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BAHAMIAN STUDENTS‟ EXPERIENCES
WITH
MATHEMATICS
IN
AMERICAN UNIVERSITIES AND COLLEGES
A DISSERTATION
SUBMITTED TO THE SCHOOL OF EDUCATION
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Nikki Rochelle Cleare
August 2011
http://creativecommons.org/licenses/by-nc/3.0/us/
This dissertation is online at: http://purl.stanford.edu/br197cx8820
© 2011 by Nikki Rochelle Cleare. All Rights Reserved.
Re-distributed by Stanford University under license with the author.
This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.
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I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Jo Boaler, Primary Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Aki Murata, Primary Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Na'ilah Nasir
Approved for the Stanford University Committee on Graduate Studies.
Patricia J. Gumport, Vice Provost Graduate Education
This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.
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ABSTRACT
This longitudinal, qualitative, multi-case study examines the mathematical
experiences of three Bahamian students pursuing science degrees in American
universities and its relationship to their previous mathematical experiences in their
Bahamian high schools. Drawing on cultural psychology and sociocultural theories
of learning, this study investigates three questions: What do students‟ perceive as the
important secondary-school mathematics experiences that have shaped their view and
interest in math, science and engineering (MSE)? What experiences do Bahamian
students pursuing degrees in MSE have with mathematics in American universities
and colleges? What relationships, if any, exist between the perceived secondary-
school mathematics preparation of Bahamian students and their experiences with
mathematics in American universities and colleges?
The three participants, who are all graduates of Bahamian high schools, were
followed during the course of their freshman year in college. Each student had
earned an A on the Bahamas General Certificate of Education mathematics
examination although their experiences with that examination differed. Each
matriculated to a full time, selective four-year university that was primarily
residential and located in metropolitan cities on the East Coast of the United States.
By using interviews and on-site observations throughout the course of the year, the
researcher investigated the relationships between the students‟ prior experiences with
mathematics and their current experiences. Then, using a theoretically based coding
scheme, the researcher coded and compared the mathematical experiences of the three
participants, developing a narrative of each participant‟s experience in the process.
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These narratives were later reviewed by the participants as a means of ensuring that
all confidential issues had been appropriately addressed in reporting the data.
This study found that the relationship between the academic preparedness of
the participants on their experiences with collegiate mathematics is not limited to
subject content only. The lessons learnt from the implied curriculum also play a role
in their experiences with collegiate mathematics. Of the three participants, two have
had to learn/develop techniques for studying because they had not been sufficiently
challenged in their previous environments to do so. Each of the cases also highlights
some powerful constructs needed for success in mathematics and describes something
which the researcher refers to as connected understanding. This type of learning
emphasizes making connections between concepts, procedures, manipulatives/tools
and multiple approaches when solving problems. As such connections were realized,
there appeared to be an increase in student confidence, agency and authority, three
constructs that were found to be inter-related.
This study contributes to an emerging area of research on student persistence
in college-level mathematics courses as well as to the ongoing discussions in
mathematics education regarding the role/importance of conceptual understanding
and how to promote it. The study makes visible some of the ways in which students‟
post-secondary experiences with mathematics are shaped by their perception of their
mathematical preparation, thus providing a basis for discussions of possible curricular
change in the Bahamas. It also suggests that issues such as confidence and
school/teacher expectations of students may play a role in the development of
conceptual understanding. Thus, this study may have significance for the larger
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mathematical community in terms of developing theory that can then be tested with
other populations of students.
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ACKNOWLEDGEMENTS
As may be the case with many dissertations in education, this study grew out of my
personal need to understand a problem I had encountered in my own educational
experience. Along the way, I have learned a great deal about the Bahamian
educational system in general, about mathematics education in the Bahamas, about
mathematics education in the United States, and about what it means to understand
mathematics. This study sits at the intersection of these ideas and I will be forever
indebted to the three Bahamian students who allowed me to step into their lives and
journey with them, not knowing what the future held or where this journey would lead
us. They were always gracious in volunteering of their time to answer my many
questions and allowed me free access into their lives. To them, I am eternally grateful.
I wish also to thank the several people who opened their homes to me, providing me
with a launch pad from which to collect my data, or a safe haven in which to write
undisturbed. Thank you therefore goes to Donna Ash, Nerissa Ash-McKinney, Ellen
Greenberg, Philippa and Fannoh Wisseh, and Edmoly Plantijn. Their generosity
allowed me to conduct this research with quality and integrity.
Special thanks goes to the three members of my reading committee, Dr. Jo Boaler, Dr.
Aki Murata, and Dr. Na‟ilah Nasir, each of whom served as an advisor to me at
various points along this journey. It was a privilege to work alongside my primary
advisor, Professor Boaler, in the Stanford Teacher Education Program because her
openness in sharing her thoughts and probing mine allowed me to learn so much about
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the rationale for the choices teachers make within the classroom. Similarly, Professor
Boaler‟s guidance of our math research team was invaluable in providing me with
experience in coding and analyzing data, and learning how to publish the results. As I
worked to develop my own style and technique for academic writing, I truly
appreciated being allowed to read several of her writings in progress because it helped
me better understand how ideas can be developed. Thanks also goes to Dr. Aki
Murata who was always available to assist me with whatever hurdles I encountered –
and there were a few – along the path to my defense. Dr. Murata never stopped
believing in me and encouraging me to persevere. To Dr. Na‟ilah Nasir whose
infectious smile and constant encouragement kept me going long after I had left the
hallowed halls of Stanford in order to collect my data. She pushed me to think deeply
about culture and the ways in which it colors our experiences. I also wish to express
my sincere gratitude to Dr. Rachel Lotan and Dr. Maryam Mirzakhani, both of whom
graciously agreed to read my dissertation and be a part of my committee as I neared
the end of my journey.
Special mention should also be made of the members of the mathematics education
research team with whom I was privileged to study – Megan Staples, Melissa Gresalfi-
Sommerfeld, Tobin White, and Karin Brodie, who allowed me to share in the joy of
the culmination of their graduate experience as I was just beginning mine; Emily
Shahan, who was always a wealth of information and a genuine source of
encouragement; Jennifer DiBrienza, for her continual support and belief that we
mothers could also complete this difficult journey; Nick Fiori and Jack Dieckman,
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whose insights and patient discussions were invaluable in helping me to design this
study; and Tesha Sengupta-Irving, for her willingness to read and critique portions of
my writing and her neverfailing kindness in drawing out the best in me intellectually.
Thanks also goes to those friends who supported and provided encouragement for me
during my years at Stanford – to my Stanford sisters whose love and support helped
me face the challenges of those first few years at Stanford – Tesha Sengupta-Irving,
Gloria Banuelos, Zanette Johnson and Laurie Stapleton; and to Eliza Spang, Matt
Ronfeldt and Martha Castellon, each of whom always took time to listen, challenge
and encourage me both during their time at Stanford and beyond.
To those who made it possible for me to remain true to my ideals as a mother while
pursuing this dream, I owe a special debt of gratitude. Thanks, therefore, goes to my
sisters-in-law, Marissa Cleare-Wilson and Monique Hepburn, who always stepped in
at a moment‟s notice to help care for my children when I needed to be away from
home collecting data or writing. To my sister, Nerissa Ash-McKinney, who always
kept things real, helping me to steer a path for my life and to find my direction
whenever my compass got a little askew. To my daughters, Angel and Aeva Cleare,
the apples of my eye, for the joy and laughter they brought into my life that always
grounded me and helped me to remember the most important things in life. And to my
husband, Frederick, for accepting the challenge of living on one income as I pursued
this goal, for allowing me the time I needed to complete this task, and for putting our
other dreams on hold while I did so.
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To my biggest supporters – my parents, Winston and Estherlean Ash – there are not
enough words to say thank you. Without their practical support this dissertation would
literally have not been possible. They provided me with financial backing so I could
travel back and forth along the East Coast and collect the data for this research with
integrity, and insisted upon caring for my children so I could find time to analyze the
data and write this dissertation. Their constant encouragement and belief in my
ability to achieve excellence pushed me to not only persevere but to give my best
whenever my faith faltered. They listened to me, argued with me, and encouraged
me, and their love always provided the guidance I needed.
Finally and most importantly, I wish to say thank you to my heavenly Father God
without whom none of this would have been possible. During the final stages of my
writing, He reminded me: “Do not be anxious about anything, but in everything, with
prayer and supplications, make your requests known to God, and the peace of God,
which surpasseth all understanding, will guard your hearts and minds in Christ Jesus”
Philippians 4:6-8. Thank you, Father, for your guidance and direction in my life.
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TABLE OF CONTENTS
CHAPTER 1: THE PROBLEM SPACE ___________________________________ 1
Introduction and background ______________________________________________ 1
Statement of the problem __________________________________________________ 6
Sociocultural Perspective __________________________________________________ 7 Rogoff's Framework __________________________________________________________ 10
Literature Base ___________________________________________________________ Mathematics Education in the Bahamas ___________________________________________ 14 Retention in Math, Science and Engineering ________________________________________ 18 Conceptual Understanding in Mathematics _________________________________________ 24
CHAPTER 2: METHODOLOGY ________________________________________ 30
Research Questions ______________________________________________________ 30
Setting and Participants __________________________________________________ 31
Research Activities ______________________________________________________ 36 Interviews___________________________________________________________________ 37
Observations ________________________________________________________________ 41 Documents and Artifacts _______________________________________________________ 44
Stages of Analysis _______________________________________________________ 44 Coding _____________________________________________________________________ 45 Contact Summary Reports ______________________________________________________ 46
Participant Check _____________________________________________________________ 47
Tensions and Reflections _________________________________________________ 48
What Follows ___________________________________________________________ 50
CHAPTER 3: LEVEL OF PREPAREDNESS______________________________ 52
Challenges of Content ____________________________________________________ 52
Accuracy of Placement ___________________________________________________ 58
Concerns with Placement _________________________________________________ 59
Study Habits ___________________________________________________________ 61 Sade's Story _________________________________________________________________ 63 Josh's Story _________________________________________________________________ 65 Brittney's Story ______________________________________________________________ 66
Student Support System __________________________________________________ 68
Prospects for Completion of Degree ________________________________________ 71
CHAPTER 4: CONNECTED UNDERSTANDING _________________________ 73
Connected Understanding ________________________________________________ 75 Connections between Math Principles and Procedures ________________________________ 75 Connections between Math Principles _____________________________________________ 82
Connections between Math Principles and Manipulatives ______________________________ 87 Connections between Math Principles and Multiple Methods ___________________________ 90
Summary ______________________________________________________________ 93
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CHAPTER 5: CONFIDENCE, AGENCY AND AUTHORITY ________________ 95
Mathematical Dispositions ________________________________________________ 95
Agency _______________________________________________________________ 101
Authority _____________________________________________________________ 105
Summary _____________________________________________________________ 110
CHAPTER 6: DISCUSSION AND CONCLUSION ________________________ 112
Addressing the research question _________________________________________ 112
Implications ___________________________________________________________ 117 Teacher Moves ______________________________________________________________ 121 When the goal is not connected understanding _____________________________________ 126
Future Directions ______________________________________________________ 131
APPENDICES ______________________________________________________ 136
Appendix 1.1: Aims of Bahamian National Curriculum ______________________ 136
Appendix 2.1: "High School" Interview Protocol ____________________________ 137
Appendix 2.2: College Interview #1 Protocol _______________________________ 139
Appendix 2.3: College Interview #2 Protocol _______________________________ 141
Appendix 2.4: College Interview #3 Protocol _______________________________ 143
Appendix 2.5: Final Interview (Summer) __________________________________ 146
Appendix 2.6: Sample of Provisional Definitions of Selected Codes for Study ____ 149
Appendix 2.7: Sample of Emergent Codes from Data ________________________ 150
Appendix 2.8: Contact Summary Form -- Interviews ________________________ 151
Appendix 2.9: Selected Sample of the Matrix of Themes Selected from Data _____ 153
Appendix 3.1: First Year Higher Level International Baccalaureatte Topics _____ 157
Appendix 3.2: Topics Covered in Brittney's College Algebra Course ___________ 158
Appendix 3.3: BGCSE Syllabus ___________________________________________ 159
Appendix 6.1: Sample of Mandatory Homework Assignment __________________ 160
REFERENCES _____________________________________________________ 162
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LIST OF TABLES
Number Title Page
Table 2.1
Table 2.2
Table 2.3:
Table 3.1
Table 3.2
Table 3.3
Table 3.4
Summary of Participants‟ High School Mathematics Programs
Summary of Participants‟ Matriculation and Degree Interests
in 2006
Fit between data collection methods, theoretical framework and
research questions
Summary of Years Taken and Grades Attained Per Participant
Entry Level Mathematics Requirements vs. Actual Placement
Per Participant
Topics covered in Precalculus course vs. National
Curriculum
High School Curriculum Demands influence development of
study habits later used in College
33
36
38
53
54
56
62
1
“Education in the Commonwealth of The Bahamas is the principal vehicle for
promoting the development of individuals and the nation as a whole. It is
essential to enhancing the quality of life of our people” (Ministry of Education,
2003a, p.2).
CHAPTER 1
THE PROBLEM SPACE
This research explores the relationship between students‟ preparation in
mathematics in Bahamian secondary schools and their subsequent experiences with
mathematics in American universities. My inquiry grew out of my own experience
transitioning between the two settings – the shocking realization that although I had
excelled in mathematics in the Bahamian system, I was still woefully underprepared
for the mathematical requirements of the American collegiate system; the questioning
as to whether my non-readiness was a reflection of my particular school or endemic to
all Bahamian schools; and the wondering that, if the latter, how did students
compensate for their lack of preparedness.
I suspected that the secondary school curriculum would play a significant role
in how students experienced mathematics in American universities. What I did not
fully appreciate at the time, however, was that, in using a sociocultural lens to study
this phenomenon, my definition of preparation would have to broaden and expand; the
role of the curriculum would go beyond the content knowledge the students had been
exposed to and would include the ways in which the student had learnt the content and
the underlying lessons, like how to study – what, when and how long, that would be
taught. As Boaler (2005) points out, “one of the important contributions of situated
theories in recent years is the notion of practice and the idea that students in
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classrooms are not only learning knowledge but ways to engage in a set of practices”
(p.2). These underlying lessons would, in turn, influence the students‟ experiences
with mathematics in their new settings. As such, the findings of this dissertation deal
minimally with the topics within the curriculum although, as will be seen in chapter
three, these did have a significant influence on where each student began their
postsecondary education.
The study itself is important for several reasons. For many Caribbean
nations endeavoring to improve the quality of life of their citizenry, establishing a
skilled local workforce able to do the work formerly performed by expatriates
and/or foreigners continues to be of prime importance in their post-colonial
development. Furthermore, with free trade becoming a more frequent practice
between nations (U.S. Dept. of Education, 2000), the need for these countries to be
able to compete with the economies of other nations is becoming a matter of some
concern (Dupuch, 2006; The Coalition for Education Reform, 2005). In the
Bahamas, as the opening quote indicates, education is viewed as a means of
addressing these concerns (Ministry of Education, 2003a) and a significant portion
of the country‟s budget – approximately 18% (Sears, 2005) – is routinely allocated
to assist with the education of its citizens. Between 1980 and 1998, 2247
Bahamian students received government scholarships to pursue post-secondary
degrees, 84% at colleges in North America (United States and Canada) (Craton,
2002; Urwick, 2002). In 2006 when this study was conducted, with a population
size of approximately 350,000, over 2000 Bahamian students attended American
colleges and universities alone (UNESCO, 2005). Compare this to the 4000
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students (50% full-time) who attended the College of the Bahamas (The College of
the Bahamas, 2003), several of whom eventually transferred to foreign colleges
and universities to complete bachelor and/or graduate degrees (Vanderpool, 1999).
Concerns have arisen, however, regarding the educational expectations of the
Bahamian National Curriculum at the secondary school level. In his speech at the
18th
National Education Conference in the Bahamas, John Rolle (2005) argued that
“the minimum quality of instruction received has to increase if The Bahamas is to
successfully alter its development strategy” (p.1). If it is indeed our hope to maintain
our edge in the banking industry and to continue attracting foreign investments, we
will not only need a workforce equipped with financial skills but one that also has
experience in computer science. Even maintaining our ability to repair the vehicles
and machinery we import (some of which are used in our agricultural production),
requires knowledge and ability to work with computers (Rolle, 2005). In order to
maintain the fragile ecosystem of our archipelagic nation that both our tourism and
fishing industries rely on, our students will need some knowledge of “hydrology,
marine biology and aquaculture” (Pratt, 2005, p.5) and hence, attention must be given
to the training they are receiving in mathematics and science. This means, therefore,
that we “need to impart higher-level skills to a rising proportion of the workforce”
(London, 2005, p.4), skills in the areas of computer science, mathematics, science and
technology (London, 2005; Rolle, 2005). Our goal, argues John Rolle (2005),
“should be to advance math and communication skills by at least two years beyond
those targeted for the BGCSE [Bahamas General Certificate of Secondary Education]”
(p.11).
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The majority of high schools in the Bahamas, however, continue to use the
Bahamas General Certificate of Education (B.G.C.S.E.) curriculum, the successor of
London‟s General Certificate of Education Ordinary Level (G.C.E. O‟level)
curriculum, as the hallmark for what students should know and be able to do upon
completion of high school. It seems only appropriate, therefore, to ask the question
which the first prime minister of the Bahamas, the Right Honorable Sir Lynden Oscar
Pindling, in addressing The Bahamas Union of Teachers almost four decades ago
asked:
“Does the education that our children get today in school have a logical
connection with what they will be doing out of school tomorrow? . . . Is
it germane to the occupations which they will have to perform? Is it
pertinent to the changes in the social conditions in which they will find
themselves in five, ten or fifteen years‟ time? . . .” (Speech given on
May 21, 1970; quoted in Craton, 2002, p. 160).
This research poses this question in the realm of mathematics by first seeking to
understand what students perceive to be the important secondary-school mathematics
experiences that shaped their view of mathematics. Thus, the first of three questions
this study explored was:
What do Bahamian students’ perceive as the important secondary-
school mathematics experiences that have shaped their views of
mathematics?
Mathematics also continues to serve as a gatekeeper into many scientific
courses at the American collegiate level (Sells, 1978; Whitely & Fenske, 1990;
Cooney et al, 1990) with most requiring “calculus as the minimal entry-level
mathematics competency” (Whitely & Fenske, 1990, p.358). In countries that operate
under the O-level system, students are generally expected to complete an additional
one to two years of study at the Advanced level (A-levels) before matriculating into
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collegiate programs. With secondary education in the Bahamas culminating at
BGCSE (the equivalent of O-levels), however, it is possible that even our high-
attaining students (those earning an A or B on the BGCSE mathematics examination)
are at a less competitive level when they matriculate to American colleges and
universities. This is worrisome when one considers that students who have taken too
few mathematics courses in high school generally take college preparatory courses in
college in order to gain access to mathematics, science or engineering (MSE)1 fields of
study and that the success rate for such students in American colleges and/or
universities hovers around 50% (Maple & Stage, 1991). In addition, even when one
considers those students who were sufficiently prepared to study collegiate
mathematics, fewer than 50 percent of American students who initially choose to
pursue MSE majors in college successfully complete degrees in these fields within
five years. The few studies that focus on the reason(s) students persist or leave MSE
fields (for example, Treisman, 1985; Tobias, 1990; Lipson & Tobias, 1991; Seymour
& Hewitt, 1997) note the powerful influence of the norms that exist within the MSE
environment. This study, therefore, with its focus on the mathematical experiences of
a select group of students, can provide additional insight regarding mathematics at the
collegiate level, as it seeks to understand:
What experiences do Bahamian students pursuing degrees in MSE have
with mathematics in American universities and colleges?
1 As the study of the experiences of math, science and engineering students is a small but growing field,
the literature is not consistent in the acronym used to reference math, science and engineering degrees.
Some refer to it as S&E for science and engineering; others use S.M.E for science, math and
engineering, while still others use MSE for math, science and engineering. Since I am particularly
interested in mathematics, I have chosen to use MSE as the default throughout this paper unless the
author being referenced specifically chose another acronym.
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Finally, this study grapples with the difficult question of what it means to
“understand mathematics” and how it relates to one‟s prior understandings of
mathematics. In particular, the study sought to answer:
What relationships, if any, exist between the perceived secondary-school
mathematics preparation of Bahamian students pursuing degrees in MSE
and their experiences with mathematics in American universities and
colleges?
Statement of the problem
To summarize, the primary objective of this study was to explore the
mathematical experiences of Bahamian students pursuing math, science and
engineering degrees in American universities and colleges. Noting that these students
would likely be underprepared to pursue collegiate-level mathematics, the study aimed
to discover what relationship, if any, existed between the students‟ perceived
secondary-school mathematics preparation and their current experiences with
mathematics in the United States by addressing three questions:
1) What do Bahamian students‟ pursuing degrees in MSE perceive as the
important secondary-school mathematics experiences that have shaped
their views of mathematics?
2) What experiences do Bahamian students pursuing degrees in MSE have
with mathematics in American universities and colleges?
3) What relationships, if any, exist between the perceived secondary-school
mathematics preparation of Bahamian students pursuing degrees in MSE
and their experiences with mathematics in American universities and
colleges?
Clearly, what the student brings with them (mathematical background, ways of
working, support system, etc) matters but, given that some students succeed while
others do not, it is likely that it is not the only thing that matters. Similarly, the new
environment in which the student finds him/herself – the particular institution with its
norms and practices, the culture of the MSE departments, the social others with whom
7
he/she encounters – matters but again, because some students succeed and others do
not at the same institutions in the same programs surrounded by the same social
others, clearly the new environment will not be the only thing that matters. What I
propose then is a theoretical perspective that allows us to look at both the individual
and the environment including the social others and available tools/resources and try
to make sense of what gets co-constructed between the two. I turn your attention,
therefore, to the socio-cultural perspective.
Socio-cultural Perspective
The theoretical underpinnings of this research draw from the general field of
cultural psychology and its overlap with the more specific and elaborated upon
sociocultural theories of learning. Within these frameworks, culture is viewed as being
“both carried by individuals and created in moment-to-moment interactions with one
another as they participate in (and reconstruct) cultural practices” (Nasir and Hand,
2006, p. 450). I sought, therefore, to understand what gets shared between each
individual student and the social others with whom they came in contact. To do so, I
had to consider how the individual student, groups with whom he/she interacts, and
the mathematical communities transform each other as they together “constitute and
are constituted by sociocultural activity” (Rogoff, 1995).
While the domain of psychology has been one that focuses on the development
of the mind and the notion of a psychic unity (Schweder, 1990; Cole, 1996), studies in
cross-cultural psychology have attempted to refine our understanding of the
developmental process by comparing cultural groups – noting what is universal across
groups and the variations in developmental trajectories between groups. Often, the
8
treatment of culture within these studies is one that is static, a stable set of norms and
symbols shared by the group, and individuals are considered to be socialized by their
societies (Dasen, 2000). Cultural psychology, however, has sought to move away
from this paradigm, recognizing that even within groups that share norms, symbols,
and similar ways of living, individuals vary. Thus, culture is not viewed as a stable
property that individuals belonging to a particular group possess; rather, it is co-
constructed, with the socialization process being one that is a negotiation between the
individual and his/her society (Dasen, 2000). Cultural psychology, therefore, attempts
to explore the process by which the sociocultural world and the individual (not
universal) human psyche creates and sustains one another. Explanations concerning
development are therefore “formulated in ways that characterize psychological
functioning of individuals within a particular cultural group at a particular point of
time in a particular social-historical context” (Dasen, 2000, p.430). Thus, while we
shall look at mathematics education in the Bahamas to understand the history and
background from which the students in this study come, it will be with a recognition
that the students‟ experiences within this culture will have been varied and therefore
what they bring with them into their new environment will be distinctly different from
each other as will the ways in which they draw upon that background, whether
consciously or subconsciously, as they interact with others and the artifacts of their
new environment at any given moment in time.
Sociocultural theory examines “the roles of social and cultural processes as
mediators of human activity and thought” (Nasir & Hand, 2006, p.458). It is a theory
concerning the learning process that holds that human consciousness (mind) can only
9
be understood through a consideration of the social and cultural life of the individual
(Lerman, 1996, 2001). As Vygotsky, the theorist considered to be the founding father
of sociocultural theory, wrote:
Every function in the child‟s cultural development appears twice: first,
on the social level, and later, on the individual level; first, between
people (interpsychological), and then inside the child
(intrapsychological). . . All the higher functions originate as actual
relations between human individuals. (Vygotsky, 1978, p.57)
This position takes on significant implications when one considers that, from birth,
children actively observe and participate with others in such a way that they are both
engaged in and contribute to the development of social practices (Rogoff, 1990). In
fact, it is through this process of active observation and participation that individuals
move from what Lave and Wenger (1991) refer to as “legitimate peripheral
participation” to central participants within the community. This move, from being on
the periphery of a community towards becoming a central part of the community, is
central to my study as the students involved in this study are in a transition process –
moving from the norms, practices, and values of the particular institutions and
communities in which they were a part of in the Bahamas to the norms, practices and
values of the particular institutions and communities in which they seek to become a
member of in America. Their “histories of engagement” (Gutiérrez & Rogoff, 2003)
and ways of understanding the world, what Gutiérrez & Rogoff (2003) refer to as their
“repertoires of practice”, play a pivotal role in how they engage with and contribute to
their receiving communities and institutions. As Gutiérrez & Rogoff (2003) point out,
“Individuals‟ background experiences, together with their interests, may
prepare them for knowing how to engage in particular forms of language and
literacy activities, play their part in testing formats, resolve interpersonal
10
problems according to specific community-organized approaches, and so
forth” (p.22).
While there are many aspects of each individual student‟s life that interacts
with and impacts their particular transition process (Terenzini, 1982) the focus of this
study was on mathematical practices in general and on the individual students‟
perspectives of their experiences with those mathematical practices in particular.
Thus, with a view of individuals as “unique „collections‟ of subjectivities and
positionings” (Lerman, 1996, p.137), I attempted to understand how each individual
student was transformed by their particular mathematical community at their
particular institution as they together “constitute[d] and [were] constituted by
sociocultural activity” (Rogoff, 1995, p.161). Such an analysis required attention to at
least three different levels of activity – the institutional level, the interpersonal level
and the individual/personal level – levels which are “inseparable, mutually
constituting planes” (Rogoff, 1995, abstract). Rogoff‟s (1995) analytical framework
provided a helpful way of thinking about such an analysis.
Rogoff’s framework
Rogoff (1995) begins by describing three planes of analysis (apprenticeship,
guided participation, and participatory appropriation) that can allow the researcher
insight into “the efforts of individuals, their companions, and the institutions they
constitute and . . . see development as grounded in the specifics and commonalities of
those efforts, opportunities, constraints and changes” (p. 159). I begin then with a
discussion of the two planes that form the background of this study.
11
Apprenticeship Plane/ Institutional Level
The apprenticeship plane is one in which the focus is on the institution and
“the specific nature of the activity involved” (Rogoff, 1995, p.143) at the particular
institution. It considers the purposes of and values associated with the activity as
defined by the community and/or institution in which it is located as well as the
societal constraints, resources, cultural tools, etc that are available to the individual.
For this study, it implied being attentive to the defining characteristics of each
university and their MSE programs and ranged from such things as competitive level
and mission of the university and/or MSE programs, the student-body composition,
size of classes, etc. For example, the match between each student‟s experience of
their particular high school institution (competitive, nurturing, demanding, monolithic,
diverse, etc.) or their familial pressures (to compete, exert oneself, admit need for
help, etc) and their university or program played a role in what was constituted
between them and their receiving environments.
Guided Participation/Interpersonal Level
Guided participation is the plane in which attention is given to the
involvement of the individual with social others – how they communicate and
coordinate their involvement and adjust to the “arrangements for each others‟ and
their own activities” (Rogoff, 1995, p.146). The „guidance‟ aspect refers to the
direction offered by social partners as well as by cultural and social values, while the
„participation‟ aspect includes observation, hands-on involvement in an activity, and
active avoidance of activities. As Lerman (1996) notes, “In different contexts, with
different sets of social relationships, individuals occupy different „positionings‟”
12
(p.147). It would seem reasonable to inquire, therefore, into those different
„positionings‟. What is the role of the various social others involved in the
mathematical practices in which the individual now finds him/herself a part, and how
are those roles similar to and/or different from the ways of engagement in the
individual students‟ former settings? When do the deliberate attempts at learning
occur, and how do the individuals engage in and influence those activities? What
opportunities are afforded them? What are the constraints? What are the incidental
comments and actions that influence the individual; when do they occur and how does
the individual engage in and influence these activities? These are all questions that
arise during guided participation (Rogoff, 1995), questions at the interpersonal level
that aided me in understanding the primary focus of my attention, the personal level.
Participatory Appropriation/Personal Level
Within Rogoff‟s framework, there is this constant awareness that the
aforementioned planes are not distinct – to understand one, requires attention to the
others, “observing both similarities and differences across sociocultural activities as
well as tracking the relations among aspects of events viewed in different planes of
analysis” (Rogoff, 1995, p.161). Attention to these first two planes, therefore, helped
me to zone in on the perspectives of the students and how he/she changed through
their involvement in an activity due to their participation in previous activities,
becoming prepared for subsequent involvement in later, related activities.
Participatory appropriation, then, is the process by which the individual
construes relations between the purposes of the activities in which they are involved
13
and the meanings associated with them by society and the social others with whom
they are in contact. As Rogoff (1990) puts it,
In the process of participation in social activity, the individual already
functions with shared understanding. The individual‟s use of this
shared understanding is not the same as what was constructed jointly;
it is an appropriation of the shared understanding by each individual
that reflects the individual‟s understanding of and involvement in the
activity (p.195).
This level of analysis required an examination of each student‟s actual involvement in
the mathematical practices -- how do they participate and what in their history of
engagement allowed them to participate in that way? Is their participation peripheral,
observing and carrying out roles they view as expected of them in learning/practicing
mathematics, or is it more central, where they see themselves as responsible for
managing the mathematical practices in which they are involved? Does their
participation change – from peripheral to central – and if so, through which processes?
How do they become aware of and familiar with the expectations, demands, and
constraints of the environment and tasks expected of them? What cultural tools
(taking notes, studying – solo or with others, tutorial sessions, reading textbooks, use
of calculators, etc) are familiar to them, which do they make use of, which do they
extend? Do disjunctures exist between their previous and present experiences and
what cultural tools do they employ to address any such disjuncture? Despite
differences at both the institutional level and interpersonal level and their own
positioning within those, are there ways of engaging that all the participants seem to
have in common and is it indicative of some constant in their histories of engagement,
some relation to the previous environments to which they belonged? What variations
can be observed in their ways of engaging both with the mathematical practices and
14
the social others with whom they must work? Focusing on these types of questions
throughout data collection helped to keep the focus trained on the individual students‟
perspective and the sense he/she is making of the mathematics experience in which
he/she is engaged.
One final note: within this framework time is not divided into separate units
of past, present, future. Rather, noting again that individuals function “within a
particular cultural group at a particular point of time in a particular social-historical
context” (Dasen, 2000, p.430) events in the present are seen as extensions of previous
events which are directed towards a goal that is yet to be accomplished. “As such, the
present extends through the past and future and cannot be separated from them”
(Rogoff, 1995, p.155). Keeping this in mind, I discuss next the three domains that
play a prominent role in understanding the perspectives of the students involved in this
study: mathematics education in the Bahamas, the retention of students in MSE
programs in the United States, and the role of conceptual understanding in
understanding mathematics.
Mathematics education in the Bahamas
A former British colony, the Bahamas achieved self-governance in 1964
followed by complete independence from Britain in 1973. Prior to majority rule in
1967, “only 6 per cent of children went on to secondary education” (Craton, 2002,
p.125) as a competitive entrance examination, limited enrollment and tuition barred
most from admission into the premier school, the Government High school, where
courses were taught that would lead to the O‟level examinations (the precursor to the
Bahamas General Certificate of Secondary Education (BGCSE) examinations)
15
(Craton, 2002; Sumner, 1995). With majority rule came the mandate to provide
equality of opportunity through education for all citizens, ensuring that no Bahamian
was denied an education because of “place of birth, residence, social status or skin
colour” (Sears, 2005). Thus began a period of unprecedented expansion within the
educational system and an increase, over the years, in the number of students who not
only completed secondary school but also sat external examinations (O‟levels or
BGCSEs) (Craton, 2002).
Mathematics continues to be taught using the strand approach whereby
concepts in number theory/computation/estimation, sets, algebra/patterns/functions,
geometry/trigonometry, measurement/mensuration, and statistics and probability are
addressed over several grade levels (Ministry of Education, 2004b). Students take
mathematics every year with their age cohort and, given the government‟s interest in
providing “a goodness of fit between what students are learning in school and the
interests and demands of their lives outside school” (Ministry of Education, 2003a,
p.6), assessment centers on “technique with application” (Ministry of Education,
2006, p.4). The first nine aims of the curriculum (see Appendix 1.1) focus on
developing mathematical literacy for all students while the remaining four aims
address other areas of mathematics (reasoning, classifying, generalizing,
interdependence of topics, etc) considered essential for developing a deep appreciation
of the subject (NCTM, 1991; National Research Council, 2001). The curriculum also
identifies topics that all students should be exposed to versus elective topics that the
student interested in pursuing more mathematics would be advised to take. These
elective concepts are interspersed throughout the various strands of mathematics but
16
are concentrated at the 11th
and 12th
grade level. Some mathematics topics which are
not mentioned in the curriculum (as required or elective), however, include: complex
numbers, exponential equations, logarithms, surds, transformations of functions,
graphs of trigonometric functions, and trigonometric identities. Instead, students who
earned a „B‟ or better on the BGCSE mathematics extended level exam2 begin the
mathematics program at The College of the Bahamas with a study of these and other
College Algebra topics (The College of the Bahamas, 2003).
The decision to include certain algebra/precalculus topics at the collegiate level
was made at a time when the needs, goals and resources of the country (Clarke, 2003)
indicated the necessity of doing so. One year after gaining independence, in 1974,
The College of the Bahamas was established. The A-level courses which had
previously only been offered in the 6th
form of private schools and the Government
High School, then became a part of the academic program of the college, thus making
available more resources for the secondary schools to focus on providing basic
education to all students rather than a select few. While some private secondary
schools continued to offer A-level coursework, as more Bahamian students began
availing themselves of American higher education, both the private secondary schools
and The College of the Bahamas began phasing out the A-level examinations.3 The
College of the Bahamas replaced these courses with credit-hour courses and the
associate degree (Urwick, 2002). This choice was appropriate for several reasons: 1)
2 Students who only sit the core level examination sit papers 1 and 2 and may earn grades ranging from
C-G. Those taking the extended level examination, intended for the above average candidate, must also
sit paper 3 and may earn grades ranging from A- G. A and B grades are given to candidates who
demonstrate a comprehensive to excellent grasp of the material, while a grade of D is considered
average performance (Ministry of Education, 2003b, 2004). 3 Two private secondary schools do, however, offer advanced placement courses, with one of the two
implementing the International Baccalaureate program.
17
both the credit-hour courses and the associate degree were transferable to many
American colleges and universities; 2) the credit system allowed for earlier response
to academic failure, requiring students to retake courses if they received a failing
grade at the end of the semester versus allowing them to continue for two years before
being assessed via the A-level examinations (Urwick, 2002), and was a less expensive
program to run; and 3) by 1983, the British universities as well as the University of the
West Indies began accepting the associate degree as an appropriate equivalent to the
A-levels (K. Bethel, personal communication, March 8, 2005). Thus it came to be that
the British O-level examinations, taken at the end of 5th
form or its grade 12
equivalent, became the national standard for completion of secondary education in the
Bahamas while topics taught at the A-levels became absorbed into the college
program.
In 1993, when the British government changed to GCSE, the Ministry of
Education in the Bahamas, in consultation with the University of Cambridge Local
Examinations Syndicate, introduced the BGCSE (Sumner, 1995). This exam was
intended to serve as an evaluation tool to examine the positive achievement (rather
than failure) of students after five or six years of secondary school education and was
intended to cater to approximately 80-85% of the student body instead of just those
who were college-bound (Sumner, 1995). Initially, therefore, all students were
required to take this examination at the end of the 12th
grade year. After a few years,
however, several of the private schools returned to their practice of allowing students
to take the examinations prior to the 12th
grade. Consequently, at the time of this
study, a few of the private schools – including the two in this study – were in various
18
stages of developing mathematical programs of study beyond the BGCSE
requirements for use in the final years of high school.
Retention in Math, Science and Engineering
The theoretical models of student retention in college (see, for example, Bean
1982; Bean & Metzner, 1985; Cabrera, Nora & Castaneda, 1993; Pascarella 1980;
Tinto 1975, 1993; Weidman, 1989) suggest that, using the university setting as the
current environment, the following areas would prove worthwhile to attend to: at the
institutional level – the size, mission, selectivity and prestige of the institution along
with its organizational structure, policies and faculty culture; at the interpersonal level
– peer-group interactions, student-faculty interactions (both formal and informal),
academic performance, and extra-curricular activities; at the individual level – role of
family, prior schooling, skills and attributes, off-campus work, finances, community
activities and non-college peers. These models are all longitudinal models that view
persistence in college to be the result of a “complex set of interactions over time”
(Cabrera et al, 1993, p.125) and have as a theoretical base “the social and academic
integration of the student with the institution” (Bean, 1982, p.23). By “integration”
they mean the ways in which the norms, goals, values and aspirations that an
individual brings with him to the collegiate setting is impacted/changed/shaped by the
institution and the individual‟s interactions with social others in the institution. The
underlying focus, therefore, is on the congruency or fit between the individual and the
institution in general.
I next sought, therefore, to find empirical studies regarding retention specific to
mathematics, science and engineering (MSE) majors. These, I found to fall into two
19
major categories: 1) studies that investigated the characteristics of leavers (sex,
ethnicity, socio-economic status, etc.) at crucial points along the MSE pipeline (during
high school, exiting high school, entering college, during college, graduating college,
etc); (see Berryman, 1983; Green, 1989; Lee & Hilton, 1988; National Science Board,
2004; OTA Reports, 1988, 1989) and 2) studies that attempted to uncover the
reason(s) students persist or leave MSE fields. It was the latter type of studies, with
its focus on student perceptions and performances, that was of primary interest to this
research. These studies are of three types: predictive models that attempt to discover a
connection between preparedness and persistence; studies grounded in the students‟
experience that often result in an analysis of institutional characteristics, both
structural and cultural, in addition to the individual characteristics; and survey studies
that attempted to make use of large national databases for their findings.
The predictive models (Hudson & Rottmann, 1981; Wollman & Lawrenz,
1984; Hudson, 1986) primarily concluded that math skills, while not the determining
factor for success, was correlated with achievement in introductory physics courses.
In addition, Hudson‟s (1986) study showed that self-help material that targeted
particular deficiencies/gaps in knowledge could make a difference in student
performance. All three studies indicated, however, that something was confounding
the data. As Hudson (1986) noted, “The supposition that the magic bullet in the form
of a pretest which will separate the students at high risk for dropout is not going to be
easy to develop” (p.49).
The survey studies, while relying primarily on questionnaires to establish their
findings, made use of large databases. Using data from the 1985 SAT questionnaire,
20
Grandy (1998) found that for the high-ability students of which her sample was
comprised, high school math/science achievement was not a factor in persistence
when math/science grades were taken into account in addition to SAT math scores.
She did note, however, that “even in this select sample, high school math/science
achievement and college grades had an impact on persistence” (Grandy, 1998, p.615),
albeit small. Indeed, the less homogenous sample used in Bonous-Hammarth‟s (2000)
study (based on data from CIRP‟s 1985 and 1989 surveys) showed that high academic
achievement prior to college and interest in SME majors upon college entrance4 were
positively associated with SME retention for both groups. Another interesting finding
of Bonous-Hammarth‟s study was that institutional selectivity, which the theoretical
models indicated may impact experience, was negatively associated with SME
persistence for the African Americans, American Indians and Chicano/Latino students
included in her sample. In addition, because neither peer group nor person-
organization fit entered the regression model for this group of students, Bonous–
Hammarth (2000) concluded that “a more sensitive measure or strategy for
measurement may be needed to assess fit among people of color and other
marginalized groups pursuing SME degrees” (p.109).
Unlike the predictive models and survey studies that sought to determine the
role of particular individual attributes on student retention in MSE, the studies
grounded in students‟ experiences sought to illuminate the students‟ perception of
what impacted their desire to leave or remain in MSE. Thus, while supporting the
previous studies‟ findings that math preparation is correlated with performance in
4 In Bonous-Hammarth‟s study “high academic achievement prior to college” was defined as
participants having a high school GPA of A or A+ and high math SAT scores, while “interest in SME
majors” was determined by participant‟s response to intent to major in 1985.
21
MSE, they also identified additional factors, such as study habits of the students and
the culture of MSE (classroom climate, pedagogy, assessment, etc.), that seemed to
affect the level of impact that mathematics preparation has upon collegiate
performance. Four studies, in particular, are especially noteworthy.
Firmly believing that the traditional methods used to support minority students
in their pursuit of MSE degrees (e.g. individualized tutoring, self-paced instruction,
study skills) were not very effective, Treisman (1985) studied the apparent difference
in performance between black and Chinese students in the freshman calculus (Math
1A) course at the University of California – Berkeley. He found a difference in the
pattern of the numbers of hours the students in both groups spent, and after observing
20 black and 20 Chinese students over a period of eighteen months, noted that the
black students often studied alone while the Chinese students were more likely to
study in groups. This led to significant differences in their approach to difficult
problems as the students studying alone tended to “search for computational errors in
their work and/or insights by reworking textbook examples” (Fullilove & Treisman,
1990, p.466) while those who worked in groups assisted each other with difficult
homework problems and turned to the teaching assistant for help with problems the
group could not solve (Fullilove & Treisman, 1990). This led Treisman to establish
the Mathematics Workshop Program (MWP), a program designed for African
American and Latino students to work together collaboratively for approximately 2 hrs
twice a week on worksheets with unusually difficult problems. When evaluated in
1984, the MWP students were found to significantly outperform their non-MWP peers
with similar SAT scores and, most intriguing, the MWP students considered to have
22
poor mathematics preparation (students with SAT scores in the lowest tercile)
outperformed non-MWP students with presumably stronger backgrounds (as measured
by SAT scores) (Fullilove and Treisman, 1990).
In another qualitative study, Tobias‟ (1990) studied the experiences of seven
post-graduate non-scientists in an introductory science course in an effort to
understand why students who were capable of doing science might choose not to. Six
of the seven participants earned grades within the top 10% of the class but only two
said they would continue in science if they had a choice. The reasons given were not
lack of interest, motivation or difficulty of the subject but rather, “the absence of
history and context, „the tyranny of technique,‟ the isolation of the learner and the
struggle to attend in a sea of inattentiveness” (Tobias, 1990, p.59). They longed to use
their creative and critical thinking skills to discuss the sophisticated concepts that
supported the techniques being taught, to gain insight into the concept itself and not
simply how to apply it. As for the role of their mathematics skills in the science
course, Tobias (1990) found that those who struggled least in their science courses
tended to exhibit strong mathematical backgrounds.
“While inadequacy in mathematics is not by itself a cause of failure to
succeed in science, it surely appears to contribute to the degree of
difficulty our otherwise very able students experienced. From this, one
policy recommendation might be that emphasis be placed on early and
continuous exposure to higher and higher levels of mathematics for the
majority of students in middle school and high school” (p.91)
Abigail Lipson performed a follow-up study to Tobias‟ study in which she noted
five main themes regarding the factors that affect student retention in the sciences
(Tobias, 1990), the first four of which pertained directly to the norms established within
the MSE environment: emphasis on competition and extrinsic rewards, and the
23
decontextualization of course topics, both of which make conceptual understanding
difficult; emphasis on weeding out students early; and unit and course requirements that
tend to limit students access to a “well-rounded liberal education”. These, combined with
the fifth theme, students‟ own difficulties in choosing a major given the plethora of
choices, often result in the “weeding out of . . . some of [science‟s] deepest thinkers and
most creative minds” (Lipson & Tobias, 1991, p.95).
Seymour & Hewitt‟s (1997) three-year longitudinal ethnographic study of 460
students attending seven different 4-yr institutions also brought the culture of the MSE
discipline into question. For students of color, she also found that the three reasons
they cited more frequently than white students for switching from S.M.E majors were:
inappropriate reasons for choosing S.M.E. (usually due to the active
influence/encouragement of others and scholarship offerings) (34.6% compared to
6.1%), conceptual difficulty with one or more S.M.E. subjects (25% compared to
5.3%), and inadequate high school preparation (25% compared to 10.7%).
The theoretical models provide a macro view of the factors involved in the
retention/persistence of a student in college – the role of the university, the type of
interpersonal interactions that appear to matter and the „external‟ influences that can
affect the individual. The empirical studies add to this picture by zooming in on the
lives of math, science and engineering students – the population of concern for this
study. These studies note that there is a correlation between high school preparation
and students‟ performances in the sciences but it is not predictive of which students
remain in MSE. What the studies do note, however, is the powerful influence of the
norms that exist within the MSE environment, particularly as it concerns ways to
24
approach learning. In particular, all of the studies grounded in students‟ experiences
either allude to or speak directly to the role of understanding. I, therefore, turn your
attention to a topic that has, over the years, received much attention in the field of
mathematics education – conceptual understanding.
Conceptual Understanding in Mathematics
Over the past century, theorists have debated what it means to understand
mathematics and the pendulum has constantly swung back and forth between skilled
performance and learning procedures with understanding. Dewey (1910), often
considered to be the father of the progressive movement, directed our attention
towards the “reflective power” of mathematics and away from repetition and
mechanical drills that did not encourage reasoning. Misinterpretations and
misapplications of his theory, however, led educators to eventually appropriate
Thorndike‟s theory of connectionism which “described in detail how skills should be
taught to maximize retention” (Hiebert & Lefevre, 1986, p.1). However, with the
successful launching of the Soviet Union‟s space satellite, Sputnik, on October 4,
1957, the mathematics curricula in the United States underwent a major renewal.
Bruner‟s bold hypothesis that “any subject can be taught effectively in some
intellectually honest form to any child at any stage of development” (Bruner, 1960,
p.33), and Schwab‟s claim that “the disciplines themselves hold images of what
learning entails” (Ball, 1993, p.374) launched us into what came to be called the “new
math” movement. Attention was then focused on “understanding the structure of
mathematics together with its unifying ideas, and not just as computational skill”
(National Research Council, 2001, p.115). This 1960s movement was not long-lived,
25
however as the general perception came to be that students were not mastering the
abstract concepts and theoretical notions of the new math program or the basic skills
that generations before them had mastered. A cry went out to “return to the basics” of
computing accurately and quickly. Thus, the new math was replaced by the back to
basics movement, a movement that ended around the late 1970s when it was
recognized that the students were not performing well on questions requiring critical
thinking and problem solving nor were they any better at “the basics”. The 1980s,
therefore, ushered in another period of change with the publication by the National
Council for Teachers of Mathematics (NCTM) of Curriculum and Evaluation
Standards for School Mathematics, a document that helped to launch what became
known as the standards-based reform movement. This document and its later
companions (Professional Standards for Teaching Mathematics, Assessment
Standards for School Mathematics, Principles and Standards for School Mathematics)
were intended to be a vision of what school mathematics should strive for. Although
first viewed as promoting problem-solving as the primary goal of mathematics, this
movement has eventually come to be understood as promoting a conceptual approach
to mathematics that serves as a framework for anchoring skill acquisition. Thus the
focus has shifted from an emphasis solely on content (what to know) to more
emphasis on process (how you come to know) (Schoenfeld, 1992). Despite its
inclusion of both concepts and skills with each supporting the other, this movement
has not been without its opponents, however. Some fear that the emphasis placed on
conceptual learning has led to a focus on topics that can be represented concretely,
thus leaving students with a false impression of what mathematics is and ill-prepared
26
to deal with the level of abstraction that will later be required for their mathematical
development (see Wu, 1999).
As mathematics education has evolved, so has research in mathematics
education. In the 1980s, in particular, researchers began to embrace a new view of
how people come to know. Previously viewed as an individualistic endeavor where
people reconstructed in their minds things from the physical world, situated theories of
learning (Lave & Wenger, 1991) added a new dimension to what was considered
important in understanding how people came to know. This new dimension was the
idea of practice – that people interact with their environment in such a way that
“meaning” and “sense-making” evolves from these interactions and is neither a result
of solely the physical world or the individual themselves. As a result, more
ethnographic and qualitative studies began to emerge where the focus was upon
understanding people in particular settings. In the world of mathematics education,
this has often meant studying students holistically in particular classroom settings
performing particular activities (rather than holding the classroom and/or activity as a
constant). For example, Boaler (1997) found that traditional and reform classrooms
provided students with access to particular kinds of mathematical knowledge.
Students from the traditional classrooms often performed well on the more procedural
type questions that were familiar to them but seemed to have more difficulty with the
conceptually oriented questions. In the classrooms that were project-based and where
discussion of mathematical ideas between teacher and students and students and
students was encouraged, students showed incredible gains in conceptual
understanding while developing agency and coming to view themselves as an
27
authority in mathematics. In Kazemi‟s (1998) study of discourse that promotes
conceptual understanding, Kazemi found that “when teachers helped students build on
their thinking, student achievement in problem solving and conceptual understanding
increased” (Kazemi, 1998, p.410). To “press” students to think conceptually, Kazemi
argued entailed requiring them to provide reasons for why the procedures worked
rather than simply stating the procedures themselves. Ball (1992; 1993) and Lampert
(2001) demonstrate this beautifully in their records of their own teaching in which
they highlight the complexities involved in teaching mathematics content to children
that builds upon their thinking and fosters community.
I find myself, therefore, standing where others have stood before (Hiebert, et
al, 1997; Rittle-Johnson & Stiegler, 1998; Rittle-Johnson, Siegler, & Alibali, 2001;
NCTM, 1989, 1991, 2000; National Research Council, 2001) believing that the
dichotomy sometimes drawn between conceptual understanding and procedural
fluency is a false one, convinced that both conceptual understanding and procedural
fluency are necessary for developing understanding in mathematics, and feeling that,
at times, we are talking of one and the same thing. What then is conceptual
understanding? Often, it is thought of as the “comprehension of mathematical
concepts, operations and relations” (National Research Council, 2001, p. 116) but it is
so much more. Hiebert & Lefevre (1986) defined it as “knowledge that is rich in
relationships. . . . a connected web of knowledge, a network in which the linking
relationships are as prominent as the discrete pieces of information” (p.3). Eisenhart
et al (1993) refers to it as “knowledge of the underlying structure of mathematics – the
relationships and interconnections of ideas that explain and give meaning to
28
mathematical procedures” (p.9). This interconnectedness, the linking relationships, is
a quality that I fear is often lost in the discussions of conceptual understanding. The
findings of this research, however, zone in on this quality, teasing apart the critical
role which this idea of connections/relationships plays in conceptual understanding.
Arising out of the students‟ experiences, we will discuss four main types of
connections that are essential to conceptual understanding. They are:
Connections between ideas and procedures
Connections between ideas and the manipulatives used
Connections between ideas and multiple approaches and between one
approach and another
Connections between ideas and ideas
Various authors from time to time have mentioned some of these connections in their
discussion of conceptual understanding. The National Research Council (2001), for
example, points out that students with conceptual understanding “organize their
knowledge into a coherent whole, which enables them to learn new ideas by
connecting those ideas to what they already know” (p.118); “see how the various
representations connect with each other, how they are similar, and how they are
different” (p. 119); and “see the connections among concepts and procedures” (p.119).
They go on to say that “having a deep understanding requires that learners connect
pieces of knowledge and that connection in turn is a key factor in whether they can use
what they know productively in solving problems” (p.118). This study, however,
differs on two important notes. By highlighting the various types of connections that
are relevant for developing conceptual understanding, this research provides a
framework from which to understand prior findings regarding conceptual
understanding – findings such as the importance of multiple approaches in developing
29
conceptual understanding. It also demonstrates the importance of connections at the
secondary and post-secondary levels.
30
CHAPTER 2
METHODOLOGY
The theoretical perspective guiding this study is one in which “knowing and
doing mathematics is an inherently social and cultural activity” (Cobb et al, 1996,
p.15), where social interaction does not merely serve as a “catalyst for otherwise
autonomous cognitive development” (Cobb et al, 1996, p.3). Rather, both what the
student brings with them (their background, academic training, etc) as well as the
environment in which they find themselves matters, but the extent to which either
matters depends on what gets co-constructed between the two.
This study sought to illuminate what the mathematical backgrounds of students
from the Bahamas afforded them as they navigated the mathematical territories of
American university systems during their freshman year of college. This chapter
describes the study design and selection criteria for participants along with the data
sources, data collection, and analyses process. It ends with a reflection on the
limitations of the study including the tensions experienced during the execution of the
study, and an outline of subsequent chapters.
Research Questions
This study addresses the following research questions:
1. What do Bahamian students’ perceive as the important secondary-school
mathematics experiences that have shaped their view and interest in math,
science and engineering?
2. What experiences do Bahamian students pursuing degrees in MSE have with
mathematics in American universities and colleges?
3. What relationships, if any, exist between the perceived secondary-school
mathematics preparation of Bahamian students and their experiences with
mathematics in American universities and colleges?
31
Setting and Participants
Beginning August 2006, my search for participants for this study intensified.
Having spent the previous year contacting universities to determine the likelihood of
their having Bahamian freshman students with an interest in an MSE degree, I traveled
to two universities in the southeastern United States who had granted me permission to
recruit students at their orientation activities. This yielded two possible participants –
a freshman and a sophomore. Having already exhausted all of the Bahamian
government high schools located in Nassau without success, I then visited two
independent high schools and a scholarship granting organization in the Bahamas.
Each provided me with a list of students from the class of 2006 who were pursuing
degrees at American universities along with some contact information – either email
contact or university attending. From this list, six students responded to my emails
regarding the study. Unfortunately, two were not pursuing MSE degrees and were
eliminated from the pool of possible participants. Two others eventually chose not to
participate in the study. Thus, by mid-October, I had successfully recruited three
freshmen students and a sophomore to participate in the study. After my first round of
observations, however, I elected to eliminate the sophomore from my study as I was
having difficulty getting her permission to gather the necessary consents from her
teachers and parents5.
The three freshmen students who became participants of this study met the
following predetermined criteria: 1) attended a Bahamian high school for at least the
5 In cases where participants were under 18 years of age, I also secured parental permission before
continuing with the study.
32
last three years of high school; 6 2) achieved grades of A or B on the Bahamas
General Certificate of Secondary Education (BGCSE) examination for mathematics;
3) attends a full-time, four-year university in the United States; and 4) intended to
pursue a career in math, science and engineering.
Attended a Bahamian high school. All three participants in this study had
been formally educated in Bahamian schools from childhood and had attended their
respective high schools for grades 10 -12. These high schools are located in Nassau,
the capital city of the Bahamas and home to approximately 60% of the country‟s
population. Two of the participants graduated from Northern Academy, one in 2006,
the other in 2004. The third participant graduated from Eastern Academy. Both
schools are K-12 schools with Northern Academy boasting a school population of
approximately 1300 students and 85 teachers while Eastern Academy serves
approximately 1000 students with 100 teachers. The participants were all in the
accelerated tracks for mathematics at their high schools which meant that, in addition
to studying the BGCSE mathematics curriculum, these students were offered one to
two years of additional study in mathematics. The student who graduated in 2004
from Northern Academy completed the BGCSE mathematics curriculum in 11th
grade
and studied Advanced level GCE topics7 in the 12
th grade. By 2006, when the other
student graduated from Northern Academy, accelerated students completed the
6 Throughout this study, all references to high school are to the Bahamian senior high school (i.e. grades
10-12). While in the United States, grades 6-8 are referred to as middle school and grades 9-12 are
considered high school, in the Bahamas, high school is considered to have begun in 7th
grade with
grades 7-9 comprising the junior high school and grades 10-12 comprising the senior high school. 7 The Advanced Level GCE, usually called the A-Levels, is a general certificate of education offered by
educational institutions in the United Kingdom, some British Commonwealth countries and British
overseas territories. While the Ordinary Level (called O-levels, the predecessor to the BGCSE)
examinations is generally taken by all secondary students just prior to completion of high school, the
Advanced Level examinations is typically only taken by those students preparing to enter university in
the respective countries and requires two years of additional study beyond the O-levels/BGCSE.
33
BGCSE mathematics curriculum in the 10th
grade and studied content topics addressed
on the SAT II in 11th
grade followed by topics in AP Calculus in the 12th
grade.
Eastern Academy offered its 2006 graduates in the accelerated track content topics
addressed on the SAT II exam in the 12th
grade. This information is summarized in
Table 2.1 below.
Names High
School
Type of
School
Size of
Student
Body
Number of
Teachers
Acceleration
beyond
BGCSE
Year
Graduated
Josh
Eastern
K-12
Approx. 1000
100
teachers
12th grade
SAT II Topics
2006
Sade
Northern
K-12
Approx. 1300
85 teachers
12th grade
A-Level
Topics
2004
Brittney
Northern
K-12
Approx. 1300
85 teachers
11th grade --
SAT II Topics
12th grade –
AP Calculus
2006
Table 2.1 Summary of Participants‟ High School Mathematics Programs
Achieved grades of A or B on the Bahamas General Certificate of Secondary
Education (BGCSE) examination for mathematics. "The BGCSE is an achievement
test” (Lightbourne, 2008) intended to assess what students “know, understand and can
do” (Ministry of Education, 2003b) upon completion of secondary school. It serves as
an indication of a students‟ grasp of “key concepts, knowledge, skills and
competencies required by the syllabus” (Ministry of Education, 2003b). The exams
are administered and graded in the Bahamas using a seven-point grading scale (A-G)
where A indicates an excellent grasp of the material and B indicates a comprehensive
grasp of the material (Ministry of Education, 2003b). Grades A-C are considered to
be “equivalent, grade for grade, to the A-C grades awarded in the previous London
Ordinary Level examination” (Ministry of Education, 2003b). In addition, beginning
34
in 1993, the University of Cambridge has been advising tertiary institutions that the
grades A-C are sufficient for matriculation into four-year college programs.
In mathematics, the BGCSE exam is given at two levels – core and extended.
Students must take the exam at the extended level in order to attain a grade of A or B
in mathematics (Sumner, 1995). The extended level involves three exam papers –
papers 1 and 2 which are also administered for the core level, and paper 3 which is
only administered for the extended level. The government advises schools to screen
students so that students take the exam that is at the appropriate level for them,
whether core or extended. Some independent schools, however, encourage their very
able students to take the core level exam as a practice exam followed by the extended
level exam the next year. This is neither necessary nor intended and some students,
therefore, opt to take only one version of the exam.
All three participants in this study earned grades of A on the extended level
exam though their paths varied. Josh, who attended Eastern Academy, opted to take
only the extended-level exam in 2005, earning an A. Sade, who graduated in 2004
from Northern Academy, also wished to take only the extended level exam but was
strongly encouraged by her school to take the core level exam in 2002 followed by the
extended level exam in 2003. She earned the highest grades possible at each level – a
C on the core and an A on the extended. Brittney, who graduated in 2006 from
Northern Academy, had a messier experience with the mathematics BGCSE exam.
She took the core level exam in 2003 and earned a C, the highest grade possible on
that exam. The following year, she took the extended level exam and again earned a
35
C. She then chose to retake the extended level exam in 2005, whereupon she earned
an A.
Attends a full-time, four-year university in the United States. Two of the
universities in this study are large private, non-profit organizations while the third is a
medium-sized public institution. All three universities are primarily residential,
located in metropolitan cities on the East Coast of the United States. All three are
recognized by the Peterson Guide as at least moderately competitive and are
considered to conduct high to very-high research activity (The Carnegie Foundation
for the Advancement of Teaching, 2007). The universities do, however, vary
considerably in their proximity to the Bahamas and in the number of Bahamian,
Caribbean and black students enrolled at each institution. Winter Heights University,
the medium-sized public institution reports a 0.005% international and 0.01% African-
American student population. Central University has a black student club which
interested members of its 0.07% international and 0.07% black student population
attend8. In addition to its black-student club, Summerland University, also boasts both
a Caribbean and Bahamian student club with 0.001% Bahamian, 0.02% Caribbean ,
0.07% international and 0.09% African-American student population. Thus, while the
choice of universities was opportunistic, based on the students who chose to
participate, this variety in student membership was most desirable as it would likely
lend more credence to any similarities noted across the cases to be studied and may
also help to explain some of the differences.
8 Winter Heights and Central University did not have statistics on its Bahamian or Caribbean student
population.
36
Intended to pursue a career in math, science and engineering. All three
participants indicated that they intended to pursue a degree in either science or
engineering. Josh enrolled as an undeclared major although he spent his first year
pursuing a biology pre-med degree. By the end of his first year he had narrowed his
interest to biomedical engineering pre-med. Brittany enrolled as a biology major
whose goal, by the end of her first year, was of becoming a neurobiologist. Sade
enrolled as a chemistry major at her liberal arts college with the hopes of entering the
chemical engineering program at the partner university by the end of her sophomore
year. A summary of this information is in Table 2.2 below.
Names High School Year
Graduated
High
School
University
Matriculated
to
Undergraduate
Student Body
Population
(nearest
thousand)
Degree
Interests
Freshman
Year 2006
Josh
Eastern
2006
Summerland
10,000
Biomedical
Engineering
Pre-Med
Sade Northern 2004 Central 10,000 Chemical
Engineering
Brittney Northern 2006 Winter
Heights
9,000 Neurobiology
Table 2.2 Summary of Participants‟ Matriculation and Degree Interests in 2006
Research Activities
In this study, interviews and observations served as the primary sources of data
collection. Beginning in August 2006, I interviewed each of the participants five
times over the course of the 2006-2007 schoolyear and observed them as they engaged
in their regular campus life for three consecutive days at three different points during
the school year – once in the fall semester and twice during the spring semester. In
addition, institutional documents and artifacts (for example: published descriptions of
progression of mathematics courses, mathematics course guidelines, and course
37
requirements for chosen major) were collected. Table 2.3 on the following page
provides a summary of each of the data sources used in this study and its alignment
both with the research questions and the theoretical framework. What follows is a
detailed discussion of each data source.
Interviews: I conducted a total of 15 interviews in five rounds between
August 2006 and August 2007. The interviews were held at various locations around
campus and in the Bahamas and were digitally recorded and later fully transcribed for
analysis. In keeping with the theoretical framework, while each interview was
primarily focused on a particular research question, the nature of this study necessarily
meant that some information might be gleaned from the interview that was relevant to
another research question. This overlap was both welcomed and encouraged.
The „high school‟ interview (see Appendix 2.1) provided insight on the first
research question: What do students’ perceive as the important secondary-school
mathematics experiences that have shaped their view and interest in math, science and
engineering? This interview was intended to occur just before the beginning of the
2006-2007 school year as it focused on the ways in which each participant both
worked with and viewed mathematics based on his/her experiences with mathematics
in the Bahamas, and the expectations of each participant as he/she commenced upon
his/her college pursuits. Of the three participants, I was able to do this with only one
participant, Josh, who joined the study before his school year began. The other two
participants joined the study after their school year9 had begun. My first meeting with
these two participants occurred when I arrived on their college campuses to observe
9 Both participants were recruited via emails in September.
38
Theoretical
Framework
Research Question Primary Data Source
Description
Secondary Data Sources
Description
Institutional
Plane
University Artifacts
size, mission and
selectivity; student
demographics; published
descriptions of progression
of mathematics courses,
mathematics course
guidelines, and course
requirements for each
participant‟s major.
Individual
Plane
What do students‟
perceive as the
important
secondary-school
mathematics
experiences that
have shaped their
view and interest in
math, science and
engineering?
„High School‟ Interview
to get a sense of :
1)the way in which the
participant has both
worked with and views
mathematics based on
his/her experiences with
mathematics in the
Bahamas
2)the expectations of the
participant as he/she
commences upon his/her
college pursuits
High School Artifacts
size, mission and
selectivity; student
demographics;
mathematics grades,
performance on BGCSE
extended mathematics
examination, SAT scores,
etc
Interpersonal
Plane
What experiences do
Bahamian students
pursuing degrees in
MSE have with
mathematics in
American universities
and colleges?
„College‟ Interviews 1-3
to explore:
1) the participants‟ current
mathematics experiences
2) what challenges and/or
satisfactions the
participants‟ encounter
and why
3) their current outlook for
the future
„Retrospective‟ Interview
to reflect upon the
participants‟ overall
experience with
mathematics during the first
year of college
Observations
to get a sense of how the
participant interacts with
his/her community
Observations and
Institutional Artifacts
to gain some familiarity
with aspects of the
community to which the
participant might refer in
the interviews.
What relationships, if
any, exist between the
perceived secondary-
school mathematics
preparation of
Bahamian students and
their experiences with
mathematics in
American universities
and colleges?
„College‟ Interviews 1-3
to explore:
1) how the participants‟
current mathematics
experiences aligned with
their earlier expectations
2) any perceived connections
to their mathematics
preparation in the
Bahamas
Observations
to gain some familiarity
with aspects of the
community to which the
participant might refer in
the interviews.
Table 2.3: Fit between data collection methods, theoretical framework and research questions
39
them. I therefore opted to wait for their return to the Bahamas during the Christmas
holidays to interview them regarding their high school mathematical experiences. For
these two participants, therefore, their first interview with me occurred after I had
observed them for three days and focused on their current university experiences.
Their second interview, conducted during their Christmas vacation in the Bahamas,
addressed the aforementioned aims of the „high school‟ interview. In this way, I
hoped to afford these participants the opportunity to focus completely on one
experience at a time while they were more in that moment.
For each participant, three „college‟ interviews10
were conducted at his/her
university following three days of observation at a time of his/her choosing. These
interviews focused on the participants‟ university experience with mathematics during
the intervening times between my visits, exploring each participant‟s ways of
interacting in their present community, how the participants‟ planned trajectories had
changed and why, and any perceived connections to their mathematics experiences in
the Bahamas. The first of these interviews (see Appendix 2.2) occurred in the fall
semester and sought to understand the participants‟ perceptions of their new
community and the ways in which they were becoming a part of that community. The
second and third interviews occurred in the spring semester. During the second
interview (see Appendix 2.3), participants were given the opportunity to share how
their first semester ended and their current outlook for the spring semester. In the
third interview (see Appendix 2.4), participants shared how the semester had been
progressing for them, ways in which it differed from their expectations as well as their
previous semester‟s experiences, and were again probed for any connections to their
10
Interviews 2-4 for Josh and interviews 1,3 and 4 for Sade and Brittney
40
mathematical experiences in the Bahamas. They were also asked to bring one of their
math tests and notes for the current semester to the interview to review how the test
had gone and why, probing specifically for any connections to their mathematical
experiences in the Bahamas. Together, these three interviews served primarily in
responding to the second research question: What experiences do Bahamian students
pursuing degrees in MSE have with mathematics in American universities and
colleges?
During the third college interview I realized that it might prove helpful to have
a final, retrospective interview with each of the participants once they had had an
opportunity to decompress from their college experience during the summer. All three
participants agreed to do so and this „retrospective‟ interview (see Appendix 2.5),
therefore, was conducted just prior to each of the participant‟s return to school in
August 2007. Two of the participants, Sade and Brittney, met with me in the Bahamas
for this interview. The third participant, Josh, was ill just prior to his return to school
and thus was unable to meet with me in the Bahamas. I met with him, therefore, at his
college campus during the opening days of school when new students and student
officers return to campus. This interview proved to be the longest of the five
interviews for all three participants. Participants were asked to reflect back upon their
first year‟s math experience at college considering what they would and would not
change about that year could they relive it and why, along with how those experiences
had been shaped by their earlier math experiences in the Bahamas. Building on past
interviews, they were asked to elaborate upon their notion of critical-thinking
questions, challenging questions, knowing the mechanics of a problem, understanding
41
concept, and making connections. Finally, to get a better sense of the mathematics the
candidates had been exposed to and/or felt comfortable with prior to their university
experience, they were asked to review the extended level BGCSE mathematics
examination they would have taken, discussing their reactions to the questions in light
of their current experiences with mathematics. This interview, therefore, served
primarily in responding to the third research question: What relationships, if any, exist
between the perceived secondary-school mathematics preparation of Bahamian
students and their experiences with mathematics in American universities and
colleges?
Observations: I observed each participant at his/her university setting for a
total of nine days over 40 weeks, in three-day intervals. My goal was to gain further
insight into how the participant interacted with his/her community and develop some
familiarity with aspects of the community and the participants‟ daily life that impacted
the participants‟ experiences with mathematics in that community. These observations
also helped inform the interview process, providing me with a reference point for
some things mentioned in the interviews and aiding me in determining when to probe
further about things observed or later referred to.
Usually each participant would meet with me before the start of their class
each day at a pre-determined location (either their dorm lobby or student center) and
we would walk to their class together, often chatting as we went. During these
conversations, the participants would often volunteer information regarding the
developments in their life since we last met, current events on campus and/or what to
expect on that particular day – what they planned to be doing, how much of the day I
42
could accompany them, etc. Sometimes we chatted about events that were unfolding
at home in the Bahamas, our families, vacations, etc. On a few occasions, their peers
accompanied us and, at those times, I generally faded to the background as they
engaged in conversations with their peers about life on campus, schoolwork, etc.
Having received permission from the various teachers concerned, I
accompanied the participants to all of the classes they had invited me to attend. For
the most part, with the exception of their science labs and one participant‟s English
class that was a tightly knit work session, this meant that I attended all of the classes
they attended during the three days I spent on campus. Since math classes on all three
campuses were generally taught on alternating days, my visits were scheduled so that I
would accompany them to their mathematics classes at least two of the three days I
was there.
My observations focused upon the participant and his/her ways of interacting.
During class time, therefore, I usually sat behind the participant and to the side as this
allowed me optimal viewing of the participant and the activities he/she engaged in
while in class. To minimize my presence, I took notes by hand in the classes where
the majority of students were taking notes by hand and used my laptop only when the
participant was also using his/her laptop. Generally speaking, this meant that I took
notes by hand in all but Josh‟s biology class. My notes would first describe the
classroom and location of seating, the time we arrived and what the participant did
prior to the start of class, followed by information regarding who the participant
interacted with, the types of notes the participant took during class, what was written
43
on the chalkboard, and the contributions the participant made to the class discussions.
I wrote as events occurred rather than in timed intervals.
Following class, I would join the participant during his/her daily activities but
no notes were taken at this time unless the participant opted to do quiet study in my
presence. I sat with participants as they spent time with their classmates between
classes, attended advising sessions with two, relaxed in one‟s dorm room, and attended
astronomy help session with another. The teachers whose classes I attended were all
aware of my purpose in attending their class and their written consent was received.
They were not, however, informed as to which student I was observing unless the
student chose to identify him/herself. The participants were also told they could
introduce me to their peers in whatever fashion was most comfortable for them and
most, if they introduced me at all, opted to introduce me as an acquaintance from
home. No information was recorded regarding the behavior of their peers except that
which the participant made mention of in an interview.
Occasionally during a participant‟s non-class time, he/she might indicate that
he/she wanted to take a nap. During those times, I would either wait for the
participant in the student center or in the lobby of their dormitory.
My observations usually ended each day after the participant‟s last class. On
one occasion I did stay with a participant through the evening hour as the participant
intended to complete math homework at that time. While it was quite informative and
allowed me an opportunity to observe both the participant‟s study patterns for math
and interactions with students in the dorm, such intense observations (from 8a.m. until
10p.m.) would have proven difficult to sustain with little gains as the participants
44
rarely used their evening hours to focus on mathematics. 11
Therefore, I chose instead
to ask each participant, after their last class of the day, what their plans were for the
evening and would then follow up with them the next morning regarding what they
actually did do in the evening. I would then use my evening hours to record, as far as
possible, events and/or conversations that occurred during the non-class times of the
day.
Documents and Artifacts: In keeping with the theoretical framework guiding
this study, I also collected data relevant to the institution and the individual‟s prior
experience. The data collected at the institutional level concerned the institutional
traits (such as size, mission, selectivity, student demographics, etc), published
descriptions of progression of mathematics courses, mathematics course guidelines,
and course requirements for each participant‟s major. In addition, I asked each
participant who had consented to do so to self-report information regarding their
mathematics grades, BGCSE mathematics exam and SAT scores.
Stages of Analysis
Two types of analyses occurred in this study – analysis of the data by case and
across case. The first analysis, by cases, began with open-coding (Miles &
Huberman, 1994) of each interview shortly after each interview was conducted with
the goal of identifying important themes to keep track of, additional questions that the
interview raised, and information central to the study which the interview failed to
address. After transcribing each interview, therefore, preliminary coding of the data
was performed followed by the production of a contact summary report designed to
11
One participant usually did mathematics homework on the weekends. The other two tended to study
very late at night.
45
organize the information obtained in the interview. What follows is a description of
each of these processes.
Coding: Shortly after completing a round of interviews/observations, the
interviews were transcribed and then coded for relevant information. The open-
coding of each interview was conducted using a list of codes generated from the
conceptual framework (See Appendix 2.6) as well as codes that emerged from the data
itself (See Appendix 2.7). The coded interviews were then reviewed a second time
for the purpose of generating a contact summary report. In these reports, I both
summarized and organized the wealth of information obtained from each interview
highlighting the themes that emerged from the interview, portions of the interview that
directly addressed any of the three research questions, what questions still needed to
be addressed, other interesting details obtained in the interview, future steps, and
methodological issues to consider. In addition, memos were generated that kept track
of any emerging hypotheses and the data to support them, as well as any comments the
participants made regarding the BGCSE mathematics curriculum used in the Bahamas.
Two logs were also kept that recorded on a timeline the participants‟ responses to two
questions that were asked in multiple interviews, the first in all but the high school
interview and the second during the last two interviews (third college interview and
the retrospective interview):
1. If you were writing to a younger sister/brother/cousin, somebody
back in the Bahamas who you really cared about and they were
preparing to come to the U.S. to go to college and would take this
course, what would you tell them about your experience?
2. If you had the opportunity to tell the government of the Bahamas
what they could do to make it easier for students to study
46
mathematics when they come over here, what would that message
be?
Contact Summary Reports: The contact summary reports (see Appendix
2.8) initially served as my guide for reentering the field to collect data. I used them to
help me determine which questions in the interview should be revised, eliminated
and/or restructured, which questions required more probing for clarity of response, and
what additional questions the data revealed might be relevant to ask. In addition,
when I began full analysis of the data after coding all 5 interviews, I used the contact
summary reports generated from each interview as a means of pulling all the relevant
information on each participant into one cohesive whole called the Combined Contact
Summary Report. Using the same outline as the contact summary report, the
Combined Contact Summary Report, organized all the information contained in the
previous 5 contact summary reports by question so that a fuller picture could be seen
of what was known of each participant. This report was then used in two ways – to
identify the themes most relevant to a particular participant (by case analysis) and to
generate a matrix of themes by participant (See Appendix 2.9) for the purposes of
cross-case analysis.
Tracking across cases required that I look for both the similarities and
differences amongst the themes that comprised each case. It raised questions of how
unique a particular theme was for a given participant, possible reasons for any
differences noted, and what the common elements told me about the three participants
being studied as a group. It also drew my attention to codes that were still being
retained in the study although not strongly supported in the data while highlighting
47
themes that I may not have initially considered to be salient but were, in fact, noticed
in more than one case.
Participant Check. After determining the similarities and differences among
the three participants, my next task was to write a case study for each participant based
on the central points that each participant‟s experience illuminated. Each participant
was then invited to read his/her own case study and to both share with me their
reactions to the case story I had written as well as identify aspects of the case that they
felt should be altered in order to maintain their confidentiality while preserving the
relevant details of the case. All three participants felt that the case studies accurately
reflected their experiences with mathematics during their freshmen year and updated
me on various aspects of their lives as they read their case study. One participant
thought that the accelerated studies might easily identify the high school and was
relieved to learn that the other school included in the study also had an accelerated
program.
Each case story served as a narrative of the themes that each individual
participant represented. My final step in the analysis process was to work in reverse,
taking each case apart one paragraph at a time – noting what the paragraph was about
and its relevance to the matrix of themes that had been previously developed. This
served as a final check-point for the across-case narrative – helping me to identify the
extent to which the major themes that had emerged were represented across the three
cases.
48
Tensions and Reflections
At the very outset of this study I encountered tensions that challenged me to
think even more deeply regarding the phenomenon I was trying to understand. I
discuss the three most significant tensions -- the size and composition of my sample,
and the negotiation of the researcher-participant relationship -- below.
Small sample of participants. In designing this qualitative case study, it had
been my hope to have had at least four participants and, ideally, six to eight
participants for, as Yin (2003) noted four to six cases would allow for the possibility
of theoretical replication. Consequently, when I was only successful in recruiting
three participants for this research, I came face to face with the issue of
generalizability. The benefit of this small sample was that I was able to spend a
significant amount of time with each participant, observing them, interviewing them,
and getting to know them both in and outside of the classroom. It would have proven
difficult to focus this much energy on one person, had I needed to travel between more
than three universities per round of data collection. Thus, what I lost in breadth, I
hope I gained in “depth of observation and analysis” (Boaler, 2008, p.592) providing
you, the reader, with a “[perspective] on learning that complement[s] and enrich[es]
the experimental research traditions” (National Research Council, 2004, p.8). The
value of this study, therefore, is in what it contributes to our understanding of how
experiences with mathematics in high school may shape and/or limit students‟ future
experiences with mathematics in university.
Composition of sample. All three students in the study attended private
secondary schools in the Bahamas which offered them mathematics beyond the
49
BGCSE level. This meant that I would be limited in how much I could say regarding
the relationship between the experiences the BGCSE curriculum affords students and
their later post-secondary experiences. What I found, however, was that this limitation
opened up the study to broader implications, aiding me in developing a broader
understanding of what experiences might entail and the role of the curriculum (both
actual and implied) within those experiences.
Negotiating the researcher-participant relationship. I expected that by asking
the participants to reflect on their experiences and probing for potentially sensitive
information from them, I might heighten their awareness of their experiences with
mathematics and to what they attributed those experiences. What I did not expect,
however, was that they might become equally sensitive to what role they played in
those experiences through their attendance patterns, attention to homework, and study
habits. Brittney generally always seemed very comfortable with me; happy to have
me visit again and experience her life. Perhaps this was influenced by the fact that her
math experience at the university had far exceeded her expectations. She was the only
participant who did not grant me permission to collect data on her grades, yet she
happily volunteered such information whenever we were together. Sade, on the other
hand, was perhaps the least comfortable with my role and often surprised me. Our
first round of observations and interview went smoothly. Of all the participants, she
opened up the most to me during our first encounter and provided me with full access
into her life – taking me up to her dorm room, having me join her and her friends for
lunch and dinner, etc. However, in our later encounters, I sometimes sensed some
hesitancy on her part. These feelings would all dissipate, however, by the end of our
50
encounter when we would sit down for the interview. It was then that she seemed to
fully relax in my presence – presenting a no-holds barred approach to the interview,
happy to discuss both her own frailties and frustrations as well as her concerns
regarding the mathematics teaching she was encountering. Josh seemed to live his
life as he usually did though occasionally he would mention to me that he hoped I had
noted a particular event in my observational notes, thus indicating his awareness of my
presence.
I also found that I tended to have lengthier interviews with Brittney and Sade
who simply shared more in that setting. This was not because Josh was less willing to
let me into his life experiences – during our walks across campus he voluntarily shared
what he was experiencing and often updated me on the happenings in his life since my
last visit without my prompting. Of the three, he was the only one to encourage me to
visit his high school math teacher (which I did); yet my interviews with him often
seemed to yield so much less information regarding his thoughts and experiences than
my interviews with the female participants or my casual conversations with him
between classes.
One final note: while the mathematical experiences of the Bahamian students
within this study may help inform our thinking about the mathematical experiences of
students in general, this is not a comparative study as no contrast group is considered.
What Follows
In the following chapters, I present the results of this study. Chapter 3 looks at
the role of the Bahamian curriculum in preparing students for mathematics study in an
American university. As we shall see in the participants‟ reports of their perceptions
51
regarding the curriculum, the role of the curriculum goes beyond the content
knowledge the students were exposed to and includes the ways in which the student
learnt the content and the underlying lessons, like how to study – what, when and how
long, that were taught. Chapter 4 then takes a close look at conceptual understanding
as highlighted in the participants‟ reports of their experiences with mathematics at the
university level. In particular, we shall look at four major connections that this study
highlights as being relevant to developing conceptual understanding. The results
section then concludes in Chapter 5 with a discussion of other attributes of
mathematical awareness (agency, authority, and confidence) that these four major
connections helped to foster within these students. Finally, Chapter 6 addresses the
conclusions and limitations of this study.
52
CHAPTER 3
LEVEL OF PREPAREDNESS
In this chapter, we shall look at the role of the Bahamian curriculum in
preparing students for mathematics study in an American university. We will consider
the content the curriculum addressed and the role it played in where the students began
their collegiate studies in mathematics. We will also discuss the students‟ perceptions
regarding these placements and of the influence the curriculum had on the views
and/or practices they developed in regards to mathematics.
Challenges of Content
Treisman is quoted as saying, “at the college level there is much to be learned
by studying successful students” (OTA Report, 1989). The three students included in
this study are such students. Each had been selected to take the accelerated courses in
mathematics at their high schools. This meant that, in addition to studying the
BGCSE mathematics curriculum, these students were offered one to two years of
additional study in mathematics. Furthermore, all three students earned an A on the
Bahamas General Certificate of Secondary Education (BGCSE) examination for
mathematics at the end of their eleventh grade. This examination, designed to be an
exit examination for twelfth graders that evaluates what they “know, understand and
can do” (Ministry of Education,2003b) after five/six years of study, is given at two
levels – core and extended. In order to achieve a grade of A or B in mathematics,
students must take the exam at the extended level. This level involves three exam
papers – papers 1 and 2, which are also administered for the core level, and paper 3
which is only administered for the extended level. Table 3.1 provides a summary of
53
the year taken and grades attained by each participant on the BGCSE mathematics
examination and SATs, along with the nature of the accelerated courses they studied
per grade level.
Table 3.1
Sade Josh Brittney
BGCSE Math Extended
Year Taken: Grade
(Year Taken for Peer
Group)
2003: A (2004)
2005: A (2006)
2004: C (2006)
2005: A (2006)
Accelerated High School
Courses Grade: Description of
course
12: Content topics
for AP and A-levels
12: Content
topics for SAT
Math
11: Content topics
for SAT Math
12: AP Calculus
Math SAT I
Date Taken: Score
12/02: 600
12/03: 710
10/05: 720*
11/04: 580
11/05: 640
520
SAT II Math Level 1 1/04: 670
5/06: 630
SAT II Math Level 2 1/06: 670*
*Scores listed in italics are for exams taken at the IB World School for entrance into college.
Of the 4,367 students that took the mathematics BGCSE examination in 2004,
only 3.2% earned a grade of A. This figure dropped to 2.2% of the students in 2005
and 2.6% in 2006 (Ministry of Education, 2010). Nationally, therefore, the three
participants in this study had been successful students of mathematics in high school.
None of them, however, commenced their post-secondary studies with the study of
Calculus, the entry level for mathematics for each of their majors. Table 3.2 shows
the entry level math requirements for each of their majors along with their actual math
placement for their freshmen year.
54
Table 3.2: Entry level math requirements vs. actual math placement per participant
Participant University Major Entry level
math course
for major
First math
course
participant
took @ univ.
Math
courses taken
prior to entry
level course
for major
Josh Summerland Biomedical
Engineering
Pre-med
Calculus I for
engineers
Precalculus Precalculus 1
Precalculus 2
Brittney Winter Heights Neurobiology Calculus I for
non-math
majors
College
Algebra
College
Algebra
Sade Central Chemical
Engineering
Calculus I Calculus II 2 years at IB
school
You may notice in Table 3.2 that Sade began her collegiate experience in
Calculus II and yet I claim that she was not prepared for the entry level mathematics
course upon completion of high school. The apparent contradiction is due to the fact
that Sade graduated in 2004 and, instead of pursuing college or work upon completion
of high school, she chose to pursue the two-year International Baccalaureate (IB)
diploma at a World School. She enrolled in the higher level mathematics course her
first year where, to her surprise, she encountered a lot of new material that did not
appear to be new for students from other countries. Topics like “completing the
square” were both new and challenging for her, while others that had seemed easy in
the BGCSE curriculum (for example, probability), were shown to be more complex.
She saw mathematical proofs (both geometric and analytical) for the first time and
found it confusing because she didn‟t know all of the needed axioms to prove the
given theorem. She was also introduced to 3-d transformations, an extension of the
55
2-d transformations included on the BGCSE curriculum, and the graphing calculator, a
tool that proved to be essential in her IB mathematics courses.12
Josh‟s experience provides a clearer sense, however, of the difference in
curricular expectations. Of the three participants, Josh received the least exposure to
topics beyond the Bahamian national curriculum. Table 3.3 on the following page
itemizes the list of topics Josh encountered in the two precalculus courses he took his
freshman year according to what was and was not included in the Bahamian national
curriculum. This list shows that while there is some overlap with course content, a
number of the topics to be covered in these courses would not be familiar to him
unless they had been addressed in his twelfth grade accelerated coursework. The list
of topics to be covered in Brittney‟s college algebra class (See Appendix 3.2) is
similar although she did not receive exposure to the trigonometric topics covered in
Precalculus II.
It is also worthwhile to note that, for those topics which the participants
recognized as overlapping with those in the Bahamian national curriculum, they felt
that the topics were covered in more depth at the collegiate level. When Josh was
asked whether he had studied the topics being covered in the first six chapters of his
textbook, he replied, “Basic encountered. Of course they go a little more in-depth
now” ( Josh, 1st college interview). Brittney gave a similar response:
“ . . . they are teaching you things that you are basically familiar with
but they are going into more detail” (Brittney, 1st college interview).
12
See Appendix 3.1 for a more complete list of the topics covered in the first year of the IB Higher
Level mathematics program.
56
Table 3.3 Topics covered in Precalculus course versus national
curriculum
Josh
Precalculus 1& 2
Included in
National Curriculum
Not Included in
National Curriculum
Precalculus
I
Review of real number system
and set operations
Review of algebraic expressions,
equations and equalities
including: polynomials, rational
expressions, integer and rational
exponents
Applications of equations and
inequalities
Function notation
Composition of functions
Graphs of linear and quadratic
functions
Finding the inverse of one-to-one
functions
Systems of linear equations in
two variables with applications
Review of algebraic
expressions, equations and
equalities including: radicals,
absolute values
Division of polynomials
The Division Algorithm and
the Fundamental theorem of
algebra
Finding the domain and range
of functions
Implicit functions
Shifts of basic functions
Operations on functions
Graphs of absolute value
functions, polynomial
functions, rational functions,
exponential functions and
logarithmic functions
Defining the inverse function
Systems of linear equations in
three variables with
applications
Precalculus
II
Law of sines
Law of cosines
Vectors and applications
Trigonometric functions and
their graphs
Trigonometric identities
Solving trigonometric
equations
Inverse trigonometric
functions
Area and simple harmonic
function
Polar coordinates, equations
and graphs
The complex plane and
DeMoivre‟s Theorem
Conic sections
57
Sade, reflecting upon her IB experience, noted that “the material [in the BGCSE
curriculum] needs to be beefed up” (Sade, 2nd
college interview). Brittney gave an
excellent example of this in her discussion of functions:
They would give you f(x) = x2 + 3. You picked your x values, plug
them x values into that, we didn‟t even call them functions at that
point, get your y values and plot „em. That was it. That was it. Like it
really was not, it had nothing to do with “This is a function. This is . .
.” Like you knew that that gave you the equation of the graph but you
really didn‟t know how to manipulate that. Like if they were to say,
you know, where, if they were to give you that and say where does the
graph, um “Where does the graph increase or decrease?” you‟d be like
“What? I don‟t know how to do that” you know. (Brittney, 3rd
college
interview)
The description she gives here is accurate. The syllabus13
calls for students to be
taught how to graph , and by
constructing tables of values, to estimate the gradient of curves by drawing tangents,
and to solve systems of linear and quadratic equations by graphical methods. Students
are not required to learn what a function is or its various defining features and, in fact,
Brittney did not.
“We didn‟t even know that f(x), that was called a function or I didn‟t,
I don‟t know maybe I was just lost but, probably was lost, yeah, but
um, I didn‟t know that that was what a function was. I didn‟t. That‟s
the equation of the graph. I did not know that those were the same
thing. But like I said I was pretty much lost for most of my math years
in high school so maybe that was slipped in somewhere and I just
missed it” (Brittney, 3rd
college interview)
As Schoenfeld (1988) points out, however, when mathematics is learnt or taught in
this disjointed way it often results in students learning to study mathematics passively,
with no expectation that they should be able to make sense of it for themselves.
13
See Appendix 3.3 for a copy of the BGCSE Syllabus.
58
Accuracy of Placement
None of the participants disputed the accuracy of their postsecondary math
placement. In fact, all three participants credited their colleges with doing an excellent
job helping students transition from high school mathematics to college mathematics.
In keeping with the literature on college transitions (see Terenzini, 1982), both
Summerland and Winter Heights, the colleges Josh and Brittney attended respectively,
had set procedures for determining which course the participants should take. At
Winter Heights, Brittney was required to complete an online placement examination in
mathematics the summer prior to her enrollment. The 30 minute examination
consisted of 25 multiple choice questions to be answered without assistance from
textbooks, calculators or other persons in order for the university to place her in the
mathematics course that was most appropriate for her current mathematical level.
Brittney‟s placement test showed that she was weak in her understanding of Algebraic
concepts and should, therefore, begin her mathematics studies with College Algebra, a
preparatory course for Calculus which focused on sets, relations, algebraic,
exponential and logarithmic functions, their graphs, and applications. At
Summerland University, the decision as to which course Josh should take was based
on his SAT results. Students with SAT scores between 600 and 650 were eligible to
take the beginning precalculus course and those with scores above 650 were eligible to
take either precalculus or the introductory calculus course. Thus, with an SAT score
of 640 for math, Josh enrolled in the beginning precalculus course, a decision he
remained unwavering in asserting was the correct choice for him.
59
Sade also praised her college for the gentle introduction they offered students
to the more sophisticated levels of mathematics.
“ . . .the way that [my university] does it seems to um go through the
basics thoroughly enough because quite frankly they, they spent like
half the term just defining what, like you know, like differentiation
was, like you know, so it was like you know, it, it was very thorough
and it should have, you should have been able to get through it, . . .
you‟d just be starting from square one, which is how the class is
designed, for you to start from square one” (Sade, retrospective
interview).
Concerns with Placement
Despite the students‟ agreements with their placements, however, both Sade
and Brittney were, nonetheless, annoyed that their secondary school math curriculum
– in particular that which was stipulated by the government – was not sufficiently
challenging to have adequately prepared them to begin college level mathematics.
“ . . . Um, and the level is, Our BGCSE exam is like 9th
grade level
math here. It is. It really is. Like the stuff that, I mean, the stuff that
they do in BGCSE you don‟t even see in the revision part of these
books. Like the stuff that you see in the revision part of these books is
like what you saw towards the end of SAT II and the beginning of AP
calc. Like they don‟t even bother touching on that BGCSE level
because as far as they are concerned that‟s like 7th
, 8th
and 9th
grade
work. Like, that‟s, that‟s like 1 + 1 = 2 to them, you know. That‟s just,
it‟s not even worth touching on because it‟s so easy, you know. So
they don‟t even bother to touch on it in the review of the lowest
possible math that they offer at the university. . .” (Brittney, 1st college
interview).
Interestingly, although Sade and Brittney had both attended Northern
Academy, their experiences with mathematics had been very different. Brittney had
had to work hard to attain success in mathematics and, although she knew within
herself that her grasp of the mathematical concepts was tenuous at best, her A on the
BGCSE, albeit a second attempt, still indicated that she had achieved “an excellent
grasp of the material” (Ministry of Education, 2003b). This, coupled with her strong
60
performance in high school, and should have, she felt, yielded a higher placement in
college.
“I‟m like in the lowest level math. Which is like you know for a
student who got an A on BGCSE math in 10th
grade, lowest level math
in college, that tells you something, doesn‟t it, you know.” (Brittney,
1st college interview)
“Coming out of high school with, you know, all these awards and a 3.9
GPA and head girl and A‟s in everything, why do I need to go into
elementary math class. No, that doesn‟t make sense. If I was
struggling in school, that‟s what that is for” (Brittney, retrospective
interview).
“ . . .how is it that I can go into college and do well in Biology,
Chemistry, everything, everything that I did in high school, perfectly,
pick it up, As straight through, but Calculus or math, I had to go in an
elementary math class with people who were flunking out of school
cause they just didn‟t care. That‟s not right.” (Brittney, retrospective
interview)
Sade, on the other hand, graduated from high school confident in her ability to
understand mathematics. She had no problems earning an A on the BGCSE extended
level examination and enjoyed the opportunity to discuss and engage in mathematics
in new ways during her final year of high school. She was shocked, therefore, by her
first year experience of mathematics at the IB World School:
“I came to IB and I had this terrible experience where I felt like I was
so behind and I guess that was another thing because I was a graduate,
a high school graduate, competing with people who were technically
still only in grade 12 and they knew more than me and I guess for me it
was like okay um so obviously we are not at the level we should be at,
you know, if we‟re graduating at under a grade, . . . like grade 12 level
that kind of thing like. It‟s just, that‟s unacceptable and I think that was
what made me extra, kind of really angry um at the point in time
specially since, like you know, and it was obviously to the point where
it was actually they could not even fathom that I hadn‟t done, done this
stuff because I said something to my, one of my good friends . . . um I
said something about „I‟ve never done this kind of stuff before‟ like
you know and he was like, “what do you mean never?”, I can‟t
remember what we were talking about but he was like „Really, you‟ve
never, ever done this?‟ I was like „No.‟ He‟s like „Really?‟ I was like
61
„no‟. He was like „Oh, okay‟ but you know like he actually, he, he
gave a reaction. He‟s usually a pretty tactful person, like you know.
He was just like, „What do you mean you„ve never?” And he was like
„But I thought you‟ve graduated high school‟, you know and I was like
„I did‟, you know. . .” (Sade, retrospective interview)
Study Habits
Sade‟s and Brittney‟s experiences gives you a brief glimpse at some of the
differences that existed in the way these three successful students experienced
mathematics during their high school years. Chapter 4 will expand upon this further.
Table 3.4 on the following page briefly summarizes, however, the ways in which the
demands each participant felt regarding the high school curriculum influenced their
development of study habits. Note that this theme considers how the institutional
level (high school curriculum) as played out on the interpersonal level (level of
demand) influenced the personal level (development of study habits). It is also
interesting to note that while the institutional level (high school curriculum) was
relatively the same for each participant, what was co-constituted at the interpersonal
level was uniquely different for each of them.
62
Table 3.4: high school curriculum demands influence development of study habits later used in
college
Neither Sade nor Josh had experienced many challenges in their high school
math classes.
“Math, math was like the class that you know you didn‟t have to
dedicate a lot of time to if you didn‟t want to and you could just kind of
like, you know, stroll your way through it. (Sade, 1st college interview)
“I was the only student allowed to sleep in class. And it was crazy
„cause I would sleep and she‟d teach a new topic and I‟d wake up and
do the work and go back to sleep and she can‟t figure out how I did it.
It just clicked for some reason. And the people that are awake still
Sade:
High school curriculum
not demanding;
Still developing study
techniques
Josh:
Manages advanced high
school curriculum with
ease
Brittney:
Demanding high school
curriculum
High School
Found high school
math non-challenging
although the student-
centered curriculum
used in 12th
grade
was interesting and
required some critical
thinking.
Did not study in high
school – only
challenging subjects
was history
Acceleration in high
school confirmed belief
in strong abilities in
mathematics;
Not necessary to study in
high school – just do
problems; if I don‟t get it,
class won‟t either so
teacher will need to
discuss
Acceleration in high
school (beginning in
grade 11) was stressful
but beneficial for
college.
Critical thinking,
required for accelerated
math in high school,
continues in college.
1st
semester
college
Attempts to pattern
study habits after
high school
classmate.
Study habits in college
based on advice of peers
(all-nighters, nodos, past
exams, etc).
Uses study habits of high
school (e.g. studying for
long hours, being
thorough, making sure
comfortable with
material) but had to
adapt when she began
studying for college
tests.
2nd
semester
college
Manages to keep up
with weekly college
homework
assignments but only
understands enough
to do assignments;
must cram for exams
Critical thinking first
needed when verifying
trig identities in 2nd
semester of college.
Falls behind in
coursework in college.
Did not fall behind in
collegiate coursework
63
trying to figure out what to do, (what is going on), and I was like okay,
going back to sleep.” (Josh, high school interview)
When the cognitive demand increased after high school, however, they both found that
they needed to develop study habits during their postsecondary experiences. As Sade
explained,
“Like at Northern Academy I didn‟t really study except for when we
did the A‟level history. I learned how to study for that. Um, so yeah, I
developed not just new study skills, I guess not a new form of studying,
I just developed a form of studying at all, definitely for math.” (Sade,
1st college interview)
What follows then is a brief description of the trajectory each student followed
in developing his/her study habits.
Sade’s story
During Sade‟s first year at the IB World School, she found that her math
course involved a lot more notetaking than she had been accustomed to and, although
the teacher worked some examples for the class, no time was allotted in class for
individual seatwork. Instead, students were “given assignments in order to um
acquaint [them]selves with the procedures” (Sade, retrospective interview). Unlike at
Northern Academy, however, those assignments were not marked and this proved to
be somewhat detrimental to Sade who initially opted not to do the assignments.
“I‟m not going to do more work than I have to, kind of thing and, at
first, that meant not doing the homework and then I realized how much
I needed to and I did buckle down and do, start doing them but, like
you know I, the damage was done, that kind of thing. I was already
kind of a little bit behind and on top of that I wasn‟t as, as well-versed
in, in the material as the other people in the class so um, you know, it
was like just a double whammy. Like I hurt myself and I was already
behind kind of thing so. . .” (Sade, retrospective interview).
64
Two things should be noted in this quote. The first is that Sade‟s prior experience had
led her to believe that it was not necessary to do ungraded assignments. She is
operating from the mindset of “getting a good grade” and focuses only on what might
help her achieve that objective. She eventually comes to realize, however, that active
avoidance of the assignments was positioning her in such a way to make it difficult to
attain her goal of “a good grade” in her new environment. While the homework
assignment did not translate directly into a grade, indirectly it had a significant impact
on her course performance because grappling with the concepts in each problem
determined how well she came to understand the course content. Perhaps it was this
that also caused her to realize that mathematics was a subject that she needed to study.
She began using techniques she had seen others in both her past and present social
networks employ. She borrowed the organizational strategy employed by her best
friend from high school and kept a table of contents in the front of her mathematics
notebook which highlighted the main topic emphasized in each lesson. She began
adding a few jokes to her notes, a strategy she observed one of her IB teachers using,
to aid her in remembering the various processes encountered. She also sought help
from the social others in her environment. In particular, she asked some of the
second-year students for help. However, these students “didn‟t teach [her] how to
study, they taught [her] how to do [her] problems” (Sade, retrospective interview). By
this, Sade meant the focus was centered on learning the procedures to solve the
problems rather than on gaining conceptual understanding. The person who was
helping her was trying to get her through the problem so she could achieve her goal –
“to get a good grade” and there was little time left for explaining the underlying
65
principles. At the end of her freshman year of college, in summing up her study
habits, Sade noted:
“Yeah, basically, like I felt around for what works for me. And I think
I‟m still in the process of feeling around, like you know” (Sade,
retrospective interview).
Josh’s story
Likewise, Josh did not view himself as having many study habits in high
school. He normally just read over material before exams (sometimes twice) for most
of his courses. In math, he rarely encountered problems that he could not solve and,
when he did, he would find a similar example in his textbook to follow. If, for some
reason, he was still unable to solve the problem, he would then ask the teacher the
following day at the beginning of class. In those instances, he was confident that if he
“didn‟t get it, the majority of the class didn‟t get it. . .” either (Josh, 1st college
interview). Unfortunately, this approach did not enable Josh to be as successful in
mathematics in his first semester college course. With the help of the social others in
his environment, in particular a senior student, however, he eventually attempted to
adopt a new approach to studying. Given his habit of studying late at night, the senior
introduced Josh to a room in the dorm that was very conducive to studying because it
was “completely in the middle of the building so you‟re not affected by light and
time” (Josh, 2nd
college interview). There, they would study together in the evenings,
each doing their own work. This guidance helped Josh to remain focused on his work
during his study hours, taking breaks only at appropriate intervals. As a result, Josh
“went through and did problems and everything” (2nd
college interview) in preparation
for his math exams.
66
Josh further developed his approach to studying mathematics during his second
semester. He began using the mandatory homework problems to determine what areas
he still needed to focus on, often rereading the corresponding section to ensure
mastery of the topic on the exam. If he was still uncomfortable with the topic, using
the technique he had developed the previous semester, he would then do the last few
assigned problems from each section, consulting his notes when necessary to review
the examples done by his teacher. However, he seldom did the suggested problems
his teacher also assigned and, he confessed, he was “usually the one that waits until 12
hours before a test and then learn like 50 formulas” (Josh, 3rd
college review). Perhaps
this statement sums up his study habits best:
“I think college is new. You have to figure out what‟s happening”
(Josh, 3rd
college interview).
Brittney’s story
Of the three participants, Brittney was the only one who did not report having
to develop a form of studying during her postsecondary studies. She was also the only
participant who had reported feeling challenged by the mathematics she had studied
during high school. Despite having done so well in mathematics in 7th
through 9th
grade that she was consistently chosen for the accelerated classes, by 10th
grade,
Brittney had developed a strong dislike for mathematics. She was frustrated by the
feeling of not understanding the procedures she was being taught and she struggled
with the extended level curriculum.
“I would do the mechanics of the problem that was on the board. I
didn‟t make the connection that he‟s going to give me a problem and
it‟s not going to be exactly like this. It‟s not going to be this problem
with different numbers. It‟s going to be a completely different
67
problem that I‟ve never seen before, have no idea how to approach, but
it‟s going to require that I use this concept” (3rd
college interview).
Although she was “capable of performing symbolic operations” (Schoenfeld, 1988,
p.6), she did not connect these operations to the “real world” objects or concepts they
represented. Her focus was on learning the correct procedures for solving a particular
type of problem without attempting to understand why the procedures worked and/or
were appropriate for the given problem. She carried this same approach to
mathematics with her into the 11th
and 12th
grades where she was exposed to advanced
coursework (SAT II topics and AP Calculus, respectively) and, by the end of 12th
grade, she reported that she “didn‟t know what on earth was going on” (Brittney, 3rd
college interview).
To cope with what felt like ever-increasing demands of her high school
mathematics program, Brittney had had to develop a method for studying
mathematics. Usually, she studied on her own, regularly double-checking her work
with her friends at school the following day and, occasionally, calling them at home if
she needed help in the evenings. She also sought help from her teacher outside of
class. She did not, however, join in group study unless she had already previously
studied the material because she believed that one needed to know the material first in
order to help others. In preparing for the BGCSE extended level mathematics
examination in grade ten, she studied the BGCSE revision guide14
and her class notes
and, in preparation to retake the examination in grade eleven, also worked out the
solutions to several past examination papers. Thus, when Brittney arrived at college,
14
R. Parsons (Ed.) GCSE Mathematics The Workbook Higher Level (3rd
ed.) Newcastle: Coordination
Group Publications, Ltd. This guide reviews each of the topics covered in the BGCSE syllabus but it
does not provide practice problems for students to solve themselves.
68
she had a well-developed program for studying and, at the end of her freshman year
noted,
“. . . my style of like studying is the same as it was in high school
except for the fact that because of the increase in workload and the
increase in information, I have to break it up more.” (Brittney, 3rd
college interview)
How ironic, it seems, that the student who understood the least amount of mathematics
and who was the least able to transfer procedures from one problem to another, was
the one student to develop a habit of mind – studying – that would hold her in good
stead in college. The cognitive dissonance along with the repeated messages that she
was viewed as a successful mathematics student was sufficient to push her to develop
a useful coping mechanism.
Student Support System
All three students benefitted significantly from the significant others in their
lives who served as a support system for them as they adapted to college in their
freshman year. While the primary support for Brittney remained grounded in her
connections to her family at home, for the other two participants the primary support
came from social others in their community with whom they shared similar cultural
practices. Interestingly, these two participants attended Summerland and Central
University, schools in which there was a base of students to draw from who shared
similar cultural practices, while Brittney was the only Bahamian student attending
Winter Heights.
In particular, Josh‟s approach to studying mathematics was most impacted by
the social others with whom he interacted and affords a clear view of participatory
appropriation in action. Throughout his first year of college, he was consistent in
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reporting that he did not study in high school; that he normally just read over materials
before exams (sometimes twice) for most of his courses. However, in his freshman
year, Josh made significant attempts to adopt different practices – to move from being
on the periphery of this activity to becoming a more central participant in engaging in
study. His first attempt came early in the first semester when he and a few like-
minded Caribbean freshmen students formed a study group and met in the stacks of
the library to work on their individual assignments. Later in that same semester, in
response to poor grades on his last two math exams, he teamed up with a senior friend
whom he had gotten to know via the Caribbean Student Club. This friend introduced
him to a quiet place to study in his dormitory. There they would study together in the
evenings, each doing their own work. This guided participation helped Josh to remain
focused on his work during his study hours, taking breaks only at appropriate
intervals. As a result, in preparation for his fourth math exam, Josh “went through and
did problems and everything” (2nd
college interview).
Josh described the Caribbean students club as his most enjoyable aspect of
college as it served as a family away from home for him. From within its ranks came
the friend Josh called “mom”, who provided him with the emotional support he
needed on good and bad days, as well as the friend who took him under her wings and
taught him how to study in college. He also received advice on courses to take or
avoid, what various professors were like, and how to prepare for the assessments of
specific professors. Josh was quite pleased, therefore, to have been chosen to serve as
a member of the Caribbean students‟ board for the following year and invested a lot of
time in preparing for various events for the club for the remainder of the year. At
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times, this impacted his schoolwork as he often spent his daytime hours attending to
phone calls and meetings and thus faced long evenings/nights of study. In looking
forward to the next school year, therefore, Josh hoped to find more balance in his life,
managing his time to accommodate all aspects of his life including his involvement in
the Caribbean Student Club, an invaluable cultural connection that had enabled him to
become a more central participant within his new community.
Finding balance also proved to be a theme in Sade‟s life. Although she had
been heavily involved in extracurricular activities both in high school and the IB
World school, she did not sign up for any extracurricular activities during her first
semester of college. Initially, this bothered her but eventually, by second semester
when she became active in the socio-political wing of the Black Student Club on
campus, she decided that her decision to wait had been best for her.
“ . . . I think I really needed that break to be able to, like you know, get
a feel for the school, like advance my social life, . . . like it was you
know it was like a good time for me to like breathe and be like „Okay,
I‟m in a new setting. This is, you know, dahdahdahdah‟ and like just
figure some stuff out kind of stuff um and where I wanted to be in this
and I mean like, obviously I‟m not done doing that but I didn‟t need
that much of a break and so like next term I got more up in it and I
think it gave me, like you know, some time to be sure of what I do
wanna get involved in, uh, have a better feeling for the campus
atmosphere so I could like, you know, like be able to know how, more
how to balance that and whatnot. . . .” (Sade, retrospective interview).
Her primary friends with whom she had advanced her social life were largely
international students who had attended sister IB schools. She did, however, later
surprise herself by becoming actively involved in the socio-political arm of the black
student club on campus. In fact, by the end of the year, her strong interest in socio-
political affairs had led to her participating in a women‟s movement against crime,
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becoming a member of a group concerned with the impact of the expansion of the
university on the lives of people in the neighboring communities and enrolling in a
political theory class for the following year. As was the case with the students in the
Lipson & Tobias‟ (1991) study, therefore, from as early as her second semester, Sade
began to struggle with her choice to earn a degree in the sciences, particularly
because the unit and course requirements of her major left little room for exploring
other interests.
“The good stuff are the things that strike you. You know there‟s gonna
be some classes that, less face it, you‟re not, I mean you might use
them in life but, you know, there never gonna be something that you
base out for a career, like a woman in culture class or like, you know, a
African dias, Diaspora through the years classes, something like that,
you know something that you know you‟re like you know what I‟m not
going to use this in my career; I‟m not about to come home and talk
about this to anybody like you know but man that sounds interesting.
So basically your, your, your stuff that just sets your heart on fire. . . .
Make sure that you have your basis covered. You don‟t want to like
take a bunch of crazy classes, and then time to take declare a major and
you‟re like, „oh, yeah, I wanna,‟ I was gonna say and it‟s like, „oh,
yeah, I wanna be a chemical engineer,‟ like you know, and you have no
chemistry. Cover your, cover your bases and do what you need to do
and, unfortunately, if you wanna do engineering, that‟s gonna be a lot
but like, save some space, save some space . . . Take classes, take
opportunities that you can‟t get anywhere else I guess is what I‟m
trying to say . . .” (Sade, retrospective interview).
Prospects for Completion of Degree
Fortunately, despite their displeasure with their initial placements, none of the
participants thought that their placement in mathematics would hinder them from
successfully completing their majors in the prescribed time15
. Because of her two
years of study at the IB World School, Sade did have some leeway within her college
15
In fact, all three participants graduated within four years – Brittney with a degree in neurobiology;
Josh with a degree in biotechnology; and Sade with a dual degree in political science and physics, the
science subject she had always enjoyed studying.
72
program to pursue some of her interests while she determined whether she would
continue her study of the sciences. Brittney‟s college algebra course counted towards
her general elective requirements, and Josh had taken both chemistry and biology in
his freshman year, courses which his major usually prescribed for the sophomore year.
This provided Josh with the time needed in his sophomore year to take the calculus
course sequence that had been specified for his freshmen year. His only concern,
therefore, was the extent to which his yearlong physics course would depend on his
calculus course as he planned to take both in his sophomore year.
“I just wish there was some way I could do math like starting a month
ago to actually get a lot in cause physics is gonna pull some of the
topics and a lot they gonna be like „Well, if you did this much math,
you do the advance topics‟” (Josh, retrospective interview).
This comment is significant because Josh prided himself on being able to do the
advance topics in his science courses whether they were optional or not.
What we see here, therefore, are three students who were, by national
standards, successful students of mathematics who were nonetheless underprepared
for the mathematics requirements of their majors both in content and, for at least two,
in habits of study. In other words, what the students brought with them to their new
environment was not sufficient. Fortunately, their universities
(apprenticeship/institutional level) identified their lack of preparedness immediately
and placed the students into the appropriate courses to develop their knowledge base.
This, in addition to the guided participation the students received from their culturally
significant student support base and, as we shall see in the next chapter, the
experiences they encountered within the classroom, aided the students in what
mathematics they ultimately came to understand (participatory appropriation).
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CHAPTER 4
CONNECTED UNDERSTANDING
In 1986, Belencky, Clinchy, Goldberger and Tarule published a book of the
various ways of knowing they had identified through their interviews with individuals
about their beliefs about knowing and learning. In particular, they used Elbow‟s
(1973) terminology of the “believing game” versus the “doubting game” to emphasize
an important difference between how people viewed and considered other people‟s
ideas and opinions in their description of separate versus connected knowing. In the
“doubting game” (separate knowing) ideas are held up for inspection and tested
against arguments to ensure there are no flaws in the reasoning, whereas in the
“believing game” (connected knowing) the focus is on understanding another‟s
perspective (Clinchy, 1996). Boaler and Greeno (2000) found that connected
knowing provided a mechanism for understanding the many voices in the discursive
classrooms they studied who found that mathematics was more accessible to them
because of the opportunities provided to them to hear others‟ thoughts and ways of
processing. They recognized, however, that while connected knowing as described by
Belencky and associates was useful in interpreting their data, this description could
probably be expanded to distinguish between attitudes towards things, texts, and ideas,
including conceptual domains. The “believing game” in this broader interpretation is
a belief that the object of study is coherent and makes sense and that, with persistence,
one might be able to achieve this sense-making oneself.
This chapter takes a close look at conceptual understanding in this vein,
providing some texture to what it means to understand the “why”? Like many, I had
74
been guilty of thinking that understanding the why meant understanding the proof of a
theorem or being able to explain where a procedure came from. Brittney and Sade,
and to a lesser extent Josh, bring to focus, however, the fact that conceptual
understanding is so much more than this. They emphasize the necessity of
understanding the connected relations not just between concepts, or between concepts
and procedures (Hiebert & Lefevre, 1986; Hiebert, 1984), but also between
manipulatives and concepts (Ball 1992; Ma, 1999), between methods and concepts,
between methods and methods (Ball, 1993), etc. They hone in on what Hiebert &
Lefevre (1986) refer to as the “linking relationships [which] are as prominent as the
discrete pieces of information” (pp. 3-4). In fact, Hiebert & Lefevre (1986) go on to
say that it is now evident that it is the relationships which hold the key in developing
mathematical competence. This chapter aims to unpack what some of these
relationships are. These connected relations I refer to as connected understanding and,
as you shall see, they extend beyond the age-old argument between concepts and
procedures to include some of the teaching strategies that research has shown (see
Boaler et al., in progress; Ma, 1999) play a vital role in aiding students in their
understanding of mathematics. Perhaps Niemi (1996) said it best when he said,
“These proposals express a new vision of mathematics achievement, in which
conceptual understanding plays a central role and mathematical knowledge is
conceived as a system of relations among mathematical symbols, concepts, operations,
activities, and situations . . .” (p.351).
75
Connected Understanding
Bruner (1960/1977) said, “Grasping the structure of a subject is understanding
it in a way that permits many other things to be related to it meaningfully. To learn
structure, in short, is to learn how things are related” (p.7 quoted in Ma, 1999, p.24).
Connected understanding is about how things – ideas, procedures and skills,
manipulatives and varying solutions – are related. In particular, we shall look at four
different types of connections that this research highlights as necessary in developing
conceptual understanding of mathematics at all levels. They are:
Connections between the mathematical principle(s) and the procedures
used in a given solution process.
Connections between mathematical ideas/principles particularly when
the ideas belong to the same/similar topic
Connections between the mathematical principle(s) and the
manipulatives used in a given solution process.
Connections between the mathematical principle(s) and multiple
approaches and between one approach and another used in a given
solution process.
Because the connections to be made are always ongoing and a person may have
developed one type of connection but not others, the reader will begin to understand
why it is then that to understand something conceptually, generally means to
understand it to a certain degree (Hiebert, et al., 1997). It is for this reason that “a
variety of experiences, such as problem solving, observation of other people‟s
activities, direct verbal instruction, and reflection” (Rittle-Johnson, Siegler, & Alibali,
2001, p.347) may be needed to facilitate knowledge change.
Connections between mathematical principle(s) and procedures
As mentioned in Chapter 1, conceptual versus procedural knowledge has been
the cause of much debate within the mathematical community across the centuries. In
76
the past, they have been treated as separate entities that often competed for attention.
Today, however, there is growing interest in the relationships between these two
entities and how they can work together in mutually beneficial ways for the learner
(Hiebert & Lefevre, 1986). What follows then is a description of some of the
complexities involved in understanding the relationship between these two entities.
On the one hand, linking concepts to procedures can unify mathematics and make
learning new material much easier (National Research Council, 2001). Brittney‟s
case demonstrates this beautifully. Linking concepts to procedures also serves to
guide the learner in determining how to adapt existing procedures to new problems
(Rittle-Johnson, Siegler, & Alibali, 2001). As both Brittney‟s high school and Sade‟s
college cases demonstrate, however, students who learn procedures without linking
them to the appropriate concepts, are more likely to struggle with assimilating new
information or adapting their current knowledge to novel situations. I begin then with
Brittney‟s discovery of the connections that exist within mathematics.
Brittney‟s first semester of math at Winter Heights University afforded her the
opportunity to gain understanding and clarity on those topics which she had found to
be difficult in high school. Suspecting that her dislike and fear of math in high school
had affected the ease with which she had been able to learn the subject, Brittney
determined to adopt a more positive attitude towards math in college. Instead of
convincing herself that she was ill as she had often done in high school,
“ . . . „How can I get out of math? Do I feel sick? . . . I‟m feeling ill‟
and really I was. I was feeling ill just from the thought of going to
math class . . .” (Brittney, retrospective interview)
Brittney determined to have a fresh start in math
77
“Alright, you know what, it‟s college, new time, new experience, like
start off with a clean slate. Don‟t let anything that‟s happened in the
past affect your, you know, outlook on math right now.” (Brittney, 3rd
college interview)
As the National Research Council (2001) points out, “Students‟ disposition toward
mathematics is a major factor in determining their educational success” (p.132). To
develop a productive disposition, however, requires more than just the will to do so. It
also requires “frequent opportunities to make sense of mathematics, to recognize the
benefits of perseverance, and to experience the rewards of sense making in
mathematics” (National Research Council, 2001, p.131). Fortunately, both Brittney‟s
college algebra and calculus teachers employed several moves that have been shown
(see Gresalfi & Cobb, 2006; National Research Council, 2001; Kazemi, 1998) to aid
students in their development of productive dispositions. They spoke positively to
their students regarding their ability to understand mathematics which helped to
significantly reduce Brittney‟s anxiety. They also related the topics they taught to
those that the students would have been exposed to in previous mathematics courses
thus helping their students to begin the arduous task of making connections for
themselves. Consequently, as Brittney began to realize that the topics taught in the
College Algebra course were similar to, but more in-depth than, those she had been
introduced to in her 11th
grade SATII course, she began to appropriate this behavior
for herself and began to make connections of her own.
“Um, but in math, it connects and I think that helps me now because I
can, I stop and I‟m like, I know how to do that part of it. Now how can
I use that part of it to figure out that other part of it” (Brittney, 3rd
college interview).
78
This recognition of the connectedness of mathematical ideas led Brittney to view
mathematics as being less about memorization and more about key ideas that could be
used to build a cohesive whole (Hiebert & Lefevre, 1986). Establishing this
connection, however, is not always easy to do. In fact, as most proponents of
procedural knowledge aiding in the development of conceptual knowledge (see
Hiebert & Lefevre, 1986; Rittle-Johnson, Siegler, & Alibali, 2001; National Research
Council, 2001) acknowledge, this link is often quite difficult. Consider Sade‟s story .
Sade came into college with a strong sense of herself as a learner of
mathematics, a student accustomed to focusing on understanding the concepts behind
the procedures. When I first observed her, I noted with interest that, in both her math
and chemistry courses, she would listen intently to what was being explained before
taking her notes and, when she did take notes, she did not simply copy the problems
off the board, but rather wrote notes of explanation to herself along the way. When I
asked her about this practice, she explained that she liked to understand the material
first before taking notes. Consequently, she also did not copy down the homework
problems the teacher solved upon request at the beginning of class. She explained
that, while she listened to her teacher‟s solutions, if it was a problem that she had not
grappled with as yet, she preferred to struggle with it on her own rather than simply
copy his solution. She found that this process of working through problems on her
own helped her to gain more clarity on the various concepts presented.
“. . .usually I find that if there is something that I‟m not entirely sure
about once, once I put it into a question, when they put it into a
question and I actually am applying it, I realize that that things kinda
like fall into place and I realize it; and if as, I, I was thinking if that
didn‟t happen for this, I would go to the help room and see if they
could explain the concept to me, or at least the question so that I could
79
understand the concept through, through the question” (Sade, lst
college interview).
Here, Sade‟s goal is to gain clarity with the underlying principles embedded in the
particular problems by using her procedural fluency to develop her conceptual
understanding (Rittle-Johnson, Siegler, & Alibali, 2001). While she was able to
achieve this objective during the first trimester, Sade found it much more difficult to
do this during her second semester course, Calculus III, a course in which she
considered seeking help several times (she went once):
“The only thing is like I don‟t know if, if math help I don‟t know if
you can go and say I don‟t understand the general concepts and like
have them explain. I think that you have to come with homework
problems and I got all the homework problems. Like I, I, I did what I
was supposed to do because like, unfortunately for this class, you don‟t
have to understand to get things right. Um, so like I, I did it and then
you know when I compared my answers like with the answers that had
the answers in the back of the book like with, with what I had like
okay, like I got it right but, but like so” (Sade, 2nd
college interview)
Sade is striving for conceptual understanding – insight regarding the underlying
mathematical principle and its relation to the associated procedures as well as to other
mathematical principles. She recognizes that to do this, to achieve conceptual
understanding, it is not enough to be able to do the correct procedures. One must also
understand the underlying principles that those procedures represent. She lacks
sufficient insight, however, regarding the concepts to use her ability to perform the
procedures to illuminate and clarify those concepts as they had her first semester. As
Hiebert & Lefevre (1986) point out, “Conceptual knowledge, as we have described it,
cannot be generated directly by rote learning. Facts and propositions learned by rote
are stored in memory as isolated bits of information, not linked with any conceptual
80
network” (p.8). It is when procedures are learned with meaning, they claim, that
procedures are linked to conceptual knowledge.
In this next excerpt, Sade elaborates further upon her difficulties, noting that, at
some level, conceptual understanding must necessarily entail linking the concept to
the formulas and procedures being used.
“Um, I find that my being able to understand the concept and make
connections between that and the formula at hand, well obviously like
it just makes it more holistic understanding of the material itself and
like usually, if I can do that, then the class will be a breeze, like you
know, . . .” (Sade, retrospective interview, italics added).
As Hiebert and Lefevre (1986) point out, while it is easy to imagine learning
procedures without linking them to concepts, it is much more difficult to imagine
learning concepts without linking it to some procedures because “procedures translate
conceptual knowledge in to something observable” (p.9). In addition, linking
procedures to the underlying principles allows the reader more flexibility in applying
procedures to novel problems (Rittle-Johnson, Siegler, & Alibali, 2001). As Sade
further elaborates,
“. . . not being able to understand it, means that when they give you a,
you know one of those longer word problems or when they um just
give you a like you know, give you „oh find yaddayaddayadda‟ and
doesn’t necessarily mention the terms or what not and it doesn‟t
necessarily um mention anything you‟re familiar with, you, you can‟t
make that connection like because you really just can‟t . . . you‟re not
able to decipher what you can use. Basically you can‟t, you can‟t even
like get into your critical thinking process at all, kind of thing” (Sade,
retrospective interview, italics added).
The latter part of this statement resonates clearly with the issues Brittney encountered
in high school and highlights how much rote learning (i.e. learning procedures without
connection to the underlying concepts) (Hiebert, 1984; Hiebert & Lefevre, 1986;
81
Schoenfeld, 1992) does not generalize to other situations. Recall again Brittney‟s
statement,
“I would do the mechanics of the problem that was on the board. I
didn‟t make the connection that [my teacher‟s] going to give me a
problem and it‟s not going to be exactly like this. It‟s not going to be
this problem with different numbers. It‟s going to be a completely
different problem that I‟ve never seen before, have no idea how to
approach, but it‟s going to require that I use this concept.” (Brittney,
retrospective interview)
Procedures that are not linked to conceptual knowledge are closely tied to the context
in which they are learned and, consequently, can only be accessed and applied to
contexts that resemble the original (Hiebert & Lefevre, 1986; Schoenfeld, 1992).
Conceptual knowledge, on the other hand, “releases the procedure from the surface
context in which it was learned and encourages its use on other structurally similar
problems.” (Hiebert & Lefevre, 1986, p.14). For Sade, it was essential for her to be
able to link the formulas used to what they represented about a given concept because
this linkage, this understanding of what the procedures represented, afforded her the
ability to adapt the procedures for use with novel and/or unfamiliar problems.
“Understanding is required for critical thinking for me. It‟s like I, I
need to be aware of the principles that I‟m trying to manipulate because
with critical thinking you are basically trying to manipulate principles
and formulas to work for you and um, if I don‟t understand what this
equation actually means and what it‟s applicable to then I can‟t really
negotiate it into a position to work for me, you know, to get the, to find
the solution, like you know. I, I can‟t, can‟t create my algorithm, I‟m
like you know so, therefore like I‟m stuck, um like and I can know all
the formulas off by heart and not be able to know how to sequence
them in order to, like you know, get what I want” (Sade, retrospective
interview)
Here, Sade hones in on why making the connections between the procedures and the
concepts they represent is so important – because it allows you to recognize when an
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equation is useful so that you can then manipulate it in an appropriate manner to solve
novel problems. Both Brittney and Josh had similar comments regarding critical
thinking.
“. . . you have to use so many different concepts and then you have to
tie one answer into the other concept and, you know, continue until you
get your final answer and you almost don‟t know when to stop.”
(Brittney, retrospective interview)
“Critical thinking is challenging and you‟ve got like a bunch of
different concepts flying around and a whole bunch of numbers”
(Brittney, retrospective interview)
“I have to sit there and analyze that”; “involves a lot” (Josh, retrospective
interview)
All three students recognize that linking procedures to their underlying
concepts aids them in selecting the correct procedures to use at the appropriate times
when problem solving. Those who are able to do this have organized their knowledge
in such a way that, instead of compartmentalizing things as is usually the case with
procedural learning, new ideas can be connected to existing ideas to form a coherent
whole (National Research Council, 2001; Thurston, 1990). In addition, linking
procedures to their underlying concepts helps one to avoid choosing inappropriate
procedures when solving problems and, as Hiebert and Lefevre (1986) point out, it
helps one to “anticipate the consequences of possible actions” (p.12) one might
choose.
Connections between mathematical principles
Beyond linking the concepts to procedures or vice versa, connections also exist
between concepts. Ma (1999) mentions the importance of such connections in her
discussion of teachers with a profound understanding of mathematics. She referred to
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this level of understanding as a knowledge package in which the center is a sequence
of related ideas that is then surrounded by a circle of linked topics which connect to
the topics in the sequence. Within the package there exists “key” pieces that carry
more weight than other parts of the package. These pieces are “key” because they are
a new idea or skill that is being introduced or because they are a concept that ties
together several important ideas that are related to the topic at hand.
To illustrate this, consider Sade‟s reflection after her Calc II class discussed
power series, a topic she had not been exposed to at Northern Academy and had
struggled with in the IB higher level mathematics course.
“I was getting, I‟m getting the concepts that he‟s talking about but then
having trouble relating it back to its basis, like where we started off
with it and everything having to do with. We started off talking about,
we got to that starting off talking about I think comparing things to the
geometric series. I‟m having trouble like relating those two together
and yeah. Yeah that‟s where we started. And also we started off, we
were also talking about um oh relating it back to, to integrating
functions that you can‟t integrate and I‟m not entirely following how
these currently relate to that” (Sade, 1st college interview, emphasis
added).
Sade has identified two “key” pieces -- geometric series and integrating
functions – that she believes are important in understanding power series conceptually.
Continuing to play the “believing game”, therefore, she actively tries to find the link
between these two topics that appear to be so different. To use Hiebert & Lefevre‟s
(1986) terminology, Sade is seen here working at the reflective level, i.e. she is trying
to find the common features that link what appears on the surface to be different
concepts together. She explained that visually/geometrically she understood how a
series of functions could approximate the original function but she found it difficult to
84
conceptualize this algebraically.16
This claim highlights the layers of
conceptualization involved in what Sade is attempting to do. She recognizes that the
concepts she is trying to understand can be viewed from several dimensions –
geometrical and algebraic – and that these dimensions themselves must therefore be
related to each other --- i.e. what can be seen in the geometrical world should have an
algebraic equivalent. She, nonetheless, struggled to gain insight into how the
algebraic equivalent for series serves to approximate an integratable function. Limits,
series, integration – all three are “conceptually powerful ideas” (Ma, 1999, p.121) in
mathematics and by attempting to understand the link between the three, Sade was
struggling to understand the topic of power series with depth.
Although Sade did eventually develop an understanding of the link between
power series and integration, she was not always successful in finding the link
between concepts. In fact, time and again she mentioned understanding each concept
individually in her Calculus III course but struggling to see how they connected with
each other:
“. . . the connections between them was a big thing for me, and like I
honestly still don‟t understand why those imaginary numbers were in
there. . . I understood the tangent, the normal, the binormal. Those
made perfect sense and those were all tied into curvature. I under, I
even understood the oscill, the oscillating planes, the oscillating circle
and and all that stuff, like that made sense to me. It made visual sense.
I mean it was perfectly visual or whatever, but it was like this concept
of curvature that was thrown in with that that I didn‟t get. So
conceptually, a lot of the concepts I really got. I didn’t entirely tie
them together” (Sade, retrospective interview, italics added).
16
In her junior year, she encountered this topic again in physics (for Fourier) and she developed
different conceptualizations of series depending on whether it was her mechanics or her quantum
course, to the point of using different symbols.
85
The focus here is on concepts – how they relate to each other, rather than on the
algorithmic procedures which she already knew how to perform. In this example,
Sade is explaining how she understood the definitions for each of the individual terms
and could perform the various calculations involving those terms, but she struggled to
see how those terms then came together to form a connected whole, something she
sensed was necessary for her to gain full comprehension of the concept of curvature.
Again she mentions understanding some of the concepts geometrically but struggling
to understand the algebraic equivalent. And again, she finds herself playing the
“believing game”, convinced that “curvature” makes sense, that the concepts involved
are coherent and connected and, therefore, she persists in her attempts to discover
those connections and make sense of the concepts as a unified whole.
“I know I don‟t understand curvature. Like I understand how to get it,
I mean it‟s a formula, I mean can‟t go wrong with that, they tell you
what to do but like I don‟t understand what the theory is. It makes no
sense to me. Um yeah, that‟s definitely a big one. Like not, not so
much a big one as it will take a toll on my grade but, God knows I
don‟t have to know what curvature is to get an A on the exam even if
the exam is entirely about curvature, is the sad part. Like, you know, I
don‟t have to understand what I‟m doing to do well; yeah, um, but that
doesn‟t change the fact that I don‟t know what curvature is” (3rd
college interview)17
It is interesting that Sade notes that she can perform well on the exam without
understanding the topic. Students can often use key words in problems to determine
the procedures to be used. Thus, they can apply the procedural rules flawlessly to
particular problems and actually perform at a level that far surpasses their conceptual
understanding of the problem (Hiebert & Lefevre, 1986). Unfortunately, as the
17
Sade eventually came to understand curvature in her Calc 4 course her sophomore year when her
teacher reviewed the concepts taught in prior courses.
86
National Research Council (2001) has noted, “When skills are learned without
understanding, they are learned as isolated bits of knowledge. Learning new topics
then becomes harder since there is no network of previously learned concepts and
skills to link a new topic to . . .” (pp.123). Sade eventually came to this same
conclusion:
“um I, you know, came to understand it at a certain, in a certain
form and it was incompatible to the way like he, you know, had it on
the midterm or whatnot or somewhere I got lost in the translation”
(Sade, 3rd
college interview)
She had learned a particular form that related to the concept (Hiebert, et al., 1997) and
could answer questions that fit that format, but when faced with problems that
deviated from that format, she was at a lost as to what to do. With more insight into
the underlying concepts, she may have been able to manipulate the procedures
appropriately to fit the requirements of the problem but her lack of conceptual
understanding left her unable to address questions.
Thus far, therefore, we have seen how, in some instances, procedural
understanding can allow the learner to gain clarity regarding the connections between
concepts they understand individually. These connections, in turn, allow the learner to
work more flexibly with the given procedures, adapting them in appropriate ways to
address novel problems. If the learner does not fully understand the individual
concepts, however, finding the link via the procedures used can be quite difficult, if
not impossible. The learner then comes to rely upon the procedure for solving
problems in that domain and may find themselves at a lost should the problems vary
from the particular format they are accustomed to. Two approaches being employed
in some mathematics classrooms in an effort to assist learners in understanding the
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concepts and making the necessary connections are manipulatives and collaborative
learning. As we shall see in the next sections, however, neither are foolproof
solutions.
Connections between mathematical principle(s) and manipulatives
“Competent learning and use of mathematics – whether in the context
of algebraic, geometric, arithmetic, or probabilistic questions or
problems – depend on the way in which people approach, think about,
and work with mathematical tools and ideas. Further, we hypothesize
that these practices are not, for the most part, explicitly addressed in
schools. Hence, whether people somehow acquire these practices is
part of what differentiates those who are successful with mathematics
from those who are not.” (Ball, 2003, pp. 32-33)
Due to the amount of manipulatives and/or concrete objects used in the teaching of
elementary mathematics, other researchers (see: Ball, 1992; Driscoll, 1981; Hiebert,
1984; Ma, 1999) have already raised concerns about how manipulatives are used in
school and to what end. Ball (1992) asserts that manipulatives “do not necessarily
change the basic orientation to mathematical knowledge and to what counts as worth
knowing. They do not necessarily provide students with conceptual understandings”
(p.47). Rather, because those who already understand conventional mathematics
“see” the concepts that they already understand being demonstrated in the
manipulatives, there is a tendency to overstate the usefulness of these materials (Ball,
1992). For manipulatives to be truly useful, the learner will need to recognize the
connection between the manipulatives and the mathematical principles they represent
(Ball, 1992; Ma, 1999).
Brittney‟s recognition of the connection between math ideas was not limited to
what was discussed in the classroom. Due to her earlier exposure to calculus and the
graphing calculator in high school, she also began making connections between the
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calculus concepts and the manipulatives (namely, the graphing calculator) she had
learnt to use:
“Like I, I was still a little uncomfortable with my calculator and then
like round midway and I was like, hold on. I remember some things
from 12th
grade. Hold up. I know I know how to do that on the
calculator. I was fooling around with buttons and I stopped and I was
like, “Yes. I bet that‟s how you do it.” And I fiddled with it and I‟m
like “Oh, I got the same answer. Okay. Now wait. Why? Oh, I see”
and so like after, I‟ve made connections myself. Like he hasn‟t taught
us how to do derivatives on the calculator or how to solve for zero on
the calculator. I knew that from my Calculus class in 12th
grade. I
didn‟t know why that worked like that but I knew it” (Brittney, 3rd
college interview).
Here Brittney speaks of having learnt in her high school calculus course how to use
her calculator to “do derivatives” and “solve for zero” without understanding the
concepts involved – rather, she knew that if she pushed certain buttons she could get
the solution to particular problems. Later, however, in her college calculus course, as
she began to understand the concept of derivatives, she realized that the work she was
doing was connected to the button-pushing activities of her high school days.
It was no small achievement for Brittney to link what her calculator was doing
to the mathematics she was learning. As Ball (1992) notes, there is a lot to attend to
with the use of manipulatives and learners “may well see and do other things” (p. 17)
than the intended with the manipulatives. Ball (1992) gives the example of Jerome
and his struggle to understand what he should attend to when he used fraction bars to
answer which is bigger – four fourths or four eighths; three thirds or five fifths. He
could „see‟ using the fraction bar the correct solution but did that mean that it was the
number of pieces shaded, the size of the shaded pieces, the amount of the bar that was
shaded or even the length of the bar that determined the correct solution? Ma (1999)
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takes this further, noting that when teachers do not have a clear understanding of the
mathematical ideas included in a topic they often use the manipulatives merely to
elucidate the procedural steps rather than the conceptual underpinnings. She uses the
example of teaching subtraction with regrouping where a teacher used manipulatives
(in this case, popsicle sticks) to demonstrate the procedure of borrowing without any
reference to the underlying concepts of place value and equivalency versus another
teacher who used the same manipulative to target the underlying concept of place
value where 5 tens 3 ones is equivalent to 4 tens 13 ones. Though less manipulatives
are used in high school and college, as Brittney‟s story highlights, the same problem
can result with the tools that are used whether it be the geometer‟s sketchpad in
geometry, balance scales in algebra, calculators and/or computer programs in calculus.
Consider Josh‟s response to whether there were ever times that he found he didn‟t
understand what a particular section was about or how to do a particular problem:
The only time that has happened is on math excel because math excel
being a computer, you have to put it in exactly how it wants it and it‟s
kind of stupid sometimes (Josh, 1st college interview).
Fortunately, Josh realized that the programs‟ refusal to accept his solution could be an
equivalency issue rather than the result of an erroneous solution. As Ball (1992)
points out, however, realizing that the two solutions can and should be congruent is
something that has to be learnt. Consider how often students have marked their
solutions as incorrect because it did not match the “back of the book” or how many
times they‟ve asked their teacher if their answer was also correct. Such “mismatches”
can quickly become sources of frustration for students who are primarily focused on
procedural learning.
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Connections between math principles and multiple methods
According to the National Research Council (2001), “A significant indicator of
conceptual understanding is being able to represent mathematical situations in
different ways and knowing how different representations can be useful for different
purposes” (p. 119). Sade‟s story illuminates this beautifully.
Sade viewed her parents‟ involvement in her high school studies as minimal,
but her home environment had clearly helped to foster within her the idea that there
were multiple approaches to solving mathematics problems. Each summer her mother
not only purchased math problem books for her but would occasionally solve some of
the problems herself. Sade vividly recalled, therefore, times when she and her mom
“would arrive at things differently and [they] would argue about it and then realize
that [they] both were right” (Sade, retrospective interview). These early encounters
planted the seed that different approaches could and often did yield the same result in
math. In addition, the stories of her mom‟s own experiences with math in college
reinforced this notion since her mom would occasionally arrive “at answers in a
different fashion from her teachers, [and] would always have to go and explain to her
teachers how she got to it.” “I remember those stories, particularly,” Sade said, “about
how [my mom] argued her way up from a D to an A . . .” (Sade, retrospective
interview). It is little surprise then that, in reflecting upon her high school experience
with math Sade said,
“So like, as far as like math, I can‟t pinpoint a time when I‟ve been
extremely lost in math at Northern Academy but there were times like
you know where we would go through a question or something and I‟d,
I‟d be like, I‟d look at it and I‟d be questioning what went on and I‟d
have to have that explained, what, what we did explained and I‟d go to
the teacher and I‟d say I don‟t understand. I‟m gonna have to say most
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of the time it was more of a I didn‟t, I would approach it a different
way than him; I don‟t entirely comprehend the way he approached it, so
I can‟t say that there was ever a time at Northern Academy where I full
out didn‟t understand” (Sade, 1st college interview).
Notice here that Sade recognized that her different approach did not mean that she did
not understand the problem. This should not be taken for granted because many
students come to view mathematics as having one path that leads to a correct result
rather than multiple paths that can produce the same and/or equivalent results. In
classrooms where only one way, usually the teacher‟s way, is shown to be the correct
approach this can prove to be a tremendous hurdle for students.
Fortunately, although much of Sade‟s high school math experience did not
reflect an attitude of multiplicity of approaches in solving problems, her math classes
in grades eleven and twelve at Northern Academy encouraged it.
“. . .me and Sasha [a classmate] were always more creative about our
approach and we had like certain ways of thinking about math
problems that meant that we did things a little bit differently and so me
and Sasha would always be at it about how we did our math problems
and [our teacher] just loved it. Like you know at some point most
teachers would stop quarrels but she‟d just be like „Hmmmhmmm,
Hmmmhmmm, Hmmmhmmm.‟ I think she was just loving it” (Sade,
retrospective interview).
Here we see Sade both accepting and valuing the differentness in the way she
approaches mathematics. Brittney also eventually came to this point. Very early in
her second semester, Brittney recognized the expectations of her calculus teacher but
this did not prevent her from continuing to approach problems in the manner that came
most naturally to her. Rather, the expectations of her teacher served as checkpoints
for her along the way – forcing her to pause in her process to convert solutions into the
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expected format and thereby allowing her to double-check the accuracy of her thought
process.
“. . . you know when they give you a question and you know that they
want you to do it a certain way but you know like three other ways to
do it so like you‟ll do it that way and then double check it with those
three other ways or, you know, there‟s an easier way to do it like . . .
I‟ll take the easy way. I‟m not gonna‟ cause you know he didn‟t
require, he didn‟t say use this . . . and then I‟d double check it with my
two or three other ways . . .” (Brittney, retrospective interview)
Here we see that Brittney‟s recognition of the connection between mathematical ideas has
led her to the realization that there are often multiple ways of doing the same problem
which, in turn, ultimately allowed Brittney to become less dependent upon the teacher to
show her how to find a solution to a problem and more apt to attempt to figure out this
process for herself.
“I was like „okay, I know I can use that other method and it‟ll do the
same thing cause it‟s related to this thing in that certain way, you know,
and so you figured out things like that” (Brittney, retrospective
interview).
Both Sade and Brittney have hit upon an important aspect of learning
mathematics. As Ma (1999) notes “Mathematicians use and value different
approaches to solving problems” (p.111). This is, in part, because different
approaches allow different views into the same problem, foregrounding one part of the
problem and illuminating a particular aspect of a principle while backgrounding
others. As we saw in the excerpts above, it provides the learner with choices
regarding the appropriate method to use to solve a given problem (Rittle-Johnson,
Siegler, & Alibali, 2001). For this reason, collaboration among learners is often a
beneficial activity because it promotes and encourages recognition of multiple
approaches and a deeper understanding of particular approaches (Boaler, 2002b;
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Kazemi, 1998). We‟ve seen already that Sade found it quite beneficial to debate the
appropriateness of her solution with a classmate in eleventh and twelfth grades.
Similarly, at the end of her freshman year, she was invited to join a study group to
prepare for her final exams. Though she would never have sought out this group on
her own, she found she benefitted tremendously from her interaction with the other
two students:
“. . . we complement each other in like what, the things we know and
the way we like to study we can like bounce off each other. We all can
force each other to do different things kind of thing” (Sade,
retrospective interview).
Summary
The emphasis of this chapter has been on the importance of connections in
developing conceptual understanding – connections between concepts and procedures,
and vice versa; connections between the different concepts; connections between
concepts and the manipulatives or tools used to solve mathematical problems; and
connections between the different approaches and what they highlight about the
various concepts. As Brittney‟s and Sade‟s cases have demonstrated establishing
these connections helps unify mathematics and make learning new material much
easier (National Research Council, 2001). Not establishing these connections can
lead a learner to rely strictly on the procedures to solve problems and unable to adapt
those procedures to solve new problems (Rittle-Johnson, Siegler, & Alibali, 2001).
Such learners are more likely to struggle with assimilating new information or
adapting their current knowledge to novel situations. A model often used by teachers
to assist students in making connections is that of manipulatives. It is important,
however, to keep the focus on making connections to the concepts when manipulatives
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are used lest they become one of the many procedures students often feel they must
learn. Done properly, manipulatives not only provide insight into the concepts but
may also be used to promote an understanding of the varying approaches that can be
applied to the same problem Finally, Brittney‟s and Sade‟s cases highlight the value
that can be derived from the awareness of multiple approaches to solving problems.
By emphasizing one aspect of a problem and connecting it to its underlying principles,
multiple approaches allow learners to identify the links between the various
mathematical principles imbedded within the problem.
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CHAPTER 5
CONFIDENCE, AGENCY AND AUTHORITY
“Understanding breeds confidence and engagement; not understanding leads
to disillusionment and disengagement” (Hiebert, et al., 1997). In chapter 4, we
discussed the types of connections that Brittney and Sade found crucial for
understanding mathematics. In this chapter, we shall look at how those connections
inspired confidence, sparked changes in their levels of agency and in who/what they
viewed as an authority in mathematics. As Schoenfeld (1992) points out, the sharp
line of demarcation that once existed between the cognitive and affective domains has
been blurred. It is now becoming clear that what happens in one domain impacts the
other domain in significant ways. Brittney‟s case, in particular, demonstrates this,
highlighting something which has often been difficult to prove – that students cling to
procedures when they lack confidence but, as they become more confident in math,
they begin to explore and go deeper into the subject (Boaler, personal communication,
November 27, 2009). I begin then, with a discussion on developing confidence in
mathematics.
Mathematical Dispositions
“I actually enjoy math because I‟m confident in it and understand it and
like I seem to look forward to going to math class this semester.”
(Brittney, retrospective interview)
“Students‟ disposition toward mathematics is a major factor in determining
their educational success” (National Research Council, 2004, p.132). In his study
regarding attitude towards mathematics, Hannula (2002) developed a framework to
describe what he identified as four different aspects of attitude: 1) a student‟s
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reaction, whether conscious or unconscious, to a mathematical activity he/she is
engaged in (called an emotion); 2) their emotional disposition, that is, the automatic
associations students have to the thought of mathematics due to previous experiences
with the subject; 3) what a student expects as a consequence of doing mathematics;
and 4) the role of mathematics in a students‟ larger goal structure.
Despite being in the accelerated mathematics class, Brittney graduated from
high school feeling that she was not very good at mathematics. As her mathematics
course workload increased in grades 10 through 12, her confidence and enjoyment of
mathematics decreased. In fact, she repeatedly described her 12th
grade math course
as a time where she “didn‟t know what on earth was going on” (Brittney, 3rd
college
interview). The accelerated program felt rushed to her. There were some topics, for
example, functions, that she did not recall having learnt, and others, for example,
vectors, that she simply did not understand. However, she was always careful to point
out that this was her experience and that there were many other students in her class,
some who seemed able to teach the class with the teacher, for whom the pace of the
course was quite appropriate.
It wasn‟t, it wasn‟t anything about the program or the school because
some people excelled in math and they were fine. It was just me and
my personal experiences with math” (Brittney, 3rd
college interview)
Here we see Brittney beginning to internalize her problems with mathematics. Using
Hannula‟s framework, Brittney‟s emotional disposition towards the subject is negative
and she does not have positive expectations regarding the outcomes of mathematics
for her.
“[Math] “just didn‟t click to me” (Brittney, 2nd
college interview);
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“ . . . I was the worst person when it came to math. Really, I really
was. Like, I hated math. I just, it was horrible” (Brittney, retrospective
interview)
In fact, Brittney began to think of math as “one of those things that, for the most part,
you can‟t figure out by yourself . . . either you get it or you don‟t. And if you don‟t
get it, you need help” (Brittney, 1st college interview). Thus, by the time Brittney
completed high school, she did not view herself as someone who could make sense of
mathematics on her own but instead relied primarily upon memorizing to cope
(Schoenfeld, 1992). Nonetheless, she was adamant in claiming that she had benefitted
from exposure to the accelerated program in mathematics. She felt it had helped her
fine-tune her reasoning skills, particularly in the areas of critical thinking and
contemplation before writing. It “kinda changed the way I thought about math”
(Brittney, 3rd
college interview), she said, encouraging her to think more deeply about
the mathematical concepts themselves rather than on a set format for solving a given
problem.
These then were some of the attributes that Brittney brought with her to college
– a fearfulness of mathematics borne from her previous struggles with the subject,
some awareness that beneath all the procedures she had struggled to master there
existed mathematical concepts, and an acceptance that she did not “get” mathematics
but, with extra diligence in practicing mathematical problems, she might overcome
this handicap. It is important to note here, however, that despite Brittney‟s negative
feelings regarding mathematics, she was still willing to work hard to be successful in
the subject. Given her interest in neurobiology, this is understandable as mathematics
was necessary for achieving her goal (Hannula, 2002).
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Brittney chose to give mathematics another chance, therefore, in college. She
determined she would have a positive attitude toward mathematics and thus try to
change her emotional reaction to mathematics when she was engaged in the subject
(Hannula, 2002). As was noted in Chapter 4 however, having a positive attitude alone
will not change one‟s disposition toward mathematics (National Research Council,
2004). Fortunately, the teachers Brittney encountered in college focused her attention
on the connections in mathematics, and once Brittney began to recognize these
connections she began making sense of mathematics for herself and experiencing the
rewards of sense making and perseverance (National Research Council, 2001).
Brittney was then able to use the critical thinking skills she had begun to develop in
high school more effectively (Herrenkohl & Wertsch, 1999 in Boaler, 2002b) and, in
so doing, began the shift from being performance –oriented, where the focus is on
gaining favorable judgment of her competence, to being learning-oriented, where the
focus is on increasing one‟s understanding, competence and mastery of subject
(Dweck, 1986). To see this shift, one need only consider Brittney‟s response to the
question “What is your greatest joy in this math class?” At the beginning of the
school year, her reply was “My first exam score” (Brittney, 1st college interview) but,
by April, her response to this same question was: “Understanding it. Actually being
able to be like, oh, I, I get this” (Brittney, 3rd
college interview).
For Brittney, therefore, the process of developing a more productive
disposition towards mathematics had been a journey. She had moved from fearing
that she would not succeed in mathematics (performance-orientation) to coming to see
herself as “an effective learner and doer of mathematics” (National Research Council,
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2001, p.131) (learning-orientation); from viewing mathematics as a series of arbitrary
steps that she was required to learn to seeing it as a subject that was not only
understandable, but that she was capable of figuring out:
“Um, well in, cause the first semester like my major concern was I‟m
not gonna understand this. Oh, my god, I‟m really scared. I‟m trying
to have a new outlook and I‟m just trying to do this and, you know, I‟m
gonna try and do it well. So at that point my main concern was, “Don‟t
fail math. You cannot fail math.” . . . Now, it‟s like, now that I have
some confidence in math I can actually take some time to try and think,
“Okay, I see how this works now, like maybe, so why does that
happen?” you know and so it‟s kinda like I have a lot more confidence
in myself so my major concern isn‟t, “Oh crap. I have to know how to
do this. I have to sit down and learn like exactly how to do this.” It‟s
“Calm down, Brittney. You know you know how to do this. How
does this work? Think about it” (Brittney, 3rd
college interview, italics
added).
In this excerpt Brittney speaks first of her negative emotional disposition towards
mathematics that was brought about by years of feeling that, despite her successes due
to her ability to memorize, she nonetheless did not actually understand the processes
she was executing. Though she determined to adopt a positive attitude, her
expectations still were that she might not succeed in mathematics. Brittney‟s growing
recognition, however, of the connections that exist in mathematics and her developing
confidence that she could discover those connections herself freed her from focusing
solely on the step-by-step details for solving problems and allowed her to step back
and look at the bigger picture. She then began asking some interesting questions: why
does that happen? How does this work? These are questions that caused her to reflect
on what she had done and determine the connections between the concepts and
procedures she had just executed. Such reflection, it is theorized, on why the
procedure works serves to strengthen existing conceptual understanding and enables
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students to learn more than those who fail to pause and reflect (National Research
Council, 2001; Rittle-Johnson, Siegler, & Alibali, 2001; Wagner, 2007) .
For Brittney, making connections and reflecting upon what she was learning
prompted a change in her relationship towards mathematics. She gained confidence
and this confidence allowed her to delve deeper in to the subject and “move to another
level of understanding” (National Research Council, 2001, p.119). In the past, when
Brittney encountered problems in mathematics that she could not readily solve, she
would give up. This was in keeping with her performance-orientation where, if one‟s
confidence was not high (as had initially been the case for Brittney) then one was not
likely to persevere when faced with challenges (Dweck, 1986). Now, however,
because of her recognition of the connections in mathematics that resulted in a shift to
a learning-orientation, she was willing to seek out solutions for herself:
“. . . it has changed my opin, my, my attitude a little because I would be
willing to be like „I‟m gonna go get a book. I‟m gonna look at it and
I‟m gonna figure it out myself because it‟s gonna drive me crazy that I
cannot do it. Before I‟d be like „I don‟t know how to do it and I don‟t
care. I don‟t need it. I don‟t wanna know it. I give up,‟ you know. But
now, I‟d be more willing to say, let me go and try to do it, you know”
(Brittney, retrospective interview).
Here we see Brittney doing as Bruner (1960/1977) suggests. She is learning to master
the fundamentals of mathematics by not only developing an understanding of the
concepts but by also developing “ an attitude towards learning and inquiry, toward
guessing and hunches, toward the possibility of solving problems on one‟s own”
(p.20 quoted in Ma, 1999, p.24). She is beginning to adopt some of the social
practices and habits of mind that mathematicians themselves employ (Pickering 1995)
and has thus begun the journey from the periphery, from being an observer of what it
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means to study mathematics, to becoming a more central participant (Lave & Wenger,
1991; Nasir & Hand, 2006).
The changes noted in Brittney concern the ways in which she approaches
solving problems in mathematics and cannot be defined simply by her knowledge of
math or by her beliefs about mathematics or by some general notion of learning
identity (Boaler, 2002). Rather, Boaler (2002) describes this change as the
development of a “disciplinary relationship” with mathematics because Brittney has
not only begun to find her own voice and discover her role in what it means to study
mathematics but, as we shall see, she also recognizes that there are times when she
must yield her thoughts to the discipline of the subject. In so doing, she has begun to
engage in what Pickering (1995) refers to as “the dance of agency” (p.116).
Agency
Boaler (2005) defined agency as “the opportunities students received to think
for themselves and make decisions” (p.5). This includes, but is not limited to,
choosing what to engage deeply on, selecting methods/approaches for problem
solving, developing methods, and determining the direction in which to take a problem
(Boaler and Greeno, 2000; Pickering, 1995; Wagner, 2007). Pickering (1995) refers
to this type of agency as human agency. Students who work with human agency will
ask questions of themselves, others, and the text regarding the problems they are
trying to solve (Boaler, 2005). These questions are not unlike those Brittney asked
herself in one of the earlier excerpts: why does that happen? How does this work? In
fact, Brittney‟s determination to “go get a book . . . look at it and . . . figure it out
[her]self” (retrospective interview) is indicative of a learner who has developed a
certain degree of human agency. Brittney links her increase in human agency in
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mathematics to becoming more confident and more sure of herself as a student of
mathematics,
“I think that‟s why I feel more confident because I know, even if I
don‟t know how to do this, I know how to figure out how to do this”
(Brittney, 3rd
college interview).
This confidence comes from having developed an understanding of mathematics
through the connections in mathematics she has begun to realize:
“Cause I understand how things work rather than just oh I know how to
do that, I know why I‟m doing it . . .” (Brittney, 3rd
college interview)
Understanding the logic behind the problems, why you‟re doing what you‟re doing,
and being able to “figure out how to do this” are common characteristics of students
who work with human agency.
Students who work with human agency are also actively engaged in
mathematics (Boaler, 2005). In discussion oriented classrooms this active
engagement may be easier to note (Boaler, 2005). Consider Sade‟s description of
working in such an environment during her eleventh and twelfth grade years:
“[Our teacher] would give us problems and we had like would would
be allowed to discuss over them and like, you know, a lot of like, a lot
of us working through things, a lot of us doing things. We weren‟t just
like taking notes from the board. As a matter of fact, we rarely ever
took notes. It was almost everything we learnt, we learnt from trying
to do it.” (Sade, 3rd
college interview)
When focused on learning mathematics, discussion-oriented classrooms often
encourage the development of human agency in students (Boaler, 2002b). This is not
to say, however, that students‟ cannot develop human agency within the more
traditional and less discussion-oriented classrooms. Although it may be more difficult,
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it is still possible. Again, consider Sade, who studied mathematics in more traditional
classrooms in grades seven through ten:
“I‟ve always thought that I had to think for math . . . that‟s one of the
reasons why I like math so much because you had to think for it. It
wasn‟t just doing, doing, like whatever they told you to do. You had to
think for it. I guess especially like the types of math that I got into I
suppose” (Sade, 3rd
college interview).
For conceptual advances to occur in mathematics, however, Pickering (1995)
argues that there must be an interchange between human agency and “disciplinary
agency” (p.29), that is, the established patterns, methods and procedures that have
been accepted as the standard for verification of ideas within the discipline. Two
statements by Brittney capture this concept beautifully,
“. . . in math, it connects and I think that helps me now because I can, I
stop and I‟m like, I know how to do that part of it. Now how can I use
that part of it to figure out that other part of it” (Brittney, 3rd
college
interview).
“But now it‟s like „Wait, I don‟t know exactly how to do this question
but I could figure it out cause I remember how to do that other thing
which works with that other thing and that other thing so I can kind of
like retrace my steps or work backwards . . .” (Brittney, 3rd
college
interview)
Brittney‟s references to knowing “how to do that part of it”, remembering “how to do
that other thing” and retracing her steps or work backwards indicates a reliance upon
proven methods within the discipline of mathematics. Sade described a similar
experience as she explained why understanding the concepts in mathematics was so
important to her:
“I‟ve always been very aware of the fact, even throughout high
school, like I was always very aware of the fact that I needed to
understand something or it‟s better to understand something, like you
know, it even helps with remembering it cause if you can understand it
then you don‟t have to memorize it because even if you‟re not thinking
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about it or you‟re not like you haven‟t drilled it into yourself when it
comes down to the paper you‟d be like, you might not, you know, it
might not be like you know oh that, that equation but you could look at
it, not having studied it sometimes, and just be like, „You know, I don‟t
know what he told us to do about this but it makes sense that I would
do this because the dot product means this and that means that I can do
it like this‟ and it might not even be the way that he said to do it but if
you understood all of the material that he put in front of you, you could
create your own way to do it, you can find, like you know, I just feel as
if you can manage to work around just about everything that kind of
thing” (Sade, retrospective interview).
In this passage, Sade is talking about using what she understands in mathematics to
solve problems she is not familiar with or does not recall how to solve. Notice,
however, that in the example she uses she references a particular terminology, “dot
product,” that has been accepted within the mathematics community to mean
something in particular. Here then is the agency of the discipline – “this means that I
can do it like this” – dot product is associated with a set of standard methods and
procedures that can be applied to the problem.
In all three excerpts above it is also equally important to note that while
referencing the standard methods and procedures that directed their thoughts
(disciplinary agency), Brittney and Sade also refer to using their own ideas to address
the novel situations they had encountered (human agency). Sade “creates her own
way” while Brittney “use[s] that part of it to figure out that other part of it”. In so
doing they each work to adapt and extend the methods of the discipline (Boaler,
2002b; Wagner, 2007) and engage in what Pickering (1995) refers to as the “dance of
agency” (p.116). It is this dance that serves as the nexus of conceptual advances – for
the discipline at large (Pickering, 1995) and the individual person (Boaler 2002, 2003).
The emphasis, therefore, is on human agency and how it works in concert with
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disciplinary agency to afford deeper insights and understandings regarding
mathematics. In discussing critical thinking, Herrenkohl and Wertsch (1999) noted
that students needed to develop not only the skills for critical thinking but also the
disposition to use those skills (in Boaler, 2002b). A similar claim is being made here
– firstly, that it is important for students to develop human agency whereby they are
willing to think and reason for themselves, but also that they, secondly, need to engage
in the “dance of agency” by using the agency of the discipline to their advantage as
they work to extend methods .
Authority
Boaler (2005) described authority as “the place, people or practices students
used to know when they were working correctly” (pp.5-6). Boaler‟s description of
“authority” seems similar to the National Research Council‟s (2001) description of
adaptive reasoning where “answers are right because they follow from some agreed
upon assumptions through a series of logical steps. Students who disagree about a
mathematical answer need not rely on checking with the teacher, collecting opinions
from their classmates or gathering data from outside the classroom. In principle, they
need only check that their reasoning is valid” (p.129). Interestingly, “conceptual
knowledge fulfills this function by playing the role of a validating critic . . . judg[ing]
the reasonableness of the answer; . . . whether the answer “makes sense.” ” (Hiebert
& Lefevre,1986).
Gresalfi and Cobb (2006) point out, however, that the level of authority with
which students work does have implications in regards to how they are “positioned
relative to one another and for the rigor of mathematical content with which they are
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likely to engage” (p.52). Those with low levels of authority tend to believe as Brittney
once did that “you can‟t figure out [math] by yourself” (Brittney, 1st college
interview). Such learners view the authority in mathematics, the keeper of knowledge,
as an external source – usually their teacher or the solution key (Boaler, 2005;
Schoenfeld, 1992). They only know or are confident in their solution once the
external source has indicated that it is a correct solution. Until then, they are unsure
that their approach is valid and will attempt to modify that approach (sometimes via
erroneous methods) if their solution does not look like the solution from the authority
(even when their answer is correct but written in an equivalent form).
Brittney‟s journey to recognizing that she could rely on herself to solve
mathematical problems and determine the accuracy of a solution began in college. In
her first semester, whether doing her math homework or preparing for an examination,
she relied solely on the notes she had taken in class rather than her textbook. Given
her anxiety regarding mathematics and knowing that her assessments were based on
the teacher‟s notes, she preferred to focus on that which had been explained to her and
she understood rather than venturing on her own to decipher the meanings in the
textbook. She also continued to opt for working alone but would attend help sessions
if she had questions; her rationale being that the TAs knew what they were doing
whereas her peers might not fully understand the concepts themselves and could end
up teaching her faulty math. Consequently, when double-checking her homework
assignment with a classmate, if a disagreement arose regarding their solutions,
Brittney always turned to the teacher to resolve the issue. For Brittney, therefore,
mathematical authority at the beginning of her first semester was firmly grounded with
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the teacher or others whom she considered to be experts in the field, and not with
herself. This stood in huge contrast to her behavior in her Biology and Chemistry
courses, courses in which she was completely within her comfort zone. In those
courses, I both observed and Brittney reported that she rarely even took notes,
choosing instead to rely primarily on the textbook and the rich details it provided
when studying for those courses.
As Brittney‟s confidence grew during her first semester, however, she began
making the shift from viewing the teacher as the sole authority in mathematics to
relying more on herself to determine the appropriateness of various methods for
solving a problem. Although the textbook still intimidated her with “pictures and
diagrams all over the place” (Brittney, 2nd
college interview) and she therefore
continued to consult her notes when preparing for tests, there were some subtle
changes by the second semester that indicated that she was beginning to claim some
authority for herself. Unlike first semester where she did all the problems for fear that
she would not understand something, by second semester she began making choices,
much like she routinely did in her science classes, regarding which assignments she
needed to do in order to understand the concept. This marks a subtle shift for Brittney
from viewing all authority in mathematics as lying with the teacher (with herself on
the periphery) to beginning to see herself as an active agent in learning mathematics.
Here again, we see participatory appropriation in action.
Some might question whether this is indeed a subtle shift. Could it be that,
having already achieved success in a college math class, Brittney had simply become
less worried about succeeding and had relaxed on studying? The evidence, however,
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does not support this. Brittney continued to study intensely for her mathematics
course.
“. . . Like, I study, I study his notes; I do my past homework questions;
I do the questions that he gives in the notes and then I sit down, close
my book, I sit down with the review sheet and I treat it like it‟s a test
and I‟m like, “Go”. And then I do it all out and then I check the
answers online and then if I make mistakes, I‟m like, “Okay, why did I
do that? How? What did I mess up? Did I approach it right?” you
know and so that‟s like, the review sheets don‟t help me study. They
just help me when, once I‟m done studying to see where I am and
maybe if I should go back and over, like look at something again, you
know. So that‟s that” (Brittney, 3rd
college interview).
Brittney‟s use of an external source, namely the review sheet, serves as a guide to aid
her on what she may need to study more thoroughly rather than the final authority
regarding what she does or does not know. Notice the questions she asks herself
when the solution key indicates that one of her answers is incorrect – How? What did
I mess up? Did I approach it right? -- These are questions which indicate that
Brittney views herself as the final authority on what is correct and, rather than simply
changing an answer, seeks to determine the source of her errors. This type of scrutiny
also extended to the tools (e.g. graphing calculator) that she may have used to obtain
her solutions:
“ . . . sometimes the calculator does things and I‟m like I know that is
wrong . . . Don‟t put them brackets, or you don‟t you know put that one
little comma or something -- completely wrong answer, throw off must
be like a, b, c and d part of that question, you know. So I think that‟s
like, I and it didn‟t, I didn’t go into Calculus like knowing that. . . so
what I usually do is I do it myself and then I check with the calculator
and I‟m like, yeah, that‟s that same thing. And then even if the
calculator shoots out something that is slightly different, I‟m like
“Wait, but that can be rewritten as that. So that‟s still right.” And then
I‟ll just go back and I‟ll double check it and I‟ll say factor that answer
and it‟ll come out with what I got or expand that answer, you see”
(Brittney, 3rd
college interview, italics added).
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There are three things to note in this excerpt. Firstly, Brittney realizes that it is
possible to get an incorrect solution from the calculator and, secondly, she had not
known this before. For many students the calculator is viewed as an infallible tool. If
their solution does not match that of the calculator, they will often change their answer
to that of the calculator because of their faith in the calculator‟s accuracy. In fact, had
this occurred in high school, Brittney would likely have given up in frustration
because of how compartmentalized her mathematical facts were. Now, however, she
recognizes that even in using a tool like the calculator, she must use her disciplinary
agency -- what she knows of the mathematical domain she is addressing in order to
determine the accuracy of the solution the calculator presents her. Interestingly,
however, as Brittney also notes in this comment, sometimes disciplinary agency was
insufficient in determining the accuracy of her results. This brings me to the third
point. At times it was also necessary for Brittney to invoke her human agency (“Wait,
but that can be rewritten as that”) and engage in the “dance of agency” (“factor that
answer . . . or expand that answer”) in order to resolve the differences between her
solution and the calculator. Here we see, therefore, authority and agency working
hand in hand (Gresalfi & Cobb, 2006). As Brittney gains more human agency and is
able to engage in the “dance of agency” more frequently, she is able to rely more on
herself to determine the accuracy of her responses – even when using additional tools
to aid her in this endeavor.
Unfortunately, it may be that the inverse is also true. As students lose the
ability to engage in the “dance of agency”, the more likely they are to rely on external
sources as the voice of authority. Consider what happened to Sade. While she began
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college confident in her math skills and accustomed to engaging in the “dance of
agency”, when she encountered difficulties making connections in her second
trimester, the level of authority with which she worked decreased.
“The only thing is like I don‟t know if, if math help I don‟t know if you
can go and say I don‟t understand the general concepts and like have
them explain. I think that you have to come with homework problems
and I got all the homework problems. Like I, I, I did what I was
supposed to do because like, unfortunately for this class, you don‟t
have to understand to get things right. Um, so like I, I did it and then
you know when I compared my answers like with the answers that had
the answers in the back of the book like with, with what I had like
okay, like I got it right but, but like so” (Sade, 2nd
college interview)
Notice here that Sade relied upon the back of the book to determine if her answer was
correct and, even when she discovered that it was, her response was “like so” . She
could not begin to ask the kinds of questions Brittney would ask, the kinds of
questions she had been accustomed to asking – so why does that happen? How does
this work? -- questions of agency and authority, because she did not understand the
concepts. For someone who has a learning-goal orientation, where the focus is on
increasing one‟s understanding, competence and mastery of subject (Dweck, 1986),
this can be a most unsatisfying and frustrating experience even when the solution is
correct.
Summary
The emphasis of this chapter has been on the role that making connections
played in developing confidence, agency and authority. As Brittney‟s confidence in
her ability to not only do but also understand mathematics began to grow, she clung
less to procedures and memorizing individual facts and focused more on the
connections between concepts, procedures, manipulatives/tools, and the various
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approaches to a given problem. This, in turn, afforded her new avenues for
determining the accuracy of a solution. Thus, Brittney came to rely less on the teacher
as the mathematical authority and more dependent upon herself as the agent for
finding mathematical solutions. We also noted, however, that human agency alone
was not solely responsible for the conceptual advances which Brittney experienced.
Rather, Brittney had to learn to engage in the “dance of agency” (Pickering, 1995,
p.116), to use the standards methods and procedures of the discipline in concert with
her own ideas in order to address novel situations. As she did this, Brittney moved
from the periphery of mathematical sense-making to becoming an active participant.
She began engaging in the practices of mathematicians. Furthermore, it was realized
that engaging in the dance of agency encourages students to look to the discipline,
rather than to external sources, for validation and verification of solution processes,
thus highlighting the interrelations between agency and who is viewed as the source of
authority. Unfortunately, as we saw with Sade, when students are unable to engage in
the dance of agency because they do not understand the concepts and are clinging to
procedures, they may turn to external sources for validation of their performance. In
such a scenario, students may lose the ability to engage as active participants and thus
find themselves on the periphery, attempting but not fully succeeding in mathematical
sense-making. Thus, confidence, agency and authority may be viewed as inter-related
constructs where the increase in one often results in an increase in the other when the
goal is connected understanding.
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CHAPTER 6
DISCUSSION AND CONCLUSION
The initial motivation for this study was simply to explore the mathematical
experiences of Bahamian students in American colleges with the hope of
understanding how well they are prepared for collegiate work and what they must then
do to persevere, if necessary. This exploration was guided by three research
questions:
1) What do Bahamian students‟ pursuing degrees in MSE perceive as
the important secondary-school mathematics experiences that have
shaped their views of mathematics?
2) What experiences do Bahamian students pursuing degrees in MSE
have with mathematics in American universities and colleges?
3) What relationships, if any, exist between the perceived secondary-
school mathematics preparation of Bahamian students pursuing
degrees in MSE and their experiences with mathematics in
American universities and colleges?
In this chapter, I will summarize the findings of this study by considering each
research question in turn. I will then discuss some implications of this research in
regards to pedagogical strategies and teacher moves. The chapter concludes with a
discussion of the future direction research could take to build upon this study.
Addressing the Research Questions
RQ1)What do Bahamian students’ pursuing degrees in MSE perceive
as the important secondary-school mathematics experiences that
have shaped their views of mathematics?
While none of the participants were able to commence their post-secondary studies
with the entry-level mathematics course for their majors, all three found that, for the
courses they had enrolled in, they had been introduced to some of the basic concepts
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in the post-secondary course during their high school years although the post-
secondary courses covered those topics in much more depth. All three students also
felt their post-secondary placements were accurate and none were worried about
completion of their degrees within the regularly allotted time frame. However, two of
the participants were displeased to find that the Bahamian mathematics curriculum had
not enabled them to begin their studies at the collegiate level for math especially since
they had excelled in their Bahamian mathematics program, and all three felt the topics
covered in their pre-collegiate studies should be included in their high school
curriculums. The third participant also showed some concern for how his
mathematical background would impact his ability to study the more advance topics in
physics during his sophomore year.
Two of the participants had not found their high school mathematics program
to be very challenging. Consequently, because they were usually able to master the
techniques and understand the concepts as they were presented, they did not find it
necessary to study for mathematics assessments. As the cognitive demand increased
in their postsecondary mathematics courses, however, they found it essential to
develop study techniques and, by the end of their freshman year, were still in the
process of perfecting those techniques. The third participant was able to perform the
mathematical procedures she was taught in high school but found that she struggled to
understand the concepts and thus mathematics became increasingly more challenging
for her during the accelerated years. To cope, she had had to develop a method for
studying mathematics which she continued to employ during her collegiate years.
Consequently, the student who felt she understood the least amount of mathematics
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when she graduated from high school had developed at least one habit of mind –
studying – that held her in good stead in college.
There were other habits of mind which at least two of the participants found
particularly relevant to their study of mathematics. The most significant of these was
the realization of the role of connections in developing conceptual understanding
which brings us, therefore, to research question 2:
RQ2) What experiences do Bahamian students pursuing degrees in
MSE have with mathematics in American universities and
colleges?
Arising out of the students‟ experiences, we discussed four main types of connections
that the participants found to be essential for understanding mathematics. They were:
Connections between ideas and procedures
Connections between ideas and the manipulatives used
Connections between ideas and multiple approaches and between one
approach and another
Connections between ideas and ideas
The emphasis here is on connections. As Guiterrez (1996) points out,
"We should not be satisfied with students reaching higher levels of
mathematics if their ultimate learning is compromised (watering down
the curriculum) or if students are merely going through the motions to
graduate. Ideal 'advancement' in mathematics means more than just
moving students through the curriculum. It means increasing students'
conceptual understanding of mathematics." (p.524)
To perform well in mathematics, both the conceptual and procedural learner must
develop some understanding of the mathematical ideas and procedures, how to use the
manipulatives and/or tools, and the multiple methods for solving a problem. What this
research showed, however, is that the difference between the procedural learner who
knows how to perform technique to solve routine problems and the conceptual learner
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who can also manipulate formulas to solve novel problems often lies in the
connections the conceptual learner has made between those ideas, procedures,
manipulatives and/or tools, and multiple methods.
Consider, for instance, a problem such as (x + 2)(x + 3). Both the procedural
learner and conceptual learner may have learnt to solve such problems using two
slightly different approaches – the distributive method = x(x+2) + 3(x + 2) and what is
commonly referred to as the “FOIL” method = xx + 3x + 2x + 2(3). The procedural
learner, however, views these approaches as two distinct methods that result in the
same solution since the starting points – what he considers when initially employing
the two methods – are different. The conceptual learner, however, realizes that the
“FOIL” method can, in fact, be derived from the distribution method. For the
procedural learner, “multiple methods” is reduced to different procedures to be
memorized whereas the conceptual learner has identified a link between the two
methods that allows him/her to move more flexibly between those methods and adapt
them, when necessary, to address new problems.
This adapting that often occurs with conceptual learning led to a discussion of
some of the characteristic changes that may be noted in students as they begin to
recognize and make use of the connections in mathematics. This brings us then to
the third research question:
RQ3) What relationships, if any, exist between the perceived
secondary-school mathematics preparation of Bahamian students
pursuing degrees in MSE and their experiences with mathematics
in American universities and colleges?
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The three changes in the participants that were most obvious in this research were the
students' level of confidence, agency, and a reliance upon self as the source of
authority for determining the accuracy of one‟s work. These three constructs were
shown to be interrelated in that development along one trajectory appears to lead to
development along the other two trajectories whereas a decrease in one often leads to
a decrease in the others. Brittney and Sade‟s stories demonstrated both the importance
of and interrelatedness of these constructs. In high school, Brittney‟s focus was not on
making connections in mathematics and, as the cognitive challenge increased in her
accelerated courses, she began to view mathematics as uninteresting and unappealing.
She began college, therefore, low in confidence, agency and authority. However, as
she began to make connections in mathematics in college, she became more confident
and thus more willing to think for herself, try new ideas and develop plans for solving
problems. Thus, her increase in confidence led to her demonstrating more human
agency which, when used in concert with the agency of the discipline, where the
principles and procedures of the discipline are used as a guide to direct one‟s thinking,
allowed her to experience conceptual advances in her work.
Sade, on the other hand, graduated high school with high levels of confidence,
agency and authority in mathematics. She enjoyed discussing her solutions to
mathematics problems with others and identifying the differences in their approaches.
Despite her struggles with some of the mathematical content in the higher level IB
World program her first year, perhaps due to an easier second year in the standard
level course, she maintained her view of herself as a very capable mathematics student
and thus enjoyed the challenges of her first semester calculus II college course. In that
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course, she focused on understanding both the individual concepts and their role in the
larger world of calculus. She struggled, however, to maintain this objective during her
second semester of calculus. Although she felt she understood the individual
concepts, she did not understand how they related to each other and this lack of
connections led to her not being able to manipulate the formulas to solve novel
problems. Thus she was less able to engage in the dance of agency and began to rely
on the textbook both to show her how to solve problems and to confirm if her
solutions were correct. Consequently, her level of authority, agency and confidence
regarding what she understood and could do in mathematics decreased.
Implications
Boaler and Greeno (2000) acknowledged that while their study illuminated
how discursive teaching enabled students to gain deeper insights into mathematics and
a more connected understanding, it did not preclude the same type of learning from
occurring in non-discursive classrooms. In fact, Boaler and Greeno (2000) noted that
the conceptual framework they used of figured worlds would need to be extended to
“accommodate examples of engaged conceptual knowing that is only weakly
supported by discourse interactions in the individual‟s immediate learning community.
We believe,” they go on to say, “that such an extension could be quite important for
mathematics education. It could involve hypothesizing a form of connected knowing
that emphasizes the knower‟s being connected with the contents of a subject-matter
domain.” (p.191) This is what I believe this study does well for, as we saw in Chapter
5, whether in traditional or reform classrooms, the students who develop conceptual
understanding of mathematics have learned to seek the connections between concepts,
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methods, manipulatives, etc. They are engaged in sense-making even when
performing procedures.
Additionally, while the collaborative approach usually emphasizes thinking
and exploration and thereby often provides various moments for students to develop
agency in mathematics, what we have seen in this study is that it is also possible to
develop human agency and engage in the dance of agency while in less collaborative
and/or discursive classrooms. For Brittney, as her confidence developed, strengthened
by the connections she was able to make in mathematics, she found herself engaging
in the dance of agency.
So what then can we say about connected understanding? Because the learner
is constantly seeking connections, connected understanding encourages thinking and
reasoning and an awareness that multiple approaches not only often exist for solving
problems but are often related in some way. In addition, the flexibility required to
recognize and develop connections between various concepts, enables the learner to
develop the ability to apply their mathematical knowledge to novel problems, thus
improving their strategic competence (National Research Council, 2001). In fact,
connected understanding appears to support well the National Research Council‟s
(2001) five strands of mathematical proficiency, thus extending this framework from
grades K-8 to include the higher levels of mathematics (both secondary and college).
These five interdependent and interwoven strands of mathematical proficiency are:
conceptual understanding – comprehension of mathematical concepts,
operations, and relations
procedural fluency – skill in carrying out procedures flexibly,
accurately, efficiently, and appropriately
strategic competence – ability to formulate, represent, and solve
mathematical problems
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adaptive reasoning – capacity for logical thought, reflection,
explanation, and justification
productive disposition -- habitual inclination to see mathematics as
sensible, useful, and worthwhile, coupled with a belief in diligence and
one‟s own efficacy.
(National Research Council, 2001, p.5)
Brittney‟s decision to work on her disposition towards mathematics coupled
with the environment in which she found herself and the social others to whom she
related, enabled her to realize and appreciate the existence of mathematical concepts.
As her conceptual understanding in turn grew to match that of and strengthen her
procedural fluency, her strategic competence, adaptive reasoning and productive
disposition all improved. What this research adds to this picture, however, is the
complexity of what it means to “understand the mathematical concepts”. As Sade‟s
experience suggests, to develop conceptual understanding, one has to go beyond
understanding each of the mathematical concepts individually and begin drawing the
connections between the various concepts, their procedures, and approaches to solving
problems. It is only through recognizing these connections that one gains the
flexibility to adapt the procedures appropriately in addressing novel problems.
Finally, connected understanding requires active engagement with the
mathematical material one is studying. Unlike the common description of
mathematics where math is associated with “certainty; knowing it, with being able to
get the right answer quickly” (Schoenfeld, 1992, p.68), students who engage in
connected understanding expect to spend time grappling with problems and
developing rather than simply following a set process as they engage in the dance of
agency. Sometimes, the dance requires them to “go get a book” (Brittney,
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retrospective interview) or argue and discuss with others (Sade, retrospective
interview) until they are eventually able to reconcile differences and find solutions.
They engage, therefore, in adaptive reasoning (National Research Council, 2001) as
they explore their ideas and reflect upon their solutions by asking “why” and seeking
to discover “how things worked” (Brittney, 3rd
college interview). In so doing, they
discover a wonderful secret about mathematics and that it is that math is
“. . . amazingly compressible: you may struggle a long time, step by
step, to work through some process or idea from several approaches.
But once you really understand it and have the mental perspective to
see it as a whole, there is often a tremendous mental compression. You
can file it away, recall it quickly and completely when you need it, and
use it as just one step in some other mental process. The insight that
goes with this compression is one of the real joys of mathematics.”
(Thurston, 1990, p.5)
Within this quote, Thurston draws on so many of the features of connected
understanding – the human agency that is required to work through an idea, the use of
several approaches in attempting to solve a problem, and making sufficient
connections to see the forest, not the trees. Like Thurston, it is my contention that
once this occurs, once students are able to make the connections within a particular
realm of mathematics, a tremendous mental compression occurs. It is this mental
compression that affords students the ability to work flexibly with novel problems
because, when they need to recall an idea from the realm in which they made
connections, they are able to “recall it quickly and completely . . . and use it as just
one step” (Thurston, 1990, p.5) in another task. This allows them to use fewer
mental resources to solve the stated problem, and thus makes more resources available
for “planning, observing relations between problems, generating new procedures, and
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reflecting on the problems and the concepts underlying them” (Rittle-Johnson, Siegler,
& Alibali, 2001, p.359).
Teacher Moves
Connected understanding does not depend, however, “solely on individual
cognitive structures” (Nasir & Hand, 2006, p.462). Rather, it is significantly affected
by the “shifting roles and relationships” (Nasir & Hand, 2006, p.462) of teachers and
students working together in a community of practice. The next two sections,
therefore, considers the guided participation the students experienced within the
classroom, highlighting some of the teacher moves which played pivotal roles in either
enabling or constraining the participants from engaging in connected understanding in
the less discursive classrooms in which the participants often found themselves.
Reflection: Day 3 -- Feb. 23rd
, 2006
The thing that impressed me most in this visit is the type of teaching I have
witnessed in Brittney‟s mathematics class. This is the second teacher I have
observed and they both seemed very attuned/sensitive to the needs of their
students – that math may be intimidating for them – and therefore employ
teaching methods that opens up the possibility for the students to enjoy
mathematics. I particularly enjoyed observing this professor – the way he
introduced limits to the students and hearing Brittney discuss in the
interview how he adapted his teaching practices upon feedback from his
class last semester regarding quizzes. I look forward to observing him again
in April.
The first teacher move, which is noted in the field note excerpt above, is in making the
classroom a welcoming environment in which students‟ fears regarding mathematics
were acknowledged. As Brittney noted,
“There‟s nothing worse than being in a class for a subject that you are
terrified of with a terrifying professor. That makes it a whole lot
worst” (Brittney, 2nd
college interview).
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For someone for whom the authority for mathematics had been the teacher, this fear
that the teacher might be unapproachable is understandable. As Gresalfi & Cobb
(2006) note, “a teacher‟s demonstration of respect is integral to the creation of a safe
classroom atmosphere wherein student thinking is the basis for instruction and
mistakes are treated as a learning opportunity” (p.53). Fortunately, Brittney‟s college
professors spoke positively to their students regarding their students‟ ability to
understand mathematics which helped to significantly reduce Brittney‟s anxiety. In
fact, Brittney quoted her Calculus I teacher as saying,
“I know half of you sitting there either hate or are terrified of math. I
am going to change that . . . I understand, you know, math can be very
intimidating . . . I understand it can be very intimidating and very
frustrating at times, so we‟re gonna take it slow. We‟re gonna make it
easy for you guys. It‟s not hard. I promise you that, you know, once
you learn to understand it, you will learn to love it” (Brittney, 2nd
college interview).
In addition to acknowledging Brittney‟s fears, her teacher conveyed an interest in and
love for mathematics and a plan to help his students develop a similar interest in and
love for learning mathematics. Sade captures the importance of these attributes in her
description of the qualities of a good math teacher
“. . . she loved math and wanted us to love math . . . she tried to get us
engaged in math” (Sade, 3rd
college interview),
and her explanation of why she enjoyed her first semester math course
“ . . . .it made you feel good, you know, like you know, that he wanted
to be there teaching made it so, made it feel like the class was
worthwhile” (Sade, 3rd
college interview).
Beyond being enthusiastic regarding their subject and believing their students could be
successful, however, Brittney‟s teachers also employed teaching methods that allowed
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for the possibility that their students would learn to enjoy mathematics (Gresalfi &
Cobb, 2006). Kazemi (1998) notes that in order to “press” students to learn
conceptually, teachers need to: “(1) emphasize students‟ effort, (2) focus on learning
and understanding, (3) support students‟ autonomy, and (4) emphasize reasoning more
than producing correct answers” (p.410). Brittney‟s college algebra teacher was
careful to relate the topics taught to those that the students would have been exposed
to in their previous mathematics courses. This teacher move, which was focused on
learning and understanding, served to enable students to not only recognize which
concepts they were already familiar with but also highlighted the fact that connections
between mathematical concepts was helpful in elucidating a concept.
Brittney‟s college algebra teacher also made use of extra-credit in-class
assignments when a concept might be particularly challenging for the students. As I
observed during my first classroom visit, students were allowed to work with each
other and consult their textbooks, notes or teacher to solve the extra-credit problems.
Such activities supported students‟ autonomy in a variety of ways as some chose to
work collaboratively with their classmates while others worked alone; several sought
the teacher‟s guidance on particular problems while a few occasionally sought help
from a seatmate. It also afforded students some opportunity to engage in the dance of
agency and to realize that authority in mathematics was not the sole domain of the
teacher. In particular, it allowed Brittney, who was able to complete the extra-credit
assignment without assistance, the opportunity to realize that she could be an authority
in mathematics as she provided assistance to the young lady seated next to her
whenever the young lady sought her help.
124
In Brittney‟s calculus course, structurally, the class was designed to meet more
frequently than the College Algebra class had met18
but for shorter sessions. Thus
we see at the apprenticeship/institutional level an attempt to meet the needs of the
students by structuring the course to allow for more time to grapple with individual
concepts. In addition, at the beginning of each class, the math idea or concept for the
day was written on the board along with a list of optional problems from the textbook
that could help students gain more clarity on the concept when necessary. Mandatory
homework assignments (see Appendix 6.1) were assigned at the end of each week and
were designed to take less time than the daily assignments. They were not repetitive,
serving instead to highlight the concepts that the students‟ should have mastery of.
There are two things to note in this strategy. First, the daily assignments were
optional – designed to afford students the opportunity to gain practice using the
concepts on varying types of problems. Consider Brittney‟s explanation of the
purpose of these problems:
“About those, how could I forget them [whispers: cause I never do
them.] Um, no I do some of them. I do them when I don‟t understand
what‟s going on. Um, what he does is he lists um homework,
homework problems and he doesn‟t collect those. He just says “This is
what you could do to prepare yourself or to try to understand the
concepts. These will prepare you for the homework assignments on
Friday which will prepare you for the tests. So if you‟re feeling
comfortable with the the homework problems that I give you then
you‟ll be comfortable with the actual homework assignment which
means you‟ll be very comfortable for the test.” So, it‟s kind of just
helping you. Um, I‟ve done about one set of those. It was probably
like the first week in. There was something that I wasn‟t too sure about
so I went and said, let me go and do these problems and try to
understand that and um, you know, they, they helped, they helped”
(Brittney, 2nd
college interview).
18
The College Algebra class met 3 times per week for 1 hour. The Calculus I course met 4 times per
week for 45 minutes.
125
This is an interesting technique for differentiating learning at the college level and
allows for guided participation of varying students at their level of need in a particular
time. For the student focused on procedural learning, the suggested problems could
provide the practice needed to gain mastery of the process for solving the more
elementary problems; for the student seeking conceptual understanding via their
procedural understanding as Brittney did that first week, the suggested problems were
an indication of where to start; and for the student who has already attained conceptual
understanding, the suggested problems could be ignored.
The second point to note was that the mandatory homework assignments were
not repetitive. Consider Brittney‟s only complaint about her first semester course:
“. . . but the thing with that is, what I didn‟t like is, you know in the
book when you get problems they‟re just the same kind of problems, it
was like that. So, you were really bored. You‟re sitting there doing the
same kind of problems for a very long time and then you switch and do
like another five of the same problems, then switch and do another five,
you know. It was just so boring and so long and drawn out. This, he
tries to avoid that. I think that‟s why he does um the page because he
does like one of each kind of problem. So you‟re not sitting there
doing the same thing over and over again. Whereas with the homework
that he says, if you‟re having problems go and do this, that‟s when you
do the same kind of thing over and over” (Brittney, 2nd
college
interview).
Josh made a similar observation in discussing what he valued in his Precalculus II
course:
“. . . [my teacher] does examples that are important, like what‟s going
to let you get the concept. But she doesn‟t really go beyond that unless
you ask. Like she will ask if it was something you didn‟t understand
but she‟s not going to waste time . . . if you don‟t need to do 10 of
these, you won‟t do 10 of these. I sound like her saying that. That‟s
something she would say. But she‟s going to make sure we learn the
concept. . .” (Josh, 2nd
college interview).
126
In both these excerpts, the students are noting the futility of doing the same type of
problem repetitively when one already understands the concept.
When the goal is not connected understanding
I turn your attention now to the ways in which these less discursive classrooms
may have constrained students from making the connections in mathematics.
Interestingly, at some point in their mathematical careers, all three participants
admitted to focusing on, as Sade put it, “good grades . . . but . . . not really learning
that well” (Sade, 3rd
college interview). We see a clear example of this in Josh‟s
comment regarding compound interest where he assessed the value of learning a
particular concept based on the amount of points allotted to the question on the
assessment.
“Never, ever learned it. Didn‟t make any sense. It was only worth 2
points. Why should I learn it . . .? If I had to do it, I could sit there and
work it out . . .” (Josh, retrospective interview).
In this comment, we get a glimpse of Josh‟s agency. Although he had only been
exposed to mathematics in more traditional settings, he remained a high achiever and
confident student of mathematics. Consider his description of his high school math
classes:
“Well I‟ll start with 10 and 11 cause that was completely different from
grade 12. The teacher would introduce the topic, go through it, she‟ll
work an example by herself and let you see her working it and then ask
the class to work one with her and then after she saw that everybody
was kind of okay she‟d assign classwork and sometimes this classwork
could be individual or in peers or in groups . . .
Well, grade 12, because it was a smaller group it was more free. It was
just 11 people in the class that was made for 30 people. We would do
worksheets where she‟d explain a topic and then we‟d do it” (Josh, high
school interview).
127
Then consider his reason for enjoying math:
“I love there being a reason for something ... I love it, being able to
work it and I know it has to happen because of this. I don‟t want it to
be „bout what‟s your interpretation” (Josh, 3rd
college interview)
Here, Josh is identifying himself as someone who loves logic and its step by step
process based on concepts and reasoning. Like Sade and Brittney, he has a profound
appreciation for understanding the underlying principles. Unlike Sade, however, who
learnt from an early age the value of multiple approaches in math, Josh does not view
math and the sciences as subjects that are open to interpretation. As Schoenfeld
(1992) notes,
“These cultural assumptions are shaped by school experience, in which
doing mathematics means following the rules laid down by the teacher;
knowing mathematics means remembering and applying the correct
rule when the teacher asks a question; and mathematical truth is
determined when the answer is ratified by the teacher. Beliefs about
how to do mathematics and what it means to know it in school are
acquired through years of watching, listening, and practicing (Lampert,
in press, p. 5)” (p.68).
Thus, Josh‟s view of math as not being open to interpretation could be because he had
not experienced mathematics taught in this way (Boaler, 2005).
There were other ways in which Josh was affected by the way he had learned
to approach mathematics (Boaler, 2005). Consider the following comments made by
Josh:
“. . . once I see [math] taught, I know it. There‟s no question if I‟m
gonna understand it. Once I see you work through a problem, I know
how to do it.” (Josh, retrospective interview)
“The only thing I think I would change at all is probably pay attention
to that one class where we learned derivatives, cause that was the only
question . . . that I know I didn‟t answer on the BGCSE, one question,
was worth one point . . . I was asleep” (Josh, retrospective interview).
128
Because paying attention and watching someone do a problem are often important
skills to master in the traditional classroom setting (Boaler, 2002b; 2005; Schoenfeld,
1992), it is not surprising to hear Josh make reference to this in the excerpts above. In
addition, Josh appears to view the authority in mathematics as an external source and
he presents himself as one who is reliant on having others show the way and is
performance-goal oriented (Dweck, 1986), a characteristic he continued to display
throughout his freshman year of college.
Brittney, on the other hand, moves from being performance-goal oriented in
high school to learning-goal oriented in college. In high school, due to her frustrations
with not understanding mathematics, Brittney focused on learning how to do the
procedures so she could secure a good grade:
“ . . . I don‟t care how this works; I just wanna be able to do it so I can
get a good grade and that doesn‟t help you in the exam because when
they throw those random questions at you and you‟re like „Crap, how
do I do this?‟ ” (Brittney, 3rd
college interview).
We do not know if Brittney‟s high school teachers attempted to draw her attention to
the connections in mathematics but, whether they did or not, Brittney had not attended
to those messages in high school. The messages she attended to in high school
pertained more to the importance of accelerating for her college career
“You, you‟re the great students. We‟re gonna push you through. If
you choose to do it again that‟s up to you. You don‟t have too. You
know, we‟re just trying to get you guys going, you know” (Brittney, 3rd
college interview).
Thus, Brittney focused on amassing knowledge and making the grade rather than on
understanding the underlying concepts.
129
It is also interesting that, in the excerpt above in which Brittney discusses her
focus on procedures, she makes reference to “random questions” on the test. Josh
made a similar complaint regarding the types of problems that are appropriate for
tests:
“Like if I have time to solve a math problem and time to work it out,
and say no this is not working and what‟s not. But you can‟t put that
on a test. That, that, it doesn‟t make any sense because you‟re going to
waste time. Students are going to waste time on that instead of being
able to complete the test. It‟s not showing what the student knows”
(Josh, 3rd
college interview).
Here again we get a glimpse of Josh‟s agency even as he grapples with his concerns
regarding performance. He is confident that he can solve difficult problems when
allotted the appropriate amount of time to grapple with the problem but does not
consider it appropriate to assess such problems in timed situations. Sade raised a
similar complaint in her second semester course, the course in which she experienced
the least agency, authority and confidence in mathematics:
“. . . but he didn‟t teach, he never taught us that that was a valid way to
answer a question like that.” (Sade, 3rd
college interview)
Although Sade possessed the prerequisite mathematics knowledge and believed she
understood the individual concepts, she struggled during her second semester to gain
connected understanding on her own.
“It‟s like, usually I have my homework there and I‟m like okay, he says
do questions 1 – 7, all of them are in this format, they have a section on
the principles I need for that and an example here, so I‟m going to
follow the format of the example and do all of 1 through 7 like that, and
like it works to get the homeworks, like good grades on the homework
but, you know, at the end of the day I‟m not really learning that well”
(Sade, 3rd
college interview).
Notice Sade‟s reliance on the textbook as the authority. Notice how she yields
completely to the discipline. Notice her complaint – “I‟m not really learning that
130
well.” Sade is struggling to accept an approach to learning mathematics in which the
focus is strictly upon procedures. Her teacher would often do one example after
another on the board, presenting mathematics knowledge to the students. In such
classrooms, authority tends to rest solely with the teacher (Boaler, 2002b) and students
experience few opportunities to engage in the dance of agency (Boaler & Greeno,
2000; Gresalfi & Cobb, 2006). As Sade explained in chapter 4, however, she wanted
to understand the connections between the examples. Consequently, although she
earned an A- in the course, similar to Boaler‟s “small and opportunist sample”
(Boaler, 2002a) and Seymour & Hewitt‟s (1997) study on why students leave the
sciences, Sade found her interest in mathematics waning because of the teaching
approach employed, an approach which, after shopping19
, she believed was more the
norm than the exception within the mathematics department.
“I definitely still feel as if I really enjoy math and really love doing it. I,
I definitely feel as if my drive to pursue math is, is dampered a little bit,
like you know like I‟m not gonna have like take a lot more initiative to
like to grab at math opportunities.” (Sade, retrospective interview)
Sade‟s second semester of calculus was the first time she had struggled to understand
the mathematical concepts primarily because of how they had been taught to her
(Boaler, 2002a).20
On this she commented:
“. . . it‟s such a bad, like just such a bad experience having to teach
yourself the math” (Sade, 3rd
college interview).
This is an interesting comment because it stands in such huge contrast to her
description of mathematics in twelfth grade where “almost everything we learnt, we
19
“Shopping” is the approx. 2 weeks at the beginning of each college semester where students are
allowed to attend classes while in session to assist them in determining which courses they which to
enroll in for the semester. 20
Sade‟s previous struggle in mathematics at the IB world school was primarily due to her lack of
prerequisite knowledge and preparation for class.
131
learnt from trying to do it” (Sade, 3rd
college interview). In the latter scenario,
Sade‟s learning was supported by the teacher moves she encountered in the classroom
whereas in the former scenario the teacher moves she encountered were constraining
her learning. Sade summed it up by noting:
“I guess it was just a lesson on how much the teacher matters” (Sade,
retrospective interview).
Future Directions
The four connections which emerged in this research as being essential to
developing conceptual understanding may not be exhaustive. The list does, however,
give insight into the complex nature of conceptual understanding and, although
Hiebert and Lefevre (1986) caution that “not all knowledge can be usefully described
as either conceptual or procedural” (p.2), the distinction, as they also claim, does
provide insights into students‟ learning that helps us to better understand both
students‟ failures and successes. Parsing out exactly what conceptual understanding is,
therefore, may assist researchers in the difficult task of determining how teachers can
aid students in their quest to make the various connections.
We have discussed briefly in this chapter some of the teacher moves which the
participants in this study either found helpful or not in enabling them to make
connections as they sought conceptual understanding. The work of the teacher,
however, was backgrounded in this study and only surfaced as it related to the
students‟ perspective (the foreground of this study). More research, therefore, on the
ways in which teachers influence what connections students make would certainly be
beneficial to the mathematics education community.
132
In addition, the rapid changes and technological advances occurring in our
world today have placed a high premium on the world market for mathematicians,
scientists and engineers. The participants included within this study were deliberately
chosen, therefore, because of their interest in studying math and/or science related
subjects. These areas of study usually demand a high level of proficiency and
knowledge in mathematics, which generally translates into calculus as the minimum
entry level requirement. It can be argued, however, that with the limited budget of a
small nation like the Bahamas and with the educational objective being to provide
“equality of opportunity through education for all citizens” (Sears, 2005; emphasis
added), it may simply not be feasible to restructure the curriculum to satisfy the needs
of those who require more than the curriculum currently offers. As Sade mentioned in
one of her interviews,
“ . . . I feel as if smart people will always make their way, make or find
a way for themselves, like you know, like at the end of the day, . . .
I‟m, I‟m not saying that‟s like you know, like I said, I‟m not saying that
they should have had to do that kind of thing but at the same time like I
said these are the people that will make, make their way” (Sade,
retrospective interview)
Underlying this statement, however, is a level of privilege unexplored as she also
spoke of boarding schools and changing colleges. Nonetheless, she may have a point.
Thus, a study of successful Bahamian students in mathematics who choose to pursue
non-science majors may provide additional information for the Bahamian government
regarding the effectiveness of the current mathematics curriculum.
Finally, there were two issues not addressed in this study that were present in
the participant‟s experience of mathematics. The first issue was that of tracking. All
three participants experienced tracking in their high school math careers. Josh saw it
133
as affording him the chance to learn at a pace more suited to him. Sade, on the other
hand, while acknowledging that it allowed her to be less bored in class and work at a
more suitable pace, was sensitive to what tracking meant for those students who were
not in the high set.
“ . . . Just, I mean look, I, I, I‟m all for it if you‟re ready for it but I‟m
just, like I said, I‟m always scared at Northern Academy that they‟re
gonna leave all these people who aren‟t as bright behind and like you
know you have a responsibility to them as well, you know . . .” (Sade,
retrospective interview)
Sade was concerned that Northern Academy‟s decision to stream by ability meant that
some students were being unintentionally left behind, taking but not ever passing the
BGCSE mathematics core examination or, in some extreme cases, receiving
exemption from the examination.21
In addition some students seemed to lack
encouragement from their teachers while the students in the advanced courses often
received additional encouragement and privileges. She also worried that those
students who were not in the accelerated courses were inadvertently receiving the
message that they could not do math with their course placement becoming a self-
fulfilling prophecy in terms of what they did achieve.
Brittney, however, is the student for whom tracking probably raised the most
questions. She was what I would describe as borderline – able to excel at math at the
procedural level but struggling to make sense of the conceptual. Was acceleration
good for her? She argued that it was.
“. . . SAT II while it had absolutely nothing to do with the BGCSE it
really tuned your reasoning skills . . .” (Brittney, 3rd
college interview).
21
A satisfactory pass in the BGCSE mathematics core examination is required for entrance into The
College of the Bahamas or to be considered as a potential applicant for many jobs in the Bahamas.
134
Was what she gained really worth the years of frustration and agony she experienced
with mathematics? What if her college experience had been similar to her high school
experience? Would she have been one of the many “second tier” (Tobias, 1990)
students who were capable of doing math and science but, for one reason or another,
had chosen not to?
The second issue was that of race, culture and identity. While this research has
focused upon the mathematical experiences of Bahamian students in American
universities and colleges, it is fair to say that several of the findings might pertain to
students of all walks of life. Yet, there is no denying that some of the students‟
experiences were likely colored by their cultural lens in ways that extended beyond
their level of preparation, development of study habits or even support systems. Josh
and Sade spoke directly to this when they shared their concerns about adjusting to the
university:
“I think this [second] semester I‟ve become a college student. Last
semester it was like still trying out everything. It was like maybe I
should be partying more, maybe I should be studying more, maybe I
should be making more friends, I should be more social, I should be
more reserve, like figuring out where exactly I belong in this whole
mix. Cause it‟s a big world. It‟s like should I hang out with all the
Caucasians in Wonder Bread land, should I hang out with all the Black
people in the boondocks, should I hang out with the Caribbean people
in the middle, should I mix with all the groups?” (Josh, 3rd
college
interview).
“I was originally didn‟t even want to be in a black group at all. . . and
I think they do a lot of great stuff and it‟s great to have that fall back
and to have like people that you can relate to, um to an extent, I guess.
I was going to say culture even like I guess they can relate to each other
and I can just kinda relate a little bit. . .” (Sade, retrospective
interview).
135
Bahamian students as well as others from the Caribbean often grapple with issues of
race, culture and identity as they transition to American institutions and begin to
realize the similarities and differences that exist between themselves, mainstream
America, and their African-American peers (Phelps, Taylor, & Gerard, 2001). This is
clearly a fascinating topic and, while it was coded for in this research, it was not fully
explored. Future research, therefore, would do well to attend to this issue in a more
prominent way within its framework.
136
APPENDICES
Appendix 1.1: Aims of Bahamian National Curriculum
(Ministry of Education, 2006)
137
Appendix 2.1: “High School” Interview Protocol, (August/December 2006)
Thank you for agreeing to let me interview you today. I am very interested in learning
about your experiences in mathematics. Today‟s interview will focus on how you
experienced mathematics in high school.
1. Before we get to that, however, why don‟t you tell me a little about yourself –
where do you live in the Bahamas (or Nassau)? How many siblings do you
have? What high school did you go to?
2. What do you think you will miss most about your high school? What won‟t
you miss?
3. Describe your high school to me.
Prompts: location and proximity to home, government/private,
secondary/senior high (if senior high, where attended junior
high)
structure – school size; class size; demographics of student body
and faculty. Secondary or senior high (if senior high
where attended junior high)
curriculum – streams; courses offered; courses taken
4. Who were your friends in high school? How would other students describe
you and the people you hung out with? How would teachers describe you and
the people you hung out with?
5. What were your math classes like?
Prompts:structure – length of class; same students 7/10 -12tracked;
number of math teachers.
Pedagogy – lecture; groupwork; individual work; question-
asking/teacher assistance
Environment – stressed; relaxed; student talk
Homework -- How much? How challenging? How often did you
work with others?
Tests –Type? How did you prepare for them – by self or with
others?
6. So, how would you describe how you felt about math in high school?
Prompts: level of enjoyment, perceived areas of strength/weakness
7. What was your favorite year of math or most positive math memory from high
school? Why?
8. What was your least favorite year of math or most negative math memory from
high school? Why?
9. When did you take the math BGCSE and what grade did you get? How many
other BGCSEs did you take?
10. What was the objective of your high school math classes? What were they
trying to teach you?
11. What type of student would you describe yourself as?
Prompts: study patterns/ways of working; grades earned; relationships
with teachers
138
College Transition
12. What was your experience applying to colleges?
Prompts: when took SATs? Assistance from high school? how many
colleges applied to? Help from parents and/or other family member?
13. Have you taken any college preparation courses or did any additional academic
work outside of high school?
14. Why did you choose this college to attend?
15. What major are you interested in pursuing and why?
16. How much mathematics do you think may be required for your major and what
are your feelings regarding mathematics?
17. What are your expectations regarding college?
Prompts: areas of anxiety; knowledge of college thru relatives and
friends
139
Appendix 2.2: College Interview #1 Protocol, (October/November 2006)
This protocol will be modified based on data collected prior to these interviews.
Thank you for agreeing to meet with me again.
3. How has this quarter in college been going?
4. What has been the best surprise for you and what have you been enjoying most
in college?
5. What has been the biggest challenge for you?
6. What courses are you taking and how was that course schedule determined?
Probe especially regarding self-selection into pre-calculus, aunt
influence? Probe also for which courses are required for major.
7. Describe your first day in your math class for me.
Prompts:affective description
structure – length of class; number of students (and reaction to it)
Pedagogy – lecture; groupwork; individual work; question-
asking/teacher assistance
Compare to high school math class.
Environment – stressed; relaxed; student talk
(compare to high school math class))
pace of class in comparison to high school math class
8. How has the math class been going for you thus far?
Prompt: level of enjoyment, greatest joy, biggest challenge, comfort
level, What things do you feel you know well in this class?
Prompt: Use textbook/class session to discuss this question.
9. Is there anything you feel you don‟t know well?
Prompt: Use textbook/class session to discuss this question.
10. How do you do your homework assignments and prepare for test?
Prompts: work alone/others (if so, who); use textbook; use notes;
tutoring; see professor
Compare to high school experience – how challenging, worked with
others? Whom turned to for help?
11. What differences exist between your college math and high school math?
What similarities? Would you change anything about your prior math
experiences if you could?
12. What was the objective of this math classes? What are they trying to teach
you?
Compare to high school math classes.
13. Are there other courses that you are taking that require mathematics? If so,
how are those courses going?
Probe for connections to mathematics class.
14. What do you do when you are not studying for classes?
Prompts: involvement in campus organizations, who friends are, work
15. Who do you talk to about your challenges or share your joys/triumphs with?
140
Probe for friends at college, adult relationships at school, parents,
family, h.s. friends etc.
Wrap-up
16. Would you say most of the students in your class are having a similar
experience to you?
17. If you were writing to a younger sister/brother/cousin, somebody back in the
Bahamas who you really cared about and they were preparing to come to the
U.S. to go to college and would take this course, what would you tell them
about your experience?
18. Is there anything else that I haven‟t asked about which you feel characterizes
or may be affecting your experience of transitioning from math home to math
here in the United States?
141
Appendix 2.3: College Interview #2 Protocol, (February 2007)
While most questions are general – to be asked of each student, some questions in this
interview were based on the responses of the previous interview. The italicized
questions, therefore, indicate questions that were intended for a particular
participant, in this case, Sade.
Thank you for agreeing to meet with me again. I‟ll start with a few questions I forgot
to ask you last semester and then going into how this semester has been going for you:
1. Who would you say most influenced your interest in mathematics and
science
-- while in high school, while in IB World, now while in college? 2. What grades did you receive last semester? Were there any surprises and/or
disappointments?
3. How has this semester in college been going?
what have you been enjoying most?
what is turning out to be the biggest challenge for you?
4. What courses are you taking and how was that course schedule determined?
Probe especially for determination of math class.
Probe also for which courses are required for major.
5. Are there other courses that you are taking that require mathematics? If so,
how are those courses going?
Probe for connections to mathematics class.
6. Was last semester your first experience with shopping for classes? Did you
shop for any classes this semester? If so, which ones?
7. Last semester you described your courses as being standard – not anything
you could throw herself into. How would you describe your courses this
semester?
8. Describe your first day in your math class for me.
Prompts:affective description
structure – length of class; number of students
Pedagogy – lecture; groupwork; individual work; question-
asking/teacher assistance
Environment – stressed; relaxed; student talk
9. How has the math class been going for you thus far?
Prompt: level of enjoyment, greatest joy, biggest challenge, comfort
level
10. What things do you feel you know well in this class?
Prompt: Use textbook/class session to discuss this question.
11. Is there anything you feel you don‟t know well?
Prompt: Use textbook/class session to discuss this question.
142
12. Would you say most of the students in your class are having a similar
experience to you?
13. Why are you taking this particular math course?
14. What differences exist between the teaching style of your current professor and
Lauder?
15. How do you do your homework assignments for this math class and prepare
for test?
Are recommended questions provided for studying? Practice exam?
Prompts: work alone/others (if so, who); use textbook; use notes;
tutoring; see professor
16. Last semester you said you’d never gotten stuck to the point where you had to
seek additional help. What about this semester?
17. What study habits or learning strategies that you developed from your high
school days are holding you in good stead now that you are in college? From
IB World? Have there been others you‟ve had to develop?
18. Since you are not taking kickboxing or music this semester, what do you do
when you are not studying for/preparing for classes?
Prompts: involvement in campus organizations, who friends are, work
19. Who do you talk to about your challenges or share your joys/triumphs with?
20. If you were writing to a younger sister/brother/cousin, somebody back in the
Bahamas who you really cared about and they were preparing to come to the
U.S. to go to college and would take this course, what would you tell them
about your experience?
21. Is there anything else that I haven‟t asked about which you feel characterizes
or may be affecting your experience of transitioning from math home to math
here in the United States?
More details from prior interview.
1. What grades did you earn on your BGCSEs? With SATs?
2. Physics is most favorite subject. How come signed up for chem. and math
immediately last semester – not physics?
3. Why chemical engineering for major?
4. When discussing converting from equation format to power series, you
mentioned not having done the proof of it. Did you do much proofs in high
school? If so, what kind.
143
Appendix 2.4: College Interview #3 Protocol, (April 2007)
While most questions are general – to be asked of each student, some questions in this
interview were based on the responses of the previous interview. The italicized
questions, therefore, indicate questions that were intended for a particular
participant, in this case, Brittney.
ASK TO BRING A MATH TEST TO INTERVIEW
Thank you for agreeing to meet with me again.
1. What have you been enjoying most in college thus far?
2. What has been the biggest challenge in college for you thus far?
3. How would you describe your math experience thus far at the university and
what are your feelings about mathematics now?
4. Does your current math class challenge you? If so, in what ways? If not, why
not?
How about last semester?
5. Would you say critical thinking is required in your math courses – this
semester, last semester, high school, BGCSE? Explain.
6. In the last interview, I asked you about who influenced your interest in science
and you gave me a chronicle of the impact of your various teachers on how
you felt towards science. Can you give me a similar chronicle regarding your
math teachers from high school to now?
7. What changed in test preparation between grade10 and grade 11?
8. How has the math class been going for you since we last spoke?
Prompt: level of enjoyment, greatest joy, biggest challenge, comfort level
9. What things do you feel you know well in your current math class?
Prompt: Use test(s) to discuss this question.
10. Is there anything you feel you don‟t know well?
Prompt: Use test(s) to discuss this question.
144
11. Would you say most of the students in your class are having a similar
experience to you?
12. Does the practice of curving happen with math?
13. In the last interview, when I asked what they were trying to teach you in this
math class, you said, “The fundamental theorems of calculus”. What does that
mean?
14. Has anything changed in the way you do your homework assignments and/or
prepare for test?
Prompts: work alone/others (if so, who); use textbook; use notes; tutoring; see
professor
15. Has your current math professor ever assigned extra credit problems? Does he
continue to use handouts occasionally for material, or was it just that once?
What about collaboration?
16. In the last interview you told me that you spoke to your teacher regarding
differences with calculator. Have you had any other reason to speak to your
teacher or seek his help regarding math? Former professor?
17. In your last interview in talking about chemistry you said:
“. . .they aren’t going to give you that one kind of problem. They’re going to
give you the theory behind that problem into a whole other kind of problem,
and it confuses you. And so, I’ve learned from experience that you can’t study
for chem. that way. . .” (p.26):
What did you mean by that? Has there been a shift in your focus – away from
simply getting a good grade to understanding the concepts?
Refer to: description of studying in high school in first interview
Description of joy in math first semester: “My first exam score” versus
second semester: “The fact that I understand it.”
18. Has anything else transpired since we last met that you think I should know
about -- with your classes, work, extracurriculas, etc? (What do you do when
you are not studying for classes?
Prompts: involvement in campus organizations, who friends are, work)
19. Who do you talk to about your challenges or share your joys/triumphs with?
20. When and/or why do you get to feeling that you wanna come home?
145
Looking back over the year
21. Do you think your grades on your math tests/transcript thus far is an accurate
reflection of what you learned in mathematics this year? (Both courses)
22. Of your two math courses, which did you prefer and why? And how do they
compare to high school math?
23. If you could relive your first year of mathematics in college, what would you
change?
24. Would you say that your study habits now are similar to the study habits you
had at the beginning of the year or have they changed/evolved and, if so, how?
Prompt: Use a particular study session to discuss this.
Probe for any connection to the ways of studying in high school.
25. Will you continue to pursue this major or are you considering other options?
26. If you were writing to a younger sister/brother/cousin, somebody back in the
Bahamas who you really cared about and they were preparing to come to the
U.S. to go to college and would take this course, what would you tell them
about your experience?
27. Is there anything else that I haven‟t asked about which you feel characterizes
or may be affecting your experience of transitioning from math home to math
here in the United States?
28. If you had the opportunity to tell the government of the Bahamas what they
could do to make it easier for students to study mathematics when they come
over here, what would that message be?
146
Appendix 2.5: Final Interview Protocol, (Summer 2007)
Thank you for agreeing to this final interview. I will begin by asking questions
regarding last semester and tying up so loose ends and then go into general questions
that ask you to reflect about the past year.
But first, tell me how your summer has been going? Chances to reconnect at
home, share some of your college life with parents/friends/siblings?
Wrapping up spring semester
1. How did last semester end? Were there any surprises and/or disappointments
last semester?
Josh: Will you be able to take the math course you were hoping to take
next year?
2. (Was math final exam written by professor or department?) How did you
prepare for the final?
Sade: did professor post solutions to practice exam?How useful was it?
Did you use math help lab?
3. Were there any:
a. challenging questions on your final exam? Tests throughout term?
b. questions that required critical thinking? Is there a difference between
a) and b)?
4. Is there a difference between understanding the mechanics of a problem,
understanding the concept, and making connections? If so, what?
Sade: “conceptually I didn’t find calc III very hard but I got in there
and I just hadn’t done enough to get through the midterm . . .”
(p.11)
5. Were grades curved in your math course? Have you experienced curving prior
to college?
6. How often did you use your calculator and why?
7. Where/How did you come to develop the study habits that you have?
Brittney: Were extended topics taught while you were preparing for
the BGCSEs?
Sade: why were you’re last two years of h.s. such great years for math
– material covered? Way taught? Teacher (different from
10th
)?
8. Sade: what did advisor say about course-load for next semester?
9. How would you describe the friendships you have made in college?
147
Brittney: a) Do you belong to any student groups/organizations?
b) Messages to self – how much connected to messages/
internalized based on messages received from home
(parents esp. mom) and teachers?
Sade: a) How and when did you and Cassie become friends? Is
she also a member of the black student club?
b) “. . .I would have gotten a totally different like, like you
know, feeling for the school if I had jumped, like jumped
right into the activities” (p.10) Please elaborate.
c) What’s the good stuff?
10. Are there any adults at your university whose advice/opinion you value?
Looking Back over the year
1. How would you describe your math experience this year and your feelings
about mathematics now?
Ask these questions directly? a. Was there anything you found difficult and to what do you attribute
those difficulties?
b. What there anything you found different and why?
c. Did you experience moments of discontinuity and, if so, how did you
deal with those moments?
d. Did you experience moments of continuity?
2. Who/what would you say has most influenced your interest in mathematics
and science? 3. Would you change anything about your high school math experience knowing
what you know now about what college math?
4. Would you change anything about your first year of math in college knowing
what you know now about the college experience?
5. What would you say is the purpose of mathematics? Why study it?
Probe for focus in mathematics: is it on getting a good grade or
understanding concept and making connections? Is there a difference?
6. How would you describe a good math teacher? a bad math teacher? What
attributes do they possess?
7. What is your outlook for this coming year of math in college? Any new
resolutions/minor changes to previous practices?
8. Will you continue to pursue this major or are you considering other options?
9. Are you concerned about meeting the requirements of this major in the allotted
time due to where you began in mathematics? Why/why not?
Have look over BGCSE exam for their year and discuss.
Ask about the impact having multiple choice questions might make.
148
10. If you were writing to a younger sister/brother/cousin, somebody back in the
Bahamas who you really cared about and they were preparing to come to the
U.S. to go to college . . .?
11. Is there anything else that I haven‟t asked about which you feel might
characterize or, characterize your experience transitioning from math home to
math here or affected how you feel about math or anything like that?
12. If you had the opportunity to tell the government of the Bahamas what they
could do to make it easier for students to study mathematics when they come
over here, what would that message be?
149
Appendix 2.6: Sample of Provisional Definitions of Selected Codes for Study
Codes Definitions
Mathematics Experience
Perceived Curriculum The mathematics topics taught (including related
years when provided)
Classroom Environment Class size and affect – inviting, stifling, non-
threatening,stressful etc
Pedagogy How mathematics was taught – lecture style,
individual seatwork, groupwork, discussion, etc.
Study Habits How participant studies –
Structure: alone, with others, section help,
tutoring, office
hours
Manner: solving assigned problems, reading
text, seeking out extension problems,
reviewing lecture notes, etc.
150
Appendix 2.7: Sample of Emergent Codes from Data
Codes Example
Student
Seeking challenge “. . . the whole prospect of being challenged, it just
felt so good. . . It felt so good to, to be free to think
and to push myself and to, like you know, try to like
press limits and whatnot. . .” (Sade, retrospective
interview).
Making connections “. . . math kind of works its way into, you know, into
each other so you can kind of work it out” (Brittney,
3rd
college interview).
Critical thinking “ . . . I can know all the formulas off by heart and not
be able to know how to sequence them in order to,
like you know, get what I want. . .”(Sade,
retrospective interview)
Test-taking strategies “I would have seen the rest of the test because I knew
the last 4 pages of logs but I never go there because
of the stupid [iterative first problem]” (Josh, 3rd
college interview)
151
Appendix 2.8: Contact Summary Form – Interviews
Contact type: Interview Site:
Participant: Bahamian Contact Date:
Today‟s Date:
1. What were the main issues or themes that struck you in this contact?
2. Summarize the information you got (or failed to get) on each of the target
questions you had for this contact.
a. What were the secondary-school mathematics experiences of Bahamian
students studying MSE at American universities and colleges?
How do students describe the curriculum content and objectives of
their secondary mathematics programs?
How do students describe the teaching approaches encountered in their
secondary mathematics programs?
What were the work patterns and learning strategies that Bahamian
students developed in their secondary mathematics programs?
Who were the social others that influenced the students‟ interest in
mathematics and science?
b. What experiences do Bahamian students pursuing degrees in MSE have with
mathematics in American universities and colleges?
How do students describe the curriculum content and objectives of
their college mathematics courses?
How do students describe the teaching approaches encountered in their
college mathematics courses?
What are the work patterns and learning strategies that Bahamian
students employ in their college mathematics courses?
Who are the social others that influence the students‟ continued interest
in mathematics and science?
c. What relationships, if any, exist between the secondary-school mathematics
preparation of Bahamian students and their experiences with mathematics in
American universities and colleges?
What do the students find difficult and to what do they attribute those
difficulties?
What do the students find different and why?
Do the students experience moments of discontinuity and, if so, how
does the student address those moments?
152
3. Anything else that struck you as salient, interesting, illuminating or important to
this contact?
4. What new (or remaining) target questions do you have in considering the next
contact with this individual?
153
Appendix 2.9: Selected sample of the matrix of themes generated from the data
TOPIC Josh Brittney Sade
Confidence Level in Math
entering college
High Confidence Low Confidence High Confidence
Study habits Begun developing after high
school (in college)
Developed in high school Began developing after high
school (at IB World)
“. . . I think I‟m still in the process
of feeling around . . .” (p.25)
All three participants consider
parental influence to be
minimal
Changes in major Undeclared (premed-biology)
to declared Bioengineering
Refined – biology to
neurobiology (wants to be
anesthesiologist)
Pragmatic – chemical
engineering for job prospects
though loves women/socio-
political issues
Math course Attendance usually attends class (missed 1
week first semester when sick; 3?
2nd
semester because he could)
Always attends class attended class 1st semester; more
sporadic attendance beginning
midterm of 2nd
semester
Adjusting to college Gradual Gradual Adjusted to college last
semester although intensity did
increase a little from last semester to
this semester. (Compare to Josh’s
April interview or observation.)
Gradual
Goal in college math course Focus on grades
Greatest enjoyment (first
semester) – getting an A on that
Shift in focus from grades to
understanding
Greatest joy – earning a 95% on her
Focus on understanding
IB World “. . . top marks. . .it
doesn‟t matter what everyone else
154
math test (compare to Brittney) first math exam (never earned so
high a grade in math before).
(Compare to Josh)
“I really don‟t pay attention to other
people like, you know, cause I‟m
focusing on my work.” (p.12)
was getting, A‟s girl . . .”
Compare to Brittney’s current
outlook
College Goal-oriented: “I‟ve
gotten more focused on myself
and, like you know, trying to better
myself, kind of thing.” (p.31)
Compare to Brittney
Second Semester
Take note of differences in
calculus courses by university:
see schools’ catalogues
took 2 semesters of precalculus –
1st semester (algebra topics); 2
nd
semester (trig) followed by Calc
I
took 1 semester college algebra
followed by Calc I and II.
No trig used in college algebra
course or Calc I
took Calc II – sequences and series
topic clearly demonstrated
expectation that student was
familiar with trigonometry
Coursework Fell behind in coursework. Has not fallen behind in
coursework. (Contrast with Sade
and Josh).
Fell behind in coursework 2nd
semester: “Had many catch ups and
trip downs and what nots” (p.9)
“I‟m not that well-prepared for class”.
(p.10)
Differences in study patterns
by course
relies on textbook to study for chem.;
relies on course notes to study for
math compare with Brittney;
contrasts with Sade
Relies on textbook to study
biology. Does not use textbook
to study for math. Purpose of
textbook: backup should she
ever miss class or run into
difficulties with professor‟s notes
or need a little extra reference
(Contrast with Josh and Sade)
1. Uses textbook to learn math; does
not have a chem. textbook
When falls behind in notetaking
for math, will copy notes with
knowledge that can read later
along with textbook to make sense
of
2. Reads only sections/examples
needed in math textbook to do h.w.
problems; reads entire chapter in
physics textbook before doing the
assignment and sometimes (more at
155
the beginning of the term) reads parts
of chapter before physics class to better
understand prof‟s lecture
3. Finds math lectures difficult to
follow; physics lectures are easy to
follow
Working with others If found something awkward in
math done at home, will compare
with friend (Christian) when arrive
to class.
Arrives to class early to compare
homework with seatmate (Megan)
(Compare to Josh.
Contrast? With Sade)
Does not like working with others –
likes to do it herself. (exception was
friend from high school whom she
worked with occasionally and learned
technique for organizing notebook
from).
Occasionally participates in a
chemistry study group though finds
her approach (whether right or wrong)
often differs from peers
When encounter difficulty with
homework more convenient to seek
answer in textbook rather than seek out
another‟s help: “If I don‟t understand
things, . . .I‟ll find that the book is
closer.”
Connection between age and
ways of explaining
In reference to second semester
teacher: Young “I don‟t know if it‟s
because she‟s so young and she
knows how to teach young people”
(p.21) Compare to Brittney and
grad teacher
Students, in my opinion, can teach
students best . . . students know how
a students‟ gonna think. . . . He was a
student. He told me the student‟s
way of doing it. He didn‟t say,
“Well, first you follow rule 1 and
Connection between age and
type of explanation (compare
with Josh) – “she was very
straightforward cause she was a
grad student so she knew, you
know.” (p.5)
Notes youthfulness of first
semester teaching contributing to
her understanding. (Compare to
Josh)
Efficiency of having a study
partner: It‟ll probably be, if they
understand it, they‟ll be able to get
it through to me before the, like
before, rather than me trying to
like sifle through the book. And
they‟d probably be able to like say
it, because if they‟ve come to
understand it, I‟m not assuming
that they‟re very much more
intelligent than I am, kind of thing,
so they‟ll be able to say it in a way
156
then you go to rule 2.” No, that‟s not
how I‟m gonna think.” (p.28)
Compare to Brittney
that it makes sense to them and
therefore will be able to, you
know, penetrate.” (p.25)
Compare highlight to Josh and
Brittney
On studying with Nicole: both
know how to simplify material;
(compare to Josh’s description
of student help)
Level/Accuracy of awareness of
other’s performance in class
others in class as having a similar
level of understanding to himself
but no evidence to support (w/
exception of Christian, he does
not interact with other students in
class; my observations of class
leads to no conclusion either
way)
First semester: “Cause no doubt,
anybody in that class, if you give
them a basic question to see if they
understand the concept, they
understand.” (p.10)
Data for second semester???
aware that she does not know
how others are performing but
not concerned because she is
doing well; assumes class is
doing fine (i.e. no-one failing).
“I‟ve been doing good so I didn‟t
care.” (p.2)
“I really don‟t pay attention to other
people‟s grades. I worry about me.”
Used class average when asked to
compare self‟s performance to
others(Compare to Sade)
probably attends class the least of the
three but more aware of others‟
feelings regarding course (some
classmates are dorm mates).
(observational data seems to support
student disconnect with course).
157
Appendix 3.1: Higher Level Mathematics Curriculum for International
Baccalaureate Program
Sade
Higher Level Mathematics*
*Derived from: Mathematics Standard Level for the IB Diploma by Smedley and Wiseman
Included in
National Curriculum
Not Included in
National Curriculum
Exponents
Function notation
Composition of functions
Linear and quadratic functions
Sine and Cosine Rule
Matrices (2 x 2)
Vectors – sums, differences, scalar
multiplication
Statistics – Mean, Median, Mode;
Cumulative Frequency;
Probability – Counting principles; Tree
Diagrams
Sequences and series
Exponents and logarithms
Binomial theorem
Complex numbers
Division of polynomials: Remainder
& Factor theorems
Real and Complex roots of
polynomials
Inverse functions
Transformations
Analyzing graphs of quadratic
functions
Polynomial functions and equations
Exponential and logarithmic functions
and their graphs
Trigonometric identities
Transformations and inverses using
trigonometric functions
Trigonometric equations
Matrices (3 x3)
Vectors – properties of scalar product
Statistics – box and whisker plots;
quartiles; expectation; relative
frequency vs. theoretical probability;
variance and standard deviation;
binomial distribution; normal
distribution; probability density
function
Limits of convergence
Gradient functions
Equations of tangents
Differentiation
Local Extrema
158
Appendix 3.2: “Brittney’s College Algebra Curriculum
Brittney
College Algebra
*Derived from: College Algebra by Beecher, Penna, and Bittinger (2nd
edition).
Included in
National Curriculum
Not Included in
National Curriculum
Review of real number system
Review of integer Exponents, Scientific
Notation, Order of Operations
Review of addition, Subtraction and
Multiplication of Polynomials
Review of factoring
Review of rational expressions, radical
notation and rational exponents
Linear equations, slope, and applications
Function notation
Composition of functions
Linear equations, functions and models
Quadratic equations, functions and
models
Linear Inequalities
Equations of lines and modeling
Functions and graphs
The algebra of functions
Symmetry and Transformations
Analyzing graphs of quadratic
functions
Polynomial functions and models
Division of polynomials: Remainder
& Factor theorems
Theorems about zeros of polynomial
functions
Polynomial and rational inequalities
Inverse functions
Exponential and logarithmic functions
and their graphs
Properties of logarithmic functions
Solving exponential and logarithmic
equations
Applications with growth and decay
159
Appendix 3.3: Function Topics on BGCSE Syllabus
160
Appendix 6.1: Sample of Mandatory Homework Assignment
161
162
REFERENCES
Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of math education.
American Federation of Teachers , Summer, 15-18, 46-47.
Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching
elementary school mathematics. The Elementary School Journal , 93 (4), 373-397.
Ball, D. L. (2003). Mathematical proficiency for all students: Toward a strategic
research and development program in mathematics education. Santa Monica, CA:
RAND.
Bean, J. P. (1982). Conceptual models of student attrition: How theory can help the
institutional researcher. In E. Pascarella (Ed.), New Directions for Institutional
Research: Studying Student Attrition (Vol. 36, pp. 17-33). San Francisco: Jossey-Bass.
Bean, J. P., & Metzner, B. S. (1985). A conceptual model of nontraditional
undergraduate student attrition. Review of Educational Research, 55(4), 485-540.
Berryman, S. E. (1983). Who will do science? Minority and female attainment of
science and mathematics degrees: Trends and causes. New York: The Rockefeller
Foundation.
Boaler, J. (2002). Experiencing school mathematics: Traditional and reform
approaches to teaching and their impact on student learning, Revised and expanded
edition. Mahwah, NJ: Lawrence Erlbaum Associates.
Boaler, J. (1997). Experiencing school mathematics: Teaching styles, sex and setting.
Buckingham, UK: Open University Press.
Boaler, J. (2002a). Paying the price for 'sugar and spice' -- shifting the analytical lens
in equity research. Mathematical Thinking and Learning , 4 (2 & 3), 127-144.
Boaler, J. (2002b). The development of disciplinary relationships: Knowledge,
practice adn identity in mathematics classrooms. For the Learning of Mathematics , 22
(1), 42-47.
Boaler, J. (2003). Studying and capturing the complexity of practice -- the case of the
"dance of agency.". In N. Pateman, B. Dougherty, & J. Zilliox (Ed.), Proceedings of
the 27th Conference of the International Group for the Psychology of Mathematics
Education held jointly with the 25th Conference of PME-NA., I, pp. 3-16. Honolulu,
Hawaii.
Boaler, J. (2005 (unpublished)). Relationship with math.
163
Boaler, J. (2008). When politics took the place of inquiry: A response to the national
mathematics advisory panel's review of instructional practices. Educational
Researcher , 37 (9), 588-594.
Boaler, J., & Greeno, J. (2000). identity, agency and knowing in mathematics worlds.
In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp.
171-200). Westport, CT: Ablex Publishing.
Boaler, J., Sengupta-Irving, T., Dieckmann, J., & Fiori, N. (in progress). The many
colors of algebra – engaging disaffected students through collaboration and agency.
Bonous-Hammarth, M. (2000). Pathways to success: Affirming opportunities for
science, mathematics, and engineering majors. The Journal of Negro Education, 69,
92-111.
Cabrera, A. F., Nora, A., & Castaneda, M. B. (1993). College persistence: Structural
equations modeling test of an integrated model of student retention. The Journal of
Higher Education, 64(2), 123-139.
Clarke, D. (2003). International comparative research in mathematics education. In A.
Bishop, M. A. Clements, C. Keitel, J. Kilpatrick & F. Leung (Eds.), Second
international handbook of mathematics education (pp. 143-184). Dordrecht: Kluwer.
Clinchy, B. M. (1996). Connected and separate knowing: Toward a marriage of two
minds. In N. R. Goldberger, J. M. Tarule, B. M. Clinchy, & M. F. Belencky (Eds.),
Knowledge, difference, and power: Essays inspired by "Women's ways of knowing"
(pp. 205-247). New York, NY, USA: Basic Books.
Cobb, P., Jaworski, B., & Presmeg, N. (1996). Emergent and sociocultural views of
mathematical activity. In L. Steffe & P. Nesher (Eds.), Theories of mathematical
Learning. Mahwah, New Jersey: Lea Lawrence Erlbaum Associates.
Cole, M. (1996). Cultural psychology: A once and future discipline. Harvard.
Cooney, M. P., Dewar, J. M., Kenschaft, P. C., Kraines, V., Latka, B., & LiSanti, B.
(1990). Recruitment and retention of students in undergraduate mathematics. The
College Mathematics Journal, 21(4), 294-301.
Craton, M. (2002). Pindling: The life and times of the first Prime Minister of The
Bahamas. Oxford: MacMillan Caribbean.
Dasen, P. R., & Mishra, R. C. (2000). Cross-cultural views on human development in
the third millennium. International Journal of Behavioral Development, 24(4), 428-
434.
164
Dewey, J. (1933). How we think: A restatement of the relation of reflectgive thinking
to the educative process. Boston: Heath.
Driscoll. (1981). Research within reach: Elementary school mathematics. Reston, VA:
National Council of Teachers of Mathematics.
Dupuch, E. (2006). Bahamas handbook and business manual. Nassau, Bahamas:
Etienne Dupuch Jr Publications Ltd.
Dweck, C. S. (1986). Motivational processes affecting learning. American
Psychologist , 41 (10), 1040-1048.
Eisenhart, M., Borko, H., Underhill, R., Brown, C., Jones, D., & Agard, P. (1993).
Conceptual knowledge falls through the cracks: Complexities of learning to teach
mathematics for understanding. Journal for Research in Mathematics Education , 24
(1), 8-40.
Engle, R. A., & Conant, F. R. (2002). Guiding principles for fostering productive
disciplinary engagement: Explaining an emergent argument in a community of
learners classroom. Cognition and Instruction , 20 (4), 399-483.
Etienne Dupuch Jr Publications Ltd. (2005). Bahamas handbook and businessman's
annual 2006. (R. D. Dawn Lomer, Ed.) Nassau, The Bahamas: Etienne Dupuch Jr.
Fullilove, R. E., & Treisman, P. U. (1990). Mathematics achievement among African
American undergraduates at the University of California, Berkeley: An evaluation of
the mathematics workshop program. Journal of Negro Education, 59(3), 463-478.
Grandy, J. (1998). Persistence in science of high-ability minority students. The
Journal of Higher Education, 69(6), 589-620.
Green, K. C. (1989). A profile of undergraduates in the sciences. American Scientist,
77, 475-480.
Gresalfi, M. S., & Cobb, P. (2006). Cultivating students' discipline-specific
dispositions as a critical goal for pedagogy and equity. Pedagogies: An international
journal , 1 (1), 49-57.
Gutiérrez. (1996). Practices, beliefs and cultures of high school mathematics
departments: understanding their influence on student advancement. Journal of
Curriculum Studies , 28 (5), 495-529.
Gutiérrez, K., & Rogoff, B. (2003). Cultural ways of learning: Individual traits or
repertoires of practice. Educational Researcher, 32(5), 19-25.
165
Hannula, M. S. (2002). Attitude towards mathematics: emotions, expectations and
values. Educational Studies in Mathematics , 49, 25-46.
Hiebert, J. (1984). Children's mathematics learning: The struggle to link form and
understanding. The Elementary School Journal , 84 (5), 496-513.
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in
mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural
knowledge: The case of mathematics. Hillsdale, NJ: Lawrence Erlbaum Associates,
Publishers.
Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., et al.
(1997). Making sense: teaching and learning mathematics with understanding.
Portsmouth, NH: Heinemann.
Hudson, H. T., & Rottmann, R. M. (1981). Correlation between performance in
physics and prior mathematics knowledge. Journal of Research in Science Teaching,
18(4), 291-294.
Hudson, H. T. (1986). A comparison of cognitive skills between completes and
dropouts in a college physics course. Journal of Research in Science Teaching, 23(1),
41-50.
Kazemi, E. (1998). Discourse that promotes conceptual understanding. Teaching
Children Mathematics , 4 (7), 410-414.
Lampert, M. (2001). Teaching problems and the problems of teaching. New Haven,
CT, USA: Yale University Press.
Lave, J., & Wenger, E. (1991). Situated learning: legitimate peripheral participation.
New York, NY: Cambridge University Press.
Lee, V., & Hilton, T. L. (1988). Student interest and persistence in science: Changes
in the educational pipeline in the last decade. The Journal of Higher Education, 59(5),
510-526.
Lerman, S. (1996). Intersubjectivity in mathematics learning: A challenge to the
radical constructivist paradigm? Journal for Research in Mathematics Education,
27(2), 133-150.
Lerman, S. (2001). Cultural, discursive psychology: A sociocultural approach to
studying the teaching and learning of mathematics. Educational Studies in
Mathematics, 46, 87-113.
Lipson, A., & Tobias, S. (1991). Why do some of our best college students leave
science? Journal of College Science Teaching, 21(2), 92-95.
166
London, E. D. (2005). The impact of 21st century global realities on education. Paper
presented at the 18th Annual National Education Conference, Nassau, Bahamas.
Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ:
Lawrence Erlbaum Associates, Inc.
Maple, S. A., & Stage, F. K. (1991). Influences on the choice of math/science major
by gender and ethnicity. American Educational Research Journal, 28(1), 37-60.
Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: an expanded
sourcebook (2nd ed.). Thousand Oaks, CA: SAGE Publications, Inc.
Ministry of Education. (2003a). Teaching in the public education system in The
Bahamas. Commonwealth of the Bahamas.
Ministry of Education. (2003b). Understanding the BGCSE seven-point grading
system. Commonwealth of the Bahamas: Testing and Evaluation Section.
Ministry of Education. (2004b). Retrieved March 2, 2005, from
http://www.bahamaseducation.com/The%20Curriculum/curriculum.asp
Ministry of Education. (2006). Bahamas General Certificate of Secondary Education
Mathematics Syllabus 2006. Nassau: Bahamas Government Printing Department.
Ministry of Education. (June 2010). Bahamas general certificate of secretary
education report on the examinations June 2010 Mathematics. Bahamas Government.
Nasir, N. S., & Hand, V. M. (2006). Exploring sociocultural perspectives on race,
culture, and learning. Review of Educational Research , 76 (4), 449-475.
National Council of Teachers of Mathematics. (1989). Principles and standards for
school matematics. Reston, VA.
National Council of Teachers of Mathematics. (1991). Principles and standards for
school mathematics. Reston, VA.
National Council of Teachers of Mathematics. (2000). Principles and standards for
school mathematics. Reston, VA.
National Research Council. (2001). Adding it up: Helping children learn mathematics.
(J. Kilpatrick, J. Swafford, & B. Findell, Eds.) Washington, DC: National Academy
Press.
National Research Council. (2004). How people learn: Brain, mind, experience, and
school. (J. D. Bransford, A. L. Brown, & R. R. Cocking, Eds.) Washington, D.C.:
National Academy Press.
167
National Science Board. (2004). Science and Engineering Indicators. Retrieved June
2, 2005, from http://www.nsf.gov/sbe/srs/seind04/start.htm
Niemi, D. (1996). Assessing conceptual understanding in mathematics:
Representations, problem solutions, justifications, and explanations. The Journal of
Educational Research , 89 (6), 351-363.
Office of Technology Assessment. (1988). Educating scientists and engineers: Grade
school to grad school (No. OTA-SET-377). Washington, DC: U.S. Government
Printing Office.
Office of Technology Assessment. (1989). Higher education for science and
engineering -- A background paper (No. OTA-BP-SET-52). Washington, DC: U.S.
Government Printing Office.
Pascarella, E. (1980). Student-faculty informal contact and college outcomes. Review
of Educational Research, 50(4), 545-595.
Phelps, R. E., Taylor, J. D., & Gerard, P. A. (2001). Cultural mistrust, ethnic identity,
racial identity, and self-esteem among ethnically diverse black university students.
Journal of Counseling and Development , 79 (2), 209-216.
Pickering, A. (1995). The mangle of practice: time, agency and science. Chicago, IL:
University of Chicago Press.
Pratt, C. (2005). Keynote address. Paper presented at the 18th Annual National
Education Conference, Nassau, Bahamas.
Rittle-Johnson, B., & Siegler, R. S. (1998). The relation between conceptual and
procedural knowledge in learning mathematics: A review. In C. Donlan (Ed.), The
development of mathematical skills (Vol. xiv, pp. 75-110). Hove, England:
Psychology Press/Taylor & Francis (UK).
Rittle-Johnson, B., Siegler, R., & Alibali, M. W. (2001). Developing conceptual
understanding and procedural skill in mathematics: An iterative process. Journal of
Educational Psychology , 93 (2), 346-362.
Rogoff, B. (1990). Apprenticeship in thinking: Cognitive development in social
context. Oxford: Oxford University Press.
Rogoff, B. (1995). Observing sociocultural activity on three planes: participatory
appropriation, guided participation, and apprenticeship. In J. V. Wertsch, P. del Rio &
A. Alvarez (Eds.), Sociocultural studies of mind. Cambridge, UK: Cambridge
University Press.
168
Rolle, J. (2005). Education and national development. Paper presented at the 18th
Annual National Education Conference, Nassau, Bahamas.
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving,
metacognition, and sense making in mathematics. In D. Grouws (Ed.), Handbook for
Research on Mathematics Teaching and Learning (pp. 334-370). New York:
MacMillan.
Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of
"well taught" mathematics courses. Educational Psychologist , 23 (2).
Schweder, R. (1990). What is cultural psychology?
Sears, A. M. (2005). A status report on Bahamian education. Paper presented at the
18th Annual National Education Conference, Nassau, Bahamas.
Sells, L. W. (1978). Mathematics -- a critical filter. The Science Teacher, 45(2), 28-29.
Seymour, E., & Hewitt, N. (1997). Talking about leaving: Why undergraduates leave
the sciences. Boulder, CO: Westview.
Sumner. (n.d.). Examination development for educational reform Bahamas new
secondary examination - BGCSE. 63-72.
Sumner, L. (1995). Examination development for educational reform Bahamas new
secondary examination -- BGCSE. The International Journal of Bahamian Studies , 7,
63-72.
The Coalition for Education Reform. (2005). Bahamian youth: The untapped
resource. Nassau, Bahamas.
The College of the Bahamas. (2003). The college catalogue. Retrieved 5/25/2006,
from www.cob.edu.bs/publications/CATALOGUE.pdf
Terenzini, P. (1982). Designing attrition studies. In E. Pascarella (Ed.), New directions
for institutional research: Studying student attrition (Vol. 36, pp. 55-72). San
Francisco, CA, USA: Jossey-Bass.
Terenzini, P. e. (n.d.). Influences affecting the development of students' cricial
thinking skills.
Thurston, W. (1990). Mathematical Education. Notices of the AMS , 37, 844-850.
Tinto, V. (1975). Dropout from higher education: A theoretical synthesis of recent
research. Review of Educational Research, 45(1), 89-125.
169
Tinto, V. (1993). Leaving college: Rethinking the causes and cures of student
attrition. Chicago: University of Chicago Press.
Tobais, S. (1990). They're not dumb, they're different: Stalking the second tier.
Tucson, AZ: Research Corporation.
Treisman, P. U. (1985). A study of the mathematics achievement of Black students at
the University of California, Berkeley. Unpublished doctoral dissertation, University
of California, Berkeley.
U.S. Department of Education. (2000). Before it's too late: A report to the nation from
the National Commission on Mathematics and Science Teaching for the 21st century
(No. EE0449P). Jessup, MD: Education Publications Center.
UNESCO. (2005). Foreign students by country of origin (for all countries). Retrieved
April 27, 2006, from stats.uis.unesco.org
Urwick, J. (2002). The Bahamian educational system: A case study in
Americanization. Comparative Education Review, 46(2), 157-181.
Vanderpool, J. D. (1999). International academic relations and small nation states: A
case study of selected British, American and Canadian initiatives in Bahamian higher
education. Unpublished dissertation, University of Toronto, Toronto.
Vygotsky, L. S. (1978). Mind in society: The development of higher psychological
processes. Cambridge, MA: Harvard University Press.
Wagner, D. (2007). Students' critical awareness of voice and agency in mathematics
classroom discourse. Mathematical Thinking and Learning , 9 (1), 31-50.
Weidman, J. C. (1989). Undergraduate socialization: A conceptual approach. In J. C.
Smart (Ed.), Higher education: Handbook of theory and research (Vol. 5, pp. 289-
322). New York: Agathon Press.
Whitely, M. A., & Fenske, R. H. (1990). The college mathematics experience and
changes in majors: A structural model analysis. The Review of Higher Education,
13(3), 357-386.
Wollman, W., & Lawrenz, F. (1984). Identifying potential "dropouts" from college
physics classes. Journal of Research in Science Teaching, 21(4), 385-390.
Wu. (1999, Fall). Basic skills versus conceptual understanding: A bogus dichotomy in
mathematics education. American Educator/American Federation of Teachers , 1-7.
Yin, R. K. (2003). Case study research: design and methods (Third ed. Vol. 5). Sage
Publications: Thousand Oaks, CA.