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.

ii

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.

iii

iv

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.

v

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

vi

mathematical community in terms of developing theory that can then be tested with

other populations of students.

vi

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

vii

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,

viii

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.

ix

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.

x

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

xi

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

xii

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

2

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

3

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).

4

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

5

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.

6

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.

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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).

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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

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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

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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;

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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

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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.

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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.

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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)

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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

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