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Cognitively guided instruction: an mplementation case study of a high performing school district
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Content
COGNITIVELY GUIDED INSTRUCTION:
AN IMPLEMENTATION CASE STUDY OF
A HIGH PERFORMING SCHOOL DISTRICT
by
William D. B. Dowdy
A Dissertation Presented to the
FACULTY OF THE USC ROSSIER SCHOOL OF EDUCATION
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF EDUCATION
August 2011
Copyright 2011 William D. B. Dowdy
ii
Acknowledgements
I offer deep gratitude to my advisor and committee chair, Dr. Larry Picus. His
advice and direction helped me to refocus my energy toward the most important elements
of research. His warmth and humor helped to defuse an intimidating process, and let me
know that he was always there to support me in my research and my writing. He is the
definition of a caring and critical professor, offering his guidance to help students reach
their potential.
Dr. Julie Slayton was also instrumental in helping my dissertation. I would like to
thank her for the objective feedback that helped improve my research. Her advice will
serve me for many years as a consumer of research, and an educational leader seeking to
evaluate curricular programs.
I also thank Dr. Gregory Franklin for serving on my dissertation committee. His
practical advice, coming from a superintendent’s perspective, helped me to understand
the needs of a school district. I appreciate his candor regarding what is required for an
instructional leader’s decisions toward program evaluation.
Kate Garfinkel was instrumental in crafting this study. As my research partner for
the first half of this project, she inspired me to become a more critical consumer of
research. Without her support, I would not have made it through some challenging times.
I offer her my deepest thanks and admiration.
iii
The staff of Green Valley USD, who shall remain nameless for the purposes of
anonymity, were very gracious in supporting this research. They opened their doors to a
stranger, and inspired me by their high quality teaching and leadership. I am a better
teacher as a result of your example. All of you serve as pillars of excellence within our
public education system.
I also wish to thank my administrators and fellow teachers at TeWinkle
Intermediate. Your support allowed me substitute days to perform the critical tasks of
collecting, analyzing, and writing about my data. Because of your strength, I was
comfortable turning my classroom over to subs on several occasions.
Finally, and most importantly, I want to thank my family. My wife’s unwavering
support over the past three years kept me sane throughout my graduate studies. She
protected my time for study, and insisted that I maintain personal balance by relaxing
with our children. Thank you Kim. Kaitlyn, Lucy, and Teddy gave me the inspiration to
budget my time, and always encouraged me to finish my work so that we could spend
time together. Karen and Don provided me countless hours of quality study time by
watching the kids, and also helped keep me focused on graduation. My mother, Linda,
and father, David, gave me the inspiration and self-esteem to become the first Doctor
within our family. I love all of you, and thank you for your continual encouragement and
enthusiasm.
iv
Table of Contents
Acknowledgements ii
List of Tables vii
List of Figures viii
Abstract ix
Chapter 1 The Problem and Its Underlying Framework 1
Background of the Problem 2
Purpose of the Study 5
Research Questions 6
Significance of the Problem 6
Methodology 6
Limitations 8
Delimitations 8
Definition of Terms 8
Organization of the Study 10
Chapter 2 Review of the Literature 12
Literature Review 13
Effective mathematics instruction for primary grades. 13
Classroom learning environment. 16
Sociocultural roles in mathematics. 19
Professional development for teachers. 23
Cognitively Guided Instruction. 26
Conclusions 31
Chapter 3 Research Methodology 33
Research Questions 35
Research Design 35
Population and Sample 40
v
Instrumentation 43
Data Collection 46
Validity and Reliability 48
Data Analysis 50
Conclusion 51
Chapter 4 Findings 53
Findings 54
District Case Study – Green Valley Unified School District 55
School Case Study – Lincoln Elementary School 61
Teacher case study – Mrs. Kennedy at Lincoln Elementary School. 69
Teacher case study – Mrs. Madison at Lincoln Elementary School. 78
Teacher case study – Mrs. Pierce at Lincoln Elementary School. 85
Lincoln performance data. 92
School Case Study – Adams Elementary School 96
Teacher case study – Mrs. Carter at Adams Elementary School. 101
Teacher case study – Mrs. Grant at Adams Elementary School. 107
Adams performance data. 116
Comparisons Between Adams Elementary and Lincoln Elementary 119
Summary of Data Supporting the Research Questions 124
Conclusion 126
Chapter 5 Discussion 127
Analysis and Implications 129
Professional development. 129
Learning environment. 133
Leadership. 137
Limitations 140
Recommendations 141
Areas for action by stakeholders. 141
Areas for future research. 143
Conclusion 144
References 146
Appendix A Instruments 151
Principal Interview Protocol 151
Teacher Interview Protocol 152
Teacher Survey Questions 154
Teacher Observation Rubric 156
vi
Appendix B Sample CGI Problems 158
Mrs. Carter, Adams Elementary 158
Mrs. Grant, Adams Elementary 159
Mrs. Kennedy, Lincoln Elementary 160
Mrs. Madison, Lincoln Elementary 162
Mrs. Pierce, Lincoln Elementary 163
vii
List of Tables
Table 1 Green Valley Unified School District Demographics 2006-2010 56!
Table 2 Adjacent School Districts' Annual Performance Index (API) for 2010 57!
Table 3 Lincoln Demographics 2006-2010 62!
Table 4 Early Use of CGI at Lincoln 65!
Table 5 Lincoln Teacher Case Study Comparison 91!
Table 6 Percent of Lincoln Second-Graders Advanced or Proficient on CST Math 93!
Table 7 Adams Demographics 2006-2010 96!
Table 8 Adams Teacher Case Study Comparison 116!
Table 9 Percent of Adams Second-Graders Advanced or Proficient on CST Math 117!
Table 10 Comparison of Second-Graders Advanced or Proficient on CST Math 121!
Table 11 Comparison of Students Advanced or Proficient on CST Math by Cohort 122!
Table 12 Comparisons of API Data 2000-2010 123!
viii
List of Figures
Figure 1 Green Valley Unified School District CST Math Proficiency 2007 Cohort 58!
Figure 2 Lincoln CST Math Proficiency 2007 Cohort 94!
Figure 3 Lincoln API Data 2000-2010 95!
Figure 4 Adams CST Math Proficiency 2007 Cohort 118!
Figure 5 Adams API Data 2000-2010 119!
ix
Abstract
No Child Left Behind legislation developed goals for every student to be
proficient in each academic subject by 2014. California’s students are far from meeting
this goal, especially in mathematics. One Southern Californian school district, renamed
Green Valley Unified School District for anonymity, began using Cognitively Guided
Instruction district-wide in 2005 for all elementary students in an effort to meet the
NCLB goals. This dissertation is a case study of five second-grade teachers in two Green
Valley schools and the degree of CGI implementation within their classrooms. This
research developed assessment tools that may be useful for others evaluating teachers’
use of CGI. This study also characterizes elements of classroom culture, professional
development, and teacher’s practice that lead toward CGI mastery. Recommendations
are made for implementing a high quality CGI program, specific for Green Valley,
however they offer guidance for other schools and districts that may use Cognitively
Guided Instruction.
1
Chapter 1
The Problem and Its Underlying Framework
A common theme among current educators is the need to close an achievement
gap between students who are highly proficient, and those who are far below basic
expectations (Marzano, 2003). A major obstacle for students caught in this gap is the
field of mathematics. This research study will explore how one school district sought to
close this gap. Green Valley Unified School District uses an elementary school
mathematics program called Cognitively Guided Instruction (CGI), which may increase
student achievement and teachers’ capacity to teach mathematics (Carpenter, Fennema,
Loef Franke, Levi & Empson, 1999; Carpenter, Fennema, Peterson & Carey, 1989;
Fennema, Carpenter, Franke, Levi, Jacobs & Empson, 1996). This study will explore the
research basis for CGI and the level of implementation within two Green Valley schools,
Adams Elementary and Lincoln Elementary.
Green Valley Unified School District is a suburban school district located in
Southern California. It has maintained high scores on state assessments for more than a
decade (California Department of Education, Assessment, Accountability and Awards
Division, 2010); however district administration wants students to do better. They
believe all students are able to reach high levels of proficiency in mathematics, and all
teachers are able to reach excellence.
2
Specifically the administration chose Algebra 1 performance as a targeted subject
for improvement, and decided to invest resources to improve elementary school
mathematics instruction to build algebra readiness. The administration saw high levels of
student mathematics proficiency at one elementary school, Lincoln Elementary, where
teachers used CGI as the cornerstone of math instruction. This led the district to
implement Cognitively Guided Instruction in all elementary schools. The district has not
performed a formal evaluation of the program to determine the level of implementation
across schools, grade levels, or teachers. District administrators made an informal
request for the University of Southern California to research the effectiveness of CGI
within Green Valley schools. This project serves to address that request.
This study compares Green Valley’s CGI program to models developed by the
pioneers of CGI and recommends areas for improvement. These recommendations are
based on observations and interviews of staff at two schools in comparison to the
research base of effective mathematics instruction and teacher professional development.
Background of the Problem
While the United States was a pioneer in the innovation of free public education
for its citizens a century ago, it is no longer the international leader in mathematics
instruction and student achievement (National Center for Education Statistics, 2009;
Stigler & Hiebert, 1999). During the 1990s, the Third International Mathematics and
Science Study (TIMSS, now renamed Trends in International Mathematics and Science
Study) showed that American fourth and eighth-grade students lagged many other
3
industrialized nations in mathematics (Mullis, Martin, Beaton, Gonzales, Kelly &Smith,
1997; Stigler & Hiebert, 1999).
The 1995 TIMSS mathematics data for fourth-grade students showed American
students were above the international average, however eleven countries were ahead of
the US (Mullis et al., 1997). Eighth-grade students lost ground, and performed below
the international average with seventeen countries ahead of them (Mullis et al., 1997).
The United States also had the lowest increase in scores between the fourth and eighth
grades of any nation assessed (Mullis et al., 1997). Stigler and Hiebert (1999) analyzed
videotaped lessons from the 1995 TIMSS and concluded there was a significant gap in
the quality of math instruction between American classrooms and leading countries of the
TIMSS study.
In addition to the United States’ lower performance on international assessments,
California students lag many other states in mathematics performance. California scored
below the national average of fourth-grade math scores on each National Assessment of
Educational Progress (NAEP) assessment from 1992 to 2009 (NCES, 2010). Forty-two
states had scores higher than California on the 2009 fourth-grade mathematics NAEP
assessment (NCES, 2010). The state has seen many students improve their NAEP
performance from below basic to higher levels over this 17-year period, however student
growth has remained stagnant since 2005 (NCES, 2010).
The results of TIMSS and NAEP motivated policy makers, educational
researchers, administrators, and teachers to study methods for reforming mathematics
instruction. In response, California leaders developed common mathematics standards in
4
all grades, and standardized assessments to monitor progress of schools and students.
The federal government implemented No Child Left Behind (NCLB) legislation in 2001,
which added punitive consequences for schools that do not show annual improvements in
these assessments. The combinations of state and federal policies placed pressure on
districts and schools to significantly improve performance of all students. A stated goal
of NCLB is for 100% of students to reach proficiency in all subjects by 2014. Green
Valley broadened its use of CGI with hopes to reach this goal. The decision to extend
CGI use was based on internal data showing very high levels of mathematical proficiency
at Lincoln Elementary.
There is consensus that the means for instructional change lies within current
education systems. A broad base of literature indicates that the knowledge and skill of
classroom teachers is one of the most important factors of student achievement (Carey,
2004; Darling-Hammond, 2002; Gordon, Kane & Staiger, 2006; Marzano, 2003; Nye,
Konstantopoulos & Hedges 2004; Sanders & Rivers, 1996). These factors can be
positive, increasing student achievement, or points of stagnation when traditional
instructional practices are continued. For students to reach the goal of NCLB, 100%
proficiency, instructional practices require change. Teachers are often not ready to
change their practices (Guskey, 2000; Darling-Hammond & Richardson, 2009). They are
relatively autonomous in their decisions about the curriculum offered, the method of
instruction, and the level of rigor placed on students (Johnson, 2002). In order to effect
greater student understanding of mathematics, and ultimately better achievement on
standardized tests, teachers must be the motivators of change. Quality professional
5
development can positively influence teachers to change their practice (Darling-
Hammond, 2002; Johnson, 2002; Marzano, 2003).
CGI seeks to address the need to improve student mathematical proficiency and
professional development of teachers (Carpenter et al., 1999). Existing research shows
that CGI is effective in raising student achievement under specific professional
development models, combined with teachers reaching higher levels of implementation
within their practice (Carpenter, Fennema, Peterson, Chiang & Loef, 1989; Fennema et
al., 1996). By choosing CGI as the primary means of teacher change and elementary
math curriculum, Green Valley hopes to see positive effects on student achievement.
Purpose of the Study
This study determines how Green Valley USD implemented CGI and how the
GVUSD model compares to effective programs established by Carpenter, et. al. (1989;
1999; Fennema et al., 1996). This study also determines to what extent teacher practices
are consistent with CGI expectations. One must determine the level of CGI
implementation before analyzing any connections between CGI and student performance
in Green Valley. Future research may determine whether there is a link between CGI and
an increase in students’ scores on California Standards Tests in mathematics; these are
beyond the scope of this study.
6
Research Questions
1. How does the implementation of the CGI professional development
program at Green Valley USD compare to the model designed by
Carpenter, et al.?
2. In what ways has the CGI model at GVUSD changed over time?
3. Are the actions of elementary teachers while teaching mathematics in the
classroom consistent with those of CGI?
Significance of the Problem
This study will help Green Valley to reflect upon teacher practice, its professional
development programs, and its implementation of CGI. Green Valley may use this study
to reevaluate it priorities for mathematics instructional programs such as CGI. The
district may also pursue further study into the effectiveness of the CGI program, which is
beyond the scope of this study. This study expands the general knowledge of CGI
implementation in comparison to Carpenter, et. al. (1989) and Fennema et al. (1996)
models. It also develops assessment instruments for evaluation of other CGI programs.
Methodology
Two schools were selected for study, Adams Elementary and Lincoln Elementary.
They represent a variety of CGI implementation within Green Valley Unified; one was
perceived by administration as an exemplar, while administration viewed the other to
have room for improvement. Informal interviews with the district’s assistant
7
superintendent of curriculum helped select the schools for study. She also provided
information regarding district priorities, an overview of CGI implementation across all
schools, and historical context to compare data collected from principals and teachers at
each school.
Within these two schools, the research team interviewed principals to learn about
the professional development model used at each site, and how it changed over time.
These interviews also provided data to determine the general level of CGI use by each
school’s faculty. Principals provided information related to the history of
implementation, areas of success, and barriers to successful implementation.
The researchers collected data from second grade teachers at both schools. Two
Adams teachers, and three Lincoln teachers received multiple classroom observations,
interviews, and a survey. This data provided information used to measure the levels of
CGI instructional practice, the history of professional development for each teacher,
additional history on CGI at the school, and the degree of peer teachers’ implementation.
A rubric, based on research by Carpenter et al. (1999), provided the criteria to measure
each teacher’s instructional practice.
A qualitative analysis of these observations, interviews, rubrics, and district
professional development records determined the level of implementation. Based on
these data, the research team provided recommendations for Green Valley to improve the
CGI program, as well as areas for future study.
8
Limitations
1. This study is limited to subjects who agree to participate voluntarily.
2. This study is limited to teachers, principals, and administrators within Green Valley
Unified School District and the 2010-11 school year.
3. This study is limited to five second-grade classrooms, in two schools, within one
school district.
4. Validity of this study is limited to the reliability of the instruments used.
Delimitations
This study confined itself to a case study of two elementary schools within Green
Valley Unified School District. This study focused on data related to the history of CGI
implementation, teacher professional development, and teacher practices within the
classroom. It did not study relationships between student achievement and the use of
CGI. Only district administrators, principals of the two schools and teachers within those
schools were included in the study. Green Valley Unified is more homogenous and
affluent than most California school districts. Green Valley’s demographics and the
study’s sample size, limit the generalizability of the study’s findings.
Definition of Terms
API – Academic Performance Index is a score to measure school-wide
performance on standardized tests used within the state of California. It ranges range
from 400-1000 with a proficiency goal of 800 for every school by 2014.
9
CGI – Cognitively Guided Instruction is a professional development model of
teaching beliefs and practice for K-5 mathematics instruction. It provides a framework
for teachers and schools to develop a challenging mathematics curriculum, through
problem design, inquiry, and reflection. It is a philosophy toward teaching mathematics,
not a specific instructional program.
Cotsen Foundation – The Cotsen Family Foundation partners with schools to
provide an in-depth, two-year mentoring program for teachers. Some Green Valley
schools participated in this program. See http://www.cotsen.org for more information.
Combination class – A combination class is an elementary classroom with two
grade levels of students. For example, a combination class may have 50% first-graders
and 50% second-graders taught by the same teacher.
CST – California Standards Tests are standardized tests given from grades 2-11
annually. They are used to determine AYP and API scores for schools, as well as
determine individual student’s proficiency in English language arts and mathematics in
all grades 2-11. Additional tests in science and history are given in upper grades.
Fosnot Math – Some Green Valley elementary schools use a supplemental math
program called Contexts for Learning Mathematics. It replaces CGI or Houghton Mifflin
during math lessons. Catherine Twomey Fosnot is the primary author of the program. It
uses problem solving to teach larger concepts related to mathematics, and has some
elements similar to CGI. See http://www.contextsforlearning.com for more information.
Houghton Mifflin Math - Green Valley’s adopted elementary mathematics
curriculum is from Houghton Mifflin Company. It is a traditional textbook and
10
workbook mathematics program. See http://www.eduplace.com/math/mw/ for more
information.
Jiji Math – Some Green Valley schools offer an enrichment program, known
colloquially as Jiji. The program does not replace CGI or Houghton Mifflin math
lessons; it is additive in nature. It is a computer-based software package called ST Math
developed by the MIND Research Institute. It teaches logical reasoning and numeracy
skills through non-verbal games, whose primary character is Jiji the penguin. See
http://mindresearch.net for more information.
NAEP – National Assessment of Educational Progress is a standardized test given
to a sample of students throughout the US. It compares the performance of students on a
national basis, since local state tests are written to local performance standards.
TIMSS – Originally named the Third International Mathematics and Science
Study, it has been renamed the Trends in International Mathematics and Science Study.
The study analyzes student performance in math and science in over 40 countries around
the world. Students test in the fourth and eighth grades for both content knowledge and
cognitive skills.
Organization of the Study
Chapter 1 of the study has presented the introduction, the background of the
problem, the statement of the problem, the purpose of the study, the questions to be
answered, the research hypotheses, the significance of the study, a brief description of the
methodology, the assumptions, limitations, delimitations, and the definitions of terms.
11
Chapter 2 is a review of relevant literature. It addresses the following topics:
Effective Mathematics Instruction for Primary Grades
Classroom Learning Environment
Sociocultural Roles in Mathematics
Professional Development for Teachers
Cognitively Guided Instruction
Chapter 3 presents the methodology used in the study, including the research
design, population and sampling procedure, the instruments and their development,
together with information on validity and reliability. Each of these sections concludes
with a rationale, including strengths and limitations of the design elements. The chapter
goes on to describe the procedures for data collection and the plan for data analysis.
Chapter 4 presents the results of the study. Chapter 5 discusses and analyzes the
results, culminating in conclusions and recommendations.
12
Chapter 2
Review of the Literature
This study will examine the quality of implementation of Cognitively Guided
Instruction (CGI) within Green Valley Unified School District. An existing body of
research on mathematics instruction for elementary grades, sociocultural theory,
professional development for teachers and the characteristics of CGI will form the basis
of comparison to Green Valley’s program. This body of research will define the
elements of a high quality math program used in grades K-5. The research will also
identify instruments for measurement of teachers’ actions within the classroom. The
traits identified within this body of literature drive the following research questions.
The research questions for this study are:
1. How does the implementation of the CGI professional development
program at Green Valley USD compare to the model designed by
Carpenter et al.?
2. In what ways has the CGI model at GVUSD changed over time?
3. Are the actions of elementary teachers while teaching mathematics in the
classroom consistent with those of CGI?
13
Literature Review
Effective mathematics instruction for primary grades.
Before we can determine how well Green Valley has implemented the CGI
program, we should first identify the characteristics of a high quality mathematics
instructional program. The National Research Council (2001) identified five strands for
mathematical proficiency. They are:
1. Conceptual understanding – A student has a firm comprehension of
mathematical concepts, their operations, and the relationships between them.
2. Procedural fluency – Students are able to be flexible, accurate, and efficient
with mathematical operations.
3. Strategic competence – Children proficient in mathematics can solve a variety
of problems, and are able to formulate and represent them in various ways.
4. Adaptive reasoning – Mathematicians have a capacity for logical thought and
communicate their justifications in a cogent manner.
5. Productive disposition – Students have a high degree of self-efficacy with
respect to mathematics. They see mathematics as being sensible and
worthwhile.
These strands are the true essence of what defines a student, or teacher, with
mathematical competence. Students should demonstrate proficiency in each strand and
teachers should foster children’s development in them. Historically, teachers have not
developed these traits equally within students (NRC, 2001), hence the desire of Green
Valley to seek philosophical changes within mathematics instruction.
14
There are many methods to effectively teach mathematics, and the National
Research Council (2005) identified three areas that support the strands previously
mentioned.
1. Teachers should fully engage students’ prior knowledge.
2. Teachers should develop students’ factual knowledge and conceptual
understanding in context.
3. Teachers should help students become self-monitoring in their knowledge.
Building on the findings of the NRC in 2005, Clarke and Clarke (2004) provide
additional examples of quality teaching practice. Their study focused specifically on
teachers of young children from kindergarten through second grade. Teacher qualities
that correlate to higher student performance are:
1. Students’ use of higher order statements, questions, and tasks that required
thought rather than rote practice.
2. The connection between different mathematical ideas and the context of their
use. Student dialogue especially emphasized these connections.
3. The use of collaborative problem solving by students, both whole group and
small group.
4. Student autonomy to develop and discuss their own ideas and methods.
The role of the teacher in Clarke and Clarke’s (2004) study is to serve as a
facilitator of student knowledge, rather than as a dispenser of knowledge. Students work
in collaboration to learn from each other; they see themselves as participatory educators.
Effective teachers help students collaborate by structuring purposeful tasks for children
15
that allow different strategies to emerge (Carpenter et al., 1999; Clarke & Clarke, 2004).
These tasks are focused on the most important mathematical ideas for study, and help
children to become clear in their understanding of the problem (Clarke & Clarke, 2004).
Teachers choose problems to maintain student involvement, and be engaging (Clarke &
Clarke, 2004; Johnson, 2002). These problems present teachable moments, which
teachers use as they occur, for example, using a student’s incorrect explanation as an
opportunity to review (Clarke & Clarke, 2004).
The traits of high-quality teachers, enumerated by the NRC (2001, 2005) as well
as Clarke and Clarke (2004), build a foundation for Cognitively Guided Instruction. This
foundation provides context to CGI philosophies, and help focus instruments used to
evaluate Green Valley’s program.
Juanita Copley (2000) focused her research on the teaching of young children,
and their mathematical needs. Rather than purely focusing on characteristics of quality
teachers, she also observed student needs within the context of the classroom. Copley
(2000) stated that mathematics education should be child centered, meaning the teachers
design curriculum around children’s needs and current knowledge. The content is
appropriate to students’ current level of knowledge so it pushes the child but does not
overwhelm him (Copley, 2000; Johnson, 2002; Mayer, 2008).
The child-centered classroom uses student solutions as a vehicle of instruction.
Student explanations are a primary form of assessment for the teacher, and serve as
means to strengthen the learning process of other students (Carpenter et al., 1999;
Copley, 2000). Students explore multiple solutions with a rich variety of resources, such
16
as manipulatives, computing technologies, response boards, or alternative texts. Students
have a sense of exploration, with the teacher guiding them appropriately (Copley, 2000).
Classroom learning environment.
The classroom environment is complex, and requires a mixture of cultures in
order to be highly effective. It should be learner-centered, knowledge-centered,
assessment-centered, and community-centered (NRC, 2005). The learner-centered
domain requires a teacher to pay close attention to the student. Care should be taken to
understand the student’s background and cultural values, (Johnson, 2002; NRC, 2005) as
they will influence a student’s conceptions or misconceptions of mathematics (Clarke &
Clarke, 2004). Students should be presented with difficulties, which can just be
managed, so they engage students but do not lead to discouragement (NRC, 2005; Mayer,
2008).
The knowledge-centered classroom focuses on the content taught, the reasons for
understanding, and the classroom’s organization to support student development (NRC,
2005). The knowledge-centered classroom also defines the level of mastery and makes
the definitions of success explicit for students (NRC, 2005; Johnson, 2002). It
emphasizes the connection of knowledge to the foundational ideas of a subject (NRC,
2005). The logical sequence of the curriculum will either support student learning, by
allowing reasonable stretches, or will suppress students’ abilities by asking too much of
their current levels. The blend of students’ abilities within a classroom will require a
17
teacher to have very individualized plans of action. This requires understanding each
child’s needs as lessons are developed.
An assessment-centered classroom uses formative, ongoing assessments that
make student thinking visible, to the teacher, and to other students (NRC, 2005). These
assessments support student learning by providing opportunities to revise and improve
thought processes (NRC, 2005; Johnson, 2002). The feedback from assessments
transforms students’ abilities to self-monitor (NRC, 2005; Mayer, 2008). Becoming self-
evaluative and building metacognition, are indicators of deep understanding (Anderson et
al., 2001). Metacognitive abilities allow students to evaluate their learning, and not rely
on external indicators (NRC, 2005). Concerning mathematics, this indicates that students
are highly proficient and achieve deep understanding, consistent with recommendations
of the National Research Council (2005).
The community-centered classroom develops a culture in which students question
the learning process, and provide explanations that guide each other and the teacher
(NRC, 2005). Students actively participate in the learning process of others, asking
questions to clarify solution strategies. Teachers ask probing questions that will extend
the solutions of students, and incorrect or naïve answers are fully explored by the class
(NRC, 2005). Teachers analyze and compare different student strategies (NRC, 2005) so
students of varying ability levels can learn from each other (Carpenter et al., 1999).
Connections between strategies help to deepen the level of cognition related to the
curriculum (Carpenter et al., 1999). Students discussing multiple solutions helps build a
18
conceptual ladder, allowing students to develop more efficient and abstract approaches
toward mathematics (NRC, 2005).
The questioning strategies used by the teacher stimulate students’ abilities to
clearly communicate thinking, and concurrently clarify reasoning to make sure it is
logical (NRC, 2005; Carpenter et al., 1999; Clarke & Clarke, 2004; Copley, 2000).
Additionally, encouraging students to have a voice in the classroom builds their
confidence and self-efficacy, which in turn allows learning to develop personal
significance and better long-term retention (Mayer, 2008; Copley, 2000; Johnson, 2002).
The deep questioning strategies used by teachers, and encouragement of peer
questioning, help students achieve strong conceptual understanding of mathematics
(Carpenter et at. 1989), consistent with best practices for math instruction (NRC, 2001).
Teachers exhibiting best practice, as Green Valley hopes to have, demonstrate
characteristics of all these elements. They exhibit beliefs that a classroom should be
child-centered. Their planning is reflective of this belief, developing lessons based on the
needs of their students, and not based on the flow of the textbook. Teachers believe it is
important for students to be the leaders within the classroom, and the teacher should
facilitate the learning experience, and not serve as the only means of instruction. The
teacher uses student assessments to provide feedback to both the teacher and the student,
with the goal for children to become self-aware of their successes and mistakes.
19
Sociocultural roles in mathematics.
Sociocultural theories state learning is a result of cultural factors, in combination
with academic content. Ethnicity, family environment, school organizational structures,
classroom policies, teacher expectations, and student interactions all contribute to the
culture of learning (Mayer, 2008; Rueda & Dembo, 1995). The classroom is a rich blend
of these components. Social interactions among students greatly influence motivation of
students, either stimulating or repressing them (Mayer, 2008; Rueda & Dembo, 1995).
Classroom expectation and norms greatly influence student motivation, partly
because of participation in classroom activities. Students with a voice in the classroom
have greater confidence and self-efficacy, which in turn allows learning to develop
personal significance and better long-term retention (Copley, 2000; Johnson, 2002;
Mayer, 2008). Students with means to participate feel included in the classroom’s
culture, hence leading to increased learning (Mayer, 2008). Conversely, if they do not
have means to participate in the classroom’s mathematical practices, they become
excluded from the classroom community (Cobb & Bowers, 1999). Students have a
strong desire to feel part of the community. They strive to become “good students” and
work to meet classroom norms of behavior (Packer & Gorcoecha, 2000). The teacher
must establish the norms that encourage student participation.
These norms are often invisible to educators and students, ingrained in the
reflexive patterns of teacher routines and strategies (Gallimore & Goldenberg, 2001).
Because of this fact, teachers should consciously develop classroom norms that will
increase student engagement. They should create situation of learning where success or
20
failure is dependent upon effort and not just abilities, where students can observe their
peers succeeding and also experience success themselves (Mayer, 2008). Students
experience greater success when the classroom is based on mastery of content,
whereupon the teacher expects students to learn the content with depth, rather than a
performance approach, meaning that a student meets particular scores on assessments
(Rueda & Dembo, 1995).
A mastery approach to learning requires students to have a complex knowledge of
content and the ability to assess one’s knowledge. Students may assess their knowledge
by becoming more self-aware of their own beliefs, determine the feasibility of these
beliefs in the context of classroom knowledge, and through self-monitoring (Anderson et
al., 2001). Written student records serve as a means for self-monitoring. Students that
compare one’s own solution to others learn to assess efficiency. These comparisons,
combined with student explanations of written work, build students’ abilities to reflect
upon their own work (Yackel & Cobb, 1996). Students build knowledge and mastery
based on interactions with peers, enhancing learning opportunities (McClain & Cobb,
2001).
Yackel and Cobb (1996) found that learning opportunities for students and
teachers are significantly influenced by the degree of peer interactions within the
classroom. Fewer peer interactions, or those of lesser depth and complexity, reduce
opportunities for students to learn. Student interactions centered on mathematical
solutions give direction to student learning, and build flexible reasoning skills (McClain
& Cobb, 2001). Opportunities for students to grapple with a variety of solutions, and
21
explain them in multiple ways, strengthen these reasoning skills (Carpenter et al., 1999;
Clarke & Clarke, 2004; NRC, 2001). Carpenter et al. (1999) found that presenting
student solutions from a variety of ability levels strengthens student’s knowledge.
Lower-performing students learn unique solutions and more efficient practices. They see
peers solving problems, and may be motivated to attempt problems that are more
challenging. Higher-performing students learn presentation skills, building fluency in the
language of mathematics. A participatory model of presentations, in which both novice
and experts have discourse, leads to increased learning (Mayer, 2008).
Discussions of mathematics still require direction from the teacher. Quality
participation does not just echo other’s solutions, but compares and contrasts them,
hoping to add new dimensions to the dialogue. The teacher defines and reinforces this
dialogue so classroom norms expect students to become additive in discussions. When
students look for alternative solutions, assess the correctness, or attempt to find more
efficient strategies, their mathematical performance increases (McClain & Cobb, 2001).
Students find opportunities to learn when others have made mistakes, perhaps from more
advanced students. The teacher can greatly influence the long-term culture of the
classroom, and students’ identities as mathematicians (Cobb & Hodge, 2011), when
students discover mistakes. Appropriate praise and encouragement builds mathematical
efficacy, thus increasing motivation, in turn supporting deeper cognition (Mayer, 2008).
Alternatively, if the teacher hastily corrects students or offers negatively received
criticism of incorrect solutions, students will lose motivation. Students’ beliefs in their
22
mathematical prowess influence their intellectual autonomy (Cobb, Yackel & Wood,
1989/2011).
Encouragement to reflect upon the current solution, and the ability to validate the
presentation, can diffuse barriers to students learning mathematics (Cobb, Yackel &
Wood, 1989/2011). Based on the learning process, students constantly reorganize their
methods of classroom participation (Cohen & Ball, 2001). Students’ involvements in
classroom mathematical activities support their sense of affiliation, guided by the teacher
(Cobb & Hodge, 2011).
The ability of teachers to stimulate student involvement requires students’ active
engagement; student engagement requires teacher stimulation (Elmore, 2004). Student
engagement increases when teachers design lessons that use cooperative learning
strategies and require social exchange, and connect curriculum to students’ personal
values (Mayer, 2008). Increased student involvement builds expectations for students to
fully engage in mathematics. A sense of peer-based accountability emerges, in which
students expect each other to contribute in meaningful ways to the body of classroom
knowledge. Internal accountability mounts as the teacher transfers the basis of
instruction to students (Elmore, 2004) through presentations of solutions and strategies,
both formally and informally.
Student interactions also inform teachers’ practice. Alternative models of thought,
often presented by students, build the knowledge base for teachers. In turn, teachers are
able become more flexible and responsive in their knowledge of mathematics content and
pedagogy (Hill & Loewenberg Ball, 2004). When teachers collaboratively share
23
knowledge of students learning, schools enhance the blend of pedagogy and assessment,
which produces better results for students, teachers, and the school (Gallimore &
Goldenberg, 2001). Teachers with opportunities to learn about students’ solutions refine
their practice, which lead to student learning improvements and higher state test scores
(Cohen & Ball, 2001).
Professional development for teachers.
In order to help teachers use best practices for mathematics instruction, it is
important to support them through quality professional development (PD). High-quality
professional development will have an enduring impact on teachers, and will change their
practices for the long term. This section will explore research on PD targeted for teachers
and proven to have lasting impacts.
Effective professional development aligns with student learning (Darling-
Hammond & Richardson, 2009; Loucks-Horsley, Stiles, Mundry, Love & Hewson, 2010;
Marzano, 2003; NRC, 2001). It is intensive and ongoing over time; it does not stop with
one workshop (Darling-Hammond & Richardson, 2009; Loucks-Horsley et al., 2010). It
connects to wider school initiatives and gives time for teachers to collaborate and reflect
upon their learning (Darling-Hammond & Richardson, 2009; Loucks-Horsley et al.,
2010; Marzano, 2003). Improvement in professional development requires continual
monitoring and evaluation (Loucks-Horsley et al., 2010; Darling-Hammond &
Richardson, 2009). Darling-Hammond and Richardson (2009) state that teachers need
training with opportunities for active learning.
24
Effective professional development also requires a lengthy timeline in order to
cause change in teacher practices. Darling-Hammond and Richardson (2009) state it
takes between 30-100 hours of training and reflection for a program to be most effective.
One-shot workshops do not work, and the PD program must extend over a longer time
period, as well as all levels of the organization (Guskey, 2000; Darling-Hammond &
Richardson, 2009; Loucks-Horsley et al. 2010; Saxe, Gearheart & Nasir, 2001). This
includes allowing teachers to observe each other and collaborate over student work
samples and problems. The goal is for continual improvement to become part of the
fabric of the organization, including students, teachers, and administrators (Guskey,
2000; Marzano, 2003; Loucks-Horsley et al. 2010).
It will be important to determine how well Green Valley school have been able to
create a consistent message within their organizations, dedicating themselves toward
continuous mathematics improvement. The professional development program should
spread throughout the school year, and ideally over multiple years.
Two particularly effective professional development models are worth noting.
Both targeted mathematics education for elementary grades, and both found positive
gains in student achievement. Fennema, Carpenter, Franke, Levi, Jacobs and Empson
(1996) used a model consistent with the above recommendations to implement
Cognitively Guided Instruction (CGI). The goal of the program sought to raise student
achievement by teachers better understanding student needs, building mathematics
content knowledge, and the reflecting on mathematics instruction pedagogy. The CGI
model lasted three years. It required participating teachers to attend a two-day workshop
25
before the start of the school year to learn about, discuss, and plan for using CGI in their
classrooms during year 1. Teachers in year 1 then attended fourteen 3-hour workshops
throughout the school year to reflect upon their practice. The second-year teachers
participated in four, 2
! -hour workshops and one 2-day reflection study workshop.
Third-year teachers attended one 3-hour reflective workshop and two 2 ! -hour
workshops throughout the school year (Fennema et al., 1996). A teacher that participated
all three years would have received approximately 88 hours of formal workshops
occurring before the school year and during the school year. In addition to the
workshops, a CGI staff member was assigned to each participating school site to visit
classrooms, guide teacher discussions, and provide support for teachers implementing the
program. The mentors met approximately once a week for year-1 teachers, once every
two weeks with year-2 teachers, and only occasionally with year-3 teachers (Fennema et
al., 1996).
Saxe, Gearheart and Nasir (2001) also developed a professional development
model, targeting elementary grade instruction of mathematics. The Integrated
Mathematics Assessment (IMA) program sought to raise student achievement, and kept
this as the focus when designing a professional development program for teachers. It
began with a five-day workshop held during the summer, and continued with 13 follow
up meeting during the school year. The researchers found this type of focused,
supportive, and long-term professional development to be effective in raising student
achievement scores.
26
The professional development program of Green Valley Unified should hold the
main elements of Fennema et al. (1996) and Saxe et al. (2001). Both of these programs
dedicated multiple training days prior to start of school for new teachers. They both
followed up with reflective workshops spread throughout the school year. Green Valley
should also differentiate its PD model for teachers of varying experience levels. The use
of mentor coaches is another key element used within Fennema et al.’s (1996) CGI
program, and should be in Green Valley’s CGI program
Cognitively Guided Instruction.
Cognitively Guided Instruction (CGI) fosters best practice in the mathematics
instruction of elementary school students, and uses the conduit of professional
development to effect this change. It is highly consistent with the literature already
reviewed. The program was a result of cognitive science research, experimental
implementation of teaching methods, and experimental designs of teacher professional
development. Thomas P. Carpenter, Elizabeth Fennema, Megan Loef Franke, Penelope
L. Peterson, Linda Levi, Susan E. Empson, Diana A. Carey, and Chi-Pang Chiang
performed the major research studies between 1988 and 1996.
The research began with knowledge about the cognition of children studied by
Carpenter in the mid 1980s. Carpenter, Fennema, Peterson, Chiang, and Loef performed
an experimentally designed study in 1989 that serves as the precursor of CGI. It
determined teachers’ specific knowledge about student’s cognition relates to student
achievement; teachers with more detailed knowledge about each student had higher
27
levels of student achievement. Additionally the study found that these teachers
emphasized word problems with greater complexity throughout the year (Carpenter et al.,
1989).
Another significant study occurred in 1996. Fennema, Carpenter, Franke, Levi,
Jacobs and Empson performed a multi-year longitudinal study on professional
development of CGI teachers. It found that fundamental changes in the beliefs of
teachers, with teachers becoming highly consistent with CGI philosophies, lead to
increased student performance. It also found the professional development model used
by Fennema et al. (1996) had lasting impacts, teachers’ changes became ingrained in
their practice. This latter study provides the framework for comparing the CGI
professional development model to Green Valley’s.
CGI will look different in every classroom, but does have common elements for
all applications (Carpenter et al., 1999). Teachers of CGI need to have a strong
pedagogical understanding of the conceptual and procedural elements of mathematics,
knowing how layers of mathematical proficiency weave together. In addition to
understanding the fabric of mathematics, teachers need to know the misconceptions that
students have related to mathematics. The prior knowledge of students should be valued
in order for students to put their knowledge into context. Teachers need to deeply
understand how students will progress from a state of low knowledge to levels of mastery
(Carpenter, Fennema, Peterson & Carey, 1988).
Students approach problems depending on their understanding of mathematics,
and their levels of sophistication for logical thought. Different problem types reveal
28
areas of weakness, or strength, for students, while at the same time progressing their
knowledge in an organized manner. Carpenter et al. (1988; 1999) create several different
categorizations for problems. These categories are based on the progression children
make going from direct models of calculation, i.e. using blocks or fingers, to counting
methods, and eventually to abstract methods using symbols, i.e. 4 + __ = 10. Teachers
need to be able to recognize the state of knowledge for a particular problem type for each
student. Then, the teacher challenges a student with problems that stretches his or her
thinking, builds on prior knowledge, and makes connections to future areas of study. The
ability of a student to make connections to previously mastered content helps organize a
progression of mathematics. The careful design of problem types to match a student’s
knowledge, progress, and needs is a critical element of CGI (Carpenter et al., 1999), and
one that takes many years to perfect (Fennema et al., 1996; Loef Franke, Carpenter, Levi
& Fennema, 2001). Green Valley’s confidence in creating appropriate CGI problems
serves as another point of evaluation.
Teachers having reached the highest levels of proficiency with CGI instruction
tend to use teacher-created problems, designed for student needs, much more than a
textbook based curriculum (Fennema et al., 1996). Word problems serve as the primary
means of delivery, and students solve them regardless of their reading levels. There is a
premise that all students are able to think logically about mathematics in context, despite
any reading deficiencies (Carpenter et al., 1999). Students choose their own means to
solve problems, and determine whether to use direct modeling, counting, or abstract
methods; they do not rely on procedural solutions. Teachers encourage students to find
29
solutions on their own, or with the help of peers, and intervene with teacher direction as a
last result. Teachers design problem sets for students so they progress from using direct
models, to counting methods, and eventually more efficient symbolic algorithms.
Students solve a wide variety of problems, often set in a story context. This
variety begins with addition and subtraction problems in the primary grades, and evolves
into multi-step, multiple operation problems containing extraneous information in the
upper grades (Carpenter et al., 1999). These problem types are highly consistent with
those recommended by the National Research Council (2001; 2005) to build student
proficiency in mathematics. The style and variety of problems used, and the absence of a
textbook-based curriculum, will form metrics to measure Green Valley teachers’ practice.
Children build their mastery by constantly, and clearly, communicating their
solution strategies. Teachers of CGI expect students to present solutions so others can
follow their thinking. This may be expressed verbally, in writing, physically, or by
drawings. The teacher does not interject his or her opinion on the solution, until the
student has fully explained the logic. Clarifying questions are asked, in a non-judgmental
manner, to fully elicit students’ strategies and opinions regarding correctness. Other
students are expected to listen carefully, and to follow the reasoning of their classmates.
Students learn from one another in a highly collaborative manner. Students themselves
become the creators of knowledge, and the teacher serves as a facilitator of the process.
The explanations offered by children are the conduit of learning for both the students and
the teacher (Carpenter et al., 1999; Peterson, Fennema, Carpenter & Loef, 1989). The
30
frequent mathematics dialogue between students is central to both CGI and sociocultural
theories. Measurements of student exchange will be an important element for evaluation.
As a result of deep understanding, students and teachers knowledge can become
generative. This means they can apply knowledge in new ways, able to solve unfamiliar
problems in new ways (Carpenter, Blanton, Cobb, Loef Franke, Kaput & McClain,
2004). For students, they will develop higher levels of cognition as they compare
preexisting models to new situations. For teachers, they will become independent of the
CGI training program, developing their own unique methods to understand children’s
cognition and find solutions to help them reach proficiency. Masterful teachers of CGI
will demonstrate this knowledge in the problems present in their classroom, the
questioning techniques to elicit student explanations, even from English Learners and
reticent students, and will have a basis for lesson planning based on student needs.
Fennema et al. (1996) and Carpenter et al. (1999) use a five point rubric to
determine the level of CGI teacher proficiency. The levels are based on four criteria, (a)
the opportunities for children to solve problems within the curriculum, (b) the level to
which children share their thinking with peers and the teacher, (c) the teacher’s elicitation
and understanding of students’ thinking, and (d) the teacher’s use of students’ thinking as
a basis to make instructional decisions. Teachers that are not exhibiting these practices, or
do so rarely, are labeled level 1. Level 2 teachers demonstrate these practices
infrequently, or with limited application. Level 3 teachers exhibit the tenets of CGI, and
attempt to use these criteria frequently, but do not fully understand their students, or may
not provide enough variety in problems or questions to elicit higher levels of student
31
comprehension. The fourth and fifth levels, 4A and 4B, are highly consistent with CGI
expectations. These teachers fully embrace the philosophies of the program, and
demonstrate proficiency in their classroom interactions with students. The difference
between 4A and 4B lies within the instructional planning. Level 4A teachers tend to base
lesson plans on groups of student needs, while level 4B teachers base them on individual
student needs (Fennema et al., 1996; Carpenter et al., 1999). Teachers that reach level 3
within their first years of CGI tend to continuously gain in proficiency, moving toward
levels 4A or 4B in subsequent years (Fennema et al., 1996). They demonstrate
maintenance of these skills, having transferred their CGI workshop knowledge into long-
term learning. Their belief structure changes so these teachers see themselves as being
action researchers, focused on understanding their students’ thinking (Fennema et al.,
1996). Green Valley has used CGI for over a decade. Measuring teachers’ beliefs and
actions in the classroom according to this rubric will be a strong indicator on the level of
implementation for CGI.
Conclusions
Green Valley Unified chose a program based on a strong research base. CGI has
many traits consistent with recommendations for best instructional practice. Students are
provided many opportunities to solve a rich variety of problems, set in contexts familiar
to students. Teachers provide and encourage student engagement, through peer-based
sharing and collaborative learning. Teachers have detailed knowledge of student thinking
and use extensive questioning techniques to make thinking evident to all classroom
32
stakeholders. This knowledge of student thought is carefully and deliberately used to
develop lessons, activities, and problems that will enrich mathematical knowledge.
These skills are developed through a three-year professional development system that
includes multiple workshops spread through each year, and peer-based coaching.
Green Valley Unified invested tremendous resources and political capital to bring
CGI to all its elementary schools. The body of research discussed provides a framework
to evaluate how well Green Valley implemented CGI. This framework guides the
development of instruments that will measure the level of implementation for individual
teachers, and provided reference for qualities of good professional development. The
program itself may be of high quality, however the means for its use determines how well
students and teachers learn. Chapter 3 outlines the plan to make these comparisons,
through data collection and analysis.
33
Chapter 3
Research Methodology
This research study is an implementation evaluation of an elementary school
mathematics program called Cognitively Guided Instruction (CGI) within Green Valley
Unified School District. CGI is a professional development program for teachers, which
helps teachers base instruction on detailed knowledge of students’ levels of cognition
(Carpenter, Fennema, Loef Franke, Levi & Empson, 1999). It is an effective
instructional program for raising student achievement (Carpenter, Fenema, Peterson,
Chiang & Loef, 1989; Fennema, Carpenter, Franke, Levi, Jacobs & Empson, 1996;
Peterson, Carpenter & Fennema, 1989), hence a reason why Green Valley chose the
program to improve student performance in mathematics, as reported by district
administration.
Green Valley is a suburban, largely upper middle-class, Caucasian school district
located in Southern California (California Department of Education, Assessment,
Accountability and Awards Division, 2010). The district implemented CGI in all
elementary schools during the past five years, and has not formally evaluated the
efficiency nor effectiveness of the program. This study serves as a first step for Green
Valley to determine the program’s effectiveness, and it begins the process of
understanding how well the district implemented program. Future studies may evaluate
the relationship between CGI and increased mathematics scores, however this initial
34
study will not go that far. This case studies of two schools represents an initial
investigation into the level of implementation of CGI math programs. The research team
chose the participating schools and teachers based on recommendations from their
respective administrators. We performed case studies of schools and teachers that were
highly consistent with CGI practices, and ones that had room for improvement. District
administrators stated that all teachers and schools exhibit characteristics of full
implementation; however, the spectrum of quality was unknown. The study team
compared Green Valley’s professional development program to a highly effective
professional development model developed by Fennema et al. (1996).
As part of this study, we conducted interviews with district administrators, two
site principals, and five teachers during 2010 to determine the means of program
implementation. An analysis of data from these interviews compared the district’s
professional development model to Fennema et al. (1996). Additionally, we observed five
teachers’ classroom instruction and conducted follow-up interviews to determine their
level of fidelity to CGI. A rubric, based on those used by Carpenter et al. (1999), will
determine the teachers’ levels of CGI practice. We used these data to evaluate the
practice of teachers, which indicates the level of implementation.
This study will not evaluate the effects of CGI on student achievement, however
student performance on the CST helped draw comparisons of teachers’ practice. The
effectiveness of CGI within the school district, and its impact on algebra readiness, is an
area for future study.
35
This chapter outlines the methods used in conducting this study. It also describes
the methods used to select the sample population, and the instruments used. This chapter
reflects on the validity and reliability of the study and gives an outline of the data
collection and analysis processes.
Research Questions
The research questions for this study are:
1. How does the implementation of the CGI professional development program
at Green Valley USD compare to the model designed by Carpenter, et al.?
2. In what ways has the CGI model at GVUSD changed over time?
3. Are the actions of elementary teachers while teaching mathematics in the
classroom consistent with those of CGI?
Research Design
Pseudonyms replaced the names of all teachers, administrators, and schools.
These names come from a list of former US presidents. They have no relationship to any
employee or school in California; any similarities are coincidental. Green Valley Unified
School District is a fictitious name. It does not exist within California. Names of
researchers are real.
The first research question evaluated how well Green Valley implemented the
CGI professional develop system in comparison to the model developed by Carpenter et
al. (1999) and Fennema et al. (1996). Carpenter et al. (1999) and Fennema et al. (1996)
36
developed a model that used a combination of teacher workshops prior to the start of
school, follow-up workshops during the school year, and mentors to provide classroom
support and teacher feedback. They implemented this model over three years, with
differentiation for teachers based on CGI experience (Fennema et al., 1996). Data that
compared Green Valley to this model included
1. The number of teachers involved in professional development,
2. The quantity of training days offered prior to the start of school,
3. The quantity of release days offered during the school year,
4. The degree of participation among all teachers within a grade level, and
5. The degree of collaboration with coaches.
Informal interviews with the Assistant Superintendent of Curriculum, Dr.
Washington, in the spring of 2010 indicated that she and the school principals held most
of this information. Teachers were able to verify data pertaining to their own personal
experience. We conducted informal interviews and e-mail exchanges with Dr.
Washington from March to June 2010. A formal structure was not required, as there is
only one assistant superintendent to interview, which lead to consistency in data
collection. Dr. Washington helped the research team select two schools for in-depth
study, one that is considered a model of CGI implementation, Lincoln Elementary and
one which is considered to need improvement, Adams Elementary.
Because two principals were under study, we used a semi-structured interview in
order to maintain consistency in data collection. These interviews provided more detail
regarding each sites’ professional development program, and teacher participation, than
37
could be offered by Dr. Washington. A copy of the interview questions is included in
Appendix A. District staff provided additional professional development records related
to teacher attendance and dates of workshops to supplement the primary data collected
from Dr. Washington.
The research team also conducted interviews and surveyed second-grade teachers
at each school. These data provided verification of data previously obtained from
administrators related to the training programs used at Adams and Lincoln. Teachers
provided information related to the quantity of workshops and mentoring received. They
also gave perspective on their perceived quality of the professional development
program, as well as recommendations for improvement. Teachers provided data to
determine the level of implementation by peer teachers within their grade level and lower
grade levels. Because we studied five different teachers, a semi-formal interview and a
formal written survey were used. Copies of these instruments are in Appendix A. The
researchers used qualitative methods to develop case studies of each teacher and school
since the primary data were interviews and observation.
The second research question considered how the professional development
model changed over time. This question used the same research methods as before. We
used the same techniques and population to collect data, and modified questions to
develop a history of the program. District administration indicated the professional
development programs varied according to school site. Implementation history data at
both sites added relevancy to discrepancies between schools. It allowed the research
team to provide more depth in recommendations for the district.
38
The third research question measured the actions of teachers within classrooms.
Carpenter et al. (1999) provided numerous examples of teacher practices that are
consistent with CGI. Carpenter et al. (1999) also provided five levels of teacher
proficiency, ranging from 1 to 4B. The data collected is consistent with information used
by Carpenter et al. (1999) to rate teachers on this scale. The data collected included
1. The frequency of students explaining their solutions,
2. The frequency of opportunities for students to solve complex problems,
3. The variety of complex problems solved,
4. The ability of teachers to elicit student explanations,
5. The amount of direct explanation offered by the teacher,
6. The use of outside resources in planning instruction, i.e., textbooks and
district curricula,
7. The degree to which teachers use student understanding to plan
instruction, and
8. The degree to which teachers understand individual student
comprehension.
The researchers performed multiple classroom observations for a subset of
second-grade teachers at Adams and Lincoln Schools. We studied two teachers at
Adams, Mrs. Carter and Mrs. Grant. Three teachers participated from Lincoln, Mrs.
Kennedy, Mrs. Madison, and Mrs. Pierce. Principals recommended the teachers for
observations, interviews, and surveys. An effort was made to represent a spectrum of
implementation at each school whereupon teachers were role models for CGI instruction
39
or in need of improvement. Two researchers, Britt Dowdy and Kate Garfinkel, scripted
the classroom observations, and conducted follow-up interviews. Each researcher
conducted the follow-up analysis separately, in an effort to differentiate each
dissertation’s results.
I conducted a case study for each teacher, to determine the level of classroom
practice related to CGI. A rubric designed by Garfinkel and myself, based on Carpenter
et al. (1999), helped me focus observation notes, and compare classroom actions to best
CGI practice. A copy of this rubric is in Appendix A. I assigned teachers a score to
measure their level of CGI practice, with 1 the lowest and 4B the highest. Patton (2002)
cautions researchers to avoid quantitative analysis on small sample sizes, hence the need
for case studies.
Each teacher participated in a semi-structured follow-up interview to add context
to classroom observations. Appendix A contains the outline for this interview. These
interviews clarified questions from classroom observations, illustrated teacher
perceptions on their understanding of individual students, and explained the basis for
planning that day’s lesson. These last two pieces of data were essential to determine
whether a teacher is level 4A or 4B. A level 4A teacher bases decision on groups of
students, while a level 4B teacher uses individuals to guide instruction (Carpenter et al.,
1999). This information was not obtained through classroom observations, hence the
requirement for follow-up interviews.
Dowdy and Garfinkel gained approval for the study before conducting interviews,
observations, or surveys of staff within Green Valley. The University of Southern
40
California IRB panel reviewed and approved the methods for this study. Additionally the
superintendent and assistant superintendent of curriculum of Green Valley USD reviewed
and approved proposals for this study. The school district did not have an IRB system in
place, however the assistant superintendent informed the school board of the study.
Principals discussed the study with participating teachers before the USC research team
began direct communication. Teachers and principals understood their participation was
voluntary, did not reflect on their professional evaluations, and would remain
anonymous.
Population and Sample
The school district of Green Valley Unified serves a middle-class community that
has a majority of Caucasian students . The largest ethnic groups are composed of 59%
White, 19% Hispanic or Latino, and 11% Asian students in 2010. By comparison,
California had a state average of 28% White, 50% Hispanic or Latino, and 8% Asian
students. Green Valley has many fewer students from socioeconomically disadvantaged
homes, with 10% versus a state average of 56%. There are 5.3% English Learners in
comparison to California’s 33%, and Green Valley has 8.6% students with disabilities
versus 11% in California (CDE, AAAD, 2010). Teachers in the district are also a
majority of Caucasian (92.5%) with 4.1% Latino and 3.1% Asian (Educational Data
Partnership, 2010).
The research team is limited to two researchers, both of whom hold full time
employment outside of this project. It would not be practical to conduct multiple
41
observations of every teacher within six elementary schools, so the team chose to
purposefully select the sample of teachers for study. The district administration reported
that CGI is functioning relatively well in most classrooms, and wants all teachers to reach
high levels of mastery. When the majority of the larger population is performing
acceptably, Patton (2002) recommends samples based on extreme cases, i.e. instances of
poor performance or high performance. These extreme cases offer greater learning
opportunities for the researchers, and end clients (Patton, 2002). District administration
also recommended that the research team perform case studies of a school which
exemplifies CGI best practice and one in which improvements need to occur. For these
reasons, the research team decided to purposefully select two schools for case study.
District administration recommended Lincoln Elementary as an exemplar and Adams
Elementary as one needing further improvement.
We selected five teachers for participation, based on input from site principals at
these two schools. The five teachers studied are experienced in the classroom and
received at least three years of CGI training. They ranged from six to fourteen years
experience teaching elementary school, and five to eleven years teaching CGI. All
teachers hold permanent teaching credentials within the state of California, none are
teaching with emergency permits. This indicates that all five teachers are knowledgeable
of CGI, elementary curricula, and elementary pedagogy. We expected all five teachers to
be highly proficient in all areas of CGI instructional practice, however this was not the
case. Chapters 4 and 5 present and explore these findings.
42
Principals recommended second-grade teachers for classroom observations and
follow-up interviews for several reasons. Cognitively Guided Instruction targets students
in grades K-5, which limited the study to elementary schools. The California Standards
Test begins in second grade, therefore CST data is not available for teachers in
kindergarten or first-grade classrooms. While students in observed classrooms have not
taken the CST, all five teachers taught second grade previously. CST proficiency data
was available to compare these teachers’ student proficiencies, which provided context
for CGI implementation. Dr. Washington, the assistant superintendent, also
recommended second-grade classrooms for study so Green Valley may perform future
studies using CST data. Mrs. Johnson, Lincoln’s principal, recommended we observe
kindergarten through second grade because the third-grade CGI program is significantly
different in approach. Lincoln third-graders rely less upon manipulatives and mainly use
algorithms or paper-based strategies. Mrs. Johnson believed it was easier for the research
team to observe variations of student thought in the lower grades.
We attempted to observe complete second-grade teams at each school, however
this was not feasible. The entire Lincoln second-grade team participated in the study.
All three teachers taught only second grade. Adams had two pure second-grade teachers
that did not participate in the study. The principal did not elaborate why; we concluded
they were likely uncomfortable with CGI and did not want to be part of a research study,
although we were not able to talk directly with these teachers. The two participating
teachers from Adams taught combination classes, one combined first and second grades,
the other second and third grades. This meant we could not make direct comparisons of
43
curricula between classrooms, since they were not all using the same pacing guides. This
helped focus the data on CGI techniques, and not mathematics content.
Principals chosen for this study were a result of the schools recommended by the
assistant superintendent. Both principals are highly experienced; Mrs. Johnson has fifteen
years at Lincoln Elementary and Mrs. Franklin has four years at Adams Elementary with
additional principal experience outside the district. According to district administration,
the principal at Lincoln was responsible for bringing CGI to the school. The principal at
Adams had experience with CGI in another district, and came to Adams during the first
few years of CGI implementation. Each principal lead the professional development of
CGI for several years, and has detailed knowledge of its history at each site.
Instrumentation
Four main instruments collected data for this project. The first was a principal
interview used to understand the context, history, and overall implementation of CGI
within the two schools. Secondly, an observation rubric measured the instructional
practice of the five observed teachers. Third, a teacher interview facilitated follow-up
questioning. Lastly, a teacher survey provided historical information for each teacher’s
professional development and daily preparation of CGI lessons.
The principal interview helped answer the first two research questions, which are
focused on the structure of the current CGI professional development program and its
history of implementation. The principal interview collected data regarding which
teachers and coaches participated in training, the year in which professional development
44
began, the number of years CGI has been in place, the agency responsible for
professional development, the use of release days for training, and the degree of
participation for teachers and coaches. Because the data to be collected is specific to
Green Valley schools, an existing interview guide does not exist in the research base.
The interview was a guide of questions, and allowed a conversational flow during the
interview. Patton (2002) notes that interview guides are useful when time is limited for
an interview, the researcher knows which data to collect in advance of the interview, the
questioner needs flexibility to follow-up based on responses given, and interviewer will
be interviewing multiple subjects. The guide provided a framework to ensure consistent
data collection. The research team also designed the questions to be open ended, taking
care to avoid dichotomous responses. Patton (2002) recognizes that open-ended
questions generate a conversational tone, which lowers the anxiety of the subject, and
may lead to higher quality data. The interviewer must also take care to maintain
neutrality in the questions, so the researcher does not influence given answers (Patton,
2002). This research team was careful to not suggest a particular type of professional
development model, or a recommended us of teacher coaches, during the interview.
Appendix A provides the principal interview guide.
The two researchers scripted ten classroom observations, two observations per
teacher. Both researchers were present during each observation. Attempts were made to
script student-teacher interactions, samples of student work, student presentations,
students’ use of number choices, and students’ use of manipulatives to solve problems.
These observations were the primary vehicle to collect data for research questions three.
45
We developed a rubric to assess the level of instruction for each lesson. Classroom
observation notes were compared to this rubric to develop rating for each teacher, and
each lesson.
This rubric ranks teachers as level 1, 2, 3, 4A, or 4B consistent with those used by
Fennema et al (1996) and Carpenter et al. (1999). Appendix A provides a copy of this
rubric. Fennema et al. (1996) and Carpenter et al.’s (1999) levels of teacher proficiency
are based on four main areas, (a) opportunities for children to solve problems, (b)
children sharing their thinking with peers and the teacher, (c) teachers’ elicitation and
understanding of children’s thinking, and (d) teacher’ use of children’s thinking as a basis
for making instructional decisions. We evaluated each lesson according to these points.
Teachers received a score for each point, and an overall cumulative score.
We based these scores on the work of Fennema et al. (1996) and Carpenter et al.
(1999). Level 1 teachers do not demonstrate use of CGI practices within the classroom.
Level 2 teachers exhibit beginning practice of CGI techniques, and use them infrequently.
Level 3 teachers use most of the CGI practices, however not with the frequency of master
teachers nor with full understanding of student cognitive needs. Teachers on levels 4A
and 4B exhibit the characteristics of master teachers. The difference between these levels
is from teacher’s knowledge of individual students, and the use of this knowledge to plan
instruction. Level 4A teachers tend to think of students’ needs as groups within the
classroom, whereas level 4B teachers think of the individual needs of students (Fennema
et al., 1996; Carpenter et al., 1999).
46
We conducted semi-structured follow-up interviews with the observed teachers.
These interviews will help to clarify questions from the observed lessons. They helped
us understand how the teacher adapted lessons to meet students’ needs, and how the
teacher prepared lessons for students based on their current level of understanding. These
elements were necessary to determine whether a teacher is reaching the highest levels of
CGI implementation. Additional questions clarified teachers’ knowledge of students, and
enriched other data related to professional development practices at school sites. We
avoided a formal interview rubric due to time constraints for teachers. The research team
also needed flexibility to respond to data collected during observations. While the main
interview themes were known in advance, additional questions developed as a result of
the classroom observations. The absence of knowledge for the researchers, time
constraints for the interview, and the need to maintain a conversational tone with subjects
are reasons Patton (2002) gives for using a less formal interview guide. A sample
interview guide appears in Appendix A.
Data Collection
The research team collected data during the summer and fall of 2010. We
interviewed the assistant superintendent of curriculum, Dr. Washington, to gain
perspective on the overall history of CGI within the district, and to gain permission to
continue the project at local school sites. The University of Southern California’s IRB
panel granted approval before we conducted research. Dr. Washington assisted with
47
recommendations for the two case study schools, initiated contact with the site principals,
released professional development records and student CST data, and approved all data
instruments for use within the district. The communication with the assistant
superintendent was ongoing via e-mail and informal telephone conversations as questions
rose.
Interviews with site principals were approximately one hour in length, and
recorded with digital audio. We interviewed each principal once, with both researchers
present. We used a semi-structured interview to provide a framework for the
conversation. Both principals received a similar set of questions in order to ensure
consistency of data. Please see Appendix A for a list of questions used. Written notes
kept during the interview helped guide follow-up questions as they came up. The research
team transcribed audio recordings for further analysis. The interviews occurred before
each school year began, partially to maintain a block of time convenient for the
principals, and partially to gain their recommendations for teacher observations.
Site principals offered recommendations for teacher observations, and assisted
with coordinating observation times. We observed each teacher twice, with both
researchers present. Each observation lasted approximately 55 minutes, and covered one
CGI math lesson. Teachers chose the time and date of observations, however most
lessons occurred in the morning. One teacher taught math in afternoons because two
students had pull-out instruction for language arts each morning. All observations
occurred between October and December 2010.
48
After each observation, a follow-up interview occurred. On 8 of 10 occasions,
this interview was either immediately after the lesson, or at the end of the same school
day. The other two interviews happened via e-mail. The two teachers received the
interview questionnaire, completed them, and e-mailed them to the research team. These
interviews were not conducted in person due to time constraints on the teacher. We
digitally recorded all eight personal interviews, and transcribed them for further analysis.
Both researchers were present during each interview. The research team reviewed the
interview protocol in advance of each interview to ensure consistency of questioning.
Before each teacher’s second interview, the researchers collaboratively reviewed data
from prior interviews and observations to determine whether additional questions were
needed that were not on the protocol. Teachers’ first interviews typically lasted 45
minutes. The second sets of interviews were approximately 20-30 minutes each. The
interviews clarified questions that arose from observations and surveys, and determined
the level of teacher planning based on students’ abilities and understanding of
mathematics.
Validity and Reliability
The instruments have different methods to achieve validity and reliability. We
aligned the observation rubrics to similar ones designed, tested, and published by
Fennema, et al. as part of their 1996 study. Professional researchers referenced these
studies numerous times over the past fourteen years, a testament to the trust bestowed on
Fennema et al. by their peers. The research team developed the principal and teacher
49
interviews. We designed questions according to criteria enumerated by Patton (2002),
and described previously. In addition, they were tested on principals and teachers not
used in this study to validate them. Both members of the research team were present
during all interviews and electronic audio files were transcribed to verify reliability of
data collection. The transcriptions provided opportunity for more detailed analysis, and
offered multiple opportunities for review, which increased the reliability of data
collection.
The research team used training videos, ones similar to those used to train
teachers within Green Valley, to build expectation for classroom observations.
Additionally the researchers practiced recording observations from these videos, and in
classrooms not affiliated with this study to gain experience taking field notes. Both
researchers recorded independent field notes during classroom observations. These notes
were shared with each other to verify a complete picture was obtained. The comparison
of notes also provided opportunities for researchers to focus on different elements of the
classroom, such as recording the use of manipulatives across the entire room, scripting
interactions in two different corners of the room, and validating common recordings such
as student presentations. Having all members of the team go through the same training,
and perform the same observation increases the quality of the data collection process
(Patton, 2002).
Triangulation of data occurred by multiple team members observing and
recording data from the same events (Patton, 2002). Triangulation also occurred by
collecting data from multiple sources, and multiple events (Patton, 2002). For example,
50
research questions one and two explore the resources available to support CGI
implementation. We collected this data from multiple sources including informal
interviews with district administration, formal interviews with two different principals,
teacher surveys, formal interviews with five teachers, and professional development logs
kept by the district office.
There are limitations in the study related to the population of teachers and
students. Green Valley Unified is a suburban, mainly Caucasian and middle-class school
district. The school district has a stable staff in comparison with urban school districts
located in Southern California. This leads to a different ability for administration and
teachers to implement the CGI program in comparison to less stable districts. The results
of this study might not be replicable in districts with less stability in leadership or
instruction. Additionally students within Green Valley have higher degrees of social
capital, a result of few minorities and relative wealth, which may influence the abilities of
teachers to build confidence in CGI. The student demographics may also influence their
mathematical achievement. Parents of Green Valley students have greater levels of
education than nearby districts, which may indicate there are different norms for
education within homes.
Data Analysis
The qualitative data collected from administrator interviews analyzed for patterns
that occur within schools, and the district. A case study outlines the history of
51
implementation, and how it compares to the methods used by Fennema et al (1996) and
Carpenter, et al (1996).
Researchers analyzed teacher observation notes for the frequency of teacher and
student tasks. Both team members collaboratively rated each teacher after each
observation. We evaluated each of the four rubric points, and used them to obtain a
cumulative score. This score indicates the degree of sophistication for CGI activities,
with a 1 being lowest and 4B being the highest. Teacher interviews combined with the
observation data facilitated the assignment of a score. The interviews determined the
level of planning made by the teacher, which differentiated a score of 4A or 4B on the
rubric.
We performed all other forms of analysis individually so the school district might
receive unique sets of recommendations with different points of view.
Conclusion
This research is a case study of two schools within Green Valley Unified School
District. Analysis of these case studies determined the level of implementation of the
Cognitively Guided Instruction program within each site. The CGI professional
development model relies on extensive training and support for teachers spread over
multiple years. Workshops and reflective meetings should be conducted throughout the
school year, facilitated by mentors (Fennema et al., 1996). The case studies revealed the
extent to which Green Valley had similar models. The schools studied served as
representatives of best CGI practice and less desirable CGI practice within the district.
52
Interviews conducted with teachers, site principals and district administrators were used
to collect the necessary data to make the comparison.
Additionally this study analyzed the actions of teachers, measured during
classroom observations, and determined whether they were consistent with CGI
expectations. This study used a teacher observation rubric, based on Fennema et al.
(1996) and Carpenter et al.’s (1999) levels of teacher practice, to measure the degree of
CGI practice.
Chapter 4 will present the findings of this study. Chapter 5 will then analyze and
interpret these findings.
53
Chapter 4
Findings
Two researchers, Kate Garfinkel and Britt Dowdy collected data over a course of
6 months to determine the level of CGI implementation within Green Valley Unified
School District. Each researcher presented different findings from the collective data; this
chapter will present Dowdy’s case studies. The staff at Adams Elementary School and
Lincoln Elementary School provided the bulk of data for this study. We observed,
interviewed, and surveyed two Adams teachers. We used the same methods to collect
data from three additional teachers at Lincoln. Interviews with principals at both sites
revealed the history and expectations of CGI at each school. Informal interviews with the
Assistant Superintendent of Green Valley provided guidance and context for this study.
The research team developed a five-point rubric to evaluate the classroom
practices of each teacher. This rubric has levels 1, 2, 3, 4A, and 4B, with 1 being the
lowest and 4B the highest. This rubric is consistent with scales used by Fennema et. al.
(1996) and Carpenter et. al (1999). See chapter 2 for a more detailed description of
Carpenter et al.’s (1999) description, or the rubric itself in Appendix A. The works of
Fennema et. al. (1996) and Carpenter et. al. (1999) have been referenced many times
during the past 15 years, and is a testament to the scale’s validity and reliability. The
research team compared data from observations, interviews, and surveys to this rubric,
54
and collaboratively developed a rating for each teacher. Support for these ratings follows
each teacher’s case study.
The research team performed practice observations using non-CGI classrooms,
and CGI training videos to validate techniques. Both researchers observed lessons
together, and compared field notes to increase validity. Both researchers were present
when conducting interviews. They recorded the audio electronically, transcribed the
notes, and kept written field notes as backup during the interview. The interview
questions were prepared in advance, and followed a semi-structured protocol to also
increase validity. The researchers independently analyzed the data to maintain separate
dissertations. Independent analyses were intentional so that Green Valley could receive
two sets of recommendations from one set of data.
Case studies present the data, organized by district, then school, and individual
teacher. Following is a presentation of CST and API data for CGI instructional years to
add context to the school culture and expectations of students and teachers. The Green
Valley district office, and California Department of Education website yielded these data.
This study does not attempt to correlate the use of CGI to increased test scores.
Findings
Research data is presented in the following sections. Each school will have a
description of the student characteristics, and an overall picture of CGI use. This picture
includes a history of the path implementing CGI. The perceptions of CGI
implementation came from interviews with principals and teachers at each school. Case
55
studies for each teacher are grouped according to school. These case studies provide an
overview of the teachers experience, their professional development related to CGI, a
picture of the classroom environment, detailed data to support rubric evaluations, and
recommendations from each teacher for CGI improvements.
District Case Study – Green Valley Unified School District
Green Valley Unified School District is unique among California school districts.
The demographics are more homogeneous than most Californian districts, with 59%
White, 19% Hispanic or Latino, and 11% Asian students in 2010 (CDE, AAAD, 2010).
By comparison, California had a state average of 28% White, 50% Hispanic or Latino,
and 8% Asian students (CDE, AAAD, 2010). See Table 1 for details on other ethnicities.
Green Valley also has fewer students with social barriers to learning compared to the
state average. California has a significantly higher percentage of socioeconomically
disadvantaged students, English Learners, and students with disabilities. See Table 1 for
a comparison. A body of research (Bennett, 2001; Brooks-Gunn & Duncan, 1997;
Gallimore & Goldenberg, 2001; Stanton-Salazar, 1997) states that students of color, low-
SES homes, or English Learners have less social capital to access higher levels of
learning, and historically perform lower than middle-class White students. Green Valley
does not see a large population of at-risk students, which was a significant factor to
consider.
56
Table 1
Green Valley Unified School District Demographics 2006-2010
Green Valley Unified School District CA
Subgroup Category 2006 2007 2008 2009 2010 2010
Black or African American 3.0% 3.2% 3.8% 3.1% 3.4% 7.1%
American Indian or Alaska
Native 0.5% 0.4% 0.4% 0.3% 0.3% 0.7%
Asian 11.4% 11.5% 11.3% 10.6% 10.5% 8.2%
Filipino 2.1% 2.3% 2.8% 2.4% 2.8% 2.7%
Hispanic or Latino 13.4% 13.7% 14.4% 16.8% 18.9% 50.3%
Native Hawaiian or Pacific
Islander 0.8% 0.7% 0.8% 0.6% 0.5% 0.6%
White 67.6% 66.7% 64.9% 61.2% 59.2% 27.9%
Socioeconomically
Disadvantaged 8.1% 8.6% 8.2% 8.9% 9.8% 55.5%
English Learners 4.2% 4.6% 4.5% 5.0% 5.3% 33.5%
Students with Disabilities 9.0% 8.7% 9.5% 8.7% 8.6% 10.9%
Students in Green Valley have API scores significantly higher than the state
average. The district-wide Annual Performance Index (API) for all schools in 2010 is
904 (CDE, AAAD, 2010), a very high number when compared to the state and other
surrounding districts. In 2010, the state API was 767 for all grades kindergarten through
12
th
, and the highest unified school district adjacent to Green Valley was 822. See Table
2 for a complete list of adjacent district’s API scores. The districts vary from elementary
districts, having only grades K-6
th
, secondary districts with 7
th
-12
th
, and unified districts
with K-12
th
.
57
Table 2
Adjacent School Districts' Annual Performance Index (API) for 2010
Green Valley Unified (K-12) 904
California (K-12) 767
Elementary (K-6) District 1 889
Elementary (K-6) District 2 821
Elementary (K-6) District 3 877
Secondary (7-12) District 1 889
Secondary (7-12) District 2 748
Unified (K-12) District 1 802
Unified (K-12) District 2 822
Unified (K-12) District 3 759
Students from largely middle-class, White homes were a factor in higher
performance, however the district also invested resources toward teacher professional
development with a goal for all students to reach high levels of proficiency. Interviews
with the assistant superintendent indicated that professional development for teachers had
expenditures exceeding $20,000 each year from 2005-2010. Principal interviews
indicated that Lincoln and Adams each allocated $5,000 or more per year toward
professional development from 2005-2010. The use of CGI is one of these investments.
Future studies may explore whether CGI has contributed to Green Valley outperforming
expectations for similar students. Elementary students are highly proficient in
mathematics across the district. Figure 1 shows that one cohort of Green Valley
maintained high levels of proficiency since the district-wide implementation of CGI in
2005. The second-grade class of 2006-2007 received CGI instruction for two years
58
before the CST administration in May 2007. Eighty percent or more of students reached
advanced or proficient scores each year through fifth grade while receiving CGI
instruction. These high proficiency rates may be a combination of many factors, student
demographics being primary among them, yet they exist during years in which CGI
instruction was provided. GVUSD administrators stated a belief that CGI instruction is a
primary cause for high proficiency. This claim was not evaluated in this dissertation.
Green Valley has high expectations of all students, and teachers. There is room for
improvement; a primary goal of the district is for 100% of students to be proficient in
mathematics.
Figure 1. Green Valley Unified School District CST Math Proficiency 2007 Cohort.
This data is for the same group of students, those beginning second grade during the
2006-2007 school year. The data follows them through 5
th
grade.
150
200
250
300
350
400
450
500
550
600
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
2nd 2007 3rd 2008 4th 2009 5th 2010
CST Mean Math Score
Percent of Students
Grade and Year
% Advanced
% Proficient
% Basic
% Below Basic
% Far Below Basic
Mean Scale Score
59
We interviewed Dr. Washington, the Assistant Superintendent of Curriculum,
regarding these goals, and Green Valley’s efforts to reach them. Interviews with
classroom teachers and principals reiterated the statements made by Dr. Washington.
There are currently several efforts in place for elementary schools to improve student
learning. These include the use of Houghton-Mifflin as the primary adopted curriculum.
It is a state approved, traditional textbook and workbook program. The program serves
as guidance for the content to be taught. The district also has a common pacing plan
according to grade levels, so teachers may collaborate on common units of study.
District-wide benchmark assessments are now in place for most elementary grades.
Teachers helped write these assessments over recent years, and data collection is still in
the early phases. The district hopes to use this data to help drive instructional decisions
both within and between schools in future years. Some schools, such as Adams
Elementary, also use Fosnot math curriculum to help bridge traditional instructional
programs to CGI. According to Adams’ staff, Fosnot was a response to teachers’
discomfort developing CGI curriculum from scratch. It provides a structured problem-
solving curriculum that has some elements of CGI, such as complex problem solving,
structured number choices to foster deep numeracy skills, requirements for students to
document their work, and explanations of mathematics pedagogy to build teacher
knowledge. It replaces Houghton Mifflin or CGI lessons during the instructional day.
Other schools, such as Lincoln, use a computer-based program, ST Math from MIND
Research Institute, to build logical thinking skills and numeracy concepts, as reported by
the Lincoln staff. It is an additive program that does not replace math lessons. With such
60
a variety of mathematics programs in place, and to different extents within the two
schools studied, connections between student performance and CGI can not be made.
These are significant reasons why this study focused on the implementation of CGI, and
not its efficacy.
Dr. Washington made it clear that Cognitively Guided Instruction serves as the
primary focus for instructional improvement. The principal at Lincoln, Mrs. Johnson,
echoed this sentiment, however Adams’ principal, Mrs. Franklin, indicated there were
several different mathematics programs in place, and that CGI was one component of
mathematics improvement. Dr. Washington and the two principals stated that GVUSD
invested tens of thousands of dollars over the past five years to help train teachers. The
district maintained these expenditures even in the midst of the worst fiscal environment
for schools in decades. District professional development records showed all elementary
school teachers received at least three years of CGI workshops, most provided by the
County Department of Education (DOE). Teachers received varying levels of support
from mentors during this time, as indicated from interviews with the five teachers and
two principals. The first two years of district-wide implementation, 2005-2007, Dr.
Washington asked Lincoln’s teachers to mentor their colleagues in sister schools. There
were many logistical barriers to facilitate this, combined with differences of school
expectations, which lead the district to use DOE mentoring services from 2007-2011.
Lincoln was the pioneer in CGI, within both Green Valley and Southern
California. The school brought CGI to the region during the late 1990s, and developed
their expertise internally. Dr. Washington informed us that Lincoln developed a unique
61
culture regarding the use of CGI, where all teachers expected each other to use the
program. Mrs. Johnson verified these statements during her interview. The school
showed tremendous results in students’ math proficiency during the early 2000s, which
Mrs. Johnson attributed to the use of CGI and strong teacher collaboration. Green Valley
recognized these results and worked to expand CGI’s use among all sister schools in
2005. Dr. Washington indicated this expansion has been slow and required consistent
expectations for teachers to use CGI. She stated that school principals communicate
these district-wide expectations to their staffs, with varying degrees of success. Some
teachers are highly resistant to use a program dictated from the district office, however
with time, more teachers are using the program, based on principals’ and Dr.
Washington’s perceptions. These administrators also believe GVUSD student
improvements on state assessments are related to CGI’s more widespread use, yet this has
not been evaluated in a systematic manner. Dr. Washington maintained consistent vision
for the use of CGI over six years, and protected its professional development
expenditures even in the midst of major budget cuts.
School Case Study – Lincoln Elementary School
Lincoln Elementary School, a K-5 school, has similar demographic trends to the
district. The students are primarily White and middle-class. The school has a larger
population of Asians, and a smaller population of Hispanics or Latinos than the district
average. Similar to the district, all at-risk populations are significantly smaller than the
state average. See Table 3 for corresponding demographic data. The demographic data
62
would indicate that Lincoln should perform similar to district averages on the CST and
API, however the school outperforms the district. This case study explores these facts
later.
Table 3
Lincoln Demographics 2006-2010
Lincoln Elementary School GVUSD CA
Subgroup Category 2006 2007 2008 2009 2010 2010 2010
Black or African
American 1.4% 2.9% 2.7% 2.5% 2.9% 3.4% 7.1%
American Indian or
Alaska Native 0.0% 0.0% 0.0% 0.0% 0.0% 0.3% 0.7%
Asian 15.5% 14.8% 16.4% 18.3% 19.0% 10.5% 8.2%
Filipino 3.6% 3.9% 5.0% 4.8% 5.1% 2.8% 2.7%
Hispanic or Latino 13.6% 14.8% 14.6% 14.6% 4.6% 18.9% 50.3%
Native Hawaiian or
Pacific Islander 0.3% 0.3% 0.3% 0.3% 0.0% 0.5% 0.6%
White 64.8% 62.9% 59.9% 58.0% 53.4% 59.2% 27.9%
Socioeconomically
Disadvantaged 8.9% 7.3% 7.4% 4.8% 8.0% 9.8% 55.5%
English Learners 4.2% 2.9% 4.0% 4.0% 4.9% 5.3% 33.5%
Students with
Disabilities 6.4% 5.7% 9.5% 7.8% 8.8% 8.6% 10.9%
The following history of Lincoln is provided from the interview of Mrs. Johnson.
It is based on her perceptions of the school, and her accounts of changes that occurred
from 1997-2011. Lincoln Elementary School has a unique history, which created an
opportunity to build CGI into the school culture from the ground up. The school was
closed during the 1980s and early1990s due to declining district enrollment. It reopened
in 1996 in an effort to protect the property from being sold to housing developers.
63
Because student enrollment was still down, Green Valley Unified opened its enrollment
to neighboring cities, most of which were not as affluent as Green Valley. Lincoln had
100% of its students enroll from these outside communities for several years. The
principal, Mrs. Johnson, came from outside the district, which was not a typical practice
for Green Valley USD. All teachers hired were new to the district as well; Mrs. Johnson
was in direct control of the hiring process. This allowed Mrs. Johnson to establish a
school culture centered on collaborative, professional development, which was not
common for schools in general or Green Valley during the late 1990s and early 2000s.
Mrs. Johnson remained the principal at Lincoln since it opened 15 years ago. She was
able to hire every teacher on the campus, and created strong expectations and support
regarding CGI.
Mrs. Johnson began using release days for teachers to observe each other monthly
during the school’s second year, 1997. These observation days were mandatory for all
teachers. Mrs. Johnson worked with teachers to create a schedule for release days, and
teachers had flexibility in using them. Each teacher must use the equivalent of two full
days per year, and must have a plan for their use, developed in collaboration with Mrs.
Johnson. Mrs. Johnson stated that this practice continued through 2011. As a result of
peer observations in 1997, teachers learned that one teacher, Mrs. Roosevelt, was using
CGI as the foundation for math instruction. Her students were performing challenging
math problems in early grade levels, and this excited the staff. Other teachers wanted to
begin using CGI within their own classrooms. In 1998, Lincoln trained 8 of 12 teachers,
in the use of CGI. No other schools in Southern California were using CGI, so Mrs.
64
Roosevelt consulted with George W. Bright, from the University of North Carolina,
Greensboro for additional training resources. Mrs. Roosevelt had worked with Dr. Bright
in graduate school before moving to California. Dr. Bright served as an editor for
numerous books published by the National Council for Teachers of Mathematics
(NCTM) and had a professional relationship with Dr. Carpenter and Dr. Fennema, the
pioneers of CGI. The original cohort of CGI teachers spent long, voluntary hours
working with the Mrs. Johnson to train themselves. Mrs. Johnson recalled they would
often meet at her house after school, and practice problems until 11:00 at night.
Students in the CGI classrooms experienced success, according to Mrs. Johnson,
which caused an additional eight teachers to begin using CGI during 1999. Some of
these teachers were newly hired due to the growth in student population. At this point,
the majority of the staff used CGI, 17 of 20 teachers, and all 20 teachers continued to
have release days for peer collaboration and observation. Table 4 provides an overview
of how many teachers were using CGI and how many were on staff during these early
years. The non-CGI teachers observed the success of CGI students, and combined with
parental pressure for children to be in CGI classrooms, requested to be trained for the
2000-01 school year. Mrs. Johnson stated that in 2000 “even my bottom-of-the-barrel
teacher wanted in, and so we began to do a yearly training with Mrs. Roosevelt in the
lead” (Personal communication, June 30, 2010). Beginning in the year 2000, all staff
members were trained by Mrs. Roosevelt in the use of CGI. Mrs. Johnson continued to
provide release days for Mrs. Roosevelt to coach each teacher on the staff, and for
teachers to collaborate as grade level teams around the use of CGI each subsequent year.
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Teachers received 2-3 days of workshops centered on the philosophies of CGI, monthly
release days for grade level teams to collaborate, and the equivalent of two full days used
to observe peers and receive coaching. Lincoln performed almost all of this professional
development at their own site, used their own PTA funds to pay for substitutes, and did
not require additional conferences or workshops. The staff and principal felt that CGI
was an essential piece of Lincoln’s curriculum, and made it a requirement for new
teachers. Mrs. Pierce and Mrs. Madison were hired in 2000, and began their training
with this cohort.
Table 4
Early Use of CGI at Lincoln
School Year # Teachers Using
CGI
# of Teachers on
Staff
1996-1997 0 4
1997-1998 1 8
1998-1999 9 14
1999-2000 17 20
2000-2001 24 24
Mrs. Johnson stated in her interview that student proficiency grew from an
already high 75% advanced or proficient in mathematics, to 85% advanced or proficient
during the early 2000’s. Lincoln was now outperforming the rest of the district, which
created some tension within Green Valley USD, according to Mrs. Johnson.
Lincoln is the orphan child because you have to understand we rose from the
depths… the out of District people. [The students that no one else wanted and a
staff that was brand new.] So that's [strike] number one next to us. [Strike]
number two next to us is that we then outperformed everybody in the District. So
everybody was sitting on their laurels and here comes this Lincoln school that
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isn't doing it the Green Valley Way, and we left them in the dust and so, strike
number two. Strike number three was that we have no Special Ed on our campus
because they will not give us an RSP teacher because we're a school of choice. …
But the truth is that when we you have the staff that doesn’t have the crutch of
identifying kids for RSP you find alternative ways to meet needs, and so we keep
kids who really should be identified. (Johnson, personal communication, June 30,
2010)
Lincoln continued to be the only school within Green Valley using CGI. The
County Department of Education took notice of the high performance on state
assessments, and the county superintendent became interested in developing CGI
programs through the DOE, according to Mrs. Johnson’s interview. As a result, Mrs.
Roosevelt became a part-time trainer for the county. The DOE also hired former Lincoln
teacher, Mrs. Truman, to serve as a full-time trainer. Thus, Lincoln created the CGI
program used by the DOE. The county program grew over time, and was not able to use
teachers with CGI classroom experience as trainers. The only experienced teachers were
at Lincoln. Mrs. Johnson stated that Lincoln teachers continued to collaborate with DOE
and felt the quality of workshops and coaching was not representative of CGI. As a
result, teachers at Lincoln felt the DOE program was not consistent with their standards,
and pulled away from an association with the DOE training program. Mrs. Truman
continued to promote other schools to observe Lincoln teachers, and over time, schools
from Santa Cruz to San Diego began regularly observing Lincoln teachers. In 2009-10,
there were 365 visitations at Lincoln (Johnson, personal communication, June 30, 2010).
Through the generated buzz surrounding Lincoln, the Cotsen Foundation recruited
the school to participate in its two-year program around 2003. Cotsen designs its
program to help good schools to become great. Upon completing the two-year training
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program, Cotsen invited Lincoln to serve as a demonstration school. In turn, this doubled
Lincoln’s budget for substitute release days, and they were able to continue internal
workshops, observations, and peer mentoring.
Yet through these years, Lincoln was not effectively used to train other Green
Valley schools. Mrs. Pierce, a Lincoln teacher observed for this study, reported that she
arrived at a Green Valley school to observe and coach four different teachers, in a
collaborative effort to train other schools. Two of the teachers were not prepared for her
visitation, and did not allow her observation. Mrs. Johnson echoed the sentiment from
other Lincoln coaches, “I spent the day making sub plans. I went down to the site… and
when I walked into the teacher’s classrooms they said, ‘This is really not a good day’”
(Johnson, personal communication, June 30, 2010). After a few years of Lincoln
teachers coaching at other sites, and experiencing these types of frustrations, they “said
we will no longer coach outside of Lincoln. We’re done” (Johnson, personal
communication, June 30, 2010). This happened around 2006.
Mrs. Johnson also made a point to attend workshops with the Lincoln teachers.
She emphasized that site administrators must participate in these workshops, frequently,
in order to understand how to lead the staff.
I have sat through every single one of those trainings, every year, over and over
and over again until it was drilled in my head. I can recite those problem types in
my sleep, because I felt that as a site administrator that because I don’t have that
continual contact daily with kids, that I'd loose it if I didn’t refresh. …I tell that to
principals all the time. You've got to sit through the training with your teachers.
You've got to feel how hard this is. You have to understand what you're looking
at. You have to know what you're dealing with. I really encourage them not to go
through it just once, but you know as many times as they can. And to hold
teachers accountable when you're in the initial phases, and you’re just starting
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year one, that has to be on your all-bets-off radar. (Johnson, personal
communication, June 30, 2010)
Mrs. Johnson expects Lincoln’s teachers to use CGI, and to work hard to master
it. Her formal evaluations will always include a CGI lesson.
You can't assume the teachers are doing it because they won't if they're given the
opportunity not to. It is hard work. It is hard work. … At our school it was, “I'm
coming in to watch you do CGI. It is do or die. You need to be doing this…. And
if it's a bust of a lesson while I'm there, give me the high sign while I'm there and
I'll get up and I'll leave, and I'll come back. But I'll come back for more CGI, or
we'll do it together. (Johnson, personal communication, June 30, 2010)
Lincoln has two other major math initiatives, MIND Research Institute software,
and the use of math walls. The MIND Research Institute has a non-verbal software
package, called ST Math, using Jiji the penguin as the main character, to teach logical
reasoning through visual puzzles. Lincoln began using the software in 1997 as a
supplement to math instruction. Today teachers have between 60-90 minutes a week to
use the computer lab for Jiji, in addition to regular math lessons. Over the past four
years, teachers have developed common math wall activities that are used approximately
four times a week, for 15-30 minutes each. These math wall activities allow rug time for
students to discuss mathematical concepts. They may vary from problem solving, to
vocabulary, to numeracy skills. They are a key component for teaching mathematical
skills, and reinforce peer teaching among students. Teachers developed math wall
activities to coincide with CGI problems of the week. They also designed the activities to
reinforce the communication and problem solving skills used in CGI lessons. These two
programs are additive in nature to the CGI curriculum in place. It is evident that Lincoln
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has made mathematics instruction a priority through the allocation of instructional time,
resources, training, and expenditures.
In summary, the principal and teachers at Lincoln expect each other to use CGI at
least twice a week. The principal expects teachers to collaborate about the successes and
failures of CGI, not just plan CGI, during grade level release time. Teachers are provided
release time for their personal development, and for team development throughout each
year. There has been consistent leadership and CGI support for 15 years. The
perceptions of the principal, three second grade teachers, and Dr. Washington, are that all
teachers at the school have strong buy-in to using CGI, however not all teachers at the
school were observed or interviewed to validate these beliefs.
Teacher case study – Mrs. Kennedy at Lincoln Elementary School.
Summary.
We observed Mrs. Kennedy on two occasions, October 11, 2010 and November
15, 2010. After each observation, we interviewed her according to an informal protocol.
Mrs. Kennedy also completed a written survey. Based on these observations, she rates a
level 3 teacher according to our rubric. She could improve by allowing students greater
opportunities to communicate their solutions and strategies to one another, maintaining
higher time on task for students, and designing number choices that are consistent with
students’ abilities.
Mrs. Kennedy has 14 years teaching experience. Eleven of these years are in the
second grade, with five years at Lincoln school. She also taught three years of
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kindergarten outside California. She has been using CGI about 4 1/2 years, all at this
school site. Her current class is purely second grade.
Math curriculum.
Mrs. Kennedy teaches CGI lessons approximately two days per week. She also
teaches Houghton Mifflin, a district adopted workbook program, two days per week, and
uses the classroom math wall four times a week, for approximately 30 minutes each time.
CGI lessons are typically on Mondays and Tuesdays. Mrs. Kennedy prefers to have
back-to-back days of instruction for the same CGI problem set. Students solve the CGI
problem on day 1 and present solutions on day 2.
Mrs. Kennedy selects students for presentation based on the written work
produced day 1, which she takes home and reviews. According to her interview, she tries
to choose different, or more efficient solution strategies compared to the rest of class. She
may choose students depending on how she wants to redirect other students in the class.
She also expects all students to solve more than one number choice.
Mrs. Kennedy took time to set up CGI problems for students, but did not
consistently assist students with their solutions. She emphasized mathematical language
within the classroom, and took time so that students understood the language used in
problems. From classroom observations, she expected students to work independently;
they were not to engage in group activities. According to interviews, her belief for
individual work was based on information she received in CGI workshops. “But I always
have to understand that it is their chance to use what they know, their chance to get
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something from someone else is from the presentation” (Kennedy, personal
communication November 15, 2010). During the second observation, the teacher
circulated quickly through the classroom, and at least two students expressed frustrations,
without having their needs addressed. Mrs. Kennedy seemed focused on managing the
class, and did not spend quality time questioning students and their solutions.
Mrs. Kennedy referred to this year’s group of students as “a wily group”
(personal communication, November 15, 2010). She still seems very concerned about
classroom management activities in the class. Her CGI expectations were recorded,
which were on a flip chart placed in front of the class. They were as follows.
Expectations during CGI
o Use the bathroom at lunch
o Work independently
o Stay in your seat
o If you are stumped, keep trying
o Use your inside voice
o Listen attentively when someone presents
These expectations focused more on classroom management, and not different
strategies, multiple number choices, or documentation of thought. The lack of these
expectations was a contributing reason that Mrs. Kennedy does not rate a level 4A or 4B
teacher, however many of her classroom practices were consistent with proficient CGI
instruction. The section regarding her rubric score explores this ranking in detail.
CGI professional development.
Mrs. Kennedy discovered CGI when she began teaching at Lincoln in 2006-07.
She described her professional development during her interview as follows. The first
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year of training provided support from a mentor teacher on the school staff, Mrs.
Roosevelt, who was regarded a master teacher throughout the county. Mrs. Kennedy also
attended two to three workshops from the County Department of Education (DOE)
during her second year, 2007-08. The support from Mrs. Roosevelt continued through
years two and three. These three years included coaching from Mrs. Roosevelt,
observations of Mrs. Kennedy’s lessons from peer teachers and the principal, and
observations by Mrs. Kennedy of demonstration lessons. Mrs. Kennedy also read
Carpenter et al.’s (1999) book, Children’s Mathematics: Cognitively Guided Instruction
during her first year at Lincoln. During the current school year, 2010-11, she did not
attend any workshops, but did observe a peer teacher’s mathematics instruction.
Her peer teachers received a more formal three-year program, which had multi-
day workshops before the school year began, and follow-up one-day workshops during
each school year. During her interview, Mrs. Kennedy was concerned that her peers had
more training and support in their professional development than she. Most teachers at
Lincoln received training internally, lead by Mrs. Roosevelt. They did not attend
sessions led by the DOE.
The DOE workshops she attended were the same as those attended by Adams
teachers, as indicated in district professional development records. Mrs. Kennedy stated
in her interview that she considered the workshops offered through Lincoln, attended by
the majority of Lincoln’s teachers from 2000-05, to be superior to those offered by DOE.
Comments from Adams teachers reinforce this conclusion. The workshops built a
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foundation of knowledge related to CGI philosophies, problem design, pedagogy, and
self-reflection.
According to Mrs. Kennedy, the second-grade teachers met collaboratively to
decide which types of CGI problems to use, the pacing of those problem types, and to
offer support to each other writing these problems. The team did not collaborate on
which number choices should be used, the rationale behind them, or the specific learning
goals of each problem. These statements contradict those of the school principal. In Mrs.
Johnson’s interview, the principal stated that team meetings should be used exclusively
for analyzing student work, reflecting upon practice, and collaboratively developing CGI
problems, not for planning purposes. Mrs. Kennedy stated that the second-grade team set
a goal to build the use of algorithms, which was a continuation of work started in first
grade. Mrs. Kennedy did not discuss her students’ knowledge of CGI coming to her
classroom, or the use of CGI by other teachers at the school site.
CGI rubric evaluation.
Overall, Mrs. Kennedy is rated a level 3 teacher. A level 3 teacher uses the tenets
of CGI philosophies frequently, but does not provide enough variety in problems and
questioning strategies to elicit higher levels of student comprehension (Fennema et al.,
1996; Carpenter et al., 1999). These characteristics describe my observations of Mrs.
Kennedy during two lessons, and two post-lesson interviews.
Students solved complex, rich problems during each lesson, similar in nature to
those presented in CGI literature and County Department of Education workshops.
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Examples of the CGI problems used are in Appendix B. The first lesson’s CGI problem
used the context of Mrs. Kennedy eating cookies throughout the day. The second
lesson’s CGI problem referred to students in the class owning Silly Bands. In both
lessons, students were able to understand the context of the problems. Mrs. Kennedy also
used warm-up story problems to set up each lesson. Students solved two to four
problems, set in a variety of contexts, during each lesson. The curriculum centered
around these word problems, and was not dependent upon a textbook or scripted plan.
Mrs. Kennedy encouraged students to solve the problems using multiple strategies. She
reviewed recently used strategies during each warm-up, but did not emphasize students
using procedural solutions. These criteria were evidence of a teacher providing quality
opportunities for children to solve problems according to the rubric in Appendix A.
Mrs. Kennedy rated a level 3, instead of a level four, in this first piece of the
rubric. She had several strong areas of practice, which include the types of problems
used, and strategies used by children. The problems she developed, which are available
in Appendix B, were similar to those in CGI literature. The curriculum appeared to be
solely made of complex problems; it did not rely upon a textbook or external pacing
guide. Students had a variety of problems to solve, and used a variety of solutions.
There was not an emphasis on particular procedures needed to solve the problems.
Despite these positive traits, key elements were missing to warrant a level 4. The amount
of time students spent solving problems is a major flaw. During the first lesson, 8 of 24
students were not working on mathematics after 20 minutes. The class as a whole was in
a low level of chaos after 30 minutes. The second lesson had similar patterns; four
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students were finished with mathematics after 20 minutes, and seven were not focused on
math after 30 minutes. During the second observation, only two students worked on
math the full 40 minutes given. Mrs. Kennedy worked with their group during the last
five minutes of the lesson. Classroom management, keeping students focused on
mathematics, is an area of improvement for Mrs. Kennedy to reach 4A or 4B. This is
ironic since her written CGI expectations were founded on classroom management.
The second point of the rubric considers children sharing their thinking with peers
and teachers, and Mrs. Kennedy rates a 2. This is the weakest area for Mrs. Kennedy,
and limits her ability to fully implement CGI. She structures CGI lessons to last two
days; students solve the CGI problem the first day and present solutions on day two. This
limits students sharing their work to one day per week. The majority of class time, in
each lesson, was solving spent problems rather than explaining and discussing solutions.
During both lessons, the teacher presented student solutions to the class. Mrs. Kennedy
provided most of the verbal explanations, and used student strategies. The presenting
students had limited opportunities to speak for themselves. Students were not fluent in
reporting their solutions; Mrs. Kennedy presented them. A fluent student would be able
to describe one’s own solution using mathematical logic and vocabulary. The rest of the
class listened attentively on each occasion, but were not encouraged to ask questions.
The last two rubric points relate to a teacher’s elicitation and understanding of
children’s thinking, and the teacher’s use of children’s thinking as a basis for making
instructional decisions. On these two points, Mrs. Kennedy rates a 3. I observed her
questioning strategies in both lessons to be very detailed with individual students. Mrs.
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Kennedy interacted with approximately 1/3 of her students in a detailed manner during
each lesson. She questioned the students in detail to clearly understand their strategy, the
rationale for the strategy, and provided encouragement for students to discover the
correct answer. The teacher sought to root out the basis for misconceptions, and asked
leading questions to reteach students one-on-one. She used this knowledge of student
understanding to develop her CGI and warm-up problems. Mrs. Kennedy reported in her
interview that she has modified problem types during recent weeks to clarify student
misconceptions for specific individuals. However she was rushed during lessons in
supporting the class as a whole. As previously mentioned, there were instances when
students approached her for help, their questions were not addressed, and they left her
frustrated.
Number choices used in her CGI problems have wider levels of difficulty in
comparison to other observed teachers. Students were able to successfully solve the first
two number choices during each lesson, a positive, however the third number choice was
beyond the abilities of her students on both observations, a negative. Properly designed
CGI problems will challenge students, and provide levels of difficulty appropriate to each
individual. In the first lesson, the number choices for addition problems were as follows,
(13, 40), double-digit with no regrouping, (138, 261), triple-digit with no regrouping, and
((5x7)+(5x23), (4x75)+(9x3)). See Appendix B for a sample of the CGI problem.
Multiplication was not part of the second-grade curriculum, especially the multi-step
versions. These number choices would be appropriate for advanced third graders or
fourth graders. There was a large gap between using numbers without regrouping, and
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multi-step multiplication, which indicated the problem was not designed to scaffold
students from less rigorous problem sets to more challenging ones.
Students who attempted the third number choice during the first lesson were not
correct in their solutions, spent less than 20 minutes reaching answers, and spent an
additional 10 minutes not focused on mathematics. The second lesson had similar
number choices, with the third set using [(10x100)+(5x100)+(4x4), 1,000+(3x333)].
Again, students choosing the third set did not solve successfully, and spent less than 20
minutes on their solutions, and were off-task for 10 minutes. Mrs. Kennedy indicated in
her interview that she want to challenge student through the number choices, however the
number choice used each lesson was beyond her students’ abilities. She was very
thorough working to understand her students, however she did not apply this knowledge
in her problem design. The gap between knowledge of students and planning for student
activities was another reason for a level 3 rating.
Teacher perspectives on CGI.
Mrs. Kennedy stated in her interviews that she would like more time to
collaborate on recordkeeping and note taking. She wanted to develop checklists and
rubrics of skills for documenting student knowledge. She believed these records could
show growth of students’ thinking, both for parents and her personal edification. Her
colleague, Mrs. Madison, who was on the same grade level team, used teacher notes and
charts to document student progress for many years, however Mrs. Kennedy was unaware
of these forms.
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Mrs. Kennedy found CGI to be very useful teaching mathematics and believes it
is effective for students “only when the teacher is well organized, good at analyzing
student work, and skilled at choosing number choices and using strategic questioning”
(personal communication, November 15, 2010). According to her interview, she loves
CGI and believes she has greatly improved her questioning strategies of students.
Teacher case study – Mrs. Madison at Lincoln Elementary School.
Summary.
The study team observed two CGI lessons taught by Mrs. Madison, the first on
November 15, 2010 and the second on November 29, 2010. Mrs. Madison was not able
to have a live interview following the observation on November 15th, so she completed a
written interview questionnaire. She was interviewed in person on November 29
th
. Mrs.
Madison also completed a written survey.
Mrs. Madison had 14 years teaching experience at the time of this study, all in
second grade. She taught three years in a different school district, and 11 years at
Lincoln. Overall, she rated a level 3 teacher, and was close to level 4A/4B. Mrs.
Madison reported the school used CGI throughout all grade levels. She believed that
teachers received quality support learning CGI, and they would continue to benefit from
additional peer mentoring. Following is a more detailed description of these observations
and interview data. Her class was composed of only second grade students.
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Math curriculum.
Mrs. Madison taught CGI lessons two consecutive days a week, and sometimes
three days a week. She reported that students typically solved the CGI problem the first
day, and presented their solutions the second day. Mrs. Madison stated that she took
home written student work from day 1, and used it to revise the lesson for day 2. We
observed students solving and presenting solutions during the same lesson on each visit.
She taught mathematics exclusively during the afternoon because two students received
language arts instruction as part of a pull-out program each morning.
She based her classroom CGI expectations on students solving problems, and
recording their strategies. These expectations were on display during the lesson, and Mrs.
Madison verbally reviewed them with students at the beginning of class, and they were
on a flip chart for students to read. Mrs. Madison’s’ expectations read as follows.
CGI Steps
1. Highlight important information.
2. Solve the word problem using a strategy. (Decomposing, manipulatives, a
picture, etc.)
3. Record a strategy. (Leave it on your desk!)
4. Write an equation (vertically) to match.
5. Solve the challenge problem or...
! Solve the first problem using a different number choice.
! Solve the first problem using a different strategy.
Mrs. Madison set an expectation for students to solve the CGI problems with
multiple strategies, or to solve additional problems that are challenging. She also
reminded students to “pick a number choice that's suitable for you” (Madison, personal
communication, November, 15, 2010).
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Mrs. Madison taught the district-adopted Houghton Mifflin program on non-CGI
days. She stated that it was a workbook program, traditional in its approach toward
problem solving. Problems were given with guidance for one solution strategy. Students
also received instruction via math wall activities four days a week, as reported by Mrs.
Madison and Mrs. Johnson, the principal. Mrs. Madison claimed that she used the math
wall on days when it would reinforce skills needed for the CGI problem given on the
same day.
Mrs. Madison reported that the second-grade team meets in June each year to
create a yearly plan for CGI. They planned the sequence of problem types by month.
Each month they met to discuss the problems used. She stated that each teacher may
have different number choices, but were typically similar. Mrs. Madison believed that
Lincoln teachers consistently used CGI two days per week through all grade levels. She
observed that many of her students “have kind of learned a lot [about how to share their
thinking] in kindergarten and first-grade. So they come in knowing how to decompose
numbers… and how to teach the rest of the kids” (Madison, personal communication,
November, 15, 2010).
CGI Professional Development
Mrs. Madison received CGI training through Mrs. Roosevelt, a mentor teacher at
her site. The year in which she was hired, 2000-01, saw nine teachers join the Lincoln
staff. The school was in its third year of CGI implementation, and developed a school-
wide training program for all teachers. According to interviews with Mrs. Madison and
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Mrs. Johnson, the training included monthly observations and debriefing during the first
few years. The coaching occurred within each teacher’s classroom. Mrs. Madison also
observed the master teacher’s classroom. Mrs. Madison thought it was beneficial to have
access to CGI research, training in CGI philosophy, and coaching within her own
classroom. “The questioning piece of CGI can be a difficult task. [Mrs. Roosevelt]
would jump in and coach us along… I think [having Mrs. Roosevelt coming to her
classroom] is a crucial piece to assist in developing as a CGI teacher” (Madison, personal
communication, November, 15, 2010). Currently CGI professional development consists
of team collaboration, and outside services such as the Cotsen Foundation and the County
Department of Education, according to Mrs. Madison and Mrs. Johnson. Teachers did
not received mentor coaching in recent years, according to interviews with Mrs. Madison,
Mrs. Kennedy, and Mrs. Pierce.
CGI Rubric Evaluation
Overall Mrs. Madison rated a level 3, emerging to 4A/4B. There was room for
improvement, especially with students’ opportunities to share with each other, which is a
cornerstone of CGI.
Children in Mrs. Madison’s classroom had opportunities to solve a variety of rich
CGI problems. I observed students solving one main CGI problem, and a second
challenge problem during each lesson. Samples of the problems are in Appendix B.
Students had choice in the number sets used, which seemed to be appropriate to their skill
levels. The first lesson was a subtraction problem with number choices (92, 56), double-
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digit with single regrouping, (874, 427), triple-digit with single regrouping, and (2x437,
4x129), groups of numbers leading to multiplication. One student chose the first set, 11
students the second set, and seven students the third set. The second lesson had two sets
of numbers for subtraction, (956, 328), triple-digit with single regrouping and (2x376,
4x51). In this problem, 11 students chose the first set and seven the second set. Students
completed their problems in the given amount of time during each lesson, and a majority
worked on either the challenge problem or the original number choice with a second
strategy.
Students solved problems for 20 minutes in each lesson, and they mainly
remained on mathematical tasks each lesson. During the second lesson, five students
finished the problem within eight minutes and began reading. Mrs. Madison did not
focus on procedural solutions; rather she encouraged students to solve problems
according to their own methods. Students used a variety of approaches such as base ten
blocks, counters, 100s charts, algorithms, and drawings to find solutions. The first
problem asked students to consider the number of turkeys in a field; the lesson occurred
just before Thanksgiving break. The second lesson presented the number of snowmen
that might melt on a local mountain. The CGI problems used a variety of contexts to help
students understand mathematics. These observations were consistent with a level 4A/4B
teacher for the first point in the rubric.
The second point in the rubric, children sharing their thinking with peers and
teachers, had room for improvement. Mrs. Madison was consistent with a level 3
teacher. My observations noted many successful traits, however students did not
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communicate with each other as effectively as a level 4A/4B classroom. The first lesson
allowed 30 minutes for student presentations; the second lesson allowed 40 minutes. The
teacher helped students explain solutions to herself, however students were expected to
work independently and were not able to explain to each other. This observation ran
counter to quality CGI. During the second observation, students seemed embarrassed to
ask their peers for help on two occasions. At the lesson’s start, Mrs. Madison directed the
class to remain in their seats and work independently. Students were not able to talk a
great deal about the problem, or their strategies, to peers. This made it difficult to
observe their level of mathematical fluency. Mrs. Madison expected children to write
about solution in multiple ways. The class expectations, on display at the front of the
room, and reviewed at the start of each lesson, stated “record your strategy” and “write an
equation (vertically) to match.” Mrs. Madison collected these written strategies at the
end of each lesson. During her interview, she reported she analyzes these written
solutions each day after class. Mrs. Madison expected students to listen to their peers
during presentations.
Mrs. Madison demonstrated an advanced ability to elicit and understand
children’s thinking, the third rubric point, and use their thinking as a basis for making
instructional decisions, the fourth point. She rated a level 4B, the highest level, on each
point. During each lesson, she constantly questioned students so she could fully
understand that particular student’s strategy. She interacted one-on-one with five to six
students during each solving session. When students presented solution, the teacher
questioned presenter’s solutions until the class clearly understood.
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Mrs. Madison used students’ written solutions to guide her instruction. She
reported that a notebook was kept to record student explanations, the number sets chosen,
and the solution’s correctness. I observed that the second lesson did not include the
easier double-digit number choice that was present in the first lesson. This may have
been a result of Mrs. Madison’s analysis, however I did not question her about this. Only
one student used the simpler number choice during the first lesson, which indicated the
class was ready for more challenging choices. During her interview, Mrs. Madison
referred to multiple students’ strengths and weaknesses. She talked about her response to
their needs both in class, and in her plan of action for the next CGI problems. She
indicated a detailed knowledge of each student’s mathematical abilities, and how each
would respond to the CGI problems used in class.
Teacher perspectives on CGI.
Mrs. Madison reported that CGI was useful for teaching students mathematics
because “it provides a learning environment in which students take what they know and
build on it” (Madison, personal communication, November 15, 2010). She believed that
CGI was an important teaching philosophy, yet challenging to master. She stated that
teacher mentoring was the most beneficial method to help her improve her technique,
especially with questioning strategies. She believed that Lincoln needed to promote
teachers observing one another, and provide feedback regarding each others questioning
strategies.
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Teacher case study – Mrs. Pierce at Lincoln Elementary School.
Summary.
The research team observed Mrs. Pierce teach two different CGI lessons. The
first was October 10, 2010 and the second on November 15, 2010. The team interviewed
her twice, each on the same day as the observation. Mrs. Pierce also completed a written
survey.
Overall, Mrs. Pierce was one of the strongest CGI teachers observed in this study.
She rated a level 4B teacher, the highest level. She believed that a very high percentage
of teachers at Lincoln are using CGI at least twice a week. She would like to see greater
reciprocity among the elementary schools sharing CGI knowledge.
Mrs. Pierce taught a total of 13 years, eleven of which have been at Lincoln. She
has been using CGI for 11 years. Lincoln introduced her to CGI, and provided her
training for using CGI. Currently she is teaching a pure second grade class.
Math curriculum.
The math curriculum in Mrs. Pierce’s class typically uses CGI problems two days
a week, the math wall four days a week, and Houghton Mifflin three days a week. CGI
lessons were typically taught on Mondays and Thursdays. Mrs. Pierce offered two
different CGI lessons during the week, while her peers typically had one CGI lesson
spread over two days. Students in her classroom are strongly encouraged to interact with
each other and use each other for learning support. She expects students to teach one
another throughout the lesson. For example during the second observation, Mrs. Pierce
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suggested that a student help a struggling neighboring student. She reported teachers at
Lincoln were
Very dedicated [to using CGI]. If I had to give a percent… in the primary
grades… I would say probably 90% of teachers do CGI. Because K does it, first
does it, second does it. The kids come in well prepared if they have been here
since kindergarten. (Pierce, personal communication, November 15, 2010)
CGI professional development.
Lincoln Elementary introduced Mrs. Pierce to CGI. The school hired
approximately nine teachers the year Mrs. Pierce joined the staff, 2000-01, and offered a
formal training program. She recalled that she attended workshops regarding the
fundamentals of CGI, and received coaching based on classroom observations. Mrs.
Roosevelt coached Mrs. Pierce for two years. Mrs. Pierce no longer received formal
coaching during 2010-11, however Mrs. Roosevelt offered informal support as needed.,
as described by Mrs. Pierce I was not able to interview Mrs. Roosevelt to determine the
level of support she offered her peers. Mrs. Pierce stated that she was comfortable using
CGI, but reported that the first few years were discouraging.
I felt like a lot of times things were going wrong, and then I finally just gave
myself permission, for the year, to try and go with the flow and experience it. It
made me feel a bit better, and better for the kids too.… The first year was rough
and I would say by year three I felt a lot more confident with the problem types. I
would anticipate what the students would do in second grade about year three and
from then on it just got better and better. (Pierce, personal communication,
November 15, 2010)
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CGI rubric evaluation.
Mrs. Pierce was very strong in her application of CGI. She rated a level 4B
teacher overall, and was one of the strongest teachers of the five observed for this study.
She had personal goals for improvement, however I observed her to execute all areas of
the rubric successfully.
Mrs. Pierce rated a level 4 on the first rubric point, opportunities for children to
solve problems. The first observation consisted of one CGI problem focused on addition.
It had number choices of (59, 34), double-digit with single regrouping, (467, 328), triple-
digit with single regrouping, and (586, 674), triple-digit with double regrouping. See
Appendix A for an example of the CGI problem used. The second lesson focused on
subtraction and used (98, 45), double-digit without regrouping, (93, 46), double-digit
with single regrouping, and (231, 152), double-digit with double regrouping, for the
number choices. These choices indicated a clear progression of skill for students from
little or no regrouping to more advanced regrouping. Unlike Mrs. Kennedy and Mrs.
Madison’s number choices, they remained focused on subtraction, and did not mix
multiplication. This separation of skills was consistent with second-grade curricular
expectations.
Students had 20 minutes in each lesson to solve the CGI problem. All students
remained engaged during the allocated time, and were able to solve the problem in a
allotted amount of time. The problems were set in contexts using teachers’ and students’
names. The first lesson referenced pumpkin carving and the unknown quantities of
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pumpkin seeds; the second used a comparison of rock collections.. Mrs. Pierce presented
different CGI problems during each lesson; meaning students solved and presented two
unique CGI problems each week.
Students actively shared their solutions with each other and the teacher during the
lesson. They collaboratively solved the problems, asked each other for help independent
of the teacher, used mathematical language and models in their explanations, and
questioned their peers’ solutions. Each lesson dedicated 30 minutes to student
presentations. The teacher asked questions to help students explain their solutions, but
allowed the student to lead the presentation. Students presented a variety of solution
strategies for each number choice. Students fluently used mathematical language, and if
they did not, Mrs. Pierce asked the class for the appropriate vocabulary. She expected all
students to communicate their thinking both orally and in writing. During the first lesson,
Mrs. Pierce instructed all students to walk around the room and review written work on
display at each desk. This presented every student an opportunity to share his or her own
thoughts, even if they were not presenting formally. Mrs. Pierce rates a level 4A/4B in
this area of the rubric.
The third rubric point, teacher’s elicitation and understanding of children’s
thinking was a level 4B. Mrs. Pierce questioned students individually during the first half
of the lesson to understand their solutions and misconceptions. According to her
interview, she used this time to decide which students would present in order to clear up
common problems, or to show unique solutions that benefitted others. Mrs. Pierce
designed the day’s problem to help students progress in their thinking, and to elicit
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specific responses from them. In her October 11 interview she stated, “I felt that the
students were able to see more efficient thinking from one another in today’s lesson.”
Mrs. Pierce uses children’s thinking as a basis for her instructional decisions, and
rates a level 4B in this area of the rubric. During her interview, she referenced specific
student’s solutions and misconceptions as a basis for problem design, and selection for
presentation. She probed with her questions so she could understand what changes were
made for future lessons.
Teacher perspectives on CGI.
The district recruited Mrs. Pierce to serve as a mentor to other schools, after she
completed four years CGI experience, and her principal thought she was proficient with
CGI. By her own account, she found this to be discouraging.
The school where I was [coaching] was all over the place in terms of what
teachers knew about CGI, and they did not know how comfortable they were.…
One experience… I got to the school site and was given the schedule of who was
doing what, and then I showed up in the teacher's classroom, and they were like
‘What, you are here? You are coaching me?’ And I was like, I could be in my
own classroom right now. The district is trying to get more on board with all
schools using CGI, but it really varies site to site. In fairness, Lincoln is for sure
the strongest site, at least in the primary grades we are all doing it. (Pierce,
personal communication, November 15, 2010)
Additionally Mrs. Pierce was frustrated by the lack of collaboration among Green
Valley schools. She stated that Lincoln invited non-Green Valley schools to visit their
classrooms, seeking how to effectively teach CGI. Some schools developed an informal
collaboration, occasionally sharing ideas and inviting Lincoln teacher for reciprocal
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visits. Her perception was Green Valley USD had not worked with Lincoln teachers in a
similar manner.
It is unfortunate that in our district we have not seen the reciprocity with other
school sites with CGI problem solving. After your door has been opened so many
times, you like to be asked into somebody else's classroom just to see. I would
love the opportunity to do some of that, and see more of that, and learn from
others. You can always learn no matter where a teacher is with CGI. (Pierce,
personal communication, November 15, 2010)
Lincoln teacher case studies comparison.
The case studies are summarized in Table 5 for comparison purposes. Dr.
Washington and the Lincoln staff perceive the school to be very high performing with
regards to Cognitively Guided Instruction. This perception built an expectation to see a
majority of level 4A or 4B teachers, however this was not the case. Only Mrs. Pierce
rated above level 3. Additionally, a school fully committed to CGI would have ongoing
coaching, peer mentoring, and collaboration of problem design. Interviews with the
teachers indicate that coaching and mentoring activities have not regularly occurred for
the past two years. Evidence from number choices in CGI problems indicated that
teachers are not effectively collaborating about student needs during problem design.
Two teachers used multiplication as part of their number choices, and one did not.
Multiplication was not part of the second-grade standards.
Mrs. Kennedy stated a need for a record-keeping system, while her grade-level
colleague, Mrs. Madison, used a system that would meet Mrs. Kennedy’s needs. The
lack of resource sharing also demonstrates that CGI collaboration is not occurring at a
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high quality level, which is counter to expectations based on administrators’ perceptions
of Lincoln.
Table 5
Lincoln Teacher Case Study Comparison
Mrs. Kennedy Mrs. Madison Mrs. Pierce
4 ! years of CGI 11 years of CGI 11 years of CGI
Overall Level 3 (Weak) Overall Level 3 (Strong) Overall Level 4B
Teaches 2 CGI lessons per week
on back to back days - First day
solving, Second day presenting
Teaches 2 CGI lessons per week
on back to back days - First day
solving, Second day presenting
Teaches 2 unique CGI lessons
per week on non-consecutive
days
Initial PD from DOE workshops
and Lincoln mentoring
Initial PD from Lincoln
workshops and Lincoln
mentoring
Initial PD from Lincoln
workshops and Lincoln
mentoring
Currently receives little coaching Currently receives little coaching Currently receives little coaching
Rubric points
1. Student time solving is low
1. Curriculum is based on
problem solving
2. Student collaboration is low
2. Student communication to
peers is low
2. Teacher presents student
solutions
3. Teacher questioning of
students is good
4. Teacher knowledge of
individuals is high
4. Number choices do not fit
student needs – Used
advanced multiplication
Rubric points
1. Student time solving is high
1. Curriculum is based on
problem solving
2. Student collaboration is low
2. Student communication to
peers is low
2. Students present solutions
3. Teacher questioning of
students is good
4. Teacher knowledge of
individuals is high
4. Number choices fit student
needs – Used simple
multiplication
4. Uses record keeping system to
document CGI student
learning
Rubric points
1. Student time solving is high
1. Curriculum is based on
problem solving
2. Student collaboration is high
2. Student communication to
peers is high
2. Students present solutions
3. Teacher questioning of
students is good
4. Teacher knowledge of
individuals is high
4. Number choices fit student
needs
Wants record-keeping system Wants more teacher mentoring Wants greater collaboration
between schools
There were discrepancies between Mrs. Kennedy’s classroom practice and her
colleagues. Mrs. Madison and Mrs. Pierce developed number choices that were
reflective of students’ abilities. These two teachers kept students on task for longer
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periods of time, and encouraged their students to actively present their own solutions to
the class. Additionally Mrs. Pierce was the only teacher that actively encouraged her
students to collaboratively solve CGI problems.
These differences may indicate that the second-grade team was not focused on
teacher practice during collaborative meetings. Or they may indicate that Mrs. Kennedy
needed additional mentoring support, since she taught fewer years at Lincoln than either
Mrs. Madison or Mrs. Pierce, therefore she received fewer years of collaborative
guidance. The tenets of ongoing, professional reflection, which are part of successful
CGI programs, were not as strong as expected. Lincoln was presented by administrators
to have strong professional dialogue, however the lack of consistent number choices,
non-communication regarding data collection forms, and differences of opinion for
student collaboration indicate these topics are not discussed by the second-grade team.
Lincoln performance data.
Lincoln outperforms the district average on standardized assessments, and
significantly outperforms the state average. Table 6 shows these trends since 2007. Mrs.
Johnson reported the school began with 75% of student advanced or proficient in 2000,
however this data was not available. The high performance of Lincoln students is another
piece of evidence to show the school’s uniqueness.
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Table 6
Percent of Lincoln Second-Graders Advanced or Proficient on CST Math
2007 2008 2009 2010
California 59% 59% 63% 62%
Green Valley USD 84% 90% 89% 92%
Lincoln Elementary 88% 94% 97% 93%
Mrs. Kennedy 70% 95% 95% 84%
Mrs. Madison 95% 89% 95% 96%
Mrs. Pierce 100% 95% 100% 92%
A 2006-2007cohort of second-graders consistently maintained high mathematics
proficiencies while at Lincoln. These students had greater than 90% advanced or
proficient results on the CST from second to fifth grades (see Figure 2.). This is higher
than proficiencies of the district-wide cohort, which was closer to 80%, presented in
Figure 1 on preceding pages.
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Figure 2. Lincoln CST Math Proficiency 2007 Cohort. This data is for the same group of
students, those beginning second grade during the 2006-2007 school year. The data
follows them through 5
th
grade.
Lincoln also maintained overall state scores well above state expectations.
California’s Annual Performance Index (API) ranges from 200-1000, with an expectation
that all schools reach the level of 800. Lincoln consistently maintained scores near or
above 950 since 2003 (see Figure 3).
150
200
250
300
350
400
450
500
550
600
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
2nd 2007 3rd 2008 4th 2009 5th 2010
CST Mean Math Score
Percent of Students
Grade and Year
% Advanced
% Proficient
% Basic
% Below Basic
% Far Below Basic
Mean Scale Score
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Figure 3. Lincoln API Data 2000-2010. Gaps in trend lines for Asian and Hispanic or
Latino students are from years these populations decreased in size. They were not
statistically significant subgroups and did not receive an API.
The other school studied, Adams Elementary, had similar and different traits
compared to Lincoln. The school had different demographic than Lincoln and the
district. Accordingly, it had different performance on state assessments. The school had
a much different path to implement CGI. The Adams teachers had a different
professional development experience, yet they still had measures of success related to
CGI instruction. Following is a case study of the school and the observed teachers. After
the Adams case study, both schools will be compared.
700
750
800
850
900
950
1000
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
School-wide Asian Hispanic or Latino
White California Expectation
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School Case Study – Adams Elementary School
Adams Elementary had different demographics than Lincoln, or the district.
There were more at-risk students, with increased numbers of Hispanic or Latino students,
students from socioeconomically disadvantaged homes, and English Learners. See Table
7 for a demographic profile. As was the case in the district profile, these numbers were
significantly smaller than the state average.
Table 7
Adams Demographics 2006-2010
Adams Elementary GVUSD CA
Subgroup Categories 2006 2007 2008 2009 2010 2010 2010
Black or African
American 4.3% 3.9% 4.5% 5.0% 5.6% 3.4% 7.1%
American Indian or
Alaska Native 0.9% 1.0% 0.7% 0.7% 0.9% 0.3% 0.7%
Asian 18.0% 15.9% 14.6% 14.6% 15.7% 10.5% 8.2%
Filipino 3.4% 4.6% 4.7% 5.7% 5.2% 2.8% 2.7%
Hispanic or Latino 24.8% 23.9% 22.9% 23.6% 27.7% 18.9% 50.3%
Native Hawaiian or
Pacific Islander 0.5% 0.2% 0.2% 0.0% 0.0% 0.5% 0.6%
White 46.6% 48.8% 50.2% 46.8% 44.4% 59.2% 27.9%
Socioeconomically
Disadvantaged 23.2% 21.0% 20.5% 20.5% 24.2% 9.8% 55.5%
English Learners 11.1% 15.0% 14.9% 19.1% 18.8% 5.3% 33.5%
Students with
Disabilities 7.5% 8.2% 8.5% 8.4% 8.2% 8.6% 10.9%
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The path of CGI implementation at Adams Elementary School was quite different
than Lincoln’s. Following is an account of Adams’ history given in Mrs. Franklin’s, the
principal’s, interview. Adams began using CGI in 2005 under a former principal. The
school used teachers from Lincoln, notably Mrs. Roosevelt, as key trainers and coaches
for three years. Mrs. Franklin took the helm in 2007 after the second year of CGI
implementation at Adams. She had prior principal experience bringing CGI to a school
in a different school district. Her prior school phased in CGI over multiple years based
on teacher buy-in. The first year began with a two strong grade levels eager for the
challenge of CGI, according to her interview. Additional grade levels jumped on board
as their peers experienced success, and parents requested CGI teachers. Mrs. Franklin’s
prior school grew into CGI similar to Lincoln, experienced success with a core group,
and let the rest of the staff request training during subsequent years. All of this training
was done through the County Department of Education, and the perception of Mrs.
Franklin was that it was well received.
Mrs. Franklin found that Adams had a much different approach to
implementation. She believed that not all teachers were supportive of CGI, and they had
varying levels of expertise and training. Mrs. Franklin noted that
When I first came, there were teachers who have said, “well I have been trained
CGI two years right now,” but in my opinion, I would have expected a higher
level of competence for somebody who had been trained for two years. (Franklin,
personal communication, June 14, 2010).
Some teachers received training at other school sites, and transferred to Adams.
Mrs. Franklin reported the transferred teachers were told by other principals that two
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years of CGI training was sufficient, which made it difficult for these teachers to engage
with additional professional development. Some teachers at Adams had received very
little peer coaching from Lincoln because of scheduling problems. Mrs. Franklin’s
perception was that fourth- and fifth-grade teachers were more resistant to using CGI due
to concerns about the quantity of curriculum to be taught, and larger class sizes. Because
Mrs. Franklin was new to Adams, she decided to keep using Lincoln’s teachers for the
third year of training, 2007-08. It had limited results, mainly due to challenges
coordinating observations and coaching between teachers of different schools.
The third year of implementation, 2008-09, Adams began using DOE trainers,
rather than Lincoln’s. Mrs. Franklin claimed she chose this path so greater consistency
would occur among teacher’s workshop training, and among peer coaching. The DOE
offered multi-day workshops targeted for year one and year two teachers before the start
of school. DOE mentors came to Adams approximately once a quarter to work with
individual grade levels. The coaches focused the team to develop CGI problems and
analyze student responses, as reported by Adams’ teacher and principal. They reviewed
questioning strategies and CGI philosophies. The team also made long term plans
regarding problem types and pacing. The DOE coaches taught CGI lessons in Adam’s
classrooms, teams would observe, and debrief following the lessons. Adams’ teachers
were positive toward the coaching sessions from DOE, and had mixed reviews of the
DOE workshops. Several teachers told Mrs. Franklin that the DOE workshops were a
waste of time. Mrs. Franklin, and teachers, wanted more on-site coaching, rather than
workshops.
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The fourth year of implementation, 2009, saw Adams teachers become
coaches,and DOE staff offered pre-service workshops. This did not prove to be effective.
Adams coaches had many changes occurring personally, weddings, babies, and new
grade assignments, which limited their abilities to regularly coach peers. The master
schedule did not allow for grade level teams to meet easily, teachers had various
schedules for computer labs, PE, etc. which did not provide for simultaneous planning
time.
At the same time, competing initiatives used up the weekly school-wide
professional development time, as reported by Adams’ teachers. These included reading
literacy and comprehension training, and a new Response to Intervention (RTI) model.
Because CGI was not a start-up program, most teachers had experience with it, and
coaches were stretched too thin, Mrs. Franklin did not prioritize CGI training.
Leadership rearranged the master schedule to prepare for team level collaboration in
2010. The Adams coaches now had fewer constraints on their time, and the DOE agreed
to bring workshops on-site, tailored to fit Adams’ needs. Mrs. Franklin expected the
professional development program to be significantly better in 2010.
The Adams coaches were the leaders for CGI professional development needs.
Mrs. Franklin stated, “I want the teachers to take ownership and I want the onsite math
coaches to be leaders. And so I have given them a lot of flexibility and leeway in
designing it” (personal communication, June 14, 2010). In June of 2009, the staff worked
with these coaches to develop a plan of action for the 2009-10 school year. Teachers,
such as Mrs. Grant, developed a plan for coaching, observations, debriefing, and training
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workshops that would use substitutes and release days. This plan did not come to
fruition, partially due to constraints previously mentioned. Teachers were not mandated
to schedule observations and coaching during 2009. According to Mrs. Franklin, coaches
Put it out there that if you want me [the coach] to come in and model a lesson in
your room, then I will .... and if teachers want to go into their classrooms, the
coaches' classrooms, to observe them, all they needed to do was sign up and go
there. (Franklin, personal communication, June 14, 2010).
In other words, teachers were provided the opportunity to observe and coach each
other, but not required to do so. Mrs. Franklin stated that she set an expectation that
teachers should be teaching CGI at least once a week during 2008, twice a week during
2009, and “hopefully three or more times a week” (personal communication, June 14,
2010) for 2010. She claims to have asked teachers how often they used CGI, and
reviewed lesson plans with them, but was not so firm that she went “with a clipboard and
checked… I am assuming that they are doing it” (personal communication, June 14,
2010).
The Adams coaches discovered teachers had many fears associated with
designing CGI lessons, according to Mrs. Franklin. The perception of Adams’ coaches,
reported to Mrs. Franklin, was that teachers were not comfortable with the rationales for
number choices in problems, and did not have the mathematical confidence to sequence
problems to scaffold knowledge. Some were resistant to implementing CGI because they
were intimidated with the flexibility of the curriculum. The Adams coaches
recommended the school adopt a bridge program to help teachers learn these skills, yet
still maintain the openness of problem solving, and require students to communicate their
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thoughts. Therefore, the school adopted the Fosnot program, which began in 2010. Each
grade level chose specific units to be taught, and individual teachers promised to teach at
least one Fosnot unit during the beginning of 2010. Mrs. Carter and Mrs. Grant have
stated that Fosnot is similar in philosophy to CGI, but it is a scripted program. When
teachers use Fosnot, they do not develop their own CGI problems during that unit of
study, which may last two weeks. Teachers and students use multiple strategies to solve
problems, communicate their solutions, and question each other to elicit understanding,
much like CGI. According to Mrs. Franklin and Mrs. Grant, the Adams coaches see the
Fosnot program as a tool to help recalcitrant teachers adopt CGI philosophies, and to
build confidence in planning complex CGI problems.
In summary, Adams was working to fully implement CGI. The school was in its
sixth year of implementation. Teachers’ training and coaching was uneven during these
years. There was other competing priorities to math instruction, such as RTI and reading
comprehension. Mrs. Franklin put it best, “We are not there yet. We are not where
Lincoln is. We are still working to do that, and part of that is also dispelling a lot of
misunderstandings about CGI” (Personal communication, June 14, 2010).
Teacher case study – Mrs. Carter at Adams Elementary School.
Summary.
Mrs. Carter was a strong CGI teacher. She rated a level 4B, the highest level.
She felt supported in her training for CGI. She would like additional coaching on her
questioning techniques to improve her practice. Mrs. Carter reported there was a mixed
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level of CGI implementation by Adams Elementary teachers. I observed Mrs. Carter
teach two CGI lessons, the first on November 5, 2010 and the second on December 16,
2010. I interviewed her the same day as each observation. Mrs. Carter also completed a
written survey to provide additional data.
Mrs. Carter had 12 years teaching experience at the time of this study, a
combination of kindergarten five years, first grade four years, second grade two years,
and this year a second- and third-grade combination class. She taught at Adams
Elementary for nine years, and CGI six years.
Math curriculum.
This year Mrs. Carter is teaching a second and third combination class. There are
13 third-grade students, “the benchmark kids” (Carter, personal communication,
November 5, 2010), and 11 second-graders, “the cream of the crop” (Carter, personal
communication, November 5, 2010). The class was assembled three weeks into the
school year. According to Mrs. Carter’s interview, the third-grade students did not begin
the year doing as much thinking, or self-starting. “[CGI] was just kind of like I made this
up” (Carter, personal communication, November 5, 2010). She stated the third graders
came into the class with less confidence using CGI; they were not able to explain their
thinking as well as second graders.
It has taken a long time just to build positive thinking… and we are not separate,
we are together, we are all here to learn from each other. [The third-graders] were
very intimidated at the beginning of the year and did not want to [share their
thoughts]… they were very nervous about it, and now they are opening up a little
bit more. (Carter, personal communication, November 5, 2010)
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The second and third graders had weekly, grade level specific, group instruction
from Mrs. Carter, because of other pull-out programs that took the other grade level. She
claimed in her interview that she allocated part of this time toward mathematics
instruction, catered toward the needs of the remaining students. Her perception was this
helped build confidence in the third-graders.
Mrs. Carter went to the computer lab, in addition to regular math lessons, one 45-
minute block and one 30-minute block per week. In the computer lab, students worked
with MIND Research Institute’s ST Math program, which is also known by the mascot
penguin, Jiji. For math lessons, Mrs. Carter taught CGI lessons two consecutive days a
week, and Houghton Mifflin on remaining days. She also taught one Fosnot unit every
few weeks. When she used Fosnot, she did not teach CGI lessons. A Fosnot unit
typically lasted two weeks. She reported that Fosnot had many elements of CGI such as
student explanation of thought, complex, open-ended problems, and varying levels of
number choices. It was a packaged program, and not designed by teachers themselves.
Mrs. Carter believed the school started using Fosnot because many teachers were not
confident with CGI.
As reported by Mrs. Carter, the second-grade team planned CGI lessons together
once a month; the third-grade team did not. Her perception for other teachers follows.
It is still very overwhelming to do [CGI] and to do Fosnot units, and to do number
sense [at the math wall], … and they are all great teaching strategies. It is kind of
like it is left to the teachers to pick and choose what they think is most effective,
and so it is very obvious which kids had CGI once at least and some kids… have
not been that exposed to it. [Students] have all been exposed to it, but at different
lengths of time, different amounts of time. (Carter, personal communication,
December 16, 2010)
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CGI professional development.
Mrs. Carter learned of CGI while teaching at Adams during the 2005-06 school
year. Adams teachers received training and support from Lincoln teachers for two to
three years. Lincoln’s Mrs. Roosevelt was the lead mentor teacher. Adams teachers
received coaching on their own lessons, based on observations from Lincoln mentors.
The County Department of Education (DOE) took over the training program, which was
off-site. Mrs. Carter’s perception was the DOE program was not as effective since
teachers were not learning from interactions within their own classrooms. Additionally,
the DOE provided a coach three times a year to meet with grade level teams. The coach
offered training and a chance for group reflection on their use of CGI. Mrs. Carter stated
the training was not able to focus on individual classroom observations or individual
teacher mentoring since it was less frequent than mentoring provided by Lincoln in prior
years.
CGI rubric evaluation.
Overall Mrs. Carter is a strong CGI teacher, rated level 4B, the highest level. She
is one of the strongest teachers evaluated in this project. I observed her to demonstrate
proficiency in all areas of our rubric. This is significant because administration expected
Adams teachers to not have implemented CGI as well as Lincoln teachers.
Children received one CGI problem each lesson, each with three number choices.
Mrs. Carter expected them to solve one number choice with two different strategies. The
number choices for the first lesson, a subtraction problem, were (47, 85), double-digit
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with regrouping, (89, 140), triple and double-digit with double regrouping, and (278,
323), triple-digit with single regrouping. The number choices were appropriate for the
students of both grade levels. It was not recorded how many students chose each number
set during first observation, however all students were engaged in their solutions the full
40 minutes. See Appendix B for a sample problem. The second lesson was a money
problem, asking students to extrapolate an allowance over several weeks. The number
choices were 2 weeks, 5 weeks, and 8 weeks. This lesson had zero students use the first
number choice, 16 use the second, and eight use the third. Again, students used the full
20 minutes to solve the problem. Mrs. Carter “purposely did not give problems where it
is just adding or taking away, where you really have to think about what it is you are
trying to find, and break apart that problem” (personal communication, November 15,
2010). She also questioned the students in multiple ways at the beginning of each lesson
so students saw the relationship between the wording of problems and other subjects.
Each problem was set in a context that helped students understand what was asked. The
first problem built upon the number of Accelerated Reader pages read; the second related
to money given in an allowance. All these elements combined for a level 4A/4B rating,
the highest level, for the first rubric point.
The second part of the rubric also rated highly, a level 4A/4B. It was evident that
students shared their thinking with peers and the teacher. Mrs. Carter required students to
share their preliminary thoughts with one another before solving the first lesson’s
problem. While she asked students to remain seated, and work quietly, they were
encouraged to help one another. Students worked in their table groups to solve problems
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collaboratively, shared manipulatives, and explained solutions to each other. Students
documented their work on handouts collected at the end of class. Mrs. Carter circulated
around the room during each lesson, and interacted with approximately half of students
each lesson. She questioned students so they explained their thinking, and helped them to
use mathematical vocabulary. The first lesson allowed for one student presentation,
which lasted 8 minutes. The second lesson shared four student solutions during a 30-
minute block. Students presented a variety of solution strategies during the second
lesson, and all students were required to use at least two different strategies in their
solutions.
Interactions between students, and with the teacher, allowed Mrs. Carter to elicit
students’ thoughts so they were clearly understood by all students, which is the third part
of the rubric. I found evidence that Mrs. Carter rated a level 4B. She designed the
curriculum to have number choices that challenged each student. Mrs. Carter anticipated
their responses in advance, which guided her clarifying questions. She guided presenters
through her questions so that seated students understood and learned from the
presentation.
Mrs. Carter designed her lessons to build on students’ prior knowledge. She
developed problems in response to individual student needs. She used the problems to
clarify misconceptions held by some, and challenge others. During her interviews, she
referred to specific students and their strategies, and added many details to their successes
and struggles. She chose students for presentation based on how each person would add
value to the remainder of class. Mrs. Carter chose student to present a unique strategy, to
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offer solutions for peers that struggled, or would help individuals learn how to
communicate mathematics clearly. She rated a level 4B for her detailed use of
individual’s knowledge in lesson design.
Teacher perspectives on CGI.
Mrs. Carter liked using CGI. She believed that CGI was useful because “it allows
the teacher to really get into their student’s mind and to discover any misconceptions
about math” (Carter, personal communication, November 28, 2010). She felt well
supported in her professional development for CGI. In the future, she would like to be
observed, and coached, on questioning techniques for students. The most beneficial part
of her training was “being able to go into classrooms and observe CGI in action. Being
able to debrief afterwards and discuss individual students thinking, where they are at, and
next steps” (Carter, personal communication, November 28, 2010).
Teacher case study – Mrs. Grant at Adams Elementary School.
Summary.
Mrs. Grant taught a combination first and second grade class at Adams
Elementary school. I observed her teaching CGI lessons on two occasions, November 5,
2010 and December 16, 2010. I interviewed her the same day as each lesson.
Additionally, Mrs. Grant completed a written survey. Overall, she rated a level 3, and
would be a level 4A/4B if students had greater opportunities to communicate their
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knowledge to each other. She reported there were varying levels of CGI instruction at
Adams.
Mrs. Grant had six years of teaching experience at the time of this study, four in
first grade, one in second grade, and one year in a first- and second-grade combination
class. All her experience was at Adams elementary school. She had five years of
experience with CGI.
Math curriculum.
This class was a combination of first- and second-grade students. The class was
created three weeks into the school year. Mrs. Grant began the year teaching a pure first
grade class. First graders selected in this combo class were “high achieving kids…They
have to be independent workers, well behaved, and really high academically” (Grant,
personal communication, November 5, 2010). The second grade students were “kind of
the middle-of-the-road average kids, but independent workers and well behaved” (Grant,
personal communication, November 5, 2010). Two second-grade girls “were placed in
my class because they are so well behaved, but academically they are struggling more
than the first grade” (Grant, personal communication, November 5, 2010).
Typically, Mrs. Grant taught CGI two days a week, Thursdays and Fridays. She
stated in her interview that she had students solve the problem on day one, took their
written work home for analysis, and chose students for presentations on day two. The
other three days, Mrs. Grant claimed she used a workshop approach toward teaching
math concepts. Parent volunteers came in these days to help. While parents worked with
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the majority of students, Mrs. Grant provided small group instruction to seven or eight
students each day. Mrs. Grant reported the class went to a computer lab for two 45-
minute sessions each week and worked on ST Math, also known as Jiji, software from the
MIND Research Institute. This was in addition to daily math lessons. Mrs. Grant also
taught Fosnot units throughout the year. A unit typically lasted two weeks, and during
this time, she did not teach CGI lessons, according to her interview.
According to Mrs. Grant, the first-grade team did not collaborate often regarding
CGI. She believed she taught with CGI more than other first-grade teachers. Her
perception is the level of high-quality collaboration eroded over recent years.
I’ll collaborate with a few other teachers in other schools in our district, and we
will e-mail and give each other problems and talk about them. We were at a point
maybe a year ago, two years ago, where we were all on the same pacing guide. So
we were all in the same chapter, and we would write a problem together and then
we would talk about what strategies we hope to see, what we would think the kids
would do. And then we would bring our student work back together at a grade
level meeting. Talk about what we’re seeing, what we need to do for the kids, and
then we would write another one, and it was fabulous. But I feel like people have
so much on their plates that that has been totally pushed aside. (Grant, personal
communication, December 16, 2010)
Mrs. Grant stated in her interview that Adams teachers put CGI problems in
mailboxes for each other, and did not discuss them. In the past, teacher had in-depth
discussion about CGI, the reasons for number choices, student responses, and common
problem types according to a pacing plan. She stated the school had a focus on Fosnot
for 2010. Mrs. Grant believed that Fosnot is similar to CGI in many aspects.
I think the biggest challenge that many teachers have to CGI, and why so many
avoided it, is writing problems that are meaningful, looking for what [teachers]
need, what do the kids need, the rationale for the numbers. That’s been our
biggest concern to the district… [Teachers are concerned] if [CGI] was beneficial
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to their class, and if it was what they should be doing in their class.… The Fosnot
units, they get all that information. (Grant, personal communication, December
16, 2010)
CGI professional development.
Mrs. Grant learned of CGI through Adams School. She reported the school
offered multiple years of professional development, through the County Department of
Education, beginning in 2006. The first three years of training were held off-site at the
DOE offices. Training was more recently held at Adams with individual grade-level
teams. A coach from the DOE met with each grade level multiple time each year. The
team was able to discuss needs for students and develop plans for CGI. Mrs. Grant stated
the grade level team also met on its own approximately once a month to plan for CGI,
however “our level of implementation in our classrooms is a pretty big range, which
makes it a little more difficult to plan on a more consistent basis” (Grant, personal
communication, November 2010). Mrs. Grant also attended workshops featuring Megan
Franke. Dr. Washington, the assistant superintendent, invited her to attend a ten-day
workshop to become a CGI coach for her school. The DOE hosted this workshop.
CGI rubric evaluation.
Mrs. Grant had an overall rating of 3, and is emerging as a level 4A/4B teacher
according to the rubric. Her major limitation was the ability of students to communicate
their thoughts and strategies within the classroom. Following are details related to this
rating.
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Using the first criterion, opportunities for children to solve problems, Mrs. Grant
rated a 4A/4B, the highest level. Mrs. Grant wrote CGI problems in a context to help
students understand the use of numbers. Examples of these problems are in Appendix B.
The first lesson asked student to use addition, and subtraction, of Silly Bands. The
second lesson used a pictograph to compare the number of fish students caught. Each
problem used student names from the class, which generated interest from the class
during group discussion. Students spent the allocated time solving mathematics, 33
minutes the first lesson, and 20 minutes the second. They remained focused on the
problem and did not move to other tasks, such as silent reading, during the lesson. Mrs.
Grant expected students to solve their number choice with two different strategies, and
students were able to find solutions during the allocated time. The first lesson used
number choices of (54, 97), double-digit with regrouping, (175, 25), triple-digit with
single regrouping, (653, 498) triple-digit with double regrouping using sets of ten, (2,346,
2,654), quadruple-digit with triple regrouping. Five students chose the first set, five the
second set, seven the third set, and four the fourth set. The second lesson used a common
number choice for all students. The problem was differentiated by the six questions
asked, some focused on addition, others subtraction, and others comparison. Students
used a variety of solution strategies, not procedural methods. They used counters, base
ten blocks, 100s charts, written algorithms, and drawings to solve the problems.
Children had more limited opportunities to share their thinking with peers and the
teacher, the rubric’s second point. Mrs. Grant rated a level 3 in this category. Students
did not publicly share their solutions during the first lesson. Mrs. Grant pulled six
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students to a conference table for 25 minutes, and had limited interactions with students
outside this group. She reported in her interviews that she rotates the students at the
conference table so all students see her over a two-week period. Mrs. Carter began using
this approach so that she would not focus all her attention on struggling students, instead
would balance her time among all students. This was not the case during the observation.
While at the conference table, she focused almost all her attention on two struggling
students, which I will call Student 1 and 2. These two were able to share their thoughts
with the teacher, but did not share with the other four students. Students 3 and 4 received
limited interactions with the teacher. She had brief exchanges to help clarify simple
questions, but did not provide an opportunity for the students to fully explain their
solutions. The teacher checked Students 5 and 6 only for correctness, and had no other
interactions with them during the 25 minutes. Mrs. Grant’s perception was she worked
with all students, however her actions indicated she exclusively worked with struggling
students.
The second lesson allowed 20 minutes for four students to share. The first
presenter did not speak for herself. Mrs. Grant explained the student’s solution to the
class, and used the student’s thoughts during the explanation. The second student used
his own voice slightly more, however the teacher did most of the presentation, again
using the student’s strategy and written work. The third and fourth students presented for
themselves, while Mrs. Grant questioned them to help focus their thoughts. The rest of
the class did not ask questions of any the presenters, and did not interact with the
presented strategies. Mrs. Grant did not prompt the students to ask questions, so this may
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have been intentional.. Mrs. Grant’s students were attentive to the four presentations; she
expected them to pay attention to solutions. Mrs. Grant asked questions to check the
class level of understanding during each presentation.
The solutions presented had a variety of strategies and represented all six
questions. Mrs. Grant referenced many details of the student solutions during
presentations. Few children shared their thinking with classmates during the time
allocated for problem solving, in either lesson. Mrs. Grant allowed students to discover
answers for themselves, and did not base instruction on procedural methods. These
students demonstrated fluency in their use of mathematical concepts and vocabulary. For
these latter reasons, Mrs. Grant was a rated a level 3 instead of a level 2.
The third and fourth rubric points both rated a level 4B. Mrs. Grant demonstrated
an advanced knowledge of individual students’ thinking, elicited their understand for
others and herself, and used this knowledge in making instructional decisions. She did
interact deeply with selected students during both lessons, asked probing questions so she
fully understood their misconceptions, guided their thoughts to alternative solutions, and
recommended revisions. As she questioned students during presentations, she did so to
make presenter’s thoughts clear to the rest of class. She elicited details from their
strategies so seated students could learn alternative solutions. Mrs. Grant stated in her
interview that she took the written solutions home, analyzed them, and chose presenters
based on impact to the class. Students chosen had unique solutions, demonstrated correct
solutions where others made mistakes, or needed help building their ability to
communicate mathematical thought. She stated that she maintains a notebook at home to
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record progress for each student, and used the notes to determine who went to the
conference table. Mrs. Grant provided the observers with written rationales for all
number choices in the first lesson. She anticipated student responses to these numbers,
chose the numbers for specific goals, and knew which students needed help with
regrouping and those that needed a challenge. During her interviews, Mrs. Grant
referenced specific students strengths and weaknesses on numerous occasions. It was
evident that she built lessons around individual student needs.
Teacher perspectives on CGI.
Mrs. Grant was enthusiastic about CGI, because it allowed her to understand her
students’ thinking, according to her interview. She found it most helpful to observe other
teachers, especially when a DOE coach taught CGI lessons to her students, and
afterwards they debriefed the lesson. Mrs. Grant would like her grade level team to
consistently use CGI, and collaborate regarding problem development and student
performance. She is not satisfied with the current DOE training.
We had [a trainer from the] Department of Education come in and work with us
last Friday. She worked with first and second grade together and we learned not
one thing out of it. It was just an entire waste of the day. We looked at a problem
that somebody else had already done, then we went in a class, and she did her
problem, and we just watched… it is sad that it got to that because formerly it was
phenomenal training. (Grant, personal communication, December 16, 2010)
Two years ago, Mrs. Grant and three other math coaches developed a school-wide
plan for CGI training at Adams. It used a common problem through all grades levels K-
5, and provided coaching and observations for each staff member. The math coaches
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surveyed the staff to determine areas of concern, and addressed them within this plan.
The plan was not implemented; Mrs. Grant did not elaborate why.
Adams teacher case studies comparison.
A summary of the Adams’ case studies is presented in Table 8. A comparison of
the two teachers indicated they are at higher levels of CGI implementation, for their
experience with CGI techniques. Mrs. Carter is a level 4B and Mrs. Grant is at a strong
level 3, with potential to evolve to level 4A or 4B. The school continued to use
professional development to train teachers through workshops and teacher coaching,
though the principal and teacher indicated there is room for improvement in the
professional development system used. From this evidence, it may be concluded the
school made efforts to improve teacher performance with CGI, and toward CGI
implementation. This is in contrast to the perception from Dr. Washington and Mrs.
Franklin that Adams did not implement CGI well, as indicated in their interviews. The
teachers outperformed expectations from administrators.
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Table 8
Adams Teacher Case Study Comparison
Mrs. Carter Mrs. Grant
6 years of CGI 5 years of CGI
Overall Level 4B Overall Level 3 (Strong)
Teaches 2 CGI lessons per week
on back to back days - First day
solving, Second day presenting
Teaches 2 CGI lessons per week
on back to back days - First day
solving, Second day presenting
Initial PD from DOE workshops
and Lincoln mentoring
Initial PD from DOE workshops
and DOE mentoring
Currently receives voluntary
coaching
Currently receives voluntary
coaching
Rubric points
1. Student time solving is high
1. Curriculum is based on
problem solving
2. Student collaboration is high
2. Student communication to
peers is high
2. Students present solutions
3. Teacher questioning of
students is good
4. Teacher knowledge of
individuals is high
4. Number choices fit student
needs
Rubric points
1. Student time solving is high
1. Curriculum is based on
problem solving
2. Student collaboration is low
2. Student communication to
peers is low
2. Teacher presents student
solutions
3. Teacher questioning of
students is good, but limited
to very few students
4. Teacher knowledge of
individuals is high
4. Number choices fit student
needs
4. Uses record keeping system to
document CGI student
learning
Wants more teacher coaching,
peer observations, and grade-
level collaboration
Not satisfied with DOE training.
Wants grade-level team to
consistently use CGI
Adams performance data.
Adams had routinely performed below the district average on CST math
proficiencies, yet was significantly above the state average (see Table 9). As shown
previously in Table 7, Adams has higher percentages of at-risk students than Green
Valley, yet significantly fewer at-risk students than California.
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Table 9
Percent of Adams Second-Graders Advanced or Proficient on CST Math
2007 2008 2009 2010
California 59% 59% 63% 62%
Green Valley USD 84% 90% 89% 92%
Adams Elementary 74% 82% 84% 86%
Mrs. Carter n/a 75% n/a 87%
Mrs. Grant n/a n/a 81% n/a
Notes: Mrs. Carter and Mrs. Grant taught either kindergarten or first grade in years with
no data. These grade levels do not take the CST.
The 2006-2007 cohort of second-grade students maintained high levels of
proficiency during their tenure at Adams. Nearly 40% of them were advanced, an
additional 30% were proficient in CST mathematics (see Figure 4). These are high levels
compared to the state, however Lincoln achieved much higher levels during the same
time. This is a topic for later comparison.
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Figure 4. Adams CST Math Proficiency 2007 Cohort. This data is for the same group of
students, those beginning second grade during the 2006-2007 school year. The data
follows them through 5
th
grade.
Adams’ overall API scores are also respectable. They stayed above 850 since
2004, and broke the 900 mark in 2010. These are well above state expectations of 800,
yet below scores of Lincoln which were near or above 950 during the same years. Figure
5 presents Adams’ API trends.
150
200
250
300
350
400
450
500
550
600
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
2nd 2007 3rd 2008 4th 2009 5th 2010
CST Mean Math Score
Percent of Students
Grade and Year
% Advanced
% Proficient
% Basic
% Below Basic
% Far Below Basic
Mean Scale Score
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Figure 5. Adams API Data 2000-2010. Gaps in trend lines for Asian, Hispanic or Latino,
and Socioeconomically Disadvantaged students are from years these populations
decreased in size. They were not statistically significant subgroups and did not receive
an API.
Now that the case studies have been presented, a comparison of the two schools is
in order. There are similarities within each respective CGI program, and room for
improvement in each school. Additionally, student performance varies between the
schools, however this may or may not be a result of CGI practices.
Comparisons Between Adams Elementary and Lincoln Elementary
This research project does not attempt to draw relationships between CGI
instruction and performance on the California Standards Test (CST) in mathematics. The
CST data helps to further illustrate characteristics of each school. Students in grades two
700
750
800
850
900
950
1000
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
School-wide Asian
Hispanic or Latino White
Socioeconomically Disadvantaged California Expectation
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through five take the CST each year at Adams and Lincoln schools. Second-grade
students at Adams school, in theory, received two years of CGI instruction (first and
second grade) before the 2007 test. Adams Elementary was in its second year of CGI
implementation during 2006-07, and Lincoln its ninth. Lincoln’s second-graders had
three years of CGI instruction before the 2007 test, however Lincoln teachers had more
years experience with CGI than Adams teachers. For these reasons, the presented data
begins in 2007.
Using Table 10 comparisons may be made regarding levels of proficiency on the
second-grade CST math exam. Students in Lincoln classrooms scored higher rates of
advanced and proficient. The perception of Lincoln’s staff was Lincoln had fidelity to the
use of CGI by all teachers, including kindergarten and first grade. Lincoln also had more
experience using CGI, 13 years total. The teachers studied from Adams reported that
there was not a consistent use of CGI by teachers in lower grades. Adams had six years
of experience with CGI, not all of which have been consistent. Table 10 presents data for
all second-grade students, including teachers not observed or interviewed.
It is of note that Lincoln lead the district average, and Adams lagged the district
average. The district and both schools significantly outperformed the state average,
between 25% and 35%. Lincoln’s leadership on CST mathematics cannot be fully
attributed to the school’s demographics; they are similar to the district as a whole. Since
the two agencies have relatively similar demographics, we would expect both to have
similar CST proficiencies, if one assumes this is the cause for variation. Other factors
have lead to Lincoln’s higher performance, and perhaps more than a decade of experience
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with CGI is the primary motivator. On a related tangent, Adams less consistent use of
CGI may be a cause for it to see mathematics proficiencies below the district average.
Perhaps other schools not part of this study have better CGI models in place. These are
not trivial variables, and are major topics worth future study by Green Valley or other
researchers.
Table 10
Comparison of Second-Graders Advanced or Proficient on CST Math
2007 2008 2009 2010
California 59% 59% 63% 62%
Green Valley USD 84% 90% 89% 92%
Adams Elementary 74% 82% 84% 86%
Mrs. Carter n/a 75% n/a 87%
Mrs. Grant n/a n/a 81% n/a
Lincoln Elementary 88% 94% 97% 93%
Mrs. Kennedy 70% 95% 95% 84%
Mrs. Madison 95% 89% 95% 96%
Mrs. Pierce 100% 95% 100% 92%
Notes: Mrs. Carter and Mrs. Grant taught either kindergarten or first grade in years with
no data. These grade levels do not take the CST.
A teacher’s CST data may vary year to year because they have different students
each year. One should also consider how the same cohort of students performs over
multiple years. This comparison shows whether similar students are maintaining high
levels of proficiency as the content becomes more complex. Table 11 provides
proficiency data by student cohorts. Data are for the entire school, which included
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teachers not participating in this study. Lincoln outperformed the district average by 4%
to 12%, and maintained consistent results year to year with little fluctuation. Adams and
the district had greater fluctuations year to year. Adams lagged the district average by
1% to 17%.
Table 11
Comparison of Students Advanced or Proficient on CST Math by Cohort
2007 2008 2009 2010
2nd Grade 3rd Grade 4th Grade 5th Grade
Green Valley USD 84% 87% 86% 80%
Adams Elementary 74% 75% 69% 75%
Lincoln Elementary 88% 96% 97% 92%
2008 2009 2010
2nd Grade 3rd Grade 4th Grade
Green Valley USD 90% 85% 88%
Adams Elementary 82% 70% 75%
Lincoln Elementary 94% 97% 98%
2009 2010
2nd Grade 3rd Grade
Green Valley USD 89% 91%
Adams Elementary 84% 92%
Lincoln Elementary 97% 96%
2010
2nd Grade
Green Valley USD 92%
Adams Elementary 86%
Lincoln Elementary 93%
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Table 12 displays the Annual Performance Index (API) over 10 years for Adams
and Lincoln schools. The API is the primary means of ranking California schools, and
the CST is the primary basis in elementary school. District-wide API data includes
secondary schools; therefore it was not chosen for comparison. California’s API may
vary from 200 to 1000 points. The state has set an expectation for all schools to reach a
score of 800. Both Adams and Lincoln have consistently exceeded this benchmark. As
of 2010, both schools are above 900, an elite level. The reader will note that Lincoln has
maintained an API near or above 950 for eight years. California did not disaggregate the
state API by grade level until 2006, therefore this data is missing from 2000-2005.
Table 12
Comparisons of API Data 2000-2010
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
Lincoln
Elementary 913 897 922 951 945 947 960 955 961 980 971
Adams
Elementary 800 819 795 839 856 862 866 858 869 865 911
California
Grades 2-6 n/a n/a n/a n/a n/a n/a 752 763 774 788 800
While this study does not attempt to correlate performance on the CST or API
scores to a school’s use of CGI, these data indicate that Green Valley USD, Adams
Elementary, and Lincoln Elementary are all high performing institutions of learning.
Students at these schools are consistently achieving at very high levels. The desire of
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these schools to implement a challenging mathematical program, while students are
demonstrating mastery, should be recognized.
Summary of Data Supporting the Research Questions
1. How does the implementation of the CGI professional development program
at Green Valley USD compare to the model designed by Carpenter, et al.?
Professional development is the cornerstone of CGI in Green Valley Unified.
Both schools invested time and money to train teachers over multiple years. Adams and
Lincoln used pre-service workshops to train teachers on CGI philosophy, mathematical
content, the mechanics of generating problems, and classroom questioning techniques.
These workshops were differentiated according to teachers’ experience. Lincoln used
workshops designed internally, while Adams used workshops developed by the County
Department of Education (DOE). Teachers at both schools received long-term mentoring
from master teachers on a regular basis. This occurred for more than five years at
Lincoln. Lincoln exclusively used internal mentors who met monthly. Adams had three
years of inconsistent coaching from Lincoln, one year of quarterly coaching from the
DOE, and one year of inconsistent internal coaching.
2. In what ways has the CGI model at GVUSD changed over time?
The CGI model changed with regard to the delivery of training. Initially Lincoln
Elementary provided training and mentoring for other schools. This lasted two or three
years; Lincoln teachers became frustrated and withdrew from training other sites.
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Lincoln continued to internally train its own teachers. Lincoln reduced the amount of
peer observation and coaching in 2009.
Adams CGI program had several evolutions. Lincoln provided training and
support for three years, with mixed success. Teachers did not consistently receive
observations and coaching. The DOE provided the fourth year of training and coaching.
These mentors observed and coached teachers quarterly in grade level teams. The DOE
provided training and support the fifth year, while Adams teachers provided the coaching.
This coaching was inconsistent and voluntary in nature.
3. Are the actions of elementary teachers while teaching mathematics in the
classroom consistent with those of CGI?
The actions of teachers observed at Adams and Lincoln are consistent with CGI.
They all rate a level 3 or 4B on our rubric. The problems used are consistent with those
offered in literature and county training documents. Teachers do not rely on a textbook
for the mathematics curriculum. They have detailed knowledge of students’ knowledge
and misconceptions. Teachers design lessons to displace student misconceptions and to
build mathematical understanding in a systematic manner. They expect students to solve
problems with a variety of strategies, progressing from direct models to algorithms.
Teachers do not interject their own opinions as students solve problems; rather they
question students to elicit self-discovery. There is inconsistent use of peer-based
learning. Two teachers routinely use student presentations and peer questioning as means
of instruction. Three teachers have more limited use of student lead instruction. Adams
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staff report that there is inconsistent use of CGI by all teachers. The Lincoln staff
reported all teachers used CGI on a regular basis.
Conclusion
Green Valley Unified, Adams Elementary, and Lincoln Elementary have very
successful mathematics programs, evident in their strong scores on state assessments.
Both schools have made significant investments in teacher training and coaching toward
CGI. Observed teachers progressed to the upper levels of our rating scale; none were at a
level 1 or 2. However, with the significant resources used, and the length of time for
implementation, teachers should consistently be at level 4B. Additionally, all teachers
should use CGI on a regular basis. Teachers using CGI, are highly consistent with their
techniques, some having room for growth. Chapter 5 will outline recommendations to
help these schools move from being high performing to extraordinarily performing
schools.
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Chapter 5
Discussion
This study evaluates the Cognitively Guided Instruction (CGI) program of Green
Valley Unified School District. Green Valley has not had a formal evaluation of the
program, hence the need to study the method of implementation as a first step. This study
serves as a basis for organizational decisions regarding CGI’s use and future research
within the district. It intends to provide data for improvement of Green Valley’s
instructional program and teachers’ professional development.
Initially, CGI began at Lincoln Elementary in 1997 and spread to other school
sites over time, with encouragement from the district office. The district began using
CGI as the primary means of K-5 math instruction for all schools in 2005. This required
an extensive teacher professional development program to prepare faculty for
implementation. This professional development program is compared to models used by
Fennema, Carpenter, Franke, Levi, Jacobs and Empson (1996) and Carpenter, Fennema,
Loef Franke, Levi, and Empson (1999), the founders of CGI. The collected data
measures the degree to which Green Valley’s CGI program is similar to these models. It
also measures how CGI has changed over time within two schools, Lincoln Elementary
and Adams Elementary. I collected data from teachers and administrators in these
schools, and used a rubric based on Fennema et al. (1996) and Carpenter et al.’s (1999)
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work to measure the congruence of teachers’ actions within the classroom to the research
base.
Chapter 4 presented data which indicated Green Valley’s CGI had many
characteristics similar to those of Carpenter et al. Teachers received training through
multi-day workshops and mentoring, which extended over many years. Much of it was
offered at local school sites, through expert teachers on staff. One teacher at Lincoln, who
brought CGI to the school upon her hire in 1997, trained her colleagues through a highly
collaborative environment that promoted peer observations. The success of Lincoln
students lead the district to use Lincoln teachers as support for sister schools’
implementations from 2005-2007. Many scheduling conflicts ensued, so the County
Department of Education (DOE) offered services from 2007-2011. Between these two
paradigms, the professional development program was lacking according to teacher
interviews. The quantity of mentoring was not consistent with Lincoln’s support; the
quality was not consistent with DOE’s
Teachers use of CGI is mixed, both within schools and among schools. The five
observed teachers use CGI in varying degrees consistent with the rubric, however not all
teachers are using CGI on a regular basis. At Lincoln, teachers are consistently using
CGI throughout the school in all grade levels. Three observed second grade teachers
rated 3, 3, and 4B on a rubric scale going from 1, least consistent, to 4B, highly
consistent. They report CGI is used at least twice weekly for their grade level, as well as
kindergarten and first grade. At Adams, teachers are not consistent in their school-wide
CGI practice. The two teachers observed rated 3 and 4B on the rubric. They report some
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students entering their classrooms did not receive CGI instruction during kindergarten or
first grade.
Analysis and Implications
Professional development.
Green Valley Unified implemented many characteristics of a high quality
professional development program. Quality programs espoused by Fennema et al. (1996)
and Saxe, Gearhart and Nasir (2001) include the use of multi-day workshops held
throughout the school year, regular mentoring with reflective sessions for neophyte
teachers, and continuity for at least three years. All of these elements presented
themselves at both schools. Interviews and district records show that the program was
widespread and consistently used at all elementary sites. The means of delivery varied
between Lincoln and Adams, as well as the teachers’ perceptions of quality. I did not
assess the quality of the professional development program, as this was beyond this
study’s scope and capacity.
Lincoln developed its own professional development program, built within a
culture of reflective collaboration as the staff was hired. The use of CGI was interwoven
into the norms and expectations of all teachers. Teachers performed peer observations
and reflected upon their practices, in general, and specific to CGI. The staff developed a
strong affinity for CGI during the early years of the school, 1997-2000, and as a result,
the principal made the use of CGI a key component of the hiring process, formal
evaluations, and strategic plans for the school. Site leadership maintained a clear vision,
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curricular expertise developed within the staff, peer teachers became instructional
leaders, and a deep mentoring program ensued that provoked teachers to reflect critically
on their practices. These actions are consistent with successful programs evaluated by
Grossman and Thompson (2004). The ability of Lincoln to develop such a level of
consistency is due, in part to the uniqueness of the school. Few schools start from scratch,
select their staff with an expectation to use one particular instructional program, and keep
the same principal for 15 years.
Adams made efforts to build a similar school culture, however keys are missing
from its professional development program. The most vexing is the mentoring program,
which has not reached the quantity or quality expected by the Fennema et al. (1996)
model. According to teacher interviews, Teacher-mentor observations were sporadic
during the first two years of implementation, due to Adams teachers’ unpreparedness on
days scheduled for observation, and Lincoln teachers’ rebuff in response. Mentors from
Lincoln felt their time was not valued, having prepared substitute plans, prepared for
observations, arriving at Adams, and the neophyte teacher subsequently asking the coach
to come a different day because she had not prepared a CGI lesson for observation.
Lincoln teachers requested to withdraw from coaching duties in other schools. The
district office, along with Adams principal, had positive relationships with the DOE.
Thus, the Department of Education began offering CGI professional development.
The DOE did not offer the frequency of mentoring described by Fennema et al.
(1996), meeting quarterly, rather than two or three times per month. Additionally,
mentoring offered by DOE was not individualistic, given collaboratively to grade level
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teams. Six years of modest quality mentoring made it difficult for teachers to reach their
full potential. The principal reported the difficulties of inheriting a “trained” staff, one
that received two years of training, believing no additional training was needed, and not
meeting the her expectations for quality.
It appeared that a flaw in both service providers was the lack of a high-quality
mentoring program. A program based on peer observations, and reflections, can increase
the internal accountability of an implemented program, which in the end will benefit
students and the school as a whole (Elmore, 2005). If Adams maintained expectations for
consistent peer-based mentoring for six years, the school might have a greater degree of
consistent CGI application among all teachers. Teachers might expect each other to use
CGI techniques regularly.
Interviews with Adams teachers and the principal indicated one barrier to CGI
implementation is the staff’s comfort with mathematics content, concepts, and pedagogy.
Hill and Loewenberg Ball (2004) found that a quality mathematics professional
development program should help build a strong conceptual understanding of
mathematics. They state that workshops focused on mathematics content alone do not
meet the needs of teachers. Hill, Schilling and Loewenberg Ball (2004) echo these
sentiments and add the importance of pedagogy to the equation. Teachers need to clearly
understand how to transition students from levels of low understanding, to rich, complex
mathematical knowledge. Often teachers’ professional development workshops do not
teach the pedagogy required to help with these transitions. They instead focus on
mathematical content or curriculum.
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The content of the DOE’s workshops offered to Adams was not researched.
Teachers found them to be of varying quality, some were received with accolade, and
other were believed to be a waste of time, as stated in teacher interviews. In either case,
the content of these sessions did not reduce the anxieties of teachers. These anxieties
include teachers’ conceptual understanding of CGI problem types and how they build
upon each other to develop student understanding. There are many levels of pedagogy
and conceptual understanding for teachers to master so they become confident with CGI,
based entirely on teacher-written problems.
It appears that DOE workshops are not correctly addressing teachers’
pedagogical needs. Adams teachers are not comfortable with the mathematical pedagogy
required to properly design CGI problems, and they all attended DOE workshops that
should have addressed these needs. For example, number choices used in these problems
require foresight to what is expected of students – will they need to use regrouping
strategies once, or twice? From teacher interviews, it was evident that some teachers
used problems without considering these impacts. In the future, a teacher survey, using
instruments similar to Hill et al. (2004), can determine professional development needs
that decrease teacher knowledge gaps. The discovered needs may or may not be part of
existing DOE workshops. Adams and Green Valley leadership must evaluate the content
of DOE workshops to determine how well they address teacher needs. Either the
workshop content or the service provider requires revision to build teacher expertise.
Lincoln developed its own workshops based on literature and videos obtained
directly from Carpenter, Fennema and Franke. The Lincoln faculty also used systemic,
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ongoing mentoring to alleviate concerns related to mathematical pedagogy. This school’s
model was the basis for the DOE training program; the expertise to build a high quality
professional development model exists within the district. Adams does not need to rely
upon the DOE to provide services in the future. The capacity to revise their CGI
workshops and mentoring program reside within the Green Valley organization.
Learning environment.
CGI problems should promote student autonomy, allowing a choice of numbers
that are challenging, yet within the grasp of students. Mathematics problems should
engage students and not lead to discouragement (National Research Council, 2005;
Johnson, 2002; Mayer, 2008). Observed teachers at Adams, and two of three teachers at
Lincoln followed this recommendation. One teacher at Lincoln Elementary developed
number choices that discouraged students from solving the most challenging problems.
Mrs. Kennedy, a teacher at Lincoln, developed number choices that were far beyond her
students’ abilities on both observations. She used advanced versions of grouping and
multiplication that were not appropriate for second graders in an effort to challenge
advanced students. Unfortunately, they were not able to solve these number sets
correctly, and did not persevere to find correct solutions. Students choosing these
number sets spent less than 20 minutes on each problem, even though 40 minutes were
given. They did not remain engaged in mathematical inquiry at a level appropriate for
CGI. This indicates that there was a major discrepancy between the teacher’s knowledge
of students and the application of that knowledge toward lesson design. This gap may
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lead to student discouragement toward solving advanced problems. Lower-level students
cannot access the complex nature of these number choices. They are not motivated to try
them, and higher-level students are not able to offer explanations. An opportunity is lost
to build student knowledge from one another.
Alternatively, Mrs. Kennedy’s grade-level colleague, Mrs. Pierce, used number
sets that seemed appropriate for both high and low students. The number choices built on
each other in a systematic manner, and were accessible to all. Lower-level students
readily understood proofs of the advanced number sets, when presented by high-level
peers. The number choices successively built levels of complexity, so that students could
relate simple solutions to more complex ones. The presentation of various student ability
levels is consistent with Carpenter et al.’s (1999) model of best practice.
The five observed teachers from both schools worked diligently to understand
student misconceptions. During lessons, each teacher questioned students to elicit
solution details, both to understand what was successful, and misconstrued. Students
were encouraged to revise their thoughts in order to achieve correct solutions. This is
consistent with the body of research for quality mathematics instruction (Clarke &
Clarke, 2004; NRC, 2005; Johnson, 2002). Presentations made student thinking clearly
visible in all classrooms. Students displayed their own work during presentations and
used their own voice during explanations, independent of teacher judgments. In each
case, the teacher questioned presenters until they made revisions to achieve correct
solutions, or seated students offered corrections.
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When students see one another’s solutions, they realize that peers are generative
in mathematical knowledge, not just communicators of information (McClain & Cobb,
2001). The role of instructor transfers from teacher to students, which sends powerful
messages regarding students’ importance within the classroom (Elmore, 2004). Students
feel valued as their participation is recognized.
These opportunities for peer sharing also diffuse barriers to learning mathematics,
and build expectations to challenge children in mathematics (Cobb, Yackel & Wood,
1989/2011). Children challenge themselves when comparing solutions among peers,
noting correctness, efficiencies, and connections between physical model algorithms.
These higher orders of thought, increase the cognitive load on students. Students build
flexible reasoning skills when comparing solutions to one another (McClain & Cobb,
2001). Because three of five classrooms do not have student actively sharing each lesson,
children are missing valuable opportunities to strengthen their reasoning skills. They are
missing chances to challenge themselves in creative ways, and do not receive stimulation
from complex solutions on a regular basis. Daily peer exchanges are an essential
component to develop complex patterns of thought in students.
The use of a consistent program year to year, in each classroom, builds the culture
of high expectations over time. Cobb and Hodge (2011) state that pervasive programs
benefit both student and teacher learning, and build a sense of affiliation within the
mathematics classroom. Students develop long-term identities concerning mathematics
success in the presence of pervasive programs, and a sense of mathematical detachment
from programs that are not consistently applied (Cobb & Hodge, 2011). I reported in
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Chapter 4 that the CGI program is not consistent within or between grade levels at
Adams Elementary. Adams teachers reported students enter their classrooms with
varying degrees of mathematical preparedness. Some students do not understand the
cultural expectations of CGI, and are not confident presenting solutions to others. They
did not receive CGI instruction in deep, meaningful ways during kindergarten or first
grade. As a result, these students have not developed a sense of identify that equates to
success in complex problem solving.
In comparison, Lincoln staff reported there is widespread and consistent use of
CGI in all grade levels. Second-grade teachers reported second-grade students are
prepared for CGI activities, because of kindergarten and first-grade teachers’ CGI usage.
Students begin the school year with expectations to develop complex solution, to share
solutions with peers, and to provide clear explanations for others. Second-grade teachers
do not create these sociomathematical norms; they reinforce and add to it.
Teachers at both schools can improve their management of peer interactions.
Peer interactions are central to creating rich learning opportunities for students; they
should occur during each CGI lesson. In contrast to this need, three of the five observed
teachers plan lessons so students present solutions only one day per week. Two of five
teachers actively discouraged students from solving problems collaboratively. These
teachers expected students to work independently, work quietly, and remain seated
during the time allocated for problem solving. This significantly decreases the
opportunities for stakeholders to learn from students. Mentoring and professional
development must emphasize the need for strong peer-interactions among students, and
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the connection to cultural norms that stimulate higher levels of learning, for both students
and the teacher.
Leadership.
School leaders set the tone for implementation of programs, and the CGI program
is no exception. The district office serves as a lens to focus teachers and principals on
what is important. It communicates the importance of a program through policies,
especially the professional development program for teachers, and provides either clear
or unclear guidance for neophyte teachers (Grossman & Thompson, 2004) to implement
programs. Green Valley’s focus on CGI has remained clear over several years. The
assistant superintendent and principals maintained expectations for teachers to use CGI
multiple times per week, and offered professional development to help teachers gain
competence. However, these expectations, supports, and policies did not see all teachers
are performing up to expectations, especially considering the program is in its fifth year
of implementation.
The district administration should provide clear guidance on the use of mentoring
programs, which are currently the weakest link of professional development. A large
expenditure of funds toward new professional development programs is not required;
using quality mentors and training them may achieve significant gains. A high frequency
of quality teacher mentoring will increase the sense of internal accountability for teachers
to use CGI. Teachers will develop a sense of urgency to use a program, not because an
administrator dictates, but because their colleague expects a professional discourse on
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needs within one’s own classroom (Grossman & Thompson, 2004). By mandating the
frequent use of mentors, and providing necessary logistical support for mentoring, such
as a well-planned master schedule and a calendar of teacher release days that takes
priority over other initiatives, teachers will expect one another to use and reflect upon the
practice of CGI. When programs become voluntary, as the Adams mentoring program
did, teachers make personal decision about their professional development, which
fragment reform efforts (Cohen & Ball, 2001).
Lincoln now has a sense of internal accountability between teachers, in large part
because of strategic, long-term, public discourse of classroom practice. They expect each
other to add to the communal knowledge of CGI through regular grade-level discussions.
External pressures also motivate teachers to use CGI and hone their practice. Parents and
students expect to enter classrooms using CGI, because they experienced previous
success in mathematics classrooms using the program. There is a high degree of
accountability for teachers to seek excellence in their practice, and it extends beyond the
principal. Adams teachers do not have the same sense of urgency. The mentoring
program is discontinuous, which has negatively reinforced the use of CGI. Peer teachers
accept the fact some teachers do not effectively use CGI. Site leadership implemented
other programs, such as a new Response to Intervention (RTI) model, reading initiatives,
and Fosnot math, while the roots of CGI still developed. Multiple program initiatives
allowed competing priorities for teachers’ attention and they chose which parts of CGI to
use. Site leadership, with the support of district administration, must focus the priorities
of programs so all teachers master and implement CGI.
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A principal that asks teachers to implement a program must have enough
knowledge to direct the staff on its use, assess teachers in a meaningful way regarding
implementation, and coach teachers so they have the necessary skills for success (Elmore,
2004). Principals need CGI training to become effective leaders of the program, so they
are able to build effective expectations for teachers. Green Valley principals attended
CGI training over an extended period. Lincoln’s principal attends workshops annually
with her teachers, and frequently participates in grade-level discussions. Adams’
principal also participated in grade-level meetings, but has not attended CGI workshops
with her teachers during the previous two years. She did attend multiple years of CGI
workshops in the past. As a result, these principals have detailed knowledge of CGI
philosophies, problem types, questioning strategies, additional resources, and models of
classroom use. Sitting through the workshops and discussions with teachers, helps keep
administrators fresh in their knowledge of CGI, since they do not interact with the
content on a daily basis. It also sends powerful signals to the faculty regarding the
importance of CGI to the school’s culture. When a principal schedules multiple days to
attend workshops, it reaffirms that the workshop is a valuable use of teachers’ time.
Principals attending multiple years of professional development validate the need for
teachers to do likewise.
Both principals indicated they use CGI lessons as part of the formal evaluation
process, however it was not evident whether they have specific tools to measure the use
of CGI within classrooms. The rubric developed in this study is a tool that administrators
and teacher-leaders can use to assess the quality of CGI instructional practice. It offers a
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model of highly effective CGI practices, based on Fennema et al. (1996) and Carpenter et
al. (1999). Teachers can use this rubric for self-assessment or to stimulate reflection in
groups. Administrators and teacher-leaders may use it to identify areas of weakness and
strength among the staff. In turn, this drives the professional growth program so that
teachers become more consistent with level 4A or 4B techniques. Schools benefit from a
public display of norms and assessments used to measure teacher practice (Elmore, 2004;
Marzano, 2003) and from a common definition of quality instruction (Grossman &
Thompson, 2004), both of may occur through regular assessment.
Limitations
The population of this study limits the extrapolation of these findings to other
schools or districts. Several characteristics of the district and schools are different from
most California schools.
1. This study used a small sample of teachers and principals. Not all grade levels were
represented, nor were all schools. Results may be different at in other grade levels
and other school sites.
2. Results for Lincoln Elementary may be unique, due to the principal’s ability hire all
teachers currently employed at the school. The school maintained consistent
leadership for 15 years, and the principal has 18 years experience as a principal. This
is in contrast to 69.4% of principals with less than 10 years experience on the job
(NCES, 2010).
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3. Green Valley is relatively homogeneous, by Californian standards. It has a majority
of Caucasian, middle-class students. The district is very high performing according
to state standardized tests, has stable leadership, and highly qualified teachers in
every classroom.
Recommendations
This study’s recommendations are specific to Green Valley Unified, and may or
may not be generalizable to other agencies. This implementation study is specific to this
district, and for review by its senior leadership. The recommendations can serve as
guidance for other districts choosing to implement CGI or evaluate their existing
program. Following are recommendations that will improve the CGI program, and
recommendations for further study within the district.
Areas for action by stakeholders.
1. Green Valley Unified School District should maintain or increase the amount of
funding dedicated toward CGI professional development. Teachers should continue
to receive multiple days of reflective workshops per year, combined with release time
for peer observation and coaching throughout each year. The current model of a
three-year professional development program should continue for any new teachers.
2. Teachers should focus their practice to increase the frequency of student collaboration
within the classroom. Students should have opportunities to present their solutions
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during each CGI lesson. Administrators will need to closely monitor this process,
offering support as required.
3. A systemic teacher-mentoring program is lacking. Administrators and teacher leaders
need to evaluate mentor teachers according to this study’s rubric. Mentor teachers
need training to understand the importance of their role in the development of student
and teacher interactions, lesson design, and continued professional growth of
neophyte teachers. Mentor teachers should observe and debrief lessons monthly for
teachers determined to be in need of improvement. Mentor and neophyte pairs should
remain intact during the three years of professional development. Four of the five
observed teachers requested greater opportunities for peer observations and coaching.
4. District and school leaders must communicate a consistent expectation regarding the
use of CGI. Teachers need a clear definition of instructional practice and
expectations for frequency of use. Expert teachers and site administrators should
collaboratively develop these expectations, for consistent articulation by the
superintendent, principals, and teacher-leaders.
5. Teachers at Adams are not consistently using CGI. The district should survey
teachers at this school, and perhaps others, to determine barriers to implementation.
These survey results will direct future professional development.
6. Teacher teams and site administrators should evaluate classroom practices regarding
the use of CGI. Teachers rated as level 4B deserve consideration to serve as lead
trainers and mentors. Green Valley may consider using internal experts to provide
all future professional development needs, rather than the County DOE. Adams
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teachers, whom received mostly DOE training support, have been dissatisfied with
the quality of support given since 2009.
7. Green Valley needs to evaluate the content of workshops offered by DOE. The
content of the current workshops do not address teacher needs for mathematical
proficiency and pedagogy. The district should survey teacher to determine their
knowledge of mathematics content and pedagogy. Hill (2004) offers instruments that
serve as models for this survey. A gap analysis would determine the content of future
workshops.
8. Sister schools need opportunities for collaboration. The district may consider some
professional development workshops to be district-wide grade-level meetings for
reflection upon practice. The district may also consider maintaining a video library of
effective lessons and a print library of student work and CGI problems. Each would
require contributions from all schools and active use at each school site.
Areas for future research.
This study was the first step in evaluating the use of CGI within Green Valley
Unified. It was able to determine the level of implementation of second grade teachers at
two schools. Each of the following topics would provide additional information to the
stakeholders of Green Valley.
1. Is there a causal relationship between the use of CGI and student performance? Are
students outperforming their expected abilities, controlled for demographic factors,
because of CGI instruction?
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2. Does CGI instruction during elementary grades influence when, or if, students take
courses beyond basic high school graduation requirements, such as Algebra 2?
3. Dos CGI instruction during elementary grades influence the performance of students
in secondary mathematics courses such as Pre-Algebra or Algebra 1?
4. Have CGI philosophies and teaching techniques affected the instruction of other
subjects, especially English Language Arts?
5. Has the use of CGI decreased the number of special education referrals, and increased
the number of GATE students?
6. What is the impact of CGI upon English Language Learners, minority students, and
students from low socioeconomic households?
Conclusion
Green Valley Unified School District has built a professional development model
that deserves recognition for its impact upon teachers and students. Many teachers are
using CGI in highly effective methods, which lead to increased student confidence and
aptitudes toward mathematics. As with any program, there is room for improvement; one
always strives for excellence. Changes in the professional development model created
some of these gaps, however the program has remained intact despite teacher misgivings.
Consistent expectations from district and site leadership helped the program to keep its
focus through the years. As a result, all observed teachers demonstrated most elements of
quality CGI instruction to be part of their practice. I hope that Green Valley is able to
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continue its path for mathematical excellence, and continues to be a role model for other
districts throughout the region.
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151
Appendix A
Instruments
Principal Interview Protocol
Questions
1. How much time do we have today for this interview?
2. Please give us an overview of how teachers use CGI within the school.
a. What did you hope to accomplish by adopting CGI?
3. Please describe the professional development of teachers related to CGI this
school year.
a. Number of days
b. Time of year (summer, during school year, etc.)
c. Who facilitated?
d. Who participated?
e. What types of activities – use of video, teacher reflection, etc.
f. Ongoing over multiple years
g. Use of teacher coaches
4. How have each of these changed over the past five years?
5. Thinking of the priorities for teachers at your school, how does CGI compare to
other initiatives? Has this changed over time, why?
6. How were you, as a principal, trained in the use of CGI at your school?
7. What are your expectations for the use of CGI throughout the school? For
individual teachers in their lessons?
8. To what extent do teachers support CGI activities within their lessons?
9. What evidence do you use in judging whether CGI is in use?
10. To what extent are teachers accountable for using CGI in their lessons?
152
Teacher Interview Protocol
1) What were your objectives for the lesson? (KNOW, DO, PROCESS) (What did
you want them to be able to do at the end? What did you want them to come away
with?)
a. How did you choose these objectives? (standards, pacing plan, team
decision, importance, etc.)
b. Are there any long-term goals or strategies that you incorporated into your
lesson?
2) How did you choose what activities to do in the lesson? (Interviewer may ask
specific questions here about what they observed.)
a. What did you want your students to do in order to reach the objectives?
b. How did you plan to get them there?
c. As you create your lesson plans, do you base the activities on the needs of
groups of students or individual students?
d. Specifically in the observed lesson, how and why did you choose the
specific problem difficulty options
3) How/when do students share their solutions? Will/did you do that for this lesson?
How will you facilitate student sharing? How will you choose which students to
present?
4) During class, how did you choose which students to directly interact with as they
worked on the problem?
5) Did you alter the lesson at any point to address a student need that surprised you?
Did you deviate at all from the lesson you had planned? Why?
6) What did the students learn from the lesson and how do you know that?
7) What parts of the lesson do you think worked well? Why?
8) Are there any objectives or parts of objectives that you think need to be further
addressed? Why?
a. What would you like to do to further address this objective?
9) What would you do differently if you were to teach this lesson again? Why?
10) How often do you use CGI?
153
a. How is CGI instruction different than your instruction during non-CGI
days?
11) What do you think your role is as a math instructor?
154
Teacher Survey Questions
Please answer the following questions to the best of your ability. Thank you very much
for your time.
1) For how many years have you been teaching?
2) For how many years have you taught this grade level? What other grades have
you taught, if any?
3) For how many years have you been at this particular school?
4) How long have you been teaching CGI?
5) What are your long-term goals for student learning of mathematics during the
year?
6) How do you usually go about designing daily mathematics lessons?
7) What professional development did you have to initially prepare you to
implement CGI?
a. Do you feel that it was effective in supporting you in teaching CGI? Why
or why not?
b. Have you completed all three years of the CGI training? If not, how much
have you completed?
8) What professional development/classroom support for CGI is currently available
to you (this school year)?
a. Do you feel that it is effective in supporting you in teaching CGI? Why or
why not?
b. What aspect of the training (could be initial or current) has made the
greatest impact on your teaching? Please explain.
c. How could you be better supported in implementing CGI in your
classroom?.
d. In which specific aspects of CGI do you think you would benefit from
additional support?
155
e. Have the professional development opportunities changed over the years?
In what ways?
9) To what degree do you plan CGI activities as a grade level/team? (How often?)
We probably discuss CGI about once a month.
10) Do you find CGI to be useful in teaching students mathematics? Why or why not?
a. Do you feel that CGI is effective for all students? Why or why not?
11) Do you feel that you implement CGI as it is intended? Why or why not?
12) Is there anything else you would like to share with us about teaching CGI or your
professional development experiences?
Teacher Observation Rubric
1 2 3 4-A 4-B
Opportunities
for Children to
Solve Problems
! Few, if any
! Children practice
repeating steps
! Limited
! One day per
week or at certain
set intervals
! May be used but
is not the focus
and is used in a
superficial way
! Sometimes used
word problems
like those
discussed in the
CGI workshop
! Engaged in rich problem solving most
of the time
! Spend most of class solving problems
! Structured curriculum around problem
solving
! Uses problems like those discussed in
the CGI workshops
! Decreased emphasis on learning
procedures
! Children solve a large variety of
mostly simple problems
! 2-4 problems solved during class
! Problems set in a variety of contexts
! Provides a variety of problem-
solving activities
! Engaged in rich problem solving all
of the time
! Curriculum made up of problem
solving
! Solved a large variety of challenging
problems
! Relationships between math and
other subjects continually stressed
Children
Sharing their
Thinking with
Peers and
Teacher
! Few, if any
! Direct instruction
! If children had
difficulty, the
teacher would
demonstrate
again with
different
numbers
! Limited
! One day per
week or at certain
set intervals
! Few children
might share their
methods but
would not be
questioned by the
teacher or peers
! Children did not
often talk about
their thinking
! Extensively reported on their thinking
and engaged in discussions about
math
! Spend most of class reporting
solutions to a variety of problems
! Talk a great deal about math both to
peers and the teacher
! Reported a variety of solution
strategies
! Fluent in their reporting
! Referred to explicit details of how
other children solved problems
! Children had time to either talk or
write about how they solved problems
! Attends to children sharing their
thinking
! Children expected to communicate
their solution strategies (orally or in
writing) and listen to other students’
strategies when shared orally
156
1 2 3 4-A 4-B
Teachers’
Elicitation and
Understanding of
Children’s
Thinking
Elicits and attends to children’s thinking
- Driven by general
knowledge of
children’s thinking
but not by
individual’s
- Basis of instruction
is on groups of
students thinking
- Driven by teacher’s
knowledge of
individual children
in the classroom
- Basis of instruction
is on each
individual student
- More detailed
knowledge of
individual students
Teachers’ use of
Children’s
Thinking as a
Basis for Making
Instructional
Decisions
! Little, if at all
! Not Evident
! Follows textbook
! Direct instruction
asking students to
follow steps
- Elicits or attends to
children’s thinking OR
uses what they share in a
limited way to make
instructional decisions
- At times asks students to
how they arrived at
answers
- Appears to be listening to
student
- Makes statements that
indicate the student was
misunderstood
- While student explains,
teacher interrupts to
correct the student
- Does not ask probing
follow up questions of
student
- Does not make the
student’s thinking clearly
evident
- Begins to elicit and
attend to children’s
thinking but does NOT
use this to make
instructional decisions
- Children are expected
to be able to report
how problems are
solved
- Children’s solutions
are valued by the
teacher and other
students
- Teacher elicit student
thinking and can
explain it
- Teachers see student
understanding as an
end to the process and
not a means for
planning
- Teacher tries to
understand student
thinking but is
incomplete
- Curriculum is based on problem solving
exclusively
- Problems are selected based on impact on
students
- Student responses to problems are
anticipated in advance
- Students are expected to communicate their
thinking
- Teachers probe with questioning until
student thinking is clearly
157
158
Appendix B
Sample CGI Problems
Mrs. Carter, Adams Elementary
159
Mrs. Grant, Adams Elementary
160
Mrs. Kennedy, Lincoln Elementary
161
162
Mrs. Madison, Lincoln Elementary
163
Mrs. Pierce, Lincoln Elementary
Abstract (if available)
Abstract
No Child Left Behind legislation developed goals for every student to be proficient in each academic subject by 2014. California's students are far from meeting this goal, especially in mathematics. One Southern Californian school district, renamed Green Valley Unified School District for anonymity, began using Cognitively Guided Instruction district-wide in 2005 for all elementary students in an effort to meet the NCLB goals. This dissertation is a case study of five second-grade teachers in two Green Valley schools and the degree of CGI implementation within their classrooms. This research developed assessment tools that may be useful for others evaluating teachers' use of CGI. This study also characterizes elements of classroom culture, professional development, and teacher's practice that lead toward CGI mastery. Recommendations are made for implementing a high quality CGI program, specific for Green Valley, however they offer guidance for other schools and districts that may use Cognitively Guided Instruction.
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Cognitively guided instruction: an mplementation case study of a high performing school district
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Rossier School of Education
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Publication Date
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