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Boston College
Lynch Graduate School of Education
Department of Teacher Education, Special Education, and Curriculum and Instruction
Curriculum and Instruction
An Investigation of Successful Mathematics Teachers Serving Students from Traditionally Underserved Demographic Groups
Dissertation
by
MICHAEL C. EGAN
submitted in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
May 2008
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© Copyright by MICHAEL C. EGAN
2008
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Signature Page
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An Investigation of Successful Mathematics Teachers Serving Students from
Traditionally Underserved Demographic Groups
By Michael C. Egan
Lillie Richardson Albert, Ph.D., Chair
Abstract
The publication of Curriculum and Evaluations Standards (NCTM, 1989)
ushered in a new era in the mathematics education community in which excellence would
be viewed as the expectation for all students rather than the domain of a privileged few.
Nineteen years later, a substantive achievement gap along lines of race and social class
persists. During this time period, researchers concerned with equity in mathematics have
focused considerable attention on seeking ways to better reach historically underserved
students in the classroom. The bulk of the related research literature has centered on
reform-oriented curricula as a potential means of closing the achievement gap. Teachers
and their pedagogy have received less attention in the literature, though scholars are
increasingly recognizing the need to investigate the role of instruction in the struggle for
equity. The present study contributes to this emerging body of research.
The goal of this study is to begin to uncover and describe promising instructional
approaches for reaching historically underserved students in the mathematics classroom.
This research rests on the assumption that practicing teachers with a sustained record of
success teaching mathematics to poor students and students of color are a valuable source
of knowledge about effective instruction. Seven middle and high school teachers drawn
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from two urban school districts participated in this study. These teachers were identified
as “successful” by virtue of nomination from their supervisors, their tenure of at least five
years, and other factors. The teachers were observed and interviewed over the course of
an academic year. Frameworks modeling the teachers’ underlying attitudes and
pedagogical styles are proposed. Fundamental findings indicate that these successful
teachers view their work as a vocation rather than an occupation, and that the teachers
value their students’ existing knowledge and seek to connect new mathematical concepts
to students’ ideas. These findings resonate with scholarship pertaining to culturally
responsive pedagogy and contribute further to this developing theory by illustrating how
culturally responsive instruction plays out in a broad range of mathematics classroom
settings.
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DEDICATION
This study is dedicated foremost to Coleen who left her sunny homeland in order to carry
me through this process. I love you and thank you.
It is also dedicated to Dana and Christen who provided motivation, laughter, and joy along the way.
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Acknowledgement
I wish to convey gratitude and respect to the teachers of “Adamstown” and
“Milltown” who permitted me to enter their classrooms and learn from them. The
generous amount of time you provided was greatly appreciated. The lessons you taught
me are appreciated even more...you have all made me a better teacher.
I am also grateful to the school and district administrators of the two districts who
put me in contact with these wonderful teachers. I am particularly indebted to the
mathematics supervisors in “Milltown” who provided additional support and follow-up
for me as I maneuvered my way through their district and schools.
Dr. Michael Schiro and Dr. Margaret “Peg” Kenney went above and beyond as
members of my dissertation committee. Each of you provided valuable feedback and
support along the way. Most doctoral students are lucky to receive helpful guidance from
one dissertation advisor: I received it from three.
The guidance I received from Dr. Lillie Albert was more than simply “helpful.”
She walked with me throughout my five years at Boston College, helping me navigate its
strange bureaucracy, teaching me to be a better scholar and teacher, locating meaningful
(and financially much needed) work opportunities for me, and providing a model for how
one can maintain one’s values while surviving and thriving in one’s career. Despite the
enormous pressures on her own time, she spent hours each week providing personal
counsel to me and others. She was literally a godsend, and she deserves a substantial
amount of credit for this particular accomplishment as well as whatever other
accomplishments I may earn in my academic career.
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TABLE OF CONTENTS
Title ......................................................................................................................... i
Copyright ................................................................................................................ ii
Signature ................................................................................................................ iii
Abstract .................................................................................................................. iv
Dedication .............................................................................................................. vi
Acknowledgement ................................................................................................ vii
CHAPTER I: INTRODUCTION.............................................................................1
Purpose of the Study, Research Questions, and Definitions of Terms ........3
Importance of the Study ..............................................................................7
Guiding Assumptions ................................................................................10
The Researcher ..........................................................................................15
Overview of the Chapters .........................................................................17
CHAPTER II: REVIEW OF THE LITERATURE ...............................................18
The Role of Teachers and Pedagogy in Student Success .........................19
What Makes for Good Instruction? Perspectives from
Quasi-Experimental Research....................................................................21
The Developing Theory of Culturally Relevant Instruction .....................26
Culturally Relevant Pedagogy in Practice: Findings from Empirical
Studies .......................................................................................................31
Discussion and Implications for the Present Study....................................37
CHAPTER III: METHODOLOGY AND PROCEDURES ..................................39
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Design of the Study ...................................................................................40
Qualitative Research: Addressing the Question “What is
Happening Here?”..........................................................................40
Drawing on the Research Traditions of Ethnography and
Grounded Theory ...........................................................................43
Access and Entry .......................................................................................49
Setting and Participants .............................................................................52
Data Collection .........................................................................................57
Observations .................................................................................59
Interviews.......................................................................................61
Archival Data ................................................................................62
Data Analysis ............................................................................................63
Limitations of the Study ............................................................................66
CHAPTER IV: ATTITUDES AND MOTIVATIONS..........................................68
Portraits of High School Teachers ............................................................70
...selfishly, I would prefer to teach in the city ...............................70
I feel like, in some way, that my job has meaning .........................75
...if you want to see a revival, then this is where it starts,
right here in the schools ................................................................79
Portraits of Middle School Teachers..........................................................84
I feel more compelled to the urban setting.....................................84
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What I’m always pushing for is for all of them to be fully
competent and excelling.................................................................89
It’s hard work, it’s exhausting, but I can’t picture myself
teaching anywhere else ..................................................................93
You have to love them ....................................................................97
Separate Stories, Unifying Themes: The Educational Outlook of
Successful Teachers .................................................................................101
Conclusion ..............................................................................................109
CHAPTER V: PEDAGOGICAL APPROACH ..................................................113
Faith and Communication: A Framework for the Pedagogy of
Effective Teachers ..................................................................................114
Strong Student Ability Assumed ............................................................120
High Expectations........................................................................120
Classroom Management Focused on Learning............................123
Presenting Challenging Mathematical Content ...........................127
Student Participation....................................................................129
Summary ......................................................................................133
Focus on What Students Know................................................................135
Valuing and Connecting to Student Knowledge..........................136
Capitalizing on Students’ Experiential Knowledge.....................143
Summary ......................................................................................146
Emphasis on Mathematical Vocabulary ..................................................148
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Making Mathematical Vocabulary a Routine ..............................150
Appreciating and Using Precise Mathematical Language ...........156
Summary ......................................................................................160
Safe Environment for Meaningful Communication ................................160
Sharing Ideas in a Comfortable Learning Environment ..............161
Valuing All Contributions............................................................164
Modeling Effective Communication............................................169
Summary ......................................................................................173
Concluding Discussion ...........................................................................173
CHAPTER VI: SUMMARY, CONCLUSIONS, AND IMPLICATIONS .................................................................................................180
Summary of the Study .............................................................................180
Importance of the Study...........................................................................182
Discussion of Findings.............................................................................183
Attitudinal and Motivational Factors ...........................................184
Pedagogical Style.........................................................................186
Building on Culturally Responsive Instruction: Focusing on
Content in Diverse Settings .........................................................191
Conclusions and Implications ..................................................................193
Limitations of the Study...........................................................................198
Recommendations for Future Research ...................................................199
Closing Comments...................................................................................201
REFERENCES ....................................................................................................202
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APPENDICES .....................................................................................................208
A. Informed Consent Form.....................................................................208
B. Interview Protocols ............................................................................212
C. Ms. Thompson’s “Derivative Song”..................................................216
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LIST OF FIGURES Figure Page 5.1 A Model for the Practices of Effective Teachers ....................................119 5.2 Ms. Etienne Word Bank...........................................................................155 5.3 Ms. Kelly Word Bank ..............................................................................155 5.4 Ms. Frederick Word Bank........................................................................155 5.5 Ms. Zimmerman Word Bank ...................................................................155 5.6 Ms. Etienne’s Communication Guidelines ..............................................171
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LIST OF TABLES Table Page 3.1 Information About Participants..................................................................54 4.1 Summary of Findings...............................................................................111
1
CHAPTER 1
INTRODUCTION
The publication of Curriculum and Evaluation Standards by the National Council
of Teachers of Mathematics (NCTM, 1989) formally ushered in a new era in mathematics
education in which excellence would be viewed as the expectation for all students rather
than the domain of a talented few. Recognizing the historical dearth of mathematical
opportunity for “women and most minorities,” the 1989 Standards document argued that
“past schooling practices can no longer be tolerated” and that equity “has become an
economic necessity” (p. 4). The subsequent Principals and Standards for School
Mathematics (NCTM, 2000) further emphasized the need for equity in mathematics
education, listing equity as the first of six principles for school mathematics.
Despite this clarion call of excellence for all in mathematics, it is clear that not all
students are served equally. Data gleaned from the administration of the National
Assessment of Educational Progress (NAEP) in 2007 reveals that traditionally
underserved students of color are persistently found on the short end of the achievement
gap. The proportion of White, Black, and Hispanic 4th grade students achieving at least
the basic level of proficiency on this exam was 91%, 64%, and 70% respectively. The
corresponding percentages for 8th grade students were 82%, 47%, and 55%.
Achievement differentials between students of lower socioeconomic status (SES) and
higher-SES backgrounds are also evident. Using eligibility for free or reduced price
school lunch as an indicator of socioeconomic status, 91% of higher-SES 4th graders
earned basic proficiency on the NAEP as compared to 70% of the lower-SES students.
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For 8th graders, high-SES students earned an 81% proficiency rate versus 55% for low-
SES pupils (National Center for Education Statistics, 2008). Tate (1997) found similar
disparities in other indicators of mathematical achievement, including college entrance
test scores and course enrollment patterns.
If we accept the NCTM’s premise that all students are capable of excellence in
mathematics, then the differential achievement patterns exhibited by various student sub-
groups reflect a failure on the part of the educational community, as well the larger
society, to adequately serve all of its charges. A multitude of factors contributes to
uneven achievement in our schools, and many of these factors are beyond the control of
educators (e.g., the unbalanced distribution of material resources across schools, the
damaging impact of poverty and crime in many communities, etc.). Though schools are
unable to directly address many impediments to equitable achievement, this does not
absolve educators of the responsibility of more adequately meeting the needs of all
students. This paper addresses how educators, classroom teachers in particular, can
improve their work with diverse students.
A true commitment to equity requires that mathematics educators place the needs
of underserved students at the forefront of the reform agenda (Stanic, 1989). The data
presented above point to the urgent need for researchers concerned with equity in
mathematics to fix their gaze on the specific needs of underserved ethnic and
socioeconomic groups. Unfortunately, this issue has received inadequate attention in the
research literature. The bulk of the existing scholarship which does pertain to SES and
race/ethnicity focuses on reform-oriented curricula as a potential means of closing the
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mathematical achievement gap. While attempts to create a more inclusive and process-
oriented curriculum are laudable and worthy of our attention, curricular materials are not
a panacea for eradicating mathematical inequality. Indeed, some evidence suggests that
reform materials of recent years may unintentionally exacerbate the achievement gap
(Lubienski, 2000). As we grapple with the issue of how to better meet the needs of
disadvantaged students, we must complement our efforts to optimize what is taught, e.g.,
the curriculum, with consideration of how the material can be effectively presented, e.g.,
the instructional practices of teachers. Established mathematics teachers with a
consistent record of success with traditionally underserved students are a promising
source of insight into the question of effective pedagogy for this population. Researchers
focusing on equity-related issues in mathematics education are beginning to recognize the
value of investigating the practices of highly effective teachers of traditionally
underserved students, and are calling for this gap in the research literature to be filled
(Boaler, 2002; Gutiérrez, 2002; Lubienski, 2002).
Purpose of the Study, Research Questions and Definition of Terms
Responding to this call for a new direction in equity research, the purpose of this
study is to begin to uncover and describe promising instructional approaches for reaching
traditionally underserved students in the mathematics classroom. Operating under the
assumption that successful practicing teachers are a valuable source of insight into the
question of effective instructional techniques, seven successful urban middle and high
school teachers were chosen to participate in this study. They were observed and
interviewed over the course of an academic year, leading to this qualitative description of
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the teachers and their work. The entire investigation was designed to address the
overriding research question, “What are the characteristics of successful1 mathematics
teachers who work primarily with traditionally underserved student groups?”
The primary research question includes several terms requiring further
clarification. These terms are italicized above and elaborated here. The characteristics
of the teachers were the primary focus of the investigation. The characteristics of
successful teachers align with the purpose of the study, which involves the identification
of promising instructional approaches for traditionally underserved student groups. The
characteristics this study sought to identify were characteristics which contribute to the
teachers’ effectiveness and may potentially be utilized by other teachers. Idiosyncratic
personality traits, such as disposition, charisma, energy, etc., were not considered as such
traits are unlikely to be adopted by other teachers. Characteristics which were
investigated included matters such as teaching style, attitudes, and overall approaches to
the work of teaching. These latter characteristics hold more potential as models others
might reflect on and initiate in their own practice. Identification of such teacher
characteristics comprised the major objective of the research. This search for useful
characteristics was focused by identifying and defining them in relation to an additional
set of research questions. These sub-questions include:
1. What are the pedagogical styles of the teachers?
2. What is the nature of their interactions with their students?
3. What are their attitudes toward their students and their work?
1 The terms “successful” and “effective” will be used interchangeably throughout this text.
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4. What motivates them to teach mathematics in general and to teach this population
of students in particular?
Each of these sub-questions is directly related to the overriding research question, hence
each helped to uncover the characteristics of successful mathematics teachers who work
primarily with traditionally underserved student groups.
The participating teachers were identified as successful based on two primary
criteria: 1) their extended commitment to working in urban schools (each teacher had
served at least five years in an urban classroom at the time of the study) and 2) via
nomination from a supervisor or other authority with direct knowledge of the teachers’
work (nominating authorities included a district mathematics supervisor, a vice principal,
and a university-based mathematics educator). Both criteria were deemed necessary for
considering the teachers “successful.” The tenure criterion ensured that the teachers had
developed reputations for effectively teaching mathematics to diverse students over an
extended period of time. Nominating supervisors were individuals most familiar with the
professional reputations of the teachers. As these authority figures were charged with the
responsibility of ensuring a quality mathematics education for a broad number of
students, the nominators were keenly aware of which teachers under their supervision
were best at meeting the needs of the students. Other criteria, including student
standardized test scores, were also considered in evaluating the “success” of the teachers.
It was difficult to establish a common indicator of “success” beyond the two primary
criteria of tenure and nomination, however. More commentary on this difficulty as well
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as further details about the process of selecting successful mathematics teachers of
traditionally underserved student groups will be presented in Chapter 3.
The participating teachers were identified as working primarily with traditionally
underserved student groups because they work in schools in which the majority of
students belong to those groups which are consistently listed on the short end of the
mathematics achievement gap: low-SES, African American, and Latino/a students. It is
well documented that these students are disproportionately concentrated in urban areas,
and it is indeed the case that the teachers in this study work in urban schools. While the
singular descriptor “urban” is certainly less cumbersome than the chosen “traditionally
underserved student groups,” the latter phrase has been purposefully employed for
several reasons. Firstly, it is the author’s firm conviction that the term “urban” has
become an overused and misallocated euphemism for underachieving, under-resourced,
dilapidated, and/or practically hopeless school settings. The negative connotation
attached to the word is not a fair representation of city-based schools. Some urban
schools are among the finest educational institutions in the country, many more are
decidedly average, and, yes, some are in desperate need of additional funding and
substantial improvement. Such uneven distribution of quality is also characteristic of
rural and suburban schools. Though it may be argued that the problem of under-
resourced schooling is particularly acute in urban areas, the term “urban” is too often
mistaken for “poor quality,” and is therefore avoided as a key descriptor in this study.
Additionally, the phrase “traditionally underserved” is favored over other oft-used terms
such as “low achieving” and “underperforming.” These latter terms suggest that the
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relative underachievement of demographic groups is in many ways the responsibility of
the children themselves. They subtly imply that the students lack sufficient work ethic,
proper attitude, or are in some other way deficient relative to the demands of the
academic curriculum. A key assumption underlying this study is a position quite contrary
to this. The term “underserved” is preferred in that it shifts responsibility to the
education system more broadly. That is, the relative underachievement of certain groups
is more closely related to our collective failure to adequately serve the needs of these
learners.
Importance of the Study
This study highlights promising mathematics instructional practices, a topic
requiring much more attention in the field. The current reform movement has produced a
great deal of insight, and no small amount of debate, about the proper form and function
of mathematics curricula for our schools. However, “Research is less definitive on what
makes for good math instruction...particularly for lower-achieving students” (Viadero,
2005). The standards movement of the last two decades is based on curricular standards
and high-stakes assessment. Consideration of effective pedagogy is strangely missing
from this mix. Stiegler & Hiebert’s (1999) analysis of videotaped classroom lessons
from the Third International Mathematics and Science Study led them to conclude that
instructional style is perhaps the key factor contributing to the comparative success of
Japanese and German mathematics students relative to their American peers. “In our
view,” they offer, “teaching is the next frontier in the continuing struggle to improve
schools. Standards set the course, and assessments provide benchmarks, but it is teaching
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that must be improved to push us along the path to success” (p. 2). This study provides a
window into what successful mathematics teaching looks like in a sample of urban
American classrooms.
The fact that the schools utilized in this study are urban schools serving
predominantly underserved demographic groups makes the findings of this study all the
more compelling. This study showcases success in schools where a casual observer
might not expect to find it. It provides evidence supporting the view that all students,
regardless of perceived disadvantages, can be expected to perform to high mathematical
standards, and it underscores the important role of the teacher in helping students meet
their potential. While this study provides preliminary evidence supporting these
important themes, the small sample of teachers limits the generalizability of the findings.
Despite this inherent weakness, this initial investigation provides promising referents
upon which to build future research. Many of the preliminary themes generated here
hold potential as quantifiable variables in future research. With continued investigation,
more robust knowledge about the characteristics of effective teaching may spring from
this study.
While insight regarding effective mathematics instruction in general may
eventually be gleaned from the findings of this report, the primary concern of this
investigation is the issue of effective pedagogy for traditionally underserved student
groups. As has been noted previously, the term “underserved” is apt in that it suggests
that the achievement gap associated with student demographics is attributable to
differential access to educational opportunity. Now more than ever, access to quality
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education, particularly in technical areas such as mathematics and technology, has
become a necessity of economic survival (Friedman, 2005). As such, equity in
mathematics education is nothing less than an issue of civil rights (Moses & Cobb, 2001).
While this is justification enough to investigate effective teaching practices for poor
students and students of color, the issue of meeting the educational needs of these
students will soon be more than a problem affecting “minority” populations.
Demographic trends indicate that children of color will comprise the majority of school-
aged children by the year 2035 (Villegas & Lucas, 2002). The performance of these
students will soon be the driving force behind America’s educational and economic
competitiveness.
Significant improvements in education can never be brought about by the results
of a single study. However, increased attention in the literature devoted to a particular
problem over time can have a positive impact. Since the 1970s, the bulk of equity-related
research appearing in academic journals has focused on issues of gender in the
mathematics classroom (Lubienski & Bowen, 2000). This literature has raised awareness
about gender issues in mathematics teaching and learning, influencing policy and
curricular development. Such heightened awareness among researchers and the
educators they influence surely contributed to our present state in which the gender gap in
school mathematics is nominal (National Center for Education Statistics, 2008; Tate,
1997). Socioeconomic status and race are now the most urgent foci for researchers
concerned with equity in school mathematics. This study represents one contribution to a
much-needed body of literature, a body of literature with potential to foster real change.
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A final contribution of this study lies in the fact that it highlights instances of
promise among underserved students and the teachers who serve them. Morris (2004)
points out that the schooling of these students is often depicted in a bleak, almost
hopeless fashion (e.g., Anyon, 1997; Kozol, 1991). While such portrayals may serve to
heighten awareness of current social injustices, they may also unintentionally bolster the
myth that historically oppressed persons are incapable of self-advancement. It is
therefore imperative that we bring increased attention to cases where success has been
found against the odds (Ladson-Billings, 1997; Morris, 2004). In focusing on the specific
realm of mathematical success, this paper provides evidential support to the conviction
held by the NCTM and conscientious teachers everywhere that all students are capable of
excellence in mathematics.
Guiding Assumptions
This belief that all students should be expected to excel underlies the decision,
mentioned earlier, to refer to African American, Latino/a, and low-SES students as
“underserved” rather than “underachieving.” While the rhetoric of universal standards
and high expectations is widespread, aggregated data suggest that this idea is not fully
employed in action. Oakes (2005) has spent decades chronicling the fact that these
underserved students are far more likely than their more privileged peers to be placed in
non-academic tracks or to be diagnosed with special needs. Low-tracked students receive
a clear message from educational institutions that they are not considered capable of
achieving, a tragic self-fulfilling prophecy (Rist, 1970). This denial of opportunity, and
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de facto vote of no confidence, is a significant contributor to the widespread achievement
disparities in mathematics and other academic areas.
Holding the learner to high expectations, then, is a key component of unleashing
student potential. Communicating expectations via institutional policy is one important
means of doing this, but the day-to-day interaction between teacher and student is an
even more potent means of communicating expectations. As such, another central
assumption of this research is that the teacher holds a primary role in fostering student
success. It was assumed that effective teachers would demonstrate, in word and deed, the
expectation that all students would perform to high standards. Though their pre- and
post-adolescent charges may inevitably frustrate them with occasional or possibly
frequent lapses, effective teachers will patiently and diligently persevere in their efforts to
push their students. Such teachers employ a variety of strategies, catered to the needs of
individual students and consistent with the personality of the teacher, in order to keep the
students on their toes. They may prompt, cajole, inspire, badger, praise, admonish,
challenge, withdraw, or utilize any number of other situation-appropriate techniques in
dealing with the students, but they will never quit; they will never cease to believe in
students individually or collectively.
Another pre-guiding assumption I had related to effective teachers of traditionally
underserved student groups is that effective teachers will find ways to connect
mathematics to students’ cultural knowledge. Effective teaching occurs, and genuine
learning results, when teachers and learners are able to build bridges from the learners’
existing bases of knowledge toward new insights and understandings. The foundational
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knowledge of the students partially consists of prior academic learning, and teachers
certainly must attempt to build off of this. An additional source of student knowledge,
one which is at least as potent as academic knowledge, is the students’ cultural
knowledge. Though the concept of culture is exceedingly complex and has been assigned
numerous meanings, Deal & Kennedy’s (1983) simplified definition captures its intended
meaning here: culture is “the way we do things around here” (p. 501). Cultural
knowledge, then, relates to knowledge drawn from one’s day-to-day environment. It
includes, among other things, modes of expression, social norms governing relationships
and interactions, social roles, moral values, and community events. Effective teachers
will be in touch with their students’ cultural knowledge, drawing on it to scaffold
learning and respecting it so as not to violate its dignity.
While the idea that effective teachers will capitalize on student culture is
presented as a guiding assumption here, this assumption is informed by a growing body
of literature related to culturally responsive (Gay, 2000) or culturally relevant (Ladson-
Billings, 1995) pedagogy in the classroom. Gay (2000) argues that effective instruction
for traditionally underserved student groups “makes academic success a non-negotiable
mandate for all students and an accessible goal…It does not pit academic success and
cultural affiliation against each other” (p. 34). Far from detracting from academic
success, cultural affiliation and the knowledge which accompanies group membership is
indeed a starting point from which to build success. More details about the concept of
culturally responsive/relevant instruction and its influence on this study will be presented
in Chapter 2.
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The terms “culturally relevant” and “culturally responsive” pedagogy have
surfaced in the educational literature in recent years, but the theoretical underpinnings of
this concept are long established. Vygotsky (1978) posited that learning occurs in a zone
of proximal development (ZPD). The ZPD represents the set of skills and competencies
that a learner can achieve with the assistance of a more experienced teacher; the learner
could not master these skills and competencies by drawing solely on his or her prior
knowledge. As the student learns, his or her repertoire of knowledge expands, and,
likewise, his or her realm of potential knowledge, or ZPD, also expands. What is
important to note here is that the ZPD is determined by the learner’s existing base of
knowledge. Instruction related solely to what the learner already knows, of course, will
fail to advance learning. Instruction which deals with ideas lying beyond the learner’s
ZPD, however, will also fail to advance learning as the student is not yet ready to connect
to such knowledge. Effective teachers, then, must be aware of students’ existing
knowledge base so that they can design learning experiences which will expand on this
knowledge just enough to foster the acquisition of new knowledge. Students’ existing
knowledge consists of both prior academic attainment (Perry, 2000) and cultural
knowledge (Ladson-Billings, 1995). Successful teachers will make academic success
“non-negotiable” via their persistently high expectations. They will make success
“accessible” by catering their instruction to the learning needs, i.e., the prior knowledge
and ZPD, of their students.
The assumed characteristics of successful teachers discussed so far are certainly
not unique to mathematics teachers. Effective mathematics teachers do possess
14
discipline-specific qualities, however (Shulman, 1987). An often reported problem
pertaining to urban mathematics education is that too many schools are unable to provide
students with subject-matter specialists. This is indeed a problem, as effective
mathematics teachers require a fairly sophisticated knowledge of the subject. A strong
knowledge of mathematics enables the teacher to approach the content from numerous
perspectives. For example, a more sophisticated mathematician will recognize algebraic,
geometric, and function-related perspectives on graph-plotting; a less sophisticated
instructor might view graph-plotting as nothing more than an exercise in computing input
and output values and mechanically plotting them. A more advanced mathematical
perspective enables the teacher to be more flexible with the content. This, in turn,
provides the teacher more options for connecting the material to student knowledge.
These guiding assumptions have been outlined in this section with the intention of
providing some transparency to the research. I entered the classrooms of the participating
teachers with these ideas in mind, and, therefore, these assumptions likely colored my
interpretations of the teachers and their work. However, the literature references listed in
this section demonstrate that these assumptions are shared by others in the educational
community. Furthermore, while these assumptions are rich in applicability, they are low
in specificity. They leave open many questions, questions which are partially answered
through this research. Such questions include: How do these teachers connect to
students’ cultural knowledge? What does it look like in practice? How do teachers
convey, and students pick up on, high expectations? In what ways do teachers draw on
15
their more sophisticated understanding of mathematics in order to connect the material to
student understanding? These and other related questions are addressed herein.
The Researcher
The assumptions I’ve formed and articulated above are derived primarily from
experience. This lived experience, coupled with my social position, likewise influenced
the way I gathered and interpreted this study’s data. Once again in the interest of
transparency, some comments on my positionality are presented below.
I am a white male, raised in a middle class family in a small town in the Ozark
Mountains of southern Missouri. I am aware of, humbled by, and grateful for the fact
that my journey through the formal education system has been paved with privileges and
advantages. My parents were first-generation college graduates, causing my siblings and
I to represent the first generation for which college attendance was assumed. Though our
small town had only one public high school, our parents saw to it that we capitalized on
the best opportunities the school had to offer. Due primarily to a combination of support
and coercion from my parents, I earned decent grades in the school’s college preparatory
classes.
On the night of my high school graduation, I was amazed at how many of my
classmates I didn’t recognize. My participation in the college prep track ensured that my
day-to-day classroom interactions were limited to a cadre of less than 30 students.
Reflecting on this separation between the academic “haves” and “have-nots” in my
school led me to conclude that only two possible interpretations existed: either the
members of my elite group were naturally more intelligent or more deserving than the
16
rest, or we had unfairly been separated from the others who were just as capable as we
were. Drawing on my grandmother’s oft-spoken admonishment, “nobody’s any better
than you, but you’re no better than anybody else,” I concluded that academic opportunity
had more to do with fortune than talent.
I went on to college where, once again, I was very fortunate to be able to attend a
well-known Catholic university. After earning a degree in mathematics I joined a two-
year Catholic volunteer program which provided teachers for urban schools in Jamaica.
My placement was at Alpha Academy, an all-girls high school in the heart of Kingston.
When I began teaching at Alpha in 1995, the school had developed a long track
record of poor performance in mathematics. Alpha’s results on the high stakes Caribbean
Examinations Council (CXC) mathematics exam had been at or below the national
average for many years. Administrators, teachers, parents, and students generally
assumed that poor mathematics performance was a given. Citing numerous hypotheses
such as socioeconomic status, female math phobia, and natural incompetence, the
unquestioned presumption was that improvement in mathematics was impossible.
Drawing inspiration from my grandmother’s maxim, I was convinced otherwise. If I
could do well in mathematics, then so could these students. I chose to remain at the
school until I could prove to others in the school community, particularly the students,
that my conviction was true.
My initial two-year commitment stretched into seven years. When I left the
school, Alpha’s CXC pass rate was listed among the top ten in the island. Alpha’s results
have continued to improve since my departure, demonstrating that the students’ success
17
did not hinge on my presence. The Alpha girls were always capable of excellence, they
just needed teachers to push them. My major contribution to the school, I believe, was
not so much my talents as a teacher, but rather the role I played in shifting expectations
for student performance.
My experience in Jamaica solidified my view that all students can reasonably be
expected to achieve in mathematics. It also convinced me that teachers have a major role
to play in seeing to it that students reach their potential. I am troubled by the fact that
mathematical achievement is so inequitably distributed among student groups, and
believe that this problem can be remedied via improved pedagogy. This motivates my
desire to inquire into effective teaching practices for underserved students.
Overview of the Chapters
This chapter has presented the research objectives of the current study as well as
arguments pertaining to the importance and timeliness of investigating successful urban
mathematics teachers’ practices. Chapter 2 includes a review of the literature related to
mathematics instruction for traditionally underserved students. Chapter 3 outlines the
research methodology and design of this study. Chapters 4 and 5 include the findings of
the study. Chapter 4 provides individualized depictions of each of the seven participating
teachers, and concludes with a commentary on attitudinal and motivational factors related
to the teachers. Chapter 5 provides a grounded theoretical model of the teachers’
pedagogical approach. Chapter 6 includes a discussion of the study’s conclusions,
implications, shortcomings, and possible future directions for this research.
18
CHAPTER 2
REVIEW OF THE LITERATURE
This chapter reviews and analyzes existing literature pertaining to effective
instruction of traditionally underserved students, with a particular focus on mathematics
instruction. It begins with a brief discussion of a growing body of research supporting
the position that teachers and their instructional practices are essential elements of student
academic success. While investigations into curriculum, assessment, and other
educational endeavors remain important, findings from this body of literature underscore
the urgent need for researchers to find and disseminate insight into effective pedagogy.
Research focusing specifically on effective mathematics instruction for
traditionally underserved students has emerged in recent decades, and this literature is
reviewed subsequently. A dichotomous set of assumptions seems to guide writers in this
area. One orientation is rooted in the psychological tradition. Research studies here
entail the administration of an instructional treatment in a quasi-experimental setting.
Significant results are reported, and authors in turn suggest that the findings are
applicable in other settings. This line of research is summarized in this review, and a
critique of its shortcomings is offered.
A second orientation found in the research on effective mathematics instruction
for underserved students is more anthropological in nature. This line of research places a
great deal of emphasis on contextual factors surrounding the educational site. The
assumption here is that effective teachers are in tune with the particular realities of their
classrooms, capitalizing on local assets and overcoming local limitations. Researchers of
19
this type typically describe and analyze the work of successful teachers, and refrain from
suggesting that a given set of practices will readily transfer to other settings. A
significant proportion of this type of literature utilizes the notion of culturally relevant
pedagogy (Gay, 2000; Ladson-Billings, 1995) as a theoretical framework.
This dissertation aligns more closely with the latter orientation. As such, the bulk
of this literature review focuses on the principles of culturally relevant pedagogy and its
implementation in the classrooms of traditionally underserved students. Following the
review and critique of quasi-experimental research, a discussion of culturally relevant
pedagogy is offered. This discussion serves to frame the remaining empirical studies
presented in this chapter, qualitative studies highlighting successful mathematics teaching
in urban settings. The lens of culturally relevant pedagogy will also be utilized in
subsequent chapters of this dissertation, informing the interpretation and analysis of data.
This chapter concludes with a summary of key insights drawn from the conceptual and
empirical literature on culturally relevant pedagogy and a discussion of its influence on
the present study.
The Role of Teachers and Pedagogy in Student Success
The tenor of major research findings pertaining to teachers’ ability to have a
positive impact on student achievement has shifted dramatically during the last forty
years. The notion that intelligence is an innate quality, measurable via the intelligence
quotient and other testing, under girded educational research for the better part of the
twentieth century. Under this viewpoint, an individual’s “natural intelligence” was
considered essentially fixed, and, it was believed, his or her academic potential could be
20
reasonably predicted (Lagemann, 2000). The influential Coleman Report of 1966 fueled
this fatalistic perspective (Coleman et al., 1966). Coleman and his colleagues argued that
a student’s family background is the most salient factor predicting his or her academic
potential, suggesting that the achievement of students could be almost predicted based on
cultural and/or economic factors alone. Deemed a “cultural deficit” approach by
subsequent scholars (Sleeter & Grant, 1994), many educators interpreted Coleman et al.’s
(1966) work as implying that the economic and cultural characteristics of poor children
and children of color explained their underachievement in school. The assumptions of
innate intelligence and cultural deficiency combined to make it appear that teachers had
very little, if any, impact on student achievement.
These assumptions continue to influence policy and practice (Oakes, 2005), but
they have been largely dismissed in the research literature. Gardner’s (1983) work on
multiple intelligences has revolutionized educators’ conceptions of the nature of
intelligence. It is now widely assumed that all individuals possess sophisticated
intelligence, though this intelligence is manifested in multiple ways. This, in turn, has
raised awareness of the complexities of teachers’ work: teachers are now called upon to
design educational experiences accommodative of diverse learning styles (Cohen et al.,
1994).
As early as the 1970’s, researchers began to dismantle Coleman et al.’s (1966)
suggestion that schooling has little power to overcome (mis)perceived deficiencies in
students’ family backgrounds in fostering achievement. Edmonds (1979) investigated
several public schools serving primarily poor students of comparable socioeconomic
21
background. Some of these schools produced high achieving students, while others
exhibited the expected low levels of achievement. In comparing the successful and
unsuccessful schools, Edmonds (1979) noted substantial differences in school practices,
beliefs, and administration. Other high performing, high poverty schools have been
found to exhibit similar characteristics as those found in Edmonds’ study (Brookover,
Beady, Flood, Schweitzer, & Wisenbaker, 1979; Taylor, 1990).
The studies cited above emphasize a combination of school organizational
elements in promoting the achievement of traditionally underserved students. Rivers &
Sanders (2002) hone in more specifically, arguing that teacher quality is the most salient
factor predicting student success. Representing a complete about-turn from the findings
of Coleman et al. (1966), Rivers & Sanders (2002) conclude that individual teachers have
a greater influence on student achievement than student ethnicity, socioeconomic status,
and previous student achievement.
What Makes for Good Instruction? Perspectives from Quasi-Experimental Research
It is now widely accepted that teachers and their instructional style make a
substantial impact on student achievement. By implication, educators of all stripes
(policymakers, administrators, curricula designers, teachers, etc.) must grapple with the
question, “How can we optimize the instruction which occurs in our classrooms?” This
question can be approached in at least two ways. One approach leans toward
standardization. That is, it is assumed that an optimal set of instructional techniques
exist, and that teachers in all classrooms should be trained to implement them. Another
approach favors diversity of instruction. Here it is assumed that optimal instruction is
22
dependent on the educational context, i.e., different classrooms and different students will
require different pedagogical approaches. The academic literature focusing on effective
mathematics instruction for underserved students includes work from both perspectives.
Findings from the former approach, consisting largely of quasi-experimental research, are
reviewed in this section.
Cardelle-Elawar (1992; 1995) proposed that metacognitive instruction is an
effective teaching approach based on research with low-SES Hispanic students in
Arizona. This method prompts the teacher to model the thinking process required in
problem-solving, engaging in “explicit discussion of not only what to learn but also how
and why” (Cardelle-Elawar, 1992). The initial study was conducted in two phases. In
the first phase, Cardelle-Elawar instructed 6th grade students using the metacognitive
method. The researcher’s presence in the classroom served two purposes: it provided an
initial experimental group of students to compare against a control, and it also provided
teachers in the school with an example of the metacognitive approach in action. In the
second phase of the study, the regular teachers instructed students under Cardelle-
Elawar’s supervision and tutelage. In both phases of the study, students receiving
metacognitive instruction performed significantly better on a standard test and on
teacher-made tests than the control group (Cardelle-Elawar, 1992). Three years later,
Cardelle-Elawar extended the study in the same school district. In the second study,
Cardelle-Elawar expanded the age range of the students to include 3rd through 8th graders.
Secondly, all teachers were trained in the metacognitive approach prior to the school year
rather than during the year as in the previous study. The second study confirmed the
23
initial findings: students of all ages in the treatment group outperformed those students
who did not receive metacognitive instruction. Following these results, Cardelle-Elawar
recommended that other teachers of low-performing students adopt the metacognitive
method (Cardelle-Elawar, 1995).
Fantuzzo, Ginsburg-Block, et al. performed numerous investigations of the
Reciprocal Peer Tutoring (RPT) technique of student organization with predominately
low-income African American children. Specifically designed for at-risk students, RPT
requires teachers to train the students in a set of peer-tutoring protocols such as
explanatory prompts, directed questions, and the use of praise. The teacher then oversees
the students as they alternate the roles of tutor and tutee, ensuring that tutors follow the
protocols. Students exposed to the RPT method consistently demonstrated superior
computational speed and accuracy relative to control-group peers (Fantuzzo, Davis, &
Ginsburg, 1995; Fantuzzo, King, & Heller, 1992; Ginsburg-Block & Fantuzzo, 1997).
The effects of RPT instruction were enhanced when used in conjunction with a student-
instigated reward structure (Fantuzzo et al., 1992) and active parental involvement
(Fantuzzo et al., 1995). Ginsburg-Block and Fantuzzo (1998) also noted a positive effect
on computational skills and word problem-solving when a controlled form of peer
collaboration was used to complement a problem-solving curriculum.
The research noted above attempts to highlight promising instructional
approaches. Other researchers have investigated the impact teachers’ use of instructional
materials can have on the achievement of traditionally underserved students in
mathematics. Bottge & Hasselbring (1993) sought to determine effective means of
24
teaching students to transfer mathematical knowledge acquired in school to new
situations. Groups of students were exposed to contextual problems in two different
instructional settings. In one setting, the teacher utilized traditional textbook word
problems. In the other, problems parallel to those found in the textbook were dramatized
in five to eight minute video clips. Students in the video group performed on a par with
traditionally instructed students in textbook-type word problems. However, students in
the video group were found to perform better when encountering contextualized
problems later in the school year and also demonstrated a superior ability to transfer their
mathematical knowledge to new situations.
Woodward et al. (1999) also compared textbook-based instruction to the use of a
video teacher. Forty-four 8th and 9th grade students labeled “at-risk” were randomly
assigned to two instructionally divergent classrooms. In the first group, called the
conceptual group, the teacher used lessons from a text emphasizing conceptual
understanding through the use of manipulative objects. For the other group, the
procedural group, students were instructed by a videotaped teacher who focused on
procedures and algorithms in working with decimals. Student performance outcomes
were tied to method of instruction: those in the conceptual group performed better at a
task involving modeling a decimal value with blocks, whereas students in the procedural
group performed better on traditional paper-and-pencil problems.
Given its quasi-experimental methodology, the research cited in this section
appropriately stops short of suggesting that a given instructional technique is optimal.
The research design dictates that an instructional treatment is administered to one group
25
of students and that the measured achievement of these students is then compared to a
control group which did not receive the treatment. Published reports do not argue that a
given instructional technique is best; rather, they argue that the presence of the treatment
is favorable to its absence. Another consequence of the research design is that
researchers assume that their findings are independent of context. Writers suggest that
their finding(s) will apply in other settings and recommend that teachers elsewhere adapt
a given instructional technique. The need to train teachers is a common theme in this
literature. The viewpoint that experts discover and disseminate knowledge about
teaching, and that teachers should in turn be trained to implement this knowledge, is
strongly conveyed.
This research may provide some useful information for educators concerned with
improving instruction for traditionally underserved students, but it is clearly
underdeveloped. My search of articles published since 1989 in peer-reviewed journals
yielded only eight quasi-experimental studies on the topic, and these articles were
produced by only four research teams. Fantuzzo, Ginsburg-Block, et al. (1992, 1995,
1997, 1998) and Cardelle-Elawar (1992, 1995) conducted their respective work in RPT
and metacognitive instruction over multiple studies, so they can be credited for
developing these instructional approaches. However, there is no cross-referencing among
the eight studies cited here. Overall, the findings of the quasi-experimental work in this
area are disparate and disconnected. This research has failed to produce a coherent and
developing body of knowledge related to effective mathematics instruction for
traditionally underserved student groups.
26
A further weakness of these studies lies in the fact that they disregard the role of
culture and cultural knowledge in the learning process. Each of the authors presents a
style of teaching which treats mathematics as a purely academic pursuit, largely detached
from students’ out-of-school lives. Such an approach runs counter to the commonly
accepted “connections” standard offered by the National Council of Teachers of
Mathematics (2000), which calls on teachers to make connections between pieces of
content within the discipline (highlighting links between geometric and algebraic content
for example) and also to connect material to other academic areas and students’ interests
and experiences. While connections to previous academic learning is likely embedded
within the recommendations of this literature (e.g., directed prompts in the Reciprocal
Peer Tutoring technique include reference to previously learned facts (Fantuzzo et al.,
1992)), this research does not offer local and/or cultural referents as a source to draw on
as teachers guide students toward understanding. This omission mistakenly overlooks a
potent source of student knowledge. A body of literature which openly values and
promotes the use of cultural knowledge as a springboard toward mathematical
achievement is reviewed in the next section.
The Developing Theory of Culturally Relevant Instruction
Calls for teachers to cater their instruction to the cultural perspectives of students,
particularly traditionally underserved students, can be found at least as far back as the
Great Depression. In 1933, Carter G. Woodson noted that dominant teaching practices
unjustly corresponded to the worldview of White America and were quite inaccessible to
African Americans. Woodson called for a dramatic shift in the educational approach to
27
African Americans, one which accounted for the unique experiences and perspectives of
this population (Woodson, 1990, as cited in Tate, 1995). Arguments of this nature, and
evidence supporting these arguments, have gained considerable momentum in the
research literature over the past 25 years. Researchers have studied and expounded upon
culturally focused instruction for numerous cultural groups, and work of this nature has
been given numerous labels, including culturally “appropriate,” “congruent,”
“responsive,” and “relevant” (Ladson-Billings, 1995). In this review, the descriptors
“culturally relevant” and “culturally responsive” will be used interchangeably in order to
describe instructional approaches which build on the unique and context-dependent
knowledge bases of students.
The education of traditionally underserved students has often been approached
from a deficit perspective. Characteristics such as racial minority status, poverty, and
relative lack of familiarity with the English language are commonly viewed as major
disadvantages for students. These students, it is argued, lack proficiency in the cultural
and linguistic requisites of school success, and schools are charged with the largely
doomed prospect of overcoming these deficiencies in the classroom. Gay (2000)
criticizes this approach as an unjust practice of “blaming the victims” (p. 44) of the
educational system’s shortcomings, a practice which can be rectified via culturally
relevant teaching. Culturally relevant instruction values the cultural and experiential
knowledge that all students bring to school. Inverting the deficit approach, culturally
relevant pedagogy views the lived experiences of traditionally underserved students as
assets to build upon rather than deficiencies to overcome (Ladson-Billings, 1994).
28
The metaphor of building bridges from students’ existing knowledge to the new
knowledge we wish them to acquire is used consistently in this literature (Gay, 2000;
Ladson-Billings, 1995). Students’ existing knowledge base is rooted both in prior school
learning and experiential learning achieved outside of school. Culturally relevant
teachers constantly seek ways of connecting academic goals to both sources of student
knowledge. While instruction is catered to the specific socio-cultural contours of a given
classroom, curricular and assessment goals remain congruent with the highest standards
of the mainstream educational system. This leads to a dilemma. Advocates of culturally
relevant pedagogy acknowledge that mainstream standards governing what is taught and
valued in schools unfairly favor more privileged students, yet they recognize that
underprivileged students must master dominant discourses if they are to advance
economically and become empowered politically (Delpit, 1986; Gutstein, 2006). As
such, traditional definitions of academic success, such as high test scores and enrollment
in advanced courses, are utilized as goals. A major challenge for the culturally relevant
educator, then, is to help ensure that diverse students value and remain connected to their
home culture even as they become adept in the discourse of the dominant culture (Gay,
2000).
Several interpersonal qualities are characteristic of culturally relevant teachers.
Ladson-Billings (1994) noted that teachers do not necessarily need to share racial,
socioeconomic or other demographic identities with their students, but they should be
active, contributing members of students’ out-of-school communities. That is, teachers
should be visible in community life outside of school, involved in various social, church,
29
and other cultural functions. This involvement enhances teachers’ ability to appreciate
and connect with students’ cultural knowledge. Additionally, Gay (2000) and Ladson-
Billings (1994) argue that teachers must have an unshakeable belief in students’ ability to
succeed. On the surface, this is an obvious point. Unfortunately, the preponderance of
deficit perspectives on underserved students’ academic abilities has caused this
fundamental precept of good teaching to be far too absent from schools serving poor
students and students of color (Ladson-Billings, 1995). As many of these students have
grown accustomed to low expectations, their transition into a culturally relevant
classroom with its high expectations can be a challenge. As such, Gay (2000) notes that
culturally relevant teachers must exercise patience and perseverance as they embark on
the burdensome path of teaching.
Numerous conceptual studies highlight the theoretical implications of culturally
relevant pedagogy for the mathematics instruction of traditionally underserved students.
Ladson-Billings (1997) focused on mathematics teaching for African American students,
suggesting that the traditional model of mathematics instruction prevalent in
contemporary classrooms favors the White middle-class values of “efficiency, consensus,
abstraction, and rationality” (p. 700). Ladson-Billings proposed an instructional model
capitalizing on the mathematical strengths of urban African American students such as
“affinity for rhythm and pattern” (p. 700).
McNair (2000) drew on the ideas of Dewey and Vygotsky in stressing the
importance of connecting mathematics to students’ everyday activities. McNair went
further in stating that such a program of instruction is even more vital in impoverished
30
urban environments than in more affluent areas. This contention is supported through
reference to a summer mathematics workshop for students that he directed. During the
workshop, McNair worked with two student groups. In one group, the students were from
a suburban area, ethnically diverse, and middle-class; in the other group students were
city-dwellers, African American, and poor. McNair noticed that students in the former
group tended to deal with contextual mathematics problems in the abstract, whereas
students in the latter group tried to connect the problems to their own lives. These
students demonstrated enthusiasm for problems that they deemed relevant and realistic,
while they avoided problems which seemed ungrounded in the “real world.”
Ladson-Billings (1997) and McNair (2000) highlighted the potential for culturally
relevant mathematics instruction to improve students’ academic achievement. Other
authors view culturally relevant mathematics pedagogy as an avenue toward political
empowerment. Stanic (1989) called for a reexamination of mathematics teaching and
curriculum, arguing that current mathematical practice is a part of the “selective
tradition” of knowledge presented in schools, which is inherently more beneficial for
children of powerful groups. Teachers must reflect on questions of whose knowledge and
whose ways to construct knowledge come to be valued in the culture of school
mathematics.
Tate (1995) gets more specific, positing that school mathematics in the United
States is Eurocentric and inappropriate for minority groups. Tate proposed an Africentric
model of teaching for African American students. Gutstein (2006), reflecting on his
work with low income Mexican-American students in Chicago, likewise offers a
31
curricular and pedagogical approach geared toward his students. Borrowing from
Ladson-Billings’ culturally relevant pedagogy (Ladson-Billings, 1995) and Freire’s
problem-posing education (Freire, 2000), Tate (1995) and Gutstein (2006) call for
teachers to center instruction on issues of injustice arising in the students’ communities.
The language of mathematics, they argue, should be presented as a powerful tool in
describing inequity and in proposing solutions to problems. The teacher’s objective is to
equip students with proficiency in mathematical discourse, an indispensable form of
knowledge required in the larger goal of preparing the student to be an active member in
a democratic society.
Culturally Relevant Pedagogy in Practice: Findings from Empirical Studies
Ladson-Billings (1995) concluded that “culturally relevant teaching must meet
three criteria: an ability to develop students academically, a willingness to nurture and
support cultural competence, and the development of a sociopolitical or critical
consciousness” (p. 483). A number of empirical research studies investigate the practices
of mathematics teachers who consciously attempt to connect mathematical material to
student culture. These teachers of traditionally underserved students, therefore, have
made some effort to meet Ladson-Billings’ (1995) second criteria of nurturing and
supporting cultural competence. The research reveals varying levels of emphasis on the
first and third criteria, however. Some writers highlight teachers whose primary goal is
developing mathematical proficiency as it is traditionally defined, while others discuss
sites where mathematical knowledge is viewed primarily as an avenue toward political
32
consciousness. These studies are reviewed in this section, beginning with the
academically-inclined work.
Carey et al. (1995) and Henderson & Landesman (1995) highlight instructional
techniques in which course content is derived primarily from student input and
experience. Carey et al. studied the use of Cognitively Guided Instruction (CGI) in a
low-income, urban Maryland context. CGI involves minimizing the use of commercial
textbooks and, instead, basing instruction on culturally relevant stories and student
backgrounds. Henderson and Landesman focused on the efforts of a group of teachers in
California working with predominantly poor students of Mexican descent. Operating
outside of traditional textbooks, these teachers based their entire program of instruction
around themes generated by the students. Once students had reached a consensus on
some general areas of interest, the teachers were faced with the daunting task of securing
materials and designing activities which would draw connections between the themes and
the district-mandated mathematics content. The teachers successfully managed to cover
the majority of this content through student-centered projects. Both studies concluded
that students were actively involved in class and demonstrated a high level of enthusiasm
for mathematics (Carey et al., 1995; Henderson & Landesman, 1995). Henderson &
Landesman also compared the achievement of thematically instructed students to a
control group via a test designed by the researchers. Thematically instructed students
performed as well as the control on measures of computational skills, but significantly
outperformed the control on measures of conceptual understanding and knowledge
transfer.
33
Fuson et al. (1997) and Khisty (1995) reveal ways that mathematics teachers have
capitalized on the structure of the Spanish language in developing a strong understanding
of mathematics among bilingual Latino students. Focusing their study on how teachers
develop concepts of place value in bilingual classrooms, Fuson et al. found that teachers
who stressed Spanish numerical words were remarkably successful in facilitating student
understanding. Spanish terms such as “cincuenta y tres,” or “five tens and three,” model
more directly the meaning of the digits in the number “53” than does the English term.
These students, hailing from poor Latino communities in Chicago, were able to formulate
algorithms for addition and subtraction independently (Fuson et al., 1997). Khisty
performed an ethnographic study in two low-SES bilingual classrooms. One teacher, a
Mexican woman, addressed the class in a dialogical nature which reflected the rhythm
and flow of conversations in the Mexican culture. The other teacher, an American,
elicited student responses in the “chorus” manner of traditional classrooms. Khisty
judged the students of the former teacher to be more enthusiastic and engaged in math
class (Khisty, 1995). Though Khisty did not set out to establish that the former teacher’s
approach boosted student achievement, other research indicates that student enthusiasm
provides fertile ground for academic achievement (Cohen, 1994).
Lubienski (2000) and Boaler (1997) each studied the effects of a student-centered,
project-based pedagogical technique in working class communities. Both studies suggest
that the constructivist approach improved student attitudes and participation in
mathematics. Boaler extended her study beyond the classroom and found that students in
a constructivist classroom were proficient in applying mathematical knowledge acquired
34
in school to numerical situations in their daily lives. While both researchers noted
improved scores on achievement tests for working class students in constructivist
classrooms, their findings differ in relation to the issue of closing the achievement gap
between middle class and working class students. Lubienski found that more affluent
students in her classroom experienced a greater boost in test scores than did the low-SES
students (Lubienski, 2000). Boaler, on the other hand, found that the students in her
study performed above the national average, and therefore well above expectations, on
Britain’s high-stakes General Certificate Examination (Boaler, 1997).
Operating under the assumption that mathematics is most effectively taught when
the curriculum is designed around student interests, Campbell (1996) and Gutiérrez
(2000) investigated urban schools which have successfully implemented such a program.
As no two classrooms are alike, a teaching approach centered on student-generated
themes could never be supported by a pre-packaged curriculum. Strong intra-school
(Gutiérrez, 2000) and inter-school (Campbell, 1996) collaboration among colleagues is a
necessary asset for teachers. As these teachers inevitably need to source materials and
instructional ideas which will accommodate their students, other like-minded colleagues
serve as one indispensible educational resource. Teachers striving to make instruction
relevant to students, then, must be prepared to shoulder the burdens of both instruction
and curricula design.
Frankenstein (1995) and Gutstein (2003; Gutstein, Lipman, Hernandez & de los
Reyes, 1997) have conducted self-studies of their own work as mathematics educators.
Each of these teacher/researchers value the practice of connecting mathematical content
35
to the lived experiences of the traditionally underprivileged students they serve. Their
work is distinguishable from the studies cited earlier in this section in that they view
culturally and experientially relevant pedagogy as an avenue toward the explicit goal of
raising the political consciousness of their students. Rather than drawing on student
language or personal interests as potential foundations for mathematical instruction, these
authors rather raise instances of injustice that students encounter as themes to be analyzed
mathematically.
Frankenstein investigated her work in an adult education program at the
University of Massachusetts-Boston (Frankenstein, 1995). Teaching a consumer
arithmetic course for predominantly working-class people, Frankenstein’s goal was to
develop a class consciousness in her students. The content of the course included tax
breaks and burdens for the wealthy and poor in America, differential access to loans,
budgetary options and constraints for the wealthiest and poorest Americans, etc. All
elements of the course were connected and presented as logical consequences of the
institutional structures of society. Examining student essays and classroom discussions,
Frankenstein concluded that she made moderate progress toward the goal of developing
consciousness of class in her students. Many students appreciated the fact that the course
had enabled them to perceive the world in a different way. Others held mixed reactions,
acknowledging the injustices of socioeconomic inequities in society while simultaneously
clinging to the belief that there are opportunities available for all Americans to work
toward economic prosperity.
36
Gutstein et al. reported on two sites in Chicago in which teachers geared their
mathematics instruction toward social justice goals (Gutstein, 2003; Gutstein, Lipman,
Hernandez, & de los Reyes, 1997). The earlier study was an ethnography of the
mathematics teachers at an elementary school serving Mexican American students
(Gutstein et al., 1997). The teachers shared a philosophy that their work was a political
activity and that their mission was to develop thinkers capable of changing society. The
teachers avoided overt discussions of political issues in class, but rather attempted to
challenge students to develop habits of mind and disciplined methods of discourse which
would later serve them in their role as politically active citizens. Student proficiency was
measured not on the production of precise answers, but rather on the ability to defend
their solutions to contextual problems. Teachers avoided judging student work as “right”
or “wrong,” but instead interrogated student work in such a way that the child would be
forced to clearly explain his or her conviction or would recognize the need for further
work in resolving the problem. Gutstein reports that this approach to teaching “fosters a
critical approach to knowledge, helps students question the authority of adult
perspectives, and promotes democratic practices in the classroom” (p. 721).
In his more recent work, Gutstein engaged in a practitioner-research project in
which he examined his own work with a group of Latino students (Gutstein, 2003).
Gutstein taught a cohort of students during both their seventh and eighth grade years. He
used an existing reform-oriented mathematics curriculum, but adapted it in such a way
that it became culturally and politically relevant to the students. Unlike the teachers in
his earlier study, Gutstein chose to explicitly explore issues such as racism and social
37
stratification from a mathematical perspective. Through analysis of student work,
surveys, and classroom observations, Gutstein concluded that his students demonstrated
slow but steady progress in their ability to connect mathematical ideas to sociopolitical
contexts in society.
Discussion and Implications for the Present Study
The empirical studies cited in the previous section made no claims that the
particular teaching practices employed in the respective research sites will readily
transfer to other settings. Instead, these studies illustrate how teachers can effectively
capitalize on the contextual features of particular classrooms in order to enhance
mathematical knowledge. Mathematics educators seeking prescriptions for “best
practices” for all classrooms will not find them in the research highlighting culturally
relevant forms of teaching. What can be found, however, are some fairly robust
principles of instruction which can, in turn, be applied appropriately in other classrooms.
These principles include the premise that poor students and students of color can succeed
in mathematics, and that teachers must believe that this is so. If teachers are to witness
success they must reach out to these students, catering instruction to the students’
particular ways of knowing. A fundamental component of student knowledge is their
cultural knowledge, their well-established understanding of “the way we do things around
here” (Deal & Kennedy, 1983, p. 501).
The bulk of the research reviewed in the previous section utilized a qualitative
case study methodology. The absence of an experimental or quasi-experimental design
in these studies implies that this research falls short of the National Research Council’s
38
(2002) “gold standard” for educational research. As mentioned earlier, however, the
quasi-experimental research which has been conducted in the area of effective
mathematics instruction for underserved students has failed to yield a coherent body of
knowledge. Two factors likely contribute to the relative weakness of this line of work:
first, too few studies have been conducted in this area, and, second, these studies neglect
to adequately account for contextual factors related to a given intervention’s level of
effectiveness. The case studies reviewed in the last section exhibited greater conceptual
clarity, with each study either explicitly or implicitly drawing on the theoretical
principles of culturally relevant pedagogy. While more development is needed in this
area of research, a valuable foundation of knowledge and insight about the role of
cultural awareness in effective mathematics instruction has been laid. This dissertation
seeks to build on this foundation.
By utilizing the lens of culturally relevant pedagogy in this current study, I
entered teachers’ classrooms with the assumption that each teacher would gear his or her
instruction toward the unique characteristics of his or her respective classroom. A
distinguishing characteristic of this current study is that I entered a broad range of
classroom sites. Investigating numerous effective teachers from diverse sites enabled me
to highlight the unique and contextually appropriate practice existing at each site, while
simultaneously locating general principles which were in play across sites. These latter
findings promise to build on and refine current understanding of culturally relevant and
academically effective instructional practices for traditionally underserved students.
39
CHAPTER 3
METHODOLOGY AND PROCEDURES
The primary research question driving this study is “What are the characteristics
of successful mathematics teachers who work primarily with traditionally underserved
student groups?” Successful teachers’ ability to connect mathematical content with
students’ cultural orientations, their pedagogical styles, inter-relationships with students,
attitudes toward the discipline of mathematics, and motivational factors for teaching are
among the characteristics this study seeks to uncover. As established in Chapter 1,
effective mathematics instruction occurs in far too few classrooms populated with
students representing traditionally underserved demographic groups. It is believed that
the wider educational community can learn important insights from practitioners who
have found success with these students. The models of instruction recorded here promise
to inform efforts to ensure that effective mathematics instruction is more widely available
to all students. This chapter describes the research process which was undertaken in the
effort to address the research question and generate conclusions.
A fundamental principle of sound research is that the research design and
methodology correspond appropriately with the research question (National Research
Council, 2002). The design and methods of the current study are described below,
coupled with explanations regarding why the chosen investigational approach is well-
suited to the question at hand. It includes the following sections: (a) design of the study,
(b) access and entry, (c) setting and participants, (d) data collection, (e) analysis, and (f) a
discussion of the study’s limitations.
40
Design of the Study
Qualitative Research: Addressing the Question “What is Happening Here?”
The literature review in Chapter 2 revealed a thin knowledge base pertaining to
effective mathematics instructional approaches for traditionally underserved students.
While the concept of culturally relevant pedagogy has been offered as a promising means
of thinking about such instruction, it has been argued that this framework requires further
development. Questions of specific detail as well as questions of general principle
remain. Important details which are underdeveloped or missing from current accounts of
culturally relevant and effective mathematics instruction for underserved students include
responses to pertinent questions such as the following: In the current climate of high-
stakes testing, how can a 7th grade teacher in an urban school present mathematical
material in such a way that his lessons resonate with the worldviews of his students while
simultaneously readying them for standardized tests? How can an algebra teacher ensure
that her students, composed primarily of English language learners, move beyond
mechanical manipulation of symbols and toward the habits of mind encouraged by the
National Council of Teachers of Mathematics’ (2000) process standards? What does
culturally relevant and mathematically effective instruction look like in a classroom
comprised of students from diverse cultural and educational backgrounds? Insight into
these and other focused questions must be built into the culturally relevant framework.
The general principles of the framework require expansion as well, however.
Each piece of empirical work cited in Chapter 2 investigated culturally relevant
mathematical instruction in a monolithic setting. For instance, some researchers
41
considered instruction tailored to African American students (e.g., Ladson-Billings, 1997;
Tate, 1995), others to Latino/a students (e.g., Henderson & Landesman, 1995; Khisty,
1995), and still others to working class students (e.g., Boaler, 1997; Lubienski, 2000).
Studies investigating culturally relevant and effective mathematics instruction more
broadly, e.g., looking across culturally distinctive sites or within singular sites including
diverse cultural representation, are missing. As American classrooms become
increasingly diverse, what general principles can mathematics teachers draw on as they
seek to connect with a rainbow of students?
The considerations presented above are not intended to suggest that the current
study will resolve these important questions once and for all, but rather to establish the
position that knowledge about effective mathematics instruction for traditionally
underserved students, and the role that culturally sensitive pedagogy plays in such
instruction, is highly tentative and requires a great deal of exploration. When such
circumstances exist, qualitative research is a well-suited mode of exploration. As stated
in its influential report, Scientific Research in Education, the National Research Council
(2002) recognized that qualitative inquiry is most appropriate in cases, such as the present
one, in which established knowledge is lacking: “In some cases, scientists are interested
in the fine details (rather than the distribution or central tendency) of what is happening
in a particular organization, group of people, or setting. This type of work is especially
important when good information about the group or setting is non-existent or scant” (p.
105). Indeed, in illustrating potential research endeavors in which qualitative research
would be optimal, the National Research Council (2002) almost anticipated the current
42
study: “For example, to better understand a high-achieving school in an urban setting
with children of predominantly low socioeconomic status, a researcher might conduct a
detailed case study or an ethnographic study of such a school” (p. 105).
The current investigation will highlight urban mathematics teachers whose
students consistently achieve at desirable levels and will seek to draw insight about
effective instruction from these teachers. Essentially, this study seeks to discover what is
happening in these model classrooms and describe these happenings for the benefit of
others. The need to describe is a fundamental rationale for qualitative inquiry. The
central research question, “What are the characteristics…?” demands a descriptive
response. Cresswell (1998) notes that “In a qualitative study, the research question often
starts with a how or a what so that initial forays into the topic describe what is going on”
(p. 17).
Cresswell (1998) offers other indicators of the appropriateness of a qualitative
design for a given research question, all of which apply to the current study. These
include a need to explore a topic in depth, to present a detailed account of the situation
being considered, and to study individuals and their interactions in their natural setting.
A final indicator is particularly resonant with the intent of the current study. Cresswell
suggests that researchers “employ a qualitative approach to emphasize the researcher’s
role as an active learner who can tell the story from the participants’ view rather than as
an ‘expert’ who passes judgment on participants” (p. 18). Again, a fundamental
assumption of this study is that worthwhile knowledge regarding effective mathematics
instruction for traditionally underserved students can be inductively gleaned from the
43
practices and insights of successful teachers themselves; it need not be formulated and
deductively applied by detached “experts.” The expertise of classroom teachers is the
desired source of knowledge for this study, and my goal as a researcher is to learn from
them.
An additional assumption of this study is that optimal teaching is context-specific.
That is, it is assumed that teachers must cater their instructional style to the particular
needs of their unique collection of students. This recognition of the importance of
context, once again, prompts the need for a qualitative approach to the research.
Erickson (1986) notes, “Interpretive methods…are most appropriate when one needs to
know more about…the specific structure of occurrences rather than their general
character and overall distribution…[Qualitative researchers ask:] What is happening in a
particular place rather than across a number of places?” (p. 121). I seek to describe how
each of the participating teachers makes mathematics meaningful and accessible to their
students, and the unique features of how this is done in each teacher’s particular setting.
Drawing on the Research Traditions of Ethnography and Grounded Theory
The previous section established the need for a qualitative approach to the current
research question. This section highlights the specific qualitative research traditions,
ethnography and grounded theory, which have influenced the design of this study.
Explicit connections between the methodologies of these traditions and the assumptions,
purposes, and goals of the current study are discussed. As this study is influenced by
larger research traditions as well as a theoretical framework of culturally relevant
pedagogy, it can be considered neither a “pure” ethnography nor an “objectivist”
44
(Charmaz, 2000, p. 510) exposition in grounded theory. Descriptions of how these two
methodological approaches were blended, notification regarding elements of the study
which either conform to or stray from a given tradition, and a rationale for all such design
decisions are provided below.
Wolcott (1994) compartmentalizes the process of qualitative research into three
categories: description, analysis, and interpretation. Description addresses the central
question of “What is happening here?”; analysis is the process of identifying features of
the data and interrelationships among them; and, one’s interpretation infuses these with
meaning, addressing the question, “What are we to make of this?” Each component is an
indispensable piece of the researcher’s account of a particular phenomenon. In this
study, the ethnographic tradition informs how the participants’ work is described, while
the systematic methods of grounded theory inform data analysis. Final interpretations
will emerge from the research experience writ large.
“The ethnographer’s task is the recording of human behavior in cultural terms”
(Wolcott, 1994, p. 116). I draw on the ethnographic tradition because I view classrooms
as cultural sites. Each classroom represents a unique set of actors (teacher and students)
representing a particular combination of racial, socioeconomic, gendered, and other
identities. The successful mathematics classroom, I propose, is one in which all actors
respect and value the local cultural make-up. That is, classroom norms regarding the
rules of interaction, expression, authority, responsibility, and any number of other social
conventions are fairly negotiated and generally accepted by all members. Details
regarding how these human interactions play out, and how mathematics instruction
45
coordinates with them, can be spelled out via the ethnographer’s “thick description”
(Geertz, 1983, as cited in Rossman & Rallis, 2003, p. 46). Field work, or investigating
cultural phenomena in its natural setting, is central to ethnographic methodology. Such
field work includes recording the outside researcher’s impressions of the social
interactions among a site’s actors via observational data, and also gathering insight into
the meaning these actors’ project onto these interactions through interviews. Each of
these ethnographic data gathering techniques were utilized in this study and are described
further in the “Data Collection Procedures” section to come.
Once again, though this study is influenced by ethnography’s cultural frame, it is
not a purely ethnographical account. Pure ethnography requires prolonged exposure to a
particular cultural site, followed by a richly detailed description of that site (Cresswell,
1998). Rather than a single, comprehensive ethnography focusing on a single setting, the
present study might be viewed as including seven “mini-ethnographies.” That is, seven
classroom sites were viewed through a wider lens providing less detailed, but nonetheless
comparatively useful, snapshots of distinct sites. Furthermore, while presenting
classrooms as cultural entities is a valuable component of this account, another valued
objective is to abstract principles of instruction which can be utilized elsewhere. This
gradual shift from highlighting the contextual particularities of the research sites toward
the generation of more general principles can be accommodated through the use of a
grounded theory approach.
The tradition of grounded theory provides systematic techniques of data analysis.
This approach includes simultaneously gathering and analyzing data, iteratively
46
generating hypotheses regarding the data and using these tentative hypotheses to inform
further data collection, and, ultimately, inductively generating a theory, grounded in the
data, which can be used to explain the studied situation. Strauss and Corbin (1994)
summarize the process as follows: “Theory evolves during actual research, and it does
this through continuous interplay between analysis and data collection” (p. 273). This
inductive approach of uncovering knowledge is well-suited to the purposes of the present
study. Here, the intention is to learn about effective mathematics teaching practices in
particular contexts from successful teachers already in place. This is analogous to mining
the field (classrooms) for knowledge, or building a theory about effective pedagogy for
traditionally underserved students from the ground up.
While I have claimed that this study utilized grounded theory in generating a
particular model of effective pedagogy, I have also acknowledged that an existing
theoretical framework, namely culturally relevant pedagogy, influenced my perspective.
The utilization of existing theory as an interpretive lens runs counter to established
definitions of grounded theory research. In their foundational text, The Discovery of
Grounded Theory, Glaser and Strauss (1967) explicitly argued that researchers must
avoid preconceptions as they enter their work. They asserted that grounded theories must
spring exclusively from data, not other theories; to use a pre-conceived theoretical
framework is to improperly “force fit” data into a particular interpretation.
As grounded theory methodology has evolved, however, researchers have
increasingly recognized that one’s personal experiences and theoretical perspectives
inevitably influence the interpretation of evidence, and that explicitly acknowledging
47
frames of reference can actually enhance the quality of research. Grounded theory
pioneer Strauss acknowledged and welcomed this change, noting that “contemporary
social and intellectual movements are entering analytically as conditions into the studies
of grounded theory researchers…When we carefully and specifically build conditions
into our theories, we eschew claims to idealistic versions of knowledge, leaving the way
open for further development of our theories” (Strauss & Corbin, 1994, p. 276). While
an epistemological shift has occurred, some researchers continue to espouse the classical
definition of grounded theory (see, for example, Glaser’s (1992) critique of Strauss’
evolving perspective). The continued debate regarding the role of personal history and
theoretical perspective in grounded theory analysis prompted Charmaz’s (2000)
categorization of the methodology into objectivist and constructivist perspectives, the
former position rejecting the imposition of existing theory into analysis and the latter
position accepting it.
The personal history statement provided in Chapter 1 and the acknowledgment of
culturally relevant pedagogy in Chapter 2 attest that the current study leans toward the
constructivist approach. In addition to utilizing culturally relevant pedagogy as an
analytical lens, I have also indicated that this framework requires development and that
the current study might contribute to that effort. This goal corresponds with Vaughan’s
(1992) notion of “theory elaboration”: “By elaboration, I mean the process of refining a
theory, model, or concept in order to specify more carefully the circumstances in which it
does or does not offer potential for explanation” (p. 175). Vaughan connects this concept
to the data-driven, inductive principles of grounded theory: “As in analytic deduction,
48
the data can contradict or reveal previously unseen inadequacies in the theoretical notions
guiding the research, providing a basis for reassessment or rejection; the data can confirm
the theory; the data can also force us to create new hypotheses, adding detail to the
theory, model, or concept, more fully specifying it” (p. 175).
While objectivist and constructivist grounded theorists hold conflicting
epistemological assumptions, the methodology of grounded theory remains consistent.
Data are organized and analyzed as they are collected. Preliminary analysis involves
coding lines of data, infusing descriptors into segments of the data so as to organize the
data into more manageable chunks. The initial codes are themselves organized into
conceptual categories. These codes and categories are continuously revised as new
insights emerge from the data analysis process. Ultimately, a unifying theory relating the
conceptual categories is developed (Charmaz, 2000; Glaser & Strauss, 1967; Strauss &
Corbin, 1990). Such theory “consists of plausible relationships proposed among
concepts and sets of concepts. (Though only plausible, its plausibility is to be
strengthened through continued research)” (Strauss & Corbin, 1994, p. 278).
The perspective that classrooms are cultural sites prompted my decision to draw
on the ethnographic tradition in describing the work and attitudes of my participating
teachers. The desire to generate knowledge about effective teaching grounded in the
participants’ work, coupled with a goal of contributing to existing notions of culturally
relevant pedagogy, led me to utilize the procedures of grounded theory during the
analytical process. Charmaz (2000) speaks of the complimentary potential of these two
traditions: “Grounded theory provides a systematic analytic approach to qualitative
49
analysis of ethnographic materials because it consists of a set of explicit strategies” (p.
522). Having rooted this study’s design in the research traditions of ethnography and
grounded theory, the subsequent sections below describe the details of how the precepts
of these traditions were applied.
Access and Entry
This study investigated a very specific set of people: successful mathematics
teachers working primarily with traditionally underserved student groups. Such a
focused target of study necessitated a purposive sampling strategy (Miles & Huberman,
1994). Purposive sampling requires the researcher “to set boundaries: to define aspects
of your case(s) that you can study within the limits of your time and means, that connect
directly to your research questions, and that probably will include examples of what you
want to study” (Miles & Huberman, 1994, p. 27). The boundaries for this study were
defined in relation to the descriptors successful and traditionally underserved student
groups found in the guiding research question. In this study, teachers were judged to
work primarily with traditionally underserved student groups if the majority of their
students represent demographic groups identified as consistently underachieving in
national assessments: poor students, African American students, and Latino/a students
(National Center for Education Statistics, 2008). All of the participating teachers worked
in urban school districts in which a substantial majority of students fit this description.
Further details about the demographic make-up of these districts are presented in the next
section.
50
The identification and recruitment of successful teachers was informed by Palmer
et al.’s (2005) rubric describing “expert” teachers. Palmer et al. proposed that expert
teachers should be recognized and/or nominated as experts by supervisors or other
knowledgeable colleagues, should have no less than three years of experience in a
particular instructional context, and their work should have a documented impact on
student performance. The first criterion, nomination by supervisors, was the primary
strategy for identifying effective teachers. I had contact with the district mathematics
curriculum supervisor for the Milltown2 Public School District, the vice principal at the
Franklin Middle School in Adamstown, and a university-based mathematics educator
who had provided professional development and observed classrooms in Soho High
School in Adamstown. These experts were asked to nominate mathematics teachers in
their district or school whom they considered to be very effective. In addition to relying
on the nominators’ professional judgment about the teachers’ effectiveness, I asked the
nominators to consider two additional criteria when selecting teachers: 1) they should
only suggest teachers who had served for at least three years, and 2) if possible, they
should select teachers whose students consistently performed well on standardized
mathematics tests.
Reliance on expert nomination was the primary strategy for identifying teachers, a
technique Goetze and LeCompte (1984) have described as “reputational case selection”
(p. 82). It also meets one of Palmer et al.’s (2005) criteria of identifying expert teachers.
Effort was made to honor Palmer et al.’s other criteria, years of service and documented
2 The names of all cities, schools, and participating teachers presented in this report are pseudonyms.
51
impact on student performance, by requesting that the nominators consider these criteria
when suggesting teachers. Each teacher who was ultimately selected to participate in the
study had taught mathematics in an urban school for at least five years, hence Palmer et
al.’s second criterion was met. I attempted to meet the third criterion by selecting
teachers whose students performed significantly above district averages on standardized
tests, but this goal was only partially accomplished. I received access to standardized test
scores of the students of three of the four participating teachers from the Adamstown
school district (the fourth teacher taught 12th grade, a grade which is not tested in the
state), but was unable to obtain these records for the three Milltown teachers. The
Adamstown test scores that were obtained provided some indication that the teachers’
students had performed well relative to their district peers on standardized tests. For
example, 90% of Ms. O’Reilly’s 10th graders earned passing scores on the standardized
test in 2006 as compared to 78% in the district. I also received data pertaining to the
achievement of Ms. Zimmerman and Ms. Etienne’s students on a district-wide midterm
exam administered in January 2007. Ms. Zimmerman’s 8th graders earned an overall
average score of 74.8% on this test as compared to the district average of 56.2%; Ms.
Etienne’s 7th graders did not perform as well on the 2007 exam, scoring at the same
average rate (47%) as the rest of the district. Clearly the value of these test results is very
limited. The tests themselves differed across grade levels and included data for only one
year. Such limited and inconsistent data is certainly insufficient for making claims about
the teachers’ impact on student achievement. Given that test score data was unavailable
for the Milltown teachers, I am unable to cite data supporting the claim that their work
52
has had a “documented impact on student performance” (Palmer et al., 2005, p. 22). This
failure to meet all three of Palmer et al.’s criteria for all seven teachers is certainly an
inconsistency within the study and a shortcoming. However, Palmer et al. acknowledged
that “Establishing standards for student performance to identify expert teaching is
technically challenging in light of the variability of student populations and their
associated instructional contexts” (p. 22). This was certainly a challenge for the present
study. While student achievement data was not available for all of the teachers, there is
still some evidence that the participants are effective teachers given that all teachers were
nominated by supervisors and have served for at least five years in the classroom.
The nominated teachers were formally invited to participate in the study. The
invitation letter included a brief description of the purposes of the study, an explanation
regarding how they were identified as fitting participants, an indication of the time
commitments which would be required of them, a statement of the potential risks and
benefits related to their participation, and a signature page used to establish their
informed consent to participate in the study (see Appendix A). In signing the informed
consent document, the teachers agreed to participate in three interviews over the course
of several months and to permit me to enter their classrooms periodically throughout the
2006-2007 academic year.
Setting and Participants
Seven teachers participated in this study. Three of the teachers worked in high
schools, and the remaining four worked in middle schools. These teachers were drawn
from two urban school districts located in the northeastern United States, the public
53
school districts of Milltown and Adamstown. The Milltown teachers represented three
different schools within the district: one taught in Milltown High School, another taught
in Copperfield Middle School, and the third taught in Sullivan Middle School. The four
Adamstown teachers represented two of the district’s schools: two teachers taught in
Soho High School and the other two teachers worked in Franklin Middle School. The
teachers taught a range of grade levels and courses, from 6th grade up to and including
12th grade calculus. The number of years the teachers spent in the classroom also varied
considerably. The least experienced classroom teachers completed their 5th year of
teaching during the 2006-2007 school year; the most experienced teacher completed her
22nd year. Five of the seven teachers were White females. Mr. Oden was a White male,
and Ms. Etienne as an Asian American. Table 3.1 lists the teachers by pseudonym and
summarizes information on each teacher’s experience level and instructional setting.
54
Teacher Name Grade Level and/or Subject(s) Taught
Years of Experience (including the ’06-’07 school year)
School School District
Andrea Thompson
12th grade calculus
6 Soho High School
Adamstown
Tina O’Reilly 10th and 11th grade geometry
6 Soho High School
Adamstown
Judy Etienne 7th grade mathematics
8 Franklin Middle School
Adamstown
Cindy Zimmerman
8th grade mathematics
5 Franklin Middle School
Adamstown
Andrew Oden 10th and 11th grade algebra II, 12th grade calculus
5 Milltown High School
Milltown
Carol Kelly 6th grade mathematics
11 Sullivan Middle School
Milltown
Christine Frederick
6th grade mathematics
22 Copperfield Middle School
Milltown
Table 3.1: Information About Participants Table 3.1 provides basic information about each teacher’s immediate school
environment, but some background on the cities and school districts in which their
schools are housed is also necessary. Erickson (1986) notes that “Considering the
relations between a setting and its wider environment helps to clarify what is happening
in the local setting itself” (p. 122). The following paragraphs briefly describe the urban
environments and school districts where these teachers work.
Milltown is a city of approximately 72,000 residents.3 It has the lowest per-
family median income in its state, with the most recent census reporting that 24.3% of its
3 This figure, and other numerical figures pertaining to the city of Milltown, is based on data drawn from
the 2000 census. This data was retrieved August 29, 2006 from the U.S. Census Bureau’s website at
55
residents live below the poverty line. In terms of the city’s racial make-up, a majority
(59.7%) of Milltown’s residents reported being of Hispanic or Latino origin during the
2000 census. This fact no doubt correlates to the fact that 64% of Milltown households
speak a language other than English in the home, the only city in the state for which the
majority of residents primarily speak a language other than English. Though the presence
of Vietnamese-born immigrants is felt in the city, most Milltowners can trace their
ancestry to the Spanish-speaking Caribbean. Racial minority groups in Milltown include
Whites (48.6%), multi-racial persons (6.2%), Blacks/African Americans (4.9%), and
Asians (2.7%). Presumably, the fact that the sum of self-reported racial categories is
greater than 100% indicates that many individuals reported more than one category.
The Milltown Public Schools operate within this context. The economic woes of
the city are magnified in the student demographic data provided by the district.4 During
the 2004-2005 school year, 84.6% of Milltown students were reported as low-income, as
compared to 27.7% in the state. The racial composition of the public schools differs
substantially from that of the city, however. Hispanic students comprise a large majority
of the student population, 85.5%. Though Whites account for nearly half of Milltown’s
overall population, only 9.1% of the district’s students are White. Asians and African
Americans represent 3.0% and 2.4% of the student population respectively. The district http://quickfacts.census.gov. Historical data on the city of Milltown was retrieved from the city’s website
on August 29, 2006. The url of this website is not revealed here in order to protect the true identity of
“Milltown.”
4 Data about the Milltown Public Schools was retrieved from the district’s website on August 29, 2006.
Again, the url of this website is not revealed here in order to camouflage the identity of “Milltown.”
56
has struggled to maintain its accreditation in recent years, failing to meet state guidelines
for Adequate Yearly Progress (AYP) in 2003 and 2004. Mathematics achievement has
been a particularly troubling area in the Milltown schools. The percentage of 4th grade,
6th grade, 8th grade, and 10th grade students scoring below the minimally acceptable level
on the 2004 state-mandated standardized 2004 mathematics test was 49%, 64%, 66%,
and 45% respectively. The corresponding “failure” rates in the state were 14%, 25%,
29%, and 15%.
Adamstown has the largest population of all cities in its state, and it ranks among
the twenty largest cities in the United States. Adamstown is home to approximately
600,000 people.5 Though less pronounced than Milltown, Adamstown still struggles
with problems of urban poverty. Approximately 20% of Adamstown’s citizens fall
below the poverty line. Like many urban centers, Adamstown attracts numerous
immigrants from throughout the world. 25% of Adamstown’s residents are foreign-born,
and 33.4% of households speak a non-English language in the home. The most populous
racial groups include Whites (54.5%), Blacks/African Americans (25.3%),
Hispanics/Latinos (14.4 %), and Asians (7.5%). This diversity is reflected in the public
school population, though the proportions are distinctively different: 42.8% of students
are African American, 33.8% are Hispanic, 13.6% are White, and 8.5% are Asian.
5 This figure, and other numerical figures pertaining to the city of Adamstown, is based on data drawn from
the 2000 census. This data was retrieved August 29, 2006 from the U.S. Census Bureau’s website at
http://quickfacts.census.gov/. Data about the Adamstown Public Schools was retrieved from the district’s
website.
57
Almost three-fourths of the Adamstown Public Schools’ (APS) students were reported as
low income during the 2005-2006 school year. The APS also fared significantly worse
on the 2004 state-mandated mathematics exams in comparison with state averages.
“Failure” rates for 4th, 6th, 8th, and 10th grade students were 31%, 54%, 47%, and 27%
respectively.
The information above illustrates some aspects of larger contexts affecting the
work of the teachers in this study. The teachers, of course, work in particular classrooms
situated in particular schools within the larger district of a given city. Chapter 4 provides
more detailed information about the particular classrooms of each teacher. Though all of
the participating mathematics teachers share the characteristic of working primarily with
traditionally underserved students, they are drawn from several different school contexts.
Miles and Huberman (1994) would describe this aspect of the study’s design as multiple-
case sampling. This is indeed a strength of the overall research design. “Multiple-case
sampling adds confidence to findings…[through it], we can strengthen the precision, the
validity, and the stability of our findings” (Miles & Huberman, 1994, p. 29).
Data Collection
Having described the sampling design, the process of identifying teachers, and the
larger context of the study, I now turn to the specific procedures which were used to
collect data. It must be stressed that in a grounded theory study, data collection and
analysis occur simultaneously: data is analyzed as it is collected, and this analysis in turn
influences how future data will be gathered. Reports on data collection and analysis are
presented separately here for the sake of clarity, however.
58
Wolcott (1992) noted that in qualitative research, essentially three forms of data
exist. These include observational data, interview data, and documentation produced by
others. The current study conforms to this rule. I observed participants during the
teaching process, interviewed them individually, and gathered archival data related to
their work.
Details regarding how these data were gathered are presented in subsequent
paragraphs. I will begin here by redirecting the reader to the research objectives, and
briefly comment on how each form of data serves as appropriate evidence for a given
objective. Firstly, the research question demands that the teachers’ status as working
with “traditionally underserved students” be established. Archival data played a central
role here. Publicly available demographic data, referred to in the previous section,
support the claim that the teachers work primarily with traditionally underserved
students. Observations of the teachers’ classrooms also revealed that the demographic
distributions of the respective school districts were reflected in the teachers’ classrooms.
The bulk of this research effort involved describing and analyzing the characteristics of
these successful teachers. Important characteristics included (1) pedagogical styles, (2)
inter-relationships with students, (3) attitudes toward students and the work of teaching,
and (4) motivational factors for teaching. Information regarding items 1, 3, and 4 were
gathered from both interview and observational data. Teachers were invited to describe
these aspects of their work via interviews. The teachers’ accounts of their own styles and
attitudes were then held in comparison to the impressions of their work that I developed
through direct observation. Item 1 was also supported via archival data. Lesson plans,
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student assignments, samples of student work, and classroom photographs all provided
some information regarding the teachers’ styles of teaching. Item 4, motivation for
teaching mathematics in an urban environment, does not lend itself to observational
impressions. Here, teachers were invited to share their perspectives via oral interviews.
With the exception of item 4, motivational factors, evidence pertaining to all other
desired teacher characteristics could be found in multiple data sources (interview,
observation, and archival record). This enabled me to triangulate the data (e.g.,
corroborate information over multiple data points), an essential component of ensuring
validity in qualitative research (Rossman & Rallis, 2003).
Observations
My impressions of classroom occurrences were recorded via observational field
notes. Each teacher was observed on five occasions, and observations were spread out
over a minimal three-month period for each teacher. The Milltown teachers were
observed between January and May of 2007 while the Adamstown observations were
spread out over the course of the entire school year. The specific scheduling of
classroom observations was ultimately determined by the teachers’ availability and the
scheduling realities of schools. For example, observation of the three Milltown teachers
did not commence until January of the 2006-2007 school year due to delays in obtaining
permission to visit their classrooms from the superintendent’s office. Certain weeks in
February and May were off limit in Milltown as these dates corresponded to state testing.
State testing, snow days, and unanticipated school events (such as assemblies involving
guest presenters, etc.) likewise influenced observation scheduling in Adamstown. While
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it would have been ideal to schedule observations evenly over the course of the school
year for all teachers in order to capture a better cross-section of their work, this simply
proved impossible. Effort was made to observe a reasonably representative sample of
each teacher’s work given these time limitations, however. Specifically, at least two
different classes for each teacher were observed (i.e., I avoided observing Ms. O’Reilly’s
8:00 am class on five occasions, but rather observed her 8:00 am class three times and her
11:30 am class twice). Spreading the observations out over at least a three month period
per teacher also enabled me to witness a wider variety of teaching relative to the school
calendar (I was able to observe Mr. Oden handle post-Christmas break lethargy in
January as well as spring’s teenage hormonal activity in April, for example).
The initial collection of observational field notes was influenced by the
ethnographic tradition of endeavoring to capture as many details about a given site as
possible. During my first two or three observations of each teacher, I endeavored to write
down as many rich details of a given classroom site as possible…the classroom’s
appearance, what students and teacher were wearing, what students were saying both
mathematically and socially, body language of various actors, the nature of distractions,
and any number of other aspects of classroom life which caught my eye. Notes on these
events were scribbled furiously into my notebook, then typed more intelligibly into a
word processor as soon after the observational session as possible. I typically typed out
more polished field notes within two or three hours of leaving the observation site so that
details of the class would remain fresh in my memory. The act of writing neater, more
organized notes from the initial scribbles enabled me to fill in some details of the
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observation which were impossible to handwrite on site. Keeping with the research
tradition of grounded theory, analysis of the refined field notes began as soon as they
were completed. These initial analyses included assigning descriptive codes to sections
of text (e.g., describing full paragraphs from the field notes with two- or three-word
descriptors) and beginning to write memos which captured patterns, connections, and
insights I was beginning to form from the collected data. More detail about the data
analysis process is provided later in this chapter, but for now it suffices to say that I
endeavored to begin “making sense” of the data as they were collected. My continually
developing impressions of the data impacted the manner in which observational field
notes were collected during the fourth and fifth observations of each teacher. That is, as I
began to formulate themes related to the teachers’ work, I began to focus on classroom
occurrences more closely related to these themes rather than attempting to continue
capturing as many classroom details as possible with little guiding direction.
Interviews
Observations contributed to my outside interpretation of the teachers’ work, while
interviews opened a window to their own reflections. Seidman (1991) notes that
“Interviewing provides access to the context of people’s behavior and thereby provides a
way for researchers to understand that behavior” (p. 3). Each teacher was interviewed
formally on three occasions, with formal interviews lasting anywhere from 30 minutes to
over an hour6. The first interview with each teacher occurred before the initial
6 Data from informal conversations held before and after class sessions were recorded in the observational
field notes.
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observation, the second interview occurred before the fourth observation and the final
interview occurred after the final observation. During these interviews, I strove to
maintain a balance between satisfying my own curiosities about the teachers’ work and
enabling the teachers to raise issues pertinent to them. This falls in line with established
precepts of constructivist grounded theory. “A constructivist approach necessitates a
relationship with respondents in which they can cast their stories in their own terms”
(Charmaz, 2000, p. 525). The use of an “interview guide” (Rossman & Rallis, 2003, p.
181) approach accommodated this effort. This entailed preparing a set of open-ended
questions prior to each interview, but then remaining open to the prospect of steering
away from the pre-prepared protocol in instances when the participants raised
unanticipated issues or perspectives. Copies of the pre-prepared interview protocols are
provided in Appendix B. The interviews were recorded with a digital voice recorder and
subsequently transcribed with a word processor. As with the observational data, analysis
of the interview transcripts began as soon as they were created.
Archival Data
The archival data gathered for this study included publicly available
documentation pertaining to the teachers’ schools and the schools’ communities, samples
of student work, classroom handouts, teacher lesson plans and classroom photographs.
The publicly available data was utilized in order to provide factual information about the
teachers’ contextual environments and also to support the claim that these teachers work
primarily with traditionally underserved student groups. These data were sourced
through websites, literature produced by schools and school districts, and historical books
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and articles pertaining to the larger community. The other artifacts drawn from the
teachers’ practice (lesson plans, photos of posters teachers chose to display, etc.)
provided data related to the teachers’ pedagogy. Information about the teachers’ craft
was also gleaned through observational and interview data. Hence, these archival records
were used primarily as a means of further illustrating or further substantiating claims
about the teachers’ approach. Many pieces of archival data can be found in the appendix
and also as visual images within the text of this report.
Data Analysis
As noted earlier, collection and analysis of data happened concurrently. As data
were collected, they were immediately converted into electronic documents. Handwritten
field notes were typed into a word processor, recorded interviews were transcribed, and
archival records were scanned and converted into PDF files. These electronic data were
imported into the HyperResearch computer program. Software such as HyperResearch
has proven to be an invaluable tool in the management and organization of data, though it
cannot relieve researchers of their analytical responsibilities (Miles & Huberman, 1994).
The manner in which I used the HyperResearch tool to facilitate the iterative data coding
process occurred as follows. Once a given interview transcript, observational record, or
other data source was entered into the program, its text was coded on a line-by-line basis
(Charmaz, 2000). That is, every sentence or phrase was assigned a one-to-three word
descriptor intended to capture its essential meaning. Whenever possible, these
descriptors were drawn from the data transcript itself, a process called “in vivo coding”
(Strauss & Corbin, 1990). In cases where in vivo coding is not appropriate, other generic
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descriptors were applied. This initial coding stage, also referred to as “open coding”
(Strauss & Corbin, 1990), represented an initial effort at collapsing the data into a more
manageable size. It also paid homage to the objectivist roots of grounded theory: that is,
in describing the data through in vivo codes and generic terms, this early stage of analysis
served as an effort to “let the data speak for itself.” Rather than imposing a preconceived
theoretical lens on the data at this stage, line-by-line coding represented an effort to
summarize data as objectively as possible (Charmaz, 2000).
In constructivist grounded theory, theoretical influences do come into play during
the second phase of analysis, “axial coding” (Strauss & Corbin, 1990). This stage
represented my preliminary attempts to seek patterns and connections within the data.
This effort to introduce structure into the data was influenced by the theoretical
frameworks I brought to the study. That is, my efforts to begin “making sense” of the
data were in many ways constrained by my (hopefully transparent) assumptions. The
axial coding process occurred as follows. Once again capitalizing on the features of
HyperResearch, I searched the existing codes for terms which seemed to be inter-related.
The software enabled me to subsume apparently related codes under more general
categories. Once a set of specific codes were organized under a general category, the
software enabled me to return to the original passage of each specific code in order to
determine the reasonableness of a given category.
The organizational schema produced through initial axial coding was always
tentative. That is, working categories describing sets of codes were constantly revisited
in light of new data. These categories were continually reevaluated and re-specified as
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new data came in. Final working categories were arrived at through this iterative process,
becoming cemented once they seemed to be saturated with data (Cresswell, 1998). That
is, once a reasonable set of categories was established which seemed to accurately
capture all of the data, and once it became apparent that the addition of new data would
not shed new light on the model, the model was considered saturated. This situation led
to the final stage of coding, selective coding, which involved building a plausible set of
relationships between the categories and concluding with an overall model, or grounded
theory, describing the data set (Strauss & Corbin, 1990).
The ultimate theoretical model describing the teachers’ pedagogy is presented in
Chapter 5. A framework describing the teachers’ attitudes and motivations is presented
in Chapter 4. These models are relatively simple frameworks involving essential
components of the teachers’ approach to their work. The models themselves provide
some evidence of the fruits of the grounded theory analysis process. The raw data
collected during this study amounted to dozens of pages of text chronicling a year’s
worth of school site visits and interviews. The analysis process sought to uncover
patterns and connections across all of these data. The transformation from raw data to
intelligible theoretical models resembled the construction of a pyramid. Dozens of pages
of raw data formed the base of the pyramid. These data were then abbreviated during the
initial coding phase into several pages of brief phrases or codes. The long list of codes
was shortened into a smaller list. Eventually, two models of the teachers and their work
were devised: a five part outline presented in Chapter 4 and a four-point theoretical
model presented in Chapter 5. The models presented in these chapters were general
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abstractions from the original data, abstractions which had direct links back to the
teachers’ interview commentary and classroom actions. The ultimate theoretical models
produced were hence developed from the ground up.
Limitations of the Study
A distinguishing feature of this investigation of successful mathematics teachers
of traditionally underserved student groups is the fact that it involves multiple cases. I
am aware of no other study which attempts to derive principles of effective and culturally
sensitive mathematical pedagogy from a diversity of research sites. The benefits of
multiple-case sampling were described earlier, but this approach also has its drawbacks.
In qualitative research, there is a trade-off between the comparative opportunities
provided by multiple-case analysis and the potential for rich description of context
afforded by single-case studies (Miles & Huberman, 1994). The present sample of seven
teachers is intended to achieve a semblance of balance between both of these important
objectives, but I recognize that either component could be improved by adjusting the
number of participants.
Every effort was made to represent the teachers’ perspectives of their own work
accurately. Establishing this insider’s perspective is built in to many of the research
techniques, and teachers were asked to review and suggest modifications to drafts of this
final account. Despite these efforts, this report ultimately represents the interpretations of
one individual…the author. Readers are encouraged to factor this consideration into their
evaluation of the study’s worth to the wider community.
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On a somewhat related note, the issue of a study’s generalizablity is always
pertinent to a piece of research. Consideration of a study’s potential to benefit others is a
primary concern. Current recommendations for scientifically-based education research
have placed added value on studies incorporating an experimental or quasi-experimental
research design (National Research Council, 2002). I acknowledge that the present study
falls short of this “gold standard” for social research. This is as it should be, however,
because such a design is simply inappropriate for the research question at hand…a
question which is of great interest to a growing number of mathematics educators
interested in equity of educational opportunity. While the findings of this study cannot
claim to be generalizable through reference to statistical procedure, it is intended to
approach related notions of generalizabiltiy utilized in qualitative research: credibility,
transferability, dependability, and confirmability (Lincoln & Guba, 1983). Borrowing
from Wolcott (1994), “To the extent that the cultural system involved in this study is
similar to other cultural systems serving the same purpose, this [study]…should produce
knowledge relevant to the understanding of such roles and cultural systems in general”
(p. 11).
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CHAPTER 4
ATTITUDES AND MOTIVATIONS
As noted in earlier chapters, the primary goal of this study is to identify the
characteristics of successful mathematics teachers who work primarily with traditionally
underserved student groups. This research goal is supported through consideration of
several related questions which were presented in Chapter 1. This chapter addresses
some of those questions, including the following: What is the nature of the teachers’
interactions with their students? What are the teachers’ attitudes toward their students
and their profession? What motivates them to teach mathematics in general and to teach
this population of students in particular?
The intention of this chapter is to describe the teachers’ overarching perspectives
toward their work. This chapter is not intended to highlight the specific details of their
teaching styles. A framework describing some of the patterns uncovered in the teachers’
pedagogical approaches will be presented in Chapter 5. It should be noted, however, that
the teachers’ philosophical beliefs, professional attitudes, and motivating factors
presented here directly impact the teachers’ choice of classroom pedagogy. The question
of, “what makes these teachers tick?”, which is addressed in this chapter, should provide
a useful backdrop for the subsequent questions of, “how do they teach and why do they
teach the way they do?” which will be addressed in the next chapter.
Statements made about the teachers’ attitudes and motivations for teaching are
largely drawn from interviews with the teachers. Clearly the subjects (the teachers)
would have the most insight into such questions, and, therefore, their words frame the
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commentary on these issues. However, in an effort to triangulate the data, evidence from
the teachers’ classrooms is also presented. In some cases, this evidence takes on the form
of direct descriptions of classroom events drawn from observational field notes. In other
cases, it was appropriate to describe more general patterns of practice which were
observed in a given teacher’s classroom over the course of the observations. These
classroom vignettes and descriptions illustrate how these teachers “walk the walk,” that
is, they are intended to show how the teachers’ self-proclaimed attitudes toward their
work and their students play out in the classroom.
In the text which follows, shorter direct quotations from the teachers will be
presented within quotation marks and longer quotations will be presented as indented
paragraphs. Observational data drawn directly from the observational field notes will be
presented in italicized print. Broader descriptions of the teachers’ practice, which is
based on patterns noted across classes, is presented in regular print, along with all other
text representing the researcher’s interpretations and perspectives.
Each teacher is unique with his or her own specific history and perspective. As
such, it is useful to focus on the teachers as individuals, considering the particular
background of each. In order to accomplish this, the chapter begins with a sequence of
written portraits about each teacher. The portraits seek to describe the personal histories
and idiosyncrasies of each teacher, with a view toward uncovering more general trends
which might apply to all of the teachers in the study. Narratives devoted to each of the
three high school teachers are presented first, followed by the four middle school
teachers. The chapter then concludes with a discussion of some of the overarching
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principles and attitudes which are espoused by all of the teachers. This more general
framework, grounded in the particular stories of the seven teachers, provides a useful set
of characteristics educators might look for when attempting to determine what type of
person might be well suited to teach mathematics in urban areas or other areas with large
proportions of traditionally underserved student groups.
Portraits of High School Teachers
...selfishly, I would prefer to teach in the city...
Andrea Thompson has been teaching mathematics at Soho High School since
2001. During her first few years at the school she taught algebra and geometry to ninth
and tenth graders; in recent years she has worked with seniors in calculus. In addition to
her teaching, Ms. Thompson has been quite active in other areas of school life. Prior to
her arrival at Soho High, the school did not stage plays nor did it have a girls’ soccer
team. Ms. Thompson helped organize an initial school play and has continued to direct
plays since. She and Tina O’Reilly, the other Soho High School teacher in this study,
initiated a girls’ soccer program at the school as well. Her tenure in the classroom and
extra-curricular involvement speak to her commitment to the school and her students.
When asked if she would be interested in teaching in another setting, she said, “I feel
very invested in Soho High School. It would be really hard to just go to a different
school.”
Ms. Thompson’s path to the high school classroom was somewhat
unconventional, and, for her, largely unexpected. She majored in mathematics at both the
undergraduate and graduate levels, and took an interest in educational applications via a
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minor concentration in education. Though one might assume that such a combination of
studies would be intended to prepare her for the classroom, Ms. Thompson’s goal as a
university student was to serve the field in curriculum development and/or research.
Despite her interest in mathematics and education, she actually looked at the prospect of
teaching with some disdain:
I didn’t think I wanted to be a teacher…I took education classes and I really liked
it, but, I think it takes you a while to think, “I’m gonna be a high school teacher.”
To a high school kid or even a college kid that’s like a step above bus driver or
cafeteria worker. You know, all those jobs that you hated when you were in high
school.
While she was enrolled in a doctoral program in mathematics, Ms. Thompson
took a summer job which led her to reconsider her attitudes toward the teaching
profession. She taught mathematics in an Upward Bound program (a summer enrichment
program for high school students), and found the experience of interacting with and
teaching adolescents both enjoyable and rewarding. Returning to her graduate studies in
the fall, she began to reconsider the direction she had been taking:
I was taking all math classes, all day, every day, and I was just starting to think,
“This has no impact. This just only exists on paper, and it’s all theory. I might as
well be majoring in, you know, crossword puzzles.”...I was ready to really be, you
know, making my mark on the world and actually contributing something other
than just studying for something that, basically, because I enjoy it. So I decided
to become a teacher.
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For Ms. Thompson, making a “mark on the world” involved more than simply
being a positive influence on adolescents by taking on the public servant role of teacher.
She felt driven to make a difference where the need was most acute. Partially due to her
exposure to university-level courses in education, she was convinced that this need was
found in urban schools.
I had some teachers who really influenced me, especially wanting to do urban
teaching. And that’s through the education classes is where I first learned about
the politics of education, the inequalities in public schools. And that’s why I
teach in the city, because of learning all that.
After devoting several years to Soho High, she remains convinced that her
teaching efforts are optimized in an urban setting. She has the sense that students in more
affluent areas will receive a quality education with or without her, but that students in the
city have no such guarantees. She says, “I’d rather be where I’m more needed. You
know, where the kids are who aren’t used to having high expectations.” This statement
reflects not only the fact that Ms. Thompson feels “more needed” in an urban high
school, but also her assumption that a worthwhile education demands that students be
held to high expectations and given a chance to meet these expectations. She states
further:
In the suburbs I feel like, you know, a lot of kids, you know, get a Harvard
sweatshirt when they’re babies. They already are going to college, it’s just a
question of where and how they’re going to get there. And here, that’s definitely
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not the case, and a lot of students it could go either way, and they just really need
that help and those resources to get there.
Ms. Thompson’s assumptions about her students run counter to several prevalent
stereotypes of urban teenagers. She rejects the notion that students in urban schools are
lazy, unmotivated, under-prepared, or simply incapable of performing in school. Her
comment above suggests that she believes her students are as capable of academic
success as anyone else, though she feels that students in urban schools are less likely to
have teachers who believe in their potential. She also rejects stereotypes pertaining to
urban teenagers’ attitudes toward adults, namely, the common assumption that they are
disrespectful and rude. Indeed, Ms. Thompson’s viewpoint turns this stereotype on its
head, as she feels that urban students are actually more respectful and easier to work with
than students in more affluent areas:
...selfishly, I would prefer to teach in the city because I think you get more respect
from the students in the city. At least in this school, or in Adamstown, teachers
seem to be pretty respected. It’s a respected job. In a lot of the suburban schools
that I’ve been in contact with that’s not the case. A lot of that is because, you
know, their parents make more money then all the teachers do, and the parents are
like, “Oh, well I would never be a high school teacher,” and parents are always
like, “Why did you fail my daughter?” You know, just from people I know that
teach in the suburbs. And that’s not to generalize every parent or every kid or
every town, but…The students at Soho High are so kind and so loving and so full
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of gratitude, that, just, in complete selfishness, like, I get a lot more respect and
gratitude here than I think I would, so…
While there is certainly no way to compare Ms. Thompson’s reception from the
students at Soho High with the reception she might receive elsewhere, observations of her
classroom do reveal the affectionate relationship she has developed with her students.
One particular class session with her senior calculus students is particularly illustrative.
The class had covered various differentiation techniques, and the plan for the day was to
review for a test on the topic which would be given the next day. One element of their
review involved the performance of a “Derivative Song” which Ms. Thompson had
composed and choreographed for the class (see Appendix C). Most of the students in the
room had already memorized the song. When the time came to perform it, seemingly all
of the students sang along with their teacher and nearly half of them walked up to the
front of the room to perform the accompanying dance with her. As an outside observer
of this unorthodox yet pedagogically useful activity in the classroom, I was struck by the
enthusiastic level of participation in the room. This playfulness between teacher and
students seemed to be one sign of the “loving” reception Ms. Thompson senses from her
students.
This playfulness also indicates Ms. Thompson’s comfort level in the classroom.
While the classroom is a place for serious work, she also permits herself and her students
to loosen up and potentially expose themselves to ribbing and ridicule. Despite her
misgivings about high school teaching as a young woman, the comfort she now displays
at the helm of a classroom indicates that teaching may be a natural fit for her. Her
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positive assumptions about students and motivations for working in an urban school
contribute to her effectiveness. As will be seen in subsequent sections, these perspectives
are shared by other effective urban math teachers.
“I feel like, in some way, that my job has meaning”
Tina O’Reilly joined the mathematics department at Soho High School at the
same time as Ms. Thompson in the fall of 2001. Ms. O’Reilly has been working with
students in grades 9-11 in the areas of algebra and geometry throughout her tenure at the
school. Like her colleague, Ms. O’Reilly contributes a great deal to school life beyond
the classroom and is strongly committed to the school. She is the head coach of the girls’
soccer team, a team she co-founded soon after her arrival in 2001. She also coaches
basketball. When asked if she could envision moving on from Soho High to another
setting, she replied, “No, no. I don’t . I think if I left the classroom, I don’t really know
what I would do.”
While Ms. Thompson took a very round-about path to teaching at Soho High, Ms.
O’Reilly’s route was very direct. She majored in mathematics and education as an
undergraduate, fully intending to enter the classroom upon graduation. Her practicum
placement during her senior year was at Soho High, and by December of that year the
principal invited her to join the faculty the following fall. She has remained at the school
ever since. Though she has not explored other possible career paths, she is convinced
that her road has been well chosen:
I feel like, at some point my job is meaningful. I’m not just going into work,
pushing some buttons on a computer, and coming home and someone else makes
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a million dollars because of some exchange I made, and, you know, he’s already
so wealthy. I feel like, in some way, that my job has meaning. I’m going into
work everyday, and something’s coming out of it.
For Ms. O’Reilly, “meaning” is derived through service to others. This
perspective is rooted in the values she inherited from childhood: “I think my whole
Catholic upbringing from, you know, elementary school has taught me to be a better
person, help everyone out, kind of lead that life.”
Much like Ms. Thompson, Ms. O’Reilly feels that an urban school represents a
context where her help is most needed. She feels that students in more affluent areas are
more likely to receive academic support and direction outside of school than students in
urban areas. As a teacher in an urban school, she believes her influence is more strongly
felt as she can provide this added level of guidance for her students:
I think I have more of an impact here, what I’m doing. I’m more influential here.
[Students at more affluent schools] come to math class every day and everyone
goes in having the mindset, “I need to get an A. I need to go to college.” You
know, and they walk through it…go through the motions.
While these words indicate Ms. O’Reilly’s assumption that students at Soho High
may lack some of the home-based advantages enjoyed by other students, she does not
feel that her students are any less prepared to succeed in school or are any less willing to
work hard in order to achieve success. She assumes that the students are serious about
their studies and are very much willing to struggle for success, but she feels that she has
an important part to play in maintaining a healthy work ethic in her students: “…you
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need to somehow have the respect of the students and be able to relate to them so that
they want to, you know, they want to do work for you, they want to be in your classroom,
they want to be behaved. Whereas, if you lose their respect and you don’t know how to
relate to them anymore is where you start having a lot of problems.”
Both the words she shared in interviews and the actions she demonstrated in the
classroom indicate that Ms. O’Reilly takes her students seriously and assumes that they
are interested in achievement. Some students inevitably get distracted in class, but Ms.
O’Reilly does not interpret this as a sign that the student is unwilling to participate.
Rather, she respectfully refocuses the student on the task at hand:
I can talk to them respectfully and say, you know, “What’s up? Why aren’t you
doing your work right now?” Like, we turn this around, then it’s less of a
confrontation. And they’ll be like, “Well, yeah, I just don’t get it.” Maybe, you
know, turn it into that conversation rather than, “Why aren’t you doing your
work?” “I don’t want to do my work.” “Do your work!” You know, one of those
battles that just goes nowhere.
In the interview passage above, Ms. O’Reilly was attempting to describe her
approach to maintaining student focus in class. This approach was demonstrated in the
classroom on several occasions, including the illustrative example below:
[The students were directed to complete] a project begun the day before in which
students would construct platonic solids out of paper….. The students at the back
two tables on the left side were…talking freely, [and] they seemed to be doing
little of the expected work…. Ms. O’Reilly then went to the talkative group in the
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back left. The students indicated that they weren’t sure what the worksheet was
asking them to do. Ms. O’Reilly explained the meaning of the terms “faces,”
“edges,” and “vertices” to them again. She held up a solid which one of them
had already created and demonstrated how to locate and count these objects on a
given solid. She then left this table, and the students immediately began to work
on the project more diligently….. One student from the back left table approached
Ms. O’Reilly and pointed out that one could determine the number of faces, edges,
and vertices for a solid just by counting them off of the flat net image drawn on
the worksheet. He argued that it is therefore unnecessary to go to the bother of
creating the solid since the answers to the worksheet could be found without
reference to the solid itself. Ms. O’Reilly pointed out that, while this approach
would work for counting the number of faces, it would not work for the edges as
many edges appearing on the flat worksheet were shared by two sides on the solid
object. Hence, if one just counted the lines on the worksheet, he would come up
with too many edges.
This student, a boy, was satisfied by her answer and then became very involved in
the task. After conferring with Ms. O’Reilly, he elected not to return to the back
left table. Rather, he took the solid he’d been working on and sat at a table
nearer the front of the room, joining a group of students who had been more
focused on the task throughout the class. Eventually he returned to his friends at
the back table, but continued to focus hard on his solid.
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This particular classroom episode is certainly not an awe-inspiring portrait of
exceptional teaching. What is striking is the consistency with which instances like this
occurred in her classroom. Similar situations occurred frequently during the observations
(as they will in any classroom), and Ms. O’Reilly always dealt with the matter by calmly
redirecting the students toward the task at hand. The students typically responded to this
gentle nudge favorably. It is not difficult to imagine a teacher handling the noisy table at
the back of the room differently. Some teachers might ignore these students, assuming
they weren’t interested in schoolwork anyway; others might jump to chastise this
disruptive element in the room, etc. Ms. O’Reilly consistently demonstrated her respect
for the students by projecting the assumption that they were serious about their work, and
that she could depend on them to complete it.
Students at Soho High who are assigned to Ms. O’Reilly receive a very different
experience than students in Ms. Thompson’s classes. It is difficult to imagine any
mathematical singing and dancing with Ms. O’Reilly, as her classroom environment is
more structured in a traditional format. However, both teachers demonstrate respect for
students and communicate a belief in student ability in both word and deed. These
principles which lie beneath their work may have more to do with the success they have
achieved than their surface-level teaching styles. These principles come into play with
the other teachers in this study as well.
...if you want to see a revival then this is where it starts, right here in the schools
Andrew Oden began teaching at Milltown High School in 2002. He has worked
primarily with students in 10th grade and above, teaching geometry, algebra II and
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calculus courses during his time at Milltown. Of the teachers in this study, Mr. Oden’s
path to the classroom was the most unusual. Born and raised in the San Francisco area,
he attended college at the University of California-Berkeley and went on to earn a post-
graduate degree in particle physics from MIT. He then moved into a productive career in
computer technology. He began as a software engineer, and was one of the original
designers of File Maker software. He steadily climbed the ladder of the computer
industry, eventually working his way into management positions and ultimately taking
the position of Vice President of Engineering with a software firm.
His toils in the computer technology field led him to earn some financial stability
for his family. Hard work and thrift enabled him to pay for his children’s college
education and pay off the mortgage on his home at a relatively young age. Standing in
the middle of his career with these financial burdens lifted, he was in a position to pursue
further luxury for himself and his family. He opted instead to leave the corporation for
the classroom, seeking something more meaningful from the daily grind:
I didn’t really like the people I worked with [in the computer industry]. The
people, my peers, and my peer executives…a lot of them are just greedy bastards.
They’re out just for themselves. And that sort of made me think, ‘Is this what it’s
all about?’ And so [entering the classroom] was kind of a deliberate decision…an
experiment, really. OK, let’s try this and let’s adjust to this new thing.
Much like the teachers at Soho High, Mr. Oden’s self-appointed mission was to
not only serve others via teaching, but to attempt to serve in an area where he perceived
the need to be greatest. He took this mission quite seriously, choosing to teach at
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Milltown High specifically because standardized test scores in the Milltown district were
the lowest in the state. As noted in the passage below, he believes that education is a
necessary component of prosperity in the modern economy and that the resource of
education must be distributed more equitably:
I don’t think we can afford to have an entire community not….it’s hard to
summarize in one sentence. A community like Milltown doesn’t have high
paying jobs like high tech…..the basis of the future economy is going to depend a
lot on skills….skilled labor and unskilled labor. And the gap has been growing
over the last 30 years between those who have a college degree and those who do
not, in terms of the kinds of money they can make for their families and so forth.
So, unless we take this idea of education very seriously and make sure that
everyone gets the tools that they need, then we’re closing out the American dream
for whole classes of individuals. And, that’s expensive for society. You now
have the problems associated with poverty and crime, and with broken
families…if you want to see a revival then this is where it starts, right here in the
schools.
Rather than paying lip service to the position that improvement is required in
urban schools, Mr. Oden took the radical action of stepping down from a more lucrative
career in order to make a real contribution to a needy school. With over five years of
service at Milltown High, he continues to demonstrate a commitment to this vision and
continues to believe that, as a teacher, he can affect students’ lives for the better. Much
like Ms. Thompson and Ms. O’Reilly, he senses that his influence is more pronounced in
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an urban setting than it might be elsewhere: “I could have gone to other districts. I live
in [a more affluent community]. I could have done this job in [my community’s] district.
Those kids don’t need me. They don’t need me. They’ll learn just fine with whoever is
there.”
Mr. Oden recognizes that many (possibly most) of his students have relatively
difficult lives outside of school, but he does not view these circumstances as reasons to
assume they are incapable of academic success. He states:
I don’t take excuses. There’s a lot of people, I think they’re well intentioned, who
say, ‘Oh, gee,’ you know, ‘these poor kids. They’ve had this or that or the
other....they can’t do this, they can’t do that.’ I don’t do that.
He also indicated in an interview that he tries to “take kids as they are,” avoiding the
impulse to make premature judgments about what they know and don’t know, what they
can do and can’t do, etc.
While Ms. Thompson expressed the view that students in urban schools might
actually be more respectful of their teachers than students in suburban areas, Mr. Oden
sees more similarities between students from varying backgrounds than differences:
Kids are kids and they’re all going through the same kinds of things. I’m not an
expert on developmental psychology, but I’m sure there are regular stages that
everybody goes through. What’s different is situations. You know, when you
work at a place like this, you walk away realizing how resilient young people are.
They haven’t really been beaten down by the world, and they have an optimism
and they have different ideas about what they care about and what they don’t care
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about, and that is protective in some way. And, they make choices….some of
them are good choices, and some of them are foolish choices, but that’s how they
learn. What’s different is the situations they’re in.
Mr. Oden’s ethic of accepting students as they are plays into his classroom
management tactics. The atmosphere in his classroom is always light. In an interview,
he commented: “When [students] come into my class, I’m not going to try to force [them]
to be something [they] are not. I understand that [they] are teenagers and that [they] are
surrounded by friends. I want [them] to feel like [they] can laugh and joke, and be
[themselves].” In every class which was observed, the students spent a good deal of time
joking with each other and with Mr. Oden. While many teachers might view such a
classroom atmosphere as a recipe for disaster, Mr. Oden’s students seemed to know the
difference between appropriate and inappropriate banter. The class remained light but
lively, jovial but productive. Students consistently focused on work when it was
appropriate to do so. The light-heartedness in the room also seemed to build a level of
comfort between teacher and students. Students asked questions often in class, seemingly
unconcerned that they might look “stupid” for asking questions.
While the stories of each of these high school teachers are certainly unique, many
shared attitudes and assumptions shine through. Each teacher’s decision to enter the
classroom was inspired by a motivation to contribute something to society, and each felt
that urban schools are places where their contributions are most needed. Each teacher
expresses and exhibits faith in the students, believing that the students possess both the
ability and the required work ethic to achieve success. Each teacher possesses a sense of
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agency, believing that they really can have a positive impact on the lives of their students.
Finally, the tenure of these teachers (each having served at least five years in the
classroom) indicates that their perspectives are not naïve visions based on youthful
optimism, but are rather sober reflections informed by prolonged work in urban schools.
Their perspective represents a reasonable take on what might be accomplished elsewhere,
including the classrooms of the middle school teachers included in this study.
Portraits of Middle School Teachers
“I feel more compelled to the urban setting” At the time of this study, Cindy Zimmerman was in the midst of her fifth year of
teaching at Franklin Middle School in Adamstown. Her undergraduate studies focused
mainly on the life sciences, so she was initially hired as a science teacher with the
expectation that she would teach some mathematics courses as well. She taught both
mathematics and science in grades 6 and 7 during her first few years, but now teaches 8th
grade mathematics exclusively.
As is the case for most of the teachers in this study, Ms. Zimmerman pursued
other career options before entering the classroom. Her initial intent as a college biology
major was to become a veterinarian, but an internship experience in a vet’s office
convinced her that this was not her calling. After graduating with a degree in science, she
took advantage of an opportunity to study for a masters degree in educational research at
Trinity College in Dublin, Ireland. She found success there as she not only earned her
degree but also was one of a select group of students afforded the opportunity to present
research at a major European conference.
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Bolstered by this promising start as an educational researcher, Ms. Zimmerman
initially explored employment options in the research field. As she considered these
options, she began to feel that she might have more credibility as an educational
researcher if she earned some experience as a school teacher first. Furthermore, the
thought of serving students directly appealed to her:
I wanted to be with kids. There’s an inherent feeling inside me to give back, to do
something. I knew I would never be happy with a position where I was just
sitting behind a desk, or….granted, I’ve never been in the corporate world, but I
don’t have a very good feeling about it for myself. I like very much hands-
on…and I understand you can be in the corporate world and be helping people
out, but I like the face-to-face relationships.
Her first teaching job was in a private, Jewish school working with second
graders.
While this experience confirmed her enjoyment of working directly with children, she
found little satisfaction working in the privileged environment of a private school. This
and other issues related to the job led her away from the independent school. She began
to work as a substitute teacher in the Adamstown Public Schools, and eventually
managed to land her current job at Franklin Middle School. She has now committed over
five years of service to the school, indicating her satisfaction working in this particular
context. As with the other teachers in this study, this satisfaction is largely derived from
feelings that she is making a real contribution in an area where it is most needed:
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I think that my work is needed most in this setting…in an urban setting….I feel as
though I can give more to these students that they may not have, versus when I
was in a private setting…not that there’s any problems with a private
setting…but, those students seem to have more resources. And everything was
granted or given, that they were going to go home and they were going to have
dinner and mom or dad or the babysitter was going to be there and then they were
going to go to practice and then they were going to do this…and that was never in
question. Where, here, those questions flow through kids’ minds everyday that
they’re going home…’Am I even going to make it home?’ And, I wanted to
provide that for the kids that need it. And I think that I feel good about teaching,
really no matter what setting I’m in, but I feel more compelled to the urban
setting.
While Ms. Zimmerman recognizes the challenges many of her students face
outside of the classroom, she does not use this as an excuse to compromise the standards
of achievement she expects of them. As a teacher, she accepts that the quality of her own
instruction has a lot to do with the achievement of her students. She feels that effective
teaching requires the establishment of mutual respect between teacher and students. She
articulates what she means by “respect” as follows:
When I meet kids, I think I’ll show them respect with the expectation that respect
is shown back to me. I’ll listen to them. I will ask them about things other than
academics. Establish some type of relationship. Let them know that they can
approach me at any time, in the appropriate manner. You’re having a bad day?
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Tell me you’re having a bad day, and I’d be willing to make some exceptions or
other arrangements for you for that day. I think once the students feel as though
they have my respect, then it changes their attitude. They also count on me, they
know that they can count on me. They know that I will be here. They know what
to expect from me in terms of my personality and also in terms of rules and
regulations, and that I’m willing to listen to reason. Recognizing them as a
human, other than just a kid in the class, makes a major difference. Supporting
them in their outside efforts, not just in the class. And they also know that I will
follow through...if I say I’m going to call home, I will call home. If I say I’m
going to see you at the end of the day, I will hunt you down at the end of the day.
Or, if I promised you a prize, you’ll get the prize. A lot of it...that they can trust,
and that they know that I’ll be there, and I’ll follow through with my words.
This passage suggests that the establishment of respect requires at least two
commitments from the teacher: making oneself available to students for consultation
(both academic and otherwise) and firmly adhering to one’s word and expectations. I
witnessed both of these consistently in Ms. Zimmerman’s classroom. Each day I was
present to either interview or observe her, a student would enter her room during non-
teaching time to discuss a problem with her. The problems were typically petty from an
adult perspective (squabbles with friends, scorned adolescent love, etc.), but were of
crisis proportions to an 8th grader. It was clear that these students trusted Ms.
Zimmerman as a counselor and confidant. Though she communicated with students on a
personal level outside of class, class time itself was focused on academic work. She
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could easily be described as a no-nonsense teacher who effectively kept students engaged
in learning throughout the class, thus supporting her contention that her expectations are
clear and that she follows through on her word. The following classroom vignette is
illustrative:
The students have arrived, and have (apparently) sat down wherever they have
pleased in the classroom. Ms. Zimmerman begins by requesting that all students
move to their assigned seats. There is quiet grumbling, but most of the students
comply. One girl does not. Ms. Zimmerman addresses this girl by name,
requesting that she move to her assigned seat. The girl gets up and moves to a
different seat, but it is still not her assigned seat. A minute later, Ms. Zimmerman
calls her by name again, saying, “Rita, I’m going to give you a demerit because
you still haven’t moved to your assigned seat.” Rita stands up, saying out loud,
“I don’t care.” Though she issued a comment of defiance, she was still
complying with Ms. Zimmerman’s wish…she started walking toward her assigned
seat after saying these words. Ms. Zimmerman responded to the “I don’t care”
comment by saying, “Well, I do care. Please move to your seat.” By the time she
had finished saying these words, Rita was in a new seat which, I assume, was the
correct seat as there were no further exchanges on this matter.
Ms. Zimmerman put a warm-up problem on the board [a problem designed to get
students thinking about mathematics while the teacher tended to administrative
duties such as taking attendance and collecting homework], and, as usual, the
students worked diligently on it. Rita’s hand went up, and Ms. Zimmerman
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walked over to her. Rita told her that she was unable to complete the homework
assignment (a comprehensive review sheet related to tomorrow’s test) because
she “didn’t understand it.” Rita is sitting very close to me…I glanced at her
review sheet and it appears as if she hasn’t attempted any of the problems. Ms.
Zimmerman tells her, “Well, you’ll have to stop by after school…we don’t have
enough time to go over all of this now.” Rita said, “I can’t stay after school
today.” Ms. Zimmerman responded, “You can come early in the morning, then. I
get here at 6 AM.” That was the end of the conversation.
This classroom episode, taken by itself, is not a particularly striking example of
exceptional teaching. However, the consistency with which Ms. Zimmerman addressed
similar discipline issues in her room is noteworthy. Rita was not permitted to hold a
privilege unavailable to the other students (e.g., she was not permitted to be the only
student allowed to sit wherever she pleased), nor was her failure to complete an
assignment overlooked or dismissed. Ms. Zimmerman addressed both of these issues
with Rita calmly and firmly, clearly communicating her expectations to Rita and others.
Instances similar to this occurred periodically in her classroom and were dealt with in a
consistent manner. This classroom approach is related to many other teachers in this
study, teachers who patiently see to it that students live up to the high standards which
are expected of them.
“What I’m always pushing for is for all of them to be fully competent and excelling...” Judy Etienne is Ms. Zimmerman’s colleague at Franklin Middle School. At the
time of this study, Ms. Etienne was in her third year teaching seventh grade mathematics
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at Franklin. She had taught for five years in California before arriving in Adamstown.
She also came into teaching after beginning her career in another field. She was raised in
a family of teachers…both of her parents and some siblings were teachers…and so
looked into a different profession as a means of expressing her individuality. She spent
the first two years of her working life as a social worker serving adults. While she found
this work rewarding, memories of her earlier experiences with kids as a camp counselor
and sibling caused her to feel that she’d prefer working with young people.
As a trained social worker, Ms. Etienne could have pursued many ways of
working with children, but she came to feel that classroom teaching would be the ideal
setting. She explains her choice to enter the profession she’d been avoiding as follows:
One of my favorite pieces, of, when I was actually working with my clients, my
adult clients, was when I was teaching them a new skill to help themselves. For
some of them it was a life skill, for some of them it was helping them find the
skills they needed for a job placement. But, breaking things down to help them
learn what they needed to was the most satisfying part of my job. And that
combined with working with kids made it really clear to me that what I enjoyed
most was teaching.
Her move to the classroom from the field of social work was facilitated by the
ongoing teacher shortage in California, where she was able to receive an emergency
credential quickly. Her first teaching position was in a Catholic school in an affluent
area. Ms. Etienne’s perceptions of this private school were similar to Ms. Zimmerman’s
take on the private school where she had taught. That is, while she enjoyed interacting
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with young people, she developed the sense that these relatively privileged students
would receive a quality education with or without her presence. When her husband’s
career prompted a family move to Adamstown, she made the conscious decision to find a
teaching job in an urban, public school:
I grew up in public schools, and I love public schools. And, after several years in
this other school, I felt like, those kids were so driven and so bright that having
any competent teacher, they would succeed. You didn’t need to be an outstanding
teacher to make these kids outstanding students. They were going to succeed. I
felt like, one, I love public schools because I grew up in public school and I
wanted to be serving public schools, and, two, that I wanted to see the challenge
for myself because I didn’t feel like I really knew if I was a good teacher or not
with that set of kids. It was like, great, my kids did a great job I don’t think that
necessarily told me if I was doing a great job as a teacher, you know?”
The private school where she began her teaching career was not only
economically affluent, but also academically selective. The students there were admitted
on the basis of prior academic achievement. As Ms. Etienne described it, it was a setting
in which teachers would be concerned if any student fell below the 85th percentile on any
standard measure of academic proficiency. This, then, became one of her greatest
challenges when she began working at Franklin Middle School: in this new setting,
students’ prior achievement was more evenly distributed. While some of her current
students enter the classroom with strong past performance and content knowledge, many
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others arrive without these skills. As she describes it, “some are coming in still
struggling with their multiplication tables, still adding on their fingers…”.
Despite this wide range of student backgrounds, Ms. Etienne’s goal is to ensure
that all students attain the competencies expected of a college-bound seventh grader:
“What I’m always pushing for is for all of them to be fully competent and excelling at the
material that we put forward.” Indeed, she is so committed to seeing to it that all students
in her classroom obtain the desired competencies that she fears she may be doing a
disservice to those students who arrive in the class with stronger mathematical
backgrounds: “Here, I find, where I’m weakest is pushing those kids [who arrive with
stronger background knowledge], and that is because I’m so concerned about my students
who aren’t where they should be.”
This humble admission of her own imperfection notwithstanding, observations of
her classroom make it clear that she genuinely endeavors to advance all of her students’
learning. She utilizes cooperative learning extensively…indeed, she even special-ordered
round tables for her classroom to foster student collaboration. As will be discussed in
greater length in the next chapter, this collaborative model encourages students to lean on
each other as “resident experts” as they wrestle with mathematical tasks. It encourages
stronger students to explain material to students with less experience. This permits
students with stronger backgrounds to further refine their thinking by re-presenting their
knowledge to others. Students with weaker background knowledge also benefit a great
deal as they can draw on both the teacher’s perspective on content as well as their peers’
perspectives as they endeavor to make sense of the mathematics.
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The effort Ms. Etienne put into refining this collaborative learning approach in
her classroom over the course of the school year was commendable, and, again, will be
discussed further in the next chapter. The fact that a teacher of her experience continues
to try to improve her practice speaks to her professionalism and care for her students.
She demonstrates in both word and deed a commitment to helping all students excel in
the classroom.
“It’s hard work, it’s exhausting, but I can’t picture myself teaching anywhere else…”
Carol Kelly has been teaching sixth grade mathematics and science at Sullivan
Middle School for over ten years. She also came into teaching from another profession,
having worked as a dental hygienist before entering the classroom. She is unique in this
study in that she is the only teacher who was born and raised in the same community
where she teaches. Her connection to the community, coupled with her love for children,
inspired her to teach in the Milltown Public Schools.
While Ms. Kelly shares community membership with her students, she differs
from the majority of them in terms of race and ethnicity. As noted in Chapter 3, the
complexion of the city of Milltown has been in flux since the city’s incorporation in the
nineteenth century. Ms. Kelly’s ancestry includes European immigrants who arrived in
Milltown during the first half of the twentieth century. Though her ethnic background
contrasts with her mostly Latino students, she sees parallels between the struggles faced
by this recent set of immigrant families and the struggles encountered in her own family
history:
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When I was a little girl [Milltown] was mostly Irish and Italian and French and
Lebanese, and now it’s mostly Hispanic. And, you know, I’m fine with that
because that’s what it’s all about...it’s the melting pot. And now they’re going
through...the people coming over here that have Hispanic heritage...are going
through probably what my parents and grandparents went through.
Cognizant of her own family’s difficulties as immigrants, she hopes that through
her teaching she can help improve the lives and economic opportunities of the students
she serves. She feels that such opportunities were not as readily available to her parents
and grandparents several decades ago. She hopes that the life paths of her current
students need not be so difficult:
...my father never spoke English. He got thrown into the Milltown Public
Schools, didn’t know a word of English. You know, sink or swim. They had no
help, I mean you sink or swim, you either make it or you don’t. And that’s how
it’s different today. Because people want to help people, and they want people
who come to the United States to be successful.
Ms. Kelly’s decade-long tenure at Sullivan Middle School speaks to her
commitment to the school and its students. Her commitment to the community of
Milltown is equally impressive. Many of the people she grew up with, particularly those
who managed to become professionals, departed the economically depressed city long
ago. Ms. Kelly, however, has maintained her desire to serve this community, and
anticipates continuing to do so in the foreseeable future:
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I can’t picture myself teaching anywhere else. Everybody’s like, ‘Why do you
stay in Milltown? It’s a tough job.’ I said, ‘Because, I can connect with them.’ I
don’t know if it’s because I was born here, lived here, I just...I don’t know. It’s
hard work, it’s exhausting, but I can’t picture myself teaching anywhere else.
Ms. Kelly’s professed ability to connect with her students is further illustrated in
the expectations she holds for them, both in terms of their classroom behavior and their
academic accomplishments. That is, while she is aware of the disadvantages the students
have in terms of their socioeconomic status and their status as English language learners,
she still recognizes that they are just as capable as her grandparents were of overcoming
these difficulties in order to achieve success in the classroom. She articulates these
expectations as follows:
If you let them know what the expectations are in your room upfront, you let them
know what the lesson’s going to be about and what you expect is a good lesson,
they’ll usually come through for that. And just talking to them…they’re on the
same level as you, you’re not talking down to them…
My observations of Ms. Kelly’s classes confirmed that her expectations were
clear to the students and that the students lived up to these expectations. She regularly
arranged students into collaborative working groups in each of the five classes I
observed. The noise level in class typically rose to a high volume during the group work
as students were permitted and encouraged to communicate with their group members on
the task at hand. It would have been easy for a student or group of students to hide in
such an atmosphere…that is, unmotivated students might be tempted to allow the
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murmur of mathematical discussion in the room drown out an unrelated side
conversation, or to simply disengage in the midst of the activity in the room with the
hope of going unnoticed. This never seemed to occur from my observations. That is, the
students knew they were expected to engage with their mathematical task, and they met
this expectation. Ms. Kelly assisted groups when requested, but never needed to police
students for misguided behavior or failure to work. The following passage from the
observation field notes illustrates both the clarity with which she communicated her
expectations and the degree to which the students met these expectations:
At the end of the task, Ms. Kelly pulled down a large laminated rubric which had
been displayed at the main whiteboard. This is the rubric that she uses to assess
students when they are given group work tasks. Among the items on the rubric
were, “Group goes straight to work without being told,” “Group focuses on the
task,” “Group members respect the ideas of each other.” She informed the class
that most groups received a score of 100 for this task, but that a few groups
violated the “Respects all ideas” tenet. “I think those groups know who they are.
We all know what to expect when we do group work....there’s no surprises in this
class.”
While it is unfortunate that during this particular class some of the students failed to meet
an expectation for collegiality, the students’ performance as measured the rubric
confirmed my own observation that all students took the task very seriously. Those
group members who didn’t appropriately respect all ideas were guilty of taking their own
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solution to a mathematical problem too seriously and failing to heed the perspectives of
others.
Ms. Kelly’s attitude toward her work in Milltown is somewhat unique in this
study as she draws a direct connection between her family’s experiences as newcomers to
the Immigrant City and the experiences of her students. Like the other teachers,
however, she clearly demonstrates a belief in her students’ abilities to perform
academically and also her own ability to assist them in doing so.
“You have to love them”
Christine Frederick has the longest tenure of the teachers presented here, having
served 22 years in a range of teaching roles, from pre-K to grade 8. She and Ms.
O’Reilly are the only teachers whose entire professional training and subsequent full-time
work has been in the classroom. She has been working at Copperfield School for four
years, currently teaching sixth grade mathematics.
Though her 22-year tenure as a teacher is impressive, she is a relative newcomer
to the position of middle school mathematics teacher. She worked in the area of early
childhood education for most of her career. When she first entered the teaching
workforce, a scarcity of jobs as well as personal family commitments forced her to move
frequently from job to job and district to district. After several years of teaching at
various elementary grade levels in numerous districts both north and south of the main
city of Adamstown, she finally settled into the city of Milltown as one of the founding
teachers at a new charter school in the city. She took on the primary responsibility of
developing a literacy curriculum at the charter school, and also established a reputation
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for being a dedicated and caring teacher. The principal at Lee School, another K-8
school in Milltown, heard of Ms. Frederick’s work and offered her a job as an 8th grade
mathematics teacher at Lee School. Ms. Frederick was attracted to the job and her K-8
certification qualified her for the position, but given her background in early childhood
and literacy education, she was concerned about her level of content knowledge in
mathematics. She chose to accept the job, serving there for two years before transferring
to Copperfield School, and committed herself to developing a strong grasp of the
mathematical content as well. This commitment has been strong, as over the past six
years she has taken graduate-level mathematics courses at a highly respected university in
Adamstown (taking full advantage of professors’ office hours to catch up on
undergraduate-level material) and has been the sole representative from her school to
participate in a content-driven professional development partnership between the
Milltown Public Schools and a group of mathematicians and mathematics educators at a
nearby institution.
Clearly Ms. Frederick has committed a great deal of time outside of the regular
school day toward improving her mathematical content knowledge. She also devotes a
large amount of additional time to her students outside of the classroom. She avails
herself to students beginning at 7:00 AM every morning (with the school day beginning
at 8:15), and also remains after school to help students, often remaining three or four
hours past the 2:40 PM dismissal. A core group of students has taken full advantage of
her generous availability, a group Ms. Frederick has dubbed her “math warriors”:
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It kind of started out as my extra help sessions that I do before and after school.
And we said, you know, look at how they’re fighting to be better, and their
parents like their kids coming early and staying late, and, at least they knew where
their kids were. And, so, they became my math warriors.
Her efforts to improve as a teacher via intensive professional development in
mathematics and her willingness to put in long hours for her students are two powerful
statements about Ms. Frederick’s care for her students. She wants to teach them as
competently as possible, hence her commitment to professional development, and she
wants them to succeed, hence her willingness to put in the extra time to assist them. Her
care stretches beyond effectively teaching content, however. Many of her students over
the years have trusted her enough to approach her with personal problems. During our
interviews, Ms. Frederick shared several stories of student hardship, hardships which
students had confided in her. She also showed me a large collection of letters she had
received from her Copperfield School students over the years, letters expressing gratitude
for her assistance with matters both personal and academic. The word “care” may be the
best descriptor for Ms. Frederick’s approach to her work, a point she sums up succinctly:
“One of the joys in my life is helping people.”
If a desire to care for and help people drives her work, then the level of poverty
among the students at Copperfield School ensures a steady outlet for her energies. Like
Ms. Kelly, Ms. Frederick acknowledges that the demands required of serving in an area
of great need can be both exhausting and frustrating. Difficult personal circumstances or
even simple adolescent irresponsibility can lead students to disappoint a teacher with high
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expectations such as Ms. Frederick. She describes how she manages to persevere as
follows:
I do believe that the teacher has to believe in [the students] first. I say to people,
‘You have to love them.’ You do, you have to love them. But I think they know
that I truly care...
The “them” she refers to includes every student in the classroom. As mentioned
earlier, she has helped motivate a subset of her students, her “math warriors,” to put in
the extra effort required for academic excellence. She does not use this group as a
reference point for separating her students into those who “want to learn” and those who
“don’t want to learn.” As noted above, she believes in all of her students and endeavors
to find ways to help advance all of them academically:
I try to reach all of them. I like to take them from where they are, and that’s
what’s been successful, and move them along, and you want to make them feel
successful first.
When Ms. Frederick began teaching middle school mathematics at Copperfield
School, her main trepidation involved her content knowledge. Since that time she has
compensated for her inexperience with content via rigorous professional development.
While her long experience as an early childhood teacher did not immediately seem to be
much of an asset for her new position, many of the qualities of a good K-3 teacher have
served her well in the middle school. She does not hesitate to reflect on and demonstrate
care and love toward her students. While these sentiments are usually expected of the
nurturing Kindergarten teacher, they are not always regarded as being as central to the
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work of teachers in the middle grades and above. Yet Ms. Frederick has found a way to
meld the caring approach of the early childhood teacher with her developing appreciation
of mathematical content into an effective approach to her work.
Separate Stories, Unifying Themes: The Educational Outlook of Successful Teachers These portraits of seven effective urban mathematics teachers provide some
preliminary insight into the question, “What are the characteristics of successful
mathematics teachers who work primarily with traditionally underserved student
groups?” Specifically, the data presented in this chapter relates to the teachers’
motivations for entering the urban mathematics classroom, their attitudes toward their
students and their profession, and the nature of their interactions with their students.
Each teacher is an individual, and it is not difficult to discern the unique
approaches the separate teachers incorporate in their practice. For example, Mr. Oden
fosters a light-hearted atmosphere in his classroom, while Ms. Zimmerman’s classroom is
more business-like. Ms. Thompson approaches content from numerous angles (even
incorporating song and dance), while her colleague Ms. O’Reilly’s approach follows a
more traditional format. The watchword in Ms. Etienne’s class is communication, for
Ms. Frederick it is care, and for Ms. Kelly it is connectedness.
Though each teacher brings his or her unique fingerprint to the work, there are
many consistent themes which emerge across their individual stories. All of the teachers
view their work as an important form of social service, and each has chosen to teach in a
school where the perceived need for their services is most acute. Each has served in such
settings for several years, and they continue to believe that their work in the classroom
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makes a positive difference in the lives of their students. An ethic of care permeates the
work of all of these teachers, though this care is manifested differently across classrooms.
The teachers profess and demonstrate respect for their students, and this respect is
reciprocated. Finally, the teachers operate under the assumption that their students are
capable of excellence in mathematics. Further discussion of each of these themes is
presented below.
The teachers seem to view their work as a vocation rather than an occupation
(Hansen, 1995). That is, each teacher felt compelled to enter the classroom as a means of
concretely expressing some of their ideals and principles, as opposed to entering the
classroom merely as a means of earning a living. For these individuals, teaching is more
than an avenue toward a paycheck. It is, instead, an avenue toward a meaningful
lifestyle. The fact that five of the seven teachers chose to enter the classroom from other
potentially more lucrative or less stressful professions speaks to this. Mr. Oden and Ms.
Kelly, former engineering executive and dental hygienist respectively, walked away from
substantial salaries in order to teach adolescents in public schools. Ms. Thompson and
Ms. Zimmerman, as trained educational researchers, as well as Ms. Etienne, as a trained
social worker, may not have been in professions offering heftier paychecks, but they did
have the opportunity to work in respected professions which would not require them to
bring work home or require them to endure the emotional taxation of facing dozens of
adolescents each day.
It should be noted that while the majority of the teachers in this study were career-
changers, and hence did not experience a teacher preparation program as undergraduate
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students, all of the teachers benefited from some combination of professional
development and graduate study geared specifically toward the teaching of mathematics.
Mr. Oden and Ms. Thompson participated in a state sponsored teacher preparation
program for mid-career professionals choosing to move into the classroom. Ms.
Frederick was prepared as an elementary school teacher. Her move into middle school
mathematics was a major shift, but she thoroughly prepared herself for that shift via
graduate study. Ms. Etienne, Ms. Kelly, and Ms. Zimmerman likewise noted in interview
comments (not recorded here) that they have benefited from professional development in
their content area. As a result of their continued education, all of the teachers felt
adequately prepared to teach mathematics. Their preparation in mathematics likely
contributed to their sense of efficacy in teaching the subject, a theme which will receive
more attention later in this section.
For these teachers, the choice to enter the classroom was a conscious decision
motivated by the desire to pursue meaningful work. This “meaning” is derived through
engaging in service for others. All seven teachers cited a desire to serve as a primary
motivational factor for entering the classroom. The five career-changers mentioned
above found the propensity to serve in other jobs deficient, and hence had a point of
reference when choosing to engage in teaching. The two teachers who spent their entire
professional career in the classroom also clearly articulated the opportunity to help others
as motivation for their chosen path.
If the teachers view their profession as a lifestyle of engaging in service, then
their decision to teach in urban schools in particular reflects their desire to work in a
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setting where the perceived need for service is greatest. A common sentiment expressed
by the teachers was that their presence was not really needed in more affluent schools: it
was assumed that the students in these schools would achieve success with or without the
teachers’ presence. Several teachers expressed this perspective as an untested
assumption, while others (Ms. Zimmerman and Ms. Etienne) had actually worked in
more affluent schools and had gained this perspective through experience.
Not only do the teachers assume that their services are most needed in urban
schools, but they genuinely believe that their contribution in the schools really does have
a positive impact on the lives of their students. This sense of efficacy, or the teachers’
belief in their own ability to affect positive change, likely contributes to the teachers’
effectiveness. Fine (1989) has commented on the power of teachers’ sense of agency in
the classroom: “educators who feel most disempowered in their institutions are most
likely to believe that ‘these kids can’t be helped’ and that those who feel relatively
empowered are likely to believe that they ‘can make a difference in the lives of these
youths’” (p. 158). The teachers’ sense of efficacy likely contributes to their belief that
the students can and should be expected to achieve, another important theme which has
been noted across the teachers and will receive further attention later in this section.
The teachers also profess and demonstrate a strong level of care for their students.
The manner in which the care is expressed is as different as the teachers themselves, but
it still remains a constant underlying the work of all seven teachers. Ms. Frederick has
reflected a great deal on the place of caring in her work, and she demonstrates her care by
spending a great deal of time with students outside of the classroom, providing both
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additional academic support as well as personal counseling. Both Ms. O’Reilly and Ms.
Kelly emphasize the importance of making the effort to relate to students, or to learn
about where individual students are coming from in order to more effectively sympathize
with and meet their particular needs. This approach resonates with research on the nature
of care in the classroom: “a caring teacher is someone who has demonstrated that she can
establish, more or less regularly, relations of care in a wide variety of situations”
(Noddings, 2001, pp. 100-101).
Though the other teachers do not express their ethic of care in words, all seven
teachers demonstrate care for students in practice. Again, “care” is manifested in
numerous ways: through patience with struggling students, time spent with students
outside of the classroom, effort to learn about the students and their interests, etc. One
strikingly similar manifestation of “care” across the seven teachers, however, is the
teachers’ common insistence that students put forth their best effort in the classroom.
Vignettes of Ms. Zimmerman and Ms. O’Reilly pushing students who had not adequately
attempted some of their mathematical work were presented earlier in this chapter.
Episodes such as these were found in all seven classrooms, however. For example, while
Mr. Oden welcomed a fair deal of joking in his classroom, he also saw to it that all
students completed their work, supporting them in their efforts to do so. This was
consistent for all teachers: while classroom management styles differed substantially,
one constant was that students were not permitted to avoid their responsibility to work
and learn. This “tough love” stance is a major component of genuine caring:
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If one supposes that caring is merely a nice attitude, an attitude that ignores poor
behavior and low achievement in favor of helping students to feel good, then, of
course, caring will be seen as antithetical to professional conduct. But this is just
wrong. A carer, faithfully receiving the cared-for over time, will necessarily want
the best for that person; that is part of what it means to care (Noddings, 2001, p.
101).
Inter-related with this theme of caring is the theme of respect for students. It was
noted above that the teachers demonstrate their care for students by insisting that students
perform in the classroom. This insistence is not manifested in a mean-spirited, “do your
work or else” fashion, but rather in a manner which is respectful to the students. The
vignette of Ms. O’Reilly calmly approaching a table of talkative boys and effectively re-
directing them toward the mathematical task is illustrative of this. Rather than dismissing
them as unmotivated or chastising them publicly, she respectfully engaged them in a
discussion, uncovered a misconception which was hindering their ability to move forward
with the work, and helped motivate them to return to the problem. Hollins (1996)
identifies this approach as a fruitful demonstration of respect, particularly in a diverse
setting such as an urban classroom: “[An] important aspect of building positive
relationships with students is for teachers to show respect, concern, and interest in their
students regardless of their cultural background. Teachers can show respect for students
by being polite and avoiding statements or actions that publicly humiliate, embarrass, or
reprimand” (p. 125).
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Another manner in which respect for students was demonstrated across
classrooms related to the way the teachers received the students. Both Mr. Oden and Ms.
Frederick articulated the idea that they strive to take students “as they are.” They
consciously make the effort to avoid potentially dangerous pre-judgments about the
students. That is, they attempt to avoid thinking about the students in terms of some of
the stereotypes pertaining to urban adolescents, students of color, poor students, students
of various ethnic and citizenship backgrounds, etc. Rather, effort is made to learn about
the students as individuals with varied histories. Similarly, other teachers such as Ms.
Thompson and Ms. O’Reilly indicate that they try to take classroom events “as they are”
as well. That is, if a particular student engages in inappropriate behavior on a particular
day, the event is viewed as isolated and is dealt with for what it is. These teachers, along
with the others not mentioned in this paragraph, tend to view student shortcomings as
short-term mistakes to be addressed, not as a general statement about the student his or
herself. This stance of dealing with students as individuals and avoiding the temptation
to allow a few unfortunate incidents taint one’s overall impression of a given student
resonates with Good’s (1987) model of the attributes of effective teachers.
It was noted earlier that the teachers possess a strong sense of self-efficacy. That
is, they believe that their work can and does make a difference in the lives of their
students. In addition to their belief in their own ability to effectively teach students, they
also share the belief that their students can effectively learn mathematics. Indeed, this
belief in student ability permeates many of the other characteristics of the teachers which
have been mentioned here. Obviously the teachers would not hold a sense of self-
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efficacy if they felt that the students’ chances of learning were slim. The teachers’
insistence that students perform in the classroom reflects the assumption that the students
can perform. Finally, the teachers’ belief that their students should be held to high
standards is yet another genuine sign of respect for the students, namely, respect for their
intelligence.
A comment from Mr. Oden comes close to capturing this belief in student ability
which is demonstrated by all of the teachers: “There’s a lot of people, I think they’re
well intentioned, who say, ‘Oh, gee,’ you know, ‘these poor kids. They’ve had this or
that or the other....they can’t do this, they can’t do that.’ I don’t do that.” Mr. Oden is
referring to many of his colleagues in the teaching profession who tend to view a
student’s past performance as an indelible indicator of the student’s future performance.
This, in turn, serves as a rationale for further underachievement. The teachers in this
study, who have expressed an attitude of taking students “as they are,” do not consider
past performance as an unavoidable “sentence” for what the student may yet achieve.
While these teachers are certainly aware of the earlier limitations of their students, they
still view all students as intelligent individuals with the potential to excel in mathematics.
This perspective relates closely to a dichotomy identified by Hollins (1996) as
“ability” versus “motivation.” For Hollins, educators who focus on “ability” consider
variables such as achievement scores and past performance as reliable indicators of what
students can be expected to do in the future. Educators who focus on “motivation” are
much less concerned with measures of “ability,” and, instead, are more invested in
finding ways of connecting with students so as to help them advance academically. The
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teachers in this study can be classified as meeting Hollins idea of “motivation.” Hollins
has argued that such teachers tend to be more successful, particularly when working with
diverse student populations. Synthesizing the literature on effective instructional
programs for diverse learners, Hollins argues:
Each [of the effective instructional programs reviewed] challenged the belief that
ability is a prerequisite condition for intellectual or academic development.
Rather, each believed instead that motivation is the key factor, and demonstrated
that motivation is tied to the meaningfulness of the curriculum content and
instructional approach (p. 118).
Conclusion
Data from this study of seven successful urban mathematics teachers sheds some
light on some of the important characteristics of these teachers, and, possibly, effective
teachers of traditionally underserved students more broadly. This chapter sought to
answer three research sub-questions related to these characteristics: What is the nature of
the teachers’ interactions with their students? What are the teachers’ attitudes toward
their students and their profession? What motivates them to teach mathematics in general
and to teach this population of students in particular? Five themes have been identified in
this chapter, and each of them contributes some insight into each of the three questions.
These themes are, 1) the teachers view their work as a meaningful form of social service,
and they are committed to serving in a setting where the perceived need for service is
greatest; 2) the teachers have a strong sense of efficacy...they are convinced that their
work makes a positive impact on student lives; 3) an ethic of care permeates the work of
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the teachers; 4) the teachers profess and demonstrate a high level of respect for their
students; and 5) the teachers profess and demonstrate the belief that all students can
achieve in mathematics. Table 4.1 briefly summarizes how these themes relate to the
three guiding research questions. Illustrative commentary from the teachers is also
included in this table.
The nature of these teachers’ interactions with students, then, can be characterized
as being imbued with care and respect. The meaning of these terms in the context of an
urban classroom has been discussed above. In brief, “care” involves not only showing
concern and interest in the students, but also seeing to it that students perform to their
ability. Showing “respect” includes viewing students as intelligent individuals and
avoiding an impulse to generalize them in relation to their various group memberships (as
adolescents of color, as poor students, as English language learners, etc.). Teachers’
attitudes toward their students includes care and respect, but also the belief that all
students should be expected to achieve at high levels in mathematics. Their attitude
toward their profession is that teaching is more vocation than occupation: it is a
meaningful avenue toward social service rather than a mere source of income. This bent
toward social service also relates to the teachers’ decision to teach mathematics in urban
settings. The teachers assume that students in urban schools have the strongest need for
dedicated teachers. As argued in Chapter 1, standardized test data do seem to support
this assumption: that is, the achievement gaps in mathematics seem to indicate that
students in urban areas have not received adequate service in the mathematics classroom.
These teachers have pro-actively attempted to address this issue through their own work.
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Research Question Related Research Finding Illustrative Teacher Comment
1. What is the nature of the teachers’ interactions with their students?
The teachers demonstrate care for their students in words and actions. The teachers profess and demonstrate respect for students.
“I do believe that the teacher has to believe in [the students] first. I say to people, ‘You have to love them.’ You do, you have to love them. But I think they know that I truly care...” Ms. Frederick “When I meet kids, I think I’ll show them respect with the expectation that respect is shown back to me. I’ll listen to them. I will ask them about things other than academics. Establish some type of relationship. Let them know that they can approach me at any time, in the appropriate manner.” Ms. Zimmerman
2. What are the teachers’ attitudes toward their students and their profession?
The teachers believe that their work has a positive impact on students’ lives. The teachers profess and demonstrate the belief that all students can achieve in mathematics.
“I think I have more of an impact here, what I’m doing. I’m more influential here.” Ms. O’Reilly “I don’t take excuses. There’s a lot of people, I think they’re well intentioned, who say, ‘Oh, gee,’ you know, ‘these poor kids. They’ve had this or that or the other....they can’t do this, they can’t do that.’ I don’t do that.” Mr. Oden
3. What motivates them to teach mathematics in general and to teach this population of students in particular?
The teachers find meaning in the act of engaging in social service, particularly where the perceived need for service is greatest.
“I’d rather be where I’m more needed. You know, where the kids are who aren’t used to having high expectations.” Ms. Thompson
Table 4.1: Summary of Findings
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The findings of this chapter provide some indication of the attitudes and
dispositions of effective urban mathematics teachers. The underlying attitudes of the
teachers in turn affect the specific instructional strategies the teachers use in their
classrooms, and these pedagogical approaches are the focus of the next chapter. The
manner in which these attitudes affect practice will be illustrated in Chapter 5. Indeed,
one particularly salient finding in this chapter, namely the observation that the teachers
operate under the assumption that all students have the potential to excel in mathematics,
has such a strong impact on the teachers’ instructional approach that it will be offered as
the foundation of a grounded theoretical model describing the teachers’ practice.
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CHAPTER 5
PEDAGOGICAL APPROACH
This investigation of the practices of successful mathematics teachers of
traditionally underserved students seeks to uncover some key components of the
participating teachers and their work. Chapter 4 addressed three of the research sub-
questions, namely: 1) What is the nature of the teachers’ interactions with their students?
2) What are the teachers’ attitudes toward their students and their profession? 3) What
motivates them to teach mathematics in general and to teach this population of students in
particular? This chapter addresses the remaining research sub-question: What are the
pedagogical styles of the teachers? As this chapter is essentially related to the teaching
styles of the teachers, further insight into the earlier question related to the teachers’
interactions with their students will also be provided.
The structure of this chapter is notably different from the structure of the
preceding one. Chapter 4 focused more on individual characteristics, providing separate
portraits of each of the seven teachers. This chapter is more focused on general patterns
found across the teachers, and hence seeks to lay out principles of instruction which
apply to all of the teachers. The organization of Chapters 4 and 5 also differ
substantially. The bulk of Chapter 4 described individual teachers, and then the chapter
concluded with a discussion of some of the common characteristics that were found
across teachers. This chapter begins with a general framework describing the teachers’
pedagogy. The general framework will then be illustrated with particular examples of
individual practice drawn from the research data. Finally, the data which is emphasized
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in this chapter differs from the data emphasized in the previous chapter. As Chapter 4
was directed at teacher attitudes, most of the data presented in that chapter was drawn
from the teachers’ own words and subsequently supported with brief examples from
practice. This chapter’s focus on teaching style necessitates that actual classroom
practice, described primarily through observational data, come to the fore. Whenever
possible, descriptions of practice will be supported by teacher commentary.
An additional distinction between Chapters 4 and 5 pertains to the degree of
interpretation provided by the researcher. Chapter 4 was primarily descriptive. The
teachers’ own commentary shaped the structure of the chapter, enabling particular
descriptions of their work to take place. This chapter is primarily interpretive. The task
of converting a year’s worth of interviews and observations into a sensible over-riding
framework requires that patterns and principles be abstracted from the data. This effort
to make sense of the complete mass of data inevitably requires a generous amount of
interpretation on the part of the researcher. The interpretations presented herein remain
grounded in the data, however, and all claims will be made with reference to the data.
The next section presents the overarching framework which has been developed to
capture the teachers’ pedagogical styles. Subsequent sections are intended to illustrate
and support this framework with reference to classroom occurrences and teacher
commentary.
Faith and Communication: A Framework for the Pedagogy of Effective Teachers
Chapter 4 concluded with the observation that the teachers in this study all
operate under the assumption that their students have the ability to excel in mathematics.
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This powerful assumption serves as the foundation of the teachers’ approach to
instruction. This belief in student ability is not a case of blind faith in the students.
Rather, the teachers’ belief is grounded in their respect for the students’ existing
knowledge. The teachers consider their students to be intelligent when they enter the
classroom, and the perceived present intelligence of the students convinces the teachers
that the potential to excel is there. Hence, the two attitudes, respect for the current
knowledge of students and belief in the students’ future potential, influence and build on
each other. Students’ present knowledge, which the teachers view as adequately
developed and valuable, serves an additional important role in the teaching and learning
process. The students’ existing knowledge serves as the starting point for instruction.
That is, teachers help students develop an understanding of new mathematical concepts
by connecting these concepts to ideas and experiences which are already familiar to the
students.
The effort to develop new ideas from existing ones is no small task. The task
requires first and foremost that the teacher receive a clear picture of what the students’
existing ideas are. Student ideas can only be revealed via effective communication. That
is, if the teacher is to uncover student ideas and subsequently build on these ideas, the
teacher must first open a window into existing student knowledge by helping the students
to effectively communicate what they know. The teachers in this study fostered such
communication via two avenues. First, the teachers heavily emphasized the use of
technical mathematical vocabulary in the classroom. The language of mathematics, with
its largely precise and relatively unambiguous terms and definitions, provides a common
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and clear language with which to discuss mathematical ideas. By repeatedly emphasizing
and modeling the use of technical vocabulary in the classroom, the teachers bolstered
student ability to effectively communicate their existing ideas, ideas which the teacher
would in turn attempt to expand.
The second avenue through which teachers fostered effective mathematical
communication involved the creation of a safe classroom environment for sharing ideas.
Uncovering students’ existing ideas is of central importance to the teachers as these ideas,
in turn, serve as the starting point for instruction. Hence, the teachers want the students
to be open about their ideas and feel comfortable sharing them. Fostering such an
environment is quite a challenge in the adolescent world of middle and high school,
however. The common insecurities of adolescents often serve to block meaningful
communication in the classroom: students may be afraid to discuss developing but
incomplete ideas for fear of looking “dumb,” students may wish to avoid being viewed as
overly enthusiastic about school work for fear that their peers will interpret this as
“nerdiness,” etc. The teachers in this study have found ways to overcome these potential
impediments to effective communication in the classroom, and hence, have opened
another means through which to uncover students’ existing knowledge. The manner in
which this was done will be illustrated later in this chapter.
The pedagogical approach of the seven effective teachers in this study, then, can
be captured via four inter-related tenets of a common framework. 1) The teachers
assume that the students are capable of excellence in mathematics. This assumption is
informed by a respect for students’ existing knowledge, and this assumption prompts
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teachers to deliver challenging mathematical content to their students. 2) Again, respect
for students’ existing knowledge causes the teachers to believe in the students’ ability to
excel. Students’ existing knowledge also serves as the departure point for instruction:
teachers attempt to develop new mathematical ideas from students’ existing base of
knowledge. 3) The centrality of students’ existing knowledge in instruction requires that
teachers have a clear picture of what the students’ ideas are. This picture comes into
focus via effective communication in the classroom. If the teacher is to build on existing
student ideas, the teacher must know what these existing ideas are. In order for the
teacher to be aware of student ideas, the students must communicate them to the teacher.
One way that the teachers help foster effective communication in the classroom is by
emphasizing technical mathematical vocabulary in instruction. This provides a shared
and relatively unambiguous language with which to effectively communicate ideas. 4) A
second way the teachers foster effective communication with a view toward uncovering
existing student knowledge involves the creation of a safe classroom environment for
sharing ideas. Simply having a powerful language to express ideas is not sufficient.
Students must also feel that they can share their developing ideas without fear of
subjecting themselves to repudiation or humiliation. The teachers have put structures in
place which promote the sharing of ideas in a safe manner.
This four-part framework is captured visually in Figure 5.1. Readers may find it
useful to refer to this diagram repeatedly as they continue reading this chapter.
Subsequent sections will more thoroughly explain the individual tenets of the framework
and provide samples of data which support and illustrate the tenets. While the tenets will
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be discussed separately in future sections, it is important to recognize their inter-
connectedness. The illustration models these connections. For example, double-sided
arrows connect the bubbles for “Strong Student Ability Assumed” and “Focus on What
Students Know.” This is due to the fact that the teachers’ respect for students’ existing
knowledge leads them to believe that students can do well in mathematics. The belief in
student ability to achieve prompts the teachers to present challenging mathematical tasks
to the students. The instructional approach to this mathematics begins with students’
existing ideas. Hence, there is a clear interplay between the two tenets shown on the left
side of the diagram. The location of the “Focus on What Students Know” bubble in the
middle of the diagram is itself significant. Students’ existing knowledge is central to all
instruction; it represents the starting point for teachers as they move forward with the task
of teaching. Uncovering this existing student knowledge requires effective mathematical
communication in the classroom, illustrated as a window in the diagram as effective
communication enables teachers to “see” what students already know. As discussed
earlier, effective communication involves two components: emphasis on mathematical
vocabulary and the development of a safe classroom environment for meaningful
communication. Double-sided arrows link each of these tenets with the central tenet
“Focus on What Students Know” indicating the interplay between them. For example,
the teachers’ desire to build instruction from existing student ideas prompts them to
uncover student ideas, or to open a window into students’ existing ideas. Hoping to
obtain a good sense of students’ mathematical thinking, the teachers help develop a
common and precise mathematical language in the classroom via emphasis on
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vocabulary. Armed with the language of mathematics, the students are then in a position
to more clearly share their mathematical ideas. This, in turn, helps teachers “know what
the students know,” which provides a starting point for instruction. Likewise, the
teachers’ desire to uncover student knowledge prompts teachers to encourage student
communication via the creation of a safe classroom environment for sharing ideas. Thus
empowered to share their thinking without fear of negative backlash, students become
increasingly willing and able to share their ideas in class. This also feeds into teachers’
awareness of student ideas and solidifies the foundation of instruction. The cycle of
teachers fostering effective communication, receiving communication from the students,
and allowing this communication to inform instruction repeats throughout the year.
Strong Student Ability Assumed
Focus on What Students Know
Effective Mathematical
Communication: A Window Into
Student Knowledge
Emphasis on Mathematical Vocabulary
Safe Environment For Meaningful Communication
Figure 5.1: A Model for the Practices of Effective Teachers
The next four sections of the chapter describe each of these teaching practices in
depth. The individual tenets of the framework will be illustrated and supported with
reference to observation, interview, and archival data. This model for effective teaching
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will also be connected to existing concepts about effective mathematics instruction for
traditionally underserved students found in the research literature.
Strong Student Ability Assumed
In the early 1970s, Rist (1970) argued that patterns of low achievement in urban
schools were the result of a self-fulfilling prophecy. Rist found that teachers and
administrators assumed that poor children and children of color were unable to perform at
high levels in schools, leading educators to create less rigorous curricular tracks for these
students. As many urban students were prevented from receiving exposure to
challenging curricula, they were effectively denied the opportunity to achieve in school.
Hence, educators’ initial assumptions about what urban students might be able to
accomplish ultimately determined what these students had the opportunity to accomplish.
Low achievement, then, was simply the inevitable result of low expectations.
High Expectations
The teachers in this study provide some evidence related to the converse of Rist’s
(1970) arguments. These teachers begin with the assumption that the students can and
will do well in mathematics. This assumption drives their approach to teaching and is,
therefore, one component of the teachers’ success. Mr. Oden captures this attitude well:
I create an expectation that, you know, we’re going to do this stuff....my mental
image of these kids is that they’re perfectly capable of all these mathematical
operations. And, my job is to find out a way of presenting it, a way of explaining
it, so that they can understand it on their own terms.
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The teachers not only hold this belief themselves, but they also communicate it to
the students in words and actions. The message to students is not only “you can do this”
but, more strongly, “you will do this.” This is a powerful idea, particularly for students
from traditionally underserved demographic groups. It runs counter to the destructive
message of low expectations, a message the students have likely encountered before
given its unfortunate prevalence in urban schools (Chenoweth, 2007).
Ms. Thompson provides an illustrative example of how these successful teachers
communicate in word and deed the message that students can and will achieve. She
indicated in an interview:
I’m always telling the kids… not just the honors, like all the kids…like, you
know, “You’re doing this. Next year you’re going to take Algebra II. And your
senior year you’re going to take this. When you get to college you’re going to do
this.”
Ms. Thompson is not saying, “Maybe you’ll go to college some day, but if you are to do
so, you’ll have to take a sequence of mathematics courses first.” This message leaves
plenty of room for doubt, but Ms. Thompson’s message is definitive. She informs
students that they will succeed in the college preparatory mathematics sequence and that
they will go on to college from there. Further, she goes beyond merely “telling” the
students this message. She integrates experiences into her classroom approach which
further underscore her assumption that all of her students are college-bound. The
following scene was observed during one of her classes in November. The students had
just completed their first task of the class, a journal-writing exercise in which they wrote
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some reflections on their understanding of the material they had covered in class that
week. The passage below describes the next few minutes of the class, which preceded
the introduction of a new mathematical topic:
Ms. Thompson then said, “OK. It’s now time for College of the Day,” and she
wrote COTD on the board. This is a periodic exercise in which the students talk
about a college they might consider attending. Today’s “College of the Day” was
Emerson College in Boston. Ms. Thompson asked if any students had visited the
college. One student had (a young woman at the front of the room), and she
described her experience visiting the college. When the student was done, Ms.
Thompson held aloft a brochure from the college, discussed some of the majors
offered, and briefly discussed what these majors entailed. She then asked if
anyone was interested in perusing the Emerson brochure…one male student
raised his hand and she passed it to him.
It should be noted that the class described above was a senior calculus class. The
fact that these students had reached so far in their mathematics studies likely indicates
that many of them were considering plans for college before arriving in Ms. Thompson’s
class. However, this was not an honors-level or AP-level course. These students were
not the “elite” of the school, so there was still room for doubt regarding their future
prospects. Ms. Thompson’s approach seeks to erase this doubt by emphasizing to
students that college is a reality for them. Her periodic “College of the Day” discussions
get students to think about the possibility of college and also expose them to some of the
different options that are out there. “College of the Day” was not a component of some
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of her sophomore and junior-level classes I observed, but she consistently delivered the
message to her underclass students that “the mathematics you do now as a sophomore
leads to the mathematics you’ll do as a junior, then a senior, then a college student.” She
is representative of all the teachers in this study in communicating to students that they
can and will succeed in school.
Classroom Management Focused on Learning
The teachers’ belief in the students’ academic ability is demonstrated in the
classroom in numerous other ways. Classroom management style is another powerful
avenue through which the teachers demonstrate their assumption that the students are
academically competent. The teachers’ classroom management styles can be described
as content-focused. That is, mathematical teaching, learning, and work are the primary
tools used for maintaining order in the classroom. Effort is made to keep students
occupied with the business of learning; the underlying management assumption is that
students who are kept focused on worthwhile academic content will not have the time or
inclination to engage in undesirable behaviors. This stands in contrast with other forms
of classroom management and discipline which are often found in schools, such as
systems of punishment and reward, removing unruly students from the classroom,
verbally lambasting students, etc. Ms. Thompson articulates the consequences of a
content-based management style versus a punishment and reward-based style. She states:
I assume…and I communicate this to them…I assume that you’re not going to
misbehave, because why would you? We have the same goal, and the goal is for
you to learn math. If you’re not doing what you’re supposed to, then you’re not
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learning that math. Why would you do something to hurt yourself? You know
what I mean? So it’s not like you’re doing these things to please me. You’re
doing these things because you should. So when a student, let’s say, cuts class. I
say to that student, “OK, well, you cut class. What happened? You missed this.
You need to make up that time with me after school because you need to learn
this material.” So, it’s a natural consequence.
Ms. Thompson’s attitudes about her students, and specifically her perspective that
the students are capable learners, underlay this commentary on her classroom
management style. Note her approach to the hypothetical “class-cutter.” She does not
view such a student as being rebellious, lazy, disrespectful, poorly behaved, etc. She
does not react to the student by having him or her write 500 sentences, or by sending the
student to the principal, or by simply ignoring the behavior because the student is
“hopeless.” Rather, she acknowledges that the student has erred, and she confronts the
problem. Her manner of dealing with the situation sends the message that cutting class in
not acceptable, but, more importantly, it sends the message that the student is able to
learn and is expected to learn.
This principle of classroom management was revealed in other classrooms as
well. In every classroom observed, there were moments in which a student or group of
students began to “misbehave,” i.e., engage in behaviors which did not include the
learning of mathematics and also served to distract other students from learning
mathematics. The occasional occurrence of such behavior is perhaps inevitable in middle
and high school classrooms. All of the teachers addressed such behavior in a consistent
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manner, namely, redirecting the offending students toward learning and doing
mathematics. One example from Ms. O’Reilly’s class was provided in Chapter 4. She
had organized her class into small groups, and the students were expected to investigate
geometric solids and nets. One table of boys became unruly. Ms. O’Reilly quietly
approached their table and asked about their progress on the task. Her gentle questioning
of the boys revealed that they had encountered a conceptual obstacle which prevented
them from making progress on the task. Ms. O’Reilly clarified the issue, and this was all
it took to get the boys to return their focus to the geometry task for the remainder of the
period.
This principle can also be illustrated from an episode drawn from Ms.
Zimmerman’s 8th grade class. Her daily routine usually involved starting the class with a
“warm-up” problem, a problem designed to get students thinking about some
mathematical ideas which would be relevant to the day’s lesson. The text below is drawn
from an observation made on a day in which the students were given three warm-up
problems relating the perimeter and area of a rectangle:
Most of the students took the warm-up problems seriously. Two boys seemed
inclined to discuss other things, but Ms. Zimmerman didn’t tolerate this for long.
Calling them each by name, she told them, in effect, that this was not the time for
such discussions, and that they needed to focus on their work. This worked for
both boys…they got down to work immediately.
Indeed, this approach of dealing with undesirable behavior by redirecting students
toward mathematical content was found in the classrooms of all seven teachers. As
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indicated in the examples above, this approach was effective in that it achieved the result
of restoring order to the classroom. Perhaps more importantly, though, is the message
sent by this disciplinary approach. The teacher communicates to students that they are
expected to learn mathematics. The teacher’s expectation that the students will learn
mathematics provides further evidence of the teacher’s belief that the students can learn
mathematics. Other approaches to classroom discipline might send other messages. For
example, the teacher who simply ignores unfocused behavior also sends the message that
students are not necessarily expected to learn, and, perhaps, that they cannot learn. The
teacher who punishes behavior by removing the student from the learning environment or
enacting a punishment unrelated to learning sends the message that learning is not
necessarily the uncompromised goal of the classroom. The successful teachers in this
study clearly communicate that learning is the entire purpose and that the students can
learn at high levels. Rejecting the attitudes of “these students are rude and need to be
punished” or “these students are hopeless and can be ignored,” the teachers instead have
an attitude of “these students have the ability to do well, and it is my responsibility to see
to it that they do…distractions, therefore, cannot be tolerated.” Gay (2002) found similar
characteristics in effective teachers of diverse students. Gay noted that effective teachers
of ethnically diverse students create “classroom environments that are conducive to
learning.” Further, Gay argued that teachers who are overly permissive of non-
academically focused student behaviors exhibit “benign neglect under the guise of letting
students of color make their own way and move at their own pace” (p. 109).
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Presenting Challenging Mathematical Content
Another manner in which the teachers demonstrate their belief in student ability is
by their curricular choices. Namely, most of the teachers purposefully presented
challenging mathematical ideas to their students, concepts which went beyond both the
content found in the students’ textbooks and the material required by state frameworks.
Mr. Oden, for example, required his 10th grade geometry students to produce formal
proofs of various theorems despite the fact that geometric proofs are no longer required
by the state frameworks and are not included in the state standardized test. Ms.
Thompson likewise introduced an added level of rigor in her geometry class, challenging
students not only to use established formulas, but to be able to derive these formulas as
well. The vignette below describes a situation in which her students were asked to
calculate the area of an equilateral triangle whose side length was known:
[The problem] involved the use of a formula (for an equilateral triangle,
bh23
= ). [One] student used the formula in her solution, correctly entering the
value of b in order to calculate h. Ms. Thompson said to the students, “This
student made good use of the formula. Does anyone know how we might find the
formula?”
A student replied, “Look it up in the book.”
Ms. Thompson laughed. “That’s certainly one way to find the formula. But let’s
say the formula wasn’t in the book: could we figure it out ourselves?”
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Several students responded to this higher-order question, and collectively these
students derived the formula using the Pythagorean Theorem. Ms. Thompson did
not tell them how to do it, but she did synthesize their contributions into a valid
demonstration of where the formula comes from.
These examples from Mr. Oden and Ms. Thompson, and similar examples which
can be drawn from the work of the other teachers, are powerful indicators of the teachers’
belief that the students can fully understand and use challenging mathematical ideas. Mr.
Oden goes beyond asking students to observe the fact that the diagonals of a
parallelogram bisect each other, or to use this property in the solution of unknown
lengths. He expects them to engage in the uniquely mathematical process of formally
proving that this property must be true for any parallelogram. Similarly, Ms. Thompson
is not satisfied that her students can look up a formula in a book and use it to correctly
solve a problem. She pushes them to derive the formula, connecting it to other more
fundamental ideas. This shows a great deal of respect for the intelligence and
mathematical ability of the students. Rather than minimizing their learning opportunities
by focusing only on the base requirements, these teachers attempt to push student
learning as far as they can. Chenoweth (2007) similarly noted that effective instruction
of traditionally underserved students involves aiming for maximal attainment as opposed
to settling for minimal, basic goals. Chenoweth’s study of successful urban schools
found that effective schools did not attempt to prepare students for the minimal
requirements assessed on standardized tests, but instead offered curricular opportunities
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extending well beyond the base expectations. The teachers described here incorporated
this strategy at the classroom level.
Student Participation
Yet another manner in which the teachers exhibit their belief in student ability
involves the respect they show for student contributions in the classroom. Student ideas
about mathematics are taken seriously. Indeed, student input on how to approach a
particular mathematics problem is considered just as valid as the teacher’s input,
provided that the student can establish the validity of his or her approach. The teachers
actively invite students to discuss alternative approaches to mathematical problems, and
highlight and celebrate valid student-generated approaches publicly. Albert (2003) has
identified this process of welcoming student input, or encouraging the development of
“student voice,” as a key component of promoting academic success among adolescents.
“Valuing student voice means allowing students to speak and, when they speak, being an
active listener in order to understand their perspectives” (p. 56).
The teachers in this study regularly demonstrated that student voice was valued in
their classrooms. The vignette below from Ms. Zimmerman’s 8th grade class is
illustrative. The students were asked to find the area of a rectangle whose perimeter was
16 and whose width was 3. She provided no hints regarding how to approach the
problem, but instead invited students to share their strategies:
[A boy] volunteered to do the first problem. He went up and demonstrated a
satisfactory solution. Ms. Zimmerman prompted him to explain and justify each
step of his work as he went. His method involved drawing a sketch of the
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rectangle, labeling two opposite sides as having width of 3, subtracting 6 from 16,
dividing the difference of 10 by 2 in order to arrive at the length of 5, and then
multiplying 5 and 3 to get an area of 15. Another boy from the front of the room
raised his hand and said, “I solved the problem a different way.” Ms.
Zimmerman asked him to share his method. He chose to start by dividing the
given perimeter of 16 by 2 to get 8, and then subtracting the given length of 3
from 8 to get 5, and then multiplying 5 x 3 to get the area of 15.
Ms. Zimmerman expressed enthusiasm for both approaches, and clearly
reiterated both of them for the whole class, all the while giving the two boys credit
for coming up with these solution paths.
At one point during the warm-up review, a student raised a question to the effect
of, “What if we did it another way? Could we do this instead?” The strategy the
student proposed wouldn’t work, but Ms. Zimmerman’s approach to the situation
was to say, “Well, let’s see if that does work.” At that point, another student
jumped in and said, “No, that wouldn’t work in this case (referring to the problem
which was being discussed at the time), because….” Ms. Zimmerman then said to
the boy who proposed the faulty strategy, “Well, I guess that won’t work here, but
I’m glad you’re thinking about other ways to solve the problem.”
Solution strategies to this problem were provided entirely by students in this
vignette. Two valid yet distinctive approaches were presented, and Ms. Zimmerman
simply validated them and reiterated them for the rest of the class. A third approach,
which happened to be faulty, was also brought up. In this vignette, Ms. Zimmerman did
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not need to point out the flaws of the third approach, as another member of the class was
able to identify and describe the shortcoming. Yet even this faulty strategy was
welcomed as an intelligent and worthwhile contribution to the class.
It is telling that Ms. Zimmerman refrained from directing the discussion of the
solution process to this problem. She entrusted the students to present solution processes
themselves, demonstrating her belief in the students’ ability both to do mathematics and
to explain their mathematical ideas to others. Ms. Zimmerman regularly encouraged
students to discuss their solution processes, and enthusiastically greeted valid strategies
which differed from a strategy that she herself might use. This was the case for the other
teachers in the study as well.
A case from Ms. O’Reilly’s room further demonstrates a teacher’s respect for her
students’ ideas, a respect which is grounded in the teacher’s belief in student ability. Ms.
O’Reilly was somewhat unique among the teachers in this study in that her pedagogical
style was more teacher-centered than the other teachers. For instance, the Ms.
Zimmerman vignette above revealed how Ms. Zimmerman stepped back and permitted
students to direct the mathematical discussion by presenting their own ideas. The other
teachers often utilized a similar strategy. Ms. O’Reilly typically avoided this approach,
however, and usually directed mathematical discussions herself. While her own
perspectives on the mathematics were privileged in the classroom, she still demonstrated
a strong respect for student ideas, as indicated in this vignette:
Ms. O’Reilly went over the problem…the problem displayed two similar triangles
(the clue that they were similar was provided because corresponding angles were
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marked as being congruent within the triangles). The first triangle was ABC with
side AC = 21 and side BC = 27. The other was triangle DEF with DF = x and
EF = 18. The task for the problem was to solve for x, which is an exercise of
setting up an equality of ratios for the two similar triangles, and also finding the
scale factor dilating the first into the second. Apparently they had already done
similarity, so the idea of scale factor is the only thing that’s new. Ms. O’Reilly
explained how to solve the problem…she showed how “x” could be found by
setting up equal ratios. Many students in the room somehow fixated on the idea
that the problem could be solved by subtracting 9. They noted that you can get
the 18 from triangle DEF by subtracting 9 from ABC’s 27. They then erroneously
concluded that the solution of x = 14 could be found by subtracting 9 from 21.
Ms. O’Reilly pointed out that 21 – 9 does not equal 14, however. A female
student then made an interesting argument. “I got the answer of 14 by
subtracting. I knew that if you subtracted 9 from 27, you’d have to subtract 7
from 21.” Ms. O’Reilly said, “Hmmm. You certainly got the right answer. I’m
not sure I understand exactly what you did, though. I hope you can explain it to
me later.”
The students broke for lunch shortly afterward. As can be expected, they hustled
out of the room in order to get to the cafeteria. About 5 minutes later, the same
student came back and explained her reasoning. The student’s chain of reasoning
was somewhat complex, but her essential argument was 9 is one-third of 27, and
that 9 was subtracted from 27 in order to arrive at the length of the
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corresponding side of the other triangle, 18. The student noted that one-third of
21, or 7, should likewise be subtracted from 21 in order to arrive at the solution x
= 14.. Ms. O’Reilly, though eating her lunch, engaged the student in
conversation and validated her idea.
While Ms. O’Reilly chose not to invite this student to share her alternative
method during class time, she still demonstrated a good deal of respect for the student’s
idea. Herein is a parallel between the earlier Zimmerman vignette and this one from Ms.
O’Reilly. When a student proposed a faulty solution in Ms. Zimmerman’s class, she
refrained from dismissing it outright, but instead permitted another student to point out a
flaw in the argument. Similarly, when a student proposed a hard-to-follow strategy in
Ms. O’Reilly’s class, Ms. O’Reilly refrained from dismissing it, but instead invited the
student to discuss her idea at a later time. The student took her up on this offer, and Ms.
O’Reilly engaged the student in discussion about the problem, and, ultimately, accepted
the student’s valid (if unclearly communicated) approach. We see in both Ms.
Zimmerman’s work and Ms. O’Reilly’s work a firm respect for student ideas. Again,
given that these teachers hold the belief that the students have strong mathematical
ability, taking students’ mathematical ideas seriously is one manner of demonstrating this
belief in action.
Summary
The successful urban mathematics teachers highlighted in this study operate
under the assumption that their students are capable of achieving in mathematics. It has
been argued that this assumption is the foundation of their approach to teaching, and
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hence has been presented as the first tenet of a framework describing their pedagogy.
This section has discussed ways in which the teachers’ belief in student ability is
manifested in the classroom. Namely, the teachers communicate to students in both word
and action not only the viewpoint that the students can succeed in mathematics, but also
the expectation that the students will succeed. The teachers’ classroom management style
springs from this assumption as well. Since the teachers believe that their students can
succeed, they view it as their responsibility to see to it that the students do succeed. This
requires them to address unfocused student behavior by redirecting students back to the
primary objective of the classroom, which is learning and doing mathematics. The
teachers’ belief in student ability is also manifested in the fact that the teachers choose to
challenge their students with higher-order mathematical tasks, tasks which exceed the
minimal requirements of the school or the state. Finally, teachers demonstrate their belief
in student ability by respecting student ideas in the classroom, viewing valid student
perspectives on mathematics as being as worthwhile as the teacher’s.
It bears repeating that the teachers in this study are experienced; each teacher has
served at least five years in the classroom. This suggests that their belief in student
ability is not based on untested ideology. One would assume that the teachers would not
hold such high regard for their students if their students had not proven over the years
that they are worthy of such respect. A major argument from this section is that the
teachers value student ideas. Again, it can be inferred from the data that this is so
because the teachers have seen over the course of their tenure that student ideas are
valuable. This ties in directly with the second tenet of the proposed pedagogical model,
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the notion that teachers focus their instruction on the valuable things that students already
know when they enter the classroom. This tenet is discussed in depth in the next section.
Focus on What Students Know
As noted in the previous section, the teachers in this study value the knowledge
students bring with them to class. Existing student knowledge is considered sufficient for
further mathematical development, and, ultimately the high level of achievement the
teachers expect of their students. Students’ present knowledge feeds into the teachers’
belief that all students have potential to succeed, and it also serves as the starting point
for continued teaching and learning.
In describing the teachers’ approach of focusing their instruction on what students
already know, it is useful to contrast this approach with another approach which is often
found in schools, particularly schools serving traditionally underserved students. The
contrasting approach can be described as “fixating on what students don’t know” or
pointing to perceived gaps in students’ past schooling and using them to excuse present
under-achievement. A fly on the wall of many school staffrooms is likely to hear
comments such as, “These students don’t know how to add, so how can we be expected
to teach them to multiply?” or “These students never learned how to operate with
numerical fractions, so how can I teach them to manipulate algebraic fractions?”
Teachers with this attitude are unlikely to genuinely believe that their students can
accomplish much in mathematics, and are in danger of contributing to another cycle of
the self-fulfilling prophecy of low expectations (Cohen & Lotan, 1995; Rist, 1970).
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Valuing and Connecting To Student Knowledge
The successful teachers in this study reject this attitude. While they are certainly
cognizant of the fact that some of their students do not have the full set of mathematical
skills one might desire for the age group (e.g., it is unfortunately true that some ninth
graders haven’t mastered their multiplication facts), they do not view the existence of
under-developed factual knowledge as an insurmountable barrier to student achievement
moving forward. Mr. Oden addresses this:
I don’t take excuses. There’s a lot of people, I think they’re well intentioned, who
say, “Oh, gee,” you know, “these poor kids. They’ve had this or that or the other.
They can’t multiply, they can’t do this, they can’t do that.” I don’t do that…What
I try to do is meet people as people. Just try to come in here and take them as
they were. Not to pass judgment on them, but to find out where they were. Not
to rely on anyone’s word for it, but just see what they could do, and then take
them the next step. And, yes, everyone’s talking about the state guidelines and all
that other stuff…but, ultimately, teaching is about connecting.
This brief quotation speaks volumes about Mr. Oden’s approach to teaching his
students, and it is representative of the approach used by the other teachers as well. His
statement again underscores his assumption that all of his students can achieve. He
doesn’t “take excuses,” or use perceived deficiencies in their existing mathematical
knowledge as a legitimate reason to hold them to low expectations. Rather, he endeavors
to “take [the students] as they [are]…and then take them the next step.” That is, he finds
out what they do know, and he tries to build on the promise that is already there toward
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further development. His concluding remark is powerful: “ultimately, teaching is about
connecting.” That is, making connections to what students already know and are already
familiar with, and connecting that existing knowledge to the new knowledge we wish for
them to acquire.
Mr. Oden’s remarks resonate with a statement provided by Ms. Thompson: “I
never like to [teach] something that’s like totally from left field, brand new….I like to
always have it build on something that they already know, because then it seems more
accessible to them.” For these teachers, the act of teaching is the act of helping students
recognize seemingly new ideas which are clearly related to the students’ existing ideas.
This requires the teachers to focus their instruction on things the students already know,
and build from there. Villegas and Lucas (2002) identified this stance as a fundamental
component of culturally responsive teaching.
[Effective teachers] see all students, including children who are poor and of color,
as learners who already know a great deal and who have experiences, concepts,
and language that can be built upon and expanded to help them learn even more.
Thus they see their role as adding to rather than replacing what students bring to
learning. They are convinced that all students, not just those from the dominant
group, are capable learners who bring a wealth of knowledge and experiences to
school (p. 37).
Clearly, then, the teachers need to have a reasonable picture of what their students
do know in order to build instruction from there. The next two sections of this chapter
will address how the teachers gain insight into the existing knowledge bases of their
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students. For now, it suffices to say that the teachers endeavor to uncover students’
existing ideas as the foundation for their work. Effective teaching and learning involves
action from more than just the teacher, however. If students are to learn new ideas, then
they also must be in touch with what they already know so that they can make
connections to new knowledge. Ms. Zimmerman makes a point of assisting students to
take inventory of and reflect more deeply about their existing knowledge:
I try to focus on metacognition. You know what you know, and you’re also
aware of what you don’t know….If [students] tell me [they] have problems with
#9, that means nothing to me….Where is it that the confusion starts? Is it a
positive or negative number? Is it changing signs? I want them to be that
specific. I do this primarily because it drives me crazy when [students claim that
they have no idea about how to approach a problem]. “I don’t get it” is not
acceptable. You need to tell me specifically, otherwise me helping you is a waste
of time.
Ms. Zimmerman’s comments reveal her perspective that teaching and learning
involve a good deal of effort from both teacher and student. Not only does the teacher
need to ascertain what the student knows as a basis for further instruction, but the student
also needs to reflect on what he or she does and doesn’t know about a given problem
before seeking further guidance from the teacher. Her statement “’I don’t get it’ is not
acceptable” is a reasonable stance given this approach to teaching. These teachers
believe in student knowledge and value student knowledge. The students are not “empty
vessels,” so it is impossible to accept a student’s claim that he or she knows nothing
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about a particular problem. He or she certainly has many relevant ideas which might
contribute to the solution of the problem, and the responsibility of connecting what he or
she does know to the problem at hand is as much the student’s responsibility as the
teacher’s.
Effective teaching and learning, then, resembles a dance between teacher and
students. Both parties must exude effort to make connections: the students must attempt
to reflect on what they know and struggle to make relevant connections to a new idea; the
teacher must pick up on the foundational knowledge students reveal and devise strategies
for helping students make those connections. Doerr (2006) uncovered a similar interplay
between effective teachers and their students during the teaching and learning process,
describing such an approach to teaching as a hermeneutic orientation. “Teachers with a
hermeneutic orientation interact with their students, listening to their ideas and engaging
with them in the negotiation of meaning and understanding” (p. 6).
The following episode from Ms. Kelly’s 6th grade classroom provides an
illustrative example of how this interaction between teacher and students can play out in
the classroom. Ms. Kelly had divided the students into five small groups and set up five
stations around the room. Each station included a large piece of newsprint with an open-
ended geometric question printed on it. Groups were instructed to spend ten minutes at a
station and to write their responses to the question on the newsprint. They would then
rotate to the next station, repeating the process until all five groups had visited all five
stations. At the end of the activity, Ms. Kelly posted the newsprints in full view of the
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entire class, and asked students to comment further on some of the ideas they had come
up with. The text below chronicles this “reporting out” phase:
They focused on the “Do all rectangles with the same perimeter have the same
area?” question first. All students answered, “No,” but their reasons for saying
“no” varied, and none were really sound arguments. Ms. Kelly highlighted some
of the promising ideas present in the students’ responses, such as, “A rectangle
might be a square, and the square will have a different area.”
The next newsprint’s question was, “Describe how you could find the area of a
circle by measuring the radius or diameter.” Most student responses mentioned
measuring the diameter and multiplying by pi. A newsprint asking how to find the
circumference by measuring radius or diameter had responses, “Multiply area by
pi.”
Ms. Kelly never chastised the students or told them they were wrong, but she did
say, “OK...this is a real eye-opener for me. I’m glad we did this, because now I
see that we need to spend some more time on this.”
At first glance it may appear that the results of Ms. Kelly’s lesson were
unsatisfactory as the students produced few mathematically correct responses. However,
Ms. Kelly’s primary intention here was not to elicit perfect responses. Rather, she was
attempting to gauge where the students were so that she could plan where to go next.
There were certainly some promising ideas to build on here, and Ms. Kelly was quick to
highlight these ideas in class. For instance, the group which recognized that the area of a
square of perimeter p must differ from the area of a non-square rectangle of perimeter p
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revealed numerous valuable insights. Though their explanation could have been clearer,
the students did show that they recognized that a square is a special type of rectangle and
that there isn’t a direct correspondence between a figure’s area and perimeter.
Additionally, the students seemed to be at least familiar with the terminology of circles,
even though they had not yet internalized the procedures for calculating a circle’s area
and perimeter. The gaps in student knowledge, which were revealed in this exercise, did
not cause Ms. Kelly to dismiss her students’ ability. Rather, it simply informed her that
she would have to “spend more time” in pushing the students’ budding ideas even
further. The students played their part in the dance as well. They took their work very
seriously, thinking about each question at each station and debating with each other how
to best respond to the questions. They then accepted the possibility of public scrutiny
when their written responses were exposed to the class and they were expected to
comment on their findings. This data suggests that the students did begin to get in touch
with some of their own ideas as they responded to the questions (putting down under-
formed ideas related to circles, for instance), thus meeting their responsibility in the
teaching and learning process.
Analysis of the data illustrates instances in which the practice of building on
student ideas resulted in the production of more accurate mathematical insights. For
example, the following vignette is drawn from Ms. Zimmerman’s 8th grade class. The
students had recently completed an extensive unit on linear functions and their graphs.
The class as a whole had done quite well on the linear functions unit and had become
adept at recognizing a linear rate of change, the contextual meaning of the y- and x-
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intercepts, the general form of a linear equation, etc. Given that the students had acquired
this knowledge, it is clear that Ms. Zimmerman had a lot to work with in the following
unit on quadratic functions. As was typical for her, she refrained from formally
introducing the topic. Instead, she began the unit by simply giving the students a
problem and asking them to share their ideas about what was happening in the problem.
The problem prompted students to consider a rectangle with a fixed perimeter of 20 units.
The students were asked to plot a graph indicating the possible areas of such a rectangle
for all possible whole-number rectangle lengths. The task began with a whole-group
discussion of the problem. Ms. Zimmerman pointed out that the class was accustomed to
graphing equations and asked if they could come up with an equation to model this
situation. The students collectively shared their knowledge related to this situation (how
to find the area of a rectangle, the relationship between length, width, and perimeter of a
rectangle, etc.) and, eventually, the students proposed that the equation A = L(10 – L) ,
where A = area and L = length, would model problem. They were then directed to work
on the problem from there, as recorded in the following excerpt from the observation
notes:
Students worked very diligently on this task. They didn’t really seem to be
working together in their groups (perhaps they are accustomed to always working
alone), but each table configuration was surrounded by students focusing
attentively and on an individual basis to the task at hand. Ms. Zimmerman
circulated around the room, calling students by name and assigning praise or
commentary as appropriate. For instance, she said things like, “I see that John
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has used some good mathematical vocabulary in his written explanation,” or, “I
was interested in something Joan said…she got a line graph, but she said, ‘I know
the graph shouldn’t be a line.’ Joan knows that this shouldn’t be a linear graph.”
Ms. Zimmerman did a masterful job of building from students’ existing academic
knowledge toward more advanced mathematical knowledge. She knew that her students
had attained a level of comfort with the processes of graphing linear equations and with
modeling contexts with algebraic equations. Capitalizing on this knowledge, she merely
needed to ask a simple question, “Can we model this with an equation?” to get students to
begin connecting their past knowledge to this new situation. The students themselves
generated the appropriate equation A = L(10 – L), and then set to work graphing it. At
least one student appropriately applied her existing knowledge in anticipating the shape
of the resulting graph. The student had somehow plotted points in a line but immediately
felt uncomfortable with her graph since the equation A = L(10 – L) was clearly not linear.
It must be emphasized that none of these insights or facts originated from the mouth of
the teacher. All of them sprang from the students’ existing knowledge…the teacher
merely reiterated valid insights produced by the students. Ms. Zimmerman’s main
contribution to the production of ideas involved her creation of a classroom environment
in which students were accustomed to making sense of mathematics by grounding new
ideas in their existing understanding.
Capitalizing on Students’ Experiential Knowledge
The vignette above showed how one teacher built on students’ existing
mathematical content knowledge. There were also numerous instances in which teachers
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connected new mathematical ideas to other forms of student knowledge. For instance,
the primary language spoken by the majority of Ms. Frederick’s 6th graders was Spanish.
Ms. Frederick made an effort to connect mathematical ideas and terminology to related
Spanish words whenever possible. When she embarked on a unit on probability, for
example, she noted that the Spanish translation of the English word “probability” is
probabilidad. She pointed out that the literal meaning of probabilidad is roughly
equivalent to the English word “maybe,” and discussed how the concept of “maybe” was
closely linked to the mathematical branch of probability which strives to measure the
likelihood of uncertain events. Ms. Thompson also connected to her Spanish-speaking
seniors when they were studying the natural logarithm. She informed them that the
notation for the natural logarithm, ln, is derived from the French language in which nouns
precede adjectives in the sentence structure.
She told the class, “In English we do the opposite than what it is done in
languages like French and Spanish. For instance, in English we say ‘the tall
girl.’ How is this said in Spanish?” Several Spanish-speaking students
responded to her question immediately by sharing a translation.
Connecting with students’ native language was one way the teachers capitalized
on students’ experiential knowledge in instruction. Teachers also capitalized on the
informal yet meaningful language students use in their daily lives, a practice which aligns
with Ladson-Billings (1994) research-based recommendations for effective instruction of
students of color. One example involves a 10th grade geometry lesson which was taught
by Ms. O’Reilly. The lesson was about the features of triangular prisms, particularly the
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number of faces, edges, and vertices on this solid. One student indicated that a triangular
prism looks like a block of cheese. For the remainder of the lesson, the terms “triangular
prism” and “cheese” were used interchangeably. Ms. O’Reilly certainly did not convey
that it was acceptable to use the term “cheese” in formal discussions, but she did
recognize that students can benefit from connecting mathematical ideas with more
accessible ideas whenever possible. If a student finds it easier to calculate the surface
area of a triangular prism by forming a mental image of a block of cheese, so be it.
The following vignette from Ms. Thompson’s room provides yet another example
of a teacher prompting students to begin thinking about mathematical ideas in reference
to other more immediately accessible concepts and words:
Today we’re going to talk about some other solids. Can anyone tell me what a
sphere is?” she asked.
A student called out, “A circle!”
“It’s a lot like a circle, isn’t it?”
Another student said, “A ball.”
“Yeah, it’s more like a ball. If we wanted to get more formal, we could define it
as a bunch of points which are all the same distance from a center. But it’s
probably easiest to think of it as a ball.”
Ms. Thompson held similar conversations about cylinders and cones. She did not
define the figures for the students, but rather permitted the students to provide
their own definitions of these figures. She then used the students’ language in
describing the figures.
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“Later in the week we’ll get into more formal definitions of these solids.”
Earlier in this chapter, Ms. Thompson was quoted as saying that she tries to avoid
introducing topics “from left field,” but instead tries to connect them to something the
students already know. The above vignette is one example of this. As she embarked on
the topic of geometric solids and their properties, she wanted students to first be able to
form an initial mental idea about these solids in relation to their existing knowledge.
Hence, in the early stages of instruction, she found it appropriate to have students think
about a sphere as a ball, or a cylinder as a soda can, etc. She also hints in this vignette
that these rough definitions are not entirely adequate and that the class will be moving
toward more precise definitions in the near future.
Summary
These successful urban mathematics teachers value existing knowledge and,
hence, utilize it in their instruction. Mr. Oden’s comment that “ultimately, teaching is
about connecting” captures this idea perfectly. The teachers’ goal is to help students
make connections from what they already know toward new ideas the teachers desire for
them to know. In short, the teachers made a concerted effort to locate the teaching and
learning process within the students’ zone of proximal development (ZPD) (Vygotsky,
1978). Vygotsky identified instruction within students’ ZPD as the optimally effective
approach to teaching. The concept of the ZPD posits that new ideas students form are
always rooted in earlier ideas, and that the set of new ideas students can be expected to
grasp must be reasonably related to their existing set of ideas (or within a proximal zone
of the existing body of knowledge). Instruction of content already located within
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students’ existing knowledge base fails to advance learning, while instruction related to
content which is too far removed from students’ existing knowledge cannot produce the
connections required for genuine learning. The instructional approach of the teachers
highlighted here might be characterized as a constant search for the students’ ZPD: a
constant effort to map out students’ existing knowledge and then attempt to teach new
concepts which connect reasonably to that knowledge.
This process requires effort from both teachers and students, as students must be
prepared to try to make these connections as well. This approach to teaching
mathematics also leads teachers to be less concerned about the production of “right”
versus “wrong” answers in the classroom. Rather, the teachers’ primary concern is that
students reveal how they are thinking about the mathematical material. This informs
teachers of how they should proceed with students in order to help them develop
increasingly accurate mathematical conceptions.
The existing knowledge that teachers attempt to build on can either be students’
existing academic knowledge or other forms of student knowledge such as their
familiarity with another language or informal concepts drawn from their experiential
world. This section illustrated cases in which each form of knowledge was utilized in
instruction. It closed with examples showing how two teachers utilized informal terms
offered by students in an effort to help them understand content. While this approach has
value when new mathematical content is initially introduced, the teachers recognize that,
ultimately, students must be comfortable with formal mathematical terminology and
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notation. The manner in which teachers address this issue is discussed in the next
section.
Additionally, while this section clarifies the teachers’ practice of focusing their
instruction on what students know, it does not address how teachers are able to determine
what the students know. In brief, the teachers manage to gain insight into existing
student ideas by fostering effective mathematical communication in the classroom. The
nature of this communication is described in the next two sections.
Emphasis on Mathematical Vocabulary
As noted in the previous section, the teachers consistently made the effort to
connect new mathematical ideas to existing student knowledge. One manner in which
this was accomplished was by relating mathematical concepts to students’ everyday
experiences. For example, the typical tenth grader doesn’t encounter or reflect on
triangular prisms on a regular basis. Some of the students in Ms. O’Reilly’s class found
it useful to think of a triangular prism as it relates to a more familiar object, a block of
cheese, so the early stages of instruction on the topic of prisms utilized the cheese image.
Such informal thinking was eventually phased out, however. Recall that each teacher had
high regard for their students’ mathematical ability. As such, the teachers saw to it that
students achieved a sophisticated level of understanding. A sophisticated grasp of
mathematics involves being able to communicate mathematically using the precise
language of the discipline. Pushing students to be able to share their ideas using the
language of mathematics was a major goal of each teacher. Ms. Zimmerman sums up
this goal as follows: “I want them to be mathematicians; I want them to be able to talk
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about math. I think it helps them, when they’re explaining their own strategies, to have
more specific language.” This view, shared by all seven teachers in the study, resonates
with other influential literature related to the importance of precise language in
mathematics. The National Council of Teachers of Mathematics (2000)
contends that “It is important to give students experiences that help them appreciate the
power and precision of mathematical language” (p. 60). Albert and McAdam (2007)
similarly point out the manner in which the use of terminology impacts conceptual
understanding of mathematics.
A common theme which emerged from analysis of data is that the teachers
regularly prompt students to explain their problem-solving strategies. The teachers
constantly tried to determine what the students knew and how the students were thinking
about mathematics because student thinking drove the teachers’ instruction. As Ms.
Zimmerman reveals here, mathematical vocabulary is a powerful tool students can use to
more clearly express their mathematical thinking. Effective verbal communication
between parties requires a shared language, therefore the teachers in this study exert a
great deal of effort in establishing a commonly accepted and understood mathematical
parlance in the classroom. Meaningful communication between teacher and student
opens a window into student knowledge, and an emphasis on mathematical vocabulary
makes the window clearer.
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Making Mathematical Vocabulary a Routine
The following episode from Ms. Kelly’s 6th grade class reveals both a strategy she
uses for emphasizing vocabulary and also demonstrates how the use of vocabulary can
clarify communication between teacher and student:
Before she set the students to work on the [group] task [of responding to
geometric questions], she stated, “And what will happen for the group that uses
the most accurate mathematical vocabulary?” The students all responded, “Extra
credit!” This was the second indicator I’ve received so far that Ms. Kelly values
students developing and using accurate mathematical terms. Earlier she had
asked a boy to summarize his solution to the problem they had worked on. He
used a lot of generic nouns as he spoke (“thing,” “it,” “stuff,” etc.). Ms. Kelly
intervened, “Wait a minute...I want you to use your mathematical vocabulary.
Can you use mathematical terms?” The boy balked at this, so Ms. Kelly asked the
class as a whole, “Can anyone suggest a mathematical term he can use to describe
what he’s saying?”
Clearly Ms. Kelly’s emphasis on the use of mathematical vocabulary had become an
expected routine among the students. Her students were quite accustomed to her reward-
based strategy of assigning extra credit to groups who use mathematical vocabulary
effectively. Analysis of the observation note above reveals this strategy, and it also
reveals how student use (or failure to use) of vocabulary serves to clarify (or confuse)
communication between teacher and student. The boy did not use mathematical
vocabulary, relying on generic pronouns to describe objects. The observation note
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doesn’t indicate if Ms. Kelly was able to follow his explanation or not, but one can easily
imagine situations in which the use of generic pronouns in communication would be
unclear (i.e., what exactly is the “thing” in question?). Regardless of whether she
understood the boy or not, Ms. Kelly didn’t tolerate his use of imprecise language. She
immediately cut him off and insisted that he express his ideas more accurately. When he
was unable to do so, she invited his classmates to assist him.
Ms. Kelly teaches in Milltown, a school district in which the primary language of
the majority of students is Spanish. In this setting, issues surrounding verbal
communication in the classroom take on an added level of urgency. Mastering
vocabulary is not only useful in promoting communication in the classroom, but it
becomes an indispensible skill as English learners strive to succeed with their high-stakes
tests. Ms. Kelly’s colleague in the Milltown schools, Ms. Frederick, explains:
[I emphasize vocabulary] for a lot of reasons. Partly because a lot of them speak
Spanish first. But the reality of it is, if they don’t understand what the vocabulary
means, then they’re not going to understand what they’re being asked to do. And,
that’s a serious issue with the kids I teach, that they don’t have a clue about what
the problem is asking them to do. If I explain pieces to them, then they go, ‘Oh
yeah!’ So, they might know the math, but they don’t know the words in the
questions, so I have to really push the vocabulary, and it’s really nice to see them
using it.
Ms. Frederick’s comment provides further evidence of an earlier theme, namely
that the teachers respect student ideas. Ms. Frederick recognizes that her students often
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“know the math,” but the only thing which prevents them from adequately answering a
given question is unfamiliarity with some of the mathematical terminology. Because of
this, she spends a great deal of instructional time developing vocabulary, as do the other
teachers in this study. In addition to serving as a communicative aid which enables
teachers to gain insight into student thinking, familiarity with mathematical vocabulary is
also valued as an indispensible tool in helping students comprehend written mathematical
questions and tasks.
The teachers utilize a number of instructional strategies for promoting vocabulary
in the classroom. We have already seen Ms. Kelly’s strategies of rewarding students with
extra credit and insisting that students rephrase their ideas more accurately in the midst of
class discussions. Another strategy, which was used extensively by the four middle
school teachers, was the use of “word banks.” A “word bank” is a prominently
displayed, handwritten collection of terms related to the current mathematical topic.
Indeed, the word banks in Ms. Zimmerman’s class were so prominent that they made an
immediate and lasting impression during the classroom observations, as recorded below:
Ms. Zimmerman has many, many posters all over the walls, most of them hand-
written creations on newsprint. Apparently, the room’s walls can’t accommodate
all the posters Ms. Zimmerman wishes to display, because she also has two
clothesline-like lengths of yarn stretched across the room with additional posters
hanging off of them. Most of the posters are there to reinforce mathematical
terms/content students have encountered before…in fact mathematical vocabulary
reminders seem to dominate….many posters are 8.5 x 11 sheets of construction
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paper with terms such as “Line of Symmetry” written on them, accompanied by a
drawing of a parabola with the line of symmetry drawn. These posters are
minimalist in terms of language….various terms such as “midpoint,”
“terminating decimal,” “exponent,” etc., are written and then, rather than
defining them with words, they are defined via a hand-drawn picture or symbolic
notation. (The symbols next to the term “exponent” are “ ”, where the “x” is
surrounded by a red square….next to this is another poster with the word “base,”
the symbol “ ”, and a big red square surrounding the “3”).
x4
x3
Some of the large newsprint posters have the heading “Word Bank”….these
posters have only terms, no definitions. Later in the class I got some new insight
into these “Word Bank” posters. Ms. Zimmerman began to construct a new
one…it had many words on it from the lessons they have been covering the last
few days on quadratic graphs and functions. They work from the “Connected
Math” series, and have recently done numerous problems which build toward the
quadratic behavior of the area of a rectangle with a fixed perimeter. As Ms.
Zimmerman reviewed the vocabulary they had encountered in recent days (using
terms such as “quadratic equation,” “maximum point,” “line of symmetry,” etc.,)
one student said, “I think we should add a new term to the list.” (referring to the
developing Word Bank poster). Ms. Zimmerman asked, “What is that?” The girl
replied, “Fixed perimeter.” Ms. Zimmerman acknowledged that this was a
relevant word for the Word Bank, and she wrote it on the poster.
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It would be impossible to overlook the word banks in Ms. Zimmerman’s
room…not only did they dominate the walls, but additional word banks were hung from
two clotheslines suspended over the students’ heads. These clear visual reminders of
relevant mathematical vocabulary were quite useful to students as they communicated
their ideas. For example, a student who wished to make a comment about “the hump of
the U-shaped graph” could easily locate the word bank and see that the better
terminology to use in this case would be “the vertex of the parabola.” Word banks were
not as ubiquitous in the classrooms of the other middle school teachers, but the teachers
saw to it that the word banks were not overlooked. Ms. Frederick began each lesson by
directing students’ attention to the relevant word bank for the day, and continued
referring students to the word bank throughout her lessons. Ms. Etienne and Ms. Kelly
likewise referred students to the word bank whenever a student failed to express an idea
with relevant vocabulary. Figures 5.2, 5.3, 5.4, and 5.5 show illustrations of the word
banks used in the four middle school classrooms.
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Figure 5.2. Ms. Etienne Word Bank Figure 5.3. Ms. Kelly Word Bank
Figure 5.4. Ms. Frederick Word Bank Figure 5.5. Ms. Zimmerman Word Bank
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Appreciating and Using Precise Mathematical Language
Word banks were not emphasized as heavily among the high school teachers. Ms.
Thompson and Ms. O’Reilly did display handwritten posters highlighting vocabulary, but
they did not refer to these posters regularly. Mr. Oden had no such posters in his
classroom. However, the high school teachers emphasized vocabulary in other ways,
ways which were mathematically appropriate for the developmental level of their
students. The earlier vignette of Ms. Thompson’s introduction of the sphere is a case in
point. She introduced the sphere by connecting it to students’ existing knowledge. The
students could relate to a ball, so Ms. Thompson indicated that, temporarily, it would be
useful to think of a sphere as a ball. As she told the students:
Yeah, it’s…like a ball. If we wanted to get more formal, we could define it as a
bunch of points which are all the same distance from a center. But it’s probably
easiest to think of it as a ball... Later in the week we’ll get into more formal
definitions of these solids.
Ms. Thompson knows, and she eventually wants her students to know, that defining a
sphere as a “ball” is not adequate. What do we mean by a “ball”? Obviously a football is
not a sphere. Even a ball which more closely resembles of sphere, such as a dodge ball,
does not satisfy the mathematical definition as its textured surface causes some points on
the surface of the ball to be further from the center than others. As Ms. Thompson views
her students as potential mathematicians, she works toward helping her students
appreciate the precise nature of mathematical definitions. In the passage above, she
honors the teaching strategy of connecting an idea to students’ current knowledge, but
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she also begins to move students toward a more developed knowledge base. She hints at
a more precise definition of a sphere (“a bunch of points which are all the same distance
from a center”), and informs students that eventually they will be expected to think of a
sphere formally.
A pedagogical challenge for all of these teachers is to provide instruction in ways
that help students appreciate as well as use precise mathematical language. An episode
from Mr. Oden’s class indicates his efforts to build a better appreciation of the power of
mathematical definitions, specifically the manner in which a mathematical definition can
capture an infinite class of objects.
Mr. Oden starts the class by reviewing some vocabulary. He has drawn three
lines on the board, one of which intersects the other two. The other two lines
appear to be parallel, but Mr. Oden has not stated that they are. Mr. Oden asks
the class, “What word describes this line which is intersecting the other two?”
Several students said, “Transversal,” out loud. Mr. Oden asked them, “We think
maybe it’s called a transversal. Do the other two lines have to be parallel for this
line to be a transversal?” Some students said “yes” and some said “no.” Mr.
Oden asked, “OK, how many think that the two lines must be parallel if this is to
be a transversal?” Several hands went up. “How many think the lines don’t need
to be parallel?” Several more hands went up. I’d estimated that at least 80% of
the 22 students in the room participated in this “election.”
The class discussion which ensued eventually led to the agreement that a
transversal is a line which intersects two other lines, period. The other lines do not need
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to be parallel, but, if they happen to be parallel, some nice congruencies emerge. Though
the distinction between defining a transversal as a line intersecting two other lines versus
defining it as intersecting two parallel lines is subtle, it is important for high school
students to appreciate and recognize this subtlety. Students utilize the congruency
properties of angles formed by parallel lines and transversals so ubiquitously that they
can easily lose sight of the fact (or, indeed, never recognize the fact) that a transversal is
not necessarily associated with parallel lines. In fact, the disagreement among Mr.
Oden’s students regarding whether the intersected lines needed to be parallel suggests
that some students were beginning to assume that the lines must be parallel. Mr. Oden’s
brief exercise in vocabulary began to push students in the direction of appreciating the
implications of a definition. A definition such as the one for “transversal” refers to an
infinite class of objects…students should recognize the membership of this class simply
by reflecting on the definition.
Anecdotal evidence drawn from the teacher observations indicates that the
teachers’ efforts to have students utilize mathematical vocabulary were paying off. As
indicated earlier, failure to use appropriate mathematical vocabulary in the middle school
classrooms was simply not tolerated. The middle school teachers insisted that their
students use technical vocabulary, and, hence, began to get students in the habit of doing
so. Informal observations of the high school students’ use of vocabulary were also
impressive. I regularly “caught” students using mathematical terminology as they
communicated with each other and their teacher. One particularly encouraging episode is
recorded below. I had arrived early for Ms. O’Reilly’s geometry class and overheard the
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following exchange among students who had recently completed an assignment related to
the angle properties of polygons:
As I was waiting for class to start I overheard a group of 3 female students sitting
to my right who were comparing homework answers. This didn’t seem to be a
case of one or more students copying answers from others. Rather, they seemed
to be genuinely interested in comparing answers. When a disagreement over one
particular answer came up, each student made a case for the solution they had
found, utilizing accurate mathematical terminology…for example, one girl said,
“I used both the interior angles and the exterior angles to arrive at the answer.”
This girl unambiguously communicated to her peers the reasoning that went into
her solution. Her reference to the interior and exterior angles of the polygon leaves no
doubt about what the girl was referring to and thinking about as she solved the problem.
This helps illustrate why the successful teachers in this study emphasize mathematical
vocabulary. It enables students to more clearly express their mathematical ideas. This
clear communication gives the teachers a more accurate sense of how the students are
thinking. As noted repeatedly, student thinking, in turn, becomes the departure point for
future instruction. In addition to informing teachers’ pedagogical strategies, mastery of
vocabulary also helps students more clearly understand the demands of a written
mathematical problem. Those teachers working primarily with Spanish-speaking
students indicated that this is especially crucial for English learners.
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Summary
This section has provided a rationale for the teachers’ emphasis on mathematical
vocabulary in the classroom. It has also described some of the specific methods the
teachers employed as they assisted their students to gain competence with the
mathematical language. Analysis of the data indicates that building fluency in the
language of mathematics is not sufficient for ensuring effective communication in the
classroom, however. In addition to having an accurate language with which to speak,
students also need to be willing to articulate their ideas publicly. The next section
describes how the teachers have fostered a classroom environment which encourages
students to share their ideas and, hence, participate fully in the teaching and learning
process.
Safe Environment for Meaningful Communication
A major finding that has emerged from this study is that the teachers strive to
build their instruction on what students already know. In order for this to occur, teachers
must open a window onto student ideas through effective communication. The
establishment of a common mathematical language is one way to ensure that student
thinking is made clear. Students must also be willing to share their thinking.
Unfortunately, at least two major factors often conspire to prevent middle and high
school students from sharing their ideas in the mathematics classroom. The first is the
emotional and social insecurity of many (perhaps most) adolescents. Students in this age
group, particularly students in their early teens, fear “losing face” in the presence of their
peers (Hollins, 1996). A student who is not completely confident with his or her
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mathematical idea may choose not to share it for fear of the embarrassment of looking
“dumb.” Students who are confident with mathematics, even enjoy it, may wish to
refrain from appearing enthusiastic about it as this may be viewed as “nerdy” to other
students.
A second factor which hinders open communication in the classroom relates to
the nature of mathematics itself and also the way it is often taught in schools.
Mathematics is somewhat unique among the subjects taught in schools in that there
usually are “right” and “wrong” answers. While the interpretation of history, for
example, is subjective and open to debate, there is no debating the value of the square
root of 225. Many students (and many adults) do not like being told (or, in the case of
mathematics, proven) that they are wrong. Oftentimes mathematics classrooms in this
country are organized in such a way that the efficient production of accurate solutions is
highly valued (Crespo, 2000). Such organization provides ample opportunity for students
to be “wrong.” Many adolescents carry the scars of repeatedly being “wrong” in their
earlier education into their middle and high school classrooms, and, hence, will be quite
hesitant to publicly share their ideas for fear of being “shot down” once again.
Sharing Ideas in a Comfortable Learning Environment
The task of prompting students to share their thinking, so central to the work of
these effective teachers, is therefore a major undertaking. It involves helping individual
students to overcome their personal insecurities (social or mathematical) and also
involves fostering a classroom environment in which the sharing of ideas is valued and
respected. The students as a whole must begin to view the sharing of ideas as the
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teachers do- an opportunity to reveal what one knows so that connections to new
knowledge can be made. This perspective does not place a higher value on “right’
answers versus “wrong” answers. All ideas reveal knowledge, and, hence, all answers
are a starting point. Helping students to view classroom discourse in this way requires
time and effort on the part of the teacher. First and foremost, the students must become
comfortable with sharing their ideas and also comfortable with the notion that the ideas
they share need not be completely formed or completely accurate. Ms. Thompson sums
this up:
We’re all in this together, learning math together. Sometimes I’ll make a mistake,
sometimes they’ll make a mistake. It’s a conversation…. I really want the kids to
feel comfortable. I want them to feel comfortable taking risks, and I want them to
feel comfortable making mistakes.
Sharing an idea in geometry class, for example, certainly is a “risk.” The student
sharing the idea potentially exposes himself or herself to scrutiny, to the judgment of
others, to the possibility of being wrong. Yet Ms. Thompson and the other teachers need
students to share their ideas as student ideas are the key to effective teaching. Hence, the
teachers must make students feel comfortable in taking these risks. Ms. Thompson’s
approach to building this level of comfort in her classroom can be somewhat unorthodox.
Consider the snapshot of one of her classes presented in Chapter 4. She had written a
“Derivative Song” to help her seniors remember various differentiation techniques (the
chain rule, the multiplication rule, etc.). This musical composition was also accompanied
by a “jazz box” dance. Ms. Thompson, unembarrassed, performed the song and dance in
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front of her roomful of high school seniors. The seniors received the performance well,
and eventually all of them memorized the song and dance. The song proved to be a hit,
but Ms. Thompson certainly opened herself up to ridicule in the event the song had
flopped. Her risk was pre-meditated, however. It was one way of building a classroom
environment in which risk-taking is accepted. Ms. Thompson comments:
So, we have an environment where I’m not afraid to make a fool of myself,
they’re not afraid to get up and do the jazz box in front of the class, then no one’s
gonna feel nervous to go up on the board and do a problem. Because, three other
kids have already done the jazz box in front of the class…how could it be worse
than that? It’s an atmosphere where sometimes people do stupid things,
sometimes people make mistakes, and that’s OK. I find that if I….sort of make a
fool of myself, they’re not afraid to make a fool of themselves.
Mr. Oden likewise works to promote an environment in which students are
comfortable sharing ideas, but his approach is different. He wants his students to feel
that they can be themselves in his classroom. Much like the host or hostess who invites
guests to remove their shoes or other gestures to help them “feel at home,” Mr. Oden
believes that if students are permitted to relax and be themselves in the classroom, they
will be more open to the teaching and learning process and the communication it
involves:
What I try to do is let them know that they can be kids, that I like them as they
are. I’m not here to have them be any different than what they are. We’re here to
do math together, and we’re hear to learn as much as we can learn.
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Observations of his classroom revealed how this approach played out with students. A
traditional principal might have been appalled by Mr. Oden’s classroom management.
Students were permitted to sit where they pleased in the classroom (including at the
teacher’s desk), joking and banter among students were quite common, and Mr. Oden
also joked with his students regularly. While the mood in his classroom was always
light, students were always engaged with mathematics. They engaged in written work
when it was time to do this, and they were certainly engaged in the sharing of ideas
involved in mathematical discussions. The light-heartedness of the room likely
contributed to the students’ willingness to communicate about mathematics. Nothing
ever seemed too serious in this room, so students never needed to fear serious
repercussions for their classroom comments.
Valuing All Contributions
Students’ comfort with classroom communication was also established in the way
teachers demonstrated their respect for student input. The teachers made it clear that the
students’ contributions were valued, regardless of the mathematical accuracy of these
contributions. Respect for student input was demonstrated in the way teachers publicly
highlighted the promising ideas which were found in a given student’s comments. That
is, when a student provided faulty solutions or explanations in regard to a mathematical
question, the teachers avoided completely dismissing the student’s contribution due to its
faultiness, but, instead, brought some of the student’s better ideas to the fore. The
teachers didn’t ignore errors either (it would not be productive to allow a student to
maintain a misconception), but they did attempt to highlight the positive aspects of a
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student’s contribution and help the student use his or her sound ideas in the effort to
better understand the problem. Consider, for example, the following vignette drawn from
Ms. Frederick’s 6th grade class. Ms. Frederick had asked the students to calculate 40% of
250, and she asked a boy to present his solution to the class and explain his work:
The boy said, “I know that 50% of 250 is 125. I then subtracted 10% from that,
which is 12.5. So I got 112.5.”
Ms. Frederick responded to his solution process enthusiastically. “There’s some
really good thinking there. Can you repeat that so that everyone can hear?”
The boy repeated his answer, and again Ms. Frederick responded positively.
“That is some really good mental math thinking. Did anyone else get 112.5?”
Many students in the class said, “No.”
Ms. Frederick then said, “So others got a different answer. Can someone come
up and show us what you did?”
A female student then went to the overhead and described a solution process
involving multiplying the decimal equivalent of 40% with 250 (250 x 0.40). The
girl worked through the multiplication algorithm, and arrived at the correct
solution of 100.
Ms. Frederick asked the class, “Did anyone else get 100?” Many students in the
class indicated that they agreed with the solution of 100.
Ms. Frederick said, “100 is correct.” She then addressed the boy who presented
the initial faulty solution. “Again, you were doing some very good thinking with
your approach. Most of your approach was valid…50% of 250 certainly is 125,
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for instance. I think you made a mistake in your thinking about ‘10%’ though.
Think about it some more, and see if you can arrive at 100.
In this vignette, Ms. Frederick avoided dismissing the boy’s incorrect solution
right away. Instead, she greeted his solution enthusiastically, noting that it contained
“some really good thinking.” This was not an empty compliment. She valued the boy’s
thoughts so much that she asked him to repeat his reasoning for the entire class to hear.
After the second presentation, she still avoided telling the boy he was wrong, opting
instead to find out if other students found a different solution. The correct solution was
eventually revealed, not by Ms. Frederick but by a second student who established the
solution with reference to a more straightforward calculation. Though it was now clear to
all that the boy was incorrect, Ms. Frederick once again publicly respected some
promising ideas he had shown, and provided him with a suggestion for how he might
improve his solution. In a subsequent interview with Ms. Frederick it was discovered
that the boy followed through with her suggestion. He recognized that the additional
10% which needed to be subtracted should have been 10% of 250, not 10% of 125, and,
hence, 125 – 25 = 100.
A similar instance of a teacher valuing a faulty student solution by highlighting
the correct aspects of the solution while addressing the incorrect aspects is found in Ms.
Zimmerman’s class. It was mentioned earlier that she first introduced her class to
quadratic graphs by having them graph the possible areas of a rectangle with a fixed
perimeter of 20 units. It had been briefly noted that the class had collectively generated
the equation A = L(10 – L) before embarking on their graph-plotting. The vignette below
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describes how this collective action occurred, and reveals the manner in which Ms.
Zimmerman helped students piece together their better ideas toward an ultimately correct
solution:
The students agreed that the area of a rectangle can be found via the formula A =
LW, but also recognized that this equation has three variables, and, hence, they
would not be able to graph it on a coordinate plane. Ms. Zimmerman reminded
the class that it was known that the perimeter of the rectangle was fixed at 20, and
then she asked, “Is there a relationship between length and width we can use?”
After some thinking, a boy indicated that he had an idea. He went up, wrote his
idea on the overhead, and explained each step. His path was slightly flawed….he
indicated that he subtracted the length, L, from the perimeter, to get “20 – L”.
He then took half of this, to get an equation of W = (20 – L) x (1/2).
Ms. Zimmerman said, “Oh…so you seemed to be thinking about the connection
between length, width, and perimeter. This reminds me of a problem we did last
week, when we were told that the perimeter of a rectangle was 16 and that its
length was 6. Think about that problem…can you find the unknown width if the
perimeter is 16 and the length is 6?”
This seemed to be an easy task for the boy who was still standing at the overhead.
He sketched the rectangle, marked each length with “6,” subtracted the sum of
the two lengths from the perimeter, “16 – 12 = 4” and divided the 4 by 2 because
the two widths had the same length. He explained each step as he went.
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Ms. Zimmerman said, “That seems a lot like the problem we’re doing now about
the rectangle with a fixed perimeter of 20. Think about what you just did, and
take a second look at the width of the rectangle with perimeter 20.”
The boy once again embarked on the more abstract task of relating a generic
width, W, to a generic length, L, for a rectangle with perimeter 20. “Oh! I
needed to subtract twice the length from the perimeter!” he said. He continued
his derivation, arriving at the equation LPW −=21 or W = 10 – L.
One of his classmates immediately recognized the usefulness of this equation for
their ultimate task of plotting a graph to represent the possible areas of a
rectangle with fixed perimeter of 20. She shouted, “I know our equation! Since
area equals length times width, and width is ten minus L, our equation should be
area equals length times ten minus L.”
The boy who recognized that an equation connecting width and length could be
generated clearly had an important idea. Indeed, his idea was the breakthrough the class
needed in order to come up with a graphable equation. His initial solution was erroneous,
however. Ms. Zimmerman did not dismiss his faulty solution. She immediately
recognized the productive thinking which was present (e.g., the boy’s knowledge of the
relationship between length, width, and perimeter). She offered the boy a support which
would help him perfect his idea, namely, giving him a related problem which might seem
easier to solve as it involved only one unknown variable. Ms. Zimmerman’s pedagogical
actions demonstrate both her practice of building instruction on what students know (in
this case, the boy’s knowledge of rectangle measurements) and also the manner in which
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she welcomes student input, even if it is faulty, as a valuable component of the teaching
and learning process.
Modeling Effective Communication
We have seen so far some strategies teachers have used to make their students
feel comfortable in sharing their ideas. Ms. Zimmerman and Ms. Frederick demonstrate
to students that their input is valuable even if it is not completely accurate
mathematically. Mr. Oden and Ms. Thompson find ways to help students feel
comfortable taking risks in the classroom. While ensuring that students feel safe and
comfortable as they communicate their ideas is important, it is also important that
students know how to effectively communicate their thinking. The educative
communication sought here involves a process of sharing ideas, allowing ideas to be
scrutinized, and then refining ideas. This type of discourse may not be familiar to
adolescents, particularly at the middle school level. The teachers in this study, and
particularly the middle school teachers, expend a great deal of effort explicitly modeling
the nature of effective communication for their students.
Ms. Etienne has long valued active communication in her classroom. Not only
does she want students to communicate with her so that she can determine how to better
teach them, but she also values students communicating with each other so that they can
use each other as resources as they engage in learning. Her instructional approach
supports such communication. Her classroom is furnished with round tables to facilitate
discussion and she regularly has students work collaboratively. Though she has had
success with this approach to teaching throughout her career, she faced many challenges
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during the 2006-2007 school year. The chemistry in her 7th grade class was not optimal
for meaningful communication. The difficulties of adolescence mentioned earlier were
very much present. Her students were in the habit of belittling each other for producing
incorrect answers in class, and the students as a whole were quite hesitant to speak up in
class. The student culture which emerged from this particular class was not optimal for
effective communication as students did not feel safe sharing their ideas. Ms. Etienne
elaborates:
I think the safety issue, as we’ve gone through this year, I feel like that determines
more and more all of the other behaviors and all of the other learning that I see.
I’m pushing it really hard now because I feel like there are a lot of different, for
whatever reason, different things that came up that made students feel like….I
mean, it’s natural, anyone is afraid to be wrong. I felt there were some things, just
dynamics, in two of my classes where students just didn’t want to share an idea
just because they were afraid. They wanted to sit back and wait for somebody
else who either has confidence, or is always right….[I want students to express
ideas in class] because a lot of times when students explain something, how they
start explaining something makes way more sense to their peers then how I would
explain it. Because, they think differently that I do. I’m coming from this big
picture [perspective] when I’m talking to them, whereas they’re just using the
space of whatever knowledge they happen to have. I find that whenever the kids
start getting better at expressing themselves, that’s when I find a lot of other kids
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starting to click and saying ‘Oh, that’s also what I thought.’ When they hear it
from each other, they seem to get it.
Ms. Etienne’s desire to have students communicate in class, coupled with her
frustration over the fact that this year’s group of students were so hesitant to share,
prompted her to actively work toward improved communication in the classroom. She
recognized that the students were simply unaccustomed to productive educational
discourse, and that, therefore, she would need to explicitly model the techniques of such
discourse for them. She spent a good portion of her class time addressing the issue of
effective communication directly, and she continued to provide students opportunities to
engage in such communication with her and with each other. She further emphasized the
principles of communication by prominently posting written conversation prompts and
discussion guidelines on the walls of the classroom. These posters were referred to in
class as often as the “word banks” described in the last section. Figure 5.6 includes
images of some of Ms. Etienne’s communication guidelines.
Figure 5.6 Ms. Etienne’s Communication Guidelines
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Observations of Ms. Etienne’s classroom over the course of the year revealed that her
efforts to assist students to communicate more effectively were paying off. As the year
progressed, students more readily shared their ideas in class, and students greeted their
peers’ contributions more cordially.
Ms. Etienne was not the only teacher who recognized the value of explicitly
modeling the norms of effective communication for middle grades students. Ms. Kelly
also recognized that many of her students simply weren’t aware of how to productively
share ideas in the classroom. She encouraged students to build their communications
skills by awarding credit for demonstrating useful communicative practices while
engaging in group work. The following observational passage drawn from her class
illustrates how effective communication practices are rewarded in the rubric she uses to
assess student work:
At the end of the task, Ms. Kelly pulled down a large laminated rubric which had
been displayed at the main whiteboard. This is the rubric that she uses to assess
students when they are given groupwork tasks. Among the items on the rubric
were, “Group goes straight to work without being told,” “Group focuses on the
task,” “Group members respect the ideas of each other.” Ms. Kelly informed the
class that most groups received a score of 100 for this task, but that a few groups
violated the “Respects all ideas” tenet. “I think those groups know who they are.
We all know what to expect when we do group work....there are no surprises in
this class.
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Summary
Ms. Kelly’s concluding comment, that the expectations for group collaboration
and communication are “no surprise” to students, indicates that effective communication
is an important part of the routine of her classroom. This can also be said for the other
teachers in this study. Some of the teachers, such as Ms. Etienne and Ms. Kelly, spent a
great deal of time modeling the nature of effective communication for their students.
Others promote productive discourse in the classroom by helping students feel
comfortable sharing their ideas and by demonstrating that student ideas are valued. All
teachers have created an environment in which students can feel safe sharing their
mathematical insights. The students are safe because their thinking will be taken
seriously, it will be valued, and it will not face ridicule or dismissal from peers of the
teacher. White (2003) has argued that “Productive classroom discourse requires that
students’ ideas are encouraged, valued, and used to shape instruction” (p. 51). This is
precisely the type of discourse promoted in the classroom’s of these seven effective
teachers. The environment assists both students and teachers to improve in their work.
Students benefit from examining their own ideas and allowing others to provide feedback
on those ideas, and teachers benefit because the insight they gain into student thinking
enables them to cater their instruction to student needs.
Concluding Discussion
A four-part framework modeling the instructional practices of seven effective
urban mathematics teachers has been proposed here. The framework suggests that the
teachers assume their students are capable of high achievement in mathematics, they
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value students’ existing knowledge and focus their instruction on this knowledge base,
they foster effective communication in the classroom by emphasizing mathematical
vocabulary, and they facilitate effective mathematical communication by creating a safe
classroom environment for communication. Teachers’ respect for student ideas is the
central component of the framework. This aspect of the teachers’ work connects to and
influences the other three aspects of their instruction. Teacher respect for students’
existing ideas informs the teachers’ belief in students’ future potential and prompts them
to deliver challenging mathematical content in the classroom. The teachers view the
students’ existing knowledge as an adequate and appropriate place to begin instruction.
As student ideas are the starting point for instruction, the teachers actively attempt to gain
a clear picture of what the students’ ideas are. Teachers open a window onto student
ideas via effective communication in the classroom. Mathematical vocabulary is
emphasized because it provides an accurate and common language for expressing ideas.
Communication is also fostered via the creation of a safe environment for sharing ideas.
A safe environment for communication helps ensure that students will be willing to
express their ideas to their teacher and peers.
The proposed framework serves as a somewhat simple heuristic for understanding
the complex and varied teaching approaches of these effective urban mathematics
teachers. Yet, an overarching description of their work might be simplified further. One
can accurately capture the pedagogy of these teachers as follows: their approach to their
students involves accepting and valuing students as they are, and collaborating with
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students toward further development in the area of mathematics. A comment by Mr.
Oden, cited earlier in this chapter, gets at the essence of the teachers and their work:
What I try to do is meet people as people. Just try to come in here and take them
as they were. Not to pass judgment on them, but to find out where they were.
Not to rely on anyone’s word for it, but just see what they could do, and then take
them the next step. And, yes, everyone’s talking about the state guidelines and all
that other stuff…but, ultimately, teaching is about connecting.
This statement points to a promising approach for working with traditionally
underserved students in the mathematics classroom. Additionally, the converse of Mr.
Oden’s statement warns against a destructive approach to urban mathematics teaching.
Consider, for instance, Mr. Oden’s practice of not “rely[ing] on anyone’s word”
regarding what his students can and cannot do. Teachers certainly have access to
information about their students which might cause them to pre-judge students’ ability.
Past report cards, conversations with students’ earlier teachers, and standardized test
scores are readily accessible data which can inform a teacher’s initial evaluation of a
student or group of students. Teachers face a choice regarding what to do with such
information. They may view past underachievement by a student or students as an
accurate predictor of future performance, and hence, hold students to low expectations
from the beginning. Alternatively, teachers may utilize these data as one source of
information about students’ background knowledge, but recognize that past performance
does not determine future potential. Hollins (1996) has argued that the former approach
is destructive in urban classrooms, while the latter can be productive. Contrasting teacher
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beliefs pertaining to “ability” versus “motivation,” Hollins found that teachers who view
past student performance as a reliable indicator for future performance are less effective
in urban classrooms than teachers who believe that motivating students to learn,
regardless of their past performance, is key to bringing about achievement. The present
study adds further support to Hollins’ argument, as the effective teachers in this study
have demonstrated a belief that all students can learn regardless of past student
achievement.
Mr. Oden’s concluding remark, “ultimately, teaching is about connecting,”
likewise captures the essence of the seven effective teachers and their work. This
“connecting” involves relating new mathematical ideas to students’ existing academic
and non-academic knowledge. It has been argued that student knowledge is the driving
force behind the teachers’ instruction. Student knowledge leads the teachers to believe in
their students’ ability, it is the starting point of instruction, and it motivates teachers to
continually uncover additional student knowledge via effective communication. This
finding resonates with the extensive literature on culturally relavant/responsive teaching
(Gay, 2000; Ladson-Billings, 1994, 1995). Gay (2000) argues, “Much intellectual ability
and many other kinds of intelligences are lying untapped in ethnically diverse students.
If these are recognized and used in the instructional process, school achievement will
improve radically” (p. 20). The teachers in this study have found ways to tap into their
students’ intelligence, and this practice is central to their effectiveness in the classroom.
One could reasonably argue that the instructional approach demonstrated by the
teachers in this study is not unique, and that the practice of valuing and building on
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students’ existing knowledge is utilized by good teachers in any setting, not necessarily
those teachers who work primarily with traditionally underserved students. Ladson-
Billings (1995) encountered a similar argument, and responded appropriately:
A common question asked by practitioners is, “Isn’t what you described just
‘good teaching’?” And, while I do not deny that it is good teaching, I pose a
counter question: why does so little of it seem to occur in classrooms populated
by African-American students? (p. 484).
While the model of teaching presented here may well be mapped onto effective teachers
in any setting, it is a model of instruction which requires particular emphasis in urban
settings. Oakes (2005) has established that poor students and students of color are far
more likely than their more affluent peers to be assigned to remedial-level or special
education classrooms. That is, these students are much more likely to be held to low
expectations; their teachers are less likely to see potential in their academic ability. The
teachers in this study provide a model suggesting that we should break the cycle of low
expectations in urban schools, and instead seek ways to build academic knowledge on the
foundation of experiential and school knowledge which all students bring with them to
the classroom.
The practice of connecting to students’ experiential or cultural knowledge is
another facet of these teachers’ instruction which sets them apart from effective teachers
in more affluent settings. Villegas and Lucas (2002) have highlighted a “demographic
gap” which exists between teachers and students in the United States. That is, students in
the United States are becoming increasingly diverse, while the teaching corps is projected
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to remain predominantly White, female, and middle-class. The teachers in this study
reflected the demographic make-up of the U.S. teaching corps as a whole (5 White
females, 1 Asian-American female, 1 White male). Hence, the day-to-day experiential
knowledge of these teachers differed substantially from that of their students. Their
awareness of being culturally different from their students prompted the teachers to
actively communicate with students and learn about them and their ideas. The need for
such an approach may be more acute for “good urban math teachers” than for other
“good math teachers.” The White middle-class teacher in the White middle-class town
teaching mostly White middle-class students may be more in touch with the day-to-day
experiential knowledge of the students than the White middle-class teacher in an urban
school. Respecting and valuing students’ cultural knowledge contributes to a teacher’s
belief in student potential to achieve. One could reasonably imagine that the “typical”
White middle-class teacher may not recognize the potential value of the “typical” urban
student’s experiential knowledge since that experiential knowledge is so decidedly
different from the teacher. Teachers may (erroneously) view urban students’ different
modes of communication, different manners of relating to adults and authority, different
styles of dress and deportment, etc., as signs that the students aren’t as academically
capable. That is, one can imagine a teacher thinking, “I wouldn’t express my ideas that
way....I wouldn’t address adults that way....this student wasn’t raised well, and probably
can’t think very well either.” Analysis of the data in this study suggests that the effective
urban teacher looks at the students’ cultural ways as assets, not deficits; the teacher does
well to find ways of connecting to students’ day-to-day (e.g., cultural) knowledge rather
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than assuming that the student needs to be culturally similar to the teacher in order to
succeed.
Findings pertaining to the characteristics of effective mathematics teachers of
traditionally underserved student groups have been presented in Chapters 4 and 5. These
findings connect with other research findings related to culturally relevant instruction.
This study contributes further to the existing literature in that it focuses specifically on
mathematics instruction occurring in two distinctive school districts: one district serving
students from a wide variety of ethnic backgrounds, the other district primarily serving
students of Latino/a origin. Further discussion of this study’s contributions, implications,
and limitations are presented in the concluding chapter.
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CHAPTER 6
SUMMARY, CONCLUSIONS, AND IMPLICATIONS
The previous two chapters provided detailed information about the findings of
this study. Chapter 4 included mini-portraits of each of the seven participants, and
concluded with an outline describing the teachers’ attitudes toward their work, their
students, and their motivations for teaching mathematics in an urban context. Chapter 5
included a grounded theoretical model describing the teachers’ instructional style. This
chapter summarizes the study, reiterating its importance in the field and the underlying
assumptions which guided the inquiry. Further discussion of the findings is provided,
including commentary on how they contribute to existing research on effective pedagogy
for traditionally underserved students. Final conclusions and their implications for the
field are addressed, as are the limitations of the study and recommendations for future
research.
Summary of the Study
The intention of this qualitative study, which integrates the research traditions of
ethnography and grounded theory, was to identify and describe promising approaches to
mathematics teaching for students from traditionally underserved student groups. It was
assumed that practicing mathematics teachers with a track record of success working in
urban classrooms would hold insight into useful instructional techniques for this
population of students. Seven successful teachers from two urban school districts
participated in this study, and the research objective was to learn about effective
instructional approaches from them. This inquiry was focused by the overriding research
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question: “What are the characteristics of successful mathematics teachers who work
primarily with traditionally underserved student groups?” Four research sub-questions
helped define and delimit the teacher characteristics uncovered in the study.
1. What are the pedagogical styles of the teachers?
2. What is the nature of their interactions with their students?
3. What are their attitudes toward their students and their work?
4. What motivates them to teach mathematics in general and to teach this population
of students in particular?
I entered the research process with several assumptions regarding the nature of
effective instruction and the types of practices I would find in the classrooms of effective
teachers. The first two assumptions are interrelated: 1) all students, regardless of
socioeconomic background or prior academic achievement, can do well in mathematics,
and 2) teachers play a primary role in helping students achieve in mathematics. It was
further assumed that teachers who are effective at helping students reach their potential
will hold all students to high expectations and will actively attempt to connect new
mathematical ideas to students’ existing academic and cultural knowledge. Each of these
pre-guiding assumptions surfaced in the research findings. While many of the findings
were anticipated by the initial assumptions, the specific manner in which the pre-
conceived principles of instruction played out in the mathematics classroom was not
predicted in advance. Additionally, one major finding, namely the manner in which the
teachers fostered communication in their classrooms, was not anticipated by the initial
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assumptions. Further discussion of the research findings and how they were derived from
and expanded on the initial assumptions is presented later in this chapter.
Importance of the Study
The effort to identify effective mathematics instructional approaches for
traditionally underserved students is both timely and important. Disparities in
achievement between poor students, students of color, and their more affluent peers have
persisted for many years despite concerted effort by educators to eradicate them.
Professional organizations such as the National Council of Teachers of Mathematics
(1989; 2000) as well as the federal government (http://www.ed.gov/nclb/landing.jhtml)
have acknowledged both the social injustice of differential achievement patterns as well
as the economic necessity of improving the quality of mathematics education for
traditionally underserved students. The No Child Left Behind act adds urgency to the
problem, demanding that educators close the achievement gap by 2014.
Although concern about the mathematics achievement gap is widespread, our
knowledge about effective instructional techniques in mathematics, particularly for
lower-achieving students, is underdeveloped (Viadero, 2005). Teachers and their
pedagogy hold a tremendous amount of leverage in terms of affecting student
achievement. Recent research has suggested that quality teaching has a greater impact on
student achievement than other variables such as students’ family background and prior
academic attainment (Rivers & Sanders, 2002). The importance of instructional style in
bringing about student achievement is further underscored in international comparative
research. Stiegler and Hiebert (1999) indicated that instructional quality was the primary
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factor separating higher achieving nations from lower achieving nations in the Third
International Mathematics and Science Study. Stiegler and Hiebert concluded that
“teaching is the next frontier in the continuing struggle to improve schools” (p. 2). This
study represents an initial expedition into that frontier, exploring in particular urban
classrooms in which mathematics teachers have managed to effectively reach their
students. The findings presented here are by no means a detailed map of the terrain of
effective instruction, but they do provide an initial sketch upon which further exploration
can be built.
Discussion of Findings
The findings of the study were presented in Chapters 4 and 5. Chapter 4 focused
on the teachers’ motivations for teaching in an urban context and their over-arching
attitudes toward their students and their work. Chapter 5 outlined how the teachers
applied these attitudes in practice via their pedagogical styles. Many of the findings
resonated with the pre-guiding assumptions noted earlier, as well as existing conceptual
and empirical research related to the concept of culturally responsive pedagogy (Gay,
2000; Ladson-Billings, 1994, 1995). However, some of the results of this study expand
on the author’s preconceptions and make unique contributions to the literature on
culturally responsive instruction. A reiteration of the findings, their connection to
existing educational ideas, and the new directions they imply are presented in this
section.
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Attitudinal and Motivational Factors
Though each of the seven teachers had a unique outlook on their work, five
factors related to the teachers’ attitudes and motivations for working in an urban setting
seemed to connect all of them. These included: 1) the teachers felt that engaging in
social service in an area of great need (urban schools) contributed to a worthwhile and
meaningful lifestyle, 2) the teachers were convinced that their work made a positive
difference in the lives of their students, 3) an ethic of care was central to the teachers’
approach to their work, 4) the teachers professed and demonstrated a high level of respect
for their students, and 5) the teachers professed and demonstrated a belief that all students
can achieve in mathematics.
These five findings might be organized into two broader categories. Findings 1
and 2 indicate that the teachers viewed their work as a vocation calling for them to
effectively serve others. Hansen (1995) recognized that effective teachers view their
work in these terms. The teachers certainly provide a valuable service to the students and
the wider community, and others in the community might applaud them for “fighting the
good fight” or for engaging in a “selfless” form of service. However, the teachers’
themselves are not so self-congratulatory. They acknowledge that their work serves
others, but also feel that their work enriches their own lives. Ms. Zimmerman
commented, “There’s an inherent feeling inside me to give back, to do something. I
knew I would never be happy with a position where I was just sitting behind a desk.” For
Ms. Zimmerman and the other teachers, the rewards of teaching stretch beyond material
compensation. The act of teaching provides service for students, but also provides a
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source of meaning and satisfaction for the teachers. The teachers also possess a strong
sense of efficacy. They are convinced that their work produces its intended goal of
effectively serving students. Ms. O’Reilly commented, “I think I have more of an impact
here, what I’m doing. I’m more influential here.” This comment reveals two attitudes
which were common among all of the teachers. First, the teachers are convinced that
their work does make a positive difference in student lives, and also that their influence in
an urban setting is likely greater than it would be in a more affluent setting. Fine (1989)
indicated that a sense of efficacy, such as that demonstrated by the teachers in this study,
is paramount to a teacher’s job satisfaction and impact on student learning.
Findings 3, 4, and 5 indicate that the teachers hold a profound respect for their
students. This respect is manifested in the teachers’ caring attitude toward their students
as well as the teachers’ belief in their students’ intelligence and potential. A “caring
attitude” in the context of mathematics teaching involved more than simply wishing the
students well. The teachers primarily demonstrated their “caring” by seeing to it that
students effectively learned mathematics and performed to high standards in the
classroom. These teachers exhibited the caring characteristics described by Noddings
(2001) in that their care for students involved wanting the best for them in terms of their
personal and academic development. The teachers’ respect for students’ current
knowledge and future potential likewise connects with research findings related to
effective pedagogy in diverse classroom settings. Hollins (1996) noted that an
“important aspect of building positive relationships with students is for teachers to show
respect, concern, and interest in their students regardless of their cultural background” (p.
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125). These teachers exhibited respect for students by valuing them as they were when
they entered the classroom, and by considering the students’ existing knowledge as
sufficient for further academic advancement. They demonstrated concern for their
students via the caring attitude noted above. The teachers’ interest in students involved
not only an interest in forming relationships with the students, but also an interest in
witnessing students achieve to their full potential.
The findings related to the teachers’ attitudes toward their work and motivations
for entering the urban classroom were unsurprising. They followed directly from the
initial assumptions about effective teachers, and they conform to existing ideas about
effective teachers of diverse students found in the research literature. While these
findings contribute nothing new to the research on effective teaching, they do lend further
support to the developing concept of culturally responsive pedagogy. These teachers are
respectful of the cultures and worldviews of their students, and view these as assets in
achieving the ultimate goal of solid achievement in mathematics. Gay (2000) indicates
that this is characteristic of effective culturally responsive teachers who make “academic
success a non-negotiable mandate for all students and an accessible goal…[they do] not
pit academic success and cultural affiliation against each other” (p. 34). This study
supports Gay’s arguments and provides specific evidence illustrating the power of
culturally responsive teaching in the mathematics classroom.
Pedagogical Style
A four-point framework modeling the instructional approach of the successful
teachers was presented in Chapter 5. Two of the four components of the framework
187
connect directly to the teacher attitudes described in Chapter 4: 1) the teachers operated
under the assumption that all of their students were capable of achieving in mathematics,
and 2) the teachers valued students’ existing ideas and capitalized on these ideas as a
basis for future instruction. Students’ existing ideas were so central to instruction that
teachers actively fostered effective communication in the classroom in order to
continually expose student thinking and utilize it as the basis for further mathematical
development. Effective communication was promoted via the remaining components of
the grounded theoretical model: 3) teachers emphasized the use of accurate
mathematical terminology in the classroom in order to create a common language for
communication, and 4) the teachers created a classroom environment in which students
could communicate their ideas in a safe and comfortable manner.
The first two tenets of the framework connect directly to the preliminary
assumptions I made about effective urban teaching and also to the published literature
related to culturally relevant/responsive pedagogy (Gay, 2000; Ladson-Billings, 1994,
1995). I was expecting to find that the successful teachers would be respectful of the
students and their varied cultures and simultaneously hold high expectations for all
students. Findings presented in Chapter 5 seem to suggest that these assumptions were
warranted, but the skeptical reader might reasonably charge that I simply highlighted data
supportive of my own pre-conceptions. My analysis of the data convinces me that these
findings are valid, and it is hoped that the presentation of the evidence in Chapter 5
persuades the reader that these findings lend further support to the theoretical construct of
culturally responsive pedagogy.
188
The components of the framework related to fostering effective communication
were not anticipated by the initial assumptions, however. Furthermore, they seem to
expand on concepts found in the literature. One of these components involved the
teachers’ emphasis on mathematical vocabulary in the classroom. The teachers’ efforts
to standardize the language used to communicate mathematical ideas might be viewed as
antithetical to a pedagogical approach which seeks to honor students’ ways of thinking
and communicating. There seems to be a tension between an ethic of validating student
ideas on the one hand and pushing them toward an established vocabulary on the other
hand. Indeed, some research has suggested that pushing students toward the norms of
English mathematical language is of secondary importance in comparison to developing
conceptual understanding via reference to students’ native language or via visual
modeling (Fuson et al., 1997). The teachers in this study insisted on a standardized
language, however, viewing it as a necessary component of effective communication.
The teachers resolved the tension between honoring existing student ideas and
establishing a standard language by building bridges from the students’ existing
vocabulary toward formal English mathematical language.
As noted in some of the data presented in Chapter 5, the teachers often utilized
students’ informal colloquial language, as well as formal Spanish terms in largely Latino
classrooms, as a starting point in discussing mathematical content. Gradually the
teachers would phase out this alternative terminology by increasingly using and
prompting students to use formal English terminology. The standard vocabulary was
constantly reinforced in the middle school teachers’ room via the ubiquitous “word
189
banks” posted on the classroom walls. The high school teachers tended to emphasize
formal vocabulary via classroom discussions. While the use and mastery of formal
vocabulary was a goal of all of the teachers, the teachers consistently made the effort to
gradually develop this terminology from student language. Hence, the overarching
principle of valuing existing student ideas while pushing to expand these ideas was
honored. The teachers in this study seem to provide an example of an approach toward
developing students’ capacity to communicate in the dominant discourse of mathematics
while simultaneously valuing and respecting students’ existing communicative styles.
This approach to mathematics instruction parallels arguments made elsewhere regarding
the importance of respecting culture while developing fluency with dominant modes of
discourse in the English classroom (Delpit, 1988).
Familiarizing students with modes of discourse also relates to the fourth
component of the pedagogical model. Like the emphasis on vocabulary, the finding that
teachers fostered a safe environment for meaningful communication was unanticipated
prior to this investigation and seems to expand on other published research findings.
Much of the literature on culturally relevant instruction suggests that teachers need to
become familiar with the culturally-informed communicative norms utilized by their
students and then utilize these norms of communication in the classroom. For example,
Khisty (1995) described the effectiveness of utilizing a participatory dialogical pattern of
discourse in a Mexican-American classroom context as this mode of communication was
most familiar to these students. Hollins (1996) reviewed several research articles related
to classroom discourse, and similarly concluded that teachers must find ways to cater
190
classroom communication to the cultural norms of the students. Some of the teachers in
this study seemed to take this approach. Mr. Oden particularly seemed to permit his
mostly Latino students to be themselves, and communicate with each other and with him
in a manner which was comfortable to them. This was not the case for all the teachers,
however. The middle school teachers as a whole, and particularly Ms. Etienne and Ms.
Kelly, actively sought to model a standard approach to the sharing of ideas in the
classroom. Experience had taught these teachers that students are not necessarily aware
of how to go about communicating their mathematical ideas effectively. Ms. Etienne and
Ms. Kelly found it worthwhile to explicitly model the process of productive discourse for
their students, providing them with prompts and cues for showing respect to the ideas of
others, handling disagreements, etc. This strategy was utilized in order to promote
academic collaboration within the classroom, a principle which is likewise valued in the
literature on culturally responsive pedagogy (Hollins, 1996). However, some of the
teachers in this study chose not to align classroom communication patterns with the
“natural” communicative patterns of their diverse students. Rather, they chose to
explicitly model an approach to communication which they felt would more effectively
promote collaboration. This practice aligns with the arguments of Cohen (1994) who
cautions that student collaboration is unlikely to be fruitful if students are not given
explicit guidelines regarding how to capitalize on each others’ talents and ideas during
groupwork. Norms for respectful communication are one component of such guidelines.
191
Building on Culturally Responsive Instruction: Focusing on Content in Diverse Settings
The preceding subsections highlighted some of the ways in which the findings of
this study both connect to and depart from the existing literature on culturally responsive
pedagogy. The overall findings suggest that the notion of culturally responsive
instruction is a promising lens through which to consider effective instruction for
traditionally underserved students. The analyses of data presented here suggest that
teachers can effectively reach their students by respecting students as they are, viewing
the students’ scholarly prospects positively, and taking advantage of students’ rich and
valuable ideas as a starting point for further instruction.
While the overall thrust of this inquiry is supportive of existing notions of
culturally relevant pedagogy, it also suggests ways to expand this concept. In particular,
my analysis of the work of these seven effective urban mathematics teachers suggests
that a classroom can be both culturally relevant and content-focused. Though perhaps
unintentional, the importance of content mastery and achieving academic success as it is
traditionally defined is often lost in the literature on meaningful instruction for diverse
students. There are many recommendations in the literature regarding strategies for
making school relevant to diverse cultures, including highlighting the contributions of
various groups, striving for inclusive representations of diverse people in the images and
literature students are exposed to, etc. (Gollnick & Chinn, 1994). This literature does not
always clarify how such signs of respect for various cultures can (or even whether they
should) lead to academic achievement. A major conclusion from this study is that
students’ culture should be honored precisely because it is a key to further academic
192
advancement. That is, the world views and non-academic experiences of students are a
rich reservoir of knowledge upon which further knowledge can be built. Effective
teachers actively seek to uncover what students know so that connections can be made
from the students’ knowledge base toward desired academic content.
The teachers in this study had definite mathematical goals for their students. That
is, the teachers had clear ideas regarding what content the students should learn. This
content was typically challenging and rigorous, extending beyond the minimal
requirements of state standards and assessments. The teachers worked toward this
content from their students’ existing knowledge base, but they did not compromise the
ultimate goal which involved student learning of the content. This approach of having a
clear and uncompromised academic goal for students contrasts with curricular
recommendations for traditionally underserved students found elsewhere in the literature
(e.g., Tate (1995), Ladson-Billings (1994)). It has been argued elsewhere that the content
itself be made immediately relevant to students lives. That is, it has been recommended
that teachers present mathematical ideas which students are likely to find interesting and
relevant. While such an approach may lead to a short-term increase in student
enthusiasm and engagement, it is unclear whether it will lead to the learning of a wide
range of mathematics in the long term. Findings from the present study suggest that
teachers can effectively bring students to mathematical content; the content need not be
altered in order to be brought to the level or interests of students.
An additional contribution of the present study relates to the wide variety of
classroom settings involved. The teachers in this study worked with a wide range of ages
193
(6th grade to 12th grade) and in ethnically and culturally distinctive settings. The teachers
in Milltown worked primarily with Latino students, while the teachers in Adamstown
worked with very diverse and evenly distributed racial groups. Despite this diversity, the
study has produced a pedagogical model describing the work of all the teachers. This
contrasts with existing empirical research on effective instruction for traditionally
underserved students which has focused on more monolithic research sites (e.g., Ladson-
Billings, 1997; Gutstein, 2003).
Conclusions and Implications
The purpose of this study has been to identify the characteristics of successful
urban mathematics teachers. The “characteristics” sought were intended to be
characteristics which other teachers and educators might benefit from or replicate. These
teacher characteristics were described and illustrated in depth in Chapters 4 and 5, and
summarized in the discussion section above. This section reiterates the findings and
suggests the implications of these findings for teachers, administrators, policy makers,
and teacher educators.
The effective teachers highlighted here viewed their work as a vocation of service
rather than a mere occupation or source of material livelihood. These teachers felt called
to teach in urban schools…serving students in this way was, for them, an avenue toward
a meaningful and worthwhile lifestyle. This attitude toward their work likely contributed
to the passion and enthusiasm they brought to the classroom. This enthusiasm in turn
contributed to their effectiveness.
194
At first glance, this notion of teaching as “vocation” or “calling” seems to border
on the metaphysical, making related practical suggestions unlikely. However, concrete
implications for both prospective teachers and educational authorities can be derived
from this notion. Prospective teachers should be prepared to reflect on their attitudes
about entering the classroom. The individual who is passionate about mathematics but
not necessarily interested in interacting with and serving young people might not be
ideally suited to the classroom. Individuals who possess both academic content
knowledge and a desire to engage in service are more likely to find sustained success in
the classroom.
Five of the seven teachers in this study entered the classroom through non-
traditional means. That is, they did not become teachers immediately after graduating
from college, but instead decided to enter the classroom at a more mature state in their
personal development. An analogy might be drawn here between this “adult-onset”
desire to teach and the period of discernment undergone by members of the clergy. The
five teachers came to realize their desire to work with young people precisely because
they found that they were not satisfied working in other areas. The two teachers who did
begin teaching directly after college were fortunate in that they arrived at their “vocation”
more directly than the others. This is certainly not the case for all 22 year-old
teachers…just as the five teachers from this study came into teaching due to a lack of
fulfillment elsewhere, many young teachers ultimately leave the classroom because it is
not suited to them. A period of discernment, then, might be worthwhile for all
prospective teachers. The medical field has institutionalized a discernment period for
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medical doctors via the residency requirement. The pre-tenure experience of a college
professor might likewise be viewed as an institutionalized period of discernment in which
the individual can determine whether or not the professoriate is his or her true “calling.”
School administrators searching for individuals with the “right stuff” for working in
urban schools might consider implementing a similar structure for new teachers. That is,
new teachers (and the schools they serve) might benefit from undergoing a structured
period of discernment in which new teachers are provided specific mentoring and
support, but also lowered professional status as “full teachers in waiting.” Those who
prosper during this discernment period, and come away convinced that they are
comfortable as teachers and feel drawn to the work, would then advance in professional
status.
In addition to viewing their work as a vocation, the other attitudinal and
motivational factors influencing the successful teachers in this study related to their
attitudes toward their students. Specifically, the teachers cared for their students,
respected them as they were, and believed that their students could perform well in
mathematics. Each of these attitudes had a positive impact on the work of the teachers.
Schools of education can play a major role in seeing to it that such attitudes are more
widespread among our teaching corps, particularly among teachers who will be working
with traditionally underserved students. Many prospective teachers hold inaccurate
assumptions about students in urban schools and what can be expected of them. These
misconceptions are informed by the popular media and some published educational
research which sends the counter-productive message that urban schools are
196
impoverished, underperforming, and even frightening places (Morris, 2004). The
teachers in this study reject this perspective. They view their students as promising
scholars with the ability and requisite background knowledge to succeed in mathematics.
This positive attitude, informed by practice, influences the teachers’ sense of agency and
effectiveness. Schools of education are in a position to debunk some of the myths of
urban schooling. This can be accomplished by providing prospective teachers direct
exposure to urban schools, enabling them to draw on direct experience rather than media
sensationalism when assessing the merits of urban schooling. Schools of education
should also provide students a balanced perspective on urban schools via required reading
lists. That is, while it is certainly worthwhile for education students to read some of the
troubling accounts of how society has short-changed many urban schools (e.g., Anyon,
1997; Kozol, 1991), students should also be aware of the many success stories associated
with urban schooling (e.g., Morris, 2004).
A major finding of this investigation is that the successful teachers found ways to
connect student culture to academic content. “Culture” has been defined simplistically
and broadly as “the way we do things around here” (Deal & Kennedy, 1983, p. 501).
While this simplistic definition fails to capture the complex meaning of the idea of
culture, it is informative in terms of helping teachers consider how to implement an
academically-directed culturally responsive pedagogy. Rather than suggesting that a
culturally-responsive pedagogy involves celebrations of various ethnic holidays, hanging
posters of Martin Luther King or Che Guevera, or other symbolic but ultimately non-
academic gestures, the work of these teachers suggests that an effective culturally
197
sensitive pedagogy involves getting to know one’s students as they are and actively
attempting to connect academic content to students’ knowledge and experience.
Mathematics teachers must have a strong background in mathematical content if they are
to be expected to make such connections, however. The role of mathematical content
knowledge has not been explicated in this research, but it is reasonable to assume that
teachers must be masters of content themselves if they are to help students connect their
existing ideas to new mathematical ones.
A final finding was that the successful teachers promoted effective
communication in the classroom by emphasizing mathematical vocabulary and by
fostering a safe environment for classroom communication. The practices employed by
the teachers surrounding vocabulary can easily transfer to other settings, and teachers
should consider adopting this practice. The word banks utilized by the middle school
teachers in this study provided a constant visual reminder to students of the concepts they
had studied and the terminology associated with these concepts. Creating a safe
environment for meaningful communication in the classroom may be trickier to
implement, but teachers should attempt to find site-appropriate means for promoting a
comfortable environment for communication nonetheless. A few different approaches
were highlighted in this study, including explicit modeling of the norms of effective
communication and relaxing classroom protocols in order to help students feel
comfortable in sharing their ideas.
198
Limitations of the Study
The recommendations listed above were derived from the results and findings of
this study. These recommendations should be considered tentatively, however, as they
are subject to many of the limitations which apply to the study as a whole. A primary
limitation of the study was its small sample size. While it has been argued that this
empirical investigation included a wider range of classrooms than what has been
exhibited in similar empirical studies, the quantity of observed classrooms is still too
small to justify the inclusion of any broad statements about urban classrooms in general.
The research setting was limited geographically (to teachers in one northeastern state) as
well as numerically (only seven teachers participated). An additional shortcoming lay in
the fact that the interpretations of the teachers and their work were ultimately made by a
single observer. Finally, my claim that the teachers were “successful” stands open to
debate. I rationalized the descriptor “successful” due to the fact that the teachers were
nominated by supervisors and had served for at least five years. I had hoped to be able to
establish that the teachers’ students had performed relatively well on standardized tests,
but I was only able to make this claim for the teachers of the Adamstown district. Even
for these teachers, however, it is impossible to really know if their teaching led to student
success on tests or whether other factors were more influential. Test scores aside, the
notion of “success” in teaching is highly subjective and open to debate. My limited
definition and substantiation of “success” limited the usefulness of this study.
I was aware of these limitations throughout the research process, and made some
effort to address them in the research design. While a larger sample size might have leant
199
added generalizability to the study, it would have also limited the detail of description
afforded for each individual teacher and each classroom. The sample size of seven
provided some balance between being able to note substantive patterns across sites on the
one hand and being able to provide rich descriptions of individual sites on the other. The
single observer/interpreter limitation was also addressed in the study design. The seven
teachers were provided draft copies of the research findings as they were written. The
teachers were encouraged to provide feedback and corrections as necessary. Indeed, Ms.
Frederick did clarify some of the information related to her, and her suggestion altered
some of the text of Chapter 4. So, while the ultimate responsibility for the findings and
interpretations presented here are my own, the seven participants kept me in check by
reviewing what I wrote about them. Finally, while the operational definition of a
“successful teacher” was limited and largely subject to my own personal values, the
identification of successful teachers was rationalized in relation to a set of published
guidelines related to the characteristics of “expert” teachers (Palmer, Stough, Burdenski,
& Gonzales, 2005). Connecting my definition of “successful” to an established rubric
provided a certain level of credibility to the definition.
Recommendations for Future Research
The study included some additional limitations, which were not addressed in the
previous section. This omission was intentional because some of the shortcomings of this
study provide useful directions for future research into the question of effective
instruction for traditionally underserved students. For instance, the only voices offering
perspectives on the nature of effective instruction in the current study were those of the
200
researcher and the seven participating teachers. A more holistic picture of the nature of
effective instruction could be developed if more interested parties were heard. A
worthwhile future research project would include perspectives from students, parents, and
administrators about the nature of effective mathematics instruction.
The findings of this inquiry have been offered as tentative, particularly because
the small sample of teachers precludes the possibility of generalizing the findings. Many
of the findings could be validated (or invalidated) more robustly via quantitative
investigation on a larger scale. The emphasis and use of mathematical vocabulary in the
classroom is one finding which readily lends itself to quantitative investigation. A wide
range of teachers and classrooms could be videotaped and the relative frequency of
teachers’ and students’ use of formal mathematical terms, informal terms, and colloquial
language in mathematical context could be coded and measured. Findings from such a
study might determine whether or not a strong correlation between an emphasis on
vocabulary in instruction and student achievement exists. Other findings in this study
also hold potential as quantifiable variables in future quantitative research. Many of the
findings presented here relate to teacher attitudes and assumptions (e.g., the effective
teachers in this study value student ideas, they assume that all student are capable of high
achievement in mathematics, etc.). The presence or absence of such attitudes and beliefs
can be identified on a larger scale via teacher surveys and then correlated with student
achievement data.
201
Closing Comments
In the opening chapter, I alluded to a piece of advice my grandmother passed on
to me when I was a boy: “Nobody’s any better than you, but you’re no better than
anybody else.” My grandmother’s advice is simply a variant on the Golden Rule, an
approach to morality which is acknowledged and valued across religions and cultures.
This simple maxim provides tremendous insight to those of us who have been fortunate
enough to learn some mathematics and endeavor to help others learn some mathematics
as well. We are certainly no better than anyone else. We must take the time to learn
from our students and figure out what they know, so that we can effectively share with
them some of the things that we know. Both we and our students will grow from the
process.
202
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Appendix A:
Informed Consent Form
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CONSENT TO PARTICIPATE IN A RESEARCH STUDY Introduction: You are being invited to participate in a research study about successful mathematics instructors and effective instruction techniques in urban contexts. The title of the study is “An Investigation of Successful Mathematics Teachers Serving Students from Traditionally Underserved Demographic Groups.” You are being invited to participate in this study because you have been identified as an effective teacher and because you work in an urban high school. You are one of eight teachers who have been invited to participate in this study. Your participation is completely voluntary. Please ask questions if there is anything you do not understand. The person conducting this study is Michael C. Egan, a doctoral candidate in the Curriculum and Instruction department at Boston College. This study serves as the topic of Mr. Egan’s doctoral dissertation. Dr. Lillie Albert of Boston College serves as the faculty chair of this dissertation study. Dr. Michael Schiro and Dr. Maureen Kenney of Boston College are members of the dissertation committee. In addition to being written as a dissertation, findings from this study may also be publicized via conference presentations, published research articles, or other forms of publication. No funding has been received for this study. Purpose: It is well documented that students from urban schools perform below national averages in measures of mathematics achievement. There are, however, many cases in which teachers in urban settings have consistently produced students who achieve significant success in mathematics. This study will attempt to draw insights into the practices of successful urban mathematics teachers. It is hoped that the results of this study will inform mathematics educators and others concerned with mathematics education of some of the characteristics of an effective teacher. Procedures: The research will be done at your school site. You will be asked to participate in three interviews, and you will be asked to permit Mr. Egan to observe at least five of your classes. Each interview will take 30 minutes to an hour. The first interview will be conducted prior to classroom observations, the second interview will be conducted after the second or third observation, and the third interview will be conducted after the final classroom observation. In the first interview, you will be asked questions pertaining to your professional and educational background, your motivations for entering the teaching profession, your motivations for working in an urban high school, and your intended career path. The second interview will focus more on issues pertaining to your practice: your views on how to effectively teach mathematics, your views on how students learn, your views on the teacher-student relationship, and your views on classroom management. The final interview will be used to discuss specific instances observed in your classroom, to reflect on the year, and to discuss interpretations of your work. Please note that these are the general parameters of the interviewing sessions. The goal of the study is to draw insights from your practice: hence, if pertinent topics unrelated to the
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categories above are raised over the course of the interviews, you may be asked to speak more on these topics. With your permission, the interviews will be digitally recorded. Recording will permit the interview/conversation to run more smoothly, and minimize the pauses which written notes would cause. The recordings will later be transcribed by the researcher. The recordings will be stored in Mr. Egan’s personal computer as well as an external storage device (USB stick or CD). When these items are not in Mr. Egan’s direct possession, they will be securely locked in his office or home. As data from this research may be used for publication purposes in the future, the recordings will be kept for a maximum of five years before being destroyed. Risks: To the best of my knowledge, the things you will be doing in this study have no more risk of harm to you than what you would experience in everyday life. Benefits: It is hoped that you will enjoy modest satisfaction in the knowledge that your work has been identified as exemplary, and that your approach to the teaching and learning of mathematics may have a modest influence on other teachers who may be exposed to the final research report. Costs: There will be no monetary cost for you to participate in this study. You will be asked to give some of your time for interviewing, and to permit the researcher to enter your classroom. Compensation: As a sign of appreciation for your willingness to participate in this study, you will be provided with lunch during each interview session. No further compensation will be made. Withdrawal from the study: You may choose to stop your participation in the study at any time. Confidentiality: Pseudonyms will be used to identify you and your school in all written components of the research process, including observational records, interview transcripts, archival data, and the final research report. Thus, your identity will be protected. As noted above, tape recordings of the interviews will be secured for no more than five years before being destroyed. Only Mr. Egan will have access to these recordings during that time. Although it happens very rarely, I may be required to show information that identifies you, like this informed consent document, to people who need to be sure I have done the research correctly. These would be people from a group such as the Boston College Institutional Review Board that oversees research involving human participants. Also, as with any other adult working in a school setting, I am required to report incidents of abuse in the event that I witness them. Questions:
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You are encouraged to ask questions now, and at any time during the study. You can reach me, Michael Egan, at (978) 204-5608 or via email at [email protected]. You may also contact my supervisor, Dr. Lillie Albert at [email protected]. If you have any questions regarding your rights as a participant in a research study, please contact the Boston College Office of Research Compliance and Intellectual Property Management, (617) 552-3345. Certification: I have read and I believe I understand this Informed Consent document. I believe I understand the purpose of the research project and what I will be asked to do. I have been given the opportunity to ask questions and they have been answered satisfactorily. I understand that I may stop my participation in this research study at anytime and that I can refuse to answer any question(s). I understand that I will not be identified in reports of this research. In providing my signature below, I am giving my consent to the researcher to tape record interviews with me. I have the right to request that the tape recorder be shut off at any time. I have received a signed copy of this Informed Consent document for my personal reference. I hereby give my informed consent and free consent to be a participant in this study. Signatures: ___________ ________________________________________________ Date Consent Signature of Participant ________________________________________________ Print Name of Participant ________________________________________________ Person providing information and witness to consent
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Appendix B:
Interview Protocols
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Michael C. Egan: First Interview Questions Note: the purpose of the pre-observational interview is to gain insight into the
professional and educational history of the interviewee, his/her motivations for entering the teaching profession, his/her motivation for working in an urban high school, andhis/
her intended career path. I will utilize an “interview guide approach” (Rossman & Rallis, 2003) for the interview: that is, the questions below are designed to get the
conversation started, but I will remain open to explore other issues which may come up over the course of the interview.
1. How long have you been teaching? 2. Have you always taught in this school? 3. What was your undergraduate major? 4. Have you pursued graduate studies? If so, what did you study in graduate school and how far did you advance in your graduate studies? 5. What aspects, if any, of your university studies (undergraduate and/or graduate) do you consider to be valuable in your teaching today? 6. How did you come in to teaching? (eg, did you enter teaching straight out of college, did you make a career change at some point, etc.) 7. What inspired you to enter the classroom? 8. Were there any individuals who influenced your decision to go into teaching? Tell me about them. 9. Are you satisfied with teaching? 10. Why did you choose to work in an urban high school? 11. What are your career goals? What do you hope to be doing in next few years? In five years? Ten years? Beyond? 12. Can you comment on your philosophical beliefs and their connection to your work?
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Michael C. Egan: Second Interview Questions Note: the purpose of the mid-year, second interview is to gain insight into the the
teacher’s reflections of her/his own practice. I will utilize an “interview guide approach” (Rossman & Rallis, 2003) for the interview: that is, the questions below are designed to get the conversation started, but I will remain open to explore other issues
which may come up over the course of the interview.
1. Who is the best teacher you ever had? Can you describe what it was that made him/her so effective? 2. Do you try to emulate that teacher? If so, how successful have you been? 3. Mathematics is widely regarded as a “hard” subject. Based on your own practice, what do teachers need to do in order to demystify this subject? 4. Some educators argue that teaching is teaching, and that an instructional technique that works, say, in a rural school with 20 students will work just as well in an urban school with 2000 students. Others argue that teaching is highly contextual: each classroom is different, and therefore teachers must alter their instructional approach in different settings. What is your viewpoint? 5. The arguments above often go beyond geography (eg urban vs. suburban vs. rural) and into issues of culture, gender, social class, etc. Again, do you feel that teachers should cater their instruction to the cultural contours of their students, or will the right technique work for any kid? 6. Based on your experience, how do students learn? 7. Given your views on student learning, how can teachers assist students to learn? 8. Describe your relationship with your students. 9. What are your views of classroom management? How do you deal with discipline
problems? If discipline is seldom a problem in your classroom, why do you think that is the case?
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Michael C. Egan: Third Interview Questions Note: by the time we reach the late-year, third interview, I will have observed the teacher
numerous times and I will have developed my own interpretations of their work. This interview strives to draw out the meaning that the teacher assigns to particular events
which I observed in his/her classroom, and provides the teacher an opportunity to react to the interpretations of their work which I propose. Adhering to the process of grounded
theory, the data collected during this latter stage in the research process should be strongly informed by the data collected earlier. As such, it is difficult to project exactly
what will be discussed in this interview, so the questions shown below are best-guess projections…the specific wording of the question is likely to change. As before, I will
utilize an “interview guide approach” (Rossman & Rallis, 2003) for the interview: that is, the questions below are designed to get the conversation started, but I will remain
open to explore other issues which may come up over the course of the interview. 1. In a recent class I observed, the following incident happened...(describe incident)…Why did you choose to (teach the material in this way, address the student(s) in this way, etc.)? Ask several questions modeled on the one above in order to gain the teacher’s perspective on significant events… 2. I’m beginning to get the impression that (a given component of the teacher’s work) can be characterized (in a particular way). What are your thoughts on this? Again, ask several questions modeled on the one above in order to gain the teacher’s perspective…
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Appendix C:
Ms. Thompson’s “Derivative Song”
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