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A case study of the Math Matters professional development program in one elementary school

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A CASE STUDY OF THE MATH MATTERS PROFESSIONAL
DEVELOPMENT PROGRAM IN ONE ELEMENTARY SCHOOL
by
Cara L. Kopecky
A Dissertation Presented to the
FACULTY OF THE ROSSIER SCHOOL OF EDUCTION
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF EDUCATION
May 2005
Copyright 2005 Cara L. Kopecky
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UMI Number: 3180442
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ACKNOWLEDGEMENTS
A special thanks to my number one supporters, my family, for your constant
love and persistent support. To my loving husband, Michael, for patiently
encouraging my seemingly endless pursuit for more degrees. My mother, Mary, for
being a phenomenal role model who has inspired me to begin on this educational
journey. Thank you to my father, Tom, for his sharing of knowledge and experience
that aided greatly in this project.
Moreover, I would like to thank my chairperson, Dr. Dennis Hocevar for
encouraging and guiding me every step of the way. I would also like to thank Dr.
Carl Cohn who has provided thoughtful advice and assistance throughout this
process. To all of those who have helped make this accomplishment possible, I am
exceptionally grateful.
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TABLE OF CONTENTS
ACKNOWLEDGEMENTS................................................................................... ii
LIST OF TABLES.................................................................................................. v
LIST OF FIGURES................................................................................................ vi
ABSTRACT .......................................................................................................... vii
Chapter Page
1. PROBLEM STATEMENT........................................................................ 1
Background........................................................................................... 4
Data Analysis........................................................................................ 9
2. LITERATURE REVIEW........................................................................... 11
Introduction........................................................................................... 11
History of Professional Development................................................. 13
Current Practices.................................................................................. 15
Content Knowledge.............................................................................. 17
Coaching............................................................................................... 18
Length of Professional Development.................................................. 19
School Improvement through Professional Development................ 20
Further Research.................................................................................. 21
Conclusions........................................................................................... 24
3. METHOD..................................................................................................... 26
Design Summary.................................................................................. 26
Instruments and Procedures................................................................ 26
Questions for Teachers............................................................... 28
Questions for Administrators........................................................ 29
Questions for Coaches.................................................................... 30
Questions for Students................................................................... 31
Questions for Parents..................................................................... 31
4. RESULTS.................................................................................................... 33
Research Questions.............................................................................. 33
Teacher Knowledge............................................................................. 33
Perceptions of Mathematics................................................................ 35
Long-term, Ongoing Professional Development............................... 36
Inclusion of Staff Members.................................................................. 38
Plains Elementary Student Results...................................................... 40
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Chapter Page
Results from Comparable Schools...................................................... 42
Conclusions........................................................................................... 50
5. SUMMARY AND RECOMMENDATIONS................................................ 53
Conclusions........................................................................................... 55
T eacher Level F actors.......................................................................... 55
Content Knowledge.............................................................................. 56
Instructional Strategies......................................................................... 57
Coaching............................................................................................... 57
Limitations............................................................................................ 58
Further Research.................................................................................. 59
REFERENCES......................................................................................................... 62
APPENDIX .......................................................................................................... 70
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V
LIST OF TABLES
1. Demographics of Plains Elementary and District..................................... 5
2. Effect Sizes for Training Outcomes by Training Components............... 19
3. Pre/Post Scores on Effectiveness Survey.................................................. 37
4. Percentage of Students Achieving Standards at Plains Elementary
1999-2002............................................................................................. 40
5. 4th Grade New Standards Exam Results in Mathematics......................... 42
6. Percentage of Students Achieving Standards at Plains Elementary
1999-2002............................................................................................. 54
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LIST OF FIGURES
1. Demographics of Plains Elementary and Providence District.............. 6
2. Pre/Post Scores on Effectiveness Survey................................................ 37
3. Percentage of Students Achieving Standards at Plains Elementary
1999-2002.......................................................................................... 41
4. Demographics of Sunnyvale Elementary............................................... 42
5. Demographics of Wayside Elementary.................................................. 43
6. Demographics of Alta Elementary.......................................................... 43
7. Demographics of Plains Elementary....................................................... 44
8. New Standards Exam Mathematics Scores in Problem Solving 45
9. New Standards Exam Mathematics Scores in Math Skills................... 45
10. New Standards Exam Mathematics Scores in Math Concepts 46
11. Plains Elementary New Standards Exam Scores................................... 47
12. Mean Test Scores of Students at Plains Elementary and
Comparable Schools in Math........................................................... 48
13. Percentage of Students at Pains Elementary Meeting or
Exceeding Standard 1999-2002....................................................... 48
14. Percentage of Students Meeting or Exceeding Standard in
Reading Analysis.............................................................................. 49
15. Percentage of Students Meeting or Exceeding Standard in
Basic Understanding......................................................................... 49
16. Percentage of Students Meeting or Exceeding Standard in
Writing Effectiveness........................................................................ 50
17. Percentage of Students Meeting or Exceeding Standard in
Writing Conventions.........................................................................
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ABSTRACT
This case study investigated whether the implementation of a long-term, on
going professional development program, Math Matters, which focused on
increasing content knowledge in mathematics and provided teachers with content
and classroom management techniques, affected student achievement on the New
Standards Review exam. The staff members who participated in the training
represented grades Kindergarten through 5th grade at Plains Elementary School in
Providence, Rhode Island.
In order to evaluate the effects of Math Matters at Plains Elementary, three
sources of data were collected: (a) yearly test scores on the New Standards Review
Exam, (b) pre/post surveys, and (c) interviews and classroom observations. The
following research questions were assessed: (a) Did an increase of a teacher’s
mathematical knowledge improve student achievement in mathematics?, (a) In what
ways did a long-term, ongoing, professional development improve a mathematics
program within the school?, and (c) Did the inclusion of all staff members in the
professional development contribute to the effectiveness of the mathematics
program?
The New Standards Review data were collected over a 4-year period, to
coincide with the school’s participation in the Math Matters training. The New
Standards Review exam was administered to students in Grades 2-5 in spring 1999,
spring 2000, spring 2001, and spring 2002. The mathematics portion of the test
includes three parts: (a) math skills, (b) problem solving, and (c) math concepts.
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From 1998 to 2002, the students at Plains increased a total of 40% in math skills,
47% in math concepts, and 40% gain in problem solving on the New Standards
Exams. These gains were not noted in any of the three comparison schools.
The participant surveys were given as a measurement to gauge overall
changes in the mathematics program. Through these surveys, it was reported that the
mathematics program at Plains Elementary improved by approximately 40% over the
course of the training and implementation of Math Matters.
In the course of classroom observations and staff interviews, it was revealed
that participants noted that the professional development resulted in changed
attitudes, improvement in content knowledge, and their participation in the training
was a significant part in the school’s success throughout the years.
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CHAPTER 1
PROBLEM STATEMENT
The purpose of this case study is to explore social, educational, and cultural
characteristics of the implementation of Math Matters professional development that
may contribute to teacher learning and student achievement. An evaluation of the
factors that lead to success for our students will not only help our students, but it will
also help schools and districts that are looking for long-term change. This case study
will lead professional developers to a place where they can create ongoing, effective
professional development for their teachers and schools. Results from such an
evaluation will lead our leaders to a concise formula for change.
Professional development is defined here as the knowledge, skills, abilities,
and the necessary conditions for teacher learning. The terms professional develop
ment, staff development, and professional learning are used synonymously in this
paper.
This study raises and addresses questions such as: What should be the
primary characteristics of professional development? What types of professional
development models are being used in successful schools? How does ongoing
professional development affect student learning? These questions will frame the
investigation into how this particular ongoing professional development affects
teacher and student learning.
Over the past few decades, our schools have gone though various changes.
These changes have included statewide and national initiatives that were created to
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improve the achievement of all students. Despite these initiatives, very little has
changed in terms of the overall success of schools.
In 1981, Secretary of Education T. H. Bell created the National Commission
on Excellence in Education to examine the quality of education in the United States.
A few years later, the results of this study was given in a report called A Nation at
Risk. The Commission stated its conclusions in brief but dramatic terms:
If an unfriendly power had attempted to impose on America the mediocre
educational performance that exists today, we might well have viewed it as
an act of war. As it stands, we have allowed this to happen to ourselves. We
have even squandered the gains in achievement made in the wake of the
Sputnik challenge. Moreover, we have dismantled essential support
systems which helped make those gains possible. We have, in effect, been
committing an act of unthinking, unilateral educational disarmament, (p. 5)
One of the major recommendations in the report was that the teacher
profession should include high standards for preparation and professional growth.
The Commission saw the educational system in our country as mediocre. Inequities
were found in expectations of students, time spent on schoolwork, and expenditures
on classroom materials. Recommendations were made to ratify these inequities, but
it was later found that the nation had not met the assigned goals from the report. It
has been twenty years and schools have made little progress.
At this time, new legislation was being created in attempt to develop a
successful school system. This legislation became known as the Goals 2000:
Educate America Act, which became law in 1994 and was amended in 1996. It
represented a vast approach for ‘improv[ing] student learning through a long-term,
broad-based effort to promote coherent and coordinated improvements in the system
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of education throughout the Nation at the State and local levels’ (Goals 2000:
Educate America Act, Title III, Sec. 302). Goals 2000 supports State efforts to
develop clear and rigorous standards for what every child should know and be able
to do, and supports comprehensive State— and district-wide planning and imple
mentation of school improvement efforts focused on improving student achievement
to those standards. It required implementation of significant changes aligned to high
standards and improved performance. Both State and Local Initiatives were passed
to increase the achievement of students under the new rigorous standards. Despite
the enormous amount of funding and time given to make students successful, current
studies show that students across America are not meeting standards.
In order to ratify these deficiencies in our schools, in 2002, George W. Bush
signed new legislation called No Child Left Behind (NCLB), which requires stronger
accountability for results from our schools and teachers. Under NCLB, states must
report on how they are reducing the achievement gap and make sure that all students
achieve academic proficiency.
This new legislation, as the others did in the past, is once again aiming to
dramatically change the way we school our children. This includes incorporating
professional development for teachers, creating new standards for our teachers and
students, and incorporating overall school reform.
Despite the millions of dollars that are being spent on school reform,
particularly in professional development, many different models of professional
learning are being used throughout the country. Some districts are experiencing
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dramatic success through their model of professional development, while others are
feeling defeated by the long years of limited success.
Most educators accept professional development as a normal part of the
improvement of the teaching practice. Although, few appear to understand it well
enough to develop ongoing, comprehensive programs for improving teachers and
schools. This resistance may be a result of the perception that professional learning
is a “quick fix,” rather than a set of conditions and interrelationships that must be
considered.
It is imperative to take a look at which models of professional development
are creating change in teachers and schools. If particular characteristics of
professional development are found in effective schools and districts, then the
sharing of this knowledge will greatly reduce the amount of spending needed to
produce effective, high achieving schools and districts.
Background
The Providence School District, located in Providence, Rhode Island, is
compromised of 54 schools and serves grade levels Pre-Kindergarten through 12th
grade. Providence School District had a student population of 26,937 in 2002. The
district employs 1,551 teachers, 542 instructional aides, 58 Guidance Counselors, 3
Librarians, 13 District-level administrators and 70 School-level Administrators.
According to their website (www.providenceschools.org), the Providence School
District’s population is compromised of 74% White, 14% Hispanic/Latino, 8%
African Americans, 3% Asian, and 1% American Indian. Providence School District
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was compromised of 5,772 English Language Learners and 21 Summer Migrant
Students.
Plains Elementary School was built in 1954 and is located east of downtown
Providence. It is found in one of the oldest neighborhoods in the city. Plains
Elementary is a Title 1, K-5 school with a student population of 315, with 152
female students and 163 male students (National Center for Educational Statistics*
2002). Eighty-nine percent of the students at Plains qualify for free and reduced
lunch while the state average is 39% (Table 1, Figure 1). The demographics of
Plains Elementary and Providence District in 1999 consist of:
Table 1. Demographics o f Plains Elementary and District
Plains elementary Providence district
% of total % of total
Demographic group 2002 # of students population # of student population
Total population 318 100.0 14,017 100.0
African American 73 23.0 2,945 21.0
Asian 26 8.2 1,117 8.0
Latino/Hispanic 132 41.5 7,770 55.4
Native-American 2 0.6 117 0.8
White 85 26.7 2,060 14.7
Females 151 47.5 6,829 48.7
Males 167 52.5 7,186 51.3
ESL students 106 33.3 2,434 17.4
Students in bilingual program 0 0.0 1,289 9.2
Free and reduced lunch 286 90.0 11,214 80.0
Special education 79 24.8 2,127 15.2
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Demographic Composition of School and District
Free and Reduced Lunch
Students in Bilingual Program
1 % of T ota I
Population
District Males
White
Native-A merican
Latino/ Hispanic
African Am erican
0 .0 % 20 .0 % 40 .0 % 60 .0 % 80 .0 % 100. 0 %
Figure 1. Demographics of Plains Elementary and Providence District
Plains Elementary has a population that was trilingual (Spanish, Portuguese,
and English), and has a large number of special needs children. On the New
Standards Reference Exam (NSRE) in 1998, only 3.7% of the 4th grade students
were at or above standard (50th percentile) in Mathematical Concepts, 18.5% were at
standard for Mathematical Procedures and 1.9% were achieving in Problem Solving
on the New Standards Reference Exam. Plains Elementary NSRE scores were the
lowest in the district in mathematics. In addition, for years it had been one of the
lowest performing schools in the State.
In the spring of 1999, the principal of Plains Elementary learned of Math
Matters and recognized the need for professional development for the school staff.
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The staff at Plains Elementary agreed to begin with a two-week academy with Math
Matters in August, 1999. The academy was followed in January by a week to work
with the staff, model the Math Matters teaching strategies in the classrooms at each
grade level, and continue the professional development of the teachers in mathe
matics, management strategies, and cognitive coaching. This format was repeated in
the years 2000, 2001 and 2002.
The primary goals of Math Matters at Plains Elementary were to:
(a) strengthen the mathematics subject matter knowledge of elementary grade
educators; (b) improve the mathematics instructional effectiveness; (c) facilitate
administrative support in elementary grades mathematics curriculum and delivery of
instruction; (d) provide opportunities for teacher research, reflection, and collegial
interaction; (e) Assist in implementation of new instructional strategies in actual
classrooms at all grade levels, and (f) Evaluate effectiveness in improving teaching
and learning of elementary grade mathematics.
The teachers in the Math Matters program at Plains Elementary participated
in approximately 100 hours of professional development in their first year. During
the following years, the professional development consisted of 20-30 hours a year
observing classrooms, peer coaching, and continued to enhance the teachers’
professional growth in mathematics and effective teaching.
One of the primary goals of Plains Elementary School was to improve
student achievement through building and extending teacher knowledge through four
basic elements: (a) content knowledge, (b) pedagogical/research knowledge,
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(c) assessment, and (d) a structural environment that supported continuous learning
(Providence School District, 2003). The school community believes that research-
based professional development for teachers is essential for quality reform. The
deeper the level of understanding a teacher has of the subject matter and its
pedagogy, the better the teaching and learning (Ma, 1999).
Math Matters was a standards-based professional development program,
which provides comprehensive, long-term professional development focused on
increasing the staffs and students’ procedural skills, conceptual understanding, and
mathematical reasoning. Originally jointly funded by the National Science
Foundation and the California Department of Education, Math Matters is now
funded by WestEd, county offices and local districts.
Math Matters was designed to be used in schools where students are
underachieving to improve the mathematics achievement of all students. It
incorporated a balanced and comprehensive professional development program in
mathematics for teachers and administrators. The core elements of Math Matters
included: (a) professional growth in mathematics, (b) the transition of mathematics
learned in the trainings to the standards-based material used in the classroom,
(c) classroom management strategies that support the inclusion of all students,
(d) cognitive coaching in the content area to promote reflective practice,
(e) strategies that deal directly with mathematical content, and (f) sessions designed
for the professional development needs of the administrators as well as quarterly
meetings to update progress. Math Matters provide opportunities for teachers and
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administrators to talk about mathematics, practice teaching strategies, develop key
lessons, and collaborate with their peers.
This case study will shed light on the following research questions:
1. Did an increase o f a teacher’ s mathematical knowledge improve
student achievement in mathematics?
2. In what ways did a long-term, ongoing, professional development
improve a mathematics program within the school?
3. Did the inclusion o f all staff members in the professional development
contribute to the effectiveness o f the mathematics program?
Data Analysis
The documents that were needed to analyze were standardized test scores and
fieldwork and observation notes. Student test scores will be quantitative data and
were taken from Grade 4. The 4th grade achievement scores were evaluated in order
to gauge the progress of students over time. By using the 4th grade scores, this
evaluation was able to take a closer look at the changes in student achievement as the
students moved through the grade levels. Pre and post questionnaire data were
analyzed to see if the difference of means was significant.
The fieldwork and observation notes followed the Patton (1987) model as
follows (p. 149):
Step 1-Assemble the raw case data
Step 2-Construct a case record
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Step 3-Write a case study narrative
The observation procedures, also qualitative, also followed Patton’s model.
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CHAPTER 2
LITERATURE REVIEW
Introduction
The majority of researchers agree that standards alone are not enough to
improve the education of our children. There tends to be many highly visible efforts
to improve achievement of students, including reduction of class size, vouchers, and
short-term professional development. All of these efforts appear to be missing one
essential element for school reform: they do not fundamentally improve what
happens inside classrooms. Jerome Bruner made the same conclusion in his book,
The Culture o f Education (1996):
It is somewhat surprising and discouraging how little attention has been paid
to the intimate nature of teaching and school learning in the debates on
education that have raged over the past decade. These debates have been so
focused on performance and standards that they have mostly overlooked the
means by which teachers and pupils alike go about their business in real-life
classrooms-how teachers teach and how pupils learn, (p. 86)
Thus, the goal of improving teacher quality must be at the forefront of the
efforts to improve student achievement (Ball, 2000; Darling-Hammond, L., 2000;
Ma, 1999; Sutton & Krueger, 2002).
Successful professional development “not only makes teachers feel better
about their practice, but also reaps learning gains for students, especially in the kinds
of more challenging learning that new standards demand,” according to Linda
Darling-Hammond (2000, p. 597). Darling-Hammond and McLaughlin also stated
that research shows that the single most important factor in students’ learning is the
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expertise of their teacher. That is, a successful professional development program is
focused on improving mathematical knowledge and abilities of teachers in order for
them to teach this knowledge to their students. Other research such as studies from
National Research Council and the Regional Alliance for Mathematics and Science
Education concurred that what the teacher knows and does is the most important
influence of what students can learn.
In addition to improving content knowledge, researchers, such as Marzano
(2003) and Lieberman (1995) agree, “teachers must have opportunities to discuss,
think about, try out, and hone new practices” (p. 593) by taking new roles and
creating a culture of inquiry.
Overall, there are 10 major goals of professional development in mathematics
on which researchers primarily agree (California Department of Education, 2000).
They believe that a successful professional development:
1. Focuses on teachers as central to student learning, yet includes all
other members of the school community
2. Focuses on individual, collegial, and organizational improvement
3. Respects and nurture the intellectual and leadership capacity of
teachers, principals, and others in the school community
4. Reflects the best available research and practice in teaching, learning,
and leadership
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5. Enables teachers to develop further expertise in subject content,
teaching strategies, uses of technologies, and other essential elements in teaching to
high standards
6. Promotes continuous inquiry and improvement embedded in the daily
life of schools
7. Is planned collaboratively by those who will participate in and
facilitate that development
8. Requires substantial time and other resources
9. Is driven by a coherent long-term plan
10. Is evaluated ultimately on the basis of its impact on teacher
effectiveness and student learning; and this assessment guides subsequent
professional development efforts (p.34).
As Fullan (2001) stated, it is more than “mere articulation of one element of a
big system to another element. It’s the recognition that elements dynamically
interact” (p. 11).
History o f Professional Development
The launching of Sputnik in 1957 created a focus on America’s school. The
U. S. government began focusing their attention on its own teachers as their first
defense against Soviet powers. The National Science Foundation was the first
beneficiary to federal funding designed to increase subject matter knowledge of
teachers. The focus has shifted at such a high degree to increasing content
knowledge that pedagogical knowledge was all, but forgotten in teacher professional
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developments. Despite a massive infusion of funding for many different programs,
in conjunction with the push for teaching standards, these programs were not able to
show improvement in teaching or in school programs. Expectations of the success of
these programs came to a halt after Gross’ (1971) and Sarason’s (1971) major studies
showed failed implementation.
During the 1970s educators were at a crossroads. Many wondered if the
school system was capable of change. By the end of the 1970s, the effective schools
movement had brought forth some evidence that schools can make a difference, even
under trying conditions. By the 1980s, researchers such as Bruce Joyce and Beverly
Showers (1988) state, “The field of staff development is evolving gradually from a
patchwork of courses and workshops into a system ensuring that education
professionals regularly enhance their academic knowledge and professional
performance” (p. 1).
In the 1990s the results of Third International Mathematics and Science
Study (TIMMS) was thought to be evidence of the ineffectiveness of American
education. The study involved a large-scale, multinational comparison of the
educational systems of 41 countries. Overall, U. S. 4th grade students performed
moderately well compared to other fourth graders worldwide. U. S. 8th and 2nd
grade students performed quite poorly. Many of the reports of TIMMS (Schmidt,
McKnight & Raizen, 1997; U. S. Department of Education, National Center for
Educational Statistics, 1998) stated that the results from TIMMS are evidence of the
dire need for reform in American education.
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Given the criticism that public education has endured over the past few
decades, many researchers believe that public education is beginning to flourish. As
Marzano (2003) states, “Public education is at the dawn of the best of times ....
Schools can have a tremendous impact on student achievement if they follow the
direction provided by the research” (p. 4).
Current Practices
The current practice in professional development in mathematics varies
greatly. Much of the early literature focuses on methods of staff development that
follow a "training" model: short-term, standardized sessions designed to impart
distinct skills and techniques. Too often the results of these sessions have fallen
short of expectations. It has been shown that there has been little transfer of
information into the classroom, despite the high expectations of trainers, reformers,
and the teachers themselves (Joyce & Showers, 1996).
Unfortunately, many of the teachers in the U. S. are severely unprepared to
teach challenging mathematics. This problem is prevalent not only in junior high
and high school, but also in elementary schools where students must learn
fundamental concepts of algebra, geometry and statistics. Without being prepared
for challenging mathematics, teachers are unable to create students who are critical
thinkers and problem solvers. These problems are most in part to insufficient
professional development (National Research Council, 2001). In effect, teacher
development must be seen as a continuum of programs and professional experiences
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that teachers can move through an extended period of time. There are a few models
of professional development that follow this type of research.
One of the largest leaders in professional development in mathematics is the
California Math Project, a section of the California Subject Matter project. There
were nine California Subject Matter Projects (CSMP) supported by the State of
California and administered by the University of California Office of the President.
All the CSMPs shared common goals and program elements in providing
professional and leadership development for K-12 teachers throughout the State.
The California Math Project’s goals are to enhance the quality of California’s K-12
mathematics education by increasing teachers’ mathematical knowledge and
pedagogical content knowledge, and by supporting and developing leadership skills.
The goal was to deepen teachers’ content knowledge of mathematics so that they are
better able to provide comprehensive mathematics instruction that is consistent with
the California Academic Content Standards and the Mathematics Framework and
increase student performance on state-mandated assessments (CSU Math Project,
2003).
Another leader in mathematics professional development was the Institute for
Learning, established at the University of Pittsburgh in 1995. The Institute provided
educators with current research about the learning process and resources to affect
student achievement. The Institute has identified nine principles that are
instrumental to teaching and learning. They are: academic rigor in a thinking
curriculum, accountable talk, clear expectations, fair and credible evaluations,
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learning as apprenticeship, organizing for effort, recognition of accomplishment,
socializing intelligence, and self-management of learning.
Content Knowledge
The idea that adequate knowledge of mathematics content and pedagogy is
necessary to teach a diverse group of learners resonates throughout the research done
over the years. The importance of teaching teachers content is best quoted by the
report by the National Commission on Teaching and America’s Future (1996) called
What Matters Most: Teaching for America’ s Future. The report states,
What teachers know and do is the most important influence on what
students learn .... Like other professionals, teachers have an expert
knowledge base which guides their practice. Unlike other professionals,
however, most teachers revise their pedagogical content knowledge— the
synthesis of their content area knowledge and general pedagogy— in
isolation. Without adequate access to the meaningful professional
development programs, collegial collaboration, or feedback on their own
performance, it is difficult for teachers to respond to the new
mathematics and science content and theories of learning included in
state and national standards, (p. 6)
Teacher subject matter knowledge is also frequently cited as essential to
student achievement (Ball, 1990; Joyce & Showers, 1996; Ma, 1999; Sparks &
Hirsh, 1997). Elmore (2002) also agrees that effective professional development is
“focused on the improvement of student learning through the improvement of the
skills and knowledge of the teacher” (p. 6). Little (1997) also suggests that the “test
of effective professional development is whether teachers and other educators come
to know more about their subjects, their students, and their practice, and to make
informed use of what they know.” Thus, researchers believe that teachers with deep
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knowledge of mathematical content are able to teach rich mathematical content to all
students. Darling-Hammond and McLaughlin (1995) concurs by affirming,
“Research shows that the single most important determinant of what students learn is
the expertise of their teacher” (p. 75).
Coaching
The impact and effectiveness of professional development seminars have
been evaluated by educators and researchers (Fullan, 1995; Joyce & Showers, 1995;
Showers, Joyce, & Bennet, 1997). Efforts to implement new teaching strategies are
found to be more successful when coaching is part of the training.
In many current professional development models, coaching is an integral
piece. The coaching of teaching happens in the workplace following the initial
training. Coaching often offers support for teachers attempting to implement a new
skill. It can also provide feedback on the technical aspects of pedagogy, and on the
congruence of practice and ideal performance. Coaching is typically put into place
to build a community of learners who continuously engage in the study of their craft.
Joyce and Showers (1988) did a study on teacher training and found that
when the professional development simply provided information, training outcomes
were significantly less than when the training included theory, demonstrations,
practice, feedback and coaching (Table 2).
In fact, without coaching, the teachers were not able to transfer the training in
formation into their classrooms. When in-class coaching was added to theory,
demonstration, practice, feedback treatment, skill continued to rise. In their study,
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Table 2. Effect Sizes for Training Outcomes by Training Components
Training outcomes
Training component Knowledge Skill Transfer of training
Information 0.63 0.35 0.00
Theory, demo., practice,
feedback, coaching
2.71 1.25 1.68
Joyce and Showers (1988) concluded that teachers can acquire new knowledge and
skill and use it in their instructional practice when provided with follow up coaching
in the workplace.
Length o f Professional Development
Many researchers believe that long-term, systematic learning is crucial to
sustained professional development (Little, 1989). In Teaching and Learning
Mathematics, Johnson (2000) believes that in conjunction with the comprehensive
approach of teacher training, long-term improvement only happens when the
professional development is given over a longer period of time. Johnson states,
“One time inservice workshops are unlikely to produce either significant or long
term change in mathematics teachers” (p. 64). It is believed that one time inservice
workshops can bring forth awareness and serve as a good beginning to long term
professional development, but are inadequate in terms of meeting the needs of
educators (Fullan, 1991; Joyce & Showers, 1980; Little, 1993). In order to create
improved student achievement, quality professional development must result in
teachers working with students to internalize the learning, which requires a lengthy
period of time (CDE, 2000).
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School Improvement through Professional Development
Michael Fullan (1993) states, “The moral purpose [of education] is to make a
difference in the lives of students regardless of background, and to help produce
citizens who can live and work productively in increasingly dynamically complex
societies” (p. 4). Many agree with Fullan that moral purpose cannot just be stated,
but it must be accompanied by strategies to implement it, and those strategies are
believed to be the leadership actions that motivate people to reach the goal of school
improvement.
Since the process of school improvement is a systemic progression (Fullan &
Miles, 1993), and professional learning changes the organizational context in which
it takes place (Sparks & Loucks-Horsley, 1990), it is important to take a compre
hensive approach that includes professional development for principals, assistant
principals, and administrative and resource staff at school and district levels. This
type of commitment from a leadership team is considered essential by researchers
such as Guskin (1996). He believes that such a commitment is required for any
successful change because of the time it takes for the change to become successful
and because of the large nature of school-wide improvement. Sutton and Krueger
(2002) agree that a successful educational system has student outcomes as a primary
focus and the necessary support is given by the administration so that all students
and teachers can achieve.
In order for the professional development to work, it has to be connected to
the practice of improvement. Elmore (2002) sums it up best by pointing out that,
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21
Practice must entail attention to what knowledge and skill educators require
to improve student learning and how people come to master that knowledge,
which incentives encourage people to engage in the difficult and uncertain
process of changing their teaching and administrative practice, and what
resources and capacities are required to support the practice of improvement.
(p. 28)
Professional development will succeed, according to Elmore, only when it engages
teachers and administrators in acquiring knowledge and skills they need to solve
problems and meet expectations for high achievement.
Further Research
Most of the current research addresses the need for long-term, ongoing
professional development. However, there is little research on the connection
between the professional development model and the effects it may have on student
standardized test scores. Since this case study will look at the changes in student test
scores, it would be valuable to look into the possibility of a connection between
long-term change in student achievement on standardized tests and the model of
professional development the staff participated. If there are correlations found
between professional development models and student achievement on standardized
test scores, some questions that need to be answered might be:
1. Are there other variables such as years of experience of the teachers
who participated in the trainings?
2. What is the rate of turnover of staff and students?
3. Are there differences in primary versus intermediate student
achievement?
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23
Overall, most of the research has focused on the characteristics of effective
professional development. What still needs to be done is an examination of what
happens after the professional developers have left the school or district. Questions
such as:
1. What happens to schools that have shown increases in student
achievement and are left to continue the job on their own?
2. Are gains in achievement only found when the trainers are still
working with a school?
3. What can schools do to maintain their achievement, both by the
students and by the teachers?
4. Is the current training adequate to leave schools to maintain on their
own?
5. How can school account for any loss in achievement after the trainers
have left?
In summary, most of the research has investigated what happens at the
beginning and middle of the professional development. The primary piece that is
missing is what teachers, schools, and districts can do after the professional
development is completed. Addressing this subject will allow our schools to
maintain the same success they create while they participated in the professional
development program.
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Conclusions
This review of the literature suggests a number of factors that should be
considered in a case study of a professional development program in mathematics at
the elementary level.
First, the success of the students remains in the hands of their teacher.
Preparing our teachers with adequate mathematical content knowledge is one of the
primary goals in professional development. Without content knowledge, teachers
are unable to teach the mathematics curriculum with the depth that is necessary to
create successful problem solvers and critical, self-directed thinkers.
Second, without coaching as an integral piece of the professional
development model, teachers are unable to implement their new knowledge into the
classroom. Researchers such as Joyce and Showers (1995) have done numerous
studies showing that professional development programs that have succeeded have
done so simply because the coaching and reflection piece was included. Giving
teachers and staff members’ feedback and support for their skills is an essential part
of a successful school and district.
Thirdly, much of the research shows that the long-term models of
professional development tend to show a higher rate of student achievement over
time. One day inservices are no longer though of as effective means to train
teachers. It is only when teachers are able to perfect their craft over time that they
show large levels of growth.
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Lastly, researchers who focused on school-wide change believe that change
happens when all parties are involved and have a stake in the process of change.
That is, if the entire staff participates and believes in the professional development,
change will occur at a more efficient rate. Staffs that are vested in the process of
change and are participating in the process are more likely to show improvements at
their school or district.
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CHAPTER 3
METHOD
Design Summary
The case study was conducted on Plains Elementary School after their long
term participation in the Math Matters professional development training. During
the 4 years, the training focused on increasing the participants’ mathematical
knowledge and classroom management techniques connecting that knowledge to the
current curriculum. All teachers participated, including the science and art teachers,
the principal, vice principal, librarian, and other staff members who interacted with
the students at Plains Elementary School. For 3 of the 4 years, the staff participated
in 2 weeks of professional development which was followed by 1 week 4 months
later. The 4th year the staff had one week followed by 4 days of professional
development 4 months later. In addition, the staff was supported with unsupervised
trainings and weekly coaching sessions.
The participants in the case study was all the students, teachers, principals,
and other staff members that participated in the Math Matters training between 1999
and 2002 at Plains Elementary School. Parents and other community members were
part of the interview process and their responses will be an important part to the
overall picture of the success of the school.
Instruments and Procedures
Overall, the data for this case study was evaluated through triangulation.
These multiple sources were interviews, test scores, and pre/post questionnaires.
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The use of the data from this particular elementary school shed light on possible
trends and changes that occur after participation in a long-term professional
development program.
The first set of data was from the New Standards Reference Examinations
(NSRE). These criterion referenced tests were given to students in Grade 4 that
attended Plains Elementary School in the years 1998, 1999, 2000, 2001 and 2002.
The mathematics test included three main components: Problem Solving, Math
Concepts, and Math Procedures. The tests were not graded on a scale, rather, for
fourth grade mathematics, there are “standards” that have been adopted by the R. I.
Board of Regents for Elementary and Secondary Education. These standards
describe the quality of work expected at 4th grade. The score for each test
represented the percentage of students in that grade level that have met or exceeded
the state standard. The data from these tests were correlated and compiled
throughout the years. These scores were collected from the Providence School
District website. These scores were reformatted into new charts and correlated by
the year and subject (i.e., problem solving, math skills and concepts). The New
Standards Reference Exam scores were used to gauge the ongoing achievement
levels of students at Plains Elementary.
In order to get a broader view of the effects of Math Matters, the teachers
completed two identical questionnaires (Appendix), one as a pre-Math Matters
survey and the other served as a post assessment of their math program. The results
of the questionnaire were evaluated as quantitative data. Responses to each question
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had a score of one to three. The scores of each section were compiled and entered
into a data table. This questionnaire yielded nine measures, one for each of the
sections: (a) Assessment, (b) Instruction, (c) Instructional Time, (d) Instructional
Resources, (e) Instructional Grouping and scheduling, (f) Classroom Management,
(g) Professional Development, (h) Administrative Practices, and (i) Community
Involvement. Each of the constructs is described in the questionnaire in the
Appendix.
The pre/post scores will be combined into two groups, one score for pre-Math
Matters, and the other score for post-Math Matters. Each of the sections had a
numerical average for all the pretest scores and a separate average for the post test
scores. The two scores were compared in each area to evaluate whether the staff felt
their math program improved after the implementation of the program.
The third piece of the data was staff interviews. In order to have accurate and
comprehensive data, these interviews were held at Plains Elementary and included
teachers, principals, school volunteers, and parents. The interviewees were asked
various questions, one-on-one in-person, relating to the effectiveness of Math
Matters at their school and how the professional development affected their own
content knowledge and teaching/leadership skills. The interviewees were asked the
following questions:
Questions for Teachers
1. How long have you been involved in Math Matter si
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2. To what extent have you implemented the different parts of Math
Matters?
3. Which Math Matters Management techniques and strategies have you
found most effective?
4. How do you feel the Math Matters head problems helps the students
learn?
5. How do you feel the daily two problem helps the students learn?
6. Describe the support have you received in order to implement
changes in you r classroom.
7. What has been the impact of Math Matters on your attitude? Content
knowledge? Skills?
8. As you consider these changes in your classroom practices, did
student performance in mathematics change?
Questions for Administrators
1. Talk to me about the impact that Math Matters has had on this school.
2. What is the overall feeling about Math Matters techniques and
strategies?
3. How do the teachers see Math Matters activities as relevant to their
needs?
4. What support have you received in order to implement changes in
your school?
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5. What has been the impact of Math Matters on your attitude? Content
knowledge? Skills?
6. Describe how has Math Matters has changed your instructional
leadership practices, and what impact has that had on teachers’ attitudes, knowledge,
and skills.
Questions for Coaches
1. How has Math Matters been implemented in classrooms?
2. What is the overall feeling about Math Matters techniques and
strategies?
3. What is the overall feeling about the head problems and two problem
approaches?
4. What Math Matters activities best address your grade level math
standards?
5. Describe the support have you received in order to implement
changes in you r school.
6. What has been the impact of Math Matters on your attitude? Content
knowledge? Skills?
7. How has Math Matters changed your coaching practices?
8. As you consider these changes in your coaching skills/practices, what
was the impact on teachers’ attitudes? Knowledge? Skills?
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Questions for Students
1. What activities to you most enjoy during your math time?
2. If you could change something about math time, what would you
change?
3. When doing homework at home, whom do you talk about what you
are doing in math? Your parents? Brothers or sisters? Another adult?
4. Name two things in math that you can do really well.
Questions for Parents
1. What do you think about the school’s math program?
2. Have you noticed any changes in your child’s learning in the last few
years?
3. What have you noticed about your child’s attitude about
mathematics?
4. What types of things would you still like to see happen at the school
in the future?
The interviews were taped and field notes were taken during and after the
interview sessions. Complete transcripts were taken and recorded. This data were
analyzed for common themes between teachers, administrators, and parents as to
how they attribute the success of the students in mathematics throughout the 4 years.
Classroom observations was also be done at Plains Elementary in various
classrooms. Throughout visitations to the school, students were interviewed about
their experiences at the school on an informal basis. Students and other staff
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members were able to communicate their experiences and concerns during this
observation period. Throughout the observation period, notes were taken and
compiled. This data were collected in order to get a global view of the school culture
and environment. Commonalities and differences were found between classroom
instruction and the intentions of the professional development program. Overall,
the classroom observations were able to show whether or not the teachers are
implementing the program in their classrooms.
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CHAPTER 4
RESULTS
Research Questions
In this case study, three major research questions were posed:
1. Did an increase o f a teacher’ s mathematical knowledge improve
student achievement in mathematics?
2. In what ways did a long-term, ongoing, professional development
improve a mathematics program within the school?
3. Did the inclusion o f all staff members in the professional development
contribute to the effectiveness o f the mathematics program?
After reviewing the interviews, observations, and field notes, the four major
themes that emerged were: an increase of mathematical knowledge allowed the
teachers to improve their instruction, the training was able to change negative
perceptions of mathematics, the length of the professional development made a
positive impact on the students and staff, and the inclusion of all staff members made
a positive impact on the culture of the school.
Teacher Knowledge
Through staff interviews and classroom observations, it was shown that the
teachers who fully participated in the Math Matters training were more likely to
teach a math lesson that contained higher-level math than those who were just
beginning the training. In Math Matters classrooms at Plains Elementary, one would
observe the students working on content focused activities, such as head problems
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and a two problem approach, on a daily basis. Students were actively engaged
throughout the day and teachers appeared to be continuously evaluating students’
progress and achievement. The staff at Plains Elementary was also observed
conversing about various lesson plans and sharing new ideas with one another.
Plains created a school culture in which the teachers and the students were
consistently working to improve their math knowledge, and as a result, the math
program.
The entire staff at Plains Elementary School was trained in Math Matters in
1999 and continued to receive training through 2002. This training included
classroom teachers, librarians, art teachers, music instructors, assistants, vice
principal and principal. During the interviews, many of the staff members
commented on how their improvement of mathematical knowledge helped increase
their students’ achievement. For example, one teacher commented, “The content
knowledge made me look at things differently, knowing and teaching it, under
standing standards was a large part of the connecting the pieces together.” Another
teacher also stated that,
The content was at the adult level, but modeled in a way that made it clear
how we are going to implement this into our classrooms. I was thinking
about things that I had never connected before. The way we were taught
allowed me to become much more comfortable with the math content and
take chances. That is when I knew it would work with the kids.
Most staff members agreed that connecting their new content knowledge to their
instruction was the key piece of increasing their students’ achievement. “I know that
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since I’m thinking more mathematically, it is translating to the kids,” stated a staff
member.
Another consistent response was the acknowledgement that once the content
knowledge of the staff increased; the staff were able to connect that knowledge to
how they were going to teach a particular concept. For example, one teacher stated
that she was working on fractions with her students and had to stop the lesson
because she knew the mathematics was not correct in the text. She took that lesson,
investigated the problem with other staff members and then returned the following
day with a deeper understanding of the mathematics, which was then translated more
effectively to the students. She said, “I know enough (math content) to know if
something is not right, I work at it until I get it right. I have to attribute it to being a
part of Math Matters. I never had that before; I never cared if the math didn’t make
sense.”
Perceptions o f Mathematics
Many staff members stated that the training changed their attitude about
mathematics. One teacher stated,
It made math not become a four-letter word with me, math used to be a part
that I had to fit in to the day, now I look forward to it and is a larger part of
the day, {Math Matters) made me think of tearing out the pages differently. I
no longer just tear out the worksheets and hand them to the kids during math
time. I am more positive about math, children can relate to it and it is more
interesting. My math time use to be 45 minutes and is now an hour and it
goes by fast and it makes more sense for me and my students.
Overall, the improvement of content knowledge appeared to be a large piece
of the teachers’ successes. As one teacher commented,
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“I was always trying to find answers that no one was able to give me on a
regular basis, so I needed something to give those answers. I feel much more
capable than I did before (Math Matters) was a dream come true for me.
Long-term, Ongoing Professional Development
These data were collected was through a pre- and post survey of the staff at
Plains Elementary School (Appendix). Twenty-five staff members participated in
the survey. There were nine categories that the staff evaluated through the survey.
These categories were: (a) assessment, (b) instruction, (c) instructional time,
(d) instructional resources, (e) instructional grouping and scheduling, (f) classroom
management, (g) professional development, (h) administrative practices, and
(i) community involvement. These categories were taken from the California
Framework for Mathematics guidelines. There were various questions under each
section and these questions were each given a value from one to three, 1 = Not Very
Effective, 2 = Somewhat Effective, and 3 = Very Effective. The staff evaluated the
effectiveness of their math program according to these guidelines before Math
Matters and after Math Matters. The ratings given by each participant (1, 2, or 3)
were combined in each of the nine categories and given a total score. The total
scores for each category on the primary evaluation were compared with the total
scores on the final evaluation in each category (Table 3, Figure 2).
Overall, the survey showed in each section that was assessed, the math
program improved over the time Math Matters was implemented at Plains
Elementary. The largest improvement was seen in the area of instruction, which
improved by 55%. Instructional grouping and scheduling, and professional
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Table 3. Pre/Post Scores on Effectiveness Survey
Pre-Math Post-Math Total
Matters Matters possible % increase
Assessment 155 285 300 43
Instruction 130 295 300 55
Instructional time 200 340 375 37
Instructional resources 312 555 600 41
Instructional grouping and scheduling 70 140 150 47
Classroom management 168 220 225 23
Professional development 180 355 375 47
Administrative practices 235 405 450 38
Community involvement 110 150 225 18
□ P re-M a th M atters
■ P o st-M a th M atte rs
Figure 2. Pre/Post-Scores on Effectiveness Survey
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38
development also saw 47% gains over time. Assessment and Instructional resources
gained by 43% and 41%, respectively. Instructional time had an increase of 37%
and administrative practices were rated with a 38% increase. Community involve
ment saw the smallest increase, 18%. According to the surveys, the overall
mathematics program improved about 40% from the first to the second survey
(Figure 3).
In addition to the data collected for this case study, the state of Rhode Island
also conducted a study on the partnership of Math Matters and Plains. Its results
stated,
The data confirm that Plains Elementary’s mathematics initiative has been
successful. The Math Matters— Plains Elementary partnership has shown a
high degree of success in developing a long-term comprehensive program
which provides teachers with opportunities to build and extend their
mathematical understanding, discuss ideas with colleagues, practice effective
instructional techniques, implement coaching, and reflect on practice. Trends
indicate that the school will be able to meet its goal of becoming an
improving high performing school. (Infoworks, 2002)
Many of their findings were consistent to those found in this study.
Inclusion o f Staff Members
Interviews with staff and volunteers throughout the school showed that their
participation in the trainings throughout the years was a significant part of the
school’s success. The inclusion of all staff members, including nonteaching staff,
enabled the school to create a culture of learning for all students that was very unique
to Plains Elementary. For example, when the students were in line to go to the
bathroom, the aide would give them a head problem to be working on in the
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meantime. Or, when the students went to art class, their art teacher was gifted at
creating two problem approaches that taught proportions or color theory. No matter
where the students went throughout the building, every adult was trained using the
same techniques and strategies, thus using consistent procedures throughout the
school.
Through school observations, it was seen that one of the largest advantage of
the training of all of the staff was the overall use of classroom management
techniques throughout the building. When the students were in a different
classroom, the expectations were consistent and the vocabulary was also used in the
same manner. Each of the teachers shared an understanding of what classroom
management should look like and were all implementing the same techniques at the
same time throughout the school.
Another major component of the program’s success came with the
involvement of the entire staff. After interviewing various staff members who were
not classroom teachers, it was clear to see that their involvement in Math Matters
was an essential piece of the puzzle. For example, when the kids went to library,
their librarian used the same classroom management techniques their classroom
teacher used. As a result, the students were clear on teacher expectations, no matter
where they were in school. In addition, when the students went to resource, the art
or science teachers could successfully integrate content strategies (e.g., head
problems and two problem approaches) into their curriculum. The students were
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clear on what is expected from them and there was a culture of understanding that
resounds through the school.
The study of Plains Elementary was unique because the school followed the
model throughout the years and saw tremendous gains. Interviewees agreed that the
administrator at Plains was consistent and had a clear vision of what long-term
professional development looked like for her site. It was also apparent that the
administrator held high expectations for her teachers and the students and was
instrumental in the partnership with Math Matters and Plains Elementary.
Plains Elementary Student Results
Some of the most important pieces of data were the test scores of the students
from 1998-2002 (Table 4, Figure 3). Teachers and staff members stated in their
interviews that these gains in student test scores are attributed to the training they
received in Math Matters. From 1998 to 2002, the students at Plains Elementary
meeting standard increased a total of 40% in math skills, 47% in math concepts, and
40 percent gain in problem solving on the New Standards Exams (Table 4).
Table 4. Percentage o f Students Achieving Standards at Plains Elementary 1999-2002
1999 2000 2001 2002
Math skills 23% 43% 55% 63%
Math concepts 2% 19% 33% 49%
Problem solving 5% 19% 32% 45%
Since the New Standards Exam was a criterion referenced exam, the
“standard” was set and every student who took the exam was scored on whether or
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41
P e r c e n ta g e o f stu d e n ts that m et or e x c e e d e d
s ta n d a r d s
Math S k ills Math C o n c e p ts P r o b le m
S o lv in g
Figure 3. Percentage of Students Achieving Standards at Plains Elementary
1999-2002
not they met that standard. Percentages were taken from those students who met or
exceeded that given standard over the 4-year period in Math Skills, Math Concepts,
and Problem Solving (Figure 3).
Results from Comparable Schools
In order to control for the treatment, the Math Matters training, comparable schools
were compared to determine whether or not the increase of test scores was seen
across the district (Table 5 ) The schools that were used in this comparison were
Wayside Elementary, Sunnyvale Elementary, and Alta Elementary Schools. All
three of the comparison schools were similar to Plains in demographics throughout
the examined years (Figures 4, 5, 6 and 7).
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Table 5. 4th Grade New Standards Exam Results in Mathematics
1999 2000 2001 2002 Total Growth
Math Skills
Plains Elementary 23% 43% 55% 63% 40%
Wayside 25% 44% 25% 32% 7%
Sunnyvale 28% 37% 48% 43% 15%
Alta 20% 27% 13% 39% 19%
Math Concepts
Plains Elementary 2% 19% 33% 49% 47%
Wayside 22% 14% 6% 13% -9%
Sunnyvale 7% 5% 25% 16% 9%
Alta 18% 3% 2% 4% -14%
Problem solving
Plains Elementary 5% 19% 32% 45% 40%
Wayside 2% 11% 3% 11% 9%
Sunnyvale 10% 5% 6% 12% 2%
Alta 1% 3% 3% 2% 1%
4 0 %
2 0%
0%
Sunnyvale Elementary Demographics
100%
8 0 %
6 0 %
1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2
■ Native
American
0 1 Asian/Pacific
Islander
□ Hispanic
■ Black
■ White
Figure 4. Demographics of Sunnyvale Elementary
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43
W ayside Elementary School
Demographics
1 0 0 %
8 0 %
6 0 %
4 0 %
2 0%
0%
III
1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2
■ Native
American
m Asian/Pacific
Islander
□ Hspanic
■ Black
■ W hite
Figure 5. Demographics of Wayside Elementary
A lta Elementary School Demographics
100%
8 0 %
6 0 %
4 0 %
20%
0%
I
1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2
■ Native
American
□ Asian/Pacific
Islander
□ Hspanic
■ Black
■ W hite
Figure 6. Demographics of Alta Elementary
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44
Plains Elem entary School Dem ographics
1 0 0 % -p i
—
-
—
8 0 % -
< • - V .
6 0 % tr
1
fy;-
4 0 % -
1
2 0 %
o % - -
1
‘ S 1 !
. . . « . ....
1 9 9 9 2 0 0 0 2 0 0 1 2 0 0 2
■ Native
American
H i Asian/Pacific
Islander
□ Hispanic
■ Black
■ W hite
Figure 7 . Demographics of Plains Elementary
The results showed that when compared with other nonequivalent control
schools, Plains Elementary’s total growth in mathematics was still unmatched
(Figure 8, 9, and 10).
For example, the total growth in math skills at Plains Elementary was 40%,
while the other schools growth only totaled between— 3% and 19%. Additionally, in
math concepts, Plains Elementary’s scores increased 47% from 1999. The other two
comparable schools all had a decline in their test scores in this area and Sunnyvale
increased by 9%. Lastly, in problem solving, Plains Elementary’s students showed
an increase of 40% from 1999-2002. The only school of the three compared
increased their scores by only 9% (Table 5).
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P rob lem Solving - P e rcen ta g e o f stu d en ts who
m et or e x c e e d e d sta n d a rd s
50%
45%
40%
35%
30%
25%
20%
15%
10%
5%
0%
E
P la in sB em e rrtary V W yade B em en ta ry S u n n y v a le
B em en ta ry
A lta B e m e n ta ry
□ 2001
0 2002
Figure 8. New Standards Exam Mathematics Scores in Problem Solving
M ath Skills- Percentage of students who met or
exceeded standards
PlainsBementary V\6yside Bement ary Sunnyvale Bementary AltaBementary
□ 2001
□ 2002
Figure 9. New Standards Exam Mathematics Scores in Math Skills
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46
M ath Concepts- Percentage of students who met
or exceeded standards
60%
□ 1999
■ 2000
□ 2001
□ 2002
50%
40%
30%
2 0%
10%
0%
P lain sB em en tary V\&yside B em en tary S u n n y v aleB em en tary A ltaB em en tary
Figure 10. New Standards Exam Mathematics Scores in Math Concepts
This comparison indicated that when compared to other similar schools,
Plains Elementary was the only elementary school that made significant gains in the
math portion of the New Standards Exam. Plains was also the only school in the
district that received the treatment, the professional development program.
In order to create a contrast, a comparison of the growth in the Mathematics
and Language Arts sections of the New Standards Exam for the same years, 1999-
2002 were compared (Figure 11).
The mean scores from the Language Arts section for Plains included an
increase of 47% from 1999-2002. For the other comparative schools (Wayside,
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47
NSRE Scores at Rains Berrertary
70%
60%
50%
40%
30%
20%
10%
0%
—
--- —— ' r~y — # .......... 4 -^r-
1999
• Language Ats Mean
Scores
M ath Skills
IV b th Concepts
h *— Problem Soling
2000 2001 2002
Figure 11. Plains Elementary New Standards Exam Scores
Sunnyvale and Alta Elementary), their combined mean scores had an increase of
34%. In total, there was only a 12% difference in the mean scores on the Language
Arts section for Plains’s students compared to those attending the other three
comparable schools (Figure 12).
In comparing the four schools , Plains Elementary, Wayside Elementary,
Sunnyvale Elementary, and Alta Elementary in Language Arts, it is clear that there
were no steady gains achieved by any of the schools during this time (Figures 13, 14,
15, and 16).
The gains in the writing sections of the exams were made by Plains
Elementary in 2002 were attributed to a recent school-wide focus on writing
instruction. Overall, the comparison schools were not able to show significant,
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48
M ean N S R E S c o r e s a t E lem en tary S c h o o ls
60%
"O
cd
" C3
50%
40%
cti
"to
c n 30%
"55 20%
CD
^ 10%
o%
1999 2000 2001 2002
—♦— IV fean Rains
Bementary
\
— ■— IV fean Scores of
VN&yside,
Sunnyvale, and
Alta
Figure 12. Mean New Standards Review Exam scores of students at Plains
Elementary and comparable schools in math.
Reading: Analysis and Interpretation
60%
50%
CO
40%
CD
f ^
To
30%
' o
20%
10%
0%
M 1999
m 2 0 0 0
□ 2 0 0 1
□ 2002
Plains
Elem entary
W ayside
Elementary
Sunnyvale
Elementary
A k a Elem entary
Figure 13. Percentage of Students Meeting or Exceeding Standard in Reading
Analysis and Interpretation
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49
Reading: Basic Understanding
CO
c
CD
80%
70%
60%
50%
_ 40%
"to
° 30%
20%
10%
0%
R ains B em entary Surryvale A lta Bem entary
Bem entary
■ 1 9 9 9
■ 2 0 0 0
□ 2 0 0 1
□ 2 0 0 2
Figure 14. Percentage of Students Meeting or Exceeding Standard in Basic
Understanding
W riting: Effectiveness
80%
70%
60%
C D
“a
50%
" to
40%
M —
□
30%
X ?
ON
20%
10%
0%
PlainsBementary V\6yg'de
Bementary
Sunnyvale
Bementary
AltaBementary
■ 1999
D 2000
□ 2001
□ 2002
Figure 15. Percentage of Students Meeting or Exceeding Standard in Writing
Effectiveness
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50
V W iting Conventions
70%
60%
C O
50%
£=
C O
40%
13
C O
M —
o
30%
•vO
o N
20%
10%
0%
Plains Bementary Wfeyside
Bementary
Sirryvale
Bementary
■ 1999
B 2000
□ 2001
□ 2002
Alta Bemertary
Figure 16. Percentage of Students Meeting or Exceeding Standard in Writing
Conventions
steady growth over time in Language Arts. What the data indicates is small, sudden
growth in certain areas during a specific year. Unlike the growth shown
Mathematics sections at Plains Elementary, the student achievement in Language
Arts at all four schools remains largely irregular and not able to withstand the high
levels of achievement over time.
Conclusions
As a result of the findings, it is clear that there are patterns that indicate an
increase in student mathematical achievement on the New Standards Exam from
1998-2002. These patterns of increasing student achievement occurred during the
time that Math Matters was implemented at Plains Elementary. Through interviews
and surveys, the staff articulated that the only variable that changed during those
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51
years was the professional development. The staff at Plains stated that there were
little or no changes in staff or student population. The school maintained the same
administrator through those years who was instrumental in the overall success of the
teachers and students.
When Math Matters began at Plains Elementary School, only 3.7% of the
students were at or above standard in Mathematical concepts, 18.5% were at
standard for math skills and 1.9% were achieving in problem solving on the SAT9
exam. By 2002, 49% of the students were at or above standard in math concepts,
63% in math skills and 45% in problem solving. The students at Plains Elementary
not only achieved far beyond their scores four years prior, but also outscored most of
the other schools in the district. In problem solving, the students at Plains were the
highest scorers of all schools (Title 1 and Non-Title schools) in the state of Rhode
Island by 2002. That was a huge achievement for a school where only 5% of its
students were at standards in 1999.
The teachers stated that the success of the professional development was in
part due to the nature of its long-term process. They felt that if Math Matters had
been done over a shorter period of time, its effectiveness would have been
compromised. During classroom/school observations, it was clear to see that the
teachers and students were fluent in the techniques taught in Math Matters. Plains
Elementary was a place where one could observe students creating head problems,
teachers utilizing wait time with their students, and most of all a community of
learners who had come together and successfully implemented the Math Matters
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techniques and strategies. As a result, it became an active place of learning where
students and teachers were constantly learning and growing in their knowledge of
mathematics.
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53
CHAPTER 5
SUMMARY AND RECOMMENDATIONS
Plains Elementary School participated in many professional development
programs throughout the years. There were various initiatives that centered upon
increasing student achievement in both language arts and mathematics. Despite
many attempts to create school wide change, the staff at Plains Elementary was
receptive to a new program called Math Matters that was to begin at their school in
1999. The entire staff at Plains decided that they wanted to be a part of a long-term
investment that was aimed at increasing their content knowledge in mathematics and
increasing the achievement of their students.
Math Matters is a standards-based professional development program, which
provides comprehensive, long-term professional development focused on increasing
the staffs and students’ procedural skills, conceptual understanding, and mathe
matical reasoning. Math Matters was designed to be used in schools where students
are underachieving to improve the mathematics achievement of all students. It
incorporates a professional development program for teachers and administrators.
The core elements of Math Matters includes: (a) professional growth in
mathematics, (b) the transition of mathematics learned in the trainings to the
standards-based material used in the classroom, (c) classroom management strategies
that support the inclusion of all students, (d) cognitive coaching in the content area to
promote reflective practice, (e) strategies that deal directly with mathematical
content, and (f) sessions designed for the professional development needs of the
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54
administrators as well as quarterly meetings to update progress. Math Matters
provide opportunities for teachers and administrators to discuss mathematics,
practice teaching strategies, develop key lessons, and collaborate with their peers.
For 4 years the staff at Plains remained committed to the vision of school-
wide, long-term change in mathematics. Principals, art teachers, librarians, and even
teacher’s aides were all trained to become teachers that inspired mathematical
thinking and continuously challenged their students at every opportunity. As a
result, the students at Plains Elementary achieved levels that were previously seen as
unattainable. Within the 4 years, the students at Plains went from the lowest
achieving in the district to one of the highest scoring schools in the city in
mathematics.
One of the most significant findings was the increase of test scores of the
students at Plains Elementary. From 1998 to 2002, the students at Plains Elementary
meeting standard increased a total of 40% in math skills, 47% in math concepts, and
40% gain in problem solving on the New Standards Exams (Table 6).
Table 6. Percentage of Students Achieving Standards at Plains Elementary 1999-2002
1999 2000 2001 2002
Math skills 23% 43% 55% 63%
Math concepts 2% 19% 33% 49%
Problem solving 5% 19% 32% 45%
Teachers and staff members stated in their interviews that these gains in student test
scores are attributed to the training they received in Math Matters.
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55
The interviews, observations, and field notes showed four major themes in
terms of school-wide change: an increase of mathematical knowledge allowed the
teachers to improve their instruction, the training was able to change negative
perceptions of mathematics, the length of the professional development made a
positive impact on the students and staff, and the inclusion of all staff members made
a significant influence on the culture of the school.
Conclusions
This case study focused on what occurred those years at Plains Elementary in
terms of student achievement.
1. Why was it that the staff at Plains was successful while other schools
in the community were not able to achieve the same success?
2. What were some factors that created success?
3. What characteristics of the Math Matters program helped to improve
student achievement?
4. What can other schools take from the experiences of the staff and
students at Plains in planning their own professional development programs?
5. How can schools use the Math Matters model to improve student
achievement?
Teacher Level Factors
The bottom line is when it comes to student achievement; what a teacher
knows and does is the most important influence on what students learn. As Marzano
(2003) points out, 67% of the variance in student achievement has to do with the
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56
individual teacher, while about 20% of the variance is due to school-wide factors.
The research has shown throughout the years that effective teachers have a profound
effect on the achievement of their students. In terms of teaching students mathe
matics, that means that in order to be effective, the teacher is equipped with content
knowledge and classroom management and instructional strategies.
Following that model, the teachers at Plains were given the content
knowledge that enabled them to construct meaningful and effective lessons to their
students. They were also given management strategies that gave them the
opportunity to focus on instruction rather than on the behavior. Lastly, they were
given instructional strategies that they were able to use daily in their classrooms to
encourage student interaction and continuous feedback.
Content Knowledge
The staff at Plains Elementary was taught mathematical content throughout
the trainings from 1999-2002. In addition, the participants were taught how students
learn the subject matter and how to successfully integrate it with the curriculum they
were using. Teachers and staff members were given a stronger understanding of the
conceptual basis of mathematics and reported that they were able to teach all
students and provided students with mathematical examples in a variety of ways.
During those years, teachers and staff members were able to communicate and
discuss with one another their own perceptions of the content and engaged in
discussions regarding the mathematics. This type of learning is essential to
successful instruction. In fact, teacher subject matter knowledge is frequently cited
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57
as imperative to student achievement (Ball, 1990; Joyce & Showers, 1996; Ma,
1999; Sparks & Hirsh, 1997). Linda Darling-Hammond (2000) affirmed this belief
by saying, “Research shows that the single most important determinant of what
students learn is the expertise of their teacher” (p. 75).
Instructional Strategies
Educators agree that an effective teacher uses more effective instructional
strategies (Good, 1983; Wang, Haertel, & Walberg, 1993). In fact, teachers’
choice of instructional strategies can have a drastic effect on student achievement
(Marzano, Gaddy, & Dean, 2000). Through the training with Math Matters, the
staff was taught strategies that fell under four major categories: (a) management,
(b) involvement, (c) feedback, and (d) focus. The management and involvement
strategies that were taught helped teachers increase the amount of time students are
actively engaged in learning, which positively affects student learning (Cotton, 1990;
Evertson, & Harris, 1992). The focus and feedback strategies help teachers improve
their questioning techniques and create an active learning environment (Cotton,
1989; Marzano, 2000). After learning these strategies, the staff was encouraged to
use the techniques in their classrooms.
Coaching
The implementation of the instructional strategies was successful mainly,
because the staff received support from their administrators, fellow staff members,
and instructional coaches. All of these supporters observed classrooms and gave
constructive feedback and continued support for the implementation of the strategies.
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58
At Plains Elementary, the coach regularly modeled instruction and provided
feedback to the teacher. The “cognitive coaching” model developed by Costa and
Garmston (2002) was used to shape teachers thinking and to improve their problem
solving abilities. At Plains, the Math Matters coaches helped teachers think through
their teaching strategies as well as the mathematics content and determine what
strategies were critical and what mathematical concepts were essential.
As Joyce and Showers have continually pointed out (Joyce & Showers, 1985/
1995/1997/2001), professional development programs that have succeeded have
done so simply because the coaching and reflection piece was included. They found
that teachers can acquire new knowledge and skill and use it in their instructional
practice when provided with follow up coaching in the workplace. By introducing
the coaching model at Plains Elementary, benefits accrued not only to the coaching
recipients but to the school as a whole by improving the school climate and fostering
interest in change.
Limitations
This case study has various limitations, including the fact that the school and
its participants were not randomly selected. That is, the administrators at Plains
Elementary decided to bring in Math Matters to train their staff. As a result, they
were used in the case study because of their participation in the program.
A randomly selected control group was not used, which limits some validity
of the study. If this study was to be replicated, it would enhance the study if the
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59
researcher was able to use a randomly selected control group. Using that type of
model would allow the researcher to account for various factors and circumstances.
Lastly, another limitation of this study was the inability to watch the staff
members on a continual basis throughout the years. The school was located in a
different state than the researcher, which created some limitations for the researcher
in terms of continuous observation. Due to the large distance, the observations and
interviews were not impromptu, which might have created a different viewpoint for
the researcher.
Further Research
Despite many connections made in this case study between the characteristics
of the Math Matters professional development and the increase of student
achievement at Plains Elementary School, more research still has to be done in order
to broaden the scope of change. Research concerning how achievement changes
based on standardized tests needs to be evaluated.
If there are correlations found between professional development models and
student achievement on standardized test scores, some questions that need to be
answered might be: Are there other variables such as years of experience of the
teachers who participated in the trainings? Are there differences in primary versus
intermediate student achievement? For example, there are many schools in the U. S.
that do not experience positive change in student achievement on standardized tests,
even after participating in professional development. If researchers know what
characteristics tend to raise achievement, then districts across the nation should be
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60
able to get that information and use it in order to make significant changes for their
students.
The community at Plains Elementary appeared to have a significant effect on
the ability of the school to promote change. Knowing this, research can be done to
find out how this type of inclusion of the community affects school-wide change.
Answers to questions such as:
1. How does the role of the parents and community members play a part
in student achievement?
2. What were common characteristics of community members/parents
personality factors that played a role in the increase of achievement?
3. Could this model be replicated throughout other districts and
communities?
If a simple model was created and implemented that instructed schools how to
successfully include and utilize their community, more widespread change would
occur in our districts.
Looking at the long term effects Math Matters had on Plains, it would be
worthwhile to examine how the same students progress through middle and high
school math.
1. Did their experiences at Plains have a positive effect on their later
math achievement?
2. How does the student that came from Plains differ from those of the
other neighborhood schools in terms of preparedness for Algebra?
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3. Can the students who made significant gains in elementary school
continue that level of success throughout middle and high school?
Plains Elementary is one of the few schools that despite leadership and staff
changes, the enthusiasm for teaching their students mathematics has not changed
since their training. The staff remains committed to the vision of creating an
environment that demands critical thinking and creates positive experiences in
mathematics for its students and its teachers. If its past is indicative of its future,
Plains Elementary will remain an inspiration to others and will continue its journey
of creating tomorrow’s future.
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62
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APPENDIX
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71
How effective is your mai
^ —i ^ a ^ a
In an effective and well-designed mathematics program students move steadily
from what they already know to mastery of skills, knowledge, and
understanding. Their thinking progresses from an ability to explain what they
are doing, to an ability to justify how and why they are doing it, to a stage where
they can derive formal proofs. The quality of instruction is a key factor in
developing student proficiency in mathematics. In addition, several other factors
or program components play an important role.
Directions: Consider each of the following categories and specific criteria as
outlined in the Framework. Use the three-point scale shown below and measure
your classroom, school, or district math program for its level of effectiveness.
1 = Not Very Effective 2 = Somewhat Effective 3 = Very effective
I. Assessment: Assessment should be the basis for instruction, and different
types of assessment interact with the other components of an effective mathematics
program.
a. Assessment is aligned with and guides instruction.
Students are assessed frequently to determine whether
they are progressing steadily toward achieving the
Standards, and the results of this assessment are useful
in determining instructional priorities and modifying
curriculum and instruction.
1 2 3
b. Assessment looks at the same balance (computational
and procedural skills, conceptual understanding, and
problem solving) emphasized in instruction.
1 2 3
c. Assessments serve different purposes and are designed
accordingly. Those purposes include, but are not limited
to, student placement; monitoring student progress in a
mathematics program; or specific and detailed
information about which Standards have or have not
been achieved. Both formative and summative
assessments are used to serve these purposes.
1 2 3
d. Assessment can improve instruction when teachers use
the result of assessment to analyze what students have
learned and to revisit difficult concepts.
1 2 3
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72
II. Instruction: Arguably the quality of instruction is the most important single
component of an effective mathematics program. International comparisons show a
high correlation between the quality of mathematics instruction and student
achievement.
a. Teachers possess an in-depth understanding of the
Content Standards and the mathematics they are
expected to teach and continually strive to increase their
content knowledge.
1 2 3
b. Teachers are able to select researched-based
instructional strategies that are appropriate to the
instructional goals and to students’ needs.
1 2 3
c. Teachers effectively organize instruction around goals
that are tied to the Standards and direct students’
mathematical learning.
1 2 3
d. Teachers use the results of assessment to guide
instruction.
1 2 3
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73
III. Instructional Time: Study after study has demonstrated the relationship
between time on task and student achievement (Stigler, Lee, & Stevenson, 1987,
p. 1283). Priority must be given to the teaching of mathematics, and instructional
time must be protected from interruptions.
a. Every day all students receive at least 50-60 minutes of
mathematics instruction, not including homework.
Additional instructional time is allocated for students
who are, for whatever reason, performing substantially
below grade level in mathematics.
1 2 3
b. All students are encouraged to take mathematics courses
throughout high school.
1 2 3
c. Learning time is extended through homework that
increases in complexity and duration as students mature.
Homework should be valued and reviewed.
1 2 3
d . During the great majority of allocated time, students are
active participants in the instruction (engaged in
thinking about mathematics or doing mathematics
1 2 3
e. Instructional time for mathematics is maximized and
protected from interruptions, such as calls from the
office and extracurricular activities.
1 2 3
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74
IV. Instructional Resources: All teachers need high-quality instructional
resources, but new teachers especially depend on well-designed resources and
materials that are aligned with the Standards.
a. Instructional resources focus on the grade-level
Standards.
1 2 3
b. Instructional resources are factually and technically
accurate and address the content outlined in the
Standards.
1 2 3
c. Instructional resources emphasize depth of coverage.
The most critical, highest-priority Standards are
addressed in the greatest depth. Ample practice is
provided.
1 2 3
d. Instructional resources are organized in a sequential,
logical way. The resources are coordinated from level to
level.
1 2 3
e. Instructional options for teachers are included, e.g.,
direct instruction or discovery-oriented, as well as
means of assessing student progress
1 2 3
f. Resources balance basic computational and procedural
skills, conceptual understanding, and problem solving
and stress the interdependency of all three.
1 2 3
g. Resources provide ample opportunities for students to
explain their thinking, verbally and in writing, formally
and informally.
1 2 3
h. Resources supply ideas or tools for accommodating
diverse student performance within any given
classroom. They offer suggestions for re-teaching a
concept, providing additional practice for struggling
students, or condensing instruction so that advanced
students can concentrate on new material.
1 2 3
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75
V. Instructional Grouping and Scheduling: Research shows that
what students are taught has greater effect on achievement than how they are
grouped. The first focus of educators should always be on the quality of instruction.
Grouping and scheduling are tools that educators can use to improve learning, not
goals in and of themselves.
a. An effective mathematics program: (1) utilizes
grouping options in accordance with variability within
individual classrooms, and (2) maintains or changes
grouping strategies in accordance with student
performance on regular assessments.
1 2 3
b. Cooperative group work is used judiciously;
supplementing and expanding on initial instruction
either delivered by teachers or facilitated through
supervised exploration. When cooperative group work is
used, it should lead toward students’ eventual
independent demonstration of mastery of the Standards
and individual responsibility for learning.
1 2 3
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76
VI. Classroom Management! Potentially the primary management tool
for teachers is the mathematics curriculum itself. When students are actively engaged
in focused, rigorous mathematics, fewer opportunities for inappropriate behavior
arise. When students are successful and their successes are made clear to them, they
are more likely to become intrinsically motivated to work on mathematics.
a. Teachers are positive and optimistic about the prospects
for all students to achieve.
1 2 3
b. Classrooms have a strong sense of purpose. Both
academic and social expectations are clearly understood
by teachers and students alike. Academic expectations
relate directly to the Standards.
1 2 3
c. Intrinsic motivation is fostered by assisting students to
develop a deep understanding of mathematics,
encouraging them to expend the effort needed to learn,
and organizing instruction so they experience
satisfaction when they master a difficult concept or
skill.
1 2 3
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77
VII. Professional Development: The preparation of teachers and support
for their continuing professional development are critical to the quality of our
schools. Research from other countries suggests that student achievement can
improve when teachers are able to spend time together planning and evaluating
instruction.
a. Teachers have received excellent preservice training, are
knowledgeable about mathematics content, and are able
to use a wide variety of instructional strategies.
1 2 3
b. Continuing teacher in-service focuses on (1) enhancing
teachers’ own mathematics proficiency and (2) providing
pedagogical tools that help teachers to ensure that all
students meet or exceed grade-level standards.
1 2 3
c. Staff development is a long-term, planned investment,
strongly supported by the administration and designed to
ensure that teachers continue to develop skills and
knowledge in mathematics content and instructional
options.
1 2 3
d. As with students, staff development actively engages
teachers in mathematics and mathematics instruction. In
addition to active involvement during classroom-style
staff development, teachers have the opportunity to
interact with students and staff developers during in-class
coaching sessions.
1 2 3
e. Teachers are given time and opportunities to work
together to plan mathematics instruction. Districts and
schools find creative ways in order for this to happen.
1 2 3
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78
VIII. Administrative Practices: Administrative support for mathematics
instruction can help remind all those involved in education that reform efforts are not
effective unless they contribute to increased achievement. Administrators can help
teachers maintain a focus on high-quality instruction.
a. Mathematics achievement is among the highest priorities
at the school.
1 2 3
b. Long-term and short-terms goals for the school, each
grade levels, and individuals are outlined clearly and
reviewed frequently.
1 2 3
c. Scheduling, grouping, and the allocation of personnel are
shaped by a determination that all students will meet or
exceed the Mathematics Standards.
1 2 3
d. Principals demonstrate a strong sense of personal
responsibility for achievement within their schools.
1 2 3
e. Administrators consider using Mathematics Specialists to
teach most or all of the mathematics classes or coach other
teachers.
1 2 3
f. Administrators and teachers collaborate on developing
school-wide management systems and school-wide efforts
to showcase mathematics for students, parents, and other
members of the community.
1 2 3
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79
IX. Community Involvement: Mathematics education is everybody’s
business. Parents, community members, and business and industry can all make
significant contributions.
a. Parents are encouraged to be involved in education and
are assisted in supporting their children’s learning in
mathematics. Parent input is encouraged, valued, and used
for program planning.
1 2 3
b. Materials are organized so that parents, siblings, and/or
community members can provide extended learning
experiences.
1 2 3
c. The community is used as a classroom that offers
abundant examples of how and why mathematics is
important in our lives, our work, and our thinking.
1 2 3
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Asset Metadata

Creator Kopecky, Cara L. (author)

Core Title A case study of the Math Matters professional development program in one elementary school

School Rossier School of Education

Degree Doctor of Education

Degree Program Education

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

Tag education, elementary,Education, Mathematics,Education, Teacher Training,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Hocevar, Dennis (committee chair), Cohn, Carl (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-386095

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Document Type Dissertation

Rights Kopecky, Cara L.

Type texts

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

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

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