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.

10

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.

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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

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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

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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

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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.

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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

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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.

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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

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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

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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.

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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

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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.

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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 eganme@bc.edu. You may also contact my supervisor, Dr. Lillie Albert at albertli@bc.edu. 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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