EXPERT COACHING: EFFECTS ON A TEACHER,
HER STUDENTS AND THE COACH
by
Lawrence Eugene Linnen
B. S., Montana State University Billings, 1966
M. E., Southeastern Oklahoma State University, 1973
M. E., Lesley College, 1988
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Educational Leadership and Innovation
2004
2004 by Lawrence Eugene Linnen
All rights reserved.
This thesis for the Doctor of Philosophy
degree by
Lawrence Eugene Linnen
has been approved
by
William Juraschek
Mark A. Clarke
Date
James to;
Linnen, Lawrence Eugene (Ph.D., Educational Leadership and Innovation)
Expert Coaching: Effects on a Teacher, Her Students and the Coach
Thesis directed by Professor William Juraschek
ABSTRACT
This dissertation is a multilevel action research study of one middleschool
mathematics teachers responses to coaching. The study examined effects of the
coaching, by the researcher, on the teachers pedagogy, in particular her
communication with her students and their views of mathematics. Also explored
were the effects of the coaching on the coach.
Data consisted of a series of video and audio taped classroom observations in two of
the teachers classes, individual and followup interviews with the teacher, and field
notes. Several patterns of interaction emerged: a gradual shift in the nature of
student presentations, a pattern of questioning and discussion that monitored student
understanding, and a pattern of consistent mathematical inquiry.
Four main conclusions emerged from the present study. First, the collaboration
between the teacher and researcher proved to be a powerful tool that helped shape
and frame classroom instruction. Second, although the teacher held views of
mathematics and its instruction that were predominately based on symbolic rules
and procedures, interactions between particular views of mathematics and its
instruction were many and varied, and particular views of mathematics frequently
complemented each other. Third, consistent efforts by the teacher to probe and
promote student understanding, and efforts stirred by the teachers and students
natural curiosity, led to a classroom characterized by mathematical inquiry, which in
turn shaped instruction purposefully designed to promote mathematical inquiry.
Fourth, the teacher handled disturbances in her classroom in a manner that enhanced
her overall effectiveness in managing her classroom.
This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
Signed
iv
DEDICATION
This dissertation is dedicated to the teacher and her students noted in the present
study. You confirmed my beliefs about mathematics education and inspired me to
expand my own work in supporting mathematics education reform.
ACKNOWLEDGEMENTS
From my first day as a teacher, I have felt honored and privileged to be able
to work with children and teachers to help them shape their mathematical and
pedagogical thinking. But the credit for my understanding of mathematics and its
instruction goes to those same students and teachers who willingly and unselfishly
shared their understandings with me. The interactions with students and colleagues
inspired my development as a teacher and a coach of teachers with one goal in
mind, the education of children. In short, these interactions over a long career in
education shaped my views of mathematics and its instruction.
I thank all of you who touched my personal and professional life, but
especially those listed below:
To Bill Juraschek, Jean Klanica, Patty Quan, and Gene Maier, whose
commitment to reforming and informing mathematics education has inspired me
in my own quest to improve teaching and learning for students.
To my son, Chris, and my late father, Clarence, and mother, Elsie, who knew
that there were no limits to my learning.
Finally, and most importantly, I must thank my wife, Diana. Ultimately, it is she
who supported me in pursuing, persevering, and completing this work.
CONTENTS
Figures.......................................................xv
Tables........................................................xvi
CHAPTER
1. INTRODUCTION................................................1
Purpose of Study.........................................4
Significance of Study.................................. 4
Conceptual Framework.....................................5
Research Questions.......................................9
Definitions..............................................9
MultiLevel Action Research...................... 9
Change...........................................10
Views of Mathematics.............................10
Classroom Rhythms................................11
Disturbance......................................11
Interaction......................................11
Problematic Problems.............................12
Classroom Community..............................12
Methodology.............................................13
Vll
Source of Subject Population.....................13
Criteria and Method for Including/Excluding Subjects.13
Using Student Populations........................14
Sources of Data..................................14
Data Analysis....................................15
Organization of the Dissertation.......................15
2. LITERATURE REVIEW.........................................18
Overview...............................................18
Views of Mathematics and its Instruction...............19
Teachers Views of Mathematics...................20
Students Views of Mathematics...................29
My Views of Mathematics..........................32
Interactions and Communication.........................37
Action Research and Systems Influences.................42
Systems Theory...................................47
MultiLevel Action Research......................48
Chapter Summary........................................50
3. METHODOLOGY...............................................52
Overview: An Ethnography of One Teacher................52
vui
Research Setting
52
MultiLevel Action Research........................54
Site and Population Selection......................56
Background and Role of Researcher....,.............57
Data Collection: Sources and Procedures...................58
Data Sources.......................................59
Data Analysis Procedures...........................60
Coding: Views of Mathematics..................... 61
Secondary Codes: Views of Mathematics..............62
Descriptive Coding: Classroom Rhythms..............64
Summary: Data Collection...........................68
Data Display and Conclusions..............................68
Credibility of Conclusions................................69
Internal Reliability...............................69
Chapter Summary...........................................71
4. THE SETTING: CLASSROOM, SCHOOL CULTURE,
AND INITIAL RESULTS..........................................72
Overview..................................................72
Demographics..............................................73
Testing Results....................................74
ix
Preparing for Observations........................76
The Physical Setting .............................76
How Class Began...................................77
Results: Rhythms of School Life.........................79
Frequencies: Classroom Rhythms....................79
Classroom Rhythms: Disturbances...................81
Classroom Rhythms: Routines.......................84
Classroom Rhythms: Background.....................89
Chapter Summary.........................................91
5. VIEWS OF MATHEMATICS:
CLASSROOM CONVERSATIONS......................................92
Overview................................................92
Results: Frequencies of Views of Mathematics............92
Analyses: Classroom Interactions Level of Scale.........94
Symbolic Rules and Procedures.....................94
Science Applications.............................101
Mathematics as Language..........................103
Patterns, Form and Relationships.................106
Analyses: Teacher/Researcher Interpersonal Level of Scale.113
Checking for Understanding.......................115
x
Changing Student Roles............................127
Chapter Summary..........................................129
6. THE RESEARCHER AS OBSERVER AND COACH.......................131
Overview.................................................131
Analyses: Interpersonal Level of Scale...................132
Janes Interventions..............................132
Janes Compassion.................................135
Sharing Interpretations That Differ...............137
Linking Mathematics and Language..................138
Analyses: District Constraints Level of Scale............139
More on Janes Interventions......................139
Connecting Mathematics and Science................140
A Change: Researchers Role.......................147
Chapter Summary..........................................150
7. FOLLOWUP INTERVIEW AND OBSERVATIONS........................151
Overview: Followup Interview with Jane..................151
Question 1: What role does questioning play
in your instruction this year?...........................152
Question 2: How do you see these math views
influencing your teaching this year?.....................154
xi
Question 3: We had a conversation about the
number line. How has that influenced your
instruction this year?.......................................155
Question 4: How do you allow for think time this year?.......157
Question 5: Have you tried a strategy of
asking kids to draw sketches this year?......................162
Question 6: Do you regularly ask
kids for estimates this year?................................164
Question 7: This year have you seen times when
the discussion of mathematics takes an unexpected
turn? If so, did you evaluate the mathematics
covered? Did the mathematics you wanted to cover
come out of these unexpected events?......................165
Question 8: Have you purposefully provided time
this year for conceptual development? If so,
what did that look like? If not, what were some
factors that influenced the decision not to
provide such time?........................................166
Question 9: How are you responding to kids when
they ask you is the answer right?.........................168
Question 10: (Discussion on the use of CMP)...............169
Question 11: How have your concerns about next
grade level needs changed or not changed? How
does your knowledge of science applications
influence your instruction this year?..........................173
Question 12: How would you characterize the
responsibility of your students this year?
What have you done to promote that?............................174
Question 13: We had conversations about that.
You know getting kids to ask other kids, what do
you think about what Ive done. Have you done
some of that this year with the kids?...................175
Followup Observations: Other Teachers..................177
8. WHAT DID I LEARN?...........................................181
Conclusion 1: Action Research...........................182
Changes I Noticed in the Teacher..................182
Changes That Did Not Sustain Over Time............185
What I Learned About Myself.......................186
Conclusion 2: Views of Mathematics......................189
Insight 1: Mirroring The Teachers Views..........192
Insight 2: Acknowledging Students Views..........194
Insight 3: Not Challenging the Teachers Views....195
Conclusion 3: Curiosity and Mathematical Inquiry........198
Conclusion 4: Classroom Rhythms.........................202
Implications for Further Study..........................205
Chapter Summary.........................................206
APPENDIX
A. INFORMED CONSENT STUDENTS...............208
XIII
B. INFORMED CONSENT TEACHER................209
C. INFORMED CONSENT PARENTS................210
D. TRANSCRIPTION DATA BY PAGES,
WORDS, AND PASSAGES........................211
E. SAMPLE WORKSHEET OF CODING................212
F. PASSAGES FOR DISCUSSION IN FOLLOWUP
INTERVIEW..................................213
BIBLIOGRAPHY .....................................221
xiv
FIGURES
1.1 Conceptual Framework.................................................8
3.1 Analytical Framework.................................................55
xv
TABLES
Table
3.1 Sources and Frequency of Data Collected.............................59
3.2 Codes for Views of Mathematics......................................62
3.3 Codes for Classroom Rhythms.........................................68
4.1 Ethnicity at Makefield Middle School, 20022003.....................73
4.2 2002 State Test Results in Mathematics Makefield Middle School...75
4.3 How Class Began....................................................78
4.4 Number of Passages Classroom Rhythms..............................80
5.1 Number of Passages Views of Mathematics...........................93
5.2 Checking for Understanding.........................................118
In Appendix D: Transcription Data by Pages, Words, and Passages
Table D.l: Observation Transcription Data..............................210
Table D.2: Interview Transcription Data................................210
CHAPTER 1
INTRODUCTION
My experiences as a classroom teacher, an educational consultant, and a
districtlevel mathematics coordinator, spread over thirtyseven years, have
provided me with opportunities to view mathematics instruction from several
perspectives. I have viewed my own teaching of mathematics, the teaching of
others, and how these endeavors fit into the systems determined by schools and the
districts in which they reside. As part of a collaborative project between a university
and a middle school, I was invited into one teachers classroom to act as a coach to
help the teacher improve her teaching. In light of my background as a classroom
teacher, teaching coach and district mathematics coordinator, the situation offered a
rich source of insights into the system comprising teacher, students, coach and
school culture.
As a classroom teacher, I participated in a National Science Foundation
grant that provided my colleagues and me with opportunities to visit each others
classrooms. Following each visit, we collaboratively debriefed our classroom
observations. This collaboration provided us with an approach to mathematics
education that involved teaching, reflecting on the teaching with other colleagues,
adjusting our instruction based on our collaborative findings, and then repeating this
1
whole process. Such a process would likely be termed action research by many
(Argyris, Putnam, & Smith, 1985; Koba, Clarke, & Mitchell, 2000; Zmuda, Kuklis,
& Kiline, 2004), although we did not characterize it as such. Our observations and
subsequent debriefings became a catalyst for crafting my philosophy as a
mathematics teacher.
As an educational consultant, I have provided staff development for many
teachers in many schools and school districts throughout the United States. This
staff development has primarily been focused on curriculum and pedagogy
developed by The Math Learning Center and The Interactive Mathematics Program,
both acknowledged curricular leaders in innovative approaches to mathematics
education. The consulting and what I learned about how teachers view the
mathematics they teach provided a catalyst for crafting my philosophy as a
mathematics educator. Ultimately, these consulting experiences have helped shape
my theories and ideas for mathematics teaching and learning for adults.
As a district coordinator of mathematics education and curricula, I am
expected to pay attention to district and national trends in mathematics achievement,
to instigate and lead focused discussions on what these trends mean to mathematics
education, and to provide teacher training and coaching designed to improve the
teaching of mathematics and student achievement, as measured by district and state
expectations.
2
My goal as district mathematics coordinator is to create a climate where a
teacher can begin to change his or her instruction and improve instruction. This
process involves interactions between the teacher, his or her students, and me, as a
district leader. This sort of teaching and coaching requires systemic change,
In the classroom,
In the school,
At home, and
In the district.
As mathematics teachers seek to improve their instructional practices, they naturally
choose or discard ideas based upon what they believe is best for their students.
These beliefs may or may not be consistent with their students beliefs about
mathematics and its instruction. For instance, a teacher may believe that
mathematics is largely the study of patterns, but his or her students may believe
mathematics is largely a set of rules to master.
Clarke (2001) confirmed that a systems approach necessarily involved levels
of scale that needed consideration. He asserted, Teaching results in change, not
only in individuals, but also in organizations and, ultimately, in society (Clarke,
2001, p. 12). Thus it makes sense to pay attention to the components of a system
that are directly and indirectly affected by change. Clarke (2003b) warned us
though that, Things are the way they are because living systems tend to function
toward stability; they resist change (p. 11).
3
Purpose of Study
The primary purpose of the present study is to use an experienced eye to
investigate the interactions between a teachers and her students views of
mathematics and how these interactions influence her choice of teaching practices,
as well as any changes that emerge from these interactions in the students, the
teacher, and me in my role as teaching coach.
Significance of Study
A number of researchers have investigated teacher beliefs and dispositions
about teaching and learning (L. M. Anderson & HoltReynolds, 1995; Andrews &
Hatch, 1999; Archer, 1999; Cooney, Shealy, & Arvold, 1998; Nathan & Koedinger,
2000b; Vacc & Bright, 1999) and many have investigated student beliefs and
dispositions about learning (Franke & Carey, 1997; Koch, 1998; Kouba &
McDonald, 1991; Lester, 1989; Nathan & Koedinger, 2000a; Osborn, 1999;
Schunk, 1996b). Few, however, have investigated the interaction of the two
(Thompson, 1992), and there is little research exploring the extent to which
teachers and students beliefs interact to influence choices of teaching practices
(Cohen & Hill, 2002; Thompson, 1992).
The present study broadened the research base on this interaction of
teachers and students beliefs and views. The present study provided more evidence
4
on how these beliefs and views influenced instruction. Furthermore, the present
study enhanced and extended existing research by describing the changes that occur
as a result of interaction between a teacher, her students, and a coach. As Clarke
(2003b) stated, Learning is change over time through engagement in activity
(p.41). It is the investigation of changes over time, in the teacher, in her students,
and in me, that forms the crux of this study, and ultimately, the significance.
Conceptual Framework
The present study focuses on a teacher, her students, and me as her
instructional coach. The attempts to understand the complexities of these foci are
informed and guided by the following common principles:
The complexity of living entities requires us to acknowledge
that they can only be influenced, not unilaterally controlled.
Because of the complexity, our understanding of what is
going on is necessarily limited.
Of particular importance in this regard is the status of
problems and the relationship between problems and
solutions; problems are seen as the local occurrences of larger
systemic functioning, and solutions as efforts to nudge people
and institutions toward healthier conditions.
Todays problems took many generations to develop, and the
solutions will not be quick in coming.
This accounts for the futility of formulaic or programmatic
responses; prescriptions and direct action have limited effect
because living organisms respond in ways that reveal habitual
rhythms developed over time.
And finally, all living phenomena are interconnected and our
efforts at changing one are necessarily going to impact and be
impacted by others. (Clarke, 2003a)
5
These principles challenge the conventional structures and organizational
approaches of the major educational institutions in the United States. Bateson
(2000) pointed out that, Epistemological error is often reinforced and therefore
selfvalidating. You can get along all right in spite of the fact that you entertain at
rather deep levels of the mind premises which are simply false (p. 488). For
example, many schools, instead of improving teaching (Stigler & Hiebert, 1999),
typically try to fix underachievement by implementing a new curriculum, buying
software purported to improve student achievement, or changing graduation
requirements to include more academic credits, and so putting increased pressure on
students, parents, and communities. All of these perceived fixes for
underachievement allow teachers to perpetuate their ingrained beliefs without
considering change.
A component of the present study is to characterize the research setting and
to observe and record the routines and rhythms of daily classroom life in a middle
school. These occurrences that permeate almost every classroom are not isolated
from instruction; they become a part of the instruction as the teacher and students
adapt to and deal with the routines and rhythms. For the present study, these
routines and rhythms include background knowledge of students, routines and
constraints, and disturbances to the flow of planned activities.
6
The present study looks at change by also examining views of mathematics,
as observed and interpreted from the classroom and from subsequent conversations.
Particular views of mathematics and its instruction include those of the teacher and
her students. On purpose, these views also include my own views of mathematics
and its instruction, because my views influence my coaching of others.
A third focus of the present study is to examine my role as observer and
coach throughout the process. My interactions, as well as those of the teacher and
students, become topics for examination and discussion.
Finally, classroom instruction is the overarching focus of this study, and in
particular how it is influenced by the other three foci. Because of this influence, I
represent this interaction with a think cloud, to represent the thinking necessary
with respect to the other foci. In other words, how do views, rhythms, and my work
influence instruction, and how does instruction influence the other three?
The present study reflects my effort, using Clarkes (2003a) principles, to
weave views of mathematics, the rhythms of a classroom, and my work as observer
and coach into a systemic look at the interactions between selected students, their
teacher, and me, as well as a look at the influence of these interactions on
instruction. The main ideas and the asserted relationships between them are in
Figure 1.1.
7
Figure 1.1: Conceptual Framework
The Setting
Demographics
Classroom Rhythms
The Researcher
As Observer
As Instructional
Coach
Views of Mathematics as Classroom Conversations:
Students
Teacher
Researcher
Three main components, The Setting, Views of Mathematics as Classroom
Conversations, and The Researcher, depicted in rectangles, al contribute to and
define the fourth component, Classroom Instruction. In other words, classroom
instruction is the interaction of these three main components. Likewise, Classroom
Instruction necessarily, and on purpose, interacts with the three main components.
Thus, each component of this conceptual framework, including instruction,
contributes to and influences the other components, as represented by the dual
pointing Interaction arrows and the three concentric arrows, which illustrate the
mixing and blending of all the components. The think cloud represents thinking
8
about instruction, or how instruction is informed and framed by the setting, views
of mathematics, and the researcher, as well as how instruction informs and frames
the other three components of instruction.
Research Questions
The overarching question of this study is: What insights can be gleaned as an
experienced eye views the instruction in one classroom from a systems perspective?
Specifically, the questions that guided the present study are:
What are the interactions between students views of mathematics and their
teachers views of the same mathematics?
How do the teachers views of mathematics and their interactions with those
of her students influence instruction?
How does instructional coaching influence the teacher being coached, her
students, and the coach?
Definitions
The following definitions explain important terms, which are found in the
present study and are defined especially for the present study.
MultiLevel Action Research
For the present study, action research is defined as, a collaborative form
9
of systematic inquiry that can be used by teachers, administrators and other
educators. Action researchers use the tools of research to investigate their own
practices in their own schools and classrooms. The process gives educators tools to
discover and implement what works better for their students (Park & Greenleaf,
2002, p. 38). The research is analyzed at different levels of scale, hence the term,
multilevel.
Change
For the present study, change is defined as difference over time (G.
Bateson, 2000; Clarke, 2003b). For example, students might provide explanations
of problems that reveal thinking about why a particular solution was reasonable,
whereas before they only provided an account of the procedures involved in solving
the problem.
Views of Mathematics
Views of mathematics are those beliefs about mathematics and its
instruction held by individuals or institutions as evinced by actions or words, and
inferred from the data. The views for the present study include mathematics as rules
and procedures, mathematics as patterns, mathematics as language, and mathematics
viewed as applications in science.
10
Classroom Rhythms
For purposes of the present study, classroom rhythms are defined as those
happenings in the classroom that are patterns in the daily activities of the students
and teacher. These rhythms might be informed by, or happen as a result of, study
participants backgrounds. The classroom rhythms also include routines and
constraints, and disturbances.
Disturbance
A disturbance is often defined as a disruptive event or occurrence, and it is
often characterized as negative. However, disturbances in systems theory generally
refer to events that cause wobbles in the system. For the present study a
disturbance is defined as an event or occurrence, planned or unplanned, during
some interaction that affects the interaction in some way.
Interaction
Interaction is defined as a combined or reciprocal action of two or more
things that have an effect on each other and work together.1 For the purposes of the
present study, an interaction is defined as a reciprocal information exchange with
information transmitted from one person to another either verbally or through
1 Encarta World English Dictionary 1999 Microsoft Corporation.
11
actions. The transmission of information is likewise revealed verbally or through
actions.
Problematic Problems
The teacher began several classes, particularly in period six, with what she
termed problematic problems. These are typically problems from a previous
homework assignment that students questioned or wondered about. The teacher in
this study, Jane, generally solicits students to put these problems on the chalkboard
and discuss them. Sometimes Jane presents the problems herself for students.
Classroom Community
Community is part of a social ecological construct that might be described
as an interdependent and complex web of reciprocal relationships sustained and
informed by their purposeful actions (Lambert et al., 1995). For the present study,
classroom community, as defined by Lambert et al. (1995) refers to Janes two
mathematics classes, Jane, and her students in those classes. Throughout the study, I
gradually become part of this classroom community through communication of
instructional possibilities and changes that affect Jane, her students, and me using
multilevel action research.
12
Methodology
An ethnographic approach is used in this qualitative study to investigate two
of a teachers classrooms of students. One class is an eighth grade algebra class and
the other is an eighth grade mathematics class.
Source of Subject Population
Makefield2 Middle School teacher, Jane, and her 6th and 7th period classes of
math students agreed to participate in the present study. Jane had demonstrated
confidence in her teaching of mathematics at the middle school level, as verified by
her principal and conversations prior to the study. She also participated in a
weeklong training session in mathematics teaching, which I instructed in 1992. She
mentioned to me that she was quite excited to be a participant in the present study,
and she indicated that respect for me was established in that earlier training.
Criteria and Method for Includine/Excluding Subjects
The teacher, Jane, was included in the development and had consented to all
aspects of the study. The study and observations did not begin until all students
informed consent forms were returned to Jane. Three students in the algebra class
did not agree to be taped by any means, but they agreed to participate. Of note is the
2 All school names and individual names are pseudonyms.
13
fact that Jane had all permission forms returned, which is not typical as return rates
are often quite low. The significance of this finding is in Janes influence she had
with her class and the respect the students had for Jane as teacher.
Thus, only audiotapes were used for the algebra class and the tape recorder
was turned off if any of those students were speaking. Late in the study, two of the
three students who initially declined to be taped changed their minds and agreed to
be audiotaped. All students in period seven agreed to be videotaped. No
advertisement for subjects was needed for the present study.
Using Student Populations
All participating students and their parents signed an informed consent form
prior to the start of the study, and the participation was constantly reevaluated as the
study progressed. Students were allowed at any time to withdraw from participation
with no penalty, although no student chose to withdraw from participation. The
consent forms (See Appendices AC) outlined the risks and benefits to study
participants, and the risks were minimal.
Sources of Data
Data were gathered from multiple sources during mathematics class time and
during teacher interviews after class time. I videotaped Janes seventh period eighth
grade mathematics class, and audiotaped her sixth period algebra class. All post
14
observation interviews were audiotaped. Field notes were also taken during the
observations.
Data Analysis
Qualitative data techniques as suggested by Miles and Huberman (1994)
were used to analyze the data. Analysis includes three concurrent processes: data
coding and reduction, observation and interview analyses, and conclusion drawing
and verification. Interviews were initially transcribed; then I assigned codes to
passages and sentences that emerged based on the conceptual framework. Next I
created tables to organize the coded data and to help make patterns more visible. I
used computer software including NVivo, Microsoft Word and Excel to record and
sort coded data, thereby aiding me in identifying patterns and drawing conclusions.
Organization of the Dissertation
Chapter 2 presents a review of the literature relevant to the present study,
including the current state of mathematics education in the United States and
philosophies and findings of several key educational researchers, e.g. Dewey
(1997a, 1997b, 1997c) and Vygotsky (1978, 1986). Included also are discussions of
literature pertaining to views of mathematics, the interaction of those views, and
discussions of literature pertaining to action research. Chapter 2 ends with a brief
15
discussion on the need for research on the extent to which teachers and students
conceptions interact during instruction.
Chapter 3 describes the methodology used in the present study, including
multilevel action research, selection processes and outcomes, data collection
methods and analysis procedures, and a discussion on the credibility of the
conclusions.
Chapter 4 describes the setting including demographics and physical
characteristics of the school and its neighborhood, along with frequencies of
particular classroom rhythms, and interpretations of these frequencies.
Chapters 5 through 7 present the results from the analyses of the data, along
with visual displays of the data as reduced. Specifically, Chapter 5 presents data
concerning particular views of mathematics and its instruction with analyses at the
primary level of scale, Classroom Interaction, and a secondary level of scale,
Interpersonal. Chapter 6 presents data concerning changes that occurred in the
teacher, her students, and me, with analyses at two secondary levels of scale,
Interpersonal Interaction and District Constraints. Chapter 7 provides analyses of
data gathered in a followup interview with the teacher in the present study along
with accounts of observations of three teachers in my district for comparisons.
16
Chapter 8 provides a summary of the key findings from the study and
examines the findings by how they address and answer the research questions.
Implications for practice are discussed along with possibilities for further research.
17
CHAPTER 2
LITERATURE REVIEW
Overview
The purpose of this study is to examine the interactions between a teacher,
her students, and me regarding our views of mathematics and its instruction. To
conduct this examination the present study focuses on the interactions between the
teacher and her students, how these interactions influence instruction, and the
instructional changes that these interactions prompt with respect to the teacher, her
students, and me. The impetus for the present study resulted initially from
Thompsons (1992) suggestions for further research on teacher beliefs, student
beliefs, and on the interaction or lack of interaction between students and teachers.
In the first section of this chapter, Views of Mathematics and Its Instruction,
I provide a review of literature on views of mathematics and its instruction, not only
from the perspectives of a teacher and his or her students, but also from my
perspective. This section ends with definitions of four views of mathematics that are
used in the initial analyses for the present study. In the second section, Interaction
and Communication, I provide reviews of literature on interactions of views and
communication of mathematics, including suggestions from the National Council of
Teachers of Mathematics (NCTM). In the third section, Action Research and
18
System Influences, I provide a review of the literature on action research and
connections to systems theory, both the philosophy concerning why one might want
to conduct such research and the thoughts of other researchers on conducting such
research. Next I provide a review of literature to establish a case for conducting
action research by examining the findings through several lenses, i.e., multilevel
action research.
Views of Mathematics and Its Instruction
Many of us have strong beliefs about how school mathematics should look
and how it should be taught (Aichele & Coxford, 1994; Andrews & Hatch, 1999;
Archer, 1999; Bauch, 1984; Nathan & Koedinger, 2000b; Stigler & Hiebert, 1999;
Varrella & BurryStock, 1996; Weissglass, 1994). Many mathematics teachers,
parents, and administrators hold the belief that school mathematics should be taught
just like it was taught for them and that not everyone needs the same mathematics to
be successful (Gage, 1963; Noddings, 1994). After all, it worked for them. Why
shouldnt it work now for their children? This belief suggests that the primary
purpose of schools is to provide students with knowledge, in other words fill then
heads with knowledge. Others feel this traditional approach does not provide
opportunities for students to grow as thinkers and learners (Brooks & Brooks, 1993;
19
Cangelosi 2003; Stigler & Hiebert, 1999). Ritchhart (2002) summarized his
thoughts clearly when he stated,
Rather than working to change who students are as thinkers and
learners, schools for the most part work merely to fill them up with
knowledge. Although some may see intelligence as a natural by
product of schooling, in reality the curriculum, instruction, and
structure of schools do little to promote intelligence and may even
impede it in some cases, (p. 7)
Hiebert and LeFevre (1986) suggested there are two views of knowledge in
mathematics, conceptual and procedural. They defined conceptual knowledge as
that which contains relationships to other pieces of knowledge. In contrast,
procedural knowledge consists of formal, or symbolic, representation of
mathematics and the algorithms, or rules, for doing mathematical tasks (Hiebert &
LeFevre, 1986). But Hiebert and LeFevre (1986) warn, Students are not fully
competent in mathematics if either kind of knowledge is deficient or if they both
have been acquired but remain separate entities (p. 8).
Particular views of mathematics comprise one focus of the present study, so
the next three sections present more reviews of literature as they relate to teachers,
students, and my own views of mathematics and its instruction.
Teachers Views of Mathematics
Clearly, teachers instruction is strongly influenced by their beliefs in how
school, and particularly school mathematics, should look, and it often looks quite
20
similar to instruction the teachers got when they were students (Stigler & Hiebert,
1999). Freire (2003) claimed the teachers doing the telling and the students listening
fundamentally characterized the teacherstudent relationship, either in or out of
school. Friere (2003) warned of the risk of such instruction, saying, Narration
(with the teacher as narrator) leads the students to memorize mechanically the
narrated content (pp. 7172). Simon, Tzur, Heinz, Kinzel, and Smith (2000)
questioned,
What pedagogical approaches are teachers of mathematics
developing to meet the challenges posed by current mathematics
education reforms, particularly the challenges of adapting their
teaching to perceived mandates to reduce the role of teachers
showing and telling? What are the perspectives (meaningmaking
systems) that underlie such adaptations of teaching practice? An
understanding of the perspectives teachers hold while they struggle
to participate effectively in reforming mathematics teaching can
contribute to mathematics educators efforts to work more
effectively with teachers in transition, (p. 579)
It should come as no surprise that recent research (Schmidt & Kennedy, 1990) has
also suggested that teachers' beliefs about subject matter influence what they choose
to teach and how they choose to teach it. Cooney et al., (1998) suggested that
understanding the structure of teachers beliefs, as systems of beliefs, would provide
a certain dimensionality to what people believe (p. 331). Teachers likely did not
need research to tell them that their beliefs would influence the scope and structure
of their instruction or that understanding these beliefs would likely lead to improved
21
instruction. Nonetheless, research has revealed much about what teachers believe
and conceive about mathematics and its instruction. For instance, Archer (2000)
found that primary teachers tended to see mathematics as linked to students'
everyday lives, while secondary teachers tend to see mathematics as selfcontained,
and their role is to guide students through its structure. Raymond (1997) suggested
that although her study suggested, beliefs about the nature of mathematics were
more strongly linked to actual teacher practice than were pedagogical beliefs, there
was not sufficient evidence to confirm or refute this assertion (p. 573). However,
Raymond (1997), in her discussion about a particular teacher said, Thus, her
traditional practices were more heavily influenced by her deeply held beliefs about
mathematics content than her surface beliefs about mathematics teaching and
learning (p. 573).
Another study (Atweh, Bleicher, & Cooper, 1998) examined the social
context of two classrooms that differed in the socioeconomic backgrounds and
genders of their students. The researchers found from interviews with the two
teachers that,
Their perceptions of their students needs and abilities differed.
Ivor saw his students as potential mathematicians with
considerable ability and as needing the mathematical knowledge
required for university study. Jeff saw his students as having
limited ability and as needing mathematics for financial and
commercial transactions. (Atweh et al., 1998, p. 79)
22
A study in South Carolina (Spillane, 2002), conducted between 199296,
examined the relationship between local teachers knowledge and beliefs about
disadvantaged students and the teachers beliefs about teaching, learning, and class
management as they implemented reform policies. Results indicated a wide gap
between the rhetoric of reform and local practice. Initially, teachers rejected
proposals for intellectually challenging content and instruction for these
disadvantaged students and focused on increasing their basic skills. The study
found that the interaction between teacher knowledge of instruction and beliefs
about disadvantaged students changed for some teachers over time, and that these
teachers found their disadvantaged students interested in learning and motivated to
learn. As is often the case, teachers leave training sessions for implementation with
enthusiasm and beliefs that the reform will work. My experience has shown,
though, this initial enthusiasm and belief in the success of the reform often declines
to skepticism and sometimes apathy after attempts at changing practices meet with
resistance from students, and sometimes other colleagues. A high school teacher
that participated in an algebra workshop that I presented in spring 2002, told me that
she liked the activities we did, when she tried them with her students, they balked at
using the manipulatives required for the activities. AlMusawi (2001) found that
even student teaching did not significantly change student teachers beliefs about
teaching and learning, although the student teachers did leave the experience with a
23
more realistic view of the profession. I experienced this lack of belief change when
I had a student teacher who worked with two classes that were using Interactive
Math Program (Fendel, Resek, Alper, & Fraser, 1991), a reform mathematics
curriculum that was designed to support a constructivist approach to teaching. The
student teacher adapted well to a role of facilitating learning for students rather than
a role of direct instruction. Every day she stood back and let the students explore
the mathematics. She asked appropriate questions that helped focus the students
inquiry. But, when her university supervisor observed her, she directed the
instruction in a periodlong lecture. When I asked her about why she lectured, she
said she thought her supervisor wanted to see her teach. Clearly her belief about
teaching, and what teaching should look like had not changed, in spite of the
experience with the reform approach.
It seems that for the most part teachers hold to beliefs that teaching should
be similar to how they were taught (Stigler & Hiebert, 1999). And some have
asked, So what is wrong with that? Ravitch (2000) in citing a survey by NCTM
said,
The NCTM had some inkling that trouble might lie ahead when it
commissioned a survey of opinion about some of its precepts; for
example, overwhelmingly majorities of math supervisors (87
percent) and education professors (83 percent) agreed that students
best learn to solve complex mathematics problems when they are
given these problems even before they know all the basic skills
24
they might need, but only 23 percent of high school principals and
41 percent of PTA members concurred, (p. 440)
In fact, Ravitch (2000) argued that the NCTM (1989) standards were more
about teaching than standards. Papert (1980) acknowledged, It is deeply embedded
in our culture that the appreciation of mathematical beauty and the experience of
mathematical pleasure are accessible only to a minority, perhaps a very small
minority, of the human race (p. 190). Both Noddings (1994) and Devlin (1994)
questioned that mathematics should be for all students, a strong belief of NCTM
(1989). Noddings (1994) argument was that while many jobs required
mathematical proficiency for job entry, few of them actually required mathematics
to perform the jobs. Devlin (1994) claimed that,
Sadly, the level of abstraction in mathematics, and the consequent
need for notations that can cope with that abstraction, means that
many, perhaps most, parts of mathematics will remain forever
hidden from the nonmathematician; and even the more accessible
parts the parts described in books such as this one may be at
best dimly perceived, with much of their inner beauty locked away
from view. (p. 5)
A colleague of mine provided an anecdote that illustrated what learning for
all meant to him. He asked his students in a graduate level course, if they believed
all students could learn mathematics. The initial responses were all affirmative, but
then some of the teachers began to qualify their answers. One teacher jokingly said
all but his fifth period class. Another teacher expressed doubt about some of her
25
special education students. Several teachers said that some students simply do not
want to learn, or they are so distracted that they cannot learn. My colleague then
replied by saying, Then you dont believe that all students can learn. If I believe
that all students can learn, which Ive already stated, then I am not restricting my
efforts only to those that I feel are capable. One of my primary goals in teaching
mathematics is to provide opportunities for students to learn by using mathematical
activities that can be accessed by all students, regardless of their level of
understanding, e.g., wellformed stories involving a sequence of events (Stigler and
Hiebert, 1999) that are frequently a part of Japanese classrooms. Freire (2003)
claimed that, Students, as they are increasingly posed with problems relating to
themselves in the world and with the world, will feel increasingly challenged and
obliged to respond to that challenge (p. 81).
It is not as though teachers have not thought about their practice. My
experience suggests that good teachers are constantly looking for ways to improve.
But beliefs, dispositions, and realities of everyday school life often get in the way of
that improvement. FerriniMundy (1998) said, Teachers struggled mightily to
enact some of the ideas and practices they believed promising while remaining very
concerned that they might be taking time away from students' learning of important
basic skills (p. 134). Weinert and Helmke (1995) said that many studies had
shown that,
26
Instruction in which the teacher actively presented information to
students and supported individual learning processes was more
effective than instruction in which the teachers only role was to
provide external conditions that would make individual or social
learning success possible, (p. 138)
But, teachers often hold beliefs and perceptions about students and
mathematics that may or may not fit with what students believe. Beliefs and
dispositions tend to change with experience. That some students are at risk of being
misidentified is likely an understatement. Obiakor (1999) claimed that students who
behaved, looked, spoke, and learned differently were at risk of misidentification,
inaccurate assessment, misclassification, misplacement, and ineffective instruction
due to traditional school expectations. Andrews and Hatch (1999) found that,
The evidence indicates that teachers hold simultaneously a variety
of not necessarily consistent conceptions of, and beliefs about,
mathematics and its teaching. However, the degrees to which such
conceptions and beliefs are held vary subtly, and the evidence
suggests that such variations are likely to be consequences of
deeper philosophical beliefs privileging a dominant perspective.
The indications are that teachers' dominant pedagogic beliefs are
not inconsistent with their dominant perspectives on mathematics.
(p. 12)
In spite of the roadblocks for change, many teachers are adjusting then
views of how mathematics can be effectively taught. The NCTM (2000) standards,
despite the objections of some (Noddings, 1994; Ravitch, 2000), suggest a change in
teaching. This change has not been easily or effectively implemented for many
27
mathematics teachers, but many are trying to make the adjustments. McCombs
(2001) found that
A teacher confronted with the awareness that prior instructional
practices aren't working with a new group of students is most
likely to change those practices to more learnercentered
approaches if (a) he or she learns that this group of students has a
higher level of prior knowledge about the topic being covered than
prior groups of students and (b) a valued colleague has worked
with similar students successfully using new instructional practices
that give the students more choice and control over the
instructional process, (p. 188)
But, as stated earlier, resistance to changing beliefs may not be solely the
fault of teachers. Senge et al., (1999) warned that, "From a systems viewpoint, it is
not the people who are resistingrather, it is a system functioning to maintain its
internal balances, as all living systems function" (p. 558). Koch (1998) stressed that
we must remember,
Nowhere is reform deeper, more personal, or more threatening
than with teachers. The teachers are the ones who must use the
new materials; understand the teaching implications; change how
they interact with students, other teachers, and administrators; and
defend their actions to all who are involved with the school system.
(p. H8)
But as with all of education, we have much to learn. Nathan and Koedinger
(2000b) concluded that,
As research into mathematical learning and instruction continues,
teachers and members of the research community will be provided
with greater understanding of students mathematical conceptions
and development. And as studies of teachers knowledge and
28
beliefs continue, enhanced programs of teacher preparation and the
development of theoretically and empirically rooted approaches to
classroom instruction are to be expected, (p. 187)
Lambdin (1998) suggested that, "Mathematics educators are moving from
viewing mathematics as a fixed and unchanging collection of facts and skills to
emphasizing the importance in mathematics learning of conjecturing,
communicating, problem solving, and logical reasoning" (p.98).
Students Views of Mathematics
Lester (1989) conducted study that was designed to: (1) assess 7thgraders'
metacognitive beliefs and processes and investigate how they affect problem
solving behaviors; and (2) explore the extent to which these students can be taught
to be more strategic and become aware of their own problem solving behaviors.
Lester (1989) found that familiarization, a part of the cognitivemetacognitive
framework, which included orientation, organization, execution, and verification,
had the most important effect on students' problem solving.
Whether students use metacognition or not may be a key to understanding
their beliefs and dispositions. Schunk (1996a) asserted that, students need to
monitor their understanding of main ideas, but if they do not understand what a
main idea is or how to find one, the monitoring is pointless (p. 207). Such
metacognition, if verbalized, will likely lead to altered learning. While instructing
29
students deficient in division skills, Shunk (1996b) also found that some students
verbalized clear statements (e.g., "check," "multiply," and "copy"), others
constructed their own verbalizations, a third group verbalized the statements, and
their own verbalizations, and students in a fourth condition did not verbalize. He
found that selfconstructed verbalizations, alone or combined with the statements,
led to the highest division skill. The metacognition accompanied by a conversation
of the thinking produced the best results in this case.
Not surprisingly, students beliefs and dispositions do not always coincide
with teachers beliefs and dispositions. Nathan and Koedinger (2000b) found that,
Results from the analyses of students problemsolving strategies
lead us to suggest specific ways that students algebraic reasoning
differs from the views of development of algebraic reasoning
commonly held by teachers and researchers and the views
presented in popular algebra textbooks, (p. 169)
Many texts tend to present procedural ways of approaching problem solving,
which leaves many students with limited understanding of why the procedures
work. Also, students exposed to different instructional styles often struggle. Koch
(1998) said that, sometimes students have a difficult time initially accepting new
roles that give them more responsibility for their own learning (p. 120). But,
Carter and Norwood (1997) found that, teachers who hold beliefs more closely
aligned with the NCTM Standards use practices in teaching that influence student
30
beliefs and serve to accentuate intellectual development (p. 66). Carter and
Norwood (1997) further asserted,
That students can derive pleasure from working hard is not
surprising, if they are given the types of problems that challenge
and interest them. These students also believe that they will do
well in mathematics if they understand it. Problems that are
conceptually oriented kindle beliefs that understanding is
important in mathematics, (p. 66)
McLeod (1991) felt that beliefs, emotions, and attitudes were critical factors
in research on the affective domain in mathematics education, and that attitudes
toward mathematics appeared to develop in two different ways. Attitudes may
result from the automatizing of a repeated emotional reaction to mathematics. A
second source of attitudes is the assignment of an already existing attitude to a new
but related task (McLeod, 1991, pp. 6869).
If we knew more about students beliefs and dispositions, then, at the very
least, the job of instruction would be more informed. And, it is likely that many of
our revelations about students beliefs and dispositions might be surprising. For
example, the soontobe teacher in the study by Thomas and Montgomery (1998)
wondered, I am still not sure why children continued to express their needs for
teachers who are gentle, caring, understanding, and fun loving. We were asking
them about good teaching and they were describing good feeling (p. 380). The
children were characterizing good teaching as that which satisfied emotional needs,
31
yet their teachers were likely characterizing good teaching as that which focused on
instructional practice or student achievement. So this is evidence of a probable
misalignment of priorities. Thomas and Montgomery (1998) suggested, Perhaps
we all need to refocus our ideas regarding good teaching according to the emotional
needs of children (p. 380).
Lappan (1999) wrote,
To create mathematics programs that can foster powerful
mathematical learning for all students, we must make fundamental
changes in students opportunities to learn. These changes include
modernizing the curriculum, improving classroom instruction, and
assessing student progress in a way that informs and supports the
continued mathematical learning of each student, (p. 576)
This necessarily implies an assessment of students views of mathematics to
effectively inform instruction and ultimately influence learning.
Mv Views of Mathematics
The NCTM Standards (1989) had just been released and my principal had
decided we would no longer teach mathematics in segregated classrooms, or as
many describe as a tracked format. I believed that students should have a voice in
the classroom, but I also believed that students could learn best if they were placed
in groups of similar academic proficiency. My principal said that research was
suggesting that tracking was not effective, especially at the middle school level.
Oakes (1992), who cited much research that had been conducted prior to the 1980s,
32
said, The best research evidence on tracking supports the increasingly clear and
consistent (if not yet universally accepted) conclusion that this common way of
organizing students for instruction is, in most instances, neither equitable nor
effective" (p. 12). Oakes (1985) did acknowledge, though, that math classes were
different from English classes.
While the topics of math classes differed considerably and the
differences in the conceptual difficulty of these topics is dramatic 
 students at all levels of math classes were expected to perform
about the same kinds of intellectual processes. That is, at all levels,
a great deal of memorizing was expected, as was a basic
comprehension of facts, concepts, and procedures.
(Oakes, 1985, p. 78)
However, Ireson and Hallam (2001) found, Attainment in mathematics is
influenced by ability grouping, whereas attainment in science and English is not (p.
35). In mathematics, the pupils attaining higher levels at the end of primary school
make greater progress in sets, whereas pupils whose attainment is low at the end of
primary school make more progress in mixed ability classes (Ireson & Hallam,
2001, pp. 3435).
Nevertheless, I held fast to my belief that in mathematics tracking made
sense and was effective. My principal was firm in his edict, though, and another
teacher and I enrolled in a workshop titled, Visual Math. It was in this workshop
that I saw the power of allowing students to construct meaning for mathematics
through handson activities, and through the communication that I already
33
incorporated in my instruction. The revelation happened for me in this workshop
when I found a solution to a fraction problem strictly by using manipulatives. I was
so excited that I volunteered to present my solution for others at the overhead
projector. While walking up to the overhead, I experienced panic when I realized I
had not checked my solution with the traditional algorithm. But at the overhead, I
merely demonstrated how I used the manipulatives, and I realized that I really did
not need the algorithm. My thought was exactly as follows, If I can do this, then
think what kids could do.
Based on this and other experiences over 37 years, I now believe that a
mathematics teacher must provide students with opportunities to engage in
worthwhile mathematics, opportunities to talk to other students and people about
their learning, and opportunities and strategies for recording their mathematical
thinking so that others can make sense of it. Explicitly, I believe that effective
mathematics instruction should include,
1) Opportunities for learners, both adults and children, to engage in
mathematics in ways that allow prior knowledge to emerge, and ways that
nourish natural curiosity by providing handson experiences with
mathematical concepts,
2) Opportunities for learners to discuss and communicate their learning with
other peers and adults, and
34
3) Opportunities for learners to record their findings in a variety of ways,
including sketches, diagrams, words, and the abstract language of
mathematics.
These three components of what I describe as good mathematics instruction
are consistent with much current research on the teaching and learning of
mathematics (Brooks & Brooks, 1993; Dewey, 1997b; Kilpatrick, 2001; Lave &
Wenger, 1991; Marzano, 2003; Stigler & Hiebert, 1999; ThayerBacon, 1992;
Vygotsky, 1986). Armstrong (1998) discussed characteristics of genial classrooms,
a term chosen by him because of its association with genius, and claimed,
I believe that all genial classrooms share at least five
characteristics that guide their instruction regardless of content or
grade level. These characteristics are (1) freedom to choose, (2)
openended exploration, (3) freedom from judgment, (4) honoring
of every student's experience, and (5) belief in eveiy student's
genius, (p. 60)
Armstrongs five characteristics are in harmony with my three descriptions of
opportunities necessary for effective mathematics instruction. My idea of an
effective teacher is also compatible with Mary Catherine Batesons (2000)
description of her father, Gregory Bateson. She said, "He seemed dedicated to that
other task of teaching, helping students discover what they don't know and offering
only vague hints of how to find out and fit in, so that the extended process would
make it part of them" (M. C. Bateson, 2000, p. 235).
35
Initially, though, I use the following definitions, which serve as the basis for
analyses of views of mathematics in the present study. Juraschek (2002)
characterized four different views of mathematics claiming,
Teachers who think of mathematics as mostly a collection of
symbolic rules about numbers very likely will emphasize this in
their teaching. They will show students how to follow the rules by
modeling the procedures, providing guided practice with feedback,
and testing for proficiency. Teachers who associate mathematics
with applications in science will try to integrate mathematics
lessons and science lessons. They will provide activities to show
how mathematics is used to model and to analyze science
concepts. Teachers who see mathematics as a language to help
describe and understand our environment will see connections
between learning language and learning mathematics. These
teachers will link learning mathematics with a whole language' or
language experience' approach and will encourage students to talk
and write about their world using mathematical models. These
teachers will emphasize making sense of the symbols and figures,
that is, constructing meaning. Teachers who think of mathematics
as the study of patterns, form and relationships will emphasize
recognition of patterns and the processes of mathematics. They
also will interrelate mathematical concepts in their classrooms to
encourage students to see connections and generate their own
patterns, (p. 1)
These four views of mathematics are not necessarily independent. In
contrast, Hiebert and Lefevre (1986) considered two only views of mathematics
knowledge, conceptual and procedural, and warned, not all knowledge fits nicely
into one class or the other. Some knowledge lies at the intersection. Heuristic
strategies for solving problems, which are themselves objects of thought, are
examples (p. 9). Similar thinking applies to Juracheks (2002) characterizations. As
36
a person engages in mathematics, he or she may hold one view, several views, or
combinations of views.
Interactions and Communication
Thompson (1992) suggested that, virtually nothing is known about whether
students' views of the subject matter influence teachers' instructional decisions and
actions as well as their views of the subject (p. 142). Cohen and Hill (2002) cited a
few studies that examined the relations between teachers and students learning, but
found it had been relatively unusual for researchers to investigate these relations.
When they did so they found it was even more unusual to find evidence that
teachers learning influenced students learning. Cohen and Hill (2002) found a few
recent studies to be consistent with their results. One such study analyzed teacher
learning, practice, and student achievement data collected from four QUASAR
project schools, and found that students had higher scores when teachers had more
opportunities to study a coherent curriculum designed to enhance both teacher and
student learning (Cohen & Hill, 2002, p. 36). QUASAR (Quantitative
Understanding Amplifying Student Achievement and Reasoning) was a multiyear
project sponsored by the Ford Foundation to change mathematicsteaching practices
at six urban middle schools.
37
The mathematics education community in the United States has been
strongly influenced by the NCTM Principles and Standards for School Mathematics
(2000) and other NCTM documents (1989,1991,1995) that preceded these
Principles and Standards. The NCTM Standards (2000) address communication in
every standard, including how students can better communicate their mathematical
understandings to others. Evidence of the impact of the NCTM Standards (1989,
2000) can be found in the development of mathematics standards by all state
education departments in the United States. Lappan (1999) concluded,
The (NCTM) Standards have influenced more than a decade of
curriculum development with goals of higher mathematical
expectations for students, an inquiryinvestigative approach to
teaching, appropriate use of technology in learning mathematics, a
central role for applications or uses of mathematics, and a more
comprehensive view of assessment, (p. 568)
Snead (1998) declared, Thus, supporting evidence exists that teachers who
have reflected on their classroom experiences have come to many of the same
conclusions that the creators of the Standards have advocated (p. 293). Textbook
publishers have hastened to address these standards by aligning their standards
based curriculum, including pedagogical and instructional implications, to the
standards. The United States Department of Education has published a list of
mathematics curricula that, in their opinion, are promising or exemplary.3 One
3 See www.mathematicallvsane.com for NSFsponsored promising and exemplary math curricula.
38
of these curricula, Connected Math, was used for one of the classes in the present
study. Riordan and Noyce (2001) discussed research on Connected Math, and
presented that, in five Minneapolis schools fully implementing Connected
Mathematics, researchers found that most eighthgrade students significantly
outscored their counterparts in comparison sites on the State Basic Standards Tests
(p. 376). Connected Math, like other promising or exemplary curricula, e.g.
Math Alive (Foreman & Bennett, 1996) or the Interactive Mathematics Program
(Fendel et al., 1991) are characterized by activities that lend themselves to
substantial communication between the teacher and students, and between the
students and other students.
The thirteen NCTM Standards (2000) stress the importance of
communication, suggesting that instructional programs from prekindergarten
through grade 12 should enable all students to:
1. Organize and consolidate their mathematical thinking through
communication,
2. Communicate their mathematical thinking coherently and
clearly to peers, teachers, and others,
3. Analyze and evaluate the mathematical thinking and strategies
of others, and
4. Use the language of mathematics to express mathematical ideas
precisely, (p. 268)
Its noteworthy, though, that these four suggestions by NCTM about communication
imply an orderly classroom and respectful interaction.
39
Dewey (1997a) said, the communication which insures participation in a
common understanding is one which secures similar emotional and intellectual
dispositions like ways of responding to expectations and requirements (p. 4). But
Dewey (1997b) also warned, The belief that all genuine education comes about
through experience does not mean that all experiences are genuinely or equally
educative (p. 25).
Vygotsky (1978) believed that "the most significant moment in the course of
intellectual development, which gives birth to the purely human forms of practical
and abstract intelligence, occurs when speech and practical activity, two previously
completely independent lines of development, converge" (p. 24). Vygotskys
(1978) experiments demonstrated two important facts:
(1) A child's speech is as important as the role of action in attaining
the goal. Children not only speak about what they are doing;
their speech and action are part of one and the same complex
psychological function, directed toward the solution of the
problem at hand.
(2) The more complex the action demanded by the situation and
the less direct its solution, the greater the importance played by
speech in the operation as a whole, (pp. 2526)
It is this speech that allows others to know how one is thinking, yet many
teachers do not include such interaction in their classrooms (Freire, 2003; Stigler &
Hiebert, 1999). This disposition to avoid discussion may hinge on an understanding
of how mathematics is best learned, on how mathematics has typically been taught,
40
or simply lack of control in the classroom. The discourse in most United States
classrooms has long been little more than a speech on a topic and onesided at that
(Kilpatrick, 2001; Stigler & Hiebert, 1999). Teaching mathematics for many is
simply showing students what to do and how to do it, with very little interaction of
adult and student thinking. Kilpatrick (2001) suggested that, discourse should not
be confined to answers only but should include discussion of connections to other
problems, alternative representations and solution methods, the nature of
justification and argumentation, and the like (p. 426). Coulter (2001) said,
Habermas suggested that humans actions could be grouped into
two general orientations to act: strategic and communicative. In
strategic action, people attempt to influence the objective world,
while in communicative action, people try to come to an
understanding with others about something in the objective world,
social world, or subjective world that will help them to coordinate
action with others, (p. 91)
It is this coordinating of action that defines interaction, and this necessarily requires
systemic change. But interaction, by and of itself, does not guarantee effectiveness
of instruction or student interest. Leroys (2000) case study of a student named
Cathy, her teacher, and her struggles with reading and writing revealed,
The teacher's influence was likely weaker than it could have been
due to her lack of involvement with Cathy's reading and writing
while it was in progress. As Vygotsky (1978) suggested, the
process of scaffolding is a dynamic process in which one must
always stay in tune with the child to provide assistance as
necessary while the child is actively involved in a task. This does
not mean that teachers need to engage in onetoone teaching with
41
all children. But they do need to arrange for social support in small
group and wholeclass activities, as well as to interact with
individuals during literacy events. As Cathy's case shows,
interacting with the children before and after they do their reading
and writing is insufficient for maintaining their active interest in it.
(p. 89)
Communication involves interaction between teachers and students. One focus of
the present study is to investigate how this communication and interaction
influences instruction.
Action Research and Systems Influences
My intent in the present study is to examine the interactions between
particular views of mathematics, both teachers and students views and to examine
the influences these interactions have on instruction. In order to best to examine
these issues, it is necessary to acknowledge and study the intricacies of human
interaction. So, my own involvement in the study is as an observer of classroom
interactions, but also as a coach and participant in the process. My work currently
involves collaboration and interaction with mathematics teachers at all grade levels
of public school education. Much of what I do for teachers in my district is akin to
what I do in the present study. In my district I provide staff development, I model
lessons and activities for teachers, I observe teachers and students in classrooms,
and I provide feedback and consultation for district teachers. In the present study,
the teacher and I work together collaboratively to better understand not only her
42
teaching and her interactions with students, but also to better understand my work
with teachers and students.
Because the present study involves the interactions between human beings,
and because it relies heavily on the examination of conversations, or
communication, between these human beings, a qualitative method is the most
appropriate. Ethnographies are generally characterized by investigation of a small
bounded study site, by residence of the researcher at the site, by participant
observation as the primary data collection strategy, by the creation of a data base of
field notes, and by an emphasis on interpretive description and explanation of the
details and phenomena related to the subjects of investigation (LeCompte &
Preissle, 1993). All of these criteria of ethnographies are consistent with the
components of the present study, hence the present study is characterized as an
ethnography. Not all researchers necessarily agree with the effectiveness of
qualitative, or ethnographic, studies. Eisenhart (2001) expressed the following
concerns for qualitative studies:
I continue to receive letters from colleagues around the country
who complain about dissertation chairs and hiring and tenure
decisions that go against ethnographic researchers because their
methodology is not considered valuable or scientific. In some
places a backlash against its popularity is occurring. Some
programs within the National Science Foundation (NSF) recently
adopted a requirement that educational research projects be
reliable and rigorous, criteria that many quantitative researchers
believe ethnographies cannot meet. Pending Congressional
43
legislation (e.g., the Castle bill) suggests that the federal
government may not be far behind. Reactions like these can be
seen as a response to the range of varied methods that are now
categorized as qualitative research in education, (p. 19)
Eisenhart (2001) further suggested, Such studies are necessarily limited by
the researchers ability to participate in various settings, the amount of time the
researcher can devote and the researchers areas of special interest and expertise (p.
18). My reasoning for conducting this project using action research is based in part
on the potential for me to address Eisenharts (2001) limits by participating through
coaching the teacher, by the flexibility of my work schedule to make time for
observations, and by the fact that the teacher and I are both involved in math
education.
A number of researchers have defined and examined the details and
distinctiveness of the action research process. The name itself, action research,
implies that the present study involves both research and action, but not as
disjointed entities. The research and action are in concert with each other, and they
are ongoing. In other words, research and action may indeed occur simultaneously.
The research may prompt action and an action may prompt research. Eisenhart
(2001) said,
To be involved directly in the activities of people still seems to be
the best method we have for learning about the meaning of things
to the people we hope to understand. Only by watching carefully
what people do and say, following their example, and slowly
44
becoming a part of their groups, activities, conversations, and
connections do we stand some chance of grasping what is
meaningful to them. (p. 23)
Action research, at least in the present study, requires direct involvement in the
activities of the teacher and her students, as well as involvement in the activities of
the researcher. Feldman (2003), in his article on validity and quality in selfstudy
said,
The selfstudy of teacher education practices is also moral work
because it has a normative, teleological component we dont
want to just study our practice, we want to improve it in a
particular direction that will affect what happens in our colleges,
universities, and schools, (p. 27)
One key to action research is the collaboration that occurs, or can occur, through
investigations of the culture, the phenomena that occur, and the social structures of
those under scrutiny. Park and Greenleaf (2002) conducted a study titled, The
Constructing Knowledge Project, and defined action research as,
A collaborative form of systematic inquiiy that can be used by
teachers, administrators and other educators. Action researchers
use the tools of research to investigate their own practices in their
own schools and classrooms. The process gives educators tools to
discover and implement what works better for their students.
(P 38)
Johnson (1993) said the action research is characterized by spiraling cycles of
problem identification, systematic data collection, reflection, analysis, datadriven
action taken, and, finally, problem redefinition (p. 1). These spiraling cycles are
45
described in Chapter 3 of the present study, as they relate specifically to this same
study. Koch, McQueen, and Scott (1997) said, denoting AR as action research,
The origins of this research approach rest on sociopsychological
studies of social and worklife issues. AR is often uniquely
identified by its dual goal of both improving the organisation
participating in the research project, usually referred to as client
organisation, and at the same time generating knowledge, (p. 3)
Clarke (2003a) stated, Action research is an approach to inquiry that
provides a theoretical umbrella and an action agenda for community and
organizational development and individual change (p. 5). All of these definitions
and clarifications of action research point to the need for a systemic and
collaborative approach for conducting studies of human beings. But essential to the
present study is the examination not only of the interactions between the teacher and
her students, but also of the emerging changes in my own knowledge and
understanding of my practice. Argyris et al., (1985) argued,
To put it most succinctly, action scientists engage with participants
in a collaborative process of critical inquiry into problems of social
practice in a learning context. The core feature of this context is
that it is expressly designed to foster learning about one's practice
and about alternative ways of constructing it. (p. 237)
My role as participant and researcher in the present study thus becomes an
important factor in addressing the issues of interactions, because these interactions
impact the teacher, her students, and my work as a district leader. Schon (1983)
emphasized,
46
The inquirer's relation to this situation is transactional. He shapes
the situation, but in conversation with it, so that his own models
and appreciations are also shaped by the situation. The phenomena
that he seeks to understand are partly of his own making; he is in
the situation that he seeks to understand, (pp. 150151)
In other words, the teacher and I engage in examination of both her practice and
mine. Action research thus surfaces as the most logical and valid approach for such
an examination.
Systems Theory
One cannot conduct action research without paying attention to the system
that surrounds and defines the setting of the study. The setting for the present study
is an urban middle school in a large metropolitan area of Colorado. Jenlink (1995)
warned, Changing complex social systems like schools involves negotiating
multiple intercolliding systems that interrelate and contribute to the total identity of
the system as well as provide a source of identity for individuals within the system
(p. 45). Another key idea in systems thinking is that of changing the present system
to a more efficient or effective system. This necessarily and typically involves
personal and organizational change. McCombs (2001) claimed,
Personal change in one's perceptions, values, attitudes, and beliefs
results from transformations in thinking. These transformations in
thinking most often result from critical connections made in one's
own understanding, knowledge, and ways of thinking, as well as
from critical connectionspersonal relationshipswith others of
significance in the learning environment, (p. 188)
47
Covey (1989) said, The key to the ability to change is a changeless sense of who
you are, what you are about and what you value (p.108). As Clarke (2003b) said,
Learning is change over time through engagement in activity (p.41). Rogers
(1995) spoke of the importance of changes that met the clients needs, and said,
One of the main roles of a change agent is to facilitate the flow of
innovations from a change agency to an audience of clients. For
this type of communication to be effective, the innovations must be
selected to meet the clients needs, (p. 336)
So, my search for changes over time necessarily must be linked to both my needs
and those of the other participants.
MultiLevel Action Research
Action research links closely to systems theory. Baskerville (1999) stated,
Action researchers clearly recognize that human activities are systematic, and that
action researchers are intervening in social systems (p. 5). The setting itself, in this
case a teachers classroom, is layered with complexities that are best studied not
only as pieces of a large picture, but also as these pieces weave together and
influence this same large picture. Baskerville (1999) declared,
Action researchers are among those who assume that complex
social systems cannot be reduced for meaningful study. They
believe that human organizations, as a context that interacts with
information technologies, can only be understood as whole entities.
A key implication of this assumption is that the factoring of a
48
social setting, like an organization and its information technology,
into variables or components, will not lead to useful knowledge
about the whole organization, (p. 2)
Hence multilevel action research has potential to be the most logical
approach for a complete study of these complexities. Research on systems thinking
and multilevel action research confirms my choice for a research framework. Capra
(1996) asserted that dissecting a system into isolated elements literally obliterates
systemic properties, and also said,
Another key criterion of systems thinking is the ability to shift
one's attention back and forth between systems levels. Throughout
the living world we find systems nesting within other systems, and
by applying the same concepts to different systems levelsfor
example, the concept of stress to an organism, a city, or an
economywe can often gain important insights, (p. 37)
Clarkes (2003a) description of multilevel action research is characterized
by looking at problems through various lenses. Clarke (2003a) stated, Because the
problems we are interested in are complex and manifest themselves in many
contexts, we believe it is necessary to work simultaneously at different levels of
scale; hence the addition of the phrase multilevel (p. 4). It is precisely these
levels of scale that provide a vehicle for achieving coherence between classrooms,
community, and university (Clarke, 2003a).
The present study concerns the interactions between a mathematics teacher,
her students, and the coach viewed through the lenses of teacher, coach, and district
49
coordinator. Such interaction involves action, i.e., change required at different levels
to support changes in classroom instruction. Such change implies that the school,
the district, and the community, at least the parents, need to change in order to
support classroom change. Evans (1996) maintained, "the key factor in change is
what it means to those who must implement it, and that its primary meanings
encourage resistance: it provokes loss, challenges competence, creates confusion,
and causes conflict" (p. 21). The likelihood of resistance looms high. The teacher
might be willing to consider change, but if even one of her students, his or her
parents, the school administration, or the school district resists, then change itself is
at risk of occurring. As Clarke (2003b) stated,
I cant change myself without affecting others. And because I
cannot change others unilaterally, I will need to engage them in
some principled interactions. This may precipitate crises of various
magnitude, as I discover where people stand and the relative
importance (to me) of my change efforts versus the good opinions
of my friends and colleagues, (p. 40)
Chapter Summary
Evidence has revealed that the research on students and teachers beliefs
and conceptions on mathematics and its instruction is substantial. In contrast, the
research on the interactions between these beliefs and conceptions has been limited
or, in the views of some researchers, virtually nonexistent. Thus, several researchers
have agreed that further investigation into the effects of and interaction between
50
these beliefs is warranted. The literature supports that complex social systems, such
as the setting for the present study, can likely be best studied using action research.
The intent of the present study is to investigate and examine the interactions of ONE
teacher and HER students to help determine whether they influence one another,
which is just what Thompson (1992) suggested.
51
CHAPTER 3
METHODOLOGY
Overview: An Ethnography of One Teacher
The research methodology is an ethnography of one teacher, her interactions
with her students and me, and the influences these interactions have on her
instruction, as well as their effect on my own learning. Because of the complex
nature of human interactions, the methodology for analysis is multilevel action
research. This means that the analyses are conducted with respect to a primary
perspective, classroom interactions, and two secondary perspectives, interpersonal
interactions and school district constraints.
Observations span two months with sixteen visits to two middle school
mathematics classrooms, both taught by one teacher. The research includes
classroom observations, interviews with the teacher, the use of a camcorder and
audio recorder, unstructured interviews with students, written field notes,
collaborative discussions, and analyses of recorded classroom discourse.
Research Setting
The study involves students, one teacher, the researcher and the various
communities that belong to and interact with each other. These communities vary
as the interactions vary, but are comprised of the classroom, peer communities of
52
I
students and the teacher, a community of the teacher and the researcher, the school,
and the community of families that have students attending this school. Lambert et
al., (1995) defined a community as:
an interconnected and complex web of reciprocal relationships
sustained and informed by their purposeful actions. Complexity is
manifest in the diversity of the system; and the more diverse, the
more rich and complex. Such communities are flexible and open
to information provided through feedback spirals, as well as
unexpected fluctuations and surprises that contain possibilities.
The coevolution, or shared growth, of the participants in this
community is propelled by the joint construction of meaning and
knowledge and involves continual creation and adaptation, (p. 42)
The community in the present study has all the attributes and idiosyncrasies as the
one described directly above. It is complex and diverse, and the members of the
community are interconnected through work, school, and the environment they live
in.
Baskerville (1999) stated that the fundamental contention of the action
researcher was that complex social processes could be studied best by introducing
changes into these processes and observing the effects of these changes (p. 3). The
interactions between students views of mathematics and their teachers views of the
same mathematics, and how the interactions influence instruction form the complex
social processes described by Baskerville (1999). So, because of the complex nature
of a middle school classroom, the setting for the present study is ideal for an action
research investigation.
53
MultiLevel Action Research
As part of their research endeavors, my doctoral lab4 crafted a platform for
theory and action, and a principled basis for collegial critique of teaching and
learning in a variety of settings. The basic steps of this procedure as used in the
present study are detailed as follows and also represented in Figure 3.1 below:
1. Patterns of activity that characterized the functioning of the people
and organizations were described.
2. Particular events or situations that characterized this pattern, and
were representative of the way the system functions, were
identified.
3. Factors that contributed to the maintenance of this pattern were
analyzed. Factors were identified at different levels of scale, which
required research of both contemporary and historical phenomena 
 institutional and legal documents, biographical analyses, and
interviews with stakeholders.
4. The functioning of the systems was described at three levels of
scale the level that was the primary focus of research, and two
other levels. One was at a smaller level, interactions between the
teacher and her students and interaction between the teacher and
me. The other one was at a higher level, constraints of the
teachers district and those of my district. This was done in order
to gain a systemic perspective. In the present study, the analysis of
classroom interaction was the primary focus, with analyses of
interpersonal interactions and district constraints as secondary foci.
5. A detailed analysis of the area of primary interest was conducted
that both described the status quo and explored avenues of action
for change
6. Changes at the secondary levels of scale that would need to occur
to sustain the changes identified in #5 were also explored.
4 Laboratory of Learning and Activity (Lola) University of Colorado at Denver
54
Figure 3.1: Analytical Framework
MultiLevel Action Research:
A Methodological Framework
The arrows represent the cyclic nature of this process, and though the process likely
leads from one component to the next, e.g., from step 3 to step 4, nothing about the
55
process prevents an alteration of this flow of the cycle. For example, following an
analysis at a particular level of scale in step 5, its possible that one could back up
and revisit the meaning of a particular level in step 4. Most likely, though, the flow
of the cycle proceeds from step to step, in order, with the cycle starting all over
again following step 6.
Site and Population Selection
This investigation was conducted at Makefield Middle School. Negotiating
entry to this site was enhanced by several factors. A partnership between my
university and Makefield has existed for over two years. In fact two of my
university advisors were already working with Jane and the technology teacher at
Makefield. These two professors were also in the second year of a threeyear grant
investigating ways to improve the academic engagement of African American and
Latino students. Furthermore, the Laboratory of Learning and Activity (Lola) had
an after school club established at Makefield. Also, a personal connection existed
between Jane and me, because Jane had participated in a mathematics workshop
about seven years ago facilitated by me. Thus mutual respect was already
established, and Jane agreed to participate in the present study. Jane indicated in our
first meeting that she would like for me to model some lessons for her students.
Although I had offered to present lessons for some classes on some days, time
56
constraints, both of Jane and me, prevented this modeling from occurring. The
discussions, characterized by clinical reflections and comments rather than
judgmental reflections and comments, helped maintain access.
Access to the student piece of the present study was unknown when I first
began observations. My experience however, suggested that student access would
be enhanced through my interaction with the students. I initially kept a low profile
and interacted rarely with the students. My responsibility was to film, record audio,
and take notes. Later, based on discussions with Jane, I interacted more frequently
with the students in ways decided upon by Jane and me. Jane had informed me that
her students were sweet kids who are game for something new, so they should give
us a good time.
Background and Role of Researcher
I was a secondary mathematics teacher for 34 years and a mathematics
department coordinator for 24 of those years. Since 19901 have been conducting
training workshops, nationally and locally, for mathematics teachers seeking to
improve their mathematics and instructional skills. I am currently the K12
Mathematics Coordinator for a major suburban school district in Colorado. My
current job involves training of teachers in mathematics instruction, consulting with
teachers and administrators in mathematics pedagogy, classroom visitations and
57
observations, modeling of mathematics instruction for students and teachers, and
other administrative duties.
During the present study my initial role was to observe two mathematics
classes taught by Jane. I recorded my observations of teacher/student interactions,
but was primarily concerned with observations of Jane. My role included coaching
during the regular conferences Jane and I had in which we discussed observational
data.
Data Collection: Sources and Procedures
Study participants included one teacher and two classes of her students. Data
collection from students occurred only with those who submitted an informed
consent form signed by the individual and his or her parents or guardians. One class
agreed to be videotaped and the other class agreed only to be taped only by audio
means, because three students declined to be taped at all. These three students
agreed, however, to participate in the study. Norms for all participants were
identified for behavior, participation, sharing information, talking to others about
study events, and withdrawal from participation. Expectations for these norms
included preserving confidentiality and safety as primary goals.
Data were recorded from 3/31/03 to 5/22/03 using observations of classroom
events and interviews with Jane, field notes, audiotape, videotape, and computer
58
entry. Data were not collected from the three students who declined to be taped. One
of the three was never present in class when I observed. In situations where the
other two students were speaking, the tape recorder was turned off. Observations
were focused on what students and teachers said, and how they responded to
comments or observations of others. Interviews included questions about the
observations that emerged from the observations.
Data Sources
The sources of data for the present study are the observations, written field
notes, video and audiotapes, and the interviews I had with the teacher. These
sources and their frequencies are pictured in Table 3.1:
Table 3.1: Sources and Frequency of Data Collected
Source of Data Teacher Period 6 Period 7
Interviews 5
Observation of mathematics
classes 16 9 7
Artifacts:
Field notes V V
Video tapes V V
Audio tapes V V V
Note. Checks indicate artifacts were collected. Dashes indicate no data obtained
59
I
Data Analysis Procedures
The observations and interviews were digitally recorded using videotapes for
the eighth grade mathematics class and audiotapes for the eighth grade algebra
class. All media were transferred to a computer and then transcribed. The video
recordings were transcribed while viewing them and the audio recordings were
transcribed using DSS Player software designed for use with the Olympus voice
recorder used for the recording. Transcriptions were initially recorded rising
Microsoft Word, saved as an RTF (rich text file), and then transferred to QSR
NVivo qualitative software. Using NVivo, all individual transcripts were reduced by
major then secondary codes.
Noteworthy to researchers intending to use NVivo or other software was the
fact that the software became inaccessible during the coding process until I rebuilt
some software directories. The NVivo program would still run, but I could not save
any work with it and I could not find the files I had previously saved. I discovered
the solution to recovering the data after several weeks of investigation. The solution
required a fix in the Windows environment, not in the NVivo program. Fortunately,
I had saved most NVivo reports in Word format, so I did not have to start over with
the data reduction. I transferred the data from the Word documents into an Excel
spreadsheet, which allowed for sorting and filtering. Because of the problems
60
encountered with NVivo, I chose to use Word, Excel, and FileMaker Pro as primary
software analysis tools for the remainder of the data analysis.
Coding: Views of Mathematics
The particular views of mathematics that form the foundation for the present
study (See Table 3.2) are based on Jurascheks (2002) characterization of four
different views of mathematics, presented and defined in Chapter 2. These views
are:
Symbolic Rules and Procedures
Applications in Science
Language
Patterns, form and relationships
These four views are classified as major codes for the purpose of data reduction.
Data reduction refers to the process of selecting, focusing, simplifying, abstracting,
and transforming the data that appear in writtenup field notes or transcriptions
(Miles & Huberman, 1994, p. 10). No assumptions are made about difference of
abstraction or value for these four views of mathematics.
61
Table 3.2: Codes for Views of Mathematics
Major Code Secondary Code
View of mathematics as symbolic or procedural Follow rules/Model procedures Guided practice w/feedback Testing proficiency
View of mathematics as applications in Science Integrate or connect math and science Model or analyze science concepts in activities
View of mathematics as language Connections with math and language Making sense or constructing meaning
View of mathematics as the study of patterns. Forms, and relationships Pattern recognition or analysis Connections
View of mathematics as language Connections with math and language Making sense or constructing meaning
Each of these views is further coded for particular aspects of each view, followed by
the secondary code descriptions. The abbreviations I used for the coding process, in
parentheses for each major and secondary code, follow:
Secondary Codes: Views of Mathematics
Major Code Symbolic Rules and Procedures (MVSymprc)
Secondary Codes:
1. Following rules and modeling procedures Talk or actions that
evince rules or procedures in mathematics or its instruction
(FollowRP)
62
2. Guided practice with feedback Planned or occurring practice or
feedback (Practice)
3. Testing proficiency Questions that are asked to assess learning,
either formal or informal (Assess)
Major Code Applications in Science (MVSciAp)
Secondary Codes:
1. Integrate or connect math and science Talk or actions that suggest
mathematics and science were integrated, connected, or linked
(ModelMS)
2. Model or analyze science concepts in activities Modeling or using
mathematics to explain science or science to help explain
mathematics (ConnectMS)
Major Code Mathematics as Language (MVLang)
Secondary Codes:
1. Connecting mathematics and language Approaches or conversation
involving connections or links between mathematics and language
(ConnectML)
2. Making sense of math symbols and figures Talk or actions that
clarified mathematics through language (MakeSenseML)
63
Major Code Patterns, Form and Relationships
Secondary Codes:
3. Pattern recognition or analysis Conversations or actions that reveal
observed patterns and their relationships (RecogAnalyze)
4. Connections Conversations or actions that compared or connected
patterns in mathematics to other mathematical notions
(ConnectCompare)
Descriptive Coding: Classroom Rhythms
Most observations and interviews with Jane also revealed a number of
discussions that described events and situations that were not necessarily directly
related to views of mathematics. These discussions were mostly about daily
classroom routines, or information about students and schools, and were
characterized and classified as classroom rhythms.
The following are examples of these classroom rhythm codes (See table 3.3)
and include my coding abbreviations in parentheses.
1. Disturbances (RhyDisturbance)
Stigler and Hiebert (1999) discuss classroom interruptions that
consistently occur in United States classrooms, but occur rarely, if ever, in
Japanese classrooms. They cite one instance where a public address system
64
announcement interrupts a classroom to the dismay of the Japanese
delegation of observers. However, an interruption often implies a negative
occurrence, which is not always the case in classrooms. Some
interruptions are merely everyday occurrences that are a part of normal
classroom events, or classroom rhythms. I chose to use the word
disturbance as a descriptor of classroom events that occur regularly, but
the meaning of the word is not to be taken as necessarily positive or
negative. In fact a disturbance may be negative, positive, or neutral,
depending on the circumstances that contribute to the disturbance. For
example, a student, teacher, or other school staff member may need
information or service related or unrelated to the daily lesson, so they
disturb an ongoing discussion or conversation to present their views or
needs. Disturbance is defined by the dictionary in Microsoft Word as a break
or gap in a process that would normally be continuous. To me, disturbance is
a general descriptor of events that just happen, rather than events that happen
to disrupt, disturb or interrupt the normal flow of a lesson. In systems theory,
a disturbance is an event that changes the wobble of the system.
Disturbances in the present study did wobble the system of classroom
instruction.
65
I describe two types of disturbances that surfaced in the observations:
disturbances that were lesson related or connected, and disturbances that
were school related or connected. One could argue that any disturbance was
related to both a lesson and school because the disturbance affected or
influenced the lesson, which was part of what happened at school. However,
as a secondary code, I defined a lesson related disturbance (LessonRel) as
one that was initiated by a student or Jane, and occurred during a lesson. I
defined a school related disturbance (SchoolRel) as one initiated by someone
or something outside of the classroom, such as a fire drill, a public address
system announcement during class, or a phone call for Jane during class.
2. Routines and events (RhyRoutine)
I defined routines and events to be those occurrences in classroom
life that surfaced frequently, happened randomly, or were schoolimposed
requirements. Some examples of these follow:
Many teachers have classroom routines that they often
follow and repeat frequently, such as beginning a
lesson with a warmup activity, offering suggestions
or directions for students, joking with students, and
separating students into working groups.
Teachers are also expected to handle discipline and
classroom management issues and problems as they
arise.
Attendance and other recordkeeping tasks are
generally school and district expectations, a necessary
66
part of a teachers daily activities. In fact attendance
record keeping is mandated for teachers by state law.
A secondary code called record keeping (RecordKp), includes
recording of grades, correcting papers during class time, or assignment of
student homework or class work constituted. This teacher also precedes
many lessons with an activity generally related to the lesson, called a warm
up activity. Such an activity and other routines are coded as classroom
routines (ClsRoutine). Student management or discipline problems are
coded under the category of discipline (Discipline).
3. Background information (RhyBkgmd)
Definition: Teachers and students are often privy to information that
informs them of personal and/or school and non schoolrelated situations
with peers, families, and communities. Some discussions were about
students performance in and out of the classroom or about situations in
other schools. I developed the following secondary codes for this
background information: personal information (Personal), and schoolrelated
information (SchoolRelRB).
67
i
Table 3.3: Codes for Classroom Rhythms
Major Code Secondary Code
Disturbances Lesson related School related or initiated by school situation
Routines and events Record keeping Classroom routines Student management or discipline
Background information Personal information School related information
Summary: Data Collection
The present study includes classroom observations and individual interviews
of Jane. All observations and interviews were recorded electronically, transcribed,
and then crafted into written documents. Participants engaged in normal classroom
activity with mathematics learning and instruction. Jane used the existing middle
school mathematics curriculum, a high school level algebra text and Connected
Math for the eighth grade class, to guide the learning activities and instruction.
Connected Math is a National Science Foundation funded middle school
mathematics curriculum.
Data Display and Conclusions
Data display was the phase of analysis where I organized the data into visual
displays, e.g. figures and tables. Summaries and syntheses of emerging patterns
68
appear in tables displayed in Chapter 4. These displays allow for visual examination
that point to common themes, similarities, and major differences.
Conclusion drawing and verification was designed to build a logical
presentation of evidence through patterns, themes, and relationships by comparing
and contrasting the evidence. Connections were made between the events and
experiences of the participants and my own observations and interpretations. I also
linked my findings from the observations and interviews with relevant literature to
provide a connection of my understandings and interpretations with theory.
Credibility of Conclusions
Reliability and validity add to the credibility and trustworthiness of any
research study. The constructs of reliability and validity, traditionally statistical
constructs in quantitative research, can be applied to qualitative studies (Krathwohl,
1998; LeCompte & Preissle, 1993; Miles & Huberman, 1994). However, I was
interested mainly in whether my coding of sentences and passages from the
transcribed data were consistent with how another might code the same data, which
is referred to as internal reliability.
Internal Reliability
To determine whether multiple observers would agree about what has
happened (LeCompte & Preissle, 1993), a colleague and I examined several
69
passages from the transcriptions and independently coded them for discussion. We
then independently coded about 90 lines of text from transcriptions of nine class
periods, and three teacher interviews. We agreed that our comparison coding would
be based on the primary code for any particular passage. By that I mean, that a
passage that was primarily representative of a patterns view of math, even though
parts of the same passage might have been doublecoded as well to represent math
as language, would be for the purposes of this comparison coded as a patterns view
of math.
I compared our independent coding decisions using Cohens kappa5 to
determine the index for interrater reliability. Essentially, Cohens kappa is a
statistical measure that reveals a level, or percent, of consistency between how two
independent persons rate, or categorize, data. Results are considered good if the two
raters agree in their categorization 90% of the time. The significant number
resulting from the statistical calculation is the Kappa number. This kappa number,
or index, translated to about 92.7%, which means my colleague and I agreed on
approximately 92.7% percent of the coding decisions.
5 See: http://www.kokemus.kokugo.juen.ac.jp/service/kappae.html
70
Chapter Summary
I began this chapter with descriptions of the research study designed as an
ethnography using multilevel action research. Action research permits and
encourages looking at classrooms as a myriad of interactions between the students,
Jane, and me. I described the research design as a platform for theory and action,
and a basis for collegial critique of teaching and learning in a variety of settings. I
discussed the site and population setting and discussed my background and role as a
researcher in this setting. Next, I provided discussion of the data collection methods
and the data sources. The data analysis procedures were described, including the
coding used and how I reduced the data for analysis. Finally, I discussed how
closely a colleague and I agreed on how I coded the data from the observations and
interviews.
71
CHAPTER4
THE SETTING: CLASSROOM, SCHOOL CULTURE,
AND INITIAL RESULTS
Overview
Results of my investigations and observations of the mathematics
community at Makefield Middle School are presented in this chapter. First, in a
section titled, Demographics, I describe the research setting, including
demographics of the school and state mathematics testing results for the school, how
I prepared for the observations, and the physical setting. In section two of this
chapter, Results Rhythms of School Life, I present data on daily on classroom
occurrences that initially seem to have little or no connection to the daily lesson,
including frequencies of observed data pertaining to classroom rhythms, and
observations of these rhythms. These data illustrate how such occurrences are
woven into the daily interplay of classroom activities. All these data are presented to
help the reader establish a sense of the demographics of this particular urban middle
school, frequencies of occurrences of particular events, particular phenomena, and
patterns that emerged from the observations.
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Demographics
Makefield is a Rocky Mountain region urban public middle school for
students in grades six through eight. Most students at the school are African
American or Hispanic. Most of the homes in the neighborhood were built in the
early twentieth century. The houses are being remodeled and wealthier residents are
solicited and encouraged to move into the neighborhood. Almost 75% of the
neighborhood students qualify for free or reduced school lunches.
Table 4.1 illustrates the ethnicity composition of Makefield Middle School.
Over 90% of the ethnicity at Makefield is either African American or Hispanic, with
76% of the students being African American. In contrast, the district has 19%
African American and 56% Hispanic. Caucasians make up 7% of Makefields
student population, while the district is 20% Caucasian.
Table 4.1: Ethnicity at Makefield Middle School, 20022003
Ethnicity School % District %
African American 76% 19%
American Indian Less than 1% 1%
Asian Less than 1% 3%
Hispanic 15% 56%
Caucasian 7% 20%
During the course of the present study, the majority of students were from
lowincome families, as evidenced by the fact that in fall 2002, 76% of the total
number of students qualified for free or reduced school lunches during the 2002
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2003 school year. There were 493 students who were enrolled at Makefield Middle
School in 20022003, and the average daily attendance was 462. During the school
year there were six incidents of substance abuse involving drugs, two incidences of
substance abuse involving tobacco, no incidences of substance abuse involving
alcohol, and 209 other minor violations of Code of Conduct. Student dropouts,
calculated for the 20022003 school year, were at 1.5%. I cite these data to provide a
small snapshot of the milieu that many Makefield students bring to the classroom.
Testing Results
Mandatory state testing, the Colorado Student Assessment Program (CSAP),
places students into four categories of proficiency: unsatisfactory, partially
proficient, proficient, and advanced. Public reporting of these data promotes a high
stakes atmosphere that surrounds the testing. Such high stakes testing generally
increases time spent on test format practice and lowers staff morale (Shepard,
Taylor, Kinnen, & Rosenthal, 2003). Schools are judged from unsatisfactory to
excellent based to a large part on students scoring on this annual test. Makefield
was rated as a low school in the 20022003 school year. For the 20022003 school
year four other nearby district middle schools were rated as low, and three as
unsatisfactory that same year, so Makefield is not alone in the low categorization by
the state. Based on state test results for the 2002 testing year, of 214 sixth graders at
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Makefield Middle School, only 7% were judged as proficient in math. Similar
results were found at the 7th and 8th grade levels, with 6% of 225 seventh graders
and 8% of eighth graders judged as proficient in math. The complete 2002 results
for eighth grade at Makefield are in Table 4.2 below.
Table 4.2: 2002 State Test Results in Mathematics Makefield Middle School
Math Results Students % Unsatisfactory % Partially Proficient % Proficient % Advanced
6a2002 214 60 25 7 0
7th 2002 225 56 27 6 1
8th 2002 220 59 24 8 2
6* 2003 156 51 28 7 1
7th 2003 150 69 25 5 0
8th 2003 175 65 22 6 1
These test results are by no means the last word on how students at this
school, or for that matter any other school, can perform, but they are reported to the
general public this way. Shepard et al., (2003) recommended,
More attention should be paid to content standards and to both
professional development and curriculum materials that support
instructional improvements. Less attention should be paid to
raising test scores per se and to evaluating the quality of
schooling only on the basis of CSAP results, (p. 54).
The present study is not about CSAP or its scores, so the presentation of this
information is strictly for informational purposes. However, a reality of public
schooling in Colorado is that teachers are consistently reminded of their
responsibilities to improve test scores, and so consequently, their instruction is
influenced by these expectations. In particular, Jane and the other teachers at
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influenced by these expectations. In particular, Jane and the other teachers at
Makefield, as the CSAP evidence shows, were placed in a tough situation. They had
to deal with students that did not tend to do well on state tests.
Preparing for Observations
The teacher, Jane, and I agreed on two classes that would constitute the
observed classes for the present study. Jane handed out permission forms to both of
these classes. All students in period six, an eighth grade algebra class, returned their
permission forms by the end of March 2003, but three students only agreed to
participate and not be taped. Therefore, no video was used for period six and
audiotaping was suspended when any of these three students spoke. All students in
period seven, a regular eighth grade math class, had returned their permission forms
by midApril 2003, and all agreed to be taped by video. Jane only taught eighth
grade students during the period of the present study.
The Physical Setting
Janes classroom is located on the third floor of a school building that was
constructed in 1929. The ceiling is about 16 feet high. The wall across from the
outside wall contains bulletin boards to the side of an elegant wood builtin cabinet
with shelves. The cabinet has doors with paned windows. The large windows on the
outside wall, situated over radiators stretching to the height of the ceiling, can be
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opened from the bottom. The front wall has chalkboards for almost the length of the
wall, but two small 4 feet by 2 feet bulletin boards are placed to each side of the
chalkboards. The back wall also has two slate chalkboards that extend almost the
length of the wall with a small bulletin board between the two chalkboards. Jane
positions her desk near the front of the room and near the windows. A cart with an
overhead projector is usually near the center of the front of the room, and a screen
can be pulled down in front of the chalkboard. A small table with plastic bins for
paper is placed underneath the chalkboards in the front. Fourteen small tables, about
2 feet by 4 feet, serve as student desks, and are arranged in groups of two tables per
group, with four groups on one side of the room and three groups on the other side
of the room. All students usually sit facing the front of the room, except dining
certain group activities where students join others to work. During my observations,
the number of students in each group ranged from two to four. Jacob, a student in
period six, frequently sat by himself, but occasionally chose to work with groups of
students.
How Class Began
The first pattern that emerged was in how the classes typically began. How
the classroom activities began varied according to the period but was generally
characterized by class discussions of mathematics. Sometimes Jane presented a
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problem for the students to think about, and other times the students began working
on mathematics, but open discussion of the mathematics, whether or not the students
were working in groups, generally involved the whole class. Discussing previous
homework problems, which Jane labeled as problematic problems, was apparent
particularly in Janes algebra class and to a lesser extent in period seven. If the
beginning activity did not take the whole period, the remainder of the class time was
usually spent by having students work on problems or assignments related to the
current curriculum. Data on how class begins is in Table 4.3 below.
Table 4.3: How Class Began
Observation Period 6 % of Period Period 7 % of Period
3/31/03 PP 78% No observation
4/17/03 PP 45% Homework quiz 13%
4/24/03 PP 50% Homework discussion 63%
4/29/03 Teacher circulating 100% No Observation Schedule conflict
5/6/03 Quiz/PP 80% Multiplication quiz 20%
5/13/03 Fraction Chart/PP 61% Homework discussion 75%
5/15/03 PP 58% Fraction discussion 63%
5/21/03 Return quizzes/PP 18% Drawing Activity 100%
5/22/03 PP 44% Chart Activity 100%
PP is a label for problematic problems. If I define a significant amount of
class time to be that equal to or over 50% of the class time, then 11 of the 16
beginning class activities took a significant amount of time. This observation of
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significant amount of time is not meant to be judgmental. In fact, students, Jane, and
occasionally I were consistently involved in class and individual discussions about
mathematics, whether the beginning activity lasted a long time or not. In other
words, the importance of noting how Janes classes begin is better characterized by,
When class begins, discussion about mathematics begins,
In both periods, Jane occasionally explained how to do certain problems by
demonstrating them on the overhead or chalkboard. Sometimes, Jane stepped in
during a students presentation to offer advice or to take over the presentation of the
particular problem, particularly if a student made an error in his or her presentation.
Results Rhvthms of School Life
During the course of each day, events occur that sometimes have little to do
with the intended lesson or the mathematics of that lesson. At other times, these
events are indeed a part of, and contribute to, the intended lesson. Table 4.4 presents
data pertaining to classroom rhythms.
Frequencies: Classroom Rhvthms
. Table 4.4 below shows the numbers of passages I coded for each identified
code and the percent of the total number of passages for each identified code.
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Table 4.4: Number of Passages Classroom Rhythms
Primary Code Secondary Code Period 6 Period 7 Interviews Total %of Total
RhyDisturbance LessonRel 6 6 12 4%
RhyDisturbance SchoolRel 3 1 1 5 1%
Total % RhyDist > 5%
RhyRoutine RecordKp 32 14 1 47 14%
RhyRoutine ClsRoutine 121 119 240 70%
RhyRoutine Discipline 2 2 0 4 1%
Total % RhyRout > 85%
RhyBkgmd Personal 0 0 24 24 7%
RhyBkgmd SchoolRelRB 0 0 12 12 3%
Passages > 344 Total % RhyBkgmd > 10%
Passages I coded for classroom rhythms generally consisted of a
conversation between Jane and her student(s) or between Jane and me. I provided
detailed descriptions of the coding for classroom rhythms in Chapter 3.
As displayed in Table 4.4, disturbances in the classroom comprised about
5% of the total number of passages, classroom routines comprised almost 85% of
the total number of passages, and emerging background information comprised over
10% of the total number of passages classified as classroom rhythms. These
classroom rhythms did not appear to be directly related to views of mathematics,
although some background information influenced how Jane dealt with certain
individuals. However, the classroom rhythms were as much a part of the instruction
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as the particular views of mathematics, because both emerged throughout every
classroom period.
Classroom Rhythms: Disturbances
I first describe the disturbances that occurred during the observations. These
observed disturbances, although a common part of everyday school life in many
United States classrooms, only comprised 5% of the total number of classroom
rhythms passages in the present study.
Lesson Related Disturbances. This type of disturbance comprised 3% of the
total number of classroom rhythms passages. Students interrupting Jane during the
lesson with studentperceived needs made up most of these disturbances. The
following is a conversation from April 29 in period 6 between Jane and a student
who asked to take some papers to another teacher:
Jane: What kind of papers do we need over there, honey?
Student [Nardi]: Progress reports.
Jane: No, youll have to wait until after class. Im sure she
doesnt want you in there.
(Observation Period 6,4/29/03)
Nardi had asked Jane for permission to leave class and deliver some papers
to another teacher. Jane quickly assessed the request, which I characterized as a
disturbance, and made a decision not to let him leave class for this errand. Jane
frequently listened to student requests and made quick decisions on how to respond
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to such requests. Three excerpts below from other passages serve to illustrate some
of the disturbances, for which Jane accommodated and made quick decisions.
Student [Charlotte] Do you have any tissue?
Jane: I dont. Youll have to go to the bathroom, darling.
(Observation Period 6, 5/13/03)
Student [Kelly] : Can I open a window?
Jane: You can open anything you want, dear. You know that.
(Observation Period 6, 5/6/03)
Student [Antonio]: Do you have a bandaid?
Jane: Ill go look. Just a second. Uh, Brit? [Then Jane gets back to
Antonio] top right hand drawer, honey, if you need a band
aid.
(Observation Period 6, 5/6/03)
Requests such as those above were present in all observed class periods, and
Jane handled all of them with a quick response. Jane sent Charlotte to the bathroom,
gave Kelly permission to open a window, and told Antonio where he could find a
bandaid, even as Jane fielded another question from Brit. Sometimes she honored
requests, but often she asked a student to wait for a more appropriate time to take
care of his or her request, particularly if the request involved leaving the room for an
errand.
School Related Disturbances. These schoolrelated disturbances only
comprised 2% of the total number of classroom rhythms passages. Taking students
out of class for assemblies or field trips places a burden on many teachers at
Makefield, although most of the teachers agree that these experiences are important
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for the students. The following excerpt is representative of another disturbance that
was characterized as school related:
Students from Janes team came into room for a scavenger hunt,
and some disruption occurred.
Jane: (Warning the visitors) Be silent guys. Ignore us.
(Observation Period 6, 3/31/03)
Jane asked her students to ignore the intruders on the scavenger hunt, and,
except for a few comments between friends, Janes 6th period students did indeed
ignore the scavengers. Over the years, Ive observed many classrooms where such
an interruption would halt the lesson, but the result in this case was that the intruders
did not interfere with Janes lesson.
The next excerpt is an example of a disturbance that actually caused Jane to
leave her room for a brief time.
Student [Jake]: I think somebodys giving that substitute
problems, cause she came through the door and its still
open. So you should probably check on that.
Jane: OK. Ill do that. Just one sec.
(Observation Period 6,4/29/03)
The conversation above illustrates two points. First, Jake noticed that a
substitute was possibly having problems with his or her class, and he reported this to
Jane. Second, Jane recognized the potential problem and went out into the hallway
to check on it. This really did interrupt the lesson, because Jane left the room, but it
illustrates one of many disturbances that teachers handle on a daily and hourly basis.
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Thus, although events like the ones above are classified as disturbances, they are
really just a part of the normal classroom rhythms in Janes daytoday classroom
life, and, in fact, in most teachers daily lives.
Classroom Rhythms: Routines
Next, I describe and analyze the classroom rhythms associated with
classroom routines, i.e., record keeping, routines that characterized the classroom,
and discipline situations.
Record Keeping. The passages that are characterized as record keeping
comprise 14% of the total number of classroom rhythms passages, which suggest a
pattern that part of Janes class was spent announcing assignments, grading papers
or quizzes, or handing back corrected papers. Jane characterized many homework
assignments as pushups. The pushups were exercises that Jane wanted her
students to practice on. For example, Jane said, Now kids I need you to do this
pushup time, and then were going to put you in your groups right away
(Observation Period 6, 5/21/03). Jane frequently ended her assigning of
homework with endearing statements. One through nine all. You have a lovely
time kids (Observation Period 6, 3/31/03). Jane asked her students to do nine
problems for homework, and she suggested that they have fun doing the homework.
Jane also frequently set the stage for further inquiry into mathematical topics when
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