University of Southern California Dissertations and Theses

Cognitively guided instruction: A case study in two elementary schools

(USC Thesis Other)

Cognitively guided instruction: A case study in two elementary schools

PDF

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content
COGNITIVELY GUIDED INSTRUCTION:
A CASE STUDY IN TWO ELEMENTARY SCHOOLS

by

Kate Garfinkel

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

May 2011

Copyright 2011 Kate Garfinkel
ii

Acknowledgements

I would never have been able to complete my dissertation without the guidance of
my committee members, encouragement from friends and colleagues, and support from
my husband and family.
My dissertation committee has generously given their time and expertise to better
my work. I would like to thank my dissertation chair, Dr. Larry Picus, for his guidance,
support, patience, and constant availability throughout this process. My thanks and
appreciation to Dr. Julie Slayton for her thoughtful, detailed feedback and for challenging
me to think critically. Thank you to Dr. Gregory Franklin for his contributions to my
dissertation and for welcoming me into his district.
I would like to thank Ilda Jimenez y West for all of her writing and moral support
and reminding me of my accomplishments every step of the way. Thanks to Britt Dowdy
for working with me through the first part of this project. I must acknowledge as well the
Assistant Superintendent of Curriculum & Instruction, the principals, and the teachers
who participated in this study; without them this dissertation would not exist.
I would like to thank my friend Steve Sher for being a sounding-board and always
offering support. I hope that our discussions about learning theory, teacher education,
student learning, and mathematics never end.
Generous and loving thanks to my parents, Judy Politzer and Howard Garfinkel,
my brothers, Brian Garfinkel and Joel Garfinkel, my grandmother, Laura Garfinkel, my
great aunt, Jean Caro, and all of my family and friends back in Pennsylvania who have
supported and encouraged me from afar.
At this time I remember my grandfather, Joseph Garfinkel, and my grandmother,
Ruth Politzer, who would have been so proud to see me graduate as Dr. Kate Garfinkel.
Lastly I would like to thank my husband, Jesus Barajas, for being patient with me
even when I was impatient with myself, for tolerating my piles of books and papers all
over our tiny house, for playing his music quietly, and for taking over my chores so I
could have more time to write; his patience, understanding, and love as I have worked
through this process has been vital to my success. On to our next adventure!
iii

Table of Contents
Acknowledgements ..................................................................................................................ii
List of Tables ............................................................................................................................ v
List of Figures.......................................................................................................................... vi
Abstract ...................................................................................................................................vii
Chapter 1 The Problem and its Underlying Framework ........................................................ 1
Statement of the Problem .......................................................................................... 10
Purpose of the Study .................................................................................................. 12
Research Questions.................................................................................................... 13
Importance of the Study ............................................................................................ 13
Methodology .............................................................................................................. 14
Assumptions ............................................................................................................... 15
Limitations ................................................................................................................. 15
Delimitations .............................................................................................................. 16
Definition of Terms ................................................................................................... 17
Chapter 2 Review of the Literature ....................................................................................... 20
Learning and Teaching Mathematics ....................................................................... 21
Professional Development ........................................................................................ 28
Cognitively Guided Instruction ................................................................................ 39
Conclusion.................................................................................................................. 59
Chapter 3 Research Methodology ......................................................................................... 61
Research Questions.................................................................................................... 61
Research Design ........................................................................................................ 62
Conclusion.................................................................................................................. 82
Chapter 4 Results ................................................................................................................... 84
CGI Philosophy.......................................................................................................... 84
Findings for the First Research Question ................................................................. 85
Findings for the Second Research Question .......................................................... 119
Conclusion................................................................................................................ 127
Chapter 5 Conclusions ......................................................................................................... 129
Conclusions for the First Research Question ......................................................... 130
Conclusions for the Second Research Question .................................................... 134
Recommendations for the District .......................................................................... 138
Recommendations for Further Study...................................................................... 141
Conclusion................................................................................................................ 142
iv

References............................................................................................................................. 144
Appendices .................................................................................................................................
Appendix A .............................................................................................................. 153
Appendix B .............................................................................................................. 156
Appendix C .............................................................................................................. 158
Appendix D .............................................................................................................. 161
Appendix E............................................................................................................... 162
Appendix F ............................................................................................................... 164

v

List of Tables
Table 2.1: Cognitively Guided Instruction Workshop Description by Year ..................... 56
Table 3.1: GVUSD Student Demographics ......................................................................... 66
Table 3.2: Teacher Participant Information ......................................................................... 73
Table 3.3: Levels of Cognitively Guided Instruction ......................................................... 74
Table 4.1: Teacher Ratings for each CGI Critical Feature and Overall
Level of CGI Instruction ............................................................................... 87
Table 4.2: The Problems that Ms. Q Assigned to her Students during
Observation #1 .............................................................................................. 90

vi

List of Figures
Figure 1.1: Fourth and Eighth Grade NAEP Mathematics Average.................................... 5
Figure 1.2: California Fourth and Eighth Grade NAEP Mathematics
Scores versus National Fourth and Eighth Grade NAEP
Mathematics Scores ........................................................................................ 7
Figure 2.1: Lewin's (1935) Model of Teacher Change ....................................................... 30
Figure 2.2: Guskey's (1986) Model of Teacher Change ..................................................... 31
Figure 2.3: The Cognitively Guided Instruction Framework ............................................. 43

vii

Abstract

This study investigated the ways in which the elementary mathematics teaching
philosophy, Cognitively Guided Instruction (CGI), was reflected in the teaching practice
of a sample of teachers who subscribe to the philosophy. Cognitively Guided Instruction
relies on teachers basing their instructional decisions on the knowledge of their students’
mathematical thinking. The research questions for this study were (1) to what extent do
teachers’ instructional practices reflect the critical features of CGI? and (2) what aspects
of CGI professional development did the teachers perceive to be the most effective in
supporting the integration of CGI philosophy into their teaching practice? Five second
grade teachers within two schools in the same district participated in this study. Data
were collected through principal interviews and teacher observations, interviews, and
questionnaires. Evidence was found that the critical features of CGI were present in the
teaching practices of the teacher participants but the critical features were observed to
varying degrees. The majority of the teacher participants reported that CGI professional
development that provided immediate feedback that was directly connected to current
classroom practice (e.g., CGI coaches) was the most supportive of their CGI enactment.

1

Chapter 1
The Problem and its Underlying Framework

This study examines the enactment of an elementary school mathematics teaching
philosophy that was initiated with the intent to improve student mathematics achievement
in a school district. The Cognitively Guided Instruction (CGI) elementary mathematics
teaching philosophy has been shown to improve student achievement by improving
instruction through enhancing teachers’ knowledge of students’ learning (Carpenter,
Fennema, Peterson, Chiang, & Loef, 1989). Many researchers have demonstrated the
strong positive link between student achievement and effective instruction (Sanders &
Rivers, 1996; Nye, Konstantopoulos, & Hedges, 2004; Darling-Hammond, 2000;
Haycock, 1998) as well as between effective instruction and high-quality professional
development opportunities (Guskey, 1986, 2000; Loucks-Horsley, Stiles, Mundry, Love,
& Hewson, 2010). This study investigates the extent to which the principles of CGI are
currently evident in classroom practice as well as the professional development activities
that the observed teachers found to be the most effective in supporting their CGI practice.

Background of the Problem

A high-quality high school education can open doors for many students and pave
the way to advanced degree attainment and higher earnings. However, not all high school
courses equally influence later success. High school mathematics courses, in particular,
2

can have a powerful impact on students’ accomplishments after high school graduation
(Adelman, 1999, 2006; Rose & Betts, 2001, 2004). A 1999 study by Adelman concluded
that ―finishing a course beyond the level of Algebra 2 more than doubles the odds that a
student who enters postsecondary education will complete a bachelor’s degree‖ (p. 3). A
follow-up study by Adelman (2006) concluded that ―the highest level of mathematics
reached in high school continues to be a key marker in pre-collegiate momentum, with
the tipping point of momentum toward a bachelor’s degree now firmly above Algebra 2‖
(Adelman, 2006, p. xix). Adelman (2006) emphasized the importance of taking high-
level mathematics classes, not only for the influence on higher education, but for its
impact after graduation from college:
It’s not merely getting beyond Algebra 2 in high school any more: The world
demands advanced quantitative literacy, and no matter what a student’s
postsecondary field of study—from occupationally-oriented programs through
traditional liberal arts— more than a ceremonial visit to college-level mathematics
is called for. (Adelman, 2006, p. 108)
Rose and Betts (2001, 2004) studied links between high school mathematics
coursework, the probability of graduating from college, and potential future earnings. The
authors found that when controlling for occupation, demographics, and educational
attainment factors, there existed a positive relationship between taking advanced
mathematics courses, the probability of college graduation, and future earning power.
―The math courses that students take in high school are strongly related to students’
earnings around 10 years later... algebra and geometry [are] systematically related to
3

higher earnings for graduates a decade after graduation‖ (Rose & Betts, 2004, p. 17).
Rose and Betts (2001, 2004) additionally found that it is not only the number of math
courses taken that matters, but the ―extent to which students take the more demanding
courses, such as algebra/geometry‖ (Rose & Betts, 2001, p. xx). High level mathematics
courses, such as calculus, were found to have the largest effects while lower level courses
had progressively smaller effects on graduation rates and future earnings (Rose & Betts,
2001).
Student success in high school mathematics courses is dependent upon a solid
foundation of mathematical knowledge from elementary and middle school (National
Mathematics Advisory Panel (NMAP), 2008). In preparation for Algebra, elementary
mathematics curricula should ―simultaneously develop conceptual understanding,
computation fluency, and problem-solving skills‖ (NMAP, 2008, p. xix). Students cannot
be expected to complete higher level mathematics courses without these prerequisite
knowledge and skills; elementary and middle school mathematics courses must provide
this groundwork.
In general, K-12 schools in the United States ―are not doing a good job in
preparing students, especially minority and disadvantaged students, to excel in school and
to be successful in the labor market‖ (Rose & Betts, 2001, p. v). When compared both
nationally and internationally, elementary and middle school students in the United States
fared poorly in the basic mathematical skills necessary for success in high school
mathematics courses (NMAP, 2008; Stigler & Hiebert, 1999). Without basic math skills,
4

student success in high school math is limited. The future of students across the United
States is jeopardized when their achievement in mathematics is trivialized.
Student mathematics achievement, when compared nationally over time, has
become stagnant. The National Assessment of Educational Progress (NAEP) studies
educational trends within the United States using student achievement tests. The 2009
NAEP results described a trend of mathematics achievement within the United States that
appeared to be leveling out; historically, student math achievement showed increased
improvement each year but, in recent years, the increase in student math achievement has
become steady and minimal (see Figure 1.1). Thirty states showed no significant change
in mathematics test scores between 2007 and 2009 (National Center for Education
Statistics (NCES), 2009a). This leveling out is not occurring at maximum student
achievement levels; rather it is far below what has been determined to be proficient.
When examining the figure below, note that, for fourth graders, a score of 249 is
considered proficient and for eighth graders, a score of 299 is considered proficient.

5

NAEP has compared the fourth and eighth grade test scores of more than 160,000
students across the United States since 1990. Fourth grade student test scores in 2009
indicated that only 45% of students were scoring at levels that were proficient or above.
While this proportion has increased since 1996, when testing accommodations were first
permitted, there has only been a 9% increase in the percentage of students proficient or
above in the last six years. Eighth grade proficiency scores showed perhaps a brighter
picture. In 2009, 42% of eighth graders performed at proficient or above, significantly
higher than in all previous years (NCES, 2009a). While this percentage is lower than that
of fourth graders, there has been a greater increase over time. These proficiency levels
Figure 1.1. NAEP mathematics scores for fourth and eighth graders from 1990 to 2009. For
fourth graders, a score of 249 is considered proficient, for eighth graders, a score of 299 is
considered proficient. Data from U.S. Department of Education, Institute of Education Sciences,
National Center for Education Statistics, National Assessment of Educational Progress (NAEP),
various years, 1990 – 2009 Mathematics Assessments.

Figure 1.1. Fourth and Eighth Grade NAEP Mathematics Average
6

remain troubling because, ―what appears to matter most for increasing both earnings and
the probability of graduating from college is that students progress beyond basic courses
such as vocational math and pre-algebra toward more advanced topics‖ (Rose & Betts,
2001, p. 78); if students have low achievement levels in fourth and eighth grade they will
most likely not have the necessary preparation for Algebra 2 in the following years.
NAEP additionally reports on the achievement levels in each of the individual 50
states. When examining student achievement in California, it is apparent that students are
faring far more poorly than the national average. In the 2009 NAEP, while the percentage
of California fourth graders scoring proficient or above did increase (by one percentage
point) from 2007 to 2009, only 31% of students scored proficient or above; only three
states/jurisdictions scored significantly lower than California’s students, the District of
Columbia, Mississippi, and Alabama. California eighth graders’ performance is not
significantly different. The percentage of eighth graders scoring proficient or above
decreased (by one percentage point) from 2007 to 2009; 23% of California eighth grade
students scored proficient or above (NCES, 2009b; NCES, 2009c). NAEP indicates that
fourth and eighth graders in California are performing well below their peers nation-wide
as seen in Figure 1.2.

7

Fourth Grade NAEP Mathematics Average
Nation vs. California

Eighth Grade NAEP Mathematics Average
Nation vs. California

Figure 1.2. Comparison of California fourth and eighth graders NAEP performance with the
performance of fourth and eighth graders nationally. Source: U.S. Department of Education,
Institute of Education Sciences, National Center for Education Statistics, national Assessment of
Educational Progress (NAEP), various years, 1992 – 2009 Mathematics Assessments.

Research has shown that effective instruction can be a powerful influence on
student achievement (e.g., Gordon, Kane, & Staiger, 2006; NMAP, 2008; Nye, et al.,
2004; Sanders & Rivers, 1996). In fact, teacher effectiveness has been found to have such
Figure 1.2. California Fourth and Eighth Grade NAEP Mathematics Scores
versus National Fourth and Eighth Grade NAEP Mathematics Scores
8

a tremendous influence on student achievement that no student characteristic (such as
race, poverty level, parent’s education level, etc.) has such an impact (Carey, 2004). The
most effective teachers facilitate student achievement gains for students at all
achievement levels (Sanders & Rivers, 1996) no matter their race or family
circumstances: ―In the hands of our best teachers, the effects of poverty and institutional
racism melt away, allowing these students to soar to the same heights as young
Americans from more advantaged homes‖ (Haycock, 1998, p. 11). Keeping this in mind,
however, it is important to note that the National Mathematics Advisory Panel has
declared that ―the delivery system [instruction] in mathematics education – the system
that translates mathematical knowledge into value and ability for the next generation – is
broken and must be fixed‖ (NMAP, 2008, p. 11). As teacher instruction can have a
powerful impact on student achievement and the instructional system in the United States
has been declared to be ―broken,‖ it is no wonder that students are not performing at high
achievement levels.
The Third International Mathematics and Science Study (TIMSS) compared the
math achievement of fourth and eighth grade students in 36 and 48 countries, respectively
(Stigler & Hiebert, 1999). In addition to collecting student test score data, the TIMSS also
collected data to compare instruction in classrooms internationally. The TIMSS recorded
videotapes of instruction in eighth grade classrooms in the United States, Germany, and
Japan. Upon analysis of these videotapes, American students were found to be ―at a clear
disadvantage in their opportunities to learn‖ (Stigler & Hiebert, 1999, p. 65). Instruction
in the United States was characterized as ―learning terms and practicing procedures‖
9

because of the abundance of definitions presented to students and the vast amount of time
students spent practicing skills (Stigler & Hiebert, 1999). When compared with their
German and Japanese counterparts, U.S. students were found to be exposed to less
challenging mathematics–mathematical content was, on average, at a mid-seventh grade
level while the mathematical content in German and Japanese ―lessons were at the high
eighth- and beginning ninth-grade levels, respectively‖ (Stigler & Hiebert, 1999, p. 57).
Mathematics classrooms in the U.S. tended to cover more topics less deeply than in other
countries. Research (Stigler & Hiebert, 1999; NMAP, 2008) consistently demonstrates
that math instruction in the United States includes less advanced topics presented in a
more fragmented and prescriptive way when compared internationally.
Moreover, across the United States there exist substantial differences in teacher
effectiveness (Nye, et al., 2004). These teacher differences are profoundly impacting the
achievement of our students. While there is considerable variation in teacher
effectiveness, there are ways to reduce that variation and improve the quality of
instruction in order to improve student achievement. Through effective professional
development opportunities, teachers can gain specialized content knowledge and learn to
reflect on their practices in order to improve their classroom instruction (Loucks-Horsley,
et al., 2010). Unfortunately, not all teachers are exposed to the same high-quality
professional development programs (Darling-Hammond, Wei, Andree, Richardson, &
Orphanos, 2009). A promising strategy for improving student achievement may be in
ensuring the effectiveness of teachers by providing high-quality professional
development opportunities (Nye, et al., 2004). ―Ultimately, the success of U.S. public
10

education depends upon the skills of the 3.1 million teachers managing classrooms in
elementary and secondary schools around the country‖ (Gordon, et al., 2006, p. 5).

Statement of the Problem

It is clear from looking at performance data that student mathematics achievement
in the United States is in need of improvement (NCES, 2009a; NMAP, 2008). A
promising strategy to improve student achievement is to improve teacher effectiveness
(NMAP, 2008; Nye, et al., 2004). The issue that follows is in determining how to inform
the beliefs and actions of teachers in order to best improve teacher effectiveness.
Teachers often teach in the ways in which they were taught as students
(Loughran, 2006; Stigler & Hiebert, 1999; Toll, Nierstheimer, Lenski, & Kolloff, 2004).
These methods, such as ―teaching as telling‖ and a focus on skills rather than concepts,
are generally outdated and ineffective (Fuson, Kalchman, & Bransford, 2005; Loughran,
2006; Stigler & Hiebert, 1999). However, studies have shown that it is very difficult to
change the ways that teachers teach (e.g. Guskey, 1986, 2000; Loucks-Horsley, et al.,
2010; Loughran, 2006; Toll, et al., 2004). Recent research in teacher education has
focused on how to best alter the beliefs and actions of teachers in both pre-service and in-
service professional development programs in order to improve instruction and student
achievement (Guskey, 1986; Loucks-Horsley, et al., 2010; Loughran, 2006).
Guskey (1986) investigated professional development in schools and he found
that the majority of teachers participate in professional development because they want to
11

improve their teaching. Recent research by Webster-Wright (2009) additionally supports
this finding. Unfortunately, professional development programs are often characterized
by ―disorder, conflict, and criticism‖ (Guskey, 1986, p. 5) and additionally occur in
isolation from teachers’ classroom responsibilities (Darling-Hammond, et al., 2009;
Guskey, 1986, 2000; Loucks-Horsley, et al., 2010). Many professional development
programs fail because ―they do not take into account two critical factors: what motivates
teachers to engage in staff development, and the process by which change in teachers
typically takes place‖ (Guskey, 1986, p. 6; Webster-Wright, 2009).
Effective professional development activities must provide teachers with
practical, concrete ideas that can be used in the classroom immediately (Guskey, 1986;
Loucks-Horsley, et al., 2010). Guskey (1986) provides three principles of effective
development. The first is to recognize that change is a difficult and gradual process for
teachers. Second, teachers must receive regular feedback on the learning progress of their
students. Third, teachers must be provided with continued support throughout the year
(Guskey, 1986). These principles are further supported by more recent research by
Hammerness, Darling-Hammond, and Bransford (2005) and Loucks-Horsley, et al.
(2010). It can be very difficult and resource-consuming to change the beliefs and actions
of teachers. It is necessary, however, in order to improve student achievement in the
United States.

12

Purpose of the Study

The purpose of this study is to examine one districts’ approach to addressing the
quality of mathematics instruction in order to improve students’ academic achievement in
math. Thus, this study focuses on teachers’ approaches to math instruction and student
learning as they enact a philosophy of mathematics teaching. In addition, the study
examines the districts’ support of teachers as they enact this philosophy.
Blue River Elementary School (BRES) was the first school in Green Valley
Unified School District (GVUSD) to embrace the philosophy of Cognitively Guided
Instruction (CGI), which focuses on cultivating specific teacher beliefs about learning,
teaching, and applying effective instructional strategies in the classroom. Five years later,
after BRES students experienced improvements in achievement on standardized
mathematics tests, CGI was expanded to the other five district elementary schools. Green
Valley Unified School District adopted CGI in order to improve student mathematics
achievement across the district.
CGI has been accepted and used differently in each of the elementary schools;
this study focuses on two specific elementary schools in the GVUSD. This study
determines the degree to which the critical features of CGI are present in mathematics
lessons as well as the teachers’ perceived impact of CGI professional development on
their teaching practice. Because effective integration of the CGI philosophy necessitates
that teachers’ beliefs and actions are aligned with those of CGI, the researcher
investigated the current classroom practice of second grade teachers in an attempt
13

determine the alignment of the teachers’ actions to those of the critical features of CGI.
Additionally, the researcher explored the aspects of CGI professional development that
the teachers in this study perceived to have had the greatest impact on their CGI
classroom practice.

Research Questions

1. To what extent do teachers’ instructional practices reflect the critical features of
CGI?
2. What aspects of CGI professional development did the teachers perceive to be the
most effective in supporting the integration of the CGI philosophy into their
practice?

Importance of the Study

This study provides an investigation of the consistency of teachers’ instructional
actions to those of CGI. Two specific elementary schools were examined where teachers
in those schools were educated in the principles that guide CGI instruction. The degree to
which the critical features of CGI were present in the classroom practice of these teachers
was examined. Additionally, the CGI professional development that these teachers
perceived to be the most effective in supporting their adoption and use of the CGI
philosophy was explored. It is difficult to change the ways that teachers teach (Guskey,
14

1986, 2000; Loucks-Horsley, et al., 2010; Loughran, 2006; Toll, et al., 2004); the two
schools in this study serve as examples of how difficult it is to align teacher actions to
those of CGI.

Methodology

This study consisted of principal interviews and teacher observations, interviews,
and questionnaires. The principal of each elementary school included in the study was
interviewed in order to provide background information about the adoption of CGI at
each school site. Teacher observations were conducted with second grade teachers at
each of the schools to explore possible evidence of the critical features of CGI in their
classroom practice. The interviews with the teacher participants served as a follow-up to
the observations so that the impetus behind their actions could be explored.
Questionnaires were distributed to each teacher who participated in the study to gather
information about their past CGI professional development experiences as well as those
experiences they found to be the most influential to developing their CGI practice.
15

Assumptions

For purposes of this study, it was assumed that principal and teacher responses in
interviews and on the questionnaire were honest. Additionally, it was assumed that the
observed lessons were typical examples of classroom practice for each teacher observed.
This study attempted to determine the typical actions of teachers during ―CGI lessons‖ in
the two elementary schools where the study took place.

Limitations

Participation in this study was voluntary and relied on the willing participation
and honesty of both the principals and teachers who were interviewed and completed
questionnaires. When teachers were observed in their classrooms, it was important that
they did not alter their practice for our observation; the goal was to see a typical
mathematics lesson so that our observation was a true example of that teacher’s
instruction. It is not possible to know how consistent the lessons that were observed for
this study were with each teacher’s typical classroom practice. Since only two
observations took place in each classroom and the teachers knew that their CGI practice
was being studied, it was impossible to gain a complete view of each teacher’s typical
classroom practice (Patton, 2002).
Additionally, the researchers did not observe the professional development that
the teachers in this study were provided, nor did the researchers investigate how CGI had
16

evolved at each school. This study simply considered what CGI looked like in second
grade classrooms in the two schools at a moment in time; the reasoning behind why the
classrooms looked the way they did is beyond the scope of this study.

Delimitations

This study was limited by the amount of time and man power available to
complete classroom observations and teacher interviews. Because there were only two
researchers collecting data for this study, the number of classrooms that were observed,
as well as the number of observations, were fewer than would have been ideal; each
teacher was observed two times during ―CGI mathematics lessons.‖ The researchers were
not able to observe the teachers as they taught other subjects or taught ―non-CGI‖
lessons. As CGI is a philosophy, its principles may have been evident in the general
teaching practice of the observed teachers; unfortunately, that is impossible to determine
in this study.
Second grade teachers were observed at the recommendation of one of the school
principals and the teachers were voluntary participants; the sample was not chosen
randomly nor was it based on specific criteria determined by the researchers, thus the
ability to generalize the findings to other grade levels within and across schools in the
same district may be limited (Patton, 2002). Further, this study is limited to two schools
within one school district and thus may not be representative of all schools and districts
that use or have used the principles of CGI.
17

Definition of Terms

1. API: Academic Performance Index. In California, the API was established
by law in 1999 in an attempt to create an academic accountability system
for K-12 public schools. The API is calculated using students’
performance scores on state assessments in multiple content areas. The
API for a school can range from a low of 200 to a high of 1000 (California
Department of Education (CDE), 2010).
2. CGI: Cognitively Guided Instruction. This is an elementary school
mathematics philosophy that focuses on the instructional decisions that
teachers make in the classroom based on the knowledge and beliefs that
teachers have of their students (Fennema, Carpenter, & Franke, 1992).
Chapter 2 provides a more complete description of CGI.
3. CGI Implementation: While CGI is a philosophy of teaching elementary
mathematics, the district and schools that participated in this study
referred to it as if it were a program that was used at specific times. For
example, the teachers taught ―CGI lessons‖ twice per week where they
claimed to use the CGI philosophy in their teaching, ―non-CGI lessons‖
during the rest of the week when they used different curricula, and
continually referred to their ―CGI implementation.‖ While CGI is a system
of beliefs meant to be integrated into mathematics teaching practice
(Carpenter, Fennema, Franke, Levi, & Empson, 1999), the teachers and
18

administrators in the schools often spoke of CGI as if it was separate from
general mathematics instruction. It is unclear if the language by which
they refer to CGI was self-created or if it was part of the professional
development they received. The researcher will use the term ―CGI
Implementation‖ in chapters 4 and 5 because that is the way that CGI is
referred to in the schools; she is aware that this vocabulary is most likely
not intended to be used to describe the adoption of the CGI philosophy by
the developers of CGI.
4. NAEP: National Assessment of Educational Progress. NAEP is conducted
by the U.S. Department of Education and provides national results of
student achievement, instructional practices, and the school environment.
The results are based on samples of students in 4
th
, 8
th
, and 12
th
grades and
provide a common measurement for all states. NAEP is administered
periodically in mathematics, reading, science, writing, the arts, civics,
economics, geography, and U.S. history (NCES, 2010).
5. NMAP: National Mathematics Advisory Panel. In 2006, the President of
the United States created NMAP to advise national educational decisions
with the goal ―to foster greater knowledge of and improved performance
in mathematics among American students … with respect to the conduct,
evaluation, and effective use of the results of research relating to proven-
effective and evidence-based mathematics instruction‖ (NMAP, 2008, p.
7). The Panel compiled the findings that resulted from 20 months of
19

research into Foundations for Success: The Final Report of the National
Mathematics Advisory Panel (2008) into improvement recommendations
for mathematics curriculum, learning processes, teachers and teacher
education, instructional practices, instructional materials, assessment, and
research policies (NMAP, 2008).
6. Professional Development: Professional development is ―a systematic
attempt to bring about change–change in the classroom practices of
teachers, change in their beliefs and attitudes, and change in the learning
outcomes of students‖ (Guskey, 1986, p. 5). Professional development can
take many forms, such as workshops, conferences, learning communities,
coaching, and college courses (Desimone, 2009).
7. TIMSS: Trends in International Mathematics and Science Study.
Compared the mathematics and science achievement of students in 41
nations and collected observation videos from eighth grade classrooms in
German, Japan, and the United States (Stigler & Hiebert, 1999).
20

Chapter 2
Review of the Literature

There exists substantial agreement that mathematics education in the United
States is in need of improvement (Hill & Ball, 2004; NMAP, 2008; Stigler & Hiebert,
1999; Adelman, 2001, 2004). Extensive research has shown that mathematics instruction
will improve by providing teachers with effective professional development programs
that influence their beliefs, actions, and instructional practices (Darling-Hammond, et al.,
2009; Carpenter, et al., 1999; Guskey, 1986, 2000). Cognitively Guided Instruction (CGI)
is a philosophy of teaching elementary mathematics that is introduced to teachers through
a professional development model that has been shown to have positive effects on
improving classroom instruction and thus improving student achievement (Carpenter, et
al., 1989; Fennema & Carpenter, 1989; Fennema, Carpenter, Franke, Levi, Jacobs, &
Empson, 1996).
This chapter will describe what researchers have discovered about how students
learn mathematics and how teachers effectively teach mathematics. The aspects of
professional development programs that successfully produce teachers with these skills
will be discussed as well. Finally, a detailed description of the CGI philosophy and
framework will be presented.

21

Learning and Teaching Mathematics

The mathematics education that students are exposed to in the United States is not
adequately preparing them for college and the work force (Rose & Betts, 2001; Stigler &
Hiebert, 1999). Many researchers have studied how students learn in general (e.g.,
Bransford, Brown, & Cocking, 2000), and particularly, how students learn mathematics
(e.g., Fuson, et al., 2005; Carpenter, 1985; Loucks-Horsley, et al., 2010). In addition to
studying how students learn math, researchers have studied instructional practices that
best facilitate the learning process (e.g., Bransford, et al., 2000; Fuson, et al., 2005;
Loucks-Horsley, et al., 2010). The next section will provide an overview of research on
effective instructional practices for teaching mathematics and proceed to improving
instructional practices through professional development.
Mathematics classrooms across the United States have been plagued with
traditional teaching methods that have tended to emphasize students’ acquisition of skills
and competencies rather than their conceptual understanding (Fuson, et al., 2005).
Students come to school with intuitive, informal mathematical frameworks and a desire
to make meaning from what they know and learn (NMAP, 2008; Carpenter, et al., 1999).
Unfortunately, ―mathematics instruction often overrides students’ reasoning processes,
replacing them with a set of rules and procedures that disconnects problem solving from
meaning making‖ (Fuson, et al., 2005, p. 217) which encourages students to accept the
belief that that their ―sense making is irrelevant‖ (Fuson, et al., 2005, p, 217). In many
mathematics classrooms, ―procedural knowledge is often divorced from meaning making,
22

[so that] students do not use metacognitive strategies when they engage in solving
mathematics problems‖ (Fuson, et al., 2005, p. 217). Students often learn math as a set of
steps to solve problems rather than learning to become a critical problem solver, which
often leads to students believing that math is boring or that they just are not good at it
(Fuson, et al., 2005).
Becoming an effective mathematics teacher means learning how to analyze
student understanding and build on it with an end goal in mind (Fuson, et al., 2005;
Griffin, 2005). One researcher of mathematics teaching in primary grades put it this way:
You need to know where you are now (in terms of the knowledge children in your
classroom have available to build upon). You need to know where you want to go
(in terms of the school year). Finally, you need to know what is the best way to
get there (in terms of the learning opportunities you will provide to enable all
children in your class to achieve your stated objectives). (Griffin, 2005, p. 257)
Teaching math effectively is not easy; it takes effort over a long period of time
and is hard work (Fuson, et al., 2005; Loucks-Horsley, et al., 2010). As will be made
evident later in this chapter, these three ideas of knowing what students understand, what
they need to understand, and how to connect this gap sit at the very core of the CGI
philosophy.

23

Principles of learning.

The National Research Council conducted an extensive review of research on
student learning (Fuson, et al., 2005). Through their broad research, Fuson, Kalchman,
and Bransford (2005) synthesized three principles of learning that guide instruction in
any subject and at any level. They believe that current classroom instruction is poor
because teachers ―rarely [teach] in a way that makes use of the three principles‖ (Fuson,
et al., 2005, p. 217). The three principles are: (1) ―Teachers must engage students’
preconceptions,‖ (2) ―Understanding requires factual knowledge and conceptual
frameworks,‖ and (3) ―A metacognitive approach enables student self-monitoring‖
(Fuson, et al., 2005, p. 219).
The first principle, teachers must engage students’ preconceptions, assumes that
―people possess resources in the form of informal strategy development and
mathematical reasoning that can serve as a foundation for learning more abstract
mathematics‖ (Fuson, et al., 2005, p. 219). Students enter the classroom with the ability
to solve problems and think mathematically, teachers must build on this knowledge to
help students to make connections between what they know and what they are expected
to learn.
This first principle brings two instructional challenges to light (Fuson, et al.,
2005). The first is teaching mathematics so that students understand that it is about
solving important problems and not simply computing and following rules. The second
challenge is in linking students’ informal mathematical and problem solving knowledge
24

with the formal mathematics of the classroom. Through their research, the authors
devised three ways to overcome these challenges, which all address students’
preconceptions (Fuson, et al., 2005). The first solution is to allow students to use their
own problem solving strategies first, and then teachers may guide their thinking toward
more efficient strategies. This allows students to incorporate their own knowledge into
solving the problem. The second solution is to encourage math talk to clarify ideas and
compare approaches. Students should have the opportunity to talk about their problem
solving methods with the teacher and peers to learn to explain their reasoning and
compare their methods with those of other students. The last solution is for teachers to
design instructional activities that effectively bridge the gap between informal and formal
problem solving and address preconceptions by carefully planning the problems that are
posed for students to solve. Students often see the problem solving skills and
mathematical understandings that they have as separate from what they learn in school.
Students cannot usually bridge this gap between informal and formal mathematical
knowledge alone, they need guidance from the teacher (Fuson, et al., 2005).
The second principle of learning is that understanding requires factual knowledge
and conceptual frameworks (Fuson, et al., 2005). There is a relationship between
students’ knowledge of procedures and facts and their conceptual understanding that
must remain balanced. The current issue in the United States is that students gain factual
and procedural knowledge in school without conceptual understanding (Stigler &
Hiebert, 1999; NMAP, 2008; Fuson, et al., 2005). Deep conceptual understanding,
however, is impossible without a basis of factual and procedural knowledge; they work
25

together, hand in hand. Factual and procedural knowledge must form the basis for
students’ conceptual understanding and teachers must facilitate the process of student
thinking from facts to concepts. Conceptual knowledge, procedural fluency, and effective
organization of knowledge are all necessary for effectively learning mathematics (Fuson,
et al., 2005).
The challenge for teachers, then, ―is to help students build and consolidate
prerequisite competencies, understand new concepts in depth, and organize both concepts
and competencies in a network of knowledge‖ (Fuson, et al., 2005, p. 232). Using student
discussion can help students to clearly articulate their understandings, see new methods
of solving problems, and develop a deeper comprehension of the mathematics. Teachers
who are knowledgeable of student learning paths can direct math talk and student
thinking toward valued knowledge networks and conceptual frameworks (Fuson, et al.,
2005).
The third and last principle of learning is that a metacognitive approach enables
student self-monitoring (Fuson, et al., 2005). The experiences that students have in
mathematics classrooms can ―have strong effects on their beliefs about themselves, as
well as their abilities to remember information and use it spontaneously to solve new
problems‖ (Fuson, et al., 2005, p. 238). Because ―wisdom can’t be told,‖ students need
guidance to reflect on their experiences so that they are able to see ―their ideas as
instances of larger categories of ideas,‖ (Fuson, et al., 2005, p. 238) enabling them to see
the ―big picture‖ and develop a conceptual understanding of mathematics.
26

Instruction that supports metacognition includes a focus shift from ―answers as
just right or wrong‖ to ―debugging a wrong answer‖ (Fuson, et al., 2005, p. 239). In
essence, the process of solving problems becomes just as important, if not more so, than
the answer. Teachers can encourage students to focus on the problem solving process by
teaching them how to communicate about mathematics and ensuring that they have the
confidence not only to attempt challenging problems, but to ask for help when they
become stuck (Fuson, et al., 2005). When teachers support students in constructing their
own answers to a question ―by reflecting on their own activity, teachers are encouraging
the use of metacognitive processes and allowing children to take charge of their own
learning‖ (Fuson, et al., 2005, p. 294). This is the general consensus of the ways in which
students learn mathematics (Fuson, et al., 2005), now the focus will shift to the ways in
which teachers can effectively teach to maximize student learning.

Teaching.

Loucks-Horsley, et al. (2010) conducted an extensive analysis of research on the
teaching and learning of mathematics and science. Through this analysis, they have
synthesized three general concepts that frame what is known about teachers and teaching,
which will be discussed here, in addition to characteristics of effective professional
development for teachers, which will be discussed later in this chapter.
The first conceptual frame defines the purpose of teaching as facilitating learning
(Loucks-Horsley, et al., 2010). Similar to the work of Fuson, et al. (2005), Loucks-
27

Horsley, et al. (2010) concluded that teachers must ―match learners and what they know
with the intended curriculum in ways that make learning achievable‖ (p. 61). Essentially,
teachers must be able to determine what students currently know, what students should
know, and have the knowledge and expertise to get them there; this is almost exactly
what was explained in the quotation cited earlier in this chapter by Griffin (2005). For
teachers to have this knowledge and expertise, they must have ―opportunities to develop
advanced knowledge in their content, an understanding of what they can learn by
examining student work and thinking, a diverse array of assessment strategies, and a
range of instructional strategies‖ (Loucks-Horsley, et al., 2010, pp. 61-62).
The second conceptual frame is that teaching is a profession requiring specialized
knowledge (Loucks-Horsley, et al., 2010). Teachers require three types of knowledge:
content knowledge, pedagogical knowledge, and specialized content knowledge, or, what
Shulman (1986) calls pedagogical content knowledge (Loucks-Horsley, et al., 2010).
Content knowledge is the expertise and understanding that a teacher has in a specific
content area, like mathematics. Pedagogical knowledge refers to teacher’s knowledge of
general effective instructional strategies and how students learn and develop. Specialized
content knowledge is ―an understanding of what makes the learning of specific concepts
easy or difficult for learners, an awareness of what concepts are more fundamental than
others, and knowledge of ways of representing and formulating subject matter to make it
accessible to students‖ (Loucks-Horsley, et al., 2010, p. 63). In other words, specialized
content knowledge is the combination of content and pedagogy; what a teacher knows
specifically about teaching a particular content area.
28

The final conceptual frame is that the practice of teaching is complex (Loucks-
Horsley, et al., 2010). For a teacher, ―developing rote and factual knowledge is simpler
than developing in-depth understanding of science and mathematics concepts‖ (Loucks-
Horsley, et al., 2010, p. 64-65). Teaching in ways that promote conceptual understanding
and metacognition by using the learning principles of Fuson, et al. (2005) is complicated
and time-consuming because it requires a complex cycle of planning, implementing,
assessing, and reflecting (Loucks-Horsley, et al., 2010). Teachers must understand what
their students know by watching and listening to them, facilitate discourse, create
opportunities for all students to learn, and ―have a repertoire of strategies for responding
appropriately to the variety of knowledge and experience brought by their students, and
work to ensure equal access to equitable teaching for all students by using proven
strategies‖ (Loucks-Horsley, et al., 2010, p. 65).
Teaching effectively is difficult (Fuson, et al., 2005; Loucks-Horsley, et al.,
2010). Teachers must facilitate the learning of their students by using their specialized
knowledge to constantly reflect on and improve their instructional practice (Loucks-
Horsley, et al., 2010). As student learning is influenced by teacher instruction, teacher
instruction is influenced by professional development.

Professional Development

Thomas Guskey, a well-known researcher and developer of professional
development for teachers, defines professional development as ―a systematic attempt to
29

bring about change – change in the classroom practices of the teacher, change in their
beliefs and attitudes, and change in the learning outcomes of students‖ (1986, p. 5).
However, changing the classroom practices and beliefs of teachers can be a daunting task
(Guskey, 1986, 2000; Loucks-Horsley, et al., 2010; Loughran, 2006; Stigler & Hiebert,
1999; Toll, et al., 2004).
Many people enter the teaching profession because as students they were
successful in traditional, didactic classrooms, where learning was memorizing and
mathematics standards were different than they are today (Loucks-Horsley, et al., 2010;
Loughran, 2006). Their experiences as students serve as a ―script‖ that they follow in
their own teaching (Stigler & Hiebert, 1999). Unfortunately, this ―script‖ is outdated and
ineffective (Loucks-Horsley, et al., 2010; Loughran, 2006; Stigler & Hiebert, 1999).
Effective professional development activities support the change process that requires
teachers to act and think in new ways (Loucks-Horsley, et al., 2010).
Guskey (1986) describes a framework for teacher learning that was originally
described by Lewin (1935) and explains how he has altered this framework to take into
account his own research. Lewin (1935) developed a model of teacher change consistent
with Figure 2.1.

30

Figure 2.1. From ―Staff Development and the Process of Teacher Change,‖
by T.R. Guskey, 1986, Educational Researcher, 15(5), p. 7.

Lewin (1935) describes the process of teacher change as initiated by staff
development. The staff development changes the beliefs and attitudes of teachers by
exposing them to new ideas or information. This change in beliefs and attitudes translates
into altered instructional practices which then results in a change in student achievement.
This framework is also consistent with Desimone (2009) who proposed a similar model
as a ―core theory of action‖ for investigating the effects of professional development on
teachers and students.
Guskey (1986) has altered this framework, based on his own and others’ research
as is illustrated in Figure 2.2.

Staff
Development
Change in
Teachers’
Beliefs and
Attitudes
Change in
Teachers’
Classroom
Practices
Change in
Student
Achievement
Figure 2.1. Lewin's (1935) Model of Teacher Change
31

Figure 2.2. From ―Staff Development and the Process of Teacher Change,‖
by T.R. Guskey, 1986, Educational Researcher, 15(5), p. 7.

Guskey’s (1986) model proposes that staff development alters teachers’
classroom practices which changes student achievement. This change in student
achievement then causes a change in the beliefs and attitudes of the teacher because the
teacher can see the direct effects of her altered practice on student achievement.
The major difference between Lewin’s (1935) and Guskey’s (1986) model is the
placement of change in teachers’ beliefs and attitudes in the teacher change sequence.
The model of Lewin (1935) relies on the staff development activities to change teachers’
beliefs and attitudes. Guskey’s (1986) model relies on staff development to alter the
actions of teachers while the change in teachers’ beliefs and attitudes will not occur until
the effects of the changed actions are seen in student achievement. The professional
development model developed for CGI is based upon Guskey’s (1986) model of teacher
change (Franke, Carpenter, Fennema, Ansell, & Behrend, 1998; Fennema, et al., 1996).
Whether Lewin’s (1935) model, Guskey’s (1986) model, or a different model of
professional learning is assumed, many current professional development programs are
Change in
Teachers’
Beliefs and
Attitudes
Staff
Development
Change in
Teachers’
Classroom
Practices
Change in
Student
Achievement
Figure 2.2. Guskey's (1986) Model of Teacher Change
32

ineffective in improving the actions of teachers in the classroom because they are
disconnected from practice, sporadic, disorganized, and consist of short-term conferences
or workshops that teachers do not find useful in improving their practice (Darling-
Hammond, et al., 2009; Desimone, Porter, Garet, Yoon, & Birman, 2002; Webster-
Wright, 2009). ―Simply providing [teachers with] different and more collegial forms of
professional development is not the answer‖ to improving instruction (Loucks-Horsley, et
al., 2010, p. 68). When comparing the professional development opportunities for United
States public school teachers to the opportunities offered to teachers in other countries,
the U.S. is behind in providing ―opportunities to participate in extended learning
opportunities and productive collaborative communities‖ (Darling-Hammond, et al.,
2009, p. 6). Effective professional development must reflect the nature of the discipline
and guide teachers to ―construct knowledge in the same ways as do effective learning
experiences for students‖ (Loucks-Horsley, et al., 2010, p. 76).
There are a number of research studies that have attempted to define specific
aspects of effective professional development programs for teachers of mathematics
(Bransford, et al., 2000; Desimone, et al., 2002; Franke, Carpenter, Fennema, Ansell, &
Behrend, 1998; Franke, Carpenter, Levi, & Fennema, 2001; Garet, Porter, Desimone,
Birman, & Yoon, 2001; Hill & Ball, 2004). In addition to these, there are recent studies
that have attempted to synthesize the research on effective professional development
programs (Darling-Hammond, et al., 2009; Loucks-Horsley, et al., 2010; Whitcomb,
Borko, & Liston, 2009; National Research Council (NRC), 2001). In another publication,
Guskey (2000) describes effective professional development for in-service teachers as a
33

process that is intentional, ongoing, and systematic; this process is also supported by
Darling-Hammond, et al. (2009), in their review of research. This simple framework will
now be used to compare and contrast further research on professional development; the
research findings can be situated within Guskey’s (2000) effective professional
development processes of intentional, ongoing, and systematic.

Professional development as intentional.

Guskey (2000) considers intentional professional development to be deliberate
and guided by a clear vision of purpose. Professional development must have a clear
focus on learners and learning that emphasizes individual and organizational change
(Guskey, 2000). Garet, Porter, Desimone, Birman, and Yoon (2001) conducted a survey
using a national probability sample of teachers to collect detailed information about
specific professional development activities. The teachers ―self-reported increases in
knowledge and skills in several different areas and changes in classroom practice‖ (Garet,
et al., 2001, p. 920). The authors analyzed the responses from 1,027 mathematics and
science teachers and synthesized the results to create a list of six key elements of
effective professional development. In a similar survey-based study, Desimone, Porter,
Garet, Yoon, & Birman (2002) used the same key features as in the study by Garet, et al.
(2001), but focused on how professional development that incorporated the key features
changed teachers’ practices over three years. They found that the six key elements of
professional development were effective in improving teaching practice (Desimone, et
34

al., 2002), providing additional support for the key elements of mathematics professional
development described in the previous study by Garet, et al. (2001). These elements will
be discussed below as they appropriately relate to Guskey’s (1986) framework.
Two key elements of effective professional development determined by Garet, et
al. (2001) and Desimone, et al. (2002) can be considered to focus on professional
development as intentional. One key feature is that of coherence; professional
development programs must be aligned with the goals of teachers, state standards, and
developing a professional community. The second key feature is content focus;
deepening teachers’ knowledge in mathematics is an important factor in effective
mathematics teaching (Garet, et al., 2001). Professional development as coherent and
content focused provides a clear, deliberate intent for the structure and design of the
program.
Hill and Ball (2004) studied California’s Mathematics Professional Development
Institutes (MPDIs). The MPDIs, at the time, were the largest content-focused professional
development program in the United States, serving over 23,000 K-12 teachers within the
first three years of the program (Hill & Ball, 2004). The focus of this study was to find
out whether ―elementary school teachers can learn mathematics for teaching in a
relatively traditional professional development setting – the summer workshop
component‖ (Hill & Ball, 2004, p. 331).The researchers administered pre- and post-
assessments to the teacher participants before and after a one-to-three week summer
institute focused on developing content knowledge for teaching mathematics (Hill &
Ball, 2004). Hill and Ball (2004) found that one successful aspect of the program was its
35

focus specifically on mathematical content. This clear purpose supports the importance of
intentionality in professional development.
Loucks-Horsley, et al., (2010) conducted a synthesis of research on professional
development practices and found nine aspects of effective professional development.
Although they were not categorized according to the research of Guskey (2000), they will
be discussed here as they fit into his framework. Three of the aspects of effective
professional development discussed by Loucks-Horsley, et al. (2010) can be considered
intentional. The first is that professional development must show ―direct alignment with
student learning goals and needs‖ (Loucks-Horsley, et al., 2010, p.70); this supports the
need for a clear purpose. The second, ―experiences [should be] driven by a well-defined
image of effective classroom learning and teaching‖ (Loucks-Horsley, et al., 2010, p. 70)
supports the need for a clear vision for professional development. The third element that
supports intentionality in a professional development program by dictating the need for a
clear goal is that it includes ―experiences that provide opportunities for teachers to build
their content and pedagogical content knowledge and skills and examine and reflect on
practice critically‖ (Loucks-Horsley, et al., 2010, pp. 70-71). Effective professional
development must have a clear purpose and deliberate structure, making its design
intentional (Guskey, 2000; Loucks-Horsley, et al., 2010).

36

Professional development as ongoing.

Ongoing professional development is long-term, focuses on change over a period
of time (often for a number of years) and focuses on teachers as life-long learners
(Guskey, 2000). The duration of the professional development is important in both the
number of hours of participation in an activity as well as the span of time that the activity
takes place (Garet, et al., 2001; Desimone, et al., 2002). In the study by Hill and Ball
(2004), teachers who participated in content-focused professional development programs
that had a longer duration tended to show higher gains in student achievement. The
authors do caution, however, that what teachers spend time doing is also important (Hill
& Ball, 2004); the activity must provide opportunities for active learning on the part of
the teachers where they engage in meaningful analysis of teaching and learning (Garet, et
al., 2001; Desimone, et al., 2002).
Two of the aspects of effective professional development determined by Loucks-
Horsley, et al. (2010), can be included in the category of ongoing: ―professional
development must be intensive, ongoing, and connected to practice;‖ and ―there must be
support for teachers to deepen their professional expertise throughout their career and
serve in leadership roles‖ (pp. 70-71). When professional development is ongoing,
teachers often become accustomed to reflecting on and analyzing their practice which
encourages the idea of teachers as life-long learners who continuously work to improve
their teaching practice (Loucks-Horsley, et al., 2010; Guskey, 2000). Because teaching is
difficult (Fuson, et al., 2005; Loucks-Horsley, et al., 2010), ongoing professional
37

development provides support for teachers to develop their teaching practices over time
with the guidance of experienced personnel; through this ongoing practice, teachers are
supported and encouraged to become life-long learners (Guskey, 2000; Loucks-Horsley,
et al., 2010).

Professional development as systematic.

Professional development that is systematic takes all levels of the school as an
organization into account, focusing both on teachers as individuals and the school as a
whole (Guskey, 2000). Systematic professional development also includes the type of
activity that the teachers are experiencing (Guskey, 2000). Traditional professional
development activities such as short-term workshops or conferences are not considered
systematic; rather, reform activities such as study groups, mentoring opportunities, task
forces, internships, action research, or lesson study are systematic because they take place
over long periods of time and are aimed at improving the professional community of
teachers (Garet, et al., 2001; Desimone, et al., 2002). Systematic professional
development includes collective participation of groups of teachers from the same grade
level, content area, or school, as opposed to participation of individual teachers from
many schools (Garet, et al., 2001; Desimone, et al., 2002).
The elements of effective professional development compiled by Loucks-Horsley,
et al. (2010) were placed into four categories that pertain to systematic professional
development: ―experiences that are research based and engage teachers as adult learners
38

in the learning approaches they will use with their students,‖ ―provide time and
opportunities for teachers to collaborate and build strong working relationships,‖ ―are
connected to other school initiatives,‖ and ―involve continuous monitoring and evaluation
of the activity‖ (p. 70-71). These aspects connect teacher learning in professional
development activities to their teaching actions in the classroom and encourage school-
wide, long-term efforts to improve teaching and student learning (Loucks-Horsley, et al.,
2010).
Effective professional development programs can only succeed ―with
simultaneous attention to changing the system within which teachers and other educators
work‖ (Loucks-Horsley, 2010, p. 77). With this change in mind, it is important to
consider both individual and systemic change (Loucks-Horsley, et al., 2010). Professional
development programs must focus on influencing the beliefs and actions of teachers
while also creating a professional community of educators within the school who work
collectively to improve student achievement. System-wide change such as this takes time;
―it can take three to five years for teachers to fully implement a new practice or program‖
(Loucks-Horsley, 2010, p. 77); participants must be patient and committed to long-term
improvement. When new professional development programs are implemented, it is
crucial that Guskey’s (2000) three principles of effective professional development are
considered (intentional, ongoing, and systematic) so that the program can truly create the
change that is necessary for improved student achievement (Darling-Hammond, et al.,
2009). Guskey’s (2000) three principles of effective professional development are
39

evident in the design of Cognitively Guided Instruction professional development, as will
be shown in the following section.

Cognitively Guided Instruction

Cognitively Guided Instruction (CGI) is a philosophy of teaching elementary
school mathematics that focuses on students’ thinking. It is based on the premise that
―children enter school with a great deal of informal or intuitive knowledge of
mathematics that can serve as the basis for developing understanding of the mathematics
of the primary school curriculum‖ (Carpenter, et al., 1999, p. 4). When students are
guided to make connections between their informal, intuitive knowledge and the formal
knowledge of the classroom, their understanding of factual and procedural knowledge
improves as well as their problem solving abilities (Fuson, et al., 2005; Carpenter, et al.,
1999; Carpenter, et al., 1989; Fennema, et al., 1996; Peterson, Fennema, Carpenter &
Loef, 1989).
Cognitively Guided Instruction was developed at the Wisconsin Center for
Education Research at the University of Wisconsin – Madison in the late 1980s by
Elizabeth Fennema, Thomas Carpenter, Penelope Peterson, and Megan Franke and is
primarily based on research by Thomas Carpenter (1985) that focused on the ways in
which students conceptually learn addition and subtraction. Cognitively Guided
Instruction began as a research project designed to study ―the impact of research-based
knowledge about children’s thinking on teachers and their students‖ (Fennema, et al.,
40

1992, p. 1). The project grew to include the investigation of effective professional
development opportunities for teachers, childrens’ thinking in grades K-3, CGI in urban
schools, and CGI in preservice teachers’ education (Fennema, et al., 1992).
Cognitively Guided Instruction does not prescribe instruction and it is not a
curriculum; it is a process of teaching and interacting with students that bases
mathematics instruction on students’ thinking (Fennema, Carpenter, Levi, Franke, &
Empson, 1999). The CGI philosophy is based on three Big Ideas: ―(1) Young children
have a rich informal knowledge of mathematics that can serve as a basis for developing
understanding for mathematics, (2) Mathematics instruction should be based on what
children understand about mathematics, and (3) When teachers understand a research-
based framework of children’s thinking, they can make instructional decisions about what
and how to teach that expand children’s knowledge about mathematics‖ (Fennema, et al.,
1999, p. 1). The goal of CGI is not to tell teachers how to teach, but to expose them to the
research on how students learn mathematics so that they are able to make effective
instructional decisions (Fennema, et al., 1999).
Teachers who apply the philosophy of CGI in their teaching focus on student
understanding because ―when children understand something, they can use that
knowledge in numerous ways, they remember it longer, and they can use it to learn more
mathematics‖ (Fennema, et al., 1999, p. x). Teachers base their instruction on three basic
assumptions about how children learn mathematics: ―Children have knowledge of
mathematics, which should be used as a basis for instructional decisions;‖ ―children
develop mathematical understanding and acquire skills by solving a variety of problems
41

in any way that they choose;‖ and ―children gain understanding and inform other children
and their teachers as they communicate their mathematical thinking‖ (Fennema, et al.,
1999, p. xii). These three assumptions are supported by additional research on childrens’
learning of mathematics (e.g., Fuson, et al., 2005; Loucks-Horsley, et al., 2010).
In CGI classrooms, the basic assumptions of how students learn mathematics are
translated into classroom instruction by allowing students to decide how they will solve
problems, focusing on problem solving as the center of instruction, and having children
communicate their reasoning behind how they solved problems to their teacher and peers
(Fennema, et al., 1999). These teaching methods allow students to connect what they are
learning to prior knowledge, experience solving a variety of problems, and discuss
problem solving methods with their peers. CGI classrooms are places where students
explore mathematics in a supportive environment and are able to create their own
meaning of basic mathematical concepts that will allow them to build on that knowledge
in the future and have a solid base for algebraic reasoning.
Teachers must be knowledgeable of both mathematics content and children’s
mathematical thinking so that they can design problems that are both challenging and
solvable for students (Fennema, et al., 1992). The role of the CGI teacher is to understand
students’ thinking and actively make ―decisions about what experiences those children
should have to ensure that their knowledge and abilities in mathematics [grow]‖
(Fennema, et al., 1999, p. xi). As one CGI teacher explained, ―My role is just helping.
Not showing them how, but giving them different opportunities and stretching them as
they go‖ (Fennema, et al., 1999, p. xi).
42

There are four core teacher beliefs that have been found to correlate positively
with children’s learning. First, successful CGI teachers believe that instructional
decisions should be based on children’s learning. Teachers should teach in ways that best
enable their students to learn new material. The second belief is that children come to
school with informal knowledge that allows them to solve math problems without
instruction. The third belief is that the role of the teacher is to create a learning
environment ―where children can construct their own knowledge rather than where the
teacher is a transmitter of knowledge‖ (Fennema, et al., 1992, p. 8). The fourth belief is
that children can learn to solve math problems before they learn procedural skills.
Becoming a successful CGI teacher takes time as it often includes modifying the
knowledge and beliefs of teachers (Fennema, et al., 1992).
Teachers are exposed to CGI through ongoing professional development that is
structured around a research-based framework of children’s mathematical thinking when
solving addition and subtraction problems. Addition and subtraction problems can be
categorized by type of problem and difficulty level to help teachers choose the most
appropriate problems for their students. Carpenter, et al. (1999) provide a framework that
categorizes addition and subtraction problems in this way (see Figure 2.3).

43

Figure 2.3. The Cognitively Guided Instruction Framework

Figure 2.3. Cognitively Guided Instruction framework developed to explain and give examples
of the various types of addition and subtraction problems that elementary students are exposed to
in mathematics classrooms. This framework is used to provide teachers with a structure with
which to categorize types of problems. From ―Children’s Mathematics: Cognitively Guided
Instruction,‖ by T.P. Carpenter, E. Fennema, M.L. Franke, L. Levi, & S.B. Empson, 1999, p. 12.
Copyright 1999 by the authors.

The framework helps teachers to understand their own students’ thinking while
support through the professional development program guides teachers in deciding how
to use that knowledge in making instructional decisions (Carpenter, et al., 1999). In CGI
professional development, teachers learn to ―ask questions that elicit children’s thinking,
Problem
Type

Join (Result Unknown)
Connie had 5 marbles.
Juan gave her 8 more
marbles. How many
marbles does Connie
have altogether?
(Change Unknown)
Connie has 5 marbles.
How many more marbles
does she need to have 13
marbles altogether?
(Start Unknown)
Connie had some
marbles. Juan gave her
5 more marbles. Now
she has 13 marbles.
How many marbles did
Connie have to start?
Separate (Result Unknown)
Connie had 13 marbles.
She gave 5 to Juan. How
many marbles does
Connie have left?

(Change Unknown)
Connie had 13 marbles.
She gave some to Juan.
Now she has 5 marbles
left. How many marbles
did Connie give to Juan?
(Start Unknown)
Connie had some
marbles. She gave 5 to
Juan. Now she has 8
marbles left. How
many marbles did
Connie have to start
with?
Part-
Part-
Whole
(Whole Unknown)
Connie has 5 red marbles and 8 blue
marbles. How many marbles does she
have?
(Part Unknown)
Connie has 13 marbles. 5 are red and
the rest are blue. How many blue
marbles does Connie have?
Compare (Difference Unknown)
Connie has 13 marbles.
Juan has 5 marbles. How
many more marbles does
Connie have than Juan?
(Compare Quantity
Unknown)
Juan has 5 marbles.
Connie has 8 more than
Juan. How many marbles
does Connie have?
(Referent Unknown)
Connie has 13 marbles.
She has 5 more marbles
than Juan. How many
marbles does Juan
have?
44

listen to what children report, and build their instruction on what is heard‖ (Fennema, et
al., 1992, p. 8).
The CGI professional development workshops begin ―with teachers’ preexisting
knowledge about children’s thinking and continue by aiding teachers to build a structured
body of knowledge that enables them to understand the thinking of each child in the
classroom‖ (Fennema, et al., 1999, p. xii). It is important to note that just as CGI
instruction must build on students’ preexisting knowledge of mathematics, so must CGI
professional development build on teachers’ preexisting knowledge of children’s
thinking. The assumptions of the CGI professional development program reflect the
assumptions of the CGI classroom:
Teachers’ understanding of their students’ thinking enables them to make
instructional decisions. This, in turn, enables their students to learning with
understanding.
Teachers have intuitive understandings of children’s thinking that can serve
as the basis for acquiring a more structured understanding of children’s
thinking.
Teachers can understand the thinking of each child in their classroom when
they have a structured understanding of children’s thinking.
Teaching for understanding is too complex to be prescribed. Thus, teaching
is a process of problem solving. The problem solving that teachers must do in
order to encourage their students to learn with understanding is facilitated by
knowledge of children’s thinking, by communication with other teachers
45

about their students’ problem solving, and by working with children in
classrooms. (Fennema, et al., 1999, p. xii)
The CGI professional development program, like CGI teaching, does not have a
prescribed method of implementation. Rather, schools can structure CGI professional
development in a variety of ways that are most appropriate for their school site (Fennema
& Carpenter, 1989; Fennema, et al., 1999). Often, CGI professional development
programs include a multi-day workshop with ongoing classroom support and shorter
workshops throughout the year (Fennema, et al., 1999). In addition to studying the
research base of CGI and analyzing the framework, CGI workshops often include solving
mathematical problems, reflecting on beliefs and practices of instruction, and watching
and analyzing video tapes of children solving problems and explaining their thinking, as
well as videos of CGI teachers teaching in their classrooms (Fennema, et al., 1999). The
success of CGI in the classroom is dependent upon the extent to which the knowledge
and beliefs of the teacher are consistent with those of CGI.

Supporting research for Cognitively Guided Instruction.

Multiple research studies of Cognitively Guided Instruction (CGI) have found
many benefits for student learning. A synthesis of the benefits of CGI finds that children
who experience CGI in their classroom were more confident in their math abilities, had a
higher level of mathematical understanding, were more flexible in their choice of solution
strategies, and had increased fluency in reporting their mathematical thinking (Fennema,
46

et al., 1992). When CGI students were compared with those in traditional mathematics
classes, CGI students could solve a larger variety of problems and were able to recall
number facts at a higher rate (Fennema, et al., 1992).
Two major studies, one published in 1988 and one in 1996, were conducted by the
developers of CGI. The first major study was conducted in 1986 – 1987 in Madison,
Wisconsin and four smaller communities in the surrounding area (Carpenter, Fennema,
Peterson, & Carey, 1988). Forty first-grade teachers from 27 schools volunteered to
participate in a month-long in-service workshop in mathematics during the summer of
1986 (Carpenter, et al., 1988). For the entire group of teachers, the mean number of years
teaching was 10.9 years, the mean number of years teaching first grade was 5.62, and two
of the teachers were in their first year of teaching (Carpenter, et al., 1988). Three
published articles used this group of teachers as the subjects of their study and were
considered to be baseline studies (Fennema, Franke, Carpenter, & Carey, 1993). These
baseline studies focused on such aspects as teachers’ content knowledge of mathematics
(Carpenter, et al., 1988), teachers’ content beliefs in mathematics (Peterson, Fennema, et
al., 1989), and teachers’ knowledge of student mathematical knowledge (Peterson,
Carpenter, & Fennema, 1989). The culminating study investigated how teachers use their
knowledge of students’ knowledge and the impact on student achievement (Carpenter, et
al., 1989). Additionally, a case study was published that followed one of the teachers
from the CGI treatment group over a four year period (Fennema, Franke, et al., 1993).
For a more detailed description of each of these studies, please see Appendix A.
47

The culminating study by Carpenter, et al. (1989) was conducted to investigate
whether teachers’ instruction and their students’ achievement would be influenced by
providing teachers with research-based knowledge on children’s thinking in a content
domain, specifically addition and subtraction concepts. The authors hypothesized that
―knowledge about differences among problems, children’s strategies for solving different
problems, and how children’s knowledge and skills evolve would affect directly how and
what teachers did in classrooms‖ (Carpenter, et al., 1989, p. 500). The altered actions of
teachers in their classrooms included teachers’ ability to assess students and match
instruction to the needs of their students to therefore improve the meaningful learning and
problem solving abilities of their students.
The teacher participants of the study were randomly assigned to the control or
treatment group by school (Carpenter, et al., 1989). The treatment group attended a CGI
workshop during the first four weeks of summer vacation that was led by experts in the
CGI philosophy. The workshop focused on helping teachers understand how children
develop concepts of addition and subtraction and allowed them to explore how they
would use this knowledge in their classroom. Teachers learned how to ―classify
problems, to identify the processes that children use to solve different problems,‖
(Carpenter, et al., 1989, pp. 504-505), to relate those processes to the different types of
problems, to ask appropriate questions and listen to children’s responses as an effective
way to analyze children’s thinking, and to familiarize teachers with available curricular
resources. After the workshop, the researchers met with the teachers in the fall to discuss
what they had done with CGI up to that point. Additionally, a staff member was available
48

to the teachers throughout the year to respond to any CGI questions (Carpenter, et al.,
1989).
The control group teachers had two 2-hour workshops held in September and
February. These workshops focused on the importance of problem solving and using non-
routine problems to encourage students to participate in problem solving activities but did
not incorporate the principles of CGI in any way (Carpenter, et al., 1989).
Each teacher in the sample was observed for four separate weeks from November
through April by two trained observers at a time. Teachers were asked to complete three
knowledge measures based on their ability to predict their target students’ number fact
strategies, problem solving strategies, and problem solving abilities. Additionally,
teachers were given an assessment to determine their beliefs about the teaching and
learning of addition and subtraction problems (Carpenter, et al., 1989).
Students in the classrooms of the sample teachers were given a standardized
achievement pre-test at the beginning of the year while the post-test at the end of the year
included a standardized achievement test of computational and problem solving skills and
a researcher-developed problem solving test. Twelve students (six boys and six girls)
were randomly chosen from each class to serve as a target sample. The target students
were individually interviewed to determine the strategies they used to solve specific
problems, their recall of number facts, their confidence in solving addition and
subtraction problems, their beliefs about learning and teaching mathematics, and their
attention and understanding during math classes (Carpenter, et al., 1989).
49

Teachers who participated in the CGI group were found to believe more
consistently that problem solving should be the focus of math instruction and they spent
more class time on problem solving and less time teaching number facts. While student
performance in both the control and treatment classes did not differ significantly on the
computation test, CGI students had a higher level of number fact recall. When looking at
classes that scored, on the whole, at the lower end of a pre-test that assessed simple
addition and subtraction problem solving abilities, classes that used CGI scored higher on
the post-test than did the control classes; lower-level CGI classes showed a greater
improvement over time than lower-level control classes (Carpenter, et al., 1989).
This study found that CGI classrooms produced a number of benefits for
individual students. Students in CGI classes, on average, scored higher on the complex
problem solving post-test, used correct problem solving strategies significantly more in
the problem solving interview than did control students, were more confident in their
ability to solve math problems, were more cognitively guided in their thinking, reported
greater understanding of mathematics, and performed better on both problem solving and
number fact assessments (Carpenter, et al., 1989).
Villasenor and Kepner (1993) conducted a quasi-experimental study to build upon
the previous work by Carpenter, et al. (1989). The authors posed that previous CGI
studies did not include classrooms in large, urban districts with significant minority
populations. Villasenor and Kepner (1993) used the same research question and similar
study design as Carpenter, et al. (1989) but focused their study on disadvantaged minority
students from urban areas. The design of the study by Villasenor and Kepner used a
50

smaller sample size (only 24 teachers were part of the study; 12 teachers were in the
treatment group and 12 were in the control group), the teachers who participated were
from urban schools with a minimum of 50% of students who were classified as African-
American, Hispanic, or Native American, and the CGI workshop only lasted one week as
opposed to four weeks in the previous study. The results of this study support the results
of the study by Carpenter, et al. (1989) in that findings included altered instructional
practices of CGI teachers who spent more time focused on problem solving and word
problems that students could relate to and altered their teaching practices to include
activities that were more conceptually based. Students in CGI classrooms improved their
abilities in problem solving situations, number fact recall, and use of advanced strategies
when problem solving (Villasenor & Kepner, 1993).
The second major CGI study was conducted by Fennema, et al. (1996) and
investigated teachers’ instruction and beliefs and their impact on student achievement as
teachers participated in CGI professional development that focused how to use childrens’
mathematical thinking to make instructional decisions. This four-year, longitudinal study
began in 1990 and included 21 teachers of first, second, and third grade and their
students.
Baseline data were collected in the spring of 1989 that included teachers’ beliefs
about teaching and learning mathematics and student standardized mathematics
achievement scores. For the next three years, teachers participated in a professional
development program focused on understanding childrens’ mathematical thinking,
teachers were supported in their classrooms as they applied what they learned in the
51

professional development, and the teachers were observed, interviewed, and given a CGI
belief questionnaire so that changes in their beliefs and instruction could be monitored
throughout the study. This study found that when teachers develop an understanding of
children’s mathematical thinking, their instruction changes and this is directly correlated
with changes in their own students’ achievement. The authors concluded that ―starting
with an explicit, robust, research-based model of children’s thinking, as we did, enabled
almost all teachers to gain knowledge, change their beliefs about teaching and learning,
and improve their mathematics teaching and their students’ mathematics learning‖
(Fennema, et al., 1996, p. 433). A more detailed description of the implementation of this
study can be found in the next section, Cognitively Guided Instruction Professional
Development.
Carpenter, Franke, Jacobs, Fennema, and Empson (1998) utilized the same
sample of teachers and students as Fennema, et al., 1996, but instead of focusing on the
actions of the teachers, it concentrated on the development of children’s understanding of
the mathematics being taught and compared the understandings of students in CGI
classrooms with those in the control group. The study found that children who used
invented strategies to solve addition and subtraction problems before they were taught
standard algorithms (the CGI treatment group) had better conceptual understanding of the
base-ten number system and were more successful in applying that knowledge to new
situations than were students who learned standard algorithms initially (the control
group). A study by Fennema, Carpenter, Jacobs, Franke, & Levi (1998) again used the
same students from the classes of the teachers in the Fennema, et al. (1996) study, but
52

this time the focus was on gender differences of students when examining problem
solving and computational strategies. This study found that there were ―no gender
differences when solving number fact, addition/subtraction, or non-routine problems,‖
but, there were ―strong and consistent gender differences in the strategies used to solve
problems, with girls tending to use more concrete strategies like modeling and counting
and boys tending to use more abstract strategies that reflected conceptual understanding‖
(Fennema, et al., 1998, p. 11).
More recent studies by this same group of authors have focused on the transition
from elementary mathematics to algebra (Carpenter & Levi, 2000), how CGI can be used
to facilitate this transition (Carpenter, Franke, & Levi, 2003), and how the specific
questioning practices of CGI teachers can be developed to support the mathematical
understanding of students through classroom dialogue (Franke, Webb, Chan, Ing, Freund,
& Battey, 2009).
A 2008 study by Slavin and Lake examined research on ―all types of math
programs that are available to elementary educators today‖ with the intent to ―place all
types of programs on a common scale‖ (Slavin & Lake, 2008, p. 429). The authors
separated the programs into three major categories: mathematics curricula, computer-
assisted instruction, and instructional process programs. This last category focuses on
―changing what teachers do with the curriculum they have, not changing the curriculum‖
(Slavin & Lake, 2008, p. 459); CGI falls within this category. The authors reviewed 87
articles that met their criterion for inclusion in the study, of those, 36 evaluated
instructional process programs. When compared to research on other elementary
53

mathematics programs, instructional process programs were found to have impressive
effects (four of the top five most effective programs fell into this category) and were
particularly well supported by high-quality research. CGI, specifically, was found to have
moderate evidence of effectiveness, and based on the study by Carpenter, et al. (1989),
had an overall effect size of +0.24, placing CGI in the top 10 of the programs that had the
strongest evidence for effectiveness of all of the mathematics programs used in this study
(Slavin & Lake, 2008).

Cognitively Guided Instruction professional development.

The CGI professional development program was designed to address previous
findings that while teachers often have intuitive knowledge of childrens’ mathematical
thinking, this knowledge is often fragmented, disorganized, and does not guide the
planning of instruction (Carpenter, et al., 1988). A number of CGI studies have found a
positive relationship among teachers’ beliefs and knowledge of their students’
mathematical understandings and student problem solving achievement (Peterson,
Fennema, et al., 1989; Carpenter, et al., 1989; Peterson, Carpenter, & Fennema, 1989).
By providing teachers with research on students’ mathematical thinking and opportunities
to examine and reflect on their practice, student achievement can be improved. This
research base led the developers of CGI to design a professional development program
that would enable teachers to develop this critical understanding of their students’
thinking (Fennema & Carpenter, 1989).
54

The implementation of CGI professional development is based on the premise
that ―the teaching-learning process is too complex to specify in advance, and as a
consequence teaching essentially is problem solving‖ (Fennema & Carpenter, 1989, p. 1).
Thus, the professional development program aims to help teachers make informed
decisions rather than prescribe a particular way of teaching (Fennema & Carpenter,
1989). Teacher participants are guided to think about the knowledge of their students and
are given evidence that students benefit from active involvement in the construction of
their mathematical knowledge. This constructivist idea is extended to the teachers who
participate in the CGI program as learners who construct meaning from their own
knowledge and experience (Fennema & Carpenter, 1989). Effectively integrating CGI
principles into classroom practice ―takes time, support, and thoughtful discussions among
the participants before it is understood and adopted as an approach to mathematics
instruction‖ (Fennema & Carpenter, 1989, p. 2). It is imperative that teachers and
administrators understand the extent of the commitment necessary for the successful
adoption of CGI; it is a multi-year process that necessitates a serious commitment of time
and resources.
Just as there is no one way to enact CGI in the classroom, there is no one way to
implement the CGI professional development program; the implementation will depend
on the specificities of the school environment (Carpenter, Fennema, Franke, Levi, &
Empson, 2000; Fennema, et al., 1999; Fennema & Carpenter, 1989). While there is no
prescription for ―correct‖ implementation, experimental studies of CGI have all adopted a
similar structure for how teachers are exposed to and learn the principles on which CGI is
55

based (Carpenter, et al., 1989; Fennema, et al., 1996; Villasenor & Kepner, 1993).
Additionally, CGI implementation guides have been created to describe, not prescribe,
CGI professional development (Fennema & Carpenter, 1989; Fennema, et al., 1999). A
commonality between all of the implementation descriptions is that they use intensive
multi-day (and/or multi-week) summer workshops with ongoing in- and out-of-the-
classroom support throughout the academic year to focus on the development of teachers’
understanding of their students’ thinking and how to use that understanding to guide
classroom instruction. The professional development structure of the Fennema, et al.
(1996) study, specifically, will be described and compared with current research on
effective professional development.
In the 1996 study conducted by Fennema, et al., the CGI professional
development program was implemented and studied over the course of three years. In
each year of the study, teachers were supported to varying degrees with the explicit goal
of helping teachers ―develop an understanding of their own students’ mathematical
thinking and its development and how their students’ thinking could form the basis for
the development of more advanced mathematical ideas‖ (Fennema, et al., 1996, p. 406).
The CGI professional development program consisted of two main features: workshops
and in-classroom support.
As this study (Fennema, et al., 1996) was situated in previous CGI research, the
professional development program implemented in this particular situation was
developed based on five themes (Fennema, et al., 1996, p. 407): (1) Children can learn
important mathematical ideas when they have opportunities to engage in solving a variety
56

of problems, (2) Individuals and groups of children will solve problems in a variety of
ways, (3) Children should have many opportunities to talk or write about how they solve
problems, (4) Teachers should elicit children’s thinking, and (5) Teachers should
consider what children know and understand when they make decisions about
mathematics instruction. These themes were integrated throughout the professional
development program.
Each year of the CGI program consisted of specific workshops to support the CGI
teachers (see Table 2.1).

Table 2.1
Cognitively Guided Instruction Workshop Description by Year
Note. Description of the workshop participation provided for teachers in each year of the CGI
professional development program. Adapted from ―A Longitudinal Study of Learning to use
Children’s Thinking in Mathematics Instruction,‖ by E. Fennema, T.P. Carpenter, M.L. Franke, L.
Levi, V.R. Jacobs, & S.B. Empson, 1996, Journal for Research in Mathematics Education, 27(4),
p. 109.

All content was addressed in the first year of the program where teachers were exposed to
the principles and research based model of CGI. The first aspect of this model provided
Year Workshop Participation
Year 0 2.5-day workshop in late spring

Year 1 2-day workshop prior to the beginning of the school year, fourteen
3-hour workshops during the academic year

Year 2 Four 2.5-hour workshops and one 2-day reflection workshop

Year 3 One 3-hour reflection workshop and two 2-hour review workshops

57

teachers with an ―integrated perspective of basic number concepts and operations and
how children usually think about them‖ (Fennema, et al., 1996, p. 406) which was based
on previous research by Carpenter (1985). The second aspect described the classes of
problems organized by the type of action or relationship ―so that knowledge of a few
general rules is sufficient to generate the complete range of problems‖ (Fennema, et al.,
1996, p. 406). The third aspect prepared teachers to gauge the difficultly of specific
problems based on childrens’ thinking about the problems. This last aspect of the model
relied on videotapes of children solving word problems. These videotapes provided a
basis for discussion of childrens’ thinking as they solved problems. The model just
described was not necessarily explicitly provided to the Year 1 teachers; often it was
derived together after watching and analyzing the videotapes of students solving
problems.
Workshops in the following years primarily served to help ―teachers review and
reflect on the content and their students’ thinking, and on encouraging the teachers to
reflect on using knowledge of their students’ thinking in making instructional decisions‖
(Fennema, et al., 1996, p. 409). All workshops were held after school or during release
time. Teachers were also encouraged to share ideas for communicating with parents
about CGI.
This study (Fennema, et al., 1996) focused on 21 teachers of first, second, and
third grade from three different elementary schools. Each school was provided with one
CGI staff member and one mentor teacher. The staff member and mentor teachers were
trained to focus their interactions with teachers directly on children’s thinking and its use
58

in the classroom. Their responsibilities included ―participating in the workshops, visiting
the classrooms, engaging the teachers in discussions, and generally providing support as
the teachers learned to base instruction on their students’ thinking‖ (Fennema, et al.,
1996, p. 409).
In the first year of program implementation, staff members visited each teacher’s
classroom about once per week. In the second year it decreased to about once every two
weeks and in the third year staff members visited classrooms only occasionally. Mentor
teachers visited classrooms more regularly throughout the three years. The mentor
teachers were chosen because of their use of children’s thinking in their classrooms and
because of their participation in previous CGI studies. Mentor teachers were given a
stipend and one released day per week for mentoring. School principals were encouraged
to visit CGI classrooms and ask students questions about solving problems. Additionally,
they were provided with informational meetings and half-day workshops for the first two
years of program implementation (Fennema, et al., 1996).
While not explicitly described in their studies, the structure of the CGI
professional development program is supported by research on aspects of effective
professional development. Guskey (2000) explains that effective professional
development must be intentional, ongoing, and systematic. The model used in the 1996
study by Fennema, et al., described above, is situated in these three themes because it has
a clear focus on student thinking and changing the beliefs of teachers, provides an
extensive introduction workshop (four weeks), includes in- and out-of-classroom support
over time (usually at least three years), focuses on teachers and students in order to
59

change the system as a whole, and is implemented with the intent of reaching a long-term
goal.
A case study by Franke, et al. (1998) focused on three teachers from the original
1996 study to investigate the possible self-sustaining, generative change that resulted
from participating in the CGI treatment group. The teacher participants were interviewed
and observed, both formally and informally, over the four years that the study took place.
This study found that when teachers participate in professional development that focuses
on student thinking (CGI), this provides a basis for ―teachers to engage in ongoing
practical inquiry directed at understanding their own students’ thinking and thus, provides
a basis for teachers to engage in self-sustaining, generative growth‖ (Franke, et al., 1998,
p. 79). Finally, a study by Franke, et al. (2001) followed-up with the teachers from the
original 1996 study four years after its completion to determine if teachers’ beliefs and
instructional practices remained consistent with those of CGI. This study found that, four
years after the conclusion of the initial study, teachers continued to implement principles
of CGI by creating teacher learning communities that focused on children’s mathematical
thinking (Franke, et al., 2001).

Conclusion

Research has demonstrated that there are effective teaching methods that address the
ways in which students learn mathematics (Fuson, et al., 2005; Loucks-Horsley, et al.,
2010). Through professional development that is intentional, ongoing, and systematic,
60

teachers can adopt instructional strategies that improve student achievement, thus
changing teachers’ beliefs about learning and teaching (Guskey, 1986). The CGI
professional development program uses these three features of effective professional
development to facilitate teachers’ understanding and use of childrens’ thinking in their
mathematics instruction.
The following chapter will describe the methodology used to address the research
questions. Principal interviews and teacher observations, interviews, and questionnaires
were used to describe the critical features of CGI evident in classroom practice and the
implementation of CGI professional development in two elementary schools within the
same school district.
61

Chapter 3
Research Methodology

It is difficult to change the beliefs and actions of teachers (Guskey, 1986, 2000;
Loughran, 2006; Toll, et al., 2004; Loucks-Horsley, et al., 2010) to alter classroom
instruction and thus improve student achievement. Green Valley Unified School District
(GVUSD) adopted the Cognitively Guided Instruction (CGI) elementary mathematics
philosophy in an effort to improve student achievement through improved classroom
instruction. The enactment of this philosophy was analyzed to investigate whether the
principles of CGI were evident in current classroom practice and what aspects of CGI
professional development the teachers found to be the most effective in supporting their
CGI practice. This chapter includes the research questions, research design, population
and sample, instrumentation, and procedures for data collection and analysis.

Research Questions

The research questions for this study are:
1. To what extent do teachers’ instructional practices reflect the critical
features of CGI?
2. What aspects of CGI professional development did the teachers perceive
to be the most effective in supporting the integration of CGI philosophy
into their practice?
62

Research Design

The data collection for this study combined methods used primarily in two
previous Cognitively Guided Instruction (CGI) studies. The methods used in these
previous studies were relevant because they have been shown to be valid, reliable, and
have been used in a number of additional studies by various researchers (Peterson,
Fennema, et al., 1989). The two studies used are the 1996 study by Fennema, et al. and
the 1989 study by Peterson, Fennema, et al. The current study was formative in nature as
the goal was to determine the degree to which the principles of CGI were evident in
classrooms in the two elementary schools; this study did not determine program
effectiveness (Patton, 2002). Data limitations made a causal relationship difficult or
impossible to determine so program effectiveness was not a concern of this particular
study, although it could be a topic of future research. This was a qualitative research
study (Creswell, 2009) as it was a non-experimental study that consisted of interviews of
school principals and teacher observations, interviews, and questionnaires. The two
researchers conducting this study collaborated on the instrumentation and data collection;
separate analyses were completed by each researcher.
The first research question was addressed through classroom observations of five
second grade teachers from two elementary schools in the district. Each teacher
participant was observed twice teaching a CGI mathematics lesson. During the classroom
observations, the researchers took on the role of complete observer and descriptive field
notes were taken to document the actions of each teacher (Creswell, 2009). Additionally,
63

an observation rubric from a previous CGI study (Fennema, et al., 1996) was adapted and
used to rate the extent to which the critical features of CGI were present in each teacher’s
actions during the observed lessons. After each lesson, the researchers rated the teacher
using the rubric and came to an agreement as to the degree to which each critical feature
was observed in that lesson. These rubric ratings were then further evaluated and revised
after all data had been collected and analyzed.
The teacher interviews served two purposes. First, the interviews worked in
conjunction with the observations. Because the researchers could not spend a great deal
of time observing each teacher, it would have been impossible to gain a thorough
understanding of the teacher and her practice without an interview. Interviews are a
―necessary tool of the part-time observer‖ (Patton, 2002, p. 317). One goal of the teacher
interviews was to understand the thinking behind the instructional activities that were
seen during the observation. The teacher was asked specific questions about what had
been seen in the observation, what her goals of the lesson were, why she had made the
instructional decisions that she had, and how she used students’ thinking to influence her
instruction (Fennema, et al., 1996). These questions were purposefully open-ended as
they would vary depending on what was observed in the classroom. An interview guide
was developed to ensure that teachers were asked similar questions about their practice
but that there was also flexibility to vary questions according to what was observed in the
classroom (Patton, 2002). The second purpose of the interview was to provide
information about the past and current professional development and classroom support
64

opportunities that were available to each teacher to address the second research question;
this will be discussed further in the following paragraph.
The second research question was addressed through a questionnaire and
interviews with each observed teacher. The questionnaire served to provide background
information from each teacher such as the number of years since they had been
introduced to CGI, the professional development they have had and currently experience,
and which activities they found the most effective in supporting their development and
use of the CGI critical features. The researchers used the interview to ask follow-up
questions about their professional development experiences after the observation had
taken place and the questionnaires had been collected and examined.
The district Assistant Superintendent of Curriculum & Instruction provided an
overview of CGI at the district level. Information gathered from the district Assistant
Superintendent of Curriculum & Instruction was collected through informal
conversations and email. Additionally, interviews were conducted with school principals
to collect general information about the history of CGI in each school. The interviews
with the principals consisted of experience and background questions (Patton, 2002) with
the primary purpose being to determine the past and present support of CGI at the school
site. A standardized open-ended interview protocol (Patton, 2002) was developed to
ensure that both principals were asked the same questions and the same information
would be gathered from each school.

65

Population and sample.

Green Valley Unified School District requested an investigation of the CGI math
program that was used in its elementary schools. The district was interested in gathering
information about how teachers were supported in their use of CGI and if teachers were
actually using CGI principles in their classroom instruction (Assistant Superintendent of
Curriculum & Instruction, personal communication, April 15, 2010). There was a point in
time when only one of the six elementary schools had implemented CGI; currently, all
elementary schools have adopted CGI. Because of time and resource constraints, the
researchers could not investigate all of the six elementary schools in the district. Blue
River Elementary School (BRES) was chosen for this study because it had adopted CGI
in a way that was the most consistent with CGI research and the teachers’ math lessons
were thought to be the most aligned with the philosophy of CGI (Assistant
Superintendent of Curriculum & Instruction, personal communication, April 15, 2010).
Green Valley Elementary School (GVES), on the other hand, had experienced more of a
piece-meal adoption of CGI which was more typical of other elementary schools in the
district (Assistant Superintendent of Curriculum & Instruction, Personal Communication,
April 15, 2010). Table 3.1 provides a comparison of student demographics for the state,
county, district, and each elementary school in this study.

66

Table 3.1
GVUSD Student Demographics

State County
School
District GVES BRES
Hispanic 49.0% 44.7% 13.0% 22.7% 15.3%
White 27.9% 32.8% 61.3% 46.1% 54.4%
Asian 8.4% 14.0% 10.4% 14.4% 18.6%
African
American
7.3% 1.7% 3.2% 4.2% 2.9%
Filipino 2.7% 1.8% 2.4% 5.6% 3.7%
American
Indian
0.7% 0.5% 0.3% 0.8% 0.0%
Pacific Islander 0.6% 0.6% 0.6% 0.0% 0.0%
Multiple/No
Response
3.4% 4.0% 8.6% 6.2% 5.2%
English
Learners
24.2% 27.9% 2.7% 12.9% 2.9%
Free/Reduced
Price Meals
52.6% 42.0% 8.3% 19.8% 5.0%
Note. Displays the student population demographics, including the percentage
of students who are English Learners and who qualify for Free/Reduced Price
Meals, for the state, county, school district, Green Valley Elementary School,
and Blue River Elementary School for the 2008-2009 school year. Adapted
from Ed-Data: Education Data Partnership, 2010.

Green Valley Unified School District was located in an area considered to be an
―urban fringe of a large city‖ (Ed-Data, 2010) in the southwestern United States. The
district was composed of six elementary schools, two middle schools, one high school,
and one continuation school for a total student enrollment of 9,475 in the 2008-2009
academic year (Ed-Data, 2010). The student population in the school district for the
2008-2009 academic year was approximately 61% White, 13% Hispanic, 10% Asian, 3%
African American, 2% Filipino, 1% Pacific Islander, <1% American Indian, and the non-
response rate was 9%. English Learners accounted for approximately 2.7% of the
population where over half of this population were native Spanish speakers.
67

Approximately 8.3% of students qualified for Free/Reduced Price Meals. The district
employed 414 teachers, of whom 99.3% were fully credentialed. The average class size
in the district was 27.3 students (Ed-Data, 2010).
Blue River Elementary School (BRES) was a year-round school with a total
student population of 596. The API of BRES was 980 and it was ranked in 10
th
place
across the state. Within the student population, 54% were White, 19% were Asian, 15%
were Hispanic, 4% were Filipino, 3% were African American, and 5% chose not to
respond. English Learners were considered 2.9% of the population. Approximately 5% of
students qualified for Free/Reduced Price Meals. The average class size in BRES was
23.1 students and all of the 28 teachers were fully credentialed (Ed-Data, 2010).
Blue River Elementary School has a very different history than the other
elementary schools in GVUSD and was the first school in the district to adopt CGI. In
1996, BRES first opened as a year-round school of choice. The principal, Principal B,
was new to the district, the entire teacher force was brand new, and 100% of the 115
students were from outside of the district (Principal B, personal communication, June 30,
2010). In the second year of BRES’s operation, a new teacher who had studied at the
University of Wisconsin with the developers of CGI was hired. The professional
development offered in the school at the time included teacher peer observations and
when other teachers saw her math lessons, they were intrigued by the high degree of
conceptual understanding that her students demonstrated. This sparked the beginning of
CGI interest at BRES and the teachers requested professional development in the
principles of CGI. In the first few years, CGI professional development was voluntary but
68

soon all of the teachers had adopted it and CGI training became mandatory for all newly
hired teachers (Principal B, personal communication, June 30, 2010).
The initial voluntary CGI training workshop was five days long and was led by
this new teacher, who also served as a CGI coach. As a CGI coach, she observed the
BRES teachers in their classrooms and helped them to develop their teaching practice to
be consistent with CGI. Additionally, she invited the teachers into her classroom to
observe and question her teaching. She continued to lead summer workshops for the
BRES teachers and served as a classroom CGI coach until other teachers felt they were
proficient in their use of CGI and could also serve as classroom coaches. The summer
workshops were developed into a series of three separate trainings that took place over
three years. Soon, all of the teachers at BRES had been through the three-year training
and were coached by BRES teachers (Principal B, personal communication, June 30,
2010).
In the third year of CGI professional development the math scores on the CAT 6
standardized exam were released and BRES outperformed every school in the district
(Principal B, personal interview, June 30, 2010). Over the years, math achievement
school-wide continued to increase and many people, including administrators, teachers,
and parents, held CGI responsible for the high math achievement scores (Assistant
Superintendent of Curriculum & Instruction, personal communication, April 15, 2010).
Soon, CGI spread to the County Board of Education and other schools first in the
county, then in the Southern California area (Principal B, personal communication, June
30, 2010). BRES became the center of CGI implementation in Southern California and at
69

the time of the study continued to host hundreds of visitors every year who wanted to see
CGI ―in action‖ (Principal B, personal communication, June 30, 2010). At the time of this
study, teachers at BRES were expected to use CGI at least two times per week. Principal
B explained that she would have liked to see CGI used three to five times per week
because ―the more you use it the better your students will do‖ (personal communication,
February 20, 2011), but that two times per week was a minimum. Teachers at BRES
additionally used Math Wall; the district-adopted math curriculum, Houghton Mifflin
California Math (2009); and the MIND Institute (Music, Intelligence, Neural
Development) computer software (Principal B, personal communication, June 30, 2010).
At the time of this study, the formal summer workshops and coaching had ended,
but BRES continued to provide in-house coaching to teachers when requested (Principal
B, personal communication, June 30, 2010). The BRES teachers were provided regular
collaboration time to discuss CGI and plan future lessons. They met yearly for long-range
planning, monthly to review the problem types that would be covered that month, and
weekly for more immediate planning. Each grade level team had scheduled one hour of
collaboration time per week to help them stay ―on the same page‖ about what they were
doing in CGI (Principal B, personal communication, June 30, 2010). They discussed
problem types and wrote problems together each week but the number choices were
developed by each individual teacher for her specific students.
Green Valley Elementary School (GVES) had an API of 865 and had one of the
most diverse student populations in the district, where students were approximately: 46%
White, 23% Hispanic, 14% Asian, 6% Filipino, 4% African American, 1% American
70

Indian, and 6% chose not to respond. Within the student population, 12.9% of students
were English Learners where the most common first languages were Spanish and Korean.
Almost 20% of the students qualified for Free/Reduced Price Meals. The average class
contained 22.3 students and all of the 31 teachers in the school were fully certified (Ed-
Data, 2010). The student population of GVES was somewhat more diverse both
racially/ethnically and socioeconomically than BRES. Including GVES in this research
study provided a broader view of CGI adoption across schools with varying student
populations.
The adoption of CGI in GVES began in 2006 (Assistant Superintendent of
Curriculum & Instruction, personal communication, February 25, 2011) and Principal G
was hired in the same year (Principal G, personal communication, June 14, 2010). The
impetus for the program adoption was a mandate from the district superintendent; in the
words of Ms. A, a teacher participant in this study, ―the district brought it to us‖ (personal
communication, December 16, 2010). The initial training involved two years of
classroom coaching and two one-week workshops over the summer (Principal G,
personal communication, June 14, 2010). In the first two years the teachers at GVES
were trained by teacher coaches from BRES but soon the coaches from BRES realized
that it was too difficult to balance teaching in their own classroom and training teachers
at a different school site (Principal B, personal communication, June 30, 2010). The
County Department of Education (CDE) took over the training and hired a former BRES
teacher to lead it (Principal B, personal communication, June 30, 2010).
71

In her second year as principal, Principal G felt that the degree to which her
teachers demonstrated use of the principles of CGI was inconsistent and she decided to
more fully commit her school to CGI by providing additional CGI professional
development. She arranged for all teachers in the school to complete the mandatory two
years of training and made the third year of training optional (Principal G, personal
communication, June 14, 2010).
Over the years, additional math programs had been adopted at GVES and
included Math Wall; the district-adopted math curriculum, Houghton Mifflin California
Math (2009); the MIND Institute computer program; and Fosnot units (Fosnot, 2007).
Teachers are currently expected to implement CGI at least two times per week in their
classrooms (Principal G, personal communication, June 14, 2010).
At the time of the study, CGI professional development at GVES included
mandatory coaching sessions three to four times per year by a CGI coach from the CDE.
Each grade level team met with the coach to create problems, discuss mathematics and
issues in enacting CGI, and watch the coach teach a lesson that they designed (Principal
G, personal communication, June 14, 2010). The teachers also met in grade level teams
once per month for CGI planning where they shared problems they had created and
discussed their current CGI experiences. Additionally, there were four teacher-leaders
who served as on-site CGI coaches and were available at a teacher’s request (Principal G,
personal communication, June 14, 2010).
This study utilized a sample of five second-grade teachers who were observed
teaching in their classrooms and then interviewed, three teachers were from BRES and
72

two teachers were from GVES. The sample of teachers observed was chosen because
they all taught second grade at one of the two elementary schools and agreed to
participate in this study. The researchers attempted to observe one grade level across both
schools so that their lessons could be more easily compared. The principal at BRES
recommended that second grade teachers be included in the study (Principal B, personal
communication, September 13, 2010). At BRES, there were four second grade
classrooms and three teachers agreed to participate in this study (the fourth second grade
teacher, who declined to participate in the study, was a long-term substitute). At GVES,
there were three second grade classrooms and two second grade combination classes, one
first/second grade combination and one second/third grade combination. The two
teachers who agreed to participate in this study taught each of the combination classes (it
is unclear as to why the other second grade teachers declined to participate in this study).
During the observations at GVES, both teachers were engaged in teaching mathematics
classes that included the second grade mathematics standards. The number of years since
each teacher was first introduced to CGI as well as the grade level taught at the time of
this study can be seen in Table 3.2.

73

Table 3.2
Teacher Participant Information
BRES GVES
Ms. P Ms. Q Ms. R Ms. A Ms. B

Grade level taught at
time of study

2

2

2

2/3

1/2
Number of years since
introduction to CGI
11 4.5 11 6 5
Note: Displays the grade level taught and years since introduction to CGI at the time of the
study for each teacher participant. This data was collected through the teacher
questionnaires.

Instrumentation.

The interview protocol used when interviewing the principals was developed by
the researchers (see Appendix B). The protocol was meant to elicit responses from the
principal that described the history of the adoption of CGI in the school and explained
how it had changed over the years. The principal interviews provided the researchers with
background knowledge of CGI in each school.
The researchers developed an observation rubric (see Appendix C) to aid in
assessing the degree to which the critical features of CGI were present in classroom
practice. The observation rubric was designed to provide the researchers with a list of
observable teacher actions that were either seen or not seen during the observations. The
observation rubric was based on the levels of CGI instruction developed by Fennema, et
al. (1996). The levels of CGI instruction can be seen in Table 3.3.

74

Table 3.3
Levels of Cognitively Guided Instruction
Level Description
1
Provides few, if any, opportunities for children to engage in
problem solving or to share their thinking.

2
Provides limited opportunities for children to engage in problem
solving or to share their thinking. Elicits or attends to children’s
thinking or uses what they share in a very limited way.

3
Provides opportunities for children to solve problems and share
their thinking. Beginning to elicit and attend to what children
share but doesn’t use what is shared to make instructional
decisions.

4-A
Provides opportunities for children to solve a variety of
problems, elicits their thinking, and provides time for sharing
their thinking. Instructional decisions are usually driven by
general knowledge about his or her students’ thinking, but not
by individual children’s thinking.

4-B
Provides opportunities for children to be involved in a variety of
problem solving activities. Elicits children’s thinking, attends to
children sharing their thinking, and adapts instruction according
to what is shared. Instruction is driven by teacher’s knowledge
about individual children in the classroom.
Note. This figure describes each of the levels (1 – 4-B) of teachers’ instructional
consistency with Cognitively Guided Instruction. The teacher observation and interview
will facilitate their placement on this scale. From ―A Longitudinal Study of Learning to
use Children’s Thinking in Mathematics Instruction,‖ by E. Fennema, T.P. Carpenter,
M.L. Franke, L. Levi, V.R. Jacobs, & S.B. Empson, 1996, Journal for Research in
Mathematics Education, 27(4), p. 412.

The descriptions of each CGI instructional level were separated into four CGI
critical features: opportunities for children to solve problems, children sharing their
thinking with peers and teacher, teachers’ elicitation and understanding of childrens’
thinking, and teachers’ use of childrens’ thinking as a basis for making instructional
decisions (Fennema, et al., 1996). Each of these critical features was separated into four
levels (1 – 4) that aligned with the levels of CGI instruction (Fennema, et al., 1996).
75

Because the chief distinction between level 4-A and 4-B was in whether the teacher
planned with groups of students or individual students in mind, only the last critical
feature, teachers’ use of childrens’ thinking as a basis for making instructional decisions,
distinguished between levels 4-A and 4-B (Fennema, et al., 1996).
The teacher interview served to clarify the teacher’s thinking behind her
instructional decisions and provided background information about her experiences with
CGI. Most of the questions for the interview were derived from previous CGI research
(Fennema, et al., 1996). The teachers were asked about specific incidents that were
observed during the lesson for which the researchers required clarification, what her
goals for the lesson were, why she had made certain instructional decisions, and how her
students’ thinking had influenced her instruction. The researchers developed the teacher
interview protocol to ensure consistency across the interviews; this interview protocol
can be viewed in Appendix D.
The teacher questionnaire was used to gather information about the past
professional development experiences of the teachers and what professional development
they found to be the most effective. Because time with each teacher was limited, a
questionnaire was developed that each teacher completed on her own and returned to the
researchers at her convenience; the researchers requested that it be returned before the
second interview so that follow-up questions could be asked, if necessary. When creating
the questionnaire, the researchers attempted to limit the questions to short-answer
responses so that the teachers did not feel burdened by the task. Questions that the
researchers thought might involve a long answer or would likely elicit a follow-up
76

question were reserved for the teacher interview protocol. The teacher questionnaire can
be found in Appendix E.
Peterson, Fennema, et al. (1989) developed the CGI Belief Scale survey to assess
the pedagogical content beliefs of teachers. It was planned that this survey would be
given to all teachers in both elementary schools to assess the degree to which teachers’
beliefs were consistent with those of CGI. When a CGI developer was contacted to obtain
a copy of the CGI Belief Scale, she provided the document but indicated that ―some of
the items are not sensitive and some do not make as much sense in the current
environment‖ (M. Franke, personal communication, September 2, 2010). The researchers
then decided not to use the CGI Belief Scale as part of this study. Without the CGI Belief
Scale, it was not possible to determine the teachers’ true beliefs about student learning.
While the teachers were asked questions about their beliefs in the interviews, the teachers
were aware that this study focused on CGI, which may have influenced their responses;
their responses to the questions may not have been representative of their true beliefs
(Patton, 2002). For this reason, the degree to which the beliefs of the teachers were
consistent with those of CGI was not a major part of this study.

Data collection.

Data were collected in a variety of ways through interviews, questionnaires, and
observations. The interviews with the principals were audio recorded and then
transcribed. During each interview, the principal was asked for permission to have the
77

session audio recorded; neither principal declined. The purpose of the interview was to
gain background information about the adoption of CGI in that particular school (Patton,
2002). The result of the interview was used to describe how the school adopted, and
continues to use, CGI.
Teacher observations were conducted by the two researchers of this study.
Teachers were observed two times in the fall of 2010 and both researchers were present
at all classroom observations. While in each classroom, the researchers took detailed field
notes as an observer (Creswell, 2009; Patton, 2002) to document what was seen. When
possible, verbatim notes were recorded as the teacher interacted with her students.
Following each observation, the teacher was given a CGI level using the observation
rubric that was based on the level of CGI instruction observed in her classroom.
An interview with each teacher was attempted as soon as possible after each
observation. For two of the teachers, an interview after the first observation was not
possible so the researchers designed a personalized ―interview‖ to be completed
electronically that included specific questions regarding that particular classroom
observation in addition to the general teacher interview questions. An in-person interview
was conducted for both of these teachers after the second observations.
Both researchers were present for all of the in-person interviews. Each interview
was audio recorded and written notes were taken; the audio recordings were later
transcribed. The interviews lasted from 15 to 40 minutes. All of the in-person interviews
took place on the same day as the classroom observation and were either conducted
immediately after the observation or after school that same day.
78

After the first observation, each teacher was provided with a copy of the
questionnaire that included a self-addressed, stamped envelope for return when
completed. The teachers were additionally offered to be emailed an electronic copy of the
questionnaire in the case that they preferred to complete it on their computer. It was
requested that the questionnaire be completed and returned before the second interview
so that researchers could ask follow-up questions during the second interview. Four of the
five teachers returned their questionnaire before the second interview. The one teacher
who returned her questionnaire after the second interview offered that she could be
contacted through email with any follow-up questions; the researcher determined this to
be unnecessary.

Analysis.

Information from both the classroom observations and teacher interviews served
the researchers as they placed each teacher in the CGI instructional level continuum
(Fennema, et al., 1996). At the conclusion of each observation and interview, the
researchers discussed what they had seen and agreed on the degree to which each critical
feature was evident in classroom practice according to the observation rubric. Using the
observation rubric ratings, an average score was determined and each teacher was then
assigned a CGI instructional level. The observation rubric allowed the researchers to use
a consistent scale on which to determine the CGI instructional levels of the observed
79

teachers. The CGI instruction level corresponds to the instructional levels determined by
Fennema, et al. (1996), in Table 3.3 on page 74.
Fennema, et al. (1996) described instruction by teachers in each of the levels in
their study. The instruction of teachers who were categorized at a Level 1 primarily
involved direct instruction where the teacher demonstrated a procedure and then allowed
children time to practice the procedure (Fennema, et al., 1996). The teacher saw
themselves as a ―demonstrator of procedures and organizer of instructional environments
who enabled children to practice what had been demonstrated‖ (Fennema, et al., 1996, p.
415). Level 1 teachers allowed the textbook to dictate the mathematical content as well as
its sequencing and emphasis (Fennema, et al., 1996).
Level 2 teachers engaged students in activities that were similar to those of Level
1 teachers. The primary difference was that Level 2 teachers included some components
of CGI although the components were often used in a superficial way. Level 2 teachers
sometimes (about once per week) asked children to solve word problems similar to those
used in CGI and allowed some sharing of solutions and thinking, but the teachers did not
question the children or explicitly listen to their responses. The word problems selected
by the teacher seemed inappropriate for the children in the class as they were often too
easy or too difficult (Fennema, et al., 1996).
Fennema, et al. (1996) described Level 3 teachers as providing classroom
experiences for their students in which students were ―engaged in rich problem solving
most of the time, they extensively reported on their thinking and engaged in discussions
about mathematics, and their thinking was valued by their teachers and peers‖ (Fennema,
80

et al., 1996, p. 418). Level 3 teachers spent more time listening to students and engaged
students in mathematics problem solving throughout the school day as opportunities
arose (Fennema, et al., 1996).
Teachers at Level 4 were found to vary to subtle degrees and were thus
categorized into two groups – Level 4-A and Level 4-B (Fennema, et al., 1996). All
Level 4 teachers asked students to solve a large variety of word problems, expected
students to solve problems using the method of their choice (and sometimes using more
than one method), required students to share their thinking with peers and/or the teacher,
and made continual connections between mathematics and other subjects (Fennema, et
al., 1996). Level 4-A teachers based instructional decisions on the needs of groups of
students while Level 4-B teachers based instructional decisions on more detailed
knowledge of individual children’s thinking. While students in the classrooms of Level 4-
B teachers were engaged in common activities, the teacher was able to plan and teach in
ways that addressed the thinking of each child (Fennema, et al., 1996).
The rubric ratings for each teacher were based on the classroom observations but
there were times when the interview data was necessary. For example, the impetus
behind the teachers’ instructional decisions was not always evident during the
observation so interview data was used. After each teacher was rated by the researchers
in each of the four critical features, an initial overall CGI instructional level was assigned
based on the average score for each critical feature.
After each day of observations and interviews, the field notes were typed and the
interviews were transcribed. The field notes were then coded by CGI critical feature so
81

that concrete support for each element could be documented. This concrete support was
placed into an observation rubric for each teacher so that the CGI instructional level
could more accurately be determined. A CGI instructional level based on this concrete
support was established for each teacher. Finally, using the three CGI instructional levels
that had been assigned (the first two immediately after the two observations and
interviews, the third after the data had been coded and analyzed), an ultimate CGI
instructional level for each teacher was determined based on concrete evidence of the
degree to which each critical feature was observed in each classroom. Upon determining
the CGI instructional levels, the observed teachers both within and across schools were
compared.
The teacher interview transcriptions were coded by CGI critical feature also;
those data were used to compare the observed actions of each teacher to that teacher’s
perceived actions. The interview data were also used to supplement the data collected
from the questionnaire because some of the teachers were asked about their professional
development experiences during the second interview as follow-up to their responses on
the questionnaire. The questionnaire data were organized by topic so that the professional
development histories and responses to what was perceived as the most effective
professional development could be compared.

82

Validity and reliability.

Validity in a qualitative study refers to relying on multiple strategies to check for
the accuracy of the findings (Creswell, 2009). The use of principal interviews, teacher
observations and interviews, and teacher questionnaires provided triangulation of the
study (Creswell, 2009; Patton, 2002). Triangulation was important because it checked for
consistency across the collected data (Creswell, 2009; Patton, 2002).
Reliability in a qualitative study indicates that the data collection approach was
consistent across researchers and study participants (Creswell, 2009). To address
reliability, the researchers were both present for each classroom observation and
debriefed afterwards to agree on the observation rubric ratings. The teacher interviews
used an interview protocol so that similar questions were asked of each teacher and some
of the teacher interview questions were taken from a previous study, Fennema, et al.
(1996). The principal interview was created by the researchers, tested, and revised before
use. Finally, the teacher questionnaire was standard for each teacher participant.

Conclusion

This study focused on the evidence of CGI principles in second grade classrooms
in two elementary schools in the GVUSD. Teacher observations were used to determine
the degree to which the critical features of CGI were evident in the actions of the
observed teachers. Each teacher was assigned a CGI instructional level based on the
83

concrete evidence of the degree to which the CGI critical features were present in
classroom practice. The professional development that the teachers perceived to be the
most effective in supporting them in developing their use of these CGI features was
additionally investigated through the teacher questionnaire and interviews. Principal
interviews provided background information on the adoption of and teacher support for
CGI in each school. Chapter 4 will provide the results from the data that were collected in
order to answer the research questions for this study.
84

Chapter 4
Results

This chapter will discuss the findings for each of the study research questions. In
order to contextualize the findings, first a description of the CGI philosophy that was
evident in the two schools will be given and discussed. Second, the findings in relation to
the first research question, to what extent do teachers’ instructional practices reflect the
critical features of CGI, will be offered. Third, the findings in relation to the second
research question, what aspects of CGI professional development the teachers perceived
to be the most effective in supporting the integration of the CGI philosophy into their
practice, will be presented.

CGI Philosophy

During the data collection for this study, all of the participants used similar
language when discussing their enactment of CGI which may point to an incomplete
understanding of the CGI philosophy. Cognitively Guided Instruction is not a program or
curriculum to be implemented; rather, it is a philosophy of mathematics teaching
(Carpenter, et al., 1999). The Oxford dictionary online (2010) defines philosophy as ―a
theory or attitude held by a person or organization that acts as a guiding principle for
behavior.‖ Cognitively Guided Instruction is a philosophy, once believed in, it should
guide the instructional decisions in the mathematics classroom. Even though some of the
85

teachers referred to CGI as a philosophy, it was treated in the two schools more like a
program to be implemented; it was used only on certain days of the week (during ―CGI
lessons‖) as a supplement to the Houghton Mifflin California Mathematics (HMCM)
curriculum. The teachers often referred to ―CGI lessons,‖ ―CGI implementation,‖ and the
―CGI program,‖ and differentiated between their ―CGI days‖ and ―non-CGI days.‖ The
language that the teachers used when talking about their CGI practice indicates that the
philosophy of CGI may have been fundamentally misunderstood.

Findings for the First Research Question

This section will describe the findings for the first research question, the extent to
which teachers’ instructional practices reflect the critical features of CGI. Each element
of the observation rubric will be described and related to the practice of the observed
teachers. In some cases, the classroom actions of the teachers were inconsistent with their
stated CGI beliefs during the interview; in those cases, the discrepancies will be
discussed.
The developers of CGI state that ―there is no optimal way to organize a CGI class.
Whatever organization enables a teacher to get the children to solve problems and to
listen to the students’ problem solving strategies is the optimal organization for that
teacher‖ (Carpenter, et al., 1999, p. 87). Each of the teachers in this study were aware that
this was the case and had arranged their CGI lessons in the ways that they had found to
work the best for them. Through the interviews, the teachers all remarked that they had
86

altered their use of CGI as they became more experienced and knowledgeable of the CGI
principles, student mathematical development, and their own individual students. This
study provides a snapshot of the degree to which the critical features of CGI were present
in each teacher’s classroom at the time of the observations.
Every teacher participant was assigned a score for each of the four critical
features based on the actions of that teacher (see the observation rubric, Appendix C).
The first three critical features were ranked on a scale from 1 – 4 and the fourth critical
feature was ranked on a scale of 1 – 4-B. The highest ranking for the fourth critical
feature, teachers’ use of children’s thinking as a basis for making instructional decisions,
was separated into two rankings (4-A and 4-B) because of subtle differences between
whether teachers based their instructional decisions on groups of or individual students.
Using the assigned scores, an overall level of CGI instruction was determined for each
teacher based on a scale of 1 – 4-B, where 4-B would indicate an exemplary use of the
CGI critical features. The following sections will provide a presentation of the findings
organized by critical feature.
Table 4.1 displays the ratings of each teacher for each critical feature and for their
overall level of CGI instruction.

87

Table 4.1
Teacher Ratings for each CGI Critical Feature and Overall Level of CGI
Instruction
BRES GVES
Ms. P Ms. Q Ms. R Ms. A Ms. B
Opportunities
for children to
solve
problems
2/3 3 2/3 2/3 2/3

Children sharing
their thinking
with peers and
teacher
3 2 2/3

3 2/3

Teachers’
elicitation and
understanding
of children’s
thinking
4 2 3

4 4

Teachers’ use of
children’s
thinking as a
basis for
making
instructional
decisions

3 2 2

3/4-A 3
Overall level of
CGI
instruction
3 2 2/3

3/4-A 3
Note. The number scores are on a scale from 1 – 4 for the first three critical features
and 1 – 4-B for the last critical feature. The overall CGI instructional level was on a
scale from 1 – 4-B based on the rankings for each of the critical features.

The overall CGI level of instruction ranged from 2 to 3/4-A across the two schools in this
study; BRES teachers ranked from 2 to 3 and GVES teachers ranked from 3 to 3/4-A.
While the enactment of CGI in BRES was considered to be the most consistent with the
CGI philosophy (Assistant Superintendent of Curriculum & Instruction, personal
communication, April 15, 2010), of the teachers who participated in this study, the
teachers at GVES ranked on the higher end of the spectrum compared to the teachers at
88

BRES. This suggests that the teachers at GVES were able to apply what they understood
about CGI at least as well as the teachers at BRES, who were considered to be in a more
supportive CGI environment (Assistant Superintendent of Curriculum & Instruction,
April 15, 2010). The overall CGI instructional level for each teacher was based on the
rankings in each of the four critical features. Each critical feature will now be explored
and the teachers’ rankings will be discussed.

Opportunities for children to solve problems.

In order for a teacher to achieve the highest rating in this category (4), she would
have to provide a variety of challenging problem solving activities so that students were
engaged in rich problem solving for the entire math class, implement a curriculum
comprised of problem solving, and continually emphasize the relationships between
mathematics and other subjects. Providing students with opportunities to solve a variety
of challenging problems is important because it helps them to develop mathematical
concepts, skills, and problem solving strategies in a realistic setting as opposed to
learning isolated bits of information (Carpenter, et al., 1999). Each of these elements will,
in turn, be compared to the findings of the study in the following paragraphs. Of the five
teachers observed, none of them ranked as a 4 in this category. One teacher, Ms. Q,
ranked as 3 and the other four teachers ranked as 2/3.
The first element to be examined is the level of challenge and the variety of
problems presented to the students to be solved. The variety of problems presented will
89

first be discussed followed by the level of challenge of the problems. The primary
difference between Ms. Q and the other teachers was the variety of problems that she
gave to her students. While the other teachers gave their students one main problem for
the day and some gave an additional warm-up or challenge problem (the main problems
of the day given by each teacher can be found in Appendix F), Ms. Q gave at least four
warm-up problems of various types and one main problem for the day. For example, the
problems that Ms. Q assigned to her students during the first observation can been seen in
Table 4.2.

90

Table 4.2
The Problems that Ms. Q Assigned to her Students during Observation #1
Problem
What students are asked
to do
Warm-up #1
19 + 13

(Join Result-Unknown)

Solve using three strategies: first
by decomposing, then by using
base 10, and finally by using an
algorithm.
Warm-up #2
Ms. Q ate 2 cookies at lunch. Then she
ate some more after dinner. She ate 6
cookies in all that day. How many
cookies did she eat after dinner?
(Join Change-Unknown)
Write an equation for this
problem.
Warm-up #3
Haley found 16 seashells and Ireland
found 14. How many did they find in all?
(Join Result-Unknown)
Write an equation for this
problem.
Warm-up #4
Caitlin bought 3 Hex Pets. Then her mom
gave her some more. Now Caitlin has 7
Hex Pets in all. How many Hex Pets did
Caitlin’s mom give her?
(Join Change-Unknown)
Determine which of the
previous two problems is the
most like this one.
Write an equation for this
problem.
Warm-up #5
(6, 28)
Mr. Danny picked up ____ pieces of trash
before recess. After recess, he picked up
some more. He picked up ____ pieces of
trash in all. How many pieces of trash did
Mr. Danny pick up after recess?
(Join Change-Unknown)
Solve the problem.
Main Problem
for the day
(13, 40) (138, 261)

((5x7)+(5x23), (4x75)+(9x3))

Mr. Danny picked up ____ pieces of trash
before recess. After recess, he picked up
some more. He picked up ____ pieces of
trash in all. How many pieces of trash did
Mr. Danny pick up after recess?
(Join Change-Unknown)
Solve the problem using
two strategies.

Ms. Q asked her students to solve both Join Change-Unknown and Join Result-Unknown
problems (see Figure 2.3 on page 39) in addition to asking them to compare problems to
91

each other and write equations to represent the problems. Although Ms. Q provided her
students with a greater variety of problems than the other teachers provided to their
students, she was not ranked a 4 because her students were not engaged in problem
solving for most of the class time and most of the problems would not be considered
challenging by CGI terms. Challenging CGI problems are meant to ensure that ―each
child is actively involved in deciding how best to resolve a mathematical situation. It is
when children decide upon a strategy to represent a mathematical situation and
implement that strategy that problem solving takes place‖ (Carpenter, et al., 1999, p. 97).
In the first four problems that Ms. Q assigned to her students, she prescribed the method
they should use for solving them. In the last warm-up and the main problem students
were able to use any strategy that they preferred. While all of the teachers allowed their
students to solve the main problem of the day using a strategy of their choice, none of the
teachers presented their students with a large variety of challenging problems that would
consider them to be ranked as a 4.
The next element considered is the degree to which students were engaged in
problem solving for the entire time provided. While all of the teachers provided time that
was dedicated to problem solving and all of the teachers asked their students to solve at
least one problem using two different strategies and record one, in some of the classes
students worked for the entire time on mathematics and continually challenged
themselves while in other classes the students became easily distracted and only
completed the minimal amount of work necessary for the assignment. For the most part,
students in Ms. P’s class and Ms. A’s class were able to persevere and worked for the
92

entire time on problem solving. Neither Ms. P nor Ms. A gave students options of
activities to engage in other than trying new strategies or new number choices for the
problem they were working on. Students in the other classes often recorded one strategy
and moved on to the next task, whether or not they had solved using two strategies. Ms.
Q, Ms. R, and Ms. B often had to redirect students to solve using two strategies as they
did not remain engaged in mathematics problem solving for the entire time. These three
teachers also provided additional activities that were not math-related to complete after
the problem was finished; these activities may have served as incentives to finishing the
problem rather than encouraging the students to spend their time focused on mathematics.
The opportunities for engagement in problem solving were present in all of the
classrooms but the ways that the teachers organized the lesson and their expectations for
student problem solving at times encouraged students to not engage in problem solving
for the entire time provided.
The conclusions for the third element of whether or not the teachers implemented
a curriculum comprised of problem solving remains questionable because the researchers
did not observe any non-CGI mathematics lessons. Both of the schools relied principally
on the district-adopted elementary mathematics curriculum, Houghton Mifflin California
Math (HMCM) and also used Math Wall, the MIND Institute’s ST Math + Music
program; teachers at GVES additionally used units of instruction from Contexts for
Learning Mathematics, Level 1 (Fosnot, 2007). All of the teachers stated that they used
the HMCM curriculum as a basis for their mathematics content and standards for each
lesson; the teachers planed ―CGI lessons‖ two days per week around the HMCM content.
93

Three of the teachers indicated in their interviews that ―non-CGI days‖ consisted
primarily of direct instruction by the teacher and student work from the HMCM
workbook. The HMCM curriculum and workbook were not analyzed for this study so it
was not possible to conclude that simply because students were completing workbook
pages they were not engaged in problem solving. Problem solving in CGI does not have
to only consist of story problems, the problems could be set in other formats like writing
number sentences, or simply solving an addition problem; the important thing is that
students have an opportunity to choose a solving strategy and implement that strategy
(Carpenter, et al., 1999). It was also impossible to know if teachers used the CGI
principles in their teaching even when they were not implementing ―CGI lessons.‖ For
example, Ms. A stated that she used direct instruction but later commented that:
I teach math very CGI-ish with just keeping open ended questions and allowing
kids to teach each other… why do you think that? So it’s a lot of just the way I
teach CGI in math is like that and I like it a lot… (Ms. A, personal
communication, December 16, 2010)
Ms. A indicated that she tried to incorporate what she considered to be CGI lesson
features, particularly student questioning and peer interaction, into her math lessons on
non-CGI days.
Because this study did not investigate lessons on non-CGI days, nor did it analyze
other math programs used, it was impossible to judge whether or not the teachers’
curriculum was based on problem solving. However, the teachers did refer to specific
lessons as ―CGI lessons‖ so it may be possible to conclude that students did not spend
94

time every day on problem solving as defined by CGI and were only given these
opportunities a few times per week, on ―CGI days.‖
The final element in this category is the degree to which the relationships between
mathematics and other subjects were continually emphasized. Two teachers, Ms. P and
Ms. R, related what they were doing in their math lesson to other subjects, but these
relationships were by no means emphasized continually. Both Ms. P and Ms. R wrote
problems involving themes they had covered in class that prompted short discussions
relating the math problem to what they were learning about those themes. For example,
the first observation of Ms. P took place in October; the students had carved pumpkins
the day before and had counted the number of seeds in each pumpkin. The problem that
Ms. P wrote was about counting pumpkin seeds and this prompted a conversation about
how large the pumpkins would be if they had various numbers of seeds. For each number
choice, Ms. P asked the students how big they thought a pumpkin would be that had that
many seeds. As another example, in the second observation Ms. R prompted her students
to briefly discuss capitalization rules when she wrote a problem about a trip to Big Bear.
T: Ms. R went to Big Bear during winter break. On Monday, she made ____
snowmen. On Tuesday, ____ snowmen melted. How many showmen were
left? Why did I capitalize Big Bear?
S1: It’s the name of a place.
T: What kind of noun would it make it?
S2: Proper noun.
95

This was the extent to which they discussed non-math related topics. Neither Ms. P nor
Ms. R spent much time connecting math to other topics the students were learning about,
but these brief examples are more than was evident in the observations of the other
classrooms. Since the researchers did not observe entire days of teaching, it was
impossible to know if the teachers made connections to math content throughout the
school day.
To earn the highest rating in this category would be very difficult at these two
schools because of the way that CGI has been integrated into teaching practice. The
mathematics curriculum was not obviously structured around problem solving and rich
problem solving was reported to be expected only a few days each week, during ―CGI
lessons.‖ Teacher participants in this study did incorporate various aspects of this critical
feature, opportunities for children to solve problems, into their teaching, it was not
enough, however, to earn the highest rating.

Children sharing their thinking with peers and teacher.

For a teacher to rank highly in this category (4), her students would have to
communicate their problem solving strategies and listen to other students’ sharing their
strategies, and her students would talk a great deal about math to each other and to the
teacher. It is important for children to share their thinking because in encourages
understanding; ―in order to be able to report, they have to understand what they have
done‖ (Carpenter, et al., 1999, p. 98). Additionally, having children report on their
96

strategies allows the teacher to assess their thinking while also providing students with
opportunities to hear a variety of solution strategies (Carpenter, et al., 1999). In this
category two teachers, Ms. P and Ms. A, were ranked a 3, two teachers, Ms. R and Ms. B
were ranked as 2/3, and one teacher, Ms. Q, was ranked as 2. The following section will
discuss each of the elements of this critical feature in turn.
The first element of this critical feature, expecting students to communicate their
strategies (either orally or in writing) and listen to other students sharing their strategies,
was found to some degree in each of the classrooms. All teachers provided time for
students to record their thinking on paper and collected student work at the end of the
lesson. Students were expected to communicate their strategy orally to the teacher when
asked, but a limited number of students (three to six students) were actually questioned
by the teacher in each lesson. During sharing time, all of the teachers, except Ms. Q,
chose two to four students to share their strategies with their peers. Students were chosen
to present so that all of the number choices were explained and different strategies were
demonstrated. While students reported their thinking, the rest of the class was expected to
listen to the students’ explanations. Students prepared to listen to their classmates by
turning their chairs to face the student presenter or moving to the rug.
During the transition to sharing time, only three of the teachers explicitly stated
that they wanted their students to listen to their peers and/or why students were sharing to
the class at all. Ms. R announced to her class in the second observation to ―sit up nice and
tall and show the presenters respect‖ (personal communication, November 29, 2010); she
was emphasizing that she wanted her students to at least appear as though they were
97

listening but did not encourage her students to further engage in sharing in any way. Ms.
A asked her students to think about if the student presenter was using a strategy that they
used, and if not, could they use the strategy the next time they were asked to solve a
similar problem. Her students were prompted to engage in the sharing activity by
comparing the shared strategy to their own and deciding whether the shared strategy
might work for them in the future (this example is further discussed below). When Ms. B
began sharing time during the second observation, she explained to her students that:
especially for those of you who had questions, this is where you get to learn from
your friends. During CGI you get to solve the way that makes sense to you…
right now is your chance to listen and learn from your classmates. (Ms. B,
personal communication, December 16, 2010)
Ms. B made it clear to her students that they were sharing so that they could learn from
each other; she was the only teacher who the researchers observed explicitly explaining
to her students what her reasoning was for sharing time. This did not mean that the
students of the other teachers were not clear as to why sharing time was important, it
simply was not evident during the observations for this study.
Ms. Q was not observed during sharing time because she used a two-day CGI
lesson where her students worked on the problem one day and shared their strategies the
following day. She described what her sharing time looked like and it sounded to be
consistent with what was observed for the other teachers; she tried to choose students to
share each of the number choices and use at least two different solving strategies. During
the observed lessons, student work was shared from the warm-up problems, however Ms.
98

Q explained the students’ strategies herself; the students did not share their own work to
the class. This is problematic for CGI enactment because Ms. Q was explaining how to
solve the problem rather than questioning or listening to students (Peterson, Carpenter, &
Fennema, 1989). Even though she was explaining strategies that students used, she was
assuming that she understood their thinking since she did not question them or elicit an
explanation.
The second element of this critical feature involved students talking a great deal
about math to each other and to the teacher. This element was what really differentiated
the teachers in this critical feature and excluded any of them from ranking as a 4. While
students worked on the problem for the day, the degree to which students engaged with
each other varied. The classroom with the least student interaction was Ms. R, who did
not want her students talking during this time. Ms. R’s students worked independently
and silently as they solved the problem; they were discouraged from interacting with each
other. She did, however, provide students with a few minutes to share their strategies
with their neighbors during the second observation. While students did talk quietly with
their neighbor during this sharing time and it appeared that they were discussing their
problem solving strategy, the quality of this sharing was unclear as it was difficult for the
researchers to monitor all of the student interactions. On the other end of the spectrum,
Ms. P encouraged her students to discuss the problem they were solving and ask their
peers for help when they had questions. For instance, in the second observation, Ms. P
noticed that one boy was struggling with the strategy that he chose and Ms. P suggested
that his neighbor, who was using a similar strategy, might be able to help him.
99

In all of the classes where sharing time was observed, the student presenter and
teacher mostly interacted with each other while the rest of the class was expected to
listen; there was very little, if any, participation from the other students that contributed
to a meaningful discussion of mathematics. There were two instances, one in Ms. P’s
class and one in Ms. A’s class, where a discussion of mathematics was prompted during a
student presentation. The discussion that took place in Ms. A’s class is described below.
In the following example from her first observation, Ms. A, who scored a 3 in this
category, had her students all sitting on the rug in front of an easel for sharing time and a
student, Sam, was going to explain how he solved the following problem:
Victoria has already read ______ pages in her AR book. How many more pages
does she need to read if the book has ______ pages?
(47, 85) (89, 140) (278, 323)
The students had already worked on this problem either individually or with their group
members. They were asked to solve this problem using two different strategies and to
record one strategy on their paper. Before convening on the rug, the students had a few
minutes to explain their problem solving strategies to their neighbor in a pair-share. Sam
was about to begin his explanation for the class.
T: Think about if Sam is using a strategy that you use and if not, think about
if you could use this strategy next time. Sam, what number choice did you
use?
Sam: I did 47 and 85.
100

T: [Reads the problem with 47 and 85 substituted in the blanks.] Sam, talk
out loud and I’ll record what you say on the easel. Sam told me that he did
the first number choices in his head and then I asked him to record his
thinking.
Sam: I started at 47 then I added 3 ones so I got 50.
T: Why do you think he added 3 ones to get to 50? [Addresses the class.]
Alex: Because it’s close to 50. [Ms. A writes 47 + 1 + 1 + 1 = 50 on the easel
and circles each 1. She reminds students that 50 is a friendly number
because it ends in a zero. She prompts Sam to continue.]
Sam: I added a 10 to get to 60 then another 10 to get to 70 then another 10 to get
to 80. [Ms. A records his thinking on the easel and circles each 10 that he
adds.] Then I add 5 ones to get to 85. [Ms. A begins to write 80 + 1 + 1…
and then asks him if she can just write 80 + 5.]
Sam: Yes. [Ms. A writes 80 + 5 = 85 and circles the 5. She now has this on the
easel:
47 + 1 + 1 + 1 = 50
50 + 10 = 60
60 + 10 = 70
70 + 10 = 80
80 + 5 = 85 ]
Sam: So far it’s 38.
T: Where did you get 38 from?
101

Sam: I added 5 + 3 and got 8. [Ms. A writes 5 + 3 = 8 in the lower left corner of
the easel so the students can see that it is separate from the previous
equations. She circles the 8.]
Sam: Then I added 10 plus 10 plus 10 equals 30. [Ms. A writes 10 + 10 + 10 =
30 under the previous equation and circles 30.] Then I added the 3 ones to
the whole thing.
T: So you took the 30 and added another 3 to it?
Sam: Yes. [Sam has mistakenly added the 3 an extra time.]
T: Do you see where you got the 38 from? [Many of the students sitting on
the carpet raise their hands to help. At this point the teacher walks Sam
through what she just said again, showing him where the 3 was already
added.] So, where do we have 38? [Sam sees his mistake and shows where
the 38 comes from correctly (from adding all of the circled numbers,
above). Ms. A now asks for comments or compliments from the students
for Sam.]
Nicole: I like how he added 10 each time and you can write it all together. [Ms. A
asks Nicole to explain more and from her explanation, Ms. A writes 50 +
10 = 60 + 10 = 70 + 10 = 80 across the top of the easel.]
T: Is this true? [There is a chorus of yeses and nos from the class] Is 50 + 10
equal to 60 + 10 equal to 70 + 10 equal to 80?
102

Chris: No. 50 + 10 is not equal to 60 + 10. [Ms. A says that that is right and they
will discuss number sentences further on another day but that it was a very
good and interesting problem.]
T: What strategy did Sam use?
Sarah: Adding.
T: Did he take the two numbers and add them together?
Mary: No, he took the 47…
T: He started with 47 then what did he do?
David: 47 plus 3 more and got 50 and added 10 more…
T: What number sentence would match his thinking? [Ms. A calls on another
student who directs her to write 47 + = 85 on the easel.]
David: Counting on!
T: Right! [The bell rings for recess] We’ll continue going over the other
students’ work after recess.
This example provides evidence of a student communicating his solution strategy
orally and a discussion about mathematics. As Sam shared his solution strategy, Ms. A
continually asked questions until his thinking was clear to her and the other students. She
sometimes asked him why he made certain problem solving choices and where his
numbers were coming from. At times Ms. A asked Sam to clarify his thinking while at
other times she asked other students to explain what they thought he was thinking. When
other students shared their thinking, like Nicole, Ms. A questioned them to so that their
thinking was also clear. Ms. A could have questioned each student to a higher degree in
103

order to understand their individual thinking more completely, however this was the best
example of students being questioned and interacting with each other during sharing
time.
While Sam was explaining his strategy, Ms. A expected the other students in the
class to be paying attention to his thinking. At the very beginning of the example, Ms. A
asked the students to ―think about if Sam is using a strategy that you use and if not, think
about if you could use this strategy next time.‖ Prompting students to compare Sam’s
strategy with their own may encourage students to pay closer attention to Sam’s
explanation, reflect on their own strategy, and help them to learn new problem solving
strategies from their peer (Carpenter, et al., 1999). However, Ms. A did not follow-up
with students to find out if any of them actually did use a strategy similar to Sam’s or if
they thought they might use that strategy in the future.
While this episode does not provide an example of deep discussion about
mathematics, it does offer the richest example of student interaction and potential
mathematics discussions that was seen in this study. During Sam’s explanation there
were multiple opportunities for discussion about mathematics although Ms. A did not
take full advantage of them during this lesson. For example, when Sam became confused
as to how he got 38, Ms. A explained his error to him when perhaps a richer discussion
may have taken place if another student had been called on for an explanation (many
student hands were up and they looked eager to provide their own explanations). Also,
Nicole began a discussion about whether 50 + 10 = 60 + 10 = 70 + 10 = 80 was a true
statement. When the students were asked what they thought, Ms. A was met by
104

disagreement among the students. While this discussion did not last long and only one
other student provided an explanation, Ms. A realized that this was a topic that she would
need to further address with the students at another time. It seemed as though the limited
amount of time available for students to share their thinking kept Ms. A from eliciting
more elaborate discussions from the students.
During the interviews, all of the teachers said that they saw importance in students
explaining their thinking to them so they understood their students’ thought processes.
The teachers expressed frustration that they could not interact with each student every
day, but most of them had a system set up in an attempt to ensure that they worked with
each student regularly. For example, Ms. P, Ms. R, and Ms. B all used clip boards to keep
notes as they interacted with students. These teachers recorded who they met with, the
strategy they were using, and if they were having any difficulties. Ms. R discussed a
binder that she updated every day with this information as well as the number choice
difficulty level that each student chose and which students presented during sharing time.
This way she could keep a record of which students she met with, their understanding,
and the degree to which they challenged themselves with the number choices they chose.
She was also able to ensure that she interacted with every student and that each student
had an opportunity to present his/her solution strategy over a period of time.
All of the teachers stated that they found value in students learning from each
other and explaining their work to their peers. However, the degree to which students
shared their thinking with other students was limited in most of the classrooms. Only two
to four students had an opportunity to share their work with their peers during sharing
105

time and the degree to which students were engaged in that sharing was questionable.
While students were sharing in front of the class, the interaction was primarily limited to
the teacher and the student presenter. It was rare that a question was asked to the class
that might engage them in a discussion about mathematics. While all of the teachers
stated in the interviews that they valued their students discussing mathematics and
learning from each other, the evidence that this was actually happening in their
classrooms was very limited.
In all of the observations there was evidence that students were expected to
communicate their thinking; all students were expected to write about their problem
solving strategy, however, only two to four students shared their strategy with the class.
During sharing time, it was clear that students were expected to listen to their peers’
explanations but the degree of student engagement varied and there was often little
participation from the class at this time. There was no evidence in any of the classes that
students talked a great deal about math. While there were a few isolated incidents of
mathematics discussion, students engaged very little in conversations about their
strategies or mathematics either with peers or the teacher.

Teachers’ elicitation and understanding of children’s thinking.

For a teacher to rank highly in this category (4), she would elicit and attend to
children’s thinking by questioning students until their thinking was clear and the teacher
could explain her students’ thinking. It is important for teachers to have an accurate
106

picture of their students’ thinking and mathematical development so that they can provide
an appropriate learning environment for their students by selecting problems that were
challenging and engaging (Carpenter, et al., 1999). Unlike the previous critical feature
which focused on children communicating their problem solving strategies, this critical
feature focuses on the degree to which the teacher understands the thinking of her
students. In this category, three teachers, Ms. P, Ms. A, and Ms. B ranked as 4, one
teacher, Ms. R ranked as 3, and one teacher, Ms. Q, ranked as 2.
This critical feature had the most variation between scores and it had the largest
number of highest scores. The teachers who scored 4s in this category spent time eliciting
student thinking by asking students probing questions until their thinking was clear.
Additionally, when students were stuck or displayed misunderstandings about the
mathematics, the teachers questioned them in such a way as to help them to clarify their
thinking as opposed to giving them the right answer. The questions that these teachers
asked often included: How did you figure that out? How do you know? Why did you do
this? Can you show me…? Why is this different? What does that mean? What if we…?
These teachers often questioned their students until they were sure they understood the
problem, making sure that they could explain their thinking from beginning to end. This
was different than the lower-scoring teachers, who often did not question their students
until their thinking was complete. For example, the following is a typical dialogue
between Ms. Q, who ranked as a 2, and her students. When Ms. Q approached, Steve had
the following equation written on his paper:
138 + 199 = 261.
107

T: Lets’ see your mathematical thinking today Steve. How can you prove this
is right?
Steve: I just know it’s right.
T: I saw you use different strategies in the same problem. First you added
something from here to here, then you did it here to here… prove your
answer is right by doing an algorithm. If you did it correctly it should
wind up 261, right? [Ms. Q writes the algorithm on his paper:
138
+ 199

Steve nods his head and begins working; Ms. Q moves to another
student.]
Ms. Q did not question Steve nor does she prompt him to figure out what to do next. It
remains unknown how Steve came to his answer, what misunderstandings he may have
had that led him to this answer, and if he really understood why he was being prompted
to use an algorithm to check his answer. In contrast, note the next example from Ms. B,
ranked as a 4.
In the following example, students in Ms. B’s class were working independently
on solving the main problem for the day:
Nicole had ___ silly bands that her mom bought her at the store. Madison had ___
silly bands that she got from trading with her friends. How many bands did Nicole
and Madison have together?
(54; 97) (175; 25) (653; 498) (2,346; 2,654)
108

Ms. B approached a student who appeared to be struggling. Andrew was working with a
hundreds board, had circled 54, and had stopped working. When Ms. B approached, he
said that he would now count 97.
T: Is 54 or 97 greater? Think back to counting on, do you start with the
greatest or least number? [Andrew realizes that 97 is greater so he should
start counting from there. He erases the circle he drew around 54 and
draws a circle around 97.]
T: Is it more efficient as a mathematician to count by ones or tens?
Andrew: Tens.
T: If you start at 97 and count 10 more, what will be the next number?
Andrew: The hundreds board only goes up to 100.
T: [The teacher puts a piece of paper below the hundreds board and extends
the grid onto the paper so he can count ten more. Andrew looks confused.]
If I start at 35 [points to 35 on a hundreds board similar to the one below]
and count 45, 55, 65, what am I counting by?

109

1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100

Andrew: Five. [Andrew did not understand that if you moved one row vertically
from 35 to 45 you were adding 10. He was using the last digit of the
number as the quantity that he was counting by.]
T: What if I count here [points to 4] 4, 14, 24, 34. What am I counting by?
Andrew: Fours.
T: What about 10, 20, 30, 40?
Andrew: Tens
T: What about 3, 13, 23, 33?
Andrew: Threes.
T: OK. Let’s count from three to 13 to see what we’re counting by. [She
points to 3 on the hundreds board and moves her finger to the next number
as she counts to 13.] 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
Andrew: 10!
110

T: How about from 13 to 23?
Andrew: [Uses his finger to point to each number as he counts.] 10!
T: OK. So let’s go to 4, 14, 24. What are we counting by?
Andrew: 10! [Andrew sees that he is counting by 10s now, not by 4s.]
T: OK. So if we start at 97 should we count by 1s or 10s? As a
mathematician, what would be more efficient?
Andrew: Tens.
T: If we start at 97 and add 10, where will we land?
Andrew: [Thinks for a few seconds.] 107!
[The teacher seemed satisfied with Andrew’s progress and moved to
another student. She checked back in with Andrew after a few minutes.]
In this interaction, Ms. B questioned Andrew until his thinking was clear to her
and she could identify where his understanding of counting by tens was breaking down.
Once she understood his misunderstanding, she was able to help him to correct his
thinking by continued questioning; she did not simply give him the answer, she was able
to elicit it from him and clarify his thinking. Andrew was able to complete the problem
on his own after this interaction with Ms. B. However, there were opportunities for Ms. B
to further elicit Andrew’s thinking which she did not take advantage of. For example, she
did not ask him why 97 was greater than 54 or why it was more efficient to count by tens.
These questions could have provided more detailed evidence of Andrew’s understanding
of the problem. It is not known whether Ms. B had previously assessed Andrew’s
111

understanding of these basic concepts which may explain why she did not ask further
probing questions. In her interview, Ms. B further discussed her interaction with Andrew:
I keep going to Andrew because it was such a perfect example, him thinking he
was counting by fours and then I went to sixes and then I went to sevens and then
I went to tens to show… and he still wasn’t getting it. It’s like, knowing ok, what
do I need to do right now to make him understand… and that’s something we do
on the math wall all the time. So it was real eye opening for me to see that
Andrew still didn’t understand that until right now… it’s like in Andrew’s mind
you’re only counting by tens when you start at 10… something that was so simple
for me that maybe I would get frustrated as a teacher before [CGI], like, ok, why
don’t you get this, we do this all the time, well, obviously he didn’t get it so
knowing like, to be patient and understand where they’re all at in their different
thinking and to kind of facilitate that and guide them to it. (Ms. B, personal
communication, November 5, 2010)
Ms. B has shown an example of a teacher who elicited her student’s thinking to find a
misconception, attended to his thinking through questioning so that he corrected his
misconception, and was able to later explain his thinking.
In the interviews, all of the teachers discussed specific students’ thinking in detail
and that they believed that it was important to understand their students’ thinking. The
three highest scoring teachers explained that CGI had really helped them to understand
more clearly what their students’ thought process was; Ms. P put it this way:
112

I feel like it’s almost given me, like, if you could open up a child’s head and look
inside their brains and see the wheels spinning it’s helped me to really understand
how they’re thinking, how they think incorrectly, and how to kind of navigate
them to make correct connections in math. (Ms. P, personal communication,
October 11, 2010)
The two lowest-scoring teachers did not discuss why or how student questioning was
helpful to them. For example, Ms. Q explained that if her students did not show their
work, she would question them the next day so that ―it was clear to her what their
counting strategy was‖ but she did not elaborate on how she used that information.
Additionally she commented that ―there are days that I find that I have to decide just to
stop by for a question to make sure they’re on task versus stop by to really understand
their thinking and have them explain‖ (Ms. Q, personal communication, November 15,
2010). This was evidence that Ms. Q does, at times, use student elicitation in a superficial
way. The findings from the interviews were fairly consistent with what was seen during
the observations. The high-scoring teachers talked in detail about their students’
understandings and their impetus for questioning students while the same was not true to
the same degree in the interviews with the lower-scoring teachers.
The degree to which teachers elicited and attended to childrens’ thinking varied
among the five teachers observed. The three highest scoring teachers, who all ranked as a
4, asked probing questions of their students that helped them to understand students’
thought processes and further their students’ thinking. During the interviews these
teachers provided detailed descriptions of students’ thinking as well as why they valued
113

understanding student thinking. This critical feature was evident in the ways that they
interacted with their students as well as the ways that they talked about their students’
mathematical understanding (Carpenter, et al., 1999).

Teachers’ use of children’s thinking as a basis for making instructional
decisions.

Teachers who scored highly in this category selected problems based on their
impact on students and anticipated students’ responses to the problems. Basing
mathematics instruction on children’s thinking was necessary in order to challenge
students to engage in problem solving that was appropriate for them at their
developmental level (Carpenter, et al., 1999). The highest score was separated into two
categories for this critical feature, 4-A and 4-B. Teachers who would be classified as a 4-
A would base instructional decisions on their knowledge of groups of students and where
those groups of students were struggling (Fennema, et al., 1996). Teachers who would be
classified as a 4-B would base instructional decisions on their knowledge of individual
students and considered the needs of individuals as they planned and taught their lessons
(Fennema, et al., 1996). Of the teachers in this study, Ms. A ranked as a 3/4-A, Ms. P and
Ms. B both ranked as 3, and Ms. Q and Ms. R both ranked as 2.
When ranking for this critical feature, it was necessary to analyze both the
observation and interview data. It was not always possible to determine, simply from the
observation, the impetus behind each teacher’s instructional decisions. All of the teachers
114

stated that the specific topic they were working on with their students was dictated by the
unit they were working on in a specific curriculum. While it is understood that many
schools follow a specific curriculum, CGI emphasizes that ―the important criteria for
making decisions about what and how to teach [should be] children’s understandings‖
(Fennema, et al., 1996, p. 408). In order for a teacher to be ranked as a 4-A or 4-B for this
critical feature, her instruction would have to be made up of problem solving and a major
influence on the topics and problems would be her children’s thinking. It was not clear
that any of the teachers based their instruction on their knowledge of their students’
thinking to this extent; none of the teachers in this study were ranked as a 4-A or 4-B for
this reason.
All of the teachers had a relatively similar focus on problem solving during their
lessons and their problems and number choices were mostly comparable (except for Ms.
Q). However, Ms. A ranked the highest for this critical feature because of the ways that
she talked about how she planned her lessons around student thinking. Most of the other
teachers did have a rationale for their problem and number choices, but Ms. A explained
her impetus for planning in much more detail and seemed to plan her lessons to a higher
degree around her students’ thinking. For example, when asked about how she goes about
planning classroom activities, she explained that
I have to sit and think about how a lot of kids are doing this counting right now,
they’re counting on using their fingers, I have to get them to record that and then
also push them on, how am I going to get them on the next level … I’d have to sit
and think about how to change the problem, if I need to change the problem to get
115

them up to flexible thinking. Definitely change the number choices, make them a
little more challenging … and a larger difference between the two number choices
… I’ve changed it [curriculum] a little bit, you know, according to what you’re
kids show you that they need … (Ms. A, personal communication, November 5,
2010, italics added for emphasis)
In this example, it was evident that when Ms. A planed her lessons, she tried to bridge the
gap between what her students knew and could do and what she wanted her students to
know and be able to do. She talked about planning her lessons according to what her
students showed her that they needed, which is the basis of CGI instruction. Ms. A also
spoke in detail during the interview about her rational for the number choices and the
strategies that she hoped to see from her students, then she related these to what she saw
one of her students actually doing:
I chose the numbers in the problem … the first number was close to a friendly
number, a [multiple of] 10, to see if they could get there first before they added on
and the problem that I had … could either be counting on for the strategy or
starting at the larger number and counting back but either way I wanted them to
get to a friendly number first and then count by tens. And some kids are doing
that but, um, Mary … she started at 89, which was the first number, and tried to
get to the 140 so she went 89-99-109, so she didn’t get to 90, which is fine …
when she got to 139 she added that one more because they’ve been practicing
counting by tens starting at any number, not just a friendly number… (Ms. A,
personal communication, November 11, 2010)
116

Ms. A wrote the number choice for this problem based on its impact on student thinking
and spent time anticipating her students’ responses. She described a student, Mary, who
solved the problem using a variation of Ms. A’s predicted strategy and Ms. A was able to
connect Mary’s strategy with another strategy they had been working on in class. Ms. A
was the best example, out of all the teachers observed and interviewed, of a teacher
basing instructional decisions on students’ thinking.
The information gathered in the interviews aligned with what was seen in the
observations, for the most part, except in the case of Ms. Q. Ms. Q stood out from the
other teachers because of the complicated number choices she assigned. For example, in
the first observation she assigned her students the following problem:
Mr. Danny picked up ____ pieces of trash before recess. After recess, he picked
up some more. He picked up _____ pieces of trash in all. How many pieces of
trash did Mr. Danny pick up after recess?
(13, 40) (138, 261) ((5x7)+(5x23), (4x75)+(9x3))
While the first two number choices were similar to those of the other teachers, the third
number choice was much more difficult (keep in mind that the students would solve this
problem using repeated addition because they have not yet been taught multiplication). In
the first number choice, students would have to subtract and regroup one time. In the
second number choices, although the numbers were larger, students still only had to
regroup one time. In the third number choice, students had to multiply and add before
they subtracted which still only contained one set of regrouping. A goal of this lesson was
for students to practice regrouping strategies (which was a Houghton Mifflin unit
117

objective) (Ms. Q, personal communication, October 11, 2010), but all of the number
choices required the same amount of regrouping.
When asked about her rationale for writing a number choice like the third one, she
replied that she wanted to ―challenge students already proving success in
addition/subtraction with regrouping; wanted to see if students would use commutative
property of multiplication to change order of the numbers being multiplied for
efficiency…‖ (Ms. Q, personal communication, October 11, 2010). While Ms. Q did
have a rationale for her number choices, she did not discuss her students’ thinking or the
strategies they might use to the degree that Ms. A did. In Ms. Q’s class, very few students
attempted this third number choice and those who did were not able to finish the main
problem because they were so busy solving for the preliminary numbers that would go
into the blanks. It was clear that Ms. Q wanted to challenge her students, but she did not
seem to have a clear understanding of her students’ thinking that would have allowed her
to write more appropriately challenging number choices for her students.
Some of the teachers, particularly Ms. Q, seemed to struggle with the balance
between focusing on problem solving strategies and the difficulty of the number choices.
In her interview, Ms. A addressed this dilemma:
I wanted to focus on strategies, different strategies versus trying to solve bigger
numbers. I just varied [the numbers] so there could be those kids that want to go
to the higher number choices they can, and I’m sure I could have gone higher for,
like, there’s a few kids that I could have gone higher for today. (Personal
communication, November 5, 2010)
118

Carpenter, et al. (1999) explain that teachers may write specific number choices for many
reasons; for example, simple number choices may allow children to focus on their
reporting strategies while other number choices may challenge students to use more
mature problem solving strategies. The most important thing for teachers to do as they
write problems is to ―understand the way children think, understand what makes
problems easier or more difficult to solve, and then make decisions that enable children
to engage in successful problem solving with problems that are neither too easy nor too
difficult‖ (Carpenter, et al., 1999, p. 103). All of the teachers discussed the impetus for
their number choices, however, Ms. Q’s number choices appeared to be the least
successful for her students.
This critical feature was based on the degree to which teachers selected problems
based on their impact on students and anticipated students’ problem solving strategies.
Ms. A scored the highest in this category because of the depth about which she talked of
her students’ thinking and how she used that to write problems and plan lessons. None of
the teachers ranked as a 4-A or 4-B in this category because they often chose their lesson
objectives based on a prescribed curriculum, not on the knowledge of their students.
However, the teachers, for the most part, did write the problem story and number choices
based on their knowledge of their students. Ms. Q was an exception to this because her
highest number choice was inappropriate for the students in her class, which was evident
in that very few students chose this number choice and those that did likely were not able
to finish the problem.
119

All of the teachers in this study showed evidence of the critical features of CGI to
some degree in their classrooms. The overall scores for the teachers in this study ranged
from 2 to 3/4-A; none of the teachers scored the highest score (4-B). As the CGI
instruction varied within and between schools, so did the CGI professional development.
The professional development and classroom support provided to teachers as they
experimented with and came to understand CGI could have made the difference between
successful integration and abandonment (Franke, et al., 1998, 2001); the following
section will discuss the CGI professional development opportunities provided for the
teacher participants at both BRES and GVES.

Findings for the Second Research Question

This section will address the findings for the second research question which
explores the aspects of CGI professional development that the teachers in this study
perceived to have been the most influential on their CGI practice. This section provides a
description of the CGI professional development offered at each school followed by the
CGI professional development activities that the teachers in this study felt were the most
effective in supporting their current CGI practice.
All of the teachers who participated in this study experienced CGI professional
development that was similar to what the CGI literature recommends (Carpenter, et al.,
1989; Fennema, et al., 1996): intensive, multi-day summer workshops with ongoing
classroom support. Each teacher participated in three years of formal CGI training which
120

included multi-day summer workshops and ongoing classroom support provided by CGI
coaches. The summer workshops focused on the philosophy of CGI, the mathematical
content knowledge necessary for teaching elementary mathematics, the strategies that
students may use to solve the problems, and the interaction between CGI teachers and
their students through watching video clips of CGI lessons. None of the teachers who
participated in this study had been exposed to CGI prior to their experiences at BRES or
GVES.

Blue River Elementary School.

For the teachers at BRES, all of the initial professional development took place on
campus and coaching was provided by on-site teacher/coaches. All of the teachers in this
study participated in the three-year formal CGI professional development program. The
coaches led the CGI summer workshops and supported the teachers throughout the year.
During the school year, the coaches observed teachers in their classrooms and provided
feedback. Additionally, teachers were able to observe the coaches as they taught a CGI
lesson either in their classroom or in the coach’s classroom about every other month.
Initially, the professional development and coaching was led by the teacher who had
attended the University of Wisconsin and brought CGI to BRES, but as time progressed,
other teachers who had been trained also became coaches (Principal B, personal
communication, June 30, 2010).
121

For the first few years of CGI adoption, monthly meetings took place where
teachers would get together to talk about the mathematics they were teaching, what
strategies they saw their students using, and how they could bridge their students’
thinking from one level to the next. They would also write lessons together and reflect on
those lessons together after they had been taught. Over the years, as teachers felt that they
were becoming more proficient in CGI, these meetings discontinued (Principal B,
personal communication, June 30, 2010).
In their interviews, the teachers at BRES said that they engaged in weekly CGI
peer collaboration sessions where they ―discuss and plan problems together, review
childrens’ strategies, and support each other‖ (Ms. P, personal communication, October
11, 2010) which they found helpful in sustaining their CGI practice. They were no longer
formally coached because they were ―far enough into the program‖ (Ms. P, personal
communication, October 11, 2010), but the initial CGI coach remained a teacher on
campus and was available whenever the teachers felt that they needed support.
While the teachers at BRES agreed that they felt supported in their CGI practice,
it was unclear if they continued to be challenged to improve their CGI practice. The
teachers rarely participated in CGI professional development at the time of the study and
the weekly collaboration time and availability of a coach when they felt it was necessary
does not suggest an environment where CGI enactment is challenged. While perhaps in
the initial adoption of CGI, the professional development provided opportunities where
the teachers were encouraged to improve and reflect on their practice, this seems to have
122

diminished over time. The constant desire to improve one’s practice is a tenant of CGI
(Carpenter, et al., 1999) and effective professional development (Guskey, 2000).

Green Valley Elementary School.

For the teachers at GVES, most of the initial professional development workshops
took place off campus (at the CDE site) and most of the coaching was provided by off-
site CDE coaches. Both of the teachers in this study completed the three-year formal CGI
professional development program. Similarly to the coaches at BRES, the CDE coaches
would observe teachers as they taught a CGI lesson, provide feedback, and teach CGI
lessons for the teachers to observe. The CDE coaches also met with each grade level team
three to four times per year to write CGI lessons, teach a demonstration lesson, and
reflect on what they saw. After the first few years, a group of teachers at GVES became
on-site CGI teacher/coaches who were available in addition to the CDE coaches
(Principal G, personal communication, June 14, 2010)
As the CGI professional development activities with the CDE continued, the
GVES CGI teacher/coaches provided additional professional development and classroom
support on campus. This group of teacher/coaches was given freedom to design how they
wanted to coach the staff (Principal G, personal communication, June 14, 2010). For a
few years this group was very active and designed professional development workshops
over the summer and throughout the year. One of the summer workshops consisted of the
teachers at every grade level writing a problem of the same type so that the continuum of
123

difficulty from kindergarten through fifth grade could be seen. This helped the teachers to
see what types of math problems their students had had in previous grades and what they
would be expected to do in following grades (Ms. B, personal communication, December
16, 2010). In the last two years this group has become mostly inactive due to other
commitments, but they remained available for coaching (in addition to the CDE coaches)
if a teachers requested it (Principal G, personal communication, June 14, 2010).
At the time of the study, the CGI professional development consisted of meeting
in grade level teams with a CDE coach three to four times per year to write problems
together, discuss implemented lessons, and observe the coach teaching CGI lessons.
Additionally, the teachers met once per month to write CGI problems and number
choices. There were mixed responses to the professional development that was available.
Ms. A said that the existing professional development was useful because they were able
to discuss ―whatever our team needs support with‖ and were ―given great resources and
great ideas‖ (personal communication, November 5, 2010). Ms. B gave more mixed
reviews and at one point said that the professional development was ―going well‖
because they could ―target their needs when they meet as a grade level‖ (Ms. B, personal
communication, November 5, 2010) while at another time she said that the professional
development provided by the CDE coach was ―an entire waste of a day‖ (Ms. B, personal
communication, December 16, 2010).
While the feedback on the current professional development was mixed, the CGI
professional development at GVES may have provided an opportunity for teachers to be
challenged to improve their CGI practice. The degree to which their CGI practice was
124

challenged remains unclear, but the teachers did participate in mandatory professional
development multiple times per year that both teachers indicated was focused on their
individual or grade level needs.

Perceived most effective professional development.

There were certain features of their CGI professional development, either initial
or continuing, that the teachers agreed were beneficial for developing their CGI
classroom practice. Since this study does not include an investigation into the past or
current professional development opportunities, it was impossible to determine what they
actually looked like; this study relies on how the teachers perceived their professional
development opportunities.
Four out of five of the teachers who participated in this study agreed that the CGI
coaches provided the most effective CGI professional development. Common reasons for
why the coaches were the most beneficial included that the professional development
supported them as they learned to use the CGI critical features in their classrooms and it
was connected to their everyday practice. Research supports this finding that effective
professional development must provide immediate feedback that is directly connected to
daily classroom practice (Darling-Hammond, et al., 2009; Guskey, 1986, 2000;
Hammerness, et al., 2005, Loucks-Horsley, et al., 2010). There was a difference,
however, in the preferred use of the coaches between the teachers at each school site;
some teachers preferred having the coaches observe their lessons and provide feedback
125

while other teachers preferred observing the coaches teaching CGI lessons. These two
perspectives will be further address in the following paragraphs.
Two of the BRES teachers felt that the most effective professional development
was having CGI coaches observe their CGI lessons and provide feedback. They found
this to be the most beneficial because they could get immediate feedback, have the coach
focus on specific aspects of their teaching that they struggled with, and discuss/plan
future lessons. For Ms. R, the most difficult part of CGI has been in effectively
questioning her students so she found it particularly helpful to have a coach to provide
that support in her classroom (Ms. R, personal communication, November 15, 2010). In
the observations, Ms. R did question her students and was able to elicit her students’
thinking, but she did, at times, interrupt her students’ explanations and asked leading
questions. Ms. R is currently seeking out the CGI coach on campus to help her to
improve her questioning of students (Ms. R, personal communication, November 15,
2010).
The two GVES teachers found it the most beneficial to watch the CGI coach as
she taught a CGI lesson. They appreciated observing the CGI coaches teaching a lesson
because it allowed them to see ―CGI in action‖ (Ms. A, personal communication,
November 5, 2010). After the observations, the teachers and coach convened to discuss
what they saw, why the coaches made specific instructional decisions, and planed for
future lessons. They found this to be beneficial because they were able to see what CGI
lessons could look like, understood the impetus behind the coach’s instructional
decisions, and reflected on their own CGI practice.
126

Ms. Q stood out from the other teachers because of the aspect of professional
development that she found to be the most effective. For her, the most effective aspect of
CGI professional development took place primarily at the workshops where she learned
about the strategies that her students would use to solve problems. Ms. Q did say that she
found the coaching session helpful and used the CGI coach observations to ―confirm‖
that what she was doing was ―on track;‖ she compared what she saw the coaches doing
with what she perceived her own CGI lessons to look like (Ms. Q, personal
communication, October 11, 2010). While the other teachers used the CGI coaching
sessions as opportunities to improve their own practice, Ms. Q seemed to use it simply to
acknowledge that what she was doing was ―on track.‖ Ms. Q was one of the lowest
ranking teachers in this study in terms of the degree to which the CGI critical features
were seen in her lessons and it was unclear as to if or how often she was actually
observed in her classroom teaching CGI lessons and what feedback was provided to her.
As there is no one correct way to implement CGI professional development
(Carpenter, et al., 1999), it is important to note that four of the five teachers in this study
found it beneficial to work with CGI coaches. While some teachers preferred to observe
others implementing CGI, other teachers appreciated having the coaches observe them
and provide feedback. Either way, the coaching element of CGI professional
development appeared to be an integral piece to CGI adoption for these teachers. Long-
term, focused guidance, like coaching, that is directly connected to classroom practice
can have a positive impact on the actions of teachers (Darling-Hammond, et al., 2009;
127

Guskey, 1986, 2000; Hammerness, et al., 2005, Loucks-Horsley, et al., 2010), which is
consistent with the experiences of the teachers in this study.

Conclusion

The findings for the first research question, which explores the extent to which the
teachers’ instructional practices reflected the critical features of CGI, were based
primarily on classroom observations while the teacher interviews provided insight into
the choices that the teachers made in their classrooms. This study has found that the
sample teachers from BRES and GVES displayed the critical features of CGI to varying
degrees. Aspects of the CGI critical features were evident in the classroom observations
but there was no teacher who showed evidence of complete CGI enactment; the CGI
instructional levels for the teacher participants in this study ranged from 2 to 3/4-A. None
of the teachers could be ranked at the highest CGI instructional level because they all
chose their lesson objectives based on a pre-determined curriculum, not on children’s
thinking, and they implemented ―CGI lessons‖ that were separate from other mathematics
lessons, indicating that the CGI philosophy was not fully assumed.
The second research question investigated which aspects of CGI professional
development the teachers in this study found to be the most effective in supporting the
integration of CGI philosophy into their practice. The data for this research question were
collected through teacher questionnaires and interviews. The professional development
that each teacher in the study participated in was aligned with the three-year training
128

program of summer workshops and continued classroom support along with additional
classroom support in subsequent years that the developers of CGI have recommended
(Carpenter, et al., 1989; Fennema, et al., 1996). The majority of the teachers perceived
that the aspect of professional development that was most beneficial for them was in
working with the CGI coaches. Whether they preferred to have the CGI coaches observe
them teaching or observe the CGI coaches teaching, the coaches were the most helpful
because they provided immediate feedback that was directly connected to current
classroom practice (Darling-Hammond, et al., 2009; Guskey, 1986, 2000; Hammerness,
et al., 2005, Loucks-Horsley, et al., 2010).
The two research questions on which this study was based were answered from
the interview, observation, and questionnaire data collected. Chapter 5 will provide an
overview of the study conclusions, additional findings from this study, implications of the
findings, and recommendations for further research.
129

Chapter 5
Conclusionsi

This study focused on two elementary schools in GVUSD school district, BRES
and GVES, and sought to answer the following research questions: (1) To what extent do
teachers’ instructional practices reflect the critical features of CGI? and (2) What aspects
of CGI professional development did the teachers perceive to be the most effective in
supporting the integration of the CGI philosophy into their practice? To collect data to
address these questions, five second grade teachers who volunteered to participate in this
study were observed teaching ―CGI‖ mathematics lessons, interviewed, and completed
questionnaires. The observations served to provide evidence of the CGI critical features
in their classroom practice. Teacher interviews and questionnaires provided background
information about the professional development experiences of the teachers and indicated
which aspects of the CGI professional development they perceived to be most effective in
supporting their practice. Principal interviews provided an overview of the CGI
professional development offered at each school. This qualitative study found that the
critical features of CGI were evident to varying degrees in each of the observed
classrooms and that the professional development perceived to be the most effective was
offered by CGI coaches who provided support directly related to current classroom
practice.
The following sections provide the conclusions that were drawn for each of the
research questions in this study. Each research question will be discussed in turn and
130

limitations relating to that research question will be included in the analysis. Then,
recommendations will be made for both the district and for further study.

Conclusions for the First Research Question

The first research question explored the extent to which teachers’ instructional
practices reflected the critical features of CGI. Observation data were used to rank each
teacher based on the degree that the critical features of CGI were evident during CGI
mathematics lessons. On a five point scale, none of the five teachers in this study ranked
at the highest CGI instructional implementation level (4-B); one teacher ranked as 2, one
teacher ranked as 2/3, two teachers ranked as 3, and one teacher ranked a 3/4-A. In
general, the teachers at GVES scored more highly (3, 3/4-A) than the teachers at BRES
(2, 2/3, 3). This is surprising because the teachers at BRES are known throughout the
district for their enactment of CGI and the supportive CGI environment in which they
work; GVES does not have this reputation (Assistant Superintendent of Curriculum &
Instruction, April 15, 2010) but showed stronger evidence of CGI enactment in the
sample teachers’ classrooms.
Common weaknesses in the two schools were the use of a pre-determined
curriculum to dictate lesson objectives and the language used to discuss CGI. The
curriculum, HMCM, was chosen and mandated by the district for the teachers to
implement. The teachers in this study used this curriculum to determine lesson objectives
and sequencing for their mathematics lessons. The CGI principles rely on teachers using
131

knowledge of their students’ thinking to guide these instructional decisions (Fennema, et
al., 1996); no teacher in this study was found to base their lesson objectives and
sequencing on the knowledge of their students’ thinking.
When discussing CGI with district administration, principals, and teachers, a
common language was used that may have served to undermine the philosophy of CGI.
The participants in this study often referred to the ―implementation‖ of CGI, the CGI
―program,‖ and differentiated between ―CGI lessons‖ and ―non-CGI lessons.‖ This
language indicated that CGI was seen as a program to be implemented rather than a
system of beliefs about teaching. If one truly believed in the philosophy of CGI, the
critical features of CGI should be evident in their general mathematics teaching practice.
It was not clear as to how the use of this language came about; perhaps it was presented
as part of the professional development provided to the teachers or initiated by teachers
or administrators within the schools. The use of this language could indicate a superficial
understanding of CGI (as a program to be implemented rather than a belief system) or a
fundamental misunderstanding of what CGI really is. The teachers in this study could not
be ranked at the highest level of implementation because the language they used to
discuss CGI and their instructional choices did not indicate a complete understanding and
enactment of CGI as a philosophy of mathematics teaching.
The CGI instructional level scores were based on observed actions of the teachers
in the classroom. The CGI philosophy relies on both the actions and beliefs of teachers;
while it is unknown if the true beliefs of the teachers were aligned with those of CGI, the
teacher interviews provided some insight as to how the teachers viewed their own CGI
132

practice and the degree to which their beliefs may have been aligned with those of CGI.
Studies have shown that ―teachers’ practices often differ from the kinds of practices they
espouse, and that they frequently describe their own practices as more consistent with
reform ideals than outside observers believe to be the case‖ (Kennedy, 2004, p. 3; Cohen,
1990; Liljedahl, 2008). There is evidence that the beliefs of the teachers in this study
were more aligned to the CGI philosophy than their actions would suggest.
Each of the teachers was asked whether she believed that she used the principles
of CGI in her mathematics teaching as intended by the program developers. All of the
teachers responded that they did and every teacher explained that there is no one correct
way of implementing CGI (Carpenter, et al., 1999). Ms. A, the highest ranked teacher,
put it this way: ―what’s great about CGI is there’s not one correct way to implement it. It
isn’t a program. It allows each teacher to implement it the way that it works best for them
and their students‖ (personal communication, November 5, 2010). The idea that there is
no one correct way to enact CGI in a classroom is consistent with the CGI philosophy
(Carpenter, et al., 1999).
Ms. B, who ranked 3, was the only teacher to validate her claim of CGI enactment
with specific support; however, for the most part, this support was not substantiated in
her observation. In her interview, Ms. B said that she had her students work on a variety
of problems; from her observations, this was not evident as she did not provide her
students with a variety of problems types and she ranked as 2/3 for opportunities for
children to solve problems. Additionally, she explained that she focused on the student
sharing time as the most important part of her CGI lessons and during this time she
133

―highlight[ed] many different teaching points‖ (Ms. B, personal communication,
November 5, 2010). While the researchers only witnessed her sharing time during one
observation, it was rushed with very little student questioning or explanation; she earned
a 2/3 for children sharing their thinking with peers and teacher. Ms. B believed that there
was more evidence of her CGI enactment in her teaching practice than was actually
observed for this study. In fact, all of the teachers stated that they believed they were
using the principles of CGI in their classrooms and that they found CGI to be effective in
their teaching and in student learning of mathematics; however, the evidence of CGI
enactment from the observations supported this to varying degrees. It is not uncommon
for disparities to exist between teachers’ perceived actions and their actual actions in the
classroom (Cohen, 1990; Kennedy, 2004; Liljedahl, 2008); the teachers in this study
support this research.
There were a number of factors that limited the findings for the first research
question. First, due to time and manpower constraints, the researcher did not know what
―non-CGI‖ math lessons looked like, nor was there a clear picture of what instruction
over the course of a day looked like for any of the teachers in this study. Additionally, it
was difficult to predict ―typical‖ classroom practice when only two observations were
conducted for each teacher. It was possible that the teachers enacted elements of CGI in
other lessons or content areas that could have provided evidence of a more complete
adoption of the CGI critical features. Second, the sample of teachers in this study
volunteered to participate and it cannot be assumed that they are representative of the
entire population of teachers at these two schools (Patton, 2002). While it was clear that
134

CGI was enacted to varying degrees for the sample teachers, it is unknown what the
general CGI practice is across each school.
The conclusion drawn for the first research question was that the extent to which
the critical features of CGI were evident in the classroom practice of the five second-
grade teacher participants varied both across and within the two schools. In general, the
teachers at GVES scored more highly than the teachers at BRES. There was evidence that
the teachers who participated in this study believed that they enacted CGI principles in
their mathematics instruction to a higher degree than was evident in their classroom
observations.

Conclusions for the Second Research Question

The second research question investigated the aspects of CGI professional
development that the teachers perceived to be the most effective in supporting the
integration of the CGI philosophy into their teaching practice. The majority of teachers
perceived that support from coaches was the most effective because it provided
immediate feedback and was connected to current classroom practice (Darling-
Hammond, et al., 2009; Guskey, 1986, 2000; Hammerness, et al., 2005, Loucks-Horsley,
et al., 2010). Four of the five teachers found this support to be the most effective
professional development for them. One teacher, Ms. Q, reported that learning the
problem solving strategies that students were likely to use was the most helpful. Ms. Q
was the lowest ranking teacher in this study (2) and was also the teacher who had been
135

exposed to CGI for the least amount of time. Research suggests that when a teacher is
introduced to a new curriculum or teaching practice, the first concern of that teacher is
often structural and focused on the content being taught and classroom management; as
the teacher becomes more comfortable with the new approach, she is able to then shift
her focus to student learning and how she teaches the content (Loughran, 2006). Ms. Q
may be more concerned at this time with understanding the mathematics that she is
teaching as opposed to how she is enacting CGI in her classroom. Besides Ms. Q, there
were no other patterns in the CGI instructional level rankings of the teachers and their
perceived effective professional development.
All of the teachers in this study had been participating in CGI professional
development for at least four years; it had been 11 years for both Ms. P and Ms. R. While
there was a trend in the professional development that the teachers perceived to be the
most effective, it is interesting to note that none of the teachers ranked at the highest level
on the CGI instructional level rubric even though they have been participating in CGI
professional development for so many years. Research suggests that it could take three to
five years for teachers to fully enact a new program (Loucks-Horsley, et al., 2010) and
four of the five teachers have been participating in CGI professional development for at
least five years. This leads to the question of how effective the professional development
actually was in supporting and challenging the teachers’ CGI instructional practice.
Because the actual professional development activities were not observed as part of this
study, it is impossible to draw any conclusions about their quality.
136

Franke, et al. (1998) and Franke, et al. (2001) determined that for teachers’
enactment of the CGI principles to continue to develop over time, teachers must engage
in ―professional development that supports their ongoing learning and simultaneously
provides opportunities for teachers to create collaborations with their colleagues‖
(Franke, et al., 2001, p. 686). At BRES, the initial professional development did seem to
be supportive and challenging to practice (Principal B, personal communication, June 30,
2010), but the teachers have not been supported in ongoing learning as their CGI
professional development had mostly discontinued over the years. While the teachers did
meet once per week to discuss CGI lessons at the time of this study, it is unclear if this
time was spent engaging in conversation that challenged current practice; without a
supportive environment where teachers’ practices are questioned and challenged, CGI
enactment is ―likely to erode over time‖ (Franke, et al., 1998, p. 67).
At GVES, it appeared that the professional development that was offered initially
may not have been as supportive and challenging to practice as that at BRES, but it
seemed to have been more consistent over the years. At the time of this study, teachers at
GVES continued to attend CGI coaching sessions three to four times per year and
planned CGI lessons together monthly. Because the researchers did not attend any of the
professional development or planning sessions at either school, the quality of these
meetings is unknown. Professional development that supports and challenges the CGI
practice of teachers helps them to continue to improve their CGI enactment in the
classroom (Franke, et al., 1998; 2001); the authors caution, however, that
―institutionalizing collaborative work or mandating practical inquiry will not work‖
137

(Franke, et al., 2001, p. 685). The professional development offered at the two schools
may have become more of a practice in routine than an opportunity for professional
learning.
A major limitation of the findings for the second research question was that the
researcher did not attend any of the current CGI professional development activities
available to the teachers or any of the CGI collaboration sessions. It is impossible to
know the true quality of these experiences without being present; the researcher had to
rely on the perceptions of the teachers. Additionally, the researcher did not attend any of
the CGI workshops that were offered for teachers in the initial stages of CGI adoption.
While these workshops would not necessarily be the same as those that the teachers in
this study participated in, they could have offered insight into the CGI professional
development experience.
The conclusion drawn for the second research question was that most of the
teachers in this study perceived that teacher coaches were the most effective aspect of
professional development for their CGI practice. This is consistent with research that has
found that effective professional development provides teachers with immediate feedback
that is directly connected to classroom practice (Darling-Hammond, et al., 2009; Guskey,
1986, 2000; Hammerness, et al., 2005, Loucks-Horsley, et al., 2010). Teachers at both
schools continued to be supported through teacher collaboration and coaching; there was
evidence that these professional development activities served to sustain, rather than
challenge, CGI enactment.

138

Recommendations for the District

While this study did not investigate the relationship between CGI enactment and
CGI professional development, research indicates a positive correlation between high
quality professional development and effective classroom instruction for both CGI
enactment (Fennema & Carpenter, 1989; Fennema, et al., 1996; Franke, et al., 1998,
2001) and general teaching practice (Guskey, 1986, 2001; Loucks-Horsley, et al., 2010;
Loughran, 2006). The recommendations for the district rely on this relationship. It is
important to keep in mind that the sample in this study was small and the CGI enactment
of the general teacher population is unknown; the researcher is assuming that the CGI
instruction observed for this study was typical of practice within each school.
Based on the finding that the teachers who participated in this study enacted CGI
to varying degrees that ranged from 2 and 3/4-A according to the CGI instructional
levels, the school administration must determine the importance of improving teachers’
enactment of CGI in classroom practice. If it is decided that creating an environment for
teachers that supports their ongoing professional learning and integration of the CGI
philosophy into classroom practice is a priority, the administration must understand that it
will take considerable dedication of both time and resources (Fennema & Carpenter,
1989).
The structure of the CGI professional development offered in the two schools is
very similar to that proposed by Fennema, et al. (1996); however, the findings suggested
that the core difference between the professional development programs is in the quality
139

of support that was and continues to be provided. Effective professional development is
not about gaining a new set of fixed teaching skills or learning how to use a particular
instructional program (Franke, et al., 1998, 2001; Webster-Wright, 2009). It ―entails
teachers making changes in their basic epistemological perspectives, their knowledge of
what it means to learn, as well as their conceptions of classroom practice …
conceptualizing teacher change in terms of teachers becoming ongoing learners‖ (Franke,
et al., 1998, p. 67). Making deep changes such as these only happen when beliefs are
reformed through ―new understandings and experimentation with new behavior‖
(Loucks-Horsley, et al., 2010, p. 76).
Cognitively Guided Instruction professional development experiences must
provide opportunities for teachers to explore their own mathematical understandings,
learn about their students’ mathematical thinking, and have a safe and supportive
environment in which to experiment with integrating the principles of CGI into their
teaching practice (Fennema, et al., 1996). Franke, et al. (1998, 2001) propose that
successful CGI teachers must engage in self-sustaining, generative change in which
teachers come to understand the nature of their own learning, engage in inquiry into their
own teaching practices to understand why and under what conditions student learning
takes place, and have a support network of colleagues who employ these same practices
in their own classrooms. There was no indication that the teachers in this study engaged
in practices such as these and it is likely that significant changes in teacher practice,
professional development, and support from administration would need to take place in
140

order for CGI enactment to improve and for teachers to engage in practices that support
self-sustaining, generative change.
Research by Webster-Wright (2009) suggests that current systems of professional
development often ―focus on delivering content rather than enhancing learning‖ (p. 702)
which may result in superficial enactment of new information or ideas (Cohen & Ball,
2001; Webster-Wright, 2009). The speculation that aspects of CGI had been superficially
enacted in the two schools and that the professional development offered did not support
teachers in ongoing learning is not an unusual finding when professional development is
analyzed across professions. In general, professional development activities assume that
professionals need ―developing‖ (Webster-Wright, 2009, p. 713) and they do not take
into account the types of activities that foster authentic professional learning (Webster-
Wright, 2009). Professional development activities can be better understood when
participants are given opportunities to describe instances where they feel they have
learned; it is not the opportunities for professional development that result in learning but
the ways in which the participants make meaning out of those experiences (Fennema &
Carpenter, 1989; Franke, et al., 1998, 2001; Webster-Wright, 2009).
The charge to the school and district administrators, then, is to create a
professional development structure where teachers are challenged and supported in
engaging in self-sustaining, generative change. Change such as this can take years to
occur and requires ongoing commitment long after initial changes are evident; resources
such as time for teachers to engage in professional development activities and peer
collaboration and reflect on their teaching practices as well as monetary funds for
141

providing teachers with the professional support they need to engage in such change will
be required as long-term dedication to supporting teachers in their enactment of CGI
principles is undertaken (Fennema & Carpenter, 1989; Loucks-Horsley, et al., 2010).
Part of this process will be in recognizing how the teachers come to understand and
internalize the professional development opportunities that they participate in (Fennema
& Carpenter, 1989; Franke, et al., 1998, 2001; Webster-Wright, 2009). Just as CGI
enactment does not look the same in all classrooms (Carpenter, et al., 1999), CGI
professional development will not look the same in all schools (Carpenter, Fennema,
Franke, et al., 2000; Fennema, et al., 1999; Fennema & Carpenter, 1989); the challenge is
creating an environment where all teachers are provided the necessary professional
development to best support them in their CGI enactment and engage them in activities
that support self-sustaining, generative change.

Recommendations for Further Study

A recommendation for further research is to investigate the types of professional
development activities that best support teachers in their ongoing CGI enactment with
attention to their actual classroom practices. The current study suggested that teachers
perceived that the most effective professional development activities were led by
classroom coaches. This relationship could be further explored to determine whether the
coaches actually did provide the greatest support for teachers’ classroom actions.
142

Additionally, further investigation is needed into what made differences in
individual teachers’ enactment of the CGI principles. The teachers from each of the two
elementary schools reportedly participated in similar CGI professional development
experiences; why did some teachers embrace the philosophy of CGI more than others and
what accounted for the differences (and similarities) in the enactment of each of the
critical features? The outcome of a study such as this could inform the professional
development activities and teacher support programs that are available to teachers as they
adopt and integrate CGI into their teaching practice.
The general limitations of this study provide areas of potential further research. A
similar study conducted with a greater number of teacher observations, increased time in
classrooms, or a larger sample of teacher participants could provide further insight into
the first research question. Additionally, a study where professional development and
collaboration opportunities were observed, professional development was observed over
a long period of time, or teachers were followed as they progressed through the
introduction, enactment, and integration of CGI principles would all contribute to the
conclusions drawn for the second research question.

Conclusion

This study initially set out to gather information for the district administration
about whether teachers were actually using CGI principles in their classroom practice and
what professional development activities supported the teachers’ enactment of CGI.
143

These inquiries served as the basis for the research questions used in this study. While
there was evidence of CGI enactment, the degree to which the critical features of CGI
were used in classroom practice varied. The professional development activities that the
majority of teachers perceived to be the most effective were coaching sessions that
provided teachers with immediate feedback that was directly related to classroom
practice. It is recommended that the schools and district examine ways to better support
teachers in their CGI enactment and in engaging them in activities that support self-
sustaining, generative change. This study provides evidence that perhaps increased
interaction with CGI coaches could positively influence CGI enactment because the
teachers in this study reported that the coaches were effective in supporting their current
CGI practice; however, further research must be conducted in this area to make any
concrete recommendations.
144

References
Adelman, C. (1999). Answers in the Toolbox: Academic Intensity, Attendance Patterns
and Bachelor’s Degree Attainment. Washington, D.C.: U.S. Government Printing
Office. Retrieved November 21, 2008 from http://www.learndoearn.org/For-
Educators/Answers%20in%20the%20Toolbox.pdf
Adelman, C. (2006). The Toolbox Revisited: Paths to Degree Completion From High
School Through College. Washington, D.C.: U.S. Department of Education.
Bransford, J.D., Brown, A.L, & Cocking, R.R. (Eds.). (2000). How people learn: Brain,
mind, experience, and school. Washington, D.C.: National Academy Press.
California Department of Education (CDE). (2010). Retrieved on June 30, 2010 from
http://www.cde.ca.gov/ta/ac/ap/
Carey, K. (2004). The real value of teachers: Using new information about teacher
effectiveness to close the achievement gap. Thinking K-16, 8(1), 3–42.
Carpenter, T.P. (1985). Learning to add and subtract: An exercise in problem solving. In
E.A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple
research perspectives (pp. 17–40). Hillsdale, NJ: Lawrence Erlbaum Associates.
Carpenter, T.P., Fennema, E., & Franke, M.L. (1996). Cognitively guided instruction: A
knowledge base for reform in primary mathematics instruction. The Elementary
School Journal, 97(1), 3–20.
Carpenter, T.P., Fennema, E., Franke, M.L., Levi, L., & Empson, S.B. (1999). Children’s
mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann and
The National Council of Teachers of Mathematics, Inc.
Carpenter, T.P., Fennema, E., Franke, M.L., Levi, L., & Empson, S.B. (2000).
Cognitively guided instruction: A research-based teacher professional
development program for elementary school mathematics (Research Report No.
003). Retrieved from Educational Resources Information Center website:
http://www.eric.ed.gov/PDFS/ED470472.pdf

145

Carpenter, T.P., Fennema, E., Fuson, K., Hiebert, J., Human, P., Murray, H., Olivier, A.,
& Wearne, D. (1994, April). Teaching mathematics for learning with
understanding in the primary grades. Paper presented at the annual meeting of the
American Educational Research Association, New Orleans, LA.
Carpenter, T.P., Fennema, E., Levi, L., Franke, M.L., & Empson, S.B. (2000). Children’s
mathematics: Cognitively guided instruction: A guide for workshop leaders.
Portsmouth, NH: Heinemann.
Carpenter, T.P., Fennema, E., Peterson, P.L., & Carey, D.A. (1988). Teachers’
pedagogical content knowledge of students’ problem solving in elementary
arithmetic. Journal for Research in Mathematics Education, 19(5), 385–401.
Carpenter, T.P., Fennema, E., Peterson, P.L., Chiang, C., & Loef, M. (1989). Using
knowledge of children’s mathematics thinking in classroom teaching: An
experimental study. American Educational Research Journal, 26(4), 499–531.
Carpenter, T.P., Franke, M.L., Jacobs, V.R., Fennema, E., & Empson, S.B. (1998). A
longitudinal study of invention and understanding in children’s multidigit addition
and subtraction. Journal for Research in Mathematics Education, 29(1), 3–20.
Carpenter, T.P., Franke, M.L., & Levi, L. (2003). Thinking mathematically: Integrating
arithmetic & algebra in elementary school. Portsmouth, NH: Heinemann.
Carpenter, T.P. & Levi, L. (2000). Developing conceptions of algebraic reasoning in the
primary grades (Research Report No. 00-2). Retrieved from Educational
Resources Information Center website:
http://www.eric.ed.gov/PDFS/ED470471.pdf
Cohen, D. K. (1990). A revolution in one classroom: The case of Mrs. Oublier.
Educational Evaluation and Policy Analysis, 12(3), 311–329.
Cohen, D. K. & Ball, D. L. (2001). Making change: Instruction and its improvement. Phi
Delta Kappa International, 83(1), 73–77.
Creswell, J.W. (2009). Research design: Qualitative, quantitative, and mixed methods
approaches. Los Angeles: Sage.
Darling-Hammond, L. (2000). Teacher quality and student achievement: A review of
state policy evidence. Arizona State University: Education Policy Analysis
Archives. http://epaa.asu.edu/epaa/v8n1
146

Darling-Hammond, L. & Bransford, J. (Eds.) (2005). Preparing Teachers for a Changing
World: What Teachers Should Learn and be Able to do. San Francisco, CA:
Jossey Bass.
Darling-Hammond, L., Wei, R.C., Andree, A., Richardson, N., & Orphanos, S. (2009).
Professional learning in the learning profession: A status report on teacher
development in the United States and abroad. Oxford, OH: National Staff
Development Council.
Desimone, L.M. (2009). Improving impact studies of teachers’ professional development:
Toward better conceptualizations and measures. Educational Researcher, 38(3),
181–199.
Desimone, L.M., Porter, A.C., Garet, M.S., Yoon, K.S., & Birman, B.F. (2002). Effects
of professional development on teachers’ instruction: Results from a three-year
longitudinal study. Educational Evaluation and Policy Analysis, 24(2), 81–112.
Ed-Data: Education Data Partnership. (2010). Retrieved on June 17, 2010 from
http://www.ed-
data.org/Navigation/fsTwoPanel.asp?bottom=%2Fprofile.asp%3Flevel%3D06%2
6reportNumber%3D16
Fennema, E. & Carpenter, T.P. (1989). A program implementation guide for cognitively
guided instruction (CGI). Madison, WI: Wisconsin Center for Education
Research, School of Education, University of Wisconsin-Madison.
Fennema, E., Carpenter, T.P, & Franke, M.L. (1992). Cognitively Guided Instruction.
NCRMSE Research Review: The Teaching and Learning of Mathematics, 1(2), 5–
9, 12.
Fennema, E., Carpenter, T.P., Franke, M.L., & Carey, D.A. (1993). Learning to use
children’s mathematics thinking: A case study. In R.B. Davis & C.A. Maher
(Eds.), Schools, mathematics, and the world of reality (pp. 93–118). Boston, MA:
Allyn and Bacon.
Fennema, E., Carpenter, T.P., Franke, M.L., Levi, L., Jacobs, V.R., & Empson, S.B.
(1996). A longitudinal study of learning to use children’s thinking in mathematics
instruction. Journal for Research in Mathematics Education, 27(4), 403–434.
147

Fennema, E., Carpenter, T.P., Jacobs, V.R., Franke, M.L., & Levi, L.W. (1998). A
longitudinal study of gender differences in young children’s mathematical
thinking. Educational Researcher, 27(5), 6–11.
Fennema, E., Carpenter, T.P, & Lamon, S.J. (Eds.). (1991). Integrating research on
teaching and learning mathematics. New York: State University of New York
Press.
Fennema, E., Carpenter, T., Levi, L., Franke, M.L., and Empson, S.B. (1999). Children’s
mathematics: Cognitively Guided Instruction: Professional development
materials. Portsmouth, NH: Heinemann.
Fennema, E., Franke, M.L., Carpenter, T.P., & Carey, D.A. (1993). Using children’s
mathematical knowledge in instruction. American Educational Research Journal,
30(3), 555–583.
Fosnot, C. (2007). Contexts for Learning Mathematics, Level 1. Portsmouth, NH:
Heinemann.
Franke, M.L., Carpenter, T.P., Fennema, E., Ansell, E., & Behrend, J. (1998).
Understanding teachers’ self-sustaining, generative change in the context of
professional development. Teaching and Teacher Education, 14(1), 67–80.
Franke, M.L., Carpenter, T.P., Levi, L., & Fennema, E. (2001). Capturing teachers’
generative change: A follow-up study of professional development in
mathematics. American Educational Research Journal, 38(3), 653–689.
Franke, M.L. & Kazemi, E. (2001). Learning to teach mathematics: Focus on student
thinking. Theory into Practice, 40(2), 102–109.
Franke, M.L., Webb, N.M., Chan, A.G., Ing, M., Freund, D., & Battey, D. (2009).
Teacher questioning to elicit students’ mathematical thinking in elementary
school classrooms. Journal of Teacher Education, 60(4), 380–392.
Fuson, K.C., Kalchman, M., & Bransford, J.D. (2005). Mathematical understanding: An
introduction. In M.S. Donovan & J.D. Bransford (Eds.), How students learn:
History, mathematics, and science in the classroom (pp. 217–256). Washington,
D.C.: The National Academies Press.
148

Garet, M.S., Porter, A.C., Desimone, L., Birman, B.F., & Yoon, K.S. (2001). What
makes professional development effective? Results from a national sample of
teachers. American Educational Research Journal, 38(4), 915–945.
Gordon, R., Kane, T.J., & Staiger, D.O. (2006). Identifying effective teachers using
performance on the job. The Hamilton Project and The Brookings Institution.
Griffin, S. (2005). Fostering the development of whole-number sense: Teaching
mathematics in the primary grades. In M.S. Donovan & J.D. Bransford (Eds.),
How students learn: History, mathematics, and science in the classroom (pp.
217–256). Washington, D.C.: The National Academies Press.
Guskey, T.R. (1986). Staff development and the process of teacher change. Educational
Researcher, 15(5), 5–12.
Guskey, T.R. (2000). Evaluating professional development. Thousand Oaks, California:
Corwin Press.
Hammerness, K., Darling-Hammond, L., & Bransford, J. (2005). How teachers learn and
develop. In L. Darling-Hammond & J. Bransford (Eds.), Preparing teachers for a
changing world (pp. 358–389). San Francisco, CA: Jossey-Bass.
Haycock, K. (1998). Good teaching matters… A lot. Thinking K-16, 3(2), 3–14.
Hill, H.C. (2004). Professional development standards and practices in elementary school
mathematics. The Elementary School Journal, 104(3), 215–231.
Hill, H.C. & Ball, D.L. (2004). Learning mathematics for teaching: Results from
California’s mathematics professional development institutes. Journal for
Research in Mathematics Education, 35(5), 330–351.
Hill, H.C., Rowan, B., & Ball, D.L. (2005). Effects of teachers’ mathematical knowledge
for teaching on student achievement. American Educational Research Journal,
42(2), 371–406.
International Association for the Evaluation of Educational Achievement (IEA). Trends
in International Mathematics and Science Study (TIMSS), 2007. Retrieved from
http://nces.ed.gov/timss/results07_math07.asp
149

Kazemi, E. & Hubbard, A. (2008). New directions for the design and study of
professional development: Attending to the coevolution of teachers’ participation
across contexts. Journal of Teacher Education, 59(5), 428–441.
Kennedy, M.M. (2004). Reform ideals and teachers’ practical intentions. Education
Policy Analysis Archives, 12(13).
Lieberman, A. & Mace, D.P. (2010). Making practice public: Teacher learning in the 21
st

century. Journal of Teacher Education, 61(1-2), 77–88.
Liljedahl, P. (2008). Teachers’ insights into the relationship between beliefs and practice.
In J. Maaß & W. Schlöglmann (Eds.), Beliefs and Attitudes in Mathematics
Education: New Research Results (pp. 44–54). Rotterdam, NL: Sense Publishers.
Lewin, K. (1935). A dynamic theory of personality. New York: McGraw Hill.
Loucks-Horsley, S., Stiles, K.E., Mundry, S., Love, N., & Hewson, P.W. (2010).
Designing professional development for teachers of science and mathematics. (3
rd

ed.) Thousand Oaks, California: Corwin Press.
Loughran, J. (2006). Developing a pedagogy of teacher education. USA and Canada:
Routledge.
Nathan, M.J. & Koedinger, K.R. (2000). Teachers’ and researchers’ beliefs about the
development of algebraic reasoning. Journal for Research in Mathematics
Education, 31(2), 168–190.
National Center for Education Statistics (NCES). (2009a). The Nation’s Report Card:
Mathematics 2009 (NCES 2010-451). Institute of Education Sciences, U.S.
Department of Education, Washington, D.C.
National Center for Education Statistics (NCES). (2009b). The Nation’s Report Card:
Mathematics 2009: Snapshot State Report, California Grade 8. (NCES 2010-
451). Institute of Education Sciences, U.S. Department of Education, Washington,
D.C.
National Center for Education Statistics (NCES). (2009c). The Nation’s Report Card:
Mathematics 2009: Snapshot State Report, California Grade 4. (NCES 2010-
451). Institute of Education Sciences, U.S. Department of Education, Washington,
D.C.
150

National Center for Education Statistics (NCES). (2010). Retrieved June 30, 2010 from
http://nces.ed.gov/nationsreportcard/about/.
National Center for Research in Mathematics and Science Education (1992). NCRMSE
research review, 1(2). Retrieved from the National Center for Improving Student
Learning and Achievement in Mathematics and Science website:
http://ncisla.wceruw.org/publications/newsletters/NCRMSE/VOL1NUM2.pdf
National Mathematics Advisory Panel (NMAP). (2008). Foundations for Success: The
Final Report of the National Mathematics Advisory Panel, U.S. Department of
Education: Washington, DC.
National Mathematics Advisory Panel (NMAP). (2008a). Task group reports of the
National Mathematics Advisory Panel: Report of the task group on teachers and
teacher education, U.S. Department of Education: Washington, DC.
National Research Council. (2001). Adding it up: Helping children learn mathematics. J.
Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study
Committee, Center for Education, Division of Behavioral and Social Sciences and
Education. Washington, DC: National Academy Press.
Nye, B., Konstanopoulos, S., & Hedges, L.V. (2004). How large are teacher effects?
Educational Evaluation and Policy Analysis, 26(3), 237–257.
Patton, M.Q. (2002). Qualitative research & evaluation methods. Thousand Oaks: Sage
Publications.
Peterson, P.L., Carpenter, T.P., & Fennema, E. (1989). Teachers’ knowledge of students’
knowledge in mathematics problem solving: Correlational and case analyses.
Journal of Educational Psychology, 81(4), 558–569.
Peterson, P.L., Fennema, E., & Carpenter, T.P. (1988). Using knowledge of how students
think about mathematics. Educational Leadership, 46(4), 42–46.
Peterson, P.L., Fennema, E., & Carpenter, T.P. (1991). Using children’s mathematical
knowledge. In B. Means (Ed.), Teaching advanced skills to educationally
disadvantaged students (pp. 68–101). Menlo Park, CA: SRI International.
Peterson, P.L., Fennema, E., Carpenter, T.P., & Loef, M. (1989). Teachers’ pedagogical
content beliefs in mathematics. Cognition and Instruction, 6(1), 1–40.
151

Philosophy. (2010). In Oxford Dictionaries online. Retrieved from
http://oxforddictionaries.com/view/entry/m_en_us1277561#m_en_us1277561
Rose, H. and Betts, J. (2001). Math matters: The links between high school curriculum,
college graduation, and earnings. San Francisco, California: Public Policy
Institute of California.
Rose, H. and Betts, J. (2004). The effect of high school courses on earnings. Review of
Economics & Statistics. May, 2004, 497–519.
Sanders, W.L. & Rivers, J. C. (1996). Cumulative and residual effects of teachers on
future student academic achievement. University of Tennessee Value-Added
Research and Assessment Center, November 1996.
Shulman, L.S. (1986). Those who understand: Knowledge growth in teaching.
Educational Researcher, 15(2), 4–14.
Slavin, R.E. & Lake, C. (2008). Effective programs in elementary mathematics: A best-
evidence synthesis. Review of Educational Research, 78(3), 427–515.
Stigler, J. W. & Hiebert, J. (1999). The Teaching Gap: Best Ideas from the World’s
Teachers for Improving Education in the Classroom. New York, New York: The
Free Press.
Toll, C.A., Nierstheimer, S.L., Lenski, S.D., & Kolloff, P.B. (2004). Washing our
students clean: Internal conflicts in response to preservice teachers’ beliefs and
practices. Journal of Teacher Education, 55(2), 164–176.
Vace, N.N. & Bright, G.W. (1999). Elementary preservice teachers’ changing beliefs and
instructional use of children’s mathematical thinking. Journal for Research in
Mathematics Education, 30(1), 89–110.
Villasenor, A. & Kepner, H.S. (1993). Arithmetic from a problem-solving perspective:
An urban implementation. Journal for Research in Mathematics Education, 24(1),
62–69.
Walker, E. (2007). Rethinking professional development for elementary mathematics
teachers. Teacher Education Quarterly, Summer 2007, 113–134.
Webster-Wright, A. (2009). Reframing professional development through understanding
authentic professional learning. Review of Educational Research, 79(2), 702–739.
152

Whitcomb, J., Borko, H., & Liston, D. (2009). Growing talent: Promising professional
development models and practices. Journal of Teacher Education, 60(3), 207–
212.
Woodward, J. (2004). Mathematics education in the United States: Past to present.
Journal of Learning Disabilities, 37(1), 16–31.
153

Appendices
Appendix A
CGI Baseline and Case Study Descriptions

The following studies were completed with the same sample of teachers and
students that was used in the culminating study by Carpenter, et al. (1989). The first three
studies that are described (Carpenter, et al., 1988; Peterson, Fennema, et al., 1989;
Peterson, Carpenter, & Fennema, 1989) were baseline studies to collect data that would
be used in the culminating study (Carpenter, et al., 1989). The final case study (Fennema,
Franke, et al., 1993) monitored one of the teachers from the CGI treatment group over a
four year period to investigate the effects of the CGI professional development program
over time.
The first baseline study was conducted by Carpenter, et al. (1988) and
investigated the pedagogical content knowledge of elementary school teachers and how
this knowledge was related to student achievement. The researchers found that the
teachers in this study could identify the basic differences between types of addition and
subtraction problems but that they struggled with identifying differences between
difficult problems and in articulating the reasoning for the distinctions they made
between the problems; teachers did not seem to have a logical framework for classifying
problems. The researchers concluded that
teachers do not traditionally make instructional decisions based on the strategies
that children use to solve different problems, whereas they do make decisions
about whether to include particular problems based on their assessment of
154

whether the problems would be too difficult for their students… teachers
traditionally have not had a sufficiently rich knowledge base to plan for
instruction based on a careful assessment of the processes that students use to
solve problems. (Carpenter, et al., 1988, p. 399)
The second baseline study by Peterson, Fennema, et al. (1989) explored
relationships among the pedagogical content beliefs and pedagogical content knowledge
of teachers and students’ achievement in mathematics. The study found that there were
―significant positive relationships among teachers’ beliefs, teachers’ knowledge, and
students’ problem-solving achievement‖ (Peterson, Fennema, et al., 1989, p. 1). The
pedagogical content beliefs about mathematics, curriculum, and instruction varied widely
among teachers in this study. Students who were in the classrooms of teachers who were
considered to use cognitively based strategies were found to have high word problem
solving abilities and mastered computational skills and number facts while they
developed their skills in problem solving.
The third baseline study by Peterson, Carpenter, and Fennema (1989) examined
―the relationship of teachers’ knowledge to teachers’ instruction and to students’
mathematics problem solving‖ (p. 559). This study found significant positive
relationships between the knowledge that teachers had of their students and the extent to
which teachers listened to students’ problem solving solutions and questioned their
problem solving processes. The conclusions drawn here support past research (Peterson,
Fennema, et al., 1989) that ―teachers’ pedagogical content knowledge and beliefs about
155

student knowledge influence teachers’ classroom practice, which in turn influences their
students’ learning and achievement‖ (Peterson, Carpenter, & Fennema, 1989, p. 568).
The culminating study by Carpenter, et al. (1989) is described in further detail in
Chapter 3.
The final study (Fennema, Franke, et al., 1993) utilized the same sample of
teachers and students as the previous studies. Fennema, Franke, et al. (1993) wrote a case
study of one of the teachers who had been in the treatment (CGI) group. This study found
that children in a classroom where the teacher used knowledge of her students’
mathematical thinking (the basis for CGI) were able to perform at high levels of
mathematics ability and were able to exceed the standards recommended by the National
Council of Teachers of Mathematics. The teacher’s curriculum, expectations of students,
classroom structure, and assessment strategies were described in this study and indicate
the importance of using research-based knowledge in the classroom (Fennema, Franke, et
al., 1993).

156

Appendix B
Principal Interview Protocol

CGI Implementation Study for Green Valley Unified School District
Principal Interview Protocol
Britt Dowdy & Kate Garfinkel
University of Southern California

Introduction
We are doctoral students from the University of Southern California. We are
researching the implementation of CGI throughout the school district. We would like to
learn about the specifics of CGI at your school, and would like some help in selecting
teachers to assist with additional data needs.

We would like to record this interview, may we do so?

Questions
1. Please give us an overview of how teachers use CGI within the school.
a. What did you/the district hope to accomplish by adopting CGI?
2. 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
3. How have each of these changed over the past five years?
4. Thinking of the program priorities for teachers at your school, how does CGI
compare to other initiatives? Has this changed over time, why?
5. How were you, as a principal, trained in the use of CGI at your school?
6. What are your expectations for the use of CGI throughout the school? For
individual teachers in their lessons?
7. To what extent do teachers support CGI activities within their lessons?
8. What evidence do you use in judging whether CGI is in use?
9. To what extent are teachers accountable for using CGI in their lessons?
10. We would like to observe two teachers, preferably third grade. We would like to
see the extremes of CGI implementation. Whom would you recommend? Why?
11. How would you like us to proceed with establishing observation times?
157

12. We also want to survey all K-3 teachers regarding CGI. How would you like us to
proceed?
13. As we have future questions arise, how should we best contact you?

Thank you for your time.

158

Appendix C
Teacher Observation Rubric

Level of CGI Instruction
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

159

1 2 3 4-A 4-B
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
Students reported
on their thinking
and participated in
discussions about
math
- Spend most of
class reporting
solutions to a
variety of
problems
- 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
Extensively
reported on their
thinking and
engaged in
discussions about
math
- Children
expected to
communicate
their solution
strategies
(orally or in
writing) and
listen to other
students’
strategies
when shared
orally
- Talk a great
deal about
math both to
peers and the
teacher

160

1 2 3 4-A 4-B
Teachers’
Elicitation and
Understanding
of Children’s
Thinking
Little, if at
all
- Not
Evident
- Follows
textbook
- Direct
instructio
n 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
misundersto
od
- 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
Elicits and attends to
children’s thinking
- Teachers probe with
questioning until student
thinking is clearly

Teachers’ use
of Children’s
Thinking as a
Basis for
Making
Instructional
Decisions
Instructional decisions
driven by teachers’
knowledge of children’s
thinking
- Problems are selected
based on impact on
students
- Student responses to
problems are anticipated
in advance

Driven by
general
knowledge
of children’s
thinking but
not by
individual’s
- Basis of
instructio
n is on
groups of
students’
thinking

Driven by
teacher’s
knowledge
of
individual
children in
the
classroom
- Basis of
instructi
on is on
each
individu
al
student
- More
detailed
knowled
ge of
individu
al
students

Derived from Fennema, et al., 1996, p. 412

161

Appendix D
Teacher Interview Protocol

CGI Implementation Study for Green Valley Unified School District
Teacher Interview Protocol
Britt Dowdy & Kate Garfinkel
University of Southern California

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?
3) During class, how did you choose which students to directly interact with as they
worked on the problem?
4) Did you alter the lesson at any point to address a student need that surprised you?
a. Did you deviate at all from the lesson you had planned? Why?
5) What did the students learn from the lesson and how do you know that?
6) What parts of the lesson do you think worked well? Why?
7) 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?
8) What would you do differently if you were to teach this lesson again? Why?
9) How often do you use CGI?
a. How is CGI instruction different than your instruction on non-CGI days?
10) How were you first introduced to CGI?
11) How dedicated would you say your school is to CGI?
12) Are you held accountable for using CGI in your classroom?
13) Do you think CGI has helped your students? In what ways?
14) What do you think your role is as a math instructor?

Note: Some questions from this interview protocol were taken from sample teacher interview questions in a study by
Fennema, et al., 1996.
162

Appendix E
Teacher Questionnaire

CGI Implementation Study for Green Valley Unified School District
Teacher Questionnaire
Britt Dowdy & Kate Garfinkel
University of Southern California

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?
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?)
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?
163

12) Is there anything else you would like to share with us about teaching CGI or your
professional development experiences?

Thank you very much for your time!
Please contact us if you have any questions or concerns.

Britt Dowdy Kate Garfinkel
bdowdy@nmusd.us garfinke@usc.edu

164

Appendix F
CGI Lesson Main Problems

The problems in this table are the main problems that each teacher gave to her students
during each of two observations. Students were asked to solve for one of the number
choices using two strategies and record one of the strategies on his/her paper.

BRES Observation #1 Observation #2
Ms. P

(59, 34) (467, 328) (586, 674)

Ms. P and Ms. Q were carving pumpkins
together this weekend. Ms. P counted
____ seeds in her pumpkin. Ms. Q counted
____ seeds in the pumpkin she carved.
How many pumpkin seeds do Ms. P and
Ms. Q have altogether?

(98, 45) (93, 46) (231, 152)

Jaylen went rock collecting during recess
and found ___ rocks. He gave ___ rocks
to Josh B. How many rocks does Jaylen
have left?

Ms. Q

(13, 40) (138, 261)
((5x7)+(5x23), (4x75)+(9x3))

Mr. Mr. Danny picked up ____ pieces of
trash before recess. After recess, he picked
up some more. He picked up _____ pieces
of trash in all. How many pieces of trash
did Mr. Danny pick up after recess?

(58, 29) (136, 298)
[(10x100) + (5 x 110) + (4 x 4),
1000 + (3 x 333)]

Christopher had some silly bands. He
gave ____ to Fred. Now he has ____
silly bands left. How many Silly Bands
did Christopher start with?

Ms. R

(92, 56) (874, 427) (2x437, 4x129)

There were ___ wild turkeys eating in a
field. Then, __ turkeys flew away into the
woods. How many turkeys were left eating
in the field?

(956, 328) (2x376, 4x51)

Ms. R went to Big Bear during winter
break. On Monday, she made ___
snowmen? On Tuesday, ___ snowmen
melted. How many snowmen were left?

165

GVES Observation #1 Observation #2
Ms. A

(47, 85) (89, 140) (278, 323)

Victoria has already read ___ pages in
her AR book. How many more pages
does she need to read if the book has
___ pages?

(2 weeks) (5 weeks) (8 weeks)

Jordan gets 4 quarters, 3 dimes, 4 nickels,
and 4 pennies in one week for her
allowance. How much money will Jordan
have in ____ weeks?

Ms. B

(54, 97) (175, 25) (653, 498)

(2,346; 2,654)

Nicole had ___ Silly Bands that her
mom bought her at the store. Madison
had ___ Silly Bands that she got from
trading with her friends. How many
Silly Bands did Nicole and Madison
have together?

Use the pictograph below to solve the
math problems.

Michael

Emily

Joseph

Jennifer

Kevin

= 6 fish

1. How many fish did the boys catch
altogether?
2. How many fish did Emily and Jennifer
catch altogether?
3. How many more fish did Emily catch
than Joseph?
4. How many more fish did the girls catch
than the boys?
5. Who caught fewer fish, Emily or
Jennifer?

Abstract (if available)

Abstract This study investigated the ways in which the elementary mathematics teaching philosophy, Cognitively Guided Instruction (CGI), was reflected in the teaching practice of a sample of teachers who subscribe to the philosophy. Cognitively Guided Instruction relies on teachers basing their instructional decisions on the knowledge of their students’ mathematical thinking. The research questions for this study were (1) to what extent do teachers’ instructional practices reflect the critical features of CGI? and (2) what aspects of CGI professional development did the teachers perceive to be the most effective in supporting the integration of CGI philosophy into their teaching practice? Five second grade teachers within two schools in the same district participated in this study. Data were collected through principal interviews and teacher observations, interviews, and questionnaires. Evidence was found that the critical features of CGI were present in the teaching practices of the teacher participants but the critical features were observed to varying degrees. The majority of the teacher participants reported that CGI professional development that provided immediate feedback that was directly connected to current classroom practice (e.g., CGI coaches) was the most supportive of their CGI enactment.

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

Cognitively guided instruction: an mplementation case study of a high performing school district

PDF

Bridging theory and practice: developing children’s mathematical thinking through cognitively guided instruction

PDF

An examination of teacher-centered Explicit Direct Instruction and student-centered Cognitively Guided Instruction in the context of Common Core State Standards Mathematics…

PDF

Resource use and instructional improvement strategies at the school-site level: case studies from ten southern California elementary schools

PDF

Critical reflection among school psychologists: an examination of content, cognitive style, and cognitive complexity

PDF

Districtwide instructional improvement: a case study of a high school in the Los Coyotes High School District

PDF

Authentic professional learning: a case study

PDF

Quality literacy instruction in juvenile court schools: an evaluation study

PDF

Resource allocation in successful schools: case studies of California elementary schools

PDF

A story of achievement in areas where others fail: a case study of secondary school reform in mathematics at Pacific North High School

PDF

Developing mathematical reasoning: a comparative study using student and teacher-centered pedagogies

PDF

Allocation of educational resources to improve student learning: case studies of California schools

PDF

Moving from great to greater: Math growth in high achieving elementary schools - A gap analysis

PDF

Self-reflective practices and procedures to systematically examine reading comprehension instruction

PDF

Instructional coaching, educational technology, and teacher self-efficacy: a case study of instructional coaching programs in a California public K-12 school district

PDF

Transformational technology in K-12 schools: an elementary case study

PDF

"Thinking in the middle" what instructional approaches are employed by urban middle school teachers to effect changes in African American students' learning outcomes and performances?

PDF

School-wide implementation of the elements of effective classroom instruction: lessons from high-performing high-poverty urban schools

PDF

What is the relationship between self-efficacy of community college mathematics faculty and effective instructional practice?

PDF

High school reform to improve mathematics achievement

Asset Metadata

Creator Garfinkel, Kate (author)

Core Title Cognitively guided instruction: A case study in two elementary schools

School Rossier School of Education

Degree Doctor of Education

Degree Program Education

Publication Date 04/06/2011

Defense Date 03/23/2011

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

Tag CGI,Cognitively Guided Instruction,elementary mathematics,mathematical thinking,mathematics instruction,OAI-PMH Harvest,professional development,teaching philosophy

Place Name California (states), school districts: Green Valley Unified School District (geographic subject), USA (countries)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Picus, Lawrence O. (committee chair), Franklin, Gregory A. (committee member), Slayton, Julie (committee member)

Creator Email garfinke@usc.edu,kategarfinkel@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m3716

Unique identifier UC1100261

Identifier etd-Garfinkel-4365 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-446766 (legacy record id),usctheses-m3716 (legacy record id)

Legacy Identifier etd-Garfinkel-4365.pdf

Dmrecord 446766

Document Type Dissertation

Rights Garfinkel, Kate

Type texts

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

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu