of
Education for believing in me! May God bless my family, friends, teachers, and
colleagues!

vii
TABLE OF CONTENTS
DEDICATION ii

ACKNOWLEDGEMENTS iv

LIST OF TABLES ix

ABSTRACT x

CHAPTER 1: OVERVIEW OF THE STUDY 1
Statement of the Problem 5
Purpose of the Study 6
Research Questions 7
Significance of the Study 7
Assumptions 8
Limitations 9
Delimitations 9
Definitions 10
Organization of the Study 11

CHAPTER 2: REVIEW OF THE LITERATURE 12
Introduction 12
Background of Algebra Achievement 14
Understanding of Mathematics 20
Professional Development 24
Professional Learning Community 33
Lesson Study 39

CHAPTER 3: METHODOLOGY 47
Introduction 47
Purpose 48
Research Methodology Overview 48
Research Design 49
Sample and Population 50
Participants 51
Overview of the District and School 51
Instrumentation 53
Data Collection Procedures 53
Interviews 54
Observations 57
Document Analysis 57
Data Analysis 59

viii
Figure 3.1: Creswell’s Model for Qualitative Data Analysis 60
Ethical Considerations 60

CHAPTER 4: DATA ANALYSIS 62
Introduction 62
Findings for Research Question One 64
Findings for Research Question Two 74
Findings for Research Question Three 90
Summary 105

CHAPTER 5: SUMMARY, CONCLUSIONS, AND IMPLICATIONS 106
Introduction 106
Summary of the Study 107
Conclusions 109
Implications 114
Recommendations for Future Research 118

REFERENCES 119

APPENDICES
Appendix A: Interview Protocol for Teachers 127
Appendix B: Interview Protocol for Principal 129
Appendix C: Interview Protocol for Assistant Superintendent of
Instruction 131
Appendix D: Interview Protocol for Math Coach 133

ix
LIST OF TABLES
Table 3.1: Kraft Unified School District Students by Ethnicity,
2008-09 52
Table 3.2: Kraft Unified School District Student Data for Special
Programs, 2008-09 53
Table 3.3: Research Design Chart 54
Table 4.1: Data for Research Question One 65
Table 4.2: Data for Research Question Two 75
Table 4.3: Data for Research Question Three 90
Table 4.4: Professional Development Offered by the District 98

x
ABSTRACT
The purpose of this study was to examine the extent to which one Southern
California school district supported the fifth-grade teachers’ capacity to teach their
students concepts and skills related to algebra. Moreover, the study further evaluated
the school principal’s perception and fifth-grade teachers’ experiences to determine
whether the professional development offered by the district supports the fifth-grade
teachers in gaining math knowledge and effective teaching strategies to teach algebra
content.
In this investigation a qualitative case study research method was used. The
desire to gain a rich, in-depth knowledge of the different algebra professional
development for fifth-grade school teachers led to the decision to investigate a single
elementary school. The selection criteria used to identify the school-site for this case
study were: 1) The elementary school had a 2008-2009 API of 800 or above, and/or
consistent growth in API scores for the past two academic years; 2) Student
enrollment of at least 500; 3) The elementary school had a current Similar Schools
Ranking of 8 or above; 4) The elementary school had under 50% of White students;
and 6) The elementary school had computers and classrooms with internet.
The analysis revealed several findings including: 1) The school district needs
to allow more opportunities for teachers to be involved with their professional
development planning and being facilitators; 2) Professional development that
focuses on Cognitive Guided Instruction (CGI) had limitations for fifth-grade
teachers; 3) The district needs to provide more professional development that focuses

xi
on the MIND program given that teachers expressed their high regards for this
program; and 4) Serious consideration needs to be given to the culture of the school
in order to determine whether the professional development that is offered to
teachers is meeting the needs of the students. Conclusions include a discussion of
the study’s implications for various stakeholders and several recommendations for
future studies are provided.

1
CHAPTER 1
OVERVIEW OF THE STUDY
In the United States, it is a struggle for students to successfully pass
mathematics courses. Stigler and Hiebert (1997) note that 78% of mathematics
teacher in the United States do not develop topics in depth but just state them, and
96% of the time students only practice computations rather than develop conceptual
understanding. Algebra is considered a gateway that leads to more rigorous math and
science courses in high school (EdSource, 2009). Algebra has also been recognized
by both educators and policymakers to be a gatekeeper course that assists in college
preparation as well as in the work force (Choike, 2000). Not only are colleges
requiring that students possess some level of algebra knowledge prior to enrollment,
but some states in the United States, such as California, are now requiring that
students successfully complete one year of Algebra as a requirement to graduate
from high school (EdSource, 2009). Furthermore, in California, high school students
must pass the California High School Exit Exam (CAHSEE) in order to graduate
from high school which includes an Algebra I component (EdSource, 2009). Thus,
the need to be proficient in algebra is now becoming more indispensable. Despite
such expectations from high schools and colleges, there continues to be a large
percentage of students who fail to graduate from high school as a consequence of not
passing high school algebra (U. S. Department of Education, 2008).
Educators and policymakers agree that too many students are repeating
algebra, sometimes multiple times, and it has become a concern since the lack of

2
success in this course prevents students from completing college preparatory science
courses that require algebra (EdSource, 2009). Data from the 2007 National
Assessment of Educational Progress (NAEP) indicate that California 8
th
grade
students generally score below the national average on the national math
achievement exam which heightens the concern that students in the public school
system are not adequately prepared to enroll in an Algebra I course. According to
NAEP, 32% of students are at or above the “proficient” level in Grade 8, but only
23% of students reach the proficient level at Grade 12 (U.S. Department of
Education, 2008). In the United States, the current science and engineering
workforce faces a future in which there will be accelerating retirements, and for
many years the U.S. has depended on the talent from abroad. For a variety of
reasons, educators and policymakers have continued to focus on how to best prepare
students in our schools (U.S. Department of Education, 2008), including the fact that
for the United States to remain competitive in the global world it is necessary to
strengthen the mathematical knowledge of Americans. Given that the U.S. needs to
produce students who are well trained in mathematics, it is necessary for American
students to have a solid the foundation of algebra in order for them to successfully
achieve in higher mathematics courses.
In the context of the No Child Left Behind Act of 2002 (NCLB), a new
emphasis has been brought towards teacher preparation, teacher qualifications, and
teaching practices. Pressure from professional organizations for teacher knowledge
and skills, discussions of merit pay for effective teachers (Hershberg & Lea-Kruger,

3
2006), assessments to measure teacher effectiveness, teacher licensing and
credentialing discussions (Wilson, Floden, & Ferrini-Mundy, 2001), and the
standards movement are paving the way to more delicate analysis of the current U.S.
education system. Many questions in education continue to arise regarding what can
be done to improve the quality of education that American students are currently
receiving. Teacher training most certainly plays a key role in producing effective
teachers. The main questions in teacher education are: What should teachers know
and be able to do so that students will know and be able to do (Cochran-Smith,
2001)? Furthermore, Darling-Hammond (2000) explains that the United States
needs qualified teachers, equitable funding, and educational resources for all students
so that the United States will remain competitive in international comparisons of
student achievement and produce competitive citizens for the global markets. With
the growing numbers of high-minority and high-poverty student populations in the
school system, the needs of students today look different than the needs of students
in the past (Hollins & Guzman, 2005). There is a growing concern that students are
not being prepared to become successful citizens in the 21
st
century. Minority and
high-poverty students with low test scores reflect the majority of the student
population in urban schools (Garcia, 2002). In addition, low-income and minority
children are often denied the educational system’s expectations, resources, and the
best teachers (Haycock, 2004). These students need high quality teachers that have
high expectation, rigor in the curriculum, language acquisition support, problem
solving and critical thinking skills, technology skills and culturally relevant

4
pedagogy to help close the achievement gap between current student performance
and actual student potential.
The low achievement of mathematics students in American schools continues
to bring attention to what teaching methods need to be employed in the classroom to
help the students succeed in mathematics. Mathematics teachers need content
knowledge in the field as well as effective pedagogy in order to promote student
learning. Darling-Hammond (2000) notes that a fully credentialed teacher in the
classroom is the strongest predictor of positive student achievement. A fully
credentialed teacher in a mathematics classroom increases the likelihood that the
teacher has been exposed to effective pedagogy to be implemented in the classroom.
Furthermore, the National Council of Supervisors of Mathematics (NCSM, 2007)
states that mathematics teachers and leaders need a strong school and district support
system that will provide intensive and sustained professional learning in order to
make sure that all students have access to the best instruction in mathematics
possible. The shortage of qualified teachers in high-poverty schools is a contributor
to the unsuccessful experience of students and is a concern of many such schools and
districts. Each year there is a larger pool of teachers who leave the profession than
those who enter. Ingersoll (2003) noted that in 1999, nearly 290,000 teachers left the
profession while there was only an induction of 230,000 new teachers.
It is necessary for mathematics teachers to be not only qualified with a degree
in this discipline but to also have the appropriate teaching credentials. Teachers need
mathematical knowledge that combines mathematics and pedagogical procedures

5
(Fennema & Franke, 1992). It is possible that new teachers may be immersed in
schools where there is resistance to change and where support for reform-based
practices of teaching would not be welcomed. Furthermore, the expectations now
established by school districts to cover certain material at a specific pace can force
teachers to rely on more traditional teaching practices that require less
implementation time (Huffman, Thomas, & Lawrenz, 2008). Morris (2006) notes
that teachers need to have the ability to analyze their own teaching in terms of its
effects on learning, and it is through this analysis that teachers can improve their own
teaching. Through the teachers’ reflections of their teaching, they can become
metacognitive beings in the classroom allowing them to make informed decisions
about their students’ learning progress.
Statement of the Problem
The ultimate goal of education is to provide students with academic
knowledge that will prepare them to function effectively and independently in our
society. Teachers need to come to the realization that it is sometimes necessary to
depart from the traditional lecture approach and implement a different pedagogy in
the classroom, even if it means giving up the illusion of control. Professional
development training certainly is a venue for most teachers to learn of teaching
strategies that can allow students to successfully learn mathematics. Civil and Planas
(2004) state that students who are empowered by their experiences at school are able
to develop their ability, motivation, and confidence to succeed academically.

6
For many years now the low performance of students in algebra has been a
concern in the United States. The United States has ranked low in mathematics
achievement in comparison to some of the richest nations in the world (U.S.
Department of Education, 2008). Consequently, educators and policymakers
continue to strive to find strategies to increase performance in mathematics. A
number of strategies have been implemented by mathematics teachers in the United
States, but the percentage of students who successfully pass algebra continues to be
low. Algebra is a gateway to college as well as many careers, and school districts
across the U.S. have now begun to require a successful completion of algebra in
order to receive a high school diploma and/or mandatory state exams have a
component in mathematics that includes algebra.
Purpose of the Study
In order to increase student success in algebra and in other math content,
there is a need to closely examine the type of mathematics professional development
that school districts offer to their teachers to help them increase their mathematical
knowledge and provide them with effective teaching methods. Teacher preparation
in the classroom has a major role in the students’ learning of mathematics. This
preparation includes having a rich knowledge of mathematics as well as the capacity
to implement effective teaching strategies.
The purpose of this study was to examine the extent to which a school district
supported the fifth-grade teachers’ capacity to teach their students concepts and skills
related to algebra. Moreover, the study furthered evaluated the school principal’s

7
perception and fifth-grade teachers’ experiences to determine whether the
professional development offered by the district supported the fifth-grade teachers in
gaining math knowledge and effective teaching strategies to teach algebra content. It
is important for administrators and teachers to reflect on the current state of
professional development that is being offered by the district and the school, and
what changes are needed, if any, for it to be more effective.
Research Questions
This study will answer the following three questions:
1. What is the extent to which the school district supports teachers’
capacity to teach fifth-grade students concepts and skills related to
algebra through professional development?
2. What is the school principal’s perception on how professional
development supports the teaching of the foundations of algebra for
fifth grade?
3. What are the fifth-grade teachers’ perceptions on professional
development as it relates to teaching mathematics concepts and skills
related to algebra at this grade level?
Significance of the Study
Educators and policymakers have continued to focus on how to best prepare
students in our schools (U.S. Department of Education, 2008) For the United States
to remain competitive in the global world, it is necessary to the strengthen the
mathematical knowledge of Americans. By evaluating the type of professional

8
development that is being provided and its impact on the success of fifth-grade
teachers with their students during algebra instruction, effective teaching practices
and strategies that are instrumental for students’ understanding of algebra can be
identified as well as the math content knowledge that is required for teachers to have.
Poor and minority students are assigned to ineffective teachers two times
more often than other students (Darling-Hammond, 2000). Students who are
assigned to classrooms with ineffective teaching for consecutive years have lower
achievement and gains than students in highly effective teaching classrooms
(Darling-Hammond, 2000). Analyzing and determining what type of professional
development leads to successfully learning algebra will provide the students taught
by teachers who participate in such staff developments exposure to effective teaching
strategies.
Assumptions
During this study the following assumptions have been made:
1. The participation of the assistant superintendent, school principal,
teachers, and the math coach have been entirely voluntary.
2. Responses to all interview questions and questionnaires have been
authentic.
3. The instrumentations that were designed and utilized were effective in
eliciting and providing the information sought.

9
Limitations
A more comprehensive study and conclusion was not possible due to time
constraints. The researcher had a limited timetable in which to perform the research
and have conclusive results. Consequently, the results obtained from the study were
influenced by the time limitation. This study was limited to data obtained during a
short period of time and does not give a comprehensive view on the variables over a
long period of time.
Financial constraints on behalf of the researcher did not allow for a more
detailed and thorough study. The researcher had no control over what professional
development training could be offered or employed with the teachers. Furthermore,
teacher participants in this study may have viewed this study as an evaluation of their
professionalism or effectiveness in the classroom, and this may have influenced the
data that were gathered.
Delimitations
This study focused on the professional development of one school in the
district selected, thus limiting the number of participants, and consequently the
results are only applicable to this particular school and cannot be generalized. The
researcher focused on the professional development that fifth-grade teachers
participated in, and assumptions have been made that this training had some effect
on the fifth-grade students given math instruction by these teachers. No classified
staff members or community members, including parents, were interview or
surveyed.

10
Definitions
1. AYP: Adequate Yearly Progress. This is the federal government’s accountability
measure used under NCLB (2001). States, districts, and schools are given a target
growth every year that they must meet in order to attain the common Math and ELA
proficiency goal.
2. CST: California Standards Test. Per the California Department of Education,

The California Standards Tests in English-language arts, mathematics,
science, and history-social sciences are administered only to students in
California public schools. Except for a writing component that is
administered as part of the grade 4 and 7 English-language arts tests, all
questions are multiple choice. These tests were developed specifically to
assess students’ performance on California’s Academic Content Standards.
The State Board of Education adopted these standards that specify what all
California children are expected to know and be able to do in each grade of
course (CDE STAR website).

3. NCLB: No Child Left Behind Act (2001). This is the reauthorization of the
Elementary and Secondary Education Act. Currently, it is the federal government
reform and accountability model for education.
4. Free /Reduced Price Meals—A program which offers free or reduced price meals
to students. Information about students enrolled in the program is used as a proxy
for a socio-economically disadvantaged population (Education Data Partnership,
2010).
5. Unified School District—A unified school district includes both elementary and
high school students (Education Data Partnership, 2010).

11
Organization of the Study
This dissertation is organized into five chapters. Chapter One gives an
overview of the problem being studied and includes the statement of the problem, the
purpose of the study, the significance of the study, the research questions, limitations
of the study, the delimitations, the assumptions and the operational definitions of
terms that are used in this study. Chapter Two is the review of relevant literature that
relates to this particular investigation, and Chapter Three discusses the methodology
for this study. Chapter Four analyzes the findings of the study and discusses the
major themes that were found. Lastly, Chapter Five summarizes the study, presents
the conclusions and discusses the implications of the study. Recommendations for
future research are also included in this chapter.

12
CHAPTER 2
REVIEW OF THE LITERATURE
Introduction
The superiority of American schools over other school systems in the world
has been questionable in the wake of the Sputnik challenge in 1957. The United
States has since sought to pass educational reforms that focused on Algebra, science
and technology given that other countries are matching and surpassing the
educational attainments of American education (National Commission on Excellence
in Education, 1983). The continual effort of the U.S. to improve student
performance in mathematics and science is to ensure that Americans are keeping up
with the most advanced breakthroughs in science and technology. It continues to be
evident that other countries in the world have the capacity to outpace the U.S. Such
was the case during Sputnik when many believed that Russia would outpace the
world in science and technology.
With the diversity that is found in American schools, there continues to be
discussions about the achievement gap between low-income and minority students
and other young Americans. Hollins and Torres-Guzman (2005) note that the nation
faces the challenge of providing high quality schooling to students of color, low
income students, English language learners, and students in urban and rural settings
who are underserved by the educational system. The achievement gap between
African American and White students was cut in half between 1970 and 1988 and by
one-third between Hispanic and White students, but since this time, the gap has

13
widened (Haycock, 2001). It has been well documented that schools with a large
population of poor and minority students are assigned to ineffective teachers two
times more often than other students, and these students who are assigned to
classrooms with ineffective teaching for consecutive years have lower achievement
and gains than students in highly effective teaching classrooms (Darling-Hammond,
2000).
In mathematics, the achievement gap is most disheartening. During the
school year 2007-2008, about 25 percent of secondary math teachers who taught in
schools with at least half of African American students lacked certification or a
college major in mathematics compared to 8 percent of math teachers who taught in
schools with at least half White students (NCES, 2001). Furthermore, twelve percent
of secondary mathematics teachers had neither a major nor a certification in math.
For schools with at least 50 percent White enrollment, 8 percent of mathematics
teachers were not qualified with a certification or a major in math, while African
Americans in schools with the same enrollment were taught by teachers where 25
percent were not qualified. Qualified teachers are the most important factor in
student achievement (Wright, Horn, & Sanders, 1997; Darling-Hammond, 1996). If
teachers are not competent to teach mathematics, this will be reflected by the lack of
teachers’content knowledge and ineffective teaching practices. Civil and Planas
(2004) state that students who are empowered by their experiences at school are able
to develop their ability, motivation, and confidence to succeed academically.

14
Therefore, it is imperative that teachers in mathematics classrooms be able to provide
all students with rich experiences.
On the 2007 Trends in International Mathematics and Science Study
(TIMSS), both 4
th
and 8
th
grade level Asian students in the United States scored
higher than students of any other race/ethnicity, and 4
th
grade Asian students in the
United States scored higher in mathematics than students from all other participating
jurisdictions except Hong Kong, Singapore, and Chinese Taipei (NCES, 2001). In
2008, Asians also had the highest average AP exam score (3.08), while the lowest
was for African Americans (1.91), and Hispanic students in 2005 had the lowest
percentage of completed courses in geometry, algebra II, and statistics among
Whites, African Americans and Asian/Pacific Islander students (EdSource, 2009).
Algebra has been recognized as an essential course that builds the foundation
for higher level mathematics curriculum and as a gatekeeper for college entrance.
The next section in the chapter discusses the background of algebra achievement.
Background of Algebra Achievement
The academic achievement of U.S. students continues to decline, particularly
in mathematics courses such as algebra. In the early 1980s, the average achievement
of high school students was lower on most standardized tests than the time when
Sputnik was launched, and in the previous decade, American students were never
first or second on 19 academic tests in international comparisons (National
Commission on Excellence in Education, 1983). Furthermore, between 1975 and
1980, remedial mathematics courses offered at four year colleges increased by 72

15
percent. In the early 1980’s, such courses constituted one-quarter of all mathematics
courses taught in four year colleges. A Nation at Risk (National Commission on
Excellence in Education, 1983) revealed the failures of U.S. schools and aroused
political and educational leaders to reevaluate the curriculum in American schools.
Consequently, in order for U.S. schools to be competitive with other industrialized
countries, it was necessary for radical changes to occur both with the current
curricula being taught and with how school programs were being administered. This
period marked the beginnings of the transformations that followed through a series
of reforms and the creation of new legislation.
Algebra is considered a gateway that leads to more rigorous math and science
courses in high school (EdSource, 2009). Algebra has also been recognized by both
educators and policymakers to be a gatekeeper course that assists in college
preparation as well as in the work force (Choike, 2000). Not only are colleges
requiring that students possess some level of algebra knowledge prior to enrollment,
but some states in the United States, such as California, are now requiring that
students successfully complete one year of algebra as a requirement to graduate from
high school (EdSource, 2009). Furthermore, in California, high school students must
pass the California High School Exit Exam (CAHSEE) in order to graduate from
high school which includes an Algebra I component (EdSource, 2009). Thus, the
need to be proficient in algebra is now becoming more indispensable. Despite such
expectations from high schools and colleges, there continues to be a large percentage

16
of students who fail to graduate from high school as a consequence of not passing
high school algebra (U. S. Department of Education, 2008).
Educators and policymakers agree that too many students are repeating
algebra, sometimes multiple times. It has become a concern since the lack of success
in this course prevents students from completing college preparatory science courses
that require algebra (EdSource, 2009). Data from the 2007 National Assessment of
Educational Progress (NAEP) indicate that California 8
th
grade students generally
score below the national average on the national math achievement exam which
heightens the concern that students in the public school system are not adequately
prepared to enroll in an Algebra I course. According to NAEP, 32% of students are
at or above the “proficient” level in Grade 8, but only 23% of students reach the
proficient level at Grade 12 (U.S. Department of Education, 2008). In the United
States, the current science and engineering workforce faces a future in which there
will be accelerating retirements, and for many years the U.S. has depended on the
talent from abroad. For many reasons, educators and policymakers have continued
to focus on how to best prepare students in our schools (U.S. Department of
Education, 2008), including the fact that for the United States to remain competitive
in the global world it is necessary to strengthen the mathematical knowledge of
Americans. Consequently, the U.S. needs to produce students who are well trained
in mathematics. It is necessary for American students to have a solid foundation of
algebra in order for them to successfully achieve in higher mathematics courses and
attain training in fields like science, engineering or mathematics.

17
NAEP mathematics assessment measures students’ abilities in five content
areas: number sense, properties, and operations; measurement; geometry and spatial
sense; data analysis, statistics, and probability; and algebra and functions. Only about
39 percent of 4
th
graders scored at or above the Proficient level in 2009. The highest
percentage of 4
th
graders scoring at a minimum of Proficient level was Asians/Pacific
Islanders (60%), followed by White students (51%), but Hispanic students only had
22% and Black students had 16%. Similarly, in 2009, 54% of Asian/Pacific Islander
students scored at least at the Proficient level in 8
th
grade followed by Whites (44%),
Hispanics (17%), and African Americans (12%). On the TIMSS 2007 4
th
grade
exam, the U.S. scored behind eight other countries including Hong Kong SAR,
Singapore, Chinese Taipei, Japan, Kazakhstan, Russian Federation, England and
Lativa. On the same exam for 8
th
graders in 2007, the U.S. scored behind Chinese
Taipei, Republic of Korea, Singapore, Hong Kong SAR, Japan, Hungary, England,
and the Russian Federation. The U.S. is clearly not the leading world country in
mathematics and it might have been at one time.
Algebra is a gatekeeper for access to college required mathematics courses
and can become a barrier to college entrance given the current trend in student
achievement in mathematics (U.S. Department of Education, 2008; Steen, 1999).
Algebra is a foundation course for higher mathematics courses taken in high school,
such as calculus. Students who have an interest in majoring in mathematics or
science are required to complete advanced mathematics courses like calculus. The
completion of Algebra II in high school has indicated that students are more than

18
twice as likely to graduate from college in comparison to those students with less
mathematical preparation (U.S. Department of Education, 2008). NAEP (2008)
shows that in 1998, 2000, and 2005 the percentage of high school graduates who
completed geometry, algebra II, trigonometry, statistics, pre-calculus or calculus was
still lower for Latinos and African American compared to that of White students.
NAEP (2008) indicates that graduates of postsecondary institutions have more stable
employment and higher salaries that those adults without such degrees. With the
shocking performance of U.S. students in algebra, attaining a college degree
continues to be questionable.
Schools in the United States today are being called upon to ensure that all
students meet higher academic standards. Federal and state policies continue to
focus on the students’ performance in mathematics, with the expectation that all
students will demonstrate proficiency in this content area. With the adoption of
standardized testing in public schools in the United States, schools are now faced
with the challenge of teaching curricula that focuses on the content that is measured
on the standardized assessments (Levister, 2005) versus giving the students the
opportunities to acquire and master skills that are essential in the current global
economy. Current American students will delve into a workforce that will require a
different set of skills compared to those that were required in the past.
Competition in the global economy is inevitable, and countries all over the
world aim to be the top in both science and mathematics. According to the Trends in
Mathematics and Science Study (TIMSS), fourth and eighth grade students in the

19
U.S. have made some gains in math and science but continue to trail other countries
such as Singapore, Chinese Taipei, and the Russian Federation (Scott, 2004; TIMSS,
1999). The declining quality of American education over the years has brought
upon many responses from the federal government, including passing the No Child
Left Behind (NCLB) Act. NCLB requires all states to develop grade level standards
with required assessments and adopt curriculum aligned to the specific standards
(Department of Education, 2006). Part of the desired outcome from the
implementation of NCLB is that, in order for the United States to maintain its
competitiveness in the world, students must be competent in mathematics, science
and English (Department of Education, 2006).
With the current diversity that is found in the classrooms in U.S. schools, it is
necessary for teachers to be able to implement various teaching strategies during
algebra lessons. Fleischner and Manheimer (1997) note that implementing a single
method for teaching mathematics is inappropriate in a classroom that has a diverse
student population. The shortage of qualified mathematics teachers has existed for
many decades and continues to be a concern for many American schools today. In
the early 1980s, the shortage of mathematics teachers existed in 43 states and half of
the newly employed mathematics teachers were not qualified (National Commission
on Excellence in Education, 1983). Comparing teacher education to other
professions and to other countries’ standards, teacher education in the U.S. has been
thin, uneven, poorly financed, and unlike doctors, accountants, lawyers, or architects,
all teachers do no have the same type of training (Darling-Hammond, 1996).

20
Understanding of Mathematics
In the United States, research has found that students learning mathematics
are behind other countries such as Germany and Japan, and the knowledge that takes
place is very limited compared to these other countries (Hiebert & Stigler, 2004;
Stigler & Hiebert, 1999, TIMMS, 2004). This limitation is partly due to the
disconnection that exists between procedural and conceptual knowledge (Hiebert,
1986). Stigler and Hiebert (1999) note that mathematics teachers in the U.S. usually
focus on definitions and procedures instead of mathematical reasoning and problem
solving, like in other leading countries such as Japan. Conceptual teaching is not just
the conveying of facts but the opportunity for students to connect mathematical facts,
ideas, and strategies (Hiebert & Stigler, 2004). Other countries are leading the way
with mathematics learning, with domestic and international assessments of
achievement showing that students in the U.S. are learning less mathematics than
possible and learning with not as much deep understanding (Silver & Kenney, 2000;
Gonzales, et al., 2000). Less than 1% of the time did eighth graders in the U.S. have
the opportunity to explore and discuss mathematical relationships during problem
solving compared with their international peers, and the emphasis on conceptual
problems varied from 17% of the problems in the U.S. to 54% in Japan (Hiebert &
Stigler, 2004; Hiebert et al., 2003).
Conceptual mathematics problems allow students to develop a deeper
understanding of the content, but teachers need to allow students to study the
connections or relationships imbedded in the problems. Research has found that in

21
the U.S. when teachers work on conceptual problems, they tend to do the work for
the students and during the discussion ignore the conceptual component of the
problem, unlike other high-achieving countries like Germany or Japan (Hiebert et al.,
2003; Stigler & Hiebert, 1999; Stigler & Hiebert, 2004). Students in the U.S. spend
most of the time practicing skills and less time on conceptual understanding, and this
has been consistent with many past reports regarding mathematics teaching in this
country (Fey, 1979). Hiebert and Stigler (2004) present a mathematics problem
differentiating two distinct methods in approaching one concept, stressing that one
way for students to gain conceptual understanding is through the search of patterns
which then leads to developing generalizations such as mathematical formulas. The
importance for students to develop both skills and understanding is shared by various
researchers (Kilpatrick, Swafford, & Findell, 2001; Hiebert & Stigler, 2004).
The TIMMS 1999 Video Study results reveal the importance for students to
have opportunities to solve challenging problems that require them to develop
conceptual understanding through the construction of mathematical relationships and
not just to focus on skills as is most typical for 8
th
grade students in the U.S. (Hiebert
& Stigler, 2004; Stigler & Hiebert, 1999). This Video Study included seven
countries: Australia, the Czech Republic, Hong Kong SAR, Japan, the Netherlands,
Switzerland, and the United States. On the average TIMSS eight-grade mathematics
assessments for 1995 and 1999, the U.S. scored the lowest both times and the top
two countries were Japan and Hong Kong followed by the Netherlands (Hiebert et
al., 2003). Moreover, it was found that the success of Japanese mathematics

22
students was associated with the fact that the average time spent per mathematics
problem was 15 minutes compared to the U.S. who spent an average of 5 minutes per
problem. Allotting more time per problem allowed the Japanese students to engage
in a variety of learning experiences, such as proving or verifying mathematical
statements, analyzing new problems and developing new solution methods rather
than repeating previous problems or procedures (Hiebert, et al., 2003; Stigler &
Hiebert, 1999). Engaging in authentic problem solving takes more time, and the
amount of time spent on a problem can indicate the nature of the activity in which
students are participating (Hiebert & Wearne, 1993; Carpenter et al., 1989; Cobb et
al., 1991). Moreover, Hiebert and Wearne (1993) state that spending more time in
solving a mathematics problem encourages more student reflection and allows for
constructing relationships between mathematical ideas. Hiebert and Wearne (1993)
also state that teachers that focused on conceptual understanding allowed students to
construct knowledge, spend more time on one mathematics problem which allowed
teachers to ask more questions requesting students to describe and explain alternative
strategies, had longer discussions, and showed higher levels of performance.
Moreover, the higher achieving countries spent less time reviewing old content and
more time working on new mathematical ideas, while the U.S. spent more than 50%
of the time reviewing. The TIMMS 1999 Video Study concluded that teachers in
countries with higher achievement attended more to the conceptual development of
the mathematics than U.S. teachers (Hiebert et al., 2003). Conceptual instruction has
been shown to be related to improved student achievement (Gamoran, Porter,

23
Smithson, & White, 1997). Modeled by instruction in high achieving countries like
Japan and Singapore, conceptual instruction foster active thinking and problem
solving as students investigate, conjecture, and solve problems with less reliance on
computation, memorization, or other rote procedural learning methods (Smith,
Desimone, & Ueno, 2005). Walhberg (1997) points out that there are many
limitations with the traditional lecture method, including the absence of developing
mathematical autonomy and the lack of ownership of the mathematical ideas being
presented, and that the lecture method should be used to introduce students to basic
concepts as a springboard for the students’ own discoveries.
Classroom discourse in a mathematics classroom is another important
element that needs to be part of learning since it allows for more opportunities for
student learning. Classroom discourse invites students to become more active
participants and, from a social-cognitive and social-constructivist perspective, there
is richness in student interaction in mathematics classrooms (Cobb, Yackel, &
Wood, 1992; Hatano, 1988). When students work collaboratively on a mathematics
problem they can notice different elements of a problem and construct different
relationships that create opportunities to finding solutions (Noddings, 1985;
Schoenfeld, 1989). During classroom discourse, the teacher needs to guide the
discussion with appropriate questions and responses. Students need to be challenged
to support their responses or define their position so that they can engage in “deeper
reflective, integrative thought than if they are asked to recall facts or rules” (Hiebert
& Wearne, 1993, p. 397). Wolfe and Brandt (1998) state that learning is a process of

24
active construction and that the brain is constantly seeking connections between the
new and the unknown.
Hiebert and Stigler (2004) argue that mathematics teaching requires a process
of change that includes teachers engaging students in thinking about key
mathematical relations in particular problems carefully selected, and that teachers
must reflect with thorough details to the students' reactions in order to improve the
effectiveness of their lessons. Furthermore, it is critical for teachers to analyze their
students’ work and understand their thinking in order to implement adequate
teaching methods (Kazemi & Franke, 2004; Hiebert and Stigler, 2004). During
classroom discourse it is also important for teachers to become insightful to students’
ideas and be more aware of what students are thinking and learning (Ball & Cohen,
1999). By working with the students’ ideas rather than giving teacher commands,
teachers respect and validate the ideas of students as worthy of consideration and
hence, students’ autonomy and initiative increases (Cohen, 1994). These types of
changes in the weekly routines of the teachers initiate changes in the school’s culture
of teaching.
Educators are realizing that learning demands both practicing skills as well as
conceptual understanding for students to achieve in mathematics. Change in
mathematics instruction can be supported by professional development.
Professional Development
The low achievement of mathematics students continues to be a prevailing
problem in American schools. Teaching needs to be viewed as a cultural activity so

25
that changes in teaching take place gradually and steadily over time through small
daily changes in teaching and not solely through more workshops (Hiebert & Stigler,
2004; Stigler & Hiebert, 1999). The United States and other countries around the
world have initiated educational reform movements in order to promote higher
learning performance for all students, and the No Child Left Behind Act (NCLB) of
2001 mandates that teachers be provided with high quality professional development
(PD). Educational reform has called for the use of professional development to
improve mathematics education (Elmore, 1996; National Commission on
Mathematics and Science Teaching, 2000; National Education goals Panel, 1994),
and the need for PD is great since there is a large number of mathematics teachers
with an emergency certification or who teach out of their field of expertise (National
Commission on Teaching and America’s Future, 1996). Changes in classroom
practice are essential in the reform movements, and it will require a lot of learning on
behalf of teachers where there will be a need for support and guidance (Putnam &
Borko, 1997). Various studies indicate that teacher instruction is the most important
factor to impact student achievement, and by improving the effectiveness of teachers,
education can be improved more than by any other single factor (Wright, Horn &
Sanders, 1997; Darling-Hammond, 1996). Different definitions exist about what
constitutes PD. Professional development for teachers has been defined by Killion
(2002) as learning opportunities that are practical to improve teacher practice and,
for the purposes of this dissertation, this definition will be used.

26
Teachers play a major role in the students’ learning and improving teachers’
knowledge, skills, and character through professional development is a critical step
in improving student achievement (King & Newmann, 2002). Teacher quality is the
single most important influence on student achievement and school success, more so
than all other factors that affect achievement such as socioeconomic status, class
size, family background or school context (Sanders & Horn, 1998; Sanders, 1998;
Wright, Horn, & Sanders, 1997; Darling-Hammond, 1996). Educators and other
critics have stated that traditional professional development has failed to prepare
teachers to teach diverse populations (Ladson-Billings, 1999; Zeichner & Hoeft,
1996). As a result, there is a need for professional development to be relevant to the
teachers, and the knowledge that is learned at these trainings needs to be
implemented adequately in the classroom. Loucks-Horsley (1998) states that the
quality of professional development in education highly differs and usually falls
short of fulfilling the needs of the schools. It is essential that teachers implement
research-based instructional methods, share their experiences with their colleagues
and be willing to receive their feedback in order to ensure that they are promoting
effective teaching strategies in the classroom. Hattie and Timperley (2007) note that
feedback is “information provided to correct, encourage, clarify or offer alternative
strategies regarding aspects of one’s performance or understanding to ultimately
enhance student learning” (p. 81). It is important that teachers engage in dialogue
and discussion amongst colleagues so that it will give them the opportunity to
recognize that learning takes place with inquiry and discussion. Darling-Hammond

27
(1997) states that learning is most promising when teachers are active, have
intellectual engagement with reflection, place their main focus on content specific
knowledge and pedagogy, and systematically are engaged in analyzing student
learning. This is further confirmed by Cochran-Smith (2001) who states that a
teacher needs to be reflective and knowledgeable about the content taught to the
students. It’s also important to recognize that programs of professional development
do not necessarily apply to every school environments (Guskey, 2003).
The ultimate goal of professional development is to improve student learning
outcomes (Guskey, 2003). Most school districts in the U.S. invest little in ongoing
professional development for experienced teachers, and much of the spending on the
limited resources is for unproductive “hit-and-run” workshops (Darling-Hammond,
1996). Therefore, professional development is usually not seen by educators as an
instrumental component of teaching, and participants are often found to be
uninterested in the training since their interests were not considered by the organizers
of these events and often times the activities in these sessions may be irrelevant to
the teachers or of little use. Professional development needs to be part of an ongoing
learning process over the course of a teacher’s career instead of a one time learning
experience (Borko, 2004; Stigler & Hiebert, 1999). Furthermore, with the heavy
teaching load that most U.S. teachers have, they are left with almost no regular time
for collaboration with their colleagues or to learn about new teaching strategies.
This is contrary to teachers in many European and Asian countries who spend
between 15 and 20 hours per week working collaboratively, refining lessons and

28
learning about new teaching methods (Darling-Hammond, 1996). Site-based
educators are keenly aware of critical contextual characteristics of the school, and
they, along with district personnel, need to collaborate carefully to optimize the
effectiveness of professional development (Guskey, 2003). School districts need to
put more value on professional staff development and provide sufficient funding for
such activities in order to promote better student learning in the classroom and aim
for higher student achievement. The National Institute for Science Education (NISE)
and the Educational Testing Services (ETS) found that effective professional
development certainly requires time, and the time spent in professional development
needs to be well organized, carefully structured, and purposefully directed (Guskey,
2003). It is important to recognize that teachers will need time to deepen their
understanding of what is gained during professional development which will also
require analyzing the students’ work and developing new approaches to instruction.
Tucker (1996) notes the importance for everyone involved to understand the purpose
and goals being established. It is necessary for school districts, organizers of
professional development trainings and its participants to have effective
communication and actions regarding the value that professional development can
bring to teachers and the students, and professional development experiences should
be researched-based. As noted by Guskey (2003), collaboration amongst educators
needs to be structured and purposeful in order to bring its intended benefits, where
the goals for improving student learning will guide the efforts.

29
With the diversity that is currently found in most U.S. schools, professional
development training needs to focus on providing teachers with a variety of research-
based teaching strategies that they can employ in the classroom in order to meet the
needs of the students (Hiebert & Stigler, 2004). Wenglinsky (2002) states that the
identification of specific teaching strategies that can be applied to all content areas is
an important step in improving student achievement. Although class size and student
background play an important role in the student’s education, there is a more
significant correlation between student achievement and teacher classroom practices
(Darling-Hammond, 1997; Educational Testing Services, 2000). King (2002) states
that professional development needs to focus on strategies that have been identified
by teachers as essential in improving their classroom teaching. Teachers are the
most reliable sources of identifying the needs of their students and determining what
teaching strategies would be most appropriate to foster student learning. Some forms
of professional development that can assist teachers in mathematics classrooms are
studying how other teachers present problems to students or by analyzing videos of
teaching and focusing on how teachers teach a variety of mathematics problems
(Hiebert & Stigler, 2004)
The characteristics of effective professional development are difficult to
identify given that many researchers use different criteria to determine
“effectiveness,” and the research evidence that identifies most of these characteristics
is inconsistent. Guskey (2003) found in his examination of characteristics of
effective professional development that enhancement of teachers’ content and

30
pedagogical knowledge is very important. Teacher reflection is also an important
component of effective teaching that needs to be stressed more in professional
development trainings. Arens and Delandshere (2003) note that the use of portfolios
provides the opportunity for teachers to think critically about their own teaching and
can serve as a tool to demonstrate their growth over time, which then leads to self-
improvement. Improvements in student learning in mathematics necessitates that
effective professional development allows teachers to understand at a deeper level
the content that is being taught as well as the ways that students learn the content
(Guskey, 2003).
Professional development experiences fit into one of two categories-
traditional or reform. Workshops, university courses, observations, and conferences
are considered traditional forms of PD (Desimone, Porter, Garet, Yoon, & Birman,
2002; Porter, Garet, Desimone, Yoon, & Birman, 2000; Wei et al., 2009). Study
groups, lesson study, peer observations, collegial planning and working, professional
learning communities, mentoring/coaching, and networks are reform forms of PD
(Desimone et al., 2002; Wei, 2009; Guskey, 2002). The latter type of PD activities
are referred to as “reform” due to the way that the traditional, isolated, cellular model
of teaching are changed (Collinson & Cook, 2004; Gallimore & Goldenberg, 2001).
Different types of professional development exist and examples of high-
quality enhancement PD for teachers have been recommended. Loucks-Horsley,
Stiles, Mundry, Love, and Hewson (2010) have identified numerous professional
development approaches for mathematics teachers which can be grouped into four

31
categories: (1) immersion in content, standards, and research; (2) examining teaching
and learning; (3) aligning and implementing curriculum; and (4) professional
development structures. The first cluster focuses on enhancing teachers’ in-depth
understanding of mathematics content and processes, how students learn
mathematics and the pedagogical content knowledge required to teach mathematics.
The second cluster encourages collaborative learning experiences amongst teachers
to reflect on their students’ learning and personal teaching practices. Furthermore, it
gives the teachers the opportunity to grapple with authentic issues encountered in the
classroom. Examples of activities that align with the second cluster include lesson
study, action research, examining student work and thinking, demonstration lessons,
coaching and mentoring. The third cluster includes instructional materials selection
and curriculum implementation as strategies where teacher learning is focused on
learning, reflecting, and sharing information related to teaching and learning in the
context of new curriculum selection or implementation. Strategies that can be
included in the fourth cluster include study groups, workshops, seminars,
professional networks or online professional development in which the other
strategies can be embedded. These professional development approaches offered by
Loucks-Horsley, Stiles, Mundry, Love, and Hewson (2010) have been shown to have
a positive affect on teachers’ knowledge and instructional practice. Examining
practice in teaching and learning can assist teachers in improving their knowledge of
mathematics and modify their instructional practice (Barnett, 1998).

32
Improved student outcomes needs to be the intended goal of teachers and
administrators in seeking to provide continuous support for classroom instruction
through professional development training. Guskey (2000) has identified five levels
that are critical to effectively evaluate professional development: Level 1-
Participants’ Reactions; Level 2-Participants’ Learning; Level 3-Organization
Support and Change; Level 4-Partcipants’ Use of New Knowledge and Skills; and
Level 5-Student Learning Outcomes. At Level 1 the goal is to obtain the overall
feeling of the participants, such as addressing whether or not they felt that the time at
the PD was utilized wisely or whether the activities were well planned or
meaningful. Level 2 is intended to measure the knowledge and skills that the
participants gained, while in Level 3 the focus is on the organization’s characteristics
and attributes necessary for implementing and being successful with the training
received at the PD. Level 4 measures whether the new knowledge and skills that the
participants learned at the PD made a difference in their professional practice which
requires that the teachers practice the new ideas in their classrooms. Finally, Level 5
analyzes how the PD activity affected the students. Information at Level 5 can guide
improvements for future professional development, including its design,
implementation, and follow-up (Guskey, 2000; Guskey, 2002). Guskey (2002) states
that in planning PD to improve student learning, it is necessary during the planning
stages of the PD to start with Level 5, student learning outcomes desired, and work
down to Level 1. This assertion is connected to Fishman, Marx, Best, and Tal (2003)
who state that the most important measure to determine the effectiveness of a

33
professional development program is whether teacher performance provides
evidence of improved student learning. Research indicates that substantive
pedagogical change entails extensive professional development over time (Supovitz
& Turner, 2000). Research that links teacher and student learning is a difficult
relationship to establish but essential in improving student achievement (Loucks-
Horsley & Matsumoto, 1999). Ongoing professional development that encourages
active engagement of exploring new issues and problems, connections between
teachers’ work and their students’ learning, creating cognitive dissonance,
constructing new understanding, and participating in collaborative discussions to
improve professional practice (Ball & Cohen, 1999) are elements that are
incorporated in professional learning communities. This type of PD is discussed in
the next section.
Professional Learning Community
A professional learning community (PLC) is a school model that consists of a
group of members that understand the linkage between student learning and learning
with colleagues (Lambert, 2003). PLCs succeed in a collaborative culture providing
a nurturing professional learning environment (Eaker & Keating, 2008) and are
centered on focused inquiry and reflection of practice (DuFour, DuFour, Eaker, &
Many, 2006). Student data is gathered to determine areas that need the most
meticulous attention in the professional learning community so that students can
perform successfully (DuFour, DuFour, Eaker & Many, 2006). As student
performance data is collected and analyzed, teachers will be able to understand

34
principles and not just strategies when they attribute learning gains or weaknesses to
instruction, and student learning will be better understood as members of the PLC
prepare instruction that is responsive to the students’ and community’s needs
(DuFour & Eaker, 1998). In a professional learning community, the members take
responsibility to learn new content, strategies, or approaches to remedy the areas of
student weakness (Hord, 2009). Profound changes in school culture need to take
place in order for professional learning communities to come about. Three major
cultural shifts occur with a PLC. The first is a change in orientation where the focus
is now on learning versus teaching. The second is teachers going from working in
isolation to collaboration. The third is a focus on outcomes rather than inputs such
that improvements in professional practice are driven by evidence of student learning
(Eaker & Keating, 2008). The professional learning community will lend itself to
open a discourse on effective teaching and standards as well as university theory and
pedagogy as the school transforms into a learning organization (Senge, 1990). This
reform can help teachers learn and transfer pedagogies and theories of learning to
classroom practice as well as influence practicing teachers at the school with the
proper support offered by the school district. Teachers who work together create a
forum for debate and improving understanding which heightens teachers’ capacity to
grow, such as in a professional learning community, but it also gives rise to more
opportunities to discuss concepts, skills, and problems that arise during the time that
is designated as part of PD as well as share common curriculum materials, course
offerings, and assessments (Garet, Porter, Desimone, Birman, & Yoon, 2001).

35
Members of a PLC prioritize the students’ learning needs, and immediate
attention is given to one particular area at a time by collectively taking responsibility
to continuously learn new content, strategies, or approaches to address the problem
areas (Hord, 2009). This aligns with the more balanced approach to teaching, which
places more emphasis on understanding subject matter, which implies that teachers
must learn more about the subjects they teach and how these subjects are learned by
the students (Garet, Porter, Desimone, Birman, & Yoon, 2001). In any profession,
the continual deepening of knowledge and skills is an integral component, and
teaching is no exception (Shulman & Sparks, 1992; National Board of Professional
Teaching Standards, 1989). The PLC models constructivist learning as staff
members with their school leaders use student data in making decisions about what
needs to be learned, how to go about the learning, how to transfer and apply it to
their classrooms and how to evaluate its effectivenss (Hord, 2009). In traditional PD
sessions that consist of a workshop or seminar with an outside guest, typically the
presenter addresses issues that they feel are problems rather than allowing the
teachers to voice their concerns in order to derive solutions during this activity (Hill,
2004). This is not the case with a PLC.
By working to change the school culture through professional learning
communities into a learning organization, the teachers will be the change agents and
impetus to reform (Senge, 1990). The teachers participating in the professional
learning community will be able to implement effective teaching methods in the
classroom and influence other teachers at the school site to make the change so that

36
effective teaching will produce gains in academic achievement for all students,
particularly poor and minority students in urban schools that desperately need
effective teachers (Haycock, 1998). Hill (2004) states that an improved professional
development program needs to be based on outcome measures that include growth in
teachers’ content knowledge for teaching mathematics, knowledge of students’
mathematical learning, or their use of particular instructional or assessment practices.
The professional learning community model of PD incorporates all the elements
stated by Hill (2004). Working collaboratively as part of the PLC, teachers are
constantly seeking strategies that will aid students in understanding the content being
taught in the classroom. Desimone, Porter, Garet, Yoon, and Birman (2002) found
that PD that focused on particular instructional practices increased the teachers’ use
of such practices in the classroom, and active learning opportunities during PD
increased the effect of the professional development on teachers’ instruction. To
sustain the professional learning community, the teachers will organize the PLC
around gathering student data, sharing the data with all participants of the PLC,
engaging in collective analysis of the gathered data, developing new strategies to
meet the objectives more effectively, and monitoring the results of the strategies that
are being implemented (DuFour & Eaker, 1998).
Teacher inquiry also plays an important role in teacher growth. Collaborative
learning communities are imperative for the success of teacher inquiry that acts as a
force in school renewal (Cochran-Smith, 2004; Dana & Yendol-Silva, 2003).
Participating in a professional learning community not only gives rise to teacher

37
development but can also cultivate teacher researchers (Clark, 2001). Professional
learning communities give opportunities for dialogue and create an environment in
which teachers feel safe to ask questions--uncertainty is valued and supported, and
community perspectives are valued over an individual perspective (Snow-Gerono,
2005).
PLCs are so valuable for teacher inquiry due to the fact that American
traditions of teaching have not promoted or embraced the notion of teachers taking
up a position of uncertainty based in their search of questions (Snow-Gerono, 2005).
The structure of a PLC prevents teachers from being afraid to lose their credibility as
a consequence of announcing to their colleagues their uncertainty regarding their
instructional practices. Inquiry becomes a collaborative, more attainable experience
when teachers are supported and assisted with their questions (Snow-Gerono, 2005).
The dialogue that takes place during inquiry promotes dissensus (Trimbur, 1992).
Dissensus is dialogue where teachers are able to disagree and critique various
elements of teaching and learning in a manner that embraces problem-posing as a
means for professional development (Snow-Gerono, 2005). Dissensus creates a
space for teachers to engage in productive experiences where learning and growth
occur in relation to uncertainty, even when no consensus is reached (Snow-Gerono,
2005).
Student achievement can be improved through professional development and
teacher inquiry and dialogue, but another important component that has the potential
to create effective teaching is the practice of reflection. Zeichner and Liston (1996)

38
state that “reflective teaching entails a recognition, examination, and rumination over
the implications of one’s beliefs, experiences, attitudes, knowledge, and values as
well as the opportunities and constraints provided by the social conditions in which
the teacher works” (p. 20). Bensimon (2004) states that although teachers may be
aware of the disparities in educational outcomes, getting them to learn how to reflect
on their own practices may be contributors to the problem that demands a different
kind of learning experience. Jay and Johnson (2002) discuss a typology that profiles
three dimensions of reflective thought for teachers to use as a tool to make necessary
changes in classroom instruction.
Descriptive reflection involves describing a situation such as a classroom
concern or finding meaning in a matter so as to recognize significant features, extract
and study its causes and consequences in order to envision a change (Jay & Johnson,
2002; Schon 1983). Once the problem has been identified, comparative reflection
follows. The teacher needs to think about the matter from a number of different
frames or perspectives to gain new insights or better understandings to the problem
and minimize neglecting important meaning (Jay & Johnson, 2002; Schon, 1983).
Comparative reflection expands the teacher’s understanding of a situation but only a
narrative view of the situation is attained and so the third dimension of reflection is
needed. Critical reflection gives the teacher the opportunity to choose amongst the
possible actions to take or integrate what the teacher has discovered into a new
understanding of the problem (Jay & Johnson, 2002; Schon, 1983).

39
Teachers need to have the ability to analyze their own teaching in terms of its
effects on learning, and it is through this analysis that teachers can improve on their
own teaching (Morris, 2006). Professional learning communities are an example of
professional development that aims to enrich teachers with effective instructional
strategies through analysis of data, dialogue and reflection. The next section
discusses lesson study which is another type of practice that has shown to improve
mathematics achievement, and it also incorporates some of the same features of a
PLC.
Lesson Study
Sperling and Freedman (2001) note that a central concern of the schooling
enterprise in a subject like mathematics is being able to teach students how to think
and engage in problem solving via reasoning and critique. Educational researchers
over the years have investigated various factors that may affect student learning. In
efforts to improve classroom instruction, U.S. schools often use professional
development activities. Garet, Porter, Desimone, Birman, and Yoon (2001) indicate
that focus on content knowledge, opportunities for active learning, and coherence
with other learning activities increases a teacher’s knowledge and skills and change
in classroom practice, and that sustained, coherent professional development has the
greatest impact on classroom teaching. Furthermore, it was found that effective
professional development training gave opportunities for teachers to be active
learners, focused on content, and was integrated into the regular school schedule.
The Japanese Lesson Study is one successful professional development model that

40
incorporates these characteristics and can serve as a model of teaching to improve
mathematics instruction.
Since 1999, lesson study, a Japanese form of professional development that
concentrates on collaborative analysis of live classroom lessons, has spread rapidly
in the United States (Lewis, Perry, & Murata, 2006). Lesson study (jugyoukenkyuu)
is a Japanese professional development process that gives teachers the opportunity to
systematically examine their teaching practice so that they can become more
effective teachers (Fernandez & Chokshi, 2002; Lewis, 2000; Stigler & Hiebert,
1999). It encompasses a family of strategies for instructional improvement that share
the common feature of live classroom observation by a group of teachers that are
working collaboratively to collect data on teaching and learning and to analyze their
data (Lewis, 2002; Lewis & Tsuchida, 1997; Wang-Iverson & Yoshida, 2005).
Lewis, Perry, and Hurd (2009) define lesson study as a system of
collaborative learning from instruction that uses investigation, planning, research
lesson, and reflection which create changes in the knowledge and beliefs of teachers
as well as on the professional community and teaching-learning resources. Features
of lesson study as described by Lewis, Perry, and Hurd (2009) include:
1. Investigation
a. Students’ current characteristics are considered
b. Long term goals for student learning and development are
considered

41
c. Content area of study includes: key concepts, existing curricula,
standards, learning trajectory, and research
2. Planning
a. Selection or development of research lesson
b. Trying tasks in order to anticipate student solutions
c. Writing up instructional plans, including goals for student learning
and development, anticipated student thinking, data collection points,
rationale for lesson design, and connection to long-term goals
3. Research Lesson
a. Conducting research lesson
b. Team members observe and collect data during live research lesson
4. Reflection
a. Sharing and discussion of data from research lesson in post-lesson
colloquium
b. Implications for lesson redesign, for teaching-learning more
broadly, and for understanding of students and subject matter are
drawn out by team members
c. Written summary of what was learned from the cycle to consolidate
the learning
d. Revise and re-teach the lesson
The process of lesson study is to bring teachers together to work
collaboratively by developing a small number of “study lessons” that are guided by

42
an overarching goal that they have identified and which they would like to attain
with their students (Fernandez & Chokshi, 2002; Fernandez, 2002). The study
lessons, also known as research lessons, share five special characteristics: (1) lessons
are observed by other teachers; (2) lessons are planned for a long time, typically in
collaboration; (3) lessons are designed to bring to life a particular goal or vision of
education; (4) lessons are recorded; and (5) lessons are discussed (Lewis, 2000). The
written lesson plans describe in detail the design of the lesson. Next, one of the
teachers in the group will teach the lesson while the other members are observing
and taking note of the experience. Once the lesson has been taught, the reactions
from the observations and the lesson itself are discussed which often leads to
modifying the lesson, and then it is re-taught by another member of the group
(Fernandez & Chokshi, 2002). Typically it takes about 10 to 15 hours of group
meetings over a period of several weeks with multiple teaching separated by a few
days for a typical study lesson (Fernandez, 2000). Having other colleagues not only
developing lessons but evaluating the teaching that is taking place allows educators
to closely examine the teaching and learning process and its rationale. This
sometimes leads teachers to reshape their own practice and lessons as a result of their
discussion, and widely shared practices about teaching and learning may begin to
change.
Collegiality and collaboration, as demonstrated by lesson study, amongst
teachers can also serve to promote effective professional development. Guskey
(2003) states that working together, reflecting on teaching practices, exchanging

43
ideas, and sharing strategies are valued by educators at all levels. Through lesson
study, teachers will be exposed to various teaching practices that can meet the
learning needs of the students and increase their success in mathematics. Evaluation
of teacher performance in the classroom by other teachers during instruction, such as
in the Japanese lesson study model, needs to be more common and done consistently
in the U.S. Not only will instructional practices utilized in the classroom be
evaluated for their effectiveness, but a system that develops teachers and creates
knowledge about teaching that is relevant to classrooms can be shared by members
of the teaching profession (Stigler & Hiebert, 1999). Classroom observation by
teachers’ colleagues for the purpose of professional learning is rare in the United
States (Darling-Hammond, 1997; Darling-Hammond & Ball, 1998), but there is
evidence that this element of professional learning can lead to more successful lesson
implementation, as is demonstrated by lesson study.
The lesson plan incorporated as part of the lesson study needs to be related to
a particular unit and the curriculum at large in order for the lesson to consider the
students’ past and future learning experiences (Fernandez & Chokshi, 2002). It is
further recommend that the teachers in this type of group collaboration may also hold
an open house periodically so that teachers can share their lesson study with invited
guests. Ideas initiated by teachers at research lesson may spread to other colleagues
and could eventually be made part of the national curriculum. Fernandez (2002)
states that to enrich and elevate the work of lesson study groups, it is recommended
that an outside advisor participate in the various stages of the lesson study, such as in

44
key meetings or during the lesson presentation to give feedback. Moreover, school
districts can hire “instructional superintendents” to take the role of lesson study
advisor and deliver professional development that connects to the daily efforts of
teachers. Outside guests such as university professors, district personnel, and
policymakers invited to witness the research lesson can see how students grapple
with the new subject matter or with goals such as “autonomy” and take this
information to create national change (Fernandez, 2002). These advisors need to be
selectively chosen according to their strong content, pedagogical, and/or curricular
knowledge that they can share with the lesson study group (Fernandez, 2002).
Through these elements of lesson study as described by Lewis, Perry, and
Hurd (2009) and Fernandez and Chokshi (2002), the thinking of the teachers
becomes visible, and there is a development of community norms and tools needed
for instructional improvement. For example, if the goal is to develop students’
critical thinking skills, then the group involved with the lesson study can plan all the
lessons that would answer the question, “How does one create and teach lessons that
encourage students to think critically?” (Fernandez & Chokshi, 2002). Teachers
would greatly benefit from participating in this type of experience since the features
that are part of the lesson study can give rise to the recognition that students’ ways of
learning can reveal their thinking, that particular worksheets may not indicate
learning, that students learn from organizing data, strengthen commitment to
instructional improvement and ownership of improvement work, strengthen
emphasis on inquiry, increase responsibility to colleagues and students, and provide

45
opportunity to revise lesson plans that promote better student thinking (Lewis, Perry,
& Hurd, 2009). All these elements are part of the changes in teachers’ knowledge
and beliefs, professional community and in teaching-learning resources. Rethinking
the lessons and outcomes of the lesson study gives the teachers the opportunity to
ensure that the students are indeed the learners. Goals set out by the lesson study
group can also be derived through the collective dialogue of teachers in the lesson
study of the students’ weaknesses as well as reviewing their past test scores. Lewis,
Perry, and Hurd (2009) state that the “Willingness to try teaching outside one’s
comfort zone may be supported by the structure of lesson study, in which team
members share responsibility for the lesson, so that it its ‘our’ lesson not ‘your’
lesson” (p. 14). Furthermore, Stigler and Hiebert (1999) contend that the
collaborative nature of lesson study balances the self-critiquing of individual
participants with the notion that improved teaching is a collaborative effort and not
the responsibility of any individual. This type of approach to teaching will certainly
give teachers the opportunity to engage with their colleagues and to learn from all of
their experiences and knowledge. Moreover, lesson study offers a public, shareable,
and verifiable knowledge base for teaching (Hiebert, Gallimore, & Stigler, 2002).
Several components of Japanese Lesson Study have not been successfully
implemented in the U.S. Fernandez (2002) states that some of the challenges faced
by U.S. teachers attempting to implement the Japanese Lesson Study include lack of
time and interest on behalf of teachers to participate in lesson study, fear of making
one’s teaching public, and adopting a research stance. Moreover, teachers who

46
engage in lesson study need to pose rich research questions, design a classroom
experiment where the research questions can be investigated with evidence, and be
able to interpret and generalize their findings (Fernandez, 2002). In the study
presented by Fernandez (2002), many of these elements of lesson study were found
to be troublesome and need to be overcome if lesson study is to translate successfully
to U.S. schools. Furthermore, it may be difficult for teachers to find the time that is
required to observe the research lesson during its implementation given that the
structure of most U.S. schools do not lend themselves to allow time for teachers to
leave their classroom.
Levine and Marcus (2007) note that teachers engaged in joint
experimentation, inquiry, and discussion with regard to new approaches of teaching
are more likely to understand the approach and be able to use it regardless if external
supports disappear. The expectations that are now established by school districts to
cover certain material at a specific rate can force teachers to rely on more traditional
teaching practices that require less implementation time (Huffman, Thomas, &
Lawrence, 2008). If changes are to occur in mathematics classrooms that will
promote higher student achievement, then it is time to implement professional
development training that focuses more on content knowledge, collaboration,
inquiry, and reflection.

47
CHAPTER 3
METHODOLOGY
Introduction
In the United States it is a struggle for students to successfully pass
mathematics courses. Stigler and Hiebert (1997) note that 78% of mathematics
teachers in the United States do not develop topics in depth but just state them and
96% of the time students only practice computations rather than develop conceptual
understanding. Algebra is considered a gateway that leads to more rigorous math
and science course in high school (EdSource, 2009), and algebra has also been
recognized by both educators and policymakers to be a gatekeeper course that assists
in college preparation as well as in the work force (Choike, 2000). Not only are
colleges requiring that students possess some level of algebra knowledge prior to
enrollment, but some states in the United States, such as California, are now
requiring that students successfully complete one year of algebra as a requirement to
graduate from high school (EdSource, 2009). Furthermore, in California, high
school students must pass the California High School Exit Exam (CAHSEE) in order
to graduate from high school which includes an Algebra I component. Thus, the need
to be proficient in algebra is now becoming more indispensable. Despite such
expectations from high schools and colleges, there continues to be a large percentage
of students who fail to graduate from high school as a consequence of not passing
high school algebra (U. S. Department of Education, 2008).

48
Purpose
In order to increase student success in algebra and other math content, there
is a need to closely examine the type of mathematics professional development that
school districts offer to their teachers to help them increase their mathematical
knowledge and provide them with effective teaching methods. Teacher preparation
in the classroom has a major role in the students’ learning of mathematics. This
preparation includes having a rich knowledge of mathematics as well as the capacity
to implement effective teaching strategies.
The purpose of this study was to examine the extent to which a school district
supported the fifth-grade teachers’ capacity to teach their students concepts and skills
related to algebra. Moreover, the study furthered evaluated the school principal’s
perception and fifth-grade teachers’ experiences to determine whether the
professional development offered by the district supported the fifth-grade teachers in
gaining math knowledge and effective teaching strategies to teach algebra content. It
is important for administrators and teachers to reflect on the current state of
professional development that is being offered by the district and the school, and
what changes are needed, if any, to have it be more effective.
Research Methodology Overview
The purpose of this chapter is to describe the research design, sample and
population, discuss the instrumentation used, present the data collection plan,
proposed data analysis, and ethical considerations of this study. The intent of this
study was to examine the different professional development that was offered by a

49
school district and to determine the perceptions of the fifth-grade teachers and
administrators on the adequacy of the staff development as it relates to algebra. One
elementary school in the Kraft Unified School District participated in this study to
answer the three established research questions.
This study will answer the following three questions:
1. What is the extent to which the school district supports teachers’ capacity
to teach fifth-grade students concepts and skills related to algebra through
professional development?
2. What is the school principal’s perception on how professional
development supports the teaching of the foundations of algebra for fifth
grade?
3. What are the fifth-grade teachers’ perceptions on professional
development as it relates to teaching mathematics concepts and skills related
to algebra at this grade level?
Research Design
In this investigation a qualitative case study research method was used. A
case study is defined by Merriam (2009) as an “in-depth description and analysis of a
bounded system” (p. 40) where a bounded system represents a single unit or entity
enclosed by boundaries. The fact that this investigation will be a descriptive case
study will result in the end product to be a rich description of the phenomenon under
study (Merriam, 2009). Through the use of a case study the researcher is allowed to

50
uncover knowledge that is definite, contextual, and which is developed and based on
populations determined by the researcher (Merriam, 2009).
The desire to gain a rich, in-depth knowledge of the different algebra
professional development for fifth-grade school teachers led to the decision to
investigate a single school site. Patton (2002) notes that this type of study can be
closely aligned to applied research as it sought to “contribute to knowledge that will
help people understand the nature of the problem” (p. 217).
Sample and Population
The selection criteria used to identify the school-site for this case study were:
1) the elementary school had a 2008-2009 API of 800 or above, and/or consistent
growth in API scores for the past 2 academic years; 2) student enrollment of at least
500; 3) the elementary school had a current Similar Schools Ranking of 8 or above;
4) the elementary school had at least 19% or more students qualifying for free and
reduced lunch; 5) the elementary school had under 50% of White students; and 6) the
elementary school had computers and classrooms with internet. Purposeful sampling
was used to identify a school for this case study. Patton (2002) notes that purposeful
sampling provides insight into and in-depth understanding of a specific phenomenon.
Various resources were used to locate a school that met the criteria for this
study, including the California Department of Education (CDE) Education Data
Partnership (Ed-Data) and Data Quest websites. Similar School Rankings, school
demographic characteristics, test scores, ethnic makeup and other information on
school performance like API and AYP data were retrieved from these websites.

51
From these resources, a school was identified as having met the criteria discussed
above.
Participants
This study involved administrators, teachers, and one math coach from the
school. The assistant superintendent of instruction, the school principal, four fifth-
grade teachers and one math coach directly informed this study through semi-
structured interviews. The semi-structured interviews consisted of a mix of “more or
less structured interview questions” (Merriam, 2009, p. 89) that were used flexibly.
The semi-structured interviews were comprised of a list of questions that were
explored in no pre-determined order, with particular data that was required from all
the respondents (Merriam, 2009). The teachers were from the elementary school
chosen for this study and were self-selected volunteers. A wide range of teaching
experience was represented by these teachers.
Overview of the District and School
The elementary school selected for this study was Kerns Elementary School.
Kerns Elementary School is located within Kraft Unified School District (KUSD) in
Southern California. Based on Ed-Data: Fiscal, demographic, and performance
data on California’s K-12 Schools (Education Data Partnership, 2010), KUSD has
nearly 9,500 students in six elementary schools, two middle school, one
comprehensive high school, and one continuation high school. Many of these
schools have been honored as Distinguished Schools and National Blue Ribbon
Schools. The student population at KUSD consists of American Indian or Alaska

52
Native (0.3%), Asian (10.4%), Pacific Islander (0.6%), Filipino (2.4%), Hispanic or
Latino (13.0%), African American (3.2%), White (61.3%) and other (8.6%).
Additionally, KUSD has students who qualify for special programs: English
Learners (2.7%), Free/Reduced Meals (8.3%), and Compensatory Education (1.6%).
Table 3.1 describes the student ethnicity of KUSD and Table 3.2 describes the
makeup of the special programs.
Table 3.1: Kraft Unified School District Students by Ethnicity, 2008-09
Enrollment Percent of Total
American Indian or
Alaska Native

30

0.3%
Asian 989 10.4%
Pacific Islander 59 0.6%
Filipino 230 2.4%
Hispanic or Latino 1,233 13.0%
African American 306 3.2%
White 5,812 61.3%
Multiple or No Response 816 8.6%
Total 9,475 100%
Source: Education Data Partnership (2010)
Kerns Elementary School has a student population of 612 students that
consists of American Indian or Alaska Native (0.8%), Asian (14.4%), Filipino
(5.6%), Hispanic or Latino (22.7%), African American (4.2%), White (46.1%) and
other (6.2%). Students at this school who qualify for special programs are: English
Learners (12.9%), Free/Reduced Meals (19.8%), and Compensatory Education
(12.1%). This school had 69.5% of its students from the following subgroups score
proficient or above in the mathematics component of the California Standards Test:
Hispanic or Latino, White, Socio-Economically disadvantaged and English learners.

53
Table 3.2: Kraft Unified School District Student Data for Special Programs, 2008-
09
Number of Students Percent Enrollment
English Learners 252 2.7%
Free/Reduced Price Meals 782 8.3%
Compensatory Education 148 1.6%
Source: Education Data Partnership (2010)
Instrumentation
The researcher used qualitative research to develop instruments for this case
study. Interview data, document review data and data obtained through participation
in the school’s professional development were captured by the researcher through the
chosen instruments. The method of triangulation was used in this study to ensure
internal validity and strengthen the results of this investigation by the use of multiple
data sources (Merriam, 2009; Patton, 2002). Semi-structured interview protocols for
administrators and teachers, observations, and document analysis were the
instruments used in this research study. The interview protocols were refined
through prior testing with other teachers in a different school district to ensure
validity and adequate insight into the three research questions.
Data Collection Procedures
The data collection process consisted of semi-structured interviews,
observations, and document analysis which were guided by the research questions
(see Table 3.3). The researcher met with all of the participants prior to the
interviews to discuss the study, the interview and observation processes, and how
confidentiality would be ensured. All the participants were volunteers and gave their

54
approval for participation in the research study in writing or in person. The next
section describes the procedures for data collection.
Table 3.3: Research Design Chart
Research Question Data Needed Data Sources Instrumentation
What is the extent
to which the school
district supports
teachers’ capacity
to teach fifth-grade
students concepts
and skills related to
algebra through
professional
development?

- Evidence of the
different type of PD
that is offered to
fifth-grade teachers
that support math and
algebra

- Professional
development training
- Assistant
superintendent
- School-site teachers
- School-site math
coach
- Website
- Memos and other
documents related to
PD

- Interview protocol
- Document Review
- Observations
What is the school
principal’s
perception on how
professional
development
supports the
teaching of the
foundations of
algebra for fifth
grade?
- Evidence on how
the school principal
supports PD for fifth-
grade teachers as it
relates to math and
algebra topics

- Professional
development trainng
- School-site teachers
- School-site math
coach
- Email

- Interview protocol
- Document Review
- Observations
What are the fifth-
grade teachers’
perceptions on
professional
development as it
relates to teaching
mathematics
concepts and skills
related to algebra at
this grade level?
- Evidence that
supports the teachers’
beliefs whether the
professional
development offered
by the district is
adequate and
successful.

- Professional
development training
- School-site teachers
- School-site math
coach
- Interview protocol
- Document Review
- Observations

Interviews
The intent of the qualitative interview was to allow the researcher to take part
into the other person’s perspective which is assumed to be meaningful, knowable and
able to be made explicit (Patton, 2002). To conduct the interviews the researcher

55
used different standardized open-ended interview protocols: Teacher Interview
Protocol (Appendix A), Principal Interview Protocol (Appendix B), Assistant
Superintendent of Instruction Protocol (Appendix C), and the Math Coach Protocol
(Appendix D).
Patton (2002) notes that the development of an interview protocol or guide
ensures that the same lines of inquiry are followed with all respondents to minimize
the variation of different interview questions. Furthermore, Merriam (2009) notes
that the interview guide of a semi-structured interview includes a mixture of more
and less structured questions. The use of standardized open-ended interview
protocols allowed the researcher to obtain data that addressed the research questions.
The primary focus of the interview with the superintendent of instruction was
to determine what professional development the school district was offering to fifth-
grade teachers to support their capacity to teach algebra content and to find out
whether the professional development enhanced their mathematical knowledge and
provided them with effective teaching tools. The focus of the principal interview
was similar to the assistant superintendent’s interview, but from the principal’s
perspective the researcher wanted to examine the awareness that the principal had
regarding the appropriateness of the professional development for the fifth-grade
teachers in terms of enriching the mathematical knowledge of the teachers and
giving them tools to more effectively teach algebra.
The primary focus of the teacher interviews was to evaluate the fifth-grade
teachers’ experiences with the school district professional development as it relates

56
to algebra and determine how relevant the professional development was for the
teachers in terms of increasing their math content knowledge and providing them
with tools to enrich the learning experiences of the students. Furthermore, school
districts and administrators have the responsibility to ensure not only that appropriate
professional development is being provided to the teachers but it is equally important
for districts to ensure that the fifth-grade teachers are continuously collaborating
throughout the academic year in order to provide the students with a valuable
experience in algebra. Teachers, administrators and school districts have the
responsibility to meet the students’ academic needs.
All interviews were conducted at the single site with the teachers, math
coach, school principal, and the assistant superintendent of instruction in the private
setting of the participant’s classroom or office in order to offer the most familiar and
comfortable environment. Every effort was carried out by the researcher in assuring
the participants that the reporting of the study’s results would be completely
anonymous. Interview protocols were provided to the participants on the day of the
interview and the researcher recorded the interviews using a digital recorder and
generated written field notes. As noted by Patton (2002), the taking of field notes is
not an option. Moreover, field notes offer descriptions of the observations made and
include the researcher’s “insights, interpretations, beginning analyses, and working
hypotheses about what is happening in the setting and what it means” (Patton, 2002,
p. 304). The duration of each interview was between 35 and 60 minutes and was
conducted from October 2010 to December 2010.

57
Observations
The purpose of the observation was to obtain direct, personal observation of
the type of professional development that was being offered to the fifth-grade
teachers with respect to mathematics. The observations allowed the researcher to
gain a better insight of what actually takes place during professional development
that focuses on mathematics and gave the researcher a better understanding of how
teachers, math coaches, and the school principal collaborated during professional
development. Patton (2002) notes that through direct observations, the researcher is
better able to understand and capture the environment where the investigation is
being done as well as be open, discovery-oriented and inductive, and things that may
typically escape awareness of the participants are observed by the researcher.
Observations were conducted at the single school-site during the school’s
professional development. Careful notes were taken during the math professional
development about the math content that was being focused on, the new strategies
that were being discussed, the strategies that had previously been used to teach other
math content, as well as the type of collaboration that was taking place between the
teachers, the math coach, and the school principal.
Document Analysis
Document review was another strategy used by the researcher to collect data
for the study. The analysis of district and school documents allowed the researcher
to gather information that cannot otherwise be gathered through interviews or
observations such as information that occurred before this study, private

58
interchanges, and goals or decisions that might otherwise remain unknown to the
researcher (Patton, 2002). District and school-site documents were reviewed to
gather information on the duration, type, and purpose of professional development
that have been offered to teachers of the school under investigation and those that
will be offered in the near future. Documents that were analyzed included the
district’s Annual District Staff Development Plan for the 2010-2011 academic year,
school principal’s emails, Kerns’ staff meeting agendas, the Pine County Department
of Education (PCDE) professional development announcements, school and district
memos, school calendar of meetings, and documents pertaining to CGI, Fosnot, and
the MIND program. Given that the professional development already exists at the
school district in which the study took place, it was valuable to document and
understand the context for which the professional development was developed for
and so it was necessary to review some history. Patton (2002) states that under such
circumstances some important questions that will need to be addressed in relation to
this study include questions that pertain to the creation of the current professional
development trainings as well as how the target population has changed over time,
the staffing patterns over time, the board members’ involvement at the various stages
of the professional development program history, and what some of the critiques of
the professional development program have been. The response to these and other
questions may play a key role in understanding the current level of effectiveness of
the professional development and how it can be improved in the future so that
teachers are receiving valuable professional development that benefit students.

59
Data Analysis
Creswell’s (2009) data analysis stages were used to analyze the data for this
case study. Creswell (2009) notes that qualitative data analysis includes the
following steps: 1) organize and prepare data for analysis by transcribing interviews
and compiling field notes; 2) read data to determine general tone; 3) code data into
discernible chunks; 4) generate categories or themes for the data; 5) determine
representation of categories or themes; and 6) interpret data (see Figure 3.1). Data
from the interviews were transcribed, and field notes taken during the school’s
professional development and document reviews were typed up. Written notes were
also made during the interviews about aspects of the data that were not captured by
the audiotape such as facial expressions or gestures. The transcribed data was coded
initially according to the three research questions that guided this study by assigning
a different color to each question. From the coded data common themes arose which
were generalized into the phenomenon of this study.

60
Figure 3.1: Creswell’s Model for Qualitative Data Analysis

Ethical Considerations
The purpose of this study was to examine the extent to which the school
district supported the fifth-grade teachers’ capacity to teach their students concepts
and skills related to algebra. Moreover, the study furthered evaluated the school
principal’s perception and fifth-grade teachers’ experiences to determine whether the
professional development offered by the district supports the fifth-grade teachers in
gaining math knowledge and effective teaching strategies to teach algebra content.
The Institutional Review Board (IRB) at the University of Southern California
approved this study, including the data collection instruments and research
methodology that were used, and prior to conducting the study a written or verbal

61
consent was received from all of the participants. The anonymity of the participants
was kept at all times to ensure the confidentiality of the participants’ responses and
all participants were treated in an ethical manner. Participation was completely
voluntary and no individual was placed at harm. The greatest ethical care has been
taken with the use of all existing public and private records, written observations,
and interviews.

62
CHAPTER 4
DATA ANALYSIS
Introduction
This chapter presents the findings from a qualitative study at Kerns
Elementary School. The study was comprised of a review of documents that were
provided by the school district, school principal, and documents gathered during the
observation by the researcher in professional development sessions. Furthermore,
interview responses of four fifth-grade teachers, two administrators, and one math
coach are also included in this study. The administrators interviewed consisted of the
school principal and the district assistant superintendent of instruction. Interviews
were conducted based on four protocols, the Teacher Interview Protocol (Appendix
A), Principal Interview Protocol (Appendix B), Assistant Superintendent of
Instruction Protocol (Appendix C), and the Math Coach Protocol (Appendix D). The
triangulation of the research was completed upon analyzing the data collected
through the review of the documents, interviews and the researcher’s participation in
the school’s professional development.
Kerns Elementary School, the school selected for this case study, was
selected in the Kraft Unified School District (KUSD). The sample for the qualitative
case study was selected based on the following criteria for the school site: 1) the
elementary school had a 2008-2009 API of 800 or above, and/or consistent growth in
API scores for the past 2 academic years; 2) student enrollment of at least 500; 3) the
elementary school had a current Similar Schools Ranking of 8 or above; 4) the

63
elementary school had at least 19% or more students qualifying for free and reduced
lunch; 5) the elementary school had under 50% of White students; and 6) the
elementary school had computers and classrooms with internet. Patton (2002) notes
that purposeful sampling provides insight into and in-depth understanding of a
specific phenomenon.
Through the interviews, document analysis and the researcher’s participation
in the school’s professional development the following research questions were
addressed:
1. What is the extent to which the school district supports teachers’ capacity
to teach fifth-grade students concepts and skills related to algebra through
professional development?
2. What is the school principal’s perception on how professional
development supports the teaching of the foundations of algebra for fifth
grade?
3. What are the fifth-grade teachers’ perceptions on professional
development as it relates to teaching mathematics concepts and skills related
to algebra at this grade level?
The three research questions required that the administrators and teachers
reflect on the current state of professional development that is being offered by the
district and the school, and what changes were needed, if any, to have it be more
effective. Educational reform has called for the use of professional development to
improve mathematics education (Elmore, 1996; National Commission on

64
Mathematics and Science Teaching, 2000; National Education Goals Panel, 1994).
Changes in classroom practice are vital in the reform movements, and they will
require a lot of learning on behalf of teachers where there will be a need for support
and guidance (Putnam & Borko, 1997). Teachers have a major impact in the
students’ learning, and improving teachers’ knowledge, skills, and character through
professional development is a critical step in improving student achievement (King
& Newmann, 2002).
This chapter is organized into four sections. Each of the first three sections
addresses one research question. The last section discusses the themes that emerged
from the research questions and provides a summary. Followed by each research
question is a table that gives an overview of what data were need to answer the
research question, the sources that were consulted to gather the data, and the
instrumentation used to extract the data from those sources.
Findings for Research Question One
Research Question 1: What is the extent to which the school district supports
teachers’ capacity to teach fifth-grade students concepts and skills related to algebra
through professional development?
Research question one focused on Kraft Unified School District’s range of
professional development opportunities that are offered to fifth-grade teachers to
enhance their knowledge and skills to effectively teach mathematics concepts and
skills that are related to algebra at this grade level. Walhberg (1997) points out the
many limitations that exist with the traditional lecture method, including the absence

65
of developing mathematical autonomy and the lack of ownership of the mathematical
ideas being presented. The lecture methods should be used to introduce students to
basic concepts as a springboard for the students’ own discoveries. The fifth-grade
teachers need to be provided with opportunities to explore different teaching
strategies and models to more effectively teach mathematics. Wolfe and Brandt
(1998) state that learning is a process of active construction, and that the brain is
constantly seeking connections between the new and the unknown. The data
collected to answer the first research question involved interviews of administrators,
teachers and school-site math coach and a review of documents such as district staff
development memos, district sponsored professional development announcements,
and school staff development agendas. The researcher’s participation in the school’s
professional development training also contributed to the efforts in answering the
first research question. Table 4.1 shows the data needs and sources for this research
question.
Table 4.1: Data for Research Question One
Research Question Data Needed Data Sources Instrumentation
What is the extent
to which the school
district supports
teachers’ capacity
to teach fifth-grade
students concepts
and skills related to
algebra through
professional
development?

- Evidence of the
different types of PD
that are offered to fifth-
grade teachers that
support math and
algebra

- Professional
development training
- Assistant
superintendent
- School-site teachers
- School-site math coach
- Website
- Memos and other
documents related to PD

- Interview
protocol
- Document
Review
- Observations

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For the school year 2010-2011, the KUSD has as their top priority the goal of
increasing the academic excellence for all of their students by implementing best
practices and programs based on the analysis of benchmark results in all academic
areas, including mathematics. The district’s assistant superintendent of instruction,
Dr. Jones, stated,
We expect all of our fifth graders to be proficient or higher on district
benchmark assessments. About 80% of our fifth graders are proficient but
the goal is to increase that number so that all fifth graders are proficient.
Period. I aspire to every fifth grader being proficient on CST and
benchmarks. You have to fundamentally believe every kid can be proficient.
You have to be committed to doing everything it takes. It’s that part that we
haven’t figured out yet, what it takes for every kid. So the responsibility for
us is what do we try next.

The school principal, Dr. Johnson, also shared the same sentiment about
student success in mathematics. Dr. Jones also noted that students who successfully
learn mathematics, including algebra-readiness skills such as knowing what a
variable is and solving simple equations in fifth grade, will be prepared for algebra in
middle school. During administrative meetings between school principals and the
assistant superintendent of instruction, principals are asked to consult with their
teachers to determine their interests in terms of professional development. Then
KUSD decides what needs to be offered to teachers for professional development.
KUSD currently offers fifth-grade teachers Cognitively Guided Instruction (CGI),
the Music, Intelligence, Neural Development (MIND) program, and Fosnot as their
major professional development training in mathematics.

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Currently, Kraft Unified School District is training all elementary school
teachers in the district, including Kerns Elementary School, on Cognitively Guided
Instruction to teach mathematics. CGI is an approach to teach mathematics by
focusing not on rote practice but on the students’ mathematical thinking where
teachers are trained to use CGI strategies to engage students in mathematical
dialogue (Carpenter, Fennema, Franke, Levi & Empson, 1999; Carpenter, Franke, &
Levi, 2003). CGI empowers students to explain their reasoning, justify their
strategies used to derive their solutions, and thus allow them to build a deeper
collective understanding of mathematics. This type of teaching is supported by
Stigler and Hiebert (1999) who note that mathematics teachers in the U.S. usually
focus on definitions and procedures instead of mathematical reasoning and problem
solving like in other leading countries such as Japan. This training offered by the
school district is a three year journey where the participants receive four days of
professional development along with four additional days of school site coaching
with the district’s county department of education staff. All elementary school
teachers are expected to be CGI trained. The assistant superintendent of instruction,
Dr. Jones, noted about the district faculty regarding both CGI and other professional
development training:
You don’t get to be a teacher here without being trained. If you were a new
teacher who said no, we probably wouldn’t keep you. I say we are a district
committed to staff development. You are our currency and this is what we
expect. If they weren’t to go, I can’t imagine we would keep them. But they
go because I explain it’s our priority that they be highly skilled so our kids
benefit from the best and it’s not okay for every other teacher to know it and
you not, that it will not facilitate collaboration.

68

During the three year CGI training, the teachers are able to consult with CGI
researchers, experiment with CGI problem types and student-centered instructional
practices, and make mathematical inquiries that bridge arithmetic to algebraic
thinking. The training is meant to enhance the teachers’ knowledge and skills that
will promote better learning experiences for the students. CGI is based on a
conceptual teaching and understanding of mathematics which is supported by
Hiebert and Stigler (2004) who state that conceptual teaching is not just the
conveying of facts but the opportunity to connect mathematical facts, ideas and
strategies. Once teachers complete the three year training, they can get further
support by mathematics coaches through the Pine County Department of Education
(PCDE). Kraft Unified School District works in collaboration with PCDE to offer
professional development trainings throughout the year both at the school-site and
off-campus. Although not all fifth-grade teachers at Kerns Elementary School have
completed the three year training, the district still provides several CGI training
sessions for fifth-grade teachers through PCDE throughout the year.
Unlike some math coaches in other elementary schools within the district, the
math coaches at Kerns Elementary School are not fully competent to take on full
responsibility in adequately assisting their teachers on CGI or other advanced topics
in algebra at the fifth-grade level, but the district recognizes the importance of
having competent mathematics coaches within the school. Dr. Jones stated,
What’s really important is that we hired the county and I sent all the teachers
for their three years of training. Then we hired the county to do our coaching

69
and what we are trying to do now to build capacity is we want every site to
have their own coaches so every elementary school will have teacher experts
on site who are released to coach their own teachers. So we are building that.

She made it clear that a fully trained math coach is supposed to be an expert
in CGI, have the capacity to ask reflective questions and share effective teaching
strategies. Dr. Jones noted that Kerns Elementary School is not yet fully ready with
“expert” coaches like some of the other district schools, but that the district would
expand the training of their coaches in the near future.
The math coach interviewed at Kerns Elementary stated that the school
principal sets up dates regarding CGI training with PCDE a year in advance so that
PCDE mathematics coaches work with the fifth-grade teachers 3 to 4 times per year
for half day (3.5 hours). During such trainings, the focus is sometimes agreed upon
by the teachers in advance so that the PCDE math coach can make any necessary
preparations. Sometimes the teachers will work on a lesson in a classroom, meet in a
group to discuss concerns or topics in using CGI, or the coach may perform a lesson
with the group of teachers and then debrief. All fifth-grade teachers interviewed
noted that although there may be some input from some of the faculty on CGI
training, most of the CGI training is more appropriate for the lower grades. More
will be discussed later when addressing the third research question.
Besides coaching, the district also offered through the PCDE the CGI
summer academy in August 2010 for teachers who were in any stage of the three
year CGI training program. Furthermore, CGI training sessions were also offered in
September 2010 but were more specific to the year level of the CGI program in

70
which the teacher was in. In December 2010 and the spring of 2011, PCDE offered
a total of three mathematics symposiums for the teachers to attend. The main focus
of the symposiums was on topics like Arithmetic to Algebra, Fractions, and
Relational Thinking. Not all fifth-grade teachers interviewed attended since some of
them have already been through the three required years of CGI training. Some
teachers and the math coach in the study expressed their disappointment with the
workshops offered by PCDE. The math coach stated,
Department of Education is more broad concepts and I think one of the
downfalls of that is that they lump K-5 together and you’ve got fifth-grade
teachers and kindergarten teachers all going to the same workshop and I
mean, that’s not going to work.

Although PCDE provides training off-site, they are also responsible for
coaching CGI at Kerns Elementary, and the assistant superintendent of instructions
praised their good work. Dr. Jones confirmed that the district firmly believes that
CGI is an outstanding program that is supporting the fifth-grade teachers’
professional development. She stated,
To be an outstanding math teacher you really have to have a fundamental
understanding about how kids learn math. So for me what has to happen is to
have a system in place that guarantees that all elementary school teachers
have a fundamental understanding of math and they might think they do, so
what system do you have in place to guarantee that they do? So our system is
CGI training so that way, even though CGI is training you how to be a
teacher, three years of CGI four days a year plus coaching, I’ve increased my
odds greatly that every teacher really understands math because now they’ve
had a lot of professional development and the basics. CGI is all about
problem types and teaching kids the concepts of math and you can’t teach it
if you don’t know it. So the staff development really is teaching the teacher,
then they’re doing it with kids.

71
Kraft Unified School District also offers the fifth-grade teachers with training
in the Music, Intelligence, Neural Development program that was developed by the
MIND Research Institute and offered for approximately two hours at the beginning
of the school year. This program is composed of two components, the Spatial-
Temporal Animation Reasoning (STAR) and the Fluency MIND. KUSD notes that
MIND is a visual approach that utilizes a learner’s spatial temporal reasoning
abilities to understand, explain and solve multi-step mathematics problems that are
aligned to the state standards. This program eliminates the language barrier that
some students encounter when learning mathematics. Not only does it engage the
students at every level of mathematics, but it continues to be an effective learning
tool that is used in conjunction with the textbook. MIND’s use of Jiji, the penguin,
through a game metaphor assists students who have had difficulty with traditional
approaches to learning mathematics. The games offered through the STAR
component expect the students to solve mathematical puzzles, while deepening the
students’ understanding of mathematics and building on their problem solving skills.
Teacher 1 at Kerns Elementary noted,
We have the capability to adjust Jiji to follow the order we teach the material.
So just prior or during the unit or lesson they might have some Jiji time
where they are doing what we are learning in class. So that coupled with
homework, they seem to be progressing easier.

During the 2010-2011 academic year, the district began offering the Fluency
component of MIND. This part of the MIND program has been designed to assist
students retrieve basic math facts quickly, accurately, and with little effort in order to

72
free up the students’ working memory to avoid getting slowed down by basic
computations. Through this Fluency component of MIND, the students are gaining a
solid conceptual understanding of addition, subtraction, multiplication and division.
As noted by the fifth-grade teachers, the math coach, the school principal and Dr.
Jones, Fluency is only used for student intervention for a handful of students who are
still struggling with the basic mathematics facts, but they need to implement STAR
twice a week for a total of 90 minutes.
Another professional development that the district offers to the fifth-grade
teachers focuses on Fosnot. At Kerns Elementary, the fifth-grade teachers use
Fosnot to cover one unit in math over a period of ten days. The focus is to consider a
particular math topic and look at the big ideas in depth by creating strategies and
models that are related to the unit chosen. This math program realizes that less
content will be covered when the math idea is examined in depth, but currently the
fifth-grade teachers are responsible to cover only one math unit using Fosnot for the
entire academic year. The professional development offered for Fosnot has been the
focus of the Tuesday math meetings which are scheduled once a month for an hour at
the school site. The school coaches were responsible for guiding and facilitating the
fifth-grade teachers to create a Fosnot unit on place value and division. The teachers
collaborated during two math Tuesday meetings and have now completed their unit.
For the remainder of the school year, they were responsible for implementing their
unit and then analyzing the results as a grade level. This professional development
was intended to assist the fifth-grade teachers in being able to facilitate a deeper

73
understanding of place value and division for their students using various strategies
and models. The long term goal is that the teachers will be able to use Fosnot to
create lessons with other topics including those in algebra. The PD offered for
Fosnot was continuous during the entire academic year, and the teachers were
expected to reflect on the unit they put together as well as design a plan for
implementation.
Grade level meetings after school are another means for offering fifth-grade
teachers professional development specifically to discuss algebra readiness. These
types of meetings take place about once a month for a period of one hour. According
to Dr. Jones, at all the grade level meetings last year, the fifth-grade teachers came
together after school to discuss CGI problems that were later used with their
students. Then they would reconvene to discuss student work samples to determine
how they would proceed to address the students’ performance in order to make
improvements. She noted, “Looking at kids’ work helps you know what part they
don’t understand in mathematics. Every fifth-grade teacher in the district has
opportunities to collaborate each year.” Currently, grade level meetings for fifth-
grade teachers at Kerns Elementary take place every Wednesday to have discussions
on algebra or other mathematics-related topics as well as discussions that pertain to
other academic disciplines. Compared to last year, the grade level meetings have
lost the focus to be solely on mathematics.
Moreover, the district releases all fifth-grade teachers once a year to visit the
district office and analyze the student results from the common district benchmarks.

74
KUSD expects that this event will promote further collaboration amongst the fifth-
grade teachers and give them the opportunity to duplicate the best practices being
implemented by their colleagues.
The researcher also reviewed the district’s Annual District Staff Development
Plan for the 2010-2011 academic year and determined that besides the CGI training
program, the MIND program training, and the FOSNOT training offered by the
district, only one out of the seventeen professional development activities planned
for the year was related to mathematics. This one activity focused on CGI training
for teachers who were still in the process of completing the three year program, so
some of the fifth-grade teachers who have already completed the CGI program were
unable to benefit from this one other professional development training offered by
the district. One out of seventeen activities focusing on mathematics is about 5.88%
of the school year’s professional development trainings which does not validate Dr.
Jones’ assertion during the interview that 20% of the district’s PD trainings focus on
algebra readiness or mathematics.
Findings for Research Question Two
Research Question 2: What is the school principal’s perception on how professional
development supports the teaching of the foundations of algebra for fifth grade?
Research question two focused on the evaluation by the school principal at
Kerns Elementary of the professional development related to teaching the
foundations of algebra in fifth grade available at Kerns Elementary School that is

75
either provided by the school district or at the school site. Data were collected from
a variety of sources as listed below in Table 4.2.
Table 4.2: Data for Research Question Two
Research Question Data Needed Data Sources Instrumentation
What is the school
principal’s perception
on how professional
development supports
the teaching of the
foundations of algebra
for fifth grade?
- Evidence on how
the school principal
supports PD for fifth-
grade teachers as it
relates to math and
algebra topics

- Professional
development training
- School-site
teachers
- School-site math
coach
- Email

- Interview
protocol
- Document
Review
- Observations

Dr. Johnson, the principal at Kerns Elementary, recognized that a drop in
mathematics achievement in the past warranted an examination of the school’s
curriculum as well as the pacing. Through collaboration between the school
principal and the staff, Kerns Elementary made some changes to its curriculum and
how the math content would be paced during the school year. Overall, Kerns
Elementary had a 46 point gain in the state Academic Performance Index (API) score
this past year. Kerns Elementary has the lowest API score in the district, and the
performance of the fifth-grade students in mathematics also remains lower than other
fifth graders in the same district. Dr. Johnson concluded that there was a need to
have a systematic approach in place for teacher collaboration through professional
development in order to improve the students’ academic performance.
Dr. Johnson noted that most people in education want to see results quickly
but that building teacher capacity can only be done incrementally and that this takes
time. This aligns with what researchers have noted about teaching needing to be

76
viewed as a cultural activity so that changes in teaching take place gradually,
steadily, over time through small daily changes in teaching and not solely through
more workshops (Hiebert & Stigler, 2004; Stigler & Hiebert, 1999). Professional
development at Kerns Elementary School is a team effort between the teachers, the
principal and the district. The principal stated that both the district and the school
are very supportive of the fifth-grade teachers as it relates to professional
development training in mathematics. She stated,
The district provides funding for teachers to get training through the PCDE
and it’s differentiated so it’s based on the teacher’s needs. Any kind of
materials that the school asks for, the district supports us. The school is the
same way. If a teacher tells me they need math manipulatives, I’ve never said
no.

The type of professional development that is offered by the district is
influenced in part by the feedback that the school principal provides. At Kerns
Elementary, Dr. Johnson expects her staff to collaborate with her in organizing and
facilitating the various math professional development trainings that are offered at
the school. The continuous dialogue between them also has an influence on what the
school principal communicates to the district in order that the needs of the teachers
are addressed by the trainings that PUSD offers. Dr. Johnson further noted the
importance of differentiating for the teachers and has made it clear to the teachers
that it is their responsibility to inform the principal of their needs given that they are
the ones who know their students best. She can only offer adequate professional
development for mathematics based on what she knows and what she thinks the
teachers need.

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Dr. Johnson clearly sees the important role that professional development has
in preparing fifth-grade teachers to teach general mathematics and algebra-related
topics, but she also acknowledges the importance of working with the teachers’
belief system since she feels that the teachers need to be in a particular state of
readiness in order to build their capacity for mathematics. The principal’s
impression is that teachers who are forced to attend conferences or other type of
professional development trainings that they don’t particularly have an interest in or
do not see a value in the experience will not enrich their knowledge or teaching of
mathematics. Therefore, the principal has no expectations for the fifth-grade
teachers to attend professional development trainings other than those offered by the
district which include training on CGI, Fosnot, and MIND or other sessions offered
by the school throughout the year. She puts the responsibility on the teachers to be
“smarter” in math. She stated,
If you don’t know something, you’ve gotta learn it. And if you think you
know something, you have to review it and see if there might be better ways
or stronger research practices that you could be using. More effective
practices.

According to Dr. Johnson, it’s important for principals to differentiate the
type of professional development that is offered to the teachers. The fifth-grade
teachers at her school are aware that they will be supported with money if they
decide to attend additional professional development trainings, need a particular
textbook or if they want to go visit a middle school class. The principal
acknowledged that she will grant such requests because she trusts that her fifth-grade

78
teachers will likely enhance their learning and teaching of mathematics with such
experiences. Some of the teachers interviewed disagreed with Dr. Johnson’s remark.
They stated that they did not feel like they could go to her to request funds for
additional professional development trainings related to math and receive her
approval. Regarding the asking of funds for math conferences, one teacher noted,
Oh I probably wouldn’t get that. No way. If I wanted a type of training or
materials for my class, I can forget that. If it were one hundred dollars and I
could justify the heck out of it, I might get it. But I honestly don’t know
because unfortunately I’ve brought things up like that before and it opens ups
a can of worms. Like, well if I let you go to a conference, I have to let
everyone else go.

There seems to be a discrepancy between the principal and the fifth-grade
teachers in understanding that the principal is willing to support the teachers by
providing them with more opportunities to attend professional development trainings
if the trainings would enrich their understanding of mathematics or help them learn
more effective teaching strategies related to algebra or other math content.
The problem that Dr. Johnson feels that most fifth-grade teachers encounter
is the fact that they don’t want to become too involved with many professional
development events other than those expected by the district or the school because
they are too busy or prefer not to leave their classrooms. Contrary to this notion,
several of the fifth-grade teachers interviewed did acknowledge their satisfaction
interacting with middle school math teachers during one professional development
activity since it gave them the opportunity to discuss important algebra readiness
topics that were supposed to be emphasized in fifth grade. They would be

79
enthusiastic to have more of these meetings due to their relevance if the district
would give them more than just one opportunity during the academic year. The
teachers expressed that this was one of the more valuable professional development
activities that they have participated in. One teacher stated,
Definitely the one with the sixth-grade teachers. That PD stood out because I
felt like it was effective. I felt the notes I took helped me. I knew what we
were doing in certain areas was not enough. I needed to reevaluate in certain
areas and figure out why or why not they were retaining. So hearing from
them is helping me know what I need to spend more time cementing. I just
wish there was more.

Even though all four teachers interviewed stated that they usually avoid as
much as possible leaving their classrooms with substitutes, Dr. Johnson recognized
the value that some events could bring by missing a day of instruction when she
stated,
You’re going to bring so much more back that the short period of time that
when you’re not with the kids is not going to hurt them. We just need to
make sure that when you are out of the classroom, it’s for something
meaningful and that you are going to bring back something meaningful to the
kids.

Another valuable professional development support that is being offered to
the fifth-grade teachers at Kerns Elementary, which Dr. Johnson firmly believes in,
is coaching. Coaching for fifth-grade teachers is sponsored by the district through
PCDE and through two math coaches at the school-site who are responsible for
fourth and fifth grade. Throughout the academic year, Dr. Johnson tries to organize
the coaching offered by the PCDE by grouping some grade levels. She states,

80
The math coach and I try to organize it that way. So the PCDE math coach
and I keep in touch and I tell her what I think my staff needs or what certain
grade level needs from my point of view. Then she works with the teachers.

These coaching sessions take place for three hours during designated days
throughout the school year. This type of arrangement allows for the fifth-grade
teachers to have an opportunity to work with the fourth-grade teachers during the
coaching sessions offered by the PCDE in order for them to share their perspectives
on what math content needs to be more emphasized at both grade levels. Then, fifth-
grade teachers know what to expect and are able to present lessons that are
meaningful to the students as they work with their students on algebra readiness
topics such equations and the meaning of a variable.
The coaching sessions offered by PCDE mostly focuses on CGI. Dr. Johnson
feels very positively about her fifth-grade teachers working on CGI with the PCDE
coaches since this program focuses on assisting teachers with their students to
develop a conceptual understanding of mathematics. She does admit that CGI has a
lot of limitations for fifth-grade students since it lacks focus on math topics that
relate to algebra topics covered in fifth grade. As will be discussed later in the
chapter, the fifth-grade teachers have made it clear to the district and to the school
principal that the first two years of CGI training focuses on math topics that are too
elementary for fifth-grade algebra and other math-related topics.
During the 2010-2011 school year, the fifth-grade teachers will have received
a total of three coaching sessions by PCDE, two of which the fourth and fifth-grade

81
teachers will work jointly. Dr. Johnson did admit that during the previous school
year there wasn’t a system in place where there was consistency of PD training, so
this year more PCDE coaching and Tuesday staff meetings, which will be discussed
later, would be implemented. The importance of coaching for the principal was
evident not only during the interview but also during the researcher’s observation at
a professional development training being facilitated by a school coach where the
school principal was seen taking notes and then walking around to discuss some of
the ideas with a few of the faculty.
Although the coaches at Kerns Elementary do not have any formal training as
do other school-site coaches within the same district, as noted by both the school
principal and the assistant superintendent of instruction, they have the longest CGI
experience and have a comfortable knowledge base of mathematics. Dr. Johnson
notes that these coaches are probably the best human resources that the school has
for mathematics PD training at the school site. This is of great value since the school
principal believes that for PD to be effective in preparing the fifth-grade teachers to
teach algebra readiness skills or fifth-grade mathematics in general, coaching is
essential, and it has to be ongoing throughout the school year. Her strong belief in
coaching is confirmed by her statement, “You can’t just have seminars and theory
based workshops.” Dr. Johnson also encourages all teachers to seek out any of the
coaches during the year, but as noted by most of the teachers who were interviewed,
they are more comfortable working with any of the other fifth-grade teachers, and
they constantly have informal interactions with each other throughout the week to

82
discuss different issues related to their students’ learning of algebra or other related
mathematics content.
Kerns Elementary now also has in place Tuesday staff meetings every week
for one hour, and once a month the meeting is devoted to mathematics. This year the
math Tuesday meeting’s goal is for the school-site math coaches to guide each grade
level through a math unit using Fosnot. The coaches selected to work on a third-
grade Fosnot unit during a couple of the Tuesday staff meetings to model what the
other grade levels were supposed to do once they selected a particular topic in
mathematics. The four fifth-grade teachers collaboratively developed a Fosnot math
unit on place value and division over a period of two math Tuesday meetings with
the guidance of the coaches. Then, the teachers were going to implement their unit
and analyze the results.
The researcher observed during a math Tuesday meeting that all four fifth-
grade teachers cheerfully shared their strategies on how they would implement the
activity that they had created as well as some effective teaching strategies they were
going to use with other topics in the upcoming chapters of their curriculum. During
this meeting, the researcher also noticed that the principal, Dr. Johnson, was
interacting with the faculty optimistically, praising them for the good work that they
had all produced. Dr. Johnson’s belief in the valuable experience that is gained
through collaboration and coaching was apparent since all the teachers at this
professional development meeting were genuinely engaged in their discussions about

83
mathematics, and they had collectively created a math activity during previous
Tuesday math meetings.
This year Kerns Elementary will have about five math Tuesday staff
meetings. Sometimes these math Tuesday meetings do not work out to once a month
as is intended by the school principal since some months are short or parent
conferences take place. The principal recognizes the importance of planning ahead,
and so at the beginning of the current academic year, she gave the school coaches a
half-day to meet and plan the Tuesday math meetings. In June the principal and her
staff will review the Tuesday math meetings that took place during the school year
and hope to have a better plan in place for the following academic year.
Dr. Johnson believes that working on Fosnot units drives the teachers to
identify the big ideas and then identify the mathematics. By doing so she feels that
the teachers will need to see the big ideas and then formulate the equivalent concepts
using algebraic equations, and consequently this teaches the teachers and gives them
the necessary resources to teach students the mathematics. She states,
If you are a teacher that’s not very confident, or thinks you know it, in
studying this you find out that there are some little pieces you might not
know or have not thought about. So this is also kind of my way of building
math knowledge.

Fosnot units give the teachers a resource to plan their lesson. The fifth-grade
teachers at Kerns Elementary also participate in CGI. Part of CGI requires that
teachers anticipate the strategies the students will be using in solving the math
problems, and the Fosnot units assist the fifth-grade teachers to foresee this.

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The school principal believes that the Tuesday staff meetings to work on
Fosnot units continue to be successful and having the extra support from the PCDE
math coach for CGI has also contributed to the fifth-grade teachers’ professional
development in mathematics. This gives them the tools to enrich their students’
math learning. Moreover, the principal noted that she serves as a trainer as well
during the Tuesday staff meetings by providing “PD points” whenever possible. For
instance, she indicated that she does not lead a 45 minute PD session but instead
notes important points that teachers can fix immediately. As an example, she
indicated that during a coaching session between the PCDE coach and the fifth-grade
teachers, a valuable piece that was shared was the misrepresentation of the equal sign
in mathematics and how from a student’s perspective the ideas of two elements
having the same value was not the same as being the same. Then, the principal
shared this learning experience with the rest of her staff during a Tuesday math
meeting. Dr. Johnson stated,
I’m not doing a 45 minute session on the equality sign. I’m just doing small
pieces of information. Small, digestible pieces of information because that is
such an easy fix. The teacher can go to their class the very next day and all
they need to adjust is the word “value”. So that to me is that I’m a trainer in
that sense, but I’m not leading a 45 minute session in algebra readiness. So
it’s a PD point but it’s brief. It’s not overwhelming. You do a lot of that
over time and you’ve done a lot by the end of the year.

Kerns Elementary also offers the fifth-grade teachers professional
development on the MIND program which they are expected to use with the students
for math instruction. This program was noted by all four fifth-grade teachers, the
school-site coach, the school principal and the assistant superintendent of instruction

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to be a fun and effective program that the fifth-grade students are enjoying while
successfully learning math. Dr. Johnson noted that the teachers usually receive a
refresher at the beginning of each academic year since the faculty are already
familiar with this software. This school year they received an actual PD training
since the program went from software based to web based. Now that the school has
adopted MIND Fluency for the first time, the fifth-grade teachers received an
additional hour of training at the beginning of the year for a total of two hours of PD
on MIND. The principal confirmed that the MIND ST software really helps the
students with algebra readiness and that although the MIND Fluency component is
not directly related to algebra readiness, MIND Fluency focuses on assisting the
students with their math facts. Having a deficit with such facts will slow down the
students’ understanding when the fifth-grade teachers start discussing problems that
incorporate fifth-grade algebra topics such as problems that incorporate more than
one unknown variable. Fifth-grade teachers are expected to implement MIND for a
total of 90 minutes per week, and the principal is confident that her staff is receiving
appropriate training with MIND every year.
Kerns Elementary also has in place grade level meetings where fifth-grade
teachers meet every Wednesday after school for about an hour as part of their
professional development. During these meetings the teachers may spend some time
on math related topics but also discuss other academic issues. Dr. Johnson’s goal in
designing the weekly grade level meetings is for the teachers to have another
opportunity to continuously interact with their colleagues and share strategies as well

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as knowledge that can benefit their teaching and their students’ learning. The school
principal firmly believes in teacher collaboration.
Some of the professional development that Kerns offers to their fifth-grade
teachers such as CGI, MIND or Fosnot, focuses on problem solving skills and
conceptual understanding of algebra and mathematics. The school principal noted
the importance of both the computation component of algebra skills for fifth-grade
students as well as learning to build their problem solving skills. She stated,
We’re not just interested in computation. We are interested in problem
solving. Not just in giving the algebraic equation with an unknown variable
so they can solve for it. What we want to see is if there’s a problem so they
can create their own equation with an unknown variable, then solve from
there and we want to see that.

During the study it was evident that both Dr. Jones and Dr. Johnson have a
firm belief that some of the teachers lack the mathematical knowledge needed to
facilitate the students’ learning. Dr. Johnson confirmed this belief when she said, “I
think there’s some lack of knowledge. Mathematical knowledge with some of the
teachers. They don’t know the math as well as they should.” Dr. Jones similarly
stated, “I think sometimes teachers in elementary get a multiple subject. They don’t
even know what they don’t know because if you aren’t a math person and haven’t
been trained in math, that was never your favorite area.” The researcher confirmed
Dr. Jones’ and Dr. Johnson’s assertion about the lack of mathematical knowledge
that some teachers have during an observation of a Tuesday math meeting when one
of the fifth-grade teachers was not able to distinguish the difference between a
mathematical equation and a mathematical expression in algebra.

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Dr. Johnson and Dr. Jones trust that with the combination of CGI, MIND,
Fosnot, the Tuesday math meetings and the grade level meetings along with the
support from the PCDE and school coaches, the teachers are gaining more
mathematical knowledge and learning how to implement effective teaching strategies
with their students. The importance for students to develop both skills and
understanding is shared by various researchers (Kilpatrick, Swafford, & Findell,
2001; Hiebert & Stigler, 2004). Moreover, Dr. Johnson noted that her fifth-grade
teachers need to be scholars about the math they are teaching in the classroom and
need to have an in-depth understanding of what it is. She further notes the
importance for teachers to take the time to study the math, because no matter how
much professional development the teachers are getting with CGI, Fosnot or MIND,
it is important for the teacher to understand the level of understanding in which the
students find themselves so that the students are guided step by step until they reach
mastery in mathematics.
Despite the fact that the school district and the school offer many
opportunities for mathematics professional development, the principal at Kerns
Elementary would prefer to have her fifth-grade teachers spend more time on
learning math content rather than on instructional strategies where teachers would be
like students who are receiving explicit instruction on particular topics such as
equivalence. As an example, she discussed the idea of learning more about
equivalence and stated,

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Having explicit instruction on what is equivalence. What does it mean? Let’s
define it. Does everybody have the same definition of equivalence? When
we say that a student has mastered the concept of equivalence, what do we
mean? I think that would get at whether teachers are really understanding
what equivalence is.

She would most prefer to have as part of professional development for fifth-
grade teachers some experts who can point out teachers’ common pitfalls when
teaching certain topics as a result of poor understanding of the mathematics. The
school principal noted an experience during a coaching session where one upper
grade teacher was able to discuss teaching strategies but was unable to understand
why an algorithm works and the mathematical concepts behind the algorithms. The
concern here is that some teachers may be teaching the algorithm since their
understanding of the mathematics is at a surface level, and they are not teaching with
a conceptual understanding. The principal added that a conceptual approach to teach
mathematics creates a more powerful learning experience for the students, and the
students are then able to derive the algorithms on their own such as the associative
and commutative properties of multiplication which later can be extended to algebra
concepts.
Currently, the school site coaches do not have the capacity to facilitate such
guidance and the principal fears that the lack of mathematics understanding may
result in some teachers teaching by just simply regurgitating what the book says. Dr.
Johnson stressed the importance of not lowering expectations regardless of what the
current situation is. Doing so would set up students for long-term failure. Instead,
they need to figure out why the student’s performance is where it is. Collaboration

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between teachers and the professional development experiences being offered by
PUSD and Kerns Elementary show great promise, as discussed by the assistant
superintendent of instruction and the school principal. It is evident that the principal
at Kerns Elementary strives to enrich the mathematics professional development
experiences of her fifth-grade teachers. As mentioned earlier, Dr. Johnson has a
vision for the future and plans ahead. During the researcher’s interview with Dr.
Johnson, the researcher was able to review an email sent in early August that the
principal provided noting her communication with one of the math coaches regarding
the planning of the Tuesday math meetings that would take place during the
academic year and their focus. Her participation in helping the teachers grow in the
area of mathematics is evident. One teacher commented, “She does email us
opportunities that she gets for free. So she does want us to go. She is promoting PD
in the ways she can.”
Dr. Johnson and Dr. Jones recognize the value of math professional
development and are continuously seeking ways to help the fifth-grade teachers at
Kerns Elementary. The district currently supports their fifth-grade teachers with
math professional development that is intended to focus on conceptual instruction.
Conceptual instruction has been shown to be related to improved student
achievement (Gamoran, Porter, Smithson, & White, 1997). The next research
question will discuss the findings about the current state of professional development
through the experiences of the fifth-grade teachers.

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Findings for Research Question Three
Research Question 3: What are the fifth-grade teachers’ perceptions on professional
development as it relates to teaching mathematics concepts and skills related to
algebra at this grade level?
Kraft Unified School District and the principal at Kerns Elementary
collaborate during the year to provide the fifth-grade teachers appropriate
professional development that will prepare the teachers to teach fifth-grade
mathematics, which includes algebra topics such as simplifying simple algebraic
expressions, graphing ordered pairs and finding appropriate equations when solving
problems. This research question focused on evaluating the four fifth-grade
teachers’ perspectives to determine whether they feel that the current professional
development is successfully meeting their needs to be fully prepared to teach algebra
and other mathematics topics taught at the fifth-grade level by increasing their math
knowledge as well as learning to use effective teaching strategies. Data were
collected from a variety of sources as listed in Table 4.3.
Table 4.3: Data for Research Question Three
Research Question Data Needed Data Sources Instrumentation
What are the fifth-
grade teachers’
perceptions on
professional
development as it
relates to teaching
mathematics
concepts and skills
related to algebra at
this grade level?
- Evidence that supports
the teachers’ beliefs
whether the professional
development offered by
the district is adequate
and successful.

- Professional
development training
- School-site teachers
- School-site math
coach
- Interview
protocol
- Document
Review
- Observations

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Kraft Unified School District has been offering CGI training as one of their
math professional development trainings for fifth-grade teachers through a three-year
preparation program. Besides the three-year program, which some of the fifth-grade
teachers have already completed, there is CGI coaching provided by PCDE
throughout the academic year. The fifth-grade teachers at Kerns Elementary
acknowledged that the first two years of the CGI training are focused on the lower
grades but that the third year has more appropriate training for their grade level.
Teacher 1 stated that this training has not helped her become more prepared to teach
algebra related concepts. Her lack of enthusiasm for CGI was noted when she stated,
We get information presented to us but have no time to practice it, bounce
ideas off each other, visit classrooms. We are told there will be days
available, but then those never come about. Not even at our school. Because
we’re upper grade, the upper grades were complaining because the first two
years of the three-year program was mostly geared toward primary so last
year they tried to revamp it and incorporate more for the upper grade but it’s
still not up to par. They do have a lot, it’s not horrible but we can’t use a lot
of the stuff they give us.

Teacher 2 described CGI training as a professional development experience
that really stands out due to the lack of effectiveness for fifth-grade teachers. She
expressed her disappointment when she said, “When CGI was implemented with
upper grades, we hated it. We felt like the fifth grade didn’t use manipulatives. We
felt unheard, and we just didn’t feel that CGI was fitting with us.”
Teacher 3 also noted that they receive CGI training during the school year
through math coaches sponsored by PCDE as a method to more effectively teach
their students since it’s not rote practice, not algorithms but more of a thinking-based

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approach. In spite of this, she also made reference about CGI being difficult to
implement in the classroom since it lacks the capacity to assist them with most of the
algebra related content. Furthermore, she expressed the same sentiment as her
colleagues when she stated, “We go to these trainings and either they are so boring
or you’re sitting and listening the whole time like a lecture format and you can’t see
how it will work in your own class and you never use it.” Teacher 4 had a similar
impression about CGI when she stated,
When I first started CGI training, very honestly I felt it was a huge waste of
time. Once I was there I felt nothing was applicable to the upper grade. But
they did kind of hear us because we made those comments as upper grade
teachers. So they are trying to get the focus more on things that are
applicable to upper grade. My personal opinion is that I’m not so sure it
helps the kids in the upper grade.

Despite the fact that the fifth-grade teachers expressed their disappointment
with CGI training, given that it mostly focused on the lower grades for the first two
years of the program, most of them also shared that they were able to make some
adaptations to implement CGI to some degree and saw some of its benefits. Teacher
1 commented that CGI forces teachers to slow down and discuss the math ideas from
a conceptual approach rather than immediately going straight to an algorithm.
Teachers are now able to learn the “why” behind the mathematical concepts.
Teacher 2 noted that one of the math coaches has taught them how to use it in their
class, how to get the students to think outside the box, and how to get the students to
communicate their mathematics understanding in their own way. Teacher 3 also
shared something positive thoughts about CGI when she stated,

93
It’s more thinking based. The process, the steps, really getting students to
look at the big picture instead of just being able to do a specific problem that
applies to everything. For me, it’s helped me look at math a different way
and to teach it a different way. So there is some benefit to it, it’s just not as
applicable.

Hiebert and Stigler (2004) argue that mathematics teaching requires a process
of change that includes teachers engaging in thinking about key mathematical
relations in particular problems carefully selected and that teachers must reflect with
thorough details to the students’ reactions in order to improve the effectiveness of
their lessons.
Kraft Unified School District provides CGI training for the fifth-grade
teachers but there’s no accountability set upon the teachers at Kerns Elementary to
ensure that its teachers are actually implementing lessons that incorporate CGI. The
fifth-grade teachers at Kerns Elementary expressed both their dissatisfaction with a
lot of the CGI training and some positive changes that it brought to their instruction
as noted above. Overall, though, they felt that CGI was not an adequate program
that allowed them to build their mathematical knowledge and discover effective
strategies that would promote a more enriching learning experience during algebra-
related topics.
For the second year now, Fosnot has been implemented at Kerns Elementary
School. Fifth-grade teachers received about two hours of professional development
related to Fosnot last academic year. This year, Fosnot is the topic of focus during
the math Tuesday meetings which are targeted to assist the fifth-grade teachers in
enhancing their instruction of mathematics and algebra-related topics. The fact that

94
the fifth-grade teachers are expected to teach a ten day unit using Fosnot every
academic year makes them nervous. During this period of time they have to throw
out their regular math curriculum from the textbook and substitute it with one unit
using Fosnot on a chosen topic. With ten less days of regular instruction, teachers
feel pressured to cover all the necessary content to prepare for the CST by focusing
on only one lesson for an extended number of days. This is surely no surprise.
Levister (2005) notes that with the adoption of standardized testing in public schools
in the United States, schools are now faced with the challenge of teaching curricula
that focuses on the content that is measured on the standardized assessments.
Regardless of this sentiment, the fifth-grade teachers seemed pleased with
Fosnot. Teacher 3 commented,
The Fosnot unit we did a little bit last year, and it is more algebra. What
we’ve chosen is an algebra lesson and it’s one lesson for ten days geared
more like CGI, so it’s a problem that could be solved in half-hour but the
goal of the unit is to really go in depth and talk about it and look at other
ways of solving it and having the kids really evaluate their own thinking and
that kind of stuff.

The fifth-grade teachers recognized how Fosnot can benefit the students’
learning of algebra and have collaborated with the school-site math coaches during
the math Tuesday monthly meetings for the past three sessions in developing a better
understanding on how to implement the big ideas using Fosnot. They collaboratively
developed strategies and models appropriate for fifth-grade mathematics. One
teacher commented,
According to the math coach team it will make our teaching easier. Easier
for them to learn the concepts. They’re hoping that based on the Fosnot units

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they will be able to learn the strategies and hopefully apply that to other parts
of math and that’s what I mean by easier.

Furthermore, being trained and implementing a program like Fosnot gives the
students more time which encourages more student reflection and allows for
constructing relationships between mathematical ideas (Hiebert & Wearne, 1993).
Wearne (1993) also stated that teachers that focused on conceptual understanding
allowed students to construct knowledge, spent more time on one mathematics
problem which allowed teachers to ask more questions requesting students to
describe and explain alternative strategies, had longer discussions, and showed
higher levels of performance.
The math coaches at Kerns have received more training on Fosnot, and they
take the lead during this professional development. Several fifth-grade teachers
commented that the onsite math coaches are giving professional development for
teachers to implement Fosnot multiple times this academic year, but that the coaches
are doing more facilitating than actual training. The coaches are facilitating how and
what teachers might do to prepare to teach the units in their own classes. The
principal at Kerns Elementary provided the researcher a copy of an email that she
and one of the coaches shared during the summer prior to the beginning of the school
year where they both discussed the dates and plans for the several professional
development sessions on Fosnot that would take place. It was apparent that the
school principal had a genuine interest in providing the teachers with valuable
professional development experiences, and the fifth-grade teachers recognized how

96
important math is to her. One teacher noted when referring to the high value that
both the district and the principal place on math PD, “It’s really important to them.
Especially with our principal. She’s wonderful. You can ask her questions. She
knows the studies, she knows the theories, but she also knows how to bring it home.
She’s amazing to me.”
During the professional development on Fosnot at Kerns Elementary, the
researcher saw that there was positive collaboration between the fifth-grade teachers.
They were attentively listening to each other’s ideas and gave constructive feedback.
It was also evident that the teachers had previously shared their teaching strategies
on algebra when one teacher said at this professional development, “Kids love your
method,” when she referred to graphing in algebra. Overall, the fifth-grade teachers
continue to see professional development training on Fosnot as a good opportunity
for them to learn strategies and create models that will assist them with the big math
ideas presented at the fifth-grade level including algebra-related topics. Currently,
the fifth-grade teachers continue to rely on the knowledge and guidance of their
school math coaches but hope that eventually they will become more independent
and be able to lead more discussions on the topic within their own grade level.
Another professional development offered by the school district to fifth-grade
teachers that focuses on mathematics is MIND STAR and MIND Fluency. This is
the second year MIND STAR has been available to fifth-grade teachers and the first
year of MIND Fluency. The school offered two hours of MIND professional
development training at the beginning of the year, and the fifth-grade teachers have

97
access to their site math coaches if they need further assistance with this program.
Last year the district offered about two hours of professional development related to
MIND STAR and learning the basics of the program, but the teachers felt that this
year the training was more focused on how to use and implement it in the classroom.
One teacher stated, “MIND itself is cool because we can align it with what we are
working on in class. It’s fun, it’s interactive and it helps them cement what we do in
class.” The teachers are encouraged not to help the students too much by giving
them the answers but instead are expected to ask questions and prompt them.
Another teacher stated,
I would say that the one PD that stands out to me would be the MIND
institute because I think that is making a difference in what my kids are doing
in math. That training doesn’t really train us how to teach, but they do train
us on some of the components so that we can relate what the kids are
learning in the ‘games,’ how we can take those and incorporate into the
concepts we are teaching. We just now have the ability to bring up the Jiji
games in class. So for instance, my kids in Jiji are working on factoring and
fractions because I can arrange the Jiji curriculum to match what I’m doing in
class. So I can bring up these games and show them how it relates to what
we are doing in math and help them tie it in. Jiji is spatial relations but
nothing is labeled as fractions or factors. They are just playing games. So I
can bring up the game and show them how it is the same thing we are doing
in math. So they see it and have touched it on the computer so I can now
relate it to the concepts we are learning in math.

Moreover, this school year the district has added MIND Fluency as an
intervention for students who are weak in basic math skills. The teachers
interviewed mentioned how some students are still having trouble with basic math
facts, and the implementation of MIND Fluency has certainly been a great asset.
MIND Fluency is geared towards helping students with learning how to retrieve

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basic math ideas such as multiplication facts. Although this is the first year of
MIND Fluency, the teachers are already seeing good results using it. Both the
students and the fifth-grade teachers have had good experiences using both
components of MIND. Overall the fifth-grade teachers feel that even though MIND
is in the beginning stages of implementation, the program has shown great promise
for them to implement it with their algebra and other math-related topics. Their hope
is that as time progresses, if the district expects them to continue using MIND, then
they will continue providing relevant professional development training that will
enable them to offer maximum use of this program with their students during their
lessons on algebra and other related math topics taught in the fifth grade.
Table 4.4 summarizes the reaction that the district, school principal and
teachers had regarding the professional development that the district is offering for
fifth-grade teachers as it relates to algebra and math related topics. A plus sign
means that it was a PD that was favorable, a negative sign means that the PD was
unfavorable or not enough of it was available for fifth grade, and a plus with a minus
sign means that the PD needed some changes but had some positive elements.
Table 4.4: Professional Development Offered by the District
PD District School Principal Teachers
CGI + ± __
Fosnot + + +
MIND + + +
Staff Meetings + + +
School-site
Coaching
± ± ±
Other __ __ __

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All four fifth-grade teachers expressed that their role in professional
development is that of a participant only. They do not participate in the planning,
organization, or anything of that sort. Teacher 1 said, “Teachers don’t have a big
impact in terms of training.” Teacher 2 stated, “For me, my role is to attend. I have
never facilitated. I have never planned it.” Teacher 3 said, “I’m a follower. I’m not
on the committee.” Teacher 4 added, “As far as math PD, I’m more just attending. I
don’t have anything to do with the facilitating or planning.” Several of the teachers
expressed that they have given the district some feedback during the evaluation of
the professional development trainings that they attended but have not received much
from the district in response to address their concerns.
The sentiment of the teachers in general was that both the district and the
school need to take into account the student population that Kerns Elementary serves
rather than training fifth-grade teachers on programs that are widely used by other
schools in the district. They expressed some dissatisfaction with the district. One
teacher exclaimed,
I think that professional development has to be the reality of the classroom
and the school. If they don’t, you can come in with all the ideology you want
but you have to know the reality. So if you are a presenter who was a former
teacher, I found them to be more effective than someone who has written
books but hasn’t been in the classroom.

A second teacher had similar remarks and said,
I think their heart is in the right place, they want us to get better at math. Yet
they are not giving the time, not giving us adequate training, so it’s just kind
of like a half effort. Yes, we value math, but we don’t value it enough to give
you the resources or the training. So it’s kind of frustrating.

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With the same frustration a third teacher added,
Sometimes I wish they just listened to us. Often times when teachers want to
talk, it’s necessary because we are letting out what is driving us crazy. So I
guess what I’m trying to say is, I wish we were allowed to voice our
frustration more and be heard. Then have them help us with what our
frustrations are. Like how can we solve what we know as teachers is holding
us back. I would love that. I guess what I’m saying is to just have more
honest open discussion where you’re not feeling like, oh, they are looking at
you like you’re complaining.

The fourth teacher stated,
Ask the staff what they think they need support in. I need more support in
certain areas and they are not asking me. If it wasn’t during school I’d be
much more excited to go because I’m not having to worry about missing,
writing lesson plans. I feel we should be forced to take college level classes
to stay fresh. And I understand that the CGI teachers are professors that are
coming to speak to us, but I feel there should be specifics to upper and lower
grades. And choosing better speakers. I feel the teachers need to have more
input you know because we’re in here every day and we know what we need.
More cross grade collaboration, less time out of class. Lessons would be
more engaging if they were geared toward what I want to learn.

It was evident that the math coach who was interviewed shared the same feeling
when she stated,
I’m not sure of how the district goes about getting PD but I think they don’t
do enough talking to us to find out what we need. They need to talk to us and
find out what we want. I think they are going about it backwards.

The teachers’ and coach’s reactions to the current situation is supported by
Guskey (2003) who states that site-based educators are keenly aware of critical
contextual characteristics of the school, and they along with the district need to
collaborate carefully in order to optimize the effectiveness of the professional

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development. Guskey (2003) states the importance of recognizing that programs of
professional development do not necessarily apply to every school environment.
In order for the teaching quality to improve, it is important for the school
district to listen to the voices of their teachers and provide the most appropriate
professional development. Some of the fifth-grade teachers gave some
recommendations during the interviews. Teacher 1 commented on how she very
much would like for KUSD to have math professional development like the Cotsen
program that she participated in a few years ago through the Cotsen Family
Foundation. During this two-year program she was given the opportunity to visit
other school sites in other school districts and see how CGI was being implemented
effectively as a tool to complement the textbook, experienced different teaching
styles and got great ideas from coaches that mentored her weekly. This program set
goals for the participants to grow in content knowledge and in pedagogy through
videotaping, reflection, inquiry, and coaching. Classroom observations by teachers’
colleagues for the purpose of professional learning is rare in the United States
(Darling-Hammond, 1997; Darling-Hammond & Ball, 1998), but there is evidence
that this element of professional learning as well as reflection and inquiry can lead to
more successful implementation as is demonstrated by lesson study and professional
learning communities which are discussed in Chapter 2. The district needs to
seriously consider this type of professional development that was recommended by
Teacher 1 since it was an enriching professional development that focused on
improving the teaching of mathematics. Teacher 2 and Teacher 3 discussed how

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they really found the grade-level curriculum meetings that fifth-grade teachers have
with sixth-grade teachers at the local middle school during the school year to be
beneficial and expressed their desire to do it more often. During these meetings,
they discussed the students’ deficiencies and how the fifth-grade teachers can be
more effective in the classroom. Morris (2006) notes that teachers need to have the
ability to analyze their own teaching in terms of its effects on learning, and it’s
through this analysis that they can improve their own teaching.
Three themes emerged from the data analysis. First, although student
academic enrichment is a district priority through the implementation of best
practices and programs, not all elementary schools in the district have trained on-site
coaches to lead staff development in mathematics. Four out of the six district
elementary schools currently have trained coaches in the area of mathematics.
Unfortunately, Kerns Elementary is one of the two elementary schools that lack
trained teachers to coach mathematics. The current designated math coaches are not
adequately trained and can only give limited assistance at the school site. Kerns
Elementary expects their fifth-grade teachers to implement the MIND program 90
minutes per week through two 45 minutes sessions for the entire school year, but
their only training is provided at the beginning of the year for about two hours and
the only resources for the entire year are themselves or two math coaches onsite who
have limited training. All four teachers interviewed admitted that their use of any of
the two on-site math coaches typically takes place during their monthly staff
meetings only and that there aren’t any opportunities for teachers to be observed by

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the coaches during the teaching of a mathematics lesson due to time constraints and
difficulty of finding substitute teachers.
The second theme that emerged from the study was the importance of
collaboration. Dr. Jones mentioned that the district fosters collaboration. She states,
There’s a lot of collaboration that happens when you put all your teachers in
the same CGI training at the county because when they go to year one
training every first year teacher is there. When they go to year two every
second year teacher is there. So everybody in the district collaborates. When
we do grade level meetings all teachers in the district collaborate. When we
release all fifth grade teachers to share data after second district benchmark,
they are collaborating. So we do a lot of releasing as a district.

Through the three year CGI program training, the Fosnot professional
development where the fifth-grade teachers had to create appropriate strategies and
models on a particular math unit during the Tuesday math meetings, the MIND
program training, and the grade-level meetings on Wednesdays, the district has been
promoting the importance of collaboration between faculty. Each of these trainings
requires that the fifth-grade teachers collaborate with each other.
The third theme that arose from this study is the questionability of the
effectiveness of CGI for fifth grade. The district is under the impression that CGI
training is the solution to improving students’ achievement in algebra readiness.
This is evident from Dr. Jones statement:
The reason I love CGI so much is it teaches teachers math. Because if you
really get math, you’ll be a better teacher and I don’t know how to tell an
elementary teacher that they don’t know enough. That’s insulting. But by
putting them through three years of CGI training I have a guarantee that they
have a fundamental understanding of math and they’ve been exposed to how
to implement a really good quality program.

104
Although the district continues to provide CGI training for all fifth- grade
teachers who have not yet completed the three-year program as well as coaching
through the PCDE, the teachers interviewed in the study have unequivocally
expressed their dissatisfaction with CGI given that this program mainly focuses on
content that is related to the lower grades. The principal also made reference on the
limitations of CGI and its training due to the fact that the on-site coaches are not
“experts” in CGI and some of the components of CGI are more challenging for fifth-
grade teachers to incorporate during the instruction algebra. The district
superintendent of instruction even commented in the direction that indicated some
doubt on her mind about the effectiveness of CGI despite the fact that she firmly
believes that this is a big factor in the increase of student achievement in algebra and
mathematics as a whole. She stated,
We evaluate professional development by looking at data. Even if it goes up
we have to create a culture of asking why the data is the way it is. Regardless
if it went up or down you have to determine cause. If it went up, maybe the
pacing was good, better materials, placed kids properly, whatever it is.
Because if you don’t know why, you are just lucky or unlucky and that’s not
good enough. So every year regardless of your data is, teachers get together
in teams and discuss why it is. If it’s good we say do that more. If it’s bad
they know why and say don’t do that, what are you going to do instead?

So although achievement of fifth-grade students has improved in
mathematics, it is doubtful whether CGI really has any positive impact on the
students’ success with math. It is evident from the teachers’ interviews that they have
expressed their concerns about the appropriateness of CGI for their fifth-grade

105
students, but the district has done very little to address such concerns. Dr. Jones
clearly stated,
Our teachers are pretty honest with us and sometimes they say information
was good but presenter is horrible, then we get a new presenter. Some didn’t
like CGI but too bad. So there are a few things we kept anyway.
As noted previously, the district typically chooses the presenters, and the
fifth-grades teachers expressed their desire to have presenters that were more attuned
to the actual experiences that take place in a fifth-grade classroom.
Summary
Chapter Four reviewed district data evidencing the various forms of
professional development related to algebra that are offered to fifth-grade teachers at
Kerns Elementary and the perceptions of the district, the principal and the teachers to
these trainings. Data for this study were collected through interviews, document
reviews, and the researcher’s participation in the district’s professional development.
Findings were analyzed and discussed with connections made with the literature
from Chapter Two: Review of the Literature. A summary of the study, conclusions
and implications are presented in Chapter Five.

106
CHAPTER 5
SUMMARY, CONCLUSIONS, AND IMPLICATIONS
Introduction
The importance of algebra in the United States is a topic of conversation that
continues to grow. Educators and policymakers have recognized that success in the
21
st
century requires the need for students to be academically prepared in algebra
since algebra has been recognized to be a gatekeeper course that assists in college
preparation as well as in the work force (Choike, 2000). Data from the 2007
National Assessment of Educational Progress (NAEP) indicate that 8
th
grade students
score below the national average on the national math achievement exam which
heightens the concern that students in the public system are not adequately prepared
to enroll in an Algebra I course. Preparation for algebra in middle school begins in
the elementary grade levels. This study examined professional development in
algebra for fifth-grade teachers in one school in a district that aims to increase the
academic excellence for all students in all areas, but in particular recognizes the
importance of providing an enriching mathematics learning experience for all
students.
This chapter summarizes the study, presents the conclusions and discusses
the implications of the data collected. The summary section of this study will review
the methodology that was used to gather the data as well as the research questions
that guided the study. A review of the study’s findings will be presented in the
discussion followed by the implications and recommendations for future research.

107
Summary of the Study
This study examined professional development in algebra for fifth-grade
teachers by evaluating the extent to which the school district supports the teachers at
the school where the study was conducted. The study further considered the school
principal’s perception and fifth-grade teachers’ experiences to determine whether the
professional development offered by the district supports the fifth-grade teachers in
gaining knowledge and effective teaching strategies to teach algebra content. The
sample for the qualitative study was selected based on the following criteria for the
school site: 1) the elementary school had a 2008-2009 API of 800 or above, and/or
consistent growth in API scores for the past 2 academic years; 2) students enrollment
of at least 500; 3) the elementary school had a current Similar Schools Ranking of 8
or above; 4) the elementary school had at least 19% or more students qualifying for
free and reduced lunch; 5) the elementary school had under 50% of White students;
and 6) the elementary school had computers and classrooms with internet.
The study involved reviewing documents that were provided by the school
district, school principal, and documents gathered during the researcher’s
observations of the school’s professional development. Furthermore, interview
responses of the district assistant superintendent of instruction, the school principal,
four fifth-grade teachers, and one math coach are also included in this study.
Interviews were conducted based on three protocols, the Teacher Interview Protocol
(Appendix A), Principal Interview Protocol (Appendix B), the Assistant
Superintendent of Instruction Protocol (Appendix C) and the Math Coach Protocol

108
(Appendix D). For document reviews, the documents that were analyzed included
the district’s Annual District Staff Development Plan for the 2010-2011 academic
year, school principal’s emails, Kerns’ staff meeting agendas, the Pine County
Department of Education (PCDE) professional development announcements, school
and district memos, school calendar of meetings, and documents pertaining to CGI,
Fosnot, and the MIND program. The triangulation of the research was completed
upon analyzing the data collected through the review of the documents, interviews,
and the researcher’s participation in the school’s professional development.
Data collected through interviews, document reviews, and the researcher’s
participation in the school’s professional development were based on three guided
research questions:
1. What is the extent to which the school district supports teachers’ capacity
to teach fifth-grade students concepts and skills related to algebra through
professional development?
2. What is the school principal’s perception on how professional
development supports the teaching of the foundations of algebra for fifth
grade?
3. What are the fifth-grade teachers’ perceptions on professional
development as it relates to teaching mathematics concepts and skills related
to algebra at this grade level?
The three research questions required that the administrators and teachers
reflect on the current state of professional development that is being offered by the

109
district and the school, and what changes were needed, if any, to have it be more
effective.
Conclusions
Research Question 1: What is the extent to which the school district supports
teachers’ capacity to teach fifth-grade students concepts and skills related to algebra
through professional development?
The interviewees’ responses, data from document analysis and the
researcher’s attendance at the school’s professional development highlighted the
range of professional development opportunities that are offered to fifth-grade
teachers at Kerns Elementary School by Kraft Unified School District (KUSD) to
enhance their knowledge and skills to effectively teach mathematics concepts and
skills that are related to algebra at this grade level. Principals are expected by the
school district to consult with the teachers so that during the administrative meetings
that take place during the year between the assistant superintendent of instruction
and the school administrator there can be a discussion to determine what would be
some appropriate professional development trainings that would provide an
enriching experience for the teachers in order for the district to provide better
learning experiences for the students with algebra content. KUSD currently offers
fifth-grade teachers Cognitively Guided Instruction (CGI), the MIND program, and
Fosnot as their major professional development in mathematics. The district is
currently collaborating with PCDE to provide professional development in CGI
through workshops and coaching.

110
Professional development on CGI targets to provide teachers with the ability
to teach algebra and other math related topics for the fifth-grade level using a
conceptual approach so that they can empower students to explain their reasoning,
justify their strategies used to derive their solutions and consequently give them the
opportunity to build a deeper collective understanding of mathematics. The MIND
program is a visual approach that utilizes a learner’s spatial temporal reasoning
abilities to understand, explain and solve multi-step mathematics problems that are
aligned to the state standards. MIND’s use of Jiji, the penguin, through a game
metaphor assists students who have had difficulty with traditional approaches to
learning mathematics. During the professional development that focuses on MIND,
the goal is to give teachers the ability to become familiar with the program and
implement its games according to the math topics being covered in class so that
students can solve mathematical puzzles, while deepening the students’
understanding of mathematics and building on their problem solving skills. This
year the teachers are also being trained in the Fluency component of MIND as an
intervention program to assist students who are struggling with basic math facts.
Another professional development offered to fifth-grade teachers, which
takes place during the Tuesday math meetings at Kerns Elementary, focuses on the
use of Fosnot to cover one unit in math over a period of ten days. The focus is to
consider a particular math topic and look at the big ideas in depth by creating
strategies and models that are related to the unit chosen. During the week, another
professional development offered is the grade level meetings for fifth-grade teachers

111
to have discussions on algebra or other mathematics related topics as well as
discussions that pertain to other academic disciplines. Compared to last year, the
grade level meetings have lost the focus to be solely on mathematics.
The district is big on coaching but unlike some math coaches in other
elementary schools within the district, the math coaches at Kerns Elementary School
are not fully competent to take on full responsibility in adequately assisting their
teachers on CGI or other advanced topics in algebra at the fifth-grade level. The
district is in the process of expanding their coach training in the near future. After
reviewing the district’s Annual District Staff Development Plan for the current
academic year, it was found that only one out of seventeen professional development
activities were focused on mathematics.
Kraft Unified School District continues to offer many opportunities for the
fifth-grade teachers to enhance their knowledge and skills as it relates to algebra and
other mathematics topics. Nonetheless, it is necessary to carefully consider both the
school principal’s and the fifth-grade teachers’ perspectives on the current
professional development in algebra and other math content in order to determine its
adequacy in providing the teachers with math knowledge and effective teaching
strategies.
Research Question 2: What is the school principal’s perception on how professional
development supports the teaching of the foundations of algebra for fifth grade?
Interview responses, document analysis and the researcher’s attendance at the
school’s professional development were used to determine the school principal’s

112
evaluation of the current staff trainings with respect to algebra for the fifth-grade
level. Some of the professional development offered to fifth-grade teachers at Kerns
Elementary focused on CGI, MIND or Fosnot, and their focus is on problem solving
skills and conceptual understanding of algebra and mathematics. The school
principal noted the importance of learning both the computational component of
algebra skills for fifth-grade students as well as building their problem solving skills.
Furthermore, she trusts that professional development that combines CGI, MIND,
Fosnot, the weekly staff meetings along with the support from the PCDE and school-
site coaches is giving the fifth-grade teachers the opportunities to gain more
mathematical knowledge and to learn how to implement effective teaching strategies
with their students. Despite the fact that Dr. Johnson feels very positively about her
fifth-grade teachers working on CGI with the PCDE coaches, she admitted that CGI
has limitations for fifth-grade students since it lacks focus on math topics that relate
to algebra content covered in fifth grade.
Moreover, the importance of teachers collaborating with each other and with
the coaches was stressed by Dr. Johnson. She noted that most of the professional
development offered to the fifth-grade teachers was guided by collaboration between
professionals. The school principal believes that for PD to be effective in preparing
the fifth-grade teachers to teach algebra content or other math related topics in fifth
grade, coaching and collaboration are essential, and it has to be ongoing throughout
the school year. Dr. Johnson also feels the responsibility of contributing to her

113
teachers’ growth and provides “PD points” as often as she can throughout the
academic year.
Research Question 3: What are the fifth-grade teachers’ perceptions on professional
development as it relates to teaching mathematics concepts and skills related to
algebra at this grade level?
Data collected from interviews, document analysis and the researcher’s
participation in the school district’s professional development were used to evaluate
the four fifth-grade teachers’ perspectives in order to determine whether they feel
that the current professional development is successfully meeting their needs to be
fully prepared to teach algebra and other mathematics topics taught at the fifth-grade
level by increasing their math knowledge as well as learning to use effective teaching
strategies.
The overall sentiment of the fifth-grade teachers was that the district provided
several opportunities for professional development but that the PD that focused on
CGI was least effective in trying to enhance their math knowledge or provide
effective teaching strategies. The fifth-grade teachers at Kerns Elementary
acknowledged that most of the training they’ve had or continue to have on CGI
focuses on the lower grades which makes it challenging to implement this program
since it lacks the capacity to assist them with most of the algebra related content. The
goal with CGI is for teachers to learn and teach algebra or other math related topics
from a conceptual approach but professional development for Fosnot and MIND
have been more successful for them.

114
The fifth-grade teachers see professional development on Fosnot as a good
opportunity for them to learn strategies and create models that will assist them with
the big math ideas, including content in algebra. Currently, the fifth-grade teachers
continue to rely on the knowledge and guidance of their school math coaches as they
learn more about Fosnot but hope that eventually they will become more
independent and rely on their grade level colleagues.
Professional development with both components of MIND were seen by the
fifth-grader teachers as very promising. With MIND, not only are they able to
provide their students with opportunities to solve mathematical puzzles to deepen the
students’ understanding of mathematics and build on their problem solving skills, but
now they can also provide an intervention program that helps students who are
having difficulty with basic math facts. The fifth-grade teachers have access to math
coaches on-site to request help for any math related concern but they expressed
hesitance in seeking their help due to personality factors or time conflicts. Moreover,
they expressed an honest desire for the district to seriously consult with the teachers
to determine their needs rather than offer district-wide professional development
without considering the student population of each classroom.
Implications
Districts and schools continue to face challenges with low success rates of
algebra students. The low performance of mathematics students in American
schools continues to bring attention in regards to what the most effective teaching
methods should be used in the classroom as well as the recognition that mathematics

115
teachers need content knowledge in the field as wells as effective pedagogy in order
to promote student learning. The National Council of Supervisors of Mathematics
(NCSM) argues that mathematics teachers and leaders need a strong school and
district support system that will provide intensive and sustained professional learning
to ensure that all students will have access to a quality mathematics instruction.
Numerous strategies have been employed by teachers in the United States, but the
percentage of students who successfully pass algebra continues to be low. Algebra is
a gateway to college and many careers including mathematics, medicine,
engineering, and business. The concern for students not being prepared to become
successful citizens in the 21
st
century keeps rising.
This study has implications for various stakeholders including policymakers,
district leaders, teachers, and educational researchers. For policymakers, the study
provides evidence that there is a need for districts to be supported financially in order
for the schools to receive the highest quality professional development that is
supported by research and not resulting from what is popular or new on the market.
For district leaders, the study reveals how district administrators need to fully
consider the needs of each school when making decisions to determine the kind of
professional development that will be offered. Professional development needs to be
appropriate for the teachers in order to give them the knowledge and tools to enhance
their teaching of algebra and other math content. District leaders need to understand
the importance of collaboration, accountability, and on-going professional
development during the academic year. Offering a vast amount of professional

116
development does not necessarily imply that the teachers’ experiences will be
enlightening. For teachers, the study provides a reminder that they have the
responsibility to make their concerns be heard by school administrators and district
leaders. Both teachers and district leaders need to recognize that they have the joint
responsibility in being active participants in both the design and implementation of
professional development. Another implication for teachers is the realization that
students can benefit from non-traditional teaching methods and teaching
mathematics from a conceptual approach leads to more enriching learning
experiences. For educational researchers, the study presents evidence that not all
programs intended to assist with mathematics learning can be appropriate for all
grade levels or benefit every teacher.
School districts need to allow more opportunities for teachers to be more
involved with their professional development planning and being facilitators. By
holding teachers accountable, they can find more value with their professional
development and become stewards of their own learning. The limitations of CGI
were expressed by all participants in the study and some of the participants also
commented that after the three years of CGI training some of the teachers do not see
any value in continuing to pursue professional development with CGI given that the
district has no policy set up that will hold the faculty accountable for further
engagement with CGI. Teachers are expected to use CGI during their classroom
instruction but unless an administrator is closely monitoring, the teacher may not be
implementing this program. Both administrators and teachers need to be held

117
accountable for implementing and evaluating the different types of professional
development that are offered by the district.
This study further suggested that the district needs to provide more
professional development that focuses on the MIND program. Teachers expressed
their high regards for this program since it gives students opportunities to solve math
puzzles and interact with various games that assist them in developing problem
solving skills as well as gaining a deeper understanding of algebra and other math
content. Moreover, teachers have found their students actively engaged during the
implementation of the MIND program and having positive conversations about it
with their peers.
Serious consideration needs to be given to the culture of the school in order
to determine whether the professional development that is offered to teachers is
meeting the needs of the students. The district, school administrators, and the
teachers need to carefully evaluate each program to determine whether its
effectiveness or lack of is due to the program itself, the academic or pedagogical
background of the teachers, and/or the implementation of the professional
development related to the program. Then they can all collaborate to design and
implement professional development that will address their concerns. Although the
administrators, the teachers and the math coach noted that time was one of the
biggest factors that limited the amount of professional development that could
reasonably be expected by the district, it’s also important for districts to focus on the
structuring of the school day so that teachers are receiving continuous and on-going

118
professional development throughout the academic year as well as providing them
with opportunities to constantly reflect on their teaching practices.
Recommendations for Future Research
Although this qualitative research study examined the extent to which the
school district supports its fifth-grade teachers’ capacity to teach algebra through
professional development and the perspectives of the school principal and fifth-grade
teachers with regards to the different professional development offered by the
district, recommendations for future research exists:
• How does a school district evaluate the differences between its high
performing and low performing schools in mathematics achievement if
both schools are receiving the same type of professional development?
• How might district leaders, school administrators, and teachers
collaborate more effectively to assess the individual needs of the student
population it serves in order to provide more adequate professional
development in algebra or mathematics?
• How might district leaders measure the impact of resource allocations for
mathematics professional development that focuses on algebra?
• What are the differences, if any, on the perspectives of teachers who use
Cognitively Guided Instruction (CGI) for teaching algebra in a low
performing school versus a high performing school? To what factors are
these differences attributed to?

119
REFERENCES
Adams, D., & Hamm, M. (1996). Cooperative learning: Critical thinking and
collaboration across the curriculum. Springfield, IL: Charles C. Thomas
Publisher.

Arens, S. A., & Delandshere, G. (2003). Examining the quality of the evidence in
preservice teacher portofolios. Journal of Teacher Education, 54(1), 57-73.

Ball, D. L., & Cohen, D. K. (1999). Developing practice, developing practitioners:
Toward a practice-based theory of professional education. In G. Sykes & L.
Darling-Hammond (Eds.), Teaching as the Learning Profession: Handbook
of Policy and Practice (pp. 3-32). San Francisco, CA: Jossey-Bass.

Barnett, C. (1998). Mathematics teaching cases as a catalyst for informed strategic
inquiry. Teaching and Teacher Education, 14(1), 81-93.

Bensimon, E. (2004). The diversity scorecard: A learning approach to institutional
change. Change, 36(1), 45-52.

Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C.P., & Loef, M. (1989).
Using knowledge of children’s mathematical thinking in classroom teaching:
An experimental study. American Educational Research Journal, 26, 499-
531.

Carpenter, T. P., Fennema, E., Franke, M. L., & Empson, S. B. (1999). Children’s
mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.

Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking Mathematically:
Integrating arithmetic and algebra in elementary school. Portsmouth, NH:
Heinemann.

Civil, M., & Planas, N. (2004). Participation in the mathematics classroom: Does
every student have a voice? For the Learning of Mathematics, 24(1), 7-13.

Clark, C. M. (2001). Talking shop: Authentic conversation and teacher learning.
New York, NY: Teachers College Press.

Cobb, P., Wood, T., Yackel, E., Nicholls, J., Wheatley, G., Trigatti, B., & Perlwtiz,
M. (1991). Assessment of a problem-centered second grade mathematics
project. Journal for Research in Mathematics Education, 22, 3-29.

120
Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the
representational view of mind in mathematics education. Journal for
Research in Mathematics Education, 23, 2-33.

Cochran-Smith, M. (2001). Constructing outcomes in teacher education: Policy,
practice and pitfalls. Educational Policy Analysis Archives, 9(11).

Cochran-Smith, M. (2004). Walking the road: Race, diversity, and social justice in
teacher education. New York. NY: Teachers College Press.

Cohen, E. G. (1994). Designing groupwork: Strategies for the heterogeneous
classroom. New York, NY: Teachers College Press.

Dana, N. F., & Yendol-Silva, D. (2003). The reflective educator’s guide to
classroom research: Learning to teach and teaching to learn through
practitioner inquiry. Thousand Oaks, CA: Corwin Press Inc.

Darling-Hammond, L. (1997). The right to learn. San Francisco. CA: Jossey-Bass.
Darling-Hammond, L., and Ball, D.L. (1998). Teaching for high standards: What
policymakers need to know and be able to do. New York. NY: National
Commission on Teaching and America’s Future and Consortium for Policy
Research in Education.

Darling-Hammond, L. (2000). Reforming teacher preparation and licensing:
Debating the evidence. Teachers College Record, 102(1), 28-56.

Desimone, L. M., Porter, A. C., Garet, M. S., Yoon, K. S., & Birman, B. F. (2002).
Effects of professional development on teachers’ instruction: Results from a
three-year longitudinal study. Educational Evaluation and Policy Analysis,
24(2), 81-112.

DuFour, R., DuFour, R., Eaker, R., & Many, R. (2006). Learning by doing: A
Handbook for professional communities at work. Bloomington, IN: Solution
Tree.

DuFour, R., & Eaker, R. (1998). Professional learning communities at work: Best
practices for enhancing student achievement. Bloomington, IN: Solution
Tree.

Eaker, R., & Keating, J. (2008). A shift in school culture: Collective commitments
focus on change that benefits student learning. Journal of Staff Development,
29(3), 14-17.

121
Earley, P.C., Northcraft, G. B., Lee, C., & Lituchy, T. R. (1990). Impact of process
and outcome feedback on the relation of goal setting to task performance.
Academy of Management Journal, 33(1), 87-105.

Elmore, R. F. (1996). Getting to scale with good educational practice. Harvard
Educational Review, 66(1), 1-26.

Fernandez, C. (2002). Learning from Japanese approaches to professional
development: The case of lesson study. Journal of Teacher Education,
53(5), 393-405.

Feuer, M. J., Towne, L., and Shavelson, R. J. (2002). Scientific culture and
educational research. Educational researcher, 31(8), 4-14.

Fey, J. T. (1981). Mathematics teaching today: Perspective from three national
surveys. Reston, VA: National Council of Teachers of Mathematics.

Garet, M. S., Porter, A. C., Desimone, L., Birman, B. F., & Yoon, K. S. (2001).
What makes professional development effective? Results from a national
sample of teachers. American Educational Research Journal, 38(4), 915-945.

Gokhale, A. A. (1995). Collaborative learning enhances critical thinking. Journal of
Technology Education, 7(1), 1-8.

Gonzales, P., Calsyn, C., Jocelyn, L., Mak, K., Kastberg, D., Arafeh, S., Williams, T.
& Tsen, Winnie (2000). Pursuing excellence: Comparisons of international
eighth- grade mathematics and science achievement from a U.S. perspective,
1995 and 1999. (NCES 2001-028). Washington, DC: U.S. Department of
Education, National Center for Education Statistics.

Guskey, T. R. (2000). Evaluating professional development. Thousand Oaks, CA:
Corwin.

Guskey, T. R. (2002). Does it make a difference? Evaluating professional
development. Educational Leadership, 59(6), 45-51.

Hatano, G. (1988). Social and motivational bases for mathematical understanding. In
G. B. Saxe & M. Gearhart (Eds.), Children’s mathematics (pp. 55-70). San
Francisco, CA: Jossey-Bass.

Haycock, K. (1998). Good teaching matters: How well-qualified teachers can close
the gap. Thinking K-16, 3(2), 3-14.

122
Hiebert, J., Gallimore, R., & Stigler, J. W. (2002). A knowledge base for the
teaching profession: What should it look like and how can we get one?
Educational Researcher, 31(5), 3-15.

Hiebert, J., Gallimore, R., Gamier, H., Givvin, K. B., Hollingsworth, H., Jacobs, J.,
Chui, A. M., Wearne, D., Smith, M., Kersting, N., Manaster, A., Tseng, E.,
Etterbeek, W., Manaster, C., Gonzales, P., & Stigler, J. W. (2003).
Understanding and improving mathematics teaching: Highlights from the
TIMSS 1999 Video Study. The Phi Delta Kappan, 84(10), 768-775.

Hiebert, J. & Stigler, J. W. (2004). A world of difference: Classrooms abroad
provide lesson in teaching math and science. National Staff Development
Council, 25(4), 10-15.

Hiebert, J., & Wearne, D. (1993). Instructional tasks, classroom discourse, and
students’ learning in second-grade arithmetic. American Educational
Research Journal, 30(2), 393-425.

Hill, H. C. (2004). Professional development standards and practices in elementary
school mathematics. The Elementary School Journal, 104(3), 215-231.

Hollins, E. R., & Torres Guzman, M. (2005). Studying teacher education: The report
of the AERA panel on research and teacher education. Chapter 8: Research
on preparing teachers for diverse populations. Lawrence Erlbaum Associates:
Mahwah, NJ.

Hord, S. M. (2009). Professional learning communities: Educators work together
toward a shared purpose-improved student learning. JSD, 30(1), 40-43.

Huffman, D., Thomas, K., & Lawrenz, F. (2008). Science and mathematics
instruction in a reform-based teacher preparation program. School Science
Math, 108(4), 137-148.

Jay, J. K. & Johnson, K. L. (2002). Capturing complexity: A typology of reflective
practice for teacher education. Teaching and Teacher Education, 18, 73-85.

Kazemi, E. & Franke, M. L. (2004). Teacher learning in mathematics: Using student
work to promote collective inquiry. Journal of Mathematics Teacher
Education, 7, 203-235.

Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping
children learn mathematics. Washington, DC: National Academy Press.

123
Ladson-Billings, G. J. (1999). Preparing teachers for diverse student populations: A
critical race theory perspective. Review of Research in Education, 24(24),
211-247.

Lambert, L. (2003). Leadership capacity for lasting school improvement.
Alexandria, VA: ASCD.

Levine, T. H. & Marcus, A. S. (2007). Closing the achievement gap through teacher
collaboration: Facilitating multiple trajectories of teacher learning. Journal of
Advanced Academics, 19(1), 116-138.

Lewis, C. (2002). Does lesson study have a future in the United States? Nagoya
Journal of Education and Human Development, 1(1), 1-23.

Lewis, C. C., Perry, P. R., & Hurd, J. (2009). Improving mathematics instruction
through lesson study: A theoretical model and North American case. Journal
of Math Teacher Education, 6(2), 107-137.

Lewis, C., Perry, R., & Murata, A. (2006). How should research contribute to
instructional improvement? The case of lesson study. Educational
Researcher, 35(3), 3-14.

Lewis, C., & Tsuchida, I. (1997). Planned educational change in Japan: The shift to
student-centered elementary science. Journal of Education Policy, 12(5),
313-331.

Loucks-Horsley, S., & Matsumoto, C. (1999). Research on professional development
for teachers of mathematics and science: The state of the scene. School
Science and Mathematics, 99(5), 258-271.

Loucks-Horsley, S., Stiles, K. E., Mundry, S., Love, N., & Hewson, P. W. (2010).
Designing Professional Development for Teachers of Science and
Mathematics. Thousand Oaks, CA: Corwin.

Merriam, S. B. (2009). Qualitative research: A guide to design and Implementation.
San Francisco, CA: Jossey-Bass.

Morris, A. K. (2006). Assessing pre-service teachers’ skills for analyzing teaching.
Journal of Mathematics Teacher Education, 9, 471-505.

National Board for Professional Teaching Standards (1989). Toward high and
rigorous standards for the teaching profession. Washington, DC: Author.

124
National Commission on Excellence in Education (1983). A nation at risk: The
imperative for educational reform. The Elementary School Journal, 84(2),
112-130.

National Commission on Mathematics and Science Teaching (2000). Before it’s too
late: A report to the nation from the national commission on mathematics
and science teaching for the 21
st
century. Washington, DC: Author.

National Commission on Teaching and America’s Future (1996). What matters most:
Teaching for America’s future. New York, NY: Author.

National Education Goals Panel (1994). National education goals report: Building a
nation of leaders (3
rd
ed.). Washington, DC: Government Printing Office.

Pardoe, I. (2004). Multidimensional scaling for selecting small groups in college
courses. The American Statistician, 58(4), 317-321.

Pintrich, P.R. (2003). A motivational science perspective on the role of student
motivation in learning and teaching contexts. Journal of Educational
Psychology, 95(4), 667-686.

Sanders, W. (1998). Value-added assessment. The School Administrator, 55(11), 24-
27.

Sanders, W., & Horn, S. (1998). Research finding from the Tennessee Value-Added
Assessment System (TVAAS) database: Implications for educational
evaluation and research. Journal of Personnel Evaluation in Education,
12(3), 247-256.

Schon, D. A. (1983). The reflective practitioner: How professionals think in action.
New York, NY: Basic Books, Inc.

Senge, P. (1990). The leader’s new work: Building learning organizations. Sloan
Management Review, Fall 1990, 7-23.

Shulman, L. & Sparks, D. (1992). Merging content knowledge and pedagogy: An
interview with Lee Shulman. Journal of Staff Development, 13(1), 14-16.

Smith, T., Desimone, L. M., & Ueno, K. (2005). “Highly Qualified” to do what? The
relationship between NCLB teacher quality mandates and the use of reform-
oriented instruction in middle school mathematics. Educational Evaluation
and Policy Analyis, 27(1), 75-109.

125
Snow-Gerono, J. L. (2005). Professional development in a culture of inquiry: PDS
teachers identify the benefits of professional learning communities. Teaching
and Teacher Education, 21, 241-256.

Sperling, M. & Freedman, S. (2001). Research on writing. In V. Richardson (Eds.),
Handbook of Research on Teaching (pp. 370-389). Washington, D.C.:
American Educational Research Association

Stigler, J. W., & Hiebert, J. (1997). Understanding and improving classroom
mathematics instruction: An overview of the TIMSS video study. In D.
Huffman, K. Thomas, & F. Lawrenz (2008). Science and mathematics
instruction in a reform-based teacher preparation program. School Science
Math, 108(4), 137-148.

Silver, E. A., & Kenney, P. A. (Eds.) (2000). Results from the seventh mathematics
assessment of the National Assessment of Educational Progress. Reston,
VA: National Council of Teachers of Mathematics.

Stigler, J. W. & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s
teachers for improving education in the classroom. New York, NY: Free
Press.

Supovitz, J. A., & Turner, H. M. (2000). The effects of professional development on
science teaching practices and classroom culture. Journal of Research in
Science Teaching, 37(9), 963-980.

Trimbur, J. (1992). Consensus and difference in collaborative learning. In P.
Shannon (Ed.). Becoming Political: Readings and writings in the politics of
literacy education (pp. 208-222). Portsmouth, NH: Heinemman.

Wahlberg, Melanie (1997). Lecturing at the “Bored.” The American Mathematical
Monthly, 104(6), 551-556.

Wang-Iverson, P., & Yoshida, M. (2005). Building our understanding of lesson
study. Philadelphia, PA: Research for Better Schools.

Wolfe, P., & Brandt, R. (1998). What do we know from brain research? Educational
Leadership, 56(3), 8-13.

126
Zeichner, K. M., & Hoeft, K. (1996). Teacher socialization for cultural diversity. In
J. Sikula, T. J. Buttery & E. Guyton (Eds.), Handbook of research on teacher
education, 2
nd
edition, (pp. 525-547). New York, NY: Macmillan.

Zeichner, K. M., & Liston, D. P. (1986). Reflective teaching: An introduction.
Mahwah, New Jersey: Lawrence Erlbaum Associates.

127
APPENDIX A

INTERVIEW PROTOCOL FOR TEACHERS
1. How has the performance of 5
th
grade students changed in algebra/math over the
past several years in your school? How has the school/district addressed these
changes in the students’ math performance? Was there any success with these
efforts? Do they continue to this day-or what happened to the efforts?

2. Regarding students’ math performance in 5
th
grade related to algebra, are there
any formal or informal goals for what you or the district are trying to accomplish?
- What are these goals?
- What do you aspire to? In what time frame?
- How will you, the school, and the district know if it is successful?
- How big is the gap between where you are now and where you aspire to
be?
- What is keeping the school/district from achieving perfect success?

3. Please describe how the school and school district use professional development
(PD) to help teachers with their algebra/math instruction in 5
th
grade. Who is in
charge of planning the PD, how often it occurs and when, duration per PD, activities,
etc.?

4. Have you experienced an emphasis in teaching algebra skills during PD?

5. What is your role in your school/district in relation to PD (attending, facilitating,
planning, evaluating, other) in general and as it relates to algebra readiness?
Explain. Is PD in math instruction a priority at the school/district? Why or why not?

6. What kind of PD have you participated in to build your capacity in algebra/math
skills as a 5
th
grade teacher? (what, when, how often, duration, etc.) Is there
collaboration between the teachers and between teachers and administrators? Is the
PD in algebra/math continuous?

7. Describe a memorable math teaching strategy (related to algebra if possible)
shared during a PD by a presenter/facilitator. What was most effective about the
experience? Were you able to implement something that you learned from the
presenter into your own classroom during math instruction? Was it easy or difficult?
How practical are the PD ideas that focus on algebra skills for you to implement in
the classroom? Explain.

8. What do you believe is necessary for PD to be effective in preparing you and the
students to teach and learn algebra skills?

128

9. If possible, what would you change, if anything, about the way your
school/district conducts professional development for algebra? Please explain.

10. Describe a component or event that stands out regarding PD related to algebra in
your school/district.

11. What is the school’s/district’s plan for planning PD and following up with PD
related to algebra? How does the school/district evaluate the PD/staff training in
algebra/math?

12. Please describe your thoughts on the budget for PD in your school/district in
general and specifically as it relates to math PD (algebra).

13. How does the administration at the school/district support the teachers in being
more prepared to teach algebra skills in 5
th
grade?

14. Students may remember their math skills for a longer period of time and may be
able to apply these skills in various contexts if they are taught using a conceptual
approach. Does the PD in algebra focus on teaching math from a conceptual
approach? If not, would you prefer that it did? Why?

15. Would you like to share anything else about PD related to algebra readiness or in
general that I have not asked you?

129
APPENDIX B
INTERVIEW PROTOCOL FOR PRINCIPAL
1. How has the performance of 5
th
grade students changed in algebra/math over the
past several years in your school? How has the school/district addressed these
changes in the students’ math performance? Was there any success with these
efforts?

2. Regarding students’ math performance in 5
th
grade related to algebra, are there any
formal or informal goals for what you or the district are trying to accomplish?
- What are these goals?
- What do you aspire to? In what time frame?
- How will you/the school/the district know if it is successful?
- How big is the gap between where you are now and where you aspire to
be?
- What is keeping the school/district from achieving perfect success?

3. How does the administration at the school/district support the teachers in being
more prepared to teach algebra skills in 5
th
grade?

4. Please describe how your school/district use professional development (PD) to
help teachers with their algebra instruction in 5
th
grade.

5. How do you address concerns with teacher capacity as it relates to algebra
instruction? What strategies do you have in place to support algebra instruction in 5
th

grade?

6. Who determines/organizes the type of PD that is offered at the school? What is
your role in your school/district in relation to PD (attending, facilitating, planning the
PD?) Is PD in math instruction a priority at the school/district? Why or why not?

7. What kind of PD does your staff participate in to build teacher capacity in algebra
skills for 5
th
grade teachers? (when, how often, duration) Is there collaboration
between the teachers and between teachers and administration? Is the PD in algebra
/math continuous?

8. Students may remember their math skills for a longer period of time and may be
able to apply these skills in various contexts if they are taught using a conceptual
approach. Does the PD in algebra focus on teaching math from a conceptual
approach?

130
9. Sometimes teachers recognize that they need assistance teaching some math
topics. What do you do to create a climate to ensure that you engage all teachers in
PD that relates to algebra/math when it is offered by the school/district or by seeking
the math coaches/committee on campus?

10. Please describe how the school/district uses student data to guide PD/staff
training in algebra skills for 5
th
grade.

11. How does the school/district evaluate PD training in algebra/math, in general?

12. What do you believe is necessary for PD to be effective in preparing the 5
th

grade teachers and the students to teach and learn algebra skills?

13. If possible, what would you change, if anything, about the way your
school/district conduct PD for algebra readiness? Please explain.

14. Please describe your thoughts on the budget for PD in your school/district in
general and specifically as it relates to algebra/math staff training.

15. Please describe a component or event that stands out regarding PD related to
algebra in your school/district. Describe a memorable math related experience at a
PD/staff training given by a presenter/facilitator.

16. Would you like to share anything else about PD related to algebra/math or in
general that I have not asked you?

131
APPENDIX C
INTERVIEW PROTOCOL FOR ASSISTANT SUPERINTENDENT OF
INSTRUCTION

1. Regarding students’ math performance in 5
th
grade related to algebra readiness,
are there any formal or informal goals for what the district is trying to accomplish?
- What are these goals?
- What do you aspire to? In what time frame?
- How will you/the school/the district know if it is successful?
- How big is the gap between where you are now and where you aspire to
be?
- What is keeping the school/district from achieving perfect success?

2. How does the assistant superintendent at the school district support the teachers in
being more prepared to teach algebra skills in 5
th
grade?

3. Please describe how your school district uses professional development (PD) to
help teachers with their algebra instruction in 5
th
grade.

4. How do you address concerns with teacher capacity as it relates to algebra/math
instruction? What strategies do you have in place to support algebra /math
instruction in 5
th
grade?

5. Who determines/organizes the type of PD that is offered by the district? By the
schools?

6. Do you attend any of the PD for math/algebra offered by the district or the
schools? What is your role in your school district in relation to PD (attending,
facilitating, planning the PD)?

7. Is PD/staff training in math instruction a priority at the school district? Why or
why not?

8. What kind of PD does your staff participate in to build teacher capacity in algebra
skills for 5
th
grade teachers? (when, how often, duration)

9. What does the school district doing to foster collaboration between the teachers
and between teachers and administration? Is the PD in algebra/math continuous?

10. Do you feel that the PD that relates to 5
th
grade math/algebra is adequate and
sufficient with what the elementary schools and the district currently offer? Why or
why not?

132

11. Students may remember their math skills for a longer period of time and may be
able to apply these skills in various contexts if they are taught using a conceptual
approach. Does the PD in algebra focus on teaching math from a conceptual
approach?

12. Sometimes teachers recognize that they need assistance teaching some math
topics. What do you do to create a climate to ensure that you engage all teachers and
school principals in PD that relates to algebra when it is offered by the school/district
or by seeking the math coaches on campus?

13. Please describe how the school district uses student data to guide PD in algebra
skills for 5
th
grade.

14. How does the school district evaluate PD in algebra, in general? How are
teachers held accountable for what they learn in math/algebra PD? Are evaluations
anonymous?

15. What do you believe is necessary for PD/staff training to be effective in
preparing the 5
th
grade teachers and the students to teach and learn algebra skills?

16. What role does the district expect the school principals to play in PD as it relates
to math/algebra ? Who determines the math/algebra PD/staff training that will be
offered at the elementary schools for fifth grade?

17. What percentage of PD/staff training offered by the school district is focused on
math/algebra at the elementary grade levels? In particular, in 5
th
grade?

18. If possible, what would you change, if anything, about the way your school
district conducts PD for algebra readiness? Please explain.

19. Please describe your thoughts on the budget for PD/staff training in your school
district in general and specifically as it relates to algebra/math staff training.

20. Describe a component or event that stands out regarding PD/staff training related
to math/algebra in your school district?

21. Would you like to share anything else about PD related to algebra/math or in
general that I have not asked you?

133
APPENDIX D
INTERVIEW PROTOCOL FOR MATH COACH
1. How has the performance of 5
th
grade students changed in algebra/math over the
past several years in your school? How has the school/district addressed these
changes in the students’ math performance? Was there any success with these
efforts?
Do they continue to this day-or what happened to the efforts?

2. Regarding students’ math performance in 5
th
grade related to algebra, are there
any formal or informal goals for what you or the district are trying to accomplish?
- What are these goals?
- What do you aspire to? In what time frame?
- How will you/the school/the district know if it is successful?
- How big is the gap between where you are now and where you aspire to
be?
- What is keeping the school/district from achieving perfect success?

3. Please describe how your school/district use professional development (PD) to
help teachers with their algebra/math instruction in 5
th
grade. Who is in charge of
planning the PD, how often it occurs and when, duration per PD, activities, etc.?

4. Have you experienced an emphasis in teaching algebra skills during PD?

5. What is your role in your school/district in relation to PD (attending, facilitating,
planning, evaluating, other) in general and as it relates to algebra? Explain. Is
PD/staff training in math instruction a priority at the school/district? Why or why
not?

6. What kind of PD have you participated in to build your capacity in algebra/math
skills as a 5
th
grade teacher? (what, when, how often, duration, etc.) Is there
collaboration between the teachers and between teachers and administrators? Is the
PD in algebra/math continuous?

7. Describe a memorable math teaching strategy (related to algebra readiness if
possible) shared during a PD by a presenter/facilitator. What was most effective
about the experience? Were you able to implement something that you learned from
the presenter into your own classroom during math instruction? Was it easy or
difficult? How practical are the PD ideas that focus on algebra readiness skills for
you to implement in the classroom? Explain.

134
8. What do you believe is necessary for PD to be effective in preparing you and the
students to teach and learn algebra skills?

9. If possible, what would you change, if anything, about the way your
school/district conducts professional development for algebra readiness? Please
explain.

10. Describe a component or event that stands out regarding PD related to algebra in
your school/district.

11. What is the school’s/district’s plan for planning PD and following up with PD
related to algebra? How does the school/district evaluate the PD in algebra/math?

12. Please describe your thoughts on the budget for PD/staff training in your
school/district in general and specifically as it relates to math PD (algebra).

13. How does the administration at the school/district support the teachers in being
more prepared to teach algebra skills in 5
th
grade?

14. Students may remember their math skills for a longer period of time and may be
able to apply these skills in various contexts if they are taught using a conceptual
approach. Does the PD in algebra/math focus on teaching math from a conceptual
approach? If not, would you prefer that it did? Why?

15. Do you feel that you have adequate training as a math coach to assist the 5
th

grade teachers with algebra/math skills?

16. How much training have you received in algebra/math? How often does this
training occur?

17. How often do 5
th
grade teachers come to you for math assistance? What types of
assistance to they often seek?

18. How do you address concerns with teacher capacity as it relates to algebra/math
instruction? What strategies do you have in place to support algebra/math instruction
in 5
th
grade?

19. Would you like to share anything else about PD related to algebra or in general
that I have not asked you?

Abstract (if available)

Abstract The purpose of this study was to examine the extent to which one Southern California school district supported the fifth-grade teachers’ capacity to teach their students concepts and skills related to algebra. Moreover, the study furthered evaluated the school principal’s perception and fifth-grade teachers’ experiences to determine whether the professional development offered by the district supports the fifth-grade teachers in gaining math knowledge and effective teaching strategies to teach algebra content.

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Asset Metadata

Creator Morales, Eduardo (author)

Core Title An examination of professional development in algebra for fifth-grade teachers

School Rossier School of Education

Degree Doctor of Education

Degree Program Education (Leadership)

Publication Date 04/19/2011

Defense Date 03/22/2011

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag algebra,education,fifth-grade teachers,math,OAI-PMH Harvest,professional development

Place Name California (states), USA (countries)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Garcia, Pedro E. (committee chair), Castruita, Rudy M. (committee member), Rousseau, Sylvia G. (committee member)

Creator Email emorales@elcamino.edu,mathmorales@yahoo.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m3760

Unique identifier UC1305343

Identifier etd-Morales-4489 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-455640 (legacy record id),usctheses-m3760 (legacy record id)

Legacy Identifier etd-Morales-4489-0.pdf

Dmrecord 455640

Document Type Dissertation

Rights Morales, Eduardo

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu